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Solution Manual For Financial Management Theory and Practice 4CE Eugene F. Brigham Michael C. Ehrhardt Jerome Gessaroli Richard R. Nason Chapter 1-24

Chapter 1 An Overview of Financial Management and the Financial Environment ANSWERS TO END-OF-CHAPTER QUESTIONS 1-1

a. A proprietorship, or sole proprietorship, is a business owned by one individual. A partnership exists when two or more persons associate to conduct a business. In contrast, a corporation is a legal entity created by provincial or federal laws. The corporation is separate and distinct from its owners and managers. b. In a limited partnership, limited partners’ liabilities, investment returns, and control are limited, while general partners have unlimited liability and control. The primary benefit of a limited liability partnership (LLP) is the protection it offers partners to liability exposure from their other partners’ professional negligence. Individual partners still maintain unlimited liability for their own negligence or negligence of those they directly supervise. A professional corporation (PC) has most of the benefits of incorporation but the participants are not relieved of professional (malpractice) liability. c. Shareholder wealth maximization is the appropriate goal for management decisions. The risk and timing associated with expected earnings per share and cash flows are considered in order to maximize the price of the firm’s common stock. d. A money market is a financial market for debt securities with maturities of less than one year (short-term). The New York money market is the world’s largest. Capital markets are the financial markets for long-term debt and corporate stock. The New York Stock Exchange and Toronto Stock Exchange are examples of capital markets. Primary markets are the markets in which newly issued securities are sold for the first time. Secondary markets are where securities are resold after initial issue in the primary market. The New York Stock Exchange and Toronto Stock Exchange are secondary markets.

e. In private markets, transactions are worked out directly between two parties and structured in any manner that appeals to them. Bank loans and private placements of debt with insurance companies are examples of private market transactions. In public markets, standardized contracts are traded on organized exchanges. Securities that are issued in public markets, such as common stock and corporate bonds, are ultimately held by a large number of individuals. Private market securities are more tailor-made but less liquid, whereas public market securities are more liquid but subject to greater standardization. Derivatives are claims whose value depends on what happens to the value of some other asset. Futures and options are two important types of derivatives, and their values depend on what happens to the prices of other assets, say Tim Hortons shares, Japanese yen, or pork bellies. Therefore, the value of a derivative security is derived from the value of an underlying real asset. f. An investment banker is a middleman between businesses and savers. Investment banks assist in the design of corporate securities and then sell them to savers (investors) in the primary markets. A financial intermediary buys securities with funds that it obtains by issuing its own securities. An example is a common stock mutual fund that buys common shares with funds obtained by issuing shares in the mutual fund. g. A mutual fund is a corporation that sells shares in a fund and uses the proceeds to buy stocks, long-term bonds, or short-term debt instruments. The resulting dividends, interest, and capital gains are distributed to the fund’s shareholders after the deduction of operating expenses. Different funds are designed to meet different objectives. Money market funds are mutual funds that invest in short-term debt instruments, typically with maturity dates of less than one year. h. Production opportunities are the returns available within an economy from investment in productive assets. The higher the production opportunities, the more producers would be willing to pay for required capital. Consumption time preferences refer to the preferred pattern of consumption. Consumers’ time preferences for consumption establish how much consumption they are willing to defer, and hence save, at different levels of interest.

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i. A foreign trade deficit occurs when businesses and individuals in a country import more goods from foreign countries than are exported. Trade deficits must be financed, and the main source of financing is debt. Therefore, as the trade deficit increases, the debt financing increases, driving up interest rates. Interest rates, however, must be competitive with foreign interest rates; if the central bank attempts to set interest rates lower than foreign rates, foreigners will sell bonds, decreasing bond prices, resulting in higher rates. Thus, if the trade deficit is large relative to the size of the overall economy, it may hinder the central bank’s ability to combat a recession by lowering interest rates. Sole proprietorship, partnership, and corporation are the three principal forms of business organization. The advantages of the first two include the ease and low cost of formation. Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc.

The advantages of the corporation include limited liability, indefinite life, ease of ownership transfer, and access to capital markets. The disadvantages of a sole proprietorship are (1) difficulty in obtaining large sums of capital, (2) unlimited personal liability for business debts, and (3) limited life. The disadvantages of a partnership are (1) unlimited liability, (2) limited life, (3) difficulty of transferring ownership, and (4) difficulty of raising large amounts of capital. The disadvantages of a corporation are (1) double taxation of earnings and (2) requirements to file federal reports for registration, which are expensive, complex, and time consuming. 1-3

A firm’s fundamental, or intrinsic, value is the present value of its free cash flows when discounted at the weighted average cost of capital. If the market price reflects all relevant information, then the observed price is also the intrinsic price. If material information is withheld from investors, the firm’s intrinsic value may differ from its actual market value.

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a. Corporate philanthropy is always a sticky issue, but it can be justified in terms of helping to create a more attractive community that will make it easier to hire a productive workforce. This corporate philanthropy could be received by shareholders negatively, especially those shareholders not living in its headquarters’ city. Shareholders are interested in actions that maximize share price, and if competing firms are not making similar contributions, the ―cost‖ of this philanthropy has to be borne by someone—the shareholders. Thus, the stock price could decrease. b. Companies must make investments in the current period in order to generate future cash flows. Shareholders should be aware of this and, assuming a correct analysis has been performed, they should react positively to the decision. The Mexican plant is in this category. Assuming that the correct capital budgeting analysis has been made, the stock price should increase in the future.

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Earnings per share in the current year will decline due to the cost of the investment made in the current year and no significant performance impact in the short run. However, the company’s intrinsic value and its share price should increase due to the significant cost savings expected in the future.

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In a well-functioning economy, capital will flow efficiently from those who supply capital to those who demand it. This transfer of capital can take place in three different ways: 1. Direct transfers of money and securities occur when a business sells its shares or bonds directly to savers, without going through any type of financial institution. The business delivers its securities to savers, who in turn give the firm the money it needs. 2. Transfers may also go through an investment banking house that underwrites the issue. An underwriter serves as a middleman and facilitates the issuance of securities. The company sells its stocks or bonds to the investment bank, which in turn sells

these same securities to savers. The businesses’ securities and the savers’ money merely ―pass through‖ the investment banking house. 3. Transfers can also be made through a financial intermediary. Here, the intermediary obtains funds from savers in exchange for its own securities. The intermediary uses this money to buy and hold the businesses’ securities. Intermediaries literally create new forms of capital. The existence of intermediaries greatly increases the efficiency of money and capital markets. 1-7

Financial intermediaries are business organizations that receive funds in one form and repackage them for the use of those who need funds. Through financial intermediation, resources are allocated more effectively, and the real output of the economy is thereby increased.

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a. If transfers between the two markets were costly, interest rates would be different in the two areas. Area Y, with the relatively young population, would have less in savings accumulation and stronger loan demand. Area O, with the relatively old population, would have more savings accumulation and weaker loan demand as the members of the older population have already purchased their houses, and are less consumption oriented. Thus, supply/demand equilibrium would be at a higher rate of interest in Area Y. b. Yes. Nationwide branching would reduce the cost of financial transfers between the areas. Thus, funds would flow from Area O with excess relative supply to Area Y with excess relative demand. This flow would increase the interest rate in Area O and decrease the interest rate in Y until the rates were roughly equal, the difference being the transfer cost.

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The immediate effect would be to lower interest rates.

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A primary market is the market in which corporations raise capital by issuing new securities. An initial public offering (IPO) is a share issue in which privately held firms go public. Therefore, an IPO would be an example of a primary market transaction.

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The two stock markets today are the Toronto Stock Exchange (TSX) and the TSX Venture Exchange. The TSX trades shares of primarily senior Canadian issuers, while the TSX Venture Exchange trades shares of junior or speculative corporations.

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MINI CASE The detailed solution for the Mini Case is also available in spreadsheet format, in the file Brigham4Ce_MiniCase_Ch01.xlsx, and is available on the textbook’s website. Assume that you have recently graduated and have just reported to work as an investment advisor at the brokerage firm of Penman and Ross Inc. One of the firm’s clients is Aleksandar Milic, a professional tennis player who has just come to Canada from Serbia. Milic is a highly ranked tennis player who would like to start a company to produce and market apparel with his signature. He also expects to invest substantial amounts of money through Penman and Ross. Milic is very bright, and therefore he would like to understand in general terms what will happen to his money. Your boss has developed the following set of questions that you must answer to explain the Canadian financial system to Milic. a.

Why is corporate finance important to all managers?

Answer: Corporate finance provides the skills managers need to (1) identify and select the corporate strategies and individual projects that add value to their firm; and (2) forecast the funding requirements of their company, and devise strategies for acquiring those funds. b.

What are the organizational forms a business might have as it evolves from a start-up to a major corporation? List the advantages and disadvantages of each form.

Answer: The three main forms of business organization are (1) sole proprietorship, (2) partnership, and (3) corporation. In addition, several hybrid forms are gaining popularity. These hybrid forms are the limited partnership, the limited liability partnership, and the professional corporation. The proprietorship has three important advantages: (1) it is easily and inexpensively formed, (2) it is subject to few government regulations, and (3) the business pays no corporate income taxes. The proprietorship also has three important limitations: (1) it is difficult for a proprietorship to obtain large sums of capital; (2) the proprietor has unlimited personal liability for the business’s debts, and (3) the life of a business organized as a proprietorship is limited to the life of the individual who created it. The major advantages of a partnership are its low cost and ease of formation. The disadvantages are similar to those associated with proprietorships: (1) unlimited liability, (2) limited life of the organization, (3) difficulty of transferring ownership, and (4) difficulty of raising large amounts of capital. The tax treatment of a partnership is similar to that for proprietorships, which is often an advantage. Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc. 2-5

The corporate form of business has three major advantages: (1) unlimited life, (2) easy transferability of ownership interest, and (3) limited liability. While the corporate form offers significant advantages over proprietorships and partnerships, it does have two primary disadvantages: (1) corporate earnings may be subject to double taxation and (2) setting up a corporation and filing many required reports is more complex and time consuming than for a proprietorship or a partnership. In a limited partnership, the limited partners are liable only for the amount of their investment in the partnership; however, the limited partners typically have no control. The limited liability partnership form of organization combines the limited liability advantage of a corporation with the tax advantages of a partnership. Professional corporations provide most of the benefits of incorporation but do not relieve the participants of professional liability. c.

How do corporations go public and continue to grow? What are agency problems? What is corporate governance?

Answer: A company goes public when it sells shares to the public in an initial public offering. As the firm grows, it might issue additional shares or debt. An agency problem occurs when the managers of the firm act in their own self-interests and not in the interests of the shareholders. Corporate governance is the set of rules that control a company’s behaviour towards its directors, managers, employees, shareholders, creditors, customers, competitors, and community. d.

1. What should be the primary objective of managers?

Answer: The corporation’s primary goal is shareholder wealth maximization, which translates to maximizing the price of the firm’s common stock. d.

2. Do firms have any responsibilities to society at large?

Answer: Firms have an ethical responsibility to provide a safe working environment, to avoid polluting the air or water, and to produce safe products. However, the most significant cost-increasing actions will have to be put on a mandatory rather than a voluntary basis to ensure that the burden falls uniformly on all businesses.

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3. Is share price maximization good or bad for society?

Answer: The same actions that maximize stock prices also benefit society. Stock price maximization requires efficient, low-cost operations that produce high-quality goods and services at the lowest possible cost. Stock price maximization requires the development of products and services that consumers want and need, so the profit motive leads to new technology, to new products, and to new jobs. Also, stock price maximization necessitates efficient and courteous service, adequate stocks of merchandise, and well-located business establishments—factors that are all necessary to make sales, which are necessary for profits. d.

4. Should firms behave ethically?

Answer: Yes. Results of a recent study indicate that the executives of most major firms in the United States and Canada believe that firms do try to maintain high ethical standards in all of their business dealings. Furthermore, most executives believe that there is a positive correlation between ethics and long-run profitability. Conflicts often arise between profits and ethics. Companies must deal with these conflicts on a regular basis, and a failure to handle the situation properly can lead to huge product liability suits and even to bankruptcy. There is no room for unethical behaviour in the business world. e.

What three aspects of cash flows affect the value of any investment?

Answer: (1) amount of expected cash flows; (2) timing of the cash flow stream; and (3) riskiness of the cash flows. f.

What are free cash flows?

Answer: Free cash flows are the cash flows available for distribution to all investors (shareholders and creditors) after paying expenses (including taxes) and making the necessary investments to support growth. FCF = g.

Sales revenues

–

Operating Operating – – costs taxes

Required investments in operating capital

What is the weighted average cost of capital?

Answer: The weighted average cost of capital (WACC) is the average rate of return required by all of the company’s investors (shareholders and creditors). It is affected by the firm’s capital structure, interest rates, the firm’s risk, and the market’s overall attitude toward risk.

How do free cash flows and the weighted average cost of capital interact to determine a firm’s value?

Answer: A firm’s value is the sum of all future expected free cash flows, converted into today’s dollars. Value 

FCF1 (1  WACC)1



FCF2 (1  WACC) 2

 ....

FCF

(1  WACC) 

Who are the providers (savers) and users (borrowers) of capital? How is capital transferred between savers and borrowers?

Answer: Households are net savers. Nonfinancial corporations are net borrowers. Governments are also net borrowers, although the government is a net saver when it runs a surplus. Capital is transferred through (1) direct transfer (e.g., corporation issues commercial paper to insurance company); (2) an investment banking house (e.g., IPO, seasoned equity offering, or debt placement); or (3) a financial intermediary (e.g., individual deposits money in bank, bank makes commercial loan to a company). j.

What do we call the price that a borrower must pay for debt capital? What is the price of equity capital? What are the four most fundamental factors that affect the cost of money, or the general level of interest rates, in the economy?

Answer: The interest rate is the price paid for borrowed capital, while the return on equity capital comes in the form of dividends plus capital gains. The return that investors require on capital depends on (1) production opportunities, (2) time preferences for consumption, (3) risk, and (4) inflation. Production opportunities refer to the returns that are available from investment in productive assets: the more productive a producer firm believes its assets will be, the more it will be willing to pay for the capital necessary to acquire those assets. Time preference for consumption refers to consumers’ preferences for current consumption versus savings for future consumption: consumers with low preferences for current consumption will be willing to lend at a lower rate than consumers with a high preference for current consumption. Inflation refers to the tendency of prices to rise, and the higher the expected rate of inflation, the larger the required rate of return. Risk, in a money and capital market context, refers to the chance that the future cash flows will not be as high as expected—the higher the perceived default risk, the higher the required rate of return. Risk is also linked to the maturity and liquidity of a security. The longer the maturity and the less liquid (marketable) the security, the higher the required rate of return, other things constant.

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What are some economic conditions (including international aspects) that affect the cost of money?

Answer: The cost of money will be influenced by things such as Bank of Canada (BoC) policy, fiscal deficits, business activity, and foreign trade deficits. If the BoC increases growth in the money supply, the cost of money will initially decline. If the federal government shows a deficit, interest rates are likely to increase. The cost of money will typically rise before a recession and decline thereafter. The cost of money for an international investment is also affected by country risk, which refers to the risk that arises from investing or doing business in a particular country. This risk depends on the country’s economic, political, and social environment. Country risk also includes the risk that property will be expropriated without adequate compensation, as well as new host country stipulations about local production, sourcing or hiring practices, and damage or destruction of facilities due to internal strife. The cost of money for an international investment is also affected by exchange rate risk. When investing overseas, the security usually will be denominated in a currency other than the dollar, which means that the value of the investment will depend on what happens to exchange rates. Changes in relative inflation or interest rates will lead to changes in exchange rates. International trade deficits/surpluses affect exchange rates. Also, an increase in country risk will also cause the country’s currency to fall. l.

What are financial securities? Describe some financial instruments.

Answer: Financial securities are pieces of paper with contractual obligations. Short-term securities (i.e., they mature in less than a year) are instruments with low default risk such as treasury bills, banker’s acceptances, commercial paper, and Eurodollar deposits. Commercial loans (which have maturities up to 7 years) have rates that are usually tied to the prime rate (i.e., the rate that banks charge their best customers) or LIBOR (the London Interbank Offered Rate), which is the rate that banks in the U.K. charge one another. Government of Canada bonds have maturities from 2 to 30 years; they are free of default risk. Mortgages have maturities up to 30 years. Corporate bonds have maturities up to 40 years. Corporate bonds are subject to default risk. Some preferred shares have no maturity date; some do have a specific maturity date. Common shares have no maturity date, and are riskier than preferred shares. m.

List some financial institutions.

Answer: Commercial banks, trust companies, credit unions, life insurance companies, mutual funds, and pension funds are examples of financial institutions.

What are some different types of markets?

Answer: A market is a method of exchanging one asset (usually cash) for another asset. Some types of markets are physical assets versus financial assets; spot versus future markets; money versus capital markets; and primary versus secondary markets.

Chapter 2 Financial Statements, Cash Flow, and Taxes ANSWERS TO END-OF-CHAPTER QUESTIONS 2-1

a. The annual report is a report issued annually by a corporation to its shareholders. It contains basic financial statements, as well as management’s opinion of the past year’s operations and the firm’s future prospects. A firm’s balance sheet is a statement of the firm’s financial position at a specific point in time. It specifically lists the firm’s assets on the left-hand side of the balance sheet, while the right-hand side shows its liabilities and equity, or the claims against these assets. An income statement is a statement summarizing the firm’s revenues and expenses over an accounting period. Net sales are shown at the top of the statement, after which various costs, including income taxes, are subtracted to obtain the net income available to common shareholders. The bottom of the statement reports earnings per share. b. Common shareholders’ equity (net worth) is the capital supplied by common shareholders—capital stock, paid-in capital, retained earnings, and, occasionally, certain reserves. Paid-in capital is the amount that the shareholders paid when they

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bought newly issued shares. Retained earnings is the portion of the firm’s earnings that have been saved rather than paid out as dividends. c. The statement of changes in equity shows how much of the firm’s earnings were retained in the business rather than paid out in dividends. Note that retained earnings represents a claim against assets, not assets per se. Firms retain earnings primarily to expand the business, not to accumulate cash in a bank account. The statement of cash flows reports the impact of a firm’s operating, investing, and financing activities on cash flows over an accounting period. d. Depreciation is a non-cash charge against tangible assets, such as buildings or machines. It is taken for the purpose of showing an asset’s estimated dollar cost of the capital equipment used up in the production process. Amortization is a non-cash charge against intangible assets, such as copyrights. EBITDA is earnings before interest, taxes, depreciation, and amortization. e. Operating current assets are the current assets used to support operations, such as cash, accounts receivable, and inventory. It does not include short-term investments. Operating current liabilities are the current liabilities that are a natural consequence of the firm’s operations, such as accounts payable and accruals. It does not include notes payable or any other short-term debt that incurs interest charges. Net operating working capital is operating current assets minus operating current liabilities. Total net operating capital is the sum of net operating working capital and operating longterm assets, such as net plant and equipment. Operating capital also is equal to the net amount of capital raised from investors. This is the amount of interest-bearing debt plus preferred shares plus common equity minus short-term investments. f. Accounting profit is a firm’s net income as reported on its income statement. Net cash flow, as opposed to accounting net income, is the sum of net income plus non-cash adjustments. NOPAT, net operating profit after taxes, is the amount of profit a company would generate if it had no debt and no financial assets. Free cash flow is the cash flow available for distribution to investors after the company has made all investments in fixed assets and working capital necessary to sustain ongoing operations. Return on invested capital is equal to NOPAT divided by total net operating capital. It shows the rate of return that is generated by assets. g. Market Value Added is the difference between the market value of the firm (i.e., the sum of the market value of common equity, the market value of debt, and the market value of preferred shares) and the book value of the firm’s common equity, debt, and preferred shares. If the book values of debt and preferred shares are equal to their market values, then MVA is also equal to the difference between the market value of equity and the amount of equity capital that investors supplied. Economic Value Added represents the residual income that remains after the cost of all capital, including equity capital, has been deducted. Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc. 2-11

h. A progressive tax means the higher one’s income, the larger the percentage paid in taxes. Taxable income is defined as gross income less a set of exemptions and deductions that are spelled out in the instructions to the tax forms individuals must file. Marginal tax rate is defined as the tax rate on the last unit of income. Average tax rate is calculated by taking the total amount of tax paid divided by taxable income. i. Capital gain (loss) is the profit (loss) from the sale of a capital asset for more (less) than its purchase price. Ordinary corporate operating losses can be carried back for 3 years or forward for 20 years to offset taxable income in a given year. 2-2

The four financial statements contained in most annual reports are the balance sheet, income statement, statement of changes in equity, and statement of cash flows.

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No, because the $20 million of retained earnings would probably not be held as cash. The retained earnings figure represents the reinvestment of earnings by the firm. Consequently, the $20 million would be an investment in all of the firm’s assets.

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The balance sheet lists assets the company owns at any one point in time. The assets can change daily, such as inventories that are bought and sold daily. A company’s balance sheet can look different at different times of the year. The income statement reflects the company’s performance over a period of time. Though it can cover any time period, the income statement is usually prepared quarterly, semi-annually, or annually.

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Operating capital is the amount of interest-bearing debt, preferred shares, and common equity used to acquire the company’s net operating assets. Without this capital a firm cannot exist, as there is no source of funds with which to finance operations.

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NOPAT is the amount of net income a company would generate if it had no debt and held no financial assets. NOPAT is a better measure of the performance of a company’s operations because debt lowers income. In order to get a true reflection of a company’s operating performance, one would want to take out debt to get a clearer picture of the situation.

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Free cash flow is the cash flow actually available for distribution to investors after the company has made all the investments in fixed assets and working capital necessary to sustain ongoing operations. It is the most important measure of cash flows because it shows the exact amount available to all investors.

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If the business were organized as a partnership or a proprietorship, its income could be taken out by the owners without being subject to double taxation. Also, if you expected to have losses for a few years while the company was getting started, if you were not incorporated, and if you had outside income, the business losses could be used to offset your other income and reduce your total tax bill. These factors would lead you to not incorporate the business.

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SOLUTIONS TO END-OF-CHAPTER PROBLEMS 2-1

Corporate yield = 4.25%; T = 27% AT yield = 4.25%(1 – 0.27) = 4.25%(0.73) = 3.1%.

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NI = $3,500,000; EBIT = $7,000,000; T = 30%; Interest = ? Set up an income statement, plug in the given values, and work from the bottom up. EBIT Interest EBT Taxes (30%) NI

$7,000,000 2,000,000 5,000,000 1,500,000 $3,500,000

EBT =

Interest = EBIT – EBT = $7,000,000 – $5,000,000 = $2,000,000. 2-3

NI = $7,900,000; EBIT = $13,000,000; T = 21%; Interest = ? Set up an income statement, plug in the given values, and work in the order of the steps shown below.

(3)

EBIT = $13,000,000 (Given) −Interest = 3,000,000 = EBIT – EBT = $13,000,000 – $10,000,000 = $3,000,000.

(1)

EBT = $10,000,000 NI = EBT(1− T)  EBT =

NI (1-T)

$7,900,000

=$10,000,000

0.79

(2) −Taxes (21%) = 2,100,000 = EBT(T) = $10,000,000)(0.21) = $2,100,000. NI = $7,900,000 (Given) More directly, use algebra to determine: Interest = EBIT – [NI/(1 – T)] = $13,000,000 − $7,900,000/(1− 0.21) = $3,000,000. 2-4

EBITDA = $25,000,000; NI = $14,000,000; Int = $2,000,000; T = 30%; DA = ?. Set up an income statement, plug in the given values, and work from the bottom up. EBITDA DA EBIT Int EBT Taxes (30%) NI

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$25,000,000 3,000,000 22,000,000 2,000,000 20,000,000 6,000,000 $14,000,000

EBITDA – DA = EBIT; DA = EBITDA – EBIT EBIT = EBT + Int = $20,000,000 + $2,000,000 (Given) $14,000,000 $14,000,000 = (1-T) 0.7 (Given)

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NI = $3,100,000; DEP = $250,000; AMORT = 0; NCF = ? NCF = NI + DEP + AMORT = $3,100,000 + $250,000 = $3,350,000.

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NI = $70,000,000; R/EY/E = $900,000,000; R/EB/Y = $855,000,000; Dividends = ? R/EB/Y + NI – Div = R/EY/E $855,000,000 + $70,000,000 – Div = $900,000,000 $925,000,000 – Div = $900,000,000 Div = $25,000,000.

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Cash Provided (Used) Operating Activities Net income Adjustments: Noncash adjustments: Amortization Due to changes in working capital Increase in accounts receivable Decrease in inventories Increase in accounts payable Net cash provided by operating activities

$18,000

4,000 (4,000) 8,000 4,000 $30,000

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NOPAT = EBIT(1 – T) = $4,000,000(1 – 0.25) =$3,000,000.

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Total net operating capital = Net fixed assets + net operating working capital = Net fixed assets + (Operating CA – Operating CL) = $15,000,000 + ($10,000,000 – $3,000,000) = $22,000,000.

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Free cash flow = NOPAT – net investment in total operating capital = NOPAT – (Total net operating capital in current year – total net operating capital in previous year) = $16,000,000 – ($12,000,000 − $10,000,000) = $14,000,000.

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Interest Income = $400 Capital Gain = $10,000/$50 = 200 shares × $2/share = $400 Interest Income $400 × 16.67% = $66.68 $400 × 20.5% = 82.00 Total tax = $148.68 $400 – $148.68 =

Capital Gain $400 × 16.67% × 0.5 = $400 × 20.5% × 0.5 = Total tax =

$33.34 41.00 $74.34

$400 – $74.34 = $325.66

$251.32

2-12 Sales COGS (65%) Gross profit Other operating expenses Depreciation EBT Taxes (26%) Earnings After Tax + Depreciation Cash Flow

$5,000,000 3,250,000 1,750,000 700,000 60,000 990,000 257,400 732,600 60,000 $ 792,600

$5,000,000 3,250,000 1,750,000 700,000 150,000 900,000 234,000 666,000 150,000 $ 816,000

The difference between the cash flows is $23,400 ($816,000 – $792,600). Company Y has larger cash flows because it claimed higher depreciation expense. Since depreciation is a tax-deductible item, it creates a tax shield. The depreciation tax shield for Y is $39,000 ($150,000 × 0.26), while the depreciation tax shield for Company X is $15,600 ($60,000 × 0.26). The difference in tax shields of $23,400 accounts for the difference in cash flows. 2-13

EBIT = $750,000; DEP = $200,000; 100% Equity; T = 27%; NI = ?; NCF = ? First, determine net income by setting up an income statement: EBIT Interest EBT Taxes (27%) NI

$750,000 0 750,000 202,500 $547,500

NCF = NI + DEP = $547,500 + $200,000 = $747,500. 2-16

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a. Income Statement Sales revenue Costs except depreciation Depreciation Earnings before tax Taxes (27%) Net income Add back depreciation Net cash flow

$12,000,000 9,000,000 1,500,000 1,500,000 405,000 1,095,000 1,500,000 $ 2,595,000

b. If depreciation doubled, taxable income would fall to zero and taxes would be zero. Thus, net income would decrease to zero, but net cash flow would rise to $3,000,000. Berndt would save $405,000 in taxes, thus increasing its cash flow: ∆CF = T(∆Depreciation) = 0.27($1,500,000) = $405,000. c. If depreciation were halved, taxable income would rise to $2,250,000 and taxes to $607,500. Therefore, net income would rise to $1,642,500, but net cash flow would fall to $2,392,500. d. You should prefer to have higher depreciation charges and higher cash flows. Net cash flows are the funds that are available to the owners to withdraw from the firm and, therefore, cash flows should be more important to them than net income. 2-15

NOPAT = EBIT(1 – Tax rate) = $80 million (1 – 0.30) = $56 million FCF = NOPAT – Net investment in operating capital FCF = $56 million – $30 million FCF = $26 million Market Value of 10% of shares = 0.10($22 × 10 million) = $22 million $26 million is the free cash flow available to investors. The after-tax cost of paying $10 million to creditors in interest is $7 million ($10 million × (1 – 0.30) and therefore $19 million is available to repurchase shares. Since $22 million is needed to repurchase 10% of its shares, Marine Tech will not be able to fully carry out its repurchase plan.

2-16

NOPAT = EBIT(1 – T) = $80,000(1 – 0.25) =$60,000.

2-17

NOWC = Operating CA – Operating CL = (Cash + AR + INV) – (AP + Accruals) = ($90 + $1,200 + $900) – ($600 + $200) = $1,390 million.

2-18

Net investment in operating capital = (NOWC + Op LT assets) – (Total net Op Cap. in previous year) = ($13 + $51) – ($50) = $14 million.

2-19

a. EBIT x (1 – Tax rate) Net operating profit after taxes (NOPAT)

$1,008 × 0.75 $ 756

Cash + Accounts receivable + Inventories Operating current assets

b. 2021 550 2,750 1,650 4,950

2020 $ 500 2,500 1,500 4,500

Accounts payable + Accruals Operating current liabilities

1,100 550 1,650

1,000 500 1,500

Operating current assets – Operating current liabilities Net operating working capital (NOWC)

4,950 1,650 $3,300

4,500 1,500 $3,000

Net operating working capital (NOWC) + Net plant and equipment Total net operating capital

2021 $3,300 3,850 $7,150

2020 $3,000 3,500 $6,500

NOPAT – Investment in total net operating capital Free cash flow

2021 $756 650 $106

2-18

e. NOPAT ÷ Total net operating capital Return on invested capital (ROIC)

2021 $ 756 7,150 10.57%

f. Uses of FCF After-tax interest payment = Reduction (increase) in debt = Payment of dividends = Repurchase (Issue) stock = Purchase (Sale) of short-term investments = Total uses of FCF =

2021 $ 90 -284 202 88 10 $106

2-20 a. MVA = Market Value of Stock – Equity capital supplied by shareholders = ($60)(120 shares) – ($4,312 + $1,464) = $1,424.

EVA = (Operating capital)(ROIC – WACC) = ($7,150)(0.1057 – 0.12) = –$102.245.

2-21

Rhodes Corporation: Statement of Cash Flows for 2021 (Millions of Dollars) Cash Provided or (Used) Operating Activities Net income Adjustments: Noncash adjustments: Depreciation Due to changes in working capital Increase in accounts receivable Increase in inventories Increase in accounts payable Increase in accruals Net cash provided by operating activities Investing Activities Cash used to acquire fixed assets Change in short-term investments Net cash used by investing activities Financing Activities Increase in notes payable Increase in long-term debt Purchase of shares Payment of dividends Net cash used by financing activities Summary Increase in cash Cash at beginning of year Cash at end of year

2-20

$666

380 (250) (150) 100 50 796 (730) (10) (740) 184 100 (88) (202) (6) 50 500 $550

2-22

Prior Years Profit earned Carry-back credit Adjusted profit Tax previously paid (40%) Tax refund: Taxes previously paid

2018

2019

2020

$150,000 150,000 0

60,000

$ 60,000

Total cheque from the Canada Revenue Agency= $60,000 +$60,000 +$60,000=$180,000. Future Years

2021

2022

2023

2024

2025

Estimated profit Carryforward credit Adjusted profit Tax (at 40%)

($650,000) 0 $0 $0

$150,000 150,000 $ 0 $ 0

$150,000 50,000 $100,000 $ 40,000

$150,000 $150,000 0 0 $150,000 $150,000 $ 60,000 $ 60,000

2-23

NOPAT = EBIT(1 – T) = $1,750,000(0.75) = $1,312,500. NOWC21 = Operating CA – Operating CL = ($4,830,000 – $380,000) – ($3,700,000 – $1,050,000) = $1,800,000. NOWC22 = $4,980,000 – ($3,878,000 – $903,000) = $2,005,000. Operating Capital21 = Net plant and equipment + NOWC = $3,670,000 + $1,800,000 = $5,470,000. Operating Capital22 = Net plant and equipment + NOWC = $4,320,000 + $2,005,000 = $6,325,000. FCF = NOPAT – Net investment in operating capital = $1,312,500 – ($6,325,000 – $5,470,000) = $457,500.

ROIC =

NOPAT $1,312,500 = = 0.2075 = 20.75% Operating Capital $6,325,000

EVA=(Operating capital)(ROIC –WACC)=($6,325,000)(0.2075 – 0.13)=$490,188. MVA = Market value of stock – Equity capital supplied by shareholders = $20 x 400,000 – $4,922,000 = $3,078,000.

2-22

2-24

Bristle Brush-Off Corporation Statement of Cash Flows, Year Ended December 31, 2022 ($000s)

Cash Provided or (Used) Operating Activities Net income Adjustments: Noncash adjustments: Depreciation Due to changes in working capital Increase in accounts receivable Increase in inventory Increase in accounts payable Increase in accruals Net cash provided by operating activities Investing Activities Cash used to acquire fixed assets Proceed from sale of short-term investments Net cash used by investing activities Financing Activities Repayment of notes payable Payment of cash dividends* Net cash used by financing activities Summary Decrease in cash Cash at beginning of year Cash at end of year



$ 822

750 (190) (360) 310 15 1,347 (1,400) 380 (1,020) (147) (200) (347) (20) 60 $ 40

$1,100 + $822 − $1,722 = $200

SOLUTIONS TO SPREADSHEET PROBLEMS 2-25 and 2-26 The detailed solutions for the spreadsheet problems are in the file Ch 02 Build a Model Solution.xlsx and are available on the textbook’s website.

2-24

MINI CASE The detailed solution for the Mini Case is also available in spreadsheet format, in the file Brigham4Ce_MiniCase_Ch02.xlsx and is available on the textbook’s website. Jenny Cochran, a graduate of the University of Ottawa with 4 years of experience as an equities analyst, was recently brought in as assistant to the chairman of the board of Computron Industries, a manufacturer of computer components. During the previous year, Computron had doubled its plant capacity, opened new sales offices outside its home territory, and launched an expensive advertising campaign. Cochran was assigned to evaluate the impact of the changes. She began by gathering financial statements and other data. Balance Sheets Assets Cash and cash equivalents Short-term investments Accounts receivable Inventories Total current assets Gross fixed assets Less: Accumulated depreciation Net plant and equipment Total assets

2021 $

$4,900

Liabilities and equity Accounts payable Notes payable Accruals Total current liabilities Long-term bonds Total liabilities Common stock Retained earnings Total equity Total liabilities and equity

50 10 520 820 1,400 4,820 1,320 3,500

2020 $

60 100 400 620 1,180 3,900 1,000 2,900

$4,080

400 250 240 890 1,100 1,990 1,000 1,910 2,910

300 50 200 550 800 1,350 1,000 1,730 2,730

4,900

$ 4,080

3-26

Income Statement Net sales Cost of goods sold (excluding depreciation) Depreciation and Other operating expenses Total operating costs Earnings before interest and taxes (EBIT) Less interest Pre-tax earnings Taxes (25%)

2021 $ 6,000 4,800 320 420 5,540 460 108 352 88

2020 $ 5,500 4,300 290 350 4,940 560 68 492 123

Net Income

Other Data Stock price Shares outstanding (millions) Common dividends (millions) Tax rate Weighted average cost of capital (WACC)

2021 $30.00 100 $84 25% 10%

264

369

2020 $50.00 100 $90 25% 10%

Statement of Cash Flows Operating Activities Net income before preferred dividends Noncash adjustments Depreciation Due to changes in working capital Change in accounts receivable Change in inventories Change in accounts payable Change in accruals Net cash provided by operating activities

$ 264 320 (120) (200) 100 40 404

Investing activities Cash used to acquire fixed assets Change in short-term investments Net cash provided by investing activities

(920) 90 (830)

Financing Activities Change in notes payable Change in long-term debt Payment of cash dividends Net cash provided by financing activities

200 300 (84) 416

Net change in cash and cash equivalents Cash and cash equivalents at beginning of the year

(10) 60

Cash and cash equivalents at end of the year

2021

What effect did the expansion have on sales and net income? What effect did the expansion have on the asset side of the balance sheet? What effect did it have on liabilities and equity?

Answer: Sales increased by $500 million (9% growth), but net income fell by $105 million. Current assets and net plant and equipment each grew by over 20%. Large increases in debt funded the expansion, causing a 59% increase in interest payments.

What do you conclude from the statement of cash flows?

Answer: Net CF from operations was positive but was dragged down by a large net increase in working capital. Net CF from investing was negative even though the firm sold shortterm investments. This was because the expenditures in fixed assets were so high. Net CF from financing shows heavy borrowing. Even after borrowing, the cash account fell. c.

What is free cash flow? Why is it important? What are FCF’s five uses?

Answer: FCF is the amount of cash available from operations for distribution to all investors (including shareholders and debtholders) after making the necessary investments to support operations. A company’s value depends upon the amount of FCF it can generate. 1. Pay interest on debt. 2. Pay back principal on debt. 3. Pay dividends. 4. Buy back stock. 5. Buy nonoperating assets (e.g., marketable securities, investments in other companies)

3-28

What is Computron’s net operating profit after taxes (NOPAT)? What are operating current assets? What are operating current liabilities? How much net operating working capital and total net operating capital does Computron have?

Answer: NOPAT = EBIT(1 – Tax rate) Current year: NOPAT = 460(1 - 0.25) = $345. Previous year: NOPAT = $420. Operating current assets are the CA needed to support operations and include cash, inventory, and receivables. Operating current assets exclude short-term investments because these are not a part of operations. Operating current liabilities are those resulting as a normal part of operations and include accounts payable and accruals. Operating current liabilities exclude notes payable because this is a source of financing, not a part of operations. NOWC = operating CA – operating CL Current year: NOWC = ($50 + $520 + $820) - ($400 + $240) = $1,390 - $640 = $750. Previous year: NOWC = $580, Total operating working capital = NOWC + net fixed assets. Current year: Operating capital = $750 + $3,500 = $4,250. Previous year: Operating capital = $3,480.

What is Computron’s free cash flow (FCF)? What are Computron’s ―net uses‖ of its FCF?

Answer: FCF = NOPAT - Net investment in capital = $345 - ($4,250 - $3,480) = $345 - $770 = -$425. Uses of FCF: After-tax interest payment = Reduction (increase) in debt = Payment of dividends = Repurchase (Issue) stock = Purchase (Sale) of short-term investments = Total uses of FCF = f.

$81 −$500 $84 $0 −$90 −$425

Calculate Computron’s return on invested capital (ROIC). Computron has a 10% cost of capital (WACC). What caused the decline in the ROIC? Was it due to operating profitability or capital utilization? Do you think Computron’s growth added value?

ANSWER:

ROIC = NOPAT/TOTAL NET OPERATING CAPITAL.

Current year: ROIC = $345/$4,250 = 8.1%. Previous year: ROIC = 12.1%. Current year: Operating profitability = 5.8%. Previous year: Operating profitability Current year: Capital utilization = 70.8%. Previous year: Capital utilization

3-30

= $345/$6,000

= 7.6%.

= $4,250/$6,000

= 63.3%.

The current ROIC dropped from the previous year. This decline was due to worse operating profitability (5.8% versus 7.6%) and worse capital utilization (CR ratio of 70.8% versus a CR ratio of 63.3%). The ROIC is less than the WACC of 10%. Investors did not get the return they require. Note: high growth usually causes negative FCF (due to investment in capital), but that’s OK if ROIC > WACC. g.

Cochran also has asked you to estimate Computron's EVA. She estimates that the after-tax cost of capital was 10 percent in both years.

ANSWER:

EVA = NOPAT- (WACC)(CAPITAL).

Current year: EVA = $345 - (0.1)($4,250) = $345 - $425 = -$80. Previous year: EVA = $420 - (0.10)($3,480) = $420 - $348 = $72. h.

What happened to Computron's market value added (MVA)?

Answer: MVA = market value of the firm - book value of the firm. Market value = (# shares of stock)(price per share) + market value of debt. Book value = total common equity + book value of debt. If the market value of debt is close to the book value of debt, then MVA is market value of equity minus book value of equity. Assume market value of debt equals book value of debt. Current year: Market value of equity = (100)($30.00) = $3,000. Book value of equity = $2,910. MVA = $3,000 - $2,910 = $90. Previous year: MVA = 100($50.00) - $2,730 = $2,270.

Chapter 3 Analysis of Financial Statements ANSWERS TO END-OF-CHAPTER QUESTIONS 3-1

a. Liquidity ratios show the relationship of a firm’s cash and other current assets to its current liabilities. The current ratio is found by dividing current assets by current liabilities. It indicates the extent to which current liabilities are covered by those assets expected to be converted to cash in the near future. The quick, or acid test, ratio is found by taking current assets less inventories and then dividing by current liabilities. b. Asset management ratios are a set of ratios that measure how effectively a firm is managing its assets. The inventory turnover ratio is cost of sales divided by inventories. Days’ sales outstanding is used to appraise accounts receivable and indicates the length of time the firm must wait after making a sale before receiving cash. The average payables period measures the length of time it takes for a company to pay its suppliers. It is found by dividing payables by average cost of sales per day. The fixed assets turnover ratio measures how effectively the firm uses its plant and equipment. It is the ratio of sales to net fixed assets. Total assets turnover ratio measures the turnover of all the firm’s assets; it is calculated by dividing sales by total assets. c. Financial leverage ratios measure the use of debt financing. The debt ratio is the ratio of total debt to total assets. The times-interest-earned ratio is determined by dividing earnings before interest and taxes by the interest charges. This ratio measures the extent to which operating income can decline before the firm is unable to meet its annual interest costs. The EBITDA coverage ratio is similar to the times-interestearned ratio, but it recognizes that many firms lease assets and also must make sinking fund payments. It is found by adding EBITDA and lease payments, then dividing this total by interest charges, lease payments, and sinking fund payments. d. Profitability ratios are a group of ratios that show the combined effects of liquidity, asset management, and debt on operations. The profit margin on sales, calculated by dividing net income by sales, gives the profit per dollar of sales. Basic earning power is calculated by dividing EBIT by total assets. This ratio shows the raw earning power of the firm’s assets, before the influence of taxes and leverage. Return on total assets is the ratio of net income available to common shareholders to total assets. Return on common equity is found by dividing net income available to common shareholders by common equity.

3-32

e. Market value ratios relate the firm’s stock price to its earnings and book value per share. The price/earnings ratio is calculated by dividing price per share by earnings per share—this shows how much investors are willing to pay per dollar of reported profits. The price/cash flow ratio is calculated by dividing price per share by cash flow per share. This shows how much investors are willing to pay per dollar of cash flow. Market/book ratio is simply the market price per share divided by the book value per share. Book value per share is common equity divided by the number of shares outstanding. f. Trend analysis is an analysis of a firm’s financial ratios over time. It is used to estimate the likelihood of improvement or deterioration in its financial situation. Comparative ratio analysis is when a firm compares its ratios to other leading companies in the same industry. This technique is also known as benchmarking. g. The DuPont equation is a formula which shows that the rate of return on assets can be found as the product of the profit margin times the total assets turnover. Window dressing is a technique employed by firms to make their financial statements look better than they really are. Seasonal factors can distort ratio analysis. At certain times of the year, a firm may have excessive inventories in preparation for a ―season‖ of high demand. Therefore, an inventory turnover ratio taken at this time as opposed to after the season will be radically distorted. 3-2

The emphasis of the various types of analysis is by no means uniform, nor should it be. Management is interested in all types of ratios for two reasons. First, the ratios point out weaknesses that should be strengthened; second, management recognizes that the other parties are interested in all the ratios and that financial appearances must be kept up if the firm is to be regarded highly by creditors and equity investors. Equity investors are interested primarily in profitability, but they examine the other ratios to get information on the riskiness of equity commitments. Long-term creditors are more interested in the debt ratio, TIE, and fixed-charge coverage ratios, as well as the profitability ratios. Shortterm creditors emphasize liquidity and look most carefully at the liquidity ratios.

3-3

Given that sales have not changed, a decrease in the total assets turnover means that the company’s assets have increased. Also, the fact that the fixed assets turnover ratio remained constant implies that the company increased its current assets. Since the company’s current ratio increased and yet its quick ratio is unchanged means that the company has increased its inventories.

3-4

Differences in the amounts of assets necessary to generate a dollar of sales cause asset turnover ratios to vary among industries. For example, a steel company needs a greater number of dollars in assets to produce a dollar in sales than does a grocery store chain. Also, profit margins and turnover ratios may vary due to differences in the amount of expenses incurred to produce sales. For example, one would expect a grocery store chain to spend more per dollar of sales than does a steel company. Often, a large turnover will be associated with a low profit margin (the elements of the DuPont analysis), and vice versa.

3-5

a. Cash, receivables, and inventories, as well as current liabilities, vary over the year for firms with seasonal sales patterns. Therefore, those ratios that examine balance sheet figures will vary unless averages (monthly ones are best) are used. b. Common equity is determined at a point in time, say, December 31, 2021. Profits are earned over time, say, during 2021. If a firm is growing rapidly, year-end equity will be much larger than beginning-of-year equity, so the calculated rate of return on equity will be different depending on whether end-of-year, beginning-of-year, or average common equity is used as the denominator. Average common equity is conceptually the best figure to use. In public utility rate cases, people are reported to have deliberately used end-of-year or beginning-of-year equity to make returns on equity appear excessive or inadequate. Similar problems can arise when a firm is being evaluated.

3-6

Firms within the same industry may employ different accounting techniques, which make it difficult to compare financial ratios. More fundamentally, comparisons may be misleading if firms in the same industry differ in their other investments. For example, comparing PepsiCo and Coca-Cola may be misleading because, apart from their soft drink business, Pepsi also owns other businesses such as Frito-Lay, Gatorade, and Quaker Oats.

3-34

SOLUTIONS TO END-OF-CHAPTER PROBLEMS 3-1

DSO = 20 days; ADS =

$20,000; AR = ? DSO =

Accounts receivable Sales 365

20 =

Accounts receivable $20,000

Accounts receivable = $400,000

3-2

A/E = 3.2; D/A = ?

  D  1  = 1A  A    E D  1  = 1  A  3.2  D = 1 – 0.3125 = 0.6875 = 68.75% A 3-3

3-4

TA = $35,000,000; CL = $5,000,000; LT debt = $10,000,000; CE = $20,000,000; Shares outstanding = 2,500,000; P0 = $20; M/B = ? Book value per share =

$20,000,000 = 8.0 2,500,000

Market to ook

$20 = 2.5 8.0

EPS = $1.50; CFPS = $3.00; P/CF = 8.0; P/E = ? P/CF = 8.0 P/$3.00 = 8.0 P = $24.00. P/E = $24.00/$1.50 = 16.0.

3-5 Average daily COGS =

$2,800,000 = $7,671/day 365

Increase in payables = $7,671/day × 6 days = $46,026.

3-6

ROA = 12%; PM = 5%; ROE = 20%; S/TA = ?; A/E = ? ROA = NI/A; PM = NI/S; ROE = NI/E ROA = PM  S/TA NI/A = NI/S  S/TA 12% = 5%  S/TA S/TA = 2.40. ROE = PM  S/TA  TA/E NI/E = NI/S  S/TA  TA/E 20% = 5%  2.4  TA/E 20% = 12%  TA/E TA/E = 1.67.

3-7

CA = $3,000,000;

CA - Inv CA = 1.5; = 1.0; CL CL

CL = ?; Inv = ? CA =1.5 CL $3,000,000 =1.5 CL 1.5CL=$3,000,000 CL=$2,000,000 CA - Inv =1.0 CL $3,000,000 - Inv =1.0 $2,000,000 $3,000,00 - Inv = $2,000,000 Inv 1,000,000

3-36

3-8

We are given ROA = 4%, ROE = 7%, and Sales/Total assets = 1.2. From DuPont equation: ROA = Profit margin  Total assets turnover 4% = Profit margin × 1.2 Profit margin = 4%/1.2 = 3.33%. For the debt ratio: ROE = ROA  EM 7% = 4%  EM EM = 7%/4% = 7/4 = TA/E. Take reciprocal: E/TA = 4/7 = 0.571%; therefore, D/A = 1 – 0.571 = 0.429 = 42.9%.

3-9

Present current ratio =

$1,312,500 = 2.5. $525,000

Minimum current ratio =

$1,312,500 +  NP = 2.0. $525,000 +  NP

$1,312,500 + ∆NP = $1,050,000 + 2∆NP ∆NP = $262,500. Short-term debt can increase by a maximum of $262,500 without violating a 2 to 1 current ratio, assuming that the entire increase in notes payable is used to increase current assets. Since we assumed that the additional funds would be used to increase inventory, the inventory account will increase to $637,500, and current assets will total $1,575,000. Quick ratio = ($1,575,000 – $637,500)/$787,500 = $937,500/$787,500 = 1.19.

3-10

TIE = EBIT/INT, so find EBIT and INT Interest = $600,000  0.08 = $48,000 Net income = $3,000,000  0.03 = $90,000 Pre-tax income = $90,000/(1 – T) = $90,000/0.75 = $120,000 EBIT = $120,000 + $48,000 = $168,000 TIE = $168,000/$48,000 = 3.5

3-11 ROE19

$207,000/$2,600,000

7.96%

25.87%

ROE20

$222,000/$3,000,000

7.40%

27.66%

ROE21

$245,000/$4,000,000

6.13%

28.45%

$2,600,000/$2,000,000 ×

1.30

$2,000,000/$800,000 ×

$3,000,000/$2,250,000 ×

1.33

$2,250,000/$800,000 ×

$4,000,000/$3,000,000 ×

1.33

2.50

2.81

$3,000,000/$860,000 ×

3.49

The ROE is 25.9%, 27.7% and 28.5% (rounded) for years 2019, 2020, and 2021, respectively. The company’s ROE growth is not sustainable as the ROE increase is due to taking on more debt as a way of boosting its returns. In fact, while asset turnover is consistent, the profitability of the company’s operations has decreased each year. 3-12 Current ratio =

Current assets $468,000 = = 2.6 Current liabilities Current liabilities Current liabilities =

$468,000 = $180,000 2.6

Current assets + x $468,000 + x = = 2.0 $180,000 + x Current liabilities + x $468,000 + x = 2.0($180,000 + x) $468,000 + x = 2x + $360,000 x = $108,000 The company can increase its inventory by $108,000 without its current ratio falling below 2.0.

3-38

3-13 BEP = EBIT/Total assets = 8% EBIT = 8% × Total assets = 8% × $15 million = $1.2 million ROA = Net Income/Total assets = 4.8% Net Income = 4.8% × Total assets = 4.8% × $15 million = $720,000 EBIT $1,200,000 Interest ? EBT ? Tax (30%) Net income $720,000 EBT = $720,000/(1 – T) = $720,000/(1 – 0.3) = $1,028,571 Interest = EBIT – EBT = $1.200,000 – $1,028,571 = $171,429 TIE = EBIT/Interest = $1,200,000/$171,429 = 7.0. 3-14

EBIT

ROA = 9%; BEP = 14%, Assets = $14 million; TIE = Interest expense = ? EBIT = 14% Assets EBIT = 14% × Assets = 0.14 × $14,000,000 = $1,960,000 BEP =

Net income = 9% Assets Net income = 9% × Assets = 0.09 × $14,000,000 = $1,260,000 ROA =

EBIT (Interest expense) EBT (Taxes) Net income

$1,960,000 ? EBT = Net income + Taxes 30% of EBT $1,260,000 70% of EBT

0.70EBT = $1,260,000 EBT = $1,260,000/0.70 = $1,800,000

Interest expense = EBIT – EBT = $1,960,000 – $1,800,000 = $160,000 TIE= 3-15

EBIT $1,960,000 = =12.25x Interest expense $160,000 AR

DSO = Sales

⁄365

$750,000

= Sales

⁄365

=55

$750,000 = 55(Sales/365) Sales = $4,977,273 15% less sales: $4,977,273(0.85) = $4,230,682 New level of receivables (at 35 days) = $4,230,682/365 × 35 = $405,682. 3-16

1. Sales = (1.5)(Total assets) = (1.5)($400,000) = $600,000. 2. Cost of goods sold = (Sales)(1 – 0.25) = ($600,000)(0.75) = $450,000. 3. Accounts receivable = (Sales/365)(DSO) = ($600,000/365)(36.5) = $60,000. 4. Inventory = COGS/3.75 = $450,000/3.75 = $120,000.

5. Accounts payable = Average daily COGS × APP = $450,000/365 × 89.22 = $110,000. Total liabilities and equity – Total liabilities – Retained earnings = $400,000 – $160,000 – $100,000 = $140,000.

6. Common stock =

7. Cash + Accounts receivable = (0.80)(Accounts payable) Cash + $60,000 = (0.80)($110,000) Cash = $88,000 – $60,000 = $28,000. 8. Fixed assets = Total assets – (Cash + Accts rec. + Inventories) = $400,000 – ($28,000 + $60,000 + $120,000) = $192,000.

3-40

3-17 Current assets Current liabilities

= 2.5

$810,000 Current liabilities

= 2.5

Current liabilities = $324,000. Current assets - Inventories =1.1 Current liabilities

$810,000- Inventories = 1.1 $324 ,000 Inventories = $453,600. Current Marketable Accounts = Cash + + + Inventorie s assets Securities receivable

$810,000 = $120,000 + Accounts receivable + $453,600 Accounts receivable = $236,400. COGS = 4.0 Inventory COGS = 4.0 $453,600

COGS= $1,814,400 Gross rofit = 0.10 Sales Sales - COGS = 0.10 Sales Sales - $1,814,400 = 0.10 Sales ,0

,000

DSO = 3-18

Accounts receivable $236,400 = = 42.8 days. Sales/ 365 $2,016,000/365

a. (Dollar amounts in thousands.) Firm

3-42

Industry Average

Current assets Current liabilities

$2,925,000 $1,453,500

2.01

2.0

Accounts receivable Sales/ 365

$1,575,000 $20,548

77 days

35 days

COGS Inventory

$6,375,000 $1,125,000

5.67

6.7

Sales Fixed assets

$7,500,000 $1,350,000

5.56

12.1

Sales Total assets

$7,500,000 $4,275,000

1.75

3.0

Net income Sales

$131,858 $7,500,000

1.76%

1.2%

Net income Total assets

$131,858 $4,275,000

3.1%

3.6%

Net income Common equity

$131,858 $1,752,750

7.5%

9.0%

Total debt Total assets

$1,395,384 $4,275,000

32.6%

30.0%

Total liabilities Total assets

$2,522,250 $4,275,000

59%

60.0%

b. For the firm, ROE = PM  T.A. turnover  EM = 1.76%  1.75 

$4,275,000 = 7.51%. $1,752,750

For the industry, ROE = 1.2%  3  2.5 = 9%. Note: To find the industry ratio of assets to common equity, recognize that 1 minus the liabilities-to-assets ratio = common equity/total assets. So, common equity/total assets = 1 – 60% = 40%, and 1/0.40 = 2.5 = total assets/common equity. c. The firm’s days’ sales outstanding is more than twice as long as the industry average, indicating that the firm should tighten credit or enforce a more stringent collection policy. The total assets turnover ratio is well below the industry average so sales should be increased, assets decreased, or both. While the company’s profit margin is higher than the industry average, its other profitability ratios are a bit lower compared to the industry—net income could be higher given the amount of equity and assets. However, the company seems to be in an average liquidity position and financial leverage is similar to others in the industry.

3-19

Here are the firm’s base case ratios and other data as compared to the industry:

Quick Current Inventory turnover Days’ sales outstanding Fixed assets turnover Total assets turnover Return on assets Return on equity Profit margin on sales Debt-to-asset ratio EPS Stock price P/E ratio M/B ratio

$511,000/$602,000 $1,405,000/$602,000 $3,580,000/$894,000 $439,000/$11,753 $4,290,000/$431,000 $4,290,000/$1,836,000 $108,408/$1,836,000 $108,408/$829,710 $108,408/$4,290,000 $504,290/$1,836,000 $4.71 $23.57 $23.57/$4.71 $23.57/$36.07

Firm = 0.85 = 2.33 = 4.00 = 37.35 days = 9.95 = 2.34 = 5.90% = 13.07% = 2.53% = 27.47%

= 5.00 = 0.65

Industry 1.0 2.7 7.0 32 days 13.0 2.6 9.1% 18.2% 3.5% 21.0% n.a. n.a. 6.0 3.5

Comment Weak Weak Poor Poor Poor Poor Bad Bad Bad High --Poor Poor

The firm appears to be badly managed—all of its ratios are worse than the industry averages, and the result is low earnings, a low P/E, P/CF ratio, a low stock price, and a low M/B ratio. The company needs to do something to improve.

SOLUTION TO SPREADSHEET PROBLEM 3-20 The detailed solution for the spreadsheet problem is in the file Ch 03 Build a Model Solution.xlsx and is available on the textbook’s website.

3-44

MINI CASE The detailed solution for the Mini Case is also available in spreadsheet format, in the file Brigham4Ce_MiniCase_Ch03.xlsx, and is available on the textbook’s website. The first part of the case, presented in Chapter 2, discussed the situation of Computron Industries after an expansion program. A large loss occurred in 2021, rather than the expected profit. As a result, its managers, directors, and investors are concerned about the firm’s survival. Jenny Cochran was brought in as assistant to Computron’s chair, who had the task of getting the company back into a sound financial position. Cochran must prepare an analysis of where the company is now, what it must do to regain its financial health, and what actions to take. Your assignment is to help her answer the following questions. Provide clear explanations, not yes or no answers. Use the recent and projected financial information shown. Balance Sheets Assets Cash and cash equivalents Short-term investments Accounts receivable

2022E $

60 50 530

2021 $

50 10 520

2020 $

60 100 400

Inventories Total current assets

660 1,300

820 1,400

620 1,180

Net fixed assets

3,700

3,500

2,900

Total assets

$5,000

$4,900

$4,080

Liabilities and equity Accounts payable Notes payable

2022E $ 330 100

2021 $ 400 250

2020 $ 300 50

Accruals Total current liabilities

270 700

240 890

200 550

Long-term bonds Total liabilities Common stock (100,000 shares)

1,100 1,800

1,100 1,990

800 1,350

1,000

Retained earnings

2,200

1,910

1,730

Total common equity

3,200

2,910

2,730

Total liabilities and equity

$5,000

$4,900

$4,080

Note: ―E‖ denotes the ―estimated forecast.‖

Income Statements Net sales Cost of goods sold (Excluding depr.)

2022E $6,600 5,210

2021 $6,000 4,800

2020 $5,500 4,300

Depreciation

370

320

290

Other operating expenses Earnings before interest and taxes (EBIT)

400 620

420 460

350 560

Less interest Pre-tax earnings

100 520

108 352

68 492

Taxes (25%)

130

123

Net income

$ 390

$ 264

$ 369

Note: ―E‖ denotes the ―estimated forecast.‖ Also, Computron has no amortization. Other Data Per Share Information EPS DPS Book value per share

2022E

2021

2020

$3.90 $1.00 $32.00

$2.64 $0.84 $29.10

$3.69 $0.90 $27.30

Additional Information Dividends Additions to retained earnings Year-end shares outstanding Year-end common stock price Lease payments Tax rate

$100 $290 100 $49.00 $20 25%

$84 $180 100 $30.00 $20 25%

$90 $279 100 $50.00 $20 25%

Note: ―E‖ denotes the ―estimated forecast.‖

Answers and Solutions: 4-46

Ratio Analysis Profit margin Operating profit margin Basic earning power ROA ROE Inventory turnover Days’ sales outstanding Fixed assets turnover Total assets turnover Current Quick Debt ratio Debt-to-equity ratio Equity multiplier TIE EBITDA coverage Price/earnings (P/E) Market/book

2022E

2021 4.4% 7.7% 9.4% 5.4% 9.1% 6.2 31.6 1.7 1.224 1.6 0.7 27.6% 0.46 1.7 4.3 6.3 11.4 1.0

2020 6.7% 10.2% 13.7% 9.0% 13.5% 7.4 26.5 1.9 1.348 2.1 1.0 20.8% 0.31 1.5 8.2 9.9 13.6 1.8

Industry Average 7.2% 10.4% 15.6% 10.8% 15.4% 9.0 28.0 3.0 1.5 2.5 1.4 15.0% 0.22 1.5 13.0 17.2 16.8 2.6

Why are ratios useful? What three groups use ratio analysis and for what reasons?

Answer: Ratios facilitate comparison of (1) one company over time and (2) one company versus other companies. Ratios are used by managers to help improve the firm’s performance, by lenders to help evaluate the firm’s likelihood of repaying debts, and by shareholders to help forecast future earnings and cash flows.

Calculate the projected profit margin, operating profit margin, basic earning power (BEP), return on assets (ROA), and return on equity (ROE). What can you say about these ratios?

Answer: Projected Profit Margin = Net Income/Sales = $390/$6,600 = 5.9%. Projected Operating Profit Margin = EBIT/Sales = $620/$6,600 = 9.4%. Projected Basic Earning Power = EBIT/Total Assets = $620/$5,000 = 12.4%. Projected ROA = Net Income/Total Assets = $390/$5,000 = 7.8%. Projected ROE = Net Income/Common Equity = $390/$3,200 = 12.2%.

Profitability Ratios Profit margin Operating profit margin Basic earning power ROA ROE

2022E 5.9% 9.4% 12.4% 7.8% 12.2%

2021 4.4% 7.7% 9.4% 5.4% 9.1%

2020 6.7% 10.2% 13.7% 9.0% 13.5%

Industry Average 7.2% 10.4% 15.6% 10.8% 15.4%

The projected profitability ratios are improved but still are below the industry average.

Answers and Solutions: 4-48

Calculate the projected inventory turnover, days’ sales outstanding (DSO), fixed assets turnover, and total assets turnover. How does Computron’s utilization of assets stack up against other firms in its industry?

Answer: Projected Inventory Turnover = COGS/Inventory = ($5,210 + $370)/$660= 8.5. Projected DSO = Receivables/(Sales/365) = $530/($6,600/365) = 29.3 Days. Projected Fixed Assets Turnover = Sales/Net Fixed Assets = $6,600/$3,700 = 1.8. Projected Total Assets Turnover = Sales/Total Assets = $6,600/$5,000 = 1.32. Asset Management Ratios Inventory turnover Days’ sales outstanding Fixed assets turnover Total assets turnover

2022E 8.5 29.3 1.8 1.32

2021 6.2 31.6 1.7 1.22

2020 Industry 9.0 7.4 28.0 26.5 3.0 1.9 1.5 1.35

All the asset management ratios are projected to improve, but they still are not as good as the industry’s ratios. d.

Calculate the projected current and quick ratios based on the projected balance sheet data. What can you say about the company’s liquidity position and its trend?

Answer: Projected Current Ratio = Current Assets/Current Liabilities = $1,300/$700 = 1.9. Projected Quick Ratio = (Current Assets – Inventory)/Current Liabilities = ($1,300 – $660)/$700 = 0.9. Liquidity Ratios Current ratio Quick ratio

2022E 2021 1.9 1.6 0.9 0.7

2020 Industry 2.1 2.5 1.0 1.4

Calculate the projected debt ratio, the debt-to-equity ratio, equity multiplier, times-interest-earned ratio, and EBITDA coverage ratio. How does Computron compare with the industry with respect to financial leverage? What can you conclude from these ratios?

Answer: Projected Debt Ratio = Total Debt/Total Assets = ($100+ $1,100)/$5,000 = 24.0%. Projected Debt-to-Equity Ratio

= Total Debt/Common Equity = ($100 + $1,100)/$3,200 = 0.38.

Projected Equity Multiplier

= Total Assets/Common Equity = $5,000/$3,200 = 1.56

Projected TIE = EBIT/Interest = $620/$100 = 6.2 times. Lease   Loan Lease     /  Interest   Proj. EBITDA Cov. =  EBITDA  Payments   Repayments Payments  

= ($620 + $370 + $20)/($100 + $20) = 8.4.

Answers and Solutions: 4-50

Debt Management Ratios Debt ratio Debt-to-equity Earnings multiplier Times interest earned EBITDA coverage

2022E 2021 24.0% 27.6% 0.38 0.46 1.56 1.7 6.2 4.3 8.4 6.3

2020 Industry 20.8% 15.0% 0.31 0.22 1.5 1.5 13.0 8.2 17.2 9.9

The company definitely has greater than average leverage, making it more risky than the average firm in the industry. Its projected ratios for 2022 appear to show an improvement in its high leverage, though this trend will need to continue for several more years before it comes closer to the overall industry’s averages. f.

Calculate the projected price/earnings ratio and market/book ratio. Do these ratios indicate that investors are expected to have a high or low opinion of the company?

Answer: EPS = Net Income/Shares Outstanding = $390/100 = $3.90. Projected Price/Earnings = Price Per Share/Earnings Per Share = $49.00/$3.90 = 12.6. Projected BVPS = Common Equity/Shares Outstanding = $3,200/100 = $32.00. Projected Market/Book = Market Price Per Share/Book Value Per Share = $49.00/$32.00 = 1.5.

Market Value Ratios Earnings per share Price/earnings (P/E) Book value per share Market/book

2022E $3.90 12.6 $32.00 1.5

2021 $2.64 11.4 $29.10 1.0

2020 $3.69 13.6 $27.30 1.8

Industry Average n/a 16.8 n/a 2.6

All the market value ratios are projected to improve, but the P/E ratio and market/book still are below the industry average.

Perform a common size analysis and percentage change analysis. What do these analyses tell you about Computron?

Answer: For the common size balance sheets, divide all items in a year by the total assets for that year. For the common size income statements, divide all items in a year by the sales in that year. Common Size Balance Sheets Assets 2022E 2021 Cash and cash equivalents 1.2% 1.0% Short-term investments 1.0% 0.2% Accounts receivable 10.6% 10.6% Inventories 13.2% 16.7% Total current assets 26.0% 28.6% Net fixed assets 74.0% 71.4% Total assets 100.0% 100.0%

2020 1.5% 2.5% 9.8% 15.2% 28.9% 71.1% 100.0%

Industry 1.5% 24.9% 11.5% 12.1% 50.0% 50.0% 100.0%

Common Size Balance Sheets Liabilities and equity Accounts payable Notes payable Accruals Total current liabilities Long-term bonds Total liabilities Common stock Retained earnings Total common equity

2022E 6.6% 2.0% 5.4% 14.0% 22.0% 36.0% 20.0% 44.0% 64.0%

2021 8.2% 5.1% 4.9% 18.2% 22.4% 40.6% 20.4% 39.0% 59.4%

2020 7.4% 1.2% 4.9% 13.5% 19.6% 33.1% 24.5% 42.4% 66.9%

Total liabilities and equity

100.0%

Answers and Solutions: 4-52

Indust ry 6.8% 3.0% 10.2% 20.0% 12.0% 32.0% 27.2% 40.8% 68.0% 100.0 %

Common Size Income Statements 2022E 2021 Net sales 100.0% 100.0% COGS except depr. 78.9% 80.0% Depreciation 5.6% 5.3% Other expenses 6.1% 7.0% EBIT 9.4% 7.7% Less interest 1.5% 1.8% Pre-tax earnings 7.9% 5.9% Taxes (25%) 2.0% 1.5% Net income 5.9% 4.4%

2020 100.0% 78.2% 5.3% 6.4% 10.2% 1.2% 8.9% 2.2% 6.7%

Industry 100.0% 69.0% 3.3% 17.3% 10.4% 0.8% 9.6% 2.4% 7.2%

Computron has a much higher proportion of net fixed assets than the industry. Computron’s total debt is 24% (the combined percentages of notes payable and longterm bonds) of its assets, which is higher than the industry’s combined debt percentage of 15%. Computron’s profit margin is less than the industry ratio because Computron has much higher cost of goods sold, even though it has lower other costs.

For the percent change analysis, divide all items in a row by the value in the first year of the analysis. Percentage Change Income Statements 2022E 2021 Net sales 20.0% 9.1% Cost of goods sold 21.2% 11.6% Depreciation 27.6% 10.3% Other expenses 14.3% 20.0% –17.9% EBIT 10.7% Less interest 47.1% 58.8% – 28.5% Pre-tax earnings 5.7%

2020 0% 0% 0% 0% 0% 0% 0%

Taxes (25%)

5.7%

–28.5%

Net income

5.7%

–28.5%

Percent Change Balance Sheets Assets 2022E Cash and cash equivalents 0.0% Short-term investments –50.0% Accounts receivable 32.5% Inventories 6.5% Total current assets 10.2% Net fixed assets 27.6% Total assets 22.5%

2021

2020

–16.7% –90.0% 30.0% 32.3% 18.6% 20.7% 20.1%

0% 0% 0% 0% 0% 0% 0%

Percent Change Balance Sheets Liabilities and equity Accounts payable Notes payable Accruals Total current liabilities Long-term bonds Total liabilities Common stock Retained earnings Answers and Solutions: 4-54

2022E 10.0% 100.0% 35.0% 27.3% 37.5% 33.3% 0.0% 27.2%

2021 33.3% 400.0% 20.0% 61.8% 37.5% 47.4% 0.0% 10.4%

20 20 0% 0% 0% 0% 0% 0% 0% 0%

Total common equity Total liabilities and equity

17.2% 22.5%

6.6% 20.1%

0% 0%

We see that, in the most recent year, total assets grew by 20.1% relative to the baseline (with inventory growing by 32.3%). However, actual sales only grew by 9.1%, which indicates trouble, with unsold inventory causing much of the problem. Other expenses also grew rapidly, causing a steep decline in net income (–28.5%). Part of this decline is due to the 58.8% increase in interest expense. For the projected year, sales are projected to grow by a factor of 20.0% relative to the baseline year, which is about the same as total assets are projected to grow (22.5%). Note that net income is projected to grow by 5.7% from the baseline year. Much work remains, but the projections reflect the turnaround plans.

Use the extended DuPont equation to provide a breakdown of Computron’s projected return on equity. How does the projection compare with the previous years and with the industry’s DuPont equation?

Answer: DuPont Equation: ROE = Profit  Total Assets  Equity Margin Turnover Multiplier ROE

Net income Equity

Net income Total assets



Total assets Equity

Net income Sales



Sales Total assets



Total assets Equity

Profit margin



Total assets turnover



Equity multiplier

ROE2020 = (6.7%)(1.35)(1.5) = 13.5% ROE2021 = (4.4%)(1.22)(1.7) = 9.1% ROE2022E = (5.9%)(1.32)(1.56) = 12.2% ROEInd = (7.2%)(1.5)(1.47) = 15.9 Computron’s profit margin and total assets turnover ratio were below the industry average. It had a higher equity multiplier because it had more debt, which boosted its ROE. Otherwise, its ROE would have been even lower than the industry ROE.

Answers and Solutions: 4-56

What are some potential problems and limitations of financial ratio analysis?

Answer: Some potential problems are listed below: 1. Comparison with industry averages is difficult if the firm operates many different divisions. 2. Different operating and accounting practices distort comparisons. 3. Sometimes hard to tell if a ratio is ―good‖ or ―bad.‖ 4. Difficult to tell whether company is, on balance, in a strong or weak position. 5. ―Average‖ performance is not necessarily good. 6. Seasonal factors can distort ratios. 7. ―Window dressing‖ techniques can make statements and ratios look better. j.

What are some qualitative factors that analysts should consider when evaluating a company’s likely future financial performance?

Answer: Top analysts recognize that certain qualitative factors must be considered when evaluating a company. These factors, as summarized by the American Association of Individual Investors (AAII), are as follows: 1. Are the company’s revenues tied to one key customer? 2. To what extent are the company’s revenues tied to one key product? 3. To what extent does the company rely on a single supplier? 4. What percentage of the company’s business is generated overseas? 5. Competition 6. Future prospects 7. Legal and regulatory environment

ANSWERS TO END-OF-CHAPTER QUESTIONS 4-1 a. PV (present value) is the value today of a future payment, or stream of payments, discounted at the appropriate rate of interest. PV is also the beginning amount that will grow to some future value. The parameter I is the periodic interest rate that an account pays. The parameter INT is the dollars of interest earned each period. FVN (future value) is the ending amount in an account, where N is the number of periods the money is left in the account. PVAN is the value today of a future stream of equal payments (an annuity) and FVAN is the ending value of a stream of equal payments, where N is the number of payments of the annuity. PMT is equal to the dollar amount of an equal or constant cash flow (an annuity). In the EAR equation, M is used to denote the number of compounding periods per year, while INOM is the nominal, or quoted, interest rate. b. The opportunity cost rate (I) of an investment is the rate of return available on the best alternative investment of similar risk. c. An annuity is a series of payments of a fixed amount for a specified number of periods. A single sum, or lump sum payment, as opposed to an annuity, consists of one payment occurring now or at some future time. A cash flow can be an inflow (a receipt) or an outflow (a deposit, a cost, or an amount paid). We distinguish between the terms ―cash flow‖ and ―PMT.‖ We use the term ―cash flow‖ for uneven streams, while we use the term ―PMT‖ for annuities, or constant payment amounts. An uneven cash flow stream is a series of cash flows in which the amount varies from one period to the next. The PV (or FVN) of an uneven payment stream is merely the sum of the present values (or future values) of each individual payment. d. An ordinary annuity has payments occurring at the end of each period. A deferred annuity is just another name for an ordinary annuity. An annuity due has payments occurring at the beginning of each period. Most financial calculators will accommodate either type of annuity. The payment period must be equal to the compounding period. e. A perpetuity is a series of payments of a fixed amount that last indefinitely. In other words, a perpetuity is an annuity where n equals infinity. Consol is another term for perpetuity. Consols were originally bonds issued by England in 1815 to consolidate past debt.

Answers and Solutions: 4-58

f. An outflow is a deposit, a cost, or an amount paid, while an inflow is a receipt. A time line is an important tool used in time value of money analysis; it is a graphical representation that is used to show the timing of cash flows. The terminal value is the future value of an uneven cash flow stream. g. Compounding is the process of finding the future value of a single payment or series of payments. Discounting is the process of finding the present value of a single payment or series of payments; it is the reverse of compounding. h. Annual compounding means that interest is paid once a year. In semiannual, quarterly, monthly, and daily compounding, interest is paid 2, 4, 12, and 365 times per year, respectively. When compounding occurs more frequently than once a year, you earn interest on interest more often, thus increasing the future value. The more frequent the compounding, the higher the future value. i. The effective annual rate is the rate that, under annual compounding, would have produced the same future value at the end of 1 year as was produced by more frequent compounding, say, quarterly. The nominal (quoted) interest rate, INOM, is the rate of interest stated in a contract. If the compounding occurs annually, the effective annual rate and the nominal rate are the same. If compounding occurs more frequently, the effective annual rate is greater than the nominal rate. The nominal annual interest rate is also called the annual percentage rate, or APR. The periodic rate, IPER, is the rate charged by a lender or paid by a borrower each period. It can be a rate per year, per 6month period, per quarter, per month, per day, or per any other time interval (usually 1 year or less). j. An amortization schedule is a table that breaks down the periodic fixed payment of an installment loan into its principal and interest components. The principal component of each payment reduces the remaining principal balance. The interest component is the interest payment on the beginning-of-period principal balance. An amortized loan is one that is repaid in equal periodic amounts (or ―killed off‖ over time). 4-2

The opportunity cost rate is the rate of interest one could earn on an alternative investment with a risk equal to the risk of the investment in question. This is the value of I in the TVM equations, and it is shown on the top of a time line, between the first and second tick marks. It is not a single rate—the opportunity cost rate varies depending on the riskiness and maturity of an investment, and it also varies from year to year depending on inflationary expectations.

4-3

True. The second series is an uneven payment stream, but it contains an annuity of $400 for 8 years. The series could also be thought of as a $100 annuity for 10 years plus an additional payment of $100 in Year 2, plus additional payments of $300 in Years 3 through 10.

4-4

True, because of compounding effects—growth on growth. The following example demonstrates the point. The annual growth rate is I in the following equation: $1(1 + I)10 = $2 The term (1 + I)10 is the FVIF for I%, 10 years. Using a financial calculator, input N = 10, PV = –1, PMT = 0, FV = 2, and I = ?. Solving for I, you obtain 7.18%.

4-5

For the same stated rate, daily compounding is best. You would earn more ―interest on interest.‖

Answers and Solutions: 4-60

SOLUTIONS TO END-OF-CHAPTER PROBLEMS 4-1

0 4% 1 | | PV = 5,000 FV5

2 |

3 |

4 |

5 | FV5 = ?

= $5,000(1.04)5 = $5,000(1.216653) = $6,083.27.

Alternatively, with a financial calculator, enter the following: N = 5, I = 4, PV = –5000, and PMT = 0. Solve for FV = $6,083.26. b.

FV5

= $10,000(1.04)5 = $10,000(1.216653) = $12,166.53.

With a financial calculator, enter the following: N = 5, I = 4, PV = –10000, and PMT = 0. Solve for FV = $12,166.53. c.

FV5

= $15,000(1.04)5 = $15,000(1.216653) = $18,249.80.

With a financial calculator, enter the following: N = 5, I = 4, PV = –15000, and PMT = 0. Solve for FV = $18,249.79. d.

FV5

= $20,000(1.04)5 = $20,000(1.216653) = $24,333.06.

With a financial calculator, enter the following: N = 5, I = 4, PV = –20000, and PMT = 0. Solve for FV = $24,333.06. 4-2

0 7% | PV = ?

5 |

10 |

15 |

20 | FV20 = 20,000

With a financial calculator, enter the following: N = 20, I = 7, PMT = 0, and FV = 20000. Solve for PV = $5,168.38. b.

With a financial calculator, enter the following: N = 20, I = 7, PMT = 0, and FV = 30000. Solve for PV = $7,752.57.

With a financial calculator, enter the following: N = 20, I = 7, PMT = 0, and FV = 40000. Solve for PV = $10,336.76.

With a financial calculator, enter the following: N = 20, I = 7, PMT = 0, and FV = 50000. Solve for PV = $12,920.95.

4-3

0 I=? | PV = 50,000

18 | FV18 = 1,000,000

With a financial calculator, enter the following: N = 18, PV = –50000, PMT = 0, and FV = 1000000. Solve for I = 18.11%. b. With a financial calculator, enter the following: N = 18, PV = –100000, PMT = 0, and FV = 1000000. Solve for I = 13.65%. c. With a financial calculator, enter the following: N = 18, PV = –200000, PMT = 0, and FV = 1000000. Solve for I = 9.35%. d. With a financial calculator, enter the following: N = 18, PV = –300000, PMT = 0, and FV = 1000000. Solve for I = 6.92%. 4-4

0 2.5% | PV = 1

N=? | FVN = 3

$3 = $1(1.025)N. With a financial calculator, enter the following: I = 2.5, PV = –1, PMT = 0, and FV = 3. Solve for N = 44.49 years. b. With a financial calculator, enter the following: I = 4, PV = –1, PMT = 0, and FV = 3. Solve for N = 28.01 years. c. With a financial calculator, enter the following: I = 7, PV = –1, PMT = 0, and FV = 3. Solve for N = 16.24 years. d. With a financial calculator, enter the following: I = 10, PV = –1, PMT = 0, and FV = 3. Solve for N = 11.53 years. 4-5

0 12% |

1 |

PV = 42,180.53 5,000

2 | 5,000



N–2 |

N–1 |

N |

5,000

FV = 250,000

Using your financial calculator, enter the following: I = 12; PV = –42180.53; PMT = −5000; FV = 250000; N = ? Solve for N = 11. It will take 11 years to accumulate $250,000.

Answers and Solutions: 4-62

4-6

Ordinary annuity: 0 7% 1 2 | | | 300 300

3 | 300

4 5 | | 300 300 FVA5 = ?

With a financial calculator, enter the following: N = 5, I = 7, PV = 0, and PMT = –300. Solve for FV = $1,725.22. Annuity due: 0 7% 1 | | 300 300

2 | 300

3 | 300

4 | 300

5 |

With a financial calculator, switch to ―BEG‖ and enter the following: N = 5, I = 7, PV = 0, and PMT = 300. Solve for FV = $1,845.99. Don’t forget to switch back to ―END‖ mode. 4-7

0 8% |

1 | 100

2 | 100

3 | 100

4 | 200

5 | 300

PV = ?

6 | 500 FV = ?

Using a financial calculator, enter the following: CF0 = 0; CF1 = 100; Nj = 3; CF4 = 200 (Note: Calculator will show CF2 on screen.); CF5 = 300 (Note: Calculator will show CF3 on screen.); CF6 = 500 (Note: Calculator will show CF4 on screen.); and I = 8. Solve for PV = $923.98. To solve for the FV of the cash flow stream with a calculator that doesn’t have the NFV key, do the following: Enter N = 6, I = 8, PV = –923.98, and PMT = 0. Solve for FV = $1,466.24.

4-8

Using a financial calculator, enter the following: N = 60, I = 1, PV = –20000, and FV = 0. Solve for PMT = $444.89. 



EAR = 1  NOM  – 1.0 

M 

= (1.01) – 1.0 = 12.68%. Alternatively, using a financial calculator, enter the following: NOM% = 12 and P/YR =12. Solve for EFF% = 12.6825%. Remember to change back to P/YR = 1 on your calculator. Brigham, Financial Management, 4Ce, Solutions Manual © 2023 Cengage Learning Canada, Inc. 4-63

4-9

–500

FV = ?

$500(1.06) = $530.00.

–500 c.

FV = ?

0 6%

PV = ?

500

$500(1/1.06) = $471.70.

PV = ?

4-10

$500(1.06)2 = $561.80.

$500(1/1.06)2 = $445.00.

500

–500 b.

10 $500(1.06)10 = $895.42.

FV = ?

0 12%1

–500

FV = ?

0 6% 1

| $500(1/1.06)

PV = ?

| $500(1/1.12)

PV = ?

7% | –200

= $279.20.

= $160.99.

500

12%

4-11

$500(1.12)10 = $1,552.92.

500

? | 400

With a financial calculator, enter I = 7, PV = –200, PMT = 0, and FV = 400. Then press the N key to find N = 10.24 ≈ 10. Answers and Solutions: 4-64

10%

–200

400

With a financial calculator, enter I = 10, PV = –200, PMT = 0, and FV = 400. Then press the N key to find N = 7.27 ≈ 7.

18%

–200

400

With a financial calculator, enter I = 18, PV = –200, PMT = 0, and FV = 400. Then press the N key to find N = 4.19 ≈ 4.

100%

–200

400

With a financial calculator, enter I = 100, PV = –200, PMT = 0, and FV = 400. Then press the N key to find N = 1.00.

4-12

10%

10 |

400

400 FV = ?

With a financial calculator, enter N = 10, I = 10, PV = 0, and PMT = –400. Then press the FV key to find FV = $6,374.97.

0 5%

200

200 FV = ?

With a financial calculator, enter N = 5, I = 5, PV = 0, and PMT = –200. Then press the FV key to find FV = $1,105.13.

0 |

400

400 FV = ?

With a financial calculator, enter N = 5, I = 0, PV = 0, and PMT = –400. Then press the FV key to find FV = $2,000.

d. To solve part d using a financial calculator, repeat the procedures discussed in parts a, b, and c, but first switch the calculator to ―BEG‖ mode. Make sure you switch the calculator back to ―END‖ mode after working the problem. (1)

0 10% 1

400

FV = ?

With a financial calculator set to ―BEG‖ mode, enter N = 10, I = 10, PV = 0, and PMT = –400. Then press the FV key to find FV = $7,012.47. (2)

0 5% 1

200

200 200

5 |

FV = ?

With a financial calculator set to ―BEG‖ mode, enter N = 5, I = 5, PV = 0, and PMT = –200. Then press the FV key to find FV = $1,160.38. (3)

5 |

400

FV = ?

With a financial calculator set to ―BEG‖ mode, enter N = 5, I = 0, PV = 0, and PMT = –400. Then press the FV key to find FV = $2,000.

4-13

0 10% 1

10 |

PV = ?

400

With a financial calculator, enter N = 10, I = 10, PMT = –400, and FV = 0. Then press the PV key to find PV = $2,457.83.

Answers and Solutions: 4-66

0 5% 1

PV = ?

200

With a financial calculator, enter N = 5, I = 5, PMT = –200, and FV = 0. Then press the PV key to find PV = $865.90.

PV = ?

400

With a financial calculator, enter N = 5, I = 0, PMT = –400, and FV = 0. Then press the PV key to find PV = $2,000.

d. (1)

10%

400 400 PV = ?

400

With a financial calculator set to ―BEG‖ mode, enter N = 10, I = 10, PMT = –400, and FV = 0. Then press the PV key to find PV = $2,703.61. (2)

0 5% 1

200 200 PV = ?

200

With a financial calculator set to ―BEG‖ mode, enter N = 5, I = 5, PMT = –200, and FV = 0. Then press the PV key to find PV = $909.19. (3)

0 0% 1

400 400 PV = ?

400

With a financial calculator set to ―BEG‖ mode, enter N = 5, I = 0, PMT = –400, and FV = 0. Then press the PV key to find PV = $2,000.00.

Answers and Solutions: 4-68

4-14

Cash Stream A 0 8% 1 2 3

Cash Stream B 0 8% 1 2 3 4

PV = ? 800

300

300 300

100

PV = ? 100 300 300

300 800

With a financial calculator, simply enter the cash flows (be sure to enter CF0 = 0), enter I = 8, and press the NPV key to find NPV = PV = $1,524.66 for the first problem. Override I = 8 with I = 0 to find the next PV for Cash Stream A for part b. Repeat for Cash Stream B to get NPV = PV = $1,352.92. b. PVA = $800 + $300 + $300 + $300 + $100 = $1,800. PVB = $100 + $300 + $300 + $300 + $800 = $1,800. 4-15

These problems can all be solved using a financial calculator by entering the known values shown on the time lines and then pressing the I button. a.

I=?

–749

+700

With a financial calculator, enter N = 1, PV = 700, PMT = 0, and FV = –749. Then press the I key to find I = 7%. b.

I=?

–700

+749

With a financial calculator, enter N = 1, PV = –700, PMT = 0, and FV = 749. Then press the I key to find I = 7%. c.

0 |

I=?

–201,229

+85,000

With a financial calculator, enter N = 10, PV = 85000, PMT = 0, and FV = –201229. Then press the I key to find I = 9%. d.

0 I=?

–2,684.80

+9,000 –2,684.80

With a financial calculator, enter N = 5, PV = 9000, PMT = –2684.80, and FV = 0. Then press the I key to find I = 15%.

4-16

0 10% | –2,000

1 |

2 |

3 |

4 |

5 | FV = ?

With a financial calculator, enter N = 5, I = 10, PV = –2000, and PMT = 0, and then press FV to obtain FV = $3,221.02.

0 5% 1

–2,000

10 |

FV = ?

With a financial calculator, enter N = 10, I = 5, PV = –2000, and PMT = 0, and then press FV to obtain FV = $3,257.79. c.

0 2.5% 4

–2,000

20 |

FV = ?

With a financial calculator, enter N = 20, I = 2.5, PV = –2000, and PMT = 0, and then press FV to obtain FV = $3,277.23. d.

0 0.833%12 |

–2,000

With a financial calculator, enter N = 60, I = .833333, PV = –2000, and PMT = 0, and then press FV to obtain FV = $3,290.62.

Answers and Solutions: 4-70

4-17

0 5% |

PV = ?

2,000

With a financial calculator, enter N = 10, I = 5, PMT = 0, and FV = –2000. Then press the PV key to find PV = $1,227.83. Alternatively,

   1   PV = FVN  1 I    M  = $2,000( b.

0 |

2.5%

)

( )

)

= $2,000(

= $2,000(PVIF5%, 10) = $2,000(0.6139) = $1,227.83.

20 |

PV = ?

2,000

With a financial calculator, enter N = 20, I = 2.5, PMT = 0, and FV = –2000. Then press the PV key to find PV = $1,220.54, or ( )

PV = $2,000(

)

0 0.833% 1

  

PV = ?

= $2,000(

)

= $1,220.54.

2,000

With a financial calculator, enter N = 12, I = .83333, PMT = 0, and FV = –2000. Then press the PV key to find PV = $1,810.42, or ( )

PV = $2,000(

)

= $2,000(

)

= $1,810.42.

4-18

0 5%

750

  

750

750 FV = ?

Enter N = 4  2 = 8, I = 10/2 = 5, PV = 0, PMT = –750, and then press FV to get FV = $7,161.83. b. Now the number of periods is calculated as N = 4 × 4 = 16, I = 10/4 = 2.5, PV = 0, and PMT = –375. The calculator solution is $7,267.58. Note that the solution assumes that the nominal interest rate is compounded at the annuity period. c. The annuity in part b earns more because some of the money is on deposit for a longer period of time and thus earns more interest. Also, because compounding is more frequent, more interest is earned on interest. 4-19

a. Universal Bank: Effective rate = 7%. Regional Bank: Effective rate

= 1  

0.06  4  – 1.0 = (1.015) – 1.0 4 

= 1.0614 – 1.0 = 0.0614 = 6.14%. With a financial calculator, you can use the interest rate conversion feature to obtain the same answer. You would choose Universal Bank. b. If funds must be left on deposit until the end of the compounding period (1 year for Universal and 1 quarter for Regional), and you think there is a high probability that you will make a withdrawal during the year, the Regional account might be preferable. For example, if the withdrawal is made after 10 months, you would earn nothing on the Universal account but (1.015)3 – 1.0 = 4.57% on the Regional account.

Answers and Solutions: 4-72

4-20

a. With a financial calculator, enter N = 4, I = 10, PV = –15000, and FV = 0, and then press the PMT key to get PMT = $4,732.06. Then go through the amortization procedure as described in your calculator manual to get the entries for the amortization table. Year 1 2 3 4

Payment $ 4,732.06 4,732.06 4,732.06 4,732.07* $18,928.25

Repayment of Principal $ 3,232.06 3,555.27 3,910.79 4,301.88 $15,000.00

Interest $1,500.00 1,176.79 821.27 430.19 $3,928.25

Remaining Balance $11,767.94 8,212.67 4,301.88 0

*The last payment must be larger to force the ending balance to zero. b. Here the loan size is doubled, so the payments also double in size to $9,464.12: enter N = 4, I = 10, PV = –30000, and FV = 0, and then press the PMT key to get PMT = $9,464.12. c. The annual payment on a $30,000, 8-year loan at 10% interest would be $5,623.32: enter N = 8, I = 10, PV = –30000, and FV = 0, and then press the PMT key to get PMT = $5,623.32. Because the payments are spread out over a longer time period, more interest must be paid on the loan. The total interest paid on the 8-year loan is 8($5,623.32) – $30,000 = $14,986.56 versus interest of 4($9,464.12) – $30,000 = $7,856.48 on the 4-year loan. 4-21

0 I=? 1

–2

6 (in millions)

With a calculator, enter N = 5, PV = –2, PMT = 0, FV = 6, and then solve for I = 24.57% ≈ 25%. b. The calculation described in the quotation fails to take account of the compounding effect. It can be demonstrated to be incorrect as follows: $2,000,000(1.40)5 = $2,000,000(5.37824) = $10,756,480, which is much greater than $6 million. Thus, the annual growth rate is less than 40%; in fact, it is about 25%, as shown in part a.

4-22

| I=? |

–4

8 (in millions)

With a calculator, enter N = 10, PV = –4, PMT = 0, FV = 8, and then solve for I = 7.18%. 4-23

0 I=?

70,000

–8,273.59

20   

–8,273.59

With a calculator, enter N = 20, PV = 70000, PMT = –8273.59, and FV = 0. Then solve for I = 10.1%. 4-24

0 7%

4 |

PV = ?

–10,000

With a calculator, enter N = 4, I = 7, PMT = –10000, and FV = 0. Then press PV to get PV = $33,872.11. b.

(1) At this point, we have a 3-year, 7% annuity of $10,000 whose present value is $26,243.16: N = 3, I = 7, PMT = –10000, and FV = 0. Then press PV to get PV = $26,243.16. You can also think of the problem as follows: (Beginning balance)(1 + I) – PMT = Ending balance $33,872.11 (1.07) – $10,000 = $26,243.16. (2)

4-25

Zero after the last withdrawal.

| 6%

12,000

–1,950

?   

–1,950

With a calculator, enter I = 6, PV = 12000, PMT = –1950, and FV = 0. Press N to get N = 7.91  8 years. Therefore, it will take approximately 8 years to pay back the loan.

Answers and Solutions: 4-74

4-26

| 12%

1,250

? FV = 10,000

With a financial calculator, get a ―ballpark‖ estimate of the years by entering I = 12, PV = 0, PMT = –1250, and FV = 10000, and then pressing the N key to find N = 5.94 years. This answer assumes that a payment of $1,250 will be made 94/100ths of the way through Year 6. Now find the FV of $1,250 for 5 years at 12%; N = 5, I = 12, PV = 0, and PMT = –1250. Press FV to get FV = $7,941.06. Compound this value for 1 year at 12% to obtain the value in the account after 6 years and before the last payment is made; it is $7,941.06(1.12) = $8,893.99. Thus, you will have to make a payment of $10,000 – $8,893.99 = $1,106.01 at Year 6, so the answer is: it will take 6 years, and $1,106.01 is the amount of the last payment. 4-27

PV = $10,000/0.05 = $200,000. PV = $10,000/0.10 = $100,000. When the interest rate is doubled, the PV of the perpetuity is halved.

4-28

0 8.24%

4 |

PV = ?

1,050

Discount rate: Effective rate on bank deposit: EAR = (1 + 0.08/4)4 – 1 = 8.24%. Find PV of above stream at 8.24%: with a financial calculator, N = 4, I = 8.24, PMT = 50, and FV = 1000, then press the PV key to get PV = $893.26. They are not worth $900 to you.

4-29

This can be done with a calculator by specifying an interest rate of 5% per period for 20 periods with 1 payment per period to get the payment each 6 months: N = 10  2 = 20, I = 10%/2 = 5, PV = –10000. FV = 0. Solve for PMT = $802.43. Set up an amortization table as below:

Period 1 2

Beg Bal. $10,000.00 9,697.57

Payment $802.43 802.43

Pmt. of Interest $500.00 484.88 $984.88

Pmt. of Principal $302.43

End Bal. $9,697.57

You can also work the problem with a calculator having an amortization function. Find the interest in each 6-month period, sum them, and you have the answer. Even simpler, some calculators, such as the HP-10BII, allow you to find the interest paid during a particular period of time. 4-30

First, find PMT by using a financial calculator: N = 7, I = 10, PV = –1000000, and FV = 0. Solve for PMT = $205,405.50. Then set up the amortization table: Year 1 2

Beginning Balance Payment $1,000,000.00 $205,405.50 894,594.50 205,405.50

Interest $100,000.00 89,459.45

Principal $105,405.50 115,946.05

Ending Balance $894,594.50 778,648.45

Fraction that is principal = $115,946.05/$205,405.50 = 0.5645 = 56.45%.

4-31

a. Effective monthly rate

= 1  

0.045   2 

2 / 12

– 1.0 = (1.0225)0.166667 – 1.0

= 1.00371532 – 1.0 = 0.00371532 = 0.371532%. Find PMT by using a financial calculator: N = 25 × 12 = 300, I = .371532, PV = –210000, and FV = 0. Solve for PMT = $1,162.29. b. Total payments over 5 years = 60 × $1,162.29 = $69,737.40 Remaining payments after 5 years: 300 – 60 = 240 Calculate the value of the mortgage after 5 years: N = 240, I = .371532, PMT = 1162.29, and FV = 0. Solve for PV = $184,371.90. Total principal paid over 5 years: $210,000 – $184,371.90 = $25,628.10. Total interest paid over 5 years: $69,737.40 – $25,628.10 = $44,109.30.

Answers and Solutions: 4-76

4-32

For 20 years: Find PMT by using a financial calculator: N = 20 × 12 = 240, I = .371532, PV = –210000, and FV = 0. Solve for PMT = $1,323.85. For 15 years: Find PMT by using a financial calculator: N = 15 × 12 = 180, I = .371532, PV = –210000, and FV = 0. Solve for PMT = $1,602.02.

4-33

Effective rate for loan A = 1  .01112 – 1.0 = 1.1403 – 1.0 = 0.1403 = 14.03%. Loan B cannot have a higher effective rate than 14.03%. 0 i=?

–$1.1403

Find the period interest rate: PV = 1, PMT = 0, N = 4, and FV = –1.1403. Solve for I = 3.337. 3.337% × 4 = 13.35%

4-34

0 i=3.5%

–$10,000

–$3,000

2 |

3 |

–$3,000

4 |

–$3,000

5 |

–$3,000

Effective rate = (1.03.5)2 – 1 = 1.0712 – 1 = 0.0712 = 7.12% Find the future value: PV = –10000, PMT = –3000, N = 5, and I = 7.12 Solve for FV = $31,397.92. 4-35

Bank A: N = 2, PV = 50000, PMT = 0, FV = –54800, and CPT for I = 4.69 = 4.69%. Bank B: N = 4, PV = 50000, PMT = –13900, FV = 0, and CPT for I = 4.3859 EAR = (1.043859)2 – 1 = 0.0896 = 8.96%. Bank C: N = 24, PV = 50000, PMT = –100, FV = –50000 and CPT for I = 0.20 EAR = (1.0020)12 – 1 = 0.024266 = 2.4266%. Bank C offers the lowest interest rate.

4-36

a. Begin with a time line: 6-mos. 0 Years 0

2 1

4 2

6 3

8 4 |

10 5

12 6 |

14 7 |

16 8 |

100 100 100 100 100 FVA

Since the first payment is made today, we have a 5-period annuity due. The applicable interest rate is I = 8/2 = 4 per period, N = 5, PV = 0, and PMT = –100. Setting the calculator on ―BEG,‖ we find FVA (Annuity due) = $563.30. That will be the value at the 5th 6-month period, which is t = 2.5. Now we must compound out to t = 8, or for 5.5 years at an EAR of 8.16%, or 11 semiannual periods at 4%. $563.30  16 – 5 = 11 periods @ 4%  $867.17, or $563.30  8 – 2.5 = 5.5 years @ 8.16%  $867.17. b. 0 |

1 4

PMT PMT PMT PMT PMT

10 years 40 quarters   

FV = 2,200

The time line depicting the problem is shown above. Because the payments only occur for 5 periods throughout the 40 quarters, this problem cannot be immediately solved as an annuity problem. The problem can be solved in two steps: (1)

Discount the $2,200 back to the end of Quarter 5 to obtain the PV of that future amount at Quarter 5.

(2)

Then solve for PMT using the value solved in Step 1 as the FV of the fiveperiod annuity due.

Step 1: Input the following into your calculator: N = 35, I = 3, PMT = 0, FV = 2200, and solve for PV at Quarter 5. PV5 = $781.84. Step 2: The PV found in Step 1 is now the FV for the calculations in this step. Change your calculator to the BEGIN mode. Input the following into your calculator: N = 5, I = 3, PV = 0, FV = 781.84, and solve for PMT = $142.97.

Answers and Solutions: 4-78

4-37

Cost of boat in 10 years: $65,000(1.02)10 = $79,234.64

Raj will have $47,621.70 at the end of 4 years: N = 4, PV = –10000, PMT = –8000, I = 6%, CPT for FV = $47,621.70. He will not make any deposits for the next three years, but will earn interest on his money: N = 3, PV = 47621.70, PMT = 0, I = 6%, and CPT for FV = $56,718.21. He will have $56,718.21 in his account after 7 years. He will need to make three annual payments of $3,669.54 to reach his goal: N = 3, PV = –56718.21, I = 6%, FV = 79234.64, and CPT for PMT = $3,669.54. 4-38

Here we want to have the same effective annual rate on the credit extended as on the bank loan that will be used to finance the credit extension. We need to find the effective annual rate (EAR) the bank is charging first. Then, we can use this EAR to calculate the nominal rate that should be quoted to the customers. Bank EAR: EAR = (1 + INOM/M)M – 1 = (1 + 0.08/12)12 – 1 = 8.30%. Nominal rate that should be quoted to customers: 8.30% = (1 + INOM/4)4 – 1 1.0830 = (1 + INOM/4)4 1.0201 = 1 + INOM/4 INOM = 0.0201(4) = 8.04%.

4-39

Information given: 1. Will save for 10 years, then receive payments for 25 years. 2. Wants payments of $40,000 per year in today’s dollars for first payment only. Real income will decline. Inflation will be 5%. Therefore, to find the inflated fixed payments, we have this time line: 0 5% |

5 |

40,000

10 |

FV = ?

Enter N = 10, I = 5, PV = –40000, PMT = 0, and press FV to get FV = $65,155.79. 3. He now has $100,000 in an account which pays 8%, annual compounding. We need to find the FV of the $100,000 after 10 years. Enter N = 10, I = 8, PV = –100000, PMT = 0, and press FV to get FV = $215,892.50. 4. He wants to withdraw, or have payments of, $65,155.79 per year for 25 years, with the first payment made at the beginning of the first retirement year. So, we have a 25year annuity due with PMT = 65,155.79, at an interest rate of 8%. (The interest rate is 8% annually, so no adjustment is required.) Set the calculator to ―BEG‖ mode, then enter N = 25, I = 8, PMT = 65155.79, FV = 0, and press PV to get PV = $751,165.35. This amount must be on hand to make the 25 payments. 5. Since the original $100,000, which grows to $215,892.50, will be available, we must save enough to accumulate $751,165.35 – $215,892.50 = $535,272.85. 6. The $535,272.85 is the FV of a 10-year ordinary annuity. The payments will be deposited in the bank and earn 8% interest. Therefore, set the calculator to ―END‖ mode and enter N = 10, I = 8, PV = 0, FV = 535272.85, and press PMT to find PMT = $36,949.61.

Answers and Solutions: 4-80

4-40

Information given: The nominal time line is shown below, with a different payment each period and a FV of a nominal $1 million:

0 |

PMT1 PMT2 PMT3 PMT4 PMT5

PMT6

  

PMT30 FV = $1 million

The key is to ―rewrite‖ this as a real time line (i.e., a time line based on today’s purchasing power). First, the purchasing power of $1 million in 30 years with an inflation rate of 2% per year is: $1,000,000/(1 + Inflation)N = $1,000,000/(1 + 0.02)30 = $552,070.89. This is a growing annuity, with a nominal rate of 7% and an inflation rate of 2%. You should use the real rate in the calculator: rr = [(1 + rNOM)/(1 + Inflation)] – 1.0 = [1.07/1.02] – 1.0 = 0.0490196 = 4.90196%. So the ―real‖ time line expressed in today’s purchasing power is: 4.90196%

PMT PMT PMT

PMT

29   

PMT

30 |

FV = $552,070.89

Set the financial calculator to the ―BEG‖ mode, N = 30, I = 4.90196, PV = 0, FV = 552070.89, and press PMT for PMT = $8,055.48. Thus, an initial payment of $8,055.48 that grows at 2% each year for 29 more payments, invested at a rate of 7% per year, will accumulate to $1 million at Year 30.

4-41 12/31/20 | 45,000.00

0 1 2 12/31/21 12/31/22 12/31/23 7% | | | 46,350.00 47,740.50 49,172.72 250,000.00 40,000.00 Payment will be made

3 12/31/24 | 50,647.90

Step 1: Calculate salary amounts (2020–2024): 2020: $45,000 2021: $45,000(1.03) = $46,350.00 2022: $46,350(1.03) = $47,740.50 2023: $47,740.50(1.03) = $49,172.72 2024: $49,172.72(1.03) = $50,647.90 Step 2: Compound backpay, pain and suffering, and legal costs to 12/31/22 payment date: $45,000(1.07)2 + $336,350(1.07)1 = $51,520.50 + $359,894.50 = $411,415.00. Step 3: Discount future salary back to 12/31/22 payment date: $47,740.50 + $49,172.72/(1.07)1 + $50,647.90/(1.07)2 = $47,740.50 + $45,955.81 + $44,237.84 = $137,934.15. Step 4: City must write a cheque for $411,415.00 + $137,934.15 = $549,349.15.

Answers and Solutions: 4-82

4-42

Step 1: Determine the annual cost of university. The current cost is $15,000 per year, but that is escalating at a 5% inflation rate: University Year 1 2 3 4

Current Cost $15,000 15,000 15,000 15,000

Years from Now 5 6 7 8

Inflation Adjustment (1.05)5 (1.05)6 (1.05)7 (1.05)8

Cash Required $19,144.22 20,101.43 21,106.51 22,161.83

Now put these costs on a time line: 13 |

14 |

15 |

16 |

17 |

18 19 20 21 | | | | –19,144 –20,101 –21,107 –22,162

How much must be accumulated by age 18 to provide these payments at ages 18 through 21 if the funds are invested in an account paying 6%, compounded annually? With a financial calculator, enter: CF0 = 19144, CF1 = 20101, CF2 = 21107, CF3 = 22162, and I/YR = 6. Solve for NPV = $75,500.00. Thus, the father must accumulate $75,500 by the time his daughter reaches age 18. Step 2: The daughter has $7,500 now (age 13) to help achieve that goal. Five years hence, that $7,500, when invested at 6%, will be worth $10,037: $7,500(1.06)5 = $10,036.69 ≈ $10,037. Step 3: The father needs to accumulate only $75,500 – $10,037 = $65,463. The key to completing the problem at this point is to realize that the series of deposits represent an ordinary annuity rather than an annuity due, despite that the first payment is made at the beginning of the first year. The reason it is not an annuity due is there is no interest paid on the last payment that occurs when the daughter is 18. Using a financial calculator, N = 6, I/YR = 6, PV = 0, and FV = –65463. PMT = $9,385.

SOLUTION TO SPREADSHEET PROBLEM 4-43

The detailed solution for the spreadsheet problem is in the file Ch 04 Build a Model Solution.xlsx and is available on the textbook’s website.

Answers and Solutions: 4-84

MINI CASE The detailed solution for the Mini Case is also available in spreadsheet format, in the file Brigham4Ce_MiniCase_Ch04.xlsx, and is available on the textbook’s website. Assume that you are nearing graduation and that you have applied for a job with a local bank. As part of the bank’s evaluation process, you have been asked to take an examination that covers several financial analysis techniques. The first section of the test addresses discounted cash flow analysis. See how you would do by answering the following questions. a.

Draw time lines for (1) a $100 lump sum cash flow at the end of Year 2, (2) an ordinary annuity of $100 per year for 3 years, and (3) an uneven cash flow stream of –$50, $100, $75, and $50 at the end of Years 0 through 3.

Answer: (Begin by discussing basic discounted cash flow concepts, terminology, and solution methods.) A time line is a graphical representation that is used to show the timing of cash flows. The tick marks represent ends of periods (often years), so time 0 is today; time 1 is the end of the first year, or 1 year from today; and so on. 0

Lump sum

year

100

Annuity 0

100

Uneven cash flow stream –50 100

A lump sum is a single flow; for example, a $100 inflow in Year 2, as shown in the top time line. An annuity is a series of equal cash flows occurring over equal intervals, as illustrated in the middle time line. An uneven cash flow stream is an irregular series of cash flows that do not constitute an annuity, as in the lower time line. –50 represents a cash outflow rather than a receipt or inflow.

1. What is the future value of an initial $100 after 3 years if it is invested in an account paying 10% annual interest?

Answer: Show dollars corresponding to question mark, calculated as follows: 0 |

10%

100

FV = ?

After 1 year: FV1 = PV + I1 = PV + PV(I) = PV(1 + I) = $100(1.10) = $110.00. Similarly: FV2 = FV1 + I2 = FV1 + FV1(I) = FV1(1 + I) = $110(1.10) = $121.00 = PV(1 + I)(1 + I) = PV(1 + I)2. FV3 = FV2 + I3 = FV2 + FV2(I) = FV2(1 + I) = $121(1.10) = $133.10 = PV(1 + I)2(1 + I) = PV(1 + I)3. In general, we see that:

FVN = PV(1 + I)N, FV3 = $100(1.10)3 = $100(1.3310) = $133.10.

Note that this equation has 4 variables: FVN, PV, I, and N. Here we know all except FVN, so we solve for FVN. We will, however, often solve for one of the other three variables. By far the easiest way to work all time value problems is with a financial calculator. Just plug in any 3 of the 4 values and find the 4th. Finding future values (moving to the right along the time line) is called compounding. Note that there are 3 ways of finding FV3: using a regular calculator, financial calculator, or spreadsheets. For simple problems, we show only the regular calculator and financial calculator methods. (1)

Regular calculator: 1. $100(1.10)(1.10)(1.10) = $133.10. 2. $100(1.10)3 = $133.10.

Mini Case: 4-86

(2)

Financial calculator: This is especially efficient for more complex problems, including exam problems. Input the following values: N = 3, I = 10, PV = –100, PMT = 0, and solve for FV = $133.10.

2. What is the present value of $100 to be received in 3 years if the appropriate interest rate is 10%?

Answer: Finding present values, or discounting (moving to the left along the time line), is the reverse of compounding, and the basic present value equation is the reciprocal of the compounding equation: 0 |

10%

PV = ?

100

FVN = PV(1 + I)N transforms to:

 1  FVN PV = = FVN   N (1  I) 1 I 

= FVN(1 + I)–N

thus: PV = $100 

1   = $100(PVIFI,N) = (0.7513) = $75.13.  1.10 

The same methods used for finding future values are also used to find present values. Using a financial calculator, input N = 3, I = 10, PMT = 0, FV = –100, and then solve for PV = $75.13.

We sometimes need to find how long it will take a sum of money (or anything else) to grow to some specified amount. For example, if a company’s sales are growing at a rate of 20% per year, how long will it take sales to double?

Answer: We have this situation in time line format: 0 20% |

3.8

–1

Say we want to find out how long it will take us to double our money at an interest rate of 20%. We can use any numbers, say $1 and $2, with this equation: FVN = $2 = $1(1 + I)N = $1(1.20)N. (1.2)N N Ln(1.2) N N

= $2/$1 = 2 = Ln(2) = Ln(2)/Ln(1.2) = 0.693/0.182 = 3.8.

Alternatively, we could use a financial calculator. We would plug I = 20, PV = –1, PMT = 0, and FV = 2 into our calculator, and then press the N button to find the number of years it would take 1 (or any other beginning amount) to double when growth occurs at a 20% rate. The answer is 3.8 years, but some calculators will round this value up to the next highest whole number. The graph also shows what is happening.

Mini Case: 4-88

FV 2

1 3.8

Year

If you want an investment to double in 3 years, what interest rate must it earn?

Answer:

1(1 + I)

1(1 + I)2

2 1(1 + I)3

–1 FV = $1(1 + I)3 $1(1 + I)3 (1 + I)3 1+I 1+I

= $2. = $2. = $2/$1 = 2. = (2)1/3 = 1.2599 = 25.99%.

Use a financial calculator to solve: enter N = 3, PV = –1, PMT = 0, FV = 2, then press the I button to find I = 25.99%. Calculators can find interest rates quite easily, even when periods and/or interest rates are not even numbers, and when uneven cash flow streams are involved. (With uneven cash flows, we must use the ―CFLO‖ function, and the interest rate is called the IRR, or ―internal rate of return‖; we will use this feature in capital budgeting.) e.

What is the difference between an ordinary annuity and an annuity due? What type of annuity is shown below? How would you change it to the other type of annuity? 0

100

Answer: This is an ordinary annuity—it has its payments at the end of each period; that is, the first payment is made 1 period from today. Conversely, an annuity due has its first payment today. In other words, an ordinary annuity has end-of-period payments, while an annuity due has beginning-of-period payments. The annuity shown above is an ordinary annuity. To convert it to an annuity due, shift each payment to the left, so you end up with a payment under the 0 but none under the 3.

1. What is the future value of a 3-year ordinary annuity of $100 if the appropriate interest rate is 10%?

Answer:

0 |

10%

100

100 110 121 $331

Go through the following discussion. One approach would be to treat each annuity flow as a lump sum. Here we have FVAN = $100(1) + $100(1.10) + $100(1.10)2 = $100[1 + (1.10) + (1.10)2] = $100(3.3100) = $331.00. Using a financial calculator, N = 3, I = 10, PV = 0, PMT = –100. This gives FV = $331.00. f.

2. What is the present value of the annuity?

Answer:

0 |

1 10%

100

90.91 82.64 75.13 248.68 The present value of the annuity is $248.68. Using a financial calculator, input N = 3, I = 10, PMT = 100, FV = 0, and press the PV button. Spreadsheets are useful for time lines with multiple cash flows. The following spreadsheet shows this problem: 1 2 3

A 0

B 1 100

C 2 100

D 3 100

248.69

The Excel formula in cell A3 is = NPV(10%,B2:D2). This gives a result of 248.69. Note that the interest rate can be either 10% or 0.10, not just 10. Also, note that the range does not include any cash flow at time zero.

Mini Case: 4-90

Excel also has special functions for annuities. For ordinary annuities, the Excel formula is = PV(interest rate, number of periods, payment). In this problem, = PV(10%,3,–100), gives a result of 248.69. For the future value, it would be = FV(10%,3,–100), with a result of 331. f.

3. What would the future and present values be if the annuity were an annuity due?

Answer: If the annuity were an annuity due, each payment would be shifted to the left, so each payment is compounded over an additional period or discounted back over one less period. To find the future value of an annuity due use the following formula: FVAN(Annuity Due) = FVAN(1 + I). In our situation, the future value of the annuity due is $364.10: FVA3(Annuity Due) = $331.00(1.10)1 = $364.10. This same result could be obtained by using the time line: $133.10 + $121.00 + $110.00 = $364.10. The best way to work annuity due problems is to switch your calculator to ―BEG‖ or beginning or ―DUE‖ mode, and go through the normal process. Note that it’s critical to remember to change back to ―END‖ mode after working an annuity due problem with your calculator. This formula could be used to find the present value of an annuity due: PVAN(Annuity Due) = PVAN(1 + I) = PMT(PVIFAI,N)(1 + I). In our situation, the present value of the annuity due is $273.56: PVA3(Annuity Due) = $248.69(1.10)1 = $273.56. The Excel function is = PV(10%,3,–100,0,1). The fourth term, 0, tells Excel there are no additional cash flows. The fifth term, 1, tells Excel it is an annuity due. The result is $273.56. A similar modification gives the future value: = FV(10%,3,–100,0,1), with a result of 364.10.

What is the present value of the following uneven cash flow stream? The appropriate interest rate is 10%, compounded annually. 0

4 years

100

300

–50

Answer: Here we have an uneven cash flow stream. The most straightforward approach is to find the PVs of each cash flow and then sum them as shown below: 0 10% 1

4 years

100

300

–50

90.91 247.93 225.39 (34.15) 530.08 Note that the $50, year 4 outflow remains an outflow even when discounted. There are numerous ways of finding the present value of an uneven cash flow stream. But by far the easiest way to deal with uneven cash flow streams is with a financial calculator or a spreadsheet. Calculators have a function, which on the HP 17B is called ―CFLO,‖ for ―cash flow.‖ Other calculators could use other designations such as cf0 and CFi, but they explain how to use them in the manual. You input the cash flows, so they are in the calculator’s memory, then input the interest rate, I, and then press the NPV or PV button to find the present value. Spreadsheets are especially useful for uneven cash flows. The following spreadsheet shows this problem: 1 2 3

A 0

B 1 100

C 2 300

D 3 300

E 4 -50

530.09

The Excel formula in cell A3 is = NPV(10%,B2:E2), with a result of 530.09.

Mini Case: 4-92

1. Define (a) the stated, or quoted, or nominal rate, (INOM), and (b) the periodic rate (IPer).

Answer: The quoted, or nominal, rate is merely the quoted percentage rate of return. The periodic rate is the rate charged by a lender or paid by a borrower each period (periodic rate = INOM/M). h.

2. Will the future value be larger or smaller if we compound an initial amount more often than annually, for example, every 6 months, or semiannually, holding the stated interest rate constant? Why?

Answer: Accounts that pay interest more frequently than once a year, for example, semiannually, quarterly, or daily, have future values that are higher because interest is earned on interest more often. Virtually all banks now pay interest daily on passbook and money market fund accounts, so they use daily compounding. h.

3. What is the future value of $100 after 5 years under 12% annual compounding? Semiannual compounding? Quarterly compounding? Monthly compounding? Daily compounding?

Answer: Under annual compounding, the $100 is compounded over 5 annual periods at a 12.0% periodic rate: INOM= 12%.

 I  FVN = PV1  NOM  M  



= $100 1  

0.12   1 

1*5

= $100(1.12)5 = $176.23.

Under semiannual compounding, the $100 is compounded over 10 semiannual periods at a 6.0% periodic rate: INOM = 12%.

 I  FVN = PV1  NOM  M  

 

= $100 1 

0.12   2 

2*5

= $100(1.06)10 = $179.08.

quarterly: FVN = $100(1.03)20 = $180.61. monthly: FVN = $100(1.01)60 = $181.67. daily: FVN = $100(1+ 0.12/365)365*5 = $182.19.

4. What is the effective annual rate (EFF%)? What is the EFF% for a nominal rate of 12%, compounded semiannually? Compounded quarterly? Compounded monthly? Compounded daily?

Answer: The effective annual rate is the annual rate that causes the PV to grow to the same FV as under multi-period compounding. For 12% semiannual compounding, the EFF is 12.36%:

 1  I NOM  EAR = Effective Annual Rate =    1.0.  M  M

IF INOM = 12% and interest is compounded semiannually, then:  

EAR = 1 

0.12  2   1.0 = (1.06) – 1.0 = 1.1236 – 1.0 = 0.1236 = 12.36%. 2 

For quarterly compounding, the effective annual rate is: (1.03)4 – 1.0 = 12.55%. For monthly compounding, the effective annual rate is: (1.01)12 – 1.0 = 12.68%. For daily compounding, the effective annual rate is: (1 + 0.12/365)365 – 1.0 = 12.75%. i.

Will the effective annual rate ever be equal to the nominal (quoted) rate?

Answer: If annual compounding is used, then the nominal rate will be equal to the effective annual rate. If more frequent compounding is used, the effective annual rate will be above the nominal rate.

Mini Case: 4-94

1. Construct an amortization schedule for a $1,000, 10% annual rate loan with 3 equal installments. 2. What is the annual interest expense for the borrower, and the annual interest income for the lender, during Year 2?

Answer: To begin, note that the face amount of the loan, $1,000, is the present value of a 3year annuity at a 10% rate: 0 |

10%

–1,000

PMT

 1   1   1  PVA3 = PMT   + PMT    + PMT  1 I  1 I  1 I 

$1,000 = PMT(1 + I)–1 + PMT(1 + I)–2 + PMT(1 + I)–3 = PMT(1.10)–1 + PMT(1.10)–2 + PMT(1.10)–3. We have an equation with only one unknown, so we can solve it to find PMT. The easy way is with a financial calculator. Input N = 3, I = 10, PV = –1,000, FV = 0, and then press the PMT button to get PMT = 402.1148036, rounded to $402.11. Now make the following points regarding the amortization schedule: 

The $402.11 annual payment includes both interest and principal. Interest in the first year is calculated as follows: 1st year interest = I  Beginning balance = 0.1  $1,000 = $100.



The repayment of principal is the difference between the $402.11 annual payment and the interest payment: 1st year principal repayment = $402.11 – $100 = $302.11.



The loan balance at the end of the first year is: 1st year ending balance



= Beginning balance – Principal repayment = $1,000 – $302.11 = $697.89.

We would continue these steps in the following years. The interest for Year 2 is 10% × $697.89 = $69.79.



Notice that the interest each year declines because the beginning loan balance is declining. Since the payment is constant, but the interest component is declining, the principal repayment portion is increasing each year.



The interest component is an expense that is deductible to a business and it is taxable income to the lender. If you buy a house, you will get a schedule constructed similar to ours, but longer, with 25  12 = 300 monthly payments if you get a 25-year, fixed-rate mortgage. Also, mortgages in Canada are compounded semiannually.



The payment may have to be adjusted by a few cents in the final year to take care of rounding errors and make the final payment produce a zero ending balance.



The lender received a 10% rate of interest on the average amount of money that was invested each year, and the $1,000 loan was paid off. This is what amortization schedules are designed to do.



Most financial calculators have amortization functions built in.

Suppose on January 1 you deposit $100 in an account that pays a nominal, or quoted, interest rate of 11.33463%, with interest added (compounded) daily. How much will you have in your account on October 1, or after 9 months?

Answer: The daily periodic interest rate is rPer = 11.3346%/365 = 0.031054%. There are 273 days between January 1 and October 1. Calculate FV as follows: FV273 = $100(1.00031054)273 = $108.85. Using a financial calculator, input N = 273, I = 0.031054, PV = –100, and PMT = 0. Pressing FV gives $108.85. An alternative approach would be to first determine the effective annual rate of interest, with daily compounding, using the formula: EAR = 1  

0.1133463   365 

365

– 1 = 0.12 = 12.0%.

(Some calculators, e.g., the HP 10BII and 17B, have this equation built in under the ICNV [interest conversion] function.) Thus, if you left your money on deposit for an entire year, you would earn $12 of interest, and you would end up with $112. The question, though, is this: how much will be in your account on October 1?

Mini Case: 4-96

Here you will be leaving the money on deposit for 9/12 = 3/4 = 0.75 of a year. 0

0.75

–100

FV = ?

1 12% |

112

You would use the regular set-up, but with a fractional exponent: FV0.75 = $100(1.12)0.75 = $100(1.088713) = $108.87. This is slightly different from our earlier answer, because n is actually 273/365 = 0.7479 rather than 0.75. Fractional time periods Thus far all of our examples have dealt with full years. Now we are going to look at the situation when we are dealing with fractional years, such as 9 months. In these situations, proceed as follows: 

As always, start by drawing a time line so you can visualize the situation.



Then think about the interest rate—the nominal rate, the compounding periods per year, and the effective annual rate. If you have been given a nominal rate, you may have to convert to the EAR, using this formula: M

 I  EAR = 1  NOM   1 M   

If you have the effective annual rate—either because it was given to you or after you calculated it with the formula—then you can find the PV of a lump sum by applying this equation:

1   PV = FVN    1  EAR 



Here N can be a fraction of a year, such as 0.75, if you need to find the PV of $1,000 due in 9 months, or 450/365 = 1.2328767 if the payment is due in 450 days.



If you have an annuity with payments different from once a year, say every month, you can always work it out as a series of lump sums. That procedure always works. We can also use annuity formulas and calculator functions, but you have to be careful.

1. What is the value at the end of Year 3 of the following cash flow stream if the quoted interest rate is 10%, compounded semiannually? 0

2 |

100 Answer:

0 5% |

100

1 |

3 years

100

2 |

100

100 110.25 = 100(1.05)2 121.55 = 100(1.05)4 331.80

Here we have a different situation. The payments occur annually, but compounding occurs each 6 months. Thus, we cannot use normal annuity valuation techniques. There are two approaches that can be applied: (1) treat the cash flows as lump sums, as was done above, or (2) treat the cash flows as an ordinary annuity, but use the effective annual rate: M

 I   0.10  EAR =  1 + NOM  - 1 =  1 +  - 1 = 10.25%. M  2    Now we have this 3-period annuity: FVA3 = $100(1.1025)2 + $100(1.1025)1 + $100 = $331.80. You can plug in N = 3, I = 10.25, PV = 0, and PMT = –100, and then press the FV button to find FV = $331.80.

Mini Case: 4-98

2. What is the PV of the same stream?

Answer:

0 5% |

1 |

2 |

100 90.70 82.27 74.62 247.59

100

3 |

100

PV = 100(1.05)–4

To use a financial calculator, input N = 3, I = 10.25, PMT = 100, FV = 0, and then press the PV key to find PV = $247.59. l.

3. Is the stream an annuity?

Answer: The payment stream is an annuity in the sense of constant amounts at regular intervals, but the intervals do not correspond with the compounding periods. This kind of situation occurs often. In this situation the interest is compounded semiannually, so with a quoted rate of 10%, the EAR will be 10.25%. Here we could find the effective rate and then treat it as an annuity. Enter N = 3, I = 10.25, PMT = 100, and FV = 0. Now press PV to get $247.59. l.

4. An important rule is that you should never show a nominal rate on a time line or use it in calculations unless what condition holds? (Hint: Think of annual compounding, when INOM= EFF% = IPER) What would be wrong with your answer to questions l-(1) and l-(2) if you used the nominal rate (10%) rather than the periodic rate (INOM /2 = 10%/2 = 5%)?

Answer: INOM can only be used in the calculations when annual compounding occurs. If the nominal rate of 10% was used in question l-(1) the future value would be understated by $331.80 – $331.00 = $0.80. If the nominal rate of 10% was used to discount the payment stream the present value would be overstated by $248.69 – $247.59 = $1.10.

Suppose someone offered to sell you a note calling for the payment of $1,000, 15 months from today. They offer to sell it to you for $850. You have $850 in a bank time deposit which pays a 6.76649% nominal rate with daily compounding, which is a 7% effective annual interest rate, and you plan to leave the money in the bank unless you buy the note. The note is not risky—you are sure it will be paid on schedule. Should you buy the note? Check the decision in three ways: (1) by comparing your future value if you buy the note versus leaving your money in the bank, (2) by comparing the PV of the note with your current bank account, and (3) by comparing the EFF% on the note versus that of the bank account.

Answer: You can solve this problem in three ways—(1) by compounding the $850 now in the bank for 15 months and comparing that FV with the $1,000 the note will pay, (2) by finding the PV of the note and then comparing it with the $850 cost, and (3) finding the effective annual rate of return on the note and comparing that rate with the 7% you are now earning, which is your opportunity cost of capital. All three procedures lead to the same conclusion. Here is the time line:

0 |

–850 (1)

1 |

1.25 |

1,000

FV = $850(1.07)1.25 = $925.01 = amount in bank after 15 months versus $1,000 if you buy the note. (Again, you can find this value with a financial calculator. Note that certain calculators like the HP 12c perform a straight-line interpolation for values in a fractional time period analysis rather than an effective interest rate interpolation. The value that the HP 12c calculates is $925.42.) This procedure indicates that you should buy the note. Alternatively, 15 months = (1.25 years)(365 days per year) = 456.25 ≈ 456 days. FV456 = $850(1.00018538)456 = $924.97. The slight difference is due to using N = 456 rather than N = 456.25.

(2)

PV = $1,000/(1.07)1.25 = $918.90. Since the present value of the note is greater than the $850 cost, it is a good deal. You should buy it. Alternatively, PV = $1,000/(1.00018538)456 = $918.95.

Mini Case: 4-100

(3)

FVN = PV(1 + I)N, so $1,000 = $850(1 + I)1.25. Since we have an equation with one unknown, we can solve it for I. You will get a value of I = 13.88%. The easy way is to plug values into your calculator. Since this return is greater than your 7% opportunity cost, you should buy the note. This action will raise the rate of return on your asset portfolio. Alternatively, we could solve the following equation: $1,000 = $850(1 + I)456 for a daily I = 0.00035646, With a result of EAR = EFF% = (1.00035646)365 – 1 = 13.89%.

WEB EXTENSION 4B: CONTINUOUS COMPOUNDING AND DISCOUNTING SOLUTIONS TO PROBLEMS 4B-1

FV15 = $15,000e0.06(15) = $36,894.05.

4B-2

PV = FVN/eIN = $200,000/e0.09(7) = $200,000/1.8776= $106,518.36.

4B-3

Daily compounding: FV2 = PV(1 + 0.07/365)365(2) = $1,000(1.15026)

= $1,150.26

Continuous compounding: FV2 = PVeIN = $1,000e0.069(2) = $1,000(1.14798) Difference between accounts 4B-4

= $1,147.98 $ 2.28

Calculate the growth factor using PV and FV, which are given: FVN = PVeIN; $40,000 = $20,000eI6 eI6 = 2.0. Take the natural logarithm of both sides: I(6)ln e = ln 2.00. The natural log of e = 1.0. Inputs: 2.0. Press LN key. Output: LN = 0.6931. I(6)ln e = ln 2.0 I(6) = 0.6931 I = 0.1155 = 11.55%.

4B-5

Determine the effective annual rates and then choose c. a. 10.25% annually = 10.25%. 2

 0.10  b. 10.0% semiannually = 1   – 1.0 = 0.1025 = 10.25%. 2   c. 9.8% continuously = e0.098 – 1.0 = 0.10296 = 10.30%.

4B-6 $11,572.28 = PVe0.09(15) $11,572.28 = PVe1.35 PV = $11,572.28/e1.35 = $11,572.28/3.85743 = $3,000. 4B-7

 I  = 1  NOM  2  

(0.03)(10)

 I  e = 1  NOM  2  

0.3

e0.3/20= 1 + 1.01511

I NOM 2

=1+

I NOM 2

I NOM = 0.01511 2 INOM = 0.0302 = 3.02%.

4B-8

Step 1: Calculate the FV of the $2,000 deposit at 8% with continuous compounding: Using ex key: Inputs: X = 0.24; press ex key. Output: ex = 1.27125. FVN = $2,000e0.08(3) = $2,000(1.27125) = $2,542.50. Step 2: Calculate the PV or initial deposit: Input the following data into your calculator: N = 6; I/YR = 4.5; PMT = 0; FV = 2542.50; and then solve for PV = $1,952.37.

5-1

a. The operating plan provides detailed implementation guidance designed to accomplish corporate objectives. It details who is responsible for what particular function, and when specific tasks are to be accomplished. The financial plan details the financial aspects of the corporation’s operating plan. In addition to an analysis of the firm’s current financial condition, the financial plan normally includes a sales forecast, the capital budget, the cash budget, pro forma financial statements, and the external financing plan. A sales forecast is merely the forecast of unit and dollar sales for some future period. Of course, a lot of work is required to produce a good sales forecast. Generally, sales forecasts are based on the recent trend in sales plus forecasts of the economic prospects for the nation, industry, region, and so forth. The sales forecast is critical to good financial planning. b. A pro forma financial statement shows how an actual statement would look if certain assumptions are realized. With the forecasted financial statement method, many (but not all) items on the income statement and balance sheets are assumed to increase proportionally with sales. As sales increase, these items that are tied to sales also increase, and the values of these items for a particular year are estimated as percentages of the forecasted sales for that year. c. Spontaneous liabilities are the first source of expansion funds as these accounts increase automatically through normal business operations. Examples of spontaneous liabilities include accounts payable, accrued wages, and accrued taxes. No interest is normally paid on these spontaneous liabilities; however, their amounts are limited due to credit terms, contracts with workers, and tax laws. Therefore, spontaneous liabilities are used to the extent possible, but there is little flexibility in their usage. Note that notes payable, although a current liability account, is not a spontaneous liability since an increase in notes payable requires a specific action between the firm and a creditor. A firm’s profit margin is calculated as net income divided by sales. The higher a firm’s profit margin, the larger the firm’s net income available to support increases in its assets. Consequently, the firm’s need for external financing will be lower. A firm’s payout ratio is calculated as dividends per share divided by earnings per share. The less of its income a company distributes as dividends, the larger its addition to retained earnings, i.e., its retention ratio. Therefore, the firm’s need for external financing will be lower.

d. Additional funds needed (AFN) are those funds required from external sources to increase the firm’s assets to support a sales increase. A sales increase will normally require an increase in assets. However, some of this increase is usually offset by a spontaneous increase in liabilities as well as by earnings retained in the firm. Those funds that are required but not generated internally must be obtained from external sources. Although most firms’ forecasts of capital requirements are made by constructing pro forma income statements and balance sheets, the AFN formula is sometimes used to forecast financial requirements. It is written as follows: Spontaneous Increase in Additional Required = asset – liability – retained funds increase needed earnings increase

AFN = (A*/S0)S

– (L*/S0)S

– MS1(RR)

Capital intensity is the dollar amount of assets required to produce a dollar of sales. The capital intensity ratio is the reciprocal of the total assets turnover ratio. The sustainable growth rate is the maximum growth rate the firm could achieve without having to raise any external capital. A firm’s self-supporting growth rate can be calculated as follows: Self-supporting g =

M(RR)(S0 ) *

A - L* - M(RR)(S0 )

e. ―Lumpy‖ assets are those assets that cannot be acquired smoothly, but require large, discrete additions. For example, an electric utility that is operating at full capacity cannot add a small amount of generating capacity, at least not economically. A firm has excess capacity when its sales can grow before it must add fixed assets such as plant and equipment. When economies of scale occur, the ratios are likely to change over time as the size of the firm increases. For example, retailers often need to maintain base stocks of different inventory items, even if current sales are quite low. As sales expand, inventories may then grow less rapidly than sales, so the ratio of inventory to sales declines. f. Financing feedback effects are incorporated in the forecasted financial statement method when the effects of external capital are included in the analysis. Interest on notes payable and long-term debt will increase (decrease) as notes and debt are issued (retired). Dividends of additional common and preferred shares issued must be included in the analysis. This will lower the addition of retained earnings. Likewise, if common and preferred shares are repurchased then dividends will decline and this will increase the addition of retained earnings. When financial feedback effects are present, this makes the AFN forecast inaccurate if these feedback effects are not considered.

5-2

Accounts payable, accrued wages, and accrued taxes increase spontaneously and proportionately with increased sales. Retained earnings also increase spontaneously, but not proportionately.

5-3

The equation gives good forecasts of financial requirements if the ratios A*/S and L*/S, the profit margin, and payout ratio are stable. This equation assumes that ratios are constant. This would not occur if there were economies of scale, excess capacity, or when lumpy assets are required. In those cases, the forecasted financial statement method should be used.

5-4

The five factors that impact a firm’s external financing requirements are sales growth, capital intensity, spontaneous liabilities-to-sales ratio, profit margin, and retention ratio.

5-5

The self-supporting growth rate is the maximum rate a firm can achieve without having to raise external capital. The self-supporting growth rate is calculated using the AFN equation, setting AFN equal to zero, replacing the term ΔS with the term g × S0, and replacing the term S1 with S0 × (1 + g). Once the AFN equation is rewritten with these modifications, you can now solve for g. This ―g‖ obtained is the firm’s self-supporting growth rate.

5-6

a. +. b. +. It reduces spontaneous funds; however, it may eventually increase retained earnings. c. +. d. +. e. –. f. –.

SOLUTIONS TO END-OF-CHAPTER PROBLEMS 5-1

AFN = (A0*/S0)∆S – (L0*/S0)∆S – (M)(S1)(1 – payout rate) $6,800,000 $2,040,000 = ($17,000,000) $2,550,000 – ($17,000,000) $2,550,000 – 0.05($19,550,000)(1 – 0.30) = (0.40)($2,550,000) – (0.12)($2,550,000) – ($977,500)(0.7) = $1,020,000 – $306,000 – $684,250 = $29,750.

5-2

$9,350,000

$2,040,000

AFN = ($17,000,000) $2,550,000 – ($17,000,000) $2,550,000 – 0.05($19,550,000)(1– 0.30) = (0.55)($2,550,000) – $306,000 – $684,250 = $1,402,500 – $306,000 – $684,250 = $412,250.

The capital intensity ratio is measured as A0*/S0. This firm’s capital intensity ratio is higher than that of the firm in Problem 5-1; therefore, this firm is more capital intensive—it would require a larger increase in total assets to support the increase in sales. 5-3

AFN = (0.40)($2,550,000) – (0.12)($2,550,000) – ($977,500)(1.0) = $1,020,000 – $306,000 – $977,500 = ($263,500). Under this scenario the company would have a higher level of retained earnings, which would lead to having more than enough internal financing to support the company’s growth.

5-4

S0 = $5,000,000; A0* = $2,500,000; CL = $700,000; NP = $300,000; AP = $500,000; Accruals = $200,000; M = 7%; payout ratio = 80%; A0*/S0 = 0.50; L0* = (AP + Accruals)/S0 = ($500,000 + $200,000)/$5,000,000 = 0.14. AFN = (A0*/S0)∆S – (L0*/S0)∆S – (M)(S1)(1 – payout rate) = (0.50)∆S – (0.14) ∆S – (0.07)(S1)(1 – 0.8) = (0.50)∆S – (0.14)∆S – (0.014)S1 = (0.36)∆S – (0.014)S1 = 0.36(S1 – S0) – (0.014)S1 = 0.36(S1 – $5,000,000) – (0.014)S1 = 0.36S1 – $1,800,000 – 0.014S1 $1,800,000 = 0.346S1 $5,202,312 = S1. Sales can increase by $5,202,312 – $5,000,000 = $202,312 without additional funds being needed. This is a self-supporting growth rate of 1.01%.

5-5

Total liab. = Accounts + Long-term + Common + Retained payable stock earnings and equity debt

$2,170,000 = $560,000 + Long-term debt + $625,000 + $395,000 Long-term debt = $590,000. Total liab. = Accounts payable + Long-term debt = $560,000 + $590,000 = $1,150,000.

b. Assets/Sales (A0*/S0) = $2,170,000/$3,500,000 = 62%. L0*/Sales = $560,000/$3,500,000 = 16%. 2022 Sales = (1.35)($3,500,000) = $4,725,000. New long-term debt = AFN – New common stock = (A0*/S0)(∆S) – (L0*/S0)(∆S) – (M)(S1)(1 – payout) – New common stock = (0.62)($1,225,000) – (0.16)($1,225,000) – (0.05)($4,725,000)(0.55) – $195,000 = $759,500 – $196,000 – $129,938 – $195,000 = $238,562. Alternatively, using the forecasted financial statement method:

2021 Total assets Current liabilities Long-term debt Total liabilities Common stock Retained earnings Total common equity Total liabilities and equity

$2,170,000 560,000 590,000 1,150,000 625,000 395,000 1,020,000

Forecast Basis % 2022 Sales 0.62 0.16

Additions (New Financing, R/E)

195,000* 129,938**

$2,170,000

AFN = Additional long-term debt =

2,929,500 – 2,690,938 =

2022 $2,929,500 756,000 590,000 1,346,000 820,000 524,938 1,344,938 $2,690,938 $ 238,562

* Given in problem that firm will sell new common stock = $195,000. ** PM = 5%; Payout = 45%; NI2022 = $3,500,000 × 1.35 × 0.05 = $236,250. Addition to RE = NI × (1 – Payout) = $236,250 × 0.55 = $129,938.

5-6 Cash Accounts receivable Inventories Net fixed assets Total assets Accounts payable Notes payable Accruals Long-term debt Common stock Retained earnings Total liabilities and equity

2021 $ 100.00 200.00 200.00 500.00 $1,000.00 $

50.00 150.00 50.00 400.00 100.00 250.00

 2  2  2 + 0.0

= = = =

 2 = 150.00 + 360.00 =  2 =

+ 40

$1,000.00 AFN

2022 $ 200.00 400.00 400.00 500.00 $1,500.00 $ 100.00 510.00 100.00 400.00 100.00 290.00 $1,500.00 $ 360.00

Capacity sales = Sales/0.5 = $1,000/0.5 = $2,000. Target FA/S ratio = $500/$2,000 = 0.25. Target FA = 0.25($2,000) = $500 = Required FA. Since the firm currently has $500 of fixed assets, no new fixed assets will be required. Addition to RE = M(S1)(1 – Payout ratio) = 0.05($2,000)(0.4) = $40.

5-7

Cash Accounts receivable Inventories Net fixed assets Total assets Accounts payable Notes payable Accruals Long-term debt Common stock Retained earnings Total liabilities and equity

2021 $ 100.00 200.00 200.00 500.00 $1,000.00 $

50.00 150.00 50.00 400.00 100.00 250.00

 2 329 for 2022 350 for 2022 + 0.0

= = = =

 2 + 239.00  2

= = =

+ 40

$1,000.00 AFN

2022 $ 200.00 329.00 350.00 500.00 $1,379.00 $ 100.00 389.00 100.00 400.00 100.00 290.00 $1,379.00 $ 239.00

Accounts receivable for 2022 = $2,000/365 × 60 = $329. Inventory = $1,600/4.57 = $350 Capacity sales = Sales/0.5 = $1,000/0.5 = $2,000. Target FA/S ratio = $500/$2,000 = 0.25. Target FA = 0.25($2,000) = $500 = Required FA. Since the firm currently has $500 of fixed assets, no new fixed assets will be required. Addition to RE = M(S1)(1 – Payout ratio) = 0.05($2,000)(0.4) = $40. b.

Change in incremental borrowing is $121 ($360 – $239) and the change in interest expense is $121 × 0.09 = $10.89 annually.

5-8

a. AFN = (A*/S)(S) – (L*/S)(S) – MS1(1 – d) =

$122.5 $14.35 $17.5 ($70) – ($70) – ($420)(0.6) $350 $350 $350

= $10.67 million. b.

M(1 - d)(S0 ) A* - L* - M(1 - d)(S0 )

0.041(0.6)($350) $122.5 - $17.5 - 0.041(0.6)($350)

$8.61 $96.39

= 8.93%

Cavuto Limited Pro Forma Balance Sheet December 31, 2022 (Millions of Dollars)

2021 3.5 26.0 58.0 87.5 35.0 $122.5

Cash Receivables Inventories Total current assets Net fixed assets Total assets

Accounts payable Notes payable Accruals Total current liabilities Mortgage loan Common stock Retained earnings Total liab. and equity

AFN =

9.0 18.0 8.5 35.5 6.0 15.0 66.0 $122.5

Forecast Pro Forma Basis % after 2022 Sales Additions Pro Forma Financing Financing 0.0100 $ 4.20 $ 4.20 0.0743 31.20 31.20 0.1657 69.60 69.60 105.00 105.00 0.100 42.00 42.00 $147.00 $147.00 0.0257 0.0243

10.33*

$ 10.80 18.00 10.20 39.00 6.00 15.00 76.33 $136.33

+10.67

$ 10.80 28.67 10.20 49.67 6.00 15.00 76.33 $147.00

$ 10.67

* PM = $14.35/$350 = 4.1%. Payout = $5.74/$14.35 = 40%. NI = $350  1.2  0.041 = $17.22. Addition to RE = NI – DIV = $17.22 – 0.4($17.22) = 0.6($17.22) = $10.33.

5-9 The first column shows the financial statement accounts and the method used for

projections. Projected sales are equal to the actual sales that grow at the specified growth rate. Projected operating costs are equal to a percentage of the projected sales, with the percentage equal to the ratio of actual operating costs to sales in the actual year. Projected cash, receivables, inventories, and fixed assets are determined in the same manner as projected operating costs. Projected interest is equal to the product of the interest rate and the total debt in the actual year because the LOC is added at the end of the projected year. Income Statements (Thousands of Dollars) Sales = $36,000(1.15) Op. costs = ($34,000/$36,000)($41,400) Earnings before interest and taxes Interest = 10%($2,100 + $3,500) Pre-tax earnings Taxes = 25%(Pre-tax earnings) Net income Dividends = 40%(Net income) Addition to retained earnings

Actual $36,000 34,000 2,000 160 1,840 460 1,380 552 $ 828

Projected $41,400 39,100 2,300 560 1,740 435 1,305 522 $ 783

Balance Sheets (Thousands of Dollars) Cash= ($1,080/$36,000)($41,400) Receivables= ($6,480/$36,000)($41,400) Inventories= ($9,000/$36,000)($41,400) Total current assets Net fixed assets= ($12,600/$36,000)($41,400) Total assets

Actual $1,080 6,480 9,000 16,560 12,600 $29,160

Projected $1,242 7,452 10,350 19,044 14,490 $33,534

Accounts payable= ($4,320/$36,000)($41,400) Accruals= ($2,880/$36,000)($41,400) Line of credit (see calculation below) Notes payable = carry over Total current liabilities Long-term bonds = carry over Common stock = carry over Retained earnings REActual + Add. to REProj. Total liabilities and equity

$4,320 2,880 0 2,100 9,300 3,500 3,500 12,860 $29,160

$4,968 3,312 2,511 2,100 12,891 3,500 3,500 13,643 $33,534

LOC = TA – (AP + Accr. + Notes pay. + LT bonds + com. stk + RE) a. b. c. d. e. f. g. h.

$2,511

What is the projected value for earnings before interest and taxes? $2,300. What is the projected value for pre-tax earnings? $1,740. What is the projected net income? $1,305. What is the projected addition to retained earnings? $783. What is the projected value of total current assets? $19,044. What is the projected value of total assets? $33,534. What is the projected sum of accounts payable, accruals, and notes payable? $10,380 What is, if any, the additional funds needed? $2,511.

5-10

Garlington Technologies Inc. Pro Forma Income Statement December 31, 2022

Sales Operating costs EBIT Interest EBT Taxes (40%) Net income

2021 $3,600,000 3,279,720 320,280 18,280 302,000 120,800 $ 181,200

Dividends: Addition to RE:

$ 108,000 $ 73,200

Forecast Basis 1.10  Sales21 0.911  Sales22

Additions

0.13 × Debt21

Set by management

2022 $3,960,000 3,607,692 352,308 20,280 332,028 132,811 $ 199,217 $ 112,000 $ 87,217

Garlington Technologies Inc. Pro Forma Balance Statement December 31, 2022

Cash Receivables Inventories Total current assets Fixed assets Total assets

2021 $ 180,000 360,000 720,000 1,260,000 1,440,000 $2,700,000

Accounts payable $ 360,000 Notes payable 156,000 Accruals 180,000 Total current liabilities 696,000 Common stock 1,800,000 Retained earnings 204,000 Total liab. and equity $2,700,000

Forecast Basis % 2022 Sales 0.05 0.10 0.20

Additions

0.40 0.10 0.05

87,217*

AFN Effects

With AFN 2022 $ 198,000 396,000 792,000 1,386,000 1,584,000 $2,970,000

$ 396,000 156,000 +128,783 198,000 750,000 1,800,000 291,217 $ 2,841,217

$ 396,000 284,783 198,000 878,783 1,800,000 291,217 $2,970,000

AFN =

$ 128,783

2022 $ 198,000 396,000 792,000 1,386,000 1,584,000 $2,970,000

*See income statement.

b. I.

Current ratio = $1,386,000/$878,783 = 1.58× Return on equity = $199,217/($2,091,217) = 9.5% Earnings per share = $199,217/108,000 = $1.84 Dividends per share = $112,000/108,000 = $1.04

II.

$1,386

Current ratio = $396 + $198 + Notes payable = 1.7 Notes payable + $396 + $198 =

$1,386 1.7

Notes payable = $221.294 Increase in notes payable = $221,294 – $156,000 = $65,294 III.

Current ratio = $1,386,000/($396,000 + $198,000 + $221,294) = 1.70× New financing required Increase in note payable New equity

$128,783 65,294 $ 63,489

Return on equity = $199,217/($2,091,217 + $63,489) = 9.2% New shares = $63,489/$18 = 3,528 (rounded up) Earnings per share = $199,217/(108,000 + 3,528) = $1.79 Dividends per share = $112,000/(108,000 + 3,528) = $1.00 IV.

Return on equity = $199,217($2,091,217 + $128,783) = 9.0% New shares = $128,783/$18 = 7,155 Earnings per share = $199,217/(108,000 + 7,155) = $1.73 Dividends per share = $112,000/(108,000 + 7,155) = $0.97 Yes, the financing choice matters as dividends will not grow as fast (or decline) if additional stock is sold. ROE will also not grow as fast (or remain the same).

SOLUTION TO SPREADSHEET PROBLEM 5-11

The detailed solution for the spreadsheet problem is in the file Ch 05 Build a Model Solution.xlsx and is available on the textbook’s website.

MINI CASE The detailed solution for the Mini Case is also available in spreadsheet format, in the file Brigham 4Ce_MiniCase_Ch05.xlsx, and is available on the textbook’s website. Betty Simmons, the new financial manager of Okanagan Chemicals Ltd. (OCL), a B.C. producer of specialized chemicals for use in fruit orchards, must prepare a financial forecast for 2022. OCL’s 2021 sales were $2 billion, and the marketing department is forecasting a 25% increase for 2022. Simmons thinks the company was operating at full capacity in 2021, but she is not sure about this. The 2021 financial statements, plus some other data, are shown below. Assume that you have recently been hired as Simmons’s assistant, and that your first major task is to help her develop the forecast. She asked you to begin by answering the following set of questions.

Financial Statements and Other Data on OCL (Millions of Dollars) A. 2021 Balance Sheet of

% of

Sales Sales Cash and securities Accounts receivable Inventories Total current assets Net fixed assets Total assets

20 240 240 500 500

$1,000

1% 12 12 25

Accounts payable and accruals Notes payable Total current liabilities Long-term debt Common stock Retained earnings Total liabilities and equity

$ 100 100 200 100 500 200 $1,000

B. 2021 Income Statement Sales Cost of goods sold Sales, general, and administrative costs Earnings before interest and taxes Interest Earnings before taxes Taxes (40%) Net income Dividends (40%) Addition to retained earnings

C. Key Ratios Profit margin Return on equity Days’ sales outstanding (365 days) Inventory turnover Fixed assets turnover Debt/assets Times interest earned Current ratio Return on invested capital (NOPAT/Operating capital) a.

% of sales $2,000.00 1,200.00 700.00 100.00 10.00 90.00 36.00 54.00 21.60 $ 32.40

60% 35

OCL 2.70% 7.71% 43.8 days 5.00× 4.00× 20.00% 10.00× 2.50×

Industry 4.00% 15.60% 32.00 days 6.60× 5.00× 24.00% 9.40× 3.00×

6.67%

14.00%

Describe three ways that pro forma statements are used in financial planning.

Answer: Three important uses are: (1) forecast the amount of external financing that will be required, (2) evaluate the impact that changes in the operating plan have on the value of the firm, (3) set appropriate targets for compensation plans. b.

Explain the steps in financial forecasting.

Answer: (1) forecast sales, (2) project the assets needed to support sales, (3) project internally generated funds, (4) project outside funds needed, (5) decide how to raise funds, and (6) see effects of plan on ratios and stock price.

Answers and Solutions: 6-118

Assume that (1) OCL was operating at full capacity in 2021 with respect to all assets, (2) all assets must grow proportionally with sales, (3) accounts payable and accruals will also grow in proportion to sales, and (4) the 2021 profit margin and dividend payout will be maintained. Under these conditions, what will the company’s financial requirements be for the coming year? Use the AFN equation to answer this question.

Answer: OCL will need $184.5 million. Here is the AFN equation: AFN = (A*/S0)∆S – (L*/S0)∆S – M(S1)(RR) = (A*/S0)(g)(S0) – (L*/S0)(g)(S0) – M(S0)(1 + g)(1 - DPR) = ($1,000/$2,000)(0.25)($2,000) – ($100/$2,000)(0.25)($2,000) – 0.0270($2,000)(1.25)(0.6) = $250 – $25 – $40.5 = $184.5 million. d.

How would changes in the following items affect the AFN? (1) sales increase; (2) the dividend payout ratio increases; (3) the profit margin increases; (4) the capital intensity ratio increases; (5) OCL begins paying its suppliers sooner. (Consider each item separately and hold all other things constant.)

Answer: 1. If sales increase, more assets are required, which increases the AFN. 2. If the payout ratio were increased, then less earnings would be retained, and this would increase the need for external financing, or AFN. Note that if the firm is profitable and has any payout ratio less than 100%, it will have some retained earnings, so if the growth rate were zero, AFN would be negative, i.e., the firm would have surplus funds. As the growth rate rose above zero, these surplus funds would be used to finance growth. At some point, i.e., at some growth rate, the surplus AFN would be exactly used up. This growth rate where AFN = $0 is called the ―sustainable growth rate,‖ and it is the maximum growth rate that can be financed without outside funds, holding the debt ratio and other ratios constant. 3. If the profit margin goes up, then both total earnings and retained earnings will increase, and this will reduce the amount of AFN. 4. The capital intensity ratio is defined as the ratio of required assets to total sales, or A*/S0. Put another way, it represents the dollars of assets required per dollar of sales. The higher the capital intensity ratio, the more new money will be required to support an additional dollar of sales. Thus, the higher the capital intensity ratio, the greater the AFN, other things held constant. 5. If OCL begins paying sooner, this reduces spontaneous liabilities, leading to a higher AFN.

Briefly explain how to forecast financial statements using the forecast financial statement approach. Be sure to explain how to forecast interest expense.

Answer: Project sales based on forecasted growth rate in sales. Forecast some items as a percentage of the forecasted sales, such as costs, cash, accounts receivable, inventories, net fixed assets, accounts payable, and accruals. Choose other items according to the company’s financial policy: debt, dividend policy (which determines retained earnings), and common stock. Given the previous assumptions and choices, we can estimate the required assets to support sales and the specified sources of financing. The additional funds needed (AFN) is required assets minus specified sources of financing. If AFN is positive, then you must secure additional financing. If AFN is negative, then you have more financing than is needed and you can pay off debt, buy back stock, or buy short-term investments. Interest expense is based on the daily balance of debt during the year. There are three ways to approximate interest expense. You can base it on (1) debt at end of year, (2) debt at beginning of year, or (3) average of beginning and ending debt. Basing interest expense on debt at end of year will overestimate interest expense if debt is added throughout the year instead of all on January 1. It also causes circularity called financial feedback: more debt causes more interest, which reduces net income, which reduces retained earnings, which causes more debt, etc. Basing interest expense on debt at beginning of year will underestimate interest expense if debt is added throughout the year instead of all on December 31. But it doesn’t cause the problem of circularity. Basing interest expense on average of beginning and ending debt will accurately estimate the interest payments if debt is added smoothly throughout the year. But it has the problem of circularity. A solution that balances accuracy and complexity is to base interest expense on beginning debt, but use a slightly higher interest rate. This is easy to implement and is reasonably accurate. See Ch 05 Mini Case Feedback.xlsx for an example basing interest expense on average debt.

Answers and Solutions: 6-120

Now estimate the 2022 financial requirements using the forecast financial statement approach. Assume that (1) each type of asset, as well as payables, accruals, and fixed and variable costs, will be the same percentage of sales in 2022 as in 2021; (2) the payout ratio is held constant at 40%; (3) external funds needed are financed 50% by notes payable and 50% by long-term debt (no new common stock will be issued); (4) all debt carries an interest rate of 10%; and (5) interest expense should be based on the balance of debt at the beginning of the year.

Answer: Income Statement (In Millions of Dollars) Sales Cost of goods sold Selling, general & admin expenses EBIT Less interest EBT Taxes (40%) Net income Dividends Add. to retained earnings

Actual 2021 Forecast Basis $ 2,000.0 Growth 1.25 1,200.0 % of Sales 60.00% 700.0 % of Sales 35.00% 100.0 10.0 Interest rate × Debt21 90.0 36.0 $ 54.0 $ 21.6 $ 32.4

Forecast 2022 $ 2,500.0 1,500.0 875.0 125.0 20.0 105.0 42.0 $ 63.0 $ 25.2 $ 37.8

Balance Sheet (In Millions of Dollars) Assets Cash Accounts receivable Inventories Total current assets Net plant and equipment Total assets

2021

Forecast Basis

20.0 240.0 240.0 500.0 500.0 $ 1,000.0

% of Sales % of Sales % of Sales

% of Sales

2022 Forecast Without AFN

AFN

1.00% $ 12.00% 12.00%

25.0 300.0 300.0 625.0 25.00% 625.0 $ 1,250.0

Liabilities and Equity Accounts payable & accruals $ 100.0 % of Sales 5.00% $ 125.0 Notes payable 100.0 Carry-Over 100.0 $93.6 Total current liabilities 200.0 225.0 Long-term debt 100.0 Carry-Over 100.0 $93.6 Total liabilities 300.0 325.0 Common stock 500.0 Carry-Over 500.0 Retained earnings 200.0 RE21+ RE22 237.8 Total common equity 700.0 737.8 Total liabilities and equity $ 1,000.0 $ 1,062.8 Required Assets = Specified Sources of Financing = Additional Funds Needed (AFN)

Answers and Solutions: 6-122

2022 Forecast With AFN $

25.0 300.0 300.0 625.0 625.0 $ 1,250.0

125.0 193.6 318.6 193.6 512.2 500.0 237.8 737.8 $ 1,250.0

$1,250.0 $1,062.8 $ 187.2

Why does the forecast financial statement approach produce a somewhat different AFN than the equation approach? Which method provides the more accurate forecast?

Answer: The difference occurs because the AFN equation method assumes that the profit margin remains constant, while the forecasted balance sheet method permits the profit margin to vary. The balance sheet method is somewhat more accurate, but in this case the difference is not very large. The real advantage of the balance sheet method is that it can be used when everything does not increase proportionately with sales. In addition, forecasters generally want to see the resulting ratios, and the balance sheet method is necessary to develop the ratios. In practice, the only time we have ever seen the AFN equation used is to provide (1) a ―quick and dirty‖ forecast prior to developing the balance sheet forecast and (2) a rough check on the balance sheet forecast. Calculate OCL’s forecasted ratios, and compare them with the company’s 2021 ratios and with the industry averages. Calculate OCL’s forecast free cash flow and return on invested capital (ROIC).

Answer: Key Ratios

Actual Forecast 2021 2022 Industry

Profit margin 2.70% 2.52% 4.00% ROE 7.71% 8.54% 15.60% DSO 43.80 43.80 32.00 Inventory turnover 5.00 5.00 6.60 Fixed asset turnover 4.00 4.00 5.00 Debt/assets 20.00% 30.98% 24.00% TIE 10.00 6.25 9.40 Current ratio 2.50 1.96 3.00 Free Cash Flow

Operating Gross Investment in – Operating Capital Cash Flow = NOPAT – Net Investment in Operating Capital FCF = NOPAT – (Operating Capital2022 – Operating Capital2021) = $125(1 – 0.4) + [($625 – $125 + $625) – ($500 – $100 + $500) = $75 – ($1,125 – $900) = $75 – $225 = –$150.

Note: Operating Capital = Net Operating Working Capital + Net Fixed Assets. ROIC = NOPAT/Operating Capital = $75/$1,125 = 0.067 = 6.67%.

Based on comparisons between OCL’s days sales outstanding (DSO) and inventory turnover ratios with the industry average figures, does it appear that OCL is operating efficiently with respect to its inventory and accounts receivable? Suppose OCL was able to bring these ratios into line with the industry averages and reduce its SGA/Sales ratio to 33%. What effect would this have on its AFN and its financial ratios? What effect would this have on free cash flow and ROIC?

Answer: The DSO and inventory turnover ratio indicate that OCL has excessive inventories and receivables. The effect of improvements here would reduce asset requirements and AFN. See the results below based on the spreadsheet Ch 05 Mini Case.xlsx.

Inputs Before DSO 43.80 Accounts Receivable/Sales 12.0% Inventory Turnover 5.00 Inventory/Sales 12.0% SGA/Sales 35.0%

After 32.01 8.77% 6.60 9.09% 33.0%

Outputs AFN FCF ROIC ROE

$15.7 $33.5 10.8% 12.3%

$187.2 –$150.0 6.7% 8.5%

Suppose you now learn that OCL’s 2021 receivables and inventories were in line with required levels, given the firm’s credit and inventory policies, but that excess capacity existed with regard to fixed assets. Specifically, fixed assets were operated at only 75% of capacity. 1. What level of sales could have existed in 2021 with the available fixed assets?

Answer: Full Capacity Sales =

Actual sales $2,000 = = $2,667. % of capacity at which 0.75 fixed assets were operated

Since the firm started with excess fixed asset capacity, it will not have to add as much fixed assets during 2022 as was originally forecasted.

Answers and Solutions: 6-124

2. How would the existence of excess capacity in fixed assets affect the additional funds needed during 2022?

Answer: We had previously found an AFN of $187.2 using the balance sheet method. The fixed assets increase was 0.25($500) = $125. Therefore, the funds needed will decline by $125. k.

The relationship between sales and the various types of assets is important in financial forecasting. The forecasted financial statements approach, under the assumption that each asset item grows at the same rate as sales, leads to an AFN forecast that is reasonably close to the forecast using the AFN equation. Explain how each of the following factors would affect the accuracy of financial forecasts based on the AFN equation: (1) economies of scale in the use of assets, and (2) lumpy assets.

Inventories

Base Stock

Sales

Answer: 1. Economies of scale in the use of assets mean that the asset item in question must increase less than proportionately with sales; hence it will grow less rapidly than sales. Cash and inventory are common examples, with possible relationship to sales as shown below:

Cash

Sales

Inventories

Sales

2. Lumpy assets would cause the relationship between assets and sales to look as shown below. This situation is common with fixed assets.

Fixed assets

Answers and Solutions: 6-126

Sales

Chapter 6 Bonds, Bond Valuation, and Interest Rates ANSWERS TO END-OF-CHAPTER QUESTIONS 6-1

a. A bond is a promissory note issued by a business or a governmental unit. Government of Canada bonds are issued by the federal and provincial governments; federal government bonds are not exposed to default risk. Corporate bonds are issued by corporations and are exposed to default risk. Different corporate bonds have different levels of default risk, depending on the issuing company’s characteristics and on the terms of the specific bond. Foreign bonds are issued by foreign governments or foreign corporations. These bonds are exposed not only to default risk, but also to an additional risk if the bonds are denominated in a currency other than the investor’s home currency. b. The par value is the nominal or face value of a stock or bond. The par value of a bond generally represents the amount of money that the firm borrows and promises to repay at some future date. The par value of a bond is often $1,000, but can be $5,000 or more. The maturity date is the date when the bond’s par value is repaid to the bondholder. Maturity dates generally range from 10 to 40 years from the time of issue. A call provision may be written into a bond contract, giving the issuer the right to redeem the bonds under specific conditions prior to the normal maturity date. A bond’s coupon, or coupon payment, is the dollar amount of interest paid to each bondholder on the interest payment dates. The coupon is so named because bonds used to have dated coupons attached to them that investors could tear off and redeem on the interest payment dates. The coupon interest rate is the stated rate of interest on a bond. c. In some cases, a bond’s coupon payment may vary over time. These bonds are called floating-rate bonds. Floating-rate debt is popular with investors because the market value of the debt is stabilized. It is advantageous to corporations because firms can issue long-term debt without committing themselves to paying a historically high interest rate for the entire life of the loan. Zero coupon bonds pay no coupons at all, but are offered at a substantial discount below their par values and hence provide capital appreciation rather than interest income.

d. Most bonds contain a call provision, which gives the issuing corporation the right to call the bonds for redemption. The call provision generally states that, if the bonds are called, the company must pay the bondholders an amount greater than the par value, a call premium. Retractable bonds give investors the right to sell the bonds back to the corporation at a price that is usually close to the par value. If interest rates rise, investors can retract the bonds and reinvest at the higher rates. A sinking fund provision facilitates the orderly retirement of a bond issue. This can be achieved in one of two ways: The company can call in for redemption (at par value) a certain percentage of bonds each year, or the company may buy the required amount of bonds on the open market. e. Convertible bonds are securities that are convertible into common shares, at a fixed price, at the option of the bondholder. Bonds issued with warrants are similar to convertibles. Warrants are options that permit the holder to buy stock for a stated price, thereby providing a capital gain if the stock price rises. Income bonds pay interest only if earnings can cover the interest. These securities cannot bankrupt a company, but from an investor’s standpoint they are riskier than ―regular‖ bonds. The interest rate of a real return bond is based on an inflation index such as the consumer price index (CPI), so the interest paid rises automatically when the inflation rate rises, thus protecting the bondholders against inflation. f. Bond prices and interest rates are inversely related; that is, they tend to move in the opposite direction from one another. A fixed-rate bond will sell at par when its coupon interest rate is equal to the going rate of interest, rd. When the going rate of interest is above the coupon rate, a fixed-rate bond will sell at a ―discount‖ below its par value. If current interest rates are below the coupon rate, a fixed-rate bond will sell at a ―premium‖ above its par value. g. The current yield on a bond is the annual coupon payment divided by the current market price. YTM, or yield to maturity, is the rate of interest earned on a bond if it is held to maturity. Yield to call (YTC) is the rate of interest earned on a bond if it is called. If current interest rates are well below an outstanding callable bond’s coupon rate, the YTC may be a more relevant estimate of expected return than the YTM, since the bond is likely to be called.

Answers and Solutions: 6-128

h. The shorter the maturity of the bond, the greater the risk of a decrease in interest rates when the bondholder reinvests the principal. The risk of a decline in income due to a drop in interest rates is called reinvestment rate risk. Interest rates fluctuate over time, and people or firms who invest in bonds are exposed to risk from changing interest rates, or interest rate risk. The longer the maturity of the bond, the greater the exposure to interest rate risk. Interest rate risk relates to the value of the bonds in a portfolio, while reinvestment rate risk relates to the income the portfolio produces. No fixed-rate bond can be considered totally riskless. Bond portfolio managers try to balance these two risks, but some risk always exists in any bond. Another important risk associated with bonds is default risk. If the issuer defaults, investors receive less than the promised return on the bond. Default risk is influenced by both the financial strength of the issuer and the terms of the bond contract, especially whether collateral has been pledged to secure the bond. The greater the default risk, the higher the bond’s yield to maturity. i. Corporations can influence the default risk of their bonds by changing the type of bonds they issue. Under a mortgage bond, the corporation pledges certain assets as security for the bond. All such bonds are written subject to an indenture, which is a legal document that spells out in detail the rights of both the bondholders and the corporation. A debenture is an unsecured bond and, as such, it provides no lien against specific property as security for the obligation. Debenture holders are, therefore, general creditors whose claims are protected by property not otherwise pledged. Subordinated debentures have claims on assets, in the event of bankruptcy, only after senior debt as named in the subordinated debt’s indenture has been paid off. Subordinated debentures may be subordinated to designated notes payable or to all other debt. j. Agency bonds are issued by government agencies such as Crown corporations. Agency bonds carry the same government guarantee as regular federal or provincial government bonds. Bond issues are normally assigned quality ratings by major rating agencies, such as Moody’s Investors Service and Standard & Poor’s Corporation. These ratings reflect the probability that a bond will go into default. Aaa (Moody’s) and AAA (S&P) are the highest ratings. Rating assignments are based on qualitative and quantitative factors including the firm’s debt/assets ratio, current ratio, and coverage ratios. Because a bond’s rating is an indicator of its default risk, the rating has a direct, measurable influence on the bond’s interest rate and the firm’s cost of debt capital. Junk bonds are high-risk, high-yield bonds issued to finance leveraged buyouts, mergers, or troubled companies. Most bonds are purchased by institutional investors rather than individuals, and many institutions are restricted to investmentgrade bonds, securities with ratings of Baa/BBB or above.

k. The real risk-free rate is that interest rate that equalizes the aggregate supply of, and demand for, riskless securities in an economy with zero inflation. The real risk-free rate could also be called the pure rate of interest since it is the rate of interest that would exist on very short-term, default-free federal government securities if the expected rate of inflation were zero. It has been estimated that this rate of interest, denoted by r*, has fluctuated in recent years in Canada in the range of 2% to 4%. The nominal risk-free rate of interest, denoted by rRF, is the real risk-free rate plus a premium for expected inflation. The short-term nominal risk-free rate is usually approximated by the government of Canada Treasury bill rate, while the long-term nominal risk-free rate is approximated by the rate on government of Canada bonds. Note that while government bonds are free of default and liquidity risks, they are subject to risks due to changes in the general level of interest rates. l. The inflation premium is the premium added to the real risk-free rate of interest to compensate for the expected loss of purchasing power. The inflation premium is the average rate of inflation expected over the life of the security. Default risk is the risk that a borrower will not pay the interest and/or principal on a loan as they become due. Thus, a default risk premium (DRP) is added to the real risk-free rate to compensate investors for bearing default risk. Liquidity refers to a firm’s cash and marketable securities position, and to its ability to meet maturing obligations. A liquid asset is any asset that can be quickly sold and converted to cash at its ―fair‖ value. Active markets provide liquidity. A liquidity premium is added to the real risk-free rate of interest, in addition to other premiums, if a security is not liquid. m. Interest rate risk arises from the fact that bond prices decline when interest rates rise. Under these circumstances, selling a bond prior to maturity will result in a capital loss, and the longer the term to maturity, the larger the loss. Thus, a maturity risk premium must be added to the real risk-free rate of interest to compensate for interest rate risk. Reinvestment rate risk occurs when a short-term debt security must be ―rolled over.‖ If interest rates have fallen, the reinvestment of principal will be at a lower rate, with correspondingly lower interest payments and ending value. Note that long-term debt securities also have some reinvestment rate risk because their interest payments have to be reinvested at prevailing rates. n. The term structure of interest rates is the relationship between yield to maturity and term to maturity for bonds of a single risk class. The yield curve is the curve that results when yield to maturity is plotted on the Y-axis with term to maturity on the Xaxis. o. When the yield curve slopes upward, it is said to be ―normal,‖ because it is like this most of the time. Conversely, a downward-sloping yield curve is termed ―abnormal‖ or ―inverted.‖

Answers and Solutions: 6-130

6-2

False. Short-term bond prices are less sensitive than long-term bond prices to interest rate changes because funds invested in short-term bonds can be reinvested at the new interest rate sooner than funds tied up in long-term bonds.

6-3

The price of the bond will fall and its YTM will rise if interest rates rise. If the bond still has a long term to maturity, its YTM will reflect long-term rates. Of course, the bond’s price will be less affected by a change in interest rates if it has been outstanding a long time and matures shortly. While this is true, it should be noted that the YTM will increase only for buyers who purchase the bond after the change in interest rates and not for buyers who purchased previous to the change. If the bond is purchased and held to maturity, the bondholder’s YTM will not change, regardless of what happens to interest rates.

6-4

If interest rates decline significantly, the values of callable bonds will not rise by as much as those of bonds without the call provision. It is likely that the bonds would be called by the issuer before maturity, so that the issuer can take advantage of the new, lower rates.

6-5

From the corporation’s viewpoint, one important factor in establishing a sinking fund is that its own bonds generally have a higher yield than do government bonds; hence, the company saves more interest by retiring its own bonds than it could earn by buying government bonds. This factor causes firms to favour the second procedure. Investors also would prefer the annual retirement procedure if they thought that interest rates were more likely to rise than to fall, but they would prefer the government bond purchases program if they thought rates were likely to fall. In addition, bondholders recognize that, under the government bond purchase scheme, each bondholder would be entitled to a given amount of cash from the liquidation of the sinking fund if the firm should go into default, whereas under the annual retirement plan, some of the holders would receive a cash benefit while others would benefit only indirectly from the fact that there would be fewer bonds outstanding. On balance, investors seem to have little reason for choosing one method over the other, while the annual retirement method is clearly more beneficial to the firm. The consequence has been a pronounced trend toward annual retirement and away from the accumulation scheme.

SOLUTIONS TO END-OF-CHAPTER PROBLEMS 6-1

With your financial calculator, enter the following: N = 7; I/YR = YTM = 8%; PMT = 0.06  1,000 = 60; FV = 1000; PV = VB = ? PV = $895.87. Alternatively, VB = $60((1 – 1/1.087)/0.08) + $1,000(1/1.087) = $895.87

6-2

With your financial calculator, enter the following: N = 7; PV = –930; PMT = 0.08  1,000 = 80; FV = 1000; I/YR = YTM = ? YTM = 9.41%.

6-3

With your financial calculator, enter the following to find the current value of the bonds, so you can then calculate their current yield: N = 12; I/YR = YTM = 8; PMT = 0.06  1,000 = 60; FV = 1000; PV = VB = ? PV = $849.28. Current yield = $60/$849.28 = 7.06%. Alternatively, VB = $60((1 – 1/1.0812)/0.08) + $1,000(1/1.0812) = $849.28. Current yield = $60/$849.28 = 7.06%.

Answers and Solutions: 6-132

6-4

r* = 3%; I1 = 2%; I2 = 5%; I3 = 5%; MRP = 0; rT-2 = ?; rT – 3 = ? r = r* + IP + DRP + LP + MRP. Since these are government securities, DRP = LP = 0. rT – 2 = r* + IP2 IP2 = (2% + 5%)/2 = 3.5% rT– 2 = 3% + 3.5% = 6.5%. rT – 3 = r* + IP3 IP3 = (2% + 5% + 5%)/3 = 4% rT – 3 = 3% + 4% = 7%.

6-5

rG – 10 = 6%; rC – 10 = 9%; LP = 0.5%; DRP = ? r = r* + IP + DRP + LP + MRP. rG – 10 = 6% = r* + IP + MRP; DRP = LP = 0. rC – 10 = 9% = r* + IP + DRP + 0.5% + MRP. Because both bonds are 10-year bonds, the inflation premium and maturity risk premium on both bonds are equal. The only difference between them is the liquidity and default risk premiums. rC – 10 = 9% = r* + IP + MRP + 0.5% + DRP. But we know from above that r* + IP + MRP = 6%; therefore, rC – 10 = 9% = 6% + 0.5% + DRP 2.5% = DRP.

6-6

r* = 3%; IP = 3%; rT– 2 = 6.3%; MRP2 = ? rT– 2 = r* + IP + MRP = 6.3% rT– 2 = 3% + 3% + MRP = 6.3% MRP = 0.3%.

6-7

The problem asks you to find the price of a bond, given the following facts: N = 8 × 2 = 16; I/YR = 8.5/2 = 4.25; PMT = 50; FV = 1000. With a financial calculator, solve for PV = $1,085.80.

6-8

With your financial calculator, enter the following to find YTM: N = 8  2 = 16; PV = –1110; PMT = 0.07/2  1,000 = 35; FV = 1000; I/YR = YTM = ? YTM = 2.65%  2 = 5.30%. With your financial calculator, enter the following to find YTC: N = 3  2 = 6; PV = –1110; PMT = 0.07/2  1,000 = 35; FV = 1050; I/YR = YTC = ? YTC = 2.30%  2 = 4.60%.

Answers and Solutions: 6-134

6-9 a. Input N = 10, I = 2.75, PMT = –30, FV = –1000, PV = ?. PV = $1,021.60. Input N = 10, I = 2.75, PMT = 0, FV = –1000, PV = ?. PV = $762.40. b. First bond with 10-year maturity (N = 20) and all else the same, PV = $1,038.07. First bond with 15-year maturity (N = 30) and all else the same, PV = $1,050.62. Second bond with 10-year maturity (N = 20) and all else the same, PV = $581.25. Second bond with 15-year maturity (N = 30) and all else the same, PV = $443.14. c. There is greater discounting in the zero-coupon bond price because there are no semiannual payments provided. The return must be made up from the difference between the purchase price and maturity price. 6-10

a. Calculator solution: 1. Input N = 5, PV = –829, PMT = 90, FV = 1000, I/YR = ?. I/YR = 13.98%. 2. Change PV = –1104, I/YR = ? I/YR = 6.50%. b. Yes. At a price of $829, the yield to maturity, 13.98%, is greater than your required rate of return of 12%. If your required rate of return were 12%, you should be willing to buy the bond at any price below $891.86.

6-11

N = 4; PV = –1000; PMT = 70; FV = 1090; I/YR = ? Solve for I/YR = 8.97%.

6-12

a. Using a financial calculator, input the following: N = 20, PV = –1100, PMT = 60, FV = 1000, and solve for I/YR = 5.1849%. However, this is a periodic rate. The nominal annual rate = 5.1849%(2) = 10.3699% ≈ 10.37%. b. The current yield = $120/$1,100 = 10.91%. c.

YTM = Current Yield + Capital Gains (Loss) Yield 10.37% = 10.91% + Capital Loss Yield –0.54% = Capital Loss Yield.

d. Using a financial calculator, input the following: N = 8, PV = –1100, PMT = 60, FV = 1060, and solve for I/YR = 5.0748%. However, this is a periodic rate. The nominal annual rate = 5.0748%(2) = 10.1495% ≈ 10.15%. 6-13

The problem asks you to solve for the YTM, given the following facts: N = 12, PMT = 60, and FV = 1000. In order to solve for I/YR we need PV. However, you are also given that the current yield is equal to 6.4%. Given this information, we can find PV. Current yield 0.064 PV

= Annual interest/Current price = $60/PV = $60/0.064 = $937.50.

Now, solve for the YTM with a financial calculator: N = 12, PV = –937.5, PMT = 60, and FV = 1000. Solve for I/YR = YTM = 6.78%.

Answers and Solutions: 6-136

6-14

The problem asks you to solve for the current yield, given the following facts: N = 7 × 2 = 14, I/YR = 10.5883/2 = 5.2942, PV = –1020, and FV = 1000. In order to solve for the current yield, we need to find PMT. With a financial calculator, we find PMT = $55.00. However, because the bond is a semiannual coupon bond, this amount needs to be multiplied by 2 to obtain the annual interest payment: $55.00(2) = $110.00. Finally, find the current yield as follows: Current yield = Annual interest/Current price = $110/$1,020 = 10.78%.

6-15

The bond is selling at a large premium, which means that its coupon rate is much higher than the going rate of interest. Therefore, the bond is likely to be called—it is more likely to be called than to remain outstanding until it matures. Thus, it will probably provide a return equal to the YTC rather than the YTM. So, there is no point in calculating the YTM—just calculate the YTC: N = 8, PV = –1410, PMT = 70, FV = 1090, and then solve for I/YR. The periodic rate is 2.35%, so the nominal YTC is 2 × 2.35% = 4.7%. This would be close to the going rate, and it is about what the firm would have to pay on new bonds.

6-16

10-year, 10% coupon 10-year zero 20-year zero

Price at 8% $1,135.90 456.39 208.29

t 0 1 2 3 4

Price of Bond C $1,121.56 1,094.02 1,064.66 1,033.36 1,000.00

Price at 7% $1,213.19 502.57 252.57

% Change 6.8% 10.1 21.3

6-17 Price of Bond Z $ 774.25 825.39 879.91 938.04 1,000.00

6-18

r = r* + IP + MRP + DRP + LP. r* = 0.02. IP = [0.03 + 0.04 + (5)(0.035)]/7 = 0.035. MRP = 0.0005(6) = 0.003. DRP = 0. LP = 0. r = 0.02 + 0.035 + 0.003 = 0.058 = 5.8%.

6-19

First, note that we will use the equation rt = 3% + IPt + MRPt. We have the data needed to find the IPs: IP5 =

25% 8% + 5% + 4% + 4% + 4% = = 5%. 5 5

IP2 =

8% + 5% = 6.5%. 2

Now we can substitute into the equation: r2 = 3% + 6.5% + MRP2 = 10%. r5 = 3% + 5% + MRP5 = 10%. Now we can solve for the MRPs, and find the difference: MRP5 = 10% – 8% = 2%. MRP2 = 10% – 9.5% = 0.5%. Difference = 2% – 0.5% = 1.5%.

Answers and Solutions: 6-138

6-20

To find the purchase price using a financial calculator, input N = 26, I = 3, PMT = 40, FV = 1000, and PV = ?. PV = $1,178.77. To find the selling price one year later, input N = 24, I = 3.75, PMT = 40, FV = 1000 and PV = ?. PV = $1,039.11. Capital loss = $1,039.11 – $1,178.77 = –$139.66

6-21

To find the total cost to Castle today using a financial calculator, input N = 20, I = 3.75/2, PMT = 0, FV = 18000000, PV = ?. PV = $12,414,238.

6-22

To find the call price using a financial calculator, input N = 7 × 2 = 14, I = (2.0 + 0.5)/2 = 1.25, PMT = 60/2 = 30, FV = 1000, PV = ?. PV = $1,223.48.

6-23

In order to solve for the price you are willing to pay today (time 0), you need the required rate of return, 10%, the coupon payment, $90, number of years you’ll hold the bond, 5, and the price you’ll sell it for in 5 years = ?. You can calculate the price in year 5, since you know the required return from years 6–20, 8.5%, number of years, 15, coupon payment, $90, and face value in year 20, $1,000. Input FV = 1000, N = 15, I/YR = 8.5%, PMT = 90, PV = ? = 1,041.52. Now you can calculate what you would pay at time = 0. Input FV = 1,041.52, N = 5, I/Y = 10, PMT = 90, PV = ? = $987.87.

6-24 a. 7% coupon bond: FV = 1000, N = 5, PMT = 70, I/Y = 7.5, PV = ? = 979.77 $50,000,000/0.97977 = $51,033,000. Zero coupon bond: FV = 1000, N = 5, PMT = 0, I/Y = 7.5, PV = ? = 696.56 $50,000,000/0.69656 = $71,782,000. Floating rate bond: $50,000,000 b. C0 = –1,000 C1 = 45 C2 = 55 C3 = 60 C4 = 65 Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc. 6-139

C5 = 1,065; IRR = 5.74% = YTM

Answers and Solutions: 6-140

c. The yield to maturity of the 7% coupon and zero coupon bonds reflects a higher maturity risk premium as compensation for greater interest rate risk. The floating rate bond has little interest rate risk because LIBOR changes along with market interest rates. This results in a lower maturity risk premium and a lower required rate of return. d. Benefits

Drawbacks

6-25

7% coupon -Fixed borrowing cost (no cost if rates rise).

Zero coupon -No annual interest payments to make. (preserves cash flow).

Floating rate -Lower cost of financing (assuming normal yield curve).

-Fixed borrowing cost (no benefit if rates drop).

-Large ―bullet‖ payment at maturity. -Potentially more difficult to sell.

-Interest rate risk now borne by the company. -Uncertain cash flows.

a. The bonds now have an 8-year or 16-semiannual-period maturity, and their value is calculated as follows: Calculator solution: Input N = 16, I/YR = 2.5, PMT = 35, FV = 1000, PV = ? PV = $1,130.55. b. Calculator solution: Change inputs from Part a to I/YR = 4.5, PV = ?. PV = $887.66. c. The price of the bonds will decline toward $1,000, hitting $1,000 (plus accrued interest) at the maturity date in 8 years (16 six-month periods).

6-26

a. Find the YTM as follows: N = 10, PV = –1150, PMT = 70, FV = 1000, I/YR = YTM = 5.05%. b. Find the YTC, if called in Year 5 as follows: N = 5, PV = –1150, PMT = 70, FV = 1070, I/YR = YTC = 4.82%. c. The bonds are selling at a premium, which indicates that interest rates have fallen since the bonds were originally issued. Assuming that interest rates do not change from the present level, investors would expect to earn the yield to call. (Note that the YTC is less than the YTM.)

d. Similarly from above, YTC can be found, if called in each subsequent year. If called in Year 6: N = 6, PV = –1150, PMT = 70, FV = 1060, I/YR = YTC = 4.93%. If called in Year 7: N = 7, PV = –1150, PMT = 70, FV = 1050, I/YR = YTC = 5.02%. If called in Year 8: N = 8, PV = –1150, PMT = 70, FV = 1040, I/YR = YTC = 5.09%. If called in Year 9: N = 9, PV = –1150, PMT = 70, FV = 1030, I/YR = YTC = 5.15%. According to these calculations, the latest investors might expect a call of the bonds is in Year 7. This is the last year that the expected YTC will be less than the expected YTM. At this time, the firm still finds an advantage to calling the bonds, rather than seeing them to maturity.

Answers and Solutions: 6-142

6-27

Real Years to Risk-Free Maturity Rate (r*) IP** MRP rT = r* + IP + MRP 1 2% 7.00% 0.2% 9.20% 2 2 6.00 0.4 8.40 3 2 5.00 0.6 7.60 4 2 4.50 0.8 7.30 5 2 4.20 1.0 7.20 10 2 3.60 1.0 6.60 20 2 3.30 1.0 6.30 **The computation of the inflation premium is as follows:

Year 1 2 3 4 5 10 20

Expected Average Inflation Expected Inflation 7% 7.00% 5 6.00 3 5.00 3 4.50 3 4.20 3 3.60 3 3.30

For example, the calculation for 3 years is as follows: 7% + 5% + 3% = 5.00%. 3

Thus, the yield curve would be as follows (includes yield curves as required for Parts b and c): Interest Rate (%)

Years to maturity

Answers and Solutions: 6-144

b. The interest rate on the Royal Bank bonds has the same components as the government bonds, except that the Royal Bank bonds have default risk, so a default risk premium must be included. Therefore: rRoyal = r* + IP + MRP + DRP For a strong company such as Royal Bank, the default risk premium is fairly small for short-term bonds. However, as time to maturity increases, the probability of default, although still small, is sufficient to warrant a default premium. Thus, the yield risk curve for the Royal Bank bonds will rise above the yield curve for the government bonds. In the graph, the default risk premium was assumed to be about 2 percentage points on the 20year Royal Bank bonds. The return should equal 6.3% + 2% = 8.3%. c. Rogers Communications bonds would have significantly more default risk than either the government securities or Royal Bank bonds, and the risk of default would increase over time due to possible financial deterioration. In this example, the default risk premium was assumed to be 1.2 percentage points on the 1-year Rogers bonds and 3 percentage points on the 20-year bonds. The 20-year return should equal 6.3% + 3% = 9.3%.

SOLUTION TO SPREADSHEET PROBLEM 6-28 The detailed solution for the spreadsheet problem is in the file Ch 06 Build a Model Solution.xlsx and is available on the textbook’s website.

Answers and Solutions: 6-146

MINI CASE The detailed solution for the Mini Case is also available in spreadsheet format, in the file Brigham3Ce_MiniCase_Ch06.xlsx, and is available on the textbook’s website. Sam Strother and Shawna Tibbs are vice presidents of Great White North Investment Management and co-directors of the company’s Pension Fund Management Division. An important new client has requested that Great White North present an investment seminar to its executive committee. Strother and Tibbs, who will make the actual presentation, have asked you to help them by answering the following questions. a.

What are the key features of a bond?

Answer: 1. Par or face value. We generally assume a $1,000 par value, but par can be anything, and often $5,000 or more is used. With registered bonds, which is what are issued today, if you bought $50,000 worth, that amount would appear on the certificate. 2. Coupon rate. The dollar coupon is the ―rent‖ on the money borrowed, which is generally the par value of the bond. The coupon rate is the annual interest payment divided by the par value, and it is generally set at the value of r on the day the bond is issued. 3. Maturity. This is the number of years until the bond matures and the issuer must repay the loan (return the par value). 4. Issue date. This is the date the bonds were issued. 5. Default risk is inherent in all bonds except treasury bonds—will the issuer have the cash to make the promised payments? Bonds are rated from AAA to D, and the lower the rating the riskier the bond, the higher its default risk premium, and, consequently, the higher its required rate of return, r.

What are call provisions and sinking fund provisions? Do these provisions make bonds more or less risky?

Answer: A call provision is a provision in a bond contract that gives the issuing corporation the right to redeem the bonds under specified terms prior to the normal maturity date. The call provision generally states that the company must pay the bondholders an amount greater than the par value if they are called. The additional sum, which is called a call premium, is typically set equal to one year’s interest if the bonds are called during the first year, and the premium declines at a constant rate of INT/n each year thereafter. A sinking fund provision is a provision in a bond contract that requires the issuer to retire a portion of the bond issue each year. A sinking fund provision facilitates the orderly retirement of the bond issue. The call privilege is valuable to the firm but potentially detrimental to the investor, especially if the bonds were issued in a period when interest rates were cyclically high. Therefore, bonds with a call provision are riskier than those without a call provision. Accordingly, the interest rate on a new issue of callable bonds will exceed that on a new issue of noncallable bonds. Although sinking funds are designed to protect bondholders by ensuring that an issue is retired in an orderly fashion, it must be recognized that sinking funds will at times work to the detriment of bondholders. On balance, however, bonds that provide for a sinking fund are regarded as being safer than those without such a provision, so at the time they are issued sinking fund bonds have lower coupon rates than otherwise similar bonds without sinking funds.

Mini Case: 6-148

c. How does one determine the value of any asset whose value is based on expected future cash flows? Answer:

CF1

CF2

CF3

n   

CFn

PV CF1 PV CF2 The value of an asset is merely the present value of its expected future cash flows: n

VALUE = PV =

CF1 CF2 CF3 CFn CFt + + + ... + = . 1 2 3 n (1 + r ) (1 + r ) (1 + r ) (1 + r ) t = 1 (1 + r ) t

If the cash flows have widely varying risk, or if the yield curve is not horizontal, which signifies that interest rates are expected to change over the life of the cash flows, it would be logical for each period’s cash flow to have a different discount rate. However, it is very difficult to make such adjustments; hence it is common practice to use a single discount rate for all cash flows. The discount rate is the opportunity cost of capital; that is, it is the rate of return that could be obtained on alternative investments of similar risk. For a bond, the discount rate is rd.

How is the value of a bond determined? What is the value of a 10-year, $1,000 par value bond with a 10% annual coupon if its required rate of return is 10%?

Answer: A bond has a specific cash flow pattern consisting of a stream of constant interest payments plus the return of par at maturity. The annual coupon payment is the cash flow: PMT = (Coupon rate)  (Par value) = 0.1($1,000) = $100. For a 10-year, 10% annual coupon bond, the bond’s value is found as follows: 0 |

10%

100

  

100

100 + 1,000

90.91 82.64 . . .

38.55 385.54 $ 1,000.00 Expressed as an equation, we have:

VB =

$100

$100 $1,000 + . . . + + (1 + rd )1 (1 + rd )10 (1 + rd )10

= $90.91 + . . . + $38.55 + $385.54 = $1,000. The bond consists of a 10-year, 10% annuity of $100 per year plus a $1,000 lump sum payment at t = 10: PV Annuity = $ 614.46 PV Maturity Value = 385.54 Value of Bond = $1,000.00 The mathematics of bond valuation is programmed into financial calculators, which do the operation in one step, so the easy way to solve bond valuation problems is with a financial calculator. Input N = 10, rd = I/YR = 10, PMT = 100, and FV = 1000, and then press PV to find the bond’s value, $1,000. Then change n from 10 to 1 and press PV to get the value of the 1-year bond, which is also $1,000.

Mini Case: 6-150

1. What would be the value of the bond described in part d if, just after it had been issued, the expected inflation rate rose by 3 percentage points, causing investors to require a 13% return? Would we now have a discount or a premium bond?

Answer: With a financial calculator, just change the value of rd = I/YR from 10% to 13%, and press the PV button to determine the value of the bond: 10-year = $837.21. Using the formulas, we would have, at r = 13%, VB(10-YR) = $100 ((1 – 1/(1 + 0.13)10)/0.13) + $1,000 (1/(1 + 0.13)10) = $542.62 + $294.59 = $837.21. In a situation like this, where the required rate of return, r, rises above the coupon rate, the bonds’ values fall below par, so they sell at a discount. e.

2. What would happen to the bonds’ value if inflation fell and rd declined to 7%? Would we now have a premium or a discount bond?

Answer: In the second situation, where rd falls to 7%, the price of the bond rises above par. Just change rd from 13% to 7%. We see that the 10-year bond’s value rises to $1,210.71. VB(10-YR) = $100 ((1 – 1/(1 + 0.07)10)/0.07) + $1,000 (1/(1 + 0.07)10) = $702.36 + $508.35 = $1,210.71. Thus, when the required rate of return falls below the coupon rate, the bonds’ value rises above par, or to a premium. Further, the longer the maturity, the greater the price effect of any given interest rate change.

3. What would happen to the value of the 10-year bond over time if the required rate of return remained at 13%, or if it remained at 7%? (Hint: With a financial calculator, enter PMT, I/YR, FV, and N, and then change (override) N to see what happens to the PV as the bond approaches maturity.)

Answer: Assuming that interest rates remain at the new levels (either 7% or 13%), we could find the bond’s value as time passes, and as the maturity date approaches. If we then plotted the data, we would find the situation shown below:

Bond Value ($)

1372 1211

rd = 7%.

rd = 10%.

837

rd = 13%.

775 30

Years remaining to Maturity At maturity, the value of any bond must equal its par value (plus accrued interest). Therefore, if interest rates (and hence the required rate of return) remain constant over time, then a bond’s value must move toward its par value as the maturity date approaches, so the value of a premium bond decreases to $1,000, and the value of a discount bond increases to $1,000 (barring default).

Mini Case: 6-152

1. What is the yield to maturity on a 10-year, 9% annual coupon, $1,000 par value bond that sells for $887.00? That sells for $1,134.20? What does the fact that a bond sells at a discount or at a premium tell you about the relationship between rd and the bond’s coupon rate?

Answer: The yield to maturity (YTM) is that discount rate which equates the present value of a bond’s cash flows to its price. In other words, it is the promised rate of return on the bond. (Note that the expected rate of return is less than the YTM if some probability of default exists.) On a time line, we have the following situation when the bond sells for $887: 0

9   

90 1,000

PVCF1 . .

rd = ?

PVCF2 PVCFM SUM = PV = 887 We want to find r in this equation:

VB = PV =

INT 1

(1 + r )

+ ... +

INT (1 + r )

M (1 + r )

We know N = 10, PV = –887, PMT = 90, and FV = 1000, so we have an equation with one unknown, rd. We can solve for rd by entering the known data into a financial calculator and then pressing the I/YR = rd button. The YTM is found to be 10.91%. We can tell from the bond’s price, even before we begin the calculations, that the YTM must be above the 9% coupon rate. We know this because the bond is selling at a discount, and discount bonds always have r > coupon rate. If the bond were priced at $1,134.20, then it would be selling at a premium. In that case, it must have a YTM that is below the 9% coupon rate, because all premium bonds must have coupons that exceed the going interest rate. Going through the same procedures as before—plugging the appropriate values into a financial calculator and then pressing the r = I button, we find that at a price of $1,134.20, r = YTM = 7.08%.

2. What are the total return, the current yield, and the capital gains yield for the discount bond? (Assume the bond is held to maturity and the company does not default on the bond.)

Answer: The current yield is defined as follows:

Current yield =

Annual coupon interest payment . Current price of the bond

The capital gains yield is defined as follows:

Capital gains yield =

Expected change in bond's price . Beginning-of-year price

The total expected return is the sum of the current yield and the expected capital gains yield: Expected Expected Expected capital = + . total return current yield gains yield

For our 9% coupon, 10-year bond selling at a price of $887 with a YTM of 10.91%, the current yield is:

$90 = 0.1015 = 10.15%. $887 Knowing the current yield and the total return, we can find the capital gains yield: Current yield =

YTM = Current yield + Capital gains yield And Capital gains yield = YTM – Current yield = 10.91% – 10.15% = 0.76%. The capital gains yield calculation can be checked by asking this question: ―What is the expected value of the bond 1 year from now, assuming that interest rates remain at current levels?‖ This is the same as asking, ―What is the value of a 9-year, 9% annual coupon bond if its YTM (its required rate of return) is 10.91%?‖ The answer, using the bond valuation function of a calculator, is $893.87. With this data, we can now calculate the bond’s capital gains yield as follows: Capital gains yield = ( V B1 - V B0 )/ V B0 = ($893.87 – $887)/$887 = 0.0077 = 0.77%

Mini Case: 6-154

This agrees with our earlier calculation (except for rounding).

How does the equation for valuing a bond change if semiannual payments are made? Find the value of a 10-year, semiannual payment, 10% coupon bond if nominal rd = 13%.

Answer: In reality, virtually all bonds issued in Canada have semiannual coupons and are valued using the setup shown below: 0

1 2

2 4

INT/2

2N – 1

N YEARS 2N SA PERIODS

INT/2

INT/2 M

  

PV1 . . .

PVN PVM VBOND = sum of PVs We would use this equation to find the bond’s value: 2N

VB = 

INT/ 2

t = 1 (1 + r d / 2 )

M (1 + rd / 2 )2 N

The payment stream consists of an annuity of 2N payments plus a lump sum equal to the maturity value. To find the value of the 10-year, semiannual payment bond, semiannual interest = annual coupon/2 = $100/2 = $50 and N = 2 (years to maturity) = 2(10) = 20. To find the value of the bond with a financial calculator, enter N = 20, rd/2 = I/YR = 5, PMT = 50, FV = 1000, and then press PV to determine the value of the bond. Its value is $1,000. You could then change rd = I/YR to see what happens to the bond’s value as r changes, and plot the values—the graph would look like the one we developed earlier. For example, if rd rose to 13%, we would input I/YR = 6.5 rather than 5%, and find the 10-year bond’s value to be $834.72. If rd fell to 7%, then input I/YR = 3.5 and press PV to find the bond’s new value, $1,213.19. We would find the values with a financial calculator, but they could also be found with formulas. Thus: V10-YEAR = $50 ((1 – 1/(1 + 0.05)20)/0.065) + $1,000 (1/(1 + 0.05)20) = $50(12.4622) + $1,000(0.37689) = $623.11 + $376.89 = $1,000.00.

Mini Case: 6-156

At a 13% required return: V10-YEAR = $50 ((1 – 1/(1 + 0.065)20)/0.065) + $1,000 (1/(1 + 0.065)20) = $834.72. h.

Suppose a 10-year, 10%, semiannual coupon bond with a par value of $1,000 is currently selling for $1,135.90, producing a nominal yield to maturity of 8%. However, the bond can be called after 5 years for a price of $1,050. 1. What is the bond’s nominal yield to call (YTC)?

Answer: If the bond were called, bondholders would receive $1,050 at the end of year 5. Thus, the time line would look like this: 0

5 |

50 1,050

PV1 . .

PV4 PV5C PV5CP 1,135.90 = sum of PVs The easiest way to find the YTC on this bond is to input values into your calculator: N = 10; PV = –1135.90; PMT = 50; and FV = 1050, which is the par value plus a call premium of $50; and then press the rd = I/YR button to find I/YR = 3.765%. However, this is the 6-month rate, so we would find the nominal rate on the bond as follows: rNOM = 2(3.765%) = 7.5301% ≈ 7.5%. This 7.5% is the rate brokers would quote if you asked about buying the bond. You could also calculate the EAR on the bond: EAR = (1.03765)2 – 1 = 7.672%. Usually, people in the bond business just talk about nominal rates, which is okay so long as all the bonds being compared are on a semiannual payment basis. When you start making comparisons among investments with different payment patterns, though, it is important to convert to EARs.

Mini Case: 6-158

2. If you bought this bond, do you think you would be more likely to earn the YTM or the YTC? Why?

Answer: Since the coupon rate is 10% versus YTC = rd = 7.53%, it would pay the company to call the bond, get rid of the obligation to pay $100 per year in interest, and sell replacement bonds whose interest would be only $75.30 per year. Therefore, if interest rates remain at the current level until the call date, the bond will surely be called, so investors should expect to earn 7.53%. In general, investors should expect to earn the YTC on premium bonds, but to earn the YTM on par and discount bonds. (Bond brokers publish lists of the bonds they have for sale; they quote YTM or YTC depending on whether the bond sells at a premium or a discount.) i.

Write a general expression for the yield on any debt security (rd) and define these terms: real risk-free rate of interest (r*), inflation premium (IP), default risk premium (DRP), liquidity premium (LP), and maturity risk premium (MRP).

Answer: rd = r* + IP + DRP + LP + MRP. r* is the real risk-free interest rate. It is the rate you would see on a riskless security if there were no inflation. The inflation premium (IP) is a premium added to the real risk-free rate of interest to compensate for expected inflation. The default risk premium (DRP) is a premium based on the probability that the issuer will default on the loan, and it is measured by the difference between the interest rate on a Government of Canada bond and a corporate bond of equal maturity and marketability. A liquid asset is one that can be sold at a predictable price on short notice; a liquidity premium is added to the rate of interest on securities that are not liquid. The maturity risk premium (MRP) is a premium that reflects interest rate risk; longer-term securities have more interest rate risk (the risk of capital loss due to rising interest rates) than do shorter-term securities, and the MRP is added to reflect this risk.

Define the nominal risk-free rate (rRF). What security can be used as an estimate of rRF?

Answer: The real risk-free rate, r*, is the rate that would exist on default-free securities in the absence of inflation: rRF = r* + IP. The nominal risk-free rate, rRF, is equal to the real risk-free rate plus an inflation premium, which is equal to the average rate of inflation expected over the life of the security. There is no truly riskless security, but the closest thing is a short-term Canadian Treasury bill (T-bill), which is free of most risks. The real risk-free rate, r*, is estimated by subtracting the expected rate of inflation from the rate on short-term treasury securities. It is generally assumed that r* is in the range of 1 to 4 percentage points. The long-term government bond rate is used as a proxy for the long-term riskfree rate. However, we know that all long-term bonds contain interest rate risk, so the long-term government bond rate is not really riskless. It is, however, free of default risk. k.

What is a bond spread and how is it related to the default risk premium? How are bond ratings related to default risk? What factors affect a company’s bond rating?

Answer: A bond spread is often calculated as the difference between a corporate bond’s yield and a Treasury security’s yield of the same maturity. Therefore: Spread = DRP + LP. Bonds of large, strong companies often have very small LPs. Bonds of small companies often have LPs as high as 2%. Bond ratings are based upon a company’s default risk. They are based on both qualitative and quantitative factors, some of which are listed below. 1. Financial performance—determined by ratios such as the debt, TIE, FCC, and current ratios. 2. Provisions in the bond contract: A. Secured vs. Unsecured debt B. Senior vs. Subordinated debt C. Guarantee provisions D. Sinking fund provisions E. Debt maturity Mini Case: 6-160

3. Other factors:

Mini Case: 6-162

A. Earnings stability B. Regulatory environment C. Potential product liability D. Accounting policy l.

What is interest rate (or price) risk? Which bond has more interest rate risk, an annual payment 1-year bond or a 10-year bond? Why?

Answer: Interest rate risk, which is often just called price risk, is the risk that a bond will lose value as the result of an increase in interest rates. Below are values for a one- and tenyear, 10% annual coupon bond at various interest rates:

rd 5% 10 15

Maturity Change 4.8%

1-Year $1,048 1,000 957

–4.3%

10-Year $1,386 1,000 749

Change 38.6% –25.1%

A 5 percentage point increase in r causes the value of the 1-year bond to decline by only 4.3%, but the 10-year bond declines in value by more than 25%. Thus, the 10year bond has more interest rate price risk.

Bond Value Interest Rate Price Risk for 10 Percent Coupon Bonds with Different Maturities ($) 1,800

1,400

1,000

1-Year 5-Year 10-Year 20-Year 30-Year

600

Interest Rate (%)

The graph above shows the relationship between bond values and interest rates for a 10%, annual coupon bond with different maturities. The longer the maturity, the greater the change in value for a given change in interest rates, rd.

What is reinvestment rate risk? Which has more reinvestment rate risk, a 1-year bond or a 10-year bond? Why?

Answer: Reinvestment rate risk is defined as the risk that cash flows (interest plus principal repayments) will have to be reinvested in the future at rates lower than today’s rate. To illustrate, suppose you just won the lottery and now have $500,000. You plan to invest the money and then live on the income from your investments. Suppose you buy a 1-year bond with a YTM of 10%. Your income will be $50,000 during the first year. Then, after 1 year, you will receive your $500,000 when the bond matures, and you will then have to reinvest this amount. If rates have fallen to 3%, then your income will fall from $50,000 to $15,000. On the other hand, had you bought 30-year bonds that yielded 10%, your income would have remained constant at $50,000 per year. Clearly, buying bonds that have short maturities carries reinvestment rate risk. Note that long maturity bonds also have reinvestment rate risk, but the risk applies only to the coupon payments, and not to the principal amount. Since the coupon payments are significantly less than the principal amount, the reinvestment rate risk on a long-term bond is significantly less than on a short-term bond. n.

How are interest rate risk and reinvestment rate risk related to the maturity risk premium?

Answer: Long-term bonds have high interest rate risk but low reinvestment rate risk. Shortterm bonds have low interest rate risk but high reinvestment rate risk. Nothing is riskless! Yields on longer-term bonds usually are greater than on shorter-term bonds, so the MRP is more affected by interest rate risk than by reinvestment rate risk. o.

What is the term structure of interest rates? What is a yield curve?

Answer: The term structure of interest rates is the relationship between interest rates, or yields, and maturities of securities. When this relationship is graphed, the resulting curve is called a yield curve.

Mini Case: 6-164

At any given time, how would the yield curve facing an AA-rated company compare with the yield curve for Government of Canada bonds? At any given time, how would the yield curve facing a BB-rated company compare with the yield curve for Government of Canada bonds?

Answer: The yield curve normally slopes upward, indicating that short-term interest rates are lower than long-term interest rates. Yield curves can be drawn for government securities or for the securities of any corporation, but corporate yield curves will always lie above government yield curves, and the riskier the corporation, the higher its yield curve. The spread between a corporate yield curve and the government curve widens as the corporate bond rating decreases.

Mini Case: 6-166

Briefly describe bankruptcy law. If a firm were to default on the bonds, would the company be immediately liquidated? Would the bondholders be assured of receiving all of their promised payments?

Answer: When a business becomes insolvent, it does not have enough cash to meet scheduled interest and principal payments. A decision must then be made whether to dissolve the firm through liquidation or to permit it to reorganize and thus stay alive. The decision to force a firm to liquidate or to permit it to reorganize depends on whether the value of the reorganized firm is likely to be greater than the value of the firm’s assets if they were sold off piecemeal. In a reorganization, management along with a court-appointed trustee creates a proposal that unsecured creditors vote on. The proposal may call for a restructuring of the firm’s debt, in which case the interest rate may be reduced, the term to maturity may be lengthened, or some of the debt may be exchanged for equity. The point of the restructuring is to reduce the financial charges to a level that the firm’s cash flows can support. If the firm is deemed to be too far gone to be saved, it will be liquidated and the priority of claims would be as follows: • • • • • • • • • • • • • •

Those of suppliers, who can repossess goods delivered within 30 days prior to bankruptcy Costs for environmental damage Trustee expenses Unremitted payroll taxes and deductions Unpaid wages, up to $2,000 per worker Those of secured creditors whose loans are secured by some specific asset, mortgage, lien, etc. Bankruptcy administration costs Municipal taxes Those for rent, up to 3 months prior to bankruptcy Those of creditors who first filed claims Injury claim costs to employees not covered under Workers’ Compensation Other unsecured creditors Preferred shareholders Common shareholders If the firm’s assets were worth more ―alive‖ than ―dead,‖ the company would be reorganized. Its creditors, however, would expect to take a ―hit.‖ Thus, they would not expect to receive all their promised payments. If the firm was deemed to be too far gone to be saved, it would be liquidated.

WEB EXTENSION 6C: THE PURE EXPECTATIONS THEORY AND ESTIMATION OF FORWARD RATES SOLUTIONS TO PROBLEMS 6C-1 rT1 = 5%; 1rT1 = 6%; rT2 = ? (1 + rT2)2 = (1.05)(1.06) (1 + rT2)2 = 1.113 1 + rT2 = 1.055 rT2 = 5.5%. 6C-2 Let X equal the yield on 2-year securities 4 years from now: (1.07)4(1 + X)2 = (1.075)6 (1.3108)(1 + X)2 = 1.5433  1.5433    1.3108 

1/2

1+X = 

X = 8.5%. 6C-3 a.

(1.045)2 = (1.03)(1 + X) 1.092/1.03 = 1 + X X = 6%.

b. For riskless bonds under the expectations theory, the interest rate for a bond of any maturity is rN = r* + average inflation over N years. If r* = 1%, we can solve for IPN: Year 1: r1 = 1% + I1 = 3%; I1 = expected inflation = 3% – 1% = 2%. Year 2: r1 = 1% + I2 = 6%; I2 = expected inflation = 6% – 1% = 5%. Note also that the average inflation rate is (2% + 5%)/2 = 3.5%, which, when added to r* = 1%, produces the yield on a 2-year bond, 4.5%. Therefore, all of our results are consistent.

6C-4 r* = 2%; MRP = 0%; r1 = 5%; r2 = 7%; X = ? X represents the one-year rate on a bond one year from now (Year 2). (1.07)2 = (1.05)(1 + X) 1.1449 1.05

=1+X

X = 9%. 9% = r* + I2 9% = 2% + I2 7% = I2. The average interest rate during the 2-year period differs from the 1-year interest rate expected for Year 2 because of the inflation rate reflected in the two interest rates. The inflation rate reflected in the interest rate on any security is the average rate of inflation expected over the security’s life.

Chapter 7 Risk, Return, and the Capital Asset Pricing Model ANSWERS TO END-OF-CHAPTER QUESTIONS 7-1

a. Risk is the chance that some unfavourable event will occur. For instance, the risk of an asset is essentially the chance that the asset’s cash flows will be unfavourable or less than expected. Stand-alone risk is only a part of total risk and pertains to the risk an investor takes by holding only one asset. A probability distribution is a listing, chart or graph of all possible outcomes, such as expected rates of return, with a probability assigned to each outcome. When in graph form, the tighter the probability distribution, the less uncertain the outcome. 

b. The expected rate of return ( r ) is the expected value of a probability distribution of expected returns. c. A continuous probability distribution contains an infinite number of outcomes and is graphed from – and +.

d. The standard deviation (σ) is a statistical measure of the variability of a set of observations. The variance (σ2) of the probability distribution is the sum of the squared deviations about the expected value adjusted for deviation. The coefficient of variation (CV) is equal to the standard deviation divided by the expected return; it is a standardized risk measure that allows comparisons between investments having different expected returns and standard deviations. e. A risk-averse investor dislikes risk and requires a higher rate of return as an inducement to buy riskier securities. A realized return is the actual return an investor receives on his or her investment. It can be quite different from the expected return. f. A risk premium is the difference between the rate of return on a risk-free asset and the expected return on Stock i, which has higher risk. The market risk premium is the difference between the expected return on the market and the risk-free rate. g. CAPM is a model based upon the proposition that any stock’s required rate of return is equal to the risk-free rate of return plus a risk premium reflecting only the risk remaining after diversification. 

h. The expected return on a portfolio, r p, is simply the weighted-average expected return of the individual stocks in the portfolio, with the weights being the fraction of total portfolio value invested in each stock. The market portfolio is a portfolio consisting of all stocks. i. Correlation is the tendency of two variables to move together. A correlation coefficient (ρ) of +1.0 means that the two variables move up and down in perfect synchronization, while a coefficient of –1.0 means the variables always move in opposite directions. A correlation coefficient of zero suggests that the two variables are not related to one another; that is, they are independent. j. Market risk is that part of a security’s total risk that cannot be eliminated by diversification. It is measured by the beta coefficient. Diversifiable risk is also known as company specific risk, that part of a security’s total risk associated with random events not affecting the market as a whole. This risk can be eliminated by proper diversification. The relevant risk of a stock is its contribution to the riskiness of a well-diversified portfolio. k. The beta coefficient, b, is a measure of a stock’s market risk, or the extent to which the returns on a given stock move with the stock market. The average stock’s beta would move on average with the market so it would have a beta of 1.0. l. The Security Market Line (SML) represents, in a graphical form, the relationship between the risk of an asset as measured by its beta and the required rates of return 7-170 Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc.

for individual securities. The SML equation is essentially the CAPM, ri = rRF + bi(rM – rRF). m. The slope of the SML equation is (rM – rRF), the market risk premium. The slope of the SML reflects the degree of risk aversion in the economy. The greater the average investor’s aversion to risk, then the steeper the slope, the higher the risk premium for all stocks, and the higher the required return. n. Most people don’t behave rationally in all aspects of their personal lives, and behavioural finance assumes that investors have the same types of psychological behaviours in their financial lives as in their personal lives. Anchoring bias is the human tendency to ―anchor‖ too closely on recent events when predicting future events. Herding is the tendency of investors to follow the crowd. When combined with overconfidence, anchoring and herding can contribute to market bubbles.

7-2

a. The probability distribution for complete certainty is a vertical line. b. The probability distribution for total uncertainty is the X axis from – to +.

7-3

Security A is less risky if held in a diversified portfolio because of its lower beta and negative correlation with other stocks. In a single-asset portfolio, Security A would be more risky because σA > σB and CVA > CVB.

7-4

a. No, it is not riskless. The portfolio would be free of default risk and liquidity risk, but inflation could erode the portfolio’s purchasing power. If the actual inflation rate is greater than that expected, interest rates in general will rise to incorporate a larger inflation premium (IP) and the value of the portfolio will decline. b. No, you would be subject to reinvestment rate risk. You might expect to ―roll over‖ the Treasury bills at a constant (or even increasing) rate of interest, but if interest rates fall, your investment income will decrease. c. A Government of Canada bond that provided interest with constant purchasing power (that is, an indexed bond) would be close to riskless.

7-5

The risk premium on a high beta stock would increase more. RPj = Risk Premium for Stock j = (rM – rRF)bj. If risk aversion increases, the slope of the SML will increase, and so will the market risk premium (rM – rRF). The product (rM – rRF)bj is the risk premium of the jth stock. If bj is low (say, 0.5), then the product will be small; RPj will increase by only half the increase

in RPM. However, if bj is large (say, 2.0), then its risk premium will rise by twice the increase in RPM. 7-6

According to the Security Market Line (SML) equation, an increase in beta will increase a company’s expected return by an amount equal to the market risk premium times the change in beta. For example, assume that the risk-free rate is 6%, and the market risk premium is 5%. If the company’s beta doubles from 0.8 to 1.6, its expected return increases from 10% to 14%. Therefore, in general, a company’s expected return will not double when its beta doubles.

7-7

Yes, if the portfolio’s beta is equal to zero. In practice, however, it may be impossible to find individual stocks that have a nonpositive beta. In this case it would also be impossible to have a stock portfolio with a zero beta. Even if such a portfolio could be constructed, investors would probably be better off just purchasing Treasury bills, or other zero beta investments.

SOLUTIONS TO END-OF-CHAPTER PROBLEMS 7-1

Investment $20,000 35,000 Total $55,000

Beta 0.7 1.3

($20,000/$55,000)(0.7) + ($35,000/$55,000)(1.3) = 1.08. 7-2

rRF = 6%; rM = 11%; b = 0.6; rs = ? rs = rRF + (rM – rRF)b = 6% + (11% – 6%)0.6 = 9%.

7-3

rRF = 5%; RPM = 7%; rM = ? rM = 5% + (7%)1 = 12% = rs when b = 1.0. rs when b = 1.7 = ? rs = 5% + 7%(1.7) = 16.9%. 

7-4

r = (0.1)(–30%) + (0.2)(–5%) + (0.4)(10%) + (0.2)(16%) + (0.1)(40%) = 7.2%.

σ2 = (–30% – 7.2%)2(0.1) + (-5% – 7.2%)2(0.2) + (10% – 7.2%)2(0.4) + (16% – 7.2%)2(0.2) + (40% – 7.2%)2(0.1) σ2 = 294.36; σ= 17.16%. CV =

17.16% = 2.38. 7.2%



7-5

r m= (0.3)(12%) + (0.4)(5%) + (0.3)(15%) = 10.1%. 

r j= (0.3)(13%) + (0.4)(4%) + (0.3)(12%) = 9.1%.

b. σM = [(0.3)(12% – 10.1%)2 + (0.4)(5% – 10.1%)2 + (0.3)(15% – 10.1%)2]1/2 = 18.69% = 4.32%. σJ = [(0.3)(13% – 9.1%)2 + (0.4)(4% – 9.1%)2 + (0.3)(12% – 9.1%)2]1/2 = 17.49% = 4.18%.

7-6

c. CVM =

4.32% = 0.43. 10.1%

CVJ =

4.18% = 0.46. 9.1%

rA = rRF + (rM – rRF)bA 12% = 5% + (10% – 5%)bA 12% = 5% + 5%(bA) 7% = 5%(bA) 1.4 = bA.

b. rA = 5% + 5%(bA) rA = 5% + 5%(2) rA = 15%.



7-7

r A = (0.1)(–3%) + (0.2)(2%) + (0.4)(7%) + (0.2)(12%) + (0.1)(17%) = 7%. 

r B = (0.1)(–10%) + (0.2)(1%) + (0.4)(8%) + (0.20)(16%) + (0.1)(24%) = 8%.

σA = [(0.1)(–3% – 7%)2 + (0.2)(2% – 7%)2 + (0.4)(7% – 7%)2 + (0.2)(12% – 7%)2 + (0.1)(17% – 7%)2]1/2 = 30% = 5.48%. σB = [(0.1)(–10% – 8%)2 + (0.2)(1% – 8%)2 + (0.4)(8% – 8%)2] + (0.2)(16% – 8%)2 + (0.1)(24% – 8%)2]1/2 = 80.6% = 8.98%. CVA =

5.48% = 0.78. 7%

CVB =

8.98% = 1.12 8%



7-8

r P = (0.4)(7%) + (0.6)(8%) = 7.6%

b. p = w2A2A + (1wA)22B + 2wA (1wA)ABAB (0 4)(0 )(0 40)( 4 = √(0 4) ( 4 ) (0 ) ( ) = √4 0 0 4 √4

7-9

)(

)

a. rUT = rRF + (rM - rRF)bUT = 5% + (12% - 5%)1.4 = 14.8%. b. 1. rRF increases to 6%: The slope of the SML is equal to the market risk premium, RPM, which does not change. From Part a, the market risk premium is 7%: rM – rRF = 12% – 5% = 7%. Using 6% for the risk-free rate and 7% for the market risk premium, the required return on the stock is: rUT = rRF + (RPM)bUT = 6% + (7%)1.4 = 15.8%. 2. rRF decreases to 4%: rUT = rRF + (RPM)bUT = 4% + (7%)1.4 = 13.8%.

c. 1. rM increases to 14%: If the risk-free rate does not change but the required return on the market does change, then the market risk premium changes. For rRF = 5% and rM = 14%, the new market risk premium is 9%: RPM = rM – rRF = 14% – 5% = 9%. The required return on the stock is: rUT = rRF + (RPM)bUT = 5% + (9%)1.4 = 17.6%. 2. rM decreases to 11%: The new market risk premium is 6%: RPM = rM – rRF = 11% – 5% = 6%. The required return on the stock is: rUT = rRF + (rM – rRF)bUT = 5% + (6%)1.4 = 13.4%.

7-10

Old portfolio beta =

$190,000 $10,000 (b) + (1.00) $200,000 $200,000

1.20 = 0.95b + 0.05 1.15 = 0.95b 1.21 = b. New portfolio beta = 0.95(1.21) + 0.05(1.80) = 1.24. Alternative Solutions: 1. Old portfolio beta = 1.20 = (0.05)b1 + (0.05)b2 + ... + (0.05)b20 1.20 = (bi)(0.05) bi = 1.20/0.05 = 24. New portfolio beta = (24 – 1.0 + 1.8)(0.05) = 1.24. 2. bi excluding the stock with the beta equal to 1.0 is 24 – 1.0 = 23, so the beta of the portfolio excluding this stock is b = 23/19 = 1.21. The beta of the new portfolio is: 1.21(0.95) + 1.80(0.05) = 1.24.

b. An average stock would have a beta of 1, therefore the return would be ri = 2% + 5% × 1 = 7%. c. A security with a 0 beta has a return not correlated with the market at all. If there is no market risk associated with a 0 beta security, its return should be risk-free, in this case 2%. d. If the economy worsens, investors will become more pessimistic. They generally will revise their return requirements upward for holding stocks. In other words, they will require greater returns; therefore, the 6.5% market risk premium would be more suitable. e. Stock A: ri = 2% + 6.5% × 1.4 = 11.1%. Stock B: ri = 2% + 6.5% × 0.8 = 7.2%. Stocks A and B would be expected to fall. f. The required return on investments, including stocks, would increase by 0.5%. The new required return on A would be ri = 2.5% + 5% × 1.4 = 9.5% and B would be ri = 2.5% + 5% × 0.8 = 6.5%. The stock prices of A and B would both fall if inflation expectations increased. 7-12

^r p = wA^r A + (1  wA) ^r B = 0.30(12%) + 0.70(18%) = 16.20%



p = w2A2A + (1wA)22B + 2wA (1wA)ABAB = (0.32)(0.42) + (0.72)(0.62) + 2(0.3)(0.7)(0.2)(0.4)(0.6) = 0.0144 + 0.1764 + 0.02016 = 0.21096 = 0.459 = 45.9%

7-13

Portfolio beta= $9,000,000 1.75+ $9,000,000 -0.50+ $9,000,000 1.0+ $9,000,000 0.75 = 0.72

$1,000,000

$1,500,000

$2,500,000

$4,000,000

rp = rRF + (rM – rRF)(bp) = 5% + (10% – 5%)(0.72) = 8.6%. Alternative solution: First compute the return for each stock using the CAPM equation [rRF + (rM – rRF)b], and then compute the weighted average of these returns. rRF = 5% and rM – rRF = 5%. Stock A B C D

Investment Beta r = rRF + (rM – rRF)b $ 1,000,000 1.75 13.75% 1,500,000 (0.50) 2.50 2,500,000 1.00 10.00 4,000,000 0.75 8.75

Weight 0.11 0.17 0.28 0.44

Total

$ 9,000,000

1.00

rp = 13.75%(0.11) + 2.5%(0.17) + 10%(0.28) + 8.75%(0.44) = 8.6%. 7-14 a. Total portfolio = $75 + $125 + $200 = $400 $75/$400 × 12.5% = $125/$400 × 6.75% = $200/$400 × 9% = Portfolio expected return =

2.34% 2.11% 4.5% 8.95%

b. $75/$400 × 1.9 = $125/$400 × 0.75 = $200/$400 × 1.2 = Portfolio beta =

0.356 0.234 0.6 1.19

r̂P  3%  1.19(5%)  8.95% Since rp = r̂P the stocks plot along the SML and are priced correctly. c.

( ) 4 Since the expected return of 14% < the required return of 14.5%, the investor should not purchase the stock.

d. Total portfolio = $75 + $125 + $200 + $100 = $500 $75/$500 × 12.5% = $125/$500 × 6.75% = $200/$500 × 9% = $100/$500 × 14.5% = Portfolio expected return =

1.88% 1.69% 3.6% 2.9% 10.1%

7-15 a.

r̂M = 0.1(5%) + 0.2(9%) + 0.4(12%) + 0.2(16%) + 0.1(20%) = 12.3%.

rRF = 4%. (given) Therefore, the SML equation is: ri = rRF + (rM – rRF)bi = 4% + (12.3% – 4%)bi = 4% + (8.3%)bi. Next, determine the fund’s beta, bF. The weights are the percentage of funds invested in each stock: A = $70/$200 = 0.35. B = $50/$200 = 0.25. C = $30/$200 = 0.15. D = $30/$200 = 0.15. E = $20/$200 = 0.10. bF = 0.35(0.7) + 0.25(1.4) + 0.15(2.5) + 0.15(2.6) + 0.10(2.8) = 0.245 + 0.35 + 0.375 + 0.39 + 0.28 = 1.64. Next, use bF = 1.64 in the SML determined above. r̂F = 4% + (12.3% – 4%)1.64 = 4% + 13.61% = 17.61%.

b. rN = Required rate of return on new stock = 4% + (12.3% – 4%)1.2 = 14.0%. An expected return of 15% on the new stock is above the 14% required rate of return on an investment with a risk of b =1.2. Since rN = 14% < r̂N = 15%, the new stock should be purchased. c. The expected rate of return that would make the fund indifferent to purchasing the stock is 14%. 7-16

First, calculate the beta of what remains after selling the stock: bp = 1.3 = ($100,000/$2,000,000)0.9 + ($1,900,000/$2,000,000)bR 1.3 = 0.045 + (0.95)bR bR = 1.321. bN = (0.95)1.321 + (0.05)1.4 = 1.33.

7-17

We know that bR = 1.50, bS = 0.75, rM = 13%, rRF = 7%. ri = rRF + (rM – rRF)bi = 7% + (13% – 7%)bi. rR = 7% + 6%(1.50) = 16.0% rS = 7% + 6%(0.75) = 11.5 4.5%

7-18

The answers to a, b, c, and d are given below:

2017 2018 2019 2020 2021 Mean Std Dev CV

r̄ A (20.00%) 42.00 20.00 (8.00) 25.00

r̄ B (5.00%) 15.00 (13.00) 50.00 12.00

Portfolio (12.50%) 28.50 3.50 21.00 18.50

11.80 25.28 2.14

11.80 24.32 2.06

11.80 16.34 1.38

e. A risk-averse investor would choose the portfolio over either Stock A or Stock B alone, since the portfolio offers the same expected return but with less risk. This result occurs because returns on A and B are not correlated (ρAB = 0.13). 

7-19

a. rFSP = 4% + (5%)0.8 = 8%.

Req. return of 8% = r FSP = 8%; correctly valued. 

rRPP = 4% + (5%)1.6 = 12%. Req. return of 12% < r RPP= 14%; undervalued. 

rFGP = 4% + (5%)1.4 = 11%. Req. return of 11% > r FGP= 10%; overvalued. 

rSPEC= 4% + (5%)2.0 = 14%. Req. return of 14% = r SPEC= 14%; correctly valued. 

rDT = 4% + (5%)1.2 = 10%. Req. return of 10% < r DT= 11%; undervalued. b. bp = 0.25(0.8) + 0.25(1.6) + 0.25(2.0) + 0.25(1.2) = 1.4. rp = 4% + (5%)1.4 = 11%. 7-20

a. rX = 3% + (6%)1.50 = 12.0%. rY = 3% + (6%)0.46 = 5.76%. b. bp = 0.80(1.50) + 0.20(0.46) = 1.292 ≈ 0.1.29. rp = 3% + (6%)1.29 = 10.74%.

Alternatively, rp = 0.80(12.0%) + 0.20(5.76%) = 10.75% (difference due to rounding). c. Stock X is undervalued, because its expected return exceeds its required rate of return.

7-21

a. The arithmetic average return for Stock X is calculated as follows:

r Avg 

(14.0  23.0  ...  18.2)  10.6%. 7

The arithmetic average rate of return on the market portfolio, determined similarly, is 12.1%. For Stock X, the estimated standard deviation is 13.1%:

X 

(14.0  10.6) 2  (23.0  10.6) 2  ...  (18.2  10.6) 2  13.1%. 7 1

The standard deviation of returns for the market portfolio is similarly determined to be 22.6%. The results are summarized below: Stock X Market Portfolio 10.6% 12.1% 13.1 22.6

Average return, r Avg Standard deviation, σ

Several points should be noted: (1) σM over this particular period is higher than the historic average σM of about 15%, indicating that the stock market was relatively volatile during this period; (2) Stock X, with X = 13.1%, has much less total risk than an average stock, with Avg = 22.6%; and (3) this example demonstrates that it is possible for a very low-risk single stock to have less risk than a portfolio of average stocks, since σX < σM. b. Since Stock X is in equilibrium and plots on the Security Market Line (SML), and 



given the further assumption that r X  r X and r M  r M —and this assumption often does not hold—then this equation must hold: r X  rRF  (r  rRF )b X .

This equation can be solved for the risk-free rate, rRF, which is the only unknown:

10.6  rRF  (12.1  rRF ) 0.56 10.6  rRF  6.8  0.56 rRF 0.44rRF  10.6  6.8 rRF  3.8 / 0.44  8.6%. c. The SML is plotted below. Data on the risk-free security (bRF = 0, rRF = 8.6%) and Security X (bX = 0.56, r X = 10.6%) provide the two points through which the SML can be drawn. rM provides a third point. r(%) k(%)

rXkX= =10.6% 10.6% kM = 12.1%

rRF = 8.6%10 kRF = 8.6

1.0

2.0

Beta

d. In theory, you would be indifferent between the two stocks. Since they have the same beta, their relevant risks are identical, and in equilibrium they should provide the same returns. The two stocks would be represented by a single point on the SML. Stock Y, with the higher standard deviation, has more diversifiable risk, but this risk will be eliminated in a well-diversified portfolio, so the market will compensate the investor only for bearing market or relevant risk. In practice, it is possible that Stock Y would have a slightly higher required return, but this premium for diversifiable risk would be small.

SOLUTION TO SPREADSHEET PROBLEM 7-22

The detailed solution for the spreadsheet problem is in the file Ch 07 Build a Model Solution.xlsx and is available on the textbook’s website.

MINI CASE The detailed solution for the Mini Case is also available in spreadsheet format, in the file Brigham3Ce_MiniCase_Ch07.xlsx, and is available on the textbook’s website. Assume that you recently graduated and landed a job as a financial planner with Cicero Services, an investment advisory company. Your first client recently inherited some assets and has asked you to evaluate them. The client owns a bond portfolio with $1 million invested in zero coupon government bonds that mature in 10 years. The client also has $2 million invested in the stock of Blandy Inc., a company that produces meat-and-potatoes frozen dinners. Blandy’s slogan is ―Solid food for shaky times.‖ Unfortunately, economic times are uncertain and there is much discussion on what interest rates may be one year from now. Your first task is to determine the risk of the client’s bond portfolio. After consulting with the economists at your firm, you have specified five possible scenarios. For each scenario, you have estimated the probability of the scenario occurring and the impact on interest rates and bond prices if the scenario occurs. Given this information, you have calculated the rate of return on 10-year zero coupon government bonds for each scenario. The probabilities and returns are shown here:

Scenario Worst Case Poor Case Most Likely Good Case Best Case

Probability of Scenario 0.10 0.20 0.40 0.20 0.10 1.00

Return on a 10-Year Zero Coupon Treasury Bond During the Next Year −14% −4% 6% 16% 26%

You have also gathered historical returns for the past 10 years for Blandy and Gourmange Corporation (a producer of gourmet specialty foods), and the stock market.

Historical Stock Returns Year 1 2 3 4 5 6 7 8 9 10 Average return: Standard deviation: Correlation with the market: Beta:

Market 30% 7 18 −22 −14 10 26 −10 −3 38 8.0% 20.1% 1.00 1.00

Blandy 26% 15 −14 −15 2 −18 42 30 −32 28 ? ? ? ?

Gourmange 47% −54 15 7 −28 40 17 −23 −4 75 9.2% 38.6% 0.678 1.30

The risk-free rate is 4% and the market risk premium is 5%. a.

What are investment returns? What is the return on an investment that costs $1,000 and is sold after 1 year for $1,060?

Answer: Investment return measures the financial results of an investment. They may be expressed in either dollar terms or percentage terms. The dollar return is $1,60 – $1,000 = $60. The percentage return is $60/$1,000 = 0.06 = 6%. b.

Graph the probability distribution for the bond returns based on the five scenarios. What might the graph of the probability distribution look like if there were an infinite number of scenarios (i.e., if it were a continuous distribution and not a discrete distribution)?

Answer: Here is the probability distribution for the five possible outcomes:

A continuous distribution might look like this:

Use the scenario data to calculate the expected rate of return for the 10-year zero coupon government bonds during the next year. 

Answer: The expected rate of return, r , is expressed as follows: 

r =  Pi r i . i =1

Here pi is the probability of occurrence of the ith state, ri is the estimated rate of return for that state, and n is the number of states. Here is the calculation: 

r = 0.1(–14.0%) + 0.2(–4.0%) + 0.4(6.0%) + 0.2(16.0%) + 0.1(26.0%) = 6.0%.

What is stand-alone risk? Use the scenario data to calculate the standard deviation of the bond’s return for the next year.

Answer: Stand-alone risk is the risk of an asset if it is held by itself and not as a part of a portfolio. Standard deviation measures the dispersion of possible outcomes, and, for a single asset, the stand-alone risk is measured by standard deviation.

The variance and standard deviation are calculated as follows: n

σ =  pi ( r i - r̂i )2 2

i =1 n

σ = √σ2 =√∑ Pi (ri

̂ri )2

i=1

σ2 = [(0.1) (–0.14 – 0.06)2 + (0.2) (–0.04 – 0.06)2 + (0.4) (0.06 – 0.06)2 + (0.2) (0.16 – 0.06)2 + (0.1) (0.26 – 0.06)2] 2 σ = 0.0120 σ=

2

0.0120 = 0.1095 = 10.95%.

Your client has decided that the risk of the bond portfolio is acceptable and wishes to leave it as it is. Now your client has asked you to use historical returns to estimate the standard deviation of Blandy’s stock returns. (Note: Many analysts use 4 to 5 years of monthly returns to estimate risk and many use 52 weeks of weekly returns; some even use a year or less of daily returns. For the sake of simplicity, use Blandy’s 10 annual returns.)

Answer: The formulas are shown below: T

rˉAvg =

 rt

t 1

T T

 (rt  rAvg ) 2

Estimated σ = S =

t 1

T 1

Using Excel, the past average returns and standard deviations are:

Average return: Standard deviation of returns:

Market 8.0% 20.1%

Blandy 6.4% 25.2%

Gourmange 9.2% 38.6%

Your client is shocked at how much risk Blandy stock has and would like to reduce the level of risk. You suggest that the client sell 25% of the Blandy stock and create a portfolio with 75% Blandy stock and 25% in the high-risk Gourmange stock. How do you suppose the client will react to replacing some of the Blandy stock with high-risk stock? Show the client what the proposed portfolio return would have been in each year of the sample. Then calculate the average return and standard deviation using the portfolio’s annual returns. How does the risk of this two-stock portfolio compare with the risk of the individual stocks if they were held in isolation?

Answer: To find historical returns on the portfolio, we first find each annual return for the portfolio using the portfolio weights and the annual stock returns. The percentage of a portfolio’s value that is invested in Stock i is denoted by the ―weight‖ wi. Notice that the sum of all the weights must equal 1. With n stocks in the portfolio, its return each year will be: r‾p = w1r‾1 + w2r‾2 + . . . + wnr‾n n

=  w i ri i 1

The portfolio return each year will be: ̅ , (̅ (̅ , ) ̅ , 0 (̅ 0 (̅ , ) , )

, )

Following is a table showing the portfolio’s return in each year. It also shows the average return and standard deviation during the past 10 years.

Year 1 2 3 4 5 6 7 8 9 10 Average return: Standard deviation of returns:

Stock Returns Gourmange

Blandy 26% 15% –14% –15% 2% –18% 42% 30% –32% 28% 6.4% 25.2%

47% –54% 15% 7% –28% 40% 17% –23% –4% 75% 9.2% 38.6%

Portfolio 31.3% –2.3% –6.8% –9.5% –5.5% –3.5% 35.8% 16.8% –25.0% 39.8% 7.1% 22.2%

Notice that the portfolio risk is actually less than the standard deviations of the stocks making up the portfolio. The average portfolio return during the past 10 years can be calculated as average return of the 10 yearly returns. But there is another way—the average portfolio return over a number of periods is also equal to the weighted average of the stock’s average returns: n

r‾Avg,p =  w i rAvg, i i 1

This method is used below: r‾Avg,p = 0.75(6.4%) + 0.25(9.2%) = 7.1% Note, however, that the only way to calculate the standard deviation of historical returns for a portfolio is to first calculate the portfolio’s annual historical returns and then calculate its standard deviation. A portfolio’s historical standard deviation is not the weighted average of the individual stocks’ standard deviations! (The only exception occurs when there is zero correlation among the portfolio’s stocks, which would be extremely rare.)

Explain correlation to your client. Calculate the estimated correlation between Blandy and Gourmange. Does this explain why the portfolio standard deviation was less than Blandy’s standard deviation?

Answer: Loosely speaking, the correlation (ρ) coefficient measures the tendency of two variables to move together. The formula, shown below, is complicated, but it is easy to use Excel to calculate the correlation. T

 (ri, t  ri, Avg )(rj, t  rj, Avg )

Estimated ρi,j = R =

t 1

T T    (ri, t  ri, Avg ) 2    (r j, t  r j, Avg ) 2  t 1  t 1    

Using Excel, the correlation between Blandy (B) and Gourmange (G) is: Est. ρB,G = 0.11 A correlation coefficient of +1 means that the stocks always move together; a correlation coefficient of −1 means that the stocks always move opposite to one another. A correlation coefficient of 0 means that there is no relationship between the stocks’ movements. The correlation coefficient of 0.11 means that sometime when Blandy is up, Gourmange is down, and vice versa. This makes the total risk of the portfolio less than the risk of holding either stock by itself. h.

Suppose an investor starts with a portfolio consisting of one randomly selected stock. As more and more randomly selected stocks are added to the portfolio, what happens to the portfolio’s risk?

Answer: The standard deviation gets smaller as more stocks are combined in the portfolio, while rp (the portfolio’s return) remains constant. Thus, by adding stocks to your portfolio, which initially started as a one-stock portfolio, risk has been reduced.

In the real world, stocks are positively correlated with one another; if the economy does well, so do stocks in general, and vice versa. Correlation coefficients between stocks generally range from +0.5 to +0.7. The average correlation between stocks is about 0.35. A single stock selected at random would on average have a standard deviation of about 35%. As additional stocks are added to the portfolio, the portfolio’s standard deviation decreases because the added stocks are not perfectly positively correlated. However, as more and more stocks are added, each new stock has less of a risk-reducing impact, and eventually adding additional stocks has virtually no effect on the portfolio’s risk as measured by σ. In fact, σ stabilizes at about 20% when 40 or more randomly selected stocks are added. Thus, by combining stocks into well-diversified portfolios, investors can eliminate almost one-half the riskiness of holding individual stocks. (Note: It is not completely costless to diversify, so even the largest institutional investors hold less than all stocks. Even index funds generally hold a smaller portfolio which is highly correlated with an index such as the S&P 500 rather than hold all the stocks in the index.) The implication is clear: investors should hold well-diversified portfolios of stocks rather than individual stocks. (In fact, individuals can hold diversified portfolios through mutual fund investments.) By doing so, they can eliminate about half of the riskiness inherent in individual stocks.

1. Should portfolio effects influence how investors think about the risk of individual stocks?

Answer: Portfolio diversification does affect investors’ views of risk. A stock’s stand-alone risk as measured by its σ or CV, may be important to an undiversified investor, but it is not relevant to a well-diversified investor. A rational, risk-averse investor is more interested in the impact that the stock has on the riskiness of his or her portfolio than on the stock’s stand-alone risk. Stand-alone risk is composed of diversifiable risk, which can be eliminated by holding the stock in a well-diversified portfolio, and the risk that remains is called market risk because it is present even when the entire market portfolio is held.

2. If you decided to hold a one-stock portfolio and consequently were exposed to more risk than diversified investors, could you expect to be compensated for all of your risk; that is, could you earn a risk premium on that part of your risk that you could have eliminated by diversifying?

Answer: If you hold a one-stock portfolio, you will be exposed to a high degree of risk, but you won’t be compensated for it. If the return were high enough to compensate you for your high risk, it would be a bargain for more rational, diversified investors. They would start buying it, and these buy orders would drive the price up and the return down. Thus, you simply could not find stocks in the market with returns high enough to compensate you for the stock’s diversifiable risk.

According to the Capital Asset Pricing Model, what measures the amount of risk that an individual stock contributes to a well-diversified portfolio? Define this measurement.

Answer: Market risk, which is relevant for stocks held in well-diversified portfolios, is defined as the contribution of a security to the overall risk of the portfolio. It is measured by a stock’s beta coefficient. The beta of Stock i, denoted by bi, is calculated as:

   bi =  i  iM  M  A stock’s beta can also be estimated by running a regression with the stock’s returns on the y axis and the market portfolio’s returns on the x axis. The slope of the regression line gives the same result as the formula shown above.

What is the Security Market Line (SML)? How is beta related to a stock’s required rate of return?

Answer: Here is the SML equation: ri = rRf + RPM bi. ri = rRf + (rM − rRf)bi. The SML asserts that because investing in stocks is risky, an investor must expect to get at least the risk-free rate of return plus a premium to reflect the additional risk of the stock. The premium for a stock begins with the premium required to hold an average stock (RPM) and is scaled up or down depending on the stock’s beta. l.

Calculate the correlation coefficient between Blandy and the market. Use this and the previously calculated (or given) standard deviations of Blandy and the market to estimate Blandy’s beta. Does Blandy contribute more or less risk to a well-diversified portfolio than does the average stock? Use the SML to estimate Blandy’s required return.

Answer: Using the formula for correlation or the Excel function, CORREL, Blandy’s correlation with the market (ρB,M) is: ρB,M = 0.481

    0.252  bi =  i iM   (0.481) = 0.6  0.201   M  Blandy’s beta is less than 1, so it contributes less risk than an average stock. Suppose the risk-free rate is 4% and the market risk premium is 5%. The required rate of return on Blandy is ri = rRF + bi (RPM) ri = 4% + 0.60(5%) = 7%

Show how to estimate beta using regression analysis.

Answer: Betas are calculated as the slope of the ―characteristic‖ line, which is the regression line showing the relationship between a given stock’s returns and the stock market’s returns. The graph below shows this regression as calculated using Excel.

1. Suppose the risk-free rate goes up to 7%. What effect would higher interest rates have on the SML and on the returns required on high-risk and low-risk securities?

Answer: The SML is shifted higher, but the slope is unchanged.

Here we have plotted the SML for betas ranging from 0 to 2.0. The base case SML is based on r RF = 4% and r M = 5%. If interest rates increase by 3 percentage points, with no change in risk aversion, then the entire SML is shifted upward (parallel to the base case SML) by 3 percentage points. Now, r RF = 7%, r M = 12%, and all securities’ required returns rise by 3 percentage points. Note that the market risk premium, r m − r RF , remains at 5 percentage points. n.

2. Suppose instead that investors’ risk aversion increased enough to cause the market risk premium to increase to 8%. (Assume the risk-free rate remains constant.) What effect would this have on the SML and on returns of high- and low-risk securities?

Answer: When investors’ risk aversion increases, the SML is rotated upward about the yintercept, which is rRF. Suppose rRF remains at 4%, but now rM increases to 12%, so the market risk premium increases to 8%. The required rate of return will rise sharply on high-risk (high-beta) stocks, but not much on low-beta securities.

Your client decides to invest $1.4 million in Blandy stock and $0.6 million in Gourmange stock. What are the weights for this portfolio? What is the portfolio’s beta? What is the required return for this portfolio?

Answer: The portfolio’s beta is the weighted average of the stocks’ betas: bp

= 0.7(bBlandy) + 0.3(bGour.) = 0.7(0.60) + 0.3(1.30) = 0.81.

There are two ways to calculate the portfolio’s expected return. First, we can use the portfolio’s beta and the SML: rp = rRF + bp (RPM) = 4.0% + 0.81%(5%) = 8.05%. Second, we can find the weighted average of the stocks’ expected returns: n

rp =  w i ri i 1

Jordan Jones (JJ) and Casey Carter (CC) are portfolio managers at your firm. Each manages a well-diversified portfolio. Your boss has asked for your opinion regarding their performance in the past year. JJ’s portfolio has a beta of 0.7 and had a return of 8.5%; CC’s portfolio has a beta of 1.4 and had a return of 9.5%. Which manager had better performance? Why?

Answer: To evaluate the managers, calculate the required returns on their portfolios using the SML and compare the actual returns to the required returns, as follows: Portfolio Manager Portfolio beta = Risk-free rate = Market risk premium = Portfolio required return = Portfolio actual return = Over (Under) Performance =

JJ 0.7 4% 5%

CC 1.4 4% 5%

7.50%

11.00%

8.50%

9.50%

1.00%

-1.50%

Notice that JJ’s portfolio had a higher return than investors required (given the risk of the portfolio) and CC’s portfolio had a lower return than expected by investors. Therefore, JJ had the better performance.

Chapter 8 Stocks, Stock Valuation, and Stock Market Equilibrium ANSWERS TO END-OF-CHAPTER QUESTIONS 8-1

a. A proxy is a document giving one person the authority to act for another, typically the power to vote shares of common stock. If earnings are poor and shareholders are dissatisfied, an outside group may solicit the proxies in an effort to overthrow management and take control of the business, an undertaking known as a proxy fight. A takeover is an action whereby a person or group succeeds in ousting a firm’s management and taking control of the company. The preemptive right gives the current shareholders the right to purchase any new shares issued in proportion to their current holdings. When granted, the preemptive right enables current owners to

maintain their proportionate share of ownership and control of the business. Dualclass shares are sometimes created by a firm to give greater voting rights to existing owners. Generally, when special classifications of stock are used, one type is designated ―Class A,‖ another as ―Class B,‖ and so on. Class A might be entitled to multiple votes for every share held. Class B might have limited voting rights. b. Some companies are so small that their common shares are not actively traded; they are owned by only a few people, usually the companies’ managers. The stock in such firms is said to be closely held. In contrast, the shares of larger companies are owned by many investors, most of whom are not active in management. Such shares are said to be publicly owned stock. c. Estimated value ( P̂0 ) is the present value of the expected future cash flows. The market price (P0) is the price at which an asset can be sold. d. The required rate of return on common stock, denoted by rs, is the minimum acceptable rate of return considering both its riskiness and the returns available on 

other investments. The expected rate of return, denoted by rs , is the rate of return expected on a stock given its current price and expected future cash flows. If the stock is in equilibrium, the required rate of return will equal the expected rate of return. The realized (actual) rate of return, denoted by r s , is the rate of return that was actually realized at the end of some holding period. Although expected and required rates of return must always be positive, realized rates of return over some periods may be negative. e. The capital gains yield results from changing prices and is calculated as (P1 – P0)/P0, where P0 is the beginning-of-period price and P1 is the end-of-period price. For a constant growth stock, the capital gains yield is g, the constant growth rate. The dividend yield on a stock can be defined as either the end-of-period dividend divided by the beginning-of-period price, or the ratio of the current dividend to the current price. Valuation formulas use the former definition. The expected total return, or expected rate of return, is the expected capital gains yield plus the expected dividend yield on a stock. The expected total return on a bond is the yield to maturity. f. Normal, or constant, growth occurs when a firm’s earnings and dividends grow at some constant rate forever. One category of nonconstant growth stock is a ―supernormal‖ growth stock that has one or more years of growth above that of the economy as a whole, but at some point the growth rate will fall to the ―normal‖ rate. This occurs, generally, as part of a firm’s normal life cycle. A zero growth stock has constant earnings and dividends; thus, the expected dividend payment is fixed, just as a bond’s coupon payment. Since the company is presumed to continue operations indefinitely, the dividend stream is a perpetuity. A perpetuity is a security on which 8-200 Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc.

the principal never has to be repaid. g. Preferred stock is a hybrid—it is similar to bonds in some respects and to common stock in other respects. Preferred dividends are similar to interest payments on bonds in that they are fixed in amount and generally must be paid before common stock dividends can be paid. If the preferred dividend is not earned, the directors can omit it without throwing the company into bankruptcy. So, although preferred stock has a fixed payment like bonds, a failure to make this payment will not lead to bankruptcy. Most preferred stocks entitle their owners to regular fixed dividend payments. h. Equilibrium is the condition under which the expected return on a security is just 

equal to its required return, r = r, and the price is stable. The Efficient Markets Hypothesis (EMH) states that (1) stocks are always in equilibrium and (2) it is impossible for an investor to consistently ―beat the market.‖ In essence, the theory holds that the price of a stock will adjust almost immediately in response to any new developments. In other words, the EMH assumes that all important information regarding a stock is reflected in the price of that stock. Financial theorists generally define three forms of market efficiency: weak-form, semistrong-form, and strongform. Weak-form efficiency assumes that all information contained in past price movements is fully reflected in current market prices. Thus, information about recent trends in a stock’s price is of no use in selecting a stock. Semistrong-form efficiency states that current market prices reflect all publicly available information. Therefore, the only way to gain abnormal returns on a stock is to possess inside information about the company’s stock. Strong-form efficiency assumes that all information pertaining to a stock, whether public or inside information, is reflected in current market prices. Thus, no investors would be able to earn abnormal returns in the stock market. 8-2

True. The value of a share of stock is the PV of its expected future dividends. If the two investors expect the same future dividend stream, and they agree on the stock’s riskiness, then they should reach similar conclusions as to the stock’s value today.

8-3

A perpetual bond is similar to a no-growth common stock and to a share of preferred stock in the following ways: 1. All three derive their values from a series of cash inflows—coupon payments from the perpetual bond, and dividends from both types of stock. 2. All three are assumed to have indefinite lives with no maturity value (M) for the perpetual bond and no capital gains yield for the stocks.

8-4

The first step is to find the value of operations by discounting all expected future free cash flows at the weighted average cost of capital. The second step is to find the total corporate value by summing the value of operations, the value of nonoperating assets, and the value of growth options. The third step is to find the value of equity by subtracting the value of debt and preferred stock from the total value of the corporation. The last step is to divide the value of equity by the number of shares of common stock.

SOLUTIONS TO END-OF-CHAPTER PROBLEMS 8-1

D0 = $1.50; g1-3 = 5%; gn = 10%; D1 through D5 = ? D1 = D0(1 + g1) = $1.50(1.05) = $1.5750. D2 = D0(1 + g1)(1 + g2) = $1.50(1.05)2 = $1.6538. D3 = D0(1 + g1)(1 + g2)(1 + g3) = $1.50(1.05)3 = $1.7364. D4 = D0(1 + g1)(1 + g2)(1 + g3)(1 + gn) = $1.50(1.05)3(1.10) = $1.9101. D5 = D0(1 + g1)(1 + g2)(1 + g3)(1 + gn)2 = $1.50(1.05)3(1.10)2 = $2.1011.

8-2

D1 = $1.90; g = 4%; rs = 10%; P̂0 = ? P̂ 0 =

8-3

D1 rs

$1.90 = $31.67 g 0.10 0.04 =



P0 = $22; D0 = $1.20; g = 10%; P̂1 = ?; r s= ?

P̂1 = P0(1 + g) = $22(1.10) = $24.20. 

D1 $1.20 (1.10) +g= + 0.10 P0 $22 $1.32 = + 0.10 = 16.00%. $22

rs =



r s = 16.00%. 8-4

Dps = $2.61 Pps = $29; rps = ? rps =

Dps $2.61 = = 0.09 = 9% Pps $29

8-5

D0 = 2.00

D2 D3 P̂2 Step 1: Calculate the required rate of return on the stock: rs = rRF + (rM – rRF)b = 4.5% + (4%)1.8 = 11.7%. Step 2: Calculate the expected dividends: D0 = $2.00 D1 = $2.00(1.14) = $2.28 D2 = $2.00(1.14)2 = $2.60 D3 = $2.60(1.07) = $2.78 Step 3: Calculate the PV of the expected dividends: PVDiv = $2.28/(1.117) + $2.60/(1.117)2 = $2.04 + $2.08 = $4.12. Step 4: Calculate P̂2 :

P̂2 = D3/(rs – g) = $2.78/(0.117 – 0.07) = $59.15. Step 5: Calculate the PV of P̂2 : PV = $59.15/(1.117)2 = $47.41. Step 6: Sum the PVs to obtain the stock’s price:

P̂0 = $4.12 + $47.41 = $51.53. Alternatively, using a financial calculator, input the following: CF0 = 0, CF1 = 2.28, and CF2 = 61.75 (2.60 + 59.15) and then enter I = 11.7 to solve for NPV = $51.53.

8-6

Value of operations = Vop = PV of expected future free cash flow Vop =

8-7

FCF (1  g) $400,000 (1.05) = = $6,000,000. WACC  g 0.12  0.05

a. ̂0 = P

D1 $4.80(1.04) = = $99.84 rs g 0.09 – 0.04

b. D1 = $4.80(1.04) = $4.99 D2 = $4.99(1.04) = $5.19 D3 = $5.19(1.04) = $5.40 D4 = $5.40(1.04) = $5.62 D5 = $5.62(1.04) = $5,84 Using a financial calculator, enter: CF0 = 0, CF1 = 4.99, CF2 = 5.19, CF3 = 5.40, CF4 = 5.62, CF5 = 5.84, and then enter I = 9 to solve for NPV = $20.89. $20.89/$99.84 = 20.9%.

8-8

The problem asks you to determine the constant growth rate, given the following facts: P0 = $28, D1 = $0.84, and rs = 9%. Use the constant growth rate formula to calculate g: 

rs=

D1 +g P0

0.09 =

$0.84 +g $28

g = 0.06 = 6%.

8-9

The problem asks you to determine the value of P̂3 , given the following facts: D1 = $3, b = 0.8, rRF = 5.2%, RPM = 6%, and P0 = $40. Step 1:

Calculate the required rate of return: rs = rRF + (rM – rRF)b = 5.2% + (6%)0.8 = 10%.

Step 2:

Use the constant growth rate formula to calculate g: 

D1 +g P0 $3 0.10 = +g $40 g = 0.025 = 2.5%.

Step 3:

Calculate P̂3 :

P̂3 = P0(1 + g)3 = $40(1.025)3 = $43.076 ≈ $43.08. Alternatively, you could calculate D4 and then use the constant growth rate formula to solve for P̂3 : D4 = D1(1 + g)3 = $3.00(1.025)3 = $3.2307. P̂3 = $3.2307/(0.10 – 0.025) = $43.0756  $43.08. 8-10

Vps = Dps/rps; therefore, rps = Dps/Vps. Dividend = 0.07 × $25 = $1.75 a. b. c. d.

rps = $1.75/$18 = 9,72%. rps = $1.75/$23 = 7.61%. rps = $1.75/$25 = 7.0% rps = $1.75/$35 = 5.0%.

8-11 P̂ 0 =

D1 D0 (1 + g) $3 [1 + (–0.04)] $2.88 = = = = $16.94 rs – g rs – g 0.13 – (–0.04)] 0.17

8-12

D0 (1 + g) = $13.2 rs – g $0.50(1 + g) = $13.25 0.10 – g 0.5 + 0.5g = 1.325 – 13.25g 13.75g = 0.825 g = 6%

8-13

D3 = $1.60 D4 = $1.60(1.40) = $2.24 D5 = $2.24(1.40) = $3.14 D6 = $3.14(1.03) = $3.23 0

D1 = $0

D2 = $0

D3 = 1.60 D4 = 2.24 D5 = 3.14 D6 = 3.23 +40.38= 3.23/(0.11 – 0.03) 43.52

CF0 = 0; CF1 = 0; CF2 = 0; CF3 = 1.60, CF4 = 2.24 CF5 = 43.52; I = 11% and calculate for NPV = $28.47.

8-14

a. ri = rRF + (rM – rRF)bi. rC = 4% + (13% – 4%)0.4 = 7.6%. rD = 4% + (13% – 4%)(–0.3) = 1.3%. Note that rD is below the risk-free rate. But since this stock is like an insurance policy because it ―pays off‖ when something bad happens (the market falls), the low return is not unreasonable. b. In this situation, the expected rate of return is as follows: 

r c= D1/P0 + g = $1/$30 + 4% = 7.3%. However, the required rate of return is 7.6%. Investors will seek to sell the stock, dropping its price to the following:

P̂C =

$1 = $27.78. 0.076  0.04 

At this point, r c=

$1 + 4% = 7.6%, and the stock will be in equilibrium. $27.78

8-15

D0 = $1, rS = 6% +5% = 11%, g1 = 60%, g2 = 40%, gn = 5%. 0

rs = 11%

g1 = 60%

1.60

g2 = 40%

g1 = 5%

2.24 2.352 + 39.20 = 2.352/(0.11 – 0.05) = 41.44

1.44 33.63 $35.07 8-16

Calculate the dividend stream and place the numbers on a time line. Also, calculate the price of the stock at the end of the nonconstant growth period, and include it, along with the dividend to be paid, at t = 5, as CF5. Then, enter the cash flows as shown on the time line into the cash flow register, enter the required rate of return as I = 16, and then find the value of the stock using the NPV calculation. Be sure to enter CF0 = 0, or else your answer will be incorrect. D0 = 0; D1 = 0, D2 = 0, D3 = 0.50 D4 = 0.50(1.8) = 0.90; D5 = 0.50(1.8)2 = 1.62; D6 = 0.50(1.8)2(1.07) = $1.7334. P̂0 = ? 0 rs = 16%

3 g = 80% 4

0.50

0.90

1.62 19.26 20.88

0.32 0.50 9.94 $10.76 = P̂0

g = 7%

6 |

P̂5 = D6/(rs – g) = 1.7334/(0.16 – 0.07) = 19.26. This is the price of the stock at the end of Year 5. CF0 = 0; CF1-2 = 0; CF3 = 0.5; CF4 = 0.9; CF5 = 20.88; I = 16%. With these cash flows in the CFLO register, press NPV to get the value of the stock today: NPV = $10.76.

8-17 Vps =

6% $25 = $ 0 00. 0.05

. Vps =

6% $25 = $15.00. 0.10

8-18

g = $1.1449/$1.07 – 1.0 = 7%. Calculator solution: Input N = 1, PV = -1.07, PMT = 0, FV = 1.1449, I/YR = ? I = 7.00%.

$1.07/$21.40 = 5%. 

c. 8-19

r s= D1/P0 + g = $1.07/$21.40 + 7% = 5% + 7% = 12%.

a. 1. P̂0 =

$2(1  0.05) $1.90 = = $9.50. 0.20 0.15  0.05

2. P̂0 = $2/0.15 = $13.33. 3. P̂0 =

$2.10 $2(1.05) = = $21.00. 0.10 0.15  0.05

4. P̂0 =

$2.20 $2(1.10) = = $44.00. 0.05 0.15  0.10

b. 1. P̂0 = $2.30/0 = Undefined. 2. P̂0 = $2.40/(–0.05) = –$48, which is nonsense. These results show that the formula does not make sense if the required rate of return is equal to or less than the expected growth rate. c. No.

8-20

a. rs = rRF + (rM – rRF)b = 5% + (8% – 5%)1.5 = 9.5%. P̂0 = D1/(rs – g) = $2.25/(0.095 – 0.05) = $50.00. b. rs = 4% + (7% – 4%)1.5 = 8.5%. P̂0 = $2.25/(0.085 – 0.05) = $64.29. c. rs = 4% + (6.5% – 4%)1.5 = 7.75%. P̂0 = $2.25/(0.0775 – 0.05) = $81.82. d. New data given: rRF = 4%; rM = 6.5%; g = 5.5%, b = 1.3. rs = rRF + (rM – rRF)b = 4% + (6.5% – 4%)1.3 = 7.25%. P̂0 = D1/(rs – g) = $2.26/(0.0725 – 0.055) = $129.14.

8-21 The value of any asset is the present value of all future cash flows expected to be generated from the asset. Hence, if we can find the present value of the dividends during the period preceding long-run constant growth and subtract that total from the current stock price, the remaining value would be the present value of the cash flows to be received during the period of long-run constant growth. D1 = $2.00  (1.25)1 = $2.50 D2 = $2.00  (1.25)2 = $3.125 D3 = $2.00  (1.25)3 = $3.90625

PV(D1) = $2.50/(1.12)1 PV(D2) = $3.125/(1.12)2 PV(D3) = $3.90625/(1.12)3  PV(D1 to D3)

= $2.2321 = $2.4912 = $2.7804 = $7.5037

Therefore, the PV of the remaining dividends is $58.8800 – $7.5037 = $51.3763. Compounding this value forward to Year 3, we find that the value of all dividends received during constant growth is $72.18. [$51.3763(1.12)3 = $72.18] Applying the constant growth formula, we can solve for the constant growth rate: P̂3 = D3(1 + g)/(rs – g)

$72.18 = $3.90625(1 + g)/(0.12 – g) $8.6616 – $72.18g = $3.90625 + $3.90625g $4.75535 = $76.08625g 0.0625 = g 6.25% = g.

8-22

g=6%

D0 = 1.50

D3 P̂3

a. D1 = $1.5(1.06) = $1.59. D2 = $1.50(1.06)2 = $1.69. D3 = $1.5(1.06)3 = $1.79. b. PV = $1.59(0.8850) + $1.69(0.7831) + $1.79(0.6930) = $3.97. Calculator solution: Input 0, 1.59, 1.69, and 1.79 into the cash flow register, input I/YR = 13, PV = ? PV = $3.97. c. $27.05(0.6930) = $18.75. Calculator solution: Input 0, 0, 0, and 27.05 into the cash flow register, I/YR = 13, PV = ? PV = $18.75. d. $18.75 + $3.97 = $22.72 = Maximum price you should pay for the stock. e. P̂0 =

D 0 (1  g) D1 $1.59 = = = $22.71. rs  g rs  g 0.13  0.06

f. The value of the stock is not dependent upon the holding period. The value calculated in Parts a through d is the value for a 3-year holding period. It is equal to the value calculated in Part e except for a small rounding error. Any other holding period would produce the same value of P̂0 ; that is, P̂0 = $22.71.

8-23

a. End of Year: 0 r = 12% 1 |

g = 15%

D0 = 1.75 Dt D1 D2 D3 D4 D5

6 | g = 5%

= D0(1 + g)t = $1.75(1.15)1 = $2.01. = $1.75(1.15)2 = $1.75(1.3225) = $2.31. = $1.75(1.15)3 = $1.75(1.5209) = $2.66. = $1.75(1.15)4 = $1.75(1.7490) = $3.06. = $1.75(1.15)5 = $1.75(2.0114) = $3.52.

b. Step 1 5

PV of dividends = 

. t ( 1  r ) t 1 s PV D1 = $2.01(PVIF12%,1) = $2.01(0.8929) = $1.79 PV D2 = $2.31(PVIF12%,2) = $2.31(0.7972) = $1.84 PV D3 = $2.66(PVIF12%,3) = $2.66(0.7118) = $1.89 PV D4 = $3.06(PVIF12%,4) = $3.06(0.6355) = $1.94 PV D5 = $3.52(PVIF12%,5) = $3.52(0.5674) = $2.00 PV of dividends = $9.46 Step 2 P̂5 

$3.52(1.05) $3.70 D6 D (1  g n )  5 = = = $52.80. 0.07 rs  g n rs  g n 0.12  0.05

This is the price of the stock 5 years from now. The PV of this price, discounted back 5 years, is as follows: PV of P̂5 = $52.80(PVIF12%,5) = $52.80(0.5674) = $29.96.

Step 3 The price of the stock today is as follows:

P̂0

= PV dividends Years 1 through 5 + PV of P̂5 = $9.46 + $29.96 = $39.42.

This problem could also be solved by substituting the proper values into the following equation: 5

5 D (1  g ) t  D s 6   1  . P̂0 =  0     t  rs  g n   1  rs  t 1 (1  rs )

Calculator solution: Input 0, 2.01, 2.31, 2.66, 3.06, 56.32 (3.52 + 52.80) into the cash flow register, input I/YR = 12, PV = ? PV = $39.43. c. First Year (t = 0) D1/P0 = $2.01/$39.42 Capital gains yield* Expected total return

= 5.10% = 6.90% = 12.00%

Sixth Year (t = 5) D6/P5 = $3.70/$52.80 Capital gains yield Expected total return

= 7.00% = 5.00% = 12.00%

*We know that r is 12%, and the dividend yield is 5.10%; therefore, the capital gains yield must be 6.90%. The main points to note here are as follows: 1. The total yield is always 12% (except for rounding errors). 2. The capital gains yield starts relatively high, then declines as the nonconstant growth period approaches its end. The dividend yield rises. 3. After t = 5, the stock will grow at a 5% rate. The dividend yield will equal 7%, the capital gains yield will equal 5%, and the total return will be 12%.

8-24

a. Part 1. Graphical representation of the problem: Nonconstant growth 1

Normal growth 3

D0 PVD1 PVD2 PV P̂2 P0

(D2 + P̂2 )

D∞

∞

D1 = D0(1 + gs) = $4.00(1.30) = $5.20. D2 = D0(1 + gs)2 = $4.00(1.30)2 = $6.76.

P̂2 =

D3 D (1  g n ) $6.76(1.05) = 2 = = $141.96. 0.10 - 0.05 rs  g n rs  g n

P̂0 = PV(D1) + PV(D2) + PV( P̂2 ) =

D1 D2 P̂2   (1  rs ) (1  rs ) 2 (1  rs ) 2

= $5.20(0.9091) + $6.76(0.8264) + $141.96(0.8264) = $127.63 Calculator solution: Input 0, 5.20, 148.72(6.76 + 141.96) into the cash flow register, input I/YR = 10, PV = ? PV = $127.64.(rounding) b. P̂ 1 =

P̂ 2 + D2 $141.96 + $6.76 = = $135.20. 1 + rs 1 + 0.10

c. Expected dividend yield: D1/P0 = $5.20/$127.63 = 4.07%. $135.20 $1 P̂  P Capital gains yield = 1 0 = = 5.93%. $1 P0 Total return = Dividend yield + capital gains yield = 4.07% + 5.93% =10.00%.

Notice that this is the same as the stock’s required return, rs.

d. Expected dividend yield: D2/P1 = $6.76/$135.20 = 5.00%. P̂

Capital gains yield = 2P

P1 1

= 5.00%.

Total return = Dividend yield + capital gains yield = 5.00% + 5.00% =10.00%. Notice that this is the same as the stock’s required return, rs. Notice also that Year 1 had a bigger capital gains yield because the dividend was growing faster in Year 1.

8-25

FCF = NOPAT – Net investment in operating capital = $300,000,000 – $50,000,000 = $250,000,000 Firm value = =

FCF WACC  g $250,000,000 0.09 0.04 $250,000,000 0.05

= $5,000,000,000. This is the total firm value. Now find the market value of its equity. Market ValueTotal = Market ValueEquity + MarketValueDebt $5,000,000,000 = Market ValueEquity + $1,000,000,000 Market ValueEquity = $4,000,000,000. This is the market value of all the equity. Divide by the number of shares to find the price per share. $4,000,000,000/500,000,000 = $8.00.

SOLUTION TO SPREADSHEET PROBLEM 8-26

The detailed solution for the spreadsheet problem is in the file Ch 08 Build a Model Solution.xlsx and is available on the textbook’s website.

MINI CASE The detailed solution for the Mini Case is also available in spreadsheet format, in the file Brigham3Ce_MiniCase_Ch08.xlsx, and is available on the textbook’s website. Sam Strother and Shawna Tibbs are senior vice presidents of First Strategies Investment Counsel (FSIC). They are co-directors of the company’s pension fund management division, with Strother having responsibility for fixed income securities (primarily bonds) and Tibbs responsible for equity investments. A major new client has requested that FSIC present an investment seminar to its Executive Committee, and Strother and Tibbs, who will make the presentation, have asked you to help them. To illustrate the common stock valuation process, Strother and Tibbs have asked you to analyze Temp Force Company, an employment agency that supplies keyboarders and computer programmers to businesses with temporarily heavy workloads. You are to answer the following questions. a.

Describe briefly the legal rights and privileges of common shareholders.

Answer: The common shareholders are the owners of a corporation and, as such, they have certain rights and privileges as described below. 1. Ownership implies control. Thus, a firm’s common shareholders have the right to elect its firm’s directors, who in turn select the officers who manage the business. 2. Common shareholders may have the right, called the preemptive right, to purchase any additional shares sold by the firm.

1. Write out a formula that can be used to value any stock, regardless of its dividend pattern.

Answer: The value of any stock is the present value of its expected dividend stream: P̂ 0 =

D1 1

(1+rs )

D2 2

(1+rs )

D3 3

(1+rs )

D∞ . (1+rs )∞

However, some stocks have dividend growth patterns that allow them to be valued using short-cut formulas. b.

2. What is a constant growth stock? How are constant growth stocks valued?

Answer: A constant growth stock is one whose dividends are expected to grow at a constant rate forever. ―Constant growth‖ means that the best estimate of the future growth rate is some constant number, not that we really expect growth to be the same each and every year. Many companies have dividends that are expected to grow steadily into the foreseeable future, and such companies are valued as constant growth stocks. For a constant growth stock: D1 = D0(1 + g), D2 = D1(1 + g) = D0(1 + g)2, and so on. With this regular dividend pattern, the general stock valuation model can be simplified to the following very important equation:

P̂0 =

D1 D (1  g) = 0 . rs  g rs  g

This is the well-known ―Gordon,‖ or ―constant-growth‖ model for valuing stocks. Here, D1 is the next expected dividend, which is assumed to be paid 1 year from now, rs is the required rate of return on the stock, and g is the constant growth rate. b.

3. What happens if a company has a constant g that exceeds its rs? Will many stocks have expected g > rs in the short run (i.e., for the next few years)? In the long run (i.e., forever)?

Answer: The model is derived mathematically, and the derivation requires that rs > g. If g is greater than rs, the model gives a negative stock price, which is nonsensical. The model simply cannot be used unless (1) rs > g, (2) g is expected to be constant, and (3) g can reasonably be expected to continue indefinitely. Stocks may have periods of supernormal growth, where gs > rs; however, this growth rate cannot be sustained indefinitely. In the long-run, g < rs. 9-220 Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc.

Assume that Temp Force has a beta coefficient of 1.2, that the risk-free rate (the yield on government bonds) is 5.0%, and that the market risk premium is 4%. What is the required rate of return on the firm’s stock?

Answer: Here we use the SML to calculate Temp Force’s required rate of return: rs = rRF + (rM – rRF)bTemp Force = 5% + (9% – 5%)(1.2) = 5% + (4%)(1.2) = 5% + 4.8% = 9.8%.

Assume that Temp Force is a constant growth company whose last dividend was $2.00 and whose dividend is expected to grow indefinitely at a 5% rate. 1. What is the firm’s expected dividend stream over the next 3 years?

Answer: Temp Force is a constant growth stock, and its dividend is expected to grow at a constant rate of 5% per year. Expressed as a time line, we have the following setup. Just enter 2 in your calculator; then keep multiplying by 1 + g = 1.05 to get D1, D2, and D3: 0 r = 9.8% s

g = 5% D0 = 2.00

2.10

2.205

2.315

1.91 1.83 1.75 . . .

2. What is the firm’s current stock price?

Answer: We could extend the time line on out forever, find the value of Temp Force’s dividends for every year on out into the future, and then the PV of each dividend, discounted at r = 9.8%. For example, the PV of D1 is $1.91; the PV of D2 is $1.83; and so forth. Note that the dividend payments increase with time, but as long as rs > g, the present values decrease with time. If we extended the graph on out forever and then summed the PVs of the dividends, we would have the value of the stock. However, since the stock is growing at a constant rate, its value can be estimated using the constant growth model:

P̂0 = d.

$2.10 D1 $2.10 = = = $43.75. (rs  g ) (0.098  0.05) 0.048

3. What is the stock’s expected value 1 year from now?

Answer: After one year, D1 will have been paid, so the expected dividend stream will then be D2, D3, D4, and so on. Thus, the expected value one year from now is $45.94:

P̂1 =

D2 $2.205 $2.205 = = = $45.94. ( rs  g) (0.098  0.05) 0.048

4. What are the expected dividend yield, the capital gains yield, and the total return during the first year?

Answer: The expected dividend yield in any year n is Dividend yield =

Dn , P̂n 1

While the expected capital gains yield is Capital gains yield =

Dn ( P̂n  P̂n 1 ) =r– . Pn 1 P̂n 1

Thus, the dividend yield in the first year is 4.8%, while the capital gains yield is 5%: Total return = 9.8% Dividend yield = $2.10/$43.75 = 4.8% Capital gains yield = 5.0%

Now assume that the stock is currently selling at $43.75. What is the expected rate of return on the stock?

Answer: The constant growth model can be rearranged to this form: 

r s=

D1 g. P0

Here the current price of the stock is known, and we solve for the expected return. For Temp Force: 

r s = $2.10/$43.75 + 0.05 = 0.048 + 0.05 = 9.8%.

What would the stock price be if its dividends were expected to have zero growth?

Answer:

If Temp Force’s dividends were not expected to grow at all, then its dividend stream would be a perpetuity. Perpetuities are valued as shown below: 0 r = 9.8% |

g = 0%

2.00

1.82 1.66 1.51 . . .

P0 = 20.41 P0 = PMT/r = $2.00/0.098 = $20.41. Note that if a preferred stock is a perpetuity, it may be valued with this formula.

Now assume that Temp Force is expected to experience supernormal growth of 30% for the next 3 years, then to return to its long-run constant growth rate of 5%. What is the stock’s value under these conditions? What is its expected dividend yield and capital gains yield in Year 1? In Year 4?

Answer: Temp Force is no longer a constant growth stock, so the constant growth model is not applicable. Note, however, that the stock is expected to become a constant growth stock in 3 years. Thus, it has a nonconstant growth period followed by constant growth. The easiest way to value such nonconstant growth stocks is to set the situation up on a time line as shown below: 0 r = 9.8% | s g = 30%

2.600

g = 30%

3.380

2.368 2.804 3.319 72.619 81.110

g = 30%

4.394

g = 5%

4.614

= $96.13 =

Simply enter $2 and multiply by (1.30) to get D1 = $2.60; multiply that result by 1.3 to get D2 = $3.38, and so forth. Then recognize that after Year 3, Temp Force becomes a constant growth stock, and at that point P̂3 can be found using the constant growth model. P̂3 is the present value as of t = 3 of the dividends in Year 4 and beyond. With the cash flows for D1, D2, D3, and P̂3 shown on the time line, we discount each value back to year 0, and the sum of these four PVs is the value of the stock today, P0 = $81.11. The dividend yield in Year 1 is 3.2%, and the capital gains yield is 6.6%: Dividend yield =

$2.600 = 0.032 = 3.2%. $81.11

Capital gains yield = 9.8% – 3.2% = 6.6%. During the nonconstant growth period, the dividend yields and capital gains yields are not constant, and the capital gains yield does not equal g. However, after Year 3, the stock becomes a constant growth stock, with g = capital gains yield = 5.0% and dividend yield = 9.8% – 5.0% = 4.8%.

Answer:

Is the stock price based more on long-term or short-term expectations? Answer this by finding the percentage of Temp Force’s current stock price based on dividends expected more than 3 years in the future (using the original assumptions).

$38.26  87% $43.75

Stock price is based more on long-term expectations, as is evident by the fact that 87% of Temp Force’s stock price is determined by dividends expected more than three years from now. 0 r = 9.8% s

g = 5% D0 = 2.00

2.10

2.205

2.315

2.431

P̂3

(

)

38.26 43.75

Suppose Temp Force is expected to experience zero growth during the first 3 years and then to resume its steady-state growth of 5% in the fourth year. What is the stock’s value now? What is its expected dividend yield and its capital gains yield in Year 1? In Year 4?

Answer: Now we have this situation: 0 r = 9.8%

2.00

g = 0%

g = 0% 2.00

2.00

1.82 1.66 1.51 33.05 38.04 = P̂0

g = 0%

2.00

4 g = 5%

2.10

= 43.75 =

During year 1: $2.00 = 0.0526= 5.26%. $38.04 Capital gains yield = 9.8% – 5.26% = 4.54%.

Dividend yield =

Again, in Year 4 Temp Force becomes a constant growth stock; hence g = capital gains yield = 5.0% and dividend yield = 4.8%.

Finally, assume that Temp Force’s earnings and dividends are expected to decline by a constant 5% per year, that is, g = –5%. Why would anyone be willing to buy such a stock, and at what price should it sell? What would be the dividend yield and capital gains yield in each year?

Answer: The company is earning something and paying some dividends, so it clearly has a value greater than zero. That value can be found with the constant growth formula, but where g is negative: P0 =

D (1  g) D1 $1.90 $2.00(0.95) = 0 = = = $12.84. 0.148 rS  g rS  g 0.098  (0.05)

Since it is a constant growth stock: g = Capital gains yield = –5.0%, hence: Dividend yield = 9.8% – (–5.0%) = 14.8%. As a check: Dividend yield =

$1.90 = 0.148 = 14.8%. $12.84

The dividend and capital gains yields are constant over time, but a high (14.8%) dividend yield is needed to offset the negative capital gains yield.

What is market multiple analysis?

Answer: Analysts often use the P/E multiple (the price per share divided by the earnings per share) or the P/CF multiple (price per share divided by cash flow per share, which is the earnings per share plus the dividends per share) to value stocks. For example, estimate the average P/E ratio of comparable firms. This is the P/E multiple. Multiply this average P/E ratio by the expected earnings of the company to estimate its stock price. The entity value (V) is the market value of equity (# shares of stock multiplied by the price per share) plus the value of debt. Pick a measure, such as EBITDA, sales, customers, eyeballs, etc. Calculate the average entity ratio for a sample of comparable firms. For example, V/EBITDA, V/customers. Then find the entity value of the firm in question. For example, multiply the firm’s sales by the V/sales multiple, or multiply the firm’s number of customers by the V/customers ratio. The result is the total value of the firm. Subtract the firm’s debt to get the total value of equity. Divide by the number of shares to get the price per share. There are problems with market multiple analysis. (1) It is often hard to find comparable firms. (2) The average ratio for the sample of comparable firms often has a wide range. For example, the average P/E ratio might be 20, but the range could be from 10 to 50. How do you know whether your firm should be compared to the low, average, or high performers? l.

Temp Force recently issued preferred stock. It pays an annual dividend of $1.60, and the issue price was $25 per share. What is the expected return to an investor on this preferred stock? 

Answer:

r ps =

D ps Vps

$1.60 $25 = 6.4%. =

Why do stock prices change? Suppose the expected D1 is $2, the growth rate is 4%, and rs is 12%. Using the constant growth model, what is the price? What is the impact on the stock price if g is 5% or 6%? If rs is 11% or 13%?

Answer: Using the constant growth model, the price of a stock is P0 = D1/(rs – g). If estimates of g change, then the price will change. If estimates of the required return on stock change, then the stock price will change. Notice that rs = rRF + (rpM)bi, so rs will change if there are changes in inflation expectations, risk aversion, or company risk. The following table shows the stock price for various levels of g and rs. rs 11% 12% 13% n.

g 4% 28.57 25.00 22.22

g 5% 33.33 28.57 25.00

g 6% 40.00 33.33 28.57

What does market equilibrium mean?

Answer: Equilibrium means stable, with no tendency to change. Market equilibrium means that prices are stable—at its current price, there is no general tendency for people to want to buy or to sell a security that is in equilibrium. Also, when equilibrium exists, the expected rate of return will be equal to the required rate of return: 

r = D1/P0 + g = r = rRF + (rM – rRF)b.

If equilibrium does not exist, how will it be established?

Answer: Securities will be bought and sold until the equilibrium price is established.

What is the efficient markets hypothesis, what are its three forms, and what are its implications?

Answer: The EMH in general is the hypothesis that securities are normally in equilibrium, and are ―priced fairly,‖ making it impossible to ―beat the market‖ over the long term. Weak-form efficiency says that investors cannot profit from looking at past movements in stock prices—the fact that stocks went down for the past few days is no reason to think that they will go up (or down) in the future. This form has been proven pretty well by empirical tests, even though people still employ ―technical analysis.‖ Semistrong-form efficiency says that all publicly available information is reflected in stock prices, hence that it won’t do much good to pore over annual reports trying to find undervalued stocks. Strong-form efficiency says that all information, even inside information, is embedded in stock prices. This form does not hold—insiders know more, and could take advantage of that information to make abnormal profits in the markets. Trading on the basis of insider information is illegal.

Chapter 9 The Cost of Capital

ANSWERS TO END-OF-CHAPTER QUESTIONS 9-1

a. The weighted average cost of capital, WACC, is the weighted average of the after-tax component costs of capital—debt, preferred stock, and common equity. Each weighting factor is the proportion of that type of capital in the optimal, or target, capital structure. The after-tax cost of debt, rd(1 – T), is the relevant cost to the firm of new debt financing. Since interest is deductible from taxable income, the after-tax cost of debt to the firm is less than the before-tax cost. Thus, rd(1 – T) is the appropriate component cost of debt (in the weighted average cost of capital). b. The cost of preferred stock, rps, is the cost to the firm of issuing new preferred stock. For perpetual preferred, it is the preferred dividend, Dps, divided by the net issuing price, Pn. Note that no tax adjustments are made when calculating the component cost

of preferred stock because, unlike interest payments on debt, dividend payments on preferred stock are not tax deductible. The cost of common equity, rs, is the cost to the firm of reinvesting earnings. The cost of new common equity, re, is the cost to the firm of equity obtained by selling new common stock. It is, essentially, the cost of retained earnings adjusted for flotation costs. Flotation costs are the costs that the firm incurs when it issues new securities. The amount actually available to the firm for capital investment from the sale of new securities is the sales price of the securities less flotation costs. Note that flotation costs consist of (1) direct expenses such as printing costs and brokerage commissions, (2) any price reduction due to increasing the supply of stock, and (3) any drop in price due to informational asymmetries. c. The target capital structure is the relative amount of debt, preferred stock, and common equity that the firm desires. The WACC should be based on these target weights. d. There are considerable costs when a company issues a new security, including fees to an investment banker and legal fees. These costs are called flotation costs. The cost of new external common equity is higher than that of common equity raised internally by reinvesting earnings. Projects financed with external equity must earn a higher rate of return, since the project must cover the flotation costs. 9-2

The WACC is an average cost because it is a weighted average of the firm’s component costs of capital. However, each component cost is a marginal cost; that is, the cost of new capital. Thus, the WACC is the weighted average marginal cost of capital.

9-3

Effect On rd(1 – T)

WACC a. The corporate tax rate is lowered.

b. The Bank of Canada tightens credit.

c. The firm uses more debt.

0 or +

e. The firm expands into a risky new area.

f. Investors become more risk averse.

d. The firm doubles the amount of capital it raises during the year.

9-4

Stand-alone risk views a project’s risk in isolation, hence without regard to portfolio effects; within-firm risk, also called corporate risk, views project risk within the context of the firm’s portfolio of assets; and market risk (beta) recognizes that the firm’s shareholders hold diversified portfolios of stocks. In theory, market risk should be most relevant because of its direct effect on stock prices.

9-5

If a company’s composite WACC estimate were 10%, its managers might use 10% to evaluate average-risk projects, 12% for those with high-risk, and 8% for low-risk projects. Unfortunately, given the data, there is no completely satisfactory way to specify exactly how much higher or lower we should go in setting risk-adjusted costs of capital.

SOLUTIONS TO END-OF-CHAPTER PROBLEMS 9-1

a. rd(1 – T) = 9%(1 – 0) = 9.0%. b. rd(1 – T) = 9%(0.80) = 7.2%. c. rd(1 – T) = 9%(0.65) = 5.85%.

9-2

rd(1 – T) = 8.4%(0.75) = 6.3%.

9-3

Vps = $25; Dps = $1.75; F = 0%; rps = ? rps =

D ps Vps (1  F)

$1.75 $25(1  0.0) = 7%. =

$1.50 $25(0.06) = = 5.79%. $27.00(1  0.04) $25.92

9-4

rps =

9-5

P0 = $36; D1 = $3.00; g = 5%; rs = ? rs =

D1 + g = ($3.00/$36.00) + 0.05 = 13.33%. P0

9-6

rs = rRF + bi(RPM) = 0.03 + 0.9(0.045) = 7.05%.

9-7

30% debt; 5% preferred stock; 65% equity; rd = 6%; T = 30%; rps = 5.8%; rs = 12%. WACC = (wd)(rd)(1 – T) + (wps)(rps) + (ws)(rs) WACC = 0.30(0.06)(1 – 0.30) + 0.05(0.058) + 0.65(0.12) = 9.35%.

9-8

40% debt; 60% equity; rd = 9%; T = 25%; WACC = 9.9%; rs = ? WACC = (wd)(rd)(1 – T) + (ws)(rs) 9.9% = (0.4)(9%)(1 – 0.25) + (0.6)rs 9.9% = 2.70% + 0.6rs 7.2% = 0.6rs rs = 12%.

9-9

N = 60, PV = –515.16, PMT = 30, and FV = 1000, to get I = 6% = periodic rate. The nominal rate is 6%(2) = 12%, and the after-tax component cost of debt is 12%(0.75) = 9%.

9-10

a. rs =

$2.91 D1 +g= + 4% = 7.7% + 4% = 11.7%. $38 P0

b. rs = rRF + (rM – rRF)b = 5.0% + (9.0% – 5.0%)1.3 = 5% + (4.0%)1.3 = 5% + 5.2% = 10.2%. c. rs = Bond rate + Risk premium = 8% + 4% = 12%. d. The bond-yield-plus-risk-premium approach and the dividend growth approach both resulted in higher cost of equity values than the CAPM method. The firm’s cost of equity should be estimated to be about 11.3%, which is the average of the three methods. 9-11

a. $6.50 = $4.42(1 + g)5 (1 + g)5 = $6.50/$4.42 = 1.471 (1 + g) = 1.471(1/5) = 1.080 g = 8%. Alternatively, with a financial calculator, input N = 5, PV = –4.42, PMT = 0, FV = 6.5, and then solve for I = 8.02%  8%. b. D1 = D0(1 + g) = (0.40)($6.50)(1.08) = $2.81 c. rs = D1/P0 + g = $2.81/$36 + 8% = 15.81%.

9-12

D1 +g P0 $3.60 0.09 = +g $60.00

rs =

0.09 = 0.06 + g g = 3%. b. Current EPS $5.40 Less: Dividends per share 3.60 Retained earnings per share 1.80 Rate of return  0.09 Increase in EPS 0.162 Current EPS 5.400 Next year’s EPS $5.562 Alternatively, EPS1 = EPS0(1 + g) = $$5.40(1.03) = $5.562. 9-13

P0 = $30; D1 = $3.00; g = 5%; F = 10%; re = ? re = [D1/(1 – F) P0] + g = [3/(1 – 0.10)(30)] + 0.05 = 16.1%.

9-14

Enter these values: N = 15, PV =1000(1 – 0.03) = 970, PMT = –60(1 – .30) = –42, and FV = –1000, to get I = 4.48%, which is the after-tax component cost of debt.

9-15

a. Cost of debt: –930 PV, 10 N, 80 PMT, 1000 FV and CPT I/Y = 9.10% –1000(1 – 0.03) PV, 10 N, 91 PMT, 1000 FV and CPT I/Y = 9.58% Cost of equity: rs = 5.0% + 1.5(4.0%) = 11%

D + E = 100% 2E + E = 100% 3E = 33.3% ≈ 33% D = 100% – 33.3% = 66.7% ≈ 67% WACC = 0.33(11%) + 0.67(9.58%)(1 – 0.30) = 8.12% b. Cost of preferred stock: ($25 × 7% = $1.75)/$25(1 – 0.05) = 7.37% WACC: 0.33(11%) + 0.20(7.37%) + 0.47(9.58%)(1 – 0.30) = 8.26% c. WACC rises because the company is substituting more expensive financing (preferred shares at 7.37%), with its cheaper debt costs (at 6.71%).

9-16

a. CAPM: Average beta: (1.35 + 1.45 + 1.10)/3 = 1.30 rs = rrf + RPM(b) = 3.05% + (5.5%)1.30 = 10.20% Cost of debt + subjective risk premium: Better to use the upper end of the 3–5% risk premium due to higher investor risk aversion. rs = 7% + 5% = 12% Average of the above two methods: (10.20% + 12%)/2 = 11.10% Add 1–3% premium due to the lack of liquidity in the company’s shares. Given the above information, either 2% or 3% is appropriate. 11.10% + 3.0% = 14.10% b. The company should not accept the project in the new business unit. Its expected return of 12.5% is less than what investors require, 14.10% for that type of business.

9-17

a. Common equity needed: 0.50($30,000,000) = $15,000,000. b. Cost using rs: rs = (D1/P0) + g = 4% + 8% = 12% WACC = Wsrd(1 – T) + Wsrs = (0.50)(8.0%)(1 – 0.30) + (0.50)(12%) = 8.8%

c. rs and the WACC will increase due to the flotation costs of new equity.

9-18

a. There are two methods for determining the market value weights: (1) use the target capital structure or (2) take the market value of both debt and equity. Since the target capital structure is given, that should be used. Also, the fact that the stock price is valued very high and may not be sustainable provides another justification for using the target weightings. b. Short-term debt should not be included in the WACC since it is not part of the permanent capital structure, that is, it is paid off during the year. Long-term debt cost: –1030 PV, 30 N, 45 PMT, 1000 FV and CPT I/Y = 4.32 × 2 = 8.64% –1,000(1 – 0.02) PV, 30 N, 86.40/2 PMT, 1000 FV and CPT I/Y = 4.44 × 2 = 8.88% rd ltm = 8.88%. Cost of equity: CAPM: rs = 5.0% + 0.9(11.0% – 5.0%) = 10.4%. The current 10-year bond rate should be used as the question provides data on expected market returns (not historical returns). DCF: rs = [D1/P0] + g = [$1.50(1 + 0.02)/$21] + 0.02 = 9.3%. Average rs = (10.4% + 9.3%)/2 = 9.9% The issue here is what growth rate ―g‖ to use. Since g represents the expected longterm growth rate going forward, and the problem states that the company is entering a maturity phase, using a past growth rate of 10% is not appropriate. Given that company growth is expected to mirror the economy, the expected overall economic growth rate of 2% is probably most appropriate. g = ROE(RR) can also be calculated since we have the data. g = 18%(1 – 0.30)=12.6%. 12.6% is also based on past growth and is also much higher than a standard long-run growth rate for a mature company, so it too should not be used. WACC = (Dltm/(Dltm+ S))(rd ltm)(1 – T) + (S/(Dltm + S))(rs) = 0.40(8.88%)(1 – 0.30) + 0.60(9.9%) = 8.43% ≈ 8.4%.

9-19

The book and market value of the notes payable are $10,000,000. Bond value: Using a financial calculator, input N = 20, I/YR = 10, PMT = 60, and FV = 1000 to arrive at a PV = $659.46. The total market value of the long-term debt is 30,000($659.46) = $19,783,800. There are 1 million shares outstanding, and the stock sells for $60 per share. Therefore, the market value of the equity is $60,000,000. The market value capital structure is thus: Short-term debt Long-term debt Common equity

$10,000,000 19,783,800 60,000,000 $89,783,800

11.14% 22.03 66.83 100.00%

9-20 Establish a set of market value capital structure weights. The short-term debt is taken at par value; therefore its market value is $15,000,000. The long-term debt trades at 105 of par, therefore its market value is 105% × $1,000 × 60,000 = $63,000,000. The market value of the equity is 20,000,000 shares × $18/share = $360,000,000. Short-term debt

$ 15,000,000

3.4%

Long-term debt

63,000,000

14.4%

Common equity

360,000,000

82.2%

$438,000,000

100.0%

Establish the cost rates for the various capital structure components: Since short-term debt is valued at par and no changes in interest costs are expected: Short-term debt: rd stm = 6.0% Long-term debt: To find the market yield on long-term debt, enter into a financial calculator: –1050 PV, 30 N, 45 PMT, 1000 FV and CPT I/Y = 4.20 × 2 = 8.40% rd ltm = 8.40%.

Common equity: Information is provided to estimate rs two ways: CAPM and dividend growth. Both are estimates. CAPM: We use rrf = 10-year government bond rate = 5.5%. For RPM, we use 5%. The beta given is 0.9. Using these values we estimate rs: rs = 5.5% + (5)(0.9) = 10.0%. Dividend Growth: We first calculate this year’s earnings per share: $26,000,000/20,000,000 = $1.30. Since the growth in earnings is expected to be 6%, and 40% of earnings will be paid out in dividends, we can estimate rs: rs =

(0.40)($1.30)(1.06) + 0.06 = 9.06% $18

Based on an average, we estimate rs to be (10.0% + 9.06%)/2 = 9.53%. Calculate the WACC: WACC = (Wd stm)rd stm(1 – T) + (Wd ltm))rd ltm(1 – T) + (Ws)rs = 0.034(6.0%)(1- 0.35) + 0.144(8.40%)(1 – 0.35) + 0.822(9.53%) = 8.75%.

9-21

Several steps are involved in the solution of this problem. Our solution follows: Step 1. Establish a set of market value capital structure weights. In this case, A/P and accruals should be disregarded because they are not sources of financing from investors. Instead of being incorporated into the WACC, they are accounted for when calculating cash flows. For this firm, short-term debt is used to finance seasonal goods, and the balance is reduced to zero in the off-season. Therefore, this is not a source of permanent financing and should be disregarded when calculating the WACC. Debt: The long-term debt has a market value found as follows: 40

$40 $1,000 = $1,106.78,  t (1.035) 40 t 1 (1.035 )

V0 = 

or 1,106.78($20,000,000) = $22,135,507 in total. Notice that short-term debt is not included in the capital structure for this company. We usually include short-term debt in the total debt figure for calculating weights because, in the absence of any other information, we assume the short-term debt will be rolled over from year to year. In this case, however, the company does not use short-term debt as a permanent source of financing. Indeed, as stated in the problem, the short-term debt balance is zero off-season. In such a situation neither the lender nor the company believes that the debt balance will be rolled over from year to year as the loan is closed out each off-season and so it is not considered a component of the capital structure. Preferred Stock: The preferred has a value of Pps =

$0.45 = $27.69. 0.065 / 4

There are $3,000,000/$25 = 120,000 shares of preferred outstanding, so the total market value of the preferred is 120,000($27.69) = $3,322,800.

Common Stock: The market value of the common stock is 4,000,000($20) = $80,000,000. Therefore, here is the firm’s market value capital structure, which we assume to be optimal: Long-term debt Preferred stock Common equity

$ 22,135,507 3,322,800 80,000,000 $105,458,307

20.99% 3.15 75.86 100.00%

We would round these market value weights to 21% debt, 3% preferred, and 76% common equity. Step 2. Establish cost rates for the various capital structure components. Debt cost: rd= 7%. Preferred cost: Annual dividend on new preferred = 6.5%($25) = $1.625. Therefore, rps = $1.625/$25(1 – 0.05) = $1.625/$23.75 = 6.8%. Common equity cost: There are three basic ways of estimating rs: CAPM, dividend growth approach, and risk premium over own bonds. None of the methods is very exact. CAPM: We would use rRF = government bond rate = 5%. For RPM, we would use 2.5% to 3.5%. For beta, we would use a beta in the 1.3 to 1.7 range. Combining these values, we obtain this range of values for rs:

Highest: rs = 5% + (3.5)(1.7) = 10.95%. Lowest: rs = 5% + (2.5)(1.3) = 8.25%. Average: rs = (10.95% + 8.25%)/2 = 9.6%.

Dividend Growth: The company seems to be in a rapid, nonconstant growth situation, but we do not have the inputs necessary to develop a nonconstant rs. Therefore, we will use the dividend growth approach, but temper our growth rate; that is, think of it as a long-term average g that may well be higher in the immediate than in the more distant future. We could use as a growth estimator this method: g = ROE(Retention ratio) = 24%(0.5) = 12%. It would not be appropriate to base g on the 30% ROE, because investors do not expect that rate. Finally, we could use the analysts’ forecasted g range, 4% to 8%. The dividend yield is D1/P0. Assuming g = 6%,

$1(1.06) D1 = = 5.3%. $20 P0 One could look at a range of yields, based on P in the range of $17 to $23, but because we believe in efficient markets, we would use P0 = $20. Thus, the dividend growth approach suggests an rs in the range of 9.3% to 13.3%: Highest: rs = 5.3% + 8% = 13.3%. Lowest: rs = 5.3% + 4% = 9.3%. Midpoint: rs = 5.3% + 6% = 11.3%. Generalized risk premium. Highest: rs = 7% + 5% = 12%. Lowest: rs = 7% + 3% = 10%. Average: rs = (12% + 10%)/2 = 11%. Based on the three midpoint estimates, we have rs in this range: CAPM 9.6% Dividend growth 11.3% Risk Premium 11% Average: (9.6% + 11.3% + 11%)/3 = 10.6%

Step 3. Calculate the WACC: WACC = Wd(rd)(1 – T) + Wps(rps) + Ws(rs) = 0.21(rd) + 0.03(rps) + 0.76(rs). It would be appropriate to calculate a range of WACCs based on the ranges of component costs, but to save time, we shall assume rd = 7.0%, rps = 6.8%, and rs = 10.6%. With these cost rates, here is the WACC calculation: WACC = 0.21(7.0%)(1 – 0.30) + 0.03(6.8%) + 0.76(10.6%) = 9.3%.

SOLUTION TO SPREADSHEET PROBLEM

9-22

The detailed solution for the spreadsheet problem is in the file Ch 09 Build a Model Solution.xlsx and is available on the textbook’s website.

MINI CASE The detailed solution for the Mini Case is also available in spreadsheet format, in the file Brigham4Ce_MiniCase_Ch09.xlsx, and is available on the textbook’s website. During the past few years, Harry Davis Industries has been too constrained by the high cost of capital to make many capital investments. Recently, though, capital costs have been declining, and the company has decided to look seriously at a major expansion program that has been proposed by the marketing department. Assume that you are an assistant to Leigh Jones, the financial vice president. Your first task is to estimate Harry Davis’s cost of capital. Jones has provided you with the following data, which she believes may be relevant to your task: 1. The firm’s tax rate is 35%. 2.

The current price of Harry Davis’s 8% coupon, semiannual payment, noncallable bonds with 15 years remaining to maturity is $1,091.96. Harry Davis does not use short-term interest-bearing debt on a permanent basis. New bonds would be privately placed with no flotation cost.

3. The current price of the firm’s 6%, $25 par value, quarterly dividend, perpetual preferred stock is $19.74. Harry Davis would incur flotation costs equal to 5% of the proceeds on a new issue. 4. Harry Davis’s common stock is currently selling at $50 per share. Its last dividend (D0) was $2.00, and dividends are expected to grow at a constant rate of 5% in the foreseeable future. Harry Davis’s beta is 1.2, the yield on government bonds is 4%, and the market risk premium is estimated to be 5%. For the bond-yield-plus-risk-premium approach, the firm uses a 4 percentage point risk premium. 5. Harry Davis’s target capital structure is 30% long-term debt, 10% preferred stock, and 60% common equity. To structure the task somewhat, Jones has asked you to answer the following questions.

1. What sources of capital should be included when you estimate Harry Davis’s weighted average cost of capital (WACC)?

Answer:

The WACC is used primarily for making long-term capital investment decisions, i.e., for capital budgeting. Thus, the WACC should include the types of capital used to pay for long-term assets, and this is typically long-term debt, preferred stock (if used), and common stock. Short-term sources of capital consist of (1) spontaneous, noninterest-bearing liabilities such as accounts payable and accruals and (2) shortterm interest-bearing debt, such as notes payable. If the firm uses short-term interestbearing debt to acquire fixed assets rather than just to finance working capital needs, then the WACC should include a short-term debt component. Noninterest-bearing debt is generally not included in the cost of capital estimate because these funds are netted out when determining investment needs, that is, net rather than gross working capital is included in capital expenditures.

2. Should the component costs be figured on a before-tax or an after-tax basis?

Answer: Shareholders are concerned primarily with those corporate cash flows that are available for their use, namely, those cash flows available to pay dividends or for reinvestment. Since dividends are paid from and reinvestment is made with after-tax dollars, all cash flow and rate of return calculations should be done on an after-tax basis. a.

3. Should the costs be historical (embedded) costs or new (marginal) costs?

Answer: In financial management, the cost of capital is used primarily to make decisions that involve raising new capital. Thus, the relevant component costs are today’s marginal costs rather than historical costs.

What is the market interest rate on Harry Davis’s debt and its component cost of debt?

Answer: Harry Davis’s 8% bond with 15 years to maturity is currently selling for $1,091.96. Thus, its yield to maturity is 7%: 0

–1,091.96

30 |

  

40 1,000

Enter N = 30, PV = –1091.96, PMT = 40, and FV = 1000, and then press the I/Y button to find rd/2 = i = 3.5%. Since this is a semiannual rate, multiply by 2 to find the annual rate, rd = 7%, the pre-tax cost of debt. Since interest is tax deductible, the government, in effect, pays part of the cost, and Harry Davis’s relevant component cost of debt is the after-tax cost: rd(1 – T) = 7.0%(1 – 0.35) = 7.0%(0.65) = 4.55%. Optional Question: Should flotation costs be included in the estimate? Answer: The actual component cost of new debt will be somewhat higher than 4.55% because the firm will incur flotation costs in selling the new issue. However, flotation costs are typically small on public debt issues and, more importantly, most debt is placed directly with banks, insurance companies, and the like, and in this case flotation costs are almost nonexistent. Optional Question: Should you use the nominal cost of debt or the effective annual cost? Answer: Our 7% pre-tax estimate is the nominal cost of debt. Since the firm’s debt has semiannual coupons, its effective annual rate is: (1.035)2 – 1.0 = 1. 0712 – 1.0 = 0.0712 = 7.12%. However, nominal rates are generally used. The reason is that the cost of capital is used in capital budgeting, and capital budgeting cash flows are generally assumed to occur at year end. Therefore, using nominal rates makes the treatment of the capital budgeting discount rate and cash flows consistent.

What is the firm’s cost of preferred stock?

Answer: Since the preferred issue is perpetual, its cost is estimated as follows: rps =

D ps Pps (1  F)

$1.50 0.06($25) = = 0.080 = 8.0%. $18.75 $19.74(1  0.05)

Note that (1) flotation costs for preferred shares are significant, so they are included here, (2) since preferred dividends are not deductible to the issuer, there is no need for a tax adjustment, and (3) we could have estimated the effective annual cost of the preferred, but, as in the case of debt, the nominal cost is generally used. d.

1. What are the two primary ways companies raise common equity?

Answer: A firm can raise common equity in two ways: (1) by retaining earnings and (2) by issuing new common shares. d.

2. Why is there a cost associated with reinvested earnings?

Answer: Management may either pay out earnings in the form of dividends or else retain earnings for reinvestment in the business. If part of the earnings is retained, an opportunity cost is incurred: shareholders could have received those earnings as dividends and then invested that money in stocks, bonds, real estate, and so on. d.

3. Harry Davis doesn’t plan to issue new shares of common stock. Using the CAPM approach, what is Harry Davis’s estimated cost of equity?

Answer: rs = 0.04 + (0.05)1.2 = 10%. e.

1. What is the estimated cost of equity using the dividend growth approach? 

Answer:

rs =

D (1  g) $2.00 (1.05) D1 g =  0.05 = 9.2%. = 0 $50 P0 P0

2. Suppose the firm has historically earned 15% on equity (ROE) and retained 35% of earnings, and investors expect this situation to continue in the future. How could you use this information to estimate the future dividend growth rate, and what growth rate would you get? Is this consistent with the 5% growth rate given earlier?

Answer: Another method for estimating the growth rate is to use the retention growth model: g = ROE(1 – Payout ratio) In this case g = 0.15(0.35) = 5.25%. This is consistent with the 5% rate given earlier. e.

3. Could the dividend growth approach be applied if the growth rate was not constant? How?

Answer: Yes, you could use the DCF using nonconstant growth. You would find the PV of the dividends during the nonconstant growth period and add this value to the PV of the series of inflows when growth is assumed to become constant. f.

What is the cost of equity based on the bond-yield-plus-risk-premium method?

Answer: rs = Company’s own bond yield + Risk premium. First find the YTM of the bond: Enter N = 30, PV = –1,091.96, PMT = 40, and FV = 1000, and then press the I button to find r/2 = I = 3.5%. Since this is a semiannual rate, multiply by 2 to find the annual rate, r = 7%. The assumed risk premium is 4%, thus rs = 0.07 + 0.04 = 11%. g.

What is your final estimate for the cost of equity, rs?

Answer: The final estimate for the cost of equity would simply be the average of the values found using the above three methods. CAPM Dividend growth BOND YIELD + R.P. AVERAGE

10% 9.2 11 10.1%

What is Harry Davis’s weighted average cost of capital (WACC)?

Answer: WACC= wdrd(1 – T) + wpsrps + wce(rs) = 0.3(0.07)(0.65) + 0.1(0.08) + 0.6(0.101) = 0.082 = 8.2%. i.

What factors influence a company’s WACC?

Answer: There are factors that the firm cannot control and those that it can control that influence WACC. Factors the Firm Cannot Control: Level of Interest Rates Market Conditions Tax Rates

}

Factors the Firm Can Control: Capital Structure Policy Dividend Policy Investment Policy j.

Should the company use the composite WACC as the hurdle rate for each of its divisions?

Answer: No. The composite WACC reflects the risk of an average project undertaken by the firm. Therefore, the WACC only represents the ―hurdle rate‖ for a typical project with average risk. Different projects have different risks. The project’s WACC should be adjusted to reflect the project’s risk.

What procedures are used to determine the risk-adjusted cost of capital for a particular division? What approach is used to measure a division’s beta?

Answer: The following procedures can be used to determine a division’s risk-adjusted cost of capital: (1)

Subjective adjustments to the firm’s composite WACC.

(2)

Attempt to estimate what the cost of capital would be if the division were a stand-alone firm. This requires estimating the division’s beta.

The following approach can be used to measure a division’s beta: Pure play approach. Find several publicly traded companies exclusively in the project’s business. Then, use the average of their betas as a proxy for the project’s beta. (It’s hard to find such companies.) l.

What are three types of project risk? How is each type of risk used?

Answer: The three types of project risk are: Stand-alone risk Corporate risk Market risk Market risk is theoretically best in most situations. However, creditors, customers, suppliers, and employees are more affected by corporate risk. Therefore, corporate risk is also relevant. Stand-alone risk is the easiest type of risk to measure. Taking on a project with a high degree of either stand-alone or corporate risk will not necessarily affect the firm’s market risk. However, if the project has highly uncertain returns, and if those returns are highly correlated with returns on the firm’s other assets and with most other assets in the economy, the project will have a high degree of all types of risk.

Explain why new common stock that is raised externally has a higher percentage cost than equity that is raised internally by reinvesting earnings.

Answer: The company is raising money in order to make an investment. The money has a cost, and this cost is based primarily on the investors’ required rate of return, considering risk and alternative investment opportunities. So, the new investment must provide a return at least equal to the investors’ opportunity cost. If the company raises capital by selling stock, the company doesn’t get all of the money that investors put up. For example, if investors put up $100,000, and if they expect a 15% return on that $100,000, then $15,000 of profits must be generated. But if flotation costs are 20% ($20,000), then the company will receive only $80,000 of the $100,000 investors put up. That $80,000 must then produce a $15,000 profit, or a 15/80 = 18.75% rate of return versus a 15% return on equity raised as retained earnings. n.

1. Harry Davis estimates that, if it issues new common stock, the flotation cost will be 15%. Harry Davis incorporates the flotation costs into the dividend growth approach. What is the estimated cost of newly issued common stock, taking into account the flotation cost?

Answer:

D0 (1 + g) + g P0 (1 - F) $2.00(1.05) = + 5.0% $50(1 - 0.15) $2.10 = + 5.0% = 9.9%. $42.50

re =

2. Suppose Harry Davis issues 30-year debt with a par value of $1,000 and a coupon rate of 7%, paid annually. If flotation costs are 2%, what is the after-tax cost of debt for the new bond issue?

Answer: Using a financial calculator, N = 30, PV = (1 – 0.02)(1000) = 980, PMT = –70, FV = –1000. The resulting I is 7.16%, which is the before-tax cost of debt. The aftertax cost = 7.16%(1 – 0.35) = 4.65%.

What four common mistakes in estimating the WACC should Harry Davis avoid?

Answer: 1. Don’t use the coupon rate on the firm’s existing debt as the pre-tax cost of debt. Use the current cost of debt. 2. When estimating the risk premium for the CAPM approach, don’t subtract the current long-term government bond rate from the historical average return on stocks. For example, the historical average return on stocks has been about 12.7%. If inflation has driven the current risk-free rate up to 10%, it would be wrong to conclude that the current market risk premium is 12.7% – 10% = 2.7%. In all likelihood, inflation would also have driven up the expected return on the market. Therefore, the historical return on the market would not be a good estimate of the current expected return on the market. 3. Don’t use book weights to estimate the weights for the capital structure. Use the target capital structure to determine the weights for the WACC. If you don’t have the target weights, then use market value rather than book value to obtain the weights. Use the book value of debt only as a last resort. 4. Always remember that capital components are sources of funding that come from investors. If it’s not a source of funding from an investor, then it’s not a capital component.

Chapter 10 The Basics of Capital Budgeting: Evaluating Cash Flows ANSWERS TO END-OF-CHAPTER QUESTIONS

10-1

a. Capital budgeting is the whole process of analyzing projects and deciding whether they should be included in the capital budget. This process is of fundamental importance to the success or failure of the firm as the fixed asset investment decisions chart the course of a company for many years into the future. The payback, or payback period, is the number of years it takes a firm to recover its project investment. Payback may be calculated with either raw cash flows (regular payback) or discounted cash flows (discounted payback). In either case, payback does not capture a project’s entire cash flow stream and is thus not the preferred evaluation method. Note, however, that the payback does measure a project’s liquidity, and hence many firms use it as a risk measure.

b. Independent projects can be accepted or rejected individually. Mutually exclusive projects cannot be performed at the same time. We can choose either Project 1 or Project 2, or we can reject both, but we cannot accept both projects. c. The net present value (NPV) and internal rate of return (IRR) techniques are discounted cash flow (DCF) evaluation techniques. These are called DCF methods because they explicitly recognize the time value of money. NPV is the present value of the project’s expected future cash flows (both inflows and outflows), discounted at the appropriate cost of capital. NPV is a direct measure of the value of the project to shareholders. The internal rate of return (IRR) is the discount rate that equates the present value of the expected future cash inflows and outflows. IRR measures the rate of return on a project, but it assumes that all cash flows can be reinvested at the IRR rate. d. The modified internal rate of return (MIRR) assumes that cash flows from all projects are reinvested at the cost of capital as opposed to the project’s own IRR. This makes the modified internal rate of return a better indicator of a project’s true profitability. The profitability index is found by dividing the project’s PV of future cash flows by its initial cost. A profitability index greater than 1 is equivalent to a positive NPV project. e. An NPV profile is the plot of a project’s NPV versus its cost of capital. The crossover rate is the cost of capital at which the NPV profiles for two projects intersect. f. Capital projects with non-normal cash flows have a large cash outflow either sometime during or at the end of their lives. A project has normal cash flows if one or more cash outflows (costs) are followed by a series of cash inflows. A common problem encountered when evaluating projects with non-normal cash flows is multiple IRRs, meaning there is more than one. g.

The mathematics of the NPV method imply that project cash flows are reinvested at the cost of capital while the IRR method assumes reinvestment at the IRR. Since project cash flows can be replaced by new external capital, which costs r, the proper reinvestment rate assumption is the cost of capital, and thus the best capital budget decision rule is NPV. h. The equivalent annual annuity method is a method of comparing mutually exclusive projects that have unequal lives. This method converts the project’s NPV into an annual annuity over the project’s life. One can then compare the annual NPV each project generates. Not all projects maximize their NPV if operated over their engineering lives and therefore it may be best to terminate a project prior to the end of its potential life. The economic life is the number of years a project should be operated to maximize its NPV and is often less than the maximum potential life. Capital rationing occurs when management places a constraint on the size of the firm’s capital budget during a particular period.

10-2

Projects requiring greater investment or that have greater risk should be given more detailed analysis in the capital budgeting process.

10-3

The NPV is obtained by discounting future cash flows, and the discounting process actually compounds the interest rate over time. Thus, an increase in the discount rate has a much greater impact on a cash flow in Year 5 than on a cash flow in Year 1.

10-4

This question is related to Question 10-3 and the same rationale applies. With regard to the second part of the question, the answer is no; the IRR rankings are constant and independent of the firm’s cost of capital.

10-5

The NPV and IRR methods both involve compound interest, and the mathematics of discounting requires an assumption about reinvestment rates. The NPV method assumes reinvestment at the cost of capital, while the IRR method assumes reinvestment at the IRR. MIRR is a modified version of IRR that assumes reinvestment at the cost of capital.

10-6

The regular payback period is considered flawed for three reasons. (1) It does not take into account the time value of money. (2) It does not consider the cash flows after the payback period. (3) No specific payback period policy a company may have necessarily maximizes shareholder wealth. The discounted payback period does take into account the time value of money, but still is flawed due to (2) and (3).

10-7

Generally, the failure to employ common life analysis in such situations will bias the NPV against the shorter project because it ―gets no credit‖ for profits beyond its initial life, even though it could possibly be ―renewed‖ and thus provide additional NPV.

SOLUTIONS TO END-OF-CHAPTER PROBLEMS 10-1

NPV = –$40,000 + $9,000[(1/I) – (1/(I × (1 + I)N)] = –$40,000 + $9,000[(1/0.11) – (1/(0.11 × (1 + 0.11)7)] = $2,409.77. Financial calculator solution: Input CF0 = –40000, CF1–7 = 9000, I/YR = 11, and then solve for NPV = $2,409.77.

10-2

Financial calculator solution: Input CF0 = –40000, CF1–7 = 9000, and then solve for IRR = 12.84%.

10-3

MIRR: PV Costs = $40000. FV Inflows: PV 0 11% 1 | | 9,000

40,000

2 | 9,000

3 | 9,000

4 | 9,000

MIRR= 11.93%

5 | 9,000

6 | 9,000

FV 7 | 9,000 9,990 11,089 12,309 13,663 15,166 16,834 88,049

Financial calculator: Obtain the FVA by inputting N = 7, I/YR = 11, PV = 0, PMT = 9000, and then solve for FV = $88,049. The MIRR can be obtained by inputting N = 7, PV = –40000, PMT = 0, FV = 88049, and then solving for I/YR = 11.93%.

10-4

PV = $9,000[(1/I) – (1/(I × (1 + I)N)] = $9,000[(1/0.11) – (1/(0.11 × (1 + 0.11)7)] = $42,410. Financial calculator: Find present value of future cash flows by inputting N = 7, I/YR = 11, PMT = –9000, FV = 0, then solve for PV = $42,410. PI = PV of future cash flows/Initial cost = $42,410/$40,000 = 1.06.

10-5 Year 0 1 2 3 4 5 6 7

CF –40,000 9,000 9,000 9,000 9,000 9,000 9,000 9,000

Cumulative CF –40,000 –31,000 –22,000 –13,000 –4,000 5,000 14,000 23,000

The cumulative cash flows turn positive in Year 5, so the payback period will be 4 years plus the part of Year 5 that is required to return the investment: Payback = 4 + ($4,000/$9,000) = 4.44 years. Because the future cash flows are identical, we can also find the payback period by dividing the cost by the cash flow: $40,000/$9,000 = 4.44 years.

10-6

The project’s discounted payback period is calculated as follows: Discounted CF Cumulative Year Annual CF (@11%) Discounted CF 0 –40,000 –40,000.00 1 9,000 8,108.11 (31,891.89) 2 9,000 7,304.60 (24,587.29) 3 9,000 6,580.72 (18,006.57) 4 9,000 5,928.58 (12,077.99) 5 9,000 5,341.06 (6,736.93) 6 9,000 4,811.77 (1,925.16) 7 9,000 4,334.93 2,409.77

The discounted payback period is 6 +

$1,925 .16 years, or 6.44 years. $4,334 .93

10-7

a. Project A: Using a financial calculator, enter the following: CF0 = –15000000 CF1 = 5000000 CF2 = 10000000 CF3 = 20000000 I = 10; NPV = $12,836,213. Change I = 10 to I = 5; NPV = $16,108,952. Change I = 5 to I = 15; NPV = $10,059,587. Project B: Using a financial calculator, enter the following: CF0 = –15000000 CF1 = 20000000 CF2 = 10000000 CF3 = 6000000 I = 10; NPV = $15,954,170. Change I = 10 to I = 5; NPV = $18,300,939. Change I = 5 to I = 15; NPV = $13,897,838. b. Using the data for Project A, enter the cash flows and solve for IRRA=43.97%. The IRR is independent of the WACC, so IRR doesn’t change when the WACC changes. Using the data for Project B, enter the cash flows and solve for IRRB = 82.03%. Again, the IRR is independent of the WACC, so IRR doesn’t change when the WACC changes.

10-8

Truck: NPV = –$17,100 + $5,100(PVIFA14%,5) = –$17,100 + $5,100(3.4331) = –$17,100 + $17,509 = $409 (Accept) Financial calculator: Input the appropriate cash flows into the cash flow register, input I/YR = 14, and then solve for NPV = $409. Financial calculator: Input the appropriate cash flows into the cash flow register and then solve for IRR = 14.99% ≈ 15%. MIRR: PV Costs = $17,100. FV Inflows: PV 0 14% |

17,100

FV 5

5,100

5,100 5,814 6,628 7,556 8,614 33,712

MIRR = 14.54% (Accept)

Financial calculator: Obtain the FVA by inputting N = 5, I = 14, PV = 0, PMT = 5100, and then solve for FV = $33,712. The MIRR can be obtained by inputting N = 5, PV = –17100, PMT = 0, FV = 33712, and then solving for I = 14.54%. Pulley: NPV = $22,430 + $7,500(3.4331) = $22,430 + $25,748 = $3,318. (Accept) Financial calculator: Input the appropriate cash flows into the cash flow register, input I/YR = 14, and then solve for NPV = $3,318. Financial calculator: Input the appropriate cash flows into the cash flow register and then solve for IRR = 20%. MIRR: PV Costs = $22,430.

FV Inflows: PV 0 14% |

1 | 7,500

2 | 7,500

3 | 7,500

22,430

4 | 7,500

FV 5 | 7,500 8,550 9,747 11,112 12,667 49,576

MIRR = 17.19% (Accept) Financial calculator: Obtain the FVA by inputting N = 5, I/YR = 14, PV = 0, PMT = 7500, and then solve for FV = $49,576. The MIRR can be obtained by inputting N = 5, PV = 22430, PMT = 0, FV = 49576, and then solving for I/YR = 17.19%.

10-9

Electric-powered: NPVE = $33,000 + $7,890 [(1/i) – (1/(i*(1 + i)n)] = $33,000 + $7,890[(1/0.09) – (1/(0.09*(1 + 0.09)6)] = $33,000 + $7,890(4.4859) = $33,000 + $35,394 = $2,394 Financial calculator: Input the appropriate cash flows into the cash flow register, input I = 9, and then solve for NPV = $2,394. Financial calculator: Input the appropriate cash flows into the cash flow register and then solve for IRR = 11.40%. Gas-powered: NPVG = $28,500 + $6,650 [(1/i) – (1/(i*(1 + i)n)] = $28,500 + $8,000 [(1/0.09) – (1/(0.09*(1 + 0.09)6)] = $28,500 + $8,000(4. 4859) = $28,500 + $29,831 = $1,331. Financial calculator: Input the appropriate cash flows into the cash flow register, input I = 9, and then solve for NPV = $1,331. Financial calculator: Input the appropriate cash flows into the cash flow register and then solve for IRR = 10.55%. The firm should purchase the electric-powered forklift because it has a higher NPV than the gas-powered forklift. The company receives a higher rate of return (11.40% > r = 9%) on a larger investment.

10-10 Financial calculator solution, NPV: Project S Inputs

I/YR

3000

PMT

7400

PMT

= –10,814.33

Output

NPVS = $10,814.33 – $10,000 = $814.33. Project L Inputs

I/YR

= –26,675.34

Output

NPVL = $26,675.34 – $25,000 = $1,675.34. Financial calculator solution, IRR: Input CF0 = –10000, CF1 = 3000, Nj = 5, IRRS = ? IRRS = 15.24%. Input CF0 = –25000, CF1 = 7400, Nj = 5, IRRL = ? IRRL = 14.67%. Financial calculator solution, MIRR: Project S Inputs

3000

I/YR

PMT

= –19,058.54

Output PV costsS = $10,000. FV inflowsS = $19,058.54.

Inputs

5 N

Output

I/YR

–10000

19058.54

PMT

= 13.77 MIRRS = 13.77%.

Project L Inputs

Output

7400

I/YR

PMT

–47,011.07

PV costsL = $25,000. FV inflowsL = $47,011.07. Inputs 5 N

Output

I/YR

–25000

47011.07

PMT

= 13.46 MIRRL = 13.46%.

PIS = $10,814.33 = 1.081. PIL = $26,675.34 = 1.067. $10,000

$25,000

Thus, NPVL > NPVS, IRRS > IRRL, MIRRS > MIRRL, and PIS > PIL. The scale difference between Projects S and L results in the IRR, MIRR, and PI favouring S over L. However, NPV favours Project L; thus, L adds more value to the firm so L should be chosen.

10-11 Because both projects are the same size, you can just calculate each project’s MIRR and choose the project with the higher MIRR. (Remember, MIRR gives conflicting results from NPV when there are scale differences between the projects.) Project J:

0 1 12% | | –5,000 1,000

5,000

2 | 1,500

3 | 2,000

4 | 4,000.00 2,240.00 1,881.60 1,404.93 9,526.53

17.49% = MIRRJ

$5,000 = $9,526.53/(1 + MIRRJ)4. Project K:

0 1 12% | | –5,000 4,500

5,000

2 | 1,500

3 | 1,000

4 | 500.00 1,120.00 1,881.60 6,322.18 9,823.78

18.39% = MIRRK

$5,000 = $9,823.78/(1 + MIRRK)4. Thus, since MIRRK > MIRRJ, Project K should be chosen. Alternative step: You could calculate NPVs, see that Project K has the higher NPV, and just calculate MIRRK. NPVJ = $1,054.28 and NPVK = $1,243.19. 10-12 a. Purchase price Installation Initial outlay

$ 6,200,000 400,000 $ 6,600,000

CF0 = 6200000; CF1-5 = 1900000; I = 10; NPV = ? NPV = $602,495; IRR = 13.51%. b.

Ignoring environmental concerns, the project should be undertaken because its NPV is positive and its IRR is greater than the firm’s cost of capital.

c. Environmental effects could be added by estimating penalties or any other cash outflows that might be imposed on the firm to help return the land to its previous state (if possible). These outflows could be so large as to cause the project to have a negative NPV—in which case the project should not be undertaken. 10-13 a.

r 0.0% 10.0 12.0 14.8 18.0 20.7 25.8 30.0

NPVA $1,288 $479 $366 $228 $94 $0 −$149 −$245

NPVB $820 $372 $308 $229 $150 $93 $0 −$62

IRR of project A = 21% IRR of project B = 26%

At r = 10%, Project A has the greater NPV, specifically $478.83 as compared to Project B’s NPV of $372.37. Thus, Project A would be selected. At r = 17%, Project B has an NPV of $173.70, which is higher than Project A’s NPV of $133.76. Thus, choose Project B if r = 17%. d. Here is the MIRR for Project A when r = 10%: PV costs = $400 + $528/(1.10)1 + $219/(1.10)2 + $150/(1.10)3 + $325/(1.10)7 = $1,340.47 TV inflows = $1,100(1.10)3 + $820(1.10)2 + $990(1.10)1 = $3,545.30. Now, MIRR is that discount rate which forces the PV of $3,545.30 in 7 years to equal $1,340.47: $1,340.47 = $3,545.30/(1 + MIRR)7 MIRRA = 14.91%. Here is the MIRR for Project B when r = 10%: PV costs = 650. TV of inflows: Financial calculator settings are N = 7, I/YR = 10, PV = 0, PMT = 210, and solve for FV = –1992.31. Similarly, $650 = $1,992.31/(1 + MIRR)7 MIRRB = 17.35%. At r = 17%, MIRRA = 18.76%. MIRRB = 21.03%.

e. To find the crossover rate, construct a Project ∆, which is the difference in the two projects’ cash flows: Year Project ∆ = CFA – CFB 0 $250 1 −738 2 −429 3 −360 4 890 5 610 6 780 7 −535

IRR∆ = Crossover rate = 14.76%. Projects A and B are mutually exclusive, thus, only one of the projects can be chosen. As long as the cost of capital is greater than the crossover rate, both the NPV and IRR methods will lead to the same project selection. However, if the cost of capital is less than the crossover rate, the two methods lead to different project selections—a conflict exists. When a conflict exists, the NPV method must be used. Because of the sign changes and the size of the cash flows, Project ∆ has multiple IRRs. Thus, the IRR function for some calculators will not work (it will work, however, on a BAII Plus). The HP can be ―tricked‖ into giving the roots by selecting an initial guess near one of the roots. After you have keyed Project Delta’s cash flows into the CFj register of an HP-10B, you will see an ―Error-Soln‖ message. Now enter 10; STO; IRR/YR and the 14.76% IRR is found. Then enter 100; STO; IRR/YR to obtain IRR = 246.02%. Similarly, Excel can also be used.

10-14 a. Year 0 1 2–20

Plan B ($10,000,000) 1,750,000 1,750,000

Plan A ($10,000,000) 12,000,000 0

Incremental Cash Flow (B – A) $ 0 (10,250,000) 1,750,000

If the firm goes with Plan B, it will forgo $10,250,000 in Year 1, but will receive $1,750,000 per year in Years 2–20. b. If the firm could invest the incremental $10,250,000 at a return of 16.07%, it would receive cash flows of $1,750,000. If we set up an amortization schedule, we would find that payments of $1,750,000 per year for 19 years would amortize a loan of $10,250,000 at 16.0665%. Financial calculator solution: Inputs

19 N

–10250000

1750000

PMT

Output

= 16.0665

c. Yes, assuming (1) equal risk among projects, and (2) that the cost of capital is a constant and does not vary with the amount of capital raised. d. See graph. If the cost of capital is less than 16.07%, then Plan B should be accepted; if r > 16.07%, then Plan A is preferred. NPV (Millions of Dollars) 25

B 20

Crossover Rate = 16.07%

A IRRB = 16.7% IRRA = 20%

Cost of Capital (%)

10-15 a. Financial calculator solution: Plan A Inputs

8000000

PMT

= –68,108,510

Output

NPVA = $68,108,510 – $50,000,000 = $18,108,510. Plan B Inputs

3400000

PMT

= –28,946,117

Output

NPVB = $28,946,117 – $15,000,000 = $13,946,117. Plan A Inputs

20 N

Output

–50000000

8000000

PMT

–15000000

3400000

PMT

= 15.03 IRRA = 15.03%. Plan B

Inputs

20 N

Output

= 22.26 IRRB = 22.26%.

b. If the company takes Plan A rather than B, its cash flows will be (in millions of dollars): Year 0 1 2 . . . 20

Cash Flows from A ($50) 8 8 . . . 8

Project ∆ Cash Flows ($35.0) 4.6 4.6 . . . 4.6

Cash Flows from B ($15.0) 3.4 3.4 . . . 3.4

So, Project ∆ has a ―cost‖ of $35,000,000 and ―inflows‖ of $4,600,000 per year for 20 years. Inputs

4600000

PMT

= –39,162,393

Output

NPV = $39,162,393 – $35,000,000 = $4,162,393. Inputs

20 N

Output

–35000000

4600000

PMT

= 11.71 IRR = 11.71%.

Since IRR∆ > r, we should accept ∆. This means accept the larger project (Project A). In addition, when dealing with mutually exclusive projects, we use the NPV method for choosing the best project.

c. NPV (Millions of Dollars) 125

A Crossover Rate = 11.7%

100

IRRA = 15.03%

IRRB = 22.26% 25

Cost of Capital (%) -25



IRR = 11.7%

-50

10-16

The EAA of plane A is found by first finding the PV: N = 5, I/YR = 12, PMT = 30, FV = 0; solve for PV = $108.143. The NPV is $108.143 − $100 = $8.143 million. We convert this to an equivalent annual annuity by inputting N = 5, I/YR = 12, PV = –8.143, FV = 0, and solve for PMT = EAA = $2.259 million. For plane B, we already found the NPV of $9.256 million. We convert this to an equivalent annual annuity by inputting N = 10, I/YR = 12, PV = -9.256, FV = 0, and solve for PMT = EAA = $1.638 million. Plane A has the higher equivalent annual annuity and should be accepted.

10-17

The EAA of machine A is found by first finding the PV: N = 5, I/YR = 12, PMT = .9, FV = 0; solve for PV = −3.244. The NPV is $3.244 − $2.5 = $0.744 million. We convert this to an equivalent annual annuity by inputting N = 5, I/YR = 12, PV = −0.744, FV = 0, and solve for PMT = EAA = 0.206 ≈ $206,000. For machine B, the NPV = $0.863 million. We convert this to an equivalent annual annuity by inputting N = 9, I/YR = 12, PV = −0.863, FV = 0, and solve for PMT = EAA = 0.162 ≈ $162,000. Accept machine A.

10-18

The EAA of machine 190-3 is found by converting its NPV to an equivalent annual annuity by inputting N = 3, I/YR = 14, PV = −11,981.99, FV = 0, and solve for PMT = EAA = 5,161.02. The EAA of machine 360-6 is found by converting its NPV to an equivalent annual annuity by inputting N = 6, I/YR = 14, PV = −22,256.02, FV = 0, and solve for PMT = EAA = 5,723.30. Both new machines have positive NPVs, hence the old machine should be replaced. Choose Model 360-6.

10-19 a.

The project’s expected cash flows are as follows (in millions of dollars): Time Net Cash Flow 0 ($ 4.4) 1 27.7 2 (25.0)

We can construct the following NPV profile: NPV (Millions of Dollars) Maximum NPV at 80.5%

Discount Rate (%) 10

-1

80.5

IRR1 = 9.2%

420

IRR2 = 420%

-2

-3

-4

NPV approaches -$4.4 as the cost of capital approaches 

-4.4

Discount Rate 0% 9 10 50 100 200 300 400 410 420 430

NPV ($1,700,000) (29,156) 120,661 2,955,556 3,200,000 2,055,556 962,500 140,000 70,204 2,367 (63,581)

The table above was constructed using a financial calculator with the following inputs: CF0 = –4400000, CF1 = 27700000, CF2 = 25000000, and I = discount rate to solve for the NPV. 10-276 Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc.

b. If r = 8%, reject the project since NPV < 0. But if r = 14%, accept the project because NPV > 0. c. Other possible projects with multiple rates of return could be nuclear power plants where disposal of radioactive wastes is required at the end of the project’s life, or leveraged leases where the borrowed funds are repaid at the end of the lease life. d. Here is the MIRR for the project when r = 8%: PV costs = $4,400,000 + $25,000,000/(1.08)2 = $25,833,470.51. TV inflows = $27,700,000(1.08)1 = $29,916,000.00. Now, MIRR is that discount rate which forces the PV of the TV of $29,916,000 over 2 years to equal $25,833,470.51: $25,833,470.51 = $29,916,000(PVIFr,2). Inputs

2 N

Output

–25833470.51

29916000

PMT

–23636688.21

31578000

PMT

= 7.61 MIRR = 7.61%.

At r = 14%, Inputs

2 N

Output

= 15.58 MIRR = 15.58%.

PV costs = $4,400,000 + $25,000,000/(1.14)2 = $23,636,688.21. TV inflows = $27,700,000(1.14)1 = $31,578,000. Now, MIRR is that discount rate which forces the PV of the TV of $31,578,000 over 2 years to equal $23,636,688.21: $23,636,688.21 = $31,578,000(PVIFr,2). Yes. The MIRR method leads to the same conclusion as the NPV method. Reject the project if r = 8%, which is greater than the corresponding MIRR of 7.61%, and accept the project if r = 14%, which is less than the corresponding MIRR of 15.58%. Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc. 10-277

10-20 Step 1: Determine the PMT: 0 12% | –1,000

1 | PMT

10 | PMT



The IRR is the discount rate at which the NPV of a project equals zero. Since we know the project’s initial investment, its IRR, the length of time that the cash flows occur, and that each cash flow is the same, then we can determine the project’s cash flows by setting it up as a 10-year annuity. With a financial calculator, input N = 10, I/YR = 12, PV = –1000, and FV = 0 to obtain PMT = $176.98. Step 2: Since we’ve been given the WACC, once we have the project’s cash flows we can now determine the project’s MIRR. Calculate the project’s MIRR: 0 10% 1 | | –1,000 176.98

2 | 176.98



 (1.10)8  (1.10)

1,000

10.93% = MIRR

9 | 176.98

10 | 176.98  1.10 194.68 . . . 379.37 417.31 TV = 2,820.61

FV of inflows: With a financial calculator, input N = 10, I/YR = 10, PV = 0, and PMT = –176.98 to obtain FV = $2,820.61. Then input N = 10, PV = –1000, PMT = 0, and FV = 2820.61 to obtain I/YR = MIRR = 10.93%. 10-21 The PV of costs for the conveyor system is ($496,401), while the PV of costs for the forklift system is ($461,171). Thus, the forklift system is expected to be ($496,401) – ($461,171) = $35,230 less costly than the conveyor system, and hence the forklift trucks should be used. Financial calculator solution: Input: CF0 = –250000, CF1 = –65000, Nj = 5, I = 10, NPVC = ? NPVC = –496401. Input: CF0 = –120000, CF1 = –90000, Nj= 5, I = 10, NPVF = ? NPVF = –461171. 10-278 Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc.

10-22 a. Payback A (cash flows in thousands):

Period 0 1 2 3 4

Annual Cash Flows ($25,000) 5,000 10,000 15,000 20,000

Cumulative ($25,000) (20,000) (10,000) 5,000 25,000

PaybackA = 2 + $10,000/$15,000 = 2.67 years. Payback B (cash flows in thousands): Annual Period Cash Flows 0 ($25,000) 1 20,000 2 10,000 3 8,000 4 6,000

Cumulative $25,000) (5,000) 5,000 13,000 19,000

PaybackB = 1 + $5,000/$10,000 = 1.50 years.

b. Discounted payback A (cash flows in thousands):

Period 0 1 2 3 4

Annual Cash Flows ($25,000) 5,000 10,000 15,000 20,000

Discounted @10% Cash Flows Cumulative ($25,000.00) ($25,000.00) 4,545.45 (20,454.55) 8,264.46 (12,190.09) 11,269.72 (920.37) 13,660.27 12,739.90

Discounted PaybackA = 3 + $920.37/$13,660.27 = 3.07 years. Discounted payback B (cash flows in thousands):

Period 0 1 2 3 4

Annual Discounted @10% Cash Flows Cash Flows Cumulative ($25,000) ($25,000.00) ($25,000.00) 20,000 18,181.82 (6,818.18) 10,000 8,264.46 1,446.28 8,000 6,010.52 7,456.80 6,000 4,098.08 11,554.88

Discounted PaybackB = 1 + $6,818.18/$8,264.46 = 1.83 years. c. NPVA = $12,739,908; IRRA = 27.27%. NPVB = $11,554,880; IRRB = 36.15%. Both projects have positive NPVs, so both projects should be undertaken. d. At a discount rate of 5%, NPVA = $18,243,813. At a discount rate of 5%, NPVB = $14,964,829. At a discount rate of 5%, Project A has the higher NPV; consequently, it should be accepted. e. At a discount rate of 15%, NPVA = $8,207,071. At a discount rate of 15%, NPVB = $8,643,390. At a discount rate of 15%, Project B has the higher NPV; consequently, it should be accepted.

f. Year 0 1 2 3 4

Project ∆ = CFA – CFB $ 0 (15) 0 7 14

IRR∆ = Crossover rate = 13.5254% ≈ 13.53%. g. Use three steps to calculate MIRRA @ r = 10%: Step 1: Calculate the NPV of the uneven cash flow stream, so its FV can then be calculated. With a financial calculator, enter the cash flow stream into the cash flow registers, then enter I = 10, and solve for NPV = $37,739,908. Step 2: Calculate the FV of the cash flow stream as follows: Enter N = 4, I = 10, PV = –37739908, and PMT = 0 to solve for FV = $55,255,000. Step 3: Calculate MIRRA as follows: Enter N = 4, PV = –25000000, PMT = 0, and FV = 55255000 to solve for I = 21.93%. Use three steps to calculate MIRRB @ r = 10%: Step 1: Calculate the NPV of the uneven cash flow stream, so its FV can then be calculated. With a financial calculator, enter the cash flow stream into the cash flow registers, then enter I = 10, and solve for NPV = $36,554,880. Step 2: Calculate the FV of the cash flow stream as follows: Enter N = 4, I = 10, PV = –36554880, and PMT = 0 to solve for FV = $53,520,000. Step 3: Calculate MIRRB as follows: Enter N = 4, PV = –25000000, PMT = 0, and FV = 53520000 to solve for I = 20.96%. According to the MIRR approach, if the two projects were mutually exclusive, Project A would be chosen because it has the higher MIRR. This is consistent with the NPV approach.

10-23 a. NPV of termination after Year t: NPV0 = $22,500 + $22,500 = 0. Using a financial calculator, input the following: CF0 = 22500, CF1 = 23750, and I/YR = 10 to solve for NPV1 = $909.09 ≈ $909. Using a financial calculator, input the following: CF0 = 22500, CF1 = 6250, CF2 = 20250, and I/YR = 10 to solve for NPV2 = $82.64 ≈ $83. Using a financial calculator, input the following: CF0 = 22500, CF1 = 6250, Nj = 2, CF3 = 17250, and I/YR = 10 to solve for NPV3 = $1,307.29 ≈ $1,307. Using a financial calculator, input the following: CF0 = 22500, CF1 = 6250, Nj = 3, CF4 = 11250, and I/YR = 10 to solve for NPV4 = $726.73 ≈ $727. Using a financial calculator, input the following: CF0 = 22500, CF1 = 6250, Nj = 5, and I/YR = 10 to solve for NPV5 = $1,192.42 ≈ $1,192. The firm should operate the truck for 3 years, NPV3 = $1,307. b. No. Salvage possibilities could only raise NPV and IRR. The value of the firm is maximized by terminating the project after Year 3.

10-24 The MIRR is the discount rate that equates the terminal value with the present value. The first step to solve the problem is to calculate the present value of the negative cash flows. PV of time = 0 cash flow ($2,000) = ($2,000.00) PV of time = 2 cash flow ($400) = (324.65) (–$400/(1 + 0.11)2 Total PV of negative cash flows = ($2,324.65) The second step is to find the value at time = 4 of the $2,324.65. Time 4 value = $2,324.65(1 + 0.080)4 = $3,162.66. The terminal value of all positive cash flow must equal this value. $3,162.66 = $840(1 + 0.11)3 + x(1 + 0.11) + $1,055 $3,162,66 = $1,148.81 + $1,055 + 1.11x 1.11 x = $958.85 x = $863.83 The cash flow at time 3 is $863.83. 10-25 Facts: 5 years remaining on lease; rent = $2,000/month; 60 payments left, payment at end of month. New lease terms: $0/month for 9 months; $2,600/month for 51 months. WACC = 12% annual (1% per month). a.

0 |

1 | –2,000

2 | –2,000

59 | –2,000



60 | –2,000

PV cost of old lease: N = 60; I/YR = 1; PMT = –2000; FV = 0; PV = ? PV = –$89,910.08. 0 |

1 | 0



9 | 0

10 | –2,600



59 | –2,600

60 | –2,600

PV cost of new lease: CF0 = 0, CF1-9 = 0; CF10-60 = –2600; I/YR = 1 NPV = –$94,611.45. The store owner should not accept the new lease because the present value of its cost is $94,611.45 – $89,910.08 = $4,701.37 greater than the old lease.

b. At t = 9, the FV of the original lease’s cost = -$89,910.08(1.01)9 = -$98,333.33. Since lease payments for months 0–9 would be zero, we can calculate the lease payments during the remaining 51 months as follows: N = 51; I/YR = 1; PV = 98333.33; and FV = 0. Solve for PMT = –$2,470.80. Check: 0 |

1 | 0



9 | 0

10 59 60 |  | | –2,470.80 –2,470.80 –2,470.80

PV cost of new lease: CF0 = 0; CF1-9 = 0; CF10-60 = –2470.80; I/YR = 1. NPV = –$89,909.99. Except for rounding; the PV cost of this lease equals the PV cost of the old lease. c. Period 0 1–9 10–60

Old Lease 0 –2,000 –2,000

New Lease 0 0 –2,600

Lease 0 –2,000 600

CF0 = 0; CF1–9 = –2000; CF10-60 = 600; IRR = ? IRR = 1.9113%. This is the periodic rate. To obtain the nominal cost of capital, multiply by 12: 12(0.019113) = 22.94%. Check: Old lease terms: N = 60; I/YR = 1.9113; PMT = –2000; FV = 0; PV = ? PV = –$71,039.17. New lease terms: CF0 = 0; CF1-9 = 0; CF10-60 = –2600; I/YR = 1.9113; NPV = ? NPV = –$71,038.98. Except for rounding differences, the costs are the same.

SOLUTION TO SPREADSHEET PROBLEM 10-25 The detailed solution for the spreadsheet problem is in the file Ch 10 Build a Model Solutionxlsx and is available on the textbook’s website.

MINI CASE The detailed solution for the Mini Case is also available in spreadsheet format, in the file Brigham4Ce_MiniCase_Ch10.xlsx, and is available on the textbook’s website. You have just graduated with a business degree from a large university, and one of your favourite courses was ―Today’s Entrepreneurs.‖ In fact, you enjoyed it so much you have decided you want to ―be your own boss.‖ While you were studying, your grandfather died and left you $1 million to do with as you please. You are not an inventor, and you do not have a trade skill that you can market; however, you have decided that you would like to purchase at least one established franchise in the fast-food area, maybe two (if profitable). The problem is that you have never been one to stay with any project for too long, so you figure that your time frame is 3 years. After 3 years you will go on to something else. You have narrowed your selection down to two choices: (1) Franchise L, Lisa’s Soups, Salads, & Stuff, and (2) Franchise S, Sam’s Fabulous Fried Chicken. The net cash flows shown below include the price you would receive for selling the franchise in Year 3 and the forecast of how each franchise will do over the 3-year period. Franchise L’s cash flows will start off slowly but will increase rather quickly as people become more health conscious, while Franchise S’s cash flows will start off high but will trail off as other chicken competitors enter the marketplace and as people become more health conscious and avoid fried foods. Franchise L serves breakfast and lunch, while Franchise S serves only dinner, so it is possible for you to invest in both franchises. You see these franchises as perfect complements to each another: You could attract both the lunch and dinner crowds and the health-conscious and not so health-conscious crowds without the franchises directly competing against each another. Here are the net cash flows (in thousands of dollars):

0 1 2 3

($100) 10 60 80

Expected Net Cash Flow YearFranchise L Franchise S ($100) 70 50 20

Depreciation, salvage values, net working capital requirements, and tax effects are all included in these cash flows. You also have made subjective risk assessments of each franchise, and concluded that both franchises have risk characteristics that require a return of 10%. You must now determine whether one or both of the franchises should be accepted. a.

What is capital budgeting?

Answer: Capital budgeting is the process of analyzing additions to fixed assets. Capital budgeting is important because, more than anything else, fixed asset investment decisions chart a company’s course for the future. Conceptually, the capital budgeting process is identical to the decision process used by individuals making investment decisions. These steps are involved: 1. Estimate the cash flows—interest and maturity value or dividends in the case of bonds and stocks, operating cash flows in the case of capital projects. 2. Assess the riskiness of the cash flows. 3. Determine the appropriate discount rate, based on the riskiness of the cash flows and the general level of interest rates. This is called the project cost of capital in capital budgeting. 4. Evaluate the cash flows. b.

What is the difference between independent and mutually exclusive projects?

Answer: Projects are independent if the cash flows of one are not affected by the acceptance of the other. Conversely, two projects are mutually exclusive if acceptance of one impacts adversely the cash flows of the other; that is, at most one of two or more such projects may be accepted. Put another way, when projects are mutually exclusive, it means that they do the same job. For example, a forklift truck versus a conveyor system to move materials, or a bridge versus a ferry boat.

1. Define the term ―net present value (NPV).‖ What is each franchise’s NPV?

Answer: The net present value (NPV) is simply the sum of the present values of a project’s cash flows: n

NPV = 

CFt

t t  0 (1  r )

Franchise L’s NPV is $18.79: 0 |

10%

(100.00) 10 9.09 49.59 60.11 18.79 = NPVL

NPVs are easy to determine using a calculator with an NPV function. Enter the cash flows sequentially, with outflows entered as negatives; enter the cost of capital; and then press the NPV button to obtain the project’s NPV, $18.78 (note the penny rounding difference). The NPV of Franchise S is NPVS = $19.98. c.

2. What is the rationale behind the NPV method? According to NPV, which franchise or franchises should be accepted if they are independent? Mutually exclusive?

Answer: The rationale behind the NPV method is straightforward: if a project has NPV = $0, then the project generates exactly enough cash flows to (1) recover the cost of the investment and (2) enable investors to earn their required rates of return (the opportunity cost of capital). If NPV = $0, then in a financial (but not an accounting) sense, the project breaks even. If the NPV is positive, then more than enough cash flow is generated, and conversely if NPV is negative. Consider Franchise L’s cash inflows, which total $150. They are sufficient to (1) return the $100 initial investment, (2) provide investors with their 10% aggregate opportunity cost of capital, and (3) still have $18.78 left over on a present value basis. This $18.78 excess PV belongs to the shareholders—the debtholders’ claims are fixed, so the shareholders’ wealth will be increased by $18.78 if Franchise L is accepted. Similarly, shareholders gain $19.98 in value if Franchise S is accepted. If Franchises L and S are independent, then both should be accepted, because they both add to shareholders’ wealth, hence to the stock price. If the franchises are mutually exclusive, then Franchise S should be chosen over L, because S adds more to the value of the firm. Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc. 11-289

3. How would the NPVs change if the cost of capital changed?

Answer: The NPV of a project is dependent on the cost of capital used. Thus, if the cost of capital changed, the NPV of each project would change. NPV declines as r increases, and NPV rises as r falls. d.

1. Define the term ―internal rate of return (IRR).‖ What is each franchise’s IRR?

Answer: The internal rate of return (IRR) is that discount rate that forces the NPV of a project to equal zero: 0

CF0 CF1 CF2 PVCF1 PVCF2 PVCF3 0 = SUM OF PVs = NPV.

CF3

IRR

Expressed as an equation, we have: n

IRR:

t = $0 = NPV.  (1  IRR )t

t 0

Note that the IRR equation is the same as the NPV equation, except that, to find the IRR, the equation is solved for the particular discount rate, IRR, that forces the project’s NPV to equal zero (the IRR) rather than using the cost of capital (r) in the denominator and finding NPV. Thus, the two approaches differ in only one respect: in the NPV method, a discount rate is specified (the project’s cost of capital) and the equation is solved for NPV, while in the IRR method, the NPV is specified to equal zero and the discount rate (IRR) that forces this equality is found.

Franchise L’s IRR is 18.1%: 0 18.1%

–100.00 10 60 80 8.47 43.02 48.57 $ 0.06 ≈ $0 if IRRL = 18.1% is used as the discount rate. therefore, IRRL ≈ 18.1%. A financial calculator is extremely helpful when calculating IRRs. The cash flows are entered sequentially, and then the IRR button is pressed. For Franchise S, IRRS ≈ 23.6%. Note that with many calculators, you can enter the cash flows into the cash flow register, also enter r = i, and then calculate both NPV and IRR by pressing the appropriate buttons. d.

2. How is the IRR on a project related to the YTM on a bond?

Answer: The IRR is to a capital project what the YTM is to a bond. It is the expected rate of return on the project, just as the YTM is the promised rate of return on a bond. d.

3. What is the logic behind the IRR method? According to IRR, which franchises should be accepted if they are independent? Mutually exclusive?

Answer: IRR measures a project’s profitability in the rate of return sense: if a project's IRR equals its cost of capital, then its cash flows are just sufficient to provide investors with their required rates of return. An IRR greater than r implies an economic profit, which accrues to the firm’s shareholders, while an IRR less than r indicates an economic loss, or a project that will not earn enough to cover its cost of capital. Projects’ IRRs are compared to their costs of capital, or hurdle rates. Since Franchises L and S both have a hurdle rate of 10%, and since both have IRRs greater than that hurdle rate, both should be accepted if they are independent. However, if they are mutually exclusive, Franchise S would be selected, because it has the higher IRR.

4. How would the franchises’ IRRs change if the cost of capital changed?

Answer: IRRs are independent of the cost of capital. Therefore, neither IRRS nor IRRL would change if r changed. However, the acceptability of the franchises could change—L would be rejected if r were above 18.1%, and S would also be rejected if r were above 23.6%.

1. Draw NPV profiles for Franchises L and S. At what discount rate do the profiles cross?

Answer: The NPV profiles are plotted in the figure below. Note the following points: 1. The y-intercept is the project’s NPV when r = 0%. This is $50 for L and $40 for S. 2. The x-intercept is the project’s IRR. This is 18.1% for L and 23.6% for S. 3. NPV profiles are curves rather than straight lines. To see this, note that these profiles approach cost = –$100 as r approaches infinity. 4. From the figure that follows, it appears that the crossover point is between 8% and 9%. The actual value is approximately 8.7%. One can calculate the crossover rate by (1) going back to the data on the problem, finding the cash flow differences for each year, (2) entering those differences into the cash flow register, and (3) pressing the IRR button to get the crossover rate, 8.68% ≈ 8.7%.

60 50

Crossover Point = 8.7%

40 30 20

IRRS = 23.6%

Discount Rate (%)

0 0

2 3 .6

-1 0

IRRL = 18.1% r NPVL NPVS 0% $50 $40 5 33 29 10 19 20 15 7 12 20 (4) 5 e.

2. Look at your NPV profile graph without referring to the actual NPVs and IRRs. Which franchise or franchises should be accepted if they are independent? Mutually exclusive? Explain. Are your answers correct at any cost of capital less than 23.6%?

Answer: The NPV profiles show that the IRR and NPV criteria lead to the same accept/reject decision for any independent project. Consider Franchise L. It intersects the x-axis at its IRR, 18.1%. According to the IRR rule, L is acceptable if r is less than 18.1%. Also, at any r less than 18.1%, L’s NPV profile will be above the x axis, so its NPV will be greater than $0. Thus, for any independent project, NPV and IRR lead to the same accept/reject decision. Now assume that L and S are mutually exclusive. In this case, a conflict might arise. First, note that IRRS = 23.6% > 18.1% and, therefore, regardless of the size of r, project S would be ranked higher by the IRR criterion. However, the NPV profiles show that NPVL > NPVS if r is less than 8.7%. Therefore, for any r below the 8.7% crossover rate, say r = 7%, the NPV rule says choose L, but the IRR rule says choose S. Thus, if r is less than the crossover rate, a ranking conflict occurs.

1. What is the underlying cause of ranking conflicts between NPV and IRR?

Answer: For normal projects’ NPV profiles to cross, one project must have both a higher vertical axis intercept and a steeper slope than the other. A project’s vertical axis intercept typically depends on (1) the size of the project and (2) the size and timing pattern of the cash flows—large projects, and ones with large distant cash flows, would generally be expected to have relatively high vertical axis intercepts. The slope of the NPV profile depends entirely on the timing pattern of the cash flows—longterm projects have steeper NPV profiles than short-term ones. Thus, we conclude that NPV profiles can cross in two situations: (1) when mutually exclusive projects differ in scale (or size) and (2) when the projects’ cash flows differ in terms of the timing pattern of their cash flows (as for Franchises L and S). f.

2. What is the ―reinvestment rate assumption,‖ and how does it affect the NPV versus IRR conflict?

Answer: The reinvestment rate assumption is behind the ranking conflicts between projects of different sizes and/or different timing patterns of cash flows. All DCF methods implicitly assume that cash flows can be reinvested at some rate, regardless of what is actually done with the cash flows. Discounting is the reverse of compounding. Since compounding assumes reinvestment, so does discounting. NPV and IRR are both found by discounting, so they both implicitly assume some discount rate. Inherent in the NPV calculation is the assumption that cash flows can be reinvested at the project’s cost of capital, while the IRR calculation assumes reinvestment at the IRR rate. f. 3. Which method is the best? Why? Answer: Whether NPV or IRR gives better rankings depends on which has the better reinvestment rate assumption. Normally, the NPV’s assumption is better. The reason is as follows: a project’s cash inflows are generally used as substitutes for outside capital, that is, projects’ cash flows replace outside capital and, hence, save the firm the cost of outside capital. Therefore, in an opportunity cost sense, a project’s cash flows are reinvested at the cost of capital. To see this graphically, think of the following situation: assume the firm’s cost of capital is a constant 10% within the relevant range of financing considered, and it has projects available as shown in the graph below:

Percent 25

IRRA = 25%

IRRB = 20%

IRRC = 15% IRRD = 12% MCC

IRRE = 8%

IRRF = 5%

Dollars Raised and Invested

What projects will be accepted, by either NPV or IRR? Projects A, B, C, and D. If the same situation exists year after year, at what rate of return will cash flows from earlier years’ investments be reinvested? Capital budgeting decisions are made in this sequence: (1) the company would say, ―We can take on A, B, C, and D and finance them with 10% money, so let’s do it.‖ (2) Then, it would get cash flows from earlier years’ projects. What would it do with those cash flows? It would use them in lieu of raising money that costs 10%, so it would save 10%. Therefore, 10% is the opportunity cost of the cash flows. In effect, cash flows are reinvested at the 10% cost of capital. Note, however, that NPV and IRR always give the same accept/reject decisions for independent projects, so IRR can be used just as well as NPV when independent projects are being evaluated. The NPV versus IRR conflict arises only if mutually exclusive projects are involved.

1. Define the term ―modified IRR (MIRR).‖ Find the MIRRs for Franchises L and S.

Answer: MIRR is the discount rate that equates the present value of the terminal value of the inflows, compounded at the cost of capital, to the present value of the costs. Here is the setup for calculating Franchise L’s modified IRR: 0 r =10% 1 | | 10% PV of costs = (100.00) 10

80.00 66.00 12.10 TV of inflows = 158.10

MIRR = ?

PV of TV = 100.00 = $100 =

$158.10 (1  MIRR ) 3

. n

PV costs =

TV (1  MIRR )

= 

 CIFt (1  r ) n  t

COFt

= t 1 (1  MIRR ) n t  0 (1  r ) t

After you calculate the TV, enter N = 3, PV = –100, PMT = 0, FV = 158.1, and then press i to get the answer, MIRRL = 16.5%. Similarly, MIRRS = 16.9%. Thus, Franchise S is ranked higher than L. This result is consistent with the NPV decision. g.

2. What are the MIRR’s advantages and disadvantages vis-à-vis the regular IRR? What are the MIRR’s advantages and disadvantages vis-à-vis the NPV?

Answer: MIRR is a better rate of return measure than IRR for two reasons. (1) It correctly assumes reinvestment at the project’s cost of capital rather than at its IRR. (2) MIRR avoids the problem of multiple IRRs—there can be only one MIRR for a given project. MIRR does not always lead to the same decision as NPV when mutually exclusive projects are being considered. In particular, small projects often have a higher MIRR, but a lower NPV, than larger projects. Thus, MIRR is not a perfect substitute for NPV, and NPV remains the single best decision rule. However, MIRR is superior to the regular IRR, and if a rate of return measure is needed, MIRR should be used.

As a separate project (Project P), you are considering sponsoring a pavilion at the upcoming World’s Fair. The pavilion would cost $800,000, and it is expected to result in $5 million of incremental cash inflows during its first year of operation. However, it would then take another year, and $5 million of costs, to demolish the site and return it to its original condition. Thus, Project P’s expected net cash flows look like this (in millions of dollars): Year 0 1 2

Net Cash Flows ($0.8) 5.0 (5.0)

The project is estimated to be of average risk, so its cost of capital is 10%. h.

1. What are normal and non-normal cash flows?

Answer: Normal cash flows begin with a negative cash flow (or a series of negative cash flows), switch to positive cash flows, and then remain positive. They have only one change in sign. (Note: Normal cash flows can also start with positive cash flows, switch to negative cash flows, and then remain negative.) Non-normal cash flows have more than one sign change. For example, they may start with negative cash flows, switch to positive, and then switch back to negative. Projects with normal cash flows have outflows, or costs, in the first year (or years) followed by a series of inflows. Projects with non-normal cash flows have one or more outflows after the inflow stream has begun. Here are some examples: Inflow (+) or Outflow (–) in Year Normal

Non-normal

0 – – –

1 2 3 4 5 + + + + + – + + + + – – + + +

– – –

+ + +

+ + – + – –

2. What is Project P’s NPV? What is its IRR? Its MIRR?

Answer: Here is the time line for the cash flows, and the NPV: 0

10%

–800,000

5,000,000

–5,000,000

NPVP = –$386,776.86. We can find the NPV by entering the cash flows into the cash flow register, entering i = 10, and then pressing the NPV button. However, calculating the IRR presents a problem. With the cash flows in the register, press the IRR button. An HP-10B financial calculator will give the message ―error-soln.‖ This means that Project P has multiple IRRs. An HP-17B will ask for a guess. If you guess 10%, the calculator will produce IRR = 25%. If you guess a high number, such as 200%, it will produce the second IRR, 400%.1 The MIRR of Project P = 5.6%, and is found by computing the discount rate that equates the terminal value ($5.5 million) to the present value of cost ($4.93 million). NPV 500

375

250

125

Cost of Capital, r (%) -125

100

200

300

400

500

600

-250

-375

Looking at the figure, if you guess an IRR to the left of the peak NPV rate, the lower IRR will appear. If you guess IRR > peak NPV rate, the higher IRR will appear.

3. Draw Project P’s NPV profile. Does Project P have normal or non-normal cash flows? Should this project be accepted?

Answer: You could put the cash flows in your calculator and then enter a series of i values, get an NPV for each, and then plot the points to construct the NPV profile. We used a spreadsheet program to automate the process and then to draw the profile. Note that the profile crosses the x-axis twice, at 25% and at 400%, signifying two IRRs. Which IRR is correct? In one sense, they both are—both cause the project’s NPV to equal zero. However, in another sense, both are wrong—neither has any economic or financial significance. Project P has non-normal cash flows; that is, it has more than one change of signs in the cash flows. Without this non-normal cash flow pattern, we would not have the multiple IRRs. Since Project P’s NPV is negative, the project should be rejected, even though both IRRs (25% and 400%) are greater than the project’s 10% cost of capital. The MIRR of 5.6% also supports the decision that the project should be rejected. i.

What does the profitability index (PI) measure? What are the PIs of Franchises S and L?

Answer: The PI is equal to the present value of all future cash flows divided by the initial cost. It measures the ―bang for the buck.‖ PIS = $119.98/$100 = 1.1998. PIL = $118.78/$100 = 1.1878.

1. What is the payback period? Find the regular and discounted paybacks for Franchises L and S.

Answer: The payback period is the expected number of years required to recover a project’s cost. We calculate the payback by developing the cumulative cash flows as shown below for Franchise L (in thousands of dollars):

Year 0 1 2 3

Expected NCF Annual Cumulative ($100) ($100) 10 (90) 60 (30) 80 50

Payback is between t = 2 and t = 3

| –100

| 10 –90

| 60 –30

| 80 +50

Franchise L’s $100 investment has not been recovered at the end of Year 2, but it has been more than recovered by the end of Year 3. Thus, the recovery period is between 2 and 3 years. If we assume that the cash flows occur evenly over the year, then the investment is recovered $30/$80 = 0.375 ≈ 0.4 into Year 3. Therefore, paybackL = 2.4 years. Similarly, paybackS = 1.6 years. The discounted payback for L (as shown in (j 3)) is 2.7 years. Similarly, discounted paybackS = 1.9 years. j.

2. What is the rationale for the payback method? According to the payback criterion, which franchise or franchises should be accepted if the firm’s maximum acceptable payback is 2 years, and if Franchises L and S are independent? If they are mutually exclusive?

Answer: Payback represents a type of “breakeven” analysis: the payback period tells us when the project will break even in a cash flow sense. With a required payback of 2 years, Franchise S is acceptable, but Franchise L is not. Whether the two projects are independent or mutually exclusive makes no difference in this case.

3. What is the difference between the regular and discounted payback periods?

Answer: Discounted payback is similar to payback except that discounted rather than raw cash flows are used. Setup for Franchise L’s discounted payback, assuming a 10% cost of capital: Year 0 1 2 3

Expected Net Cash Flows Raw Discounted ($100) ($100.00) 10 9.09 60 49.59 80 60.11

Cumulative ($100.00) (90.91) (41.32) 18.79

Discounted PaybackL = 2 + ($41.32/$60.11) = 2.69 = 2.7 years. Similarly, discounted paybackS = 1.9 years.

4. What is the main disadvantage of discounted payback? Is the payback method of any real usefulness in capital budgeting decisions?

Answer: Regular payback has two critical deficiencies: (1) it ignores the time value of money, and (2) it ignores the cash flows that occur after the payback period. Discounted payback does consider the time value of money, but it still fails to consider cash flows after the payback period; hence it has a basic flaw. In spite of its deficiency, many firms today still calculate the discounted payback and give some weight to it when making capital budgeting decisions. However, payback is not generally used as the primary decision tool. Rather, it is used as a rough measure of a project’s liquidity and riskiness.

In an unrelated analysis, you have the opportunity to choose between the following two mutually exclusive projects: Expected Net Cash Flows Year Project S Project L 0 ($100,000) ($100,000) 1 60,000 33,500 2 60,000 33,500 3 -33,500 4 -33,500 The projects provide a necessary service, so whichever one is selected is expected to be repeated into the foreseeable future. Both projects have a 10% cost of capital.

1. What is each project's initial NPV?

Answer: The NPVs, found with a financial calculator, are calculated as follows: Input the following: CF0 = –100000, CF1 = 60000, NJ = 2, and I = 10 to solve for NPVS = $4,132.23 ≈ $4,132. Input the following: CF0 = –100000, CF1 = 33500, NJ = 4, and I = 10 to solve for NPVL = $6,190.49 ≈ $6,190. However, if we make our decision based on the raw NPVs, we would be biasing the decision against the shorter project. Since the projects are expected to be replicated, if we initially choose Project S, it would be repeated after 2 years. However, the raw NPVs do not reflect the replication cash flows. k.

2. What is each project’s equivalent annual annuity?

Answer: We begin with the NPVs found in the previous step. We then find the annuity payment stream that has the same present value as follows: For Project S, input the following: N = 2, I/YR = 10, PV = −4,132.23, FV = 0, and solve for PMT = EAA = $2,380.95. For Project L, input the following: N = 4, I/YR = 10, PV = −6,190.49, FV = 0, and solve for PMT = EAA = $1,952.92. Project S is preferred because it has a higher EAA.

You are also considering another project that has a physical life of 3 years; that is, the machinery will be totally worn out after 3 years. However, if the project were terminated prior to the end of 3 years, the machinery would have a positive salvage value. Here are the project’s estimated cash flows: Initial Investment and Operating Year 0 1 2 3

($5,000) 2,100 2,000 1,750

End-of-Year Net Salvage Cash Flows $5,000 3,100 2,000 0

Value_

Using the 10% cost of capital, what is the project’s NPV if it is operated for the full 3 years? Would the NPV change if the company planned to terminate the project at the end of Year 2? At the end of Year 1? What is the project’s optimal (economic) life?

Answer: Here are the time lines for the three alternative lives: No termination: 0 |

10%

–5,000

2,100

2,000

1,750 0 1,750

NPV = –$123. Terminate after 2 years: 0 |

10%

–5,000

2,100

2,000 2,000 4,000

NPV = $215. Terminate after 1 year: 0 | –5,000

10%

1 |

2,100 3,100 5,200

NPV = –$273. We see that (1) the project is acceptable only if operated for 2 years, and (2) a project’s engineering life does not always equal its economic life.

After examining all the potential projects, you discover that there are many more projects this year with positive NPVs than in a normal year. What two problems might this extra-large capital budget cause?

You only have a limited amount of capital to commit to projects. If you have to raise external capital to fund some of these other positive NPV projects, then you may be faced with an increasing cost of capital. This is called an increasing marginal cost of capital schedule, and it also happens to companies when they exhaust their internal sources of funds and have to go to external capital markets for their funding. This increased cost of capital may cause you to reject projects that you might otherwise accept because with your increased cost of capital, some projects may show a negative NPV when they would otherwise have a positive NPV in a normal year. Another effect of this large capital budget is that you may choose to ration capital—i.e., not fund all of the projects. This is called capital rationing, and companies and investors do this when for whatever reason they put a cap on the funds they are willing to invest in new projects.

Chapter 11 Cash Flow Estimation and Risk Analysis ANSWERS TO END-OF-CHAPTER QUESTIONS 11-1

a. Cash flow, which is the relevant financial variable, represents the actual flow of cash. Accounting income, on the other hand, reports accounting data as defined by the International Financial Reporting Standards (IFRS). b. Incremental cash flows are those cash flows that arise solely from the asset that is being evaluated. For example, assume an existing machine generates revenues of $1,000 per year and expenses of $600 per year. A machine being considered as a replacement would generate revenues of $1,000 per year and expenses of $400 per year. On an incremental basis, the new machine would not increase revenues at all, but would decrease expenses by $200 per year. Thus, the annual incremental cash flow is a before-tax savings of $200. A sunk cost is one that has already occurred and is not affected by the capital project decision. Sunk costs are not relevant to capital budgeting decisions. Within the context of this chapter, an opportunity cost is a cash flow that a firm must forgo to accept a project. For example, if the project requires the use of a building that could otherwise be sold, the market value of the building is an opportunity cost of the project.

c. Net operating working capital changes are the increases in current operating assets resulting from accepting a project less the resulting increases in current operating liabilities, or accruals and accounts payable. A net operating working capital change must be financed just as a firm must finance its increases in fixed assets. Salvage value is the market value of an asset after its useful life. Salvage values and their tax effects must be included in project cash flow estimation. d. An externality is something that is external to the project but occurs because of the project. For example, cannibalization occurs when a project’s product reduces the company’s sales of similar products.

e. Sensitivity analysis indicates exactly how much NPV or other output variables such as IRR or MIRR will change in response to a given change in an input variable, other things held constant. Sensitivity analysis is sometimes called ―what if‖ analysis because it answers this type of question. Scenario analysis is a shorter version of simulation analysis that uses only a few outcomes. Often the outcomes considered are optimistic, pessimistic, and most likely. Monte Carlo simulation analysis is a risk analysis technique in which a computer is used to simulate probable future events and thus to estimate the profitability and risk of a project. f. A risk-adjusted discount rate incorporates the risk of the project’s cash flows. The cost of capital to the firm reflects the average risk of the firm’s existing projects. Thus, new projects that are riskier than existing projects should have a higher riskadjusted discount rate. Conversely, projects with less risk should have a lower riskadjusted discount rate. This adjustment process also applies to a firm’s divisions. Risk differences are difficult to quantify, thus risk adjustments are often subjective in nature. A project’s cost of capital is its risk-adjusted discount rate for that project. g. Real options occur when managers can influence the size and risk of a project’s cash flows by taking different actions during the project’s life. They are referred to as real options because they deal with real as opposed to financial assets. They are also called managerial options because they give opportunities to managers to respond to changing market conditions. Sometimes they are called strategic options because they often deal with strategic issues. Finally, they are also called embedded options because they are a part of another project. h. Investment timing options give companies the option to delay a project rather than implement it immediately. This option to wait allows a company to reduce the uncertainty of market conditions before it decides to implement the project. Growth options allow a company to expand if market demand is higher than expected. This includes the opportunity to expand into different geographic markets and the opportunity to introduce complementary or second-generation products. An abandonment option includes the option to abandon a project if market conditions deteriorate too much. A flexibility option is the option to modify operations depending on how conditions develop during a project’s life. 11-2

Only cash can be spent or reinvested, and since accounting profits do not represent cash, they are of less fundamental importance than cash flows for investment analysis. Recall that in the stock valuation chapters we focused on dividends and free cash flows, which represent cash flows, rather than on earnings per share, which represent accounting profits.

11-3

Since the cost of capital includes a premium for expected inflation, failure to adjust cash flows means that the denominator, but not the numerator, rises with inflation, and this lowers the calculated NPV.

11-4

Capital budgeting analysis should include only those cash flows that will be affected by the decision. Sunk costs are unrecoverable and cannot be changed, so they have no bearing on the capital budgeting decision. Opportunity costs represent the cash flows the firm gives up by investing in this project rather than its next best alternative, and externalities are the cash flows (both positive and negative) to other projects that result from the firm taking on this project. These cash flows occur only because the firm took on the capital budgeting project; therefore, they must be included in the analysis.

11-5

When a firm takes on a new capital budgeting project, it typically must increase its investment in receivables and inventories, over and above the increase in payables and accruals, thus increasing its net operating working capital. Since this increase must be financed, it is included as an outflow in Year 0 of the analysis. At the end of the project’s life, inventories are depleted and receivables are collected. Thus, there is a decrease in NOWC, which is treated as an inflow.

11-6

Scenario analysis analyzes a limited number of outcomes. Although the base-case scenario may be the most likely, or expected outcome, the bad and good scenarios are frequently worst-case and best-case scenarios, that is, when everything goes bad together, or everything goes right together. It is unlikely that everything will go wrong all at once, or everything will go right all at once, and so scenario analysis can tend to overestimate the riskiness of a project. Simulation analysis, on the other hand, allows the variables being simulated to vary either together or independently, as the modeller sees fit. With enough runs of the simulation, this procedure should provide a reasonably accurate description of the possible outcomes. Note, however, that if the project is big and its failure could threaten the viability of the firm, then evaluating a worst-case scenario may very well be important! A simulation may only identify that worst-case outcome infrequently, and with a scenario analysis you can specify the worst case and see how much it financially damages the company.

11-7

In replacement projects, the benefits are generally cost savings, although the new machinery may also permit additional output. The data for replacement analysis are generally easier to obtain than for new products, but the analysis itself is somewhat more complicated because almost all of the cash flows are incremental, found by subtracting the new cost numbers from the old numbers. Similarly, differences in depreciation and any other factor that affects cash flows must also be determined.

11-8

The costs associated with financing are reflected in the weighted average cost of capital. To include interest expense in the capital budgeting analysis would ―double count‖ the cost of debt financing.

11-9

Stand-alone risk is the project’s risk if it is held as a lone asset. It disregards the fact that it is but one asset within the firm’s portfolio of assets and that the firm is but one stock in a typical investor’s portfolio of stocks. Stand-alone risk is measured by the variability of the project’s expected returns. Corporate, or within-firm, risk is the project’s risk to the corporation, giving consideration to the fact that the project represents only one in the firm’s portfolio of assets, hence some of its risk will be eliminated by diversification within the firm. Corporate risk is measured by the project’s impact on uncertainty about the firm’s future earnings. Market, or beta, risk is the riskiness of the project as seen by well-diversified shareholders who recognize that the project is only one of the firm’s assets and that the firm’s stock is but one small part of their total portfolios. Market risk is measured by the project’s effect on the firm’s beta coefficient.

SOLUTIONS TO END-OF-CHAPTER PROBLEMS 11-1

Equipment NWC investment Initial investment outlay

$ 9,000,000 3,000,000 $12,000,000

11-2

a. Equipment NWC investment Initial investment outlay

$ 43,000,000 7.000,000 $50,000,000

b. No, last year’s $850,000 expenditure is considered a sunk cost and does not represent an incremental cash flow. Hence, it should not be included in the analysis. c. The potential sale of the building represents an opportunity cost of conducting the project in that building. Therefore, the possible after-tax sale price must be charged against the project as a cost.

11-3

Operating Cash Flows: t = 1 Sales revenues Operating costs Capital cost allowance (CCA) Operating income before taxes Taxes (30%) Operating income after taxes Add back CCA Operating cash flow

$18,000,000 9,000,000 4,000,000 5,000,000 1,500,000 3,500,000 4,000,000 $ 7,500,000

11-4 Initial Land investment Building and equipment Environ. cleanup Total

($ 700,000) Cash inflows (1,000,000) (250,000) ($1,950,000)

Blueberry apple product Existing apple product Dried fruit snack Total inflows

$12,000,000 (2,000,000) 1,000,000 $11,000,000

The original land cost is a sunk cost. The current market value is an opportunity cost, which must be considered. The feasibility study cost is a sunk cost. The decrease in apple juice sales is due to cannibalization and should be considered, while the increase in fruit snack sales is due to the complementary nature of the new product and should also be considered. 11-310 Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc.

11-5 a.

Initial cost Retooling cost = Initial cost =

$3,700,000 $3,700,000

The market research, R&D, and testing are all sunk costs, and therefore they are not relevant to the capital budgeting decision. The design of the new HQ and subsequent construction costs are independent of the new project and are therefore also not relevant. b. Annual cash flows before tax Quantity sold × Contribution margin: Lost sales of existing product: Increased fixed costs:

360,000 × $15 = 190,000 × $14 =

$5,400,00 0 (2,660,00 0) (100,000) $2,640,00 0

Annual cash flows before tax:

11-6 Equipment Installation NOWC investment Initial investment outlay

($30,000) (5,000) (6,000) ($41,000)

Annual cash flows t1 – 5 = $10,000 Using a financial calculator, input N = 5, I = 10, PMT = 10000, FV = 0 to solve for PV = $37,908. Terminal year cash flow is the return of the NWC of $6,000. To calculate the present value of the terminal cash inflows, input N = 5, I = 10, FV = 6000, PMT = 0, to solve for PV = $3,726.

Project’s NPV is $634, therefore proceed with the project.

11-7

= Initial Outlay + PV Project cash flows NPV = ($41,000) + $37,908 = $634. a. The present value of the CCA tax shield is $10,000 0.2 0.30 0 0 0 0

+ +

PV of terminal cash flows $3,726

,000

b. Twenty percent straight line depreciation is the same as allocating 1/5 of the cost annually, therefore: $10,000/5 = $2,000 CCA per year for 5 years. Annual CCA tax shield = $2,000 × 0.30 = $600. To find the PV of the annual tax shield, input N = 5, I = 10, PMT = 600, FV = 0, to solve for PV = $2,274. c. The timing of the tax shield cash flows accounts for the difference in the two present values. With the straight-line depreciation, all the tax shield cash flows are accounted for in the first 5 years. Although the tax shield cash flows are higher in the first few years for the declining balance method, the cash flows go out much further than 5 years (to perpetuity), thus making those distant cash flows worth much less at time zero. 11-8

Initial outlay = $110,000 Annual after-tax cash flows t = 1 – 10 = $19,000 N = 10, I/Y = 10, PMT = 19000, FV = 0, CPT PV = ? = $116,747 NPV

= = =

11-9

Initial + PV Project investment cash flows ($110,000) + $116,747 $6,747 Buy the machine.

The IRR for Project X is the discount rate that makes the NPV = 0. Therefore, if the IRR = 11.0%, then the present value of the annual payments must equal $200,000, which is the time 0 cash flow. Enter time 0 cash flow PV = –200000, annual cash flows PMT = 38840, ending cash

flow FV = 0, and a discount rate that forces the NPV to be zero, I = 11.0%, solve for N = 8.0 years. 11-10 a. The net cost is $137,000: Price Modification Increase in NOWC Cash outlay for new machine

($120,000) (9,500) (7,500) ($137,000)

b. The operating cash flows follow: Year 1 $31,000 9,300 $21,700

1. Project cash flows before tax 2. Tax (30%) Project cash flows after tax

Year 2 $31,000 9,300 $21,700

Year 3 $31,000 9,300 $21,700

Year 4 $31,000 9,300 $21,700

Using a financial calculator, input N = 4, I = 11, PMT = 21700, FV = 0 to solve for PV = $67,323. c. The present value of the CCA tax shield is 129,500 0.2 0.30 1 + 0.5 0.11 0.11 + 0.20 1 + 0.11 = 25,064.52

0.9504504

60,000 0.2 0.30 0.11+ 0.20 –

11,612.90

1 (1 + 0.11)4 ×

0.65873097

= $16,173 d. The terminal year cash flow is $67,500: Salvage value Return of NWC

$60,000 7,500 $67,500

To calculate the present value of the terminal cash inflows, input N = 4, I = 11, FV = 67500, PMT = 0, to solve for PV = $44,464.

e. The project has an NPV of ($9,040); thus, it should not be accepted. NPV

= = =

Initial investment ($137,000) ($9,040).

+ +

PV Project cash flows $67,323

+ +

PVCCA tax shield $16,173

+ +

PV of ending cash flows $44,464

11-11 The initial investment is $780,000: Equipment cost Setup and installation Investment in NOWC Total

($600,000) (30,000) (50,000) ($680,000)

The present value of the after-tax project cash flows is $583,781: (000s) Sales Less forgone rental income Less costs Project cash flows before tax Less tax (30%) Project cash flows after tax

Year 1 $2,000 80 1,700 220 66 $ 154

Year 2 $2,000 80 1,700 220 66 $ 154

Year 3 $2,000 80 1,700 220 66 $ 154

Year 4 $2,000 80 1,700 220 66 $ 154

Year 5 $2,000 80 1,700 220 66 $ 154

Using a calculator to calculate the present value of the project cash flows after tax input N = 5, I = 10, PMT = 154000, FV = 0 and then solve for PV = $583,781. The present value of the CCA tax shield is $126,924. 630,000 0.30 0.30 1 + 0.5 0.10 600,000 15%) 0.30 ) ( ) ( 0.10 + 0.30 1 + 0.10 0.10 + 0.30 = 141,750 0.954545 – 0, 0 = $ , . (

0.30

) (

(1 + 0.10)5 0.620921

The present value of the ending cash flows is $86,929. Salvage value Return of NOWC Total ending cash flow

$ 90,000 50,000 $140,000

With a calculator, to calculate the present value of the ending cash flows enter N = 5, I = 10, FV = 140000, PMT = 0 and solve for PV = $86,929. The NPV of the project is $113,443; therefore, invest in the project. NPV = ($680,000) + $583,781 + $122,733 + $86,929 = $113,443.

)

11-12 a. Sales = 1,000($138) Costs = 1,000($105) Net before tax Taxes (25%) Net after tax

$138,000 105,000 33,000 8,250 $ 24,750

Not considering inflation, NPV is –$5,000. This value is calculated as $170,000 +

$24,750 = $5,000 0.15

Considering inflation, the real cost of capital is calculated as follows: (1 + rr)(1 + i) = 1.15 (1 + rr)(1.06) = 1.15 rr = 0.0849. Thus, the NPV considering inflation is calculated as $170,000 +

$24,750 = $121,519 0.0849

After adjusting for expected inflation, we see that the project has a positive NPV and should be accepted. This demonstrates the bias that inflation can induce into the capital budgeting process: Inflation is already reflected in the denominator (the cost of capital), so it must also be reflected in the numerator. b. If part of the costs were fixed, and hence did not rise with inflation, then sales revenues would rise faster than total costs. However, when the plant wears out and must be replaced, inflation will cause the replacement cost to jump, necessitating a sharp output price increase to cover the now higher depreciation charges.

11-13

Sales Less variable cost Contribution margin Less fixed costs Cash flow before tax Less tax Cash flow after tax + CCA tax shield1 Project cash flows

Base Case $6,750,000 3,600,000 3,150,000 1,400,000 1,750,000 525,000 1,225,000 288,000 $1,513,000

Δ Price Δ Quantity $6,412,500 $6,412,500 3,600,000 3,420,000 2,812,500 2,992,500 1,400,000 1,400,000 1,412,500 1,592,500 423,750 477,750 988,750 1,114,750 288,000 288,000 $1,276,750 $1,402,750

Δ Variable cost $6,750,000 3,780,000 2,970,000 1,400,000 1,570,000 471,000 1,099,000 288,000 $1,387,000

PV5 years, 11.5% r – Investment NPV

$5,522,265 $4,800,000 $722,265

$4,659,982 $5,119,866 $4,800,000 $4,800,000 $(140,018) $319,866

$5,062,381 $4,800,000 $262,381

For a 5% decrease in selling price NPV falls by $862,283 ($722,265 – ($140,018)). For a 5% decrease in quantity sold NPV falls by $402,399 ($722,265 – $319,866). For a 5% increase in variable cost NPV falls by $459,884 ($722,265 – $262,381). 1

CCA tax shield = (Capital cost/5) × tax rate = ($4,800,000/5) × 0.30 = $288,000

11-14 a.

The initial investment is ($890,000) + ($30,000) = ($920,000). The present value of the project’s cash flows is $763,516. Sales Delivery revenue Variable costs Project cash flows before tax Taxes Project cash flows after tax

$500,000 39,000 (300,000) 239,000 (71,700) $167,300

Using a calculator to calculate the present value of the project cash flows after tax, input N = 7, I = 12, PMT = 167300, FV = 0 and then solve for PV = $763,516. The present value of the CCA tax shield is $176,620. (

0,000 0

0 0 0 0 ) 0 0 (

0, ,

40,000 0

(

)

0 0 0 0 ) 0 0 0 4 4

( ( –

)

The present value of the ending cash flows is $31,664. Salvage value Return of NOWC Total ending cash flow

$40,000 30,000 $70,000

With a calculator, to calculate the present value of the ending cash flows enter N = 7, I = 12, FV = 70000, PMT = 0 and solve for PV = $31,664. The NPV of the project is $51,800; therefore, invest in the project. NPV = ($920,000) + $763,516 + $176,620 + $31,664 = $51,800.

b. The only item to be recalculated is the present value of the CCA tax shield. (

0,000 0

0 0 0 0 ) 0 0 ( ,

40,000 0

(

)

0 0 0 0 ) 0 0 0 4 4

( –

(

0 ,

)

) 04

,4 0 The new NPV is $74,590. NPV = ($920,000) + $763,516 + $199,410 + $31,664 = $74,590.

11-15 The initial investment is $182,500: Purchase new machine Sale of old machine Total investment

($182,500) 0 ($182,500)

The present value of the after-tax project cash flows is $175,109: Years 1–8 New cash flow before depreciation $ 74,000 Old cash flow before depreciation (27,000) Increase in earnings before depreciation 47,000 Taxes (25%) (11,750) After-tax increase in cash flow $ 35,250 Using a calculator to calculate the present value of the project cash flows after tax, input N = 8, I = 12, PMT = 35250, FV = 0 and then solve for PV = $175,109. The present value of the CCA tax shield is $30,589. (

182,500 0.30 0.25 1 + 0.5 0.12 ) ( ) 0.12 + 0.30 1 + 0.12

32,589

0.94643

= $30,843. The NPV of the project is $23,452, therefore purchase the new machine. NPV = ($182,500) + $175,109 + $30,843 = $23,452

11-16 a.

The initial investment is $124,000: Equipment cost Installation and modification Investment in NOWC

$100,000 20,000 4,000

Total

$124,000

The present value of the after-tax project cash flows is $86,868: Year 1 Year 2 Year 3 Project cash flows before tax $40,000 $40,000 $40,000 Taxes (25%) (10,000) (10,000) (10,000) Net working capital (2,000) (2,000) (2,000)

Year 4 $40,000 (10,000) (2,000)

Project cash flows after tax

$28,000

Using a calculator to calculate the present value of the project cash flows after tax, input N = 4, I = 11, PMT = 28000, FV = 0 and then solve for PV = $86,868. The present value of the CCA tax shield is $20,863. (

0,000 0 0 0 0 0 0 ,

)

( 0

0 0

)

04 04

0, The present value of the ending cash flows is $7,905: Salvage value Return of NOWC Total ending cash flow

0 12,000 $12,000

Using a calculator, calculate the present value of the ending cash flows. Enter N = 4, I = 11, FV = 12000, PMT = 0 and solve for PV = $7,905. The NPV of the project is $(8,364); therefore do not invest in the new market. NPV = ($124,000) + $86,868 + $20,863 + $7,905 = ($8,364).

b. The initial cost and present value of the project cash flows are not affected by the salvage value and therefore those costs remain the same. We need to recalculate the present value of the CCA tax shield and the present value of the ending cash flows. The present value of the CCA tax shield is $18,453: 120,000 0.30 0.25 1 + 0.5 0.11 20,000 0.30 0.25 1 ( ) ( ) ( ) ( ) 0.11 + 0.30 1 + 0.11 0.11 + 0.30 (1 + 0.11)4 =

21,951

0.95045045

3,659

0.658730974

= $18,453. The present value of the ending cash flows is $21,079: Salvage value Return of NOWC Total ending cash flows

$20,000 12,000 $32,000

With a calculator, to calculate the present value of the ending cash flows enter N = 4, I = 11, FV = 32000, PMT = 0 and solve for PV = $21,079. The NPV of the project is $2,400; therefore, invest in the new market. NPV = ($124,000) + $86,868 + $18,453 + $21,079 = $2,400. c. With no salvage value and a new CCA rate, we need to recalculate the PV of the CCA tax shield. 120,000 0.10 0.25 1 + 0.5 0.11 ( ) ( ) 0.11 + 0.10 1 + 0.11 =

14,286

0.95045045 = $13,578.

The NPV of the project is ($15,649); therefore, do not invest in the new market. NPV = ($124,000) + $86,868 + $13,578 + $7,905 = ($15,649).

11-17 Initial investment: Purchase new machine Sale of old machine Change in net operating working capital* Total investment *

($12,000) 2,500 (2,000) ($11,500)

Change in net working capital = $3,000 inventory increase – $1,000 payables increase.

Annual cash inflows: Sales increase Cost decrease Cash inflows before tax Taxes Cash flows after tax

$1,400 1,500 2,900 (870) $2,030

Calculate the present value of the cash flows after tax: N = 6, I = 12, PMT = 2030, FV = 0 and then solve for PV = $8,346. The present value of the CCA tax shield is $1,797. ( (

,000

, 00) 0 0 0 0 0 0 ) ( ) 0 0 0 0 ( , 00) 0 0 0 0 ( ) ( ) 0 0 0 ( 0 ) ,0 0 4 4

0 0

The present value of the ending cash flows is $1,621. Salvage value Return of net working capital Total ending cash flow

1,200 2,000 $ 3,200

With a calculator, to calculate the present value of the ending cash flows enter N = 6, I = 12, FV = 3200, PMT = 0 and solve for PV = $1,621. The NPV of the project is $264; therefore, invest in the project. NPV = ($11,500) + $8,346 + $1,797 + $1,621 = $264.

11-18 E(NPV) = 0.05(–$70) + 0.20(–$25) + 0.50($12) + 0.20($20) + 0.05($30) = –$3.5 + –$5.0 + $6.0 + $4.0 + $1.5 = $3.0 million. σNPV= [0.05(–$70 – $3)2 + 0.20(–$25 – $3)2 + 0.50($12 – $3)2 + 0.20($20 – $3)2 + 0.05($30 – $3)2]0.5 = $23.622 million. CV =

$23.622 = 7.874. $3.0

11-19 Year

2017 2018 2019 2020 2021 a b

Beginning UCC ($) 0 3,875 2,131.25 7,372.19 3,614.70

Additions or disposals to asset class ($) 5,000

Proceeds ($)

UCC before CCA ($)

CCA taken ($)

800 1,500

5,000 3,875 10,131.25 6,572.19 2,114.70

1,125a 1,743.75 2,759.06b 2,957.49

8,000 (800) (1,500)

$5,000 × 50% × 45% = $1,125. ($8,000 × 50% × 45%) + ($2,131.25 × 45%) = $2,759.06

Taxable income before CCA Terminal loss Taxable income Tax (30%)

$50,000.00 2,114.70 47,885.30 $14,365.59

11-20 a. Expected annual cash flows: Project A:

Probable Probability  Cash Flow = Cash Flow 0.20 $6,000 $ 1,200 0.60 6,750 4,050 0.20 7,500 1,500 Expected annual cash flow = $6,750

Project B:

Probable Probability  Cash Flow = Cash Flow 0.20 $ 0 $ 0 0.60 6,750 4,050 0.20 18,000 3,600 Expected annual cash flow = $7,650

Coefficient of variation: CV = Standard deviation / Expected value = σCF/Expected CF Project A: σA = √( $750)2 (0.20) + ($0)2 (0.60) + ($750)2 (0.20) = $474. Project B: σB= √( $7,650)2 (0.20) + ($

00)2 (0.60) + ($ 0,

0)2 (0.20) =$5,756.

CVA = $474/$6,750= 0.0702. CVB = $5,756/$7,650 = 0.7524.

b. Project B is the riskier project because it has the greater variability in its probable cash flows, whether measured by the standard deviation or the coefficient of variation. Hence, Project B is evaluated at the 12% cost of capital, while Project A requires only a 10% cost of capital. Project A: With a financial calculator, input the appropriate cash flows into the cash flow register, input I = 10, and then solve for NPV = $10,036. Project B: With a financial calculator, input the appropriate cash flows into the cash flow register, input I = 12, and then solve for NPV = $11,624. Accept Project B. c. The portfolio effects from Project B would tend to make it less risky than otherwise. This would tend to reinforce the decision to accept Project B. Again, if Project B were negatively correlated with the GDP (Project B is profitable when the economy is down), then it is less risky and Project B’s acceptance is reinforced. 11-21 Initial Investment: New machine cost Sale of existing machine Total investment

($775,000) 135,000 ($640,000)

Annual cash inflows: Cash inflows Taxes (25%) Cash flows after tax

Years 1–5 $ 185,000 (46,250) $ 138,750

Calculate the present value of the cash flows after tax: N = 5, I = 12, PMT = 138750, FV = 0 and then solve for PV = $500,163. The present value of the CCA tax shield is $108,164. ($775,000 $135,000) 0.30 ( 0.12 + 0.30

0.25

1 + 0.5 0.12 ) ( ) 1 + 0.12

= $114, 286 × 0.94642857 − $18,750

$105,000 0.3 0.25 ( ) 0.12 + 0.30

1 (1 + 0.12)5

0.56742686 = $97,524

Calculate the present value of the salvage value: N = 5, I = 12, PMT = 0, FV =105000 and solve for PV = $59,580. The NPV of the project is $17,267; therefore, invest in the project. NPV = ($640,000) + $500,163 + $97,524 + $59,580 = $17,267. 11-326 Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc.

11-22 Initial investment: New machine Sale of existing machine Installation Release of NOWC Total investment

Years 1–6 ($1,175,000) 265,000 (15,000) 7,000 ($ 918,000)

Annual cash inflows: Annual savings (cash inflow) Taxes (25%) Cash flows after tax

$264,000 (66,000) $198,000

Calculate the present value of the cash flows after tax: N = 6, I = 12, PMT = 198000, FV = 0 and then solve for PV = $814,059. The present value of the CCA tax shield is $142,487. ($1,175,000 (

$265,000) 0.30 0.12 + 0.30 = $162,500 = $142,487.

($145,000 $20,000) 0.30 0.25 1 + 0.5 0.12 )( ) ( 1 + 0.12 0.12 + 0.30 0.94642857 $22,321 0.50663

The present value of the ending cash flows is $59,782. Ending cash flows Salvage value ―new‖ equipment $145,000 Foregone salvage value on ―old‖ equipment (20,000) Return of NOWC (7,000) $118,000 Calculate the present value of the ending cash flow: N = 6, I = 12, PMT = 0, FV = 118000 and then solve for PV = $59,782. The NPV of the project is $98,328, therefore invest in the project. NPV = ($918,000) + $814,059 + $142,487 + $59,782 = $98,328.

0.25

)(

(1 + 0.

11-23 a. The resulting decision tree is: t=0

t=1

t=2

t=3 $3,000,000

($1,000,000)

P = 0.5

P = 0.80

1,500,000

($500,000)

P 0.24

NPV NPV Product $881,718 $211,612

0.24

(185,952)

(44,628)

0.12

(376,709)

(45,205)

0.40

(10,000)

(4,000)

P = 0.5

P = 0.60 100,000 ($10,000)

P = 0.20

0 P = 0.40

1.00

Exp. NPV= $117,779

The NPV of the top path is:

$3,000,000 (1.12)3

–

$1,000,000 (1.12) 2

–

$500,000 (1.12)1

– $10,000 = $881,718.

Using a financial calculator, input the following: CF0 = –10000, CF1 = –500000, CF2 = –1000000, CF3 = 3000000, and I = 12 to solve for NPV = $881,718.29  $881,718. The other NPVs were determined in the same manner. If the project is of average risk, it should be accepted because the expected NPV of the total project is positive. b. σ2NPV = 0.24($881,718 – $117,779)2 + 0.24(–$185,952 – $117,779)2 + 0.12(–$376,709 – $117,779)2 + 0.4(–10,000 – $117,779)2 = 198,078,470,853. σNPV = $445,060. CVNPV =

$445,060 = 3.78. $117 ,779

Since the CV is 3.78 for this project, while the firm’s average project has a CV of 1.0 to 2.0, this project is of high risk. 11-328 Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc.

SOLUTION TO SPREADSHEET PROBLEM 11-24 The detailed solution for the spreadsheet problem is in the file Ch 11 Build a Model Solution.xlsx and is available on the textbook’s website.

MINI CASE The detailed solution for the Mini Case is also available in spreadsheet format, in the file Brigham4Ce_MiniCase_Ch11.xlsx, and is available on the textbook’s website. Shrieves Casting Company is considering adding a new line to its product mix, and the capital budgeting analysis is being conducted by Sidney Johnson, a recent business school graduate. The production line would be set up in unused space in Shrieves’s main plant. The machinery’s invoice price would be approximately $200,000, another $10,000 in shipping charges would be required, and it would cost an additional $30,000 to install the equipment. The machinery has an economic life of 4 years and would be in Class 8 with a CCA rate of 20%. The machinery is expected to have a salvage value of $25,000 after 4 years of use. The new line would generate incremental sales of 1,250 units per year for 4 years at an incremental cost of $100 per unit in the first year, excluding depreciation. Each unit can be sold for $200 in the first year. The sales price and cost are both expected to increase by 3% per year due to inflation. Furthermore, to handle the new line, the firm’s net operating working capital would have to increase by an amount equal to 12% of sales revenues. The firm’s tax rate is 28%, and its overall weighted average cost of capital is 10%. a.

Define ―incremental cash flow.‖

Answer: This is the firm’s cash flow with the project minus the firm’s cash flow without the project. a.

1. Should you subtract interest expense or dividends when calculating project cash flow?

Answer: The cash flows should not include interest expense or dividends. The return required by the investors furnishing the capital is already accounted for when we apply the 10% cost of capital discount rate, hence including financing flows would be ―double counting.‖ Put another way, if we deducted capital costs in the table, and thus reduced the bottom line cash flows, and then discounted those CFS by the cost of capital, we would, in effect, be subtracting capital costs twice. 12-330 Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc.

2. Suppose the firm had spent $100,000 last year to rehabilitate the production line site. Should this cost be included in the analysis? Explain.

Answer: The $100,000 cost to rehabilitate the production line site was incurred last year, and presumably also deducted in part for tax purposes. Since, it is a sunk cost, it should not be included in the analysis. a.

3. Now assume that the plant space could be leased out to another firm at $25,000 per year. Should this be included in the analysis? If so, how?

Answer: If the plant space could be leased out to another firm, then if Shrieves accepts this project, it would forgo the opportunity to receive $25,000 in annual cash flows. This represents an opportunity cost to the project, and it should be included in the analysis. Note that the opportunity cost cash flow must be net of taxes, so it would be a $25,000(1 – t) = $25,000(0.72) = $18,000 annual outflow. a.

4. Finally, assume that the new product line is expected to decrease sales of the firm’s other lines by $50,000 per year. Should this be considered in the analysis? If so, how?

Answer: If a project affects the cash flows of another project, this is an ―externality‖ that must be considered in the analysis. If the firm’s sales would be reduced by $50,000, then the net cash flow loss would be a cost to the project. Note that this annual loss would not be the full $50,000, because Shrieves would save on cash operating costs if its sales dropped. Note also that externalities can be positive as well as negative.

Disregard the assumptions in Part a and calculate the annual sales revenues and costs (other than CCA). Why is it important to include inflation when estimating cash flows?

Answer: With an inflation rate of 3%, the annual revenues and costs are:

Units Unit Price Unit Cost

Year 1 1,250 $200.00 $100.00

Year 2 1,250 $206.00 $103.00

Year 3 1,250 $212.18 $106.09

Year 4 1,250 $218.55 $109.27

Sales Costs

$250,000 $125,000

$257,500 $128,750

$265,225 $273,188 $132,613 $136,588

The cost of capital is a nominal cost; i.e., it includes a premium for inflation. In other words, it is larger than the real cost of capital. Similarly, nominal cash flows (those that are inflated) are larger than real cash flows. If you discount the low, real cash flows with the high, nominal rate, then the resulting NPV is too low. Therefore, you should always discount nominal cash flows with a nominal rate, and real cash flows with a real rate. In theory, you could use either way and get the correct answer. However, there is no accurate way to convert a nominal cost of capital to a real cost. Therefore, you should inflate cash flows and then discount at the nominal rate. c.

Construct annual incremental project operating cash flows.

Answer: Sales Costs Project cash flows before tax Taxes (28%) Project cash flows after tax

Year 1 $250,000 125,000 125,000 35,000 $90,000

Year 2 $257,500 128,750 128,750 36,050 $92,700

Year 3 $265,225 132,613 132,612 37,131 $95,481

Year 4 $273,188 136,588 136,600 38,248 $98,352

Calculate the required net operating working capital for each year and the cash flow due to changes in net operating working capital.

Answer: The project requires a level of net operating working capital in the amount equal to 12% of the next year’s sales. Any increase in NOWC is a negative cash flow, and any decrease is a positive cash flow. Year 0 Sales NOWC (% of sales) CF due to NOWC e.

$30,000 (30,000)

Year 1 $250,000 30,900 (900)

Year 2 $257,500 31,827 (927)

Year 3 Year 4 $265,225 $273,188 32,783 0 (956) 32,783

Calculate the present value of the CCA tax shield. PV of the CCA tax shield: 240,000 0.20 0.28 1 + 0.5 0.10 25,000 0.20 0.28 0.10 + 0.20 1 + 0.10 0.10 + 0.20

$44,800

$39,576.

× 0.954545454

–

4,667

1 (1 + 0.10)4 ×

0.683013455

Calculate the salvage cash flow.

Answer: When the project is terminated at the end of Year 4, the equipment can be sold for $25,000, which produces a present value of $17,075 as shown below. g.

What is the project’s NPV? Should the project be undertaken?

Answer:

The initial investment is $240,000: Machine Shipping Installation

$200,000 10,000 30,000 $240,000

The present value of the project cash flows after tax is $297,342: Using a financial calculator input the yearly cash flows from t=1 of 90000, 92700, 95481, 98352 in the cash flow registers, I = 10, and solve for the PV = $297,342. 12-334 Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc.

The present value of the CCA tax shield is $39,576 (from Part e).

The present value of the NOWC is ($9,911): Input the yearly cash flows from t = 0 of –30000, –900, –927, –956, 32783, I = 10, and solve for the PV = ($9,911). The present value of the salvage value is $17,075: Input N = 4, I = 10, PMT = 0, FV = 25000 and solve for the PV = $17,075. NPV = ($240,000) + $297,342 + $39,576 + ($9,911) + $17,075 = $104,082. Proceed with the project. What does the term ―risk‖ mean in the context of capital budgeting? To what extent can risk be quantified? When risk is quantified, is the quantification based primarily on statistical analysis of historical data or on subjective, judgmental estimates? Answer: Risk throughout finance relates to uncertainty about future events, and in capital budgeting, this means the future profitability of a project. For certain types of projects, it is possible to look back at historical data and to statistically analyze the riskiness of the investment. This is often true when the investment involves an expansion decision; for example, if Canadian Tire was opening a new store, if Scotiabank was opening a new branch, or if GM was expanding its Chevrolet plant, then past experience could be a useful guide to future risk. Similarly, a company that is considering going into a new business might be able to look at historical data on existing firms in that industry to get an idea about the riskiness of its proposed investment. However, there are times when it is impossible to obtain historical data regarding proposed investments; for example, if GM were considering the development of a hydrogen powered auto, not much relevant historical data for assessing the riskiness of the project would be available. Rather, GM would have to rely primarily on the judgment of its executives, and they, in turn would have to rely on their experience in developing, manufacturing, and marketing new products. We will try to quantify risk analysis, but you must recognize at the outset that some of the data used in the analysis will necessarily be based on subjective judgments rather than on hard statistical observations. h.

i. 1.

What are the three types of risk that are relevant in capital budgeting? 2. How is each of these risk types measured, and how do they relate to one another? Answer: Here are the three types of project risk:  Stand-alone risk is the project’s total risk if it were operated independently. Standalone risk ignores both the firm’s diversification among projects and investors’ diversification among firms. Stand-alone risk is measured by either the project’s standard deviation of NPV (σNPV) or its coefficient of variation of NPV (CVNPV). Note that other profitability measures, such as IRR and MIRR, can also be used to obtain stand-alone risk estimates.  Within-firm risk is the total riskiness of the project giving consideration to the firm’s other projects, that is, to diversification within the firm. It is the contribution of the project to the firm’s total risk, and it is a function of (a) the project’s standard deviation of NPV and (b) the correlation of the project’s returns with those of the rest of the firm. Within-firm risk is often called corporate risk, and it is measured by the project’s corporate beta, which is the slope of the regression line formed by plotting returns on the project versus returns on the firm.  Market risk is the riskiness of the project to a well-diversified investor, hence it considers the diversification inherent in shareholders’ portfolios. It is measured by the project’s market beta, which is the slope of the regression line formed by plotting returns on the project versus returns on the market. i.

3. How is each type of risk used in the capital budgeting process?

Answer: Because management’s primary goal is shareholder wealth maximization, the most relevant risk for capital projects is market risk. However, creditors, customers, suppliers, and employees are all affected by a firm’s total risk. Since these parties influence the firm’s profitability, a project’s within-firm risk should not be completely ignored. Unfortunately, by far the easiest type of risk to measure is a project’s stand-alone risk. Thus, firms often focus on this type of risk when making capital budgeting decisions. However, this focus does not necessarily lead to poor decisions, because most projects that a firm undertakes are in its core business. In this situation, a project’s stand-alone risk is likely to be highly correlated with its within-firm risk, which in turn is likely to be highly correlated with its market risk. j.

1. What is sensitivity analysis?

Answer: Sensitivity analysis measures the effect of changes in a particular variable, say revenues, on a project’s NPV. To perform a sensitivity analysis, all variables are fixed Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc. 12-337

at their expected values except one. This one variable is then changed, often by specified percentages, and the resulting effect on NPV is noted. (One could allow more than one variable to change, but this then merges sensitivity analysis into scenario analysis.)

2. Perform a sensitivity analysis on the unit sales, salvage value, and cost of capital for the project. Assume that each of these variables can vary from its base-case, or expected, value by plus or minus 10% and plus or minus 20%. Include a sensitivity graph and discuss the results.

Answer: The sensitivity data is given here in tabular form: NPV Deviation From Base Case WACC Units Sold

Deviation From Base Case –20% –10% 0% 10% 20%

$123,528 113,599 104,082 94,954 86,192

$ 44,610 74,344 104,082 133,811 163,545

$100,667 102,375 104,082 105,790 107,497

Range

37,336

118,935

6,830

Salvage

Sensitivity Analysis

NPV

$180,000 $160,000 $140,000 $120,000 $100,000

WACC Units Sold Salvage

$80,000 $60,000 $40,000 $20,000 $0 -20%

-10%

10%

20%

Deviation from Base Case Value

A. The sensitivity lines intersect at 0% change and the base case NPV, $104,082. Since all other variables are set at their base case, or expected values, the zero change situation is the base case. B. The plots for unit sales and salvage value are upward sloping, indicating that higher variable values lead to higher NPVs. Conversely, the plot for cost of Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc. 12-339

capital is downward sloping, because a higher cost of capital leads to a lower NPV. C. The plot of unit sales is much steeper than that for salvage value. This indicates that NPV is more sensitive to changes in unit sales than to changes in salvage value. D. Steeper sensitivity lines indicate greater risk. Thus, in comparing two projects, the one with the steeper lines is considered to be riskier. j.

3. What is the primary weakness of sensitivity analysis? What is its primary usefulness?

Answer: The two primary disadvantages of sensitivity analysis are that (1) it does not reflect the effects of diversification and (2) it does not incorporate any information about the possible magnitudes of the forecast errors. Thus, a sensitivity analysis might indicate that a project’s NPV is highly sensitive to the sales forecast, hence that the project is quite risky, but if the project’s sales, hence its revenues, are fixed by a long-term contract, then sales variations may actually contribute little to the project’s risk. It also ignores any relationships between variables, such as unit sales and sales price. Therefore, in many situations, sensitivity analysis is not a particularly good indicator of risk. However, sensitivity analysis does identify those variables that potentially have the greatest impact on profitability, and this helps management focus its attention on those variables that are probably most important. k.

Assume that Sidney Johnson is confident of her estimates of all the variables that affect the project’s cash flows except unit sales and sales price. If product acceptance is poor, unit sales will be only 900 units a year and the unit price will be only $160; a strong consumer response will produce sales of 1,600 units and a unit price of $240. Johnson believes that there is a 25% chance of poor acceptance, a 25% chance of excellent acceptance, and a 50% chance of average acceptance (the base case). 1. What is scenario analysis?

Answer: Scenario analysis examines several possible situations, usually worst case, most likely case, and best case. It provides a range of possible outcomes.

2. What is the worst-case NPV? The best-case NPV?

3. Use the worst-, base-, and best-case NPVs and probabilities of occurrence to find the project’s expected NPV, standard deviation, and coefficient of variation.

Answer: We used a spreadsheet model to develop the scenarios (in thousands of dollars), which are summarized below: Scenario

Probability

Unit Sales

Unit Price

NPV

Best Case Base Case Worst Case

25% 50% 25%

1,600 1,250 900

$240 $200 $160

$339,568 104,082 (64,811)

Expected NPV =

$120,730

Standard Deviation =

$143,936

Coefficient of Variation = Std Dev / Expected NPV =

1.19

Are there problems with scenario analysis? Define simulation analysis, and discuss its principal advantages and disadvantages.

Answer: Scenario analysis examines several possible scenarios, usually worst case, most likely case, and best case. Thus, it usually considers only three possible outcomes. Obviously the world is much more complex, and most projects have an almost infinite number of possible outcomes. Simulation analysis is a type of scenario analysis that uses a relatively powerful financial planning software such as interactive financial planning system (IFPs) or @risk (a spreadsheet add-in). Simple simulations can also be conducted with other spreadsheet add-ins, such as Simtools. Here the uncertain cash flow variables (such as unit sales) are entered as continuous probability distribution parameters rather than as point values. Then, the computer uses a random number generator to select values for the uncertain variables on the basis of their designated distributions. Once all of the variable values have been selected, they are combined, and an NPV is calculated. The process is repeated many times, say 1,000, with new values selected from the distributions for each run. The end result is a probability distribution of NPV based on a sample of 1,000 values. The software can graph the distribution as well as print out summary statistics such as expected NPV and σNPV. Simulation provides the decision maker with a better idea of the profitability of a project than does scenario analysis because it incorporates many more possible outcomes. Although simulation analysis is technically refined, its usefulness is limited because managers are often unable to accurately specify the variables’ probability distributions. Further, the correlations among the uncertain variables must be specified, along with the correlations over time. If managers are unable to do this with much confidence, then the results of simulation analyses are of limited value. Recognize also that neither sensitivity, scenario, nor simulation analysis provides a decision rule—they may indicate that a project is relatively risky, but they do not indicate whether the project’s expected return is sufficient to compensate for its risk. Finally, remember that sensitivity, scenario, and simulation analyses all focus on standalone risk, which is not the most relevant risk in capital budgeting analysis.

1. Assume that Shrieves’s average project has a coefficient of variation in the range of 0.7–0.9. Would the new line be classified as high risk, average risk, or low risk? What type of risk is being measured here?

Answer: The project has a CV of 1.19, which is above the average range of 0.7–0.9, so it falls into the high-risk category. The CV measures a project’s stand-alone risk—it is merely a measure of the variability of returns (as measured by NPV) about the expected return. m.

2. Shrieves typically adds or subtracts 3 percentage points to the overall cost of capital to adjust for risk. Should the new line be accepted?

Answer: Since the project is judged to have above-average risk, its differential risk-adjusted, or project, cost of capital would be 13%. At this discount rate, its NPV would be $77,775, so it would still be acceptable. m.

3. Are there any subjective risk factors that should be considered before the final decision is made?

Answer: A numerical analysis such as this one may not capture all of the risk factors inherent in the project. If the project has a potential for bringing on harmful lawsuits, then it might be riskier than first assessed. Also, if the project’s assets can be redeployed within the firm or can be easily sold, then, as a result of ―abandonment possibilities,‖ the project may be less risky than the analysis indicates.

What is a real option? What are some types of real options?

Answer: Real options exist when managers can influence the size and risk of a project’s cash flows by taking different actions during the project’s life in response to changing market conditions. 1. Investment timing options 2. Growth options a. Expansion of existing product line b. New products c. New geographic markets 3. Abandonment options a. Contraction b. Temporary suspension c. Complete abandonment 4. Flexibility options.

Chapter 12 Capital Structure Decisions ANSWERS TO END-OF-CHAPTER QUESTIONS 12-1

a. Capital structure is the manner in which a firm’s assets are financed; that is, the liabilities and the equity on the balance sheet. Capital structure is normally expressed as the percentage of each type of capital used by the firm—debt, preferred stock, and common equity. Business risk is the risk inherent in the operations of the firm, prior to the financing decision. Thus, business risk is the uncertainty inherent in a total risk sense, future operating income, or earnings before interest and taxes (EBIT). Business risk is caused by many factors. Two of the most important are sales variability and operating leverage. Financial risk is the risk added using debt financing. Debt financing increases the variability of earnings before taxes (but after interest); thus, along with business risk, it contributes to the uncertainty of net income and earnings per share. Business risk plus financial risk equals total corporate risk. b. Operating leverage is the extent to which fixed costs are used in a firm’s operations. If a high percentage of a firm’s total costs are fixed costs, then the firm is said to have a high degree of operating leverage. Operating leverage is a measure of one element of business risk, but does not include the second major element, sales variability.

Financial leverage is the extent to which fixed-income securities (debt and preferred stock) are used in a firm’s capital structure. If a high percentage of a firm’s capital structure is in the form of debt and preferred stock, then the firm is said to have a high degree of financial leverage. The break-even point is that level of unit sales at which costs equal revenues. Break-even analysis may be performed with or without the inclusion of financial costs. If financial costs are not included, breakeven occurs when EBIT equals zero. If financial costs are included, breakeven occurs when EBT equals zero. c. Reserve borrowing capacity exists when a firm uses less debt under ―normal‖ conditions than called for by the tradeoff theory. This allows the firm some flexibility to use debt in the future when additional capital is needed. 12-2

Business risk refers to the uncertainty inherent in projections of future ROIC.

12-3

Firms with relatively high nonfinancial fixed costs are said to have a high degree of operating leverage.

12-4

Operating leverage affects EBIT and, through EBIT, EPS. Financial leverage has no effect on EBIT—it affects only EPS, given EBIT.

12-5

If sales tend to fluctuate widely, then cash flows and the ability to service fixed charges will also vary. Such a firm is said to have high business risk. Consequently, there is a relatively large risk that the firm will be unable to meet its fixed charges, and interest payments are fixed charges. As a result, firms in unstable industries tend to use less debt than those whose sales are subject to only moderate fluctuations.

12-6

Public utilities place greater emphasis on long-term debt because they have more stable sales and profits as well as more fixed assets. Also, utilities have fixed assets that can be pledged as collateral. Further, trade firms use retained earnings to a greater extent, probably because these firms are generally smaller and, hence, have less access to capital markets. Public utilities have lower retained earnings because they have high dividend payout ratios and a set of shareholders who want dividends.

12-7

EBIT depends on sales and operating costs. Interest is deducted from EBIT. At high debt levels, firms lose business, employees worry, and operations are not continuous because of financing difficulties. Thus, financial leverage can influence sales and costs, and hence EBIT, if excessive leverage is used.

12-8

The tax benefits from debt increase linearly, which causes a continuous increase in the firm’s value and stock price. However, financial distress costs get higher and higher as

more and more debt is employed, and these costs eventually offset and begin to outweigh the benefits of debt. 12-9

a. MM Proposition I states the relationship between leverage and firm value. Proposition I without taxes is V = EBIT/rsU. Since both EBIT and rsU are constant, firm value is also constant and capital structure is irrelevant. With corporate taxes, Proposition I becomes V = Vu + TD. Thus, firm value increases with leverage and the optimal capital structure is virtually all debt. b. MM Proposition II states the relationship between leverage and the cost of equity. Without taxes, Proposition II is rsL = rsU + (rsU – rd)(D/S). Thus, rs increases in a precise way as leverage increases. In fact, this increase is just sufficient to offset the increased use of lower cost debt. When corporate taxes are added, Proposition II becomes rsL = rsU + (rsU – rd)(1 – T)(D/S). Here the increase in equity costs is less than the zero-tax case, and the increasing use of lower cost debt causes the firm’s cost of capital to decrease, and again, the optimal capital structure is virtually all debt. c. The Miller model introduces personal taxes. The effect of personal taxes is, essentially, to reduce the advantage of corporate debt financing. d. Financial distress costs are incurred when a leveraged firm facing a decline in earnings is forced to take actions to avoid bankruptcy. These costs may be the result of delays in the liquidation of assets, legal fees, the effects on product quality from cutting costs, and evasive actions by suppliers and customers. e. Agency costs arise from lost efficiency and the expense of monitoring management to ensure that debtholders’ rights are protected. f. The addition of financial distress and agency costs to either the MM tax model or the Miller model results in a trade-off model of capital structure. In this model, the optimal capital structure can be visualized as a trade-off between the benefit of debt (the interest tax shelter) and the costs of debt (financial distress and agency costs). g. The value of the debt tax shield is the present value of the tax savings from the interest payments. In the MM model with taxes, this is just interest × tax rate/discount rate = rdDT/r, and since the discount rate here is also rd, the PV of the debt tax shield = TD.

SOLUTIONS TO END-OF-CHAPTER PROBLEMS 12-1

QBE = F/(P – V) = $500,000/($75 – $50) = 20,000.

12-2

If wd = 0.21, then ws = 1 – 0.21 = 0.79. So D/S = wd/we = 0.21/0.79. bU = b/[1 + (1 T)(D/S)] = 1.2/[1 + (1 0.25)(0.21/0.79)] = 1.0.

12-3

If the company had no debt, its required return would be: rs,U = rRF + bU RPM = 5.5% + 1.0(6%) = 11.5%. With debt, the required return is: rs,L = rRF + bL RPM = 5.5% + 1.6(6%) = 15.1%. Therefore, the extra premium required for financial risk is 15.1% – 11.5% = 3.6%.

12-4 ROEbefore =

$500,000(1 – 0.26) $370,000 = = 0.0925 = 9.25% $4,000,000 $4,000,000

Interest expense = $2,000,000 × 0.07 = $140,000 ROEafter =

($500,000 – $140,000)(1 – 0.26) $266,400 = = 0.133 = 13.3% $2,000,000 $2,000,000

12-5

VL = VU = $800 million

12-6

VL = VU + TD = $600 + 0.30 ($120) = $636 million

12-7

SAfter = (1 – wd)(VNew) = (1 – 0.4)($50) = $30 million.

12-8

VU $600

(

* *

Tc)( ( Td) (

)( (

) )

+ $240

= $600 + [1 – 0.914] $240 = $600 + 0.086 ($240) = $620.68 million

12-9

a. Original value of the firm (with no debt): Original free cash flow: FCF = NOPAT = EBIT(1

T) since no growth = $600,000(1

0.25) = $450,000

Original cost of capital: WACC = wd rd(1 T) + wsrs = 0 + (1.0)(10%) = 10%. Original value of the unlevered firm: VU =

FCF (EBIT)(1 T) ($600,000)(1 0.25) = = = $4,500,000. WACC WACC 0.10

Original stock price = VU/n0 = $4,500,000/200,000 = $22.50 Original EPS = NI/n0 = (EBIT – Interest)(1 T)/n0 = [($600,000 – $0)(0.75)]/200,000 = $2.25 b. Using its target capital structure of 30% debt: With financial leverage (wd = 30%): WACC = wd rd(1 T) + wsrs = (0.3)(7%)(1 0.25) + (0.7)(10.96%) = 9.25%. Leverage doesn’t change FCF, so the value of a levered firm is based on its new WACC. VL =

FCF (EBIT)(1 T) ($600,000)(1 0.25) = = = $4,864,865 WACC WACC 0.0925

D = wd VL = 0.30($4,864,865) = $1,459,460 c. SL = VL – D = $4,864,865 $1,459,460 = $3,405,405 $3,405,405/140,000 = $24.32 EPSNew = [(EBIT – Interest)(1 – T)]/Number Sh. =[($600,000 – 0.07 x $1,459,460)(1 0.25)]/140,000 = $2.67

12-10 a.

Here are the steps involved: (1)

Determine the variable cost per unit at present, V: Profit = P(Q) – FC – V(Q) $1,500,000 = ($210,000)(200) – $5,000,000 – V(200) 200(V) = $35,500,000 V = $177,500.

(2)

Determine the new profit level if the change is made: New profit = P2(Q2) – FC2 – V2(Q2) = $200,000(240) – $8,000,000 – ($177,500 – $20,000)(240) = $2,200,000.

(3)

Determine the incremental profit: Profit = $2,200,000 – $1,500,000 = $700,000.

(4)

Estimate the approximate rate of return on new investment: Return = Profit/Investment = $700,000/$4,000,000 = 17.5%.

Return exceeds the 16% cost of equity; this analysis suggests that the firm should go ahead with the change. b. The change would increase the breakeven point: Old:

QBE =

New: QBE =

F $5,000,000 = = 154 units. P  V $210,000  $177,500 $8,000,000 = 189 units. $200,000  $157,500

c. It is impossible to state unequivocally whether the new situation would have more or less business risk than the old one. We would need information on both the sales probability distribution and the uncertainty about variable input cost in order to make this determination. However, a higher break-even point, other things held constant, is more risky. Also the percentage of fixed costs increases: Old:

FC $5,000,000 = = 12.35%. FC  V(Q) $5,000,000  $35,500,000

New:

FC2 $8,000,000 = = 17.47%. FC2  V2 (Q 2 ) $8,000,000  $37,800,000

The change in break-even points—and also the higher percentage of fixed costs— suggests that the new situation is riskier.

12-11 a. Expected ROE for Firm C: ROEC = (0.1)(–5.0%) + (0.2)(5.0%) + (0.4)(15.0%) + (0.2)(25.0%) + (0.1)(35.0%) = 15.0%. Note: The distribution of ROEC is symmetrical. Thus, the answer to this problem could have been obtained by simple inspection. Standard deviation of ROE for Firm C (for convenience, we express returns in percentage form rather than in decimal form):

C 

0.1( 5.0  15.0) 2  0.2(5.0  15.0) 2  0.4(15.0  15.0) 2  0.2( 25.0  15.0) 2  0.1(35.0  15.0) 2

 0.1( 20) 2  0.2( 10) 2  0.4(0) 2  0.2(10) 2  0.1( 20) 2  40  20  0  20  40  120  11.0%. b. According to the standard deviations of ROE, Firm A is the least risky, while C is the most risky. However, this analysis does not take into account portfolio effects—if C’s ROE goes up when most other companies’ ROEs decline (that is, its beta is negative), its apparent riskiness would be reduced. c. Firm A’s ROE = BEP = 5.5%. Therefore, Firm A uses no financial leverage and has no financial risk. Firm B and Firm C have ROE > BEP, and hence both use leverage. Firm C uses the most leverage because it has the highest ROE – BEP = measure of financial risk. However, Firm C’s shareholders also have the highest expected ROE.

12-12 a. Original value of the firm (D = $0): We are given that the book value of assets is equal to the market value of assets, so the value is $4,000,000. Alternatively, we can calculate the value as the sum of the debt (which is zero) and the stock (500,000 shares at a price of $8 per share): V = D + S = 0 + ($8)(500,000) = $4,000,000. Original cost of capital: WACC = wd rd(1 – T) + wsrs = 0 + (1.0)(8.75%) = 8.75%. With financial leverage (wd = 40%): WACC = wd rd(1 – T) + wsrs = (0.4)(7%)(1 – 0.30) + (0.6)(9.57%) = 7.7%. Because growth is zero, FCF is equal to EBIT(1 – T). The value of operations is: VL =

FCF (EBIT)(1 T) ($500,000)(1 0.30) = = = $4,545,455. WACC WACC 0.077

Increasing the financial leverage results in an increase in the firm’s value from $4,000,000 to $4,545,455. b. Using its target capital structure of 40% debt, the company must have debt of: D = wd V = 0.40($4,545,455) = $1,818,182. Therefore, its market value of equity is: S = V – D = $4,545,455

$1,818,182 = $2,727,273.

Alternatively, S = (1 – wd)V = 0.60($4,545,455) = $2,727,273.

c. Interest expense = 0.07 × $1,818,182 = $127,273 TIE =

EBIT EBIT = Interest $127,273

. Probability 0.10 0.20 0.40 0.20 0.10

T ( 0.79) 1.57 3.93 6.29 8,64

The interest payment is not covered when TIE < 1.0. The probability of this occurring is 0.10, or 10%.

12-13 a. b bU

= bU[1 + (1 – T)(D/S)] b 1.4 1.4 = = = 0.95 1 + (1 - T)(D/S) 1 + (1 0.30)( 0.40 ) 1.47 0.60

b. rsU = rRF + (rM – rRF)bU = 6% + (5%)0.95 = 6% + 4.75% = 10.75%. c. $6 million debt: VL = VU + TD = $60 + 0.15($6) = $60.9 million. rsL = rsU + (rsU – rRF)(1 – T)(D/S) = 10.75% + (10.75% – 6%)(0.85)($6/$54.9) = 10.75% + 4.75% × (0.85)($6/$54.9) = 11.19%. $18 million debt: VL = $60 + 0.15($18) = $62.7 million. rsL = 10.75% + 4.75% × (0.85)($18/$44.7) = 12.38%. $36 million debt: VL = $60 + 0.15($36) = $65.4 million. rsL = 10.75% + 4.75% × (0.85)($36/$29.4) = 15.69%.

d. $36 million debt: VL = $60 + 0.30($36) = $70.8 million. rsL = 10,75% + 4.75% × (0.70)($36/$34.8) = 14.19%. The mathematics of MM result in the required return, and, thus, the same financial risk premium. However, the market value debt ratio has decreased from $36/$29.4 = 122% to $36/$34.8 = 103% at the higher tax rate. Hence, a higher tax rate reduces the financial risk premium at a given market value debt/equity ratio. This is because a higher tax rate increases the relative benefits of debt financing.

12-14 No leverage: Debt = 0; Equity = $14,000,000. Stat e 1 2 3

EBIT

0.2 $4,200,0 00 0.5 2,800,00 0 0.3 700,000

(EBIT – rdD)(1 – T) $2,940,000

ROES 0.21

PS(ROE ) 0.042

1,960,000

0.14

0.070

490,000

0.03 5 RÔE =

0.011 0.123

RÔE = 12.3%.  = 6.31%. CV = /RÔE = 6.31%/12.3% = 0.513. Leverage ratio = 10%: Debt = $1,400,000; Equity = $12,600,000; rd = 9%. Sta te 1

EBIT

0.2

0.5

0.3

$4,200, 000 2,800,0 00 700,000

(EBIT – rdD)(1 – T) $2,851,800 1,871,800 401,800

ROES 0.22 6 0.14 9 0.03 2 RÔE =

PS(ROE ) 0.045 0.075 0.010 0.130

RÔE = 13.0%.  = 7%. CV = 7%/13.0% = 0.538.

Leverage ratio = 50%: Debt = $7,000,000; Equity = $7,000,000; rd = 11%. State 1 2 3

EBIT

0.2 $4,200,000 0.5 2,800,000 0.3 700,000

(EBIT – rdD)(1 ROES – T) $2,401,000 0.343 1,421,000 0.203 (49,000) (0.007) RÔE =

PS(ROE) 0.069 0.102 (0.002) 0.169

RÔE = 16.9%.  = 12.6%. CV = 12.6%/16.9% = 0.746. Leverage ratio = 60%: D = $8,400,000; E = $5,600,000; rd = 14%. State 1 2 3

EBIT

0.2 $4,200,000 0.5 2,800,000 0.3 700,000

(EBIT – rdD)(1 ROES PS(ROE) – T) $2,116,800 0.378 0.076 1,136,800 0.203 0.102 (333,200) (0.060) (0.018) RÔE = 0.160

RÔE = 16.0%.  = 15.8%. CV = 15.8%/16.0% = 0.988. As leverage increases, the expected return on equity rises up to a point. But as the risk increases with increased leverage, the cost of debt rises. So after the return on equity peaks, it then begins to fall. As leverage increases, the measures of risk (both the standard deviation and the coefficient of variation of the return on equity) rise with each increase in leverage. 12-15 a. VU = VL =

$2 million EBIT EBIT = = $20 million.  WACC rsU 0.10

c. SL =

$2  0.05($10) EBIT  rd D = = $10 million. rsL 0.15

SL + D = VL = VU + TD. $10 + $10 = $20 = VL = $20 + (0)$10 = $20 million. d. WACCU = rsU = 10%. For Firm L, we know that WACC must equal rsU = 10% according to Proposition I. But, we can demonstrate this as follows: WACCL

= (D/V)rd + (S/V)rs = ($10/$20)5% + ($10/$20)15% = 2.5% + 7.5% = 10.0%.

e. VL = $22 million is not an equilibrium value according to MM. Here’s why. Suppose you owned 10% of Firm L’s equity, worth 0.10($22 million – $10 million) = $1.2 million. Your cash flow is equal to 10% of the dividends paid by the levered firm. Because it is a zero-growth firm, its dividends are equal to its net income: Dividends = Net income = EBIT – rdD = $2,000,000 – 0.05($1,000,000) = $1,500,000. Your 10% share is 0.10($1,500,000) = $150,000. Therefore, your annual cash flow is $150,000. Now consider the following strategy. You could (1) sell your stock in Firm L for 0.10($2 million) = $1.2 million. Then you could borrow an amount (at 5%) equal to 10% of Firm L’s debt, or 0.10($10 million) = $1 million. You would have $1.2 million + $1 million = $2.2 million. You could spend $2 million of this to buy 10% of Firm U’s stock, and invest the remaining $200,000 in risk-free debt. Your cash stream would now be: (a) 10% of Firm U’s dividends, which is $200,000: 0.10(EBITU) = 0.10($2 million) = $200,000; plus (b) the return on the extra $200,000 profit you invested in risk-free debt, which is $10,000: rd(profit) = 0.05($200,000) = $10,000; minus (c) the interest expense on the $1 million you borrowed, which is $50,000: rd(loan) = 0.05($1 million)] = $50,000. Your net cash flow from this strategy is $200,000 + $10,000 – $50,000 = $160,000. Since the second strategy produces $160,000 annually while your position in L produces only $150,000, all investors would prefer the second strategy. The pressure to sell L’s stock would cause its price to fall, and the pressure to buy U’s stock would cause its price to increase. This would continue until VL = VU.

12-16 a. VU =

$2(1  0.4) EBIT (1  T ) = = $12 million. rsU 0.10

VL = VU + TD = $12 + (0.4)$10 = $16 million. b. rsU = 0.10 = 10.0%. rsL = rsU + (rsU – rd)(1 – T)(D/S) = 10% + (10% – 5%)(0.6)($10/$6) = 10% + 5% = 15.0%. c. SL =

( EBIT  rd D)(1  T ) [$2  0.05($10)]0.6 = = $6 million. rsL 0.15

VL = SL + D = $6 + $10 = $16 million. d. WACCU

= rsU = 10.0%.

WACCL

12-17 a. VU =

= (D/V)rd(1 – T) + (S/V)rs = ($10/$16)5%(0.6) + ($6/$16)15% = 7.5%.

EBIT (1  TC )(1  TS ) EBIT (1  TC ) $2(0.6) = = = $12 million. rsU rsU (1  TS ) 0.10

 (1  TC )(1  Ts )  b. VL = VU + 1  D (1  Td )    (0.6)( 0.8)  = $12 + 1  $10 (0.72 )   = $12 + [1 – 0.67]$10 = $12 + 0.33($10) = $15.33 million. VL = $15.33 million. Gain from leverage = $3.33 million. c. The gain from leverage under Miller is 0.33($10) = $3.33 million. The gain from leverage in Problem 12-16 is 0.4($10) = $4 million. Thus, the addition of personal tax rates reduced the value of the debt financing. d. VU = VL = $20 million. Gain from leverage = $0.00. e. VU = $12 million. VL = $16 million. Gain from leverage = $4 million. f. VU = $12 million. VL = $16 million. Gain from leverage = $4.0 million. Note that the gain from leverage is the same as in Part (e) and will be the same value, as long as Td = Ts. 12-18 a. Without new investment: Sales VC FC EBIT Interest EBT Tax (30%) Net income

$12,960,000 10,200,000 1,560,000 1,200,000 384,000* 816,000 244,800 $ 571,200

*Interest = 0.08($4,800,000) = $384,000. 1. EPSOld = $571,200/240,000 = $2.38. With new investment: Debt Sales $12,960,000 VC (0.8)($10,200,000) 8,160,000 FC 1,800,000 EBIT 3,000,000 Interest 1,104,000** EBT 1,896,000 Tax (30%) 568,800 Net income $ 1,327,200

Stock $12,960,000 8,160,000 1,800,000 3,000,000 384,000 2,616,000 784,800 $ 1,831,200

**Interest = 0.08($4,800,000) + 0.10($7,200,000) = $1,104,000. 2. EPSD = $1,327,200/240,000 = $5.53. 3. EPSS = $1,831,200/480,000 = $3.82. EPS should improve, but expected EPS is significantly higher if financial leverage is used. (Sales  VC  F  I)(1  T) b. EPS = N (PQ  VQ  F  I)(1  T) = . N ($28.8 0 Q  $18.133 Q  $1,800,000  $1,104,000 )(0. 7) 240,000 ($10.667 Q  $2,904,000 )(0. 7) = . 240,000

EPSDebt

EPSStock

($10.667 Q  $2,184,000 )(0. 7) . 480,000

Therefore, ($10.667 Q  $2,904,000 )(0. 7) ($10.667 Q  $2,184,000 )(0.7) = 480,000 240,000 $10.667Q = $3,624,000 Q = 339,740 units.

This is the ―indifference‖ sales level, where EPSdebt = EPSstock. c. EPSOld =

($28.8 Q  $22.667 Q  $1,560,000  $384,000)( 0.7) =0 240,000 $6.133Q = $1,944,000 Q = 316,974 units.

This is the QBE considering interest charges. EPSNew,Debt =

($28.8 Q  $18.133 Q  $2,904,000)(0. 7) =0 240,000 $10.667Q = $2,904,000 Q = 272,242 units.

EPSNew,Stock =

($10.667 Q  $2,184,000)(0. 7) =0 480,000 $10.667Q = $2,184,000 Q = 204,744 units.

d. At the expected sales level, 450,000 units, we have these EPS values: EPSOld Setup = $2.38. EPSNew,Debt = $5.53. EPSNew,Stock = $3.82. We are given that operating leverage is lower under the new setup. Accordingly, this suggests that the new production setup is less risky than the old one—variable costs drop very sharply, while fixed costs rise less, so the firm has lower costs at ―reasonable‖ sales levels. In view of both risk and profit considerations, the new production setup seems better. Therefore, the question that remains is how to finance the investment. The indifference sales level, where EPSdebt = EPSstock, is 339,740 units. This is well below the 450,000 expected sales level. If sales fall as low as 250,000 units, these EPS figures would result: EPSDebt =

[$28.8(25 0,000)  $18.133(25 0,000)  $2,904,000 ](0. 7) = –$0.69. 240,000

EPSStock =

[$28.8(25 0,000)  $18.133(25 0,000)  $2,184,000 ](0. 7) = $0.70. 480,000

These calculations assume that P and V remain constant, and that the company can obtain tax credits on losses. Of course, if sales rose above the expected 450,000 level, EPS would soar if the firm used debt financing. In the ―real world‖ we would have more information on which to base the decision— coverage ratios of other companies in the industry and better estimates of the likely range of unit sales. On the basis of the information at hand, we would probably use equity financing, but the decision is really not obvious.

12-19 a. Present situation (50% debt): WACC = wd rd(1 – T) + wsrs = (0.5)(8%)(1 – 0.15) + (0.5)(11%) = 8.9%. V=

FCF ( EBIT )( 1  T ) ($ 7.25 )( 1  0.15 )   = $69.24 million. WACC WACC 0.089

70% debt: WACC = wd rd(1 – T) + wsrs = (0.7)(10%)(1 – 0.15) + (0.3)(13%) = 9.9%. V=

FCF (EBIT)( 1  T) ($7.25)(1  0.15)   = $62.25 million. WACC WACC 0.099

30% debt: WACC = wd rd(1 – T) + wsrs = (0.3)(7%)(1 – 0.15) + (0.7)(10.5%) = 9.1%. V=

FCF (EBIT)( 1  T) ($7.25)(1  0.15)   = $67.72 million. WACC WACC 0.091

12-20 a. BEA’s unlevered beta is bU = b/(1 + (1 0.8421. b. b = bU (1 + (1

T)(D/S)) = 1.0/(1 + (1

0.25)(20/80)) =

T)(D/S)).

At 40% debt: bL = 0.8421 (1 + 0.75(40%/60%)) = 1.2632. rs = 6% + 1.2632(4%) = 11.0528% c. WACC = wd rd(1 T) + wsrs = (0.4)(9%)(1 0.25) + (0.6)( 11.0528%) = 9.3317%. V=

FCF (EBIT)(1  T) ($16)(1  0.25) = $128.60 million.   WACC WACC 0.093317

12-21 Tax rate = 30% bU = 1.2

rRF = 4.0% rM – rRF = 5.0%

From data given in the problem and table, we can develop the following table:

D/A 0.00 0.20 0.40 0.60 0.80

E/A 1.00 0.80 0.60 0.40 0.20

D/E 0.0000 0.2500 0.6667 1.5000 4.0000

rd 5.0% 6.0 8.0 10.0 13.0

rd(1 – T) 3.5% 4.2 5.6 7.0 9.1

Levered betaa 1.20 1.41 1.76 2.46 4.56

rsb 10.0% 11.1 12.8 16.3 26.8

WACCc 10.00% 9.72 9.92 10.72 12.64

Notes: a These beta estimates were calculated using the Hamada equation, b = bU[1 + (1 – T)(D/S)]. b These rs estimates were calculated using the CAPM, rs = rRF + (rM – rRF)b. c These WACC estimates were calculated with the following equation: WACC = wd(rd)(1 – T) + (ws)(rs). The firm’s optimal capital structure is that capital structure which minimizes the firm’s WACC. Elliott’s WACC is minimized at a capital structure consisting of 20% debt and 80% equity. At that capital structure, the firm’s WACC is 9.72%.

12-22 a. Value of unleveraged firm, VU = EBIT(1 – T) rsU: $12 = $2(1 – 0.4) rsU $12 = $1.2/ rsU rsU = $1.2/$12 = $10.0% Therefore, rsU = WACC = 10.0% b.

Value of leveraged firm according to MM mode with taxes: VL = VU + TD. As shown in the following table, value increases continuously with debt, and the optimal capital structure consists of 100% debt. Note: The table is not necessary to

answer this question, but the data (in millions of dollars) are necessary for Part c of this problem. Debt, D $ 0.0 2.5 5.0 7.5 10.0 12.5 15.0 20.0

VU $12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0

TD $0.0 1.0 2.0 3.0 4.0 5.0 6.0 8.0

VL = VU + TD $12.0 13.0 14.0 15.0 16.0 17.0 18.0 20.0

With financial distress costs included in the analysis, the value of the leveraged firm now is VL = VU + TD – PC, where VU + TD = value according to MM after-tax model. P = probability of financial distress. C = present value of distress costs.

D $ 0.0 2.5 5.0 7.5 10.0 12.5 15.0 20.0

VU + TD $12.0 13.0 14.0 15.0 16.0 17.0 18.0 20.0

P 0.0000 0.0000 0.0125 0.0250 0.0625 0.1250 0.3125 0.7500

PC = (P)$8 $0.00 0.00 0.10 0.20 0.50 1.00 2.50 6.00

VL = VU + TD – PC $12.0 13.0 13.9 14.8 15.5 16.0 15.5 14.0

Note: All dollar amounts are in millions. Optimal debt level: D = $12.5 million. Maximum value of firm: V = $16.0 million. Optimal debt/value ratio: D/V = $12.5/$16 = 78%.

The value of the firm versus debt value with and without financial distress costs is plotted next (millions of dollars): VL = Value without financial distress costs. VB = Value with financial distress costs.

12-23

a. Vu =

EBIT $5,600,000 = = $62,222,222. rsU 0.09

VL = VU = $62,222,222. b. At D = $0: rsU = 9% = WACC = 9% At D = $10 million: rsL = rsU + (rsU – rd)(D/S) rsL = 9% + (9% – 6%) (

$10,000,000 ) = 9% + 0.57 = 9.57% $52,222,222

WACC = (D/V)rd + (S/V)rs $10,000,000 $52,222,222 =( ) 6% + ( ) 9.57% = 9.0% $ 2,222,222 $62,222,222 At D = $22 million: rsL = rsU + (rsU – rd)(D/S) $22,000,000 rsL = 9% + (9% 6%) ( ) = 9% + 1.64% = 10.64% $40,222,222 WACC = (D/V)rd + (S/V)rs $ ,000,000 $40,222,222 =( ) 6% + ( ) 10.64% = 9.0% $62,222,222 $62,222,222 Leverage has no effect on firm value, which is a constant $62,222,222 since WACC is a constant 9%. This is because the cost of equity is increasing with leverage, and this increase exactly offsets the advantage of using lower cost debt financing.

c. At D = $0: rsU = WACC = 9% VU = [(EBIT – I)(1 – T)]/rsU = [($5,600,000 – 0)(0.7)]/0.09 = $43,555,556. At D = $10 million: VL = VU + TD = $43,555,556 + 0.30($10,000,000) = $46,555,556. rsL = rsU + (rsU – rd)(1 – T)(D/S) = 9% + (9% – 6%)(0.7)($10,000,000/$36,555,556) = 9.57%. WACC = ($10,000,000/$46,555,556)(6%)(0.7) + ($36,555,556/$46,555,556)(9.57%) = 8.42%. d. At D = $0: rsU = WACC = 9% VU = [(EBIT – I)(1 – T)]/rsU = [($5,600,000 – 0)(0.7)]/0.09 = $43,555,556. At D = $10 million: rsL = 9.57% (from part c.) WACC = 8.42% (from part c.) VL = $43,555,556 + 0.30($10,000,000) = $46,555,556 At D = $22 million: VL = VU + TD = $43,555,556 + 0.30($22,000,000) = $50,155,556. rsL = rsU + (rsU – rd)(1 – T)(D/S) = 9% + (9% – 6%)(0.7)($22,000,000/$28,155,556) = 10.64%. WACC = ($28,155,556/$50,155,556)(10.64%) = 7.81%.

($22,000,000/$50,155,556)(6%)(0.7)

Firm Value

9.0

52 48

WACC

8.0 7.0

Value ($ Millions)

WACC (%)

10.0

40 0%

21%

44%

Debt to Value

The maximum amount of debt financing is 100%. At this level D = V, and hence VL= VU + TD $43,555,556 + 0.3D D – 0.3D 0.7D D

=D =D = $43,555,556 = $43,555,556 = $43,555,556 /0.7 = $62,222,222 = V.

Since the bondholders are bearing the same risk as the equity holders of the unlevered firm, rd is now 9%. Now, the total interest payment is $62,222,223(0.09) = $5.6 million, and the entire $5.6 million of EBIT would be paid out as interest. Thus, the investors (bondholders) would get $5.6 million per year, and it would be capitalized at 9%: VL =

$5,600,000 = $62,222,222. 0.09

f. (1) Rising interest rates would cause rd and hence rd(1 – T) to increase, pulling up the WACC. These changes would cause V to rise less steeply, or even to decline. (2) Increased riskiness causes rs to rise faster than predicted by MM. Thus, WACC would increase and V would decrease.

SOLUTION TO SPREADSHEET PROBLEM 12-24 The detailed solution for the problem is available in the file Solution for FM4 Ch 12 Build a Model.xlsx on the textbook’s website.

MINI CASE The detailed solution for the Mini Case is also available in spreadsheet format, in the file Brigham4Ce_MiniCase_Ch12.xlsx, and is available on the textbook’s website. Assume that you have just been hired as business manager of PizzaPalace, a pizza restaurant located adjacent to campus. The company's EBIT was $500,000 last year, and since the university's enrollment is capped, EBIT is expected to remain constant (in real terms) over time. Since no expansion capital will be required, PizzaPalace plans to pay out all earnings as dividends. The management group owns about 50% of the stock, and the stock is traded in the over-the-counter market. The firm is currently financed with all equity; it has 100,000 shares outstanding; and P0 = $25 per share. When you took your corporate finance course, your instructor stated that most firms' owners would be financially better off if the firms used some debt. When you suggested this to your new boss, he encouraged you to pursue the idea. As a first step, assume that you obtained from the firm's investment banker the following estimated costs of debt for the firm at different capital structures: Percent Financed With Debt rd 0% —20 6.0% 30 6.5 40 8.0 50 9.0 If the company were to recapitalize, debt would be issued, and the funds received would be used to repurchase stock. PizzaPalace is in the 30% corporate tax bracket, its beta is 1.0, the risk-free rate is 4%, and the market risk premium is 5%.

Provide a brief overview of capital structure effects. Be sure to identify the ways in which capital structure can affect the weighted average cost of capital and free cash flows.

Answer: The basic definitions are: (1) V = Value of Firm (2) FCF = Free Cash Flow (3) WACC = Weighted Average Cost of Capital (4) rs and rd are costs of stock and debt (5) ws and wd are percentages of the firm that are financed with stock and debt.

The impact of capital structure on value depends upon the effect of debt on WACC and/or FCF. Debt holders have a prior claim on cash flows relative to shareholders. Debt holders’ ―fixed‖ claim increases risk of shareholders’ ―residual‖ claim, so the cost of stock, rs, goes up. Firms can deduct interest expenses. This reduces the taxes paid, frees up more cash for payments to investors, and reduces after-tax cost of debt Debt increases the risk of bankruptcy, causing pre-tax cost of debt, rd, to increase. Adding debt increases the percentage of firm financed with low-cost debt (wd) and decreases the percentage financed with high-cost equity (wce). The net effect on WACC is uncertain, since some of these effects tend to increase WACC and some tend to decrease WACC. Additional debt can affect FCF. The additional debt increases the probability of bankruptcy. The direct costs of financial distress are legal fees, ―fire‖ sales, etc. The indirect costs are lost customers, reductions in productivity of managers and line workers, and reductions in credit (i.e., accounts payable) offered by suppliers. Indirect costs cause NOPAT to go down due to lost customers and drop in productivity and causes the investment in capital to go up due to increases in net operating working capital (accounts payable goes up as suppliers tighten credit). Additional debt can affect the behaviour of managers. It can cause reductions in agency costs, because debt ―pre-commits,‖ or ―bonds,‖ free cash flow for use in making interest payments. Thus, managers are less likely to waste FCF on perquisites or non-value adding acquisitions. But it can cause increases in other agency costs. Debt can make managers too riskaverse, causing ―underinvestment‖ in risky but positive NPV projects. There are also effects due to asymmetric information and signalling. Managers know the firm’s future prospects better than investors. Thus, managers would not issue additional equity if they thought the current stock price was less than the true value of the stock (given their inside information). Hence, investors often perceive an additional issuance of stock as a negative signal, and the stock price falls. b. (1)

What is business risk? What factors influence a firm's business risk?

Answer: Business risk is uncertainty about EBIT. Factors that influence business risk include uncertainty about demand (unit sales); uncertainty about output prices; uncertainty about input costs; product and other types of liability; amount of operating leverage. b. (2) What is operating leverage, and how does it affect a firm’s business risk? Show the operating break-even point if a company has fixed costs of $200, a sales price of $15, and variables costs of $10.

Answer: Operating leverage is the change in EBIT caused by a change in quantity sold. The higher the proportion of fixed costs within a firm’s overall cost structure, the greater the operating leverage. Higher operating leverage leads to more business risk, because a small sales decline causes a larger EBIT decline. Q is quantity sold, F is fixed cost, V is variable cost, TC is total cost, and P is price per unit. Operating Breakeven = QBE QBE = F/(P – V) Example: F = $200, P = $15, V = $10: QBE = $200/($15 – $10) = 40.

Now, to develop an example which can be presented to PizzaPalace’s management to illustrate the effects of financial leverage, consider two hypothetical firms: Firm U, which uses no debt financing, and Firm L, which uses $10,000 of 8% debt. Both firms have $20,000 in assets, a 30% tax rate, and an expected EBIT of $3,000.

1. Construct partial income statements, which start with EBIT, for the two firms. Answer: Here are the fully completed statements: Assets Equity

Firm U $20,000 $20,000

Firm L $20,000 $10,000

EBIT Interest (8%) EBT Taxes (30%) Net income

$ 3,000 0 3,000 900 $ 2,100

$ 3,000 800 2,200 660 $ 1,540

2. Now calculate ROE and BEP (basic earnings power) for both firms.

Answer:

Firm U 15.0% 10.5% 

BEP ROE TIE c.

Firm L 15.0% 15.4% 3.75

3. What does this example illustrate about the impact of financial leverage on ROE?

Answer: Conclusions from the analysis: 

The firm’s basic earning power, BEP = EBIT/total assets, is unaffected by financial leverage.



Firm L has the higher expected ROE: o ROEU = 10.5%. o ROEL = 15.4%.

Therefore, the use of financial leverage has increased the expected profitability to shareholders. The higher ROE results in part from the tax savings and also because the stock is riskier if the firm uses debt. 

At the expected level of EBIT, ROEL > ROEU.



The use of debt will increase ROE only if ROA exceeds the after-tax cost of debt. Here ROA = unleveraged ROE = 10.5% > rd(1 – T) = 8%(0.7) = 5.6%, so the use of debt raises ROE.



Finally, note that the TIE ratio is huge (undefined, or infinitely large) if no debt is used, but it is relatively low if 50% debt is used. The expected TIE would be larger than 3.75 if less debt were used, but smaller if leverage were increased.

Explain the difference between financial risk and business risk.

Answer: Business risk increases the uncertainty in future EBIT. It depends on business factors such as competition, operating leverage, etc. Financial risk is the additional business risk concentrated on common shareholders when financial leverage is used. It depends on the amount of debt and preferred stock financing. e.

Now consider the fact that EBIT is not known with certainty, but rather has the following probability distribution:

Bad Average Good

Economic State 0.25 0.50 0.25

Probability $2,000 3,000 4,000

EBIT

Redo the Part A analysis for Firms U and L, but add basic earning power (BEP), return on invested capital (ROIC, defined as NOPAT/Capital = EBIT(1 – T)/TA for this company), and the times-interest-earned (TIE) ratio to the outcome measures. Find the values for each firm in each state of the economy, and then calculate the expected values. Finally, calculate the standard deviations. What does this example illustrate about the impact of debt financing on risk and return? Answer: Here are the pro forma income statements:

Prob. EBIT Interest EBT Taxes (30%) NI BEP ROIC ROE TIE E(BEP) E(ROIC) E(ROE) σROIC σROE

Firm U Bad 0.25 $2,000 0 2,000 600 $1,400 10.0% 7.0% 7.0% 

Avg. 0.50 $3,000 0 3,000 900 $2,100

Good 0.25 $4,000 0 4,000 1,200 $2,800

Firm L Bad Avg. 0.25 0.50 $2,000 $3,000 800 800 1,200 2,200 360 660 $ 840 $1,540

Good 0.25 $4,000 800 3,200 960 $2,240

15.0% 10.5% 10.5%  15.0% 10.5% 10.5% 2.47% 2.47%

20.0% 14.0% 14.0% 

10.0% 7.0% 8.4% 2.5

20.0% 14.0% 22.4% 5

15.0% 10.5% 15.4% 3.75 15.0% 10.5% 15.4% 2.47% 4.95%

This example illustrates that financial leverage can increase the expected return to shareholders. But, at the same time, it increases their risk. 

Firm L has a wider range of ROEs and a higher standard deviation of ROE, indicating that its higher expected return is accompanied by higher risk. To be precise: ROE (Unleveraged) = 2.47%, and ROE (Leveraged) = 4.95%. Thus, in a stand-alone risk sense, Firm L is twice as risky as Firm U—its business risk is 2.47%, but its stand-alone risk is 4.95%, so its financial risk is 4.95% – 2.47% = 2.48%.

What does capital structure theory attempt to do? What lessons can be learned from capital structure theory? Be sure to address the MM models.

Answer: MM theory begins with the assumption of zero taxes. MM prove, under a very restrictive set of assumptions, that a firm’s value is unaffected by its financing mix: VL = VU. Therefore, capital structure is irrelevant. Any increase in ROE resulting from financial leverage is exactly offset by the increase in risk (i.e., rs), so WACC is constant. MM theory later includes corporate taxes. Corporate tax laws favour debt financing over equity financing. With corporate taxes, the benefits of financial leverage exceed the risks because more EBIT goes to investors and less to taxes when leverage is used. MM show that: VL = VU + TD. If T = 30%, then every dollar of debt adds 30 cents of extra value to the firm. Miller later included personal taxes. Personal taxes lessen the advantage of corporate debt. Corporate taxes favour debt financing since corporations can deduct interest expenses, but personal taxes favour equity financing, since no gain is reported until stock is sold, and long-term gains are taxed at a lower rate. Miller’s conclusions with personal taxes are that the use of debt financing remains advantageous, but benefits are less than under only corporate taxes. Firms should still use 100% debt. However, Miller argued that, in equilibrium, the tax rates of marginal investors would adjust until there was no advantage to debt. MM theory ignores bankruptcy (financial distress) costs, which increase as more Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc. 13-378

leverage is used. At low leverage levels, tax benefits outweigh bankruptcy costs. At high levels, bankruptcy costs outweigh tax benefits. An optimal capital structure exists that balances these costs and benefits. This is the trade-off theory. MM assumed that investors and managers have the same information. But managers often have better information. Thus, they would sell stock if it is overvalued, and sell bonds if the stock is undervalued. Investors understand this, so they view new stock sales as a negative signal. This is signalling theory. The pecking order theory states that firms use internally generated funds first, because there are no flotation costs or negative signals. If more funds are needed, firms then issue debt because it has lower flotation costs than equity and not negative signals. If more funds are needed, firms then issue equity. One agency problem is that managers can use corporate funds for non-value maximizing purposes. The use of financial leverage bonds ―free cash flow,‖ and forces discipline on managers to avoid perks and non-value adding acquisitions. A second agency problem is the potential for ―underinvestment.‖ Debt increases risk of financial distress. Therefore, managers may avoid risky projects even if they have positive NPVs. Firms with many investment opportunities should maintain reserve borrowing capacity, especially if they have problems with asymmetric information (which would cause equity issues to be costly). The ―windows of opportunity‖ theory states that managers try to ―time the market‖ when issuing securities. They issue equity when the market is ―high‖ and after big stock price run-ups. They issue debt when the stock market is ―low‖ and when interest rates are ―low.‖ They issue short-term debt when the term structure is upward sloping and long-term debt when it is relatively flat.

What does the empirical evidence say about capital structure theory? What are the implications for managers?

Answer: Tax benefits are important—$1 debt adds about $0.10 to value. This supports the Miller model with personal taxes. Bankruptcies are costly—costs can be up to 10% to 20% of firm value. Firms don’t make quick corrections when stock price changes cause their debt ratios to change—this doesn’t support the trade-off model. After big stock price run-ups, the debt ratio falls, but firms tend to issue equity instead of debt. This is inconsistent with the trade-off model, inconsistent with the pecking order theory, but consistent with the windows of opportunity hypothesis. Many firms, Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc. 13-379

especially those with growth options and asymmetric information problems, tend to maintain excess borrowing capacity. Managers should take advantage of tax benefits by issuing debt, especially if the firm has a high tax rate, stable sales, and less operating leverage than the typical firm in its industry. Managers should avoid financial distress costs by maintaining excess borrowing capacity, especially if the firm has volatile sales, high operating leverage, many potential investment opportunities, or special purpose assets (instead of general purpose assets that make good collateral). If a manager has asymmetric information regarding the firm’s future prospects, then the manager should avoid issuing equity if actual prospects are better than the market perceives. Managers should always consider the impact of capital structure choices on lenders’ and rating agencies’ attitudes.

With the above points in mind, now consider the optimal capital structure for PizzaPalace.

h. (1)

For each capital structure under consideration, calculate the levered beta, the cost of equity, and the WACC.

Answer:

MM theory implies that beta changes with leverage. bu is the beta of a firm when it has no debt (the unlevered beta). Hamada’s equation provides the beta of a levered firm: b = bu[1 + (1 – T)(D/S)]. For example, to find the cost of equity for wd = 20%, we first use Hamada’s equation to find beta: b = bU [1 + (1 – T)(D/S)] = 1.0 [1 + (1 – 0.3) (20%/80%)] = 1.18 Then use CAPM to find the cost of equity: rs = rRF + b(RPM) = 4% + 1.18 (5%) = 9.9% We can repeat this for the capital structures under consideration. wd D/S b rs 0% 0.00 1.00 9% 20% 0.25 1.18 9.9% 30% 0.43 1.3 10.5% 40% 0.67 1.47 11.35% 50% 1.00 1.7 12.5%

Next, find the WACC. For example, the WACC for wd = 20% is: WACC = wd (1 – T) rd + ws rs WACC = 0.2 (1 – 0.3) (6%) + 0.8 (9.9%) WACC = 8.76% Then repeat this for all capital structures under consideration. wd 0% 20% 30% 40% 50% h. (2)

rd 0.0% 6.0% 6.5% 8.0% 9.0%

rs 9% 9.9% 10.5% 11.35% 12.5%

WACC 9% 8.76% 8.72% 9.05% 9.40%

Now calculate the corporate value. What are the optimal capital structure and the greatest corporate value?

Answer: For example the corporate value for wd = 20% is: V = FCF/WACC FCF = NOPAT = EBIT (1 – T). In this example, NOPAT = ($500,000)(1-0.30) = $350,000. Using these values, V = $350,000/0.0876 = $3,995,434. Repeating this for all capital structures gives the following table: wd WACC 0% 9.00% 20% 8.76% 30% 8.72% 40% 9.05% 50% 9.40%

Corp. Value $3,888,889 $3,995,434 $4,013,761 $3,867,403 $3,723,404

As this shows, value is maximized at $4,013,761 with a capital structure of 30% debt.

Chapter 13 Distributions to Shareholders: Dividends and Repurchases ANSWERS TO END-OF-CHAPTER QUESTIONS 13-1

a. The optimal distribution policy is one that strikes a balance between dividend yield and capital gains so that the firm’s stock price is maximized. b. The dividend irrelevance theory holds that dividend policy has no effect on either the price of a firm’s stock or its cost of capital. The principal proponents of this view are Merton Miller and Franco Modigliani (MM). They prove their position in a theoretical sense, but only under strict assumptions, some of which are clearly not true in the real world. The bird-in-the-hand theory assumes that investors value a dollar of dividends more highly than a dollar of expected capital gains because the dividend yield component, D1/P0, is less risky than the g component in the total expected return equation rS = D1/P0 + g. The tax preference theory proposes that investors prefer capital gains over dividends, because of the time value of money and the ability to time the capital gains.

The information content of dividends is a theory holding that investors regard dividend changes as ―signals‖ of management forecasts. Thus, when dividends are raised, this is viewed by investors as recognition by management of future earnings increases. Therefore, if a firm’s stock price increases with a dividend increase, the reason may not be investor preference for dividends, but expectations of higher future earnings. Conversely, a dividend reduction may signal that management is forecasting poor earnings in the future. The clientele effect is the attraction of companies with specific dividend policies to those investors whose needs are best served by those policies. Thus, companies with high dividends will have a clientele of investors with low marginal tax rates and strong desires for current income. Similarly, companies with low dividends will attract a clientele with little need for current income.

The residual distribution model states that firms should make distributions only when more earnings are available than needed to support the optimal capital budget. An extra dividend is a dividend paid, in addition to the regular dividend, when earnings permit. Firms with volatile earnings may have a low regular dividend that can be maintained even in low-profit (or high capital investment) years, and then supplement it with an extra dividend when excess funds are available.

e. The declaration date is the date on which a firm’s directors issue a statement declaring a dividend. If a company lists the shareholder as an owner on the holder-ofrecord date, then the shareholder receives the dividend. The ex-dividend date is the date when the right to the dividend leaves the stock. This date is 1 business day prior to the holder-of-record date. If the stock sale is made prior to the ex-dividend date, the dividend is paid to the buyer. If the stock is bought on or after the ex-dividend date, the dividend is paid to the seller. The date on which a firm actually mails dividend cheques is known as the payment date. f. Dividend reinvestment plans (DRIPs) allow shareholders to automatically purchase shares of common stock of the paying corporation in lieu of receiving cash dividends. There are two types of plans—one involves only stock that is already outstanding, while the other involves newly issued stock. In the first type, the dividends of all participants are pooled and the stock is purchased on the open market. Participants benefit from lower transaction costs. In the second type, the company issues new shares to the participants. Thus the company issues stock in lieu of the cash dividend. g. In a stock split, current shareholders are given some number (or fraction) of shares for each stock owned. Thus, in a 3-for-1 split, each shareholder would receive 3 new shares in exchange for each old share, thereby tripling the number of shares outstanding. Stock splits usually occur when the stock price is outside of the optimal trading range. Stock dividends also increase the number of shares outstanding, but at a slower rate than splits. In a stock dividend, current shareholders receive additional shares on some proportional basis. Thus, a holder of 100 shares would receive 5 additional shares at no cost if a 5% stock dividend were declared. Stock repurchases occur when a firm repurchases its own stock. The higher EPS on the now decreased number of shares outstanding will cause the price of the stock to rise and thus capital gains are substituted for cash dividends.

13-2

a. From the shareholders’ point of view, an increase in the personal income tax rate would make it more desirable for a firm to retain and reinvest earnings. Consequently, an increase in personal tax rates should lower the aggregate payout ratio. b. If the depreciation allowances were raised, cash flows would increase. With higher cash flows, payout ratios would tend to increase. On the other hand, the change in tax-allowed depreciation charges would increase rates of return on investment, other things being equal, and this might stimulate investment, and consequently reduce payout ratios. On balance, it is likely that aggregate payout ratios would rise, and this has in fact been the case. c. If interest rates were to increase, the increase would make retained earnings a relatively attractive way of financing new investment. Consequently, the payout ratio might be expected to decline. On the other hand, higher interest rates would cause rd, rs, and firm’s MCCs to rise—that would mean that fewer projects would qualify for capital budgeting and the residual would increase (other things constant), hence the payout ratio might increase. d. A permanent increase in profits would probably lead to an increase in dividends, but not necessarily to an increase in the payout ratio. If the aggregate profit increase were a cyclical increase that could be expected to be followed by a decline, then the payout ratio might fall, because firms do not generally raise dividends in response to a shortrun profit increase. e. If investment opportunities for firms declined while cash inflows remained relatively constant, an increase would be expected in the payout ratio. f. Dividends are currently paid out of after-tax dollars, and interest charges from beforetax dollars. Permission for firms to deduct dividends as they do interest charges would make dividends less costly to pay than before and would thus tend to increase the payout ratio. g. This change would make capital gains less attractive and would lead to an increase in the payout ratio.

13-3

The difference is largely one of accounting. In the case of a split, the firm simply increases the number of shares and simultaneously reduces the par or stated value per share. In the case of a stock dividend, there must be a transfer from retained earnings to capital stock. For most firms, a 100% stock dividend and a 2-for-1 split accomplish exactly the same thing; hence, investors may choose either one.

13-4

a. The residual distribution policy is based on a target D/E ratio and the premise that, since new common stock is more costly than retained earnings, a firm should use all the retained earnings it can to satisfy its common equity requirement. Thus, the distribution under this policy is a function of the firm’s investment opportunities. b. Yes. A shallower plot implies that changes from the optimal capital structure have little effect on the firm’s cost of capital, hence value. In this situation, dividend policy is less critical than if the plot were V-shaped.

13-5

a. True. When investors sell their stock they are subject to capital gains taxes.

True. If a company’s stock splits 2 for 1, and you own 100 shares, then after the split you will own 200 shares. c. True. Dividend reinvestment plans that involve newly issued stock will increase the amount of equity capital available to the firm. d. False. The Income Tax Act, through the tax deductibility of interest, encourages firms to use debt and thus pay interest to investors rather than dividends, which are not tax deductible. In addition, due to the deferral of capital gains tax, the Income Tax Act encourages investors in high tax brackets to prefer firms that retain earnings rather than those that pay large dividends. e. True. If a company’s clientele prefers large dividends, the firm is unlikely to adopt a residual dividend policy. A residual dividend policy could mean low or zero dividends in some years, which would upset the company’s developed clientele. f. False. If a firm follows a residual dividend policy, all else constant, its dividend payout will tend to decline whenever the firm’s investment opportunities improve.

SOLUTIONS TO END-OF-CHAPTER PROBLEMS 13-1

60% Debt; 40% Equity; Capital Budget = $5,000,000; NI = $3,000,000; DPR = ? Equity retained for capital budget = 0.4($5,000,000) = $2,000,000. NI Less Add to RE. Earnings Remaining Payout =

$3,000,000 2,000,000 $1,000,000

$1,000,000 = 33.33%. $3,000,000

13-2

The company requires 0.40($1,200,000) = $480,000 of equity financing. If the company follows a residual dividend policy, it will retain $480,000 for its capital budget and pay out the $120,000 ―residual‖ to its shareholders as a dividend. The payout ratio will therefore be $120,000/$600,000 = 0.20 = 20%.

13-3

Equity financing = $12,000,000(0.60) = $7,200,000. Dividends = Net income – Equity financing = $15,000,000 – $7,200,000 = $7,800,000. Dividend payout ratio = Dividends/Net income = $7,800,000/$15,000,000 = 52%.

13-4

Vop = (n0 P) − Extra cash = (10,000,000 × $20) − $25,000,000 = $175,000,000. N = Vop/P = $175,000,000/$20 = 8,750,000.

13-5

P0 = $120; Split = 3 for 2; New P0 = ? P0 New =

$120 = $80. 3/2

13-6

Retained earnings = Net income (1 – Payout ratio) = $11,000,000(0.45) = $4,950,000. Proportion Equity: The debt/equity ratio = D/E = 0.33. We can also state it as follows: D = 0.33E The debt and equity constitute the entire firm value, therefore D + E = 1 Substituting 0.33E for D we get 0.33E + E = 1 1.33E = 1 E = 0.75 External equity needed: Total equity required = (New investment)(% Equity) = $20,000,000(0.75) = $15,000,000. New external equity needed = $15,000,000 – $4,950,000 = $10,050,000.

13-7

Number of shares = 2,000(2) = 4,000. EPS = $10.00/2 = $5.00. DPS = $3.00/2 = $1.50. Price = $80.00/2 = $40.00.

13-8

DPS after split = $0.75. Equivalent pre-split dividend = $0.75(5) = $3.75. New equivalent dividend = Last year’s dividend(1.09) $3.75 = Last year’s dividend(1.09) Last year’s dividend = $3.75/1.09 = $3.44.

13-9

Capital budget should be $5 million since the company will accept all independent projects whose IRR exceeds the project’s cost of capital. We know that 60% of the $5 million should be equity. Therefore, the company should pay dividends of: Dividends Payout ratio

= Net income – Needed equity = $4,200,000 – $3,000,000 = $1,200,000. = $1,200,000/$4,200,000 = 0.286 = 28.6%.

13-10 Before Distribution: Share price = ($5,000,000 × 12)/5,000,000 = $12/share. Special Dividend DPS: $15,000,000/5,000,000 shares = $3/share. Each shareholder receives a special dividend of $3. New share value: Old market value = $5,000,000 × 12 = $60,000,000 Less total dividends paid = $15,000,000 New market value = $45,000,000 New share price = $45,000,000/5,000,000 share = $9/share Each shareholder received $3 per share in dividends and holds a share worth $9 for a total value of $12/share. Stock Repurchase Plan # shares repurchased = $15,000,000/$12/share = 1,250,000 shares # shares remaining = 5,000,000 – 1,250,000 = 3,750,000 shares New share price = New market value/# shares remaining $45,000,000/3,750,000 = $12/share. Shareholders will be no worse or better off if they receive a dividend or they sell their shares under the repurchase plan or hold onto them. Both alternatives would increase Titan’s debt-equity ratio since the $15,000,000 payment would be coming out of the firm’s equity (both market and book equity).

13-11 a. Projected net income Less projected capital investments Available residual Shares outstanding

$2,000,000 800,000 $1,200,000 200,000

DPS = $1,200,000/200,000 shares = $6 = D1 b. EPS = $2,000,000/200,000 shares = $10 Payout ratio = DPS/EPS = $6/$10 = 60%, or Total dividends/Net income = $1,200,000/$2,000,000 = 60% c. Currently, P0 =

= 0.14 – 0.05 = 0.09 = $66.67 r –g s

Under the former circumstances, D1 would be based on a 20% payout on $10 EPS, or $2. With rs = 14% and g = 12%, we solve for P0:

P0 =

= 0.14 – 0.12 = 0.02 = $100 r –g s

Although PTC has suffered a severe setback, its existing assets will continue to provide a good income stream. More of these earnings should now be passed on to shareholders, as the slower internal growth has reduced the need for funds. However, the net result is a 33% decrease in the value of the shares.

13-12

a. |

$6.00

t∞ | . . .

$6.00

At t = 2 (and thereafter), net income = $1.4 million. DPS = $1,400,000/200,000 shares = $7.00 t0 |

t1 |

t2 |

t∞

$7.00

| . . .

| $7.00

P0 = ? $ .00 P1 = = $87.50 0.08 P0 = $87.50/(1 + 0.08) = $81.02 The share price would rise if the company went ahead with the new investment because the new project provides a return greater than the investors’ required return, that is, 16.7% growth vs. 8% cost of equity. The timeline for the new investment and its cash flows are as follows: t0 |

t1 | –$6.00

t2 | $1.00

–

t3 t∞ | . . . $1.00

(

$1.00

)

$75.00 + $6.02 = $81.02 b. 75 = PV, 0 = PMT, 1 = N, 8 = I/Y, CPT FV = $ 81

P1 = 8% × $81 = $6.48 4 − $0.48 (which is the extra earnings per share needed for investors to be equally well-off). $0.48 × 200,000 shares = $96,000 in extra earnings needed for investors to be equally well-off, (i.e., needs 8% growth, $1.2 million × 8% = $96,000).

13-13 Southern Petroleum: The current share price is equal to the present value of all stock repurchases made, divided by the current number of shares. The present value of all stock repurchases made is the discounted value of a constant growing perpetuity, so we can use the same formula as that used to find the present value of a constant growing dividend stream. Earnings available to repurchase shares = $20,000,000 × 50% = $10,000,000 Present value of stock repurchases = =

urrent repurchase value (1 + Growth rate) Cost of equity - Growth rate $10,000,000(1 + 0.06) 0.10 - 0.06

Current share price =

= $265,000,000

$265,000,000 = $10.60 25,000,000

In Year 1, Southern’s earnings will be $20,000,000 × 1.06 = $21,200,000. Cash used to repurchase shares = $21,200,000 × 0.50 = $10,600,000. Share price in one year = $10.60 × 1.06 = $11.236. Shares repurchased = $10,600,000/$11.236 = 943,396. Shares remaining = 25,000,000 – 943,396 = 24,056,604 Value of Southern = $11.236 × 24,056,604 = $270,300,000. In one year, each of Anthony’s shares is worth $11.236. Northern Petroleum: EPS = $20,000,000/25,000,000 =$0.80 DPS=$0.80 × 50% = $0.40 Share price =

$0.40(1+0.06) = $10.60. 0.10 - 0.06

DPS1 = $0.80 × 50%(1.06) = $0.40(1.06) = $0.424

Share price in one year = $10.60 × 1.06 = $11.236. Share priceex-dividend = $11.236 – $0.424 = $10.812. Value of Northern = $10.812 × 25,000,000 = $270,300,000. In one year, each of Anthony’s shares is worth $11.236. In one year, each of Clara’s shares is worth $10.812, plus Clara received a dividend of $0.424. Her total wealth per share = $11.236 ($10.812 + $0.424), therefore both are equally well-off.

13-14 a. 1. 2022 Dividends = (1.08)(2021 Dividends) = (1.08)($2,400,000) = $2,592,000. 2. 2021 Payout = $2,400,000/$8,000,000 = 0.3 = 30%. 2022 Dividends = (0.3)(2022 Net income) = (0.3)($11,400,000) = $3,420,000. 3. Equity financing = $7,300,000(0.60) = $4,380,000. 2022 Dividends = Net income – Equity financing = $11,400,000 – $4,380,000 = $7,020,000. 4. The regular dividends would be 8% above the 2021 dividends: Regular dividends = (1.08)($2,400,000) = $2,592,000. The residual policy calls for dividends of $7,020,000. Therefore, the extra dividend, which would be stated as such, would be: Extra dividend = $7,020,000 – $2,592,000 = $4,428,000. An even better use of the surplus funds might be a stock repurchase. b. Policy 4, based on the regular dividend with an extra dividend, seems most logical. Implemented properly, it would lead to the correct capital budget and the correct financing of that budget, and it would give correct signals to investors.

13-15 a. Capital budget = $15,000,000; Capital structure = 70% equity, 30% debt. Retained earnings needed = $15,000,000 (0.7) = $10,500,000. b. According to the residual dividend model, only $500,000 is available for dividends: NI – Retained earnings needed for cap. projects = Residual dividend. $11,000,000 – $10,500,000 = $500,000. DPS = $500,000/1,000,000 = $0.50. Payout ratio = $500,000/$11,000,000 = 4.55%. c. Retained earnings available = $11,000,000 – $2.00 (1,000,000) Retained earnings available = $11,000,000 – $2,000,000 Retained earnings available = $9,000,000. d. No. If the company maintains its $2.00 DPS, only $9 million of retained earnings will be available for capital projects. However, if the firm is to maintain its current capital structure, $10.5 million of equity is required. This would necessitate the company having to issue $1.5 million of new common stock. e. Capital budget = $15 million; Dividends = $2 million; NI = $11 million. Capital structure = ? RE available = $11,000,000 – $2,000,000 = $9,000,000. Percentage of cap. budget financed with RE =

$9,000,000 = 60%. $15,000,000

Percentage of cap. budget financed with debt =

$6,000,000 = 40%. $15,000,000

f. Dividends = $2 million; Capital Budget = $15 million; 70% equity, 30% debt; NI = $11 million. Equity needed = $15,000,000(0.7) = $10,500,000. RE available = $11,000,000 – $2.00(1,000,000) = $11,000,000 – $2,000,000 = $9,000,000. External (new) equity needed = $10,500,000 – $9,000,000 = $1,500,000. Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc. 13-393

g. Dividends = $2 million; NI = $11 million; Capital structure = 70% equity, 30% debt. RE available = $11,000,000 – $2,000,000 = $9,000,000. We’re forcing the RE available = Required equity to find the new capital budget. Required equity = Capital budget (target equity ratio) $9,000,000 = Capital budget(0.7) Capital budget = $12,857,143. Therefore, if Landry cuts its capital budget from $15 million to $12.86 million, it can maintain its $2.00 DPS, its current capital structure, and still follow the residual dividend policy h. The firm can do one of four things: (1) Cut dividends. (2) Change capital structure, that is, use more debt. (3) Cut its capital budget. (4) Issue new common stock. Realize that each of these actions is not without consequences to the company’s cost of capital, stock price, or both.

13-16 a. Regular dividend to grow at 4%: Div2022 = Div2021(1 + g) = $4.7(1.04) = $4.89 Div2023 = $4.89(1.04)2 = $5.08 Div2024 = $5.08(1.04)3 = $5.29 Div2025 = $5.29(1.04)4 = $5.50 Div2026 = $5.50(1.04)5 = $5.72 b. Residual net income = Net income – Amount of investment financed by equity Residual net income2022 = $10.0 – (0.70)($7.0) = $5.10 Residual net income2023 = $10.6 – (0.70)($7.5) = $5.35 Residual net income2024 = $11.2 – (0.70)($11.0) = $3.50 Residual net income2025 = $11.9 – (0.70)($10.0) = $4.90

Residual net income2026 = $12.7 – (0.70)($8.0) = $7.10

c. & d. Residual net income – Target dividend = Income after reg. dividend Extra dividend Total short-term investments

2021 2022 $5.1 0 $4.8 9 $0.2 1 $0.0 0 $2.0 $2.2 0 1

2023 $5.3 5 $5.0 8 $0.2 7 $0.0 0 $2.4 8

2024 $3.50

2025 $4.90

$5.29

$5.50

($1.7 9) $0.00

($0.6 0) $0.00

$0.69

$0.09

2026 $7.1 0 $5.7 2 $1.3 8 $0.3 8 $1.0 9

e. Yes, from the table in Part c/d, we see that the company can meet its targeted regular dividend based on its criteria.

SOLUTION TO SPREADSHEET PROBLEM 13-17 The detailed solution for the spreadsheet problem is in the file Ch 13 Build a Model Solution.xlsx and is available on the textbook’s website.

MINI CASE The detailed solution for the Mini Case is also available in spreadsheet format, in the file Brigham4Ce_MiniCase_Ch13.xlsx, and is available on the textbook’s website. Central Canada Steel (CCS) was formed 5 years ago to exploit a new continuous-casting process. CCS’s founders, Donald Brown and Margo Lapointe, had been employed in the research department of a major integrated-steel company, but when that company decided against using the new process (which Brown and Lapointe had developed), they decided to strike out on their own. One advantage of the new process was that it required relatively little capital in comparison with the typical steel company, so Brown and Lapointe have been able to avoid issuing new stock, and thus they own all of the shares. However, CCS has now reached the stage where outside equity capital is necessary if the firm is to achieve its growth targets yet maintain its target capital structure of 60% equity and 40% debt. Therefore, Brown and Lapointe have decided to take the company public. Until now, Brown and Lapointe have paid themselves reasonable salaries but routinely reinvested all after-tax earnings in the firm, so dividend policy has not been an issue. However, before talking with potential outside investors, they must decide on a dividend policy. Assume that you were recently hired by Pierce Westerfield Carney (PWC), a national consulting firm, which has been asked to help CCS prepare for its public offering. Martha Millon, the senior PWC consultant in your group, has asked you to make a presentation to Brown and Lapointe in which you review the theory of dividend policy and discuss the following questions. a.

1. What is meant by the term ―distribution policy‖?

Answer: Distribution policy is defined as the firm’s policy with regard to (1) the level of distributions, (2) the form of distributions (dividends or stock repurchases), and (3) the stability of distributions.

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a. 2. The terms ―irrelevance,‖ ―bird-in-the-hand,‖ and ―tax preference‖ have been used to describe three major theories regarding the way dividend payouts affect a firm’s value. Explain what these terms mean, and briefly describe each theory. Answer: ―Dividend irrelevance” refers to the theory that investors are indifferent between dividends and capital gains, making dividend policy irrelevant with regard to its effect on the value of the firm. ―Bird-in-the-hand‖ refers to the theory that a dollar of dividends in the hand is preferred by investors to a dollar retained in the business, in which case dividend policy would affect a firm’s value. The dividend irrelevance theory was proposed by MM, but they had to make some very restrictive assumptions to ―prove‖ it (zero taxes, no flotation or transaction costs). MM argued that paying out a dollar per share of dividends reduces the growth rate in earnings and dividends, because new stock will have to be sold to replace the capital paid out as dividends. Under their assumptions, a dollar of dividends will reduce the stock price by exactly $1. Therefore, according to MM, shareholders should be indifferent between dividends and capital gains. The ―bird-in-the-hand‖ theory is identified with Myron Gordon and John Lintner, who argued that investors perceive a dollar of dividends in the hand to be less risky than a dollar of potential future capital gains in the bush; hence, shareholders prefer a dollar of actual dividends to a dollar of retained earnings. If the bird-in-the-hand theory is true, then investors would regard a firm with a high payout ratio as being less risky than one with a low payout ratio, all other things equal; hence, firms with high payout ratios would have higher values than those with low payout ratios. MM opposed the Gordon-Lintner theory, arguing that a firm’s risk is dependent only on the riskiness of its cash flows from assets and its capital structure, not on how its earnings are distributed to investors. The tax preference theory recognizes that there are two tax-related reasons for believing that investors might prefer a low dividend payout to a high payout since taxes are not paid on capital gains until the stock is sold.

a. 3. What do the three theories indicate regarding the actions management should take with respect to dividend payouts? Answer: If the dividend irrelevance theory is correct, then dividend payout is of no consequence, and the firm may pursue any dividend payout. If the bird-in-the-hand theory is correct, the firm should set a high payout if it is to maximize its stock price. If the tax preference theory is correct, the firm should set a low payout if it is to maximize its stock price. Therefore, the theories are in total conflict with one another.

a. 4. What results have empirical studies of the dividend theories produced? How does all this affect what we can tell managers about dividend payouts? Answer: Unfortunately, empirical tests of the theories have been inconclusive (because firms don’t differ just with respect to payout), so we cannot tell managers whether investors prefer dividends or capital gains. Even though we cannot determine what the optimal dividend policy is, managers can use the types of analyses discussed in this chapter to help develop a rational and reasonable, if not completely optimal, dividend policy.

Discuss (1) the information content, or signalling, hypothesis, (2) the clientele effect, and (3) their effects on distribution policy.

Answer: 1. It has long been recognized that the announcement of a dividend increase often results in an increase in the stock price, while an announcement of a dividend cut typically causes the stock price to fall. One could argue that this observation supports the premise that investors prefer dividends to capital gains. However, MM argued that dividend announcements are signals through which management conveys information to investors. Information asymmetries exist—managers know more about their firms’ prospects than do investors. Further, managers tend to raise dividends only when they believe that future earnings can comfortably support a higher dividend level, and they cut dividends only as a last resort. Therefore, (1) a larger-than-normal dividend increase ―signals‖ that management believes the future is bright, (2) a smaller-than-expected increase, or a dividend cut, is a negative signal, and (3) if dividends are increased by a ―normal‖ amount, this is a neutral signal. 2. Different groups, or clienteles, of shareholders prefer different dividend payout policies. For example, many retirees, pension funds, and university endowment funds are in a low (or zero) tax bracket, and they have a need for current cash income. Therefore, this group of shareholders might prefer high payout stocks. These investors could, of course, sell some of their stock, but this would be inconvenient, transaction costs would be incurred, and the sale might have to be made in a down market. Conversely, investors in their peak earnings years who are in high tax brackets and who have no need for current cash income should prefer low payout stocks.

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3. Clienteles do exist, but the real question is whether there are more members of one clientele than another, which would affect what a change in its dividend policy would do to the demand for the firm’s stock. There are also costs (taxes and brokerage) to shareholders who would be forced to switch from one stock to another if a firm changes its policy. Therefore, we cannot say whether a policy change to appeal to one particular clientele or another would lower or raise a firm’s cost of equity. MM argued that one clientele is as good as another, so in their view the existence of clienteles does not imply that one dividend policy is better than another. Still, no one has offered convincing proof that firms can disregard clientele effects. We know that shareholder shifts will occur if policy is changed, and since such shifts result in transaction costs and capital gains taxes, policy changes should not be taken lightly. Further, dividend policy should be changed slowly, rather than abruptly, in order to give shareholders time to adjust.

c. 1. Assume that CCS has an $800,000 capital budget planned for the coming year. You have determined that its present capital structure (60% equity and 40% debt) is optimal, and its net income is forecast at $600,000. Use the residual distribution model approach to determine CCS’s total dollar distribution. Assume for now that the distribution is in the form of a dividend. Then, explain what would happen if net income were forecasted at $400,000, or at $800,000. Answer: We make the following points: a. Given the optimal capital budget and the target capital structure, we must now determine the amount of equity needed to finance the projects. Of the $800,000 required for the capital budget, 0.6($800,000) = $480,000 must be raised as equity and 0.4($800,000) = $320,000 must be raised as debt if we are to maintain the optimal capital structure: Debt Equity

$320,000 40% 480,000 60% $800,000 100%

b. If a residual exists—that is, if net income exceeds the amount of equity the company needs—then it should distribute the residual amount out as either dividends or stock repurchases. For now, we assume all payouts are in the form of dividends. Since $600,000 of earnings is available, and only $480,000 is needed, the residual is $600,000 – $480,000 = $120,000, so this is the amount that should be paid out as dividends. Thus, the payout ratio would be $120,000/$600,000 = 0.20 = 20%. c. If only $400,000 of earnings were available, the theoretical break point would occur at BP = $400,000/0.6 = $666,667. Assuming the intersection of the Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc. 14-401

investment opportunity set and marginal cost of capital was still at $800,000, the firm would still need $480,000 of equity. It should then retain all of its earnings and also sell $80,000 of new stock. The residual policy would call for a zero payment. d. If $800,000 of earnings was available, the dividend would be increased to $800,000 –$480,000=$320,000, and the payout ratio would rise to $320,000/$800,000= 40%. c. 2. In general terms, how would a change in investment opportunities affect the payout ratio under the residual payment policy? Answer: A change in investment opportunities would lead to an increase (if investment opportunities were good) or a decrease (if investment opportunities were not good) in the amount of equity needed, hence a decrease or an increase, respectively, in the residual dividend payout. c. 3. What are the advantages and disadvantages of the residual policy? (Hint: Don’t neglect signalling and clientele effects.) Answer: The primary advantage of the residual policy is that under it the firm makes maximum use of lower cost retained earnings, thus minimizing flotation costs and hence the cost of capital. Also, whatever negative signals are associated with stock issues would be avoided. However, if it were applied exactly, the residual model would result in dividend payments that fluctuated significantly from year to year as capital requirements and internal cash flows fluctuated. (1) This would send investors conflicting signals over time regarding the firm’s future prospects, and (2) since no specific clientele would be attracted to the firm, it would be an ―orphan.‖ These signalling and clientele effects would lead to a higher required return on equity, which would more than offset the effects of lower flotation costs. Because of these factors, few if any publicly owned firms follow the residual model on a year-to-year basis. Even though the residual approach is not used to set the annual dividend, it is used when firms establish their long-run dividend policy. If ―normalized‖ cost of capital and investment opportunity conditions suggest that in a ―normal‖ year the company should pay out about 60% of its earnings, this fact will be noted and used to help determine the long-run policy.

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What are stock repurchases? Discuss the advantages and disadvantages of a firm repurchasing its own shares.

Answer: A firm may distribute cash to shareholders by repurchasing its own shares rather than paying out cash dividends. Stock repurchases can be used (1) somewhat routinely as an alternative to regular dividends, (2) to dispose of excess (nonrecurring) cash that came from asset sales or from temporarily high earnings, and (3) in connection with a capital structure change in which debt is sold and the proceeds are used to buy back and retire shares. Advantages of repurchases: 1. A repurchase announcement may be viewed as a positive signal that management believes the shares are undervalued. 2. Shareholders have a choice—if they want cash, they can tender their shares, receive the cash, and pay the taxes, or they can keep their shares and avoid taxes. On the other hand, one must accept a cash dividend and pay taxes on it. 3. If the company raises the dividend to dispose of excess cash, this higher dividend must be maintained to avoid adverse stock price reactions. A stock repurchase, on the other hand, does not obligate management to future repurchases. 4. Repurchases can be varied from year to year without giving off adverse signals, while dividends may not. 5. Repurchases can be used to produce large-scale changes in capital structure. Disadvantages of repurchases: 1. A repurchase could lower the stock’s price if it is taken as a signal that the firm has relatively few good investment opportunities. On the other hand, though, a repurchase can signal shareholders that managers are not engaged in ―empire building,‖ where they invest funds in low-return projects. 2. Selling shareholders may not be fully informed about the repurchase; hence they may make an uninformed decision and may later sue the company. To avoid this, firms generally announce repurchase programs in advance. 3. The firm may bid the stock price up and end up paying too high a price for the shares. In this situation, the selling shareholders would gain at the expense of the remaining shareholders. This could occur if a tender offer were made and the price was set too high, or if the repurchase was made in the open market and Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc. 14-403

buying pressure drove the price above its equilibrium level. Describe the series of steps that most firms take in setting dividend policy in practice.

Answer: Firms establish dividend policy within the framework of their overall financial plans. The steps in setting policy are listed below: 1. The firm forecasts its annual capital budgets and its annual sales, along with its working capital needs, for a relatively long-term planning horizon, often 5 years. 2. The target capital structure, presumably the one that minimizes the WACC while retaining sufficient reserve borrowing capacity to provide ―financing flexibility,‖ will also be established. 3. With its capital structure and investment requirements in mind, the firm can estimate the approximate amount of debt and equity financing required during each year over the planning horizon. 4. A long-term target payout ratio is then determined, based on the residual model concept. Because of flotation costs and potential negative signalling, the firm will not want to issue common stock unless this is absolutely necessary. At the same time, due to the clientele effect, the firm will move cautiously from its past dividend policy, if a new policy appears to be warranted, and it will move toward any new policy gradually rather than in one giant step. 5. An actual dollar dividend, say $2 per year, will be decided upon. The size of this dividend will reflect (1) the long-run target payout ratio and (2) the probability that the dividend, once set, will have to be lowered, or, worse yet, omitted. If there is a great deal of uncertainty about cash flows and capital needs, then a relatively low initial dollar dividend will be set, for this will minimize the probability that the firm will have to either reduce the dividend or sell new common stock. The firm will run its corporate planning model so that management can see what is likely to happen with different initial dividends and projected growth rates under different economic scenarios.

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What are stock splits and stock dividends? What are the advantages and disadvantages of stock splits and stock dividends?

Answer: When it uses a stock dividend, a firm issues new shares in lieu of paying a cash dividend. For example, in a 5% stock dividend, the holder of 100 shares would receive an additional 5 shares. In a stock split, the number of shares outstanding is increased (or decreased in a reverse split) in an action unrelated to a dividend payment. For example, in a 2-for-1 split, the number of shares outstanding is doubled. A 100% stock dividend and a 2-for-1 stock split would produce the same effect, but there would be differences in the accounting treatments of the two actions. Both stock dividends and stock splits increase the number of shares outstanding and, in effect, cut the pie into more, but smaller, pieces. If the dividend or split does not occur at the same time as some other event that would alter perceptions about future cash flows, such as an announcement of higher earnings, then one would expect the price of the stock to adjust such that each investor’s wealth remains unchanged. For example, a 2-for-1 split of a stock selling for $50 would result in the stock price being cut in half, to $25. It is hard to come up with a convincing rationale for small stock dividends, like 5% or 10%. No economic value is being created or distributed, yet shareholders have to bear the administrative costs of the distribution. Further, it is inconvenient to own an odd number of shares as may result after a small stock dividend. Thus, most companies today avoid small stock dividends. On the other hand, there is a good reason for stock splits or large stock dividends. Specifically, there is a widespread belief that an optimal price range exists for stocks. The argument goes as follows: if a stock sells for about $20–$80, then it can be purchased in round lots, hence at reduced commissions, by most investors. A higher price would put round lots out of the price range of many small investors, while a stock price lower than about $20 would convey the image of a stock that is doing poorly. Thus, most firms try to keep their stock prices within the $20 to $80 range. If the company prospers, it will split its stock occasionally to hold the price down. (Also, companies that are doing poorly occasionally use reverse splits to raise their price.) Many companies do operate outside the $20 to $80 range, but most stay within it. Another factor that may influence stock splits and dividends is the belief that they signal management’s belief that the future is bright. If a firm’s management would be inclined to split the stock or pay a stock dividend only if it anticipated improvements in earnings and dividends, then a split/dividend action could provide a positive signal and thus boost the stock price. However, if earnings and cash dividends did not subsequently rise, the price of the stock would fall back to its old level, or even lower, because managers would lose credibility. Interestingly, one of the most astute investors of the 20th century, Warren Buffett, chairman of Berkshire-Hathaway, has never split his firm’s stock. Berkshire currently sells for over $400,000 per share, and its performance over the years has been absolutely spectacular. It may be that Berkshire’s market value would be higher if it Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc. 14-405

had a 6,000:1 stock split, or it may be that the conventional wisdom is wrong. What is a dividend reinvestment plan (DRIP), and how does it work?

Answer: Under a dividend reinvestment plan (DRIP), shareholders have the option of automatically reinvesting their dividends in shares of the firm’s common stock. In an open market purchase plan, a trustee pools all the dividends to be reinvested and then buys shares on the open market. Shareholders use the DRIP for three reasons: (1) brokerage costs are reduced by the volume purchases, (2) the DRIP is a convenient way to invest excess funds, and (3) the company generally pays all administrative costs associated with the operation. In a new stock plan, the firm issues new stock to the DRIP members in lieu of cash dividends. No fees are charged, and companies may offer the stock up to a 5% discount from the market price on the dividend date on the grounds that the firm avoids flotation costs that would otherwise be incurred. Only firms that need new equity capital use new stock plans, while firms with no need for new stock use an open market purchase plan.

Chapter 14 Public and Private Financing: Initial Offerings, Seasoned Offerings, and Investment Banks ANSWERS TO END-OF-CHAPTER QUESTIONS 14-1

a. A closely held corporation goes public when it sells stock to the general public. Going public increases the liquidity of the stock, establishes a market value, facilitates raising new equity, and allows the original owners to diversify. However, going public increases business costs, requires disclosure of operating data, and reduces the control of the original owners. The new issue market is the market for stock of companies that go public, and the issue is called an initial public offering (IPO). b. A public offering is an offer of new common stock to the general public; in other words, an offer in which the existing shareholders are not given any preemptive right to purchase the new shares. A private placement is the sale of stock to only one or a few investors, usually institutional investors. The advantages of private placements are lower flotation costs and greater speed, since preparation of a prospectus is not required. c. A venture capitalist is the manager of a venture capital fund. The fund raises most of its capital from institutional investors and invests in start-up companies in exchange

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for equity. The venture capitalist gets a seat on the companies’ boards of directors. Before an IPO, the senior management team and the investment banker make presentations to potential investors. They carry out this roadshow in 10 to 20 cities, with 3 to 5 presentations per day, over a 2-week period. The spread is the difference between the price at which an underwriter sells the stock in an IPO and the proceeds that the underwriter passes on to the issuing firm. In other words, it is the fee collected by the underwriter, and it usually is 5.5% of the offering price.

d. Regulation of securities markets falls under provincial and territorial jurisdiction in Canada, (unlike in the United States where there is one national regulator, the Securities and Exchange Commission [SEC]). As a result, each province and territory has its own regulator, often called a securities commission, that oversees the issuance of securities. These regulators, along with the Canadian Securities Administrators (CSA), self-regulatory bodies such as the Investment Industry Regulatory Organization of Canada (IIROC), and the stock exchanges, help to ensure stable markets, sound brokerage firms, and the absence of stock manipulation. Trading is also controlled for corporate insiders, who are the officers, directors, and major shareholders of the firm. Public companies must undertake continuous disclosure, providing the public information on all material changes to the company in a timely manner. To simplify the process for obtaining approvals to sell securities in each province and territory, each regulator, with the exception of Ontario, has agreed to the passport system. When securities are cleared for sale in the principal province, they are generally cleared in the other provinces as well. e. A prospectus summarizes information about a new security issue and the issuing company. A preliminary prospectus may be distributed to potential buyers prior to its approval by the securities commission. After the approval has become effective, the securities, accompanied by the prospectus, may be offered for sale. f. A best efforts arrangement versus an underwritten sale refers to two methods of selling new stock issues. In a best efforts sale, the investment banker is only committed to making every effort to sell the stock at the offering price. In this case, the issuing firm bears the risk that the new issue will not be fully subscribed. If the issue is underwritten, the investment banker agrees to buy the entire issue at a set price, and then resells the stock at the offering price. Thus, the risk of selling the issue rests with the investment banker. g. Project financings are arrangements used to finance mainly large capital projects such as energy explorations, oil tankers, refineries, utility power plants, and so on. Usually, one or more firms (sponsors) will provide the equity capital required by the project, while the rest of the project’s capital is supplied by lenders and lessors. The most important aspect of project financing is that the lenders and lessors do not have recourse against the sponsors; they must be repaid from the project’s cash flows and the equity cushion provided by the sponsors. 14-2

No. The real value of a security is determined by the equilibrium forces of an efficient market. Assuming that the information provided on newly issued securities is accurate, the market will establish the value of a security regardless of the opinions rendered by the securities commission, or, for that matter, opinions offered by any advisory service or analyst.

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14-3

a. Going public would tend to make attracting capital easier and to decrease flotation costs. b. The increasing institutionalization of the ―buy side‖ of the stock and bond markets should increase a firm’s ability to attract capital and should reduce flotation costs. c. Financial conglomerates can offer a variety of financial services and types of investments; thus, it seems a company’s ability to attract capital would increase and flotation costs would decrease. d. Reduction in use of the preemptive right would likely not affect a large company where percentage ownership is not as important. Indeed, the trend today seems to be for companies not to use the preemptive right. e. The introduction of shelf registrations tended to speed up regulator review time and lower the costs of floating each new issue. Thus, the company’s ability to attract new capital was increased.

14-4

Investment bankers must investigate the firms whose securities they sell, simply because, if an issue is overvalued and suffers marked price declines after the issue, the banker will find it increasingly difficult to dispose of future new issues. In other words, reputation is highly important in the investment banking industry.

14-5

An IPO increases liquidity and allows founders to liquidate some (or all) of their shares, permits founders to diversify their wealth, establishes a value for the firm (which is helpful for tax purposes if the owner dies and is helpful when selling the company to another company), increases visibility, increases credibility, and often opens potential markets.

14-410 Inc.

SOLUTIONS TO END-OF-CHAPTER PROBLEMS 14-1

a. $5 per share Gross proceeds = (3,000,000)($5) = $15,000,000. Net profit = $15,000,000 – $14,000,000 – $300,000 = $700,000. b. $6 per share Gross proceeds = (3,000,000)($6) = $18,000,000. Net profit = $18,000,000 – $14,000,000 – $300,000 = $3,700,000. c. $4 per share Gross proceeds = (3,000,000)($4) = $12,000,000. Net loss = $12,000,000 – $14,000,000 - $300,000 = −$2,300,000.

14-2

14-3

Net proceeds per share = $22(1 – 0.05) = $20.90. Number of shares to be sold = ($20,000,000 + 150,000)/$20.90 = 964,115 shares. a. Company: 15,000,000[$12/share × (1 – 0.05)] – $450,000 = $170,550,000. Underwriter: 0.05(15,000,000 shares × $12/share) = $9,000,000. b. At a closing price of $15: 15,000,000($15 – $12) = $45,000,000. The IPO investors benefited by $45,000,000 in total. At a closing price of $11: 15,000,000($11 – $12) = – $15,000,000. The company benefited by $15,000,000 in total.

14-4

The shares are currently worth $20 ($800,000/40,000). With no issuance costs, a fair price would be $20 per share. If $200,000 is raised in new equity, then 10,000 ($200,000/$20) shares would be issued.

14-5

EPS (before): Net income=$30,000,000 × 0.05 = $1,500,000. EPS = $1,500,000/3,000,000 shares = $0.50/share. EPS (after): $30,000,000(1.08)=$32,400,000 in new sales. Net income=$32,400,000 × 0.05 = $1,620,000. Total proceeds to be raised = $4,000,000/(1 – 0.055) = $4,232,804. Number of shares to be issued = $4,232,804/$14 = 302,344 shares. EPS = $1,620,000/(3,000,000+302,344) = $0.491/share. Growth in shares = 302,344/3,000,000=10.1%, therefore earnings must grow by that amount for there to be no dilution.

14-6 re (low) = rRF + RPM(b) = 3% + 4%(1.2) = 7.8% re (high = rRF + RPM(b) = 3% + 4%(1.5) = 9.0% P0 = $0.90(1.05)/(0.078 – 0.05) = $33.75 or, P0 = $0.90(1.05)/(0.09 – 0.05) = $23.63 Price range is from $23.63 to $33.75. When comparing a company to the industry, you should take into account size, share liquidity, business risk, amount of debt, and operating history. The more favourable these variables, the higher a possible price is within the range. 14-7

Average beta = (1.7 + 2.1)/2 = 1.9 re = 3% + 1.9(4%) = 10.6% P0 = $4.70(0.30)/(0.106 – 0.06) = $30.65 Amount received after flotation costs per share: $30.65(1 – 0.055) = $28.96 Number of shares to sell to raise $150,000,000: $150,000,000/$28.96 = 5,179,558

14-8 Current: BVPS = $100,000,000/16,000,000 = $6.25. EPS = $0.50 (given) Market price = $9 (given) (

–

)

et ncome urrent – ew

The problem states the P/E ratio remains unchanged so we can calculate the existing P/E ratio and use it to calculate the new share price.

New market price = 18 × $0.48 = $8.64 Book value per share EPS Market price

Current New $6.25 $6.42 $0.50 $0.48 $9.00 $8.64

Dilution No Yes Yes

New shares = 1,500,000 Minimum EBIT: $0.50 × 1,500,000 = $750,000 (after-tax) $750,000/(1 – 0.30) = $1,071,429. EBIT must increase by $1,071,429.

14-9

4 million shares (P) + 1 million shares (P) = $60 million

+ 93%(1 million shares)(P) 5 million shares (P) = $60 million + 0.93 million shares (P) 4.07 million shares (P) = $60 million P = $14.74 Net amount raised = (Gross amount raised)(1 – F) = ($14.74 × 1 million shares)(1 – 0.07) = $13,708,200 14-10

If 100 shares are outstanding, then we have the

following for Edelman:

Earnings per share Dividends per share Book value per share b.

2016 2021 $8,160 $12,000 4,200 6,000 90,000

Using the following two equations, the growth rate for EPS and DPS can be determined. (1 + gEPS)5 EPS16 = EPS21. (1 + gDPS)5 DPS16 = DPS21. Kennedy Strasburg Edelman

gEPS 8.4% 6.4 8.0

gDPS 8.4% 6.4 7.4

c. Based on the figures in Part a, it is obvious that Edelman’s stock would not sell in the range of $36 to $65 per share. The small number of shares outstanding has greatly inflated EPS, DPS, and book value per share. Should Edelman attempt to sell its stock based on the EPS and DPS above, it would have difficulty finding investors at the economically justified price.

d. Edelman’s management would probably be wise to split the stock so that EPS, DPS, and book value were closer to those of Kennedy and Strasburg. This would bring the price of the stock into a more reasonable range.

e. A 4,000-for-1 split would result in 400,000 shares outstanding. If Edelman has 400,000 shares outstanding, then we would have the following: 2016 $2.04 1.05

Earnings per share Dividends per share Book value per share f. Kennedy Strasburg Edelman

ROE 15.00% 13.64 13.33

Kennedy Strasburg Edelman

Payout Ratio 2016 2021 50% 50% 50 50 51 50

2021 $ 3.00 1.50 22.50

All three companies seem to be following similar dividend policies, paying out about 50% of their earnings. h. D/A is 43% for Kennedy, 37% for Strasburg, and 55% for Edelman. This suggests that Edelman is more risky, and hence should sell at relatively low multiples. i. Kennedy Strasburg

2021 P/E $36/$4.50 = 8.00 $65/$7.50 = 8.67

These ratios are not consistent with g and ROE; based on gs and ROEs, Kennedy should have the higher P/E. Probably size, listing status, and debt ratios are offsetting g and ROE.

j. The market prices of Kennedy and Strasburg yield the following multiples:

Kennedy Strasburg

Multiple of EPS, 2021 8.00 8.67

Multiple of Multiple of Book DPS, 2021 Value per Share, 2021 16.00 1.20 17.33 1.18

Applying these multiples to the data in Part e, we obtain the following market prices: Indicated Market Price for Edelman Stock Based on Data of: Kennedy Strasburg $ 24.00 $26.00 24.00 26.00 27.00 26.59

Based on earnings, 2021 Based on dividends, 2021 Based on book value per share 

k. r s =

D 0 (1  g) + g. P0 

Kennedy

rs= 

Strasburg

rs=

$2.25(1.084 ) + 8.4% = 15.18%. $36

$3.75(1.064 ) + 6.4% = 12.54%. $65

Edelman

P̂0 =

D1 . rg

Based on Kennedy:

P0 =

1.50(1.077 ) = $21.54. 0.152  0.077

Based on Strasburg:

P0 =

1.50(1.077 ) = $33.66. 0.125  0.077

The potential range, based on these data, is between $21.54 and $33.66 a share. The data suggest that the price would be set toward the low end of the range: (1) Edelman has a high debt ratio, (2) Edelman is relatively small and unknown. The actual price would be based on negotiations between the underwriter and Edelman; we cannot say what the exact price would be, but the price would probably be set in the $20–$23 range, with $21 being a reasonable guess.

SOLUTION TO SPREADSHEET PROBLEM 14-11 The detailed solution for the spreadsheet problem is in the file Ch 14 Build a Model Solution.xlsx and is available on the textbook’s website.

MINI CASE The detailed solution for the Mini Case is also available in spreadsheet format, in the file Brigham4Ce_MiniCase_Ch14.xlsx, and is available on the textbook’s website. Adam’s, a family-owned restaurant chain operating in Montreal, has grown to the point where expansion throughout Quebec is feasible. The proposed expansion would require the firm to raise about $18.6 million in new capital. Because Adam’s currently has a debt ratio of 50%, and also because the family members already have all their personal wealth invested in the company, the family would like to sell common stock to the public to raise the $18.6 million. However, the family does want to retain voting control. You have been asked to brief the family members on the issues involved by answering the following questions: a.

What agencies regulate securities markets?

Answer: Provincial and territorial securities commissions regulate the securities market in Canada. Some of the responsibilities include regulation of companies that sell securities, and file annual reports and other disclosure documents with the relevant commissions; prohibiting stock manipulation; controls over trading by corporate insiders; and control over the proxy statement and how it is used to solicit votes. The Ontario Securities Commission (OSC) also regulates the TSX and TSX Venture Exchange Self-regulatory organizations such as the Investment Industry Regulatory Association of Canada oversee market trading and the country’s investment dealers. The securities industry itself realizes the importance of stable markets, and therefore the various exchanges work closely with the commissions to police transactions and to maintain the integrity and credibility of the system. b.

How are start-up firms usually financed?

Answer: The first financing comes from the founders. The first external financing comes from angels, who are wealthy individuals. Existing investors may also invest in future financing rounds. When a company has sufficient assets and earnings, bank financing will also be possible. Venture capital funds are another financing source. A venture capital fund raises capital from institutional investors, usually around $30 million to $100 million. The managers of the fund are called venture capitalists. The fund invests in 10 to 12 companies, and the venture capitalist sits on their boards. Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc. 14419

Differentiate between a private placement and a public offering.

Answer: In a private placement, stock is sold directly to one investor or a small group of investors rather than being distributed to the public at large. A private placement has the advantage of lower flotation costs; however, since the stock would be bought by a small number of accredited investors, it would not be actively traded, and a liquid market would not exist. Further, there are restrictions on the resale of private placement stock. It might also be difficult to find investors willing to invest large sums in the company and yet be minority shareholders. Thus, many of the advantages listed above would not be obtained. For these reasons, a public offering makes more sense in Adam’s situation. d.

Why would a company consider going public? What are some advantages and disadvantages?

Answer: A firm is said to be ―going public‖ when it sells stock to the public for the first time. A company’s first stock offering to the public is called an ―initial public offering (IPO).‖ Thus, Adam’s will go public if it goes through with its planned IPO. There are several advantages and disadvantages to going public: Advantages to going public: 

Going public will allow the family members to diversify their assets and reduce the riskiness of their personal portfolios.



It will increase the liquidity of the firm’s stock, allowing the family shareholders to sell some stock if they need to raise cash.



It will make it easier for the firm to raise funds. The firm would have a difficult time trying to sell stock privately to an investor who was not a family member. Outside investors would be more willing to purchase the stock of a publicly held corporation that must file financial reports with the regulators and provide continuous disclosure.



Going public will establish a value for the firm.

Disadvantages to going public:  

The firm will have to provide continuous disclosure with the regulators. There is a cost involved in preparing these reports.

The firm will have to disclose operating data to the public. Many small firms do not like having to do this, because such information is available to competitors. Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc. 14421

Also, some of the firm’s officers, directors, and major shareholders will have to disclose their stock holdings, making it easy for others to estimate their net worth. 

Managers of publicly owned corporations have a more difficult time engaging in deals that benefit them personally, such as paying themselves high salaries, hiring family members, and enjoying not-strictly-necessary, but tax-deductible, fringe benefits.



If the company is very small, its stock may not be traded actively and the market price may not reflect the stock’s true value.

The advantages of public ownership would be recognized by key employees, who would most likely be granted stock options, which would certainly be more valuable if the stock were publicly traded. e.

What are the steps of an initial public offering?

Answer: Select an investment banker, file a preliminary prospectus with the provincial securities commission, choose a price range for the preliminary, or ―red herring,‖ prospectus, go on a roadshow, and set a final price in the final prospectus. f.

Would the sale be on an underwritten or best efforts basis?

Answer: Most stock offerings are done on an underwritten basis, but the price is not set until the investment banker has checked investors for interest in the stock, and has received oral assurances of commitments at a price that will virtually guarantee the success of the offering barring a major stock market collapse. Best efforts IPOs are usually used with smaller, more risky companies. g.

The estimated pre-IPO value of equity in the company is about $63 million, and there are 4 million shares of existing shares of stock held by family members. The investment bank will charge a 7% spread, which is the difference between the price the new investor pays and the proceeds to the company. To net $18.6 million, what is the value of stock that must be sold? What is the total post-IPO value of equity? What percentage of this equity will the new investors require? How many shares will the new investors require? What is the estimated offer price per share?

Answer: To net $18.6 million, the company must issue shares worth $20 million: $18.6/(1 – 0.07) = $20.

The value of equity after the IPO is equal to the value before plus the net proceeds: Post-IPO equity = $63 + $20(1 – 0.07) = $81.6 million. The new owners are investing $20 million, so they require a share of the equity that is equal to their investment: Percent required by investors = $20/$81.6 = 24.51%. Number of shares new investors require: If 4 million shares represent 75.49% (1 – 0.2451) of the company, then x shares represent 24.51% of the company. x = (4 million × 0.2451)/0.7549 = 1,298,715 shares Estimated offering price per share = $20 million/1,298,715 shares = $15.40

What is a roadshow? What is book building?

Answer: The senior management team and investment banker make presentations to potential institutional investors. They usually visit 10 to 20 cities, and make 3 to 5 presentations in each city. Management can’t say anything that is not in the prospectus, because of a ―quiet period‖ from the time it makes the registration effective until 30 days after the stock begins trading. The purpose is to prevent select investors from getting information that is not available to other investors. During the roadshow, the investment bankers ask the investors to indicate how many shares they plan on buying. The banker records this in their book. The banker hopes for oversubscription. Based on demand, the banker sets the final offer price just before the stock is issued.

Describe the typical first-day returns of an IPO and the long-term returns to IPO investors.

Answer: First-day returns average approximately 4–12%, with some stocks having much higher returns. The investment banker has an incentive to set a low price, both to make its brokerage customers happy and to make it easy to sell the issue, whereas the firm would like to set as high a price as possible. Returns over the two-year period following the IPO are generally lower than for comparable firms, indicating that the offering price is too low, but that the first-day run-up is too high.

What are the direct and indirect costs of an IPO?

Answer: In this case, the underwriter charges a 7% fee, based on the offer price. In addition, there are direct costs to lawyers, accountants, printers, etc. That can easily total $400,000. Indirect costs include the money left on the table, which is equal to the difference between the offer price and end-of-first-day price, multiplied by the number of shares. Also, much of management’s time and attention is consumed by the IPO in the months preceding the IPO.

What are equity carve-outs?

Answer: Equity carve-outs are a special type of IPO in which a public company creates a new public company from one of its subsidiaries by issuing public stock in the subsidiary. The parent usually retains a controlling interest.

In what other ways are investment banks involved in issuing securities?

Answer: Investment bankers help companies with shelf registrations in which securities are registered but not all of the issue is sold at once. Instead, the company sells a percentage of the issue each time it needs to raise capital. Investment bankers also help place private and public debt issues. They also help in seasoned equity offers, in which a public firm issues additional shares of stock.

What is meant by ―going private‖? What are some advantages and disadvantages?

Answer: Going private is the reverse of going public. Typically, the managers of a firm team up with a small group of outside investors, who furnish most of the equity capital, and purchase all of the publicly held shares of the company. The new equity holders usually use a large amount of debt financing, up to 90%, to complete the purchase. Such a transaction is called a ―leveraged buyout (LBO).‖ Going private gives the managers greater incentives and more flexibility in running the company. It also removes the burden of regulatory filings, shareholder relations, annual reports, analyst meetings, and so on. The major disadvantage of going private is that it significantly limits the availability of new capital. Since the stock is not publicly traded, a new stock issue would not be practical, and, since such firms are normally leveraged to the hilt, it is tough to find additional debt financing. For this reason, it is common for firms that have recently gone private to sell off some assets to quickly reduce the debt burden to more conventional levels to give added financial flexibility. After several years of operating the business as a private firm, the owners typically go public again. At this time, the firm is presumably operating at its peak, and it will command top dollar compared to when it went private. In this way, the equity investors of the private firm are able to recover their investment and, hopefully, make a tidy profit.

Explain how firms manage the risk structure of their debt with project financing.

Answer: Project financings are arrangements used to finance mainly large capital projects such as energy explorations, oil tankers, refineries, utility power plants, and so on. Usually, one or more firms (sponsors) will provide the equity capital required by the project, while the rest of the project’s capital is supplied by lenders and lessors. The most important aspect of project financing is that the lenders and lessors do not have recourse against the sponsors; they must be repaid from the project’s cash flows and the equity cushion provided by the sponsors.

SOLUTIONS TO APPENDIX PROBLEMS 14A-1 a. Since the call premium is 9%, the total premium is 0.09($40,000,000) = $3,600,000. The premium is not tax deductible, therefore the after-tax cost is also $3,600,000. b. The dollar flotation cost on the new issue is 0.04($40,000,000) = $1,600,000. This cost is not immediately tax deductible, and hence the after-tax cost is also $1,600,000. (Note that the flotation cost can be amortized straight line over 5 years. The value of this tax savings will be calculated in Part d.) c. The cash outlay is $5,200,000, as shown below: Old issue call premium New issue flotation cost Net cash outlay

$ 3,600,000 1,600,000 $ 5,200,000

d. The new issue flotation costs of $1,600,000 would be amortized over 5 years. Thus, $1,600,000/5 = $320,000 would be expensed each year. The tax savings from this tax deduction is (0.30)$320,000 = $96,000 per year. e. The interest on the old issue is 0.09($40,000,000) = $3,600,000 annually, or $1,800,000 semiannually. Since interest payments are tax deductible, the after-tax semiannual amount is 0.7($1,800,000) = $1,260,000. The new issue carries a 6% coupon rate. Therefore, the annual interest would be 0.06($40,000,000) = $2,400,000, or $1,200,000 semiannually. The after-tax cost is thus 0.7($1,200,000) = $840,000. Thus, the after-tax net interest savings if refunding takes place would be $1,260,000 ─ $840,000 = $420,000 semiannually.

f. The cash flows are based on contractual obligations, and hence have about the same amount of risk as the firm's debt. Further, the cash flows are already net of taxes. Thus, the appropriate interest rate is GSS's after-tax cost of debt. (The source of the cash to fund the net investment outlay also influences the discount rate, but most firms use debt to finance this outlay, and, in this case, the discount rate should be the after-tax cost of debt.) Finally, since we are valuing future flows, the appropriate debt cost is today's cost, or the cost of the new issue, and not the cost of debt floated 5 years ago. Thus, the appropriate discount rate is 0.7(6%) = 4.2% annually, or 2.1% per semiannual period. Using a financial calculator, input N = 40, I = 2.1, PMT = –420,000, FV = 0, PV = ? PV = $11,290,358 to calculate the present value of the interest savings. Input N = 5, I = 4.2, PMT = 96,000, FV = 0, PV = ? PV=$424,984 to calculate the present value of the flotation cost tax shield. g. The bond refunding would require a $5,200,000 net cash outlay, but it would produce $11,290,358 + $424,984 = $11,715,342 in net savings on a present value basis. Thus, the NPV of refunding is $6,515,342: PV of net benefits Cost Refunding NPV

$11,715,342 (5,200,000) $ 6,515,342

14A-2 Investment Outlay Call premium on the old bond ($90M × 7%) ($6,300,00 0) Flotation costs on new issue

(6,000,000 )

Extra interest paid on old issue ($90M ×

(420,000)

8%/12 × 0.70) Interest earned on short-term investment ($90M × 2%/12 × 0.70)

105,000

Total after-tax investment ($12,615,0 00) Annual Flotation Cost Tax Effects: t = 1 to 5 Annual tax savings from new issue flotation

$360,000

costs ($6M/5 × 0.30) Annual Interest Savings Due to Refunding: t = 1 to 25 Interest on old bond

($90M × 8% × 0.70)

5,040,000

Interest on new bond ($90M × 6.5% × 0.70) (4,095,000 ) Net interest savings

$945,000

PV of annual flotation cost savings (FV = 0, I = 6.5 × 0.70, N = 5,PMT = 360,000, PV =?) + PV of annual interest savings (FV = 0, I = 6.5 × 0.70, N = 25, PMT = 945,000, PV =?) +

$1,578,192

13,940,822

Total after-tax investment

(12,615,00 0)

NPV of bond refunding

= $2,904,014

South Tel should refund its outstanding bonds.

Chapter 15 Lease Financing ANSWERS TO END-OF-CHAPTER QUESTIONS 15-1

a. The lessee is the party leasing the property. The party receiving the payments from the lease (that is, the owner of the property) is the lessor. b. An operating lease is a lease that does not transfer substantially all the risks and rewards incidental to ownership of an asset to the lessee. An operating lease must have a term under 12 months or must be a low-value asset. A finance lease is a lease that transfers substantially all the risks and rewards incidental to ownership of an asset to the lessee. In a sale-and-leaseback arrangement, the firm owning the property sells it to another firm, often a financial institution, while simultaneously entering into an agreement to lease the property back from the firm. A sale-and-leaseback can be thought of as a type of financial lease. c. Off-balance sheet financing refers to the fact that leasing is a type of financing and for many years neither leased assets nor the liabilities under lease contracts appeared on the lessees’ balance sheets. Capitalizing a lease means that a lease is reported on the balance sheet. Calculate the present value of the lease’s future payments and add to the balance sheet as both an asset and a liability. d. International Financial Reporting Standard IFRS 16 sets out the conditions under which a lease must be capitalized, and the specific procedures to follow.

e. The residual value is the market value of the leased property at the expiration of the lease. The estimate of the residual value is one of the key elements in lease analysis. f.

The lessee’s analysis involves determining whether leasing an asset is less costly than buying the asset. The lessee will compare the present value cost of leasing the asset with the cost of purchasing the asset. If the present value cost of the lease is less than the cost of purchasing, the asset should be leased. The lessor’s analysis involves comparing the cost of the asset to the present value of all the cash flows derived from leasing the assets. If the cost of the asset is less than the PV of the cash flows to the lessor, the NPV to the lessor is positive, and the lessor should go ahead and lease out the asset. g. The net advantage to leasing (NAL) gives the dollar value of the lease to the lessee. It is, in a sense, the NPV of leasing versus owning.

15-2

An operating lease is a lease that does not transfer substantially all the risks and rewards incidental to ownership of an asset to the lessee. An operating lease must have a term under 12 months or must be a low-value asset. A finance lease is a lease that transfers substantially all the risks and rewards incidental to ownership of an asset to the lessee. Given the recently changed criteria for determining an operating lease, the vast majority of leases will now be classified as finance leases. An operating lease would more likely be used for office furniture.

15-3

You would expect to find that lessees, in general, are in relatively low income-tax brackets, while lessors tend to be in high tax brackets. The reason for this is that owning tends to provide tax shelters in the early years of a project’s life. These tax shelters are more valuable to taxpayers in high brackets. However, tax laws have reduced the depreciation benefits of owning, so tax rate differentials are less important now than in the past.

15-4

a. If the lessor’s tax rate increases the value of the lessor’s A tax shield on that asset will increase and the lease rate could fall. Higher tax rates will also lower the after-tax cost of borrowing (i.e., the discount rate), also pushing the lease cost downward. Somewhat offsetting the effect is more tax paid on the lease payments. b. If the lessor’s borrowing cost increases (and as a result its discount rate increases), it will lower the present value of the future lease payments, therefore pushing up the lease rates. c. If the residual value of the asset increases, it will

lower the lease rates, as the lessor will derive more value from the resale value of the asset. d. If the cost of the asset is greater, the lessor will increase lease rates to cover the higher purchase cost. 15-5

A cancellation clause would reduce the risk to the lessee since the firm would be allowed to terminate the lease at any point. Since the lease is less risky than a standard financial lease, and less risky than straight debt, which cannot usually be prepaid without a prepayment charge, the discount rate on the cost of leasing might be adjusted to reflect lower risk. (Note that this requires increasing the discount rate since cash outflows are being discounted.) The effect on the lessor is just the opposite—risk is increased. (Note that this would also require an increase in the lessor’s discount rate.)

15-6

Lease payments, like depreciation, are deductible for tax purposes. If a 20-year asset were depreciated over a 20-year life, depreciation charges would be 1/20 per year (more if a declining rate CCA method were used). However, if the asset were leased for 3 years, tax deductions would be 1/3 each year for 3 years. Thus, the tax deductions would be greatly accelerated. The same total taxes would be paid over the 20 years, but because of the high deductions in the early years, taxes would be deferred more under the lease, and the PV of the future taxes would be reduced under the lease.

SOLUTIONS TO END-OF-CHAPTER PROBLEMS 15-1 a. The initial lease liability is equal to the present value of the lease payments when discounted at the cost of debt. Set N = 3, I/YR = 12, PMT = 15000, FV = 0, and solve for PV = $36,027.47 b.

The initial right-of-use asset value is equal to the initial lease liability: $36,027.47

The expense will be shown in two items, interest expense and depreciation expense: Year 1 imputed interest= (Cost of debt)(Initial lease liability) = 12%($36,027.47) = $4,323.30 Year 1 depreciation = Asset value/Lease term = $36,027.47/3 = $12,009.16 per year

The Year 1 lease liability is Year 1 lease liability = Initial lease liability + Imputed interest – Payment = $36,027.47 + $4,323.30 – $15,000 = $25,350.77.

The Year 1 right-of-use asset = $36,027.47 – $12,009.16 = $24,018.31.

The Year 1 cash payment is $15,000, before tax and its after-tax cash payment is $10,500: ($15,000 × (1 – 0.70))

15-2

The initial lease liability is equal to the present value of the lease payments when discounted at the cost of debt. Set N = 2, I/YR = 8, PMT = 55000, FV = 0, and solve for PV = $98,079.56.

The initial right-of-use asset is equal to the initial lease liability: $98,079.56.

The Year 1 reported interest expense is equal to cost of debt multiplied by the initial lease liability: 8%($98,079.56.) = $7,846.36.

The Year 1 depreciation expense is: Year 1 depreciation expense = (Initial lease liability)/(Term of lease) = ($98,079.56)/2 = $49,039.78

The Year 1 lease liability is: Year 1 lease liability = Initial lease liability + Interest expense – Payment = $98,079.56 + $7,846.36 – $55,000 = $50,925.92

The Year 1 right-of-use asset is: Year 1 right-of-use asset = Initial right-of-use asset – Depreciation expense = $98,079.56 – $49,039.78 = $49,039.78

15-3 Asset After-tax lease pmt. CCA tax shield Net cash flows

Year 0 $ 60,000 (21,700) $ 38,300

Year 1

Year 2

($21,700) (9,000) ($30,700)

($9,000) ($9,000)

C0 = 38300, C1 = –30700, C2 = –9000, I/Y = 6.3, solve for the NPV = $1,454.65 Net advantage to leasing = $1,454.65 It is cheaper to lease than to buy. After-tax lease payment = $31,000(1 – T) = $31,000(0.7) = $21,700 CCA tax shield = CCA x T = $30,000(0.30) = $9,000 Discount rate = cost of borrowing (1 – T) = 9%(0.70) = 6.3%

15-4 Discount rate = 12%(1 – 0.25) = 9%

(thousands of dollars)

Year 0

Year 1

Year 2

Year 3

Year 4

I. Cost of owning Machinery cost

$1,500

II. Leasing cash flows Lease payment

(400)

($400)

Tax savings from lease payment

100

Residual value

(250)

Net cash flow

($300)

II. PV cost of leasing at 9%

($300)

($250)

($1,236.495)

CCA tax ownership benefit

CdT 1 + 0.5r SdT 1 ( ) ( )–( ) ( ) r+d 1+r r+d (1 + r)n

Present value of the CCA tax shield

1,500 0.30 0.25 1 + 0.5 0.09 250 0.30 0.25 1 ) ( )–( ) ( ) = $242.494 (1 + 0.09)4 0.09 + 0.30 1+ 0.09 0.09 + 0.30 III. Net Advantage to Leasing (NAL) Machinery cost

$1,500

PV of leasing cash flows

(1,236.495)

PV of CCA tax shield

(242.494)

NAL

$ 21.012

NAL ($1 rounding) = $21,012 > 0, therefore Deer Mountain Copper should lease the machinery.

15-5

a. Discount rate = 7%(1 – 0.28) = 5.04%

Year 0

Year 1

Year 2

Year 3

Year 4

Year 5

($18,300)

5,124

Maintenance cost savings

880

$880

Taxes on maintenance costs

(246)

($12,542)

$634

I. Cost of owning Copiers' cost

$75,000

II. Leasing cash flows Lease payment Tax savings from lease payment

Net cash flow

($13,176)

PV cost of leasing at 5.04%

($57,112)

CCA tax ownership benefit

Present value of the CCA tax shield= (

CdT 1 + 0.5r 75,000 0.20 0.28 1 + 0.5 0.0504 ) ( )=( ) ( ) = $16 r+d 1+r 0.0504 + 0.20 1 + 0.0504

III. Net Advantage to Leasing (NAL) Machinery cost

$75,000

PV of leasing cash flows

(57,112)

PV of CCA tax shield

(16,371)

NAL

1,517

NAL > 0, therefore PGL should lease the copiers.

b. PV of Residual Value FV = 10,000, PMT = 0, N = 5, I/Y = 5.04 PV = ? = $7,820 $10,000 0.20 0.28 1 Reduction in CCA tax shield ( )( ) = 1,749 0.0504 + 0.20 (1 + 0.0504)5 NAL = $1,517 + ($7,820) + $1,749 = ($4,554).

NAL < 0, do not lease.

c. New PV of the CCA tax shield: CdT 1 + 0.5r 75,000 0.50 0.28 1 + 0.5 0.0504 ( ) ( )=( ) ( ) = $18,619. r+d 1+r 0.0504 + 0.50 1 + 0.0504 NAL = $75,000 – $57,112 – $18,619 = –$731. NAL < 0, therefore do not lease. d. New discount rate = 5%(1 – 0.28) = 3.6% Using the same net cash flows as Part (a), calculate the PV of leasing cash flows: C0 = –13,176, C1 – C4 = –12,542, C5 = 634, I/Y = 3.6, PV = ? = $-58,603. New PV of the CCA tax shield: CdT 1 + 0.5r 75,000 0.20 0.28 1 + 0.5 0.036 ( ) ( )=( ) ( ) = $17,487 r+d 1+r 0.036 + 0.20 1 + 0.036 NAL = $75,000 – $58,603 – $17,487 = –$1,090. NAL < 0, therefore do not lease.

15-6

(Thousands of dollars)

Year 0

Year 1

Year 2

Year 3

I. Cost of owning Computer cost

$1,000

II. Leasing cash flows Lease payment

($320)

Tax savings from lease payment

108.8

Residual value Net cash flow PV cost of leasing at 9.24%a

($200) ($211.2)

($211.2)

($200)

($735)

CCA tax ownership benefit

Present value of the CCA tax shield = (

CdT 1 + 0.5r SdT 1 ) ( )–( ) ( ) r+d 1+r r+d (1 +r)n

1,000 0.45 0.34 1 + 0.5 0.0924 200 0.45 0.34 1 )–( ) = $227. ) ( ) ( (1 + 0.0924)3 0.0924 + 0.45 1 + 0.0924 0.0924 + 0.45

III. Net Advantage to Leasing (NAL) Computer equipment cost PV of leasing cash flows PV of CCA tax shield NAL

$1,000 (735) (227) $ 38

The NAL is positive and therefore Alumco should lease the computer equipment. a Discount rate = borrowing cost × (1 – T) = 14%(1 – 0.34) = 9.24% The maintenance cost is not considered as the lessee must pay for it regardless of purchase or lease. b. The cash flow lease payments are determined in advance and set under contract. Since they are similar in obligation to interest payments, they are discounted at the after-tax borrowing rate. However, the sale of a used asset 3 years in the future is uncertain. Though one can estimate the Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc. 15-437

residual value of the equipment, the amount the asset ultimately sells for may be significantly more or less. As a result, it may be appropriate to discount the residual value cash flow at a higher discount rate, reflecting the riskiness of that cash flow. 15-7

a. Cost of software: $900,000 PV of lease payments: After-tax lease payments = $335,000 After-tax cost of borrowing = 13% (Payments are at beginning of the period, that is an annuity due)

PMT = 335,000, FV = 0, N = 3, I/Y = 13, CPT PV = 893,814 PVCCATS: 0 PV of Residual Value: 0 NAL = $900,000 – $893,814 = $6,186. b. Cost of software: $900,000 PV of after-tax lease payments: After-tax lease payments = $335,000(1 – 0.35) = $217,750 After-tax cost of borrowing = 7%(1 – 0.35) = 4.55% (Annuity due) PMT = 217,750, FV = 0, N = 3, I/Y = 4.55, CPT PV = 625,233 PVCCATS: (

900,000 1 0.35 1 + 0.5 0.0455 )( ) = $294,735 1 + 0.0455 1 + 0.0455

PV of residual value: 0 NPVLease = ($625,233 + $294,735) – $900,000 = $19,968.

c. Minimum acceptable lease payment to lessor when NPV of lease = 0. PV of leasing benefits – Cost of asset = 0 (PV of after-tax lease payments + PVCCATS) – Cost of asset = 0 (PV of after-tax lease payments + $294,735) – $900,000 = 0 PV of after-tax lease payments = $605,265 Annual after-tax lease payments: (Annuity due) PV = 605,265, FV = 0, N = 3, I/Y = 4.55, CPT PMT = 210,796 Annual pre-tax lease payments: $210,796 = $324,301 (1 – 0.35)

SOLUTION TO SPREADSHEET PROBLEM 15-8

The detailed solution for the spreadsheet problem is in the file Ch 15 Build a Model Solution.xlsx and is available on the textbook’s website.

MINI CASE The detailed solution for the Mini Case is also available in spreadsheet format, in the file Brigham4Ce_MiniCase_Ch15.xlsx, and is available on the textbook’s website. Big Beta Securities Inc. has decided to acquire a new market data and quotation system for its Vancouver home office. The system receives current market prices and other information from several online data services and then either displays the information on a screen or stores it for later retrieval by the firm’s brokers. The system also permits customers to call up current quotes on terminals in the lobby. The equipment costs $1,000,000, and, if it was purchased, Big Beta could obtain a term loan for the full purchase price at a 10% interest rate. The equipment falls into Class 45 with a declining balance CCA rate of 45%. If the system was purchased, a 4-year maintenance contract could be obtained at a cost of $20,000 per year, payable at the beginning of each year. The equipment would be sold after 4 years, and the best estimate of its residual value at that time is $200,000. However, since real-time display system technology is changing rapidly, the actual residual value is uncertain. As an alternative to the borrow-and-buy plan, the equipment manufacturer informed Big Beta that Consolidated Leasing would be willing to write a 4-year lease on the equipment, including maintenance, for payments of $260,000 at the beginning of each year. Big Beta’s tax rate is 35%. You have been asked to analyze the lease-versus-purchase decision, and in the process to answer the following questions:

1. Who are the two parties to a lease transaction?

Answer: The two parties are the lessee, who uses the asset, and the lessor, who owns the asset.

2. What are the primary types of leases, and what are their characteristics?

Answer: The primary types of leases are operating, finance, and sale-and-leaseback. An operating lease is a lease that does not transfer substantially all the risks and rewards incidental to ownership of an asset to the lessee. An operating lease must have a term under 12 months or must be a low-value asset. A finance lease is a lease that transfers substantially all the risks and rewards incidental to ownership of an asset to the lessee. In a sale-and-leaseback arrangement, the firm owning the property sells it to another firm, often a financial institution, while simultaneously entering into an agreement to lease the property back from the firm. A sale-and-leaseback can be thought of as a type of finance lease.

3. How are leases presented on a firm’s income statement and balance sheet?

Answer: Virtually all leases must be capitalized as finance leases, and they are shown directly on the balance sheet. The lease is shown as a right-of-use asset on the asset side of the balance sheet and a lease liability on the liability side of the balance sheet. On the income statement, the lease expense is put into two expenses—the depreciation for the period and the imputed interest expense. If it is an operating lease, it is listed only in the financial statements’ footnotes. On the income statement the actual lease charge is expensed.

4. What effect does leasing have on a firm’s capital structure?

Answer: Leasing is a substitute for debt financing, so leasing increases a firm’s financial leverage. One dollar of lease financing displaces one dollar of debt financing. For example, suppose a firm has a target capital structure of 50% debt and 50% equity. If 50% of its assets are leased, then the remainder must be financed with equity.

Explain the rationale for the discount rate you used to find the PV.

Answer: The proper discount rate depends on (1) the riskiness of the cash flow stream and (2) the general level of interest rates. The loan payments and the maintenance costs are fixed by contract, and hence are not at all risky. The depreciation deductions are also ―locked in,‖ but the tax rate could change. Thus, depreciation cash flows (tax savings) are not totally certain, but they are relatively certain. Only the residual value is highly uncertain. On balance, and in relation to cash flows associated with such activities as capital budgeting, we conclude that the cash flows in the time line are relatively safe, so they should be discounted at a relatively low rate. In fact, they have about the same degree of riskiness as the firm’s debt cash flows (which also have some tax-rate risk, and which are also contractual in nature). Therefore, we conclude that leasing has about the same impact on the firm’s financial risk as debt financing, so the appropriate discount rate is Big Beta’s cost of debt. (Note: The larger the residual value in relation to the other flows, the less justifiable this statement is.) Further, since the cash flows are stated on an after-tax basis, the rate should be the after-tax cost of debt. Big Beta’s before-tax debt cost is 10%, and since the firm is in the 35% tax bracket, its after-tax cost is 10.0%(1 – 0.35) = 6.5%. Therefore, we use 6.5% as the discount rate. Note: When we have been engaged as consultants on lease-versus-buy decisions, the proper discount rate is often discussed. We know of no way to specify exactly how to adjust for the salvage (residual) value risk. Therefore, what we have been doing is running the analysis on a spreadsheet model and making a data table where the dependent variable is the NAL as calculated below and the independent variable is the discount rate. Then, we produce a graph that shows the range of discount rates over which the NAL is positive. This usually heads off problems over the proper discount rate.

Calculate the net advantage to leasing (NAL). Does your analysis indicate that Big Beta should buy or lease the equipment? Explain. (Thousands of dollars)

Year 0

Year 1

Year 2

Year 3

Year 4

I. Cost of owning 1.

Equipment cost

$1,000

II. Leasing cash flows 2.

Lease payment

($260)

Tax savings from lease payment

Maintenance costs saved

Taxes on maintenance costs

(7)

Residual value

Net cash flow (lines 2 – 6)

($156)

PV cost of leasing at 6.5%

($724.627)

($200) ($156)

($156)

($200)

CCA tax ownership benefit

Present value of the CCA tax shield = (

CdT 1 + 0.5r SdT 1 ) ( )–( ) ( ) r+d 1+r r+d (1 + r)n

1,000 0.45 0.35 1 + 0.5 0.065 200 0.45 0.35 1 ) ( )–( ) ( ) = $248.948. (1 + 0.065)4 0.065 + 0.45 1 + 0.065 0.065 + 0.45

III. Net Advantage to Leasing (NAL) Equipment cost

$1,000

PV of leasing cash flows

(724.627)

PV of CCA tax shield

(248.948)

NAL

26.425

The NAL is positive by $26,425, and Big Beta Securities should lease the equipment.

Now assume that the equipment’s residual value could be as low as $0 or as high as $400,000, but that $200,000 is the expected value. Since the residual value is riskier than the other cash flows in the analysis, this differential risk should be incorporated into the analysis. Describe how this could be accomplished. (No calculations are necessary, but explain how you would modify the analysis if calculations were required.) What effect would increased uncertainty about the residual value have on Big Beta’s lease-versus-purchase decision?

Answer: We account for increased risk by increasing the rate used to discount the residual value cash flow, resulting in a lower present value of the residual cash flow. This leads to a higher cost of owning, so the greater the risk of the residual value, the higher the cost of owning, and the more attractive leasing becomes. In effect, the lease transaction passes the risk associated with the residual value from the lessee/user to the lessor/owner. Of course, the lessor recognizes this, and as a result, assets with highly uncertain residual values will carry higher lease payments than assets with relatively certain residual values. However, the most successful leasing companies have developed expertise in renovating and disposing of used equipment, and this gives them an advantage over most lessees in reducing residual value risks. Further, leasing companies usually deal with a wide array of assets, so residual value estimates that are too high on one asset may be offset by estimates that are too low on another. e.

The lessee compares the cost of owning the equipment with the cost of leasing it. Now put yourself in the lessor’s shoes. In a few sentences, how should you analyze the decision to write or not write the lease?

Answer: The lessor should view ―writing‖ the lease as an investment, so the lessor should compare the return on the lease with returns available on alternative investments of similar risk. f.

1. Assume that the lease payments were actually $280,000 per year, that Consolidated Leasing is also in the 35% tax bracket, and that it also forecasts a $200,000 residual value. Also, to furnish the maintenance support, Consolidated would have to purchase a maintenance contract from the manufacturer at the same $20,000 annual cost, again paid in advance. Consolidated Leasing can obtain an expected 10% pre-tax return on investments of similar risk. What would Consolidated’s NPV of leasing be under these conditions?

Answer: The lessor must invest $1,000,000 to buy the equipment, but then it expects to receive tax benefits and lease payments over the life of the lease. Note that the CCA tax shield calculated earlier also applies to the lessor, but in this case it is a benefit due to ownership. Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc. 16-445

(Thousands of dollars)

Year 0

Year 1

Year 2

Year 3

Year 4

I. Cost of owning 1.

Equipment cost

($1,000)

II. Leasing cash flows 2.

Lease payment

$280

Tax savings from lease payment

(98)

Maintenance costs

(20)

Tax savings on maintenance costs

Residual value

Net cash flow (lines 2 – 6)

$169

PV cost of leasing at 6.5%

$772.057

$200 $169

$169

$200

CCA tax ownership benefit

Present value of the CCA tax shield = (

CdT 1 + 0.5r SdT 1 ) ( )–( ) ( ) r+d 1+r r+d (1 + r)n

1,000 0.45 0.35 1 + 0.5 0.065 200 0.45 0.35 1 ) ( )–( ) ( ) = $248.948. (1 + 0.065)4 0.065 + 0.45 1 + 0.065 0.065 + 0.45

III. Net Present Value of the Lease Equipment cost

($ 1,000)

PV of leasing cash flows

772.057

PV of CCA tax shield

248.948

NPV to the lessor

21.005

2. What do you think the lessor’s NPV would be if the lease payment were set at $260,000 per year? (Hint: The lessor’s cash flows would be a ―mirror image‖ of the lessee’s cash flows.)

Answer: With lease payments of $260,000, the lessor’s cash flows would be the ―mirror image‖ of the lessee’s NAL—the same dollars, but with signs reversed. Therefore, the lessor’s NPV would be –$26,425, the negative of the lessee’s NAL. To verify this, note that a $20,000 reduction in each lease payment would reduce the lessor’s inflows by $20,000(0.65) = $13,000 at the beginning of each year. The PV of this annuity is $47,430, so the lessor’s NPV would be $21,005 – $47,430 = –$26,425. g.

Big Beta’s management has been considering moving to a new downtown location, and they are concerned that these plans may come to fruition prior to the expiration of the lease. If the move occurs, Big Beta would buy or lease an entirely new set of equipment, and hence management would like to include a cancellation clause in the lease contract. What effect would such a clause have on the riskiness of the lease from Big Beta’s standpoint? From the lessor’s standpoint? If you were the lessor, would you insist on changing any of the lease terms if a cancellation clause were added? Should the cancellation clause contain any restrictive covenants and/or penalties of the type contained in bond indentures or provisions similar to call premiums?

Answer: A cancellation clause would lower the risk of the lease to Big Beta, the lessee, because then it would not be obligated to make the lease payments for the entire term of the lease. If its situation changed, so that Big Beta either no longer needed the equipment or else wanted to change to a more technologically advanced product, then it could terminate the lease. However, a cancellation clause would make the contract more risky for the lessor. Now the lessor bears not only the final residual value risk, but also the uncertainty of when the contract will be terminated. To account for the additional risk, the lessor would undoubtedly increase the annual lease payment. Additionally, the lessor might include clauses that would prohibit cancellation for some period and/or impose a penalty fee for early cancellation. The decision as to whether or not to include a cancellation clause would depend on who was in a better position to bear the residual value risk, the lessee or the lessor. Often lessors have more expertise at disposing of used equipment than lessees, and thus they are willing to include cancellation clauses without major increases in the required lease payments.

ANSWERS TO END-OF-CHAPTER QUESTIONS 16-1

a. A warrant is an option issued by a company to buy a stated number of shares of stock at a specified price. Warrants are generally distributed with debt, or preferred stock, to induce investors to buy those securities at lower cost. A detachable warrant is one that can be detached and traded separately from the underlying security. Most warrants are detachable. b. A stepped-up price is a provision in a warrant that increases the strike price over time. This provision is included to prod owners into exercising their warrants. c. Convertible securities are bonds or preferred stocks that can be exchanged for (converted into) common stock, under specific terms, at the option of the holder. Unlike the exercise of warrants, conversion of a convertible security does not provide additional capital to the issuer. d. The conversion ratio is the number of shares of common stock received upon conversion of one convertible security. The conversion price is the effective price per share of stock if conversion occurs. Thus, the conversion price is the par value of the convertible security divided by the conversion ratio. The conversion value is the value of the stock that the investor would receive if conversion occurred. Thus, the conversion value is the market price per share times the conversion ratio. e. A sweetener is a feature that makes a security more attractive to some investors, thereby inducing them to accept a lower current yield. Convertible features and warrants are examples of sweeteners.

16-2

The trend in stock prices subsequent to an issue influences whether or not a convertible issue will be converted, but conversion itself typically does not provide a firm with additional funds. Indirectly, however, conversion may make it easier for a firm to get additional funds by lowering the debt ratio, thus making it easier for the firm to borrow. In the case of warrants, on the other hand, if the price of the stock goes up sufficiently, the warrants are likely to be exercised and thus to bring in additional funds directly.

16-3

Either warrants or convertibles could be used by a firm that expects to need additional financing in the future—warrants, because when they are exercised, additional funds will be brought into the firm directly; convertibles, because when they are converted, the equity base is expanded and debt can be sold more easily. However, a firm that does not have additional funds requirements would not want to use warrants.

16-4

The statement is made often. It is not really true, as a convertible’s issue price reflects the underlying stock’s present price. Further, when the bond or preferred stock is converted, the holder receives shares valued at the then-existing price, but effectively pays less than the market price for those shares.

16-5

a. The value of a warrant depends primarily on the expected growth of the underlying stock’s price. This growth, in turn, depends in a major way on the plowback of earnings; the higher the dividend payout, the lower the retention (or plowback) rate; hence, the slower the growth rate. Thus, warrant values will be higher, other things held constant, the smaller the firm’s dividend payout ratio. This effect is more pronounced for long-term than for short-term warrants. b. The same general arguments as in Part a hold for convertibles. If a convertible is selling above its conversion value, raising the dividend will lower growth prospects, and, at the same time, increase the ―cost‖ of holding convertibles (or warrants) in terms of forgone cash returns. Thus, raising the dividend payout rate before a convertible’s conversion value exceeds its call price will lower the probability of eventual conversion, but raising the dividend after a convertible’s conversion value exceeds its call price raises the probability that it will be converted soon. c. The same arguments as in Part b apply to warrants.

16-6

The convertible bond has an expected return that consists of an interest yield (10%) plus an expected capital gain. We know the expected capital gain must be at least 4%, because the total expected return on the convertible must be at least equal to that on the nonconvertible bond, 14%. In all likelihood, the expected return on the convertible would be higher than that on the straight bond, because a capital gains yield is riskier than an interest yield. The convertible would, therefore, probably be regarded as riskier than the straight bond, and rc would exceed rd. However, the convertible, with its interest yield, would probably be regarded as less risky than common stock. Therefore, rd < rc < rs.

16-7

A credit default swap is priced to be a fair trade. Therefore, the expected payments by the protection buyer (the protection payments) should be equal to the expected payments by the protection seller. The payment made by the protection seller in a credit default swap is made only if the underlying company (the company that the swap is based upon) defaults. Therefore, the first factor in pricing a swap is the expected probability of default. The payment by the protection seller, if any, is equal to the fall in value of the reference loan or the reference bond, which is equal to one minus the recovery value.

Thus any increase in the probability of default, or any decrease in the recovery value, will increase the price of the credit default swap protection. a. If expected recovery values decrease, then the size of any payment upon the occurrence of a credit event will increase. This will increase the protection payment. b. If the risk of default increases, then the probability of the protection seller making a payment will increase. This will increase the protection payment. c. If the company issues more debt, then the probability that it goes bankrupt increases. Additionally, if more debt is issued and the company goes bankrupt, then there are more creditors that need to be paid. Both of these factors will tend to increase the protection payment. d. If the company issues more equity, then the probability of default will decrease and thus the protection payment will be decreased. e. If the company’s bond prices increase, then that generally implies (assuming no changes in other interest rates) that the investors view the risk of default as being lower. This will decrease the protection payment. f. If the company’s stock price increases, this signals positive news about the company, which in turn implies a lower probability of default. This will decrease the protection payment. 16-8

The main factors in pricing a CDO are (a) the general probability and severity of default of each of the securities or loans that the CDO is based upon, and (b) the correlation of default between the different securities and loans. A CDO based on riskier loans will have higher protection payments being made to the SPV. This increases the maximum possible payouts to investors in the different tranches, but also increases the probability that cash flows will have to be diverted from payments to the tranche holders to make payments on defaults to the issuing bank. If the underlying securities and loans are highly correlated, this implies that the initial default of a small number of loans might foretell the default of a large number of loans. If a large number of loans default, then the SPV will have to divert cash flows to the underlying bank, and the investors in one or more of the tranches may not receive full payment.

SOLUTIONS TO END-OF-CHAPTER PROBLEMS 16-1

Bonds with warrants: $1,000 par value 15-year 5% coupon bonds with annual payments, trading for $1,000. Straight debt: $1,000 par value 15-year bonds with 7% annual coupon, also trading for $1,000.Value of warrants = ? Straight debt yield = Coupon rate = 7% since these bonds are trading at par. Note: If the bonds were not trading at par, you’d need to calculate the yield to maturity. Bonds with warrants: $1,000 = Bond + Warrants. This bond should be evaluated at 7% (since we know the straight debt trades at par) to determine its present value. Then the value of the warrants can be determined as the difference between $1,000 and the bond’s present value. N = 15; I/YR = rd = 7; PMT = 50, FV = 1000, and solve for PV = $817.84. Value of warrants = $1,000 – $817.84 = $182.16. 16-2

Convertible bond’s par value = $1,000; Conversion price, Pc = $20; CR = ? CR =

Par value $1,000 = = 50 shares Pc $20

16-3 a. $20,000,000 × 2.78% = $556,000 per year. b. If there is no credit event, then LifeCo does not make a payment to Big City Bank. c. $20,000,000 × (1 – 0.45) = $11,000,000.

16-4 Bond A: N = 10, PMT = 80, FV = 1,000, I = 10, and solve for PV = 877.11 ($1000 – $877.11 = $122.89)/$2.46 = 50 Bond B: $4.49 × 40 = $179.60, $1,000 – $179.60 = $820.40 PV = –820.40, N = 10, PMT = 40, FV = 1,000, and solve for I = 6.5% Bond C: $3.68 × 20 = $73.60, $1,000 – $73.60 = $926.40 PV = –926.40, N = 10, FV = 1,000, I = 6, and solve for PMT = 50 = 5%

16-5

a. Exercise value = Current price – Strike price. Current Price $20 25 30 100

Strike Price $25 25 25 25

Exercise Value $0 0 5 75

b. No precise answers are possible, but some ―reasonable‖ warrant prices are as follows: Current Stock Price $10 15 20 85 c. (1)

Warrant Price $2 4 7 71

Time Value $7 4 2 1

The longer the life, the higher the warrant value.

(2)

The more variable the stock price, the higher the warrant value.

(3)

The higher the expected EPS growth rate, the higher the warrant price.

(4)

Going from 0% to 100% payout would have two possible effects. First, it might affect the price of the stock, causing a change in the formula value of the warrant; however, it is not at all clear that the stock price would change, let alone what the change would be. Second, and more important here, the increase in the payout ratio drastically lowers the expected growth rate. This reduces the chance of the stock going up in the future. This lowers the expected value of the warrant, hence the premium and the price of the warrant.

d. VPackage = $1,000 =

= VB + 50($3)

VB = $1,000 − $150 = $850. 20

$850 = 

t 1 (1  rd )



$1,000 (1  rd ) 20

= 

t 1 (1.12)



$1,000 (1.12) 20

With a financial calculator, N = 20; I/YR = 12; PV = −850; FV = 1000; solve for PMT = 99.92. Therefore, the company would set a coupon interest rate of 9.99%, producing an annual interest payment of $99.90 per $1,000 par value bond with warrants.

16-6

a. A 10% premium results in a conversion price of $42(1.10) = $46.20, while a 30% premium leads to a conversion price of $42(1.30) = $54.60. Investment bankers often use the rule of thumb that the premium over the present price should be in the range of 10% to 30%. If the firm’s growth rate is low, the premium would be closer to 10%, while a high growth rate firm would command a premium closer to 30%. b. Yes, to be able to force conversion if the market rises above the call price. If, in fact, the 10% growth estimate is correct, in 4 years the earnings would be $3(1.10)4 = $4.39. If the P/E ratio remains at 14, the stock price will go to $4.39(14) = $61.46, making forced conversion possible even if a 30% premium is set.

16-7

a. Number of years until call: Conversion Price = $1,000/55 = $18.18 P0 = $18.18/1.25 = $14.54 Conversion Value = 55 × $14.54 = $799.70 PV = –799.70, I/YR = 7.8, PMT = 0, FV = 1,250, N = ? = 5.9 ≈ 6 years b. Return to convertible bondholder: Expected stock price in year 6 = $14.54(1.08)6 = $23.07 Conversion value in year 6 = $23.07 × 55 = $1,269 PV = –1,000, PMT = 60, N = 6, FV = 1,269, I/YR = ? = 9.53%.

16-8 Issuer

Coupon

Maturity

Price

YTM

Premium

Conversion

Ratio

Price

Company A

4.50%

Dec 2022

101.00

3.81%

161%

1.13

88.15

$34.31

Company B

5.50

Dec 2025

104.00

4.51

11.8

6.41

15.60

$14.51

Company C

5.00

Dec 2023

104.10

3.28

35.7

4.17

24.00

$18.39

Company D

6.00

Dec 2026

106.00

4.75

47.3

15.75

6.35

$4.57

Company E

5.50

Dec 2022

110.00

-1.09

-16.4

_7.14

14.00

$18.42

Company F

4.60

Dec 2023

120.00

-3.04

-1.26

_0.932

107.30

$130.39

Company G

4.75

Dec 2024

102.25

4.05

46.3

12.11

8.26

$5.77

FV = 100, I/YR = 3.81/2, PMT = 4.50/2, N = 3, PV = ? = 101

PV = -104, FV = 100, PMT = 5.5/2, N = 9, I/YR = ? = 2.254 YTM = 2.254 × 2 = 4.51% Conversion price =100/6.41 = 15.60

Conversion premium = (Market price of convertible – Conversion value)/Conversion value [$104.10 – (4.17 × $18.39)]/(4.17 × $18.39) = 35.7%

Conversion price = $100/15.75 = $6.35 [$106 – (15.75× Stock price)]/(15.75 × Stock price) = 0.473 $106 – 15.75(Stock price) = 7.45(Stock price) 23.2(Stock price) = $106 Stock price = $4.57

Conversion ratio = Initial bond price/conversion price = $100/$14 = 7.14 PV = –110, FV = 100, PMT = 5.50/2, N = 3, I/YR = ? = –0.547 YTM = −0.547 × 2 = –1.09%

Conversion price = $100/0.932 = $107.30 [$120 – (0.932 × Stock price)]/(0.932(Stock price) = –0.0126 $120 – 0.932(Stock price) = –0.01174(Stock price) 0.9203(Stock price) = 120 Stock price = $130.39

Conversion premium = (Market price of convertible – Conversion value)/Conversion value [$102.25 – (12.11× $5.77)]/(12.11 × $5.77) = 46.3%

16-9

a. The premium of the conversion price over the stock price was 14.1%: $62.75/$55 – 1.0 = 0.141 = 14.1%. b. The before-tax interest savings is calculated as follows: $400,000,000(0.0875 – 0.0575) = $12 million per year. However, the after-tax interest savings would be more relevant to the firm and would be calculated as $12,000,000(1 – T). c. At the time of issue, the value of the bond as a straight bond was $669.11, calculated as follows: N = 40, I/YR = 8.75, PV = ?, PMT = 57.5, FV = 1000. Solving, PV = –669.11. Notice that this implies that the value of the conversion feature at the time of issue was $330.89 = $1,000 – $669.11. d. If interest rates had not changed, then the value of the straight bond 15 years after issue would have been $699.25, calculated as follows: N = 25, I/YR = 8.75, PV = ?, PMT = 57.5, FV = 1000. Solving, PV = –699.25. Assuming that the stock had not gone above $62.75 during the 15 years after it was issued, the bond would not have been converted. For example, if a bondholder converted the bond, the bondholder would receive about 15.9 shares of stock per bond, calculated as follows: Conversion ratio = CR = $1,000/$62.75 = 15.936255 shares. If the stock price is $32.75, then the value of the bond in conversion is 15.936255($32.75) = $521.91. Because the value in conversion is less than the value as a bond, investors would not wish to convert the bond. e. The straight-bond value would have increased from $669.11 at the time of issue to $699.25 15 years later, as calculated above, due to the fact that the bonds are closer to maturity (because a bond’s value approaches its par value as it gets closer to maturity). However, the value of the conversion feature would have fallen sharply, for two reasons. First, the stock price fell from $55 to $32.75, and a decrease in stock price hurts the value of an option. Second, the time until maturity for the conversion fell from 40 years to 25 years, and a reduction in the remaining time to exercise an option hurts its value. Therefore, the bonds probably would have fallen below the $1,000 issue price. f. Had the rate of interest fallen to 5.75%, which is the coupon rate on the bonds, then their straight-bond value would be that of a par bond, which is $1,000. This can also

be calculated as follows: N = 25, I/YR = 5.75, PV = ?, PMT = 57.5, FV = 1000. Solving, PV = –1,000. The value of the bond in conversion is 15.936255($32.75) = $521.91. Although the value of the conversion feature would have dropped in value due to the decline in stock price and the decrease in the remaining time for the conversion to be exercised, the value of the conversion feature would still have a positive value (because an option value can never be zero or below). Therefore, the bonds would probably have a price slightly above their par value of $1,000. 16-10 a.

Balance Sheet Alternative 1

Total assets

$800,000

LOC Other current liabilities Long-term debt Common stock, par $1 Paid-in capital Retained earnings Total claims

$0 $150,000 – 162,500 437,500 50,000 $800,000

Notes: Total assets are equal to the previous total assets plus the additional new assets. The additional new assets are equal to the total proceeds minus the amount used to pay off the LOC: $500,000 – $250,000 = $250,000. The new total assets are equal to the old TA ($550,000) plus the new assets ($250,000), which is $800,000. The LOC is zero because it is paid off with the new stock proceeds. Other current liabilities, long-term debt, and retained earnings are all unchanged. Number of new shares of common is equal to the new financing ($500,000) divided by the issue price ($8) = $500,000/$8 = 62,500. New additional common stock par value is equal to the number of new shares multiplied by the $1 par value: 62,500($1) = $62,500. Resulting common stock at par is sum of previous common stock at par and new additional common stock at par: $100,000 + $62,500 = $162,500. Additional paid-in capital is equal to the value of the stock issue ($500,000) minus the additional common stock at par: $500,000 − $62,500 = $437,500. There was no previous paid-in capital so this is the total paid-in capital.

Alternative 2

Total assets

$ 800,000

LOC Other current liabilities Long-term debt Common stock, par $1 Paid-in capital Retained earnings Total claims

$0 $ 150,000 – 150,000 450,000 50,000 $ 800,000

Notes: Total assets are equal to the previous total assets plus the additional new assets. The additional new assets are equal to the total proceeds minus the amount used to pay off the LOC: $500,000 – $250,000 = $250,000. The new total assets are equal to the old TA ($550,000) plus the new assets ($250,000), which is $800,000. The LOC is zero because it is paid off with new stock proceeds. Other current liabilities, long-term debt, and retained earnings are unchanged. The number of bonds that were issued is equal to the total proceeds ($500,000) divided by the par value ($1,000): $500,000/$1,000 = 500. Number of new shares of common is equal to the number of bonds (500) multiplied by the conversion ratio (100): 500(100) = 50,000. New additional common stock par value is equal to the number of new shares multiplied by the $1 par value: 50,000($1) = $50,000. Resulting common stock at par is sum of previous common stock at par and new additional common stock at par: $100,000 + $50,000 = $150,000. Additional paid in capital is equal to the value of the total proceeds ($500,000) minus the additional common stock at par: $500,000 − $50,000 = $450,000. There was no previous paid-in capital so this is the total paid-in capital.

Alternative 3

Total assets

$1,300,000

LOC Other current liabilities Long-term debt (8%) Common stock, par $1 Paid-in capital Retained earnings Total claims

$0 $ 150,000 500,000 150,000 450,000 50,000 $1,300,000

Notes: Total assets are equal to the previous total assets plus the additional new assets. The additional new assets are equal to the previous total debt proceeds ($500,000) plus the proceeds from exercising the warrants ($500,000) minus the amount used to pay off the LOC: $500,000 + $500,000 – $250,000 = $750,000. The new total assets are equal to the old TA ($550,000) plus the new assets ($750,000), which is $1,300,000. The LOC is zero because it is paid off with new stock proceeds. Other current liabilities and retained earnings are unchanged. The value of the bonds is equal to the amount that was raised when the bonds were sold, $500,000. The number of bonds that were issued is equal to the total proceeds ($500,000) divided by the par value ($1,000): $500,000/$1,000 = 500. Number of new shares of common is equal to the number of bonds (500) multiplied by the number of warrants per bond (100): 500(100) = 50,000. The total proceeds from exercise of the warrants is the number of warrants (50,000) multiplied by the strike price per warrant: 50,000($10) = $500,000. New additional common stock par value is equal to the number of new shares multiplied by the $1 par value: 50,000($1) = $50,000. Resulting common stock at par is sum of previous common stock at par and new additional common stock at par: $100,000 + $50,000 = $150,000. Additional paid-in capital is equal to the value of the total proceeds from the warrant exercise minus the common stock at par ($50,000): $500,000 − $50,000 = $450,000. There was no previous paid-in capital so this is the total paid-in capital.

b. Number of shares Total shares Percent ownership

Original 80,000 100,000 80%

Plan 1 80,000 162,500 49%

Plan 2 80,000 150,000 53%

Plan 3 80,000 150,000 53%

Total assets

Original Plan 1 Plan 2 Plan 3 $550,000 $800,000 $800,000 $1,300,000

EBIT Interest EBT Taxes (40%) Net income

$110,000 $160,000 $160,000 20,000 0 0 90,000 160,000 160,000 36,000 64,000 64,000 $ 54,000 $ 96,000 $ 96,000

$260,000 40,000 220,000 88,000 $132,000

Number of shares Earnings per share

100,000 $0.54

150,000 $0,88

d. Total liabilities TL/TA

162,500 $0.59

150,000 $0.64

Original $400,000 73%

Plan 1 $150,000 19%

Plan 2 Plan 3 $150,000 $ 650,000 19% 50%

e. Alternative 1 results in loss of control (to 49%) for the firm. Under it, he loses his majority of shares outstanding. Indicated earnings per share increase, and the debt ratio is reduced considerably (by 54%). Alternative 2 results in maintaining control (53%) for the firm. Earnings per share increase, while a reduction in the debt ratio like that in Alternative 1 occurs. Under Alternative 3 there is also maintenance of control (53%) for the firm. This plan results in the highest earnings per share ($1.10), which is from the original earnings per share. The debt ratio is reduced to 50%. Conclusions. If the assumptions of the problem are borne out in fact, Alternative 1 is inferior to 2, since the loss of control is avoided. The debt-to-equity ratio (after conversion) is the same in both cases. Thus, the analysis must centre on the choice between 2 and 3. The differences between these two alternatives, which are illustrated in Parts c and d, are that the increase in earnings per share is substantially greater under Alternative 3, but so is the debt ratio. With its low debt ratio (19%), the firm is in a good position for future growth under Alternative 2. However, the 50% ratio under 3 is not prohibitive and is a great improvement over the original situation. The combination of increased earnings per share and reduced debt ratios indicates favourable stock price movements in both cases, particularly under Alternative 3. There is the remote chance that the firm could lose its commercial bank financing under 3, since it was the bank that initiated the permanent financing suggestion. The additional funds, especially under 3, may enable the firm to become more current on its trade credit. Also, the bonds will no doubt be subordinated debentures. Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc. 16-459

Both Alternatives 2 and 3 are favourable alternatives. If the principal owner is willing to assume the risk of higher leverage, then 3 is slightly more attractive than 2. The actual attractiveness of Alternative 3 depends, of course, on the assumption that funds can be invested to yield 20% before interest and taxes. It is this fact that makes the additional leverage favourable and raises the earnings per share. 16-11 a. Stock data and stock required return: rd = 8%. P0 = $23. D0 = $2. g = 6%. rs = $2.12/$23.00 + 6% = 15.22%. Convertible bond data: Par = $1,000, 20-year. Coupon = 7%. Conversion ratio = CR = 35 shares. Call = Five-year deferment. Call price = $1,075 in Year 5, declines by $5 per year. Will be called when Ct = 1.25(Par) = $1,250. Find N (number of years) to anticipated call/conversion: We need to find the number of years that it takes $805 (35 × $23) to grow to $1,250 at a 6% interest rate. Using a financial calculator, I/YR = 6, PV = 805, PMT = 0, FV = –1250; solving, N = 7.552. So the call will be at the first year end after this, or at Year 8. We could also calculate this as: (CR)(P0)(1 + g)N = $1,250 (35)($23)(1 + 0.06)N = $1,250 $805(1.06)N = $1,250. (1.06)N = $1,250/$805 = 1.55 N ln(1.06) = ln(1.55) N(0.05827) = 0.43825 N = 0.43825/0.05827 = 7.52 ≈ 8.

Straight-debt value of the convertible at t = 0: (Assumes annual payment of coupon) At t = 0 (N = 20): N = 20, I/YR = 8, PMT = 70, FV = 1,000; solving, PV = –901.82. Alternatively, 20

$60 $1,000  t (1  0.08)20 t 1 (1  0.08)

V = = $901.82.

Repeating, we can find the straight-bond value for different values of N: V at t = 5 (N = 15): $914.41. V at t = 10 (N = 10): $932.90. V at t = 15 (N = 5): $960.07. V at t = 20 (N = 0): $1,000. Conversion value: The stock price should grow at 6%. The conversion value at Year t is equal to the expected stock price multiplied by the conversion ratio: CVt = P0(1.06)N(35). Repeating for different values of N: CV0 = $23(35) = $805. CV5 = $23(1.06)5(35) = $1,077. CV8 = $23(1.06)8(35) = $1,283. CV10 = $23(1.06)10(35) = $1,442. For the expected time of conversion (N = 8), the conversion value is: CV8 = $23(1.06)8(35) = $1,283.05. The cash flow at the time of conversion (N = 8), is equal to the conversion value plus the coupon payment: CF8 = $1,283.05 + $70 = $1,353.05. 8

b. $1,000 = 

$70 $1,283.05 .  ( 1  r ) (1  rc) c t 1

SOLUTION TO SPREADSHEET PROBLEM 16-12 The detailed solution for the spreadsheet problem is in the file Ch 16 Build a Model Solution.xlsx and is available on the textbook’s website.

MINI CASE The detailed solution for the Mini Case is also available in spreadsheet format, in the file Brigham4Ce_MiniCase_Ch16.xlsx, and is available on the textbook’s website. Paul Duncan, financial manager of EduSoft Inc., is facing a dilemma. The firm was founded 5 years ago to provide educational software for the rapidly expanding primary and secondary school markets. Although EduSoft has done well, the firm’s founder believes that an industry shakeout is imminent. To survive, EduSoft must grab market share now, and this will require a large infusion of new capital. Because he expects earnings to continue rising sharply and looks for the stock price to follow suit, Duncan does not think it would be wise to issue new common stock at this time. On the other hand, interest rates are currently high by historical standards, and with the firm’s B rating, the interest payments on a new debt issue would be prohibitive. Thus, he has narrowed his choice of financing alternatives to two securities: (1) bonds with warrants or (2) convertible bonds. As Duncan’s assistant, you have been asked to help in the decision process by answering the following questions: a.

How can knowledge of call options help a financial manager better understand warrants and convertibles?

Answer: A call option is a contract that gives the holder the right, but not the obligation, to buy some defined asset, say a stock, at a specified price within some specified period of time. A warrant is a long-term option, and a convertible has built into it an implied call option. If financial managers understand how call options are valued, they can make better decisions regarding the structuring of warrant and convertible issues.

One of the firm’s alternatives is to issue a bond with warrants attached. EduSoft’s current stock price is $20, and its investment banker estimates that the cost of a 20-year, annual coupon bond without warrants would be 10%. The bankers suggest attaching 45 warrants, each with an exercise price of $25, to each $1,000 bond. It is estimated that each warrant, when detached and traded separately, would have a value of $3. 1. What coupon rate should be set on the bond with warrants if the total package is to sell for $1,000?

Answer: If the entire package is to sell for $1,000, then Vpackage = Vbond + Vwarrants = $1,000. The 45 warrants each have an estimated value of $3, so Vwarrants = 45($3) = $135. thus,

Vbond + $135 = $1,000 = $865.

Therefore, the bonds must carry a coupon rate that will cause each bond to sell for $865. The straight-debt rate is rd = 10%, so if the coupon were set at 10%, the bonds would sell at par, not at $865. The coupon must therefore be below 10%, and it is found by solving for INT in this equation: $865 = INT(PVIFA10%,20) + $1,000(PVIF10%,20) INT ≈ $84. Alternatively, using a financial calculator, enter N = 20, I = 10, PV = –865, and FV = 1,000 to solve for PMT = $84. Therefore, the required coupon rate is 8.4%. With an 8.4% coupon, the bonds would have a value of $865, and hence the package of one bond plus 45 warrants would be worth $1,000.

2. Suppose the bonds were issued and the warrants immediately traded on the open market for $5 each. What would this imply about the terms of the issue? Did the company ―win‖ or ―lose‖?

Answer: If the warrants traded for $5 immediately after issue, then the company would have set the terms of the bonds with warrants improperly, its shareholders would have been ―losers,‖ and the purchasers of the bonds with warrants would have been ―winners.‖ The 45 warrants would be worth 45($5) = $225, and the package would actually be worth $865 + $225 = $1,090. Selling something worth $1,090 for $1,000 would impose an unnecessary cost of $90 per bond on EduSoft’s shareholders. Because the package could have been sold with a lower coupon rate, the firm could have had lower future interest payments whose present value would be smaller by $90. b.

3. When would you expect the warrants to be exercised? Assume they have a 10year life; that is, they expire 10 years after issue.

Answer: Generally, a warrant will sell in the open market at a premium above its expiration value, which is the value of the warrant if exercised. Thus, prior to expiration, an investor who wanted cash would sell his or her warrants in the marketplace rather than exercise them. Therefore, warrants tend not to be exercised until just before they expire. However, note that in order to force warrant holders to exercise and thus to bring in equity capital, some warrants contain step-up provisions, whereby the strike price (also called the exercise price) increases in steps over the life of the warrant. Since the value of the warrant falls when the strike price is increased, step-up provisions encourage holders of in-the-money warrants to exercise just prior to the timing of a step-up. Finally, note that warrant holders will tend to exercise voluntarily if the dividend on the stock rises enough. No dividend is earned on a warrant, and high dividends increase the attractiveness of stocks over warrants. The expiration value of the warrant will fall if the stock price falls, and stock prices fall when the stock goes ex dividend. If the dividend is large, warrant holders can avoid recurring large losses by exercising. b.

4. Will the warrants bring in additional capital when exercised? If so, how much, and what type of capital?

Answer: When exercised, each warrant will bring in an amount equal to the strike price, which in this case means $25 of equity capital, and holders will receive one share of common stock per warrant. Note that the strike price is typically set at 10% to 30% above the current stock price on the issue date. A high-growth firm would set the strike price toward the high end of the range, and a low-growth firm would set the price toward the bottom end.

As an alternative to the bond with warrants, Duncan is considering convertible bonds. The firm’s investment bankers estimate that EduSoft could sell a 20-year, 8.5% annual coupon, callable convertible bond for its $1,000 par value, whereas a straight-debt issue would require a 10% coupon. The convertibles would be call protected for 5 years, the call price would be $1,100, and the company would probably call the bonds as soon as possible after their conversion value exceeds $1,200. Note, though, that the call must occur on an issue date anniversary. EduSoft’s current stock price is $20, its last dividend was $1, and the dividend is expected to grow at a constant 8% rate. The convertible could be converted into 40 shares of EduSoft stock at the owner’s option. 1. What conversion price is built into the bond?

Answer: Conversion Price = PC =

Par value $1,000 = = $25. # Shares received 40

The conversion price is similar to a warrant’s strike price, and, as with warrants, the conversion price is typically set at between 10% and 30% above the stock price on the issue date. c.

2. What is the convertible’s straight-debt value? What is the implied value of the convertibility feature?

Answer: Since the required rate of return on a 20-year straight bond is 10%, the value of a 8.5% annual coupon bond is $872.30: V = $85(PVIFA10%,20) + $1,000(PVIF10%,20) = $872.30. Alternatively, using a financial calculator, enter N = 20, I = 10, PMT = 85, and FV = 1,000 to solve for PV = $872.30. However, the convertible would sell for $1,000, so the implied value of the convertible feature is $1,000 – $872.30 = $127.70. The convertibility value is analogous to the price on a warrant.

3. What is the formula for the bond’s expected conversion value in any year? What is its conversion value at Year 0? At Year 10?

Answer: The conversion value in any year is simply the value of the stock one would receive upon converting. Since EduSoft is a constant growth stock, its price is expected to increase by g each year, and hence the expected stock price is Pt = P0(1 + g)t. The value of converting at any year is CR(Pt), where CR is the number of shares received (the conversion ratio). Thus, the expected conversion value in any year is: CVt = CR(Pt) = CR(P0)(1 + g)t = 40($20)(1.08)t, And, hence, for Year 0 and Year 10, we have the following: Year 0: CV0 = 40($20)(1.08)0 = $800. Year 10: CV10 = 40($20)(1.08)10 = $1,727.14. c.

4. What is meant by the ―floor value‖ of a convertible? What is the convertible’s expected floor value at Year 0? At Year 10?

Answer: The floor value is simply the higher of the straight-debt value and the conversion value. At Year 0, the straight-debt value is $872.30 while the conversion value is $800, and hence the floor value is $872.30. At Year 10, the conversion value of $1,727 is clearly higher than the straight-debt value, and hence the conversion value sets the floor price for that year. The convertible, however, will generally sell above its floor value prior to maturity because the convertibility option carries additional value. (As discussed below, the price of the convertible can fall to the floor value if the dividends paid on the stock that would be received in conversion greatly exceed the interest paid on the bond.) c.

5. Assume that EduSoft intends to force conversion by calling the bond as soon as possible after its conversion value exceeds 20% above its par value, or 1.2($1,000) = $1,200. When is the issue expected to be called? (Hint: Recall that the call must be made on an anniversary date of the issue.)

Answer: The easiest way to find the year conversion is expected is by recognizing that the conversion value begins at $800, grows at the rate of 8% per year, and must rise to $1,200. Then, simply input into a financial calculator, I = 8, PV = 800 (or –800), and FV = 1200, and then press the n button to find the year. It is 5.27, and since the call must occur on an anniversary date, we would round up to 6, so the bond should be called after 6 years. (Note: Some calculators automatically round up; the HP-12c is one. However, the HP-17b gives the unrounded answer 5.27.) c. 6. What is the expected cost of capital for the convertible to EduSoft? Does this cost Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc. 17-467

appear to be consistent with the riskiness of the issue? Answer: The firm would receive $1,000 now, would make coupon payments of $85 for 6 years, and then would issue stock worth 40($20)(1.08)6 = $1,269.50. Thus, the cash flow stream would look like this: 0

1,000

–85

–85.00 –1,269.50 –1,354.50

The IRR of this stream, which is the cost of the convertible issue, is 11.83%. Note that EduSoft’s cost of straight debt is 10%, while its cost of equity is 13.4%: rs =

$1.00(1.08) D 0 (1  g) +g= + 8% = 13.4%. $20 P0

The firm’s convertible bond has risk that is a blend of the cost of debt and the cost of equity, so rc should fall between the cost of debt and equity. Thus, an 11.83% cost appears reasonable. d.

Duncan believes that the costs of both the bond with warrants and the convertible bond are close enough to one another to call them even, and also consistent with the risks involved. Thus, he will make his decision based on other factors. What are some of the factors that he should consider?

Answer: One factor that should be considered is the firm’s future needs for capital. If EduSoft anticipates a continuing need for capital, then warrants may be favoured, because their exercise will bring in additional equity capital without the need to retire the accompanying low-coupon debt issue. Conversely, the convertible issue brings in no new funds at conversion, and the low-coupon debt will be gone when the bonds are converted. Another factor is whether EduSoft wants to commit to 20 years of debt at this time. Conversion will remove the debt issue, while the exercise of warrants does not. Of course, if EduSoft’s stock price does not rise over time, then neither the warrants nor the convertibles would be exercised, and the debt would remain outstanding in both cases.

How do convertible bonds help reduce agency costs?

Answer: Agency costs can arise due to conflicts between shareholders and bondholders, in the form of asset substitution (or bait-and-switch.) This happens when the firm issues low-cost straight debt, then invests in risky projects. Bondholders suspect this, so they charge high interest rates. Convertible debt allows bondholders to share in upside potential, so it has a low rate. Thus, convertible debt helps reduce this agency cost. Information asymmetry occurs when a company knows its future prospects better than outside investors. Outside investors think the company will issue new stock only if future prospects are not as good as the market anticipates, so issuing new stock sends a negative signal to the market, causing the stock price to fall. A company with good future prospects can issue stock ―through the back door‖ by issuing convertible bonds. This avoids the negative signal of issuing stock directly. Since prospects are good, bonds will likely be converted into equity, which is what the company wants to issue.

Chapter 17 Working Capital Management and Short-Term Financing ANSWERS TO END-OF-CHAPTER QUESTIONS 17-1

a. Working capital is a firm’s investment in short-term assets—cash, marketable securities, inventory, and accounts receivable. Net working capital is current assets minus current liabilities. Net operating working capital is operating current assets minus operating current liabilities. b. The inventory conversion period is the average length of time it takes to convert materials into finished goods and then to sell them. It is calculated by dividing total inventory by cost of sales per day. The receivables collection period is the average length of time required to convert a firm’s receivables into cash. It is calculated by dividing accounts receivable by sales per day. The payables deferral period is the average length of time between a firm’s purchase of materials and labour and the payment of cash for them. It is calculated by dividing accounts payable by credit purchases per day (COGS/365). The cash conversion cycle is the length of time between the firm’s actual cash expenditures on productive resources (materials and labour) and its own cash receipts from the sale of products (that is, the length of time between paying for labour and materials and collecting on receivables). Thus, the cash conversion cycle equals the length of time the firm has funds tied up in current assets. c. A relaxed NOWC policy refers to a policy under which relatively large amounts of

cash, marketable securities, and inventories are carried and under which sales are stimulated by a liberal credit policy, resulting in a high level of receivables. A restricted NOWC policy refers to a policy under which holdings of cash, securities, inventories, and receivables are minimized, while a moderate current asset investment policy lies between the relaxed and restricted policies. A moderate NOWC policy matches asset and liability maturities. It is also referred to as the maturity matching, or ―self-liquidating,‖ approach. d. A cash budget is a schedule showing cash flows (receipts, disbursements, and cash balances) for a firm over a specified period. The net cash increase or decrease for the period is calculated as total collections for the period less total payments for the same period of time. e. Trade discounts are price reductions that suppliers offer customers for early payment of bills.

f. Permanent NOWC is the NOWC required when the economy is weak and seasonal sales are at their low point. Thus, this level of NOWC always requires financing and can be regarded as permanent. Temporary NOWC is the NOWC required above the permanent level when the economy is strong and/or seasonal sales are high. g. A moderate short-term financing policy matches asset and liability maturities. It is also referred to as the maturity matching, or ―self-liquidating,‖ approach. When a firm finances all of its fixed assets with long-term capital but part of its permanent current assets with short-term, nonspontaneous credit, this is referred to as an aggressive short-term financing policy. With a conservative short-term financing policy, permanent capital is used to finance all permanent asset requirements, as well as to meet some or all of the seasonal demands. h. Maturity matching (or the ―self-liquidating‖ approach) is a financing policy that matches asset and liability maturities. This is a moderate policy. i

Accruals are continually recurring short-term liabilities, especially accrued wages and accrued taxes.

j. Trade credit is debt arising from credit sales and recorded as an account receivable by the seller and as an account payable by the buyer. Stretching accounts payable is the practice of deliberately paying accounts payable late. Free trade credit is credit received during the discount period. Credit taken in excess of free trade credit, whose cost is equal to the discount lost, is termed costly trade credit. k. A line of credit is an arrangement in which a bank agrees to lend up to a specified maximum amount of funds during a designated period. A revolving credit agreement is a multi-year, committed line of credit extended by a bank or other lending institution. l. Commercial paper is unsecured, short-term promissory notes of large firms, usually issued in denominations of $100,000 or more and having an interest rate somewhat below the prime rate. A bankers’ acceptance is a term draft unconditionally guaranteed by a major bank. 17-2

False. Both accounts will record the same transaction amount.

17-3

If an asset’s life and returns can be positively determined, the maturity of the asset can be matched to the maturity of the liability incurred to finance the asset. This matching will ensure that funds are borrowed only for the time they are required to finance the asset and that adequate funds will have been generated by the asset by the time the financing must be repaid. A basic fallacy is involved in the above discussion, however. Borrowing to finance receivables or inventories may be on a short-term basis because these turn over 8 to 12 times a year. But, as a firm’s sales grow, its investment in receivables and inventories grows, even though they turn over. Hence, longer-term financing should be used to finance the permanent components of receivables and inventory investments.

17-4

From the standpoint of the borrower, short-term credit is riskier because short-term interest rates fluctuate more than long-term rates, and the firm may be unable to repay the debt. If the lender will not extend the loan, the firm could be forced into bankruptcy. A firm might borrow short-term if it thought that interest rates were going to fall and, therefore, that the long-term rate would go even lower. A firm might also borrow shortterm if it were going to need the money only for a short while and the higher interest would be offset by lower administration costs and no prepayment penalty. Thus, firms do consider factors other than interest rates when deciding on the maturity of their debt.

17-5

This statement is false. A firm cannot ordinarily control its accruals since payrolls and the timing of wage payments are set by economic forces and by industry custom, while tax payment dates are established by law.

17-6

Yes. If a firm is able to buy on credit at all, if the credit terms include a discount for early payment, and if the firm pays during the discount period, it has obtained ―free‖ trade credit. However, taking additional trade credit by paying after the discount period can be quite costly.

17-7

Commercial paper refers to promissory notes of large, strong corporations. These notes have maturities that generally vary from one day to nine months, and the return is usually below the prime lending rate. Mama and Papa Gus could not use the commercial paper market.

SOLUTIONS TO END-OF-CHAPTER PROBLEMS 17-1 Nominal cost of trade credit =

3 97

365 30 15

= 0.0309  24.33 = 0.7526 = 75.26%. Effective cost of trade credit = (1.0309)24.33 – 1.0 = 1.0984 = 109.84%. 17-2

Effective cost of trade credit = (1 + 1/99)8.11 – 1.0 = 0.0849 = 8.49%.

17-3

Net purchase price of inventory = $500,000/day. Credit terms = 2/15, net 40. $500,000  15 = $7,500,000.

17-4

a. 0.3(10) + 0.7(50) = 38 days. b. $1,500,000/365 = $4,109.59 sales per day. $4,109.59(38) = $156,164= Average receivables. c. 0.3(10) + 0.7(45) = 34.5 days. $1,500,000/365 = $4,109.59 sales per day. $4,109.59(34.5) = $141,781 = Average receivables. Sales may also decline as a result of the tighter credit. This would further reduce receivables. Also, some customers may now take discounts, further reducing receivables.

17-5

17-6

1 365 = 73.74%.  99 5

2 365 = 14.90%.  98 50

3 365 = 32.25%.  97 35

2 365 = 21.28%.  98 35

2 365 = 29.80%.  98 25

3 365 = 45.15%.  97 45 - 20 Because the firm still takes the discount on Day 20, 20 is used as the discount period in calculating the cost of nonfree trade credit.

b. Paying after the discount period, but still taking the discount, gives the firm more credit than it would receive if it paid within 15 days.

17-7 Sales per day =

$4,562,500 = $12,500 365

Discount sales = 0.5($12,500) = $6,250. A/R attributable to discount customers = $6,250(10) = $62,500. A/R attributable to nondiscount customers: Total A/R Discount customers’ A/R Nondiscount customers’ A/R

$ 437,500 62,500 $375,000

A/R $375, 000 Days sales outstanding    60 days. nondiscount customers Sales per day $6, 250 Alternatively, DSO = $437,500/$12,500 = 35 days. 35 = 0.5(10) + 0.5(DSONondiscount) DSONondiscount = 30/0.5 = 60 days. Thus, although nondiscount customers are supposed to pay within 40 days, they are actually paying, on average, in 60 days. Cost of trade credit to nondiscount customers equals the rate of return to the firm: Nominal rate = 2  365 98

60  10

= 0.0204(7.3) = 14.90%.

Effective cost = (1 + 2/98)365/50 – 1 = 15.89%.

17-8

Cash a. conversion = cycle = 50 + 35 – 25 = 60 days.

b. Average sales per day = $4,380,000/365 = $12,000 Investment in receivables = $12,000  35 = $420,000. c. COGS = 0.80 × Sales = 0.80 × $4,380,000 = $3,504,000. Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc. 17-475

Inv. conversion period =

Inv. COGS/365

Inv. $3,504,000 /365 Inv. = $480,000. 50 =

Inventory turnover = COGS/Inventory = $3,504,000/$480,000 = 7.3×. 17-9

a. Inventory turnover = COGS/Inventory 6.0 = $1,800,000/Inventory Inventory = $300,000.

Inv. COGS/365 $300,000 = $1,800,000 /365

Inventory conversion period =

Inventory conversion period = 60.8 days

Cash conversion = cycle = 60.8 + 41 – 45 = 56.8 days.

b. Total assets = Inventory + Receivables + Fixed assets = $300,000 + [($3,250,000/365)  41] + $535,000 = $300,000 + $365,068 + $535,000 = $1,200,068. Note: Inventory was calculated in Part a above. Total assets turnover = Sales/Total assets = $3,250,000/$1,200,068 = 2.71. ROA = Profit margin  Total assets turnover = 0.07  2.71 = 0.1897 = 18.97%. or ROA = (0.07 × $3,250,000)/$1,200,068 = 18.96%

COGS/Inv. $1,800,000/Inv. Inv.

=9 =9 = $200,000

$200,000 $1,800,000 /365 = 40.6 days.

Inventory conversion period =

Cash conversion cycle = 40.6 + 41 – 45 = 36.6 days. Total assets = Inventory + Receivables + Fixed assets = $200,000 + $365,068 + $535,000 = $1,100,068. Total assets turnover = $3,250,000/$1,100,068 = 2.95. ROA = $227,500/$1,100,068 = 20.68%.

17-10 $18,000 installment loan, 7% nominal rate. Effective annual rate, assuming a 365-day year = ? Add-on interest = 0.07($18,000) = $1,260. Monthly ayment = 0

$18,000 + $1,260 = $1,605 12

1 i=? | –1,605

18,000

2 |

  

–1,605

With a financial calculator, enter N = 12, PV = 18000, PMT = –1605, FV = 0, and then press I to obtain 1.056570%. However, this is a monthly rate. = (1 + rd)n – 1.0 = (1.010566)12 – 1.0 = 1.1344 – 1.0 = 0.1344 = 13.44%.

Effective annual rateAdd-on

17-11 a. Effective rate = 12%. b.

0 i=?

–50,000

50,000 – 4,375 (discount interest) 45,625

With a financial calculator, enter N = 1, PV = 45625, PMT = 0, and FV = –50000 to solve for I = 9.59% Note that, if Burman actually needs $50,000 of funds, it will have to borrow = $54,794.52. The effective interest rate will still be 9.59%.

c. Effective rate: Monthly principal payment $50,000/12 = $4,166.67 12

$50,000 = 

$4,166 .67

t 1 (1  rd )



$4,000 (1  rd )12

rd, the monthly interest rate, is 1.1326%, found with a financial calculator. Input N = 12; PV = 50000; PMT = –4166.67; FV = -4000; and I = ? = 1.1326. The precise effective annual rate is (1.011326)12 – 1.0 = 14.47%. Alternative b has the lowest effective interest rate.

17-12 a. Simple interest: 12%. b. 3-months: (1 + 0.115/4)4 – 1 = 12.0055%, or use the interest conversion feature of your calculator as follows: NOM% = 11.5; P/YR = 4; EFF% = ? EFF% = 12.01%. c. Add-on: Interest = Funds needed(rd). Loan = Funds needed(1 + rd). PMT = Loan/12. Assume you borrowed $100. Then, Loan = $100(1.06) = $106. PMT = $106/12 = $8.8333. 12

$100 = ∑

$8.8333

(1 + rd )t Enter N = 12, PV = 100, PMT = -8.8333, FV = 0, and press I to get I = 0.908032% = rd. This is a monthly periodic rate, so the effective annual rate = (1.00908032)12 – 1 = 0.1146 = 11.46%. t=1

d. Trade credit: 1/99 = 1.01% on discount if pay in 15 days, otherwise pay 45 days later. So, get 60 – 15 = 45 days of credit at a cost of 1/99 = 1.01%. There are 360/45 = 8 periods, so the effective cost rate is: (1 + 1/99)8 – 1 = (1.0101)8 – 1 = 8.3723%. Thus, the least expensive type of credit for Yonge is trade credit with an effective cost of 8.37%.

17-13 a. Current year sales are expected to be $1,600,000 × (1.25) = $2,000,000. Return on equity may be computed as follows: Tight

Moderate

Relaxed

Current assets (% of sales  Sales) Fixed assets Total assets

$ 900,000 1,000,000 $1,900,000

$1,000,000 1,000,000 $2,000,000

$1,200,000 1,000,000 $2,200,000

Debt (60% of assets) Equity Total liability and equity

$1,140,000 760,000 $1,900,000

$1,200,000 800,000 $2,000,000

$1,320,000 880,000 $2,200,000

EBIT (12%  $2 million) Interest (8%) Earnings before taxes Taxes (26%) Net income

$ 240,000 91,200 148,800 38,688 $ 110,112

$ 240,000 96,000 144,000 37,440 $ 106,560

$ 240,000 105,600 134,400 34,944 $ 99,456

14.49%

13.32%

11.30%

Return on equity

b. No, this assumption would probably not be valid in a real-world situation. A firm’s current asset policies, particularly with regard to accounts receivable, such as discounts, collection period, and collection policy, may have a significant effect on sales. The exact nature of this function may be difficult to quantify, however, and determining an ―optimal‖ current asset level may not be possible in actuality. c.

As the answers to Part a indicate, the tighter policy leads to a higher expected return. However, as the current asset level is decreased, presumably some of this reduction comes from accounts receivable. This can be accomplished only through higher discounts, a shorter collection period, and/or tougher collection policies. As outlined above, this would in turn have some effect on sales, possibly lowering profits. More restrictive receivable policies might involve some additional costs (collection, and so forth) but would also probably reduce bad debt expenses. Lower current assets would also imply lower liquid assets; thus, the firm’s ability to handle contingencies would be impaired. Higher risk of inadequate liquidity would increase the firm’s risk of insolvency and thus increase its chance of failing to meet fixed charges. Also, lower inventories might mean lost sales and/or expensive production stoppages. Attempting to attach numerical values to these potential losses and probabilities would be extremely difficult.

17-14 a. I.

Collections and Purchases: Sales Purchases Payments *November purchases = $140,000.

II. Cash Gain or Loss for Month: Receipts from sales Payments for: Purchases Salaries Rent Taxes Total payments Net cash increase (decrease) III. Cash Surplus or Loan Requirements: Cash at start of month Cumulative cash Target cash balance Cumulative surplus cash or total loans to maintain $2,000 target cash balance

December January February $160,000 $42,000 $50,000 40,000 40,000 30,000 140,000* 40,000 40,000

$160,000

$42,000

$50,000

140,000 6,000 3,000 9,000 158,000

40,000 6,000 3,000 – 49,000

2,000 ($ 7,000) $ 1,000

800 2,800 2,000

$ 2,800 ($ 4,200) (4,200) (3,200) 2,000 2,000

800 ($ 6,200) ($ 5,200)

b. If the company began selling on credit on December 1, then it would have zero receipts during December, down from $160,000. Thus, it would have to borrow an additional $160,000, so its loans outstanding by December 31 would be $159,200. The loan requirements would build gradually during the month. We could trace the effects of the changed credit policy on out into January and February, but here it would probably be best to simply construct a new cash budget.

17-15 a.

Average accounts $3,650,000 =  10 days = $10,000  10 = $100,000. payable 365 days

b. There is no cost of trade credit at this point. The firm is using ―free‖ trade credit. c.

Average payables $3,650,000 =  30 = $10,000  30 = $300,000. (net of discount) 365 Nominal cost = (2/98)(365/20) = 37.24%, or $74,489.80/($300,000 – $100,000) = 37.24%. Effective cost = (1 + 2/98)365/20 – 1 = 0.4459 = 44.59%.

d. Nominal rate =

2 365  = 24.83%. 98 40 - 10

Effective cost = (1 + 2/98)365/30 – 1 = 0.2786 = 27.86%.

17-16 Trade Credit Terms: 1/10, net 30. But the firm plans delaying payments 25 additional days, which is the equivalent of 1/10, net 55.

Nominal cost = = 00 –

00– 0

00 0 0(

)

Effective cost = (1 + 1/99)365/45 – 1 = 8.49%.

17-17 a. Size of bank loan = (Purchases/Day)(Days late)      Purchases  Days payables  =  30    Days payables  outstanding    outstandin g   = ($900,000/60)(60 – 30) = $15,000(30) = $450,000. Alternatively, one could simply recognize that accounts payable must be cut to half of its existing level, because 30 days is half of 60 days. b. Given the limited information, the decision must be based on the rule-of-thumb comparisons, such as the following: 1. Total liabilities to asset ratio = ($2,400,000 + $900,000)/$5,000,000 = 66%. Automated Dynamics’ debt ratio is 66%, as compared to a typical debt ratio of 50%. The firm appears to be somewhat undercapitalized. 2. Current ratio = $2,700,000/$2,400,000 = 1.13. The current ratio is low, but current assets could cover current liabilities if all accounts receivable can be collected and if the inventory can be liquidated at its book value. 3. Quick ratio = $700,000/$2,400,000 = 0.29. The quick ratio indicates that current assets, excluding inventory, are sufficient to cover only 29% of current liabilities, which is very poor. The company appears to be carrying excess inventory and financing extensively with debt. Bank borrowings are already high, and the liquidity situation is poor. On the basis of these observations, the loan should be denied, and the treasurer should be advised to seek permanent capital, especially equity capital.

17-18 a. Malone’s current accounts payable balance represents 60 days purchases. Daily $500 purchases can be calculated as = $8.33. 60 If Malone takes discounts, then the accounts payable balance would include only 10 days’ purchases, so the A/P balance would be $8.33  10 = $83.33. If Malone doesn’t take discounts but pays in 30 days, its A/P balance would be $8.33  30 = $250. b. Takes Discounts: If Malone takes discounts, its A/P balance would be $83.33. The cash it would need to be loaned is $500 – $83.33 = $416.67. Since the loan is a discount loan, Malone would require more than a $416.67 loan. Face amount of loan =

= $490.20.

Doesn’t Take Discounts: If Malone doesn’t take discounts, its A/P balance would be $250. The cash needed from the bank is $500 – $250 = $250. Face amount of loan =

= $294.12.

c. Nonfree Trade Credit: Nominal annual cost: Discount % 100 – Discount %

365 0.75 365 = = 13.79% Days credit Discount 99.25 20 – is outstanding period 0.75 18.25 Effective cost: (1+ ) – 1 = (1.007557)18.25 – 1 = 1.1473 – 1 = 14.73% 99.25

Bank Loan: 15% discount loan. Assume the firm doesn’t take discounts, so it needs $250 and borrows $294.12. (The cost will be the same regardless of how much the firm borrows.) 0

–294.12

294.12 –44.12 Discount interest 250.00

With a financial calculator, input the following data, N = 1, PV = 250, PMT = 0, FV = –294.12, and then solve for I = 17.65%. Just to show you that it doesn’t matter how much the firm borrows, assume the firm takes discounts and it reduces A/P to $83.33 so it needs $416.67 cash and borrows $490.20. 0

–490.20

490.20 –73.53 Discount interest 416.67

With a financial calculator, input the following data, N = 1, PV = 416.67, PMT = 0, FV = –490.20, and then solve for I = 17.65%. Because the cost of nonfree trade credit is less than the cost of the bank loan, Malone should forgo discounts and reduce its payables only to $250,000. d. Pro Forma Balance Sheet (Thousands of Dollars) Cash Accounts receivable Inventory Prepaid interestb Total current assets Fixed assets

Total assets

$2,044.1

a b

50.0 450.0 750.0 44.1 1,294.1 750.0

Accounts payable Notes payablea Accruals

$ 250.0 344.1 50.0

Total current liabilities Long-term debt Common equity Total liabilities and equity

644.1 150.0 1,250.0 $2,044.1

Notes payable = $50 + $294.1 = $344.1. Prepaid interest = $294,118(0.15) = $44.1

e. To reduce the accounts payable by $250,000, which reflects the 0.75% discount, Malone must pay the full cost of the payables, which is $250,000/0.9925 = $251,889.17. The lost discount is the difference between the full cost of the payables and the amount that is reported net of discount: Lost discount = $251,889.17 – $250,000.00 = $1,889.17. The after-tax cost of the lost discount is $1,889.17(1 – 0.40) = $1,133.50. Notice that this provides a tax shield in the amount of $1,889.17(0.40) = $755.67. The total amount of cash that Malone needs to pay down $250,000 of accounts payable is the gross amount minus the tax shield: $251,889.17 – $755.67 = $251,133.50. Face amount of loan =

$251,133.50 $251,133.50 = = $295,451.18 1- 0.15 0.85

Pro Forma Balance Sheet (Thousands of Dollars): Cash Accounts receivable Inventory Prepaid interestc Total current assets Fixed assets

$ 50.0 450.0 750.0 44.3 1,294.3 750.0

Total assets

$2,044.3

Accounts payable Notes payablea Accruals

$ 250.0 345.4 50.0

Total current liabilities Long-term debt Common equityb Total liabilities and equity

645.4 150.0 1,248.9 $2,044.3

Notes payable = $50 + $295.4 = $345.4. Common equity = Previous common equity – after-tax lost discount = $1,250 – $1.1 = $1,248.9 c Prepaid interest = $295,451(0.15) = $44.3 b

17-19 a. Collections and purchases worksheet Sales (gross) Collections During month of sale During 1st month after sale During 2nd month after sale Total collections Purchases Labor and raw materials Payments for labor and raw materials

May

June

July

August

September

October

$180,000

$360,000

$540,000

$720,000

$360,000

$90,000

18,000

18,000 135,000

36,000 135,000 27,000 $198,000

54,000 270,000 27,000 $351,000

72,000 405,000 54,000 $531,000

36,000 540,000 81,000 $657,000

36,000 270,000 108,000 $414,000

9,000 270,000 54,000 $333,000

$90,000

$90,000 $90,000

$126,000 $90,000

$882,000 $126,000

$306,000 $882,000

$234,000 $306,000

$162,000 $234,000

$90,000 $162,000

$198,000 90,000 27,000 9,000 2,700

$351,000 126,000 27,000 9,000 2,700

$531,000 882,000 27,000 9,000 2,700 63,000

$657,000 306,000 27,000 9,000 2,700

$414,000 234,000 27,000 9,000 2,700

$333,000 162,000 27,000 9,000 2,700 63,000

$128,700 $69,300

$164,700 $186,300

$983,700 ($452,700)

180,000 $524,700 $132,300

$272,700 $141,300

$263,700 $69,300

$132,000 $201,300 $90,000

$201,300 $387,600 $90,000

$387,600 ($65,100) $90,000

($65,100) $67,200 $90,000

$67,200 $208,500 $90,000

$208,500 $277,800 $90,000

$111,300

$297,600

($155,100)

($22,800)

$118,500

$187,800

Cash gain or loss for month Collections Payments for labor and raw materials General and administrative salaries Lease payments Miscellaneous expenses Income tax payments Design studio payment Total payments Net cash gain (loss) during month Loan requirement or cash surplus Cash at start of month Cumulative cash Target cash balance Cumulative surplus cash or loans outstanding to maintain $90,000 target cash balance

November December

January $180,000

b. The cash budget indicates that Bowers will have surplus funds available during July, August, November, and December. During September, the company will need to borrow $155,100. The cash surplus that accrues during October will enable Bowers to reduce the loan balance outstanding to $22,800 by the end of October.

c. In a situation such as this, where inflows and outflows are not synchronized during the month, it may not be possible to use a cash budget centred on the end of the month. The cash budget should be set up to show the cash positions of the firm on the 5th of each month. In this way the company could establish its maximum cash requirement and use these maximum figures to estimate its required line of credit. The table below shows the status of the cash account on selected dates within the month of July. It shows how the inflows accumulate steadily throughout the month and how the requirement of paying all the outflows on the 5th of the month requires that the firm obtain external financing. By July 14, however, the firm reaches the point where the inflows have offset the outflows, and by July 30 we see that the monthly totals agree with the cash budget developed earlier in Part a. Opening balance

7/2/21 $132,000

7/4/21 132,000

7/5/21 7/6/21 7/14/21 7/30/21 132,000 $132,000 $132,000 $132,000

Cumulative inflows (1/30  receipts  no. of days)

13,200

26,400

33,000

39,600

92,400

198,000

Total cash available 145,200

158,400

165,000

171,600

224,400

330,000

128,700

Net cash position

145,200

158,400

36,300

42,900

95,700

201,300

Target cash balance

90,000

Outflow

Cash above minimum needs (borrowing needs) $ 55,200 $ 68,400 ($ 53,700) ($ 47,100) $

5,700 $111,300

d. The month’s preceding peak sales would show a decreased current ratio and an increased debt ratio due to additional short-term bank loans. In the following months, as receipts are collected from sales, the current ratio would increase and the debt ratio would decline. Abnormal changes in these ratios would affect the firm’s ability to obtain bank credit.

17-20 a. 1. Line of credit: Commitment fee = (0.0025)($1,500,000)(11/12) = $ 3,438 Interest = (0.06)(1/12)($1,500,000) = 7,500 Total $10,438 2. Trade discount: Nominal  1   365  =   = 0.1229 = 12.29%. rate  99   30  Total cost = 0.1229($1,500,000)/12 = $15,363. Effective cost = (1 + 1/99)365/30 – 1 = 0.1301 = 13%. Total cost = 0.13($1,500,000)/12 = $16,250. 3. 30-day commercial paper: Interest = (0.05)($1,500,000)(1/12) = $ 6,250 Transaction fee = (0.0025)($1,500,000) = 3,750 $10,000 4. 60-day commercial paper: Interest = (0.055)($1,500,000)(2/12) = $13,750 Transaction fee = (0.005)($1,500,000) = 7,500 $21,250 Marketable securities interest received = (0.03)($1,500,000)(1/12) = –3,750 $17,500 The 30-day commercial paper has the lowest cost. b. The lowest cost of financing is not necessarily the best. The use of 30-day commercial paper is the cheapest; however, sometimes the commercial paper market is tight and funds are not available. This market also is impersonal. A banking arrangement may provide financial counselling and a long-run relationship in which the bank performs almost as a ―partner and counsellor‖ to the firm. Note also that while the use of 60-day commercial paper is more expensive than the use of 30-day paper, it provides more flexibility in the event the money is needed for more than 30 days. However, the line of credit provides even more flexibility than the 60-day commercial paper and at a lower cost.

SOLUTION TO SPREADSHEET PROBLEM 17-21 The detailed solution for the spreadsheet problem is in the file Ch 17 Build a Model Solution.xlsx and is available on the textbook’s website.

MINI CASE The detailed solution for the Mini Case is also available in spreadsheet format, in the file Brigham4Ce_MiniCase_Ch17.xlsx, and is available on the textbook’s website. Dan Barnes, financial manager of Ski Equipment Inc. (SKI), is excited but apprehensive. The company’s founder recently sold his 51% controlling block of shares to Kent Koren, who is a big fan of EVA (economic value added). EVA is found by taking the after-tax operating profit and then subtracting the dollar cost of all the capital the firm uses: EVA = NOPAT – Capital costs = EBIT (1 – T) – WACC(Capital employed). If EVA is positive, then the firm is creating value. On the other hand, if EVA is negative, the firm is not covering its cost of capital, and shareholders’ value is being eroded. Koren rewards managers handsomely if they create value, but those whose operations produce negative EVAs are soon looking for work. Koren frequently points out that, if a company could generate its current level of sales with fewer assets, it would need less capital. That would, other things held constant, lower capital costs and increase its EVA. Shortly after he took control of SKI, Kent Koren met with SKI’s senior executives to tell them of his plans for the company. First, he presented some EVA data that convinced everyone that SKI had not been creating value in recent years. He then stated, in no uncertain terms, that this situation must change. He noted that SKI’s designs of skis, boots, and clothing are acclaimed throughout the industry, but that something is seriously amiss elsewhere in the company. Costs are too high, prices are too low, or the company employs too much capital, and he wants SKI’s managers to correct the problem or else. Barnes has long felt that SKI’s working capital situation should be studied—the company may have the optimal amounts of cash, securities, receivables, and inventories, but it may also have too much or too little of these items. In the past, the production manager resisted Barnes’s efforts to question his holdings of raw materials inventories, the marketing manager resisted questions about finished goods, the sales staff resisted questions about credit policy (which affects accounts receivable), and the treasurer did not want to talk about her cash and securities

balances. Koren’s speech made it clear that such resistance would no longer be tolerated. Barnes also knows that decisions about working capital cannot be made in a vacuum. For example, if inventories could be lowered without adversely affecting operations, then less capital would be required, the dollar cost of capital would decline, and EVA would increase. However, lower raw materials inventories might lead to production slowdowns and higher costs, while lower finished goods inventories might lead to the loss of profitable sales. So, before inventories are changed, it will be necessary to study operating as well as financial effects. The situation is the same with regard to cash and receivables. Barnes began collecting the ratios shown below.

Current Quick Debt/assets Turnover of cash and securities Days’ sales outstanding (365-day basis) Inventory turnover Fixed assets turnover Total assets turnover Profit margin Return on equity (ROE) Payables deferral period November

SKI 1.75 0.83 58.76% 16.67 45.63 4.82 11.35 2.08 2.07% 10.45% 30.00

Industry 2.25 1.20 50.00% 22.22 32.00 7.00 12.00 3.00 3.50% 21.00% 33.00

December

January

February

March

April

$68,212

$65,213

$52,475

$42,909

$30,524

12,781.75

10,285.10

47,748.40

45,649.10

7,121.80 $67,651.95

6,821.20 $62,755.40

$36,472.65 44,603.75

$25,945.40 36,472.65

Collections and purchases worksheet (1) (2) (3) (4) (5)

Sales (gross) $71,218 Collections During month of sale (0.2)(0.98)(month’s sales) During first month after sale (0.7)(previous month’s sales) During second month after sale (0.1)(sales 2 months ago) Total collections (Lines 2 + 3 + 4)

Purchases (6) 0.85(forecasted sales 2 months from now) (7) Payments (1-month lag)

$44,603.75

II.

Cash Increase or Decrease for Month (8) (9) (10) (11) (12) (13) (14)

III.

$67,651.95 44,603.75 6,690.56 2,500.00

$62,755.40 36,472.65 5,470.90 2,500.00

$53,794.31

$44,443.55

$13,857.64

$18,311.85

$ 3,000.00

$16,857.64

16,857.64 1,500.00

35,169.49 1,500.00

$15,357.64

$33,669.49

Cash Surplus or Loan Requirement (15) (16) (17) (18)

Collections (from Section I) Payments for purchases (from Section I) Wages and salaries Rent Taxes Total payments Net cash increase (decrease) during month (Line 8 – Line 13)

Cash at beginning of month if no borrowing is done Cumulative cash (cash at start, + increase or – decrease = Line 14 + Line 15) Target cash balance Cumulative surplus cash or loans outstanding to maintain $1,500 target cash balance (Line 16 – Line 17)

Barnes plans to use the preceding ratios as the starting point for discussions with SKI’s operating executives. He wants everyone to think about the pros and cons of changing each type of current asset and how changes would interact to affect profits and EVA. Based on the data, does SKI seem to be following a relaxed, moderate, or restricted working capital policy?

Answer: A company with a relaxed working capital policy would carry relatively large amounts of current assets in relation to sales. It would be guarding against running out of stock or running short of cash, or losing sales because of a restrictive credit policy. We can see that SKI has relatively low liquidity and inventory turnover ratios. For example, cost of sales/inventories = 4.82 versus 7.0 for an average firm in its industry. Thus, SKI is carrying a lot of inventory per dollar of sales, which would meet the definition of a relaxed policy. Similarly, SKI’s DSO is relatively high. Since DSO is calculated as receivables/sales per day, a high DSO indicates a lot of receivables per dollar of sales. Thus, SKI seems to have a relaxed working capital policy, and a lot of current assets. Its current ratio is weaker than the industry, so SKI must also have a lot of current liabilities.

How can one distinguish between a relaxed but rational working capital policy and a situation in which a firm simply has a lot of current assets because it is inefficient? Does SKI’s working capital policy seem appropriate?

Answer: SKI may choose to hold large amounts of inventory to avoid the costs of ―running short,‖ and to cater to customers who expect to receive their equipment in a short period of time. SKI may also choose to hold high amounts of receivables to maintain good relationships with its customers. However, if SKI is holding large stocks of inventory and receivables to better serve customers, it should be able to offset the costs of carrying that working capital with higher prices or higher sales quantities, and its ROE should be no lower than that of firms with other working capital policies. It is clear from the ratio data in the first table that SKI is not as profitable as the average firm in its industry. This suggests that it simply has excessive working capital, and that it should take steps to reduce its working capital. c.

Calculate the firm’s cash conversion cycle.

Answer: A firm’s cash conversion cycle is calculated as:

Inventory Receivables Payables Cash conversion  collection  deferral  conversion period period period cycle SKI’s inventory turnover is given as 4.82, so we can calculate its inventory conversion period as:

365 365 = 75.73  76 days  Inventory turnover 4.82 SKI’s receivables collection period is equal to its DSO. Its DSO is given as 45.63 days, or approximately 46 days. We are given that its payables deferral period is 30 days, so now we have all the individual components to calculate SKI’s cash conversion cycle. 76 days + 46 days – 30 days = 92 days Thus, SKI’s cash conversion cycle is approximately 92 days.

What might SKI do to reduce its cash without harming operations?

Answer: To the extent that ―cash and securities‖ consist of low-yielding securities, they could be sold off and the cash generated could be used to reduce debt, to buy back stock, or to invest in operating assets. In an attempt to better understand SKI’s cash position, Barnes has developed a cash budget. Data for the first 2 months of the year are shown above. (Note that Barnes’s preliminary cash budget does not account for interest income or interest expense.) He has the figures for the other months, but they are not shown. e.

Should depreciation expense be explicitly included in the cash budget? Why or why not?

Answer: No, depreciation expense is a noncash charge and should not appear explicitly in the cash budget that focuses on the actual cash flowing into and out of a firm. However, a firm’s depreciation expense does impact its tax liability, and hence depreciation affects SKI’s quarterly tax payments. f.

In his preliminary cash budget, Barnes has assumed that all sales are collected and, thus, that SKI has no bad debts. Is this realistic? If not, how would bad debts be dealt with in a cash budgeting sense? (Hint: Bad debts will affect collections but not purchases.)

Answer: It is not realistic to assume zero bad debts. When credit is granted, bad debts should be expected. Collections in each month would be lowered by the percentage of bad debts. Payments would be unchanged, so the result would be that loan balances would be larger and cash surplus balances would be smaller by the difference in the collection amounts.

Barnes’s cash budget for the entire year, although not given here, is based heavily on his forecast for monthly sales. Sales are expected to be extremely low between May and September but then increase dramatically in the fall and winter. November is typically the firm’s best month, when SKI ships equipment to retailers for the holiday season. Interestingly, Barnes’s forecasted cash budget indicates that the company’s cash holdings will exceed the targeted cash balance every month except for October and November, when shipments will be high but collections will not be coming in until later. Based on the ratios shown earlier, does it appear that SKI’s target cash balance is appropriate? In addition to possibly lowering the target cash balance, what actions might SKI take to better improve its cash management policies, and how might that affect its EVA?

Answer: The company’s turnover of cash and its projected cash budget suggest that the company is holding too much cash. SKI could improve its EVA by either investing the cash in productive assets or returning the cash to shareholders. If SKI uses the cash for profitable investments, its costs will remain the same, but its operating income will rise, thereby increasing EVA. On the other hand, if the company chooses to return the cash to its shareholders, for example, by increasing the dividend or repurchasing common shares, the company’s revenues would remain the same, but its overall cost of capital would fall, thereby increasing EVA. h.

What reasons might SKI have for maintaining a relatively high amount of cash?

Answer: If sales turn out to be considerably less than expected, the company could face a cash shortfall. A company may choose to hold large amounts of cash if it does not have much faith in its sales forecast or if it is very conservative. Unfortunately, given its current pressure to perform, SKI’s management does not have the luxury to be extremely conservative. i.

Is there any reason to think that SKI may be holding too much inventory? If so, how would that affect EVA and ROE?

Answer: As pointed out in Part a, SKI’s inventory turnover (4.82) is considerably lower than the average firm’s turnover (7.00). This indicates that the firm is carrying a lot of inventory per dollar of sales. By holding more inventory per dollar of sales than is necessary, the firm is increasing its costs, which reduces its ROE. In addition, this additional working capital must be financed, so EVA is lowered too.

If the company reduces its inventory without adversely affecting sales, what effect should this have on the company’s cash position (1) in the short run and (2) in the long run? Explain in terms of the cash budget and the balance sheet.

Answer: Reducing inventory purchases will increase the company’s cash holdings in the short run, thus reducing the amount of financing or the target cash balance needed. In the long run, the company is likely to reduce its cash holdings in order to increase its EVA. SKI can use the ―excess cash‖ to make investments in more productive assets such as plant and equipment. Alternatively, the firm can distribute the ―excess cash‖ to its shareholders through higher dividends or repurchasing its shares. In addition to improving the management of its current assets, SKI is also reviewing the ways in which it finances its current assets. With this concern in mind, Dan is also trying to answer the following questions. k.

Is it likely that SKI could make significantly greater use of accruals?

Answer: No, SKI could not make greater use of its accruals. Accruals arise because (1) workers are paid after they have actually provided their services, and (2) taxes are paid after the profits have been earned. Thus, accruals represent cash owed either to workers or to the Canada Revenue Agency (CRA). The cost of accruals is generally considered to be zero, since no explicit interest must be paid on these items. The amount of accruals is generally limited by the amount of wages paid and the firm’s profitability, as well as by industry conventions regarding when wage payments are made and CRA regulations regarding tax payments. A firm cannot ordinarily control its accruals. Firms use all the accruals they can, but they have little control over the levels of these accounts.

Assume that SKI buys on terms of 1/10, net 30, but that it can get away with paying on the 40th day if it chooses not to take discounts. Also, assume that it purchases $506,985 of equipment per year, net of discounts. How much free trade credit can the company get, how much costly trade credit can it get, and what is the percentage cost of the costly credit? Should SKI take discounts?

Answer: If SKI’s net purchases are $506,985 annually, then, with a 1% discount, its gross purchases are $506,985/0.99 = $512,106. Net daily purchases from this supplier are $506,985/365 = $1,389. If the discount is taken, then SKI must pay this supplier at the end of day 10 for purchases made on day 1, on day 11 for purchases made on day 2, and so on. Thus, in a steady state, SKI will on average have 10 days’ worth of purchases in payables, so, Payables = 10($1,389) = $13,890 If the discount is not taken, then SKI will wait 40 days before paying, so Payables = 40($1,389) = $55,560 Therefore: Trade credit if discounts are not taken: Trade credit if discounts are taken: Difference:

$55,560 = total trade credit –13,890 = free trade credit $41,670 = costly trade credit

To obtain $41,670 of costly trade credit, SKI must give up 0.01($512,106) = $5,121 in lost discounts annually. Since the forgone discounts pay for $41,670 of credit, the nominal annual interest rate is 12.29%:

$5,121 = 0.1229 = 12.29% $41,670 Here is a formula that can be used to find the nominal annual interest rate of costly trade credit: Nominal cost D iscount % 365 Days =  of trade credit 1 - D iscount % D ays taken  discount period

In this situation,

1 365  = 0.0101  12.1667 = 0.1229 = 12.29%. 99 40 - 10

Note that (1) the formula gives the same nominal annual interest rate as was calculated earlier, (2) the first term is the periodic cost of the credit (SKI spends $1 to get the use of $99), and (3) the second term is the number of ―savings periods‖ per year (SKI delays payment for 40 – 10 = 30 days), and there are 365/30 = 12.1667 30day periods in a year. Therefore, we could calculate the exact effective annual interest rate as effective rate = (1.0101)12.1667 – 1 = 13.01%. If SKI can obtain financing from its bank (or from other sources) at an interest rate of less than 13.01%, it should borrow the funds and take discounts. m.

SKI tries to match the maturity of its assets and liabilities. Describe how SKI could adopt either a more aggressive or more conservative financing policy.

Answer: There are three alternative current asset financing policies: aggressive, moderate, and relaxed. A moderate financing policy matches asset and liability maturities. (Of course, exact maturity matching is not possible because of (1) the uncertainty of asset lives and (2) that some common equity must be used and common equity has no maturity.) With this strategy, the firm minimizes its risk that it will be unable to pay off maturing obligations. An aggressive financing policy occurs when the firm finances all of its fixed assets with long-term capital, but part of its permanent current assets with short-term, nonspontaneous credit. There are degrees of aggressiveness; in fact, a firm could choose to finance all of its permanent current assets and part of its fixed assets with short-term credit; this would be a highly aggressive position, and one that would subject the firm to the dangers of rising interest rates as well as to loan renewal problems. A conservative financing policy occurs when the firm finances all of its permanent asset requirements and some of its seasonal demands with permanent capital. This position is a very safe one. Therefore, an aggressive financing policy uses the greatest amount of short-term debt, while the conservative policy uses the least. The maturity matching policy falls between these two policies.

What are the advantages and disadvantages of using short-term debt as a source of financing?

Answer: Although using short-term credit is generally riskier than using long-term credit, short-term credit does have some significant advantages. A short-term loan can be obtained much faster than long-term credit. Lenders insist on a more thorough financial examination before extending long-term credit. If a firm’s needs for funds are seasonal or cyclical, it may not want to commit to long-term debt because (1) flotation costs are generally high for long-term debt but trivial for short-term debt; (2) prepayment penalties with long-term debt can be expensive, and short-term debt provides flexibility; (3) long-term loan agreements contain provisions that constrain a firm’s future actions, and short-term credit agreements are less onerous; (4) the yield curve is normally upward sloping, indicating that interest rates are generally lower on short-term than on long-term debt. Even though short-term debt is often less expensive than long-term debt, shortterm debt subjects the firm to more risk than long-term financing. The reasons for this are (1) if a firm uses long-term debt, its interest costs will be relatively stable over time; however, if the firm uses short-term debt, its interest expense will fluctuate widely; and (2) if a firm borrows heavily on a short-term basis, it may find itself unable to repay this debt, and it may be in such a weak financial position that the lender will not extend the loan, which could force the firm into bankruptcy. o.

Would it be feasible for SKI to finance with commercial paper?

Answer: It would not be feasible for SKI to finance with commercial paper. Commercial paper is unsecured, short-term debt issued by large, financially strong firms and sold primarily to other business firms, to insurance companies, to pension funds, to money market mutual funds, and to banks. Maturities are generally 270 days (9 months) or less. There is an active, liquid market for commercial paper, and, since there is relatively little default risk, commercial paper rates are generally less than the prime rate. Note, though, that issuers of commercial paper are required to have back-up lines of bank credit that can be used to pay off the paper if necessary when it matures. These back-up credit lines have a cost, and this cost must be added to the interest rate on the paper to determine its effective cost. Since only large, well-known, financially strong companies can issue commercial paper, it would be impossible for SKI to tap this market.

Chapter 18 Current Asset Management ANSWERS TO END-OF-CHAPTER QUESTIONS

18-1 The two principal reasons for holding cash are for transactions and compensating balances. The target cash balance is not equal to the sum of the holdings for each reason because the same money can often partially satisfy both motives. 18-2 The four elements in a firm’s credit policy are credit standards, (2) credit period, (3) discount policy, and (4) collection policy. The firm is not required to accept the credit policies employed by its competition, but the optimal credit policy cannot be determined without considering competitors’ credit policies A firm’s credit policy has an important influence on its volume of sales, and thus on its profitability. 18-3 The latest date for paying and taking discounts is May 10. The date by which the payment must be made is June 9. 18-4 No. Although B sustains slightly more losses due to uncollectible accounts, its credit manager may have a wise policy that is generating more sales revenues (and thus profits) than would be the case with a policy that cut those losses to zero. 18-5 The DuPont equation is: et Income ales Total Assets ales Total Assets quity Holding all else constant, if a company has lower than average current assets, it will have lower than average total assets. That will increase the total asset turnover, which shows more sales are being generated per unit of assets, which increases the ROE. The lower-than-average total assets will have an indeterminate effect on the equity multiplier, depending on whether less equity or debt is used to finance the lower amount of current assets. The equity multiplier can either rise or fall. Even if the equity multiplier falls thus reducing the the firm’s level of financial risk overall falls too.

18-6 Sales Profit a. The firm loosens its credit standards. + 0 b. The terms of trade are changed from 2/10, net 30, to 1/10, net 30. 0 – c. The terms are changed from 2/10, net 30, to 0 + d. The credit manager gets tough with past-due – –

A/R +

0 3/10, 0

net

40.

accounts. 0

Explanations: a. When a firm ―loosens‖ its credit standards it allows more customers to buy on credit. It will make more credit sales. Profit may be affected in either direction. b. The smaller cash discount will probably reduce sales, so accounts receivable will decline. The company will save money because the discounts taken will be reduced. Some of the current customers who formerly took the 2% discount may decide to pay after 30 days or even longer. The effect of this would be to increase accounts receivable, so with all these changes accounts receivable and profits could go either way. c. A less stringent credit policy in terms of the credit period should stimulate sales. The accounts receivable could go up or down depending upon whether customers take the new higher discount or delay payments for the 10 additional days, and depending upon the amount of new sales generated. d. If the credit manager gets tough with past due accounts, sales will decline, as will accounts receivable.

18-7 a. Our – by (a below)

suppliers

switch

from

delivering

train

air

freight.

b. We change from producing just-in-time to meet seasonal demand to level,

year-round

+ production. c. Competition

the

markets

which

0 we (c below)

sell

increases.

d. The rate of general inflation rises. 0 e. Interest

rates

rise;

other

things

– are (e below)

constant.

(a) Lower safety stock will be required because delivery time is shortened. (c) On the one hand, the need to stay competitive may require large inventories, but if the market gets competitive, sales may fall off and the need for inventories may diminish. (e) EOQ and inventories are lower, since carrying costs are higher. 18-8 When money is tight, interest rates are generally high. This means that near-cash assets have high returns; hence, it is expensive to hold idle cash balances. Firms tend to economize on their cash balance holdings during tight-money periods. 18-9 If production is evenly spread out throughout the year, average inventory levels will be higher than if production is seasonal or matches demand.

18-10

a. Better synchronization of cash inflows and outflows would allow the firm to keep its transactions balance at a minimum, and would therefore lower the target cash balance.

b. Improved sales forecasts would tend to lower the target cash balance. c. A reduction in the portfolio of Treasury bills marketable securities would cause the firm’s cash balance to rise if the Treasury bills had been held in lieu of cash balances. d. An overdraft system will enable the firm to hold less cash. e. If the amount borrowed equals the increase in chequewriting, the target cash balance will not change. Otherwise, the target cash balance may rise or fall, depending on the relationship between the amount borrowed and the number of cheques written. f. The firm will tend to hold more Treasury bills, and the target cash balance will tend to decline. 18-1 COGS = $10,000,000; Inv Turnover = 2. Inventory = COGS/2 $10,000,000 = = $5,000,000. 2 If COGS/I = 5, how much cash is freed up? Inventory = COGS/5 $10,000,000 = = $2,000,000. 5 Cash Freed = $5,000,000 – $2,000,000 = $3,000,000. 18-2 DSO = 17; Credit Sales/Day = $3,500; A/R = ? DSO =

A/R S/365

SOLUTIONS TO END-OF-CHAPTER PROBLEMS A

18-3 Analysis of change: Projected Projected Income Statement Statement Under Current Policy Under New

Income Effect of Credit Credit Policy

Change Credit Policy Gross sales $2,200,000 + $100,000 $ 2,300,000 Less: Discounts 0 0 Net sales + 100,000 2,300,000 Variable costs + 75,000 1,725,000 Profit before credit costs and taxes + 25,000 575,000 Credit-related costs: Cost of carrying receivables* + 13,952 29,774 Collection expense – 10,000 32,000 Bad debt losses + 24,500 57,500 Profit before taxes – 3,452 455,726 Taxes (30%) – 1,036 136,718 Net income – $ 2,416 $ 319,008

0 2,200,000 1,650,000 550,000

15,822 42,000 33,000 459,178 137,753 $ 321,425

*Cost of carrying receivables:

DSO

Sales  Variable  Cost of      per day  cost ratio  funds 

 $2,200,000  Current policy = (25)   (0.75)(0.14) = $15,822. 365  

 $2,300,000  New policy = (45)   (0.75)(0.14) = $29,774. 365   Since the change in profitability is negative ($2,416), the firm should not relax its collection efforts. Note: If you wish to consider the opportunity cost or benefit of the change in time (i.e., the change in the DSO) it takes to receive the profits from the sales, then we need to make one additional calculation. In this case, the DSO is longer with the new policy, so the company receives its profits later, thus it is a cost: Opportunity cost = (20)($2,200,000/365)(0.25)(0.14) = $4,219. The new credit policy cost of carrying receivables is now $33,993 ($29,774 + $4,219). Profit before credit costs and taxes + $ 25,000 $ 575,000 Credit-related costs: Cost of carrying receivables + 18,171 33,993 Collection expense – 10,000 32,000 Bad debt losses + 24,500 57,500 Profit before taxes – 7,671 451,507 Taxes (30%) – 2,301 135,452 Net income – $ 5,370 $ 316,055

550,000

15,822 42,000 33,000 459,178 137,753 $ 321,425

18-4 Analysis of change: Income Statement

Projected Projected Income Effect of

Statement Under Current

Credit Policy

Under New Policy Change Gross sales –$ 400,000 $ 9,600,000 Less: Discounts 0 0 Net sales – 400,000 9,600,000 Variable costs – 320,000 7,680,000 Profit before credit costs and taxes – 80,000 1,920,000 Credit-related costs: Cost of carrying receivables* – 116,778 88,373 Bad debt losses 0 0 Profit before taxes + 36,778 1,831,627 Taxes (26%) + 9,562 476,223 Net income + $ 27,216 $1,355,404

Credit Credit Policy $ 10,000,000 0 10,000,000 8,000,000 2,000,000 205,151 0 1,794,849 466,661 $ 1,328,188

*Cost of carrying receivables:

DSO

Sales  Variable  Cost of      per day  cost ratio  funds 

urrent policy ew policy The firm should change its credit terms since the change in profitability is positive.

18-5 a. 0.4(10) + 0.6(40) = 28 days. b. $912,500/365 = $2,500 sales per day. $2,500(28) = $70,000 = Average receivables. c. 30:

Customers who do not take the discount and pay on Day i. Nominal cost: 3/97  365/20 = 56.44%. ii. Effective cost: (1 + 3/97)365/20 – 1 = 1.7435 – 1 = 0.7435 = 74.35%.

d. 40:

Customers who do not take the discount and pay on Day i. Nominal cost: 3/97  365/30 = 37.63%. ii. Effective cost: (1 + 3/97)365/30 – 1 = 0.4486 = 44.86%.

e. 0.4(10) + 0.6(30) = 22 days. $912,500/365 = $2,500 sales per day. $2,500(22) = $55,000 = Average receivables. Sales may also decline as a result of the tighter credit. This would further reduce receivables. Also, some customers may now take discounts, further reducing receivables.

18-6 a. EOQ =

2( F)(S) = (C)( P)

2($15)(90,000 ) = (0.2)(1.5)

9,000,000 = 3,000 bags per

order. b. The maximum inventory, which is on hand immediately after a new order is received, is 4,000 bags (3,000 + 1,000 safety stock). At $1.50 per bag the dollar cost is $6,000. c.

Average 3,000 = + 1,000 = 1,500 + 1,000 = 2,500 bags or 2 inventory $3,750.

90,000 365 days = 30 orders per year. = 12.2  12 days. 30 3,000 The company must place an order every 12 days.

18-7 a.

Gross sales Less: Discounts Net Sales Variable costs Profit before credit costs and taxes Cost of carrying receivables* Collection expense Bad debt losses Profit before taxes Taxes (30%) Net income

Projected Income Statement Under Current Credit Policy $7,200,000 36,000 7,164,000 4,320,000 2,844,000 44,739 168,000 216,000 2,415,261 724,578 $1,690,683

Effect of Credit Policy Change

Projected Income Statement Under New Credit Policy

– $360,000 – 8,640 – 351,360 – 216,000 – 135,360

$6,840,000 27,360 6,812,640 4,104,000 2,708,640

–

6,960

37,779

+ 72,000 – 113,400 – 87,000 – 26,100 – $ 60,900

240,000 102,600 2,328,261 698,478 $1,629,783

Cost of carrying receivables:

DSO

Sales  Variable  Cost of     .  per day  cost ratio  funds 

DSO(current) = (0.20 × 10) + (0.30 × 15) + (0.50 × 50) = 31.5 days Current policy = (31.5)  $7,200,000  (0.60)(0.12) = $44,739. 

365



DSO(new) = (0.40 × 10) + (0.60 × 40) = 28 days

 $6,840,000  New policy = (28)   (0.60)(0.12) = $37,779. 365   Note: If you wish to consider the opportunity cost or benefit of the change in time (i.e., the change in the DSO) it takes to receive the profits from the sales, then we need to make one additional calculation. In this case, the DSO shortens with the new policy, so the company receives its profits quicker, Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc. 18-511

thus it is a benefit:

Opportunity benefit = (3.5)($7,200,000/365)(0.40)(0.12) = $3,314. The new credit policy cost of carrying receivables is now $34,465 ($37,779 – $3,314).

Profit before credit costs and taxes Cost of carrying receivables Collection expense Bad debt losses Profit before taxes Taxes (30%) Net income

– $135,360

$2,844,000

–

44,739 168,000 216,000 2,415,261 724,578 $1,690,683

$2,708,640

10,274

34,465

+ 72,000 – 113,400 – 83,686 – 25,106 – $ 58,580

240,000 102,600 2,331,575 699,472 $1,632,103

The company should not implement the new policy.

b. The minimum sales level acceptable is the level that would result in the same income level with the new policy as the old one. The old policy results in $1,690,683 of income, therefore we solve for the level of sales under the new policy that results in income of $1,690,683. Let S = sales (0 40)(0 0 ) * 0 0) , 0,

(0 0)

(

)

(0 0)(0

)

(0 0

)

40,000+ (

S = $7,071,706

(0.40)(0.01)S = discounts (0.60)S = variable costs ( ) (0 0)(0 ) = cost of carrying receivables $240,000 = collection expense 0.015S = bad debt losses ($7,200,000 – $7,071,706)/$7,200,000 = 0.0178 = 1.78%.

18-8 a. EOQ =

2( F)(S) = (C)( P)

2($1,000)(4,000) = 0.10($125)

640,000 = 800 units.

When 800 units are ordered each time an order is placed, total inventory costs equal $10,000: TIC = CP(Q/2) + F(S/Q) = 0.10($125)(800/2) + $1,000(4,000/800) = $12.50(400) + $1,000(5) = $5,000 + $5,000 = $10,000. Note that the average inventory of custom valves is 400 units, and that 5 orders are placed per year. Also, at the EOQ level, total carrying costs equal total ordering costs. b. 600 units: TIC = $1,000(4,000/600)

CP(Q/2)

F(S/Q)

0.10($125)(600/2)

= $3,750 + $6,667 = $10,417. Added cost = $10,417 – $10,000 = $417. 1,000 units: TIC = 0.10($125)(1,000/2) + $1,000(4,000/1,000) = $6,250 + $4,000 = $10,250. Added cost = $10,250 – $10,000 = $250. Note the following points: 

At any order quantity other than EOQ = 800 units, total inventory costs are higher than they need be.



The added cost of not ordering the EOQ amount is not large if the quantity ordered is close to the EOQ. For example, if the order size is 25% above the EOQ (1,000 units), TIC increases by only $250/$10,000 = 2.5%.



If the quantity ordered is less than the EOQ, then

total carrying costs decrease, but total ordering costs increase. At q = 600 units, carrying costs fall by $1,250 per year, but ordering costs increase by $1,667. The net result is an increase in total costs. 

If the quantity ordered is greater than the EOQ, then total carrying costs increase, but total ordering costs decrease. At Q = 1,000 units, carrying costs increase by $1,250, but ordering costs fall by only $1,000, so the net result is an increase in total costs.

c. With an annual usage of 4 units Inland’s weekly usage rate is 4,000/52  77 units. If the order lead time is 2 weeks, then Inland must reorder each time its inventory reaches 2(77) = 154 units. Then, after 2 weeks, as it uses its last valve, the new order of 800 valves arrives. d. There are two ways to view the impact of safety stocks on total inventory costs Inland’s total cost of carrying the operating inventory is $10,000 (see part a). Now the cost of carrying an additional 200 units is CP(safety stock) = 0.1($125)(200) = $2,500. Thus, total inventory costs are increased by $2,500, for a total of $10,000 + $2,500 = $12,500. Another approach is to recognize that, with a 200-unit safety stock Inland’s average inventory is now 200 = 600 units. Thus, its total inventory cost, including safety stock, is $12,500: TIC = CP(average inventory) + F(S/Q) = 0.1($125)(600) + $1,000(4,000/800) = $7,500 + $5,000 = $12,500. Inland must still reorder when the operating inventory reaches 154 units. However, with a safety stock of 200 units in addition to the operating inventory, the reorder point becomes 200 + 154 = 354 units. Since Inland will reorder when its valve inventory reaches 354 units, and since the expected delivery time is weeks Inland’s normal 77 unit usage could rise to 354/2 = 177 units per Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc. 18-515

week over the 2-week delivery period without causing a stockout. Similarly, if usage remains at the expected 77 units per week, Inland could operate for 354/77  4.6 weeks versus the normal two weeks while awaiting delivery of an order.

SOLUTION TO SPREADSHEET PROBLEM 18-9 The detailed solution for the spreadsheet problem is in the file Ch 18 Build a Model Solution.xlsx and is available on the textbook’s website

MINI CASE The detailed solution for the Mini Case is also available in spreadsheet format, in the file Brigham4Ce_MiniCase_Ch18.xlsx, and is available on the textbook’s website. Rich Jackson, a recent finance graduate, is planning to go into the wholesale building supply business with his brother, Jim, who majored in building construction. The firm will sell primarily to general contractors, and it will start operating next January. Due to the nature of the products, sales will be relatively constant with some fluctuations month to month. Sales estimates for the first 6 months are as follows (in thousands of dollars): January $190 February 210

March April

$190 210

May $190 June 210

The terms of sale are net 30, but because of special incentives, the brothers expect 30% of the customers (by dollar value) to pay on the 10th day following the sale, 50% to pay on the 40th day, and the remaining 20% to pay on the 70th day. No bad debt losses are expected, because Jim, the building construction expert, knows which contractors are having financial problems. a.

Discuss, in general, what it means for the brothers to set a credit and collections policy.

Answer: When a firm sets its credit and collections policy, it determines four things: 1. The credit period, which is the length of time buyers are given to pay for their purchases. 2. The discounts that are given for early payment. 3. The credit standards, which are the financial strength requirements for customers to purchase on credit from the firm. 4. The collection policy, which is how hard the company

will work to collect slow-paying accounts. These policies determine the level of sales and also the level of accounts receivable. Note that although sales contribute to profitability, additional accounts receivable require the investment of funds, so a firm must take into consideration both the profits from additional sales and the additional capital required to fund accounts receivable when it determines a credit policy.

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Assume that, on average, the brothers expect annual sales of 18,000 items at an average price of $100 per item. (Use a 365-day year.) 1. What is the firm’s expected days sales outstanding (DSO)?

Answer: Days sales outstanding = DSO = 0.3(10) + 0.5(40) + 0.2(70) = 37 days, vs. 30-day credit period. One would expect some customers to pay somewhat slowly, so a 37-day DSO is probably not too bad. b.

2. What is its expected average daily sales (ADS)?

Answer: Average daily sales = ADS = day. b.

18,000 ($100) = $4,931.51 per 365

3. What is its expected average accounts receivable level?

Answer: Accounts receivable (A/R) = (DSO)(ADS) = 37($4,931.51) = $182,466. Thus, $182,466 of receivables are outstanding, and the firm must raise capital to carry receivables. If collections could be speeded up, and DSO reduced, then A/R, and hence the required financing, would be reduced. b.

4. Assume that the firm’s profit margin is 25%. How much of the receivables balance must be financed? What would the firm’s balance sheet figures for accounts receivable, notes payable, and retained earnings be at the end of 1 year if notes payable are used to finance the investment in receivables? Assume that the cost of carrying receivables had been deducted when the 25% profit margin was calculated.

Answer: Although the firm has $182,466 in receivables, the entire amount does not have to be financed, since 25% of the sales price is profit. This means that 75% of the price represents costs of materials, labour, rent, utilities, insurance, and so on. Thus, the firm must finance only 0.75($182,466) = $136,849 of the receivables balance. Disregarding other assets and liabilities, its balance sheet would look like this if notes payable are used to finance receivables: Accounts receivable $182,466

Notes payable Retained earnings

$136,849 45,617 $182,466

5. If bank loans have a cost of 12%, what is the annual dollar cost of carrying the receivables?

Answer: Cost of carrying receivables = 0.12($136,849) = $16,422. In addition, there is an opportunity cost associated with not having the use of the profit component of the receivables. c.

What are some factors that influence (1) a firm’s receivables level and (2) the dollar cost of carrying receivables?

Answer: 1. As shown in Question b.3. above, receivables are a function of the average daily sales and the days sales outstanding. Exogenous economic factors such as the state of the economy and competition within the industry affect average daily sales, but so does the firm’s credit policy The days’ sales outstanding depends mainly on credit policy, although poor economic conditions can lead to a reduction in customers’ ability to make payments 2. For a given level of receivables, the lower the profit margin, the higher the cost of carrying receivables, because the greater the portion of each sales dollar Brigham, Financial Management, Learning Canada, Inc. 18-521

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that must actually be financed. Similarly, the higher the cost of the financing, the higher the dollar cost of carrying the receivables.

Assuming that the monthly sales forecasts given previously are accurate, and that customers pay exactly as was predicted, what would the receivables level be at the end of each month? To reduce calculations, assume that 30% of the firm's customers pay in the month of sale, 50% pay in the month following the sale, and the remaining 20% pay in the second month following the sale. Note that this is a different assumption than was made earlier. Also assume there are 91 days in each quarter. Use the following format to answer Parts d and e: E.O.M.

Quarterly

DSO = Month

Sales

A/R

Sales

$590

$6.48

ADS

(A/R)/(ADS) Jan$190$ 133 Feb210 185 Mar190 175 Apr210 May190 Jun210

27.0

Answer: (Note: From this point on, the solutions are expressed in thousands of dollars. Also, the table given below is developed in the solutions to Parts d and e.) At the end of January, 30% of the $190 in sales will have been collected, so (1 – 0.3)($190) = 0.7($190) = $133 will remain outstanding, that is, in the receivables account. At the end of February, 30% + 50% = 80% of January’s sales will have been collected so receivables associated with January sales will be (1 – 0.3 – f February’s in sales, 30% will have been collected, so 0.7($210) = $147 will remain outstanding. Thus, the receivables balance at the end of February will be $38 from January's sales plus 4 from February’s sales for a total of y the end of March all of January’s sales will have been collected but % of February’s sales and % of March’s sales will still be outstanding so receivables will equal 0.2($210) + 0.7($190) = $175. Following this logic, the receivables balance at the end of any month can be estimated as follows:

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A/R = 0.7(sales in that month) + 0.2(sales in previous month). E.O.M.

Quarterly

DSO = Month (A/R)/(ADS) Jan Feb Mar 27.0 Apr May Jun 27.6 e.

Sales

A/R

$190 210 190

$133 185

210 190 210

185 175

Sales

ADS

175

$590

$6.48

185

$610

$6.70

What is the firm’s forecast average daily sales for the first 3 months? For the entire half-year? The days sales outstanding is commonly used to measure receivables performance. What DSO is expected at the end of March? At the end of June? What does the DSO indicate about customers’ payments?

Answer: For the first quarter, sales totalled $190 + $210 + $190 = $590, so ADS = $590/91 = $6.48. ADS for the full halfyear is ($590 + $610)/182 = $6.59. Note that we can rearrange the formula for receivables as follows: A/R = (DSO)(ADS) A/R DSO = . ADS March:

DSO =

$175 = 27.0 days; June: $6.48

DSO =

$185 = 27.6 $6.70

days. Use of DSO for analyzing aggregate receivables may not be appropriate when sales are seasonal or fluctuate. Since receivables at a given point in time reflect recent sales, but sales in the DSO calculation represents sales in the past 12 months, a seasonal increase in sales will raise the DSO, even if customers are still paying exactly

as before.

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Construct aging schedules for the end of March and the end of June (use the format given below). Age of Account

March

(Days)

A/R

0–30 31–60 61–90

June A/R

Answer: Aging schedule: Age June

Account

March

(Days) A/R Jun 38 Apr

% 0–30 79% 31–60 May 21 61–90 0

$133

Mar

76%

$147

42 0 $175

$185

A/R

Feb Jan

24 0 100%

100

To see how these aging schedules were constructed, consider first the end-of-March schedule. At that time, % of March’s sales had been collected so % remained uncollected: February’s contribution to receivables is 0.2($210) = $42. Finally, by the end of March all of January’s sales had been collected so none of January’s sales remained outstanding Thus the receivables account totals $175 at the end of March. The aging schedule total for the 2nd quarter is $185, the difference being due to small monthly sales fluctuations. Note that if a company has seasonal sales, the aging schedule will show an increasing or decreasing total, when in fact customers have not altered their payment patterns.

Assume now that it is several years later. The brothers are concerned about the firm’s current credit terms, now net 30, which means that contractors buying building products from the firm are not offered a discount and are supposed to pay the full amount in 30 days. Gross sales are now running $1,000,000 a year, and 80% (by dollar volume) of the firm’s paying customers generally pay the full amount on Day 30, while the other 20% pay, on average, on Day 40. Of the firm’s gross sales, 2% end up as bad debt losses. The brothers are now considering a change in the firm’s credit policy. The change would entail (1) changing the credit terms to 2/10, net 20, (2) employing stricter credit standards before granting credit, and (3) enforcing collections with greater vigour than in the past. Thus, cash customers and those paying within 10 days would receive a 2% discount, but all others would have to pay the full amount after only 20 days. The brothers believe that the discount would both attract additional customers and encourage some existing customers to purchase more from the firm—after all, the discount amounts to a price reduction. Of course, these customers would take the discount and, hence, would pay in only 10 days. The net expected result is for sales to increase to $1,100,000; for 60% of the paying customers to take the discount and pay on the 10th day; for 30% to pay the full amount on Day 20; for 10% to pay late on Day 30; and for bad debt losses to fall from 2% to 1% of gross sales. The firm’s operating cost ratio will remain unchanged at 75%, and its cost of carrying receivables will remain unchanged at 12%. To begin the analysis, describe the four variables that make up a firm’s credit policy, and explain how each of them affects sales and collections. Then use the information given in Part g to answer Parts h through m.

Answer: The four variables that make up a firm’s credit policy are (1) the discount offered, including the amount and period; (2) the credit period; (3) the credit standards used when determining who shall receive credit, and how much credit; and (4) the collection policy. Cash discounts generally produce two benefits: (1) they attract both new customers and expanded sales from current customers, because people view discounts as a price reduction, and (2) discounts cause a reduction in the days’ sales outstanding, since both new customers and some established customers will pay more promptly in order to get the discount. Of course, these benefits are offset to some degree by the dollar cost of the discounts themselves. The credit period is the length of time allowed to all ―qualified‖ customers to pay for their purchases In order to qualify for credit in the first place, customers must meet the firm’s credit standards. These dictate the minimum acceptable financial position required of customers to receive credit. Also, a firm may impose differing credit limits depending on the customer’s financial strength as judged by the credit department. Finally, collection policy refers to the procedures that the firm follows to collect past-due accounts. These can range from a simple letter or phone call to turning the account over to a collection agency. How the firm handles each element of credit policy will have an influence on sales, speed of collections, and bad debt losses. The object is to be tough enough to get timely payments and to minimize bad debt losses, yet not to create ill will and thus lose customers. h.

Under the current credit policy, what is the firm’s days’ sales outstanding (DSO)? What would the expected DSO be if the credit policy change were made?

Answer: Old (current) situation: DSO0 = 0.8(30) + 0.2(40) = 32 days. New situation: DSOn = 0.6(10) + 0.3(20) + 0.1(30) = 15 days. Thus, the new credit policy is expected to cut the DSO approximately in half. i.

What is the dollar amount of the firm’s current bad debt losses? What losses would be expected under the new policy?

Answer: Old (current) situation: BDLo = 0.02($1,000,000) = $20,000. New situation: BDLn = 0.01($1,100,000) = $11,000. Thus, the new policy is expected to cut bad debt losses sharply. j.

What would be the firm’s expected dollar cost of granting discounts under the new policy?

Answer: Current situation: under the current, no discount policy, the cost of discounts is $0. New situation: of the $1,100,000 gross sales expected under the new policy, 1% is lost to bad debts, so good sales = 0.99($1,100,000) = $1,089,000. Since 60% of the good sales are discount sales, discount sales = 0.6($1,089,000) = $653,400. Finally, the discount is 2%, so the cost of discounts is expected to be 0.02($653,400) = $13,068. k.

What is the firm’s current dollar cost of carrying receivables? What would it be after the proposed change?

Answer: urrent situation: the firm’s average daily sales currently amount to $1,000,000/365 = $2,739.73. The DSO is 32 days, so accounts receivable amount to 32($2,739.73) = $87,671. However, only 75% of this total represents cash costs—the remainder is profit—so the investment in receivables (the actual amount that must be financed) is 0.75($87,671) = $65,753. At a cost of 12%, the annual cost of carrying the receivables is 0.12($65,753) = $7,890. New situation: the cost of carrying the receivables balance under the new policy would be $4,068: ($1,100,000/365)(15)(0.75)(0.12) = $4,068.

What is the incremental after-tax profit associated with the change in credit terms? Should the company make the change? (Assume a tax rate of 40%.)

New

Old

Difference

Gross sales $1,000,000 Less discounts 0 Net sales 1,000,000 Production costs 750,000 Profit before credit costs and taxes 250,000 Credit-related costs: Carrying costs 7,890 Bad debt losses 20,000 Profit before taxes 222,110 Taxes (40%) 88,844 Net income $ 133,266

Answer: The income statements and differentials under the two credit policies are shown below: New Difference Gross sales $100,000 Less discounts 0 13,068 Net sales 1,000,000 86,932 Production costs 750,000 75,000 Profit before credit costs and taxes 250,000 11,932 Credit-related costs: Carrying costs

Old $1,100,000

$1,000,000

13,068 1,086,932 825,000 261,932 4,068

7,890 20,000

(3,822) Bad debt losses (9,000) Profit before taxes

11,000 246,864

222,110

24,754 Taxes (40%) 88,844 9,902 Net income $ 14,852

98,746 $148,118

$ 133,266

Thus, if expectations are met, the credit policy change would increase the firm's annual after-tax profit by $14,852. Since there are no non-cash expenses involved here, the $14,852 is also the incremental cash flow expected under the new policy. However the new policy is not riskless If the firm’s customers do not react as predicted then the firm’s profits could actually decrease as a result of the change. The amount of risk involved in the decision depends on the uncertainty inherent in the estimates, especially the sales estimate. Typically, it is very difficult to predict customers’ responses to credit policy changes. Further, a credit policy change may prompt the company’s competitors to change their own credit terms, and this could offset the expected increase in sales. Thus, the final decision is judgmental. If the prospect of an annual $14,852 increase in net income is sufficient to compensate for the risks involved, then the change should be made. (Note: Large, national companies often make credit policy changes in a given region in an effort to determine how customers and competitors will react, and then use the information gained when setting national policy. Note also that credit policy changes may not be announced in a ―broadcast‖ sense so as to slow down competitors’ reactions m.

Suppose the firm makes the change, but its competitors react by making similar changes to their own credit terms, with the net result being that gross sales remain at the current $1,000,000 level. What would the impact be on the firm’s after-tax profitability?

Gross sales Less discounts Net sales Production costs Profit before credit Costs and taxes Credit costs: Carrying costs Bad debt losses Profit before taxes Taxes (40%) Net income

$1,000,000 11,880 988,120 750,000 238,120

3,699 10,000 224,421 89,768 134,653

Note: discounts become $1,000,000 × 99% × 0.6 × 2% = $11,880 Carrying costs become $1,000,000/365 × 15 × 0.75 × 0.12 = $3,699 Under the old terms the net income was $133,266, so the policy change would result in a slight incremental gain of $134,653 – $133,266 = $1,387.

Chapter 19 Financial Options and Applications in Corporate Finance ANSWERS TO END-OF-CHAPTER QUESTIONS 19-1

a. An option is a contract that gives its holder the right to buy or sell an asset at some predetermined price within a specified period of time. A call option allows the holder to buy the asset, while a put option allows the holder to sell the asset.

A simple measure of an option’s value is its exercise value. The exercise value is equal to the current price of the stock (underlying the option) less the strike price of the option. The strike price is the price stated in the option contract at which the security can be bought (or sold). For example, if the underlying stock sells for $50 and the strike price is $20, the exercise value of the option would be $30.

c. The Black-Scholes option pricing model is widely used by option traders to value options. It is derived from the concept of a riskless hedge. By buying shares of a stock and simultaneously selling call options on that stock, the investor will create a riskfree investment position. This riskless return must equal the risk-free rate, or an arbitrage opportunity would exist. People would take advantage of this opportunity until the equilibrium level estimated by the Black-Scholes model was reached. 19-2

The market value of an option is typically higher than its exercise value due to the speculative nature of the investment. Options allow investors to gain a high degree of personal leverage when buying securities. The option allows the investor to limit loss but amplify return. The exact amount this protection is worth is the option’s time value, which is the difference between the option’s price and its exercise value.

19-3

(1) An increase in stock price causes an increase in the value of a call option. (2) An increase in strike price causes a decrease in the value of a call option. (3) An increase in the time to expiration causes an increase in the value of a call option. (4) An increase in the risk-free rate causes an increase in the value of a call option. (5) An increase in the standard deviation of stock return causes an increase in the value of a call option.

SOLUTIONS TO END-OF-CHAPTER PROBLEMS 19-1

Exercise value = Current stock price – Strike price = $45 – $35 = $10. Time value = Option price – Exercise value = $12 – $10 = $2.

19-2

Option’s strike price = $15; Exercise value = $22; Time value = $5; V = ? P0 = ? Time value = Market price of option – Exercise value $5 = V – $22 V = $27. Exercise value = P0 – Strike price $22 = P0 – $15 P0 = $37.

19-3

Call is in-the-money if the exercise value is greater than zero: Aug 92 call = Max(105.35 – 92, 0) = $13.35; in-the-money Aug 100 call = $5.35; in-the-money Aug 105 call = $0.35; in-the-money Nov 92 call = $13.35; in-the-money Nov 100 call = $5.35; in-the-money Nov 105 call = $0.35; in-the-money All other calls are out-of-the-money.

Puts in-the-money if the exercise value is greater than zero: Aug 130 put = Max(130 – 105.35, 0) = $24.65; in-the-money Aug 110 put = $4.65; in-the-money Nov 130 put = $24.65; in-the-money Nov 110 put = $4.65; in-the-money All other puts are out-of-the-money.

i. Aug 100 call exercise value = Max(105.35 – 100, 0) = $5.35 ii. Aug 110 put exercise value = Max(110 – 105.35, 0) = $4.65 iii. Nov 92 put exercise value = Max(92 – 105.35, 0) = $0

Time value = Option price – Exercise value i. Aug 92 call exercise value = Max(105.35 – 92, 0) = $13.35 Time value = 13.80 – 13.35 = $0.45

ii. Nov 92 call exercise value = Max(105.35 – 92, 0) = $13.35 Time value = 16.95 – 13.35 = $3.60 iii. The difference in time values is due to the fact that the November call has more time remaining to expiry, therefore there is a greater chance the stock can be even more-in-the-money, therefore making the option more valuable overall. 19-4

19-5

19-6

Profit/(Loss) = (Current value – Cost) × 100 shares per contract × Number of contracts = ($7.60 – $5.80) × 100 × 5 = $900

Profit/(Loss) = [($111.40 – $105) – $5.80] × 100 × 5 = $300

Rate of return = $300/$2,900 = 10.34%

($111.40 – $105.35)/$105.35 = $6.04/$105.35 = 5.74%

i. Profit/(Loss) = [$7.00 – (Max, $96 – $110, 0)] × 100 × 12 = [$7.00 – $0] × 100 × 12 = $8,400 ii. Profit/(Loss) = [$7.00 – $0] × 100 × 12 = $8,400 iii. Profit/(Loss) = [$7.00 – (Max, $112 – $110, 0)] × 100 × 12 = [$7.00 – $2.00] × 100 × 12 = $6,000 iv. Profit/(Loss) = [$7.00 – $50] × 100 × 12 = ($51,600)

i. Profit/(Loss) = Exercise value – Put cost = [(Max, $92 – $88, 0) – $3.20] × 100 × 10 = [$4.00 – $3.20] × 100 × 10 = $800 ii. Profit/(Loss) = [(Max, $92 – $93, 0) – $3.20] × 100 × 10 = [$0 – $3.20] × 100 × 10 = ($3,200) iii. Profit/(Loss) = [(Max, $92 – $107, 0) – $3.20] × 100 × 10 = [$0 – $3.20] × 100 × 10 = ($3,200)

Profit/(Loss) = Exercise value – Call cost = [(Max, $65 – $105, 0) – $11.70] × 100 × 10 = [$0 – $11.70] × 100 × 10 = ($11,700) ii. Profit/(Loss) = [$0 – $11.70] × 100 × 10 = ($11,700) iii. Profit/(Loss) = [(Max, $108 – $105, 0) – $11.70] × 100 × 10 = [$3 – $11.70] × 100 × 10 = ($8,700) iv. Profit/(Loss) = [$29 – $11.70] × 100 × 10 = $17,300

Profit/(Loss) = [$17.10 – (Max, $105 – $92, 0)] × 100 × 5 = [($17.10 – $13] × 100 × 5

= $2,050 ii. Profit/(Loss) = [($17.10 – Max, $105 – $100, 0)] × 100 × 5 = [$17.10 – $5] × 100 × 5 = $6,050 iii. Profit/(Loss) = [$17.10 – (Max, $105 – $115, 0)] × 100 × 5 = [ $17.10 – $0] × 100 × 5 = $8,550 19-7

P = $25; X = $25; t = 0.5; rRF = 0.05; 2 = 0.09; d1 = 0.2239; d2 = 0.0118; N(d1) = 0.5886; N(d2) = 0.5047; V = ? Using the Black-Scholes option pricing model, you calculate the option’s value as: V = P[N(d1)] – Xe rRF t [N(d2)] = $25(0.5886) – $25e(–0.05)(0.5)(0.5047) = $14.7150 – $25(0.9753)(0.5047) = $2.4092  $2.41.

19-8

19-9

Put = V – P + Xe rRF t = $6.56 – $33 + $32e-0.06(1) = $6.56 – $33 + $30.136 = $3.696  $3.70.

d1 

ln (P/X)  [rRF  (σ 2 / 2)]t σ t



ln ($30 /$35)  [0.05  (0.25/ 2)](0 .333333 ) 0.5 0.33333

  0.3319 .

d2 = d1 –  (t)0.5 = -0.3319 – 0.5(0.33333)0.5 = -0.6206. N(d1) = 0.3700 (from Excel NORMSDIST function). N(d2) = 0.2674 (from Excel NORMSDIST function). V = P[N(d1)] – Xe rRF t [N(d2)] = $30(0.3700) – $35e(–0.05)(0.33333)(0.2674) = $11.1000 – $9.2043 = $1.8957  $1.90. 19-10 The stock’s range of payoffs in one year is $35 – $23 = $12. At expiration, the option will be worth $35 – $30 = $5 if the stock price is $35, and zero if the stock price is $23. The range of payoffs for the stock option is $5 – 0 = $5.

Equalize the range to find the number of shares of stock: Option range/Stock range = $5/$12 = 0.4167. With 0.4167 shares, the stock’s payoff will be either $14.58 or $9.58. The portfolio’s payoff will be $14.58 – $5 = $9.58, or $9.58 – 0 = $9.58. The present value of $9.58 at the daily compounded risk-free rate is PV = $9.58/(1+ (0.05/365))365 = $9.11. The option price is the current value of the stock in the portfolio minus the PV of the payoff: V = 0.4167($28) – $9.11 = $2.56. 19-11 The stock’s range of payoffs in six months is $18 – $13 = $5. At expiration, the option will be worth $18 – $14 = $4 if the stock price is $18, and zero if the stock price $13. The range of payoffs for the stock option is $4 – 0 = $4. Equalize the range to find the number of shares of stock: Option range/Stock range = $4/$5 = 0.8. With 0.8 shares, the stock’s payoff will be either 0.8($18) = $14.40 or 0.8($13) = $10.40. The portfolio’s payoff will be $14.40 – $4 = $10.40, or $10.40 – 0 = $10.40. The present value of $10.40 at the daily compounded risk-free rate is PV = $10.40/(1+ (0.06/365))365/2 = $10.093. The option price is the current value of the stock in the portfolio minus the PV of the payoff: V = 0.8($15) – $10.093 = $1.907 .$1.91.

19-12 a.

ln(P/X) +[rRF +(σ 2 /2)]t ln($100/$100) +[0.05 + (0.20/2)](1.00) d =   0.3354. 1 σ t 0.4472 1.00 d2 = d1 –  (t)0.5 = 0.3354 – 0.4472(1.00)0.5 = -0.1118. N(d1) = 0.6313 (from Excel NORMSDIST function). N(d2) = 0.4555 (from Excel NORMSDIST function). V = P[N(d1)] – Xe rRF t [N(d2)] = $100(0.6313) – $100e(–0.05)(1.00)(0.4555) = $63.13 – $43.33 = $19.80. b. If the stock goes to $122, the value of the option will be $22. Likewise if the stock price goes to $82, then the value of the option will be $0. The value for N is 0

If you sell one call option and create a riskless hedge by purchasing N shares, the value of the portfolio at maturity will be (using stock price at $122), (0.55)($122) – $22 = $45.10. The present value of the portfolio’s value is: 4 (

)

Current option price = Current value of stock in portfolio – PV of portfolio Current option price = (0.55)($100) – $42.90 = $12.10

SOLUTION TO SPREADSHEET PROBLEM 19-13 The detailed solution for the spreadsheet problem is in the file Ch 19 Build a Model Solution.xls on the textbook’s website.

MINI CASE The detailed solution for the Mini Case is also available in spreadsheet format, in the file Brigham3Ce_MiniCase_Ch19.xlsx, and is available on the textbook’s website. You have just been hired as a financial analyst by Triple Trice Inc., a mid-sized Ontario company that specializes in creating exotic clothing. Because no one at Triple Trice is familiar with the basics of financial options, you have been asked to prepare a brief report that the firm's executives could use to gain at least a cursory understanding of the topic. To begin, you gathered some outside materials on the subject and used these materials to draft a list of pertinent questions that need to be answered. In fact, one possible approach to the paper is to use a question-and-answer format. Now that the questions have been drafted, you have to develop the answers.

What is a financial option? What is the single most important characteristic of an option?

Answer: A financial option is a contract that gives its holder the right to buy (or sell) an asset at a predetermined price within a specified period of time. An option’s most important characteristic is that it does not obligate its owner to take any action; it merely gives the owner the right to buy or sell an asset. b.

Options have a unique set of terminology. Define the following terms: (1) call option; (2) put option; (3) exercise price; (4) strike price; (5) option price; (6) expiration date; (7) exercise value; (8) covered option; (9) naked option; (10) in-the-money call; (11) out-of-the-money call.

Answer:

A call option is an option to buy a specified number of shares of a security, at a specific price, within some future period.

A put option is an option to sell a specified number of shares of a security, at a specific price, within some future period.

Exercise price is another name for strike price, the price stated in the option contract at which the security can be bought (or sold).

The strike price is the price stated in the option contract at which the security can be bought (or sold).

The option price is the market price of the option contract.

The expiration date is the date the option matures.

The exercise value is the value of a call option if it were exercised today, and it is equal to the current stock price minus the strike price. Note: The exercise value is zero if the stock price is less than the strike price.

A covered option is a call option written against stock held in an investor’s portfolio.

A naked option is an option sold without the stock to back it up.

10.

An in-the-money call is a call option whose strike price is less than the current price of the underlying stock.

11.

An out-of-the-money call is a call option whose strike price exceeds the current stock price.

In 1973, Fischer Black and Myron Scholes developed the Black-Scholes option pricing model (OPM). 1. What assumptions underlie the OPM?

Answer: The assumptions that underlie the OPM are as follows: 

The stock underlying the call option provides no dividends during the life of the option.



No transactions costs are involved with the sale or purchase of either the stock or the option.



The short-term, risk-free interest rate is known and is constant during the life of the option.



Security buyers may borrow any fraction of the purchase price at the short-term, risk-free rate.



Short-term selling is permitted without penalty, and sellers receive immediately the full cash proceeds at today's price for securities sold short.



The call option can be exercised only on its expiration date.



Security trading takes place in continuous time, and stock prices move randomly in continuous time.

2. Write out the three equations that constitute the model.

Answer: The OPM consists of the following three equations: V = P[N(d1)] – Xe  rRF t [N(d2)]. d1 =

ln( P/X)  [rRF  ( 2 /2)]t  t

d2 = d1 –  t . Here, V P N(d1)

= current value of a call option with time t until expiration. = current price of the underlying stock. = probability that a deviation less than d1 will occur in a standard normal distribution. Thus, N(d1) and N(d2) represent areas under a standard normal distribution function. X = strike price of the option. e  2.7183. rRF = risk-free interest rate. t = time until the option expires (the option period). ln(P/X) = natural logarithm of P/X. 2 = variance of the rate of return on the stock.

3. What is the value of the following call option according to the OPM? Stock price = $27.00. Strike price = $25.00 Time to expiration = 6 months. Risk-free rate = 6.0%. Stock return variance = 0.11.

Answer: The input variables are: P = $27.00; X = $25.00; rRF = 6.0 percent; t = 6 months = 0.5 years; and 2 = 0.11. Now, we proceed to use the OPM: V = $27[N(d1)] – $25e –(0.06)(0.5)[N(d2)].

ln( $27/$25 )  [(0.06  0.11/2 )]( 0.5) (0.3317 )(0.7071) 0.0770  0.0575 = = 0.5736. 0.2345

d1 =

d2 = d1 – (0.3317)(0.7071) = d1 – 0.2345 = 0.5736 – 0.2345 = 0.3391. N(d1) = N(0.5736) = 0.7168. N(d2) = N(0.3391) = 0.6327. Therefore, V = $27(0.7168) – $25e–0.03(0.6327) = $19.3536 – $25(0.97045)(0.6327) = $19.3536 – $15.3500 = $4.0036  $4.00. Thus, under the OPM, the value of the call option is about $4.00.

What impact does each of the following call option parameters have on the value of a call option? 1. 2. 3. 4. 5.

Current stock price Strike price Option’s term to maturity Risk-free rate Variability of the stock price

Answer: 1. The value of a call option increases (decreases) as the current stock price increases (decreases). 2. As the strike price of the option increases (decreases), the value of the option decreases (increases). 3. As the expiration date of the option is lengthened, the value of the option increases. This is because the value of the option depends on the chance of a stock price increase, and the longer the option period, the higher the stock price can climb. 4. As the risk-free rate increases, the value of the option tends to increase as well. Since increases in the risk-free rate tend to decrease the present value of the option's strike price, they also tend to increase the current value of the option. 5. The greater the variance in the underlying stock price, the greater the possibility that the stock's price will exceed the strike price of the option; thus, the more valuable the option will be.

What is put-call parity?

Answer:

Put-call parity specifies the relationship between puts, calls, and the underlying stock price that must hold to prevent arbitrage: Put + Stock = Call + PV of exercise price

Explain four different ways that knowledge of financial options is useful in corporate finance.

Answer: 1. Knowledge of financial options helps a manager to recognize and price real options. 2. Financial options help a manager to manage the financial risks such as uncertainty of changes in interest rates, exchange rates, or commodity prices that a company faces. 3. Knowledge of financial options is useful in capital structure decisions and in the creation of new types of securities such as convertible debt. 4. Financial options are useful in compensation plans as a company may grant stock options to key employees in order to align their interests with the interests of the company.

Chapter 20 Enterprise Risk Management ANSWERS TO END-OF-CHAPTER QUESTIONS 20-1

a. A derivative is an indirect claim security that derives its value, in whole or in part, by the market price (or interest rate) of some other security (or market). Derivatives include options, interest rate futures, exchange rate futures, commodity futures, and swaps. b. Corporate risk management relates to the management of unpredictable events that have adverse consequences for the firm. This effort involves reducing the consequences of risk to the point where there would be no significant adverse impact on the firm’s financial position. c. Financial futures provide for the purchase or sale of a financial asset at some time in the future, but at a price established today. Financial futures exist for Treasury bills, Treasury notes and bonds, CDs, Eurodollar deposits, foreign currencies, and stock indexes. While physical delivery of the underlying asset is virtually never taken, under forward contracts goods are actually delivered. d. A hedge is a transaction that lowers a firm’s risk of damage due to fluctuating stock prices, interest rates, and/or exchange rates. A natural hedge is a transaction between two counterparties where both parties’ risks are reduced. The two basic types of

hedges are long hedges, in which futures contracts are bought in anticipation of (or to guard against) price increases, and short hedges, in which futures contracts are sold to guard against price declines. A perfect hedge occurs when the gain or loss on the hedged transaction exactly offsets the loss or gain on the unhedged position. e. A swap is an exchange of cash payment obligations, which usually occurs because the parties involved prefer someone else’s payment pattern or type. f. Commodity futures are futures contracts that involve the sale or purchase of various commodities, including grains, oilseeds, livestock, meats, fibre, metals, and wood. 20-2

If the elimination of volatile cash flows through risk management techniques does not significantly change a firm’s expected future cash flows and WACC, investors will be indifferent to holding a company with volatile cash flows versus a company with stable cash flows. Note that investors can reduce volatility themselves through (1) portfolio diversification or (2) their own use of derivatives.

20-3

The six reasons risk management might increase the value of a firm is that it allows corporations to (1) increase their use of debt; (2) maintain their capital budget over time; (3) avoid costs associated with financial distress; (4) utilize their comparative advantages in hedging relative to the hedging ability of individual investors; (5) reduce both the risks and costs of borrowing by using swaps; and (6) reduce the higher taxes that result from fluctuating earnings.

20-4

There are several ways to reduce a firm’s risk exposure. First, a firm can transfer its risk to an insurance company, which requires periodic premium payments established by the insurance company based on its perception of the firm’s risk exposure. Second, the firm can transfer risk-producing functions to a third party. For example, contracting with a trucking company can, in effect, pass the firm’s risks from transportation to the trucking company. Third, the firm can purchase derivatives contracts to reduce input and financial risks. Fourth, the firm can take specific actions to reduce the probability of occurrence of adverse events. This includes replacing old electrical wiring or using fire-resistant materials in areas with the greatest fire potential. Fifth, the firm can take actions to reduce the magnitude of the loss associated with adverse events, such as installing an automatic sprinkler system to suppress potential fires. Finally, the firm can totally avoid the activity that gives rise to the risk.

20-5

The futures market can be used to guard against interest rate risk and input price risk through the use of hedging. If the firm were concerned that interest rates would rise, it would use a short hedge, or sell financial futures contracts. If interest rates did rise, losses on the issue due to the higher interest rates would be offset by gains realized from repurchase of the futures at maturity—because of the increase in interest rates, the value of the futures would be less than at the time of issue. If the firm were concerned that the Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc. 20-547

price of an input would rise, it would use a long hedge, or buy commodity futures. At the future’s maturity date, the firm will be able to purchase the input at the original contract price, even if market prices have risen in the interim. 20-6

Swaps allow firms to reduce their financial risk by exchanging their debt for another party’s debt, usually because the parties prefer the other’s debt contract terms. There are several ways in which swaps reduce risk. Currency swaps, where firms exchange debt obligations denominated in different currencies, can eliminate the exchange rate risk created when currency must first be converted to another currency before making scheduled debt payments. Interest rate swaps, where counterparties trade fixed-rate debt for floating-rate debt, can reduce risk for both parties based on their individual views concerning future interest rates.

SOLUTIONS TO END-OF-CHAPTER PROBLEMS 20-1

If Zhao issues fixed-rate debt and then swaps, its net cash flows will be: −7% + 6.8% − LIBOR = −(LIBOR + 0.2%).

20-2

The price of the hypothetical bond is $1,000(91.50)/100 = $915.00. Using a financial calculator, we can solve for rd as follows: N = 20; PV = –915.00; PMT = 30; FV = 1000; solve for I/YR = 3.604. The annual value of rd is 3.604%  2  7.21%.

20-3

The speculator will pay on their existing swap: 0,000,000

,000

The offsetting swap is only going to pay

0,000,000

,000.

This leaves a loss of $6,000. 20-4

The speculator will pay on their existing swap: 0,000,000

The offsetting swap will pay

,000 0,000,000

This gives a profit of $4,333. 20-5

Futures contract settled at 112.75% of $100,000 contract value, so PV = 1.1275  $1,000 = $1,127.50  100 bonds = $112,750. Using a financial calculator, we can solve for rd as follows: N = 20; PV = –1127.5; PMT = 30; FV = 1000; solve for I = rd = 2.205%  2 = 4.41%. If interest rates increase to 5.41%, then we would solve for PV as follows: N = 20; I = 5.41/2 = 2.705; PMT = 30; FV = 1000; solve for PV = $1,045.11  100 = $104,511. Thus, the contract’s value has decreased from $112,750 to $104,511.

20-6

If Creeson issues floating-rate debt and then swaps, its net cash flows will be –(LIBOR + 2.5%) – 5.1% + LIBOR = –7.6%. This is less than the 7.7% rate at which it could directly issue fixed-rate debt, so the swap is good for Creeson. If Branger issues fixed-rate debt and then swaps, its net cash flows will be –6% + 5.1% – LIBOR = –(LIBOR + 0.9%). This is less than the rate at which it could directly issue floating-rate debt (LIBOR + 1%), so the swap is good for Branger. When the swap rate is 5.1%, the swap is better for both parties. If the swap rate is 5.5%, then, if Creeson issues floating-rate debt and enters into the swap, its net cash flows will be –(LIBOR + 2.5%) – 5.5% + LIBOR = -8%, which is a worse rate than it would have by issuing fixed-rate debt. If the swap rate is 5.5%, then, if Branger issues fixed-rate debt and enters into the swap, its net cash flows will be –6% + 5.5% – LIBOR = –(LIBOR + 0.5%), which would be more advantageous for it.

20-7

a. In this situation, the firm would be hurt if interest rates were to rise by June, so it would use a short hedge, or sell futures contracts. Since futures maturing in June are selling for 102.53 of par, and futures contracts are for $100,000 in Government of Canada bonds, the value of 1 contract is $102,530. This means the firm must sell 10,000,000/ $102,530 = 97.53 ≈ 98 contracts to cover the planned $10,000,000 June bond issue. Should interest rates rise by June, Zinn Company will be able to repurchase the futures contracts at a lower cost, which will help offset the loss from financing at the higher interest rate. Thus, the firm has hedged against rising interest rates.

b. The firm would now pay 11% on the bonds. With an 9% coupon rate, the bond issue would bring in only $8,804,962: N = 20; I = 11/2 = 5.5; PMT = −0.09/2  10,000,000 = −450000; FV = −10000000; and solve for PV = $8,804,962. The firm would lose $10,000,000 − $8,804,962 = $1,195,038 on the bond issue. However, the firm will make money on its futures contracts. The implied yield at the time the futures contracts were entered is found by inputting N = 20; PMT = 3000; FV = 100000; PV = –$102,530; solving for I/YR = 2.833% per six months. The nominal annual yield is 2(2.833%) = 5.666%. (Note that the futures contracts are on hypothetical 10-year, 6% semiannual coupon bonds, which are yielding 5.666%.) Now, if interest rates increased by 200 basis points, to 7.666%, the value of each futures contract will drop to $88,500, found by inputting N = 20; I = 7.666/2 = 3.833; PMT = −3000; FV = −100000; and solving for PV = $88,510. The value of all of the futures contracts will drop to $88,500(98) = $8,673,000. However, the value of the short futures position began at $102,530(98) = $10,047,940. Since Zinn Company sold the futures contracts for $10,047,940, and will, in effect, buy them back at $8,673,000, the firm will make a $10,047,940 – $8,673,000 = $1,374,940 profit on the transaction ignoring transaction costs. Thus, the firm gained $1,374,940 on its futures position, but lost $1,195,038 on its underlying bond issue. On net, it gained $1,374,940 – $1,195,038 = $179,902. c. In a perfect hedge, the gains on futures contracts exactly offset losses due to rising interest rates. For a perfect hedge to exist, the underlying asset must be identical to the futures asset. Using the Zinn Company example, a futures contract must have existed on Zinn’s own debt (it existed on Treasury bonds) for the company to have an opportunity to create a perfect hedge. In reality, it is virtually impossible to create a perfect hedge, since in most cases the underlying asset is not identical to the futures asset. 20-8

a. In this situation, the firm would be hurt if interest rates were to rise by September, so it would use a short hedge, or sell futures contracts. Since futures maturing in September are selling for 97.61 of par, and futures contracts are for $100,000 in Government of Canada bonds, the value of 1 contract is $97,610. This means the firm must sell 30,000,000/$97,610 = 307.35 ≈ 307 contracts to cover the planned $30,000,000 September bond issue.

b. If interest rates rise by 250 basis points, the cost of debt for DeWalden will now be 10%. With a 7.50% coupon rate, $30,000,000 face value of bonds will bring in proceeds of only: N = 20; I = 10/2 = 5.0; PMT = −0.075/2  30,000,000 = −1,125,000; FV = −30,000,000; and solve for PV = $25,326,671. The firm will lose $30,000,000 − $25,326,671 = $4,673,329 on the bond issue. However, the firm will make money on its futures contracts. The implied yield at the time the futures contracts were entered is found by inputting N = 20; PMT = 3000; FV = 100000; PV = –$97,610; solving for I/YR = 3.163% per six months. The nominal annual yield is 2(3.163%) = 6.326%. (Note that the futures contracts are on hypothetical 10-year, 6% semiannual coupon bonds, which are yielding 6.326%.) Now, if interest rates increase by 250 basis points, to 8.826%, the value of each futures contract will drop to $88,480.44, found by inputting N = 20; I = 8.826/2 = 4.413; PMT = −3000; FV = −100,000; and solving for PV = $81,480. The value of all of the futures contracts will drop to $81,480 (307) = $25,014,360. However, the value of the short futures position began at $97,610(307) = $29,966,270. Since DeWalden sold the futures contracts for $29,966,270, and will, in effect, buy them back at $25,014,360, the firm will make a $29,966,270 – $25,014,360 = $4,951,910 profit on the transaction ignoring transaction costs. Thus, the firm gained $4,951,910 on its futures position, but lost $4,673,329 on its underlying bond issue. On net, it gained $4,951,910 – $4,673,329 = $278,581. c. If interest rates fall by 100 basis points, the cost of debt for DeWalden will now be 6.50%. With a 7.50% coupon rate, $30,000,000 face value of bonds will bring in proceeds of: N = 20; I = 6.5/2 = 3.25; PMT = −0.075/2  30,000,000 = −1,125,000; FV = −30,000,000; and solve for PV = $32,180,902. The firm will gain $32,180,902 − $30,000,000 = $2,180,902 on the bond issue.

However, the firm will lose money on its futures contracts. Now, if interest rates fall by 100 basis points, to 5.326%, the value of each futures contract will rise to $105,174, found by inputting N = 20; I = 5.326/2 = 2.663; PMT = −3000; FV = −100,000; and solving for PV = $105,174. The value of all of the futures contracts will increase to $105,174(307) = $32,288,418. However, the value of the short futures position began at $97,610(307) = $29,966,270. Since DeWalden sold the futures contracts for $29,966,270, and will, in effect, buy them back at $32,288,418, the firm will make a $29,966,270 – $32,288,418 = –$2,322,148 loss on the transaction ignoring transaction costs. Thus, the firm gained $2,180,902 on its debt issuance, but lost $2,322,148 on its futures position. On net, it lost $2,180,902 – $2,322,148 = –$141,246. 20-9

a. Each gold contract is for 100 ounces. If the price of gold changes by $0.25/oz, then the change in value of the contract will be $0.25/oz × 100 oz = $25.00. b. The company is concerned about the price of gold falling. Thus it will sell gold futures contracts. The number of contracts to sell is 2,500 oz/100 oz. per contract = 25 contracts. c. Closing Price

Daily Change

Gain/(Loss)

1158.6 1176.1 1183.0 1172.0

17.5 6.9 −11.0

($43,750) ($17,250) $27,500

Net Gain/(Loss)

13.4

($33,500)

d. The company sold 25 contracts at a price of 1,158.6 for a total value of (25)($1,158.60)(100) = $2,896,500, and bought them back at a price of 1,161.70 for a value of (25)($1,161.70)(100) = $2,904,250 for a loss on the futures position of $7,750. However, the company would have gained on the gold that it had as the price of gold would have risen (if the futures price rose). Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc. 20-553

SOLUTION TO SPREADSHEET PROBLEM 20-10 The detailed solution for the spreadsheet problem is in the file Ch 20 Build a Model Solution.xls and is available on the textbook’s website.

MINI CASE The detailed solution for the Mini Case is also available in spreadsheet format, in the file Brigham3Ce_MiniCase_Ch20.xlsx, and is available on the textbook’s website. You have just been hired as a financial analyst by Manitoba Sunshine Limited, a mid-sized Manitoba company that specializes in creating exotic sauces from imported fruits and vegetables. The firm’s CEO, Iman Hersi, recently returned from an industry corporate executive conference in Vancouver, and one of the sessions she attended was on the pressing need for smaller companies to institute corporate risk management programs. Since no one at Manitoba Sunshine is familiar with the basics of derivatives and corporate risk management, Hersi has asked you to prepare a brief report that the firm’s executives could use to gain at least a cursory understanding of the topics. To begin, you gathered some outside materials on derivatives and corporate risk management and used these materials to draft a list of pertinent questions that need to be answered. In fact, one possible approach to the paper is to use a question-and-answer format. Now that the questions have been drafted, you have to develop the answers. a.

Why might shareholders be indifferent whether or not a firm reduces the volatility of its cash flows?

Answer: If volatility in cash flows is not caused by systematic risk, then shareholders can eliminate the risk of volatile cash flows by diversifying their portfolios. Also, if a company decided to hedge away the risk associated with the volatility of its cash flows, the company would have to pass on the costs of hedging to the investors. Sophisticated investors can hedge risks themselves and thus they are indifferent as to who actually does the hedging. b.

What are six reasons risk management might increase the value of a corporation?

Answer: There are no studies proving that risk management either does or does not add value. However, there are six reasons risk management might increase the value of a firm. Risk management allows corporations to (1) increase their use of debt; (2) maintain their capital budget over time; (3) avoid costs associated with financial distress; (4) utilize their comparative advantages in hedging relative to the hedging ability of individual investors; (5) reduce both the risks and costs of borrowing by using swaps; and (6) reduce the higher taxes that result from fluctuating earnings.

What is corporate risk management? Why is it important to all firms?

Answer: Corporate risk management is the management of unpredictable events that have adverse consequences for the firm. This function is very important to a firm since it involves reducing the consequences of risk to the point where there should be no significant adverse effects on the firm’s financial position. d.

Risks that firms face can be categorized in many ways. Define the following types of risk: (1) speculative risks; (2) pure risks; (3) demand risks; (4) input risks; (5) financial risks; (6) property risks; (7) personnel risks; (8) environmental risks; (9) liability risks; and (10) insurable risks.

Answer:

Speculative risks are those that offer the chance of a gain as well as a loss, such as buying an ownership share in a company.

Pure risks are those that only offer the prospect of losses, such as a product liability or malpractice lawsuit (from the defendant’s standpoint).

Demand risks are those associated with the demand for a firm’s products or services, such as new products developed by competitors.

Input risks are those associated with a firm’s input costs, including materials and labour.

Financial risks are those that result from financial transactions, such as interest rate and currency exchange rate risks.

Property risks are associated with destruction of a firm’s productive assets, including the threat of fire, floods, and riots.

Personnel risks are risks that result from human actions, such as theft and fraud.

Environmental risks include those risks associated with polluting the environment.

Liability risks are connected with product, service, or employee liability, such as costs incurred as a result of improper actions by employees or damages resulting from defective products.

10.

Insurable risks are those that typically can be covered by insurance.

Explain how Manitoba Sunshine might benefit by implementing enterprise risk management and the COSO framework.

Answer: In addition to the reasons given in answer (b), there are many ways in which Manitoba Sunshine might benefit by implementing enterprise risk management. Implementing enterprise risk management can improve the company’s governance, improve the company’s ability to achieve its strategic and operating objectives, help the company to maintain regulatory compliance, improve its response in case a risk does arise, and overall improve the operations of the company.

What are some actions that companies can take to minimize or reduce risk exposures?

Answer: There are several actions that companies can take to minimize or reduce their risk exposure. First, companies can transfer risk to an insurance company by paying periodic premiums. Second, companies can transfer functions that produce risk to third parties, such as eliminating risks associated with transportation by contracting with a trucking company. Third, companies can purchase derivatives contracts to reduce input and financial risk. Fourth, companies can take actions to reduce the probability of occurrence of an adverse event, such as replacing old wiring to reduce the possibility of fire. Fifth, companies can take actions to reduce the magnitude of the loss associated with adverse events, such as installing automatic sprinkler systems. Finally, companies can simply avoid the activities that give rise to risk. g.

What is financial risk exposure? Describe the following concepts and techniques that can be used to reduce financial risks: (1) derivatives; (2) futures markets; (3) hedging; and (4) swaps.

Answer: Financial risk exposure refers to the risk inherent in the financial markets due to price fluctuations. 1. A derivative is a security whose value stems, or is derived, from the values of other assets. Options and futures contracts are two types of derivatives that are used to manage security price exposure. 2. Futures markets involve contracts that call for the purchase or sale of a financial (or real) asset at some future date, but at a price that is fixed today. Thus, these markets provide the opportunity to reduce financial risk exposure. 3. Hedging is generally conducted where a price change could negatively affect a firm’s profits. A long hedge involves the purchase of futures contracts in anticipation of, or to guard against, price increases. A short hedge, or sale of futures, is made when the firm is concerned about price declines in commodities or financial securities.

4. Swaps involve the exchange of cash payment obligations on debt between two parties, usually because each party prefers the terms of the other’s debt contract. Swaps can reduce each firm’s financial risk. For example, currency exchange rate risk can be eliminated if a Canadian firm with a pound-denominated debt could swap its debt with a British firm that has an equivalent dollar-denominated debt.

Describe how commodity futures markets can be used to reduce input price risk.

Answer: Essentially, the purchase of a commodity futures contract will allow a firm to make a future purchase of the input material at today’s price, even if the market price on the good has risen substantially in the interim.

It is January and Manitoba Sunshine is considering issuing $5 million in bonds in June to raise capital for an expansion. Currently, MS can issue 10-year bonds with a 7% coupon (with interest paid semiannually), but interest rates are on the rise and MS is concerned that long-term interest rates might rise by as much as 1% before June. You looked online and found that June T-bond futures are trading at 111.25. What are the risks of not hedging and how might MS hedge this exposure? In your analysis, consider what would happen if interest rates all increased by 1%.

Answer: If MS waits until June to issue its bonds, and if interest rates rise, then MS will have to pay a higher interest rate on its debt. How much does that cost MS? One way to calculate the cost is to see how much the 10-year 7% semiannual bonds that it intended to issue would be worth at the new discount rate of 8%. Input N = 20, I/YR = 8/2, PMT = –5,000,000(7%/2) = –175,000, FV = –5,000,000 and solve for PV = $4,660,242. Since they were going to be worth $5 million if they were issued immediately, then this represents a loss of $5,000,000 – $4,660,242 = $339,758. MS can hedge this risk by selling T-bond futures contracts. The T-bond futures contracts are trading at 111.25% of par, or $111,250 for a $100,000 contract value. This means MS will need to sell 5,000,000/111,250 = 44.94 contracts. Because you must trade in whole contracts, MS will use 45 T-bond futures contracts to hedge the bond position.

T-bond futures contracts are priced off a hypothetical 10-year, 6% coupon, semiannual payment bond, and the 111.25 futures price translates to a $1,112.50 for each $1,000 face value bond. The implied yield can be calculated with a financial calculator to be (N = 20; PMT = 30; FV = 1000; PV = –1,112.50; calculate I/Y = 2.2924% semiannually, which is an annual rate of about 4.585%. If interest rates increase by 1%, then the new yield on this underlying bond will be 5.585%. The sixmonth rate is 5.585%/2 = 2.7925%. The corresponding price at this yield is found by inputting N = 20, I/YR = 2.7925, PMT = –30, FV = –1000 and solving for PV = 1,031.47. The value of each futures contract will be $103,147 at this higher interest rate. This represents a decrease of $111,250 – $103,147 = $8,103 for each contract. Since MS has sold futures contracts, this represents a profit to MS. The total profit from the futures contracts is 45($8,103) = $364,635. MS will lose $339,758 on the bonds it issues but gain $364,635 on its futures contracts. The net hedging gain is $364,635 – $339,758 = $24,877. Thus MS will end up just a little bit better off if interest rates increase by 1%. Note that if interest rates were to decrease instead, then MS would gain on the bonds it issues but lose on its futures contracts.

Chapter 21 International Financial Management ANSWERS TO END-OF-CHAPTER QUESTIONS 21-1

a. A multinational corporation is one that operates in two or more countries. b. The exchange rate specifies the number of units of a given currency that can be purchased for one unit of another currency. The fixed exchange rate system was in effect from the end of World War II until August 1971. Under the system, the US dollar was linked to gold at the rate of $35 per ounce, and other currencies were then tied to the dollar. Under the floating exchange rate system, which is currently in effect, the forces of supply and demand are allowed to determine currency prices with little government intervention. c. A country has a deficit trade balance when it imports more goods from abroad than it exports. Devaluation is the lowering, by governmental action, of the price of its currency relative to another currency. For example, in 1967 the British pound was

devalued from US$2.80 per pound to US$2.50 per pound. Revaluation, the opposite of devaluation, occurs when the relative price of a currency is increased. d. Exchange rate risk refers to the fluctuation in exchange rates between currencies over time. A convertible currency is one that can be traded in the currency markets and can be redeemed at current market rates. When an exchange rate is pegged, the rate is fixed against a major currency such as the US dollar. Consequently, the values of the pegged currencies move together over time. e. Interest rate parity holds that investors should expect to earn the same return in all countries after adjusting for risk. Purchasing power parity, sometimes referred to as the ―law of one price,‖ implies that the level of exchange rates adjusts so that identical goods cost the same in different countries. f. The spot rate is the exchange rate that applies to ―on the spot‖ trades, or, more precisely, exchanges that occur two days following the day of trade. In other words, the spot rate is for current exchanges. The forward exchange rate is the prevailing exchange rate for exchange (delivery) at some agreed-upon future date, usually 30, 90, or 180 days from the day the transaction is negotiated. Forward exchange rates are analogous to future prices on commodity exchanges. Discounts (or premiums) on forward rates occur when the forward exchange rate differs from the spot rate. When the forward rate is below the spot rate, the forward rate is said to be at a discount. Conversely, when the forward rate is above the spot rate, it is said to be at a premium. g. Repatriation of earnings is the cash flow, usually in the form of dividends or royalties, from the foreign branch or subsidiary to the parent company. These cash flows must be converted to the currency of the parent, and thus are subject to future exchange rate changes. A foreign government may restrict the amount of cash that may be repatriated. Political risk refers to the possibility of expropriation and to the unanticipated restriction of cash flows to the parent by a foreign government. h. A eurodollar is a US dollar on deposit in a foreign bank, or a foreign branch of a US bank. Eurodollars are used to conduct transactions throughout Europe and the rest of the world. An international bond is any bond sold outside the country of the borrower. There are two types of international bonds: Eurobonds and foreign bonds. A Eurobond is any bond sold in some country other than the one in whose currency the bond is denominated. Thus, a Canadian firm selling dollar bonds in Switzerland is selling Eurobonds. A foreign bond is a bond sold by a foreign borrower but denominated in the currency of the country in which the issue is sold. Thus, a US firm selling bonds denominated in Canadian dollars in Canada is selling foreign bonds. i. The euro is a currency used by the nations in the European Monetary Union that signed the Treaty of Maastricht.

21-2

The US dollar. The primary reason for using the US dollar was that it provided a relatively stable benchmark, and it was accepted universally for transaction purposes.

21-3

Under the fixed exchange rate system, the fluctuations were limited to +1% and –1%. Under the floating exchange rate system, there are no agreed-upon limits.

21-4

A dollar will buy more Swiss francs.

21-5

There will be an excess supply of dollars in the foreign exchange markets, and this will tend to drive down the value of the dollar. Foreign investments in Canada will increase.

21-6

Taking into account differential labour costs abroad, transportation, tax advantages, and so forth, corporations can maximize long-run profits. There are also nonprofit behavioural and strategic considerations, such as maximizing market share and enhancing the prestige of corporate officers.

21-7

The foreign project’s cash flows have to be converted to Canadian dollars, since the shareholders of the corporation (assuming they are mainly Canadian residents) are interested in dollar returns. This subjects them to exchange rate risk, and therefore requires an additional risk premium. There is also a risk premium for political risk (mainly the risk of expropriation). However, foreign investments also help diversify cash flows, so the net effect on the required rate of return is ambiguous.

21-8

A eurodollar is a dollar deposit in a foreign bank, normally a European bank. The foreign bank need not be owned by foreigners—it only has to be located in a foreign country. For example, a Citibank subsidiary in Paris accepts eurodollar deposits. The French citizen’s deposit at Chase Bank in New York is not a eurodollar deposit. However, if he transfers his deposit to a bank in London or Paris, it would be. The existence of the eurodollar market makes the Federal Reserve’s job of controlling US interest rates more difficult. Eurodollars are outside the direct control of the US monetary authorities. Because of this, interest rates in the United States cannot be insulated from those in other parts of the world. Thus, any domestic policies the Federal Reserve might take toward interest rates would be affected by the eurodollar market.

21-9

No, interest rate parity implies that an investment in Canada with the same risk as a similar investment in a foreign country should have the same return. Interest rate parity is expressed as: Forward exchange rate 1  rh .  Spot exchange rate 1  rf

Interest rate parity shows why a particular currency might be at a forward premium or discount. A currency is at a forward premium whenever domestic interest rates are higher than foreign interest rates. Discounts prevail if domestic interest rates are lower than Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc. 21-561

foreign interest rates. If these conditions do not hold, then arbitrage will soon force interest rates back to parity. 21-10 Purchasing power parity (PPP) assumes there are neither transaction costs nor regulations that limit the ability to buy and sell goods across different countries. In many cases, these assumptions are incorrect, which explains why PPP is often violated. An additional complication, when empirically testing to see whether PPP holds, is that products in different countries are rarely identical. Frequently, there are real or perceived differences in quality, which can lead to price differences in different countries.

SOLUTIONS TO END-OF-CHAPTER PROBLEMS 21-1

$1 = 15 Mexican pesos; $1 = 97.00 Japanese yen; Cross exchange rate, yen/peso = ? Cross Rate:

Dollar Yen Yen = .  Peso Peso Dollar

Note that an indirect quotation is given for Mexican; however, the cross rate formula requires a direct quotation. The indirect quotation is the reciprocal of the direct quotation. Since $1 = 15 pesos, then 1 peso = $0.0667. Yen/Peso = 0.0667 dollars per peso  97.00 yen per dollar = 6.47 yen per peso. 21-2

rNom, 6-month T-bills = 7%; rNom of similar default-free 6-month Japanese bonds = 5.5%; Spot exchange rate = e0: 1 Yen = $0.012; 6-month forward exchange rate = ft = ? ft (1  rh ) .  e 0 (1  rf )

rf = 5.5%/2 = 2.75%. rh = 7%/2 = 3.5%. e0 = $0.012. ft $0.012

1.035 1.0275

= $0.01209.

The 6-month forward exchange rate is 1 yen = $0.01209.

21-3

Canadian computer = $790; French computer = 550 euros; Spot rate between euro and dollar = ? Ph = Pf(e0) $790 = 550 euros(e0) 790/550 = e0 $1.436 = e0. 1 euro = $1.436 or $1 = 1/1.436 = 0.6964 euros.

21-4

Dollars should sell for 1/1.50, or 0.6667 pounds per dollar.

21-5

The price of francs is $1.22 today. A 10% appreciation will make it worth $1.342 tomorrow. A dollar will buy 1/1.342 = 0.745 francs tomorrow.

21-6

Cross rate = pesos per dollar  dollars per pound = pesos/pound = 13.0  1.58 = 20.54 pesos per pound.

21-7

Spot rate = 1 yen = $0.0098; forward rate = ft = 1 yen = $0.0098; rNom of 90-day Japanese risk-free securities = 4.2%; rNom of 90-day Canadian risk-free securities = ?

ft (1  rh )  . Spot rate (1  rf ) rf = 4.2%/4 = 1.05%; rh = ? ( 1 + rh ) 1.0105 1 + rh = 1.0105 rh = 0.0105.

rNom = 1.05%  4 = 4.2%. 21-8

$1 = 15.2 pesos; Headphones = $15.00; Price of headphones in Mexico = ? Ph = Pf(Spot rate). 1 Peso = 1/15.2 = $0.06579. $15 = Pf($0.06579) $15 = 228 pesos $0.06579

21-9

a. The Canadian company is worried that the yen will fall (depreciate) over the next 60 days as the company would be able to buy fewer dollars with the yen. b. 7,000,000 yen/100.5 yen/dollar = 69,652 dollars c. (i) 7,000,000/98 yen/dollar = 71,429 dollars. $71,429 – $69,652 = $1,777 better off by not hedging. (ii) 7,000,000/103 yen/dollar = 67,961 dollars. $67,961 – $69,652 = $(1,691) worse off by not hedging.

21-10 The Canadian dollar liability of the corporation falls from $1.27(5,000,000) = $6,350,000 to $1.22(5,000,000) = $6,100,000, corresponding to a gain of $250,000 Canadian for the corporation. However, the real economic situation might be somewhat different. For example, the loan is presumably a long-term loan. The exchange rate will surely change again before the loan is repaid. What really matters, in an economic sense, is the expected present value of future interest and principal payments denominated in Canadian dollars. There are also possible gains and losses on inventory and other assets of the firm. A discussion of these issues quickly takes us outside the scope of this textbook. 21-11 The automobile’s value has increased because the dollar has declined in value relative to the yen. The 1983 cost in yen was (245 yen per dollar)(8,000 dollars) = 1,960,000 yen. At an exchange rate of 80 yen per dollar, this is (1,960,000 yen)/(80 yen per dollar) = $24,500. 21-12 a. NZD 3,000,000/NZD 1.11 = $2,702,703. b. NZD 3,000,000/NZD 1.120 = $2,678,571. c. If the exchange rate is NZD 1.06 to $1 when payment is due in 3 months, the NZD 3,000,000 will cost: NZD 3,000,000/NZD 1.06 = $2,830,189 which is $127,486 more than the spot price today and $151,618 more than purchasing a forward contract for 90 days. 21-13 a. rNom of 90-day Canadian risk-free securities = 5%; of 90-day German risk-free securities = 5.3%; Spot rate = 1 euro = $1.30; Forward rate = ft selling at premium or discount = ?

ft (1  rh )  . Spot rate (1  rf ) rh = 5%/4 = 1.25%; rf = 5.3%/4 = 1.325%; Spot rate = $1.30.

1.0125 ft = $1.30 1.01325 ft = 0.9993. $1.30 ft = $1.2990. The forward rate is selling at a discount, since a euro buys fewer dollars in the forward market than it does in the spot. In other words, in the spot market $1 would buy 1/1.30 = 0.7692 euros, but at the forward rate $1 would buy 1/$1.2990 = 0.7698 euros; therefore, the forward currency is said to be selling at a discount. b. The 90-day forward rate is ft = $1.2990. 21-14 a. 1) $1,000,000 × 82 ¥/$ = 82,000,000 ¥. 2) ¥ 82,000,000 × 0.01 = 820,000¥. 3) Lock in ¥82,820,000. Receives ¥82,820,000/80.20 ¥/$ = $1,032,668. 4) $1,000,000

× 1.025 = $1,025,000.

5) $1,032,668 – $1,025,000 = $7,668 profit.

b. Arbitrage is possible in this case because interest rate parity is not in equilibrium. We can see this with the following calculation: According to interest rate parity: (

)

(

)

( (

00 ) 00 )

But the market forward rate (ft) = 80.20 yen/dollar. The implied forward rate is not equal to the market forward rate, therefore arbitrage can occur. The way to profit is to use the (mispriced) market forward rate to your advantage. In this case, it takes fewer yen to buy a dollar than it ought to, so the trader should buy dollars at the 1-year forward rate. As traders see this opportunity, there will be demand for 1-year Japanese securities, which will push up their price and reduce their yield, which should bring interest rate parity back into equilibrium. 21-15 D1 = 2 pounds; Exchange rate = $1.80/pound; Pound depreciates 3% against $1. Dividend grows at 5% and rs = 12%. 7 million shares outstanding. g=

1.05 – 1 = 1.9417%. 1.03

P0=

( (

) )

$3.60 0.100583

= $35.791337 Total equity = $35.791337  7 million shares = $250,539,359.

21-16 a. If a Canadian-based company undertakes the project, the rate of return for the project is a simple calculation, as is the net present value. NPV = −$1,000,000 + $700,000/1.13 + $600,000/(1.13)2 = $89,357. Rate of return: Enter into your financial calculator cash flow register CF0 = −1,000,000, CF1 = 700,000, CF2 = 600,000 and calculate IRR = 20.0% b. This analysis is from the perspective of a Korean investor, so Korea is the home country and Canada is the foreign country. The interest rate parity relationship uses direct quotes for exchange rates; direct quotes are number of units of home currency per unit of foreign currency. Therefore, a direct quote from a Korean perspective is the number of won per dollar. According to interest rate parity, the following condition holds:

 1  rKorea  =   1  rCA  For the 1-year forward rate:

Forward exchange rate Spot exchange rate

1.03 Forward exchange rate = 1.04 1050 Forward exchange rate = 0.990381 1050 Forward exchange rate = 1,039.90 won per Canadian dollar. The 2-year exchange rate is calculated as: 2

Forward exchange rate  1.0325  =  1050  1.0425  2-year forward exchange rate = 1029.95 won per Canadian dollar.

c. First, we must adjust the cash flows to reflect Nam Sung’s home currency. The expected won cash flows are Year 0: (−1,000,000 dollars)(1,050 won per dollar) = −1,050.00 million won. Year 1: (700,000 dollars)(1039.90 won per dollar) = 727.93 million won. Year 2: (600,000 dollars)(1029.95 won per dollar) = 617.97 million won. Using the won-denominated cash flows, the appropriate NPV and rate of return can be found. NPV = −1,050 + 727.93/1.13 + 617.97/1.132 = 78.15 million won. Rate of return: Enter into your financial calculator or Excel cash flow register CF0 = −1,050, CF1 = 727.93, CF2 = 617.97 and calculate IRR = 18.85%

SOLUTION TO SPREADSHEET PROBLEM 21-17 The detailed solution for the spreadsheet problem is in the file Ch 21 Build a Model Solution.xlsx and is available on the textbook’s website.

MINI CASE The detailed solution for the Mini Case is also available in spreadsheet format, in the file Brigham4Ce_MiniCase_Ch21.xlsx, and is available on the textbook’s website. Pure Juice Products Ltd. is a medium-sized producer of juice drinks with orchards in southern Ontario. Until now, the company has confined its operations and sales to Canada, but its CEO, George Gaynor, wants to expand into Europe. The first step would be to set up sales subsidiaries in Spain and Sweden, then to set up a production plant in Spain, and, finally, to distribute the product throughout the European common market. The firm’s financial manager, Ruth Schmidt, is enthusiastic about the plan, but she is worried about the implications of the foreign expansion on the firm’s financial management process. She has asked you, the firm’s most recently hired financial analyst, to develop a 1-hour tutorial package that explains the basics of multinational financial management. The tutorial will be presented at the next board of directors’ meeting. To get you started, Schmidt has supplied you with the following list of questions. a.

What is a multinational corporation? Why do firms expand into other countries?

Answer: Use the examples given here when discussing why firms ―go international.‖ 1. To seek new markets. Coca-Cola and McDonald’s have expanded around the world to seek new markets. Likewise, Sony, Toshiba, and other Japanese consumer electronics manufacturers have aggressively pushed into markets worldwide. 2. To seek raw materials. Canadian resource companies have searched around the world for years for new sources of minerals and fossil fuels. It is not surprising that a large company like Barrick Gold has mining production operations not only in Canada, but also in the United States, South America, Africa, and Australia. Currently, the company is trying to get a foothold in Russia and Pakistan. 3. To seek new technology. No one country has the lead in all technologies, so many companies are going global to ensure access to new technologies. For example, in the past several years, there have been four joint ventures between Japanese and American chip manufacturers for the sole purpose of exchanging technology.

4. To gain production efficiencies. 5. To avoid political and regulatory hurdles. The most prominent example here is the move by Toyota, Honda, Mazda, and Mitsubishi to produce cars and trucks in North America to avoid import quotas. 6. To diversify. By establishing worldwide production facilities and markets, firms can cushion the impact of adverse economic trends in any single country. b.

What are the nine major factors that distinguish multinational financial management from financial management as practised by a purely domestic firm?

Answer: 1. Different currency denominations. Cash flows in various parts of multinational corporate systems will be denominated in different currencies. Hence, an analysis of exchange rates, and the effect of fluctuating currency values, must be included in all financial analyses. 2. Language differences. The ability to communicate is critical in all business matters, and Canadian businesspeople are not as strong in learning other languages. In effect, it is easier for foreign firms to invade our markets than for us to invade theirs. It is interesting to note, though, that English has become the international business language. Many business school programs in Europe are conducted in English rather than in the host country’s language. Also, some multinational companies, such as ABB, a large Swedish firm headquartered in Zurich, have adopted English as the language of corporate communication. Although English is now spoken by most international businesspeople, knowledge of other languages remains critical to the success of multinational firms. 3. Cultural differences. Different countries, and even different regions in a single country, have unique cultural heritages that shape values and influence the role of business in the society. Such differences affect consumption patterns, defining the appropriate firm goals, attitudes toward risk taking, dealings with employees, and so on. For example, most Japanese workers view their jobs as a lifetime commitment, while many American workers view theirs as temporary until something better comes along. To give another illustration, consider PepsiCo’s move into the Japanese market by its Frito-Lay subsidiary. At first, Frito-Lay marketed popular American products such as Ruffles potato chips and Doritos corn chips. These products did poorly, and the Japanese venture almost failed, but it was saved when the company began producing a chip with soy sauce and seaweed flavouring.

4. Economic systems. At one extreme are countries with free markets (i.e., unfettered capitalism); at the other are countries with command (or planned) economies. For example, Singapore, Hong Kong, and Denmark place the least restrictions on business activities, whereas North Korea and Cuba place the most. However, most countries are a mixture. For example, Canada is very much in the free market spectrum while Belarus is towards the other end. 5. Legal systems. Some countries, such as Canada and the UK, base their legal systems on common law, which emphasizes prior court decisions and usually provides stronger protection for investors and property owners. In contrast, civil law legal systems, such as in France, rely on detailed laws and regulations enacted by the government. Different legal systems complicate matters ranging from the simple recording of business transactions to the role played by the judiciary in resolving conflicts. Such differences can even make procedures that are required in one part of the company illegal in another part. 6. Taxation. Differences in countries’ tax laws can cause strikingly different aftertax cash flows for identical transactions. 7. Government intervention. The terms under which companies compete, the actions that must be taken or avoided, the ability to curtail unprofitable operations, and the terms of trade on various transactions often are determined not in the marketplace but by direct negotiation with host governments. 8. Political risk. A foreign government might place constraints on the transfer of corporate resources or even expropriate assets within its boundaries. Also, corrupt government officials might expect gifts or bribes. 9. Terrorism and crime. Terrorist groups and criminals might threaten a company’s property or the safety of its employees. This includes extortion and kidnapping. In fact, some terrorist groups finance substantial portions of their budgets through kidnap and ransom.

Consider the following illustrative exchange rates. Canadian Dollars Required to Buy One Unit of Foreign Currency Euro 1.5000 Swedish krona 0.1580

What is a direct quotation? What is the direct quote for euros??

Answer: From a Canadian perspective, the quotes in the first column are called direct quotes because they are number of units of a foreign currency that can be purchased by 1 unit of the home currency. The direct quote for the euro is CAD per EUR = 1.50. The financial press would report this as ―the euro is trading at $1.50.‖

2. Calculate the indirect quotations for euros and kronor (the plural of krona is kronor).

Answer: Indirect quotations, which are the number of units of foreign currency that can be purchased with one Canadian dollar, are merely the reciprocal of the direct quotation. Here, the table is repeated with the indirect quotations added:

Euro Swedish krona

Direct quotation: Canadian dollars required to buy one unit of foreign currency 1.5000 0.1580

Indirect quotation: number of units of foreign currency per Canadian dollar 0.6667 6.3291

3. What is a cross rate? Calculate the two cross rates between euros and kronor.

Answer: The exchange rate between any two currencies that does not involve dollars is a cross rate. Here are the two cross rates between euros and kronor: Cross rate = = 0.6667  0.1580 = 0.1053 euros per krona. And,

Kronor Dollars  Dollars Euros = 6.3291  1.5000 = 9.4937 kronor per euro.

Cross rate =

Note that the two cross rates are reciprocals of one another. Also, note that the cross rates can be calculated by dividing either the direct or indirect quotations. Thus, there are numerous ways of calculating cross rates. c.

4. Assume Pure Juice Products can produce a litre of apple juice and ship it to Spain for $1.75. If the firm wants a 70% markup on the product, what should the apple juice sell for in Spain?

Answer: There are 0.6667 euros to the dollar, so the juice must sell for ($1.75)(1.70)(0.6667) = 1.98 euros. c.

5. Now, assume Pure Juice Products begins producing the same litre of apple juice in Spain. The product costs 1.8 euros to produce and ship to Sweden, where it can be sold for 20 kronor. What is the dollar profit on the sale?

Answer: 1.8 euros are equal to 1.8(9.4937 kronor per euro) = 17.09 kronor, so the profit on the sale in Sweden is 20 – 17.09 = 2.91 kronor. Now, there are 0.1580 dollars per krona, so the dollar profit is 2.91(0.1580) = $0.4598. c.

6. What is exchange rate risk?

Answer: The volatility inherent in a floating exchange rate system increases the uncertainty of cash flows that must be translated from one currency into another. This increase in uncertainty is exchange rate risk.

Briefly describe the current international monetary system. How does the current system differ from the system that was in place prior to August 1971?

Answer: Prior to 1971, the world operated on a fixed exchange rate system. The value of the US dollar was linked to gold at the fixed price of $35 per ounce, and the values of other currencies were then tied to the dollar. For example, in 1964, the British pound was fixed at $2.80 for 1 pound, with a 1% permissible fluctuation around this rate. Thus, the British government had to regularly intervene in the foreign exchange market to keep the pound in the range of $2.77 to $2.83. When the pound fell, the Bank of England had to buy pounds, offering either foreign currencies or gold in exchange. Conversely, if the pound reached the top of the range, the Bank of England would sell pounds. The official exchange rates were occasionally ―reset‖ to reflect changing economic conditions. The current international monetary system for about 30 industrialized nations is a free floating rate system. In this system, currency exchange rates are allowed to fluctuate in response to market conditions with a minimum of governmental intervention. Changes in currency demand can be due to trade deficits (i.e., one nation imports more from another nation than it exports, causing there to be higher relative demand for the currency of the bigger exporter). It can also be due to capital movements. For example, if interest rates are relatively high in one country, then investors might seek to purchase that country’s securities, which increases demand for that country’s currency. The IMF classifies about 40 more countries as floating because they have a little more government intervention than free floating currencies. Many countries use an exchange rate that is ―pegged,‖ or fixed, with respect to another currency. Examples of pegged currencies pegged to the US dollar are those of Belize, Barbados, and Saudi Arabia. Denmark pegs to the euro. Some countries have managed currency that behaves like a pegged currency, but the country hasn’t reported it as such. This category includes Singapore, Costa Rica, Burundi, and about 15 other countries. A few countries, including China, are in the IMF category of ―Other Managed Arrangements‖ because the currency policy doesn’t fit into any other IMF category. For example, China has a currency that is pegged to a basket of currencies; however, China doesn’t tell what currencies are in the basket. Sometimes China lets the exchange rate float; sometimes it manages the currency more actively. Some countries belong to a monetary union and have no individual currency but instead agree to use a common currency. The largest is the Economic and Monetary Union (EMU) of the European Union. Nineteen EMU countries currently use the ―euro.‖ The European Central Bank controls the monetary policy of the EMU nations using the euro. When a currency increases in value relative to another currency, it is said to appreciate. Under the fixed exchange rate system, strong currencies had to be revalued occasionally, which changed the tie to other currencies to a new, higher rate.

Conversely, a currency that loses value is said to depreciate, and such currencies had to be devalued under the old fixed rate system. e.

What is a freely convertible currency? What is a partially convertible currency? What is a nonconvertible currency? What problems arise when a multinational company operates in a country whose currency is not freely convertible?

Answer: A freely convertible currency is traded in the international currency markets with no government restrictions. It is also called a fully convertible currency and a hard currency. Countries with free floating exchange systems also have fully convertible currencies. In addition, several other floating rate currencies are fully convertible, such as the New Zealand dollar and the Israeli new shekel. A partially convertible currency is one that trades in international FOREX markets but that has significant government imposed restrictions. Restrictions include limiting (1) FOREX trading in the country to a small group of currencies; (2) amounts of its currency that may be converted to foreign currencies for purposes of investing abroad; and (3) amounts foreigners may invest inside the country. Most currencies fall into this category, including India and China. A nonconvertible currency is one that is not traded in international FOREX markets due to government restrictions. It is also called a soft currency or a blocked currency. Cuba and North Korea have nonconvertible currencies. When a country’s currency is not fully convertible, it is difficult for multinational companies to conduct business in that country, because there is no easy way to return profits earned to the company’s home country. Often, in this situation, it is necessary to engage in some kind of barter arrangement to promote investment. f.

What is the difference between spot rates and forward rates? When is the forward rate at a premium to the spot rate? At a discount?

Answer: Spot rates are the rates paid to buy currency for immediate delivery (actually, two days after the date of the trade). Forward rates are the rates paid to buy currency for delivery at some agreed-upon date in the future (say, 90 days). If the forward currency is less valuable than the spot currency, the forward rate is said to be at a discount to the spot rate. Conversely, if the forward currency is more valuable than the spot currency, the forward currency is said to sell at a premium. Firms use currency forward markets to hedge against adverse exchange rate fluctuations that might occur before a transaction is completed. To illustrate, suppose a Canadian importer buys German appliances for sale in Canada. The terms are net 90, so the importer must pay in euros in 90 days. The dollar could weaken against the euro over the period, and hence force the importer to use more dollars to buy the merchandise. To guard against this possibility, the importer could buy euros for delivery in 90 days, thus locking in the current forward rate.

What is interest rate parity? Currently, you can exchange 1 euro for 1.5100 dollars in the 180-day forward market, and the risk-free rate on 180-day securities is 6% in Canada and 4% in Spain. Does interest rate parity hold? If not, which securities offer the higher expected return?

Answer: Interest rate parity holds that investors should expect to earn the same return in all countries after adjusting for risk. What is the implied forward rate (ft), given the spot rate of 1.5000? Spot rate = 1 euro = $1.5000; rh = 6%/2 = 3.00%; rf = 4%/2 = 2.00%.

1 r t h  Spot rate 1  r f f t  1.03 1.5000 1.02 f  $1.5147 . t If interest rate parity held, then ft = $1.5147; however, ft = $1.5100, so parity doesn’t hold. f

The Canadian securities offer the higher return as calculated below: 1. Assume you convert 1,000 euros to Canadian dollars in the spot market. In the spot market, spot rate = 1.5000 dollars per euro. Convert 1,000 euros  1.5000 dollars/euro = $1,500. 2. Invest $1,500 in the 180-day Canadian security, which offers a semiannual return of 6%/2 = 3%. So, in 180 days you will receive $1,500  1.03 = $1,545. 3. Agree today to exchange the $1,545, 180 days from now, at a 180-day forward exchange rate of ft = 1.5100 dollars per euro. Your dollar return after 180 days = $1,545/1.5100 dollars per euro= 1,023.18 euros. 4. The investment’s expected 180-day return = 23.18/1,000 = 0.02318 = 2.318%, or a nominal return of 2  2.318% = 4.64%, versus a 4% annual return if you had invested in Spanish securities.

What is purchasing power parity? If grapefruit juice costs $2.00 a litre in Canada, the spot exchange rate is CAD 1.5000 = EUR 1.000, and purchasing power parity holds, what should be the price of grapefruit juice in Spain?

Answer: Purchasing power parity, sometimes referred to as the law of one price (LOP), implies that the level of exchange rates adjusts so that identical goods cost the same amount in different countries. Purchasing power parity = Ph = Pf(Spot rate) Spot rate = Ph/Pf $1.5000 = $2.00/Pf Pf = $2.00/$1.5000 = 1.33 euros. i.

What effect does relative inflation have on interest rates and exchange rates?

Answer: To illustrate, consider the situation between Japan and Canada. Japan has generally had a lower inflation rate than Canada, so Japanese interest rates have been lower than Canadian interest rates. This might tempt treasurers of Canadian multinational firms to borrow in Japan rather than in Canada. However, a foreign currency will, on average, depreciate (or appreciate) at a percentage rate approximately equal to the amount by which its inflation rate exceeds (or is less than) our own. Thus, the dollar has generally weakened against the yen over time, so it would take more and more dollars to pay back interest denominated in yen. j.

Briefly discuss the international capital markets.

Answer: Individuals buy securities issued by foreign governments and firms, and US and other foreign firms issue securities abroad. These transactions take place in the international capital markets. Here is a brief description of the major international capital markets: 1. A eurodollar is a US dollar deposited in a bank outside the United States. The major difference between a ―regular‖ dollar and a eurodollar is its location. This places eurodollars outside the direct control of US monetary authorities, so regulations such as fractional reserves and FDIC insurance premiums do not apply. Eurodollars are borrowed by US and foreign individuals, corporations, and governments that need dollars for various purposes. Since the borrower must pay back the lender in dollars, eurodollar transactions are not used to convert currencies, but rather represent another source of dollar borrowing. Interest rates on eurodollars are tied to the London Interbank Offer Rate (LIBOR), which is the rate of interest offered by the largest and strongest London banks on eurodollar deposits. LIBOR rates are generally 0.5 to 1.0 percentage points higher than the rate on comparable deposits offered by domestic banks in the United States. The eurodollar market deals mostly with short maturities, generally less than one year, although loans of up to 5 years have occurred.

2. International bonds, which are any bonds sold outside the country of the borrower, fall into two categories. Foreign bonds are bonds sold by a foreign borrower, but denominated in the currency of the country in which they are sold. Thus, when Bell Canada sells bonds in the United States denominated in US dollars, the firm is selling foreign bonds. In general, foreign bonds have to meet all the regulations of the country in which they are issued. Eurobonds are bonds sold in some country other than the one in whose currency the bond is denominated. For example, when Mercedes-Benz (a German company) sells bonds denominated in euros in Switzerland, these bonds are eurobonds. In general, countries do not apply as stringent requirements on bonds denominated in a foreign currency as they do bonds denominated in the home currency. Further, most eurobonds are issued in bearer form, so buyers have anonymity, both for tax and other purposes. For these reasons, investors are usually willing to accept somewhat lower yields on eurobonds than on foreign bonds or ―regular‖ bonds. Thus, US firms can often sell eurobonds denominated in dollars at lower cost than similar domestic issues.

To what extent do average capital structures vary across different countries?

Answer: There is some evidence that average capital structures vary among the large industrial countries. One problem, however, when interpreting these numbers is that different countries often use very different accounting conventions, which makes it difficult to compare capital structures. An American study attempts to control for differences in accounting practices. This study suggests that differences in accounting practices can explain much of the cross-country variation in capital structures. After adjusting for these accounting differences, capital structures are more similar across different countries than a previous study had suggested. l.

Briefly describe special problems that occur in multinational capital budgeting and describe the process for evaluating a foreign project. Now consider the following project: A Canadian company has the opportunity to lease a manufacturing facility in Japan for 2 years. The company must spend ¥1 billion initially to refurbish the plant. The expected net cash flows from the plant for the next 2 years, in millions, are CF1 = ¥500 and CF2 = ¥800. A similar project in Canada would have a risk-adjusted cost of capital of 10%. In Canada, a 1-year government bond pays 2% interest and a 2-year bond pays 2.8%. In Japan, a 1-year bond pays 0.05% and a 2-year bond pays 0.26%. The spot rate = 99 yen/dollar. What is the project’s NPV?

Answer: The same general principles that apply to domestic capital budgeting also apply to foreign capital budgeting. However, foreign capital budgeting is complicated by the following three primary factors: 1. Tax law differences. Foreign operations are usually taxed at the local level, and then the funds repatriated, or returned, to the parent corporation may be subject to additional taxes. 2. Political risk. Foreign governments have the right to restrict the amount of funds that can be repatriated. In extreme cases, foreign governments can even expropriate the assets owned by foreign companies without offering any compensation. 3. Exchange rate risk. Funds repatriated from foreign operations have to be converted into dollars, so foreign capital projects are subject to exchange rate risk.

The first step is to estimate the future expected exchange rates using the multiyear interest rate parity equation:

 1  rh   Expected t-year forward exchange rate = (Spot exchange rate)   1  rf 

where the exchange rates are expressed in direct quotations. The indirect spot exchange rate is 99 yen per dollar, so the direct rate is 1/99 = 0.010101 dollars per yen. The expected forward rates are:

Maturity (in years) 1 2

rh 2.0% 2.8%

rf 0.05% 0.26%

Spot rate (USD per yen) 0.010101 0.010101

Expected forward rate (USD per yen) 0.010298 0.010619

The current dollar cost of the project is ¥1,000,000,000(0.010101 $/¥) = $10.101 million. The Year 1 cash flow in dollars is ¥500,000,000 (0.010298$/¥) = $5.149 million. The complete time line and NPV are shown below. Year 0

Cash flows in yen(millions)

–1,000

500

800

Expected exchange rates

0.010101

0.010298

0.010619

Cash flows in dollars

–$10.101

$5.149

$8.495

Project cost of capital = NPV(millions) =

10% $1.601

Briefly discuss special factors associated with the following areas of multinational working capital management. 1. Cash management

Answer: Although multinational and domestic firms have the same objectives for cash management and use similar procedures, the multinational firm faces a more complex task. Since the distances involved are much greater, multinational firms tend to rely more on lockbox systems and wire transfers. Also, since multinational firms have access to more financial markets than do domestic firms, multinational companies are more likely to have global concentration banks, say in Tokyo, New York, London, and Zurich, and excess funds are transferred around the world to take advantage of the best rates available. Short-term borrowings are handled in the same way, with many more opportunities available to the firm. However, whenever the borrowing or lending takes place in a currency other than dollars, it is necessary to consider the possibility of adverse exchange rate movements. m.

2. Credit management

Answer: Granting credit is riskier for a multinational firm than for a domestic corporation because, in addition to the normal risk of default, the credit granting corporation must also worry about exchange rate fluctuations between the time the credit is given and the time the payment must be made. In addition to being riskier, credit is more important for international business, because much of the commerce on which lesserdeveloped countries depend could not occur if the seller did not grant credit. Many companies buy export credit risk insurance when granting credit to foreign customers. m.

3. Inventory management

Answer: As with other aspects of financial management, inventory management in a multinational setting is similar to but more complex than that in a purely domestic firm. For example, where should Exxon store its inventories of crude oil and refined products, and how much should be stored at each location? The answer depends on many factors, including shipping times, carrying costs, import quotas and taxes, differential taxes on inventories, and expected exchange rate movements. These factors greatly complicate inventory decisions within multinational firms.

22-1

a. Assets-in-place, also known as operating assets, include the land, buildings, machines, and inventory that the firm uses in its operations to produce its products and services. Growth options are not tangible. They include items such as R&D and customer relationships. Financial, or nonoperating, assets include investments in marketable securities and non-controlling interests in the stock of other companies. b. Operating current assets are the current assets used to support operations, such as cash, accounts receivable, and inventory. They do not include short-term investments. Operating current liabilities are the current liabilities that are a natural consequence of the firm’s operations, such as accounts payable and accruals. They do not include notes payable or any other short-term debt that charges interest. Net operating working capital is operating current assets minus operating current liabilities. Operating capital is the sum of net operating working capital and operating long-term assets, such as net plant and equipment. Operating capital also is equal to the net amount of capital raised from investors. This is the amount of interest-bearing debt plus preferred stock plus common equity minus short-term investments. NOPAT is the amount of net income a company would generate if it had no debt and held no financial assets. NOPAT is a better measure of the performance of a company’s operations because debt lowers income. In order to get a true reflection of a company’s operating performance, one would want to take out debt to get a clearer picture of the situation. Free cash flow is the cash flow actually available for distribution to investors after the company has made all the investments in fixed assets and working capital necessary to sustain ongoing operations. It is the most important measure of cash flows because it shows the exact amount available to all investors. c. The value of operations is the present value of all the future free cash flows that are expected from current assets-in-place and the expected growth of assets-in-place when discounted at the weighted average cost of capital: 

Vop(at time 0)  

FCFt

t t 1 1  WACC

The terminal, or horizon value, is the value of operations at the end of the explicit forecast period. It is equal to the present value of all free cash flows beyond the forecast period, discounted back to the end of the forecast period at the weighted average cost of capital: Vop(at time N) 

FCFN 1 FCFN (1  g)  . WACC  g WACC  g

The corporate valuation model defines the total value of a company as the value of operations plus the value of nonoperating assets plus the value of growth options. d. An agency relationship arises whenever one or more individuals, the principals, hire another individual, the agent, to perform some service and then delegate decision-making authority to that agent. Primary agency relationships exist between (1) shareholders and managers, and (2) debtholders and shareholders. e. Agency costs include all costs borne by shareholders to encourage managers to maximize a firm’s stock price rather than act in their own self-interests. The three major categories of agency costs are (1) expenditures to monitor managerial actions, such as audit costs; (2) expenditures to structure the organization in a way that will encourage desirable managerial behaviour, such as appointing outside investors to the board of directors; and (3) opportunity costs that are incurred when shareholderimposed restrictions, such as requirements for shareholder votes on certain issues, limit the ability of managers to take timely actions that would enhance shareholder wealth. f. An agency problem arises whenever a manager of a firm owns less than 100% of the firm’s common stock, creating a potential conflict of interest called an agency conflict. The fact that the manager will neither gain all the benefits of the wealth created by his or her efforts nor bear all of the costs of perquisite consumption will increase the incentive to take actions that are not in the best interests of the nonmanager shareholders. In addition to conflicts between shareholders and managers, there can also be conflicts between shareholders (through managers) and creditors. Creditors have a claim on part of the firm’s earnings stream for payment of interest and principal on debt, and they have a claim on the firm’s assets in the event of bankruptcy. However, shareholders have control (through managers) of decisions that affect the riskiness of the firm. g. Managerial entrenchment occurs when a company has such a weak board of directors and has such strong anti-takeover provisions in its corporate charter that senior managers feel there is very little chance that they will be removed. Nonpecuniary benefits are perks that are not actual cash payments, such as 22-586 Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc.

lavish offices, memberships at country clubs, corporate jets, and excessively large staffs.

h. Targeted share repurchases, also known as greenmail, occur when a company buys back stock from a potential acquirer at a higher than fair-market price. In return, the potential acquirer agrees not to attempt to take over the company. Shareholder rights provisions, also known as poison pills, allow existing shareholders in a company to purchase additional shares of stock at a lower than market value if a potential acquirer purchases a controlling stake in the company. A restricted voting rights provision automatically deprives a shareholder of voting rights if the shareholder owns more than a specified amount of stock. i. A stock option allows its owner to purchase a share of stock at a fixed price, called the strike price, no matter what the actual price of the stock is. Stock options always have an expiration date, after which they cannot be exercised. 22-2

22-3

Owner/managers benefit from higher wealth due to ownership, but they also benefit from the perks they consume, such as lavish offices, vacations, and golf club memberships. If the owner/manager is the only manager, then the owner/manager bears the full cost of the perks. But if the owner/manager owns only part of the company, the owner/manager reaps all the benefits of the perks but the cost is shared by the outside shareholders. Potential investors know this might happen, so they pay less for a minority interest in a company.

22-4

After a loan is originated, borrowers might make decisions that are harmful to the lender. For example, borrowers might invest in risky projects. From the borrower’s point of view, risky projects are like options. If the project pays off big, most of the benefits accrue to the borrowers (the creditors just get the principal back). If the project fails by a little or by a lot, the borrower doesn’t get anything. So borrowers have an incentive to take on riskier projects. Borrowers also might take on additional debt. Lenders anticipate this, and charge a higher interest rate.

22-5

Entrenched managers consume too many perquisites, such as lavish offices, excessive staffs, country club memberships, and corporate jets. They also invest in projects or acquisitions that make the firm larger, even if they don’t make the firm more valuable.

22-6

Stock options in compensation plans usually are issued with a strike price equal to the current stock price. As long as the stock price increases, the option will become valuable, even if the stock price doesn’t increase as much as investors expect.

SOLUTIONS TO END-OF-CHAPTER PROBLEMS 22-1

HV2023 = (FCF2023 (1 + gL))/(WACC – gL) = ($50(1 + 0.04))/(0.09 – 0.04) = $1,040.0 million. Vop =

FCF2022 FCF2023 HV2023 + + 1 2 (1+WACC) (1+WACC) (1+WACC)2

$20 $50 $1,040 0 + + 1 2 (1+0.09) (1+0.09) (1+0.09)2

= $935.8 million.

22-2 V_operations + ST investments Total value – Total debt – Preferred stock Intrinsic value of equity Divided by # shares Intrinsic stock price 22-3

NOPAT

$80 7 87 20 5 62 1 $62

= EBIT(1 – T) = 100(1 – 0.3) = $70.

Net operating WC21 = ($27 + $80 + $106) – ($52 + $28) = $213 – $80 = $133. Operating capital21 = $133 + $265 = $398. Net operating WC22 = ($28 + $84 + $112) – ($56 + $28) = $224 – $84 = $140. Operating capital22 = $140 + $281 = $421. FCF = NOPAT – Net investment in operating capital = $70 – ($421 – $398) = $47.0 million.

22-4

Value of operations = Vop = PV of expected future free cash flow

Vop =

22-5

FCF(1 + g) $700,000 (1.04) = = $9,100,000. WACC g 0.12 0.04

The growth rate in FCF from 2023 to 2024 is g = ($750.00 Vop at 2023 =

22-6

a. Vop =

$707.55)/$707.50 = 0.06.

$707.55 (1.06) = $37,500 (in millions). 0.08 0.06

FCF (1  g) $250,000 (1.05) = = $5,250,000. WACC  g 0.10  0.05

b. Total value = Value of operations + Value of nonoperating assets = $5,250,000 + $80,000 = $5,330,000. c. Value of common equity = Total value – Value of debt = $5,330,000 – $600,000 = $4,730,000. d. Stock price = Value of common equity/Number of common shares

22-7

= $4,730,000/250,000 = $18.92. Company A: Free cash flow = NOPAT – Net investment in operating capital = $1,000,000 – $370,000 = $630,000 Value of operations =

$630,000(1.04)  $10,920,000 0.10  0.04

Company B: Free cash flow = $1,200,000 – $700,000 = $500,000

$500,000 (1.05)  $10,500,000 0.10  0.05 Company A has a higher value due to higher free cash flow, which is due to lower investments required in operating capital. Notice A's higher value even though A has a lower growth rate. Value of operations =

22-8

a. HV2 =

$108,000 = $2,700,000. 0.12  0.08

WACC = 12% 1 |

2 g = 8%

$80,000

$100,000

$108,000

N |

  

71,428.57 (80,000/1.12) 79,719.39 (100,000/1.12)2 2,152,423.47 (2,700,000/1.12)2 $2,303,571.43

22-9

1. Vop =

$3(1  0.05) $2.85 = = $15.83 million. 0.18 0.13  0.05

2. Vop = $3/0.13 = $23.08 million. 3. Vop =

$3.15 $3(1.05) = = $39.38 million. 0.08 0.13  0.05

4. Vop =

$3.30 $3(1.10) = = $110.00 million. 0.03 0.13  0.10

22-10 a. HV3 = b.

$40 (1.07) = $713.33 million. 0.13  0.07

WACC = 13% | |

–20 ($ 17.70) 23.49 522.10 $527.89

g = 7% |

4 |

N   

Vop3 = 713.33

753.33

c. Total valuet = 0 = $527.89 + $10.0 = $537.89. Value of common equity = $537.89 – $100 = $437.89. $437 .89 Price per share = = $43.79. 10.0

22-11 a. End of Year:

g = 15% FCF0 = 1.75 FCF1

FCF2

FCF3

FCF4

g = 5% FCF5 FCF6

FCFt = FCF0(1 + g)t FCF1 = $1.75(1.15)1 = $2.0125 million. FCF2 = $1.75(1.15)2 = $2.3114 million. FCF3 = $1.75(1.15)3 = $2.6615 million. FCF4 = $1.75(1.15)4 = $3.0608 million. FCF5 = $1.75(1.15)5 = $3.5199 million. b. HV5 =

$3.5199 (1.05) FCF6 FCF5 (1  g n ) = = $52.80 million.  WACC  g n WACC  g n 0.12  0.05

This is the value of operations 5 years from now. c. PV of HV5 = $52.80/(1 + WACC)5 = $52.80/(1 + 1.12)5 = $29.96 million. Calculator solution: N = 5, I/YR = 12, PV = ?, PMT = 0, FV = −$52.80: PV = $29.96 d. Total PV of FCF1 through FCF5 = $2.0125(1 + 0.12)1 + $2.3114(1 + 0.12)2 + $2.6615(1 + 0.12)3 + $3.0608(1 + 0.12)4 + $3.5199(1 + 0.12)5 = $9.48 million Calculator solution: Input 0, 2.0125, 2.3114, 2.6615, 3.0608, 3.5199 into the cash flow register, input I/YR = 12, PV = ? PV = $9.48. e. The total value of operations today is the sum of the PV of the horizon value and the PVs of the free cash flows from Years 1 through 5: Vop,0

= PV FCF Years 1 through 5 + PV of HV5 = $9.48 + $29.96 = $39.44 million.

Calculator solution: Input 0, 2.0125, 2.3114, 2.6615, 3.0608, 56.3199 (3.5199 + 52.80) into the cash flow register, input I/YR = 12, PV = ? PV = $39.44.

22-12 a. You first need to calculate the free cash flow for 2022, which was done in Problem 22-3 (or again shown below): NOPAT= EBIT(1 – T) = 100(1 – 0.3) = $70. Net operating WC21 = ($27 + $80 + $106) – ($52 + $28) = $213 – $80 = $133. Operating capital21 = $133 + $265 = $398. Net operating WC22 = ($28 + $84 + $112) – ($56 + $28) = $224 – $84 = $140. Operating capital22 = $140 + $281 = $421. FCF= NOPAT – Net investment in operating capital = $70 – ($421 – $398) = $47.0 million. The horizon value (HV)2022 = [$47.0(1.04)]/(0.12 – 0.04) = $611 million. b.

VOp at 12/31/2021 = [$47.0 + $611]/(1+0.12) = $587.5 million.

Total corporate value = $587.5 + $66.0 = $653.5 million.

Value of equity = $653.5 – ($130.0 + $164.0) = $359.5 million. Price per share = $359.5/24 = $14.98.

22-13 Total corporate value = Value of operations + Marketable securities = $220 + $24 = $244 million. Value of equity = Total corporate value – debt – preferred stock = $244 – ($70 + $80) – $35 = $59 million. 22-14 Total corporate value = Value of operations + Marketable securities = $651 + $47 = $698 million. Value of equity = Total corporate value – Debt – Preferred stock = $698 – ($65 + $131) – $33 = $469 million. Price per share = $469/10 = $46.90. Market value of $49.10 > intrinsic value of $46.90. The stock is overvalued in the market.

22-15 a. NOPAT2022 = $108.6(1 – 0.3) = $76.02 NOWC2022 = ($5.6 + $56.2 + $112.4) – ($11.2 + $28.1) = $134.9 million. Capital2022 = $134.9 + $397.5 = $532.4 million. FCF2022 = NOPAT – Investment in capital = $76.02 – ($532.4 – $502.2) = $76.02 – $30.2 = $45.82 million. b. HV2022 = [$45.82(1.05)]/(0.10 – 0.05) = $962.22 million. c. VOp at 12/31/2021 = [$45.82 + $962.22]/(1+0.10) = $916.40 million. d. Total corporate value = $916.40 + $49.9 = $966.30 million. e. Value of equity = $966.30 – ($69.9 + $140.8) – $35.0 = $720.60 million. Price per share = $720.60/12 = $60.05.

SOLUTION TO SPREADSHEET PROBLEM 23-16 The detailed solution for the spreadsheet problem is in the file Ch 22 Build a Model Solution.xlsx and is available on the textbook’s website.

MINI CASE The detailed solution for the Mini Case is also available in spreadsheet format, in the file Brigham4Ce_MiniCase_Ch22.xlsx, and is available on the textbook’s website. You have been hired as a consultant to Kulpa Fishing Supplies (KFS), a company that is seeking to increase its value. The company’s CEO and founder, Mia Kulpa, has asked you to estimate the value of two privately held companies that KFS is considering acquiring. But first, the senior management of KFS would like you to explain how to value companies that don’t pay any dividends. You have structured your presentation around the following questions. a.

List the two types of assets that companies own.

Answer: The two types of assets are assets-in-place and nonoperating, or financial, assets. b.

What are assets-in-place? How can their value be estimated?

Answer: Assets-in-place are tangible, such as buildings, machines, and inventory. Usually they are expected to grow. They generate free cash flows. The PV of their expected future free cash flows, discounted at the WACC, is the value of operations. c.

What are nonoperating assets? How can their value be estimated?

Answer: Nonoperating assets are marketable securities and ownership of non-controlling interest in another company. The value of nonoperating assets usually is very close to the figure that is reported on balance sheets. d.

What is the total value of a corporation? Who has claims on this value?

Answer: Total corporate value is the sum of value of operations, value of nonoperating assets, and value of growth options. (No examples in this chapter have a growth option—this was discussed in Chapter 11.) Debt holders have first claim. Preferred shareholders have the next claim. Any remaining value belongs to common shareholders.

The first acquisition target is a privately held company in a mature industry. The company currently has free cash flow of $20 million. Its WACC is 10% and it is expected to grow at a constant rate of 5%. The company has marketable securities of $100 million. It is financed with $200 million of debt, $50 million of preferred shares, and $210 million of book equity.

1. What is its value of operations?

Answer:

VOp 

FCF0 (1  g)  WACC  g 

VOp 

20 (1  0.05)  $420 million  0.10  0.05

2. What is its total corporate value? What is its value of equity?

Answer:

Total corporate value = VOP + MKT. SEC. = $420 + $100 = $520 million Value of equity = Total corporate value – Debt – Pref. = $520 – $200 – $50 = $270 million

3. What is its MVA (MVA = Total corporate value – Total book value)?

Answer: MVA = Total corporate value of firm – Total book value of firm Total book value of firm = Book value of equity + Book value of debt + Book value of preferred shares MVA = $520 – ($210 + $200 + $50) = $60 million

The second acquisition target is a privately held company in a growing industry. The target has recently borrowed $40 million to finance its expansion; it has no other debt or preferred shares. It pays no dividends and currently has no marketable securities. KFS expects the company to produce free cash flows of –$5 million in 1 year, $10 million in 2 years, and $20 million in 3 years. After 3 years, free cash flow will grow at a rate of 6%. Its WACC is 10% and it currently has 10 million shares outstanding.

1. What is its horizon value (i.e., its value of operations at Year 3)? What is its current value of operations (i.e., at time zero)?

Answer:

0 rc = 10% 1

3 g = 6%

-5

$ –4.545 8.264 15.026 398.197 $416.942 = Value of Operations f.

N   

= 530 =

2. What is its value of equity on a price per share basis?

Answer: Value of equity = value of operations – debt = $416.94 – $40 = $376.94 million. Price per share = $376.94/10 = $37.69. g.

List six potential managerial behaviours that can harm a firm’s value.

Answer: Managers might: 1. Expend too little time and effort. 2. Consume too many nonpecuniary benefits. 3. Avoid difficult decisions (e.g., close plant) out of loyalty to friends in company. 4. Reject risky positive NPV projects to avoid looking bad if a project fails; take on risky negative NPV projects to try to hit a home run. 5. Avoid returning capital to investors by making excessive investments in marketable securities or by paying too much for acquisitions. 6. Massage information releases or manage earnings to avoid revealing bad news.

The managers at KFS have heard that corporate governance can affect shareholder value. What is corporate governance? List five corporate governance provisions that are internal to a firm and are under its control.

Answer: Corporate governance is the set of laws, rules, and procedures that influence a company’s operations and the decisions made by its managers. The provisions under a firm’s control are (1) monitoring and discipline by the board of directors; (2) charter provisions and bylaws that affect the likelihood of hostile takeovers; (3) compensation plans; (4) capital structure choices; and (5) accounting control systems. i.

What characteristics of the board of directors usually lead to effective corporate governance?

Answer: (1) The CEO is not also the chair of the board and does not have undue influence over the nominating committee; (2) the board has a majority of true outsiders who bring some type of business expertise to the board (and the board is not an interlocked board); (3) the board is not too large; and (4) board members are compensated appropriately (not too high, and some compensation is linked to company’s performance). j.

List three provisions in the corporate charter that affect takeovers.

Answer: These include targeted share repurchases (i.e., greenmail), shareholder rights provisions (i.e., poison pills), and restricted voting rights plans. k.

Briefly describe the use of stock options in a compensation plan. What are some potential problems with stock options as a form of compensation?

Answer: Stock options give the owner of the option the right to buy a share of the company’s stock at a specified price (called the strike price) even if the actual stock price is higher. Holders cannot usually exercise the entire option for several years (called the vesting period). They cannot exercise the option after a certain number of years (called the expiration, or maturity, date). Managers can underperform the market or their peer group, yet still reap rewards from options as long as the stock price increases to above the exercise cost. Options sometimes encourage managers to falsify financial statements or take excessive risks.

What is block ownership? How does it affect corporate governance?

Answer: Block ownership occurs when an outside investor owns a large amount (i.e., block) of a company’s shares. Large institutional investors, such as the Canada Pension Plan or Ontario Teachers’ Pension Plan, often own large blocks. Block holders often monitor managers and take an active role, leading to better corporate governance. m.

Briefly explain how regulatory agencies and legal systems affect corporate governance.

Answer: Companies in countries with strong protection for investors tend to have better access to financial markets, a lower cost of equity, increases in market liquidity, and less ―noise‖ in stock prices.

Chapter 23 Mergers, Acquisitions, and Restructuring ANSWERS TO END-OF-CHAPTER QUESTIONS 23-1

a. Synergy occurs when the whole is greater than the sum of its parts. When applied to mergers, a synergistic merger occurs when the post-merger free cash flows exceed the sum of the separate companies’ premerger free cash flows. A merger is the joining of two firms to form a single firm. b. A horizontal merger is a merger between two companies in the same line of business. In a vertical merger, a company acquires another firm that is ―upstream‖ or ―downstream‖; for example, an automobile manufacturer acquires a steel producer. A congeneric merger involves firms that are in interrelated, but not identical, lines of business. One example is banking company Citicorp’s merger with Travelers Group, a financial services business. In a conglomerate merger, unrelated enterprises combine, such as WestJet and Onex Corp. c. A friendly merger occurs when the target company’s management agrees to the merger and recommends that shareholders approve the deal. In a hostile merger, the management of the target company resists the offer. A defensive merger occurs when one company acquires another to help ward off a hostile merger attempt. A tender offer is the offer of one firm to buy the shares of another by going directly to the shareholders, frequently over the opposition of the target company’s management. A target company is a firm that another company seeks to acquire. Breakup value is a firm’s value if its assets are sold off in pieces. An acquiring company is a company that seeks to acquire another firm.

d. An operating merger occurs when the operations of two companies are integrated with the expectation of obtaining synergistic gains. These may occur due to economies of scale, management efficiency, or a host of other reasons. In a pure financial merger, the companies will not be operated as a single unit, and no operating economies are expected. e. The free cash flow to equity model, also called the residual dividend model, first calculates FCFE, which is the free cash flow payable to shareholders. FCFE is free cash flow less interest expense plus the interest tax savings, plus the net change in debt. It then discounts the FCFEs at the levered cost of equity to arrive at the value of equity in operations. You add in the value of non-operating assets and you get the value of the equity. To get the value of operations, you then add in the value of the debt.

f. With acquisition accounting, the acquiring firm is assumed to have ―bought‖ the acquired company in much the same way it would buy any capital asset. The acquiring company reports the market value of the acquired company’s identifiable assets. If the purchase price is greater than the fair value of the identifiable assets, anything extra is added to goodwill. The goodwill is not amortized; rather, there is an impairment test done. If the purchase price is less than the fair value of identifiable assets, the difference is reported as a gain on purchase. g. A white knight is a friendly competing bidder that a target management likes better than the company making a hostile offer, and the target solicits a merger with the white knight as a preferable alternative. A poison pill is a deliberate action that a company takes, which makes it a less attractive takeover target. A golden parachute is a payment made to executives that are forced out when a merger takes place. A proxy fight is an attempt to gain control of a firm by soliciting shareholders to vote for a new management team. h. A joint venture involves the joining together of parts of companies to accomplish specific, limited objectives. Joint ventures are controlled by the combined management of the two (or more) parent companies. A corporate or strategic alliance is a cooperative deal that stops short of a merger. i. A divestiture is the opposite of an acquisition. That is, a company sells a portion of its assets, often a whole division, to another firm or individual. In a spinoff, a holding company distributes the stock of one of the operating companies to its shareholders. Thus, control passes from the holding company to the shareholders directly. A leveraged buyout (LBO) is a transaction in which a firm’s publicly owned stock is acquired in a mostly debt-financed tender offer, and a privately owned, highly leveraged firm results. Often, the firm’s own management initiates the LBO. j. Arbitrage is simultaneously buying and selling the same commodity or security in two different markets at different prices and pocketing a risk-free return. In the context of mergers, risk arbitrage refers to the practice of purchasing shares in companies that may become takeover targets. 23-2

Horizontal and vertical mergers are most likely to result in governmental intervention, but mergers of this type are also most likely to result in operating synergy. Conglomerate and congeneric mergers are attacked by the government less often, but they also are less likely to provide any synergistic benefits.

23-3

A tender offer might be used. Although many tender offers are made by surprise and over the opposition of the target firm’s management, tender offers can and often are made on a ―friendly‖ basis. In this case, management (the board of directors) of the target company endorses the tender offer and recommends that shareholders tender their shares.

23-4

An operating merger involves integrating the company’s operations in hopes of obtaining synergistic benefits, while a pure financial merger generally does not involve integrating the merged company’s operations.

23-5

The FCFE and corporate valuation (CV) models both do the same thing. They value a firm’s operations and its equity. When implemented under a scenario that is consistent with the assumptions of both models, they will give the same answer. The CV model discounts free cash flows at the WACC to obtain the value of operations. You then add nonoperating assets and subtract out debt to arrive at the value of equity. In this case the value of the interest tax shield is incorporated because the WACC is used to discount the cash flows; the WACC has a reduction in the cost of debt to account for its tax shield. The FCFE model calculates free cash flows to equity as FCF – interest expense + interest tax shield + net change in debt. The model then discounts these FCFEs at the levered cost of equity to get the value of equity in operations. You add in the value of nonoperating assets to get the value of equity, and then add in the value of the debt to get the value of the entire firm.

SOLUTIONS TO END-OF-CHAPTER PROBLEMS 23-1

FCF1 = 2.60(1.05) = $2.73 million; g = 5%; b = 1.2; rRF = 4%; RPM = 5%; wd = 30%; T = 30%; rd = 7% Vops = ? P0 = ? rsL = rRF + RPM(b) = 4% + 5%(1.2) = 10%. WACC = wdrd(1 –T) + wsrs = 0.30(7%)(0.70) + 0.70(10%) = 8.47% Vops =

FCF0 (1  g) WACC  g

$2.73 0.0847 - 0.05

= $78.67 million Value of equity = Vops – Debt = $78.67 – $12 = $66.67 million Price = $66.67 million/1 million shares = $66.67/share.

23-2

FCF1 = $3.0 million, FCF2 = $3.3 million, FCF3 = $3.6 million, and FCF4 = $3.9 million; g = 5%; b = 1.2; rRF = 4%; RPM = 5%; wd = 30%; T = 30%; rd = 7% Vops = ? P0 = ? WACC was calculated in Problem 23-1 to be 8.47%. Since the horizon capital structure is the same as in Problem 23-1, the WACC is the same. Corporate valuation model horizon value = FCF4(1 + g)/(WACC – g) = 3.9(1.05)/(0.0847 – 0.05) = 4.095/(0.0347) = $118.01 million Value of operations = $3.00 + $3.30 2 + $3.60 3 + $3.90 + $118.01 4 1.0847 (1.0847)

(1.0847)

= $96.46 million Equity value to Bremer = Operations – Debt = $96.46 million – $12 million = $84.46 million or $84.46 per share since there are 1 million shares outstanding. 23-3

On the basis of the answers in Problems 23-1 and 23-2, the bid for each share should range between $66.67 and $84.46.

23-4 HVOperations;year3 = VOperations = 23-5

FCFyear4 FCFyear3 (1 + g) $30.0(1 + 0.05) = = = $630 millio (WACC g) (WACC g) (0.10 0.05)

22.00 26.60 ( 30 00 + 630 00) + + = $537.9 million 2 (1 + 0.10) (1 + 0.10) (1 + 0.10)3

a. $50,000,000 – [(1,200,000 × $22) + (800,000 × $23)] = $5,200,000. b. $5.2 × 1.1 = $5.72 total value of synergies at Year 1 PV = (2 – 5.72), N = 9, I/Y = 10, FV = 0, PMT = ? = 0.6459 If $2 million of the synergies occur in Year 1, $645,900 in annual synergies would be expected for each of the next 9 years. That is, the present value of $2 million in Year 1 and $645,900 in each of Years 2–10, equals $5.2 million at time 0. c. $50,000,000 – (1,200,000 × $22) = $23,600,000 or $29.50/share ($23,600,000/800,000).

d. Total value of offer Current value of Amsted Synergy going to Amsted shareholders Total synergy value available Synergy going to Mistral shareholders

$23 $18,400,000 18,400,000 0

$25 $20,000,000 18,400,000 1,600,000

$30 $24,000,000 18,400,000 5,600,000

5,200,000 $ 5,200,000

5,200,000 $ 3,600,000

5,200,000 $( 400,000)

$23 × 800,000 = $18,400,000 $25 × 800,000 = $20,000,000 $30 × 800,000 = $24,000,000 23-6

Calculate Interstate’s free cash flow to equity: Year 1 Year 2 Free cash flow $1,000,000 $1,140,000 Less after-tax interest 59,500 64,750 Plus increase in debt 150,000 160,000 Free cash flows to equity $1,090,500 $1,235,250

Year 3 $1,280,000 70,350 170,000 $1,379,650

Year 4 $1,420,000 76,300 180,000 $1,523,700

Calculate Interstate’s horizon value: ,

, 00( 0 0 0) (0 0 0 0 0 0)

, ,4 00 0

Calculate the present value of the free cash flows to equity: VFCFE = $1,700 +

$1,090.5 $1,235.25 $1,379.65 ( 1,523.7 + $31,388.220) + + + (1.08) (1.08)2 (1.08)3 (1.08)4

04 or $29,055,204

23-7

a. 2021 NOPAT = EBIT(1 – T) Less net investment

2022

2023

2024

2025

2026

$2.60

$3.51

$4.81

$6.44

$8.06

2.0

3.0

4.0

$0.60

$1.51

$1.81

$2.44

$4.06

in operating capital1 Free cash flow 1

Net investment in operating capital is the year-over-year change in net operating capital as

calculated below.

Operating current

$4.0

$4.5

$5.0

$5.5

$6.0

40.0

42.0

45.0

48.0

52.0

56.0

3.0

4.5

5.0

5.5

6.0

$41.0

$43.0

$45.0

$48.0

$52.0

$56.0

assets Operating longterm assets Operating current liabilities Total net operating capital

b. rs = 3% + 1.4(5%) = 10% rd = 7%

Ws 

$34M  68% ($34M  $16M)

Wd 

$16M  32% ($34M  $16M)

WACC = wd(1 – T)rd + wsrs = 0.32(1 – 0.35)(7%) + 0.68(10%) = 8.256% FCF2027 FCF2026 (1 + g) $4.06(1 + 0.04) = = $99.21 (WACC – g) (WACC – g) (0.08256 – 0.04) The value of operations is the present value of the free cash flows in the forecast period and the horizon value. HV2026 =

VOperations 

$0.60 $1.51 $1.81 $2.44 ($4.06  $99.21)      $74.50M 2 3 4 (1  0.8256) (1  0.8256) (1  0.8256) (1  0.8256) (1  0.8256)5

The value of the equity = value of operations – debt = $74.50M – $16M = $58.50M Brock would pay a maximum of

$58.50M  $11.70. 5M shares

23-8

From the previous problem we already calculated the annual free cash flows. 2021

2022

2023

2024

2025

2026

Free cash flow

$0.60

$1.51

$1.81

$2.44

$4.06

Less after-tax interest payment1

1.11

1.17

1.24

1.30

1.36

Plus change in debt2

$7.80

1.80

1.60

1.70

1.60

1.20

FCFE

7.80

1.29

1.94

2.27

2.74

3.90

After-tax interest payment = Interest payment(1 – T). The change in debt is the year-over-over increase in debt taken on in order to maintain the same capital structure weight. 2

Cost of equity is 10% as calculated in the previous problem.

HVFCFE 2026 =

The value of the equity using the FCFE method is

FCFE2027 FCF2026 (1 + g) = (rs – g) (rs – g)

VFCFE  7.80 

$3.90(1 + 0.04) = $67.60 (0.10 – 0.04)

1.29 1.94 2.27 2.74 (3.90  67.60)      $58.55M 2 3 4 (1  0.10) (1  0.10) (1  0.10) (1  0.10) (1  0.10) 5

(The difference between $58.55M here and $58.50M from the corporate valuation method is due to rounding.) Brock would pay a maximum of

$58.55M  $11.71. 5M shares

23-9

Dearborn’s market value as a stand-alone company is $34 million. If Brock acquires Dearborn and the synergies are realized, the value of Dearborn is $58.5 million; therefore, the value of the synergies is $24.5 million ($58.5 million – $34 million) or $24.55 million as a result of rounding noted above. If Brock offers $8 per share, the incremental value to Dearborn’s shareholders is $1.20 per share [$8 per share – ($34 million/5 million shares)]. The incremental value for Brock’s shareholders is ($11.70 – $8) × 5,000,000 shares = $18,500,000, or $18,500,000 /10 million shares = $1.85 per share. If Brock offers $10.25 per share, the incremental value to Dearborn’s shareholders is $3.45 per share [$10.25 per share – ($34 million/5 million shares)]. The incremental value for Brock’s shareholders is ($11.70 – $10.25) × 5,000,000 shares = $7,250,000, or $7,250,000/10 million shares = $0.725 per share.

23-10

$30 Purchase: Company A Company T (Book (Fair value) value) Current assets $ 55 $ 20* Fixed assets 95 55* Goodwill 0 0 Total assets $150 $ 75 Liabilities $ 50 $ 25 Equity 100 50 Total Liab & Eq $150 $ 75 * Identifiable assets marked up to fair value

Merged Company $ 45 150 0 $195 $ 75 120 $195

($55 + $20 – $30)

$55 Purchase: Company A Company T Merged (Book (Fair value) Company* * value) Current assets $ 55 $ 20* $ 20 ($55 + $20 – $55) * Fixed assets 95 55 150 Goodwill 0 0 5 Total assets $150 $ 75 $175 Liabilities $ 50 $ 25 $ 75 Equity 100 50 100 Total Liab & Eq $ 150 $ 75 $175 * Identifiable assets marked up to fair value. ** The $5 goodwill is the unidentifiable asset created based on the difference between the purchase price over the fair value of the identifiable assets of the acquired company. b.

EPSA = $21/100 = $0.21.

EPSB = $14/55 = $0.25.

c. Company A

Company T

Sales Costs

$100 70

$ 50 30

Merged Company $150.0 106.0

Taxable income Taxes (30%) Net income

30 9 $ 21

20 6 $ 14

44.0 13.2 $ 30.8

($70 + $30 + $5/5yrs + $5)

EPSMerged = $30.8/155 = $0.199.

SOLUTION TO SPREADSHEET PROBLEM 23-11 The detailed solution for the spreadsheet problem is in the file Ch 23 Build a Model Solution.xlsx and is available on the textbook’s website.

MINI CASE The detailed solution for the Mini Case is also available in spreadsheet format, in the file Brigham4Ce_MiniCase_Ch23.xlsx, and is available on the textbook’s website. Hager’s Home Repair Company, a regional hardware chain that specializes in ―do-ityourself‖ materials and equipment rentals, is cash rich because of several consecutive good years. One of the alternative uses for the excess funds is an acquisition. Doug Zona, Hager’s treasurer and your boss, has been asked to place a value on a potential target, Lyons Lighting (LL), a chain that operates in several provinces, and he has enlisted your help. The table below indicates Zona’s estimates of LL’s earnings potential if it came under Hager’s management (in millions of dollars). The interest expense listed here includes the interest (1) on LL’s existing debt, which is $55 million at a rate of 9%, and (2) on new debt expected to be issued over time to help finance expansion within the new ―L division,‖ the code name given to the target firm. If acquired, LL will face a 40% tax rate. Security analysts estimate LL’s beta to be 1.3. The acquisition would not change Lyons’s capital structure, which is 20% debt. Zona realizes that Lyons Lighting’s business plan also requires certain levels of operating capital and that the annual investment could be significant. The required levels of total net operating capital are listed below. Zona estimates the risk-free rate to be 7% and the market risk premium to be 4%. He also estimates that free cash flows after 2026 will grow at a constant rate of 6%. Following are projections for sales and other items. 2021 Net sales Cost of goods sold (60%) Selling/administrative expense Interest expense Total net operating capital Debt

$150.00 55.55

2022 $60.00 36.00 4.50 5.00 150.00 72.22

2023 2024 $90.00 $112.50 54.00 67.50 6.00 7.50 6.50 6.50 157.50 163.50 72.22 77.78

2025 $127.50 76.50 9.00 7.00 168.00 90.67

2026 $139.70 83.80 11.00 8.16 173.00 96.10

Hager’s management is new to the merger game, so Zona has been asked to answer some basic questions about mergers as well as to perform the merger analysis. To structure the Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc. 24617

task, Zona has developed the following questions, which you must answer and then defend to Hager’s board. a.

Several reasons have been proposed to justify mergers. Among the more prominent are (1) tax considerations, (2) risk reduction, (3) control, (4) purchase of assets at below-replacement cost, (5) synergy, and (6) globalization. In general, which of the reasons are economically justifiable? Which are not? Which fit the situation at hand? Explain.

Answer: The economically justifiable rationales for mergers are synergy and tax consequences. Synergy occurs when the value of the combined firm exceeds the sum of the values of the firms taken separately. (If synergy exists, then the whole is greater than the sum of the parts, and hence synergy is also called the ―2 + 2 = 5‖ effect.) A synergistic merger creates value, which must be apportioned between the shareholders of the merging companies. Synergy can arise from four sources: (1) operating economies of scale in management, production, marketing, or distribution; (2) financial economies, which could include higher debt capacity, lower transaction costs, or better coverage by securities’ analysts, which can lead to higher demand and, hence, higher prices; (3) differential management efficiency, which implies that new management can increase the value of a firm's assets; and (4) increased market power due to reduced competition. Operating and financial economies are socially desirable, as are mergers that increase managerial efficiency, but mergers that reduce competition are both undesirable and illegal. Another valid rationale behind mergers is tax considerations. For example, a firm that is highly profitable and consequently in the highest corporate tax bracket could acquire a company with large accumulated tax losses, and use those losses to shelter its current and future income. Without the merger, the carryforwards might eventually be used, but their value would be higher if used earlier rather than later. The motives that are generally less supportable on economic grounds are risk reduction, purchase of assets at below-replacement cost, control, and globalization. Managers often state that diversification helps to stabilize a firm’s earnings stream and thus reduces total risk, and hence benefits shareholders. Stabilization of earnings is certainly beneficial to a firm’s employees, suppliers, customers, and managers. However, if a stock investor is concerned about earnings variability, he or she can diversify more easily than can the firm. Why should Firm A and Firm B merge to stabilize earnings when shareholders can merely purchase both stocks and accomplish the same thing? Further, we know that well-diversified shareholders are more concerned with a stock’s market risk than its stand-alone risk, and higher earnings instability does not necessarily translate into higher market risk. Sometimes a firm will be touted as a possible acquisition candidate because the replacement value of its assets is considerably higher than its market value. For 24-618 Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc.

example, in the early 1980s, oil companies could acquire reserves more cheaply by buying out other oil companies than by exploratory drilling. However, the value of an asset stems from its expected cash flows, not from its cost. Thus, paying $1 million for a slide rule plant that would cost $2 million to build from scratch is not a good deal if no one uses slide rules. In recent years, many hostile takeovers have occurred. To keep their companies independent, and also to protect their jobs, managers sometimes engineer defensive mergers, which make their firms more difficult to ―digest.‖ Also, such defensive mergers are usually debt-financed, which makes it harder for a potential acquirer to use debt financing to finance the acquisition. In general, defensive mergers appear to be designed more for the benefit of managers than for that of the shareholders. An increased desire to become globalized has resulted in many mergers. To merge just to become international is not an economically justified reason for a merger; however, increased globalization has led to increased economies of scale. Thus, synergies often result—which is an economically justifiable reason for mergers. b.

Briefly describe the differences between a hostile merger and a friendly merger.

Answer: In a friendly merger, the management of one firm (the acquirer) agrees to buy another firm (the target). In most cases, the action is initiated by the acquiring firm, but in some situations the target may initiate the merger. The managements of both firms get together and work out terms that they believe to be beneficial to both sets of shareholders. Then they issue statements to their shareholders recommending that they agree to the merger. Of course, the shareholders of the target firm normally must vote on the merger, but management’s support generally assures that the votes will be favourable. If a target firm’s management resists the merger, then the acquiring firm’s advances are said to be hostile rather than friendly. In this case, the acquirer, if it chooses to, must make a direct appeal to the target firm’s shareholders. This takes the form of a tender offer, whereby the target firm’s shareholders are asked to ―tender‖ their shares to the acquiring firm in exchange for cash, stock, bonds, or some combination of the three. If 51% or more of the target firm’s shareholders tender their shares, then the merger will be completed over management’s objection. c.

What are the steps in valuing a merger?

Answer:

When using the free cash flow to equity model, the following steps are used: 1. Project the yearly cash flows available to shareholders (FCFE) until the target is expected to grow at a constant growth rate. 2. Project the horizon growth rate.

3. Calculate the cost of equity. 4. Calculate the horizon value of the value of equity due to operations. 5. alculate the firm’s equity value from operations as the present value of the horizon value of the equity and the FCFEs at the cost of equity. 6. alculate the total value of the company’s equity as the value of the equity from operations plus the value of any nonoperating assets.

Use the data developed in the table to construct LL’s free cash flows to equity for 2022 through 2026. Why are investments in net operating capital and changes to debt included when calculating free cash flow to equity?

Answer:

The easiest approach here is to calculate the free cash flows to equity for the L division, assuming that the acquisition is made. (Millions of dollars) 2021 Net sales Cost of goods sold (60%) Selling/Admin expenses EBIT Interest expense EBT Taxes Net income Less net investment in operating capital Plus net change in debt FCFE

2022 $60.0 36.0

2023 $90.0 54.0

2024 $112.5 67.5

2025 $127.5 76.5

2026 $139.7 83.8

4.5

6.0

7.5

9.0

11.0

19.5 5.0 14.5 5.8 8.7 0.0

30.0 6.5 23.5 9.4 14.1 7.5

37.5 6.5 31.0 12.4 18.6 6.0

42.0 7.0 35.0 14.0 21.0 4.5

44.9 8.2 36.7 14.7 22.0 5.0

16.67

0.0

5.6

12.9

5.4

25.4

6.6

18.2

29.4

22.4

Net investment in operating capital is deducted since these cash flows are being reinvested in the business and are not available for shareholder distribution. Changes in debt are added to FCFE since the increased debt flows to the owners. Additional debt will be issued to keep LL’s capital structure constant and LL’s cost of equity constant. LL can support additional debt because it is worth more as a subsidiary of Hager than it was worth alone. (Note that we could also calculate FCFE by finding NOPAT = EBIT(1 – T) = 19.5(1 – 0.40) = 11.7 and subtracting net investment in operating capital = Free cash flow = 11.7 – 0 = 11.7. We then subtract the after-tax interest expense = 5.0(1 – 0.40) = 3.0 and finally add the net change in debt of 16.67: 11.7 – 3.0 + 16.67 = 25.37 ≈ 25.4.)

Conceptually, what is the appropriate discount rate to apply to the cash flows developed in Part d? What is your actual estimate of this discount rate?

Answer:

Since the FCFE already accounts for the cost of debt, (i.e., interest expense has been deducted), we discount the remaining cash flows available to the shareholders by the cost of equity. This cost should be calculated based on the target’s risk, not the acquirer’s risk. Hager’s investment bankers have estimated that Lyons Lighting’s beta is currently 1.3. The horizon value should also be calculated using LL’s cost of equity. To obtain the required rate of return we need the risk-free rate and the market risk premium. Thus the required return is: rs(Lyons Lighting) = rrf + (rm – rf)bLyons Lighting = 7% + (4%)1.3 = 12.2%.

What is the estimated horizon, or continuing, value of the acquisition; that is, what is the estimated value of LL’s cash flows beyond 2026? What is LL’s value to Hager’s shareholders? Suppose another firm were evaluating LL as an acquisition candidate. Would that company obtain the same value? Explain.

Answer:

The 2026 cash flow to equity is $22.4 million and is expected to grow at a 6% constant growth rate after 2026. The horizon value is calculated below: (2026 FCFE) (1+ g) $22.4(1.06) = $382.97 millio (rs – g) (0.122 – 0.06) To calculate the value of the firm, find the present value of the horizon value and the FCFE at the cost of equity (in millions of dollars): FCFEhorizon value =

Annual FCFE Horizon value Total

2022 $25.4

2023 $6.6

2024 $18.2

2025 $29.4

$25.4

$6.6

$18.2

$29.4

2026 $22.4 383.0 $405.4

The value of Lyons’s equity is the present value of these cash flows at the cost of equity, 12.2%, which equals $287.3 million. If another firm were valuing Lyons, they would probably obtain an estimate different from $287.3 million. Most important, the synergies involved would likely be different, and hence the cash flow estimates would differ. Also, another potential acquirer might use different financing and hence estimate a different discount rate at the horizon.

Assume that LL has 20 million shares outstanding. These shares are traded relatively infrequently, but the last trade, made several weeks ago, was at a price of $11 per share. Should Hager’s make an offer for Lyons Lighting? If so, how much should it offer per share?

Answer: With a current price of $11 per share and 20 million shares outstanding, LL’s current market value is $11(20) = $220 million. Since LL’s expected value to Hager’s is $287.3 million, it appears that the merger would be beneficial to both sets of shareholders. The difference, $287.3 – $220.0 = $67.3 million, is the added value to be apportioned between the shareholders of both firms. The offering range is from $11 per share to $287.3/20 = $14.365 per share. At $11, all of the benefit of the merger goes to Hager’s shareholders, while at $14.365 all of the value created goes to LL’s shareholders. If Hager’s offers more than $14.365 per share, then wealth would be transferred from Hager’s shareholders to LL’s shareholders. As to the actual offering price, Hager’s should make the offer as low as possible, yet acceptable to LL’s shareholders. A low initial offer, say $11.50 per share, would probably be rejected and the effort wasted. Further, the offer may influence other potential suitors to consider LL’s business, and they could end up outbidding Hager’s. Conversely, a high price, say $14, passes almost all of the gain to LL’s shareholders, and Hager’s managers should retain as much of the synergistic value as possible for their own shareholders. Note that this discussion assumes that LL’s $11 price is a ―fair,‖ equilibrium value in the absence of a merger. Since the stock trades infrequently, the $11 price may not represent a fair minimum price. LL’s management should make an evaluation (or hire someone to make the evaluation) of a fair price and use this information in its negotiations with Hager’s. h.

There has been considerable research undertaken to determine whether mergers really create value and, if so, how this value is shared between the parties involved. What are the results of this research?

Answer: Most researchers agree that takeovers increase the wealth of the shareholders of target firms, for otherwise they would not agree to the offer. However, there is a debate as to whether mergers benefit the acquiring firm’s shareholders. The results of these studies have shown, on average, the stock prices of target firms increase by about 30% in hostile tender offers, while in friendly mergers the average increase is about 20%. However, for both hostile and friendly deals, the stock prices of acquiring firms, on average, remain constant. Thus, one can conclude that (1) acquisitions do create value, but (2) shareholders of target firms reap virtually all the benefits. i.

What method is used to account for mergers?

Answer: Mergers must be accounted for using the acquisition method, in which the acquiring company shows identifiable assets being acquired at the fair value of these assets, irrespective of the amount actually paid for the assets. Any amount paid over the fair value of these assets is recorded as goodwill. j.

What merger-related activities are undertaken by investment bankers?

Answer: The investment banking community is involved with mergers in a number of ways. Several of these activities are (1) helping to arrange mergers, (2) aiding target companies in developing and implementing defensive tactics, (3) helping to value target companies, (4) helping to finance mergers, and (5) risk arbitrage—speculating in the stocks of companies that are likely takeover targets. k.

What is a leveraged buyout (LBO)? What are some of the advantages and disadvantages of going private?

Answer: A leveraged buyout is a situation in which a small group of investors (which usually include the firm’s managers) borrows heavily to buy all the shares of a company. Advantages to going private include administrative cost savings, increased managerial incentives, increased managerial flexibility, increased shareholder participation, and increased financial leverage. The main disadvantage of going private is not having access to the large amounts of capital available in the equity market, making it difficult to fund a firm’s projects. l.

What are the major types of divestitures? What motivates firms to divest assets?

Answer: The three primary types of divestitures are (1) selling an operating unit to another firm, (2) setting up the business to be divested as a separate corporation and then ―spinning it off‖ to the divesting firm’s shareholders, and (3) liquidating assets outright. The reasons for divestitures vary widely. Sometimes companies need cash either to finance expansion in their primary business lines or to reduce a large debt burden. Sometimes firms divest to unload losing assets that would otherwise drag the company down, or divesting may be the result of an antitrust settlement, where the government requires a breakup.

ANSWERS TO END-OF-CHAPTER QUESTIONS 24-1

a. Real options occur when managers can influence the size and risk of a project’s cash flows by taking different actions during the project’s life. They are referred to as real options because they deal with real as opposed to financial assets. They are also called managerial options because they give opportunities to managers to respond to changing market conditions. Sometimes they are called strategic options because they often deal with strategic issues. They are also called embedded options if they are a part of another project. b. Investment timing options give companies the option to delay a project rather than implement it immediately. This option to wait allows a company to reduce the uncertainty of market conditions before it decides to implement the project. Growth options allow a company to expand if market demand is higher than expected. This includes the opportunity to expand into different geographic markets and the opportunity to introduce complementary or second-generation products. It also includes the option (an abandonment option) to abandon a project if market conditions deteriorate too much. Flexibility options allow a company to have flexibility in its operations, such as the flexibility to use different fuels or different types of machinery. c. Decision trees are a form of scenario analysis in which different actions are taken in different scenarios.

24-2

Postponing a project means that cash flows come later rather than sooner; however, waiting may allow you to take advantage of changing conditions. It might make sense, however, to proceed today if there are important advantages to being the first competitor to enter a market.

24-3

Timing options make it less likely that a project will be accepted today. Often, if a firm can delay a decision, it can increase the expected NPV of a project.

24-4

Having the option to abandon a project makes it more likely that the project will be accepted today.

SOLUTIONS TO END-OF-CHAPTER PROBLEMS

24-1

├─────┼─────┼──────    ────┤

–20

NPV = $1.074 million. b. Wait 1 year: 0

Tax imposed 50% Prob.

–20

1.8

Tax not imposed 50% Prob.

–20

4.2

r = 13%

21   

1.8   

PV @ Yr. 1 12.645

4.2

29.504

Tax imposed: NPV @ Yr. 1 = (–20 + 12.645)/(1.13) = –6.509 Tax not imposed: NPV @ Yr 1 = (–20 + 29.504)/ (1.13) = 8.4106 Expected NPV = .5(–6.509) + .5(8.4106) = 0.9508 Note though, that if the tax is imposed, the NPV of the project is negative and therefore would not be undertaken. The value of this option of waiting one year is evaluated as 0.5($0) + (0.5)($ 8.4106) = $4.205 million. Since the NPV of waiting one year is greater than going ahead and proceeding with the project today, it makes sense to wait.

24-2

–8

10% ├─────┼─────┼─────┼──────┼─────┼

NPV = $7.1631 million. b. Wait 3 years:

Year

PV @ Yr. 3

Low CF Scenario 10% Probability

-12

2.2

$8.34

High CF Scenario 90% Probability

-12

4.2

$15.92

Low CF scenario: NPV = (–12 + 8.34)/(1.1)3 = –2.75 High CF scenario: NPV = (–12 + 15.92)/(1.1)3 = 2.95 Expected NPV = 0.1(–2.75) + 0.9(2.95) = 2.38 If the cash flows are only $2.2 million, the NPV of the project is negative and, thus, would not be undertaken. The value of the option of waiting three years is evaluated as 0.10($0) + 0.90($2.95) = $2.66 million. Since the NPV of waiting three years is less than going ahead and proceeding with the project today, it makes sense to drill today.

24-3

–300

13% ├─────┼─────┼──────    ────┤

NPV = –$19.0099 million. Don’t purchase. b. Wait 1 year:

2 |

–300

r = 13% | |

50% Prob. 0 |

50% Prob. 0

–300

21   

30   

NPV @ Yr. 0 –$78.9889

45.3430

If the cash flows are only $30 million per year, the NPV of the project is negative. However, we’ve not considered the fact that the company could then be sold for $280 million. The decision tree would then look like this:

0 r = 13% 1

–300

30 + 280

50% Prob. 0 |

50% Prob. 0

–300

21   

0   

NPV @ Yr. 0 –$27.1468

45.3430

The expected NPV of waiting 1 year is 0.5(–$27.1468) + 0.5($45.3430) = $9.0981 million. Given the option to sell, it makes sense to wait 1 year before deciding whether to make the acquisition.

24-4

0 |

12%

–7,200,000

14   

600,000

Using a financial calculator, input the following data: CF0 = –7,200,000; CF1 – 15 = 600,000; I = 12; and then solve for NPV = –$3,113,481.31. b.

0 12%

1   

–7,200,000

1,900,000

Using a financial calculator, input the following data: CF0 = –7,200,000; CF1-15 = 1,900,000; I = 12; and then solve for NPV = $5,740,642.53. c. If they proceed with the project today, the project’s expected NPV = (0.5  – $3,113,481.31) + (0.5  $5,740,642.53) = $1,313,580.61. So, Alberta Enterprises would potentially go ahead with the project today. d. Since the project’s NPV with the tax is negative, if the tax were imposed the firm would abandon the project. Thus, the decision tree looks like this: NPV @ 0 1 2 15 Yr. 0 r= 12% 50% Prob. | | |    | Taxes –7,200,000 6,500,000 0 0 –$ 1,396,428.57 No Taxes | 50% Prob. –7,200,000

1,900,000

  

1,900,000 5,740,642.53 Expected NPV $ 2,172,106.98

Yes, the existence of the abandonment option changes the expected NPV of the project significantly. Given this option, the firm has even more confidence in taking on the project.

e. 0 50% Prob. Taxes No Taxes 50% Prob.

r = 12%

NPV = ? |

NPV = ?

NPV @ Yr. 0

1 |

–1,500,000 +300,000 = NPV @ t = 1

wouldn’t do

0.00

–1,500,000 +4,000,000 = NPV @ t = 1

2,232,142.86 Expected NPV $1,116,071.43

If the firm pays $1,116,071.43 for the option to purchase the land, then the NPV of the project is exactly equal to zero. So the firm would not pay any more than this for the option.

24-5 a. 0 40% Prob. Good Bad 60% Prob.

r = 10%

–20,000

28,000

–20,000

28,000

NPV = 28,595

8,000

NPV = -6,116

Expected NPV = 28,595 (0.40) – 6,116 (0.60) = $7,769 b. 40% Prob. Good Bad 60% Prob.

0 r = 10% 1

–20,000

28,000

–20,000

8,000

28,000 28,000 –20,000 (r = 6%)

28,000

The PV of the 20,000 payment in Year 2 at the risk-free rate is 20,000/(1.06)2 = 17,800. The PV of the 4 cash flows of 28,000 in years 1 through 4 at the cost of capital of 10% is 88,756. The NPV of the top row is 88,756 – 20,000 – 17,800 = 50,956. The NPV of the bottom row is still –6,116, as it was in Part a. The expected NPV is 50,956 (0.40) –6,116 (0.60) = $16,713. The expected NPV of the project without the growth option is $7,769. The value of the growth option itself is the difference: Value of growth option = $16,713 – $7,769 = $8,944

24-6

P = PV of all expected future cash flows if project is delayed. From Problem 20-1 we know that PV @ Year 1 of Tax Imposed scenario is $12.64 and PV @ Year 1 of Tax Not Imposed scenario is $29.50. So the PV is: P = [0.5(12.64) + 0.5(29.500]/1.13 = $18.64. X = $20. t = 1. rRF = 0.08. Standard Deviation of PV = [0.5(12.64 – 21.07)2 + 0.5(29.50 – 21.07)2]0.5 = 8.43 CV=8.43/21.07 = 0.40 2 = ln[(0.40)2 + 1]/1 = 0.1484 d1 = ln[18.64/20] + [0.08 + .5(.1484)](1) = 0.0098 (.1484)0.5 (1)0.5 d2 = 0.0098 – (.1484)0.5 (1)0.5 = –0.3754 From Excel function NORMSDIST, or approximated from the table in Appendix C: N(d1) = 0.5039 N(d2) = 0.3537 Using the Black-Scholes option pricing model, you calculate the option’s value as: V = P[N(d1)] – Xe rRF t [N(d2)] = $18.64(0.5039) – $20e(–0.08)(1)(0.3537) = $9.39 – $6.53 = $2.86 million.

24-7

P = PV of all expected future cash flows if project is delayed. From Problem 20-2 we know that PV @ Year 3 of Low CF Scenario is $8.34 and PV @ Year 3 of High CF Scenario is $15.92. So the PV is: P = [0.1(8.34) + 0.9(15.92]/1.103 = $11.39. X = $12. t = 3. rRF = 0.06. Standard Deviation of PV = [0.1(8.34 – 15.16)2 + 0.9(15.92 – 15.16)2]0.5 = 15.26 CV=15.26/15.16 = 1.007 2 = 0.2334. d1 = ln[11.39/12] + [0.06 + .5(.2334)](3) = 0.5712 (.2334)0.5 (3)0.5 d2 = 0.5712 – (.2334)0.5 (3)0.5 = –0.2656 From Excel function NORMSDIST, or approximated from the table in Appendix C: N(d1) = 0.7161 N(d2) = 0.3953 Using the Black-Scholes option pricing model, you calculate the option’s value as: V = P[N(d1)] – Xe rRF t [N(d2)] = $11.39(0.7161) – $12e(–0.06)(3)(0.3953) = $8.16 – $3.96 = $4.2 million.

24-8

P = PV as of time zero of all expected future cash flows if the project is repeated starting in Year 2. Note it includes both the good cash flows and the bad cash flows since as of now, we don’t know which outcome will result, and P excludes the $20,000 investment in the franchise. 0 40% Prob. Good

Bad 60% Prob.

r = 10%

28,000

8,000

Expected PV of cash flows (as of time 0) = 40,161(0.40) + 11,475(0.60) = $22,949 = P. The strike price, X, is the cost to extend the franchise at the end of Year 2, and is 24-634 Brigham, Financial Management, 4Ce, Solutions © 2023 Cengage Learning Canada, Inc.

$20,000. The time to expiration is the time you decide whether or not to extend the franchise, and is at the end of Year 2. P = $22,949 X = $20,000 t = 2. rRF = 0.06. To calculate the variance of the project’s returns using the indirect method, first calculate the standard deviation of the value at Year 2. The value is either 48,595 (probability 40%) or 13,884 (probability 60%). Expected value at Year 2: 48,595(0.4) + 13,884(0.6) = 27,768. Standard deviation = [(48,595 – 27,768)2(0.40) + (13,884 – 27,768)2(0.60)]1/2 = 17,005. Then calculate the coefficient of variation: CV = 17,005/27,768 = 0.6124. σ2 = ln(CV2 + 1)/t = ln(0.61242 + 1)/2 = 0.1592 so σ2 = 0.1592 d1 = ln[22.949/20] + [0.06 + .5(0.1592)](2) = 0.7386 (0.1592)0.5 (2)0.5 d2 = 0.7386 – (0.1592)0.5 (2)0.5 = 0.1743 From Excel function NORMSDIST, or approximated from the table in Appendix C: N(d1) = 0.7699 N(d2) = 0.5692 Using the Black-Scholes option pricing model, you calculate the option’s value as: V = P[N(d1)] – Xe rRFt [N(d2)] = $22.949(0.7699) – $20e(–0.06)(2)(0.5692) = $17.67 – $10.10 = $7.57 thousand.

SOLUTION TO SPREADSHEET PROBLEM 24-9

The detailed solution for the spreadsheet problem is in the file Ch 24 Build a Model Solution.xls and is available on the textbook’s website.

MINI CASE The detailed solution for the Mini Case is also available in spreadsheet format, in the file Brigham3Ce_MiniCase_Online Ch24.xlsx, and is available on the textbook’s website. You have just been hired as a financial analyst by Tropical Sweets Inc., a mid-sized Ontario company that specializes in creating exotic candies from tropical fruits such as mangoes, papayas, and dates. The firm’s CEO, George Yamaguchi, recently returned from an industry corporate executive conference in Vancouver, and one of the sessions he attended was on real options. Because no one at Tropical Sweets is familiar with the basics of real options, Yamaguchi has asked you to prepare a brief report that the firm’s executives can use to gain at least a cursory understanding of the topic. To begin, you gathered some outside materials on the subject and used these materials to draft a list of pertinent questions that need to be answered. In fact, one possible approach to the paper is to use a question-and-answer format. Now that the questions have been drafted, you have to develop the answers.

What are some types of real options?

Answer: 1. Investment timing options 2. Growth options a. Expansion of existing product line b. New products c. New geographic markets 3. Abandonment options a. Contraction b. Temporary suspension c. Complete abandonment 4. Flexibility options. b.

What are five possible procedures for analyzing a real option?

Answer: 1. 2. 3. 4. 5.

DCF analysis of expected cash flows, ignoring option. Qualitatively assess the value of the real option. Decision tree analysis. Use a model for a corresponding financial option, if possible. Use financial engineering techniques if a corresponding financial option is not available.

Tropical Sweets is considering a project that will cost $70 million and will generate expected cash flows of $30 million per year for 3 years. The cost of capital for this type of project is 10% and the risk-free rate is 6%. After discussions with the marketing department, you learn that there is a 30% chance of high demand, with future cash flows of $45 million per year. There is a 40% chance of average demand, with cash flows of $30 million per year. If demand is low (a 30% chance), cash flows will be only $15 million per year. What is the expected NPV?

Answer: Initial cost = $70 million Expected cash flows = $30 million per year for three years Cost of capital = 10% PV of expected CFs = $74.61 million Expected NPV = $74.61 – $70 = $4.61 million Alternatively, one could calculate the NPV of each scenario: Demand High Average Low

Probability 30% 40% 30%

Annual Cash Flow $45 $30 $15

Find NPV of each scenario: PV High: N = 3 I = 10

PV = ?

PMT = –45 PV= 111.91 NPV High = $111.91 – $70 = $41.91 million.

FV = 0

PV Average: N = 3 I = 10

PV =?

PMT =–30 PV= 74.61 NPV Average = $74.61 – $70 = $4.61 million.

FV = 0

PV Low: N = 3

PV =?

FV = 0

I = 10

PMT =–15 PV = 37.30 NPV Low = $37.30 – $70 = –$32.70 million. Find Expected NPV: E(NPV) = 0.3($41.91) + 0.4($4.61) + 0.3(–$32.70) E(PV) = $4.61.

Now suppose this project has an investment timing option, since it can be delayed for a year. The cost will still be $70 million at the end of the year, and the cash flows for the scenarios will still last 3 years. However, Tropical Sweets will know the level of demand, and will implement the project only if it adds value to the company. Perform a qualitative assessment of the investment timing option’s value.

Answer: If we immediately proceed with the project, its expected NPV is $4.61 million. However, the project is very risky. If demand is high, NPV will be $41.91 million. If demand is average, NPV will be $4.61 million. If demand is low, NPV will be –$32.70 million. However, if we wait 1 year, we will find out additional information regarding demand. If demand is low, we won’t implement the project. If we wait, the up-front cost and cash flows will stay the same, except they will be shifted out by a year. The value of any real option increases if the underlying project is very risky or if there is a long time before you must exercise the option. This project is risky and has 1 year before we must decide, so the option to wait is probably valuable.

Use decision tree analysis to calculate the NPV of the project with the investment timing option.

Answer: The project will be implemented only if demand is average or high. Here is the time line: 0 1 2 3 4 High $0 –$70 $45 $45 $45 Average $0 –$70 $30 $30 $30 Low $0 $0 $0 $0 $0 To find the expected NPV, discount the cost at the risk-free rate of 6% since it is known for certain, and discount the other risky cash flows at the 10% cost of capital. High: NPV = –$70/1.06 + $45/1.102 + $45/1.103 +$45/1.104 = $35.70 Average: NPV = –$70/1.06 + $30/1.102 + $30/1.103 +$30/1.104 = $1.79 Low: NPV = $0. Expected NPV = 0.3($35.70) + 0.4($1.79) + 0.3($0) = $11.42. Since this is much greater than the NPV of immediate implementation (which is $4.61 million), we should wait. In other words, implementing immediately gives an expected NPV of $4.61 million, but implementing immediately means we give up the option to wait, which is worth $11.42 million. f.

Use a financial option pricing model to estimate the value of the investment timing option.

Answer: The option to wait resembles a financial call option—we get to ―buy‖ the project for $70 million in 1 year if the value of the project in 1 year is greater than $70 million. This is like a call option with a strike price of $70 million and an expiration date of 1 year. X = Strike price = Cost of implementing project = $70 million. RRF = Risk-free rate = 6%. T = Time to maturity = 1 year. P = Current price of stock = Current value of the project’s future cash flows. σ 2 = Variance of stock return = Variance of project’s rate of return.

We explain how to calculate P and σ2 below. Just as the price of a stock is the present value of all the stock’s future cash flows, the ―price‖ of the real option is the present value of all the project’s cash flows that occur beyond the exercise date. Notice that the exercise cost of an option does not affect the stock price. Similarly, the cost to implement the real option does not affect the current value of the underlying asset (which is the PV of the project’s cash flows). It will be helpful in later steps if we break the calculation into two parts. First, we find the value of all cash flows beyond the exercise date discounted back to the exercise date. Then we find the expected present value of those values. Step 1: Find the value of all cash flows beyond the exercise date discounted back to the exercise date. Here is the time line. The exercise date is year 1, so we discount all future cash flows back to year 1. 0 High Average Low

2 $45 $30 $15

3 $45 $30 $15

4 $45 $30 $15

High: PV1 = $45/1.10 + $45/1.102 + $45/1.103 = $111.91 Average: PV1 = $30/1.10 + $30/1.102 + $30/1.103 = $74.61 Low: PV1 = $15/1.10 + $15/1.102 + $15/1.103 = $37.30 The current expected present value, P, is: P = 0.3[$111.91/1.1] + 0.4[$74.61/1.1] + 0.3[$37.30/1.1] = $67.82. For a stock option, σ2 is the variance of the stock return, not the variance of the stock price. Therefore, for a real option we need the variance of the project’s rate of return. There are three ways to estimate this variance. First, we can use subjective judgment. Second, we can calculate the project’s return in each scenario and then calculate the return’s variance. This is the direct approach. Third, we know the project’s value at each scenario at the expiration date, and we know the current value of the project. Thus, we can find a variance of project return that gives the range of project values that can occur at expiration. This is the indirect approach.

Following is an explanation of each approach. Subjective estimate: The typical stock has σ2 of about 12%. Most projects will be somewhat riskier than the firm, since the risk of the firm reflects the diversification that comes from having many projects. Subjectively scale the variance of the company’s stock return up or down to reflect the risk of the project. The company in our example has a stock with a variance of 10%, so we might expect the project to have a variance in the range of 12% to 19%. Direct approach: From our previous analysis, we know the current value of the project and the value for each scenario at the time the option expires (Year 1). Here is the time line:

High Average Low

Current Value Year 0 $67.82 $67.82 $67.82

Value at Expiration Year 1 $111.91 $74.61 $37.30

The annual rate of return is: High: Return = ($111.91/$67.82) – 1 = 65%. Average: Return = ($74.61/$67.82) – 1 = 10%. Low: Return = ($37.30/$67.82) – 1 = –45%. Expected Return = 0.3(0.65) + 0.4(0.10) + 0.3(–0.45) = 10%. 2 = 0.3(0.65 – 0.10)2 + 0.4(0.10 – 0.10)2 + 0.3(–0.45 – 0.10)2 = 0.182 = 18.2%. The direct approach gives an estimate of 18.2% for the variance of the project’s return.

Indirect approach: Given a current stock price and an anticipated range of possible stock prices at some point in the future, we can use our knowledge of the distribution of stock returns (which is lognormal) to relate the variance of the stock’s rate of return to the range of possible outcomes for stock price. To use this formula, we need the coefficient of variation of stock price at the time the option expires. To calculate the coefficient of variation, we need the expected stock price and the standard deviation of the stock price (both of these are measured at the time the option expires). For the real option, we need the expected value of the project’s cash flows at the date the real option expires, and the standard deviation of the project’s value at the date the real option expires. We previously calculated the value of the project at the time the option expires, and we can use this to calculate the expected value and the standard deviation.

High Average Low

Value at Expiration Year 1 $111.91 $74.61 $37.30

Expected Value value

= 0.3($111.91) + 0.4($74.61) + 0.3($37.30) = $74.61. = [0.3($111.91 – $74.61)2 + 0.4($74.61 – $74.61)2 + 0.3($37.30 – $74.61)2]1/2 = $28.90.

Coefficient of variation = CV = value/Expected value CV = $28.90/$74.61 = 0.39. Here is a formula for the variance of a stock’s return, if you know the coefficient of variation of the expected stock price at some point in the future. The CV should be for the entire project, including all scenarios: σ2 = ln[CV2 + 1]/t = ln[0.392 + 1]/1 = 14.2%.

Now, we proceed to use the OPM: V = $67.82[N(d1)] – $70e–(0.06)(1)[N(d2)]. d1 =

ln( $67.82/$70)  [(0.06  0.142/2)](1) 0.5 0.5 (.142) (1)

= 0.2641. d2 = d1 – (0.142)0.5(1)0.5 = 0.2641 – 0.3768 = –0.1127. N(d1) = N(0.2641) = 0.6041. N(d2) = N(–0.1127) = 0.4551. therefore, V = $67.82(0.6041) – $70e–0.06(0.4551) = $10.98. g.

Now suppose the cost of the project is $75 million and that the project cannot be delayed. But if Tropical Sweets implements the project, then Tropical Sweets will have a growth option. It will have the opportunity to replicate the original project at the end of its life. What is the total expected NPV of the two projects if both are implemented?

Answer: Suppose the cost of the project is $75 million instead of $70 million, and there is no option to wait. NPV = PV of future cash flows – cost = $74.61 – $75 = –$0.39 million. The project now looks like a loser. Using NPV analysis: NPV = NPV of original project + NPV of replication project = -$0.39 + –$0.39/(1 + 0.10)3 = -$0.39 + –$0.30 = –$0.69. It still looks like a loser, but you will only implement project 2 if demand is high. We might have chosen to discount the cost of the replication project at the risk-free rate, and this would have made the NPV even lower.

Tropical Sweets will replicate the original project only if demand is high. Using decision tree analysis, estimate the value of the project with the growth option.

Answer: The future cash flows of the optimal decisions are shown below. The cash flow in Year 3 for the high demand scenario is the cash flow from the original project and the cost of the replication project. High Average Low

0 –$75 –$75 –$75

1 $45 $30 $15

2 $45 $30 $15

3 $45–$75 $30 $15

4 $45 $0 $0

5 $45 $0 $0

6 $45 $0 $0

To find the NPV, we discount the risky cash flows at the 10% cost of capital, and the non-risky cost to replicate (i.e., the $75 million) at the risk-free rate. NPV high = –$75 + $45/1.10 + $45/1.102 + $45/1.103 + $45/1.104 + $45/1.105 + $45/1.106 – $75/1.063 = $58.02 NPV average = –$75 + $30/1.10 + $30/1.102 + $30/1.103 = –$0.39 NPV low = –$75 + $15/1.10 + $15/1.102 + $15/1.103 = –$37.70 Expected NPV = 0.3($58.02) + 0.4(–$0.39) + 0.3(–$37.70) = $5.94. Thus, the option to replicate adds enough value that the project now has a positive NPV.

i. Answer:

Use a financial option model to estimate the value of the project with the growth option. X = Strike price = Cost of implementing project = $75 million. RRF = Risk-free rate = 6%. T = Time to maturity = 3 years. P = Current price of stock = Current value of the project’s future cash flows. σ2 = Variance of stock return = Variance of project’s rate of return. We explain how to calculate P and σ2 below. Step 1: Find the value of all cash flows beyond the exercise date discounted back to the exercise date. Here is the time line. The exercise date is Year 3, so we discount all future cash flows back to Year 3. 0 High Average Low

4 $45 $30 $15

5 $45 $30 $15

6 $45 $30 $15

High: PV3 = $45/1.10 + $45/1.102 + $45/1.103 = $111.91 Average: PV3 = $30/1.10 + $30/1.102 + $30/1.103 = $74.61 Low: PV3 = $15/1.10 + $15/1.102 + $15/1.103 = $37.30 The current expected present value, P, is: P = 0.3[$111.91/1.13] + 0.4[$74.61/1.13] + 0.3[$37.30/1.13] = $56.02

Direct approach for estimating σ2: From our previous analysis, we know the current value of the project and the value for each scenario at the time the option expires (Year 3). Here is the time line:

High Average Low

Current Value Year 0 $56.02 $56.02 $56.02

Value at Expiration Year 3 $111.91 $74.61 $37.30

The annual rate of return is: High: Return = ($111.91/$56.02)(1/3) – 1 = 25.9%. Average: Return = ($74.61/$56.02)(1/3) – 1 = 10%. Low: Return = ($37.30/$56.02)(1/3) – 1 = -12.7%. Expected Return = 0.3(0.259) + 0.4(0.10) + 0.3(–0.127) = 8.0%. 2 = 0.3(0.259-0.08)2 + 0.4(0.10 – 0.08)2 + 0.3(–0.127 – 0.08)2 = 2.3%. This is lower than the variance found for the previous option because the dispersion of cash flows for the replication project is the same as for the original, even though the replication occurs much later. Therefore, the rate of return for the replication is less volatile. We do sensitivity analysis later.

The indirect approach: First, find the coefficient of variation for the value of the project at the time the option expires (year 3). We previously calculated the value of the project at the time the option expires, and we can use this to calculate the expected value and the standard deviation.

High Average Low

Value at Expiration Year 3 $111.91 $74.61 $37.30

Expected value value

= 0.3($111.91) + 0.4($74.61) + 0.3($37.30) = $74.61. = [0.3($111.91 – $74.61)2 + 0.4($74.61 – $74.61)2 + 0.3($37.30 – $74.61)2]1/2 = $28.90.

Coefficient of variation = CV = value /Expected value CV = $28.90/$74.61 = 0.39. To find the variance of the project’s rate or return, we use the formula below: σ2 = ln[CV2 + 1]/t = ln[0.392 + 1]/3 = 4.7%. Now, we proceed to use the OPM: V = $56.06[N(d1)] – $75e–(0.06)(3)[N(d2)]. d1 =

ln( $56.06/$75)  [( 0.06  0.047/2)]( 3) 0.5 0.5 ( 0.047) ( 3)

= –0.1085. d2 = d1 – (0.047)0.5(3)0.5 = –0.1085 – 0.3755 = –0.4840. N(d1) = N(–0.1080) = 0.4568. N(d2) = N(–0.4835) = 0.3142.

Therefore, V = $56.06(0.4568) – $75e–(0.06)(3)(0.3142) = $5.92. Total value = NPV of Project 1 + Value of growth option =-$0.39 + $5.92 = $5.5 million j.

What happens to the value of the growth option if the variance of the project’s return is 14.2%? What if it is 50%? How might this explain the high valuations of many dot.com companies?

Answer: If risk, defined by σ2, goes up, then value of growth option goes up (see the file Ch 20 mini case.xls for calculations): σ2 = 4.7%, option value = $5.92 σ2 = 14.2%, option value = $12.10 σ2 = 50%, option value = $24.09 If the future profitability of dot.com companies is very volatile (i.e., there is the potential for very high profits), then a company with a real option on those profits might have a very high value for its growth option.