Solution Manual For Fundamentals of Corporate Finance, Canadian Edition, 4th edition Jonathan Berk, by StudyGuide

Solution Manual For Fundamentals of Corporate Finance, Canadian Edition, 4th edition Jonathan Berk, Peter DeMarzo, David A. Stangeland, Andras Marosi, Jarrad Harford Chapter 1-25

Chapter 1 Corporate Finance and the Financial Manager Note:

All problems in this chapter are available in MyLabFinance. An asterisk (*) indicates problems with a higher level of difficulty.

A corporation is a legal entity separate from its owners. This means ownership shares in the corporation can be freely traded. None of the other organizational forms share this characteristic.

Owners’ liability is limited to the amount they invested in the firm. Stockholders are not responsible for any encumbrances of the firm; in particular, they cannot be required to pay back any debts incurred by the firm.

Corporations and limited liability companies. Limited partnerships provide limited liability for the limited partners, but not for the general partners.

Advantages: Limited liability, liquidity, infinite life Disadvantages: Double taxation, separation of ownership and control

Real estate corporations must pay corporate income taxes but REITs do not pay corporate tax; instead, they must pass through substantially all of the income to the trust unit holders to whom the income is taxable.

Excel Solution Plan: First find the value remaining after corporate taxes. Then determine the remainder after personal taxes. Execute: First the corporation pays the taxes. After taxes, $2 × (1 – 0.34) = $1.32 per share is left to pay dividends. Once the dividend is paid, personal tax on this must be paid leaving $1.32 × (1 – 0.18) = $1.0824 per share. Evaluate: After all the taxes are paid, you are left with $1.0824 per share.

Excel Solution Plan: First find the value remaining after corporate taxes. Then determine the remainder after personal taxes.

Execute: As a REIT, there is no corporate tax so the full $2 per share can be paid out to you as a unit holder. You must then pay personal income tax on the distribution. So you are left with $2 × (1 – 0.4) = $1.20 per share. Evaluate: After all the taxes are paid, you are left with $1.20 per share. 8.

The investment decision is the most important decision that a financial manager makes, as the manager must decide how to put the owners’ money to its best use.

The goal of maximizing shareholder wealth is agreed upon by all shareholders because all shareholders are better off when this goal is achieved.

10.

Shareholders can do the following: a. Ensure that employees are paid with company stock and/or stock options. b. Ensure that underperforming managers are fired. c. Write contracts that ensure that the interests of the managers and shareholders are closely aligned. d. Mount hostile takeovers.

11.

When your parents pay for the meal, you benefit from the food but do not take on the cost of the food. This is similar to the agency problem in corporations, when managers can benefit from taking actions in their own personal interests using money that belongs to shareholders.

12.

The agent (renter) will not take the same care of the apartment as the principal (owner), because the renter does not share in the costs of fixing damage to the apartment. To mitigate this problem, having the renter pay a deposit would motivate the renter to keep damages to a minimum. The deposit forces the renter to share in the costs of fixing any problems that are caused by the renter.

13.

There is an ethical dilemma when the CEO of a firm has opposite incentives to those of the shareholders. In this case, you (as the CEO) have an incentive to potentially overpay for another company (which would be damaging to your shareholders) because your pay and prestige will improve.

*14.

Plan: For each of parts (a) to (d) you must determine if your personal change in monetary wealth more than offsets the value to you of losing your leisure time (valued at $51,000). If it does, then you would decide to proceed with the new project. Execute: a. If you owned 100% of the company and the project were accepted, your personal shares of stock would increase in value by 100% of $1 million = $1 million. This would more than offset your personal cost of lost leisure; therefore, your decision would be to proceed with the project. b. If you owned 1% of the company and the project were accepted, your personal shares of stock would increase in value by 1% of $1 million = $10,000. This would not be

enough to offset your personal cost of lost leisure; therefore your decision would be to reject the project. c. If you owned 3% of the company and the project were accepted, your personal shares of stock would increase in value by 3% of $1 million = $30,000. In addition, you would receive a bonus of $25,000, so in total your monetary wealth would increase by $55,000. This more than offsets your personal cost of lost leisure; therefore, your decision would be to proceed with the project. d. If you accept the project your monetary wealth would increase by $25,000 + 3% of $X. For you to decide to accept the project, this must be greater than $51,000 (the value of your lost leisure). Solving for X we get the following: $25,000  0.03X  $51,000 0.03X  $26,000 X  $866,666.67

Evaluate: e. In part (a), you (as the CEO) are perfectly aligned with the owners of the company as you actually own the whole company. Thus, you receive the full benefit of the $1 million increase in equity value and this offsets the value to you of the lost leisure. In part (b), your incentives are not aligned with shareholders because the project should be accepted to maximize shareholder wealth, but you reject it because the increase in your monetary wealth does not offset the cost of your extra effort and lost leisure time. Here, the principal-agent problem results in a decision that is costly to shareholders as a whole. In part (c), your incentives are aligned with shareholders as you receive enough of a monetary benefit to offset your cost of lost leisure. In part (d), though, we can see that the bonus scheme does not always solve the principal-agent problem. Your incentives are aligned with all shareholders when the project increases the equity value by an amount greater than $866,666.67. However, if the increase in equity value is lower, you would decide to reject the project even though accepting it would maximize shareholder wealth. 15.

This will impact and hurt the customers. It will be a negative impact for the customers as they will likely get sour milk. It will also be a negative impact for shareholders because, in the long run, customers will realize that the supermarket sells sour milk and they will switch supermarkets. Thus, the value today of the future income and cash flow streams generated by the supermarket will drop because of the long-term loss of customers caused by this strategy. This will negatively impact the current stock price as stockholders anticipate these long-term negative effects.

*16.

There are many considerations for you as CEO. One is the cost–benefit analysis of constructing the SD project and reaping the savings in disposal costs—that should show whether the SD project increases shareholder value. In addition, if your bonus is tied to earnings, you may be tempted to accept the project because of your higher bonuses for each of the next 10 years. There are other considerations, though. For example, is the SD method legal? If not, then the company could face substantial fines and reputational damage by using SD. Also, SD may leak into the ground water—that could further damage SPB’s reputation, cause major lawsuits, and necessitate environmental clean-up charges. These costs would affect the cost-benefit analysis for sure. For your personal

situation, will you be retiring soon and just care about the next bonus or do you plan on working for this company over the long term? Will you hold this company’s stock for the long term or do you plan to sell it quickly before a catastrophe from the project might occur? This situation ties into the principal-agent problem in that as CEO you may accept SD due to your higher bonus and potentially even a higher short-term stock price, but the ultimate effect on SPB’s true shareholder value may drop if the project is made public, a catastrophe happens, or other negative factors outlined above are factored in. 17.

The shares of a public corporation are traded on an exchange (or ―over the counter‖ in an electronic trading system) while the shares of a private corporation are not traded on a public exchange.

18.

A primary market is where the company sells shares of itself to investors. The secondary market is where investors can buy and/or sell the company’s shares with other investors (but not the company itself).

19.

Investors always buy at the ask and sell at the bid. Since ask prices always exceed bid prices, investors ―lose‖ this difference. It is one of the costs of transacting. Since the market makers take the other side of the trade, they make up this difference.

20.

Plan: For market orders, use the ask price as what you pay when buying a stock and use the bid price as what you receive when you sell a stock. Execute: a. With a market order to buy, you pay the quoted ask on September 14 multiplied by the number of shares purchased: $14.60 per share × 500 shares = $7300. You then sell using a market order, so you receive the quoted bid on September 15 multiplied by the number of shares sold: $15.04 per share × 500 shares = $7520. Your gain is $7520 – $7300 = $220. b. Using the limit order prices, you pay $14.58 per share × 500 shares = $7290 when purchasing the shares and you receive $15.08 per share × 500 shares = $7540 when selling the shares. Your gain is $7540 – $7290 = $250. Evaluate: c. The trade-offs between using market versus limit orders are as follows: i.

Market orders are executed instantaneously; it may take some time before a counterparty accepts your limit order, or it may be the case that your limit order is never executed.

ii. Using market orders, you buy at the ask and sell at the bid, so the bid-ask spread is an implied transaction cost. Using limit orders, you can buy closer to the bid and sell closer to the ask; thus, you can avoid much or all of the bid-ask spread as a transaction cost. 21.

Plan: For market orders, use the ask price as what you pay when buying a stock and use the bid price as what you receive when you sell a stock. Execute:

a. With a market order to buy, you pay the Sept 16 quoted ask times the number of shares purchased: $15.14 per share × 1000 shares = $15,140. You then sell using a market order, so on Sept. 17 you receive the quoted bid times the number of shares sold: $15.12 per share × 1000 shares = $15,120. Your gain is $15,120 – $15,140 = –$20, or, put another way, you have loss of $20. Notice, that even though the stock price went up, you lost money because of the high bid-ask spread. b. Using the limit order prices, you pay $15.08 per share × 1000 shares = $15,080 when purchasing the shares and you receive $15.18 per share × 1000 shares = $15,180 when selling the shares. Your gain is $15,080 – $15,180 = $100. Evaluate: c. The trade-offs between using market versus limit orders are as follows: i.

Market orders are executed instantaneously; it may take some time before a counterparty accepts your limit order or it may be the case that your limit order is never executed.

ii. Using market orders, you buy at the ask and sell at the bid, so the bid-ask spread is an implied transaction cost. Using limit orders you can buy closer to the bid and sell closer to the ask, thus, avoiding much or all of the bid-ask spread as a transaction cost. Note, in this example with the market orders you lost money even though the stock price rose—this is due to the high bid-ask spread. By using the limit orders, you avoid the bid-ask spread and are able to have a gain of $100 due to the stock price rising. 22.

The financial cycle describes how money flows from savers to companies and back. In the financial cycle, (1) people invest and save their money; (2) that money, through loans and stock, flows to companies that use it to fund growth through new products, generating profits and wages; and (3) the money then flows back to the savers and investors.

23.

Insurance companies essentially pool premiums together from policyholders and pay the claims of those who have an accident, fire, or medical need, or who die. This process spreads the financial risk of these events out across a large pool of policyholders and the investors in the insurance company. Similarly, mutual funds and pension funds take your savings and spread them out among the stocks and bonds of many different companies, limiting your risk exposure to any one company.

24.

Investment banking is the business of advising companies in major financial transactions. Examples include buying and selling companies or divisions, and raising new capital by issuing stock or bonds.

25.

Mutual, pension, and hedge funds all pool money together and invest it on behalf of the investors in the fund. They differ in terms of who invests in the fund and what the primary objective is. Mutual and pension funds are most similar except that pension funds are investing retirement savings invested through the workplace with the objective of providing retirement income for those employees. Hedge funds are only open to

investments by wealthy individuals and endowments. They invest across all asset categories, usually seeking low-risk investment strategies that will generate high returns.

Chapter 2 Introduction to Financial Statement Analysis Note:

All problems in this chapter are available in MyLab Finance. An asterisk (*) indicates problems with a higher level of difficulty. For a breakdown of the difficulty ratings of all of the problems in this Solutions Manual, please see the Preface – Question Difficulty Ratings.

In a firm’s annual report, four financial statements can be found: the balance sheet, the statement of comprehensive income (including the income statement), the statement of cash flows, and the statement of shareholders’ equity. Financial statements in the annual report are required to be audited by a neutral third party, who checks and ensures that the financial statements are prepared according to GAAP (or IFRS) and that the information contained is reliable.

Each method will help find the same SEC filings. Yahoo! Finance also provides some analysis, such as charts and key statistics.

Each method will help find the same SEC filings. Yahoo! Finance also provides some analysis, such as charts and key statistics. The filings on www.sedar.com are the Canadian filings for Tim Hortons.

a. Long-term liabilities would decrease by $20 million, and cash would decrease by the same amount. The book value of equity would be unchanged. b. Inventory would decrease by $5 million, as would the book value of equity. c. Long-term assets would increase by $10 million, cash would decrease by $5 million, and long-term liabilities would increase by $5 million. There would be no change to the book value of equity. d. Accounts receivable would decrease by $3 million, as would the book value of equity. e. This event would not affect the balance sheet. f.

This event would not affect the balance sheet.

Global Corporation’s book value of equity increased by $1 million from 2020 to 2021. An increase in book value does not necessarily indicate an increase in Global’s share price. The market value of a stock does not depend on the historical cost of the firm’s assets, but on investors’ expectation of the firm’s future performance. There are many events that may affect Global’s future profitability, and hence its share price, that do not show up on the balance sheet.

$12,123 million

$2,904 million

$32,686 million

$31,758 million, $15,365

$928 million

At the end of the fiscal year, Costco had cash and cash equivalents of $6,055 million.

Costco’s total assets were $40,830 million.

Costco’s total liabilities were $27,727 million, and it had $6,487 million in debt.

The book value of Costco’s equity was $13,103 million.

Revenues in 2018 were $141,576 million Increase in Revenues 

141,576  1  9.73% 129,025

Operating Margin (2018) 

4,480  3.16% 141,576

Operating Margin (2017) 

4,111  3.19% 129,025

Net Profit Margin (2018) 

3,134  2.21% 141,576

Net Profit Margin (2017) 

2,679  2.08% 129,025

Operating margin slightly decreased; net profit margin increased compared with the year before. c.

9..

The diluted earnings per share in 2018 were $7.09. The number of shares used in this calculation of diluted EPS was 441.8 million.

Excel Solution See Table 2.5 showing financial statement data and stock price data for Mydeco Corp.

Growth rate in revenues

2018 −10.02%

2019 16.71%

2020 20.28%

2021 18.29%

−83.33% 110.00% 101.59% 70.87% b. Growth rate in net income c. Net income growth rate differs from revenue growth rate because cost of goods sold and other expenses can move at different rates compared to revenues. For example, revenues declined in 2018 by 10%; however, cost of goods sold only declined by 7%. 10..

Excel Solution A repurchase does not impact earnings directly, so any change to EPS will come from a reduction in shares outstanding. 2021 shares outstanding = 55 – 4 × 2 = 47 million,

EPS  11..

21.7  $0.46 47

Excel Solution

The equipment purchase does not impact net income directly, however the increased depreciation expense and tax savings changes net income. 2018 64.50 −27.00 37.50 −32.90 4.60 −1.61 3.0

EBITDA Minus new depreciation expense New EBIT Minus interest income New pre-tax income Minus taxes New net income 12.

2019 76.20 −38.30 37.90 −32.20 5.70 −2.00 3.7

2020 95.40 −42.40 53.00 −37.40 15.60 −5.46 10.1

2021 111.40 −42.60 68.80 −39.40 29.40 −10.29 19.1

Excel Solution If Mydeco’s costs and expenses had been the same fraction of revenues in 2018–2021 as they were in 2017, then its net profit margins would have been equal. 2017 net profit margin 

18  4.45% . 404.3

Revenues Net profit margin from 2017 New net income Shares outstanding (millions) Earnings per share 13.

2017 404.30 4.452% 18.00 55.00 $0.33

2018 363.80

2019 424.60

2020 510.70

2021 604.10

16.20 55.00 $0.29

18.90 55.00 $0.34

22.74 55.00 $0.41

26.90 55.00 $0.49

A $10 million operating expense would be immediately expensed, increasing operating expenses by $10 million. This would lead to a reduction in taxes of 35%  $10 million 

$3.5 million. Thus, earnings would decline by $10  $3.5  $6.5 million. There would be no effect on next year’s earnings. b. Capital expenses do not affect earnings directly. However, the depreciation of $2 million would appear each year as an operating expense. With a reduction in taxes of 2  35%  $0.7 million, earnings would be lower by $2  $0.7  $1.3 million for each of the next five years. *14.

Plan: Quisco Systems wishes to acquire a new networking technology and is confronted with a common business problem: whether to develop the technology itself in-house or to acquire another company that already has the technology. Quisco must perform a comprehensive analysis of each option, not just comparing internal development costs versus acquisition costs, but considering tax implications as well. Execute: a. If Quisco develops the product in-house, its earnings would fall by $500  (1  35%)  $325 million. With no change to the number of shares outstanding, its EPS would decrease by $0.05 = $325/6500 to $0.75. (Assume the new product would not change this year’s revenues.) b. If Quisco acquires the technology for $900 million worth of its stock, it will issue $900/18  50 million new shares. Since earnings without this transaction are $0.80  6.5 billion  $5.2 billion, its EPS with the purchase is 5.2/6.55  $0.794. Evaluate: Acquiring the technology would have a smaller impact on earnings. But this method is not cheaper. Developing it in-house is less costly and provides an immediate tax benefit. The earnings impact is not a good measure of the expense. In addition, note that because the acquisition permanently increases the number of shares outstanding, it will reduce Quisco’s earnings per share in future years as well.

15.

a. Net cash provided by operating activities was $5,774 million in 2018. b. Depreciation expense was $1,437 million in 2018. c.

Net cash used in new property and equipment was $2,969 million in 2018.

d. Costco raised $0 from sale of shares of its stock, while it spent $328 million on the repurchase of common stock. Costco raised –$328 million from the sale of its shares of stock (net of any purchases). 16.

Excel Solution a. The company’s cumulative earnings over these four quarters were $918.268 million. Its cumulative cash flows from operating activities were $1.186 billion. b. Fraction of cash from operating activities used for investment over the four quarters:

Four Quarters

Operating Activities

227,502

–13,935

717,635

254,534

1,185,736

Investing Activities

–196,952

–35,437

–251,331

–96,848

–580,568

CFI/CFO

86.57%

–254.30%

35.02%

38.05%

48.96%

c. Fraction of cash from operating activities used for financing over the four quarters: 4

Four Quarters

Operating Activities

227,502

–13,935

717,635

254,534

1,185,736

Financing Activities

462,718

–13,357

–526,189

–96,044

–172,872

–203.39%

–95.85%

73.32%

37.73%

14.58%

CFF/CFO 17.

Plan: Even a relatively simple transaction such as receiving an order to sell merchandise on credit and shipping the order promptly creates a series of changes within the firm. Map out the changes that would occur to a firm that engages in a relatively simple business transaction. Execute: a. Revenues: increase by $5 million b. Earnings: increase by $3 million c. Receivables: increase by $4 million d. Inventory: decrease by $2 million e. Cash: increase by $3 million (earnings)  $4 million (receivables)  $2 million (inventory)  $1 million (cash) Evaluate: We can see that even a relatively simple credit sale has impacts on revenues, earnings, accounts receivable, inventory, and eventually cash.

18.

Plan: Nokela Industries plans to purchase a capital asset. In this case it is a $40 million cyclo-converter. Any time a firm acquires a capital asset it is permitted to depreciate the asset for tax purposes. This has depreciation expense, tax expense, and cash flow effects that must be understood and analyzed. Execute: a. Earnings for the next four years would have to deduct the depreciation expense. After taxes, this would lead to a decline of 10  (1  40%)  $6 million each year for the next four years. b. Cash flow for the next four years: less $36 million ( 6  10  40) this year, and add $4 million ( 6  10) for the three following years. Evaluate: For the next four years the investment in the cyclo-converter will increase Nokela’s depreciation expense by $10 million and will reduce after-tax earnings by $6 million per year. Depreciation expense is a non-cash expense (it is an accrual that recognizes that the value of the asset, which has already been paid for, is declining in value) that the firm does not have to pay out. Since every dollar of depreciation expense

lowers Nokela’s taxable income by a dollar, its tax savings therefore are 40 cents on the dollar. The $10 million in depreciation expense in the next four years will lower Nokela’s tax bill by $4 million ($10 million  0.4) per year. 19.

Plan: The problem presents us with some raw financial information for General Electric. While useful, this raw financial information is not well suited to support financial analysis of General Electric and to answer questions such as “How has the stock market valued GE?” “How much debt does GE use relative to the equity financing that GE uses?” “How valuable, in today’s dollars, is GE?” To answer these and other questions, we must compute key ratios and current market values as opposed to historical cost values. Execute: a. Market capitalization  8.7 billion  $8  $69.6 billion

69.6 = 1.34 52 110 = 2.12 b. Book debt-equity ratio  52 110 Book debt-equity ratio  = 1.58 69.6 c. Enterprise value  69.6  110  70  109.6 (billion) Market-tobook ratio 

Evaluate: GE has a market-to-book ratio of 1.34. Over time, equity investors invested $52 billion in GE; today that equity investment is worth $69.6 billion (or 1.34 times more). This indicates that equity investors expect moderate results in the future. GE has a book debt-equity ratio of 2.12. Over time, equity investors invested $52 billion in GE, and debt investors invested $110 billion (or 2.12 times more). This would indicate that GE is very heavily financed with debt. These are book values; due to GE’s low market capitalization (in part (a) above, we calculated that GE’s equity is valued at $69.6 billion in today’s dollars), the market debt-equity ratio provides a very similar picture. GE has an enterprise value of $109.6 billion. In today’s dollars, investors value the entire company at this value. 20.

140.83  1.30 108.28

Apple’s current ratio 

Apple’s quick ratio 

Apple’s higher current and quick ratios demonstrate that it has higher asset liquidity than does Hewlett-Packard. This means that, in a pinch, Apple has more liquidity to draw on than does Hewlett-Packard.

140.83  4.99  1.25 108.28

21.

Plan: The table presents raw data about ANF and GPS. Although useful, this information does not easily tell us how the stock market values each of these firms alone and by comparison. To accomplish this, we will compute the market-to-book ratio of each firm and then compare them. Execute: a. ANF’s market-to-book ratio 

26.50  65.85  0.89 1219

26.00  381.43  2.79 3553 b. The market looks more favourably on the outlook of GPS than on Abercrombie and Fitch.

GPS’s market-to-book ratio 

Evaluate: The market values, in a relative sense, the outlook of GPS more favourably than Abercrombie and Fitch. For every dollar of equity invested in GPS, the market values that dollar today at $2.79 versus $0.89 for a dollar invested in ANF. Equity investors are willing to pay relatively more today for shares of GPS than for ANF because they expect GPS to produce superior performance in the future. 22.

a. Walmart’s gross margin 

Costco’s gross margin  b. Walmart’s net margin  Costco’s net margin 

129.10  25.10% ; 514.41

18.42  13.01% . 141.58

6.67  1.30% ; 514.41

3.13  2.21% . 141.58

c. Walmart had better gross profitability, but a worse net margin in 2018. 23.

Plan: We can use Equations 2.8, 2.9, and 2.10 to compute Local’s margins. The problem gives us the necessary inputs. Execute: a. Gross Margin 

Gross Profit 10  6   0.4, or 40% Sales 10

b. Operating Margin 

Operating Income 10  6  0.5  1  1   0.15, or 15% Sales 10

c. Net Profit Margin 

Net Income (10  6  0.5  1  1)(1  0.35)   0.0975, or 9.75% Sales 10

Evaluate: Local is profitable. You can see how the margins decrease as you move down the income statement because each successive margin takes into account more costs. 24.

Plan: Selling expenses do not affect the gross margin, but the increase in such expenses will decrease the other margins. Execute: Gross margin would not change. Operating Margin 

Operating Income 10  6  0.8  1  1   0.12, or 12% Sales 10

Net Profit Margin 

Net Income (10  6  0.8  1  1)(1  0.35)   0.078, or 7.8% Sales 10

Evaluate: Gross margin only accounts for cost of goods sold. The effect of the additional selling expenses can be seen in the reduced operating and net profit margins. 25.

Plan: Only the net profit margin accounts for interest expense, so both the gross and operating margins will be unaffected. Execute: Gross margin would not change. Operating margin would not change. Net Profit Margin 

Net Income (10  6  0.5  1  1  0.8)(1  0.35)   .0455, or 4.55% Sales 10

Evaluate: If you were focused only on the gross and operating margins, you would not see the impact of the increased interest expense, which shows up in the net profit margin. 26.

Using operating income as a multiple of interest to compute interest coverage, we have operating income = 0.10  $30 million = $3 million, so its interest coverage is $3 million/$1 million = 3 times.

27.

Plan: First, we must compute Ladders’ net income using the fact that net profit margin is net income/sales. Then we can compute the ROE as net income/book equity and the ROA as net income/book assets. Execute: First, compute Ladders’ net income: 0.05  $50 million  $2.5 million. ROE  Net Income/Book Equity  $2.5 million/$40 million  6.25% ROA  (Net Income + Interest Expense)/Book Assets  ($2.5 million + $1 million)/ ($30 million  $40 million)  5%

Evaluate: ROE measure the net income (to shareholders) as a percentage of the book value of their investment. ROA measures the net income (to shareholders) as a percentage of the book value of all the assets used to generate the income. A firm with positive book equity and some debt will always have a lower ROA than ROE. ROA and ROE will be the same for a firm with no liabilities. 28.

Excel Solution Plan: Using the information provided and Equations 2.13 to 2.16, we can compute all the efficiency ratios for JPJ. Execute: Accounts Receivable Days 

Fixed Asset Turnover 

Sales 1,000,000   .333 Fixed Assets 3,000,000

(Total) Asset Turnover 

Inventory Turnover 

Accounts Receivable 50,000   18.25 Average Daily Sales 1,000,000/365

Sales 1,000,000   0.2 Total Assets 5,000,000

Cost of Goods Sold 600,000  4 Inventory 150,000

Evaluate: These ratios allow you to evaluate how efficiently JPJ is utilizing its assets and how quickly it is collecting its accounts receivable. 29.

Excel Solution Plan: Using the 10% growth rate, we can compute the new sales number and then the 5% growth rate will give us the new assets number. We can then recompute the asset turnover ratios. Execute: Sales  1,000,000(1.10)  1,100,000 Assets  5,000,000(1.05)  5,250,000 Fixed Assets  3,000,000(1.05)  3,150,000 Fixed Asset Turnover 

Sales 1,100,000   0.35 Fixed Assets 3,150,000

(Total) Asset Turnover 

Sales 1,100,000   0.21 Total Assets 5,250,000

Evaluate: Because sales are growing faster than assets, we see that efficiency of asset utilization is increasing—the turnover ratios are higher.

*30.

Excel Solution Plan: We are given some data about Global’s financial results in 2020 and 2021. Global launched a marketing campaign in 2021 that increased sales but also decreased operating margins. We must calculate the effects of these changes on revenues, net income, and stock price. Execute: a. Revenues in 2021  1.15  186.7  $214.705 million EBIT in 2021  4.50%  214.705  $9.66 million (there is no other income) Prior to changed assumptions, EBIT in 2021  5.57%  186.70  $10.39 million. b. Net income in 2021  EBIT  interest expenses  taxes  (9.66  7.7)  (1  24%)  $1.49 million Prior to changed assumptions, net income in 2021  EBIT  interest expenses  taxes  (10.39  7.7)  (1  24%)  $2.04 million c. Share price = old P/E in 2021  new EPS in 2021 = 18  (1.49/3.6) = $7.45. Prior to changed assumptions, share price in 2021  (P/E ratio in 2021)  (EPS in 2021)  18.0  (2.04/3.6)  $10.20. Evaluate: The new aggressive marketing campaign succeeded in raising revenues by 15%. Unfortunately, operating margins fell from 5.57% to 4.50%, which reduced EBIT and net income. As a result, the stock price fell from $10.20 to $7.45. The new marketing campaign destroyed shareholder value and is, therefore, a failure.

31.

Excel Solution Plan: The table presents raw data about debt, equity, operating income, and interest expense. While useful, this information does not easily tell us how much financial leverage each of these firms alone and by comparison is using. It also does not tell us how well each firm is able to support its debt. To accomplish this, we will compute various leverage ratios of each firm and then compare them. Execute: 500  1.25 400 80 Firm B: Market debt-equity ratio   2.00 40

a. Firm A: Market debt-equity ratio 

500  1.67 300 80 Firm B: Book debt-equity ratio   2.29 35

b. Firm A: Book debt-equity ratio 

100  2.00 50 8 Firm B: Interest coverage ratio   1.14 7

c. Firm A: Interest coverage ratio 

d. Evaluate: Firm B has a lower coverage ratio and will have slightly more difficulty meeting its debt obligations than Firm A. 32.

Plan: The table presents raw data about sales, accounts receivable, and inventory data for Walmart and Target. Although useful, this information does not tell us easily how well each firm is managing its accounts receivable and inventory in general and in comparison with each other. To accomplish this, we will compute the relevant ratios of each firm and then compare them. Execute: a. Walmart: Accounts Receivable Days 

5,624  4.26  482,130   365   

Target: Accounts Receivable Days 

779  3.85  73,785   365   

b. Walmart: Inventory Turnover  Target: Inventory Turnover 

360,984  8.12 44, 469

51,997  6.05 8,601

c. Target is managing its accounts receivables more efficiently (shorter AR days) and Walmart is managing its inventory more efficiently (more AR turnover). Evaluate: Walmart is managing its accounts receivable and inventory more efficiently, as shown by the above ratios. Walmart collects its accounts receivable in 5.27 days as opposed to 34.14 days for Target. Likewise, Walmart turns over its inventory 8.05 times a year, as opposed to 6.40 times for Target. 33.

Excel Solution a. Market capitalization-to-revenue ratio =

20.6 = 0.54 for United Airlines 37.9

27.5 = 1.39 for Southwest Airlines 19.8

b. Enterprise value-to-revenue ratio 

(20.6  11.9  5.2)  0.72 for United Airlines 37.9



(27.5  3.2  3)  1.40 for Southwest Airlines 19.8

c. The market capitalization-to-revenue ratio cannot be meaningfully compared when the firms have different amounts of leverage, as market capitalization measures only the value of the firm’s equity. The enterprise value-to-revenue ratio is, therefore, more useful when the firm’s leverage is quite different, as it is here. 34.

Plan: Use the DuPont Identity to perform the analysis: Net Profit Margin  Total Asset Turnover  Total Assets/Equity. Execute: a. 3.5%  1.8  44/18  15.4% b. 4%  1.8  44/18  17.6% c. 4%  (1.8  1.2)  44/18  21.1% Evaluate: The analysis demonstrates different ways that a company can increase its overall ROE—by increasing its net profit margin or its asset turnover.

35.

a. Costco’s Net Profit Margin 

3,134  2.21% 141,576

Costco’s Asset Turnover 

141,576  3.47 40,830

Costco’s Equity Multiplier 

40,830  3.12 13,103

b. Costco’s ROE (DuPont)  2.21%  3.47  3.12  23.92% c. Costco’s revised ROE 2.21%  3.61  3.12 24.92%. Costco’s would need to increase asset turnover to more than 3.61 times.

36.

Net Profit Margin Asset Turnover Equity Multiplier Walmart’s ROE (DuPont)  1.30%  2.35  2.75  8.40% The two firms’ ROEs differ because Costco has a higher profitability, asset turnover, and equity multiplier.

37.

Excel Solution Plan: You are presented with a large amount of financial information over several years about a company. You are asked to analyze this information around issues of profitability, and book and market values of equity.

Execute: a. The book value of the equity decreased by $2.101 billion compared to that at the end of the previous year, and was negative. b. Because the book value of equity is negative in this case, the company’s market-tobook ratio and its book debt-equity ratio are not meaningful. Its market debt-equity ratio may be used in comparison. c. Negative book value of equity does not necessarily mean the firm is unprofitable. Loss in gross profit is only one possible cause. If a firm borrows to repurchase shares or invest in intangible assets (such as R&D), it can have a negative book value of equity. Evaluate: The company issued debt to buy back $2.11 billion in equity. Obviously, that resulted in a large increase in outstanding debt and a large decline in outstanding equity. This resulted in the book value of the company’s equity being negative. On the surface, a negative book value of equity would suggest an unprofitable, if not failed, firm. The reality in this case is much more complicated. 38.

a. KPMG certified Costco’s financial statements. b. The CEO (W. Craig Jelinek) and the CFO (Richard A. Galanti) certified Costco’s financial statements.

Chapter 3 The Valuation Principle: The Foundation of Financial Decision Making Note: All problems in this chapter are available in MyLab Finance. An asterisk (*) indicates problems with a higher level of difficulty. For a breakdown of the difficulty ratings of all of the problems in this Solutions Manual, please see the Preface – Question Difficulty Ratings. 1.

Plan: The benefit of the rebate is that Honda will sell more vehicles and earn a profit on each additional vehicle sold: Benefit  Profit of $6000 per vehicle  15,000 additional vehicles sold  $90 million. Execute: The cost of the rebate is that Honda will make less on the vehicles it would have sold: Cost  Loss of $2000 per vehicle  40,000 vehicles that would have sold without rebate  $80 million.

Evaluate: Thus, Benefit  Cost  $90 million  $80 million  $10 million, and offering the rebate looks attractive. (Alternatively, we could view it in terms of total, rather than incremental, profits. The benefit is $6000/vehicle  55,000 sold  $330 million, and the cost is $8000/vehicle  40,000 sold  $320 million.) 2.

Plan: If the shrimp from your Czech and Thai suppliers are of equal quality, you will buy the shrimp from the supplier who offers you the lowest price. One problem is that the Czech supplier quotes you a price in koruna and the Thai supplier quotes you a price in baht. Since you will have to convert dollars to either koruna or baht, you will buy the shrimp from the supplier who will charge you the lowest cost in dollars. Execute: Czech buyer’s offer in dollars  2,000,000 CZK/(25.50 CZK/USD)  78,431.37 USD. Thai supplier’s offer in dollars  3,000,000 THB/(41.25 THB/USD)  72,727.27 USD. Evaluate: You would buy the shrimp from the Thai supplier because the Thai shrimp are ($78,431  $72,727) = $5704 less expensive today.

Plan: There are two related, but different, decisions to analyze. In both cases, you will take the choice that will give you the highest value. Execute: a. Value of the stock bonus today  100  $63  $6300 Value of the cash bonus today  $5000 Since you can sell the stock for $6300 in cash today, its value is $6300, which is better than the cash bonus of $5000 today. Take the stock. b. Because you could buy the stock today for $6300 if you wanted to, the value of the stock bonus cannot be more than $6300. But if you are not allowed to sell the company’s stock for the next year, its value to you could be less than $6300. Its value will depend on what you expect the stock to be worth in one year, as well as how you feel about the risk involved. There is no clear-cut answer to which alternative is best, because taking the stock today and having to hold it for a year involves risk. You might decide that it is better to take the $5000 in cash than to wait for the uncertain value of the stock in one year. This would be especially true if you believed you could invest the $5000 today in another asset that would be worth more than $6300 in one year. Evaluate: In part (a), there is a clear-cut answer: Take the stock today because it is worth more than the cash bonus. In part (b), there is no clear-cut answer because you cannot directly compare $5000 cash today against the uncertain value of 100 shares of stock in one year. Different people could make different decisions based largely on their estimate of the future value of the stock.

Plan: With two different interest rates for the same thing (borrowing or saving), you can make a sure profit by borrowing at the lower rate and depositing the money at the higher rate. a. Take a loan from Bank One at 5.5% and save the money in Bank Two at 6%. b. Bank One would experience a surge in the demand for loans, while Bank Two would receive a surge in deposits. c. Bank One would increase the interest rate, and/or Bank Two would decrease its rate. Evaluate: Your actions, along with those of others, would create a surge in demand for loans at the low rate and deposits at the high rate, causing the banks to change the rates until they are equal—the arbitrage opportunity would quickly disappear.

Plan: The most Apple should be able to charge you is the benefit you receive by not having to buy the CD and saving the tracks to your iPhone. Execute: For you to be indifferent, Apple can charge $25 (the benefit to you by not having to buy and save the CD tracks). Evaluate: Apple cannot charge more than the cost of doing it yourself as then the cost would exceed your benefit and you would not use the iTunes service.

Plan: The same share of stock cannot trade for two different prices—otherwise, there would be an arbitrage opportunity (you would buy it on the exchange with the lower price and immediately sell it on the exchange with the higher price). Thus, the USD price must be the Canadian price multiplied by the exchange rate. BBNASDAQ = $10.00 CAD × (0.95 USD/1.00 CAD) = $9.50 USD Evaluate: If the price were any different, there would be an arbitrage opportunity as described in the plan statement and a flood of buy and sell orders would force the prices to be equal.

Plan: Determine the value of one tonne of shrimp to Bubba, and determine the change in value to Bubba of one tonne of shrimp because of his allergy. Execute: a. The value of one tonne of shrimp to Bubba is $10,000 because that is the market price. b. No. As long as he can buy or sell shrimp at $10,000 per tonne, his personal preference or use for shrimp is irrelevant to the value of the shrimp. Evaluate: In well-functioning markets, the price, and therefore the value, is set by the supply and demand of all suppliers and users.

Plan: Brett must determine if he should accept his neighbour’s offer to trade his almond crop for the neighbour’s walnut crop. He should calculate the value of keeping his walnut crop versus the value of trading for the walnut crop. Brett should take the alternative that makes him the best off financially. Brett must also determine if his preference for walnuts over almonds should influence his decision. Execute: a. Brett calculates the market value of the almond crop is $100,000 (1000 tonnes × $100 per tonne), while the market value of the walnut crop is $88,000 (800 tonnes × $110 per tonne). Evaluate: No, he should not make the exchange. He would not give up an asset worth $100,000 for an asset worth $88,000.

Plan: You need to compute the future value (FV) based on a 5% interest rate and a present value (PV) of $100. A 5% interest rate means that for every $1 today, you get $1.05 in a year. Execute:

 $1.05 in one year    $105 in one year $1 today  

FV  $100 today   Evaluate:

The amounts, $100 today and $105 in one year, have equivalent values to you because with $100 today, you could deposit it and have $105 in one year.

10.

Plan: You need to compute the PV of $1000 (a future value) based on a 6% discount rate, meaning every $1.06 in one year is worth only $1 today. Execute:

 $1.06 in one year  PV  $1000 in one year    $1 today     $1 today  $1000 in one year     $1.06 in one year  $1000   $943.40 today 1.06 Evaluate: If you expect to have $1000 in one year, you could borrow $943.40 today and in one year you would have exactly enough to pay off the loan with 6% interest. 11.Plan: You need to calculate the amount of interest you would have paid at 9% and the amount of interest you will pay at 7.5%, and then compute the difference. Alternatively, you can calculate the amount of interest you would pay at 1.5%. Execute: At 9%, you will pay $12,000 × 9% = $1080. At 7.5%, you will pay $12,000 × 7.5% = $900. The difference is then $1080 – $900 = $180 = $12,000 × 1.5%. 12.

Excel Solution Plan: You are presented with three alternative courses of action. You could deposit the $55 you have today. You could lend the $55 you have today to your friend and use your friend’s promised payment of $58 in one year as collateral for a loan and get cash today. Or, you could loan the money to your friend today or deposit in the bank. You should select the alternative that makes you the best off financially. Execute: If you deposit the money in the bank today, you will have a.

 $1.06 in one year  FV  $55 today     $58.30 in one year $ today  

If you lend the money to your friend for one year and borrow against the promised $58 repayment, then you could borrow b.

 $1.06 in one year  PV  $58 in one year     $54.72 today $ today  

Evaluate: From a financial perspective, you should deposit the money in the bank, as it will result in more money for you at the end of the year.

13.Plan: The discount factor is equal to the inverse of 1 + the interest rate. Execute: Given a discount rate of 6%, the discount factor will be the following:

Evaluate: The discount factor equivalent to a 6% discount rate is 0.9434. 14.Plan: The discount rate is equal to the inverse of the discount factor, minus 1. Execute: Given a discount factor of 0.9009, the discount rate will be the following:

Evaluate: Given a discount factor of 0.9009, the discount rate is 11%. 15.

From your perspective: Today

1 year

1000

1080

From the bank’s perspective: Today

1 year

1000

1080

16.

You get the TV and have to pay $1000 in one year. The discount rate is 4%:

PV 

$1000  $961.54 1.04 

17.

Plan: To value the future store credit, you can determine its present value. The net cost of the extended warranty is, thus, its price today minus the present value of the future credit. Execute: a.

PV 

$399

1  .06 

 $298.16

b. Net cost  $399  $298.16  $100.84 Evaluate: Given the assumptions, the net cost of the warranty is $100.84 after considering the future store credit. Note, though, that if you were unlikely to use the future store credit or if you were likely to forget about it, then the net cost of the warranty would be the full $399. 18.

Excel Solution Plan: You are considering three related questions. As usual, you will select the alternative that makes you the financially best off (wealthiest). Execute: a. Having $200 today is equivalent to having $200  1.04  $208 in one year. b. Having $200 in one year is equivalent to having $200/1.04  $192.31 today. c. Evaluate: Because money today is worth more than money in the future, $200 today is preferred to $200 in one year. This answer is correct even if you do not need the money today, because by investing the $200 you receive today at the current interest rate, you will have more than $200 in one year.

19.

FV = $100, n = 10, r = 0.02

PV 

$100

1.02 

 $82.03

20.PV = $500, n = 5, r = 0.015 5 FVn = $500 × (1 + 0.015) = $538.64

21.Plan: The only way to compare all of these options is to put them on equal footing: present value. Execute: a. Option ii > Option iii > Option i

100  90.91 (1.10)1 200 PVii   124.18 (1.10)5 300 PViii   115.66 (1.10)10 PVi 

b. Option iii > Option ii > Option i

100  95.24 (1.05)1 200 PVii   156.71 (1.05)5 300 PViii   184.17 (1.05)10 PVi 

c. Option i > Option ii > Option iii

100  83.33 (1.20)1 200 PVii   80.38 (1.20)5 300 PViii   48.45 (1.20)10 PVi 

Evaluate: Lower discount rates make distant cash flows more valuable, so when the discount rate is only 5%, Option iii, with the longest time until receiving the cash flow, is the most valuable. However, when the discount rate is 20%, Option iii is the least valuable.

*22.

Excel Solution Plan: You must first compute the FV and then subtract the initial investment and the simple interest, which is simply 8% of $1000 per year ($80).

Execute: a. The balance after three years is $1259.71; total compound interest is $259.71, simple interest is $240, and interest on interest is $19.71. FV  1000(1.08)3  $1259.71, total compound interest = $1259.71 – $1000 = $259.71, and total simple interest is 8% per year $1000 3 years = $240, so interest on interest is $259.71  $240.00  $19.71. b. The balance after 25 years is $6848.48; total compound interest is $5848.48, simple interest is $2000, and interest on interest is $3848.48. FV  1000(1.08)25  $6848.48, total compound interest = $6848.48 – $1000 = $5848.48, and total simple interest is 8% per year $1000 25 years = $2000, so interest on interest is $5848.48  $2000.00  $3848.48. Evaluate: As time progresses, the interest on interest becomes increasingly important, dwarfing the simple interest (which represents the interest on your initial investment or initial principal). 23.Plan: Determine the future values of the different interest rates and the different time periods. Compare the differences in future values. Execute: a. FV5  2000  1.055

 2552.56 0 1

FV  ?

2000

Given: Solve for FV:

I/Y

PMT

5.00%

2000.00

Excel Formula

($2552.56)

 FV(0.05,5,0,2000)

b. FV10  2000  1.0510

 3257.79 0 1

FV ?

2000

Given:

I/Y

PMT

5.00%

2000.00

Solve for FV: c.

Excel Formula

($3257.79)

FV(0.05,10,0,2000)

FV5  2000  1.15  3221.02 0

FV  ?

2000

Given:

I/Y

PMT

10.00%

2000.00

Solve for FV:

Excel Formula

($3221.02)

FV(0.1,5,0,2000)

*d. Evaluate: Since, with compound value, in the last five years you get interest on the interest earned in the first five years as well as interest on the original $2000, the value at the end of 10 years ($3257.79) is significantly greater than the value after five years ($2552.56). 24.Plan: Determine the present values of the different interest rates and the different time periods. Compare the differences in present values. Execute: 10,000 1.0412  6245.97

a. PV 

PV  ?

10,000

Given:

I/Y

4.00%

Solve for PV:

PMT

10,000

Excel Formula PV(0.04,12,0,10000)

6245.97

10,000 1.0820  2145.48

b. PV 

PV  ? 10,000

I/Y

Given: 20

8.00%

Solve for PV:

PMT

10,000

Excel Formula

PV(0.08,20,0,10000)

2145.48

10,000 1.026  8879.71

c. PV 

PV  ?

10,000

Given:

I/Y

2.00%

Solve for PV:

25.

PMT

10,000

Excel Formula PV(0.02,6,0,10000)

8879.71

Plan: You are being offered a choice between $5000 today and $10,000 in 10 years. One way to evaluate this decision is to determine how much the $10,000 in 10 years is worth today. In this way we can compare the $5000 today against the present value of the $10,000 in 10 years. 10,000 1.0710  5083.49

Execute: PV 

PV  ?

Given:

10,000

I/Y

7.00%

Solve for PV:

PMT

10,000

5083.49

Excel Formula PV(0.07,10,0,10000)

Evaluate: The 10,000 in 10 years is worth $5083.49 today. It is preferable to the $5000 payment today because it is worth more. 26.

FV  $1000 1.025  $500 1.025  $1563.13 2

27.Plan: What your aunt and uncle want to know is, what is the present value of $100,000 in six years, discounted at 4% annually? Most likely they do not think of this decision in these terms and do not use or understand this terminology. 100,000 1.046  79,031.45

Execute: PV 

PV  ?

100,000

Given:

I/Y

4.00%

Solve for PV:

PMT

100,000

Excel Formula PV(0.04,6,0,100000)

79,031.45

Evaluate: Your aunt and uncle would have to invest $79,031.45 today at 4% compounded annual interest for it to grow in six years into the $100,000 they will need to finance your cousin’s education. 28.Plan: Your mom is being offered a choice in how she will take her retirement benefit: either $250,000 today or $350,000 in five years. If mom wants the alternative that is going to give her the most wealth, then she should take the alternative with the highest net present value. Your job is to determine the present values of the $350,000 in five years at different interest rates. Execute: 0

PV  ?

350,000

350,000 1.05  350,000

PV 

I/Y

0.00%

Given: Solve for PV: b.

PMT

350,000

Excel Formula PV(0,5,0,350000)

350,000.00

350,000 1.085  238,204

PV 

Given:

I/Y

8.00%

Solve for PV:

PMT

350,000

Excel Formula PV(0.08,5,0,350000)

238,204.12

350,000 1.25  140,657

PV 

Given: Solve for PV:

I/Y

20.00%

PMT

350,000

Excel Formula PV(0.2,5,0,350000)

140,657.15

Evaluate: a. If the interest rate is 0%, an unlikely situation, then your mom should take the 350,000 in five years. If she takes the $250,000 today and invests it at 0% for five years, she will have $250,000 in five years. Clearly, $350,000 is better in five years than $250,000. b. If the interest rate is 8%, she should take the $250,000 today. c. If the interest rate is 20%, she should take the $250,000 today. 29.

Plan: You are being offered a choice between $100,000 today or $94,000 today plus $10,000 in three years. You need to discount the $10,000 at 8% to determine what it is worth now and then add that value to the $94,000 to be able to compare the second option against the first. Evaluate:

PV 

1  r 



10000

1.08

 7938.32

So the second option has a total value to you today (present value) of $94,000 + $7,938.22 = $101,938.32. You should take that option because it is more than the $100,000 offered in the first option.

30.

a. 5000(1.09)

 186,587.66

b. 5000(1.09)

 78,816.64

*31.Plan: Parts (a) and (b) ask for future values, so we will use Equation 3.1 to calculate those answers. Part (c) asks for the starting amount, which we can treat as a present value of the FV of $3996 in the account now. We will use Equation 3.2 to discount that value back to the beginning. a. FV  3996(1.08)7  6848.44 You would have $6858.44 at age 25, which is seven years from today. 18

FV  ?

3996

Given:

I/Y

PMT

8.00%

3,996

Solve for FV: b.

Excel Formula

(6848.44)

FV(0.08,7,0,3996)

FV  3996(1.08)47  148,779 18

3996

FV ?

You would have $148,799 at age 65, which is 47 years from today.

Given: Solve for FV:

I/Y

PMT

8.00%

3996

Excel Formula

(148,779.12)

FV(0.08,47,0,3996)

3996 1.0818  1000

c. PV 

PV  ?

3996

Your grandfather invested $1000, 18 years ago, to create the fund you have today.

Given:

I/Y

8.00%

Solve for PV:

PMT

3996

1000.00

Excel Formula PV(0.08,18,0,3996)

Evaluate: Given time value of money tools, you can calculate how much your grandfather invested in the past (using the PV equation) and how much the amount in your account will grow to at various dates in time (future value). *32.

Excel Solution Plan: You want to purchase keyboards for your firm. Your decision is made more complex because you have two suppliers and each is offering you very different prices and payment options. Your decision is further complicated because you do not want to spend any of the firm’s money for keyboards today. First, we calculate the cost in today’s dollars of the cost of each keyboard supplier’s proposal. You will purchase the keyboards from the supplier who will charge you the lowest price in today’s dollars. Once you determine which supplier’s offer you will accept, then you can determine how to finance the purchase. Execute:

10,000 Supplier 1: PV  100,000  $10   $194,339.62 Costs 1.06 10,000 Supplier 2: PV  21   $198,113.21 Costs 1.06

Evaluate: Costs are lower under the first supplier’s offer, so it is a better choice. You will purchase the keyboards from the first supplier. The firm can borrow $100,000 at 6% from a bank for one year to make the initial payment to the first supplier. One year later, the firm will pay back the bank $106,000 ($100,000  1.06) and the first supplier $100,000 ($10  10,000), for a total of $206,000. This amount is less than the $210,000 ($21  10,000) the second supplier asked for in year one.

Chapter 4 The Time Value of Money Note:

All problems in this chapter are available in MyLab Finance. An asterisk (*) indicates problems with a higher level of difficulty.

Editor’s Note: As we move forward to more complex financial analysis, the student will notice that some problems may contain a large amount of data from different time periods that require more complicated and intensive analysis. Modern information technology has

evolved in the form of financial calculators with built-in analysis functions and Time Value of Money functions that are built into computer-based electronic spreadsheet software such as Excel. The solutions for many data-intensive and computationally intensive problems will be presented in formula form with a solution, as well as with the appropriate financial calculator commands and Excel functions to produce the same correct answer. In this way, it is our expectation that students will develop proficiency in solving financial analysis problems by mathematical calculation as well as by using financial calculators and electronic spreadsheets. 1.

a. Year

PV 

10 1.10



1.10 



1.10 



1.10 



1.10 5

 106.53

b. Year

 120.92

PV 

50 1.10



1.10 



1.10 



1.10 



1.10 5

c. The present value is different because the timing of the cash flows is different. In the second set, you get the larger cash flows earlier, so it is more valuable to you and shows up as a higher PV. 2.

PV 

Year

100

–100

200

–200

100 1.15



100

1.15



200

1.15



200

1.154

 28.49

PV 

Year

10000

–2000

–3000

–3500

–3975

2,000 3,000 3,500 3,975     10,000.12 , so yes, the PV of your 2 3 1.085 1.085 1.085 1.0854

payments would just cover the loan amount (actually exceeding it by 12 cents). Thus, the maximum you could borrow would be $10,000.12. 4. Month

500

550

600

650

700

750

800

850

PV 

500 1.015



550

1.015



600

1.015



650

1.015 



700

1.015 



750

1.015 



800

1.015 



850

1.015 

 5, 023.75

Yes, the PV of your payments will exceed the loan balance. This means that your planned payments will be enough to pay off your credit card. 5. 0

4000

–1000

From the bank’s perspective, the timeline is the same except all the signs are reversed. 6. 0

–300

From the bank’s perspective, the timeline would be identical except with opposite signs. 7.

Plan: Draw the timeline and then compute the FV of these two cash flows. Execute: Timeline (since we are computing the future value of the account, we will treat the cash flows as positive—going into the account):

500

300

FV  ? FV  $500(1.03)  $300  $815 Evaluate: The timeline helps us organize our work so that we get the number of periods of compounding correct. The first cash flow will have one year of compounding, but the second cash flow will be deposited at the end of period 1, so it receives no compounding. 8.

Editor’s Note: In several previous problems we used a financial calculator to solve a time value of money problem. Problems could be solved quickly and easily by manipulating the N, I/Y, PV, PMT, and FV keys. In each of these problems there was a series of payments, of equal amount, over time; that is, an annuity. All you had to do to input this series was enter the payment (PMT) and the number of payment (N). Many financial analysis problems involve a series of equal payments, but others involve a series of unequal payments. A financial calculator can be used to evaluate an unequal series of cash flows (using the cash flow, CF, key), but the process is cumbersome because each cash flow must be entered individually. I urge each student to study the Chapter 4 Appendix: ―Using a Financial Calculator,‖ as well as instructional materials that are produced by the manufacturer of the financial calculator. Here we will solve a problem with uneven cash flows, mathematically and with a financial calculator. Plan: It is wonderful that you will receive this windfall from your investment in your friend’s business. Since the cash flow payments to you are of different amounts and paid over three years, there are different ways in which you can think about how much money you are receiving. Execute: a.

10,000 20,000 30,000   1.035 1.0352 1.0353  9662  18,670  27,058  55,390

PV 

10,000

20,000

30,000

The Texas Instruments BA II PLUS calculator has a cash flow worksheet accessed with the CF key. To clear all previous values that might be stored in the calculator, press the CF, second, and CE/C buttons. The screen should show CFo  asking for the cash flow at time 0, which in this problem is 0. Press 0, then press enter, and then the down key button. The screen should show CO1 asking for the cash flow at time 1, which in this problem is 10,000. Input 10,000 followed by the enter key, followed by the down button. The screen should show FO1  1.0 asking for the frequency of this cash flow. Since it occurs only once, it is correct, and we press the down key. The screen now has CO2 asking for the time-2 cash flow, which is 20,000, which we input, followed by the enter key and the down key. The screen now has FO2  1.0, which is correct. Enter the down key, which asks for the third cash flow, which is 30,000. Input 30,000, followed by the enter and down keys. Now press the NPV key and it will display I  asking for the interest rate, which is 3.5. Input 3.5, press the enter key and press the down key, and the screen will display NPV Then, press the CPT button and the screen should display 55,390.33, the net present value. b.

FV  55,390  1.0353  61, 412

10,000

20,000

30,000

Evaluate: You may ask, ―How much better off am I because of this windfall?‖ There are several answers to this question. The value today (i.e., the present value) of the cash you will receive over three years is $55,390. If you decide to reinvest the cash flows as you receive them, then, in three years you will have $61,412 (i.e., future value) from your windfall. 9.

Plan: Use Equation 4.3 to compute the PV of this stream of cash flows and then use Equation 4.1 to compute the FV of that present value. To answer part (c), you need to

track the new deposit made each year along with the interest on the deposits already in the bank. Execute: a. and b.

PV 

100 (1.08)



100 (1.08)



100 (1.08)3

 257.71

FV  257.71(1.08)3  324.64

Year 1: 100 Year 2: 100 1.08   100  208 1

Year 3: 208(1.08)1  100  324.64

Evaluate: By using the PV and FV tools, we are able to keep track of our balance as well as quickly calculate the balance at the end. Whether we compute it step by step as in part (c) or directly as in part (b), the answer is the same.

10.

Plan: First, create a timeline to understand when the cash flows are occurring. 0

1000

Second, calculate the present value of the cash flows. Once you know the present value of the cash flows, compute the future value (of this present value) at date 3. Execute: PV 

1000



1000 2



1000

1.05 1.05 1.053  952  907  864  2723

Given:

I/Y

5.00%

Solve for PV:

PMT

1000

Excel Formula

PV(0.05,3,1000,0)

2723.25

FV3  2723  1.053  3152

Given: Solve for FV:

I/Y

PMT

5.00%

2723.43

Excel Formula

3152.71

FV(0.05,3,0,2723.43)

Evaluate: Because of the bank’s offer, you now have two choices as to how you will repay this loan. Either you will pay the bank $1000 per year for the next three years as originally promised, or you can decide to skip the three annual payments of $1000 and pay $3152 in year 3. You now have the information to make your decision. 11.

a. The FV of $100,000 invested for 35 years at 9% is $100,000 × (1.09)35 = $2,041,397.

b. The FV of $100,000 invested for 25 years at 9% is $100,000 × (1.09)25 = $862,308. c. The difference is so large because of the effect of compounding, which is exponential. In the scenario where you invest earlier, those 10 years are critical to the compound growth you achieve on your investment. 12.

Plan: This scholarship is a perpetuity. The cash flow is $10,000 and the discount rate is 7%. We can use Equation 4.4 to solve for the PV, which is the amount you need to endow.

Execute: Timeline: 0

…

10,000

…

PV  10,000/0.07  $142,857.14 Evaluate: With a donation of $142,857.14 today and 7% interest, the university can withdraw the interest every year ($10,000) and leave the endowment intact to generate the next year’s $10,000. It can keep doing this forever. 13.

a. PV = 100,000, r = 0.09, n = 35 b. PV = 100,000, r = 0.09, n = 25 c. The large difference is due to the effect of compounding. Starting just 10 years later has a huge effect on your ability earn interest on interest and exponentially grow your savings. Plan: This is a deferred perpetuity. Here is the timeline: 0

…

11 …

…

10,000

Do this in two steps: 1.

Calculate the value of the perpetuity in year 9, when it will start in only one year (we already did this in Problem 12).

Discount that value back to the present.

Execute: The value in year 9 is 10,000/0.07  $142,857.14. The value today is

142857.14  77704.82. 1.079

Evaluate: Because your endowment will have 10 years to earn interest before making its first payment, you can endow the scholarship for much less. The value of your endowment must reach $142,857.14 the year before it starts (in nine years). If you donate $77,704.82 today, it will grow at 7% interest for nine years, just reaching $142,857.14 one year before the first payment.

14.

The timeline for this investment is 0

100

a. The value of the bond is equal to the present value of the cash flows. By the perpetuity formula, which assumes the first payment is at period 1, the value of the bond is PV 

100

0.04  £2500

b. The value of the bond is equal to the present value of the cash flows. The first payment will be received at time zero. The cash flows are the perpetuity plus the payment that will be received immediately.

PV 

100

 100 0.04  £2600

15.

The value of this opportunity is equal to present value of the cash flows. By the perpetuity formula, the value is

1, 000  r 0.05  $20, 000

PV 

16.

PV 

17.

CF CF , so 100, 000  , so CF  100, 000(0.04)  4, 000 0.04 r

Because your $100,000 donation will grow at a rate of 4% per year until the first payment, the scholarship payments will be larger. To solve this problem, you need to know how much your donation will have grown one year before the first payment. This is because the perpetuity formula always takes the value one period before the first cash flow. The timeline for this scholarship is 0

Value in year 9 = 100,000(1.04)9 = $142,331.18

PV 

18.

CF r

, so value in year 9  142, 331.18 

CF10 0.04

, so CF10  142,331.18(0.04)  5, 693.25

Plan: Draw the timeline of the cash flows for the investment opportunity. Compute the NPV of the investment opportunity at 7% interest per year to determine its value. 0

100

1000

Execute: The cash flows are a 100-year annuity, so by the annuity formula,

PV 

1000



1

0.07 1.07100  14, 269.25

I/Y

Given: 100

7.00%

Solve for PV:



PMT

1000

(14,269.25)

Excel Formula PV(0.07,100,1000,0)

Evaluate: The PV of $1000 to be paid every year for 100 years discounted to the present at 7% is $14,269.25.

19.

Plan: Prepare a timeline of your grandmother’s deposits. 0

1000

The deposits are an 18-year annuity. Use Equation 4.6 to calculate the future value of the deposits.

Execute:

FV  C 

((1  r ) N  1)  1000

((1.03)18  1)  23, 414.43

.03

I/Y

PMT

Given: 18

3.00%

1000

Solve for FV:

Excel Formula

(23,414.43)

FV(0.03,18,1000,0)

At age 18 you will have $23,414.43 in your account.

Evaluate:

The interest on the deposits and interest on that interest adds more than $5414 to the account. 20.

a. 0

First, we need to calculate the PV of $160,000 in 18 years. 160,000 (1.08)18  40,039.84

PV 

I/Y

Given: 18

8.00%

Solve for PV:

PMT

160,000

Excel Formula

PV(0.08,18,0,160000)

(40,039.84)

In order for your parents to have $160,000 in your university account by your 18th birthday, the 18-year annuity must have a PV of $40,039.84. Solving for the annuity payments, 40,039.84 1  1  1   0.08  1.0818   $4272.33

C

which must be saved each year to reach the goal. N

I/Y

Given: 18

8.00%

40,039.84

Solve for PMT:

PMT

Excel Formula

0 (4,272)

PMT(0.08,18,40039.84,0)

b. First, we need to calculate the PV of $200,000 in 18 years. 200,000 (1.08)18  50,049.81

PV 

I/Y

Given: 18

8.00%

Solve for PV:

PMT

Excel Formula

200,000 PV(0.08,18,0,200000)

(50,049.81)

In order for your parents to have $200,000 in your university account by your 18th birthday, the 18-year annuity must have a PV of $50,049.81. Solving for the annuity payments, $50,049.81 1  1  1   0.08  1.0818   $5340.42

C

which must be saved each year to reach the goal. N

I/Y

PMT

Given: 18.00 0.08 50,049.81

0.00

Solve for PMT:

*21.

Excel Formula

PMT(0.08,18,50049.81,0)

5340.42

Plan: a. Draw the timeline of the cash flows for the loan. 1

5000

To pay off the loan, you must repay the remaining balance, which is equal to the present value of the remaining payments. The remaining payments are a four-year annuity, so:

b. 4

5000 Execute:

5000  1  1   0.06  1.064   17,325.53

PV 

Given:

I/Y

6.00%

Solve for PV: b.

PV 

PMT

5000

Excel Formula

PV(0.06,4,5000,0)

(17,325.53)

5000

1.06  4716.98

Evaluate: To pay off the loan after owning the vehicle for one year will require $17,325.53. To pay off the loan after owning the vehicle for four years will require $4716.98. 22.

Plan: This is a deferred annuity. The cash flow timeline is 0

1…

… 17

18 …

… 21

0…

…0

100,000 …

… 100,000

Calculate the value of the annuity in year 17, one period before it starts, using Equation 4.5, and then discount that value back to the present using Equation 4.2.

The value of the annuity in year 17, one period before it is to start, is

PV 

CF  1  100,000  1  1  1  331,212.68 r  (1  r )n  0.08  (1.08)4 

To get its value today, we need to discount that lump sum amount back 17 years to the present:

331, 212.68 (1.08)17

 $89,516.50

Evaluate:

Even though the cash flows are a little unusual (an annuity starting well into the future), we can still value them by combining the PV of annuity and PV of a FV tool. If we invest $89,516.50 today at an interest rate of 8%, it will grow to be enough to fund an annuity of $100,000 per year by the time it is needed for university expenses.

23.

Excel Solution

Plan: This is a deferred annuity. The cash flow timeline is 0

1…

… 44

45 …

0…

…0

40,000 …

40,000

Calculate the value of the annuity in year 44, one period before it starts, using Equation 4.5 and then discount that value back to the present using Equation 4.2.

Execute:

The value of the annuity in year 44, one period before it is to start, is

PV 

CF  1 1   40,000  1  1  377,865.94   n  r  r (1  r )  0.07  (1.07)16 

To get its value today, we need to discount that lump sum amount back 44 years to the present:

377,865.94 (1.07) 44

 $19, 250.92

Evaluate:

Even though the cash flows are a little unusual (an annuity starting well into the future), we can still value them by combining the PV of annuity and PV of a FV tool. The total value to you today of CPP’s promise is less than $20,000.

*24.

Excel Solution Plan: Clearly, Mr. Rodriguez’s contract is complex, calling for payments over many years.

Assume that an appropriate discount rate for A-Rod to apply to the contract payments is 7% per year. a. Calculate the true, promised payments under this contract, including the deferred payments with interest. b. Draw a timeline of all of the payments. c. Calculate the present value of the contract. d. Compare the present value of the contract to the quoted value of $252 million. What explains the difference?

Execute: Determine the PV of each of the promised payments discounted to the present at 7%. 2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

$18M

19M

21M

19M

23M

27M

2011

2012

2013

2014

2015

2016

2017

2018

2019

2020

6.7196 5.3757 4.0317 4.0317 4.0317 4.0317 4.0317 4.0317 4.0317 4.0317 M M M M M M M M M M The PV of the promised cash flows is $165.77 million.

Evaluate: The PV of the contract is much less than $252 million. The $252 million value does not discount the future cash flows or adjust deferred payments for accrued interest.

*25.

Excel Solution a. 0

5000

The amount in the retirement account in 43 years would be

FV43 

5000

((1.10) 43  1) 0.10  $2,962, 003.46

Given: Solve for FV:

I/Y

PMT

10.00%

0.00

5000

Excel Formula

2,962,003.46

FV(0.1,43,5000,0)

b. To solve for the lump sum amount today, find the PV of the $2,962,003.46. 2,962,003.46 (1.10)43  $49,169.99

PV 

I/Y

Given: 43 10.00% Solve for PV:

PMT

Excel Formula

2,962,003 PV(0.1,43,0,2962003.46)

(49,169.99)

c. 0

2,962,003.46

–C

Solve for the annuity cash flow that, after 20 years, exactly equals the starting value of the account. 2,962,003.46 1  1  1   0.10  1.1020   347,915.81

C

I/Y

PMT

Given: 20.00 0.10 2,962,003.46 Solve for PMT:

Excel Formula

0.00 347,915.81

PMT(0.1,20,2962003.46,0)

d. 0

2,962,003.46

–300,000

We want to solve for N, which is the length of time in which the PV of annual payments of $300,000 will equal $2,962,003.46. Setting up the PV of an annuity formula and solving for N: 300,000  1  1    2,962,003.46 0.10  1.10 N   1  1   2,962,003.46  0.10  0.9873345   300,000  1.10 N  1  1  0.9873345  0.0126655 1.10 N 1.10 N  78.95456 Log(78.95456) N  45.84 Log(1.10)

Or simply use your financial calculator to get the same solution quickly.

e. If we can only invest $1000 per year, then set up the PV formula using $1 million as the FV and $1000 as the annuity payment. 1000 

1  1    1,000,000 r  (1  r ) 43 

To solve for r, we can either guess and then interpolate repeatedly (which is definitely not recommended), use a financial calculator, or use the RATE function in Excel. You can check and see that r  11.74291% solves this equation. So the required rate of return must be 11.74291%.

I/Y

Given: 43 Solve for Rate:

11.74291%

PMT

0.00

1000

1,000,000

Excel Formula

RATE(43,1000,0,1000000)

26.

Plan: This problem is asking us to solve for the rate of return (r). Because there are no recurring payments, we can use Equation 4.1 to represent the problem, and then just solve algebraically for r. We have FV  100, PV  50, n  10. Execute: FV PV(1 r)n 1 10

 100  100  50(1 r) , so r     1  0.072, or 7.2%  50  10

Evaluate: The implicit return we earned on the savings bond was 7.2%. Our money doubled in 10 years, which by the Rule of 72 meant that we earned about 72/10 = 7.2% and our calculation confirmed that. 27.

Plan: This problem is again asking us to solve for r. We will represent the investment with Equation 4.1 and solve for r. We have PV  1000, FV  5000, n  10. The second part of the problem asks us to change the rate of return going forward and calculate the FV in another 10 years. Execute: a. FV=PV(1 + r)n 1 10

 5000  5000  1000(1 r) , so r     1  0.1746, or 17.46%  1000  10

b. FV  5000(1.15)10 $20,227.79 28.

Plan: Draw a timeline and determine the internal rate of return (IRR) of the investment. Execute: 0

–5000

6000

IRR is the r that solves 6000 1 r

 5000 

6000 5000

 1  20%

Evaluate: You are making a 20% IRR on this investment.

29.

Plan: Draw a timeline to demonstrate when the cash flows will occur. Then, solve the problem to determine the payments you will receive. Execute: 0

–1000

P

C r

 C  P  r  1000  0.05  $50

Evaluate: You will receive $50 per year into perpetuity. 30.

Plan: Draw a timeline to determine when the cash flows occur. Solve the problem to determine the annual payments. Timeline (from the perspective of the bank): Execute: 0

–300,000

300, 000

C

1 0.07



1

1 1.07 30



 $24,176

which is the annual payment. N Given: 30 Solve for PMT:

I/Y

PMT

7.00% 300,000.00

Excel Formula

$24,175.92

PMT(0.07,30,300000,0)

Evaluate: You will have to pay the bank $24,176 per year for 30 years in mortgage payments.

31.

Plan: Draw a timeline and determine the interest rate you are paying. Execute: 0

–32,500

10,000

The PV of the car payments is a four-year annuity: PV 

10, 000  r

1   1  4    1  r  

Setting the PV of the cash flow stream equal to the amount borrowed (the value of the car) you get: 10, 000  r

1   32,500  1  4   1  r    

Unfortunately, you cannot solve for r algebraically. You either need to guess and interpolate repeatedly (not recommended) or use your financial calculator or Excel’s RATE function. N Given:

I/Y

Solve for Rate:

PMT

32,500.00

10000

Excel Formula RATE(4,10000,32500,0)

8.86%

You can check and see that r  8.85581% solves this equation. So the rate is 8.86%. Evaluate: You are being charged 8.86% annual rate for the loan. *32.

Excel Solution Plan: Draw a timeline and solve the problem for the breakeven number of time periods. Execute: 0

–200,000

25,000

She breaks even when the PV of the cash inflows is $200,000. The value of n that solves this is

25, 000 

 1  200, 000  1  N  0.05  1.05   1 1

1.05 



200, 000  0.05

 0.6  1.05   N

1.05 

 0.4

25, 000

0.6

 1    0.6 

log 1.05   log  N

N log 1.05    log  0.6  N

 log  0.6  log 1.05 

 10.4698

Rather than doing the math, use your financial calculator or Excel’s NPER function. N Given: Solve for Rate:

I/Y

PMT

200000

25000

Excel Formula NPER(.05,25000,200000, 0)

10.4698

Evaluate: So if she lives 10.4698 or more years she comes out ahead. *33.

Excel Solution

Plan: Draw a timeline to determine when the cash flows occur. Timeline (where X is the balloon payment): 0

–300,000

23,500

23,500 X

Note that the PV of the loan payments must be equal to the amount borrowed. Execute:

300,000 

23,500 0.07



1

1 1.07





X (1.07)30

Solving for X:  

X  300, 000 

1  30 1   (1.07) 0.07  1.0730  

23,500 

 $63,848

I/Y

Given: 30

7.00%

Solve for PV:

PMT

23,500

Excel Formula

PV(0.07,30,23500,0)

291,612.47

The present value of the annuity is $291,612.47, which is $8387.53 less than the $300,000.00. To make up for this shortfall with a balloon payment in year 30 would require a payment of $63,848.02.

Given:

I/Y

PMT

7.00%

8387.53

Solve for FV:

Excel Formula

(63,848.02 )

FV(0.07,30,0,8387.53)

Evaluate: At the end of 30 years, you would have to make a $63,848 single (balloon) payment to the bank.

*34.

Plan: Draw a timeline to demonstrate when the cash flows occur. We know that you intend to fund your retirement with a series of annuity payments and the future value of that annuity is $2 million. 22

Execute: FV  $2 million. The PV of the cash flows must equal the PV of $2 million in 43 years. The cash flows consist of a 43-year annuity, plus the contribution today, so the PV is PV 

C  1  1 C 43  0.05  1.05 

The PV of $2 million in 43 years is

2,000,000 (1.05)43 N

I/Y

Given: 43

5.00%

Solve for PV:

 $245, 408.80

PMT

2,000,000

(245,408.80)

Excel Formula

PV(0.05,43,0,2000000)

Setting these equal gives

C  1  1    C  245,408.80 0.05  (1.05)43  245,408.80 C   $13,232.50 1  1  1   1 0.05  (1.05)43  We need $245,408.80 today to have $2,000,000 in 43 years. If we do not have $245,408.80 today, but wish to make 44 equal payments (the first payment is today, making the payments an annuity due) then the relevant Excel command is PMT(rate,nper,pv,(fv),type PMT(.05,44,245,408.80,0,1)  13,232.50 Type is set equal to 1 for an annuity due as opposed to an ordinary annuity. Evaluate: You would have to put aside $13,232.50 annually to have the $2 million you wish to have in retirement. 35.

Excel Solution Plan: This problem is asking you to solve for n. You can do this mathematically using logs, or with a financial calculator or Excel. Because the problem happens to be asking how long it will take our money to double, we can estimate the answer using the Rule of 72: 72/10  7.2, so the answer will be approximately 7.2 years.

Execute:

N

 20000    10000   7.27

ln 

ln 1.10 

Using a financial calculator or Excel:

10000

20000

7.27 Excel Formula: NPER(RATE,PMT, PV, FV)  NPER(0.10,0,–10000,20000) Evaluate: If you can earn 10% per year on the $10,000, it will double to $20,000 in 7.27 years. 36.

Plan: Draw the timeline and determine the interest rate the bank is paying you.

Execute: 0

–1000

100

The payments are a perpetuity, so PV  100 . r

Setting the NPV of the cash flow stream equal to 0 and solving for r gives the IRR: NPV  0 

100 r

 1000  r 

100 1000

 10%

So the IRR is 10%. Evaluate: The bank is paying you 10% on your deposit. *37.

Excel Solution

Plan: Draw a timeline to show when the cash flows occur. Then determine how long the plant will be in production. Also, estimate the NPV of the project and, hence, whether or not it should be built.

Execute: 0

–10,000,000

1,000,000 –

1,000,000

50,000

50,000(1.05)

50,000(1.05)N – 1

The plant will shut down when 1, 000, 000  50, 000(1.05) N 1  0 (1.05) N 1 

1, 000, 000 50, 000

 20

( N  1) log(1.05)  log(20) N 

log(20) log(1.05)

 1  62.4

So the last year of production will be in year 62. We now build an Excel spreadsheet with the cash flows to the 62 years. A

1.05

0.06

4 5 6T 7 8

10000000

1000000 1000000 1000000 1000000 1000000 1000000 1000000 1000000

50000

52500

55125 57881.3 60775.3 889485 933959 980657

($10,000,000.00) 950000

947500

944875 942118.8 939224.7 110515 66040.71 19342.74

11 12 NPV

$3,995,073.97 EXCEL NPV FORMULA B10NPV(C3,C11:BL11)

The net present value of the project is computed in cell B12. Evaluate: So, the NPV  13,995, 074  10, 000  $3,995, 074, and you should build it. *38.

Excel Solution

Plan: Draw a timeline to show when the cash flows will occur. Then determine how much you will have to put into the retirement plan annually to meet your goal. Execute: 22

100

–C

100

The PV of the costs must equal the PV of the benefits, so begin by dividing the problem into two parts: the costs and the benefits. Costs: The costs are the contributions, a 43-year annuity with the first payment in one year: PVcos ts 

C  1  1  0.07  (1.07) 43 

Benefits: The benefits are the payouts after retirement, a 35-year annuity paying $100,000 per year with the first payment 44 years from today. The value of this annuity in year 43 is: PV43 

Given: Solve for PV:

I/Y

7.00%

100,000  1  1 0.07  (1.07)35 

PMT

100000

(1,294,767.23)

The value today is just the discounted value in 43 years:

Excel Formula PV(0.07,35,100000,0)

PVbenefits 

PV43 (1.07)43

100,000  1  1 43  0.07(1.07)  (1.07)35   70,581.24 

Since the PV of the costs must equal the PV of the benefits (or equivalently, the NPV of the cash flow must be zero): 70,581.24 

I/Y

Given: 43

7.00%

Solve for PV:

C  1  1   0.07  (1.07)43 

PMT

1,294,767

Excel Formula

PV(0.07,43,0,1294767.23)

(70,581.24)

Solving for C gives 70,581.24  0.07 1   1    (1.07)43   5225.55

C

Given:

I/Y

7.00%

70,581.24

Solve for PMT:

PMT

Excel Formula

0 PMT(0.07,43,70581.24,0)

(5,226)

Evaluate: You will have to invest $5225.55 annually into the retirement plan to meet your goal. 39.

Plan: The bequest is a perpetuity growing at a constant rate. The bequest is identical to a firm that pays a dividend that grows forever at a constant rate. We can use the constant dividend growth model to determine the value of the bequest.

Execute: a. 0

1000

1000(1.08)

1000(1.08)2

Using the formula for the PV of a growing perpetuity gives

 1000  PV     0.12  0.08   25,000

which is the value today of the bequest.

b. 1

1000(1.08)

1000(1.08)2

1000(1.08)3

Using the formula for the PV of a growing perpetuity gives

1000(1.08) 0.12  0.08  27,000

PV 

which is the value of the bequest after the first payment is made. Evaluate: The bequest is worth $25,000 today and will be worth $27,000 in one year’s time. *40.

Plan: The machine will produce a series of savings that are growing at a constant rate. The rate of growth is negative, but the constant growth model can still be used. Execute: The timeline for the saving would look as follows: 0

1000(1 – 0.02) 1000(1 – 0.02)2

1000

We must value a growing perpetuity with a negative growth rate of –0.02: 1000 0.05  ( 0.02)  $14,285.71

PV 

Evaluate: The value of the savings produced by the machine is worth $14,285.71 today.

41.

Plan: Nobel’s bequest is a perpetuity. The total amount is 5  $45,000  $225,000. With a cash flow of $225,000 and an interest rate of 7% per year, we can use Equation 4.4 to solve for the total amount he would need to use to endow the prizes. In part (b), we will need to use the formula for a growing perpetuity (Equation 4.7) to find the new value he would need to leave. Finally, in part (c), we can use the FV equation (Equation 4.1) to solve for the future value his descendants would have had if they had kept the money and invested it at 7% per year. a. In order to endow a perpetuity of $225,000 per year with a 7% interest rate per year, he would need to leave $225,000/0.07  $3,214,285.71.

b. In order to endow a growing perpetuity with an interest rate of 7% and a growth rate of 4% and an initial cash flow of $225,000, he would have to leave CF1 225,000 PV    7,500,000 r  g 0.07  0.04 c. FV PV(1 r)n 7,500,000(1.07)118 $21,996,168,112 Evaluate: The prizes that bear Nobel’s name were very expensive to endow—$3 million was an enormous sum in 1896. However, Nobel’s endowment has been able to generate enough interest each year to fund the prizes, which now have a cash award of approximately $1,500,000 each!

42.

Plan: The drug will produce 17 years of cash flows that will grow at 5% annually. The value of this stream of cash flows today must be determined. We can use the formula for a growing annuity (Equation 4.8) or Excel to solve this. C  2, r  0.10, g  0.05, n  17. 17 1     1.05   Execute: PV  2   1      21.86  0.10  0.05    1.10  

Since the cash flows from this investment will continue for 17 years, we decided to solve for the net present value by using the NPV function in Excel. This is shown below. The 17 cash flows are presented in columns C, D, … S. The initial cash flow of $2M is presented in cell C8 and each subsequent cash flow grows at 5% until $4.365749M is presented in year 17 in cell S8. (Note that columns G through Q are not presented.) The NPV of the project is calculated using the NPV formula NPV(C3,C8:S8) in cell B10. The NPV of the future cash flows is $21.86M. A

1+g

1.05

0.1

4 5 6

7 8

2.1

2.205

2.31525

4.157856

4.365749

9 10

NPV

$21.86

11 EXCEL NPV FORMULA NPV(C3,C8:S8)

12 13

Evaluate: The value today of the cash flows produced by the drug over the next 17 years is $21.86 million. Because the cash flows are expected to grow at a constant rate, we can use the growing annuity formula as a shortcut. 43.

Plan: Your rich aunt is promising you a series of cash flows over the next 20 years. You must determine the value of those cash flows today. This is a growing annuity and we can use Equation 4.8 to solve it, or we can also solve it in Excel. C  5, r  0.03, g  0.05 and N  20.

20 1     1.03   Execute: PV  5   1      79.82  .05  .03    1.05  

Since the cash flows from this investment will continue for 20 years, we decided to solve for the net present value by using the NPV function in Excel. This is shown below. The 20 cash flows are presented in columns C, D, … V. The initial cash flow of $5000 is presented in cell C8 and each subsequent cash flow grows at 3% until $8767.53 is presented in year 20 in cell V8. (Note that columns G through R are not presented.) The NPV of the project is calculated using the NPV formula NPV(C3,C8:V8) in cell B10. The NPV of the future cash flows is $79,824. A

1+g

1.03

0.05

E…

5 6

8.264238

8.51216 5

8.76753

5.15

5.3045

8.023532

9 10

NPV

$79.82

11 12

EXCEL NPV FORMULA NPV(C3,C8:V8)

13 Evaluate: Because your aunt will be increasing her give each year at a constant rate, we can use the growing perpetuity formula as a shortcut to value the stream of cash flows. Her gift is quite generous: it is equivalent to giving you almost $80,000 today! *44.

Excel Solution Plan: Draw a timeline to show when the cash flows will occur. Then determine how much you will have to put into the RRSP annually to meet your goal.

Execute: Age

22 Contributions

3 1

100

100(1.03)

Withdrawals

–C1

–C2

–C43

The value of the contributions must equal the value of the withdrawals, so begin by dividing the problem into two parts: withdrawals and contributions. Withdrawals: The withdrawals are the payouts after retirement, a 35-year growing annuity initially paying $100,000 with the first payment 44 years from today and growing by 3% per year. The value of this growing annuity in 43 years is PVwithdrawals 43 years from now 

$100, 000 

35  1.03      $1,841,113.43  1.07  

1  

0.07  0.03 

This PV amount is essentially what has to be accumulated by the end of 43 years. We can use this value as the future value of the contributions. Contributions: The contributions, a 43-year growing annuity with the first payment in one year with future value of $1,841,113.43. $1,841,113.43 

1.07  1.05  0.07  0.05 C1

 C1  $1,841,113.43 

 0.07  0.05 

1.07  1.05  43

 $3,611.91

Evaluate: You will have to invest $3611.91 in one year and grow these contributions by 5% per year until retirement in order to fund your retirement withdrawals and meet your goals.

Chapter 5 Interest Rates

Note: All problems in this chapter are available in MyLab Finance. An asterisk (*) indicates problems with a higher level of difficulty. 1. 6  1 of 2 years, using our rule, (1  0.2)1/ 4  1.0466. a. Since 6 months is 24 4

So the equivalent 6-month rate is 4.66%. b. Since 1 year is half of 2 years, (1.2)1/2  1.0954, so the equivalent 1-year rate is 9.54%. c. Since 1 month is 1/24 of 2 years, using our rule, (1  0.2)1/24  1.00763, so the equivalent 1-month rate is 0.763%. 2.

Excel Solution If you deposit $1 into a bank account that pays 5% per year for 3 years, you will have (1.05)3  1.15763 after 3 years. a. If the account pays 2 1/2% per 6 months, then you will have (1.025)6  1.15969 after 3 years, so you prefer 2 1/2% every 6 months. b. If the account pays 7 1/2% per 18 months, then you will have (1.075) 2  1.15563 after 3 years, so you prefer 5% per year. c. If the account pays 1/2% per month, then you will have (1.005)36  1.19668 after 3 years, so you prefer 1/2% every month.

Excel Solution Plan: Draw a timeline to fully understand the timing of the cash flows. Determine the present value of the bonus payments. Execute: 0

70,000

Because (1.06)7  1.50363, the equivalent discount rate for a 7-year period is 50.363%.

Using the annuity formula: PV 

Given:

70, 000 

1   1  (1.50363) 6   $126, 964 

0.50363 

I/Y

50.36%

Solve for PV:

PMT

70000

(126,964.34)

Excel Formula PV(0.503630258991361, 6,70000,0)

Evaluate: The present value of the bonus payments is $126,964. 4.

Plan: Determine the EAR for each investment option. Execute: For $1 invested in an account with 10% APR with monthly compounding, you will have 12

 0.1   1    $1.10471  12  So the EAR is 10.471%. For $1 invested in an account with 10% APR with annual compounding, you will have (1  0.1)  $1.10.

So the EAR is 10%. For $1 invested in an account with 9% APR with daily compounding, you will have  0.09  1    365 

365

 1.09416

So the EAR is 9.416%. Evaluate: One dollar invested at 10% APR compounded monthly will grow to $1.10471 in one year. This is greater than the values for the other two investments and is, therefore, superior. 5.

Using the formula for converting from an EAR to an APR quote k

 APR  1    1.05 k   Solving for the APR,

APR  ((1.05)1/k  1)k With annual payments k  1, so APR  5%. With semi-annual payments k  2, so APR  4.939%.

With monthly payments k  12, so APR  4.889%.

Excel Solution Plan: Determine the present value of the annuity. Execute: Using the PV of an annuity formula with N  10 payments and C  $100 with r  4.067% per 6-month interval, since there is an 8% APR with monthly compounding: 8%/12  0.6667% per month, or (1.006667)6  1  4.067% per 6 months. 1 1   1   0.04067  1.0406710   $808.39

PV  100 

Given:

I/Y

4.067%

Solve for PV:

PMT

100

Excel Formula PV(0.0406726223013221,10, 100,0)

(808.38)

Evaluate: The PV of the annuity is $808.39. 7.

Excel Solution Plan: Draw a timeline to demonstrate when the tuition payments will be needed. Then calculate the PV of the tuition payments. Execute: 0

1 2

10,000

The 4% APR (semi-annual compounding) implies a semi-annual discount rate of 4%  2%. 2 So PV 

10, 000 

1  1  8  0.02  (1.02) 

 $73, 254.81

Given: Solve for PV:

I/Y

2.000%

PMT

10000

(73,254.81)

Excel Formula PV(0.02,8,10000,0)

Evaluate: You will have to deposit $73,254.81 in the bank today in order to be able to make the tuition payments over the next four years. 8.

Plan: You need to convert the APR to something to be used in your formulas, and to an effective annual rate (EAR). Remember, you need the discount rate to match the time between cash flows when using the annuity formulas. Execute: (a)

Since the payments will be monthly, you need to convert the APR to an effective monthly rate. So, using Equation 5.2, we get 6% per year compounded monthly divided by 12 months per year = 0.5% per month (this is an effective monthly rate).

(b)

To get the EAR, we can use Equation 5.2 followed by Equation 5.1: Equation 5.2: 6%/12 = 0.5% per month effective Equation 5.1: (1.005)12 = 1 + EAR = 1.06167781, so EAR = 6.167781%. Alternatively, you could have used Equation 5.3, which combines Equations 5.1 and 5.2: (1 + 0.06/12)12 = 1 + EAR = 1.06167781, so EAR = 6.167781%.

Evaluate: A car loan rate of 6% APR with monthly compounding costs you more than 6% per year. It is equivalent to 0.5% per month, or 6.167781% per year (EAR). 9.

Plan: It is easiest to compare interest rates if they are all quoted in the same way; using an EAR for comparison purposes accomplishes this goal. When you need to use your formulas to calculate monthly payments, though, you will need the effective monthly rate. Execute: (a)

The credit card rate of 21% APR with daily compounding can be converted to an effective daily rate using Equation 5.2: 21%/365 = 0.05753425%. Then use Equation 5.1 to convert to the EAR: (1.0005753425)365 = 1 + EAR = 1.233603563, so EAR = 23.3603563%. Or use Equation 5.3: (1 + 0.21/365)365 = 1+ EAR = 1.233603563, so EAR = 23.3603563%.

(b)

Use Equation 5.1 to convert from one effective rate to an effective rate over a different time period, noting that one year is the equivalent of two six-month periods. (1.10)2 = 1+ EAR = 1.21, so EAR = 21%.

(c)

Since you will be dealing with monthly payments, you need an effective rate per month. i.

Using the effective daily rate determined from Equation 5.2 in (a) and noting there are 365 days in a year and 12 months in a year, thus, 365/12 days in a

month, we can use Equation 5.1 to calculate the effective monthly rate: (1.0005753425)365/12 = 1+ effective monthly rate = 1.017648901, so the effective monthly rate needed would be 1.7648901%. ii. Using the effective rate per six-months by the parents, we can use Equation 5.1 to calculate the effective monthly rate: (1.10)1/6 = 1+ effective monthly rate = 1.01601187, so the effective monthly rate needed would be 1.601187%. Evaluate: Your parents’ rate is better than financing using your credit card, although maybe you should think about saving and not spending beyond your means! 10.

Plan: It is easiest to compare interest rates if they are all quoted in the same way. Using an effective annual rate (EAR) for comparison purposes accomplishes this goal, but in this case we could also compare using effective monthly rates or compare using effective quarterly rates. Execute: Use Equation 5.1 to convert from the given effective rate to a new rate that can be compared. The results are summarized in the table below.

11.

Plan: Compute the discount rate from the APR formula. Execute: Using the formula for computing the discount rate from an APR quote: Discount Rate  0.05 12  0.0041666667

Evaluate: The interest rate is 0.41666667% per month. 12.

Plan: Draw a timeline for the cash flows. Given that $8000 is the present value of an annuity of payments, determine the amount of the payments. Execute: 0

8000

A 5.99% APR monthly implies a discount rate of .0599

 0.00499167

or 0.499167%.

Using the formula for computing a loan payment, C

8000 1   1  60  0.00499167  (1.00499167)  1

 $154.63

N Given:

I/Y

PMT

0.499%

8,000.0 0

Solve for PMT:

Excel Formula

0 PMT(0.00499166666666667, 60,8000,0)

(154.63)

Evaluate: Your monthly payment for the motorcycle loan is $154.63. 13.

Plan: Draw a timeline to better understand when the cash flows are occurring. Since $150,000 is the present value of an annuity of mortgage payments, determine the amount of the mortgage payments. Execute: 0

360

150,000

–C

The monthly interest rate is

1.0681/12  1  0.5497367% Solve for the annuity payments that give the PV of the mortgage ($150,000). C

$150, 000 1   1  360  0.005497367  (1.005497367)  1

 $957.66

Given: Solve for PMT:

I/Y

360

0.550%

150,000.00

PMT

Excel Formula

0 (957.68)

PMT(0.00549736708252291, 360,150000,0)

Evaluate: The monthly mortgage payment is $957.66.

14.

Plan: The original mortgage had 30  12 = 360 payments. So far, (4  12) +8 = 56 payments have been made, leaving 360 – 56 = 304 payments remaining. The timeline is as follows: 56

360

304

2356

To find out what is owed, compute the PV of the remaining payments using the loan interest rate to calculate the discount rate. Execute: Interest Rate Per Semiannual Period 

6.2

 3.1%

2 1/ 6

Effective Rate Per Month = (1+0.031)

PV 

1=0.5101168%

1   1  $363,514.33  304  0.005101168  (1.005101168)  2356

Given: Solve for PV:

I/Y

304

0.551011 68%

PMT

2356

363,514.33

Excel Formula PV(0.0055101168, 304,-2356,0)

Evaluate: You would owe $363,514.33 on the mortgage today. 15.

Plan: You need to determine the effective monthly interest rate and then determine the original loan payment. Then you can take the PV of the remaining payments to determine the principal amount owed. The cash remaining will be your sale price minus the principal amount still owed. Execute: A 7.75% APR (compounded semi-annually) implies an effective semi-annual rate of 7.75/2 = 3.875%, and this can be converted into an effective monthly rate as follows: (1.03875)1/6  1 = 0.63564616%

Timeline #1: to get payment based on original mortgage 0

360

–800,000

Using the formula for a loan payment, C

800, 000  0.0063564616

  1  1  360    1.0063564616  

 $5,663.87

Now we can compute the PV of the remaining payments to get the principal still owed. The timeline is as follows: Timeline #2: (you have already made 222 payments, or 18.5 years of payments) 222

223

224

225

360

138

5663.87

Using the formula for the PV of an annuity to determine what principal is still owed: PV 

  1  $519,382.89  1  138  0.0063564616  1.0063564616   5, 663.87

(Note, if you used the unrounded payment, the PV would be $519,382.49.) So you would keep $1,000,000  $519,382.89 = $480,617.11. Evaluate: You will have cash left over after paying off your mortgage of $480,617.11. 16.

Plan: First, determine the effective monthly interest rate followed by the monthly payment. For part (a), you can determine the total payments in the year and then compare to how the principal changed. The difference between the total payments and the principal reduction over the year will be the interest paid over the year. For part (b), you will follow a similar process but you will be comparing how the principal changed from the end of 19 years to the end of 20 years.

Execute: APR of 6.5% compounded semi-annually ÷ 2 = 3.25% per semi-annual period. The effective monthly rate is (1 + 0.0325)1/6  1 = 0.53447401%. a. Payment is

$500, 000 1 1    360  0.0053447401  1.0053447401  1

 $3132.01.

Total payments in one year are $3132.01 × 12 = $37,584.12. Loan balance at the end of one year: $3132.01 

1    $494,319 1  348  0.0053447401  1.0053447401  1

Therefore, $500,000 – 494,319 = $5681 in principal is repaid in the first year, and $37,584 – 5681 = $31,903 in interest is paid in the first year. b. Loan balance in 19 years (or 360 – 19×12 = 132 remaining payments) is $3132.01 

1 1    $296,054 1  132  0.0053447401  1.0053447401 

Loan balance in 20 years is $3132.01 

1 1    $276,901 . 1  120  0.0053447401  1.0053447401 

Therefore, $296,054 – 276,901 = $19,153 in principal is repaid, and $37,584 – 19,153 = $18,431 in interest is repaid. Evaluate: In the early part of the mortgage’s life, most of the payments go to interest; however, as the principal is reduced over time, when we get to the later part of a mortgage’s life, the interest is less and more of the payments go toward repaying principal. 17.

Excel Solution Plan: Draw a timeline and compute the sums that are indicated. Execute: a. Timeline: 0

50,000

–C

First, solve for the monthly mortgage payment at 6% APR. The 6% APR implies a monthly rate of 6%  0.50% . 12

C

50, 000 1   1  60  0.005  (1.005) 

 $966.64

N Given:

I/Y

PMT

0.500% 50,000.00

Solve for PMT:

Excel Formula

0 PMT(0.005,60,50000,0)

(966.64)

Each monthly payment is $966.64. After one month, the balance (principal) of the loan will be the PV of the 59 remaining payments. PV 

Given:

I/Y

0.500%

Solve for PV:

966.64 

1  1  $49, 283.36  59  0.005  (1.005) 

PMT

966.64

Excel Formula

PV(0.005,59,966.64,0)

(49,283.36)

Thus, $50,000  49,283.36  $716.64 is amount of the payment that went to paying the principal, while $966.64  716.64  $250 was interest. For the second month, solve for the value of the remaining 58 payments: PV 

Given: Solve for PV:

I/Y

0.500%

966.64 

1  1  $48, 563.14  58  0.005  (1.005) 

PMT

966.64

(48,563.13)

Excel Formula PV(0.005,58,966.64,0)

Thus, $49,283.36  48,563.14  $720.22 is amount of the payment that went to paying the principal, while $966.64 – 720.22  $246.42 was interest. For the first year, solve for the value of the remaining 48 payments: PV 

966.64 

1  1  48  0.005  (1.005) 

 $41,159.84

Thus, $50,000  41,159.84  $8840.16 is the amount of the payment that went to paying the principal, while ($966.64  12)  8840.416  $2759.52 was interest.

b. At the end of year 3, there are 24 payments remaining. The balance of the loan is 966.64  1  1  24  0.005  (1.005)   $21,810.17

PV 

At the end of year 4, there are only 12 payments remaining. The balance of the loan at the end of the 4th year is PV 

966.64 

1   1  (1.005)12  

0.005 

 $11, 231.33

Given:

I/Y

0.500%

Solve for PV:

PMT

966.64

(11,231.32)

Excel Formula PV(0.005,12,966.64,0)

Thus, $21,810.17  11,231.33  $10,578.84 is the amount of the payment that went to paying the principal, while ($966.64  12)  $10,578.84  $1020.84 was interest. Evaluate: A financial analyst can determine the amount of any loan payment. The financial analyst can also determine the outstanding amount on the loan at any time over its life. *18.

If the balance after the 50th payment is $9405.81, then when you make your next payment, one month will have passed and you will owe 0.5% interest on that balance: $9405.81(0.005) = $47.03. Your payments remain constant at $966.64, so $47.03 goes toward interest and $966.64 – $47.03 = $919.61 goes toward principal. When you compare this with the principal and interest breakdown of your earlier payments, you can see that, over time, as you pay down the principal of the loan, less of your payment has to cover interest and more your payment can go toward reducing the principal!

*19.

Plan: Draw a timeline to better understand the timing and amount of the cash flows. Then, determine how much faster the loan would be paid off with a one-time additional $1000 payment. 0

–600

First, calculate the remaining balance on the loan, which is the PV of the remaining payments on the loan.

Execute: The 7% APR implies a monthly rate of 7%

 0.58333%

12 PV 

1   1  36  0.0058333  (1.0058333)  600

 $19, 431.88

Given:

I/Y

0.583%

Solve for PV:

PMT

600

Excel Formula PV(0.00583333333333333,36, 600,0)

(19,431.88)

If you plan on paying an additional $1000 next month, the timeline changes to this: 0

19,431.88

–600

–1000

We want to know how long the payments will last if you include an additional $1000 in next month’s payment. The principal of the loan must equal the PV of the payments made on the loan, so we can solve for the remaining length of time on the loan: 1 1, 000   1   N  0.0058333  (1.0058333)  1.0058333 600 1   18, 437.68  1  N  0.0058333  (1.0058333)  600

19,431.88 

0.179254  1 

(1.0058333) 1 (1.0058333)

 1  0.179254  0.820746

1.0058333  1.218404 N

N

log(1.218404) log(1.0058333)

 33.96

Evaluate: Thus, you reduce the amount of time remaining on the loan by slightly more than two months (from 36 to 34).

20.

Use the annuity formula to calculate the monthly payment. Your monthly rate, based on the APR, is 0.0466/12 = 0.00388333333. PV 

CF  1  1   r  1  r n 

30, 000 

so CF =

I/Y

  1 , 1  120  0.00388333333  1.00388333333  CF

30, 000  0.00388333333

  1 1  120   1.00388333333 

PMT

Given: 120 0.388% 30,000.00

Excel Formula

Solve for PMT:

*21.

 $313.23

(313.23)

PMT(0.0038333333,120,30000,0)

Excel Solution Plan: Draw a timeline to better understand the timing and amount of the cash flows. Then determine how much a one-time additional $100 payment today will reduce your last payment. Execute: We begin with the timeline of our required payments. 0

–500

Let’s compute our remaining balance on the student loan. As we pointed out earlier, the remaining balance equals the present value of the remaining payments. The loan interest rate is 9% APR, or 9%/12  0.75% per month, so the present value of the payments is PV 

1   1  48  0.0075  1.0075  500

 $20, 092.39

Using the annuity spreadsheet to compute the present value, we get the same number: N

I/Y

PMT

0.75%

20,092.39

500

Thus, your remaining balance is $20,092.39. If you prepay an extra $100 today, you will lower your remaining balance to $20,092.39  $100  $19,992.39. Though your balance is reduced, your required monthly payment

does not change. Instead, you will pay off the loan faster; that is, it will reduce the payments you need to make at the very end of the loan. How much smaller will the final payment be? With the extra payment, the timeline changes: 0

19,992.39

–500

–(500 – X)

That is, we will pay it off by paying $500 per month for 47 months, and some smaller amount, $500  X, in the last month. To solve for X, recall that the PV of the remaining cash flows equals the outstanding balance when the loan interest rate is used as the discount rate: 19, 992.39 

1 X   1   48  48 0.0075  (1  0.0075)  1.0075 500

Solving for X gives X

19, 992.39  20, 092.39 

1.0075

X  $143.14

So the final payment will be lower by $143.14. You can also use the annuity spreadsheet to determine this solution. If you prepay $100 today, and make payments of $500 for 48 months, then your final balance at the end will be a credit of $143.14: N

I/Y

PMT

0.75%

19,992.39

500

143.14

The extra payment effectively lets us exchange $100 today for $143.14 in four years. We claimed that the return on this investment should be the loan interest rate. Let’s see if this is the case: $100  (1.0075)

 $143.14, so it is.

Thus, you earn a 9% APR (the rate on the loan). Evaluate: Your last payment will be reduced by $143.14. *22.

Excel Solution Plan: Draw a timeline to better understand the timing and amount of the cash flows. Then, determine how much faster the loan would be paid off if you increase each additional payment by $250.

20,092.39

–750

And we want to determine the number of monthly payments, N, that we will need to make. That is, we need to determine what length an annuity with a monthly payment of $750 has the same present value as the loan balance, using the loan interest rate as the discount rate. As we did in Chapter 4, we set the outstanding balance equal to the present value of the loan payments and solve for N: Execute: 750

1  

  20, 092.39   1 1    20, 092.39  0.0075  0.200924  N  750  1.0075 

0.0075 

1.0075

1 1.0075

 1  0.200924  0.799076

1.0075  1.25145 N

N

Log(1.25145)

 30.02

Log(1.0075)

We can also use the annuity spreadsheet to solve for N: N

I/Y

PMT

30.02

0.75%

20,092.39

750

Evaluate: So by prepaying the loan we will pay off the loan in about 30 months, or 2½ years, rather than the 4 years originally scheduled. Because N of 30.02 is larger than 30, we could either increase the 30th payment by a small amount or make a very small 31st payment. We can use the annuity spreadsheet to determine the remaining balance after 30 payments: N

I/Y

PMT

0.75%

20,092.39

750

13.86

If we make a final payment of $750.00  13.86  $763.86, the loan will be paid off in 30 months.

*23.

Excel Solution Plan: Draw a timeline to determine the amount and timing of the cash flows. Then determine how much quickly the loan would be paid off with payments made every 2 weeks versus every month. Evaluate: From the solution to Problem 13, the monthly payment on the mortgage is $957.66. So, if we make 957.66  $478.83 every two weeks, the timeline is 2

478.83

Now, since there are 26 two-week periods in a year 1/ 26

(1.0680)

=1.0025335

So, the discount rate is 0.25335%. To compute N, we set the PV of the loan payments equal to the outstanding balance 150, 000 

1   1  N  0.0025335  (1.0025335)  478.83

And solve for N N

1   150, 000  0.0025335  0.793604   478.86  1.0025335 

1 

1      0.206396  1.0025335  N

log(0.206396) 1   log    1.0025335 

N Given:

I/Y

 623.63

PMT

0.253% 0.00

478.83

150,000

Excel Formula NPER(0.00253350161 796617,478.83,0,150000)

Solve for NPER: 623.72)

Evaluate: So, it will take 624 payments to pay off the mortgage. Since the payments occur every two weeks, this will take 624  2  1248 weeks, or approximately 24 years. (It is shorter because there are approximately two extra payments every year.) *24.

Excel Solution Plan: Draw a timeline to determine the amount and timing of the cash flows. Then, determine how much more quickly the loan would be paid off with a double payment made January 1. Execute: The principal balance does not matter, so just pick $100,000. Begin by computing the monthly payment. The semi-annual effective rate is 6%, so the effective monthly discount rate is (1+.06)1/6  1 = 0.97587942%.

Timeline #1: 0

360

100,000

C

Execute: Using the formula for the loan payment C

100, 000  0.0097587942 1 1    360   1.0097587942 

Given:

 $1,006.39

I/Y

360

1.000%

100,000.00

Solve for PMT:

PMT

Excel Formula

0 (1,006.39)

PMT(0.0097587942,360,1 100000,0)

Next, we write out the cash flows with the extra payment: Timeline #2:

The cash flow consists of two annuities. First, the original payments. The PV of these payments is PVorg 

N   1   1      0.0097587942   1.0097587942  

1, 006.39

And, second, the extra payment every January 1. There are m such payments, where m is the number of years you keep the loan. (For the moment, we will not worry about the possibility that m is not a whole number.) Since the time period between payments is one year, we first have to compute the discount rate as an effective annual rate:

1.0097587942 12  1  12.36% Note, this is the same as (1.06) 1=12.36% 2

So, the discount rate is 12.36%.

If we use the present value annuity formula on the annual extra payments that begin in month 6 and continue each year following, and we assume there are m of them, we get a present value discounted to month 6 (or 1 year before the first extra payment). Thus, we need to adjust the present value calculated to get the PV at time zero by multiplying by (1.06), recalling that 6% is the effective semi-annual rate.  1  1  m  0.1236  1.1236    1, 006.39  1 so PVextra =  1  .06   1  m  0.1236  1.1236   PVextra, at month  6 

1, 006.39 

To find out how long it will take to repay the loan, we need to determine the number of years until the value of our loan payments has a present value at the loan rate equal to the amount we borrowed. Because the number of monthly payments is N = 12 × m, we can write this as the following expression, which we need to solve for m: 100, 000  PVorg  PVextra 100, 000 

12 m    1 1   1, 006.39   1.06   1  1     m  0.0097587942   1.0097587942   0.1236  1.1236  

1, 006.39

The only way to find m is to iterate (guess) or use Solver on Excel (see spreadsheet solution). The answer is m  19.35 years or approximately 19 years and 4.2 months. Evaluate: This means you could make 19 years and 4 months of payments as outlined on the timeline and then make a final additional payment to pay the mortgage down to zero. Because the mortgage will take about 19 years to pay off this way—which is close to two-thirds of its life of 30 years—your friend is right. *25.

Plan: Determine the principal outstanding on the old mortgage after calculating the effective monthly rate applicable to it. Then, calculate the new payments for the new mortgage after determining its relevant effective monthly interest rate. Then proceed to answer the parts. a. First we calculate the outstanding balance of the mortgage. There are 25 × 12 = 300 months remaining on the loan, so the timeline is Timeline #1: 0

300

1,402

To determine the outstanding balance, we discount at the original rate after converting it to an effective monthly interest rate. The 9% APR with semi-annual

compounding implies an effective semi-annual rate of 9% ÷ 2 = 4.5%. Converting the effective semi-annual rate to an effective monthly rate, we get (1 + 0.045)1/6  1 = 0.73631230% PV 

  1  $169,328.30  1  300  0.0073631230  1.0073631230   1402

Next, we calculate the loan payment on the new mortgage. Timeline #2: 0

360

169,328.30

–C

The effective monthly discount rate on the new loan is the new loan rate: 6.625

 0.55208333%

Using the formula for the loan payment: C

b. C 

169, 328.30  0.0055208333 360   1   1        1.0055208333  

169, 328.30  0.0055208333 300   1   1        1.0055208333  

 $1084.23

 $1156.58

  1  $169, 328.30  N  199.6 months  1  N  0.0055208333  1.0055208333  (you can use trial and error or the annuity calculator to solve for N). PV 

PV 

1402

  1  $205,259.23  1  300  0.0055208333  1.0055208333   1402

 you can keep 205,259.23  169, 328.30  $35,930.93

(Note: results may differ slightly due to rounding.) Evaluate: Any way you look at it, this seems like a good plan. You get to either make smaller payments or keep some cash up front. Plus, you can live in Florida and avoid the Winnipeg winters. 26.

Plan: Use a timeline and financial analysis to determine how much more you could borrow at the lower rate and keep the same payments.

Execute: The discount rate on the original card, 15% APR with monthly compounding, is .15  .0125  1.25% per month. 12

Assuming that your current monthly payment is the interest that accrues, it equals $25, 000 ×

0.15

= $312.50

312.50

This is a perpetuity. So the amount you can borrow at the new interest rate is this cash flow discounted at the new discount rate. The new discount rate is 12  1% . 12 So PV 

312.50 0.01

 $31, 250

Evaluate: So by switching credit cards you are able to spend an extra $31,250  25,000  $6250 . You do not have to pay taxes on this amount of new borrowing, so this is your after-tax benefit of switching cards. 27.

Plan: Draw a timeline to determine the amount and timing of the cash flows. Then determine how much you would owe at the end of 5 years. Execute: a. The payments are established as if the loan will last 15 years. Thus, the ―timeline‖ for determining the payments looks like this: 0

180

500,000

–C

C

The discount rate is 9%/12  0.75% per month. C

500, 000  0.0075 1 1    180   1.0075 

 $5, 071.33

Given:

I/Y

180

0.750%

500,000.00

PMT

FV 0

Excel Formula

Solve for PMT:

(5,071.33)

PMT(0.0075,180, 500000,0)

b. The actual timeline of payments is 0

500,000

5071.33

To solve for X, set the PV of the payments equal to $500,000: 500,000 

5071.33 

1 X  1  500,000  241,064.02  59  60 0.0075  (1.0075)  1.0075 X

 258,935.98 

1.0075 X  258,935.98  1.0075

 405, 411.15

Evaluate: Thus, the final payment on the last month is $405,411.15. 28. The real rate of interest is as follows: rr 

r i 1 i



0.07782  0.12299 1.12299

 4.02229762%

The purchasing power of your savings declined by 4.02229762 % over the year. 29.

If the rate of inflation is 5%, what nominal interest rate is necessary for you to earn a 3% real interest rate on your investment? 1 + rr =

1+ r 1+ i

implies 1 + r = (1 + rr )(1 + i )

= (1.03)(1.05) = 1.0815

Therefore, a nominal rate of 8.15% is required. 30.

Excel Solution Plan: Recall that the nominal rate of interest shows your return in terms of a currency, the real rate of interest shows your return in terms of purchasing power, and the inflation rate shows how prices have increased. Execute: a. The number of baskets that can be purchased today is equal to $100,000 ÷ $100/basket = 1000 baskets. b. The investment generates a 19% return, so you will have $100,000 × 1.19 = $119,000 in one year.

c. The basket is expected to increase in price from $100 to $116 over one year, so the expected inflation rate is 116/100 – 1 = 0.16 = 16% over the year. d. Given a return on your cash investment of 19%, you expect to have $119,000 in one year [see part (b)]. So, with $119,000 and a price per basket of $116 [see part (c)], you expect to be able to buy $119,000 ÷ $116/basket = 1025.86207 baskets in one year. So, your return in terms of purchasing power is 1025.86207 baskets in one year ÷ 1000 baskets today – 1 = 2.586207%. This is the same as your expected real rate of return from investing at 19% given expected inflation of 16%. e. With a nominal interest rate of 19% and an expected inflation rate of 16%, we can calculate the expected real rate of return as (1 + nominal) ÷ (1 + inflation) = (1 + real), so we get the following: 1.19/1.16 = 1.02586209. And the real rate of return is 2.586207%, as we found for our expected increase in purchasing power. 31.

Excel Solution PV = 100 / 1.0199 + 100 / 1.02412 + 100 / 1.02743 =$285.61. To determine the single discount rate that would compute the value correctly, we solve the following for r: PV = 285.61 = 100/(1 + r) + 100 / (1 + r)2 + 100/(1 + r)3 = $285.61. This is just a rate of return calculation. Using a financial calculator or Excel’s RATE function, r = 2.49891816%. Note that this rate is between the one-, two-, and three-year rates given.

32.

Excel Solution The yield curve is increasing. This is often a sign that investors expect interest rates to rise in the future.

33.

Plan: Since the investment in your friend’s business is risky, the 1% riskless rate from Treasuries is not relevant. The 10% rate you could expect to earn on other similarly risky investments is the correct opportunity cost of capital, so you should discount the expected value of your investment using the 10% rate to see if it is greater than the $5000 cost. Execute: $5750/(1.10) = $5227.27, so the PV of the benefit (discounted at your opportunity cost of capital) exceeds the cost, making it a good investment. Evaluate: By recognizing your cost of capital and applying it to the payoffs from the investment, you can see that the PV of benefits is worth more than the cost, so it is a good investment.

34.

This business produces an expected return of $300/$5,000 = 6%. If projects that are similar in horizon and risk are offering an expected return of 8%, then this business is not earning your opportunity cost of capital, and you should invest elsewhere.

Chapter 6 Bonds Note: All problems in this chapter are available in MyLab Finance. An asterisk (*) indicates problems with a higher level of difficulty. 1.

Plan: We can use Equation 6.1 to determine the semi-annual coupon payment on the bond and then create a timeline for the cash flows using the semi-annual coupon payment found earlier. Execute: a. The coupon payment is

Coupon Rate  Face Value Number of Coupons per Year 0.055  $1000  2  $27.50

CPN 

b. Using the semi-annual coupon payment, we can then create a timeline for the cash flows of the bond. The timeline for the cash flows for this bond is (the unit of time on this timeline is six-month periods): 0

$27.50

$27.50 + $1000

Evaluate: We can compute the coupon payment in two steps. First, we determine the annual coupon amount by multiplying the face value by the coupon rate. We then divide the annual amount by the number of coupon payments per year. Showing future cash flows on a timeline helps us recognize patterns and solve present value problems correctly and efficiently. For example, looking at the timeline, it becomes clear that the coupons comprise an annuity and the face value represents a lump sum. 2.

Plan: We can see that the bond consists of an annuity of 20 payments of $20, paid every six months, and one lump-sum payment of $1000 (face value) in 10 years (20 six-month periods). We can rearrange Equation 6.1 in order to find the coupon rate knowing the coupon payment of $20. By rearranging Equation 6.1, we come up with Coupon rate  (Coupon payment/Face value)  Number of coupon payments per year.

Execute: a. The maturity is 10 years. b. (20/1000)  2  4%, so the coupon rate is 4%. c. The face value is $1000. Evaluate: The maturity of the bond is the final repayment date of that bond, after which point payments on the bond will terminate. In this case, the bond will make 20 semiannual payments terminating in 10 years. We can find the coupon rate if we know the coupon payment, face value, and number of coupon payments per year by using and rearranging Equation 6.1. Finally, we know that the face value of the bond is the amount repaid at maturity, in this case $1000. 3.

Plan: Zero-coupon bonds are pure discount bonds. They are issued at the present value of their principal (face) value. The PV must be $10,000,000 in order to raise that much money. Thus, we need to calculate the FV of $10,000,000 at 6% for 20 years, to determine the necessary face value. Execute: FV  $10,000,000 (1.06)20  $32,071,354.72 . Evaluate: Because we are not paying any interim interest payments, we must sell the bonds at a substantial discount to face value. That means that in order to receive $10 million from the sale, we must offer over $32 million in face value.

Excel Solution Plan: We can use Equation 6.2 to compute the yield to maturity for each bond. We can then use Excel to plot the zero-coupon yield curve, which will plot the yield to maturity of investments of different maturities using the yield to maturity on the y-axis and the maturity in years on the x-axis. Execute: a. Using Equation 6.2 for the first five years to compute the yield to maturity:

1/ n

 FV  1  YTM n   n   P 

1/1

 100   YTM  4.70%  1  95.51 

1  YTM 1  

1/ 2

 100  1  YTM 2     91.05 

 YTM 2  4.80%

1/3

 100    86.38 

1  YTM 3  

 YTM 3  5.00%

1/ 4

 100    81.65 

1  YTM 4  

 YTM 4  5.20%

1/5

 100    76.51 

1  YTM 5  

N Given:

I/Y

Solve for Rate:

95.51

100

I/Y

Solve for Rate:

PMT

91.05

100

I/Y

Excel Formula RATE(2,0,91.05,100)

PMT

86.38

100

Excel Formula RATE(3,0,86.38,100)

PMT

81.65

100

Excel Formula RATE(4,0,81.65,100)

5.20% N

Excel Formula

RATE(1,0,95.51,100)

5.00%

Solve for Rate:

PMT

4.80%

Given:

I/Y

Solve for Rate:

Given:

4.70%

Given:

I/Y

Solve for Rate:

Given:

 YTM 5  5.50%

PMT

76.51

100

5.50%

Excel Formula

RATE(5,0,76.51,100)

b. The yield curve is

Yield to Maturity

Zero-Coupon Yield Curve 5.6 5.5 5.4 5.3 5.2 5.1 5 4.9 4.8 4.7 4.6 0

Maturity (Years)

c. The yield curve is upward sloping. Evaluate: The yield to maturity of the bond is the discount rate that sets the present value of the promised bond payments equal to the current market price of the bond. We can use Equation 6.2 knowing the face value, price, and year of each bond in order to find the yield to maturity. We can plot the zero-coupon yield curve using Excel, which will compare the yield to maturity of investments of different maturities. 5.

Excel Solution Plan: We can use the bond’s yield to maturity to compute the bond’s price as the present value of the face amount, where the discount rate is the bond’s yield to maturity. From the table, the yield to maturity for two-year zero-coupon risk-free bonds is 5.50%. Execute: P  100(1.055)2  $89.85.

Given:

I/Y

5.500%

Solve for PV:

PMT

100

(89.85)

Excel Formula PV(0.055,2,0,100)

Evaluate: We can compute the price of a zero-coupon bond simply by computing the present value of the face amount using the bond’s yield to maturity. 6.

Excel Solution Plan: We can compute the bond’s price as the present value of the face amount, where the discount rate is the bond’s yield to maturity. From the table, the yield to maturity for four-year zero-coupon risk-free bonds is 5.95%. Execute: P  100/(1.0595)4  $79.36.

Given:

I/Y

5.950%

Solve for PV:

PMT

100

Excel Formula PV(0.0595,4,0,100)

(79.36)

Evaluate: We can compute the price of a zero-coupon bond simply by computing the present value of the face amount using the bond’s yield to maturity. 7.

Excel Solution The risk-free interest rate for five-year bonds is 6.05%.

Plan: These are all issued by the same issuer with the same par value. They only differ on the size of the coupon payments and the maturity. Execute: a. Three-year (you get the FV sooner, so its PV must be higher). b. A 4% coupon bond—the timing is the same, but the 4% coupon bond pays interest payments while the zero-coupon bond is a pure discount bond. c. A 6% coupon bond—the timing is the same, but the coupon (interest) payments are higher for the 6% bond. Evaluate: When the timing is the same, the bond with the higher coupon payments (larger interim cash flows) must be worth more. When the timing is different, but the coupons are zero, the bond that pays off sooner must be worth more.

Plan: Given the yield, we can compute the price using Equation 6.3. First, note that a 7.6% APR is equivalent to a semi-annual rate of 3.8%. Also, recall that the cash flows of this bond are an annuity of four payments of $35, paid every six months, and one lumpsum cash flow of $1000 (the face value), paid in two years (four six-month periods). Execute: PV 

35 35 35 35  1000    2 3 4  0.076   0.076   0.076   0.076  1  1  1   1       2    2  2   2  

PV  989.06

Given: Solve for PV:

I/Y

3.800%

PMT

1,000

(989.06)

Excel Formula PV(0.038,4,35,1000)

10.

First, calculate the cash flows and put them on a timeline. A $1000 par, 4% coupon bond pays (0.04  $1000)/2  $20 every six months and then pays the $1000 par at maturity. The next payment is due in six months. Today

6 months

1 year

1.5 years

1020

Next, convert the quoted APRs into six-month rates because the cash flows come at sixmonth intervals. Finally, discount the cash flows. Execute: Convert APRs to six-month rates: 0.01/2  0.005; 0.011/2  0.0055; 0.013/2  0.0065. Discount the cash flows using the appropriate spot rate for each cash flow, remembering that the number of periods (n) refers to the number of six-month periods. PV 

$20 (1.005)



$20 (1.0055)



$1020 (1.0065)3

 $1040.05

Evaluate: The bond is trading at a premium (price greater than par), because the coupon rate of 4% is higher than the current market spot rates. The price rises until the bond’s return matches the return offered elsewhere in the market for cash flows of the same timing and risk. 11.

Excel Solution Plan: The bond consists of an annuity of 20 payments of $40, paid every six months, and one lump-sum payment of $1000 in 10 years (20 six-month periods). We can use Equation 6.3 to solve for the yield to maturity. However, we must use six-month intervals consistently throughout the equation. In addition, we can use an annuity spreadsheet in Excel (shown below) to find the bond’s yield to maturity. Also, given the yield, we can compute the price using Equation 6.3. Note that a 9% APR is equivalent to a semi-annual rate of 4.5%. Again, we can use a spreadsheet in Excel to find the new price of the bond. Execute: Using Equation 6.3 to find the bond’s yield to maturity: a.

$1034.74 

 1  

40  YTM  

 1   

YTM  2

  

 

40  1000

 YTM  1   2  

 YTM  7.5%

Using the annuity spreadsheet: N Given: Solve for

I/Y

20 3.75%

PMT

1034.74

1000

Excel Formula

RATE(20,40,1034.74,1000)

Rate: Therefore, YTM  3.75%  2  7.50% b.

PV 

40  0.09  

 0.09  1   1   2    2 

  

40  1000

 0.09  1   2  

 $934.96.

Using the spreadsheet: With a 9% YTM  4.5% per six months, the new price is $934.96.

Given:

I/Y

4.50%

Solve for PV:

PMT

1000

Excel Formula

PV(0.045,20,40,1000)

(934.96)

Evaluate: The yield to maturity is the discount rate that equates the present value of the bond’s cash flows with its price. By discounting the cash flows using the yield, we can find the bond’s price. The bond’s price has lowered to $934.96, raising the yield to maturity from 7.50% to 9% APR. The present value of the bond payments equal to the current market price of the bond is lowered by increasing the discount rate of the bond. 12.

Plan: We can compute the bond’s coupon rate by rearranging Equation 6.3 to find the coupon payment. We can also use an annuity spreadsheet in Excel to find the coupon rate. Execute: 900 

C (1  .06)



C (1  .06)

  

C  1000 (1  .06)5

 C  $36.26,

So the coupon rate is 3.626%. We can use the annuity spreadsheet to solve for the payment:

Given:

I/Y

6.00% 900.00

Solve for PMT:

PMT

Excel Formula

1000 36.26

PMT(0.06,5,900,1000)

Therefore, the coupon rate is 3.626%. Evaluate: In order to compute the coupon rate, you must know the coupon payment. In this case, we can solve for the coupon payment by rearranging Equation 6.3 to find the annual coupon payment and then dividing that number by face value in order to convert the annual payment into the coupon rate. 13.

a. discount b. premium c. premium d. par 14.

The bond’s price will go down. Mathematically, the discount rate will increase, so the PV must decrease. Economically, the market interest rates represent the opportunity cost of capital. When your other investment opportunities become more attractive, this one becomes relatively less attractive, so its price must decrease until market participants are willing to buy it.

15.

Plan: Given the increase in the bond’s yield to maturity, we can compute the price using Equation 6.3. First, note that a 7.00% APR is equivalent to a semi-annual rate of 3.5%. Also, recall that the cash flows of this bond are an annuity of 14 payments of $40, paid every six months, and one lump-sum cash flow of $1000 (the face value), paid in seven years (four six-month periods). Execute: a. Because the yield to maturity is less than the coupon rate, the bond is trading at a premium. b.

40 (1  0.035)

Given:



40 (1  0.035)

I/Y

3.50%

Solve for PV:

  

40  1000 (1  0.035)14

 $1054.60

PMT

1000

(1054.60)

Excel Formula PV(0.035,14,40,1000)

Evaluate: The yield to maturity is the discount rate that equates the present value of the bond’s cash flows with its price. By discounting the cash flows using the yield, we can find the bond’s price. The bond’s price has lowered to $1054.60, raising the yield to maturity from 6.75% to 7.00% APR. The present value of the bond payments equal to the current market price of the bond is lowered by increasing the discount rate of the bond. 16.

Excel Solution Plan: Given the bond’s yield to maturity, we can compute the price using Equation 6.3. Note that the cash flows of this bond are an annuity of 10 payments of $70, paid annually, and one lump-sum cash flow of $1000 (the face value), paid in 10 years. Execute: When it was issued, the price of the bond was P

70 (1  .06)

 

70  1000 (1  .06)10

 $1073.60.

Given:

I/Y

6.000%

Solve for PV:

PMT

1,000

(1,073.60)

Excel Formula PV(0.06,10,70,1000)

Excel Solution Plan: Since we are valuing the bond just before the first $70 coupon payment, we know that that payment is worth $70. The rest of the bond is now a nine-year bond that we can value. The value of the bond is the sum of these cash flows. Execute: Before the first coupon payment, the price of the bond is P  70 

70 (1  .06)

 ... 

70  1000 (1  .06)9

 $1138.02.

Evaluate: This bond would have a value of $1138.02. 18.

Excel Solution Plan: Since the first $70 coupon has just been paid, it no longer is reflected in the value of the bond. The bond is now a nine-year maturity bond, which can be valued. Execute: After the first coupon payment, the price of the bond will be P

70 (1  0.06)

  

70  1000 (1  0.06)9

 $1068.02.

Evaluate: The value of this bond is $1068.02. 19.

Plan: A 6% coupon bond pays (0.06  $1000)/2  $30 every six months. We can use the fact that there are 20 six-month periods left until maturity and Equation 6.3 to infer the YTM (shown as y in the equation below) for the existing bond. Your company will have to offer a coupon rate equal to that YTM to sell new bonds. Execute: $1078  $30 

1

 1 1000 1   20 20 y  1  y   1  y   

Through either trial and error, a financial calculator, or Excel, we can determine that y  0.025.

1078

1000

0.025 Excel Formula:  RATE(NPER,PMT,PV,FV)  RATE(20,30, 1078,1000)

The six-month rate is 0.025, so the semi-annually compounded APR would be 5% (2.5%  2). Because the market is currently pricing your company’s bonds to yield 5%, you would need to offer a 5% coupon rate in order to have them priced at par. Evaluate: Even though you had to offer a 6% coupon rate in the past, market conditions have changed and the coupon rate you would now have to offer would only be 5%. *20.

Excel Solution Plan: We can use a financial calculator or Excel to compute the initial price of the bond using 10 annual coupon payments, 5% yield to maturity, $6 coupon payment, and $100 future value to solve for the present value, which is the cash outflow in year 0. Next, we can use a financial calculator or Excel to compute the price that the bond sold at using only six years to maturity (annual coupon payments), 5% yield to maturity, $6 coupon payment, and $100 future value to solve for the present value, which is part of the cash flow in year 4 along with the $6 coupon payment ($1000 face value  6% coupon rate  $6 coupon payment). In order to compute the IRR, we can use the annuity spreadsheet in Excel using the purchase price as the present value, the $6 coupon payment, the sales price as the future value, and four years for the length of the investment. Execute: a. First, we compute the initial price of the bond by discounting its 10 annual coupons of $6 and final face value of $100 at the 5% yield to maturity:

Given:

I/Y

5.00%

Solve for PV:

PMT

100

Excel Formula PV(0.05,10,6,100)

(107.72)

Thus, the initial price of the bond  $107.72. (Note that the bond trades above par, as its coupon rate exceeds its yield.) Next, we compute the price at which the bond is sold, which is the present value of the bond’s cash flows when only six years remain until maturity:

Given:

I/Y

5.00%

PMT

100

Excel Formula

Solve for PV:

PV(0.05,6,6,100)

(105.08)

Therefore, the bond was sold for a price of $105.08. The cash flows from the investment are, therefore, as shown in the following timeline: Year

$107.72

Purchase Bond Receive Coupons Sell Bond

$105.08 $107.72

Cash Flows

$6.00

$111.08

b. We can compute the IRR of the investment using the annuity spreadsheet. The PV is the purchase price, the PMT is the coupon amount, and the FV is the sale price. The length of the investment N  4 years. We then calculate the IRR of investment  5%. Because the YTM was the same at the time of purchase and sale, the IRR of the investment matches the YTM. N Given: Solve for Rate:

I/Y

PMT

107.72

105.08

5.00%

Excel Formula RATE(4,6,107.72,105.08)

Evaluate: In order to find the cash flows from the investment for the first four years, we need to find the initial investment in year 0 as well as the price that the bond sold for in year 4, along with the coupon payment each year of $6. Note that the $107.72 initial investment of the bond is trading above par because the coupon rate of 6.00% exceeds the yield to maturity of 5.00%. Also, we can see that because the yield to maturity remained the same at the time of purchase and when the bond sold, the IRR of the investment is identical to the yield to maturity. 21.

Excel Solution Plan: We need to compute the price of each bond for each yield to maturity and then calculate the percentage change in the prices. We can use Equation 6.3 to compute the prices. Execute: We can compute the price of each bond at each YTM using Equation 8.5. For example, with a 6% YTM, the price of Bond A per $100 face value is P (Bond A, 6% YTM ) 

100

1.0615  $41.73

The price of Bond D is P (Bond D, 6% YTM )  8 

1 

1 

0.06 

1  100  1.0610  1.0610

 $114.72

One can also use the Excel formula to compute the price: =PV(YTM,NPER,PMT,FV). Once we compute the price of each bond for each YTM, we can compute the percent price change as Percent change 

(Price at 5% YTM )  (Price at 6% YTM ) (Price at 6% YTM )

The results are shown in the table below:

Coupon Rate Bond (annual payments)

Maturity (years)

Price at 6% YTM

Price at 5% YTM

Percentage Change

$ 41.73

$ 48.10

15.3%

$ 55.84

$ 61.39

9.9%

$ 80.58

$ 89.62

11.2%

$114.72

$123.17

7.4%

Evaluate: The 15-year zero-coupon bond is the most sensitive to the decrease in the bond’s yield to maturity, while the 10-year shows the least sensitivity. Additionally, both of the 15-year bonds are more sensitive than the 10-year bonds, proving that long-term bonds are riskier than short-term bonds. 22.

Excel Solution Bond A is most sensitive, because it has the longest maturity and no coupons. Bond D is the least sensitive. Intuitively, higher coupon rates and a shorter maturity typically lower a bond’s interest rate sensitivity.

23.

Excel Solution Plan: We must compute both the purchase price of the bond and the sale price of the bond for each separate scenario. In the first scenario (A), the yield to maturity is the same when the bond was purchased and when the bond was sold and we can compute the price of the bond with 25 years left exactly as we did for 30 years, but using 25 years of discounting instead of 30. With scenarios B and C, we must use the old yield to maturity to find the purchase price and the new yield to maturity to find the sale price. Once we have found the prices for each scenario, we can compute the IRR of each scenario just as we did in Chapter 4. The FV is the price in five years, the PV is the initial price, and the number of years is five. Execute: a. Purchase price  100/1.0630  17.41. Sale price  100/1.0625  23.30. Return  (23.30/17.41)1/5 – 1  6.00%; i.e., since YTM is the same at purchase and sale, IRR  YTM. b. Purchase price  100/1.0630  17.41. Sale price  100/1.0725  18.42. Return  (18.42/17.41)1/5 – 1  1.13%; i.e., since YTM rises, IRR  initial YTM. c. Purchase price  100/1.0630  17.41. Sale price  100/1.0525  29.53. Return  (29.53/17.41)1/5 – 1  11.15%; i.e., since YTM falls, IRR  initial YTM. d. Even without default, if you sell prior to maturity, you are exposed to the risk that the YTM may change. Evaluate: In scenario A, the bond’s yield to maturity did not change, meaning the IRR of the investment in the bond equals its yield to maturity even if the bond is sold early. If the

yield to maturity increases, then the IRR will become less than the initial yield to maturity, while the opposite will occur if the yield to maturity decreases.

24.

Plan: We can use the bond’s yield to maturity to compute the bond’s price as the present value of its face amount, where the discount rate is the bond’s yield to maturity. We can compute the credit spread by taking the yield to maturity of each security and subtracting the yield to maturity for the treasury bill. Execute: a. The price of this bond will be P

100 1  0.032

Given:

 96.899

I/Y

3.200%

Solve for PV:

PMT

100

Excel Formula PV(0.032,1,0,100)

(96.90)

b. The credit spread on AAA-rated corporate bonds is 0.032 – 0.031  0.1%. c. The credit spread on B-rated corporate bonds is 0.049 – 0.031  1.8%. d. The credit spread increases as the bond rating falls, because lower rated bonds are riskier. Evaluate: We can compute the price of a zero-coupon bond simply by computing the present value of the face amount using the bond’s yield to maturity. The credit spread on the AAA-rated corporate bonds is less than the credit spread on the B-rated corporate bonds because lower rated bonds are riskier and the credit spread increases as the bond rating falls. 25.

Plan: Given the bond’s yield to maturity, we can compute the price using Equation 6.3. Note that the cash flows of this bond are an annuity of 30 payments of $70, paid annually, and one lump-sum cash flow of $1000 (the face value), paid in 30 years. Execute: a. When originally issued, the price of the bonds was P

70 (1  0.065)

  

70  1000 (1  0.065)30

 $1065.29

Given: Solve for PV:

I/Y

6.500%

PMT

1,000

(1,065.29)

Excel Formula PV(0.065,30,70,1000)

b. If the bond is downgraded, its price will fall to P

70 (1  0.069)

  

70  1000 (1  0.069)30

 $1012.53

Given:

I/Y

6.900%

Solve for PV:

PMT

1,000

Excel Formula

PV(0.069,30,70,1000)

(1,012.53)

Evaluate: If the bond is downgraded, the yield to maturity will increase from 6.50% to 6.90%, reducing the bond’s price from $1065.29 to $1012.53. A higher discount rate decreases the present value of the bond’s future cash flows and therefore its market price. 26.

Excel Solution Plan: We can use the bond’s yield to maturity to compute the bond’s price as the present value of its face amount, where the discount rate is the bond’s yield to maturity. Using the price of the bond, we can find the total principal amount of these bonds by dividing the price of the bonds into the $10 million HMK would like to raise. In order to find the rating that would sell the bonds at par, the coupon rate of the bond must equal the yield of the bond. To find the rating of the bond given the price, we can use Equation 6.3 to solve for the yield to maturity and then match that yield to maturity to the bond ratings to find the specific rating, as well as whether or not the bonds are junk bonds. Execute: a. The price will be P

65 (1  0.063)

  

65  1000 (1  0.063)5

 $1008.36

Given: Solve for PV:

I/Y

6.300%

PMT

1,000

(1,008.36)

b. Each bond will raise $1008.36, so the firm must issue

Excel Formula

PV(0.063,5,65,1000)

$10, 000, 000 $1008.36

 9917.13  9918 bonds.

This will correspond to a principal amount of 9918  $1000  $9,918,000. c. For the bonds to sell at par, the coupon must equal the yield. Since the coupon is 6.5%, the yield must also be 6.5%, or A-rated. d. First, compute the yield on these bonds: 959.54 

65 (1  YTM)

N Given:

  

(1  YTM)5

I/Y

Solve for Rate:

65  1000

 YTM  7.5%

PMT

959.54

1,000

Excel Formula

RATE(5,65,959.54,1000)

7.50%

Given a yield of 7.5%, it is likely these bonds are BB rated. Yes, BB-rated bonds are junk bonds. Evaluate: The yield to maturity is the discount rate that equates the present value of the bond’s cash flows with its price. By discounting the cash flows using the yield we can find the bond’s price. In order for a bond to sell at par, the coupon rate must equal the yield to maturity. For these bonds the coupon rate is 6.5%, so in order to sell at par, the bonds would have to be A-rated with a yield of 6.5%. Given the yield of 7.5%, these bonds are BB rated and they are in the bottom five categories. Bonds in the bottom five categories are often called junk bonds because their likelihood of default is high. 27.

Excel Solution Plan: We can use the bond’s yield to maturity to compute the bond’s price as the present value of its face amount, where the discount rate is the bond’s yield to maturity. We can compute the credit spread by taking the yield to maturity of each security and subtracting the yield to maturity for the treasury bill. Execute: a.

P

35 (1  0.0325)

 

35  1000 (1  0.0325)10

 $1, 021.06  102.1%

Given: Solve for PV:

I/Y

3.250%

PMT

1,000

(1,021.06)

Excel Formula PV(0.0325,10,35,1000)

P

35 (1  0.041)

35  1000

  

(1  0.041)10

 $951.58  95.2%

Given:

I/Y

4.100%

Solve for PV:

PMT

1,000

(951.58)

Excel Formula

PV(0.041,10,35,1000)

c. 8.2% – 6.5% = 1.7% Evaluate: The treasury bond is priced higher and has a lower yield to maturity than the BBB-rated corporate bond due to the treasury bond’s lower credit risk. The treasury bond is free of default risk, while the BBB-rated bond has some risk of default. Lower-rated bonds are riskier (more likely to default) and the credit spread increases as the bond rating falls. 28.

Plan: We can price your debt as percent of par (per $100 par value): 6% of $100 is $6, paid as $3 every six months. In that case, your debt makes $3 payments every six months for five years and then pays $100 par or face value. The appropriate YTM is the comparable Government of Canada bond yield plus the credit spread, giving 0.0285, which is the APR. That corresponds to an effective six-month rate of 0.0285/2 = 0.01425. Execute:

Price = PV =

 3  1 100   114.58 1  10  .01425  1.01425  1.0142510

Evaluate: While your firm’s bond does require a higher yield, due to its credit risk (even as an A rated bond, there is a small risk of default) compared to the Government of Canada bond, your firm’s bond is selling at a substantial premium to its par or face value. The reason your firm’s bond is selling at a substantial premium is because its 6% coupon rate is much higher than the prevailing market interest rate for 5-year A rated corporate bonds.

Chapter 7 Valuing Stocks

Note: All problems in this chapter are available in MyLab Finance An asterisk (*) indicates problems with a higher level of difficulty. For a breakdown of the difficulty ratings of all of the problems in this Solutions Manual, please see the Preface – Question Difficulty Ratings. 1.

Multiply the number of shares by the dividend per share: 15,000  $0.18  $2700.

With cumulative voting, you are able to get proportional representation by putting all of your votes toward two directors, allowing you to elect representatives to two seats (20% of 10 seats) on the board. With non-cumulative voting, you vote on each director individually, and without a majority of the shares you cannot ensure that your representative will win any of the elections (you could lose 80% to 20% in each of the 10 individual elections).

To make this easier, assume there are 100 shares of Class A and 100 shares of Class B. You then own 10 Class A shares (10%) and 20 Class B shares (20%). Because Class B shares have 10 times the voting rights of Class A, there are a total of 100  100(10)  1100 votes. You have 10  20(10)  210 of those votes, or 210/1100  0.191 (19.1%).

Plan: We can use Equation 7.1 to solve for the price of the stock in one year given the current price of $50.00, the $2 dividend, and the 15% cost of capital. Execute: 50 

2 X

1.15 X  55.50

Evaluate: At a current price of $50, we can expect Evco stock to sell for $55.50 immediately after the firm pays the dividend in one year. 5.

Plan: We can use Equation 7.2 to solve for the total return, dividend yield, and capital gain rate. Execute:

Total Return 

5 110  100  100 100 Dividend Yield

Capital Gain Rate

 5%  10%  15% Evaluate: You expect the total return to be 15%, the dividend yield to be 5%, and the capital gain rate to be 15%.

Plan: We can use Equation 7.2 to calculate the dividend yield and rearrange Equation 7.1 to find the expected stock price right after paying the dividend of $1.75. Execute: Dividend yield = 1.50/37.50 = 4% New price = 1.75/0.04 = $43.75 (note that 1.75/43.75 = 0.04) Evaluate: You expect the dividend yield to be 4%, and the new share price to be $43.75.

Plan: We can rearrange Equation 7.1 to find the cost of capital given the current stock price of $20, the $1 dividend, and the expected stock price right after paying the dividend of $22. We can use Equation 7.2 to calculate the dividend yield and the capital gain. Execute: a.

b. Dividend yield: 5% (1/20) Capital gain: 10% (2/20) Evaluate: The cost of capital for Anle Corporation is 15% given the current stock price of $20 and the expected stock price of $22 after paying the dividend of $1 in one year. The cost of capital is then split between the dividend yield at 5% and the capital gains yield at 10%, which together creates the 15% cost of capital. 8.

Plan: We can use the constant dividend growth model (Equation 7.6) with a growth rate of 0 because this preferred stock pays a constant (non-growing) dividend. Div1  3, r  0.08, g 0. Execute: P  Div1/(r  g)  3/(0.08  0)  37.50 Evaluate: To be consistent with the promised dividend and the required return, the price would be $37.50.

Dividend Yield  Div/P  $1.50/$20  0.075 (7.5%)

10.

Plan: We can use Equation 7.1 to solve for the beginning price we would pay now (P 0) given our expectations about dividends ($2.80) and future price ($52.00) and the return we need to expect to earn to be willing to invest 10%. We can use Equation 7.1 to solve for the price for which you would expect to be able to sell a share of Acap stock in one year given our expectations about dividends ($3.00) next year and future price ($52.00) and the return we need to expect to earn to be willing to invest 10%. Given the price you would expect to be able to sell a share of Acap stock for in one year, we can solve for the price you would be willing to pay for a share of Acap stock today if you planned to hold the stock for only one year using the dividend of $2.80 and the expected return of 10%. Execute: a. P(0)  2.80/1.10  (3.00  52.00)/1.102  $48.00 b. P(1)  (3.00  52.00)/1.10  $50.00 c. P(0)  (2.80  50.00)/1.10  $48.00 Evaluate: The price you would be willing to pay for a share of Acap stock today if you held the stock for two years or one year is not affected by the amount of time you hold the stock, as can be seen in parts (a) and (c).

11.

Plan: We can use Equation 7.2 to calculate the dividend yield and the capital gain. We can then compute the total expected return by adding the dividend yield to the capital gain. Execute: Dividend yield  0.88/22.00  4% Capital gain rate  (23.54  22.00)/22.00  7% Total expected return  rE  4%  7%  11% Evaluate: The stock’s dividend yield of 4% is the expected annual dividend of the stock dividend by its current price. The dividend yield is the percentage return the investor expects to earn from the dividend paid by the stock. The capital gain of 7% reflects the capital gain the investors will earn on the stock, which is the difference between the expected sale price and the original purchase price for the stock. The sum of the dividend yield and the capital gain rate is called the total return of the stock. The total return is the expected return that the investor will earn for a one-year investment in the stock.

*12.

Plan: We can use Equation 7.6 to calculate the price per share yearly or we can use our knowledge of interest rate conversions to discount the quarterly dividends. Execute: If the dividends are paid quarterly, we can value them as a perpetuity using a quarterly discount rate of (1.15)1/4  1  3.556% ; then, P  $0.50/0.03556  $14.06.

Evaluate: You would be willing to pay $14.06 for this stock with quarterly dividends and no growth. 13.

Plan: Because the dividends are expected to grow perpetually at a constant rate, we can use Equation 7.6 to value Summit Systems. The next dividend is expected to be $1.50 × (1.06) or $1.59, the growth rate is 6%, and the equity cost of capital is 11%. Execute: P  1.59/(11%  6%)  $31.80 Evaluate: You would be willing to pay 20 times next year’s dividend of $1.59 to own Summit Systems’ stock because you are buying claim to next year’s dividend and to an infinite growing series of future dividends. In other words, with constant expected dividend growth, the expected growth rate of the share price matches the growth rate of the dividends.

14.

Plan: Because we know that Dorpac currently has an equity cost of capital of 8% and a dividend yield of 1.5%, we can use Equation 7.7 to estimate the growth rate. Execute: a. Equation 7.7 implies rE  Dividend Yield  g, so 8%  1.5%  g  6.5%. b. With constant dividend growth, share price is also expected to grow at rate g  6.5% (or we can solve this from Equation 7.2). Evaluate: In this case, Dorpac Corporation has a constant growth rate and therefore the dividends are expected to grow at the same rate as the share price; that is, 6.5%.

15.

Plan: Knowing the earnings per share for next year of $4.00 and the retention rate of 70%, we can compute the dividend next year. We can compute the dividend growth rate using the return on new investment and the retention rate (see Equation 7.12). Finally, we can compute the current stock price using Equation 7.6. Execute: Earnings next year: $4.00 per share Dividend next year: $4(1  0.7)  $1.20 per share Dividend growth rate: 10% return on new investment  70% retention rate  7% Current stock price: P0 

1.20 0.10  0.07

 $40

Evaluate: The current stock price for Laurel Enterprises is $40. *16.

Plan: Several interrelated calculations will be needed to answer parts (a), (b), and (c). Execute: a. Equation 7.12: g  retention rate  return on new invest  (2/5)  15%  6% b. P  3/(12%  6%)  $50

c. g  (1/5)  15%  3%, P  4/(12%  3%)  $44.44. No, projects are positive NPV (return exceeds cost of capital), so don’t raise dividend. Evaluate: DBF’s growth rate is 6%. Its stock will sell for $50.00. DBF should not raise the dividend because it will result in a lower stock price of $44.44. 17.

Plan: We must estimate the stock price of Cooperton Mining after the dividend cut and new investment policy. If the stock price were to fall, we would not cut the dividend. Execute: Estimate rE: rE  Div Yield  g  4/50  3%  11% New Price: P  2.50/(11%  5%)  $41.67 Evaluate: In this case, cutting the dividend will reduce the stock price to $41.67. The move to cut the dividend and to expand is not positive NPV. Do not do it.

18.

Plan: Gillette’s dividend is expected to grow at 12% per year for five years and then at 2% per year indefinitely. We should employ a two-stage growth model. First, we value the constant growth in dividends five years from now and discount it to the present. Then, we determine the value today of the five dividend payments growing at 12% from year 1 to 5. The value of the stock today is the sum of these two values. Execute: Value of the first five dividend payments: PV15 

  1.12 5    1   (0.08  0.12)   1.08   0.65

 $3.24

Value on date 5 of the rest of the dividend payments: PV5 

0.65(1.12) 4 1.02

0.08  0.02  17.39

Discounting this value to the present gives, PV0 

17.39 (1.08)5

 $11.83

So the value of Gillette is P  PV15  PV0  3.24  11.83  $15.07

Evaluate: Gillette’s stock today is worth $15.07, which is the sum of $3.24 (the present value of the first five dividends) and $11.83 (the present value of the dividends growing at 2% per year from year 6 onward).

19.

Plan: CX Enterprises’ dividend is expected to grow $1 in one year, $1.15 in 2 years, and $1.25 in 3 years. After that, its dividends are expected to grow at 4% per year forever. We should employ a two-stage growth model. First, we value the dividends for 3 years from now and discount it to the present. Then, we determine the value today of the dividend payments growing at 4%. The value of the stock today is the sum of these two values. Solution: The expected dividends are 1

…

1.15

1.25

1.30 = (1.25)(1.04)

Continue growing by 4%

Price = PV =

1.00 1.15 1.25  1   1.30        14.27 1.12 1.12 2 1.12 3  1.12 3   .12  .04 

Evaluate: Note, as we learned in Chapter 4, the growing perpetuity that starts in year 4 is only discounted back 3 years because the perpetuity formula itself already gives the value in year 3 of a perpetuity starting in year 4. CX Enterprises’ stock today is worth $14.27. 20.

Plan: Colgate’s dividend is expected to grow at 11% per year for five years and then at 2% per year indefinitely. We should employ a two-stage growth model. First, we value the constant growth in dividends five years from now and discount it to the present. Then, we determine the value today of the five dividend payments growing at 11% from years 1 to 5. The value of the stock today is the sum of these two values. Execute: PV of the first five dividends: PVfirst 5 

0.96(1.11) 

5  1.11      1.085  

1  

0.085  0.11 

 5.14217

PV of the remaining dividends in year 5: PVremaining in year 5 

0.96(1.11)5 (1.052)

0.085  0.052  51.5689

Discounting back to the present: PVremaining 

51.5689 1.0855

 34.2957

Thus the price of Colgate is

P  PVfirst 5  PVremaining  39.4378 Evaluate: Colgate’s stock today is worth $39.43, which is the sum of $5.14 (the present value of the first five dividends) and $34.29 (the present value of the dividends growing at 5.2% per year from year 6 onward). *21.

Plan: Build a spreadsheet of Halliford’s expected EPS and dividends per share and then determine the present value of the expected dividends. Execute: See the spreadsheet for Halliford’s dividend forecast: Year

25%

12.5%

$3.00

$3.75

$4.6 9

$5.27

$5.93

$6.2 3

3 Retention Ratio

100%

50%

20%

4 Dividend Payout Ratio

50%

80%

5 Div (2  4)

—

$2.3 4

$2.64

$4.75

$4.9 8

Earnings 1 EPS Growth Rate (vs. prior yr) 2 EPS

Dividends

From year 5 on, dividends grow at constant rate of 5%. Therefore, P(4)  4.75/(10%  5%) 

  $95 

Then, P(0)  2.34/1.103  (2.64  95)/1.104 

 $68.45

Evaluate: Halliford stock should sell for $68.45 today. This represents the present value of the dividends that are expected to grow at different rates over time. 22.

Plan: Using the total payout method, the equity value will be the PV of the total future payouts. Solution: Its current stock price is the PV of its total payout of $20 million per year in perpetuity:

$20 million  $153.8462 million, then divided by 10 million shares = $15.38 per share .13 If it increases its total payout to $30 million with the additional $10 million in payout through repurchase, the new stock price will be PV 

$30 million  $230.7692 million, then divided by 10 million shares = $23.08 per share, .13 so, it increases by $7.70 per share. PV 

Evaluate: It does not matter how the payout is divided between repurchases and dividends—the total payout method focuses on the total amount per year. In this case, that total amount implies a share price of $7.70. 23.

Plan: Using the total payout method, the equity value will be the PV of the total future payouts. If the sum of dividends and repurchases remains at $10 million in perpetuity, then we can calculate the equity value using the perpetuity formula: Total payout  $10 million and r  0.13. Execute: The equity value is $10 million/0.13  $76.92 million. Dividing it by the current number of shares outstanding gives us the price per share: $76.92 million/5 million shares  $15.38 per share. Evaluate: It does not matter how the payout is divided between repurchases and dividends—the total payout method focuses on the total amount per year. In this case, that total amount implies a share price of $15.38.

24.

Plan: With constant payout rates, earnings growth equals payout growth. Using the total payout model, we can value AFW’s equity using the growing perpetuity formula. Execute: Total payouts this year  $700 million  0.60  $420 million. PV 

$420 million 0.12  0.08

 $10,500 million ($10.5 billion)

AFW has 200 million shares outstanding, so the price per share is $10,500/200 = $52.50. Evaluate: Holding the payout ratio constant, the earnings growth rate implies the same payout growth rate. Given the total expected payout over time, the current price per share should be $52.50. 25.

Plan: First, determine next year’s dividends for the entire firm. Then, value the entire firm’s equity, recognizing that the equity cost of capital is 12% and the growth rate in

dividend is 8%. Second, divide the total value of the firm’s equity by the number of outstanding shares to determine the value of a single share. Execute: Total payout next year  5 billion  1.08  $5.4 billion Equity value  5.4/(12% – 8%)  $135 billion Share price  135/6  $22.50 Evaluate: The firm is expected to pay $5.4 billion in dividends next year, which would mean the firm’s total equity is $136 billion. With 6 billion shares outstanding, each share is worth $22.50.

*26.

Plan: Make several interrelated calculations to determine the answers to parts (a), (b), and (c). Execute: a. Earnings growth  EPS growth  dividend growth  4%. Thus, P  3/(10%  4%)  $50. b. Using the total payout model, P  ($1 dividend + $2 repurchase)/(10%  4%)  $50. c. Earnings growth will not change; nor will the growth in total payout as a % of earnings. However, as repurchases cause the number of shares to decline, EPS growth and, hence, dividend growth will be higher. Taking the result from part (b) for the price but using the dividend growth model instead, we would have P = Div/(rE – g), so manipulating this equation, we get Div/P = rE – g. And thus, the growth in EPS and dividends, g, will be g = rE – Div/P as follows: g  10%  1/50  8%. Evaluate: Stelco’s stock price would be $50.00 if it paid out a $3.00 dividend that would grow at 4% per year. The stock would also sell for $50.00 if Stelco paid out a $1.00 dividend and $2 in share repurchases that would, in total, grow at 4% per year. The new EPS and dividend growth rates if share repurchases are done will be 8%. (Note, it is not asked, but the stock would also be valued at $50.00 per share using the dividend growth model with a $1.00 starting dividend that would grow at 8% per year.)

27.

Plan: To compute FCF, deduct taxes from EBIT, add back depreciation, and account for any capital expenditures and changes in net working capital. Execute: EBIT

Taxes

3.5

Net Income

6.5

Add back Depr

Cap Ex

1.5

Chg NWC

0.5

 FCF

5.5

Evaluate: Being able to calculate and forecast FCF is essential as an input in a discounted free cash flow model.

28.

Plan: First, compute FCF using Equation 7.17 and then apply the growing perpetuity formula (r  0.10, g  0.04) to calculate the value of the firm as the present value of its free cash flows. Execute: FCF next year is EBIT(1  Tax Rate)  Depreciation  CapEx  Change in NWC  $1,000,000(1  0.40)  $300,000  $300,000  $50,000  $550,000. Since depreciation will always equal capital expenditures, they will always cancel each other out and so we can ignore them. Both EBIT and the change in NWC are expected to increase at a rate of 4%, so the overall FCF will also increase at a rate of 4%. Using the discounted FCF model (note that the $550,000 is for next year and does not need to be multiplied by 1  growth rate): V0 

$550,000 (0.10  0.04)

 $9,166,666.67

Evaluate: The value of the firm today must be the present value of its expected free cash flows. For this firm, a good estimate of its value today is $9,166,666.67. 29.

Plan: Use Equation 7.19 to calculate the equity value and divide it by the number of shares outstanding, yielding the price per share. Execute: By Equation 7.19: P0 

V0  Cash  Debt shares outstanding



100  6  30 2

 38

$38 per share Evaluate: The firm has an enterprise value of $100 million, but $30 million of that is debt and so must be subtracted before computing the equity value. Also, the firm has $6 million in cash, which belongs to the equity holders, so it must be added. 30.

Plan: Use Equation 7.18 to calculate the enterprise value and then Equation 7.19 to calculate the price per share. Execute: By Equation 7.18, enterprise value equals the PV of future FCF: V0  By Equation 7.19, P0 

V0  Cash  Debt shares outstanding



125  1  0 5

 25.2

$25.20 per share

10 .11  .03

 125

Evaluate: Given the information, the enterprise value can be estimated as $125 million and the firm additionally has $1 million in cash, but no debt. The price per share should be $25.20. 31.

Plan: From the present value of free cash flow (FCF), we can get enterprise value (see Equation 7.18). Then, we can determine the market value of the equity using Equation 7.16 and the price per share using Equation 7.19. Execute: a.

By Equation 7.18, enterprise value equals the PV of future FCF: V0 

200 .12  .02

 $2 billion

The market value of the equity is (Equation 7.16) Enterprise Value = Market Value of Equity + Debt  Cash And, rearranging to get market value of equity, we get Market Value of Equity = Enterprise Value + Cash  Debt So, the market value of equity is 2 billion + 15 million  500 million = $1.515 billion. And, thus, the price per share is (Equation 7.19) P0 

V0  Cash  Debt shares outstanding



$1.515 billion 20 million shares

 $75.75 share

Using the same process as in part (a) but changing the growth rate to 3%, we get the following: V0 

200 .12  .03

 $2.222 billion

So the market value of equity is 2.222 billion + 15 million  500 million = $ 1,737,222,222.22 And, thus, the price per share is P0 

V0  Cash  Debt shares outstanding



$1.7372 billion 20 million shares

 $86.86 share

The change in the stock price is $11.11 per share. This can be calculated by looking at the change in enterprise value divided by the number of shares.



$2.222 billion  $2 billion $222.2 million   $11.11 share 20 million shares 20 million shares

Evaluate: The higher growth rate raises the enterprise value. This, in turn, raises the market value of the firm. There is no change in cash or debt, so the change in enterprise value flows to the equity holders who are the residual claimants of the firm’s cash flows.

32.

Excel Solution Plan: The first step is to determine the value of Heavy Metal Corporation. Since the firm has specific cash flow estimates for years 1 to 5, these must be discounted to the present. Starting in year 6, the firm’s cash flows are expected to grow at 4%, so they can be valued using the dividend growth model. Once the value of the entire firm is estimated, the second step is to determine the value of a share of equity. Execute: a. V(5)  82  (1.04)/(14%  4%)  $852.80, which is the value of the constant growth in dividends which starts in year 6. V(0)  53/1.14  68/1.142  78/1.143  75/1.144  (82  852.8)/1.145  $681.37 This is the value of the entire firm at time 0. b. P  (681  0  300)/40  $9.53 is the value per share of equity. This is the value of the entire firm, minus the value of debt, divided by the number of shares outstanding. Evaluate: The entire firm is worth $681 million dollars. The equity of the firm is worth the entire firm value less the value of the outstanding debt ($681 million minus $300 million), or $381 million. We now divide the value of the equity by the number of outstanding shares ($381 million divided by 40 million) to determine a per-share value of $9.53.

33.

Excel Solution Plan: Use Equations 7.18 and 7.19 to compute the enterprise value and then price per share of Covan. At any point in time, the enterprise value is the PV of the remaining expected FCF. Once the two prices per share are computed, you can calculate the return. Execute: a. V0 

P0 

10 1.12



12 1.12



13 1.12



14 1.12

 1  14(1.04)    152.31 4   1.12  .12  .04 



V0  Cash  Debt 152.31  3  0   19.41 shares outstanding 8

b. At any point in time, the enterprise value is still the PV of all remaining FCF:

12 13 14  1  14(1.04)       160.59 1 2 3 1.12 1.12 1.12  1.123   0.12  0.04  V  Cash  Debt 160.59  3  0 P0  0   20.45 shares outstanding 8

V0 

r

20.45  19.41 19.41

 0.054

Evaluate: Given your forecast of Covan’s FCFs and its investment plans, you expect the stock price to rise from $19.41 to $20.54, giving you a return of 5.4%.

*34.

Excel Solution Plan: The plan is to compute numerous calculations using the spreadsheet information provided in the problem. Execute: a. V(3)  33.3/(10%  5%)  666 V(0)  25.3/1.10  24.6/1.102  (30.8  666)/1.103  567 P(0)  (567  40  120)/60  $8.11 Sora Industries has forecast cash flows for years 1, 2, and 3. After year 3, cash flows are forecast to grow at 5% per year. The total value of the firm is $567 million, which is the present value of the forecast cash flows in years 1, 2, and 3, as well as the present value of the cash flows starting in year 4, which will grow at 5% per year. The value of a single share of Sora stock is $8.11, which is the sum of the value of the firm ($576) plus the value of the cash holdings ($40) minus the value of the outstanding debt ($120) divided by the number of outstanding shares, or 60 million. b. Free cash flows change as follows: Year

8.1%

10.3%

6.0%

5.0%

433.00 468.00

516.00

547.00

574.30

2 Cost of Goods Sold

(327.6)

(361.2)

(382.9)

(402.0)

3 Gross Profit

140.40

154.80

164.09

172.3

4 Selling, General, & Admin.

(93.6)

(103.2)

(109.4)

(114.9)

5 Depreciation

(7.00)

(7.50)

(9.00)

(9.5)

6 EBIT

39.8

44.1

45.7

47.9

7 Income tax at 40%

(15.9)

(17.6)

(18.3)

(19.2)

8 Unlevered Net Income

23.9

26.5

27.4

28.8

9 Plus: Depreciation

7.00

7.50

9.00

9.5

10 Less: Capital Expenditures

(7.7)

(10.0)

(9.9)

(10.4)

11 Less: Increases in NWC

(6.3)

(8.6)

(5.6)

(4.9)

Earnings Forecast ($000s) 1 Sales

Free Cash Flow ($000s)

12 Free Cash Flow

16.9

Hence, V(3)  458, and V(0)  388. Thus, P(0)  $5.14.

15.3

20.9

22.9

c. New FCF: Year

8.1%

10.3%

6.0%

5.0%

433.00 468.00

516.00

547.00

574.30

2 Cost of Goods Sold

(313.6)

(345.7)

(366.5)

(384.8)

3 Gross Profit

154.4

170.3

180.5

189.5

4 Selling, General, & Admin.

(74.9)

(82.6)

(87.5)

(91.9)

5 Depreciation

(7.0)

(7.5)

(9.0)

(9.5)

6 EBIT

72.6

80.2

84.0

88.1

7 Income tax at 40%

(29.0)

(32.1)

(33.6)

(35.3)

8 Unlevered Net Income

43.54

48.13

50.39

52.91

9 Plus: Depreciation

7.0

7.5

9.0

9.5

10 Less: Capital Expenditures

(7.7)

(10.0)

(9.9)

(10.4)

11 Less: Increases in NWC

(6.3)

(8.6)

(5.6)

(4.9)

12 Free Cash Flow

36.5

37.0

43.9

47.1

Earnings Forecast ($000s) 1 Sales

Free Cash Flow ($000s)

Now, V(3)  941, V(0)  804, P(0)  $12.07.

*d. Increase in NWC in year 1  12% Sales(1)  18%  Sales(0) Increase in NWC in later years  12%  change in sales New FCF: Year

Earnings Forecast ($000s) 1 Sales

8.1%

10.3%

6.0%

5.0%

516.00 5467.00

574.30

433.00 468.00

2 Cost of Goods Sold (COGS)

(313.6)

(345.7)

(366.5)

(384.8)

3 Gross Profit

154.4

170.3

180.5

189.5

4 Selling, General, & Admin. (SG&A)

(93.6)

(103.2)

(109.4)

(114.9)

5 Depreciation

(7.0)

(7.5)

(9.0)

(9.5)

6 Income tax at 40%

(21.5)

(23.8)

(24.8)

(26.1)

7 Unlevered Net Income

32.30

35.75

37.26

39.13

8 Plus: Depreciation

7.0

7.5

9.0

9.5

10 Less: Capital Expenditures

(7.7)

(10.0)

(9.9)

(10.4)

11 Less: Increases in NWC

21.8

(5.8)

(3.7)

(3.3)

12 Free Cash Flow

53.4

27.5

32.6

34.9

Base case of NWC = 18%, P(0) = $8.11 COGS & SG&A change, P(0) = $9.10 COGS & NWC change, P(0) = $6.04 SG&A & NWC change, P(0) = $12.97 Evaluate: An understanding of financial data allows the financial analyst to take information presented by a company and transform it in numerous ways to assist in financial analysis and decision making.

35.

Excel Solution Plan: Value Nike under various assumptions of growth rate and discount rates.

Execute: a. $64.64 to $82.93 1 Year 2 FCF Forecast ($ million) 3 Sales 4 Growth verses Prior Year 5 EBIT 6 Less Income Tax (25%) 7 Plus: Depreciation 8 Less Capital Expenditures 9 Less Increases in NWC (5% ΔSales) 10 Free Cash Flow

2018

2019

2020

36,397.0

40,036.7 10.0% 5,204.8 1,301.2 --182.0 3,721.6 1.09 3414.30573

43,640.0 9.0% 5,673.2 1,418.3 --180.2 4,074.7 1.19 3429.623047

2018

2019

2020

36,397.0

43,676.4 20.0% 5,677.9 1,419.5 --364.0 3,894.5 1.09 3572.91651

51,101.4 17.0% 6,643.2 1,660.8 --371.2 4,611.1 1.19 3881.100859

WACC Margin Initial Revenue Growth Rate

9.0% 13% 10%

Terminal (Continuing) Value Discounted Terminal Value Sum of DCFs Implied stock price

140,147.1 83565.16646 $103,661.7 $64.64

1 Year 2 FCF Forecast ($ million) 3 Sales 4 Growth verses Prior Year 5 EBIT 6 Less Income Tax (25%) 7 Plus: Depreciation 8 Less Capital Expenditures 9 Less Increases in NWC (5% ΔSales) 10 Free Cash Flow

WACC Margin Initial Revenue Growth Rate

9.0% 13% 20%

Terminal (Continuing) Value Discounted Terminal Value Sum of DCFs Implied stock price

183,092.4 109172.0322 $133,391.5 $82.93

2021

2022

2023

2024

47,131.2 50,430.4 53,456.2 56,129.0 8.0% 7.0% 6.0% 5.0% 6,127.1 6,556.0 6,949.3 7,296.8 1,531.8 1,639.0 1,737.3 1,824.2 --------174.6 165.0 151.3 133.6 4,420.7 4,752.0 5,060.7 5,338.9 1.30 1.41 1.54 1.68 3413.61645 3366.43913 3289.10087 3183.43491

2021

2022

2023

2024

58,255.6 64,663.7 69,836.8 73,328.6 14.0% 11.0% 8.0% 5.0% 7,573.2 8,406.3 9,078.8 9,532.7 1,893.3 2,101.6 2,269.7 2,383.2 --------357.7 320.4 258.7 174.6 5,322.2 5,984.3 6,550.4 6,974.9 1.30 1.41 1.54 1.68 4109.7223 4239.43232 4257.33164 4158.93456

b. $67.60 to $79.08 1 Year 2 FCF Forecast ($ million) 3 Sales 4 Growth verses Prior Year 5 EBIT 6 Less Income Tax (25%) 7 Plus: Depreciation 8 Less Capital Expenditures 9 Less Increases in NWC (5% ΔSales) 10 Free Cash Flow

2018

2019

2020

36,397.0

41,856.6 15.0% 5,022.8 1,255.7 --273.0 3,494.1 1.09 3205.60734

47,297.9 13.0% 5,675.7 1,418.9 --272.1 3,984.7 1.19 3353.878933

2018

2019

2020

36,397.0

41,856.6 15.0% 5,859.9 1,465.0 --273.0 4,122.0 1.09 3781.61491

47,297.9 13.0% 6,621.7 1,655.4 --272.1 4,694.2 1.19 3951.024394

WACC Margin Initial Revenue Growth Rate

9.0% 12% 15%

Terminal (Continuing) Value Discounted Terminal Value Sum of DCFs Implied stock price

147,874.2 88172.52429 $108,471.4 $67.60

WACC Margin Initial Revenue Growth Rate

9.0% 14% 15%

Terminal (Continuing) Value Discounted Terminal Value Sum of DCFs Implied stock price

173,189.6 103267.2771 $127,127.5 $79.08

2021

2022

2023

2024

52,500.7 57,225.7 61,231.5 64,293.1 11.0% 9.0% 7.0% 5.0% 6,300.1 6,867.1 7,347.8 7,715.2 1,575.0 1,716.8 1,836.9 1,928.8 --------260.1 236.3 200.3 153.1 4,464.9 4,914.1 5,310.5 5,633.3 1.30 1.41 1.54 1.68 3447.73893 3481.24596 3451.49172 3358.95331

2021

2022

2023

2024

52,500.7 57,225.7 61,231.5 64,293.1 11.0% 9.0% 7.0% 5.0% 7,350.1 8,011.6 8,572.4 9,001.0 1,837.5 2,002.9 2,143.1 2,250.3 --------260.1 236.3 200.3 153.1 5,252.4 5,772.4 6,229.0 6,597.7 1.30 1.41 1.54 1.68 4055.84119 4089.34822 4048.43614 3933.99151

c. $48.82 to $83.85 1 Year 2 FCF Forecast ($ million) 3 Sales 4 Growth verses Prior Year 5 EBIT 6 Less Income Tax (25%) 7 Plus: Depreciation 8 Less Capital Expenditures 9 Less Increases in NWC (5% ΔSales) 10 Free Cash Flow

2018

2019

2020

36,397.0

41,856.6 15.0% 5,441.4 1,360.3 --273.0 3,808.0 1.11 3430.66318

47,297.9 13.0% 6,148.7 1,537.2 --272.1 4,339.5 1.23 3522.017548

WACC Margin Initial Revenue Growth Rate

11.0% 13% 15%

Terminal (Continuing) Value Discounted Terminal Value Sum of DCFs Implied stock price

107,021.2 57217.92383 $77,936.7 $48.82

2018

2019

2020

36,397.0

41,856.6 15.0% 5,441.4 1,360.3 --273.0 3,808.0 1.09 3509.71071

47,297.9 13.0% 6,148.7 1,537.2 --272.1 4,339.5 1.18 3686.192377

WACC Margin Initial Revenue Growth Rate

8.5% 13% 15%

Terminal (Continuing) Value Discounted Terminal Value Sum of DCFs Implied stock price

183,465.0 112453.9578 $134,894.9 $83.85

2021

2022

2023

2024

52,500.7 57,225.7 61,231.5 64,293.1 11.0% 9.0% 7.0% 5.0% 6,825.1 7,439.3 7,960.1 8,358.1 1,706.3 1,859.8 1,990.0 2,089.5 --------260.1 236.3 200.3 153.1 4,858.7 5,343.3 5,769.8 6,115.5 1.37 1.52 1.69 1.87 3552.6227 3519.76807 3424.08617 3269.59565

2021

2022

2023

2024

52,500.7 57,225.7 61,231.5 64,293.1 11.0% 9.0% 7.0% 5.0% 6,825.1 7,439.3 7,960.1 8,358.1 1,706.3 1,859.8 1,990.0 2,089.5 --------260.1 236.3 200.3 153.1 4,858.7 5,343.3 5,769.8 6,115.5 1.28 1.39 1.50 1.63 3803.89752 3855.55595 3837.16866 3748.46526

d. By changing parameters you get prices from $40.27 to $102.37. To get to the lowest stock price of $40.27, the initial growth was set at the low of 10%, the EBIT margin was set at the low of 12%, and the WACC was set at the high of 11%. To get to the high stock price of $102.37, the initial growth was set at the high of 20%, the EBIT margin was set at the high of 14%, and the WACC was set at the low of 8.5%. 1 Year 2 FCF Forecast ($ million) 3 Sales 4 Growth verses Prior Year 5 EBIT 6 Less Income Tax (25%) 7 Plus: Depreciation 8 Less Capital Expenditures 9 Less Increases in NWC (5% ΔSales) 10 Free Cash Flow

2018

2019

36,397.0

40,036.7 10.0% 4,804.4 1,201.1 --182.0 3,421.3 1.11 3082.26847

WACC Margin Initial Revenue Growth Rate

11.0% 12% 10%

Terminal (Continuing) Value Discounted Terminal Value Sum of DCFs Implied stock price

86,064.5 46013.59584 $63,387.1 $40.27

2018

2019

36,397.0

43,676.4 20.0% 6,114.7 1,528.7 --364.0 4,222.1 1.09 3891.29217

WACC Margin Initial Revenue Growth Rate

8.5% 14% 20%

Terminal (Continuing) Value Discounted Terminal Value Sum of DCFs Implied stock price

225,747.4 138370.7792 $164,998.6 $102.77

2020

2021

43,640.0 47,131.2 9.0% 8.0% 5,236.8 5,655.7 1,309.2 1,413.9 ----180.2 174.6 3,747.4 4,067.2 1.23 1.37 3041.502411 2973.93689

2020

2021

2022

2023

2024

50,430.4 53,456.2 56,129.0 7.0% 6.0% 5.0% 6,051.6 6,414.7 6,735.5 1,512.9 1,603.7 1,683.9 ------165.0 151.3 133.6 4,373.8 4,659.8 4,918.0 1.52 1.69 1.87 2881.1415 2765.34539 2629.34833

2022

2023

2024

51,101.4 58,255.6 64,663.7 69,836.8 73,328.6 17.0% 14.0% 11.0% 8.0% 5.0% 7,154.2 8,155.8 9,052.9 9,777.2 10,266.0 1,788.5 2,038.9 2,263.2 2,444.3 2,566.5 ----------371.2 357.7 320.4 258.7 174.6 4,994.4 5,759.1 6,469.3 7,074.2 7,524.9 1.18 1.28 1.39 1.50 1.63 4242.516375 4508.86672 4668.06783 4704.66991 4612.35931

Evaluate: A range of values for Nike is based on the growth rate and discount rate that is assumed. Frequently, a financial analyst will not work with a single assumed input (i.e., a growth rate) but with a range of inputs to observe a range of estimates of value.

36.

Plan: All three companies are in the same industry, but it is critical when using multiples that the companies be truly comparable. Jones does not have the size or the breadth of product line that Coca-Cola and Pepsi do, so the better choice is to use Coca-Cola’s information to value Pepsi. Evaluate: Coca-Cola has a price of 41.09 and an EPS of 1.89, giving it a P/E of 41.09/1.89 = 21.74. Applying that multiple to Pepsi’s EPS results in a price of 21.74 × $3.90 = $84.79.

37.

Plan: You can use the EV/EBITDA ratio to estimate the enterprise value and then use Equation 7.19 to calculate the price per share based on that estimate. Execute: Using the EV/EBITDA ratio, you estimate CSH’s enterprise value as $5 million  9  $45 million. P0 

V0  Cash  Debt shares outstanding



45,000,000  2,000,000  10,000,000 800,000

 46.25

Evaluate: Based on an EV/EBITDA ratio of 9, you would estimate a price of $46.25. This price estimate is only valid if the ratio is a reasonable one for CSH. 38.

Excel Solution Plan: For the EV/EBITDA multiples, the equity value based on each comp can be computed as (EV/EBITDA)  EBITDA  Cash  Debt. Execute: Comp 1 Comp 2 Comp 3 Comp 4 EV/EBITDA

12.5

P/E

Equity Value  (EV/EBITDA  300)  30 40

3590

3290

3740

2990

Equity Value/100

35.9

32.9

37.4

29.9

P/E × 2

The range consistent with both sets would be $34.00 to $37.40. This includes the smallest value that is within both the P/E and EV/COGS DA ranges ($34) and the highest value within both ranges ($37.40).

Evaluate: There is no reason to expect different multiples to produce completely consistent estimates. In this case, one produces estimates from $29.90 to $37.40 and the other produces estimates from $34 to $40. 39.

Excel Solution Plan: Compute various estimates of the value of a share of Nike. Execute: a. Average P/E ratio is 23.99. Multiplied by EPS of $2.57 gives $61.66. b. Range of $33.74 ($2.57 × 13.13) to $111.59 ($2.57 × 43.42). c. Average P/B ratio is 7.4634. Multiplied by book equity of $6.23 per share gives $46.50. d. Range of $7.75 ($6.23 × 1.24) to $136.35 ($6.23 × 21.89). Evaluate: Based on various assumptions, it is possible to generate a range of estimates for the value of Nike.

40.

Excel Solution Plan: Compute various estimates of the value of a share of Nike. Execute: a. Average EV/Sales is 2.17. Multiply by $36,397 million in sales to get an estimate of Nike’s EV. Add excess cash, subtract debt, and divide by shares outstanding:

$36,397(2.17)  $5245  $3810  $51.09 1573.8

P

b. Range of $11.17 to $99.43:

P

$36,397(0.4439)  $5245  $3810  $11.17 1573.8

P

$36,397(4.2599)  $5245  $3810  $99.43 1573.8

c. Average EV/EBITDA is 19.9826. Multiply by $5219 million EBITDA to get an estimate of Nike’s EV. Add excess cash, subtract debt, and divide by shares outstanding:

$5219(19.98)  $5245  $3810  $67.18 1573.8 d. Range of $13.45 to $115.41. P

P

$5219(3.7806)  $5245  $3810  $13.45 1573.8

P

$5219(34.5263)  $5245  $3810  $115.41 1573.8

Evaluate: Based on various assumptions, it is possible to generate a range of estimates for the value of Nike. *41.

Plan: Estimate the values of the stocks of David Shoes and Boots and Andras Outdoor Corporation using comparable price to earnings ratios (P/E) and the enterprise value to EBITDA multiple. Execute: Using P/E ratio: 2.30  13.3  30.59 per share  5.4 million shares  $165.2 million

Using Enterprise Value to EBITDA ratio: 30.7 million  7.4  227.2 million  125 million debt  $102.2 million

Evaluate: Because the two firms have different levels of debt in their capital structure, the Enterprise Value to EBITDA valuation method is likely to be more accurate. 42. Market Enterprise Capitalization Value (EV) Company Name Delta Air Lines (DAL) 39,057 47,060 American Airlines (AAL) 15,295 33,745 United Continental (UAL) 22,648 32,143 Southwest Airlines (LUV) 29,431 29,119 Alaska Air (ALK) 7,236 8,103 JetBlue Airways (JBLU) 5,008 5,791 SkyWest (SKYW) 3,000 5,471 Hawaiian (HA) 1,366 1,575 Source: Morningstar and Yahoo! Finance, April 2019

EV/Sales EV/EBITDA 1.1x 6.0x 0.8x 6.3x 0.8x 5.8x 1.3x 6.5x 1.0x 7.7x 0.8x 7.3x 1.7x 6.7x 0.6x 3.4x

EV/EBIT 8.6x 10.5x 9.7x 8.9x 12.3x 19.2x 11.3x 4.8x

P/E 10.2x 11.2x 11.0x 12.4x 16.6x 27.7x 11.0x 6.1x

P/Book 2.9x 1.3x 2.3x 3.0x 1.9x 1.1x 1.5x 1.4x

All the multiples show a great deal of variation across firms. This makes the use of multiples problematic because there is clearly more to valuation than the multiples reveal. Without a clear understanding of what drives the differences in multiples across airlines, it is unclear what the “correct” multiple to use is when trying to value a new airline. 43.

First, calculate the median valuation multiple for the other seven airlines:

EV/Sales: 1.0x, EV/EBITDA: 6.5x, EV/EBIT: 10.5x, P/E: 11.2x, P/Book: 1.9x Next, calculate the relevant denominator by dividing Hawaiian’s valuation multiples by Hawaiian’s Market Cap or EV. Next, multiply by the median valuation ratio above to obtain the implied EV and Market Cap. For ratios involving EV, convert to market cap by subtracting off the difference between Hawaiian’s EV and Market Cap. Finally, solve for the implied price per share by dividing the Market Cap by 50 million (note that small differences exist due to rounding):

44.

Plan: Compute the values of Summit Systems as required in the problem. Execute: a. P  1.50/(11%  3%)  $18.75 b. Given that markets are efficient, the new growth rate of dividends will already be incorporated into the stock price, and you would receive $18.75 per share. Evaluate: The value of a share of Summit Systems with a 3% growth rate is $18.75. Once the information about the revised growth rate for Summit Systems reaches the capital market, it will be quickly and efficiently reflected in the stock price.

45.

Plan: Compute Coca-Cola’s stock price and future dividend growth rate. Execute: a. P  1.24/(8%  7%)  $124 b. Based on the market price, our growth forecast is probably too high. Growth rate consistent with market price is

g  rE  Div Yield  8%  1.24/43  5.12% which is more reasonable. Evaluate: Assuming a 7% annual growth rate in dividends, Coca-Cola Company common shares should sell for $124. Given that they actually sell for $43, it is obvious that investors, as a group, are only expecting a dividend growth rate of 5.12%. Given that the market expectation of 5.12% growth is the consensus forecast of all investors, it is likely more reasonable. 46.

Plan: Calculate the drop in value of Roybus Inc. equity shares because of the fire in Taiwan. Would you expect to make a profit in trading Roybus shares based on information about the fire? Execute: a. PV(change in FCF)  180/1.13  60/1.132  206 Change in V  206, so if debt value does not change, P drops by 206/35  $5.89 per share. b. If this is public information, in an efficient market share price will drop immediately to reflect the news, and no trading profit is possible. Evaluate: After news of the fire becomes known we would expect Roybus shares to drop by $5.89 per share. We would also expect the capital market to reduce the value of Roybus shares quickly and efficiently so that profiting from this announcement by trading Roybus shares would not be a profitable strategy.

*47.

Plan: Apply the concepts in this chapter to answer the questions in the problem. Execute: a. The market seems to assess a somewhat greater than 50% chance of success because good news would produce a price of $70 and bad news would produce a price of $18. Good news seems more likely. b. Yes, if they have better information than other investors. c. Market may be illiquid—no one wants to trade if they know Kliner has better information. Kliner’s trades will move prices significantly, limiting profits.

*48.

Plan: Apply the concepts of stock valuation and market efficiency from this chapter, and the concepts of shareholder wealth maximization and the principal-agent problem from Chapter 1. Execute and Evaluate: a. If the market is semi-strong form efficient, then it will reflect publicly known information. Thus, the stock price at 9:30 a.m. should be based on the expectations at that time. With a dividend just paid of $2, the expected next quarterly dividend will be $2 × (1 + 0.003) = $2.006. The equity cost of capital is 12% EAR. This needs to be converted to an effective quarterly rate so we can value the growing perpetuity of quarterly dividends: (1 + 0.12)0.25 – 1 = 0.02873734 = 2.873734% per quarter. The

stock price should be equal to the PV of the growing perpetuity of dividends: $2.006/(0.02873734 – 0.003) = $77.94. b. If the market is semi-strong form efficient, then it will reflect the newly announced public information. Thus, the stock price at 10:30 a.m. should be based on the new expectations at that time. With a dividend just paid of $2, the expected next quarterly dividend will be $2 × (1 + 0.005) = $2.01. As in part (a), the equity cost of capital of 12% EAR is equivalent to 2.873734% effective quarterly rate. The stock price should be equal to the PV of the growing perpetuity of dividends: $2.01/(0.02873734 – 0.005) = $84.68. c. SPB’s stock price rose from $77.94 to $84.68. The return over that hour can be calculated as follows: $77.94  1  r   $84.68 84.68  1.086 77.94  r  8.6%

1  r  

Even though 8.6% is a very high return over one hour, it is consistent with the efficient market hypothesis because prices are supposed to adjust when new information is realized. d. The dividend just paid was $2. There is a 75% probability that the next dividend will be $2 × (1.003) = 2.006 and will grow at a rate of 0.3% per quarter, and there is a 25% chance that there will be no future dividends. So the stock price after the announcement should be as follows: 0.75 

$2.006  0.25  $0  0.02873734  .003

 0.75  $77.94  0.25  $0  $58.46

Compared to the stock price at 10:30 a.m., this represents a significant negative return caused by the analyst’s information.

$84.68  1  r   $58.46 $58.46  0.69 $84.68  r  0.69  1  0.31  31%

1  r  

Again, this is consistent with the efficient market hypothesis because prices are supposed to adjust when new information is realized. e. The SD project is not consistent with shareholder wealth maximization because it eroded the value of SPB’s shares. Shareholder wealth maximization requires more than just having one quarter’s earning be higher; it requires that the present value of all expected future cash flows to be accrued to shareholders is to be maximized. Clearly, the SD project failed in that regard because once it was disclosed to investors, the stock price dropped. f.

The CEO might have implemented the SD project because it increased short-term earnings and the CEO may receive a bonus based on these earnings. In addition, if the CEO did not expect to work at SPB much longer and did not hold SPB’s stock, he

or she would not suffer any long-term loss but just reap the reward of any additional bonus. This would be an example of the principal-agent problem, where the CEO’s incentives are not aligned with shareholders’ desire for shareholder wealth maximization. 49.

Plan: The transaction costs will reduce your dollar gains, thus, reducing your return. The total cost is the commission per trade times the number of trades.

Execute: Trading each stock five times results in a total of 75 trades. The total commission cost is 75  $30  $2250. With a 12% return, your dollar return on the portfolio before trading costs is $100,000(0.12)  $12,000. Subtracting trading costs, your net dollar return is $12,000  $2250  $9750 and your net percentage return is $9750/$100,000  9.75%. Trading costs reduced your return by 2.25%. Evaluate: Unless you have special information, trading is costly without increasing your expected return, and so ends up being a net loss relative to a buy-and-hold strategy. 50.

Plan: Compare the FV of the investments. Execute: $100,000(1.12)10  $310,584.82 $100,000(1.18)10  $523,383.56 The difference is $212,798.73. Evaluate: The difference in returns compounds so that even over only 10 years the cost is substantial.

Chapter 8 Investment Decision Rules Note: All problems in this chapter are available in MyLab Finance. An asterisk (*) indicates problems with a higher level of difficulty. For a breakdown of the difficulty ratings of all of the problems in this Solutions Manual, please see the Preface – Question Difficulty Ratings. 1.

Plan: Calculate the NPV by computing the present value of the $60,000 using r  0.08. Execute: NPV  $50,000  $60,000/1.08  $50,000  $55,555.56  $5555.56 Evaluate:

This is a good investment opportunity as it produces a positive NPV. 2.

Plan: Calculate the NPV by computing the present value of the two positive cash flows using r  0.09. Execute: NPV  $100,000 

$80,000 $30,000   $100,000  $98,644.90  $1355.10 1.09 (1.09)3

Evaluate: This is a bad investment opportunity as it produces a negative NPV. Even though the total of the cash flows is more than the investment, they come later in time and are not enough to overcome the time value of money at your cost of capital. 3..

Plan: Determine the net present value of the proposal.

NPV  PVBenefits  PVCosts Execute:

 $1.08 in one year  PVBenefits  $100,000 in one year     $92,592.59 today $ today  

PVCosts  $95,000 today NPV  $92,592.59  $95,000  $2407.41 today

Evaluate: No, you should not take the contract, as the NPV of the contract is negative. This would destroy value for the firm. 4.

You are preparing to produce some goods for sale. You will sell them in one year and you will incur costs of $80,000 immediately. If your cost of capital is 7%, what is the minimum dollar amount for which you need to sell the goods in order for this to be a nonnegative NPV? Solution: Setting the NPV to zero and solving for the cash flow in one year: NPV  0  $80,000 

CF1 , CF1  $80,000  (1.07)  $85,600 (1.07)

Plan: The NPV of a project is the present value of the benefits minus the present value of the costs. Compute the NPV of the project. If NPV is positive, accept the project. If NPV is negative, reject the project. If the project is accepted, then determine how much money a lender would be willing to lend against the cash flows of the project.

Execute: NPV  PVBenefits  PVCosts

  PVBenefits  $20 million in one year   $1.10 in one year  $ today    $18.18 million PVThis year's cost  $10 million today

  PVNext year’s cost  $5 million in one year   $1.10 in one year  $ today    $4.55 million today

NPV  18.18  10  4.55  $3.63 million today Accept the project. Evaluate: a. The NPV of the project is $3.63 million. b. The firm can borrow $18.18 million today and pay it back with 10% interest using the $20 million it will receive from the government (18.18  1.10  20). The firm can use $10 million of the $18.18 million to cover its costs today, and save $4.55 million in the bank earning 10% interest to cover its cost of 4.55  1.10  $5 million next year. This leaves 18.18  10  4.55  $3.63 million in cash for the firm today, the same amount as the NPV.

Excel Solution Plan: a. Draw a timeline to show when the cash flows will occur. b. Determine the NPV of the cash flows at 6% interest and 2% interest. Execute: 0

–10,000

500

1500

10,000

500 1500 10,000   1.06 1.062 1.0610  10,000  471.70  1334.99  5583.95  $2609.36

NPV  10,000 

500 1500 10,000   1.02 1.022 1.0210  10,000  490.20  1441.75  8203.48  $135.43

Evaluate: a. Since, at a 6% interest rate, the NPV is –$2609.36, which is less than zero, you would not take this investment opportunity. b. Since, at a 2% interest rate, the NPV is $135.43, which is greater than zero, you would take this investment opportunity. 7.

Excel Solution Plan: Draw the timeline of the cash flows for the investment opportunity. Compute the NPV of the investment opportunity at 2% interest per year to determine if it is an attractive investment opportunity. 0

–1000

4000

–1000

4000

4000 1000 4000   2 (1.02) (1.02) (1.02)3  1000  3921.57  961.17  3769.29  $5729.69 Evaluate: Since the investment opportunity has a positive NPV of $5729.69, Marian should make the investment. Execute: NPV  1000 

Excel Solution Plan: We can compute the NPV of the project using Equation 8.2. The cash flows are an immediate $8 million outflow followed by an annuity inflow of $5 million per year for three years and a discount rate of 8%. Execute: a. The present value of the annuity of cash inflows is $12,890,000.

Given: Solve for PV:

I/Y

8.00%

PMT

(12.89)

Excel Formula PV(0.08,3,5,0)

NPV  $8 million  $12.89 million  $4.89 million The NPV rule dictates that you should accept this contract. b. The value of the firm will increase by $4.89 million. Evaluate: The NPV rule indicates that by making the investment, your factory will increase the value of the firm today by $4.89 million, so you should undertake the project. 9.

Plan: We can compute the NPV of the project using an approach similar to Equation 8.2. The cash flows are an immediate $100 million outflow followed by a perpetuity inflow of $30 million per year, starting in year 2, and a discount rate of 8%. We can compute the IRR using a financial calculator or spreadsheet or by setting the NPV equal to zero and solving for r. After we find the IRR, we can compute the maximum deviation allowable in the cost of capital estimate to leave the decision unchanged by subtracting the cost of capital from the IRR. Execute: Timeline:

–100

 1  30  100   1.08  0.08

NPV  

 $247.22 million

The IRR solves  1  30    100  0  r  24.16%. 1 r  r

So, the cost of capital can be underestimated by 16.16% without changing the decision. Evaluate: The NPV rule indicates that by making the investment, your factory will increase the value of the firm today by $4.885 million, so you should undertake the project. The IRR is the discount rate that sets the net present value of the cash flows equal to zero. The difference between the cost of capital and the IRR tells us the amount of estimation error in the cost of capital estimate that can exist without altering the original decision. 10.

Excel Solution Plan: We can compute the NPV of agreeing to write the book ignoring any royalty payments using Equation 8.2. The cash flows are an immediate $10 million outflow followed by an annuity inflow of $8 million per year for three years and a discount rate of 10%. We can compute the NPV of the book with the royalty payments by first computing the present value of the royalties at year 3. Once we compute the royalties at year 3, we can compute the present value of the royalties today and add that number to the NPV of agreeing to write the book ignoring any royalty payments. Execute: a. Timeline: 0

–8

NPV  10 

8 

1  1   0.1  (1.1)3 

 $9.895 million

b. Timeline: 0

–8

5(1 – 0.3) 5(1 – 03)2

First calculate the PV of the royalties at year 3. The royalties are a declining perpetuity: PV3  

5 0.1  ( 0.3) 5 0.4

 $12.5 million

so the value today is PVroyalties 

12.5 (1.1)3

 $9.391

Now add this to the NPV from part (a),

NPV  9.895  9.391  $503,381 Evaluate: The NPV rule indicates that by agreeing to write the book (ignoring any royalties), you will decrease the value of the firm today by $9.895 million, and by agreeing to write the book, including royalties, you will decrease the value of the firm by only $503,381. Therefore, Bill should not undertake either project because both will decrease the value of the firm. *11.

Excel Solution Plan: We can compute the NPV using Equation 8.2, an Excel spreadsheet, or a financial calculator. We can compute the IRR using a financial calculator or Excel spreadsheet (in some cases we can directly solve for the IRR by setting the NPV equal to zero and solving for r). After we find the IRR, we can compute the maximum deviation allowable in the cost of capital estimate to leave the decision unchanged by subtracting the cost of capital from the IRR. We can compute the length of time that the development must last to change the decision by using a financial calculator, a spreadsheet, or by setting the NPV equal to zero and solving for n (the number of years). Execute: a.

 1   1  300, 000  1  1  1  6   6  10  r  (1  r )   (1  r )  r  (1  r )  200, 000  1   1  300, 000  1    1  1   6   6  0.1  (1.1)   (1.1)  0.1  (1.1)10 

NPV  

200, 000 

 $169, 482

NPV  0, so the company should take the project. b. Setting the NPV  0 and solving for r (using a spreadsheet) the answer is IRR  12.66%. So, if the cost of capital estimate is too low by more than 2.66%, the decision will change from accept to reject. c. The new timeline is 0

N1

N  10

200,000 200,000 200,000

200,000 300,000

300,000

Setting NPV = 0 and solving for N gives:

NPV  

200, 000 

0.1

   300, 000   1 1 1  0 1  1  N   N  10   (1  0.1)   (1  0.1)  0.1  (1  0.1) 

 200, 000  300, 000 300, 000     200, 000 1 1   0   10    N  0.1 0.1  (1  0.1)    (1  0.1)  0.1  

   2,000,000  (1  0.1) 

 5,000,000  1,156,629.87  

3,843,370.13  1.921685066  1.1N 2,000,000 ln 1.921685066  ln 1.1N  N  ln1.1 N

ln 1.921685066  6.85 ln1.1

d. Timeline: 0

200,000 200,000 200,000

200,000 300,000

 1   1  300, 000  1  1  1  6   6  10  r  (1  r )   (1  r )  r  (1  r )  200, 000  1   1  300, 000  1    1  1   6   6  0.14  (1.14)   (1.14)  0.14  (1.14)10 

NPV  

200, 000 

 $64.816

300,000

e. Since the cash flows have not changed, the IRR is still 12.66%. So, if the cost of capital estimate is too high by more than 1.34%, the decision will change from reject to accept.

The timeline will be the same as shown for part (c). Setting the NPV  0 and solving for N gives NPV  

200, 000 

   300, 000   1 1 1  0 1  1  N   N  10  0.14  (1  0.14)   (1  0.14)  0.14  (1  0.14) 

 200, 000  300, 000 300, 000     200, 000 1 1   0   10    N  0.14 0.14  (1  0.14)    (1  0.14)  0.14  

 3,571, 428.57  $578, 022.45 

   1, 428,571.43

N  (1  0.14) 

2,993, 406.12  2.095384286  1.1N 1, 428,571.43 ln 2.095384286  ln 1.14 N  N  ln1.14 N

ln 2.095384286  5.65 ln1.14

Evaluate: When your cost of capital is greater than your IRR, your project will have a negative NPV, and according to the NPV rule you should not accept this project. In addition, when the cost of capital is higher, it is more costly to wait for future cash flows, so the development process must end sooner in order for the project to break even. 12.

Excel Solution a.

b. The IRR is the point at which the line crosses the x-axis. In this case, it falls very close to 13%. Using Excel, the IRR is 12.72%. c. Yes, because the NPV is positive at the discount rate of 12%. d. The discount rate could be off by 0.72% before the investment decision would change.

13.

5000/500  10 months. You will recover your expenditure for the sign in 10 months.

14.

Excel Solution The IRR is 39.45%. The IRR rule agrees with the NPV rule.

15.

Excel Solution Plan: We can compute the IRR by first computing the NPV and find the rate that sets that NPV equal to zero. In order to determine how many IRRs will set NPV equal to zero we can plot NPV as a function of the discount rate. Execute: Timeline: 0

–8

IRR is the r that solves NPV  0 8 1   10   1   r  (1  r )3 

To determine how many solutions this equation has, plot the NPV as a function of r.

From the plot there is one IRR of 60.74%. Since the IRR is much greater than the discount rate, the IRR rule says write the book. Since this is a negative NPV project (from Problem 10(a)), the IRR gives the wrong answer. Evaluate: In this case, there is only one IRR (the intercept on the x-axis) and the IRR is greater than the discount rate. According to the IRR rule, Bill should accept the project;

yet, because the NPV is negative, the NPV rule states that Bill should not accept the project. Although the two rules are conflicting, the NPV rule tends to be more reliable. 16.

Excel Solution Plan: We can compute the IRR by first computing the NPV and finding the rate that sets that NPV equal to zero. In order to determine how many IRRs will set NPV equal to zero, we can plot NPV as a function of the discount rate. Execute: Timeline: 0

–8

5(1 – 0.3) 5(1.03)2

From Problem 10(b), the NPV of these cash flows is 8 1  1  5  NPV  10   1     3  r (1  r )  (1  r )3  r  0.3 

Plotting the NPV as a function of the discount rate gives

The plot shows that there are two IRRs: 7.165% and 41.568%. The IRR does give an answer in this case, so it does not work. Evaluate: In this case, there are two IRRs (the intercepts on the x-axis) and the IRR does not provide an answer, so we cannot use the IRR rule and, therefore, the IRR rule does not work.

17.

Excel Solution Plan: Setting the NPV to zero, we can solve for the IRR and we can then compute the NPV of the project using Equation 8.2. The cash flows are an immediate $50,000 cash flow followed by an annuity of $4,400 per year and a discount rate of 15%.

Execute: The timeline of this investment opportunity is 0

50,000

–4400

Computing the NPV of the cash flow stream NPV  50, 000 

4400 

 1 1  12  r  (1  r ) 

To compute the IRR, we set the NPV equal to zero and solve for r. Using the annuity spreadsheet gives N Given:

I/Y

Solve for Rate:

PMT

50,000.00 4400

0.8484%

Excel Formula

RATE(12,4400,50000,0)

The monthly IRR is 0.8484, so since

1.008484   1.106696 12

The 0.8484% monthly corresponds to an EAR of 10.67%. Smith’s cost of capital is 15%, so according to the IRR rule she should turn down this opportunity. Let’s see what the NPV rule says. If you invest at an EAR of 15%, then after one month you will have (1.15)1/12  1.011715,

so the monthly discount rate is 1.1715%. Computing the NPV using this discount rate gives NPV  50, 000 

  1 1  12  0.011715  (1.011715)  4400

 $1010.06

This is positive, so the correct decision is to accept the deal. Smith can also be relatively confident in this decision. Based on the difference between the IRR and the cost of capital, her cost of capital would have to be 15  10.67  4.33% lower to reverse the decision. Evaluate: The internal rate of return (IRR) investment rule is based on the concept that if the return on the investment opportunity you are considering is greater than the return on other alternatives in the market with the equivalent risk and maturity, you should undertake the investment opportunity. In this case, the IRR is less than the discount rate so, according to the IRR rule, she should turn down this opportunity. The NPV rule states

that when making an investment decision, take the alternative with the highest NPV. Choosing this alternative is equivalent to receiving its NPV in cash today. In this case, the NPV is positive, so the correct decision is to accept the deal. 18.

Excel Solution Plan: We can compute the NPV of the project using Equation 8.2. Setting the NPV to zero, we can then solve for the IRR and then analyze the results according to the IRR rule. Execute: a. Timeline: 0

–5

1 – 0.1 1 – 0.1

1 – 0.1

0.1

The PV of the profits is PVprofits

1 1  1   r  (1  r )10 

The PV of the support costs is PVsupport 

0.1

r NPV  5  PVprofits  PVsupport

  0.1 1  1  5   1    10   r   (1  r )   r

r  5.438761% then NPV  $721,162 r  2.745784% then NPV  0 r  10.879183% then NPV  0 b. From the answer to part (a) there are two IRRs: 2.745784% and 10.879183%. c. The IRR rule says nothing in this case because there are two IRRs. Evaluate: The internal rate of return (IRR) investment rule is based on the concept that if the return on the investment opportunity you are considering is greater than the return on other alternatives in the market with the equivalent risk and maturity, you should undertake the investment opportunity. In this case, there are two IRRs and, therefore, the IRR rule is not useful. The NPV rule states that when making an investment decision, take the alternative with the highest NPV. Choosing this alternative is equivalent to receiving

its NPV in cash today. In this case, the NPV is positive only when the cost of capital is at 6% and therefore they should accept the project at the cost of capital of 6% only. 19.

–120

–2

Computing the NPV of the cash flow stream NPV  120 

20 

 1 2  1  10  r  (1  r )  r (1  r )10

You can verify that r  0.02924 or 0.08723 gives an NPV of zero. There are two IRRs, so you cannot apply the IRR rule. Let’s see what the NPV rule says. Using the cost of capital of 8% gives NPV  120 

20 

 1 2  1  10  r  (1  r )  r (1  r )10

 2.621791

So the investment has a positive NPV of $2,621,791. In this case, the NPV as a function of the discount rate is n-shaped.

If the opportunity cost of capital is between 2.93% and 8.72%, the investment should be undertaken. Evaluate: The internal rate of return (IRR) investment rule is based on the concept that if the return on the investment opportunity you are considering is greater than the return on other alternatives in the market with the equivalent risk and maturity, you should undertake the investment opportunity. In this case, there are two IRRs and therefore the IRR rule is not useful. The NPV rule states that, when making an investment decision, take the alternative with the highest NPV. Choosing this alternative is equivalent to receiving its NPV in cash today. In this case, the NPV is positive and the investment should be undertaken if the opportunity cost of capital is between 2.93% and 8.72%. 20.

Excel Solution Plan: Because these cash flows have multiple sign changes over time, there may be more than one IRR. The project should only be accepted if the project’s NPV is positive when evaluated at the cost of capital. Execute: a. The timeline of the investment opportunity is 0

–4.55

3.5

–6

There are two sign changes so there are up to two IRRs. Upon viewing the NPV profile in part (b), we can see that there are two discount rates where the NPV is zero; thus, there are two IRRs. b. The NPV profile is shown below.

Evaluate: c. Since the NPV is greater than zero at the actual cost of capital, the investment should be accepted. 21.

Plan: Compute the NPV of the project. Execute: The timeline of the investment opportunity is Timeline: 0

Investment

To have an NPV greater than zero, the initial investment must be less than the PV of the five-year annuity of $1 million per year. NPV  0  Initial Investment 

1 

1  1  5  0.12  (1  0.12) 

Initial investment  3.6 million Evaluate: The most you could pay for the project and achieve the 12% annual return is $3.6 million. *22.

Excel Solution Plan: We can compute IRR of this investment by first computing the NPV by subtracting the stabilization costs of the project from the operating profit. We can then find the IRR of the investment by plotting the NPV as a function of the discount rate. Execute: Timeline: 0

–250

–5

PVoperating profits 

20 

 1 1  20  r  (1  r ) 

In year 20, the PV of the stabilization costs is PV20 

5 r

So the PV today is PVstabilization costs 

NPV  250 

5  

(1  r ) 20  r 

20 

 1 1 5  1  20  20   r  (1  r )  (1  r )  r 

Plotting this out gives

So no IRR exists. Evaluate: In order for a project to be profitable, IRR has to be greater than the discount rate. In this case, since there is no IRR, NPV is less than zero no matter what the discount rate; thus, the project should be rejected. 23.

NPV  500,000 

125,000  1  1    500,000  736,154  236,154 .11  1.1110 

b. As the NPV is positive, you should proceed with the project because it will increase the value of the company even though it takes longer than the preferred payback period.

24.

Plan: In order to implement the payback rule, we need to know whether the sum of the inflows from the project will exceed the initial investment before the end. We can compute the NPV of the project using Equation 8.2. Execute: Since it takes one year to make the movie, and the cost of the movie is $10 million paid up front, there will be no cash flow at date 1 on the timeline. The movie will be released one year from now and will then start generating revenues over the next year that total $5 million. We will assume these revenues are collected at the end of the year so they would be at date 2 on the timeline (the end of the second year). The complete timeline would look like the following: 0

–10

It will take about five years to pay back the initial investment, so the payback period is about five years. You will not make the movie given a requirement for a two-year payback.

æ ö 1 1 ç ÷´ NPV = -10 + + 12 4 2 rç 1+ r 1+ r ÷ø 1+ r è 5

(

= -10 +

)

(1 + r )

(

+ 2

)

(

)

æ ö 1 1 ç 1÷´ 4 2 0.10 ç 1.10 ÷ø 1.10 è 2

(

)

(

)

= -$628,321.58

The NPV is negative, so you would reject the project. The NPV agrees with the payback rule in this case. Evaluate: While simple to compute, the payback rule requires us to use an arbitrary cutoff period in summing the cash flows. Further, the payback rule does not discount future cash flows. Instead it simply sums the cash flows and compares them to a cash outflow in the present. In this case, you will not make the movie because your cutoff point is two years and it will take about five years to pay back the initial investment. The NPV of this project came back as negative, so this also agrees with the payback rule and the movie should therefore not be made.

25.

Excel Solution Plan: We can compute the IRR by rearranging Equation 8.2 so that NPV equals zero and solving for r. Once we compute r, we can compute the NPV of both projects using Equation 8.2. 0

–100

–50

a. NPVA  50  25  1 r

20 (1  r )



20 (1  r )



15 (1  r ) 4

IRR (A)  24% NPVB  100  20 

1 r

(1  r )



50 (1  r )



60 (1  r ) 4

IRR (B)  21%

NPVA  50 

1  0.05

20 (1  0.05)



20 (1  0.05)



15 (1  0.05)4

 21.57 NPVB  100 

20 1  0.05



40 (1  0.05)



50 (1  0.05)



60 (1  0.05) 4

 47.88

c. Evaluate: The IRR and NPV rank differently due to the difference in the initial investment. Investment A earns a higher rate of return on a smaller investment. 26.

Plan: Compute the timeline and the NPV and IRR for each project. Decide which one to accept. Execute: a. Timeline: 0

–10

1.5

NPVA 

2 r

1.5(1.02) 1.5(1.02)2

 10

Setting NPVA  0 and solving for r IRRA  20% NPVB 

1.5 r  0.02

 10

Setting NPVB  0 and solving for r 1.5  10  r  0.02  0.15  r  17% r  0.02

So, IRRB  17%

Based on the IRR, you always pick Project A. b. Substituting r  0.07 into the NPV formulas derived in part (a) gives NPVA  $18.5714 million NPVB  $20 million So the NPV says take B. c. Here is a plot of NPV of both projects as a function of the discount rate. The NPV rule selects A (and so agrees with the IRR rule) for all discount rates to the right of the point where the curves cross.

NPVA  NPVB 2 1.5  r r  0.02 2 r  0.02  r 1.5 1.5r  2r  0.04 0.5r  0.04 r  0.08 Evaluate: Based on the IRR, you always pick Project A. Based on the NPV, take B. So the IRR rule will give the correct answer for discount rates greater than 8%. 27.

Excel Solution Plan: Compute the NPV and the IRR of each project. Execute: The timeline of the investment opportunity is

–100

NPVA  100 

NPVB  100 

1  0.11 50 1  0.11



30 (1  0.11)



40 (1  0.11)

40 3



(1  0.11) 30 (1  0.11)

50 (1  0.11)4 20 (1  0.11) 4

 9.06

 12.62

b. Solving for the discount rate that results in zero NPV IRR(A)  14.7% IRR(B)  17.8% c. This can be solved two ways. One is to calculate the IRR of the difference in cash flows between the two projects. The other is to create NPV profiles of both investments and determine the cost of capital at which the NPVs of both projects are the same. 0

–100

–25

–10

NPV  0 

25 1 r



10 (1  r )



10 (1  r )



30 (1  r ) 4

IRR  5.567%

NPV Profiles of Investments A and B:

The profiles cross at a cost of capital of 5.567%. Evaluate: d. You should invest in B, as it has a higher NPV. 28.

Excel Solution Plan: Compute the cost of ownership of the asset as well as the cost of leasing the asset. Select the option with the lowest cost. Execute: The timeline of the investment opportunity is 0

Ownership

–40

Leasing:

NPVownership  40 

–2

–10

2 

1  1  5  0.07  (1  0.07) 

 48.2

NPVleasing 

10 

1  1  5  0.07  (1  0.07) 

 41

Evaluate: The cost of leasing is less, so the firm should lease the equipment. 29.

Plan: While you recognize that Project A has a higher IRR, you immediately notice that Project B is a much larger-scale undertaking so it is not appropriate to use IRR to compare the two mutually exclusive investments. You need to evaluate the two investments using NPV instead. Execute: The NPVs of the two projects are calculated as follows: NPVProject A  2000 

13,200  $10,000 1  .10 

NPVProject B  150,000 

181,500  $15,000 1  .10 

Evaluate: Respectfully, you respond to your boss that ranking the two projects that have different scales by IRR would lead to an incorrect decision; in fact, Project B is the better project as it has NPV of $15,000 compared to A that has NPV of only $10,000. Thus, your firm should accept Project B. 30.

Plan: The paybacks and NPVs need to be determined. We should emphasize the NPV results to the CEO as they indicate which movie type will add value to Garneau Cinemas. Since payback also seems to be a concern to our company, we should report those results too. Execute: The paybacks and NPVs need to be determined. The paybacks of the two projects can be determined by finding the year when the cumulative cash flows of each project sum to zero:

So, for the comedy, the payback is three years and for the thriller the payback is five years.

The NPVs are calculated as follows: NPVComedy  20 

12 4 4 2   $1.41 million 2 3 1  .10  1  .10  1  .10  1  .10 4

NPVThriller  40 

 20 5  1 1 25     $6.70 million 1  4  1  .10  .10  1  .10   1  .10  1  .10 6

Evaluate: The comedy has a negative NPV so it should not be undertaken even though its payback falls within Garneau Cinema’s normal payback criteria for acceptance. The thriller has a positive NPV, so that indicates that it should be undertaken. However, you need to advise your CEO that the thriller’s payback is longer than what is normally required. Based on NPV alone, the thriller should be undertaken. We would likely downplay the fact that the thriller’s payback is longer than required. However, the CEO may have other reasons why she pays attention to payback—for instance, she may be aware that longer term cash flow projections for movies tend to be very error prone and are often overestimated by the movie promoters. If this is the case, then it would be best to re-estimate the longer term cash flow projections and try to get a better estimate of the project’s NPV. 31.

Plan: We need to calculate the IRR and NPV for each project. We should immediately note that Project B is an unconventional (or borrowing) type project, so we need to be careful how to interpret the IRR result. Execute: Using our financial calculator or solving manually (since there are only two cash flows) we find the IRR of A is –50% and the IRR of B is –50%. In both cases, the IRR is less than our cost of capital of 8%. The NPVs for the two projects are calculated as follows: NPVA  200  NPVB  200 

100  $107.41 1  .08

100  $107.41 1  .08

Evaluate: It would be misleading to use the normal IRR rule for Project B as it is an unconventional project. Project A has IRR of –50% and NPV of –$107.41; both criteria correctly indicate the decision should be to reject Project A. Project B also has an IRR of –50%, but because Project B is an unconventional project, we would want it to have an IRR less than the cost of capital, and it does, so this indicates that B should be accepted. The NPV of B is positive and equals $107.41; this also indicates B should be accepted. To conclude, we should reject Project A and accept Project B. 32.

Plan: We have the IRRs for the two projects, but we should immediately note that Project A gets more of its cash flows earlier and this boosts the IRR compared to Project B, which gets more of its cash flows later. We know this can be a problem when comparing mutually exclusive projects, so we had better calculate NPV to determine which project is best.

Execute: The NPVs for the two projects are calculated as follows: NPVA  10,000 

8000 4000 3000    $2466.97 2 1  .12  1  .12  1  .12 3

NPVB  10,000 

2600 6000 8000    $2798.83 2 1  .12  1  .12  1  .12 3

Evaluate: Indeed, the different timing of the cash flows across the two investments caused the IRR results to be misleading regarding which of the two mutually exclusive projects to select. The NPV of B is clearly higher, so B is the preferred project and should be accepted. 33.

You are trying to decide between two mobile phone carriers. Carrier A requires you to pay $200 for the phone and then monthly charges of $60 for 24 months. Carrier B wants you to pay $100 for the phone and then monthly charges of $70 for 12 months. Assume you will keep replacing the phone after your contract expires. Your cost of capital is 4%. Based on cost alone, which carrier should you choose? Solution: This is an equivalent annual cost (EAC) problem applied to a monthly setting. First we need to convert the cost of capital into an effective monthly rate using Equation 5.1: Equivalent n-Period Effective Rate  1  r   1 n

So converting from an EAR to a monthly rate we get: 1 1 Equivalent -Period (or Monthly) Effective Rate  1  .04  12  1 12 So the Montlhy Effective Rate  0.00327374  0.327374%

Now we can find present value of the costs of each project and convert into equivalent monthly costs using the same method we used for EAC examples in the chapter.

é ù 60 1 ê1ú = $1,582.71 0.00327374 ê (1.00327374 )24 ú ë û ù EMCA é 1 ê1ú Equivalent Montly Cost of A (EMCA ): 1582.71= 0.00327374 ê (1.00327374 )24 ú ë û 1582.71 ´ 0.00327374 \ EMCA = = $68.68 é ù 1 ê1ú ê (1.00327374 )24 ú ë û PV of costs of A = 200 +

é ù 70 1 ê1ú = $922.40 0.00327374 ê (1.00327374 )12 ú ë û ù EMC B é 1 ê1ú Equivalent Montly Cost of B (EMC B ): 922.40= 0.00327374 ê (1.00327374 )12 ú ë û 922.40 ´ 0.00327374 \ EMC B = = $78.51 é ù 1 ê1ú ê (1.00327374 )12 ú ë û PV of costs of B = 100 +

Choose carrier A because it is cheaper once all the costs are considered. 34.

Plan: Compute the NPV and the equivalent annual annuity (EAA) of each bus. Choose the bus with the lowest costs. Execute: The timeline of the investment opportunity is 0

Old Reliable

–200

–4

Short and Sweet

–100

–2

NPVOld Reliable  200 

4 

1  1  7  0.11  (1  0.11) 

 218.85

EAA Old Reliable 

218.85 

1   1  (1  0.11) 7   

0.11

EAA Old Reliable  46.44

NPVShort and Sweet  100 

2 

1   1  (1  0.11) 4  

0.11 

 106.20

106.20 

EAAShort and Sweet 

 1 1  4   (1  0.11) 

0.11

EAAShort and Sweet  34.23

Evaluate: The annual cost of the Short and Sweet bus is less, so they should buy this bus. 35.

Plan: Compute the NPV and the equivalent annual annuity (EAA) of this investment opportunity. Determine the lowest value enhancing bid. Execute: The timeline of the investment opportunity is

Timeline:

NPV  15 

–15

–2

2 

1  1  3  0.10  (1  0.10) 

 19.97 19.97 

EAA 

 1 1  3  0.10  (1  0.10) 

EAA  8.032

Evaluate: Hassle-Free could bid as little as $8032 per year and increase its value. 36.

Plan: Compute the NPV and the equivalent annual cost (EAC) of each copier. Choose the copier with the lowest costs. Solution: This is an equivalent annual cost problem. Compare the EAC of buying and maintaining the copier to leasing it.

PV of costs  2000  EAC : 3684.95 

400  1  1    3684.95 .06  1.06 5 

EAC  1  3684.95  874.79 1   , so EAC  .06  1.06 5   1  1   5  .06 .06 1.06  

So, since buying and maintaining it is equivalent to paying $874.79 per year for five years, $874.79 is the most you would pay to lease it. 37.

Plan: Compute the NPVs of the alternatives. Select the alternative with the highest NPV. Execute: a. NPV

Use of Facility NPV/Use of Facility

$2 million

100%

$1 million

60%

1.67

$1.5 million

40%

3.75

b. They should invest in B and C. Together, these result in an NPV of $2.5 million, which is greater than the $2 million NPV earned by A alone. 38.

Plan: Compute the NPV and profitability index of each proposed investment. Select the best combination of investments. Execute: Project

NPV

Profitability Index

Parkside Acres

106,143

0.16

Real Property Estates

120,523

0.15

Lost Lake Properties

40,392

0.06

Overlook

80,131

0.53

Evaluate: The PI implies that Overlook and Parkside Acres should be selected. The alternative investment opportunities that meet the resource constraint are (i) Real Property Estates alone or (ii) Lost Lake and Overlook. These alternatives generate lower NPVs, so in this case the PI rule gives the correct answer, although as the text explains, this need not always be the case when the complete budget is not used by taking the projects in order.

39.

Plan: Compute the NPV and profitability index of each proposed investment. Select the best combination of investments. Execute: Project

NPV/Headcount

1.01

5.1

1.27

6.3

III

1.47

5.5

1.25

8.3

2.01

5.0

Evaluate: a. The PI rule selects Projects V, III, and II. These are also the optimal projects to undertake (as the budget is used up fully taking the projects in order). b. The additional constraint that only 12 research scientists are available will alter the set of projects that can be selected and the optimum number of projects to select. Specifically, Project V is the most attractive project with the highest PI. But since Project V requires 12 research scientists, selecting it would mean no other projects could be selected. (Taking Project V would also only use $30 million of the $60 million available.) The PI rule selects Projects II, II, IV, and I because these projects have the highest PI and use the entire $60 million and 12 research scientists. However, this choice of projects does not maximize NPV. This solution shows that it may be optimal to skip some projects in the PI ranking if they will not fit within the budget and resource constraints.

Chapter 9 Fundamentals of Capital Budgeting Note: All problems in this chapter are available in MyLab Finance. An asterisk (*) indicates problems with a higher level of difficulty. For a breakdown of the difficulty ratings of all of the problems in this Solutions Manual, please see the Preface – Question Difficulty Ratings. 1.

Plan: We can compute the total capitalization of the machine by adding the total cost of transporting and installing the machine to the initial cost of purchasing the machine, and this will provide us with the total cost of the machine that we must depreciate for tax purposes over the five years of the machine’s life. In order to compute the annual CCA

deduction of the machine, we can then take the total capitalization of the machine and divide it by the depreciable life of the machine. Execute: Capitalization of machine: $10,050,000 Annual CCA deduction: $10,050,000/5  $2,010,000 Evaluate: Rather than expensing the $10,050,000 it costs to buy, ship, and install the machine in the year it was bought, tax laws require you to depreciate, for tax purposes, the $10,050,000 over the depreciable life of the equipment. Assuming the equipment has a five-year depreciable life and that we use the straight-line method to calculate CCA, we would deduct $10,050,000/5  $2,010,000 per year for five years. 2.

Plan: We need four items to calculate incremental earnings: (1) incremental revenues, (2) incremental costs, (3) CCA, and (4) the marginal tax rate. Execute: Annual incremental earnings  (Revenues  Costs  CCA)  (1  tax rate) Annual incremental earnings  (4  1.2  2.01)  (1  0.35)  $513,500 Evaluate: These incremental earnings are an intermediate step on the way to calculating the incremental cash flows that would form the basis of any analysis of the project. The cost of the equipment does not affect earnings in the year it is purchased, but does so through the depreciation expense in the following five years. Note that the depreciable life, which is based on accounting rules, does not have to be the same as the economic life of the asset—the period over which it will have value.

Plan: We can compute the incremental revenues by taking the percentage increase in sales of the 100,000 units multiplied by the $20 sales price per unit. Execute: Incremental revenues  (0.20  100, 000)  $20  $400, 000. Evaluate: A new product typically has lower sales initially, as customers gradually become aware of the product. Sales will then accelerate, plateau, and ultimately decline as the product nears obsolescence or faces increased competition. Similarly, the average selling price of a product and its cost of production will generally change over time. Prices and costs tend to rise with the general level of inflation in the economy.

Plan: We can compute the level of incremental sales associated with introducing the new pizza assuming that customers will spend the same amount on either version by using the sales of the new pizza and the lost sales of the original pizza (40% of customers who switched to the new pizza multiplied by the $20 million in new sales). We can compute the level of incremental sales associated with introducing the new pizza assuming that 50% of the customers will switch to another brand by using the sales of the new pizza and the lost sales of the original pizza from the customers who would not have switched brands. Execute: a. Sales of new pizza  Lost sales of original  20  0.40(20)  $12 million.

b. Sales of new pizza  Lost sales of original pizza from customers who would not have switched brands  20  0.50(0.40)(20)  $16 million. Evaluate: More incremental sales are generated if 50% of the customers who will switch from Pisa Pizza’s original pizza to its healthier pizza will switch to another brand if Pisa Pizza does not introduce a healthier pizza than just the incremental sales associated with introducing the new pizza. A new product typically has lower sales initially, as customers gradually become aware of the product. Sales will then accelerate, plateau, and ultimately decline as the product nears obsolescence or faces increased competition. Similarly, the average selling price of a product and its cost of production will generally change over time. Prices and costs tend to rise with the general level of inflation in the economy. 5.

Plan: We need four items to calculate incremental earnings: (1) incremental revenues, (2) incremental costs, (3) depreciation, and (4) the marginal tax rate. Execute: Year

Incremental Earnings Forecast ($000s) 1

Sales of Mini Mochi Munch

9,000

7,000

Other Sales

2,000

Cost of Goods Sold

(7,350)

(6,050)

Gross Profit

3,650

2,950

Selling, General & Admin.

(5,000)

—

Depreciation

—

EBIT

(1,350)

2,950

Income Tax at 35%

473

(1,033)

Unlevered Net Income

(878)

1,918

Evaluate: These incremental earnings would form the basis of any analysis of the project. There is no cost of equipment in this project that will affect the earnings of the project. Net income is negative in the first year because the additional selling, general, and administrative costs occurred only in the first year. 6.

Plan: We can compute the incremental impact on this year’s EBIT of the drop in price by subtracting the gross profit without the price drop from the gross profit with the price drop. We can compute the incremental impact on EBIT for the next three years of a price drop in the first year from the additional sales on ink cartridges by finding the change in EBIT from ink cartridge sales, which will be the incremental impact on EBIT for years 2 and 3. Note that for year 1, we must remember to subtract the incremental impact on EBIT from the price drop in year 1. Execute:

a. Change in EBIT

 Gross profit with price drop  Gross profit without price drop



 25,000  (300  200)  20,000  (350  200)



 $500,000

b. Change in EBIT from ink cartridge sales  25,000  $75  0.70  20,000  $75  0.70 

 $262,500 Therefore, incremental change in EBIT for the next three years is Year 1:

$262,500  500,000  $237,500

Year 2:

$262,500

Year 3:

$262,500

Evaluate: A new product typically has lower sales initially, as customers gradually become aware of the product. Sales will then accelerate, plateau, and ultimately decline as the product nears obsolescence or faces increased competition. Similarly, the average selling price of a product and its cost of production will generally change over time. Prices and costs tend to rise with the general level of inflation in the economy. 7.

Plan: The difference between incremental earnings and incremental free cash flows is driven by the equipment purchased. We need to recognize the cash outflow associated with the purchase in year 0 and add back the CCA deductions (which were assumed to be the same as the depreciation expenses in Problem 1) from years 1 to 5, as they are not actually cash outflows. Execute: Free cash flows  After-tax earnings  CCA  Capital expenditures  Changes in NWC FCF (this year)  $10,050,000 FCF (for each of the next five years)  513,500  2,010,000  $2,523,500 Evaluate: By recognizing the outflow from purchasing the equipment in year 0, we account for the fact that $10,050,000 left the firm at that time. By adding back the CCA in years 1 to 5, we adjust the incremental earnings to reflect the fact that the CCA deduction is not a cash outflow.

Excel Solution Plan: We can project the net working capital needed for this operation by adding cash, inventory, and receivables and subtracting payables. Execute: Net working capital in this problem is the sum of cash, accounts receivable, and inventory (Lines 1, 2, and 3 below) less accounts payable (Line 4). Line 5 is net working capital and Line 6 is the changes in working capital from year to year. For example, net working capital in year 1 was 14 and in year 2 it grew to 19, so the increase in NWC, as computed on Line 6 for year 2, is 5. The firm must add 5 to working capital in year 2, so it represents a reduction in cash flow available to investors. Year 0 Year 1

Year 2

Year 3

Year 4 Year 5

Cash

Accounts Receivable

Inventory

Accounts Payable

Net working capital (1  2  3  4)

Increase in NWC

-4

Since an increase in NWC is equivalent to a negative cash flow, the cash flow effects are as follows:

Cash Flow

Year 0

Year 1

Year 2

Year 3

Year 4

Year 5

–14

–5

–6

–1

Evaluate: Most projects will require the firm to invest in net working capital. We care about net working capital because it reflects a short-term investment that ties up cash flow that could be used elsewhere. Note that whenever net working capital increases, reflecting additional investment in net working capital, it represents a reduction in cash flow that year. 9.

Excel Solution Plan: In order to compute the net working capital for each year we need to compute the receivables and payables for each year as a percentage of sales and COGS (receivables are 15% of sales, and payables are 15% of COGS). Execute: 0

Sales

$23,500

$26,438

$23,794

$8,566

COGS

$ 9,500

$10,688

$ 9,619

$3,483

Receivables:

$ 3,525

$ 3,966

$ 3,569

$1,285

Payables:

$ 1,425

$ 1,603

$ 1,443

$ 522

NWC:

$ 2,100

$ 2,363

$ 2,126

$ 762

$ 2,100

$ 236

$1,364

 NWC (Required Investment):

263

Excel Solution Plan: We need four items to calculate incremental earnings: (1) incremental revenues, (2) incremental costs, (3) CCA, and (4) the marginal tax rate. Earnings include non-cash charges, such as CCA, but do not include the cost of capital investment. To determine the project’s free cash flow from its incremental earnings, we must adjust for these differences. We need to add back CCA to the incremental earnings to recognize the fact that we still have the cash flow associated with it. Execute: Solution: Note—we have assumed any incremental cost of goods sold is included as part of operating expenses. a. Year

Incremental Earnings Forecast ($000s) 1

Sales

125.0

160.0

Operating Expenses

(40.0)

(60.0)

CCA

(25.0)

(36.0)

EBIT

60.0

64.0

Income tax at 35%

(21.0)

(22.4)

Unlevered Net Income

39.0

41.6

Free Cash Flow ($000s)

Plus: CCA

25.0

36.0

Less: Capital Expenditures

(30.0)

(40.0)

Less: Increases in NWC

(2.0)

(8.0)

Free Cash Flow

32.0

29.6

Evaluate: These incremental earnings are an intermediate step on the way to calculating the incremental cash flows that would form the basis of any analysis of the project. Earnings are an accounting measure of the firm’s performance. They do not represent real profits, and a firm needs cash. Thus, to evaluate a capital budgeting decision, we must determine its consequences for the firm’s available cash. 11.

Plan: Earnings include non-cash charges, such as CCA, but do not include the cost of capital investment. To determine the project’s free cash flow from its incremental earnings, we must adjust for these differences. We need to add back CCA to the incremental earnings to recognize the fact that we still have the cash flow associated with it. Execute: FCF  Unlevered net income  CCA  CapEx  Increase in NWC  250  100  200  10  $140 million. Evaluate: Earnings are an accounting measure of the firm’s performance. They do not represent real profits, and a firm needs cash. Thus, to evaluate a capital budgeting decision, we must determine its consequences for the firm’s available cash.

12.

Excel Solution This opportunity cost lowers the incremental earnings of the SPI Phone 86 project by the after-tax earnings that they would have otherwise earned had they rented out the space instead. This would be a decrease in incremental earnings of –$200,000  (1  0.40)  –$120,000 per year for the 4 years.

*13.

Excel Solution Plan: Incremental EBITDA per year: $40,000  20,000 = $20,000 Incremental CapEx: $150,000 + $50,000 = $100,000 Execute: Replacing the machine increases EBITDA by $40,000 – $20,000 = $20,000. Therefore, FCF excluding CCA tax shields will increase by (20,000) × (1 – 0.45) = $11,000 in years 1 through 10.

PV 

11,000  1  1  $67,590.24  .10  1.1010 

In year 0, the initial cost of the machine is $150,000. Selling the current machine for $50,000 generates an incremental CapEx of $150,000  $50,000 = $100,000. Thus, the year 0 cash flow is $100,000.

In addition, the incremental CapEx generates CCA tax shields. The PV of CCA tax shields is  .10  1  100,000  .40  .45  2  PV    $34,363.64 .10  .40

1  .10 

The NPV of replacing the machine is thus NPV = $100,000 + 67,590.24 + 34,363.64 = $1,953.87 Evaluate: Since the NPV ≥ 0, yes, your company should replace the old machine. Even though the decision has no impact on revenues, it still matters for cash flows because it reduces costs. Further, both selling the old machine and buying the new machine involve cash flows with tax implication. *14.

Excel Solution Plan: We can use Equation 9.9 to evaluate the free cash flows associated with each alternative. Note that we only need to include the components of free cash flows that vary across each alternative. For example, since NWC is the same for each alternative, we can ignore it. Execute: The spreadsheet below computes the relevant FCF from each alternative. Note that each alternative has a negative NPV—this represents the PV of the costs of each alternative. We should choose the one with the highest NPV (lowest cost), which in this case is to purchase the existing machine. a. See spreadsheet. Notes to calculations in spreadsheet: After-tax amounts for rent, maintenance, and other costs are calculated as follows: After-tax amount = Before-tax amount × (1  TC) For example, the after-tax rent cash flow = $50,000 × (1  0.35) = $32,500 PV CCA tax shields are calculated as follows:

Option A: PV 

CapEx  d  TC rd

 r  .08   1   150, 000  .25  .35  1   2   2      $38, 299.66

1  r 

.08  .25

1  .08 

 r  .08  1   1   CapEx  d  TC  2  250, 000  .25  .35  2  Option B: PV      $63,832.77 rd

1  r 

.08  .25

1  .08 

NPVs of free cash flows excluding CCA tax shields (FCF) are calculated as follows: 10

FCFt

t 0

(1  r )t

NPV  

b. See spreadsheet.

Evaluate: When evaluating a capital budgeting project, financial managers should make the decision that maximizes NPV. In this case, Big Rock Brewery should purchase the current machine because it has the lowest negative NPV.

15.

a. No, this is a sunk cost and will not be included directly. (But see part (f) below.) b. Yes, this is a cost of opening the new store. c. Yes, this loss of sales at the existing store should be deducted from the sales at the new store to determine the incremental increase in sales that opening the new store will generate for HBS. d. No, this is a sunk cost. e. This is a capital expenditure associated with opening the new store. These costs will therefore increase HBS’s depreciation expenses (recall that depreciation is usually used for reporting purposes but not for tax purposes). This capital expenditure affects the cash flows of the project by causing a large initial outflow (the construction costs) followed by a series of inflows as the CCA tax shields are realized. f.

Yes, this is an opportunity cost of opening the new store. (By opening the new store, HBS forgoes the after-tax proceeds it could have earned by selling the land. This loss is equal to the sale price less the taxes owed on the capital gain from the sale, which is the difference between the sale price and the book value of the land.)

g. While these financing costs will affect HBS’s actual earnings, for capital budgeting purposes we calculate the incremental earnings without including financing costs to determine the project’s unlevered net income. 16. a. Using Equations 9.1 and 9.2, the CCA deductions are as follows: Year

CapEx

15,000,000.00

UCC

7,500,000.00 13,125,000.00 9,843,750.00

CCA

1,875,000.00

3,281,250.00

7,382,812.50

5,537,109.38

2,460,937.50 1,845,703.13 1,384,277.34

b. The CCA tax shield in year t is equal to the TC × CCAt Year

35%

CCA tax shield

656,250.00 1,148,437.50

861,328.13

645,996.09

484,497.07

c. The CCA tax shield in year t has present value equal to Year

10%

CCA tax

TC  CCAt

1  r 

656,250.00

1,148,437.5

861,328.13

645,996.09

484,497.07

shield

PV of CCA tax shield

596,590.9 1

NPV of first 5 CCA tax shields

647,128.5 7

949,121.90

441,224.02

300,834.56

2,934,899.9 6

d. The PV of the perpetuity of CCA tax shields is as follows:

PV 

CapEx  d  TC rd

 r  .10   1   15, 000, 000  .25  .35  1   2 2        $3,579,545.45

1  r 

1  .10 

.10  .25

e. If the marginal corporate tax rate increases over the next five years, the PV of CCA tax shields will be higher than what is calculated in part (d). In addition, if the tax rate will increase substantially, then Spherical may be better off delaying claiming CCA amounts until later years. The increased CCA tax shield generated in the future may more than make up for the extra discounting required to compensate for the time value of money. Note that, in this case, it may be easier to make a spreadsheet where the tax rates can be adjusted on a year to year basis as we cannot use the formula in part (d) because it assumes that a constant tax rate holds. 17.

Excel Solution a. Plan: Before the free cash flows can be calculated, the CCA deductions must be determined and used instead of depreciation. Execute: The CapEx amount is $25 million. Using Equations 9.1 and 9.2, the UCC and CCA amounts are (in $thousands): Year

UCC

12,500 21,250 14,875 10,413 7289 5102 3571 2500 1750 1225

CCA

3750

6375

4463

3124

2187 1531 1071

750

525

368

The free cash flows are (in thousands): Year

Cost of machine cash flow

-25,00 0

Cash flow from change in net working capital

-10,00 0

10,00 0

Sales revenue

30,00 0

Minus cost of goods sold

18,00 0

Equals gross profit

12,00 0

Minus general, sales and administrative expense

2000

Plus overhead that would have occurred anyway

1000

Minus CCA

3750

6375

4463

3124

2187

1531

1071

750

525

368

Equals net operating income

7250

4625

6538

7876

8813

9469

9929

10,25 0

10,47 5

10,63 2

Minus income tax @ 35%

2538

1619

2288

2757

3085

3314

3475

3587

3666

3721

4713

3006

4249

5120

5729

6155

6454

6662

6809

6911

3750

6375

4463

3124

2187

1531

1071

750

525

368

Equals net income

Plus CCA Plus machine and change in net working capital cash flows

-35,00 0

10,00 0

Equals free cash flow

-35,00 0

8463

9381

8712

8243

7915

7686

7525

7413

7334

17,27 9

b. The free cash flows in the previous table should not be used to calculate the NPV as there are CCA tax shields in years 11 and onward. If only the free cash flows shown in part (a) are used for the NPV, the result will underestimate the project’s actual NPV. We can calculate the free cash flows excluding CCA effects and calculate the PV of CCA tax shields separately. This is shown below.

Year

Cost of machine cash flow

-25,00 0

Cash flow from change in net working capital

-10,00 0

10,00 0

Sales revenue

30,00 0

Minus cost of goods sold

18,00 0

Equals gross profit

12,00 0

Minus general, sales and administrative expense

2000

Plus overhead that would have occurred anyway

1000

Minus CCA

not included as analyzed separately

Equals net operating income

11,00 0

Minus income tax @ 35%

3850

Equals net income

7150

Plus CCA

not included as analyzed separately

Plus cost of machine and change in net working capital cash flows

-35,00 0

10,00 0

Equals free cash flow excluding CCA tax shields

-35,00 0

7150

17,15 0

(Note: Consistent with the tables above, all dollar values are expressed in $thousands.) NPV of FCFexcluding CCA tax shields = 35, 000 

PV of CCA tax shields =

7,150  1  17,150 1   4,992.66  .14  1.149  1.1410

25, 000  0.30  .35 .14  .30

 .14  1   2     5,599.58 1  .14

NPV = 4,992.66 + 5,599.58 = $10,592.25

Evaluate: Since the project’s NPV is positive, it should be accepted.

*18.

Excel Solution Plan: It is best to calculate the FCF excluding CCA tax shields, take the NPV of these FCFs and then add the PV of CCA tax shields to get the NPV. Then compute the NPV under different assumptions about initial revenues and growth. Then compute the NPV of the project under a range of discount rates. Execute: a. The PV CCA tax shields are as follows:

 r  .12  1  1   CapEx  d  TC  2  150, 000  .10  .35  2  PV      $22,585.23 rd

1  r 

1  .12

.12  .10

The FCFs excluding CCA tax shields, their NPV, and the total NPV including CCA tax shields are calculated in the table below ($millions):

b. Revenue

10% higher

10% lower (see spreadsheet file for details)

NPV

$86.91971

$13.46681

c. Growth Rate NPV

$50.19326

5% (see spreadsheet file for details)

$65.38491

$91.06262

d. NPV is positive for discount rates below the IRR of 19.41%.

Evaluate: Under the forecast assumptions the project has an NPV of $50.2 million and therefore should be accepted. Under various scenarios of assumed initial sales and growth the project continues to have a positive NPV, meaning that, even if the forecast assumption proves too optimistic or pessimistic, the project will still create firm value. Finally, the discount rate used in the forecast assumptions is 12%, but the project would have a positive NPV using any discount rate below 19.41%. The project is positive and the results are robust. *19.

Excel Solution

20.

Real options must have positive value because they are only exercised when doing so would increase the value of the investment. If exercising the real option would reduce value, managers can allow the option to go unexercised. Thus, having the option but not the obligation to act is valuable.

21.

The XC-900 allows Buckingham the option to expand production starting in year 3. If it would be beneficial to expand production, Buckingham will increase production with the XC-900. If it would be better if production remains the same, Buckingham is under no obligation to utilize all of the XC-900 production capacity.

22.

This provides Buckingham the option to abandon the investment.

Chapter 10 Risk and Return in Capital Markets Note: All problems in this chapter are available in MyLab Finance. An asterisk (*) indicates problems with a higher level of difficulty. For a breakdown of the difficulty ratings of all of the problems in this Solutions Manual, please see the Preface – Question Difficulty Ratings. 1.

Plan: Compute the realized return and dividend yield on this equity investment. Execute: 1+ (55 - 50)

Rdiv =

50 1 50

= 0.12 = 12%

= 2%

Rcapital gain =

55 - 50 50

= 10%

Evaluate: The realized return on the equity investment is 12%. The dividend yield is 2% and the capital gain yield is 10%. 2.

Plan: Compute the capital gain and dividend yield under the assumption the stock price has fallen to $45. Execute: a. Rcapital gain = 45 - 50 / 50 = -10%.Yes, the capital gain is different, because the

(

)

difference between the current price and the purchase price is different than in Problem 1. b. The dividend yield does not change, because the dividend is the same as in Problem 1.

Evaluate: The capital gain changes with the new lower price; the dividend yield does not change. 3.

Plan: calculate the value of the portfolio and then the change in the value of the individual stocks or the change in the value of the entire portfolio. a. Your investment in CSH is 100 × $20 = $2000; in EJH it is 50 × $30 = $1500, so your total investment is $3500. Your weights are $2000/$3500 = 0.57 and $1500/$3500 = 0.43. b. There are two ways to calculate this. You can either compute the return on each stock and multiply those returns by their weights, or you can compute the total change in the value of your portfolio: CSH: (23  20)/20 = 0.15 ; EJH: (29  30)/30 = 0.033: so the return on your portfolio is (0.57)(0.15) + (0.43)(0.033) = 7.1% Or: your investment in CSH goes from $2000 to $2300, and in EJH goes from $1500 to $1450. Your portfolio has a net gain of $300  $50 = $250. As a return, that is $250/$3500 = 7.1%. [NOTE: the calculations would always yield exactly the same answer unless you round during the process.]

Plan: Compute the future sale price that is necessary to produce a 10% return. Execute: $0.50 + ( P1 - $20)

= 10% $20 ( P1 - $20) = $2 - $0.50 P1 = $21.50

Evaluate: The selling price immediately after the dividend would need to be $21.50 for you to earn a 10% return on the investment. 5.

Plan: Compute the purchase price today that is necessary to produce a 15% return. Execute: Div1 + ( P1 - P0 ) P0

Setting the return at 15% and filling in P1 = $100 and Div1 = $2, we have

0.15  P0 

$2   $100  P0  P0

, so 0.15P0  $102  P0 , so 1.15P0  $102

$102  $88.70 1.15

Evaluate: The purchase price today would need to be $88.70 for you to earn a 15% return on the investment. 6.

Plan: Compute each period’s return as the price change + dividend divided by the initial price (see Equation 10.1). Then, compute the annual realized return as the product of 1 + each period’s return, and then  1 (see Equation 10.2):

Execute: R1 =

(11- 10) + 0.20

1.20

= 12%

R2 =

10 10 (11.1- 10.5) + 0.20 0.80 R3 = = = 7.6% 10.5 10.5

(10.50 - 11) + 0.20

R4 =

11 (11- 11.1) + 0.20 11.1

= =

-0.30 11 0.10 11.1

= -2.7%

= 0.9%

R  (1  0.12)(1  0.027)(1  0.076)(1  0.009) 1  1.183 – 1  18.3% Evaluate: The annual realized return is calculated by compounding the quarterly returns, taking both dividends and price changes into account. 7.

Excel Solution Plan: Compute each period’s return as the price change + dividend divided by the initial price (see Equation 10.1). Then, compute the annual realized return as the product of 1 + each period’s return and then  1 (see Equation 10.2): Execute: Date 1/2/2008 2/6/2008 5/7/2008 8/6/2008 11/5/2008 1/2/2009

Date 1/3/2011 2/9/2011 5/11/2011 8/10/2011 11/8/2011 1/3/2012

Price 86.62 79.91 84.55 65.4 49.55 45.25

Price 66.4 72.63 79.08 57.41 66.65 74.22

Dividend 0.4 0.4 0.4 0.4

Dividend 0.42 0.42 0.42 0.42

1+R

-7.28% 6.31% -22.18% -23.62% -8.68%

0.92715308 1.06307095 0.77823773 0.76376147 0.91321897

-46.50%

0.535006

1+R

10.02% 9.46% -26.87% 16.83% 11.36%

1.1001506 1.09458901 0.73128477 1.16826337 1.11357839

14.56% 1.14564829

Evaluate: The annual realized return is the compound return of the returns from the five shorter periods within the year, taking into account both dividends and price changes.

Excel Solution Historical Stock and Dividend Data Date

Price

Dividend

Jan 1

33.88

Feb 5

30.67

0.17

May 14

29.49

0.17

Aug 13

32.38

0.17

Nov 12

39.07

0.17

Dec 31

41.99

Plan: Calculate the realized return for each period and then compound those returns. Execute: Return from Jan 1  Feb 5 R1 =

30.67 + 0.17 33.88

- 1 = -8.973%

Return from Feb 5  May 14 R2 =

29.49 + 0.17 30.67

- 1 = -3.293%

Return from May 14 Aug 13 R3 =

32.38 + 0.17 29.49

- 1 = 10.376%

Return from Aug 13  Nov 12 R4 =

39.07 + 0.17 32.38

- 1 = 21.186%

Return from Nov 12  Dec 31 R5 =

41.99 39.07

- 1 = 7.474%

Return for the year is (1+ R1 )(1+ R2 )(1+ R3 )(1+ R4 )(1+ R5 ) - 1 = 26.55%.

Evaluate: Taking into account both dividends and price changes, the return on this stock from January 1 to December 31 was 26.55%.

Plan: Calculate the annual capital gain yield for the stock. Then deduct the capital gain from the 26.55% total return that you calculated in the previous problem in order to find the dividend yield. Execute: Capital gain yield from Jan 1  Dec 31

Rannual capital gain =

41.99 -1 = 23.94% 33.88

Dividend yield from Jan 1  Dec 31

Rannual div = Rannual return - Rannual capital gain = 26.55% - 23.94% = 2.61% Double check dividend yield: first, calculate the amount that will accumulate from the dividends by the end of the year, assuming each dividend is reinvested and will earn the return applicable for the periods remaining until December 31:

Div Dec 31 = $0.17(1- 0.03293)(1+ 0.10376)(1+ 0.21186)(1+ 0.07474) + $0.17(1+ 0.10376)(1+ 0.21186)(1+ 0.07474) + $0.17(1+ 0.21186)(1+ 0.07474) + $0.17(1+ 0.07474) = $0.885 Then, calculate the annual dividend yield: divide the amount that accumulated from the reinvested dividends by the price of the stock at the beginning of the year:

Rannual div =

$0.885 = 2.61% $33.88

Evaluate: The total return for the year is equal to the sum of the capital gain yield and dividend yield for the year: 26.55% = 23.94% + 2.61%. 10.

Excel Solution Plan: Compute the arithmetic average return using Equation 10.3. For parts (b) and (c), use Equation 10.4 and Equation 10.5 to calculate the variance and standard deviation of returns. Execute:

Average annual return 

4%  28%  12%  4%  10% 4

(4%  10%) 2  (28%  10%) 2  (12%  10%) 2  (4%  10%) 2 3  0.01867

b. Variance of returns 

Standard deviation of returns  variance  0.01867  13.66%

Evaluate: The average annual return is 10%. This is our estimate of the investment’s expected return based on its past performance. The variance of return is 0.01867. The standard deviation of returns is 13.66%. Variance and standard deviation both measure the variability of returns. Standard deviation, the square root of variance, is easier to interpret since it is expressed in units of %. 11.

Excel Solution Plan: For part (a), to compute the arithmetic average, use Equation 10.3. For part (b), to compute the geometric average, take the product of 1 + each return and then take the tenth root of that product. For part (c), realize that the total return computed in part (b), before taking the average, can be applied directly to the $100. Execute: a. Using Equation 10.3: (19.9%  16.6%  18%  50%  43.3%  1.2%  16.5%  45.6%  45.2%  3%)/10  8.05% *b. (0.801)(1.166)(1.180)(0.500)(1.433)(1.012)(0.835)(1.456)(1.452)(0.970)  1.3683 1 10

(1.3683) = 1.03186 . Subtracting the 1, we get the geometric average of 3.19%. c. In part (b) we computed the total realized return as the product of 1 + each year’s return. We would have earned that return on the $100, so the answer is $100(1.3683) = $136.83. Evaluate: The geometric average return is a better representation of what actually happened. However, the arithmetic average is a better estimate of what you can expect to happen in any given year (if you were trying to forecast the return for next year, for example). 12.

Plan: Compute the arithmetic average return using Equation 10.3. For part (b), to compute the geometric average, take the product of 1 + each return and then take the fifth root of that product. For parts (c) and (d), use Equation 10.4 and Equation 10.5 to calculate the variance and standard deviation of returns. a. (5%  2%+4%  8%  1%)/5  2.8% *b. (1.05)(0.98)(1.04)(1.08)(0.99)  1.1442 1

(1.1442) 5 - 1 = 2.73%

Var(R) =

(5% - 2.8%)2 + (-2% - 2.8%)2 + (4% - 2.8%)2 + (8% - 2.8%)2 + (-1% - 2.8%)2 5-1

= 0.00177

d. Standard dev (R) =

Var(R) = 0.00177 = 4.2%

Evaluate: The arithmetic and geometric averages are different, but not by much. The standard deviation reveals that the returns are volatile around the average. 13.

The answers are different because the arithmetic average return basically assumes that you reset your investment every year. It is the best measure to use to predict the most likely annual return next year. However, it is not the best representation of how your investment actually performed. The geometric average return does that.

14.

Plan: Given the data in Figures 10.3 and 10.4, calculate the 95% confidence intervals for the four securities mentioned. Use Equation 10.6. Execute: S&P/TSX Composite Index: = 10.32% – 2 × 16.35% to 10.32% + 2 × 16.35% = –22.38% to 43.02% S&P 500 in CAD: = 12.10% – 2 × 17.00% to 12.10% + 2 × 17.00% = –21.90% to 46.10% Long-term Government of Canada bonds: = 7.89% – 2 × 10.20% to 7.89% + 2 × 10.20% = –12.51% to 28.29% Canadian treasury bills: = 5.59% – 2 × 3.93% to 5.59% + 2 × 3.93% = –2.27% to 13.45% Evaluate: The 95% confidence interval is two standard deviations from the right and the left of the mean. For example, for long-term bonds it ranges from 12.51% to 28.29%.

15.

For this, choose the investments above that have a lower limit of the 95% confidence interval that is above 8%. These investments are corporate bonds and treasury bills.

16.

Plan: Using a 95% prediction interval, the bottom of the prediction range is two standard deviations below the average return. Compare the bottom of the prediction interval to the minimum return (40%) that you are willing to tolerate. Execute: 0.12  2(0.20)  0.12  0.40   0.28 Yes, the low end of the 95% prediction interval is 28%, which is greater than 40%. Evaluate: Despite the volatile returns, you are 95% confident that you will not suffer a 40% loss.

*17.

Plan: Calculate the expected payoff of each bank’s loans. Recognize that Bank A has a portfolio of independent loans, so we would expect diversification to reduce the volatility of its loan portfolio. Execute: a. Expected payoff is the same for both banks: Bank A = ($1 million ´ 0.95) ´ 100 = $95 million

Bank B = $100 million ´ 0.95 = $95 million b. Bank A: Variance of each loan = (1- 0.95)2 0.95 + (0 - 0.95)2 0.05 = 0.0475

Variance of 100 loans = 100 ´ 0.0475 = 4.75 Standard Deviation of 100 loans = 4.75 = 2.179

Bank B: One loan: Variance = (100 - 95)2 0.95 + (0 - 95)2 0.05 = 475 Standard Deviation = 475 = 21.79 , which is much higher than that of Bank A.

Evaluate: Even though the two banks have the same expected return on their loans, Bank A’s position is much safer because it has diversified its $100 million across 100 independent loans. 18.

A risk-averse investor would choose the economy in which stock returns are independent because this risk can be diversified away in a large portfolio.

19.

Plan: Compute the arithmetic average return using Equation 10.3. For parts (b) and (c), use Equation 10.4 and Equation 10.5 to calculate the variance and standard deviation of returns.

Execute: a. 2% + 5% - 6% + 3% - 2% + 4% = 1% 6 0% - 3% + 8% - 1% + 4% - 2% Average annual return of Stock B = = 1% 6 Standard deviation of returns for Stock A Average annual return of Stock A =

(2% - 1%)2 + (5% - 1%) 2 + (-6% - 1%)2 + (3% - 1%) 2 + (-2% - 1%) 2 + (4% - 1%)2 5

= 0.04195 Standard deviation of returns for Stock B =

(0% - 1%)2 + (-3% - 1%)2 + (8% - 1%)2 + (-1% - 1%)2 + (4% - 1%) 2 + (-2% - 1%)2 5

= 0.04195

1%+1%+1%+1%+1%+1% = 1% 6 Standard deviation of returns for the portfolio is zero (returns do not deviate from the mean). Average annual return of the portfolio = Just to confirm using the formula: =

(1% - 1%)2 + (1% - 1%)2 + (1% - 1%)2 + (1% - 1%)2 + (1% - 1%)2 + (1% - 1%)2 5

=0 c. The portfolio is less risky than the two individual stocks. It has the same expected return but a standard deviation of 0, compared to standard deviations of 4.195% for both stocks. Evaluate: Stocks A and B are perfectly negatively correlated (they always consistently and proportionately move in the opposite direction), so their risks will offset each other, creating a portfolio that is completely risk free.

Chapter 11 Systematic Risk and the Equity Risk Premium Note: All problems in this chapter are available in MyLab Finance. An asterisk (*) indicates problems with a higher level of difficulty. For a breakdown of the difficulty ratings of all of the problems in this Solutions Manual, please see the Preface – Question Difficulty Ratings.

Plan: Calculate each investment’s weight as the amount invested in it as a proportion of the total amount invested. Execute: Tidepool: 100  $40  $4000 Madfish: 200  $15  $3000 Weight on Tidepool  $4000/($4000  $3000)  0.571 Weight on Madfish  $3000/($4000  $3000)  0.429 Evaluate: You cannot tell the weights just by the number of shares; what matters is the total dollar amounts invested in each stock.

Plan: The expected return on any portfolio is the weighted average of the expected returns of the securities in the portfolio. Therefore, we will compute the weighted average return on this portfolio. Execute: E[ R p ]  (70%)(20%)  (30%)(15%)  18.5% Evaluate: The expected return on this portfolio is 18.5%.

Plan: The expected return on any portfolio is the weighted average of the expected returns of the securities in the portfolio. Therefore, we will compute the weighted average return on this portfolio. Execute: E[ RP ]  wHNL E[ RHNL ]  wKOA E[ RKOA ]   0.6  0.18    0.4  0.22   0.196 Evaluate: The expected return on this portfolio is 19.6%.

Plan: Perform the calculations to answer the questions in the problem. Execute: a. Let ni be the number of shares invested in Stock i, then nG 

200, 000  0.5

 4, 000 25 200, 000  0.25 nM   625 80 200, 000  0.25 nV   25, 000 2

The new value of the portfolio is p  30nG  60nM  3nV  $232,500

b. Return 

232,500 200,000

 1  16.25%

c. The portfolio weights are the fraction of value invested in each stock GoldFinger: Moosehead: Venture:

nG  30 232,500 nM  60 232,500 nV  3 232,500

 51.61%  16.13%  32.26%

Evaluate: a. The new value of the portfolio is $232,500. b. The return on the portfolio was 16.25%. c. If you don’t buy or sell shares after the price change, your new portfolio weights are GoldFinger 51.61%, Moosehead 16.13%, and Venture 32.26%. 5.

Plan: Compute the weights on each investment and then, matching those weights to the expected returns, compute the expected return of the portfolio using Equation 11.3. Execute: The weight on the second stock is $30,000/$70,000  0.4286. Since the weights must sum to 1, the weight on the final stock is (1  0.4286  0.20) = 0.3714. E[R]  (0.2)(0.12)  (0.4286)(0.15)  (0.3714)(0.2)  0.1626

Evaluate: The expected return of the portfolio is a weighted average of the expected returns of the stocks. The biggest weight on any individual stock in this case is the 42.86% on the stock with a 15% return. 6.

Both calculations of expected return of a portfolio give the same answer.

If the price of one stock goes up, the other stock price always goes up as well. Similarly, if one goes down, the other will also be going down.

Excel Solution Plan: Go to Chapter Resources on MyLab Finance, download the data for Table 11.3 and make the calculations required in the problem. Execute: a. Correlation between Stantec and Maple Leaf Foods (using Excel’s CORREL function): 0.2693 b. Annual standard deviation for Stantec: 25.37%, using Excel’s STDEV.S function to find the monthly standard deviation of 7.32% and then multiplying by the square root of 12 to obtain the annualized figure. Maple Leaf Foods: 24.11%, using Excel’s STDEV.S function to find the monthly standard deviation of 6.96% and then multiplying by the square root of 12 to obtain the annualized figure. c. Annual variance and standard deviation of a portfolio containing 30% Stantec and 70% Maple Leaf Foods: 2 = (0.3)2 (0.2537)2 + (0.7)2 (0.2411)2 + 2(0.7)(0.3)(0.2537)(0.2411)(0.2693) = 0.041201  = 0.2030 = 20.30% Evaluate: The correlation between the returns of Stantec and Maple Leaf Foods is 0.2693, which is low positive correlation. The annual standard deviations of the returns of Stantec are 25.37%, and Maple Leaf Foods is 24.11%. The annual standard deviation of a portfolio of these two stocks is 20.30%.

Plan: Use Equations 11.3 and 11.4 to answer parts (a), (b) and (c). Execute: a. E[ R A ]  E [ RB ] 

0.1  0.07  0.15  0.05  0.08

 0.07

5 0.06  0.02  0.05  0.01  0.02

 0.024

b. (0.1  0.07) 2  (0.07  0.07) 2  (0.15  0.07) 2  ( 0.05  0.07) 2  (0.08  0.07) 2

Var ( RA ) 

5 1

 0.00545

StdDev ( RA )  0.00545  0.0738 (0.06  0.024) 2  (0.02  0.024) 2  (0.05  0.024) 2  (0.01  0.024) 2  ( 0.02  0.024) 2

Var ( RB ) 

5 1

 0.00103

StdDev ( RB )  0.00103  0.0321

c. E[ RP ]  0.7(0.07)  0.3(0.024)  0.0562 Var ( RP )  0.7 2 (0.0738) 2  0.32 (0.0321) 2  2(0.7)(0.3)(0.0738)(0.0321)(0.46)  0.00322 StdDev ( RP )  .00322  0.0567

Evaluate: Even with most of the portfolio’s weight on the riskier stock, the diversification effect brings the overall portfolio risk down below a weighted average of the two standard deviations. 10.

Excel Solution Plan: Calculate the average return and volatility of Stock A and Stock B. Realized Returns Year

Stock A

Stock B

1998

10%

21%

1999

20%

30%

2000

2001

5%

3%

2002

8%

2003

25%

Execute: RA 

RB 

0.1  0.2  0.05  0.05  0.02  0.09

 0.035  3.5%

0.21  0.30  0.07  0.03  0.08  0.25

 0.12  12%

(0.1  0.035) 2  (0.2  0.08) 2    1 Variance of A   (0.05  0.035) 2  ( 0.05  0.035) 2   5 2 2  (0.02  0.035)  (0.09  0.035)   0.01123 Volatility of A  SD( RA )  Variance of A  .01123  10.60%

(0.21  0.12) 2  (0.3  0.12) 2    1 Variance of B   (0.07  0.12) 2  (0.03  0.12) 2   5 2 2  ( 0.08  0.12)  (0.25  0.12)   0.02448 Volatility of B  SD( RB )  Variance of B  .02448  15.65%

Evaluate: The average return on Stock A is 3.5% with a volatility of 10.60%. The average return on Stock B is 12% with a volatility of 15.65%. 11.

Excel Solution Plan: Calculate the volatility of a portfolio that is 70% invested in Stock A and 30% invested in Stock B. Execute:   (0.7) 2 (0.1060) 2  (0.3) 2 (0.1565) 2  2(0.7)(0.3)(0.1060)(0.1565)(0.48)  0.1051  10.51%

Evaluate: The volatility of a portfolio of 70% invested in Stock A and 30% in Stock B is 10.51%. 12.

Excel Solution Plan: Calculate the average monthly return and volatility for the stock of Cola Co. and Gas Co. Date

Cola Co.

Gas Co.

Jan

–10.84%

6.00%

Feb

2.36%

1.28%

Mar

6.60%

–1.86%

Apr

2.01%

–1.90%

May

18.36%

7.40%

June

–1.22%

0.26%

July

2.25%

8.36%

Aug

–6.89%

–2.46%

Sep

–6.04%

–2.00%

Oct

13.61%

0.00%

Nov

3.51%

4.68%

Dec

0.54%

2.22%

Execute: The mean for Cola Co. is 2.02%; the mean for Gas Co. is 0.79%. The standard deviation (i.e., volatility) for Cola Co. is 8.24%; the standard deviation for Gas Co. is 4.25%. See spreadsheet solution for details. Evaluate: Cola Co. has a higher mean return (2.02%) and volatility (8.24%) than Gas Co. (mean return: 0.79%, volatility: 4.25%). 13.

Excel Solution Both methods have the same result: the standard deviation (i.e., volatility) is 5.90%. See spreadsheet solution for details.

14.

Excel Solution a. Year

North Jet

South Jet

Alberta Oil

Portfolio

2012

21%

–2%

6.50%

2013

30%

21%

–5%

10.25%

2014

8.00%

2015

–5%

–2%

21%

8.75%

2016

–2%

–5%

30%

13.25%

2017

30%

13.25%

b. The lowest annual return of the portfolio is 6.5%, which is higher than the lowest annual return of the individual stocks and portfolios in Table 11.2. See spreadsheet solution for details. 15.

Stantec’s σ = 0.25; Magna’s σ = 0.30. The correlation between Stantec (STN) and Magna (MG) is 0.26, and the weights are 50% each. 2 2 2 2  P2  wSTN  STN  wMG  MG  2wSTN wMG STN  MG  STN , MG

 P2  0.52  0.25  0.52  0.30   2(0.5)(0.5)(0.25)(0.30)(0.26) 2

 P2  0.04788  P  0.21906  21.88% 16.

Plan: Use Equations 11.3 and 11.4 to compute the expected return and volatility of the indicated portfolio.

Execute: In this case, the portfolio weights are xj  xw  0.50. From Equation 11.3, E[ RP ]  x j E[ R j ]  xw E[ Rw ]  0.50(7%)  0.50(10%)  8.5%

We can use Equation 11.4: SD ( RP ) 

x 2j SD ( R j ) 2  xw2 SD ( Rw ) 2  2 x j xw Corr( R j , Rw ) SD ( R j ) SD ( Rw )

 0.502 (0.16 2 )  0.50 2 (0.20) 2  2(0.50)(0.50)(0.22)(0.16)(0.20)  14.1%

Evaluate: The portfolio would have (a) an expected return of 8.5% and (b) a standard deviation of return of 14.1%. 17.

Excel Solution Volatility Arbor Systems

Gencore

Correlation

Portfolio

(a)

40%

40.00%

(b)

40%

0.5

34.64%

(c)

40%

28.28%

(d)

40%

–0.5

20.00%

(e)

40%

–1

0.00%

Weights:

50%

The volatility of the portfolio declines as the correlation coefficient decreases. There is no benefit to diversification when the two stocks are perfectly positively correlated (see [a]). In all other cases the volatility of the portfolio is lower than that of the original stocks. The benefits from diversification are the greatest when the two stocks are perfectly negatively correlated (correlation equals 1; see [e]). When two stocks are perfectly negatively correlated, risk can be completely eliminated (with the right weights, the portfolio standard deviation can be reduced to zero, as in [e]). 18.

Excel Solution Wesley

Addison

Correlation

(a) Weights

100%

30.00%

(b) Weights

25%

75%

30.00%

50%

36.74%

Volatility:

60%

30%

0.25

Portfolio Volatility

(b) Even though Wesley is twice as volatile as Addison, adding Wesley to the portfolio does not increase portfolio risk above Addison's volatility. (c) Portfolio risk exceeds Addison's volatility when we increase Wesley's weight to 50%. However, even in this case, the portfolio volatility is less than the average volatility of the two stocks. The relatively low volatility of the portfolio is the result of the benefits of diversification (Addison and Wesley are not perfectly positively correlated). 19.

Plan: You must estimate the expected return and volatility of each portfolio created by adding Stock A or Stock B. You will select that portfolio that gives you the greatest return or the least volatility. Execute: The expected return of the portfolio will be the same (17.4%) if you pick A or B, since both A and B have the same expected return. Therefore, the choice of A or B depends on how risky the portfolio becomes when you add A or B. For A:   (0.8) 2 (0.30) 2  (0.2) 2 (0.25) 2  2(0.8)(0.2)(0.30)(0.25)(0.2)  0.2548  25.48%

For B:   (0.8) 2 (0.30) 2  (0.2) 2 (0.20) 2  2(0.8)(0.2)(0.30)(0.20)(0.6)  0.2659  26.59%

Evaluate: Since the portfolio is less risky when A is added, you should add A to the portfolio. Note that Stock A’s standard deviation is higher than that of Stock B. However, Stock A has a lower correlation with the portfolio, so adding it reduces the portfolio’s standard deviation. 20.

Plan: Stocks B and C are identical except for the fact that Stock B has a lower correlation with A than C does. Given that B and C’s standard deviations are the same, the one with the lower correlation with A will produce a lower portfolio standard deviation. Since she will be putting $100,000 in each stock, her portfolio will be 50% in each stock. Execute: Using B: E[ RP ]  0.5(0.15)  0.5(0.13)  0.14 Var ( RP )  0.52 0.52  0.52 0.4 2  2(0.5)(0.5)(0.5)(0.4)(0.2)  0.1225 StdDev( RP )  .1225  .35

You can confirm that this is lower than the standard deviation of a portfolio with A and C:

E[ RP ]  0.5(0.15)  0.5(0.13)  0.14 Var ( RP )  0.52 0.52  0.52 0.42  2(0.5)(0.5)(0.5)(0.4)(0.3)  0.1325 StdDev ( RP )  0.1325  0.364

Evaluate: By choosing the stock that has the lower correlation with A, you can achieve the goal of an expected return of 14% with a lower standard deviation than if you had chosen the stock with the higher correlation. 21.

Excel Solution Plan: Compute the total market value of the total portfolio and the weighted percent that each individual stock would be in the market portfolio. Execute: Total value of the market

 10  10 million  20  12 million  8  3 million  50  1 million  45  20 million  $1.314 billion.

Stock A B C D E

Portfolio Weight 10  10 million 1314 million 20  12 million 1314 million

8  3 million 1314 million 50 million 1314 million

 7.61%  18.26%

 1.83%  3.81%

45  20 million 1314 million

 68.49%

Evaluate: The market portfolio would have a value of $1.314 billion. Stock A would be 7.61% of the market portfolio, Stock B would be 18.26%, Stock C would be 1.83%, Stock D would be 3.81%, and Stock E would be 68.49%. 22.

Plan: Compute the weighted percent that each individual stock would be in the valueweighted portfolio, and the dollar amount invested in each stock. Execute: Total value of the market  50 million  75 million  20 million  10 million  $155 million.

Stock

Portfolio Weight

$ Investment

OGG

50 = 32.26% 155

32.26%(500,000) = $161,300

HNL

75 = 48.39% 155

48.39%(500,000) = $241,950

KOA

20 = 12.90% 155

12.90%(500,000) = $64,500

LIH

10 = 6.45% 155

6.45%(500,000) = $32,250

Evaluate: You would invest $161,300 in OGG; $241,950 in HNL; $64,500 in KOA; and $32,250 in LIH. 23.

Excel Solution Plan: Compute the total market value of the total portfolio and the weighted percent that each individual stock would be in the market portfolio. Execute: Total value of all four stocks

 13  1.00 million  22  1.25 million  43  30 million  5  10 million  $1,380.5 billion.

Stock Golden Seas Jacobs and Jacobs MAG PDJB

Portfolio Weight 13 ´ 1.00

= 0.942%

0.942%($100,000) = $942

= 1.992%

1.992%($100,000) = $1992

= 93.444%

93.444%($100,000) = $93,444

= 3.622%

3.622%($100,000) = $3,622

1380.5 22 ´ 1.25 1380.5 43 ´ 30 1380.5 5 ´ 10 1380.5

$ Investment

Evaluate: The market portfolio would have a value of $1.380.5 billion. You would invest $942 in Golden Seas, $1992 in Jacobs and Jacobs, $93,444 in MAG, and $3622 in PDJB. Note: Differences from the Excel solution are due to rounding. 24.

Nothing needs to be done. The portfolio is still value weighted but the weight of the stock that went up in price would now be higher and the other three weights would now be lower.

25.

Plan: Compute the estimated returns of BlackBerry and Air Canada based on today’s market movement. Execute: a. The best estimate of BlackBerry’s excess return today is the product of the market return and BlackBerry’s beta: BlackBerry’s estimated return b. Similarly, Air Canada’s estimated return Evaluate: BlackBerry’s estimated return is –3.0% and Air Canada’s is –1.0%.

26.

27.

Excel Solution Plan: Go to Chapter Resources on MyLab Finance and access the Excel spreadsheet. Use the approach discussed in the ―Using Excel‖ box on page XXX to estimate the slope coefficient of the data, which is our estimate of beta. Execute: a. Using the approach discussed in the ―Using Excel‖ box on page XXX, we obtain the following betas (see Excel solution for details): 1987–1991: 1.4110 1992–1996: 0.8544 1997–2001: 1.8229 2002–2006: 1.0402 Evaluate: b. Microsoft’s beta decreased in the early 1990s as Microsoft established itself as the dominant operating software company, but increased during the internet bubble in the late 1990s (when tech stocks were soaring). It has since decreased.

28.

Plan: Compute the expected return for Loblaw. Execute: Expected Return  4%  0.32  (10%  4%)  5.92% Evaluate: The expected return for Loblaw is 5.92%. Notice that the standard deviation of the market portfolio was not required for these calculations.

29.

The sign of the risk premium for a negative beta stock is negative. This is because the negative beta stock acts as ―recession insurance,‖ and, thus, investors are willing to pay for this insurance in the form of accepting a lower return than the risk-free rate.

30.

Plan: The beta of a portfolio is a weighted average of the betas of the stocks in the portfolio (with the fraction of the total investment in the portfolio held in each stock in the portfolio serving as weights). Execute:  P  (0.25)(1.2)  (0.25)(0.6)  (0.5)(1)  0.95 Evaluate: Because betas only represent non-diversifiable risk, there is no ―diversification effect‖ on beta from a portfolio. So, the beta of a portfolio is simply the weighted average of the betas of the securities in the portfolio.

31.

Plan: Compute the expected returns of Intel and Boeing as well as the portfolio beta. Then compute the expected return of the portfolio. Execute: a. Intel’s expected return  4%  1.6  (0.10  0.04)  13.6%. b. Boeing’s expected return  4%  1.0  (0.10  0.04)  10%. c. The portfolio beta  (60%)(1.6)  (40%)(1.0)  1.36. d. The portfolio’s expected return = (60%)(13.6%) + (40%)(10%) = 12.16%. Evaluate: Intel’s expected return is 13.6%, Boeing’s expected return is 10%, the portfolio beta is 1.36, and the expected return of the portfolio is 12.16%.

*32.

Plan: Compute the necessary beta. Execute: Return on the stock 

$1  $117  $100 $100

 18%

Using the CAPM, find the beta that is consistent with an 18% expected return on the stock: Expected return  18%  4.5%  beta  (6%)

beta 

18%  4.5% 6%

 2.25

Evaluate: A beta of 2.25 would be consistent with an 18% return on the stock. *33.

Plan: Compute what the expected return for a stock with a beta of 1.2 should be. You should buy the stock if the expected return is 11% or less. Execute: Expected return  5%  1.2  (6%)  12.2%. Evaluate: No, you should not buy the stock. You should expect a return of 12.2% for taking on an investment with a beta of 1.2. But since you expect this stock to return only 11%, it does not fully compensate you for the risk of the stock, and you can find other investments that will return 11% with less risk.

34.

Plan: Compute the expected return for the coffee company. Execute: Expected return  5%  0.6  (5.5%)  8.3% Evaluate: The coffee company should produce a return of 8.3% to compensate its equity investors for the riskiness of their investment.

35.

Plan: Compute the expected return and the realized return for Apple. Execute:

Expected return  4.5%  1.4  (6%)  12.9%

Realized return 

198.08  84.84 84.84

 133.47%

Evaluate: Apple’s managers greatly exceeded the required return of investors, as given by the CAPM. 36.

Plan: First, solve for the market risk premium. You know the expected return for Bay Corp, the risk-free rate, and its beta, so you can algebraically solve for the market risk premium. Using that risk premium and the desired beta, you can check that the desired expected return is consistent with the CAPM. Finally, you need to solve for the weights on the two companies’ betas that would produce a portfolio beta of 1.4. Execute: 0.112  0.04  1.2( E[ RM ]  rf )

E[ RM ]  rf 

0.112  0.04 1.2

 0.06

Using the fact that the risk premium is 0.06 and the desired portfolio beta of 1.4, we can confirm that the portfolio beta and desired expected return are consistent with each other: E[R]  0.04  1.4(0.06)  0.124 To form a portfolio with a beta of 1.4 using Bay Corp and City Corp, you need to solve for the weights: 1.4 = wB(1.2)  wC(1.8) = wB(1.2)  (1  wB)(1.8) 0.4 = 0.6wB wB = (0.4/0.6)  2/3 Put two-thirds of your money in Bay Corp and one-third in City Corp: (2/3)$50,000 = $33,333.33 should be invested in Bay Corp and (1/3)$50,000 = $16,666.67 in City Corp. Note: Since there are three assets to choose from, there are many other solutions. You could, for example, put none of your money in Bay Corp, 77.78% ($38,888.89) in City Corp and 22.22% ($11,111.11) in the risk-free asset. Another solution would be to put 6.67% ($3,333.33) of your money in Bay Corp, 73.33% ($36,666.67) in City Corp and 20% ($10,000) in the risk-free asset. Evaluate: You can achieve your goals by creating a portfolio with a beta of 1.4. One way of doing so is putting two-thirds of your money (or $33,333.33) into Bay Corp and onethird (or $16,666.67) into City Corp.

Chapter 12 Determining the Cost of Capital

Note: All problems in this chapter are available in MyLab Finance. An asterisk (*) indicates problems with a higher level of difficulty. For a breakdown of the difficulty ratings of all of the problems in this Solutions Manual, please see the Preface – Question Difficulty Ratings. 1.

Plan: Compute the weights for the WACC. Execute: Value of debt: $100 million Value of preferred stock: $20 million Market value of common equity: $50 per share  6 million shares  $300 million Total market value of firm: $100  20  300  $420 million Weights for WACC calculation: Debt :

100

 23.81%

420 Preferred Stock :

 4.76%

420 Common Equity :

300

 71.43%

420

Evaluate: The total market value of the firm is $420 million. Debt is 23.81% of the total value, preferred stock is 4.76%, and common equity is 71.43%. 2.

Plan: Compute the market value weights to compute the WACC. Execute: Book value of equity  $600 Market value of equity  $600  1.5  $900 Book value of debt  $400 Total market value of firm  $1300

Weights for WACC calculation: Debt : Common Equity :

400 1300 900 1300

 30.77%  69.23%

Evaluate: Debt is 30.77% of the capital structure and equity is 69.23%. 3.

Plan: The book value of its equity and debt are not relevant for computing the weights for its WACC. Those weights should be based on market values. We need to calculate the market value (MV) of its debt and of its equity. The MV of its equity equals the number of shares outstanding times its current price per share. The MV of its debt can be calculated by multiplying its price relative to par by the par value. Execute: a. MV equity: 1 million shares  $50 per share = $50 million b. MV debt: 101% of par  $20 million par = $20.2 million c. Equity weight: $50 million/($50 million + $20.2 million) = 0.712 Debt weight: $20.2 million/($50 million + $20.2 million) = 0.288 Evaluate: The book values would give you the wrong weights. The WACC should be based on the relative market values of the company’s capital sources, not the historical book values.

*4.

Plan: Compute the firm’s pre-tax WACC and the value of a portfolio containing 40% of the firm’s debt and 60% of the firm’s equity. Show that the expected returns are identical. Execute: The firm’s assets are to be worth either $1200 or $960 in one year (with equal probability). What does this mean for the value of the debt and equity of the firm? At a 5% interest rate, the firm will be required to repay $400  1.05 = $420 in one year. Debt represents a senior claim over equity on the firm’s assets and, regardless of which outcome occurs, the assets will be large enough in one year to fulfill the debt obligations. Therefore, the debt will be worth $420 in either case. Equity (the residual claim) will be worth either $1200  $420 = $780 or $960  $420 = $540 in one year. Expected return on assets: First, we calculate the return for each outcome. There is a 50% chance that the value of assets will increase from $1000 to $1200 in one year, a 20% return: Return 

1200  1000

 20%

1000

There is a 50% chance that the value of assets will decline from $1000 to $960 in one year, a 4% return: Return 

960  1000

 4%

1000

Then, we calculate the expected return on assets using the probabilities:

Expected Return on Assets  0.5  20%  0.5  ( 4%)  8% Expected return on a portfolio consisting of 40% debt and 60% equity: Expected return on debt: 420  400

 5%

400

Expected return on equity: As before, we first calculate the return for each outcome. There is a 50% chance that the value of equity will increase from $ 600 to $780 in one year, a 30% return: Return 

780  600

 30%

600

There is a 50% chance that the value of equity will decline from $600 to $540 in one year, a 10% return: Return 

540  600

 10%

600

Then, we calculate the expected return on equity using the probabilities:

Expected Return on Equity  0.530%  0.5 (10%) 10% Our last step is to calculate the expected return on a portfolio consisting of 40% debt and 60% equity using the portfolio weights:

Expected Return on Portfolio (40% debt & 60% equity)  0.4  5%  0.6  10%  8%  Evaluate: The expected return on the portfolio of 40% debt and 60% equity is identical to the expected return on the assets of the firm. Notice that, compared to the firm’s assets, equity both is riskier and has a higher expected return. The portfolio combines highreturn equity with low-return debt in just the right proportions to achieve the same expected return as the firm’s assets. 5.

Plan: The risk-free rate will change the equity cost of capital by the same amount for both Alcoa (AL) and Hormel Foods (HF) and can, therefore, be ignored. Compute and then compare the risk premium for each firm using the CAPM formula. Execute: According to the CAPM, a firm’s equity cost of capital is rE  rf   ( E[ RMkt ]  rf )

The risk premiums are  AL ( E[ RMkt ]  rf )  2.0(5%)  10%  HF ( E[ RMkt ]  rf )  0.45(5%)  2.25%

Thus, Alcoa has the higher equity cost of capital by 10% – 2.25% = 7.75%.

Evaluate: Alcoa has approximately four times as much systematic risk as Hormel Foods; therefore, its risk premium is about four times the size of Hormel’s. 6.

Plan: Compute Avicorp’s pre-tax cost of debt and its after-tax cost of debt. Execute: a. The pre-tax cost of debt is the YTM on the outstanding debt issue. We solve for the six-month YTM on the bond, and then compute the EAR. $95 

3 (1  YTM 6 month )



3 (1  YTM 6 month )



3  100 10 (1  YTM 6 month )

 YTM 6 month  3.6044%

EAR  (1  0.036044)  1  7.3386% 2

The pre-tax cost of debt is 3.6044% every six months, or 7.3386% per year. b. After-tax cost of debt  7.3386%  (1  40%)  4.4032%. Evaluate: Avicorp’s before-tax cost of debt is 7.3386% per year (EAR), while its aftertax cost of debt (reflecting the tax deductibility of interest) is 4.4032%. 7.

Plan: Compute Laurel’s after-tax cost of debt. Execute: The pre-tax cost of debt is the yield to maturity on the existing debt, or 7%. Thus, the effective after-tax cost of debt is 7%  (1  35%)  4.55%. Evaluate: Laurel’s before-tax cost of debt is 7%; its after-tax cost of debt is 4.55%.

Plan: Compute the cost of preferred stock for Dewyco using Equation 12.4. Execute: The cost of capital for preferred stock is: rpfd 

Div pfd Ppfd



$4 $50

 8%

Evaluate: Dewyco’s cost of preferred stock is 8%. 9.

Plan: Compute Steady Company’s cost of equity using the CAPM. Execute: Using the Capital Asset Pricing Model, Steady’s cost of equity capital is 6%  0.20  7%  7.4%. Evaluate: Steady Company’s cost of equity is 7.4%.

10.

Plan: Compute Wild Swings’ cost of equity using the CAPM. Execute: Using the Capital Asset Pricing Model, Wild Swings’ cost of equity capital is 6%  2.5  7%  23.5%. Evaluate: Wild Swings’ cost of equity is 23.5%.

11.

Plan: Compute HighGrowth’s cost of equity capital using equation 12.5 (CDGM approach). Execute: The cost of equity capital for HighGrowth Company is rE 

Div1 PE

g

$1 $20

 4%  9%

Evaluate: HighGrowth’s cost of equity capital is 9%. 12.

Plan: Compute Slow ’n Steady’s cost of equity capital using Equation 12.5 (CDGM approach). Execute: The cost of equity capital for Slow ’n Steady is rE 

Div1 PE

g

$3 $30

 1%  11%

Evaluate: Slow ’n and Steady’s cost of equity capital is 11%. 13.

Plan: Compute the cost of equity capital for Mackenzie using the CAPM and then find the dividend growth rate that would yield the same cost of equity capital using CDGM. Execute: a. Using the Capital Asset Pricing Model, rE  5.5%  1.2  5%  11.5%

rE 

Div1 PE

g

g  11.5% 

$2 $36

 g  11.5%

 5.944%

Evaluate: Mackenzie’s cost of equity using the CAPM is 11.5%, which would require a dividend growth rate of 5.944% to result in the same cost of equity using CGDM. 14.

Plan: Calculate the WACC of CoffeeCarts. Execute: WACC  (30%)(4%)  (70%)(15%)  11.7% Evaluate: CoffeeCarts’ WACC is 11.7%.

15.

Plan: Calculate the WACC of AllCity Inc. Execute: AllCity’s cost of preferred equity is $2.50/$30 = 0.0833. Using the CAPM, its cost of common equity is 0.02 + 1.1(0.07) = 0.097. Applying the WACC formula, we have: WACC = (0.5)(0.097) + (0.1)(0.0833) + (0.4) (0.06) (1  0.35) = 0.07243. Evaluate: AllCity’s WACC is 7.243%.

16.

Plan: Calculate the WACC of Pfd Company. Execute: D + P + E = $10 million + $3 million + $15 million = $28 million 10 3 15 WACC    (7%)(1  40%)    (9%)    (13%)  9.4286%  28   28   28 

Evaluate: Pfd’s WACC is 9.4286%. 17.

Plan: Make the numerous calculations required in the problem. Execute: a.

rE 

Div1

b. rpfd 

g

Div pfd Ppfd



$1 $20 $2

$28

 4%  9%

 7.143%

c. The pre-tax cost of debt is the firm’s YTM on current debt. Since the firm recently issued debt at par, then the coupon rate of that debt must be equal to the YTM of the debt. Thus, the pre-tax cost of debt is 6.5%. d. Market value of debt  $20 million Market value of preferred stock  $28 per share  1 million preferred shares  $28 million Market value of equity  $20 per share  5 million shares outstanding  $100 million Market value of assets  $20 million  $28 million  $100 million  $148 million

 20   28   100  e. WACC    (6.5%)(1  35%)    (7.143%)    (9%)  8.003%  148   148   148  Evaluate: The calculation leads to a WACC of 8.003%.

18.

Plan: Because the retail coffee company is engaging in a scale expansion (doing more of what it already does), its WACC is the appropriate discount rate for the free cash flows from the project. The expansion can be valued using the growing perpetuity formula. Execute: NPV  FCF0  V0L  $200 million 

$15 million 0.10  0.03

 $14.286 million

Evaluate: Based on the projected incremental free cash flows, the expansion should proceed because it adds $14.286 million in value to the company. 19.

Plan: Draw a timeline for the RiverRocks project and compute its NPV. Execute: Timeline: 0

–50

Using the WACC as the discount rate and solving for NPV: NPV  50 million 

10 million 1.12



20 million 20 million 15 million    1.359 million 2 3 4 (1.12) (1.12) (1.12)

Evaluate: NPV is negative; RiverRocks should not take on this project. 20.

If RiverRocks is going to acquire Raft Adventures, then it should use a discount rate that is appropriate for the risk of Raft Adventures’ cash flows. That should be the WACC of Raft Adventures, which is 15%. So RiverRocks should use 15% as the discount rate for its evaluation of the acquisition.

21.

Plan: Compute the NPV of this acquisition. Execute: The NPV of this acquisition is NPV  $100 million 

$15 million 0.15  0.04

 $36.36 million

Evaluate: The acquisition has a positive NPV of $36.36 million, indicating that it will increase the value of RiverRocks. 22.

Plan: Use competitor’s beta to estimate cost of capital for new plant. Execute: Use CAPM Project beta = 0.85 (using beta of all equity competitor) Thus, WACC = rE = 4% + 0.85(5%) = 8.25%

Evaluate: The project has the same cost of capital as Harburtin because it is in the same industry (i.e., it has the same risk) and it is financed in the same way. 23.

Plan: Determine the estimated WACC for this analysis. Execute: The cost of equity is rE  4.5%  0.26  5.5%  5.93%. WACC  (0.11)(4.8%)(1  35%)  (0.89)(5.93%)  5.62%

Evaluate: CoffeeStop should use a rate of 5.62% to evaluate the liquor division. 24.

Plan: Calculate the WACC of your computer sales division based on the information you collected about HP Inc. Use the WACC of the entire company and that of the computer sales division to determine the WACC of your software division. Execute: a. To find the WACC of the computer sales division, we will calculate the WACC of HP Inc. The equity cost of capital for HP is 4.5%  1.21  5%  10.55%. Note that $700 million (the market value of HP’s debt) = $0.7 billion. HP’s

 0.7   67  WACC is   (6%)(1  35%)    (10.55%)  10.48%.  67.7   67.7  b. The firm’s WACC should be the weighted average of the divisional WACCs. WACCCompany  12%  (40%)(10.48%)  (60%)(WACCSoftware ) WACCSoftware 

12%  (40%)(10.48%) 60%

 13.01%

Evaluate: The estimated WACC for the computer sales division is 10.48%; the estimated WACC for the software division is 13.01%. 25.

Plan: Start with the NPV you calculated in Problem 21 and adjust your result to account for the issuing costs as an additional cost of the acquisition. Execute:

NPV  36.36  7  29.36 million

Evaluate: The NPV remains positive, so they should still go ahead with the acquisition.

26.

Plan: The best way to deal with the costs of external financing is to incorporate them directly into the NPV. The NPV of the project with external financing is the original NPV minus the cost of financing. Execute: $20 million in debt  3%  $600,000 in financing costs NPV  $15 million  $600,000  $14.4 million Evaluate: Incorporating the transaction costs of seeking external financing lowers the NPV of the project by $600,000 to $14.4 million, but it is still a positive-NPV project.

27.

Plan: HydroTech has significant excess cash. Use net debt and enterprise value when calculating the weights for the cost of capital formula and then calculate WACC. Execute: Net Debt = Book Value of Debt – Cash = $50 million – $10 million = $40 million (note that the market value of debt is not available). Enterprise Value = Equity + Net Debt = $100 million + $40 million = $140 million E/V = $100 million/$140 million = 71% and D/V = $40 million/$140 million = 29% The cost of equity for HydroTech is 3%  1.2  5%  9%. WACC  (0.29)(5%)(1  35%)  (0.71)(9%)  7.33% Evaluate: HydroTech should use a rate of 7.33% to evaluate projects that (i) are average risk and (ii) will be financed the same way as HydroTech is financed.

Chapter 13 Risk and the Pricing of Options Note: All problems in this chapter are available in MyLab Finance. An asterisk (*) indicates problems with a higher level of difficulty. For a breakdown of the difficulty ratings of all of the problems in this Solutions Manual, please see the Preface – Question Difficulty Ratings. 1. a. The contract with the highest volume is 19 Jun 140 Put. b. The contract with the highest open interest is 19 Jun 130 Put. c. You buy at the ask price, and each contract is for 100 options and is quoted per option, so you will pay $9.60  100 = $960. d. You sell at the bid price, and each contract is for 100 options and is quoted per option, so you will pay $1.45  100  $145. e. An in-the-money call is one for which the current stock price is greater than the strike price. Because the stock price is 131.04, this is true for strike prices of 125 and 130.

An in-the-money put is one for which the current stock price is less than the strike price. Because the stock price is 131.04, this is true for strike prices of 135 and 140. 2.

Excel Solution Plan: Determine the payoff and profit for the long call if the stock is selling at $55 at expiration. Repeat using the $35 stock price. Then draw payoff and profit diagrams for the long call as functions of the stock price at expiration. Execute: Long call option payoff and profit at expiration a. Payoff = $55 – $40 = $15; Profit = Payoff – Call Option Premium = $15 – $5 = $10 b. Payoff = $0, option is not exercised since $35 – $40 = –$5 < 0; Profit = $0 − $5 = −$5 c. Payoff diagram:

d. Profit diagram:

Evaluate: Both the payoff and profit diagrams are represented by a flat line when the stock price at expiration is $40 or less and the option is not exercised. When the stock price at expiration is greater than $40, both the payoff and the profit increase as the stock price becomes higher. 3.

Excel Solution Plan: Determine the payoff and profit for the short call if the stock is selling at $55 at expiration. Repeat using the $35 stock price. Then draw payoff and profit diagrams for the short call as functions of the stock price at expiration. Execute: Short call option payoff and profit at expiration: a. Payoff = –($55 – $40) = –$15; Profit = Payoff + Call Option Premium = –$15 + $5 = –$10 b. Payoff = $0, the option is not exercised since $35 – $40 = –$5 < 0; Profit = $0 + $5 = +$5 c. Payoff diagram:

d. Profit diagram:

Evaluate: Both the payoff and profit diagrams are represented by a flat line when the stock price at expiration is $40 or less and the call option is not exercised by the option holder. When the stock price at expiration is greater than $40, both the payoff and the profit decrease as the stock price becomes higher. 4.

Excel Solution Plan: Determine the payoff and profit for the long put if the stock is selling at $8 at expiration. Repeat using the $23 stock price. Then draw payoff and profit diagrams for the long put as functions of the stock price at expiration. Execute: Long put option payoff and profit at expiration: a. Payoff = $10 – $8 = $2; Profit = Payoff – Put Option Premium = $2 – $2 = $0 (stock price = $8 is the break-even point) b. Payoff = $0, option is not exercised since $10 – $23 = –$13 < 0; Profit = $0 − $2 = −$2 c. Payoff diagram:

d. Profit diagram:

Evaluate: Both the payoff and profit diagrams are represented by a flat line when the stock price at expiration is $40 or greater and the put option is not exercised. When the stock price at expiration is less than $40, both the payoff and the profit increase as the stock price becomes lower. 5.

Excel Solution Plan: Determine the payoff and profit for the short put if the stock is selling at $8 at expiration. Repeat using the $23 stock price. Then draw payoff and profit diagrams for the short put as functions of the stock price at expiration. Execute: Short put at expiration: a. Payoff = –($10 – $8) = –$2; Profit = Payoff + Put Option Premium = –$2 + $2 = $0 (stock price = $8 is the break-even point) b. Payoff = $0, option is not exercised since $10 – $23 = –$13 < 0; Profit = $0 + $2 = $2 c. Payoff diagram:

d. Profit diagram:

Evaluate: Both the payoff and profit diagrams are represented by a flat line when the stock price at expiration is $40 or greater and the put option is not exercised by the option holder. When the stock price at expiration is less than $40, both the payoff and the profit decrease as the stock price becomes lower. *6.

Suppose someone enters a forward contract to buy Telus stock at $50 one year from today. The investor who agrees to buy Telus stock for $50 will incur a negative payoff if Telus turns out to be cheaper than $50 in one year, and she will earn a positive payoff if Telus turns out to be more expensive than $50 in one year. For example, if Telus stock costs $45 in a year, then the investor’s payoff will be –$5 (since she is obligated to buy Telus for $50, which is $5 more than the market price). If, however, Telus stock costs $60 in a year, then the investor will earn a $10 positive payoff (since she will buy Telus for $50, which is $10 less than the market price). Notice that the investor’s payoff in one year would be exactly the same if, instead of buying Telus stock forward, she had purchased a call option on Telus stock with a $50 exercise price while simultaneously selling a put option on Telus stock with the same $50 strike price.

To protect against a fall in the price of Costco, you can buy a put option with Costco as the underlying asset. By doing this, over the life of the option you are guaranteed to get at least the strike price from selling the stock you already have.

Put: The long put payoff keeps increasing as the stock price at expiration declines, so the highest payoff will be reached at the lowest stock price. Zero is the lowest possible stock price (due to limited liability). Therefore, the maximum payoff for a long position in a put option is the strike price; even if the stock becomes worthless, the holder of a put has the right to sell that stock for the pre-agreed price. Call: The long call payoff keeps increasing as the stock price at expiration becomes higher. Since there is no clear upper bound that would limit the increase in the stock price, there is no maximum payoff for the call.

*9.

A call option cannot be more valuable than the underlying stock. On the date when the option is exercised, the option holder will receive a payoff equal to the stock price less the strike price. An investor who buys the stock instead and sells it on the same date when the option is exercised will receive a payoff equal to the stock price. Due to its higher payoff, the stock is the more valuable investment (as long as the exercise price is positive).

10.

The American option with the longer time to expiration has all the same rights and privileges that the shorter time to expiration option has. In addition, the holder of the longer-term option has the opportunity to exercise her option after the shorter-term option has already expired. This additional benefit is usually worth something.

11.

Plan: Use the replicating portfolio approach to value a binomial call option. Execute: In this case, the stock price either rises to SU  25 × 1.20  30 or falls to SD  25 × 0.80  20. The option payoff is, therefore, either CU  30 – 25 = 5 or CD  0 (since 20 – 25 < 0). The replicating portfolio is   5 030 20  0.5 and B  0 20 × 0.51.06  9.43, meaning that a portfolio consisting of half a share of stock and 9.43 borrowed at the risk-free rate will have the exact same future cash flows as a call option with a 25 strike price. Therefore, C0  0.5 × 25 9.43  $3.07. Evaluate: The call option is worth $3.07.

12.

Plan: Use the replicating portfolio approach to value a binomial put option. Execute: The parameters of the underlying stock are the same as in Problem 11, but the payoff of the put is PU = 0 if the stock goes up (since 25 – 30 < 0) and PD = 25 – 20 = 5 if the stock goes down. Therefore, the replicating portfolio is   0 530 20  –0.5 and B  5 20 × (–0.5)1.06   meaning that a portfolio consisting of a short position in half a share of stock and 14.15 invested at the risk-free rate will have the exact same future cash flows as a put option with a 25 strike price. Therefore, P0  0.5×25 +   $1.65. Evaluate: The put option is worth $1.65.

13.

Plan: Use put-call parity to determine the value of the European call option with a strike price of $35. Execute: Put-call parity (Notation: S = stock price, C = call option price, P = put option price, and K = strike price.)

C

K 1 r

PS

C PS

K 1 r

C  2.10  33 

 3.282.

1.1

Evaluate: The European call option would have a value of $3.282. 14.

Plan: Price the call using put-call parity and determine if there is an arbitrage opportunity. Execute: (Notation: S = stock price, C = call option price, P = put option price, and K = strike price.) C PS PS

K 1 r

 $3.33  $20 

$18 (1  0.08)

 $6.66

C  $7  $6.66

Evaluate: The call is overpriced compared to the portfolio of a put, the stock, and riskfree borrowing. The arbitrage strategy involves selling the call option, buying the put, buying the stock, and borrowing $16.67 (the present value of $18). This leaves the investor with $7  $6.66 = $0.34 today, and the future cash flow from this strategy is zero. *15.

According to put-call parity (see the solutions to Problems 13 and 14 above), a European call option is equivalent to a strategy that combines a European put option (with the same strike price as the call), the underlying stock, and borrowing the present value of the strike price. We also learned that equity in a firm that borrows is analogous to investing in a call option on the value of the firm’s assets, with the face value of the firm’s zero-coupon debt as the strike price. (Notation: S = stock price, C = call option price, P = put option price, and K = strike price.) C PS

K 1 r

Consider the put-call parity equation above and remember that the C value of the call is analogous to the value of equity, and the S value of the underlying is analogous to the value of the firm’s assets. Applying the put-call parity idea to corporate equity, equity holders’ claims are equivalent to the following combination: (i) a long position in a put option on the value of the firm assets with the face value of the firm’s zero-coupon debt as the strike price, (ii) a long position in the assets of the firm, and (iii) a short position in a risk-free bond with face value equal to the face value of the firm’s zero coupon debt.

The intuition is simple: equity holders have a claim on the value of the firm’s assets, but before they can get paid on their investment, the firm’s debt must be repaid. Equity holders have limited liability, so if the amount owed is greater than the value of the firm, shareholders can simply transfer the firm’s assets to creditors (which is analogous to having a put option on the firm’s assets).

*16.

According to put-call parity (see the solutions to Problems 13 and 14 above), the underlying stock is equivalent to a strategy that combines a long position in a European call option, a short position in a European put option (with the same strike price as the call), and borrowing the present value of the strike price. We also learned that equity in a firm that borrows is analogous to investing in a call option on the value of the firm’s assets, with the face value of the firm’s zero-coupon debt as the strike price. (Notation: S = stock price, C = call option price, P = put option price, and K = strike price.)

 K   P 1  r  

S  C  



Assets Equity

 Debt

Consider the put-call parity equation above and remember that the C value of the call is analogous to the value of equity, and the S value of the underlying is analogous to the value of the firm’s assets. The value of the firm’s assets is equal to the combined value of equity and debt. Therefore, if S represents the value of assets and C represents the value of equity, then (K/(1 + r) – P) must represent the value of debt. In other words, applying the put-call parity idea to corporate finance, the value of the firm’s debt is equivalent to the following combination: (i) a long position in a risk-free bond with face value equal to the face value of the firm’s zero-coupon debt, and (ii) a short position in a put option on the value of the firm assets with the face value of the firm’s zero-coupon debt as the strike price. The intuition is simple: if the amount owed by the firm is greater than the value of the firm, then the entire face value of debt will be repaid. Note, however, that equity holders have limited liability, so if the amount owed is greater than the value of the firm, shareholders can simply transfer the firm’s assets to creditors (which is analogous to creditors being short a put option on the firm’s assets).

Chapter 14 Raising Equity Capital Note: All problems in this chapter are available in MyLab Finance. An asterisk (*) indicates problems with a higher level of difficulty. For a breakdown of the difficulty ratings of all of the problems in this Solutions Manual, please see the Preface – Question Difficulty Ratings. 1.

Plan: The implied price per share is the investment divided by the number of new shares. The post-money valuation is the implied price per share multiplied by the total number of

shares of the company. Finally, your fractional ownership will equal your shares (5 million) divided by the new total shares (5,800,000). Execute: a. Implied price per share  $1,000,000/800,000  $1.25 b. Post-money valuation  $1.25  5,800,000  $7,250,000 c. Your fractional ownership: 5,000,000/5,800,000  0.862, or 86.2% Evaluate: In order to get new funding, you gave up 13.8% of your company in exchange for $1 million in new capital. The higher the price you can get for new shares, the lower the amount of ownership you have to give up for a given amount of new funding. 2.

Plan: The post-money valuation will be the total number of shares multiplied by the price paid by the venture capitalist (VC). The percentage of the firm owned by the VC is her shares divided by the total number of shares. Your percentage will be your shares divided by the total shares, and the value of your shares will be the number of shares you own multiplied by the price the VC paid. Execute: a. After the funding round, the founder’s 8 million shares will represent 80% ownership of the firm. To solve for the new total number of shares (TOTAL): 8,000,000  0.80  TOTAL So TOTAL  10,000,000 shares. If the new total is 10 million shares, and the venture capitalist will end up with 20%, then the venture capitalist must buy 2 million shares. Given the investment of $1 million for 2 million shares, the implied price per share is $0.50. b. After this investment, there will be 10 million shares outstanding, with a price of $0.50 per share, so the post-money valuation is $5 million. Evaluate: Funding your firm with new equity capital, be it from an angel or a venture capitalist, involves a trade-off—you must give up part of the ownership of the firm in return for the money you need to grow. The higher the price you can negotiate per share, the smaller the percentage of your firm you have to give up for a given amount of capital.

Plan: Post-money valuation is implied price per share × total shares (new and old). For parts (a) and (b), you must calculate the implied price per share (investment/shares received) for each offer and then calculate the post-money valuations. For part (c), you need to calculate the new total shares outstanding and divide your 10 million shares into that total. Subtracting your new ownership fraction from your old (100%) gives you the dilution. Execute: a. Implied price per share  $3 million/1 million shares  $3

Post-money valuation  $3  11 million shares  $33 million b. Implied price per share = $2 million/500,000 shares = $4 Post-money valuation  $4  10,500,000 shares  $42 million c. Offer 1, new fractional ownership: 10 million/11 million = 0.909; dilution = 1 − 0.909 = 0.091. Offer 2, new fractional ownership: 10,000,000/10,500,000 = 0.952; dilution = 1 − 0.952 = 0.048. Difference in dilution = 0.091 − 0.048 = 0.043. Evaluate: By offering a higher implied price per share, the second offer allows you to bring in more capital while suffering lower dilution. 4.

Excel Solution Plan: The pre-money valuation will be the value of the prior shares outstanding at the price in the funding round. The post-money valuation will be the total number of shares multiplied by the price paid by the VC. The percentage of the firm owned by the VC is her shares divided by the total number of shares. Your percentage will be your shares divided by the total shares, and the value of your shares will be the number of shares you own multiplied by the price the VC paid. Round

Price ($)

Number of Shares

Series B

0.50

1,000,000

Series C

2.00

500,000

Series D

4.00

500,000

Execute: a. Before the Series D funding round, there are (5,000,000  1,000,000  500,000  6,500,000) shares outstanding. Given a Series D funding price of $4.00 per share, the pre-money valuation is (6,500,000)  $4.00/share  $26 million. b. After the funding round, there will be (6,500,000  500,000  7,000,000) shares outstanding, so the post-money valuation is (7,000,000)  $4.00/share  $28,000,000. Evaluate: Funding your firm with new equity capital, be it from an angel or a venture capitalist, involves a trade-off—you must give up part of the ownership of the firm in return for the money you need to grow. The higher the price you can negotiate per share, the smaller the percentage of your firm you have to give up for a given amount of capital. 5.

1, 000, 000



7, 000, 000

or 14.29%

of the firm, while Series C and Series D investors each own 500, 000



7, 000, 000

or 7.14%

of the firm. Evaluate: When a company founder decides to sell equity to outside investors for the first time, it is common practice for private companies to issue preferred stock rather than common stock to raise capital. Preferred stock issued by mature companies, such as banks, usually has a preferential dividend and seniority in any liquidation and, sometimes, special voting rights. Conversely, the preferred stock issued by young companies typically does not pay regular cash dividends. However, this preferred stock usually gives the owner an option to convert it into common stock on some future date, so it is often called convertible preferred stock. 6.

Excel Solution Plan: The fraction of the firm that each investor owns can be determined as a percentage of the investor’s total value to the number of total outstanding shares. Execute: You will own 5,000,000/7,000,000  71.43% of the firm after the last funding round. Evaluate: When a company founder decides to sell equity to outside investors for the first time, it is common practice for private companies to issue preferred stock rather than common stock to raise capital. Preferred stock issued by mature companies, such as banks, usually has a preferential dividend and seniority in any liquidation and, sometimes, special voting rights. Conversely, the preferred stock issued by young companies typically does not pay regular cash dividends. However, this preferred stock usually gives the owner an option to convert it into common stock on some future date, so it is often called convertible preferred stock.

First, compute the cumulative total number of shares demanded at or above any given price: Price

Cumulative Demand

14.00

100,000

13.80

300,000

13.60

800,000

13.40

1,800,000

13.20

3,000,000

13.00

3,800,000

12.80

4,200,000

The winning price should be $13.40, because investors have placed orders for a total of 1.8 million shares at a price of $13.40 or higher. 8.

For investors to place orders for 2.3 million shares, the offer price will need to be $13.20. The amount raised will be $13.20  2.3 million  $30.36 million.

9.7.

Excel Solution Plan: If the IPO price is based on a price/earnings ratio that is similar to those for recent IPOs, then this ratio will equal the average of recent deals. Thus, to compute the IPO price based on the P/E ratio, we will first take the average P/E ratio from the comparison group and multiply it by Outdoor Recreation Inc.’s total earnings. This will give us a total value of equity for Outdoor Recreation Inc. To get the per-share IPO prices, we need to divide the total equity value by the number of shares outstanding after the IPO. The fraction of the firm that each investor owns can be determined as a percentage of the investor’s total value to the number of total outstanding shares. Execute: a. With a P/E ratio of 20.0×, and 2022 earnings of $7.5 million, the total value of the firm at the IPO should be P

 20.0 x  P  $150 million

7.5

There are currently (500,000  1,000,000  2,000,000)  3,500,000 shares outstanding (before the IPO). At the IPO, the firm will issue an additional 6.5 million shares, so there will be 10 million shares outstanding immediately after the IPO. With a total market value of $150 million, each share should be worth $150/10  $15 per share. b. After the IPO, you will own 500,000 of the 10 million shares outstanding, or 5% of the firm.

Evaluate: As we found in Chapter 7, using multiples for valuation always produces a range of estimates—you should not expect to get the same value from different ratios. Based on these estimates, the underwriters will probably establish an initial price range for the stock. 10.

Plan: Underpricing is the difference between the first day closing price and the offering price. To figure out how much money the firm missed out on, multiply the dollar difference in price per share by the number of shares sold. Execute: a. Underpricing  $17  $15  $2. As a percent of the offering price, this is $2/$15  13.3%. This is also the percentage return on the first day of trading. b. Forgone money: $2 per share × 10 million shares = $20 million Evaluate: Because the true market value of the shares was $2 per share higher than the price at which they were sold through the underwriters, the firm sold them for too little, missing out on an additional $20 million in capital it could have raised by pricing them at $17 per share.

11.

Plan: Calculate the initial return on Margoles stock and analyze the underpricing. Execute: a. The initial return on Margoles Publishing stock is ($19.00 – $14.00)/$14.00  35.7%. b. Who gains from the price increase? Investors who were able to buy at the IPO price of $14/share see an immediate return of 35.7% on their investment. Owners of the other shares outstanding that were not sold as part of the IPO see the value of their shares increase. To the extent that the investors who were able to obtain shares in the IPO have other relationships with the investment banks, the investment banks may benefit indirectly from the deal through their future business with these customers. Evaluate: Who loses from the price increase? The original shareholders lose, because they sold stock for $14.00 per share when the market was willing to pay $19.00 per share.

12.

Plan: Calculate the total cost of going public. Execute: The total dollar value of the IPO was ($14.00)  (10 million)  $140 million. The total cost of going public was (0.07)  ($140 million)  $9.8 million. Evaluate: It cost Margoles Publishing $9.8 million to go public.

13.

Plan: Calculate the dollar cost of the underwriter fees. Execute: The total dollar value of the IPO was ($18.50)  (4 million)  $74 million. The spread equaled (0.07)  ($74 million) or $5.18 million. Evaluate: It cost Chen Brothers, Inc. $5.18 million in underwriter fees.

14.

The spread represents the discount to the offer price that the underwriter pays the firm for the shares, so the net price to the firm of the first option is $20.00 – (0.07)($20.00) = $18.60. The net price to the firm of the second option is $19.50 – (0.04)($19.50) = $18.720. Thus, the higher offer price with the higher discount produces the higher net price to the firm.

15.

Plan: First, we must compute the underwriting spread. Then, we can compute the lowest price possible using the underwriting spread. Execute: The proceeds to your firm, given an offer price of $17.25 per share and an underwriting spread of 7%, are (0.93)  ($17.25)  (3 million)  $48,127,500. In order for you to be indifferent between the two options, the offer price (with the 5% underwriting spread) would need to drop low enough that you raise $48,127,500. $48,127, 500  ($ X )  (0.95)  (3 million) $X 

$48,127, 500 (0.95)  (3, 000, 000)

 $16.89

The offer price can fall to $16.89 before you would prefer to pay 7% to get $17.25 per share. Evaluate: While the auction IPO does not provide the certainty of the firm commitment, it has the advantage of using the market to determine the offer price. It also reduces the underwriter’s role, and consequently, fees. Use the following information for Problems 16 through 18. The firm you founded currently has 12 million shares, of which you own 7 million. You are considering an IPO where you would sell 2 million shares for $20 each. 16.

Plan: The percentage of the firm owned is the shares owned divided by the total number of shares. The percentage will be the shares divided by the total shares, and the value of your shares will be the number of shares you own multiplied by the price and VC paid. Execute: If the firm sells 2 million primary shares at $20 each, the firm will raise $40 million and the total number of shares outstanding after the IPO will be 14 million. You will own 50% of the firm (7 million/14 million) after the IPO. Evaluate: Funding your firm with new equity capital, be it from an angel or a venture capitalist, involves a trade-off—you must give up part of the ownership of the firm in return for the money you need to grow. The higher is the price you can negotiate per share, the smaller is the percentage of your firm you have to give up for a given amount of capital.

17.

Execute: If all of the shares sold are secondary shares from your holdings, the firm raises no money from the IPO. Your percentage ownership of the firm after the IPO will be 41.67% (5 million/12 million). Evaluate: Funding your firm with new equity capital, be it from an angel or a venture capitalist, involves a trade-off—you must give up part of the ownership of the firm in return for the money you need to grow. The higher the price you can negotiate per share, the smaller the percentage of your firm you have to give up for a given amount of capital. 18.

Plan: In order to know how much money will be raised, we need to compute how many total shares would be purchased if everyone exercised their rights. Then we can multiply it by the price per share to calculate the total amount raised. Execute: In order to retain 50.1% ownership of the firm, you would need to hold (0.501)  (12 million)  6,012,000 shares. Thus, you could sell up to 988,000 shares. Evaluate: A firm’s need for outside capital rarely ends at the IPO. Usually, profitable growth opportunities occur throughout the life of the firm, and in some cases it is not feasible to finance these opportunities out of retained earnings. Thus, more often than not, firms return to the equity markets and offer new shares for sale, or seasoned equity offering.

19.

The net capital to the firm will be the gross proceeds net of the underwriter fee, or 93% of the gross proceeds. In this case, ($50)(10 million)(0.93) = $465 million.

20.

Plan: In order to know how much money will be raised, we need to compute how many total shares would be purchased if everyone exercised their rights. Then we can multiply it by the price per share to calculate the total amount raised. Execute: a. The company sold 5 million shares at $42.50 per share, so it raised ($42.50)  (5,000,000)  $212.5 million. After underwriting fees, it will keep $212.50  (1  0.05)  $201.875 million. b. The venture capitalists raised ($42.50)  (3,000,000)  $127.5 million. After underwriting fees, they will keep $127.5  (1  0.05)  $121.125 million. So, in total, the SEO was worth $201.875  $121.125  ($42.50)  (8,000,000)  0.95  $323 million. c. If the stock price dropped 3% on the announcement of the SEO, the stock price would be $41.23. The SEO would be worth ($41.23)  (8,000,000)  0.95  $313 million. Evaluate: A firm’s need for outside capital rarely ends at the IPO. Usually, profitable growth opportunities occur throughout the life of the firm, and in some cases it is not feasible to finance these opportunities out of retained earnings. Thus, more often than not, firms return to the equity markets and offer new shares for sale, or seasoned equity offering.

*21.

Plan: In order to compute the number of shares needed, we can set the total money raised equal to the underwriter spread multiplied by the offer price and the number of shares sold. In order to compute the percentage reduction in ownership, we can use the number of shares outstanding divided by the number of shares the firm must sell plus the number of shares outstanding. Execute: a. Total money raised  (1  underwriter spread)  (offer price)  (number of shares sold) 100, 000, 000  (0.95)  ($50)  ( X ) X 

100, 000, 000 0.95  50

 2,105, 263

The firm would need to sell 2,105,263 shares to raise $100 million. b. A 2% drop in the stock price would result in a stock price of (98%)  ($50)  $49. Total money raised  (1  underwriter spread)  (offer price)  (number of shares sold) 100, 000, 000  (0.95)  ($49)  ( X ) X 

100, 000, 000 0.95  49

 2,148, 228

The firm would need to sell 2,148,228 shares to raise $100 million. c. If there is no drop in the stock price upon the announcement of the SEO, the existing shareholders would own 10, 000, 000

 0.8261  82.61%.

12,105, 263

This would be a 17.39% reduction in ownership of the firm. If there is a 2% drop in the stock price, then the existing shareholders would own 10, 000, 000

 0.8232  82.32%.

12,148, 228

This would be a 17.68% reduction in ownership of the firm. Evaluate: A firm’s need for outside capital rarely ends at the IPO. Usually, profitable growth opportunities occur throughout the life of the firm, and in some cases it is not feasible to finance these opportunities out of retained earnings. Thus, more often than not, firms return to the equity markets and offer new shares for sale, or seasoned equity offering. 22.

Plan: Calculate how much money would be raised if all rights were exercised. Execute: Investors will receive a total of 10 million rights. Since it takes 10 rights to purchase 1 share, they will be able to purchase 1 million shares. At a price of $40/share,

the company will be able to raise ($40/share)  (1 million shares)  $40 million in this offering. Evaluate: If all rights were exercised, MacKenzie Corporation would raise $40 million from the offering.

Chapter 15 Debt Financing Note: All problems in this chapter are available in MyLab Finance. An asterisk (*) indicates problems with a higher level of difficulty. For a breakdown of the difficulty ratings of all of the problems in this Solutions Manual, please see the Preface – Question Difficulty Ratings. 1.

Plan: Compute the closing fees and determine the proceeds from the loan. Execute: Fees: 200,000  0.02  4,000 Total Proceeds: 200,000  4,000  $196,000 Evaluate: You will receive $196,000 after paying the closing fee on your $200,000 loan.

Plan: Compute the fees and the net proceeds you will receive from the bond offering and then recalculate the interest rate based on how much interest is paid (charged on the gross amount borrowed) divided by the net amount borrowed. Execute: a. With fees of 2%, your net is 98% of the loan amount for the first option: 0.98($500,000) = $490,000; with fees of 1%, your net is 99% of the loan amount for the second option: 0.99($500,000) = $495,000. b. Because you pay interest on the gross loan amount (the principal of the loan), your interest on each option is calculated based on $500,000. For the first option, your interest is (0.04)($500,000) = $20,000, but you only received $490,000, so your true rate of interest is $20,000/$490,000 = 0.0408, or 4.08%. For the second option, your interest is (0.045)($500,000) = $22,500, so your true rate of interest is $22,500/$495,000 = 0.0455, or 4.55%. Evaluate: The fees result in your receiving less in proceeds from the loan and cause your borrowing costs on the money you do receive to be higher.

Plan: Compute the fees and the net proceeds you will receive from the bond offering. Execute: Fees: $100, 000, 000  0.03  $3, 000, 000 Total Proceeds: $ 100,000,000  3,000,000  $97,000,000 Evaluate: You will receive $97,000,000 from the bond offering.

The Canadian government uses cash management bills, treasury bills, fixed-coupon marketable bonds (otherwise known as Government of Canada bonds), and real return bonds. Cash management bills are pure discount bonds with maturities from 1 day to less than 3 months. Treasury bills are pure discount bonds with maturities of 3 to 12 months. Fixed-coupon marketable bonds (or Government of Canada bonds) are semiannual coupon bonds with maturities from 2 to 40 years. Finally, real return bonds are bonds with coupon payment and face value amounts that adjust with the rate of inflation.

Plan: Determine how much the CPI appreciated so you can adjust the face value and then determine the new coupons based on the coupon rate and the new face value. Execute: The CPI appreciated by 300 250

 1.2

Consequently, the principal amount of the bond increased by this amount; that is, the original face value of $1000 increased to $1200. Since the bond pays semi-annual coupons, the coupon payment is

 0.03    $1,000  $18  2 

1.2  

Evaluate: Inflation caused the bond’s face value to increase, and applying the coupon rate to the new face value resulted in higher coupons, too. 6.

Plan: Determine how much the CPI depreciated so you can adjust the face value and then determine the new coupons based on the coupon rate and the new face value. Note, though, that at maturity the actual face value paid will have $1000 as a lower bound. Execute: The CPI went from 400 to 300 so we can calculate its rate of change as follows: 1 + Rate of Change = 300/400 = 0.75, so Rate of Change = –25%. Thus, the CPI depreciated by 25%.

Consequently, the principal amount of the bond, from which coupons are calculated, has decreased by this amount. That is, the original face value $1000 decreased to $750. With a coupon rate of 6%, and since the bonds pay semi-annual coupons, the coupon payment is 0.06/2  $750 = $22.50. However, we assumed that the final payment of the maturity (i.e., the principal) is protected against deflation. So, since $750 is less than the original face value of $1,000, the original amount is repaid (i.e., $1,000). If this assumption did not hold, only $750 would be repaid. Evaluate: Deflation caused the bond’s face value for which coupons are calculated to decrease, and applying the coupon rate to this new face value resulted in lower coupons. However, since the face value at maturity was protected so it would not fall below $1000, there is still the full $1000 paid in principal at maturity. 7.

Holders of the CMHC securities face payment risk because homeowners have the option to prepay their debt whenever they decide to do so. In particular, they will prepay if interest rates fall and they can obtain new debt at a lower interest rate. This is precisely when the holders of CMHC securities would like to avoid payments, since they can reinvest only at a lower interest rate.

The distinguishing feature is that income from U.S. municipal bonds is not taxed in the United States at the federal level. In Canada, interest income is taxable regardless of its source unless the security is held in a tax-free account (e.g., RRSP, TFSA, RRIF, pension fund).

Plan: You have positive earnings (net income), so without any quick ratio requirement, you could pay up to $70 million in dividends. However, you must check the quick ratio as well. Current Assets  Cash  Receivables  Inventory. Execute: Current Assets  $10 million  $8 million  $5 million  $23 million Quick Ratio  (Current Assets  Inventory)/Current Liabilities  ($23 million  $5 million)/$19 million  $18 million/$19 million  0.947 In order to pay dividends, you must raise your quick ratio to 1.1, which means your current assets net of inventory must be 1 times your current liabilities (1 × 19 = 19). So you must use $1 million of your earnings to increase your cash holdings to meet the quick ratio test. With $11 million in cash instead of $10 million, the ratio is ($11 million  $8 million)/$19 million  1. Thus, the maximum dividend would be $70 million  $1 million = $69 million. Evaluate: The quick ratio requirement forced you to use some of your net income to increase your cash holdings, thus reducing the total earnings available to pay dividends.

10.

The new firm will have an NTA-to-debt ratio that is a weighted average of the ratios of your firm and the firm you are acquiring. The weights are your relative sizes. Because you are twice the size of the other firm, you will have a weight of 2/3, and it will have a weight of 1/3 in the new firm. The ratio for the combined firm will be (2/3)(2) + (1/3)(1.2) = 1.7333, so you can acquire the firm without violating your covenant.

11.

Excel Solution a. Plan: Create a timeline of the cash flows and compute the yield to maturity. Execute: Timeline: Time

Cash Flows

$100 $6

The present value formula to be solved is 1 100    1  (1  YTM)10   (1  YTM)10 

102 

YTM 

Using the annuity calculator, YTM  5.73%. N

I/Y

102.00

Given: 10 Solve for Rate:

PMT

100

Excel Formula

RATE(10,6,102,100)

5.732%

Evaluate: The yield to maturity is 5.73%. b. Plan: Create a timeline of the cash flows and compute the yield to call. Execute: YTC: Timeline: Time

Cash Flows

$100 $6

The present value formula to be solved is 106

102 

1  YTC

 YTC 

106

 1  3.92%

102

Evaluate: The yield to call is 3.92%. c. Because the bond is trading at a premium, the likelihood of call is high, and the yield to worst is the YTC: 3.92%. 12.

Excel Solution a. Plan: Create a timeline of the cash flows and compute the yield to maturity. Execute: Timeline: Years

Periods

Cash Flows

$2.5

$100 $2.5

The present value formula to be solved is 99 

2.5 

1  100 1   6  6 i  (1  i )  (1  i )

Using the annuity calculator, i  2.68%

I/Y

PV 99.00

Given: 6 Solve for Rate:

PMT

2.5

100

Excel Formula RATE (6,2.5,99,100)

2.68%

So, since YTM are quoted as APRs, YTM  i  2  2.68%  2  5.36%

Evaluate: The yield to maturity on the bond is 5.36%. b. Plan: Create a timeline of the cash flows and compute the yield to call. Execute: YTC: Timeline: Years

Periods

Cash Flows

$2.5

$100 $2.5

The present value formula to be solved is 99 

2.5  i

1  100  1  (1  i ) 4   (1  i ) 4  

Using the annuity calculator, i  2.77%

I/Y

99.00

Given: 4 Solve for Rate:

PMT

2.5

100

2.77%

Excel Formula RATE (4,2.5,99,100)

Since YTM (and therefore YTC) are quoted as APRs, YTC  i  2  5.54%

Evaluate: The yield to call is 5.54%. c. Because the bond is trading at a discount, the likelihood of call is low, and the yield to worst is the YTM: 5.36%. 13.

Plan: Compute the conversion ratio of the bond. Execute: The conversion price is the face value of the bond divided by the conversion ratio. In this case: P

Face value Conversion ratio



$10, 000 450

P  $22.22

Evaluate: The conversion ratio for this bond is $22.22. 14.

The stock price must surpass the conversion price which is equal to $1000 ÷ 40 shares = $25/share.

15.

Plan: a. Compute the value of the bond if converted to equity. Execute: If converted, each bond is worth 107.565 shares of RealNetworks stock. At $9.70 per share, this would result in value to bondholders of (107.565)  ($9.70)  $1043.38. Evaluate: a. The converted bond would be worth $1043.38. b. The call price is 100% of face value of the bond, which is $1000. Thus, the value bondholders would receive under the call is $1000. c. Bondholders will convert into shares if you call the bonds, as the value of converting is greater than the value received under the call.

Chapter 16 Capital Structure Note: All problems in this chapter are available in MyLab Finance. An asterisk (*) indicates problems with a higher level of difficulty. 1.

Plan: In order to calculate the NPV of the project, we must first compute the free cash flows for that year by calculating the average of the two likely scenarios for cash flows that year. We can then compute the NPV using the NPV formula. Knowing the free cash flows, the discount rate, and the initial investment, we can compute the NPV as well as the equity value. We can compute the cash flows of the levered equity by computing the risk-free rate of the debt payments and subtracting that from the two likely scenarios for the cash flows used in part (a). Finally, the initial value of the project can be found by subtracting the debt payments of the project from the equity value. Execute: 1  $130,000  $180,000   $155,000, 2 $155,000 NPV   $100,000  $129,166.67  $100,000  $29,166.67. 1  0.20

E[C1 ] 

b. Equity value PV ( E[C1 ]) 

$155,000  $129,166.67. 1  0.20

c. Borrowing $100,000 at the risk-free rate requires the firm to repay $100,000(1.10) = $110,000. Equity holders will receive the project cash flow less the debt repayment: either $130,000  $110,000 = $20,000 or $180,000  $110,000 = $70,000. The initial value of the equity, by M&M, is $129,166.67 – $100,000 = $29,166.67.

Evaluate: The NPV rule states to accept a project with positive NPV, such as this project; therefore, all else equal, the company should undertake the project. 2.

Plan: First, find the total market value of the unlevered firm: (a) if investors are willing to pay $2 million for 50% of the firm then 100% of the firm must be worth twice as much. Next, (b) borrowing $1 million will not change the total value of the firm, but it will change the capital structure to $3 million equity and $1 million debt. Since you already borrowed $1 million, you only need $1 million from equity investors, reducing the fraction of the firm’s equity that needs to be sold. The last step, (c), is based on the simple idea that the fraction of the equity that is not sold to investors will remain yours.

Execute: a. Total value of equity = 2  $2 million = $4 million. b. M&M says the total value of firm is still $4 million. Debt of $1 million implies the total value of equity is $3 million. Therefore, 33% of equity must be sold to raise $1 million. c. In part (a), 50%  $4 million = $2 million. In part (b), 2/3  $3 million = $2 million. Thus, in a perfect market, the choice of capital structure does not affect the value to the entrepreneur. Evaluate: In this case, changing the capital structure does not affect the value to the owner of the firm and, therefore, the owners have more flexibility with their capital structure. 3.

Plan: We can use Equation 16.1 to compute the current market value of Acort’s equity. To determine its expected return, we will compute the cash flows to equity. The cash flows to equity are the cash flows of the firm net of the cash flows to debt (repayment of principal plus interest). Execute: a. E[Value in 1 year] = 0.8($50 million) + 0.2($20 million) = $44 million VU =

$44 million 1.10

$20 million 1.05

= $40 million.

= $19.048 million. Therefore,

V L  V U  $40 million.

E  V U  D  $40 million  $19.048 million =$20.952 million c. Without leverage:

Expected return of Acort’s equity =

$44 million $40 million

-1 = 10%,

With leverage:

Expected return of Acort’s equity =

$44 million - $20 million $20.952 million

-1 = 14.55%,

d. Without leverage:

Lowest possible realized return of Acort’s equity =

$20 million $40 million

-1 = -50%,

With leverage:

Lowest possible realized return of Acort’s equity =

$0 $20.952 million

-1 = -100%,

Evaluate: The current market value of Acort’s equity when unlevered is nearly double the current market value of Acort’s equity when levered. The expected return is greater

with debt than without, yet the lowest possible realized return of Acort’s equity is less when unlevered as opposed to levered. 4.

Excel Solution Plan: We can compute the debt payments and equity dividends for each firm using the capital structure of each firm. We know from Equation 16.1 that the value of unlevered equity equals the value of levered equity plus the value of debt. We can replicate the cash flows of an investment in ABC by buying an identical stake in both XYZ’s equity and its debt. We can replicate the cash flows of an investment in XYZ by buying an identical stake in ABC’s equity and financing it in part with debt. Execute: a. ABC FCF

Interest Expense

800 1000

XYZ

Equity Dividends

Interest expense

Equity Dividends

800

500

300

1000

500

b. VU = D + E = VL. Buy 10% of XYZ’s debt and 10% of XYZ’s equity. As a result, get 0.1  $5,000 = $500 in interest and either 0.1  $300 = $30 or 0.1  $500 = $50 in dividends. In total, receive either $50 + $30 = $80 or $50 + $50 = $100. ABC FCF

10% Equity

Replicating ABC with XYZ Debt and Equity 10% Debt

10% Equity

Total

800

1000

100

c. VL = E + D = VU  E = VL – D = VU – D (i.e., levered equity is unlevered equity combined with borrowing). Borrow 10% of XYZ’s debt (i.e. borrow 0.1  $5,000 = $500). In addition to borrowing $500, buy 10% of ABC’s equity, in part financed with the borrowed funds. As a result, get either 0.1  $800 = $80 or 0.1  $1,000 = $100 in dividends and pay 0.1  $500 = $50 in interest. In total, receive either $80 – $50 = $30 or $100 – $50 = $50. XYZ 10% Equity

Replicating XYZ with ABC and Borrowing $500 10% Equity

Pay Interest

Total

– 50

100

– 50

Evaluate: M&M Proposition I states that, in a perfect capital market, the total value of a firm is equal to the market value of the free cash flows generated by its assets and is not affected by its choice of capital structure. By adding leverage, the returns of the unlevered firm are effectively split between low-risk debt and higher-risk levered equity. Leverage increases the risk of equity even when there is no risk that the firm will default. 5.

Plan: We can use Equation 16.3 to compute the expected return of equity in both cases. Execute: a. rE = rU  (D/E)(rU  rD)  12%  0.50(12%  6%)  15% b. rE  12%  1.50(12%  8%)  18% c. Returns are higher because risk is higher—the return fairly compensates for the risk. There is no free lunch. Evaluate: As the firm borrows at the low cost of capital for debt, its equity cost of capital rises according to Equation 16.3. The net effect is that the firm’s WACC is unchanged. As the amount of debt increases, the debt becomes more risky because there is a chance the firm will default; as a result, the debt cost of capital also rises.

Plan: We can use Equation 16.3 to compute the cost of equity using the unlevered cost of capital, the D/E ratio, and the cost of debt. Execute: rE  rU 

D 0.13 (rU  rD )  0.092  (0.092  0.06)  0.0968  9.68%. E 0.87

Evaluate: WACC is equal to the unlevered equity cost of capital. As the firm borrows at the low cost of capital for debt, its equity cost of capital rises according to Equation 16.3. 7.

Plan: Because you are 100% equity financed, your current cost of equity is your total cost of capital (rE = rU = WACC = 12%). As in the previous problem, we can use Equation 16.3 to compute the cost of equity using the new capital structure and the cost of debt. Alternatively, we can use Equation 16.2 (the WACC formula) and solve for rE. Execute: D 0.3 (rU  rD )  0.12  (0.12  0.07)  0.1414  14.14%. E 0.7 E D rU  rWACC  rE  rD  0.12  rE  0.7  0.07  0.3  rE  14.14%. ED ED rE  rU 

Evaluate: As the firm borrows at the low cost of capital for debt, its shares become riskier and the equity cost of capital rises. The net effect is that the firm’s WACC remains unchanged; in perfect capital markets, the WACC is always equal to the unlevered cost of capital. We can either solve for rE by rearranging the WACC formula, or, equivalently, use Equation 16.3.

Plan: Because we know that capital structure changes do not affect the pre-tax WACC of the firm, we know that the WACC will not change. So, first we compute the WACC with the existing capital structure and then use the fact that it does not change to compute the new cost of debt (in this case it makes sense to solve part (b) and then part (a)). Execute: rWACC  rE

E ED

 rD

D  0.15  0.7  0.05  0.3  12%. ED

a. The new weights apply to changed cost of equity and cost of debt: rWACC = 0.20 ´ 0.3 + rD ´ 0.7 = 0.12 Þ rD = 8.57%.

b. After the capital structure change, your WACC remains 0.12.

Evaluate: Due to the increase in leverage, your debt has become riskier, and so your cost of debt capital has increased. Have you lowered your overall cost of capital? No—your overall cost of capital (WACC) is determined by the risk of your assets. You have merely reshuffled the claims on those assets. In the end, the costs of equity and debt adjust and your overall cost of capital does not change. Plan: We can find the net income of the firm using the EBIT, interest expense, and the corporate tax rate. We can compute the interest tax shield using Equation 16.4. Execute: a. Net income = EBIT – Interest – Taxes = ($325 million – $125 million)  (1 – 0.40) = $120 million b. Net income + Interest  $120 million  $125 million  $245 million. c. Net income = EBIT – Taxes = $325 million  (1 – 0.40) = $195 million. This is $245 million  $195 million  $50 million lower than part (b). d. Interest tax shield  $125 million  40%  $50 million. Evaluate: The gain to investors from the tax deductibility of interest payments is referred to as the interest tax shield. The interest tax shield is the additional amount that a firm would have paid in taxes if it did not have leverage, but can instead pay to investors.

10.

Plan: Deduct the extra interest expense from net income. Interest is tax deductible, so the resulting tax shield will have to be added to net income. Execute: a. Net income will fall by the after-tax interest expense of 1 million × (1 – 0.35) = $650,000, from $20.750 million to $20.10 million. b. Free cash flow is not affected by interest expenses. Evaluate: Interest is tax deductible, so the decline in net income will be less than the increase in pre-tax interest expense. The decline in net income is equal to the after-tax interest expense.

11.

Plan: Calculate Braxton’s remaining debt, interest expense, and tax shield for each year from time t = 0 to t = 5. Execute: Interest for year t is calculated as 8%  Debtt-1. Tax Shield for year t is calculated as 40%  Interestt. (All amounts are in $millions.) Year Debt Interest Tax Shield

0 35

1 28 2.8 1.12

2 21 2.24 0.90

3 14 1.68 0.67

4 7 1.12 0.45

5 0 0.56 0.22

Evaluate: As Braxton reduces its debt, the amount of the tax shield declines from $1.12 million to $0.22 million. 12.

Plan: We can use Equation 16.7 to compute the present value of the tax shield. Execute:

PV(Interest Tax Shield) = TC  D PV(Interest Tax Shield) = 35%  ($300 billion  13%) = $13.65 billion

Evaluate: We know that in perfect capital markets, financing transactions have an NPV of zero. However, the interest tax deductibility makes this a positive-NPV transaction for the firm. The total value of the levered firm exceeds the value of the firm without leverage due to the present value of the tax savings from debt. There is an important tax advantage to the use of debt financing. 13.

Plan: Since all cash flows are risk free, we can use the risk free rate to determine the value of the unlevered firm as well as the value of levered equity and debt. Execute: a. Net income = $1000  (1  40%)  $600. Thus, equity holders receive dividends of $600 per year with no risk. $600 VU = = $12,000

0.05

b. Net income = ($1000 – $500)  (1  40%)  $300. Debt holders receive interest of $500 per year. $300 $500 E= = $6,000 and D = = $10,000

0.05

c. With leverage VL  E + D = $6,000  $10,000  $16,000 Without leverage VU  $12,000 Difference  $16,000  $12,000  $4,000 d. $4,000/$10,000 = 40% = corporate tax rate Evaluate: Adding $10,000 debt to the capital structure increased firm value by an amount equal to the PV of the interest tax shield = $10,000  40% = $4,000.

14.

Excel Solution Plan: We must compute the value of the tax shield in each year and then compute the present value of the tax shields. Execute: Year

$75

$50

$25

Interest (millions)

$10

$7.5

$2.5

Tax Shield (millions.)

$3.6

$2.5

$1.5

$0.7

10%

Debt (millions)

$100

PV (Tax Shield) (millions)

Evaluate: The present value of the annual tax shields is $3.6 million + $2.5 million + $1.5 million + $0.7 million = $8.3 million. 15.

Plan: We can use Equation 16.4 to compute the interest tax shield and then calculate the present value. In the case of permanent debt, we can use Equation 16.7 instead and compute the present value of the interest tax shield in a single step. Execute: a. Interest Tax Shield= $10 million × 6% × 35% = $0.21 million b. PV(Interest Tax Shield) = $0.21 million / 0.06 = $3.5 million c. InterestTax Shield  $10 million  5%  35%  $0.175 million. PV(Interest Tax Shield) = $0.175 million / 0.05 = $3.5 million Or simply: PV(Interest Tax Shield) = TC  D = 35%  ($10 million) = $3.5 million Evaluate: If the firm has permanent debt, the interest rate has no impact on the present value of the interest tax shield.

16.

Plan: We can use Equations 16.2 and 16.8 to calculate the pre-tax and after-tax WACCs, respectively. The tax rate is 35%. E = $2 million, D  $1 million, rE  0.12 and rD  0.07. E/(E + D) = $2 million / ($2 million + $1 million) = 2/3 and D/(E + D) = 1 – 2/3 = 1/3. Execute: a.

Pre-tax rWACC = rE

E D 2 1 +r = 0.12 ´ + 0.07 ´ = 10.33%. E+D DE+D 3 3

b. After-tax rWACC = rE

E D 2 1 + rD ( 1 - TC ) = 0.12 ´ + 0.07 ´ ( 1- 0.35 ) ´ = 9.52%. E+D E+D 3 3

Evaluate: The WACCs differ only by the effect of the tax deductibility of interest, which acts only on the debt portion of the cost of capital. So, while the deduction lowers the cost of debt by 35% to 4.55%, the overall WACC only decreases by 0.82 percentage points because only one-third of Rogot’s capital is debt. 17.

Plan: We can use Equations 16.2 and 16.8 to calculate the pre-tax and after-tax WACCs, respectively. E  $15  3 million = $450 million and D = $150 million, E + D = $600 million, rE  0.1, and rD  0.04. The tax rate is 35%. Execute: a.

Pre-tax rWACC  rE

E D $450 million $150 million  rD  0.1   0.05   8.75%. ED ED $600 million $600 million

After-tax rWACC  rE

b. 

E D $450 million  rD  1  TC   0.1   0.05   1  0.35  V V $600 million

$150 million  8.31%. $600 million

Evaluate: The WACCs differ only by the effect of the tax deductibility of interest, which acts only on the debt portion of the cost of capital. So, while the deduction lowers the cost of debt by 35% to 3.25%, the overall WACC only decreases by 0.44 percentage points because only 25% of Rumolt’s capital is debt. 18.

Plan: We can calculate the D/V ratio based on the D/E ratio. We can compute the reduction in the WACC due to the interest tax shield using the last term in Equation 16.9. Execute: D DE 0.65    0.3939 E  D 1  D E 1  0.65 Reduction in rWACC due to Interest Tax Shield  rDTC

D  0.07  0.4  0.3939  1.1% ED

Therefore, WACC  Pre-tax WACC  1.10%. The reduction is 1.1%.

Evaluate: The WACCs differ only by the effect of the tax deductibility of interest, which acts only on the debt portion of the cost of capital. So, while the deduction lowers the cost of debt by 40% to 4.2%, the overall WACC only decreases by 1.1 percentage points because only 39.39% of Summit’s capital is debt. 19.

Plan: The value of Milton Industries without leverage is simply the present value of its future cash flows discounted at the unlevered cost of capital. We can combine Equations 16.5 and 16.7 to compute the value of Milton Industries with leverage. Execute: a.

VU =

$5 million

0.15

= $33.33 million

b. V L = V U + TC ´ D = $33.33 million + 0.35 ´ $19.05 million = $40 million

Evaluate: The total value of the levered firm exceeds the value of the firm without leverage due to the present value of the tax savings from debt. There is an important tax advantage to the use of debt financing. 20.

Plan: If its pre-tax WACC remains the same, as it should for a pure capital structure change, you can use Equation 16.9 to calculate the after-tax WACC, accounting for the reduction due to the interest tax shield. Execute: rWACC = Pre-tax rWACC - rDTC

D = 0.15 - 0.09 ´ 0.35 ´ 0.5 = 13.425% E+D

Evaluate: The tax deductibility of debt lowers its WACC by 1.575%. *21.

Plan: In this case, the debt is permanent so we can use Equation 16.7 to value the interest tax shields. The value of the levered firm is the value of the unlevered company plus the present value of the tax shields, and the value of equity equals the value of the levered firm minus the company’s debt. Execute: a. Assets  VU  $7.50  20 million  $150 million. b. Key insight: borrowing $50 million creates future tax shields and will also raise $50 million in cash: both of these will show up as assets on Kurz’s market value balance sheet. Assets  Existing Assets (pre-leverage)  Cash Raised PV of Interest Tax Shield Assets  $150 million  50 million  40%  50 million  $220 million c. E  Assets  Debt  220 million  50 million  $170 million.

Share price = Kurz will repurchase =

$50 million $8.5

$170 million

20 million

= $8.50

= 5.882 million shares

d. Key insight: the $50 million in cash will be spent on the repurchase, leaving no cash on the market value balance sheet. Assets  Existing Assets (pre-leverage) PV of Interest Tax Shield Assets  $150 million  40%  $50 million  $170 million Debt  $50 million E  Assets  Debt  $170 million  $50 million  $120 million Shares outstanding after repurchase: 20 million – 5.882 million = 14.118 million

Share price after repurchase =

$120 million 14.118 million

= $8.50/share

Evaluate: While total firm value has increased, the value of equity dropped after the recap. How do shareholders benefit from this transaction? In addition to holding $120 million worth of shares, shareholders will also receive $50 million cash that Kurz will pay out through the share repurchase. In total, they will end up with $170 million, a gain of $20 million over the value of their shares without leverage. 22.

Plan: Compute the NPV of the investment and the price of a Kohwe share today. Execute: a. NPV =

$10 million 0.08

b. Share price =

- $50 million = $125 million - $50 million = $75 million

$75 million = $15/share 5 million

Evaluate: The NPV of the investment is $75 million. This NPV belongs to the investors who hold the 5 million shares already outstanding. Therefore, a share of Kohwe stock should sell for $15. (In order to raise the $50 million required to actually implement the investment, Kohwe will need to sell $50 million/$15 = 3.33 million new shares to investors. After the equity issue, the company will be worth $125 million, it will have 5 million + 3.33 million = 8.33 million shares outstanding, and each share will be worth $125 million/8.33 million = $15.) 23.

Plan: Estimate the value of a Kohwe equity share, including the impact of the interest tax shield created by debt financing. Execute:

Share price =

$75 million + 0.4 ´ $50 million = $19/share 5 million

Evaluate: With more debt financing, a Kohwe equity share would sell for $19, reflecting the tax shields created by debt financing. 24.

Plan: Compute the value of a Kohwe share with financial distress costs. Execute:

$9 million - $50 million + 0.4 ´ $50 million 0.08 Share price = = $16.50/share 5 million Evaluate: Financial distress costs would reduce a debt financed Kohwe’s value from $19 to $16.50. *25.

Plan: Calculate the value of a share before and after the repurchase under the various scenarios. Execute: a. Share price: $25 billion/10 billion = $2.5 per share

b. VL = VU + TC  D = $25 billion + 0.35  $10 billion = $28.5 billion E = VL – D = $28.5 billion – $10 billion = $18.5 billion Shares repurchased: $10 billion/$2.75 = 3.64 billion (rounded to two decimals) There remain 10 billion – 3.64 billion = 6.36 billion shares outstanding after the repurchase. Share price after repurchase: $18.5 billion/6.36 billion = $2.91 Therefore, shareholders will not sell for $2.75 per share; they would want to hold out knowing that after the repurchase the shares will be worth $2.91. c. Shares repurchased: $10 billion/$3 = 3.33 billion (rounded to two decimals) There remain 10 billion – 3.33 billion = 6.67 billion shares outstanding after the repurchase. Share price after repurchase: $18.5 billion/6.67 billion = $2.77 ($2.78 with no rounding) Therefore, shareholders will want to sell for $3 per share, they would not want to hold on to their shares knowing that after the repurchase the shares would be worth only $2.78. d. $28.5 billion/10 billion = $2.85 is the fair value of the shares prior to repurchase. Shares repurchased: $10 billion/$2.85 = 3.51 billion (rounded to two decimals) There remains 10 billion – 3.51 billion = 6.49 billion shares outstanding after the repurchase. Share price after repurchase: $18.5 billion / 6.49 billion = $2.85 Shareholders will be willing to sell for $2.85 per share; the shares would be worth $2.85 after the repurchase, so there is nothing to be gained (or lost) by tendering the shares for this price. Evaluate: Rally should offer $2.85 per share. 26.

Plan: Calculate Impi’s taxable income and then determine the amount of debt that would generate sufficient interest expense to reduce the taxable income to zero. Execute: Net income of $4.5 million  Taxable income =

$4.5 million = $6.923 million 1- 0.35

(

)

Therefore, Impi can increase its interest expenses by $6.923 million, which corresponds to debt of:

Additional debt capacity =

$6.923 million = $86.54 million 0.08

Evaluate: Impi can issue $86.54 million of additional debt this year in order to maximize the tax shield.

27.

Plan: If you expect the debt to be permanent, then you can use Equation 16.7 for the PV of the interest tax shield created by the debt: D  TC. Therefore, Equation 16.5 becomes VL = VU + D  TC. The value of equity is simply the value of the levered firm minus the amount of debt (remember that equity is a residual claim; shareholders receive what is left after lenders have been paid). Execute: a. VL = VU + D  TC = $20 million + $2 million  0.4 = $20.8 million Note that we get the same result without using Equation 16.7: V L = VU +

(

TC rD ´ D rD

) = $20 million + 0.4 ( 0.07 ´ $2 million ) = $20.8 million 0.07

b. VL = E + D => E = VL – D = $20.8 million – $2 million = 18.8 million Evaluate: By taking advantage of the tax deductibility of interest, you have recaptured tax payments with a present value of $0.8 million. This increases the value of the firm by the same amount. Since no net new capital has come into the firm (you used all $2 million to repurchase shares), the value of the firm is equal to the original value plus the $800,000: $20 million +$0.8 million = $20.8 million. Of the $20.8 million total value, $2 million is owed to the debt holders and the remaining $18.8 million is equity. 28.

Plan: Compute the value of a share of Hawar equity with tax subsidies from debt, and compute the financial distress costs Hawar will incur. Execute: a. VU  $5.50  10 million  $55 million. Key insight: borrowing $20 million creates future tax shields, increasing the value of the firm. The efficient market hypothesis implies that the increase in value will be incorporated into share prices when the decision to borrow money for the share repurchase is announced: VL  $55 million  30%  20 million  $61 million

Share price =

$61 million = $6.10 10 million

b. Financial distress cost per share = $6.10 – $5.75 = $0.35 Total financial distress costs = $0.35  10 million  $3.5 million Evaluate: After issuing debt and buying back shares, Hawar stock should sell for $6.10 based on its all-equity value plus the present value of the tax shields. If the shares sell for only $5.75 then the present value of financial distress costs must be $0.35 and total financial distress costs should be $3.5 million.

29.

Plan: Compute the value of Marpor without leverage and with leverage. Execute: a. r  5%  1.1  (15%  5%)  16% VU  $16 million/0.16  $100 million. b. VL  $15 million/0.16 + 0.35  40 million  $107.75 million Evaluate: Without leverage, Marpor is valued at $100 million. With leverage it is valued at $107.75 million.

*30.

Plan: Calculate Colt’s value and the maximum amount it could borrow without incurring a loss. Execute: a. FCF = EBIT  (1 – TC) + Dep – Capex – NWC FCF = $15 million  (1 – 0.35) + $3 million – $6 million = $6.75 million VU =

$6.75 million = $450 million 0.1- 0.085

b. Interest expense of $15 million would reduce pre-tax profits from $15 million to zero  useful debt capacity = $15 million /0.08 = 187.5 million. c. No. The most they should borrow is 187.5 million; there is no interest tax shield from borrowing more. Note that more than 50% debt would imply borrowing in excess of $450 million  0.5 = $225 million. Evaluate: Colt is worth $450 million with no debt. It can issue $187.5 million of additional debt in order to maximize the tax shield. 31.

If Dynron has no debt or if in all scenarios Dynron can pay the debt in full, equity holders will only consider the project’s NPV in making the decision. If Dynron is heavily leveraged, equity holders will also gain from the increased risk of the new investment (remember that equity is like a call option on the firm’s assets and the value of an option generally increases with the volatility of the underlying asset).

32.

Plan: Use the balance sheet identity, E = VL – D as well as the fact that shareholders enjoy limited liability, guaranteeing E ≥ 0. Execute: a. If the land is not developed, the value of the firm will be the PV of $10 million (or $9.09 million). The $10 million future cash flow from the sale of undeveloped land is not sufficient to repay lenders, which means the company will default and the equity will become worthless. Since 100% of the future cash flow will be used to pay lenders, the value of the firm will be equal to the value of debt.

VL = b.

$10 million = $9.09 million = D Þ E = V L - D = 0 1.1

NPV =

$35 million - $10 million - $20 million = $2.73 million > 0 1.1

c. If the land is developed, the value of the firm will be the PV of $35 million (or $31.82 million). The $35 million future cash flow from the sale of the developed land is more than sufficient to repay lenders. The cash remaining after the debt has been repaid belongs to equity holders.

$15 million = $13.64 million 1.1 $35 million - $15 million E= = $18.18 million, or: 1.1 E = V L - D = $31.82 million - $13.64 million = $18.18 million D=

d. Equity holders will not be willing to accept the deal, because for them it is a negative NPV investment: NPVEquity = $18.18 million - $20 million = -$1.82 million < 0 Evaluate: In this case, there is an underinvestment problem: shareholders choose not to invest in a positive-NPV project because the firm is in financial distress and the value of undertaking the investment opportunity will accrue to bondholders rather than themselves. This failure to invest is costly for debt holders and for the overall value of the firm, because it is giving up the NPV of the missed opportunities. *33.

Excel Solution Plan: Compute the expected payoff of each project. Compute the expected payoff to equity holders from each project and compute the agency costs. Execute: a. E(A)  $75 million i.

E(B)  0.5  $140 million  $70 million

ii. E(C)  0.1  $300 million  0.9  $40 million  $66 million iii. Project A has the highest expected payoff for equity holders. b. E(A)  $75 million  $40 million = $35 million i.

E(B)  0.5  ($140 million  $40 million)  $50 million

ii. E(C)  0.1  ($300 million $40 million)  0.9  ($40 million  $40 million)  $26 million iii. Project B has the highest expected payoff for equity holders. c. E(A)  $0 million i.

E(B)  0.5  ($140 million  $110 million)  $15 million

ii. E(C)  0.1  ($300 million  $110 million)  $19 million iii. Project C has the highest expected payoff for equity holders.

d. The optimal project is Project A. With $40 million in debt, management will choose Project B, which has an expected payoff for the firm that is $75 million  $70 million  $5 million less than Project A. Thus, the expected agency cost is $5 million. With $110 million in debt, management will choose Project C, resulting in an expected agency cost of $75 million  $66 million  $9 million. Evaluate: The calculations show that the project to be accepted is affected by the amount of debt financing. *34.

Plan: We know that 2/3 of the total value of the firm is equal to $30 million, so the value of the firm must be $45 million. Deducting the debt from the firm value tells us how much of that value is made up of equity. Then we need to find what percentage of that equity the $10 million represents. Finally, the sale of 50% of the firm’s levered equity and the debt issue together must raise exactly $30 million. Execute: a. Market value of firm assets  $30 million/(2/3)  $45 million. With debt of $20 million, equity is worth $45 million  $20 million  $25 million, so you will need to sell $10 million/$25 million = 40% of the equity. b. Given debt D, equity is worth $45 million  D. Selling 50% of equity, together with debt, must raise $30 million: 0.5  ($45 million – D) + D = $30 million. Solve for D: D  $15 million. Evaluate: In this case, changing the capital structure does not affect the value to the owners of the firm and, therefore, the owners have more flexibility with their capital structure.

*35.

Plan: Determine the benefits of issuing debt. Note: the tax rate is TC = $350/$1000 = 35%. Execute: a. In addition to the tax benefits of leverage, debt financing can benefit Empire by reducing wasteful investment. b. Net income will fall by $1  (1 – 0.35)  $0.65. Because 10% of net income will be wasted, dividends and share repurchases will fall by $0.65  (1  0.10)  $0.585. c. Pay $1 in interest, give up $0.585 in dividends and share repurchases  Increase of $1  $0.585  $0.415 per $1 of interest. Evaluate: Issuing debt can have multiple benefits to a firm.

*36.

Plan: Do the calculations to answer the several questions asked in the problem. Execute: a. (i) Borrowing costs the firm $20 million (the PV of financial distress costs). Selling ($500 million/$13.50) = 37 million shares at a premium of ($13.50 – $12.50) = $1

per share has a benefit of $37 million. Therefore, issue equity. (Issuing debt is costly while issuing equity benefits the firm’s existing shareholders.) (ii) Borrowing costs the firm $20 million (the PV of financial distress costs). Selling ($500 million/$13.50) = 37 million shares at a discount of ($13.50 – $14.50) = –$1 per share costs the firm $37 million. Therefore, issue debt. (Issuing debt and issuing equity are both costly BUT the cost associated with debt is lower: $20 million < $37 million.) b. If IST issues equity, investors should conclude that IST shares are worth only $12.50 (IST is overpriced), and the share price should decline to $12.50. (Investors should realize that the reason IST decided to issue equity is because the true value of a share is $12.50 and IST followed the process described in part (a)(i) above.) c. If IST issues debt, investors should conclude that IST shares are worth $14.50 (IST is undervalued), and the share price should rise to $14.50. (Investors should realize that the reason IST decided to issue debt is because the true value of a share is $14.50 and IST followed the process described in part (a)(ii) above.) d. If there are no costs from issuing debt, then equity is only issued if it is overpriced. But knowing this, investors would only buy equity at the lowest possible value for the firm. Because there would be no benefit to issuing equity, all firms would issue debt. Evaluate: The decision to issue debt or equity securities involves considering numerous issues by a firm.

Chapter 17 Payout Policy Note: All problems in this chapter are available in MyLab Finance. An asterisk (*) indicates problems with a higher level of difficulty. For a breakdown of the difficulty ratings of all of the problems in this Solutions Manual, please see the Preface – Question Difficulty Ratings. 1.

Plan: Determine the ex-dividend day and the last day that an investor can purchase the stock and receive the dividend. Keep in mind that April 12, 2021, is a Monday, so one business day before April 12, 2021, is Friday, April 9, 2021. Execute: a. Ex-dividend day: April 9, 2021. If an investor buys ABC stock on April 9, then the trade will be settled two business days later, on April 13, which is one day too late. The investor will not be the shareholder of record on April 12. b. If an investor buys ABC stock on April 8, then the trade will be settled two business days later, on April 12, just in time for the investor to become the shareholder of record to receive the dividend.

Evaluate: An investor who purchases the stock on April 8 will receive the dividend; an investor who purchases the stock on April 9 will not receive the dividend. 2.

Plan: Calculate the first ex-dividend day price. Execute: Assuming perfect markets, the first ex-dividend price should drop by exactly the dividend payment. Thus, the first ex-dividend price should be $49 per share. Evaluate: In a perfect capital market, the first price of the stock on the ex-dividend day should be the closing price on the previous day less the amount of the dividend.

Plan: ECB has a market capitalization of $20 million ($20  1 million shares). If it repurchases shares at the market price, it will need to pay $20  100,000  $2 million. Execute: Its new market capitalization will be $18 million (it started with $20 million and distributed $2 million through the repurchase). Its share price will stay at $20 ($18 million/900,000 shares). Evaluate: As long as the company repurchases its shares at the market price, the price will not change after the repurchase. This is because buying shares at the market price is a zero-NPV investment—it has neither created nor destroyed value.

Plan: Compute the changes in the balance sheet and determine the new leverage ratio. Execute: a. Both the cash balance and shareholder equity will drop by $20 million. b. After the repurchase, equity will be $500  $20  $200 = $280 million and debt is still $200 million. The debt to equity ratio will be 200/280  71.4%. Evaluate: Cash and shareholder equity will both decline by $20 million on the balance sheet. The new leverage ratio will be 71.4%.

Plan: Determine the present value of the annuity of dividends. Execute: The present value of the perpetuity should be $20 million, with a discount rate of 10%. Evaluate: The dividend payments should be ($20 million × 0.10) = $2.00 million per year in perpetuity. $20 million 

0.10 X  $2 million

Plan: Make the calculations requested in the problem. Execute:

a. $1 billion/20 million shares  $50 per share b. $100 million/$50 per share  2 million shares c. If markets are perfect, then the price right after the repurchase should be the same as the price immediately before the repurchase. Thus, the price will be $50 per share. Evaluate: What will the price per share of EJH be right before the repurchase? $50 per share. How many shares will be repurchased? 2 million shares. What will the price per share of EJH be right after the repurchase? $50 per share. 7.

Plan: Make the calculations requested in the problem. Execute: a. The dividend payoff is $250/500  $0.50 on a per share basis. In a perfect capital market, the price of the shares will drop by this amount to $14.50. b. $15 c. Both are the same. Evaluate: What is the ex-dividend price of a share in a perfect capital market? $14.50. If the board instead decided to use the cash to do a one-time share repurchase, in a perfect capital market what is the price of the shares once the repurchase is complete? $15.00. In a perfect capital market, which policy (in part [a] or part [b]) makes investors in the firm better off? Both policies would leave the firm equally well off.

Plan: Determine what you have to do to maintain your same position in a firm that decides to do a share repurchase. Execute: If you sell 0.5/15 of one share you receive $0.50 and your remaining shares will be worth $14.50, leaving you in the same position as if the firm had paid a dividend. Evaluate: If you sell $0.50 of stock you would be in the same position as having received a dividend.

Plan: Determine the price of HNH stock with a $2.00 dividend. Then compute the stock price if HNH suspends the dividend payments and instead repurchases shares. Execute: a. P  $1.60/0.12  $13.33 b. P  $2/0.12  $16.67 Evaluate: HNH with a dividend will sell for $13.33. If HNH suspends the dividend and uses the $2.00 per share to repurchase shares, the stock will sell for $16.67. The increased value of the repurchase policy comes from the fact that dividends are taxed and capital gains are not.

10.

Plan: Make the calculations requested in the problem. Execute: a. The capital gains tax rate is 15% and the dividend tax rate is 15%. The tax on a $10 capital gain is $1.50, and the tax on a $10 special dividend is $1.50. The after-tax income for both will be $8.50. b. If the capital gains tax rate is 20%, the tax on a $10 capital gain is $2.00, and the after-tax income is $8.00. If the dividends tax rate is 40%, then the tax on a $10 special dividend is $4.00, and the after-tax income is $6.00. The difference in aftertax income is $2.00. Evaluate: There is no difference in after-tax income when the tax rates on capital gains and dividends are the same. There is a $2.00 difference in after-tax income if the capital gains tax rate is 20% and the dividend tax rate is 40%.

11.

a. Invest the $5 special dividend and earn interest of $0.50 per year. b. Borrow $5 today and use the increase in the regular dividend to pay the interest of $0.50 per year on the loan.

12.

Because you manage a pension fund (and, thus, pay no taxes on investment income), you would prefer to receive the special dividend of $50 million immediately. You can then choose to invest in the same one-year treasury securities and receive the following:

$50M  (1.01)  $50.5M On the other hand, if KOA invests in the one-year treasury securities, it will have to pay taxes and so will only be able to pay out the following in one year:

$50M × (1.01) tax = $50.5M – ($0.5M × 0.35) = $50.325M 13.

Excel Solution Plan: Determine the values of Kay stock at various times regarding the decision to pay or not pay a one-time dividend. Execute: a. The value of Kay will remain the same. b. The value of Kay will fall by $100 million. c. It will neither benefit nor hurt investors. Evaluate: If the board went ahead with this plan, what would happen to the value of Kay stock upon the announcement of a change in policy? The value of Kay would remain the same. What would happen to the value of Kay stock on the ex-dividend date of the one-time dividend?

The value of Kay would fall by $100 million. Given these price reactions, will this decision benefit investors? It will neither benefit nor hurt investors. 14.

Excel Solution Plan: Recalculate Problem 13 assuming a 35% corporate tax rate. Execute: a. The value of Kay would rise by $35 million. b. The value of Kay would fall by $100 million. c. It will benefit investors. Evaluate: If the board went ahead with this plan, what would happen to the value of Kay stock upon the announcement of a change in policy? The value of Kay would rise by $35 million. What would happen to the value of Kay stock on the ex-dividend date of the one-time dividend? The value of Kay would fall by $100 million. Given these price reactions, will this decision benefit investors? It will benefit investors.

15.

Excel Solution Plan: Recalculate Problem 13 assuming investors pay a 15% tax on dividends but no capital gains tax. Kay does not pay corporate taxes. Execute: Assuming investors pay 15% tax on dividends but no capital gains taxes or taxes on interest income, and Kay does not pay corporate taxes: a. The value of Kay would remain the same. b. The value of Kay would fall by $85 million. c. It will neither benefit nor hurt investors. Evaluate: If the board went ahead with this plan, what would happen to the value of Kay stock upon the announcement of a change in policy? The value of Kay would remain the same. What would happen to the value of Kay stock on the ex-dividend date of the one-time dividend? The value of Kay would fall by $85 million. Given these price reactions, will this decision benefit investors? It will neither benefit nor hurt investors.

16.

a. First, find the NPV of the project: NPV  $5M 

$5.5M  $0.0893M (1.12)

Because the NPV is negative, the enterprise value of the firm will drop by $89,300. The new enterprise value of the firm will be $55(1 million shares) – $89,300 = $55M – $89,300 = $54.9107 million. Thus the new share price will be: Share price 

$54.9107 M  $54.91 1M

b. If AMS instead decides to repurchase stock immediately, the enterprise value of the firm will fall by $5 million, but the number of shares will decrease by the number of shares the firm repurchases ($5 million/$55). So the new share price will be

Share price 

$55M  $5M $50M   $55.00 $5M 909,090.9 1M  $55

Clearly, the immediate repurchase is the better decision because the share price stays the same, where it goes down upon taking the project.

Use the following information to answer Questions 17 through 21. AMC Corporation currently has an enterprise value of $400 million and $100 million in excess cash. The firm has 10 million shares outstanding and no debt. Suppose AMC uses its excess cash to repurchase shares. After the share repurchase, news will come out that will change AMC’s enterprise value to either $600 million or $200 million.

17.

Excel Solution Plan: Calculate AMC’s share price prior to share repurchase. Execute: Because Enterprise Value  Equity  Debt  Cash, AMC’s equity value is Equity  EV  Cash  $500 million. Therefore, Share price  ($500 million)/(10 million shares)  $50 per share. Evaluate: AMC’s share price would be $50.00 before share repurchase.

18.

Excel Solution Plan: Calculate the values of AMC’s share price assuming AMC’s enterprise value goes up and declines. Execute: AMC repurchases $100 million/($50 per share)  2 million shares. With 8 million remaining shares outstanding (and no excess cash), its share price if its EV goes up to $600 million is Share price  $600/8  $75 per share And if EV goes down to $200 million, Share price  $200/8  $25 per share Evaluate: AMC’s share price would rise to $75.00 if enterprise value rose. It would fall to $25.00 if enterprise value were to fall.

*19.

Excel Solution Plan: Calculate the values of AMC’s share price assuming AMC’s enterprise value goes up and declines and AMC waits until after the news comes out to execute the repurchase. Execute: If EV rises to $600 million prior to repurchase, given its $100 million in cash and 10 million shares outstanding, AMC’s share price will rise to Share price  ($600  $100)/10  $70 per share If EV falls to $200 million Share price  ($200  $100)/10  $30 per share The share price after the repurchase will also be $70 or $30 since the share repurchase itself does not change the stock price. Evaluate: AMC’s share price would rise to $70.00 if enterprise value rose. It would fall to $30.00 if enterprise value were to fall. Note: The differences in the outcomes for Problem 18 and Problem 19 arise because, by holding cash (a risk-free asset), AMC reduces the volatility of its share price.

20.

Excel Solution If management expects good news to come out, they would prefer to do the repurchase first, so that the stock price would rise to $75 rather than $70. On the other hand, if they expect bad news to come out, they would prefer to do the repurchase after the news comes out, for a stock price of $30 rather than $25. (Intuitively, management prefers to do a repurchase if the stock is undervalued—they expect good news to come out—but not when it is overvalued because they expect bad news to come out.)

*21.

Excel Solution Based on Problem 20, we expect managers to do a share repurchase before good news comes out and after any bad news has already come out. Therefore, if investors believe managers are better informed about the firm’s future prospects, and that they are timing their share repurchases accordingly, a share repurchase announcement would lead to an increase in the stock price.

22.

Plan: A 10% stock dividend means 1 new share for every 10, so FCF will distribute 2000 new shares. Execute: a. Before the dividend, FCF’s stock price is $700,000/20,000 = $35. b. Since the stock dividend does not bring in or disgorge any money, FCF’s total market capitalization does not change—only the shares outstanding change: $700,000/22,000 = $31.8182 The investor’s 1000 shares become 1100 shares, so the total value of her investment before is 1000($35)  $35,000 and after is 1100($31.8182)  $35,000. Evaluate: The stock dividend does not leave the investor any better or worse off because she received new shares in proportion to how many she held before the stock dividend.

23.

Excel Solution Plan: Calculate the value of Host’s shares assuming a 20% stock dividend and a 3:2 stock split. Execute: a. With a 20% stock dividend, an investor holding 100 shares receives 20 additional shares. However, since the total value of the firm’s shares is unchanged, the stock price should fall to Share price  $20  100/120  $20/1.20  $16.67 per share b. A 3:2 stock split means that, for every two shares currently held, the investor receives a third share. This split is, therefore, equivalent to a 50% stock dividend. The share price will fall to Share price  $20  2/3  $20/1.50  $13.33 per share

Evaluate: A 20% stock dividend should produce a stock price of $16.67, while a 3:2 stock split should produce a share price of $13.33. 24.

Plan: Determine the stock split ratio required to produce a Berkshire Hathaway A share worth $50.00. Execute: To bring its stock price down to $50 per share, Berkshire Hathaway would need a split ratio of $120, 000 $50

 2400 to 1

Evaluate: A 2400 to 1 stock split will reduce a Berkshire Hathaway A share to $50.00. 25.

Plan: Calculate the value of an Adaptec share after the stock dividend. Execute: The value of the dividend paid per Adaptec share was (0.1646 shares of Roxio)  ($14.23 per share of Roxio)  $2.34 per share. Therefore, ignoring tax effects or other news that might come out, we would expect Adaptec’s stock price to fall to $10.55  $2.34  $8.21 per share once it goes ex-dividend. Evaluate: Adaptec’s shares should sell for $8.21 once it goes ex-dividend. (Note: In fact, Adaptec stock opened on Monday May 14, 2001—the next trading day—at a price of $8.45 per share.)

Chapter 18 Financial Modelling and Pro Forma Analysis Note: All problems in this chapter are available in MyLab Finance. An asterisk (*) indicates problems with a higher level of difficulty. For a breakdown of the difficulty ratings of all of the problems in this Solutions Manual, please see the Preface – Question Difficulty Ratings. 1.

Plan: Calculate next year’s estimated cost of goods sold (COGS). Execute: Current sales: $100,000 Cost of goods sold: $72,000 COGS percent of sales: 72% Forecasted sales: $110,000 Forecasted COGS  110,000  72%  79, 200 Evaluate: Next year’s estimated cost of goods sold is $79,200.

Excel Solution Plan: Calculate the net financing required in the coming year. Execute: Beginning shareholders’ equity  $300,000 Additions to equity  Net income  Retention ratio  50, 000  (1  0.10)  45, 000 Ending shareholder equity  $345,000 Beginning total liabilities  $120,000 Increase in non-debt liabilities  $10,000 Ending total liabilities  $130,000 Ending total liabilities and equity  $475,000 Ending assets  $500,000 Net financing required: $25,000 Evaluate: $25,000 in net financing will be required.

Excel Solution Plan: Calculate the amounts of debt and equity financing that would be needed in Problem 2 to keep the capital structure constant. Execute: Beginning debt:

100,000.00

Beginning equity:

300,000.00

Beginning non-debt equity:

20,000.00

Debt/equity ratio:

0.33

New assets:

500,000.00

New non-debt liabilities:

30,000.00

New debt  equity:

470,000.00

Amount of equity needed to maintain ratio:

313,333.33

Amount of debt need to maintain ratio:

156,666.67

Amount of debt to issue:

56,666.67

Amount of equity to issue:

(31,666.67)

Evaluate: The firm would have to issue $56,666.67 in new debt and retire $31,666.67 in equity to keep the capital structure constant. Use the following income statement and balance sheet for Jim’s Espresso for Problems 4–6: Income Statement

Balance Sheet

Sales

200,000

Assets

Costs Except Depr.

(100,000)

Cash and Equivalents

15,000

EBITDA

100,000

Accounts Receivable

2,000

Depreciation

(6,000)

Inventories

4,000

EBIT

94,000

Total Current Assets

21,000

Interest Expense (net)

(400)

Property, Plant, and Equipment

10,000

Pre-tax Income

93,600

Total Assets

31,000

Income Tax

(32,760)

Liabilities and Equity

Net Income

60,840

Accounts Payable

1,500

Debt

4,000

Total Liabilities

5,500

Shareholders’ Equity

25,500

Total Liabilities and Equity

31,000

Excel Solution Plan: Use the percent of sales method to forecast the financial line items identified in the problem. Execute: Forecasted sales  200,000  (1.10)  220,000 Forecasted value  Current % of sales  Forecasted sales (except for net income calc.)

Evaluate: Each of the financial line items will grow in proportion to forecasted sales except for net income. Net income must be calculated using interest (assumed to be constant) and the calculated tax amounts. 5.

Excel Solution Plan: Use the percent of sales method to forecast next year’s shareholders’ equity and accounts payable. Execute: a. For shareholders’ equity, we need to know how much will be added to shareholders’ equity from net income. Additions to shareholders’ equity = $66,950  (1 – 0.90) = $6695. New shareholders’ equity  $25,500  $6695  $32,195 b. Current percent of sales: 0.75%. Forecasted accounts payable  0.75%  $220,000  $1650. Evaluate: Shareholders’ equity will grow by $6695 (which is the amount of earnings retained in the business) to $32,195. Accounts payable are forecasted to grow to $1650.

Excel Solution Plan: Calculate Jim’s net new financing for next year. Execute: Pro forma financial statements for Jim’s Espresso:

Total new financing required  Total assets  Total liabilities and equity  $3745 Evaluate: Jim has excess financing, which means it can use the excess financing to repay debt or, if it does not want to increase its payout ratio, repurchase equity. 7.

Plan: By reducing its payout ratio, it will increase retained earnings, which are added to shareholders’ equity. That additional shareholders’ equity will reduce the required new financing further. Execute: For shareholders’ equity, we need to know how much will be added to shareholders’ equity from net income. Additions to shareholders’ equity = $66,950 × (1 – 0.7) = $20,085. Compared to the 90% payout ratio, these additions are $20,085 – $6695 = $13,390 more, so net new financing required will be $13,390 less than what was calculated before; for a total of –$17,135 of net new financing required. Evaluate: By reducing its payout ratio, Jim’s will have a more negative net new financing amount, which means Jim has even more excess financing, which means it can use the excess

financing to repay more debt or, if it does not want to increase its payout ratio, repurchase more equity. Problems 8 through 10 are answered together in the table below and then individually as follows. Plan: To use the percent of sales method, first calculate each relevant income statement and balance sheet entry’s percent of this year’s sales. Income and taxes cannot be calculated as a percent of sales because the interest expense remains constant. Then, forecast next year’s sales as 8% higher than this year’s. Finally, create a forecasted income statement and balance sheet by applying the percent of sales calculated in the first step and calculating the remainder of the items that are not a percent of sales. To complete the balance sheet, we need to know how much of forecast net income will be retained and added to shareholders’ equity and how much will be paid out. Problem 9 states that it will be 50%. Execute:

Excel Solution Plan: Use the percent of sales method to forecast the financial line items identified in the problem. Execute: Forecasted sales: $186.7 × $1.08 = $201.64 Forecasted value  Current percent of sales  Forecasted sales So, the answers are as follows:

Each item except net income grows proportionately to sales. Net income does not because interest expense is held constant. 9.

Excel Solution Shareholders’ Equity  Previous SE  Retained Earnings  $22.2  $1.310  $23.51

10.

Excel Solution In order to make the balance sheet balance, Global’s total liabilities and equity must equal its total assets. Its total assets are projected to be $183.71. Its projected accounts payable is $37.48 and its projected shareholders’ equity is $23.51. The remainder must be in the form of new financing of $9.52. This could be by issuing either more debt (borrowing) or more equity (issuing additional shares of stock).

11.

If Global limits itself to only $9 million in new financing, then it must cut its dividend to shareholders by 0.52 in order to make up the difference on its balance sheet.. Evaluate: By creating a pro forma income statement and balance sheet, Global is able to identify how much new financing it will need and what trade-offs with payouts to shareholders exist.

12.

Excel Solution Plan: Compute production volumes under the revised growth assumptions. Execute: 2021

2022

2023

2024

2025

2026

Market Size

10,000

10,500

11,025

11,576

12,155

12,763

Market Share

10.00%

10.25%

10.50%

10.75%

11.00%

11.25%

Production Volume

1000

1076

1158

1244

1337

1436

Production Volume (000 units)

Evaluate: In 2023, production will exceed 11,000 units and production capacity will have to be increased. 13.

Excel Solution Plan: Calculate financing needs, interest payments, and interest tax shields as KXS grows. Execute: 2021

2022

2023

2024

2025

2026

4500

24,500

24,50 0

24,500

Debt and Interest Table ($000s) Outstanding Debt New Net Borrowing

20,000

Interest on Debt

306

1666

Interest Tax Shield

107

583

Evaluate: The increase in production capacity in 2023 will require KXS to issue $20,000 in new debt financing. This will increase the amount of annual interest KXS must pay and the amount of the interest tax shield. 14.

Excel Solution Plan: Reproduce Table 18.8 under the new assumptions. Execute: 2021

2022

2023

Income Statement ($000s)

2024

2025

2026

Sales

74,890

82,344

90,341

99,056

108,555

118,916

Cost of Goods Sold

(58,414)

(64,228)

(70,466)

(77,264)

(84,673)

(92,755)

EBITDA

16,476

18,116

19,875

21,792

23,882

26,162

Depreciation

(5492)

(5443)

(7398)

(7459)

(7513)

(7561)

EBIT

10,984

12,673

12,477

14,333

16,369

18,601

Interest Expense

(306)

(1666)

Pre-tax Income

10,678

12,367

12,171

12,667

17,703

16,935

Taxes

(3737)

(4328)

(4260)

(4434)

(5146)

(5927)

Net Income

6941

8038

7911

8234

9557

11,007

Evaluate: Note that net income is forecasted to decline from 2022 to 2023 as the new production capacity, with its related increase in depreciation expense, comes on line. 15.

Excel Solution Plan: Calculate KXS’s working capital requirements through 2026. Execute: 2021

2022

2023

2024

2025

2026

Working Capital ($000s) Assets 1

Accounts Receivable

14,229

15,645

17,165

18,821

20,625

22,594

Inventory

14,978

16,469

18,068

19,811

21,711

23,783

Cash

11,982

13,175

14,455

15,849

17,369

19,027

Total Current Assets

41,190

45,289

49,688

54,481

59,705

65,404

2021

2022

2023

2024

2025

2026

11,982

13,175

14,455

15,849

17,369

19,027

11,982

13,175

14,455

15,849

17,369

19,027

29,207

32,114

35,233

38,632

42,336

46,377

2907

3119

3399

3705

4041

Liabilities 5

Accounts Payable

6 Total Current

Liabilities

Net Working Capital 7

Net Working Capital

Increase in Net Working Capital

Evaluate: Net working capital is forecasted to grow continually through 2026.

16.

Excel Solution Plan: Use Equation 18.4 to calculate the internal growth rate and Equation 18.5 to calculate the sustainable growth rate. Execute: Internal growth rate: 

Net Income Beginning Assets

 50,000   1  0   0.125 or 12.50%  400,000 

 (1  Payout Ratio) = 

Sustainable growth rate: =

Net Income Beginning Equity

 50,000   1  0  0.20 or 20%  250,000 

 (1  Payout Ratio) = 

Sustainable growth rate if it pays out 40% of its net income as a dividend: Net Income Beginning Equity

17.

 50,000   1  0.4   0.12 or 12%  250,000 

 (1  Payout Ratio) = 

Excel Solution Plan: Did KXSʼs expansion plan call for it to grow slower or faster than its sustainable growth rate? Execute: 2021

2022

2023

2024

2025

2026

Income Statement ($000) 1

Sales

74,890

88,369

103,247

119,792

138,167

158,546

Cost of Goods Sold

–58,414

–68,928

–80,533

–93,439

–107,770

–123,666

EBITDA

16,476

19,441

22,714

26,353

30,397

34,880

Depreciation

–5,492

–7,443

–7,498

–7,549

–7,594

–7,634

EBIT

10,984

11,998

15,216

18,804

22,803

27,246

Interest Expense

–306

–1,666

Pre-tax Income

10,678

11,692

13,550

17,138

21,137

25,580

Taxes

–3,737

–4,092

–4,743

–5,999

–7,398

–8,953

Net Income

6,941

7,600

8,807

11,139

13,739

–8,953

Payout ratio:

30%

Additions to shareholders’ equity:

4,858

5,320

6,165

7,797

9,617

11,639

Beginning shareholders’ equity:

74,134

79,454

85,619

93,416

1,03,034

Sustainable growth rate:

7.18%

7.76%

9.11%

10.30%

11.30%

Actual growth rate:

18.00%

16.84%

16.02%

15.34%

14.75%

Evaluate: The expansion caused them to grow at a rate faster than their sustainable growth rate. 18.

Plan: Calculate the additional debt that will have to be issued to support growth. Execute: Sustainable Growth Rate  ROE  (1  Payout Ratio)  12%  (1  25%)  9%

Beginning total assets  $1 million Ending total assets at a growth rate of 9%: $1.09 million Evaluate: Because the firm grew at its sustainable growth rate, its debt/equity ratio remains constant at 0.667 and the debt to assets ratio will be 0.40. Thus, the new debt in the capital structure will be 0.4  $1.09 million  $436,000. Since the firm started at $400,000, it will issue $36,000 in additional debt. 19.

Plan: First calculate its internal growth rate and then calculate the new debt and equity. Execute: Its payout ratio is $5000/$20,000  0.25 Net Income Beginning Assets

 20,000   1  0.25   0.0375  400,000 

 (1  Payout Ratio)  

If its assets grow at its internal growth rate, they grow to $400,000(1.0375)  $415,000. All of the $15,000 in additional assets is financed by retained earnings of $15,000 ( $20,000  $5000 dividend). That means the new equity is $315,000 and the debt that remains is $100,000, for a D/E ratio of $100,000/$315,000  0.3175.

Evaluate: If it grows at its internal growth rate, its leverage will decrease as it adds assets and equity without increasing its debt. 20.

Excel Solution Plan: Calculate KXS’s free cash flow through 2026. Execute: 2021

2022

2023

2024

2025

2026

1 Net Income

6941

8038

7911

8234

9557

11,007

2 Plus: After-Tax Interest Expense

199

1083

3 Unlevered Net Income

7139

8237

8110

9317

10,640

12,090

4 Plus: Depreciation

5492

5443

7398

7459

7513

7561

5 Less: Increases in NWC

(2907)

(3119)

(3399)

(3705)

(4041)

6 Less: Capital Expenditures

(5000)

(28,000)

(8000)

7 Free Cash Flow of Firm

7631

5773

(15,611)

5377

6448

7611

Free Cash Flow ($000s)

Evaluate: KXS should generate positive free cash flow in each year except 2023, which is the year that KXS must expand production capacity and that will require a large increase in capital expenditures. 21.

Excel Solution Plan: Value KXS assuming an EBITDA multiple of 8.5. Execute: EBITDA 2026  26,162 Continuation Value2026  26,162  8.5  $222,373

Evaluate: Based on an EBITDA multiple of 8.5, KXS would have a continuation value of $222,377. 22.

Excel Solution Plan: Compute the value of KXS under the 0.25% growth assumption and a cost of capital of 10%. Execute:

2021

2022

2023

2024

7631

5773

(15,611)

2025

2026

Free Cash Flow ($000s) Free Cash Flow of Firm

5377

6448

Continuation Value

NPV 

7631 1.10



222,373

5773 (1.10)

7611



15,611 (1.10)



5377 (1.10)



6448 (1.10)



7611  222,373 (1.10)6

NPV  151, 224

Evaluate: The value of KXS in 2021 is $151,224.

Chapter 19 Working Capital Management Note: All problems in this chapter are available in MyLab Finance. An asterisk (*) indicates problems with a higher level of difficulty. For a breakdown of the difficulty ratings of all of the problems in this Solutions Manual, please see the Preface – Question Difficulty Ratings. 1.

Plan: Calculate Homer Boats’ operating cycle. Execute: Operating Cycle  Inventory Days  Receivable Days  50  30  80 days. Evaluate: Homer Boats turns over (i.e., sells) its inventory in 50 days and collects the cash from the sale in 30 days. So it takes Homer Boats 80 days to sell a boat and collect the cash from the sale.

Plan: Calculate the cash conversion cycle for FastChips. Execute: Cash Conversion Cycle  Inventory Days  Receivable Days – Payable Days  75  30 – 90  15 days. Evaluate: FastChips turns over (i.e., sells) its inventory in 75 days and collects the cash from the sale in 30 days. So it takes FastChips 105 days to sell a computer chip and collect the cash from the sale. But FastChips has to pay its own supplier in 90 days for the computer chip. Hence, at day 90, FastChips has to come up with the cash to pay its supplier for the computer chip and then wait 15 additional days (to day 105) before its customer pays it for the computer chip.

Plan: Calculate the components (inventory days, receivable days, and payable days) of the cash conversion cycle, and then calculate the cash conversion cycle.

Execute: Inventory Days  

Inventory Balance Cost of Goods Sold/365 15,000 80,000/365

 68.44 days Receivable Days  

Accounts Receivable Balance Sales/365 30,000 100,000/365

 109.5 days

Payable Days  

Accounts Payable Balance Cost of Goods Sold/365 40,000 80,000/365

 182.5 days

Evaluate: Cash Conversion Cycle  Inventory Days  Receivable Days – Payable Days  68.44  109.5 – 182.5  4.56 days. Westerly collects cash from its customers 4.56 days before it has to pay its suppliers. 4.

Plan: Aberdeen Outboard Motors will have to invest $2,000,000 in net working capital today, and not recover that $2,000,000 investment for 10 years. Calculate the NPV of the investment. Execute: Ignoring revenues and other expenses associated with the new plant, the NPV of the $2 million investment in net working capital is simply the present value of the $2 million that the firm will recoup at the end of 10 years minus the initial $2 million investment. NPV = –$2,000,000 +

$2, 000, 000 (1.06)10

= –$883,210 Evaluate: The investment in net working capital will cost Aberdeen $883,210 in today’s dollars. 5.

Plan: Calculate the present value of the working capital investment under a 3% and a 4% growth rate assumption. Then compute the difference in present value. Execute: The cost of the working capital is the present value of the future changes in the working capital from year to year. The PV of the cost of the working capital growing at 4% is

100 1.04  100 PV    50 .12  .04

The PV of the cost of working capital growing at 3% is 100 1.03  100 PV    33.33 .12  .03

Evaluate: The change from 4% to 3% growth in working capital reduces the present value of the cost of investing in the working capital from $50 to $33.33, resulting in an increase in firm value of $16.67.

Excel Solution Plan: The problem provides much financial data about the Greek Connection and requires several calculations about its net working capital and cash conversion cycle. Execute: a. Net working capital is current assets minus current liabilities. Using this definition, the Greek Connection’s net working capital is $7250  $3720  $3530. Some analysts calculate the net operating working capital instead, which is the non-interest-earning current assets minus the non-interest-bearing current liabilities. In this case, the notes payable would not be included in the calculation since they are assumed to be interest bearing. Net operating working capital for the Greek Connection is $7250  ($1500  $1220)  $4530. b. The cash conversion cycle (CCC) is equal to the inventory days plus the accounts receivable days minus the accounts payable days. The Greek Connection’s cash conversion cycle was 41.4 days. CCC 

Inventory Average Daily COGS

CCC2011 



Accounts Receivable Average Daily Sales



Accounts Payable Average Daily COGS

$1, 300 $3, 950 $1,500   $20, 000 $32, 000 $20, 000

365 365 365  23.7 days  45.1 days  27.4 days  41.4 days

c. If the Greek Connection accounts receivable days had been 30 days, its cash conversion cycle would have been only 26.3 days: CCC  23.7 days  30 days – 27.4 days  26.3 days Evaluate: The data shows that the Greek Connection collects its account receivable in 45.1 days while the industry average receivables collection is 30 days. This means that the Greek Connection has a significantly longer cash collection cycle than its competitors. 7.

Excel Solution Plan: Calculate the cost of the trade credit if your firm does not take the discount and pays on day 30. Execute: In this instance, the customer will have the use of $97 for an additional 25 days (30  5) if he chooses not to take the discount. It will cost him $3 to do so since he must pay $100 for the goods if he pays after the 5-day discount period. Thus, the interest rate per period is: $3 $97

 0.0309  3.09%.

The number of 25-day periods in a year is 365/25  14.6 periods. So, the effective annual cost of the trade credit is EAR  (1.0309)14.6  1 



 56%

Evaluate: As we have calculated, the annual cost of not taking the trade discount is 55.94%. If the firm has other means of financing, such as a bank line of credit at 10% annually, it should take the trade discount, avoid the 56% cost, and borrow from the bank. 8.

Excel Solution Plan: Calculate the cost of not taking the trade discount. Execute: If you were to pay within the 10-day discount period, you would pay $99 for $100 worth of goods. If you wait until day 45, you will owe $100. Thus, you are paying $1 in interest for a 35-day (45  10) loan. The interest rate per period is $1 $99

 0.0101  1.01%

The number of 35-day periods in a year is 365/35  10.43 periods. So, the effective annual cost of the trade credit is EAR  (1.0101)10.43  1  11.05% Evaluate: As in Problem 7, the firm should look to its other sources of financing and only pay 11.05% if it is the lowest rate financing available. *9.

Plan: Calculate the costs and benefits of outsourcing the billing and collection functions. Outsource if benefits outweigh costs. Execute: The benefit of outsourcing the billing and collection to the other firm is equal to what Fast Reader can earn on the funds that are freed up. Since average daily collections are $1200 and float will be reduced by 20 days, Fast Reader will have an additional $24,000 ($1200  20). (Think about this as follows. Immediately after hiring the billing firm, its collection float drops by 20 days, so all collections due within the next 20 days are immediately available.) The billing firm charges $250 per month. At an 8% annual rate, the monthly discount rate is 1.081/12 – 1  6.43%, so the present value of these charges in perpetuity is 250/0.0643 . Evaluate: Thus, the costs ($3888) do not exceed the benefits ($24,000), and Fast Reader should employ the billing firm.

*10.

Plan: Calculate the costs and benefits of switching banks. Shift to the new bank if benefits outweigh costs. Execute: The electronic funds transfer system will free up $50,000 ( 5  $10,000). (Think about this as follows. Immediately after switching banks, its collections due within the next five days are immediately available.) On the other hand, Saban will have to pay a

cost because it has to hold $30,000 in a non–interest-earning account, which means it has essentially given up these funds. Evaluate: Because the benefits ($50,000) are larger than the costs ($30,000), it should switch banks. 11.

Plan: Calculate accounts receivable collection in days. Execute: If we assume all the sales were made on credit, the average length of time it takes Manana to collect on its sales is 12.2 days: Accounts Receivable Days = 

Accounts Receivable Average DailySales

$2,000,000 $60,000,000

365  12.2 days

Evaluate: It takes Manana 12.2 days on average to collect its accounts receivable. 12.

Plan: Compute an aging schedule for accounts receivable. Execute: Mighty Power Tool Company Aging Schedule

Days Outstanding

Amount Owed

Percent of Accounts Receivable

0–15

$68,000

19.3%

16–30

$75,000

21.2%

31–45

$92,000

26.1%

46–60

$82,000

23.2%

over 60

$36,000

10.2%

$353,000

100.0%

Evaluate: $36,000 of accounts receivable are over 60 days outstanding. If credit terms are net 30 days, management should investigate. 13.

Excel Solution Plan: Calculate the benefit of taking the discount. If it is more than the 12% cost of the loan, then borrow from the bank and take the discount. Execute: If Simple Simon’s takes the discount, it must pay $99 in 10 days for every $100 of purchases. If it elects not to take the discount, it will owe the full $100 in 25 days. The interest rate on the loan is

$1 $99

 1.01%

The loan period is 15 days ( 25  10). The effective annual cost of the trade credit is EAR  (1.0101)365/15 – 1  27.7% Evaluate: Since the bank loan is only 12%, Simple Simon’s should borrow the funds from the bank in order to take advantage of the discount. 14.

Excel Solution Plan: Calculate the effective annual cost of not taking the discount and paying in full in 40 versus 50 days. Execute: a. Your firm is paying $3 to borrow $97 for 25 days ( 40  15). The interest rate per period is $3 $97

 0.0309  3.09%

The effective annual rate is (1.0309)365/25  1  55.9% b. In this case, your firm is stretching its accounts payable. You are still paying $3 to borrow $97, so the interest rate per period is 3.09%. However, the loan period is now 35 days ( 50  15). The effective annual rate is reduced to 37.4% because your firm has use of the money for a longer period of time: EAR  (1.0309)365/35  1  37.4% Evaluate: The annual effective cost of not taking the discount and paying in full in 40 days is 55.9%. This falls to 37.4% if the firm can delay making the payment for 50 days. *15.

Excel Solution Plan: Calculate the firm’s cash conversion cycle over two different years. Evaluate how well the firm is managing it payables. Execute: a. The cash conversion cycle (CCC) is equal to the inventory days plus the accounts receivable days minus the accounts payable days. IMC’s cash conversion cycle for 2021 was 35.2 days, and for 2022, it was 45.6 days. CCC 

Inventory Average Daily COGS



Accounts Receivable Average Daily Sales



Accounts Payable Average Daily COGS

CCC2021 =

$6,200 $2,800 $3,600 + $52,000 $60,000 $52,000

365 365 365 = 43.5 days + 17.0 days - 25.3 days = 35.2 days

CCC2022 =

$6,600 $6,900 $4,600 + $61,000 $75,000 $61,000

365 365 365 = 39.5 days + 33.6 days - 27.5 days = 45.6 days

IMC’s cash conversion cycle has lengthened in 2022, due to an increase in its accounts receivable days. The number of days goods are held in inventory has decreased and IMC is taking longer to pay its suppliers, both of which would decrease the cash conversion cycle, all else equal. These changes were not enough to offset the increase in the amount of time it is taking IMC’s customers to pay for purchases made on credit. The lengthening of the cash conversion cycle means that IMC will require more cash. Evaluate: b. If IMC’s suppliers are offering terms of net 30 days, IMC should consider waiting longer to pay for its purchases. In 2021, it paid nearly 5 days earlier than necessary, and in 2022 it paid 2.5 days earlier. IMC could, therefore, have kept the money working for it longer since there was no discount offered for early payment. The early payment may give IMC a preferred position with its suppliers, however, which may have benefits that are not presented here. IMC’s decision on whether to extend its accounts payable days would have to take these benefits into consideration. 16.

Plan: Calculate the average days of inventory. Execute: Average Days of Inventory  

Average Inventory Balance Cost of Goods Sold/365 1,200,000 7,000,000/365

 62.57 days

Evaluate: Items stay in inventory for 62.57 days. 17.

Plan: Calculate inventory in days. Also evaluate how much the firm would have to reduce its inventory to get its inventory days to match the industry average. Execute: a. The inventory days ratio is equal to the inventory divided by average daily cost of goods sold. This was 91.25 days for Happy Valley Homecare Suppliers in this past year:

Inventory Days 

$2,000,000 $8,000,000

365  91.25 days

Evaluate: b. To reduce its inventory days to 73 days, HVHS must decrease its investment in inventory to $1,600,000: 73 days 

Inventory  Inventory  $1, 600, 000 $8, 000, 000 365

This means HVHS could reduce its investment in inventory by $400,000 ( $2,000,000  $1,600,000).

Chapter 20 Short-Term Financial Planning Note: All problems in this chapter are available in MyLab Finance. An asterisk (*) indicates problems with a higher level of difficulty. For a breakdown of the difficulty ratings of all of the problems in this Solutions Manual, please see the Preface – Question Difficulty Ratings. 1.

Excel Solution a. Plan: Calculate the firm’s working capital needs for each month and select the month with the highest working capital need. Execute: To determine Sailboats’ seasonal working capital needs, we calculate the changes in net working capital for the firm: Changes in working capital

Month 1

Accounts receivable

–$4

Inventory

–$1

–$2

Accounts payable

Change in net working capital

–$6

Evaluate: From the table it can be seen that Sailboats’ working capital needs are highest in Month 2 because its investments in accounts receivable and in inventory increased the most in that month.

b. Plan: Calculate the surplus cash position by month and identify those months when surplus cash is positive. Execute: Month ($000)

Net income

$10

$12

$15

$25

$30

$1 8

plus depreciation

minus changes in net working capital

–6

Cash flow from operations

$12

$16

$27

$33

$2 8

minus capital expenditures

Change in cash

$11

$12

$16

$26

$33

$2 8

Evaluate: Sailboats Etc. has a surplus cash position in every month as shown above. 2.

Excel Solution a. Plan: Calculate the working capital levels and changes over four quarters. Execute: Current

$4000

$10,000

$2000

$10,000

Inventory

$4000

$10,000

$2000

$10,000

Accounts payable

$2000

$5000

$1000

$5000

Working capital

$2000

$9000

$11,000

$7000

$10,000

$7000

$2000

-$4000

$3000

Accounts receivable

Changes in working capital

Evaluate: Working capital increases in Quarters 1, 2, and 4. Working capital decreases in Quarter 3. b.

Plan: Determine Emerald City’s financing needs over the four quarters.

Execute:

Net income

$4000

$10,000

$2000

$10,000

Changes in working capital

$7000

$2000

-$4000

$3000

Financing needs

$3000

$5000

$11,000

$18,000

Excess cash

Evaluate: Emerald City would need to obtain short-term financing of $3000 for the first quarter. 3.

Excel Solution Plan: Calculate the permanent and temporary working capital needs for four quarters. Execute: The net working capital for each quarter is calculated below: Quarter ($000)

Cash

$100

Accounts receivable

$200

$100

$600

Inventory

$200

$500

$900

$50

Accounts payable

$100

Net working capital

$400

$600

$1000

$650

Evaluate: The minimum level of net working capital—$400,000 in Quarter 1— represents the firm’s permanent working capital. The difference between the higher net working capital levels in each quarter and the permanent working capital needs represents the firm’s temporary working capital needs. Thus, the firm has temporary working capital needs of $200,000 in Quarter 2; $600,000 in Quarter 3; and $250,000 in Quarter 4. 4.

Excel Solution Plan: Calculate the excess cash levels over four quarters. Execute: Q1

Net working capital

400

600

1,000

650

Amount of borrowing

1,000 400

350

Excess cash

600

Evaluate: You would need $1,000,000 to finance the total working capital needs of the firm. You will have excess cash in Quarters 1, 2, and 4. 5.

Excel Solution Plan: Calculate the short-term borrowing needs over four quarters. Execute: Temporary working capital

200

600

250

Short-term financing needs

200

600

250

Evaluate: The short-term financing is equal to the temporary working capital, which is the highest in Q3 at $600,000. 6.

Excel Solution Plan: Determine the maximum amount of short-term borrowing needed if the firm enters the year with $400 in cash. Execute: Q1

Cash at beginning of quarter

400

200

100

Minimum cash balance

100

Temporary working capital needs

200

600

250

Short-term borrowing

500

250

Ending cash balance

400

200

100

Evaluate: The maximum short-term borrowing occurs in Q3, and is $500,000.

Excel Solution Plan: Calculate how much cash the firm must carry to limit short-term borrowing to $500. Execute: Q1

Cash at beginning of quarter

400

200

100

Minimum cash balance

100

Temporary working capital needs

200

600

250

Short-term borrowing

500

250

Ending cash balance

400

200

100

Evaluate: The excess cash needed is $300,000 ($400,000 starting cash balance minus $100,000 minimum cash balance). 8.

Excel Solution Plan Hand-to-Mouth (HTM) is currently cash-constrained and must make a decision about whether to delay paying one of its suppliers or taking out a loan. It owes the supplier $10,000, but the supplier will provide a 2% discount if HTM pays by today (when the discount period expires). That is, HTM can either pay $9800 today or pay $10,000 in one month when the net invoice is due. Because Hand-to-Mouth does not have the $9800 in cash right now, it is considering three options: Alternative A: Forgo the discount on its trade credit agreement, wait, and pay the full $10,000 in one month. Alternative B: Borrow the money from Bank A, which has offered to lend the firm $9800 for one month at an APR (compounded monthly) of 12%. The bank will require a (no-interest) compensating balance of 5% of the face value of the loan and will charge a $100 loan origination fee, which means Hand-to-Mouth must borrow even more than the $9800. Alternative C: Borrow the money from Bank B, which has offered to lend the firm $9800 for one month at an APR of 15% (compounded monthly). The loan has a 1% loan origination fee. Execute: Alternative A:

The effective annual cost of the trade credit is calculated as follows: Interest rate per period =

$200  2.041% $9800

The loan period is one month, so the effective annual cost is (1 + 0.02041) 12 – 1 = 0.274, or 27.4%. Alternative B: Hand-to-Mouth will need to borrow $9900 to cover its $100 loan origination fee ($9800 + $100 fee). It will also need to meet its compensating balance requirement, so in order to have $9900 available to spend (the $9800 it needs plus the $100 origination fee), it will have to borrow

$9900  $10,421.05 , which will allow it to set aside 5% of the loan (0.05 × 1  0.05 $10,421.05 = $521.05) and still have $10,421.05 – $521.05 = $9900 to pay the origination fee and have a net loan of $9800. The bank is charging 12% APR, compounded monthly, which translates into 1% per month (12%/12). Thus, the total interest Hand-to-Mouth will owe at the end of one month is 1% of the amount it borrowed: 0.01 × $10,421.05 = $104.21. Its net funds after paying the origination fee and creating the compensating balance are the $9800 it needs, so it is paying $104.21 in interest plus $100 in origination in order to get a $9800 loan. (To see this another way, the total loan balance at the end of the month will be $10,421.05 × 1.01 = $10,525.26. Deducting the bank balance of $521.05, this means HTM will pay $10,525.26 – $521.05 = $10,004.21 out of pocket, or effective interest of $10,004.21 – $9800 = $204.21 on the $9800 loan.) Thus, the one-month effective interest rate of the loan is

$204.21  0.0208 , or 2.08%. $9800

The effective annual rate is (1.0281)12 –1 = 0.281, or 28.1%. Alternative C: Again, Hand-to-Mouth will need to borrow more than $9800 in order to pay the 9800  $9898.99 in origination fee. With a 1% origination fee, it will need to borrow (1  .01) order to have $9800 available after paying the origination fee. The bank is charging 15% APR, compounded monthly, which translates into 15%/12 = 1.25% per month.

So, Hand-to-Mouth will owe 0.0125 × $9898.99 = $123.74 in interest at the end of the month. Its net funds after paying the origination fee are the $9,800 it needs, so it is paying $123.74 in interest plus an origination fee of 1% ($98.99) in order to get a $9,800 loan, for a one-month effective interest rate of

$123.74  $98.99  0.0227 , or 2.27%. $9,800

The effective annual rate is (1.0227)12 – 1 = 0.3095, or 30.95%. Evaluate: Thus, Alternative A, with the lowest effective annual rate, is the best option for Hand-to-Mouth. 9.

First, find the total cost, in dollars of interest and commitment fee: ($300,000)(6%)  ($500,000  $300,000)(0.5%)  $19,000

And, as a percentage of the amount we borrowed:

$19,000  6.33% $300,000 10.

We need to find the cost of each option as a percentage of the amount we need to borrow. First, we need to find the effective annual rate (EAR) for the interest rates for both options: 4

 0.08  First option: 1+   1  8.243% 4   4

 0.076  Second option:  1+   1  7.819% 4  

Now, we can find the effective annual rate of interest. First option: ($380,000  8.243%)  ($400,000  $380,000)(0.5%)  $31,423.40

And, as a percentage of the amount of financing needed:

$31,423.40  8.269% $380,000 Second option: Although your firm only needs $380,000, it must keep an additional $20,000 (= $400,000 × 0.05) in an account at the bank, and it must pay interest on the entire $400,000.

$400,000  7.819%  $31, 276.00 And, as a percentage of the amount of financing needed:

$31,276.00  8.231% $380,000 Thus, the second option has the lower effective annual rate of interest. 11.

Excel Solution Plan: Calculate the effective annual cost of each loan. Select the loan with the lowest cost. Execute: The loan with the 1% loan origination fee would cost the most since the loan origination fee of $10 is just another form of interest; so on a $1,000 loan, the borrower is paying $90 (= $80 + $10) in interest and will have the use of only $990 for the period, making the effective annual cost of the loan over 9% (= $90/$990  9.1%). The compensating balance requirement of 5% on a $1000 loan reduces the usable proceeds of the firm by 5% to $950, but the interest rate is still 8%, so the effective annual cost of that arrangement is

 $80  8.4%     $950  The effective annual rate is not increased by a full percentage. Evaluate: Take the loan with the compensating balance. 12.

Excel Solution Plan: Calculate the annual effective cost of each loan and select the loan with the lowest cost. Execute: The effective annual rates of each of the alternatives are calculated as follows: a. Since the APR is 6%, the monthly rate is 6%/12  0.5%. This translates to an effective annual rate of (1.005)12  1  6.2%. b. The compensating balance is $1000  0.10  $100. Therefore, the borrower will have use of only $900 of the $1000. The interest is 0.06  $1,000  $60. The interest rate per period is $60/$900  6.7%. Since this alternative assumes annual compounding, the effective annual rate is 6.7% as well. c. The interest expense is 0.06  $1000, and the loan origination fee is 0.01  $1000  $10. The loan origination fee reduces the usable proceeds of the loan to $990 since it is paid at the beginning of the loan. The interest rate per period is $70/$990  7.1%. Since the loan is compounded annually in this case, 7.1% is the effective annual rate. Evaluate: Thus, Alternative A offers the lowest effective annual cost and should be taken.

13.

Plan: Calculate the effective annual rate of the loan. Execute: In this problem, Needy must pay $400 every three months to have the use of $10,000. Thus, the interest rate per period is $400/$10,000  4%. Since there are 4 threemonth periods in a year, the effective annual rate is (1.04)4  1  17%.

Evaluate: The effective annual rate on this loan is approximately 17%. 14.

Plan: Calculate the effective annual rate on this financing. Execute: Treadwater is paying $15,000 ( $1,000,000  $985,000) to use $985,000 for three months, so the three-month interest rate is $15,000/$985,000  1.523%. There are 4 three-month periods in a year, making the effective annual rate (1.01523)4  1  6.2%. Evaluate: Treadwater is paying an annual effective rate of 6.2%.

15.

Plan: Calculate the effective annual rate on this financing. Execute: Magna is paying $26,290 ( $1,000,000  $973,710) to use $973,710 for six months. The six-month interest rate is $26,290/$973,710  2.7%. There are 2 six-month periods in one year, so the effective annual rate is (1.027)2  1  5.5%. Evaluate: Magna is paying an annual effective rate of 5.5%.

16.

Plan: Calculate the dollars in interest that Treadwater and Magna saved by raising funds from commercial paper as opposed to borrowing from a bank at the prime rate of interest. Execute: For Treadwater: 1,000,000  $980,392 1.02

For Magna:

1,000,000  $961,169 (1.02)2 Evaluate: Treadwater saved $4608 ( $985,000  $980,392) and Magna saved $12,541 ( $973,710  $961,169) by issuing commercial paper as opposed to borrowing at the prime rate. 17.

Plan: Calculate the effective annual rate on this financing. Execute: Signet’s interest expense on this loan is $129,150 ( $6,000,000  $5,870,850), and the usable proceeds are $5,870,850. The interest rate for the 4-month period is $129,150/ $5,870,850  2.2%. The effective annual rate is (1.022)3  1  6.7%. Evaluate: Signet is paying an annual effective rate of 6.7%.

18.

Plan: Calculate the effective annual rate of this financing. Execute: The monthly interest rate is 9%/12  0.75%, so Ontario Steel Corporation must pay 0.0075  $5,000,000  $37,500 in interest on the loan. Combining this with the $5000 warehouse fee makes the monthly cost of the loan $42,500. Since the fee is paid at the end of the month, Ontario Steel Corporation has use of the full $5,000,000 for the month. The interest rate per period is $42,500/ $5,000,000  0.85%. There are 12 months in a year, so the effective annual rate is (1.0085)12  1  10.7%.

Evaluate: Ontario Steel Corporation is paying an annual effective rate of 10.7%. 19.

Plan: Calculate the effective annual rate of this financing. Execute: Rasputin’s interest expense is 0.10($500,000)  $50,000. The warehouse fee is 0.01($500,000)  $5000. Because the warehouse fee must be paid at the beginning of the year, Rasputin’s usable proceeds from the loan are only $495,000 ( $500,000  $5000). The effective annual rate is $55,000/$495,000  11.1%. Evaluate: Rasputin is paying an annual effective rate of 11.1%.

20.

Excel Solution Plan: Construct the short-term financial plan. Execute:

2022Q4

2023Q1

2023Q2 2023Q3 2023Q4

Income Statement ($000) 1

Sales

4545

5000

6000

Cost of Goods Sold

(2955)

(3250)

(3900)

Selling, General, and Administrative

(455)

(1,000)

(600)

EBITDA

1135

750

1500

Depreciation

(455)

(500)

(525)

EBIT

680

250

975

Taxes

(239)

(88)

(341)

Net Income

441

162

634

Statement of Cash Flows ($000) 9

Net Income

162

634

Depreciation

500

525

Changes in Working Capital

Accounts Receivable

(136)

(300)

Inventory

Account Payable

105

Cash from Operating Activities

574

964

1,159

Capital Expenditures

(1500)

(525)

Other Investments

Cash from Investing Activities

(1500)

(525)

Net Borrowing

Dividends

Capital Contributions

Cash from Financing Activities

Change in Cash and Equivalents

(926)

439

634

Cash Balance and Short-Term Financing

($000) 24

Starting Cash Balance

1000

500

502

1136

Change in Cash and Equivalents

(926)

439

634

Minimum Cash Balance

500

Surplus (Deficit) Relative to Minimum

(426)

439

636

1270

Increase (Decrease) in Short-Term Financing

426

(437)

Existing Short-Term Financing

437

Total Short-Term Financing

426

Ending Cash Balance

500

502

1136

1770

1000

Evaluate: Whistler will need to borrow $426,000 for the first quarter of 2023, and it will pay back the loan during Q2 of 2023. This financial plan will allow Whistler to expand the business and meet minimum capital needs.

Chapter 21 Risk Management Note: All problems in this chapter are available in MyLab Finance. An asterisk (*) indicates problems with a higher level of difficulty. For a breakdown of the difficulty ratings of all of the problems in this Solutions Manual, please see the Preface – Question Difficulty Ratings. 1.

3%  $65 million

1.0375  $1.88 millon

Evaluate: To insure against this risk, $1.88 million would be a fair premium. 2.

Plan: Use the Security Market Line from Chapter 11 to calculate a required return for an investment of this level of risk. Then use this discount rate to find the present value of the risk exposure. Next calculate the premium the insurance company would charge by adding an additional 15% to the premium. The loss that Genentech would have to suffer to justify paying this premium is the loss value that occurs with net present value of 0. Execute: a. From the SML, the required return for a beta of –0.5 is

rL  5%  0.5(10%  5%)  2.5%. From Equation 21.1, Premium 

2%  $450 million

1.025  $8.78 millon

b. With 15% overhead costs, the insurance premium will be $8.78  (1.15)  $10.098 million. Buying insurance is positive NPV for Genentech if it experiences distress or issuance costs equal to 15% of the amount of the loss. That is, it must experience distress or issuance costs of 15%  450  $67.5 million in the event of a loss. In that case:

NPV (buy insurance)  –10.098  

2%  $(450  67.5) million 1.025

  0

Evaluate: The pure insurance premium for risk is $8.78 million. The insurance company prices the coverage at $10.098 million to cover its overhead costs. Genentech would have to incur insurance costs of $67.5 to justify paying this premium. 3.

Plan: Compute the fair insurance premium. Then compute the NPV of purchasing the insurance. Explain why the net present value of this insurance purchase is positive. Execute: a. From the SML, the required return for a beta of –1.5 is rL  5%  1.5(10%  5%)  2.5%. From Equation 21.1, Premium 

10%  $500,000

1  0.025  $51, 282

b. If we consider after-tax cash flows, NPV  $51,282  (1  0.40)  



10%  $500, 000  (1  0.10) 1  0.025

 $15,385

Evaluate: The gain arises because the firm pays for the insurance when its tax rate is high, but receives the insurance payment when its tax rate is low. 4.

Plan: Make the numerous calculations required in the question. Execute: a. New policies reduce the chance of loss by 9%  4%  5%, for an expected savings of 5%  $10 million  $500,000. Therefore, the NPV is NPV  $100,000  $500,000/1.05 

 $376,190

b. If the firm is fully insured, then it will not experience a loss. Thus, there is no benefit to the firm from the new policies. Therefore, NPV  $100,000 c. If the firm insures fully, it will not have an incentive to implement the new safety policies. Therefore, the insurance company will expect a 9% chance of loss. Therefore, the actuarially fair premium would be Premium  9%  $10 million/1.05 



 $857,143

d. If the insurance policy has a deductible, then the firm will benefit from the new policies because it will avoid a loss and, therefore, avoid paying the deductible 5% of the time. Let D be the amount of the deductible. Then, the NPV of the new policies is NPV  $100,000  5%(D)/1.05 Setting the NPV to 0 and solving for D we get D  $2.1 million. e. With a deductible of $2.1 million, the insurance company can expect the firm to implement the new policies. Therefore, it can expect a 4% chance of loss. In the event of a loss, the insurance will pay ($10  $2.1)  $7.9 million. Therefore, Premium  4%  $7.9 million/1.05 

 $300,952 Evaluate: With this policy, the firm will pay $300,952 for insurance, $100,000 to implement the new policies, and 4%  $2.1 million  $84,000 in expected deductibles. Thus, the firm will pay $300,952  $100,000  $84,000  $484,952 in total, which is much less than the amount it would pay for full insurance in part (c).

*5.

Plan: This problem asks us to compute several ―what if‖ scenario analyses to help understand the riskiness of the business. Execute: a. Operating profit  2 billion pounds  (Price per pound  $0.90/lb). Thus, Price (S/lb)

1.25

1.50

1.75

Operating Profit ($ billion)

0.70

1.20

1.70

b. In this case they will sell for the contract price of $1.45/lb, no matter what the spot price of copper is next year: Contract Price ($/lb)

1.45

Operating Profit ($ billion)

1.10

That is, Operating Profit  2  (1.45 – 0.90)  $1.10 billion. c. In this case, Operating Profit  1  (1.45 – 0.90)  1  (Price – 0.90). Therefore, Contract Price ($/lb)

1.45

Contract Amount

1.00 billion pounds

Spot Price ($/lb)

1.25

1.50

1.75

Operating Profit ($ billion)

0.90

1.15

1.40

d. Strategy (a) could be optimal if the firm is sufficiently profitable so that it will not be distressed even if the copper price next year is low. Equity holders will in this case bear the risk of copper price fluctuations, and there is no gain from hedging the risk. It could also be optimal if the firm is currently in or near financial distress. Then, by not hedging, the firm increases its risk. Equity holders can benefit if the price of copper is high, but debt holders suffer if the price is low. (Recall the discussion in Chapter 16 regarding equity holders’ incentive to increase risk when the firm is in or near financial distress.) Strategy (b) could be optimal if the firm is not in distress now, but would be if the price of copper next year is low and it does not hedge. Then, by locking in the price it will receive at $1.45/lb, the firm can avoid financial distress costs next year. Strategy (c) could be optimal if the firm would risk distress with operating profits of $0.7 billion from copper, but would not with operating profits of $0.9 billion. In that case, the firm can partially hedge and avoid any risk of financial distress. Evaluate: The management of the firm now has information on the risk in the copper mining business. The management has the information to evaluate this risk and decide which risk it is willing to bear and which risk it will hedge. 6.

Excel Solution Plan: Mark your position to market each day. Analyze the results. Execute: a. You have gone long 100  1,000  100,000 barrels of oil. Therefore, the marking-to-market profit or loss will equal 100,000 times the change in the futures price each day: Day

Price

Price Change

Profit/Loss

$60.00

$59.50

($0.50)

($50,000)

$57.50

($2.00)

($200,000)

$57.75

$0.25

$25,000

$58.00

$0.25

$25,000

$59.50

$1.50

$150,000

$60.50

$1.00

$100,000

$60.75

$0.25

$25,000

$59.75

($1.00)

($100,000)

$61.75

$2.00

$200,000

$62.50

$0.75

$75,000

b. Summing the daily profit/loss amounts, the total is a gain of $250,000. This gain offsets your increase in cost from the overall $2.50 increase in oil prices over the 10 days, which increases your total cost of oil by 100,000  $2.50  $250,000. c. After the second day, you have lost a total of $250,000. This loss could be a problem if you do not have sufficient resources to cover the loss. In that case, your position would have been liquidated on day 2, and you would have been stuck with the loss and had to pay the higher cost of oil on day 10. Evaluate: This exercise would help management think through the potential advantages and costs of buying spot oil as needed versus locking in future prices using futures contracts. 7.

Plan: Given Starbucks’ situation, determine the amount of annual coffee purchases that should be covered with fixed cost supply contracts. Execute: If the price of coffee goes up by $0.01 per pound, Starbucks’ cost of coffee will go up by $0.01  100 million  $1 million. But because it can charge higher prices, its revenues will go up by 60%  $1 million  $0.6 million. To hedge this risk, Starbucks should lock in the price for 40 million pounds of coffee, so that it will only suffer an increase in cost for the remaining 60 million pounds of coffee. Evaluate: Starbucks should purchase 40 million pounds of coffee using fixed price contracts.

*8.

Plan: Hans, as a producer of barley, is concerned that when he sells it, the price may have dropped. He can consider various hedging strategies but must also consider the cost of hedging and his production costs. Excecute: a. Hans can take a short position in futures on 3000 tonnes of barley for delivery in October. b. i. 3000 tonnes  $200/tonne = $600,000 ii. 3000 tonnes  $200/tonne =$600,000 iii. 3000 tonnes  $200/tonne =$600,000 c. Hans can purchase the put options on 3000 tonnes of barley. d. i. He already paid for the put options an amount of $15/tonne  3000 tonnes = $45,000. With the market price of barley below the put’s strike price, he will exercise the put to receive $200/tonne  3000 tonnes = $600,000. His total net amount received is thus $600,000  $45,000 = $555,000. ii. He already paid for the put options an amount of $15/tonne  3000 tonnes = $45,000. With the market price of barley equal the put’s strike price, he can let the puts expire and sell at the market price or he can exercise the puts and sell at the strike price (it

doesn’t matter, both give the same result) to receive $200/tonne  3000 tonnes = $600,000. His total net amount received is thus $600,000  $45,000 = $555,000. iii. He already paid for the put options an amount of $15/tonne  3000 tonnes = $45,000. With the market price of barley above the put’s strike price, he will let the puts expire and sell the barley at the market price of $250 per tonne to receive $250/tonne  3000 tonnes = $750,000. His total net amount received is thus $750,000  $45,000 = $705,000. e. i. Since the futures contracts would be marked to market each day and Hans may have a margin call if his account balance falls too low, if he won’t have funds available, then he should avoid the potential liquidity problem that might arise from using the futures hedge. ii. Given the possibility that he will not have a crop to sell, the options hedge may be preferable, as with the options contracts he is not obliged to sell the barley but with the futures contract he is obliged and the futures may result in a large loss if he is forced to offset his position at a higher futures price. iii. With the futures hedge, Hans will be guaranteed to receive $600,000 (as long as his crop can be sold) so he will cover his costs with certainty and will not bear any financial distress costs. With the options hedge, if barley prices are below

$205/tonne, he will receive less than $570,000 in net revenues (when barley is below $200/tonne, he will only receive net revenues of $555,000), so he will have a shortfall, go bankrupt, and bear significant personal financial distress costs. Thus, in this case, the futures hedge is preferred. iv. With the futures hedge, Hans will lock in a certain loss, go bankrupt, and bear the financial distress costs with certainty. His revenues minus costs will be $600,000  $700,000 = $100,000 with certainty. Thus, he should not use the futures hedge. With the options hedge, Hans will have a loss for any barley price below $248.33/tonne, but he will be solvent if barley prices are higher than that. The options hedge is better than the futures hedge as it gives him a chance of solvency, whereas the futures give him certain failure. However, the best strategy for Hans in this situation is not to hedge at all. With costs of $700,000, he will break even with a barley price of $700,000/3000 tonnes = $233.33/tonne. Note that without hedging, the barley price can be $15 less than required with the put options hedge ($15 is the cost of the put option), so there is more possibility that the final barley price will allow Hans to remain solvent. Your advice in this scenario is not to hedge. Evaluate: Depending on the circumstances, the options hedge may be better or the futures hedge may be better or not hedging at all may be better. It is important to consider all factors and outcomes before deciding what, if any, hedging strategy should be used. *9.

Plan: Alberta Oil Refinery (AOR) will need to purchase oil in the future and management is concerned that oil prices might rise. AOR’s management can hedge, but they must consider which hedging strategy is best or whether hedging is necessary at all. Excecute: a. AOR can hedge by taking a long position in futures for 500,000 barrels of oil for September delivery. b. i. 500,000 barrels  $110/barrel = $55 million ii. 500,000 barrels  $110/barrel = $55 million iii. 500,000 barrels  $110/barrel = $55 million c. AOR can purchase the call options on 500,000 barrels of oil. d. i. To purchase the call options, AOR must pay $20/barrel  500,000 barrels = $10 million. If oil is $60 per barrel, AOR will let the calls expire worthless and purchase the oil at the market price, costing $60/barrel  500,000 barrels = $30 million. The net amount paid is thus $40 million. ii. To purchase the call options, AOR must pay $20/barrel  500,000 barrels = $10 million. If oil is $110 per barrel, which is the same as the call’s strike price, AOR can purchase on the market or exercise the calls (it doesn’t matter, either way the cost is the same), costing $110/barrel  500,000 barrels = $55 million. The net amount paid is thus $65 million. iii. To purchase the call options, AOR must pay $20/barrel  500,000 barrels = $10 million. If oil is $160 per barrel, AOR will exercise the calls and pay $110/barrel  500,000 barrels = $55 million. The net amount paid is thus $65 million.

e. i. Without much cash available between April and August, AOR should avoid the futures hedge as it generates a liquidity risk due to the marking to market and potential margin calls. ii. In this case, there is the possibility that AOR won’t need all 500,000 barrels of oil. The options hedge is preferred as it does not commit AOR to purchase the oil. With the futures contract, if AOR only ends up purchasing half of the oil at a lower price, this only offsets half of the losses from the short futures position, so AOR will have additional losses from the futures on the 250,000 barrels of oil it didn’t need to purchase. iii. With net revenues of $60 million, the futures guarantee a cost of only $55 million, so financial distress is avoided with certainty. However, with the options hedge, if oil is above $100 per barrel AOR’s costs will be above $60 million (and will rise to as much as $65 million for oil prices above the options strike price of $110). Thus, the options hedge still leaves a significant chance of financial distress and associated costs. The futures hedge is preferred in this case. iv. If AOR will have net revenues of $50 million, the futures hedge guarantees a certain loss and should be avoided. The options hedge allows for a profit as long as the oil price is below $80 per barrel, so it is better than the futures hedge. However, no hedge is even better. With no hedge, as long as oil prices are below $100 per barrel, AOR will be profitable and avoid financial distress costs. Note that the options hedge requires oil to be lower because there is also the $20 cost per call option on a barrel of oil. v. In this case, AOR has a natural hedge and should not use either of the futures or options hedges. Evaluate: Depending on the circumstances, the options hedge may be better or the futures hedge may be better or not hedging at all may be better. It is important to consider all factors and outcomes before deciding what, if any, hedging strategy should be used. *10.

Plan: Calculate the duration of each investment and rank them by duration. Execute: The duration of a security is equal to the weighted-average maturity of its cash flows. For zero-coupon bonds, the maturity of the bond is the same as the duration. For annuities, the duration is about half of the maturity. Thus, the ranking is 5-year annuity, 9-year annuity, 5-year zero-coupon bond, and 9-year zero-coupon bond. Evaluate: This problem shows that maturity is not the best way to rank investments.

*11.

Plan: Calculate the effective borrowing costs and evaluate how using interest rate swaps can be of value to a firm. Execute: a. Borrow $100 million, short term, paying LIBOR  1.0%. Then, enter a $100 million notional swap to receive LIBOR and pay 8.0% fixed. Effective borrowing rate is (LIBOR  1.0%)  LIBOR  8.0%  9.0%. (Note: Borrowing long term would have cost 7.6%  2.5%  10.1%.)

b. Refinance the $100 million short-term loan with a long-term loan at 9.10%  0.50%  9.60%. Unwind the swap by entering a new swap to pay LIBOR and receive 9.50%. Effective borrowing cost is now 9.60%  (LIBOR  8.0%)  (LIBOR  9.50%)  8.10% (Note: This rate is equal to the original long-term rate, less the 2% decline in the firm’s credit spread. The firm gets the benefit of its improved credit quality without being exposed to the increase in interest rates that occurred.) Evaluate: A firm may be able to lower borrowing costs and increase financial flexibility by using interest rate swaps.

Chapter 22 International Corporate Finance Note:

Plan: Calculate how many English pounds you could buy with 500 CAD. Execute: 500CAD 

1.95CAD 1GBP

 256.41GBP.

Evaluate: You could buy 256.41 GBP with 500 CAD. 2.

Plan: Calculate how many CAD you will need to exchange to receive 500,000 EUR. Execute: 500, 000 EUR 

0.65 EUR 1CAD

 769, 230.77CAD.

Evaluate: You would have to exchange 769,230.77 CAD to receive 500,000 EUR. 3.

Excel Solution Plan: Calculate the number of Polish zloty, PLN, you must be paid in three months to receive 100,000 CAD in exchange. Also determine what the difference between the spot

and three-month forward PLN/CAD exchange rate is telling you about relative interest rates in Canada and Poland. Execute: a. You can lock in an exchange rate of 2.2595 PLN/CAD. You should require 225,950 PLN to receive 100,000 CAD. Evaluate: b. Because the spot rate is higher than the forward rate, the Polish interest rate must be lower than that of Canada.

*4.

Excel Solution Plan: Draw the graphs requested in the problem and determine which type of hedge has the least downside risk in the event of a cancellation. Execute: a.

c. In order to hedge the risk of the profits, you want to buy a put option; this protects you from a drop in the exchange rate.

Evaluate: The option hedge has the least downside risk in the event of cancellation. There exists the potential for unlimited losses while holding a forward contract if the exchange rate climbs significantly, while the losses from holding an option contract are limited to 10,000 CAD.

Plan: Calculate the present values of converting into dollars and discounting and of converting into euros and then discounting. Do these currency markets appear to be integrated? Execute: a.

5 million EUR  4.67 million EUR 1.07 1.25CAD 4.67 million EUR   5.841 million CAD. 1EUR 1.215CAD  6.075 million CAD 1EUR 1 6.075 million CAD   5.841 million CAD. 1.04

b. 5 million EUR 

Evaluate: c. According to the results of parts (a) and (b), which are identical at $5.841 million, these markets appear to be internationally integrated. 6.

Plan: You can determine the CAD value today in two ways. If both results are the same, then the markets are internationally integrated. Execute: a. b.

4 million PLN  2.04 PLN CAD  1.78 million CAD 1.10 4 million PLN



2.055 PLN CAD 1.06

 1.84 million CAD

Evaluate: c. No, the markets are not internationally integrated because the answers to parts (a) and (b) are not the same. 7.

Excel Solution Plan: Compute the forward exchange rate between the dollar and the euro. Convert the euro cash flows into dollars and discount them to determine the project’s net present value. Accept the project if NPV is positive; reject if NPV is negative. Execute: First, calculate the forward rates:

F1  (1.15USD/EUR )

(1.04) (1.06)

 1.1283USD/EUR F2  (1.15USD/EUR)

(1.04)2 (1.06)2

 1.1070USD/EUR F3  (1.15USD/EUR )

(1.04)3 (1.06)3

 1.0861USD /EUR

F4  (1.15USD /EUR)

(1.04) 4 (1.06) 4

 1.0656USD /EUR

Next, convert euro cash flows into dollars: Year

Euro Cash Flow

Exchange Rate

US Dollar Cash Flow

–15

1.1500

–17.250

1.1283

10.155

1.1070

11.070

1.0861

11.947

1.0656

12.788

Finally, the net present value is 10.154 11.070 11.947 12.788    1.085 1.0852 1.0853 1.0854  20.094 million USD.

NPV  17.250 

Evaluate: Etemadi Amalgamated should undertake the project because the net present value is positive. 8.

Excel Solution Plan: Redo Problem 7 with the new exchange rates. Execute: With the 26% drop in the spot rate, the forward rates need to be recalculated:

(1.04) (1.06)  0.83396USD/EUR

F1  (0.85USD/EUR)

(1.04)2 (1.06)2  0.81823USD/EUR

F2  (0.85USD/EUR)

(1.04)3 (1.06)3  0.80279USD/EUR

F3  (0.85USD/EUR)

(1.04)4 (1.06)4  0.78764USD/EUR

F4  (0.85USD/EUR)

Next, euro cash flows are reconverted into dollars: Euro Cash Flow

Exchange Rate

US Dollar Cash Flow

–15

0.85000

–12.750

0.83396

7.506

0.81823

8.182

0.80279

8.831

0.78764

9.452

Year

Finally, the net present value is 7.505 8.182 8.831 9.452    1.085 1.0852 1.0853 1.0854  14.852 million USD.

NPV  12.750 

Evaluate: Etemadi Amalgamated should still undertake the project because the net present value is positive. Note that this is 26% lower than the answer in Problem 7, which is consistent with the 26% drop in the spot exchange rate. 9.

Plan: Use the Law of One Price to calculate the EUR cost of capital given the CAD WACC and exchange rates. Execute: The Law of One Price tells us

1  r   FS 1  r . * EUR

* CAD

As a result, we have

S * 1  rCAD  1 F 1.2   (1  0.08)  1 1.157  12.014%.

* rEUR 

Evaluate: Our CAD-denominated WACC of 8% translates into a 12.014 EUR cost of capital. 10.

Plan: Use Equation 22.9 for the foreign cost of capital to determine the Japanese yen, JPY, cost of equity. Execute: Using Equation 22.9, we have * 1  rJPY 

1  rJPY * 1  rCAD  . 1  rCAD

As a result, we obtain * rJPY 

1  rJPY * 1  rCAD   1 1  rCAD

1  0.01  (1  0.11)  1 1  0.05  6.771%. 

Evaluate: The JPY cost of equity is 6.771%. 11.

Plan: Calculate the after-tax cost of debt in CAD. Then use Equation 22.9 for the foreign cost of capital to determine the JPY cost of debt. Execute: The after-tax cost of debt in CAD is (0.075)(1 – 0.30)  0.0525, or 5.25%. Using Equation 22.9 for the foreign cost of capital, we have * 1  rJPY 

1  rJPY * 1  rCAD . 1  rCAD

As a result, we obtain

* rJPY 

1  rJPY * 1  rCAD  1 1  rCAD

1  0.01  (1  0.0525)  1 1  0.05  1.24%. 

Evaluate: The JPY cost of debt is 1.24%. 12.

Excel Solution Plan: Use Equation 22.9 for the foreign cost of capital to determine the EUR cost of capital. Then, calculate the project’s net present value. Execute: a. Using the formula for the foreign cost of capital, we have * 1  rEUR 

1  rEUR * 1  rCAD . 1  rCAD

As a result, we obtain * rEUR 

1  rEUR * 1  rCAD  1 1  rCAD

1  0.07  (1  0.095)  1 1  0.045  12.12%. 

12 14 15 15    2 3 1.1212 1.1212 1.1212 1.12124  16.975 million EUR.

b. NPV  25 

Evaluate: Since the net present value of the project is positive, it should be accepted. 13.

Plan: Calculate Tailor Johnson’s U.S. tax liability on its Ethiopian operations given the exchange rate and the tax rates in each country. Execute: With earnings of 100 million ETB and the Ethiopian tax rate of 25%, the tax paid in Ethiopia is 25 million ETB. With an exchange rate of 0.125 USD/ETB, the earnings amount to 12.5 million USD and the Ethiopian taxes amount to 3.125 million USD. With a tax rate of 45%, the U.S. tax on Tailor Johnson’s Ethiopian income would be 0.45  12.5  5.625 million USD. However, Tailor Johnson is able to claim a tax credit of 3.125 million USD, for a net tax liability of 5.625  3.125  2.5 million USD. Evaluate: Tailor Johnson would have a net U.S. tax liability of 2.5 million USD on its profits from Ethiopian operations.

*14.

Plan: Determine the present value of deferring the U.S. tax liability on Tailor Johnson’s Ethiopian earnings for 10 years, and estimate how the exchange rate in 10 years will affect the actual amount of the U.S. tax liability. Execute: a. From Problem 13, the tax liability is 2.5 million USD. Deferred for 10 years, using the after-tax cost of debt at 5%, the present value is 2.5/1.0510  1.53 million USD. Hence, the value of deferral is 2.5  1.53  0.97 million USD. b. The earnings will need to be converted at the future exchange rate, S10, although the tax credit will still be calculated at S1  0.125 USD/ETB. Evaluate: Hence, the U.S. tax liability will be (0.45)(S10)(100)  3.125 (million USD).

15.

Plan: Calculate Qu’Appelle Enterprises’ Canadian tax liability on earnings from its operations in Poland and Sweden. Then determine the Canadian tax liability by pooling. Execute: a. The net Canadian tax liability, after claiming the credit for taxes paid in Poland, is (0.35)(80) 20 =12 million CAD. b. The net Canadian tax liability, after claiming the credit for taxes paid in Sweden, is (0.35)(100)  60  25 million CAD. However, the use of the tax credit is limited to the Canadian tax liability, so the liability is actually zero. This is an excess tax credit of 25 million CAD that is lost. c. Pooling the Polish and Swedish subsidiaries, the net Canadian tax liability is million CAD but as in part (b), the liability won’t be negative; it will actually be zero. Evaluate: By pooling, Qu’Appelle Enterprises is able to use 12 million CAD of the 25 million CAD excess tax credit from earnings in Sweden to offset the 12 million CAD net tax liability from earnings in Poland, leaving a net Canadian tax liability of 0 CAD.

*16.

Plan: Compute the Russian risk-free rate of interest. The amount of interest the Russian government must pay on its bonds above the risk-free rate is the implied credit risk spread. Execute: From covered interest parity, the forward and spot RUB/CAD exchange rates satisfy: F S

1  rRUB 1  rCAD

where rRUB and rCAD are risk-free interest rates in RUB and CAD, respectively. With this equation we can use the spot and forward exchange rates, and the risk-free CAD interest rate, to solve for the risk-free RUB interest rate: 28.5  28 

1  rRUB 1.045

which implies 1.045 1 28  6.37%.

rRUB  28.5 

Evaluate: Therefore, the implied risk-free RUB interest rate is 6.37%, implying that Russian government bonds have an implied credit spread of 7.5% – 6.37%  1.13% to compensate investors for the possibility of the Russian government defaulting. (See Example 22.5 for a similar problem. Note also that an investor can obtain a risk-free investment in rubles by exchanging RUB for CAD at the spot rate of 28 RUB/CAD, investing in Canadian treasury bills at 4.5%, and locking in a forward exchange rate of 28.5 RUB/CAD to convert the proceeds back to RUB. The rate rRUB computed above is the effective return from this transaction.) *17.

Excel Solution Plan: Compute the free cash flows of the project and calculate their net present value. Execute: The solution to this problem is in the following Excel spreadsheet:

Evaluate: The net present value is positive, indicating that the project is acceptable.

Chapter 23 Leasing Note: All problems in this chapter are available in MyLab Finance. An asterisk (*) indicates problems with a higher level of difficulty. For a breakdown of the difficulty ratings of all of the problems in this Solutions Manual, please see the Preface – Question Difficulty Ratings. 1.

Plan: We can use Equation 23.1 to compute the PV of the lease payments. Then, solve for L as the payment in an annuity due. For part (b), solve for the payment on a 200,000 loan. Execute: a. From Equation 23.1, for a five-year (60 month) lease, PV(Lease payments)  $200,000  $60,000/(1  0.05/12)60  $153,247.68 Because the first lease payment is paid up front, and the remaining 59 payments are paid as an annuity,    1 1 $ 153, 247.68  L 1  1  59    0.05 / 12  (1  0.05 / 12)  

Therefore, L  $2,879.97. b. From Equation 23.2 (see also Example 23.2) $200,000  M

 1  1 1  60  0.05/15  (1  0.05/12) 

Therefore, M  $3,774.25 Evaluate: The lease payment is less, but that is because you will not own the computer at the end of the lease. 2.

Plan: You can use Equation 23.1 to compute the PV of the residual value and then compute its FV at the end of 84 months. Execute: From Equation 23.1, PV(Residual Value)  Purchase Price  PV(Lease Payments) 

   1 1  $2 million 22,000 1  1  83    0.05 / 12  (1  0.05/12)  



 $436,974.05

The future residual value in 84 months is, therefore, Residual Value  $436,974.05  (1  0.05/12)84  $619,644.96 Evaluate: In order for the lessor to cover the cost of the asset given the lease payments, it must have a residual value of $619,644.96 at the end of the lease. 3.

Plan: Use Equation 23.1 to solve for the lease payments for part (a) with a residual value of $150,000. For part (b), the PV of the $1 is small enough to be zero, so the PV of the lease payments must cover the full $400,000. Finally, for part (c), the residual value is $80,000 for Equation 23.1. Execute: a. From Equation 23.1, for a five-year (60 month) lease with a monthly interest rate of 6%/12 = 0.5%, PV(Lease payments)  400,000  150,000/(1.005)60  $288,794.17 Because the first lease payment is paid up front, and the remaining 59 payments are paid as an annuity, 1  1   $288,794.17  L  1  1  59    0.005  1.005  

Therefore, L  $5,555.42.

b. In this case, the lessor will only receive $1 at the conclusion of the lease. Therefore, the present value of the lease payments should be $400,000: 1  1   $400,000  L  1  . 1  59   0.005 1.005   

Therefore, L  $7,694.65. c. In this case, the lessor will receive $80,000 at the conclusion of the lease. Thus, PV(Lease payments)  400,000  80,000/(1.005)60  $340,690.22 Because the first lease payment is paid up front, and the remaining 59 payments are paid as an annuity, 1  1   $340,690.22  L  1  1  59    0.005  1.005  

Therefore, L  $6,553.73. Evaluate: The lease payment is smallest when the residual value is greatest. However, this also means that the lessee’s cost to acquire the asset at the end of the lease is higher. There is a trade-off between the lease payments and the residual value. 4.

Excel Solution Plan: This is like Example 23.4. In a capital lease, property is added to balance sheet and the lease is added to debt. For an operating lease, there is no change to the balance sheet. Execute: Capital Lease: Assets

Liabilities

Cash

Debt

150

Prop., Plant, Equip.

255

Equity

125

Book D/E  150/125  1.20 Operating Lease: no change to balance sheet; Book D/E  70/125  0.56 Evaluate: The operating and capital leases both get Acme access to the warehouses but have different implications for its balance sheet. 5.

Plan: Compare the PV of the lease payments to the purchase price and the lease term to the economic life of the asset to determine whether the lease must be classified as a capital lease.

Execute: a. A four-year fair market value lease with payments of $1,150 per month.   1  1  $46,559.09 PV(Lease Payments)  1150  1  1  47    0.09/12  (1  0.09 / 12)  

This is 46,559.09/50,000  931% of the purchase price. Because it exceeds 90% of the purchase price, this is a capital lease. b. A six-year fair market value lease with payments of $790 per month. The lease term is 75% or more of the economic life of the asset (75% × 8 years = 6 years), and so this is a capital lease. c. A five-year fair market value lease with payments of $925 per month.    1 1 1  $44,895 PV(Lease Payments)  925   1   59    0.09 / 12  (1  0.09 / 12)  

This is 44,895/50,000  89.8% of the purchase price. Because it is less than 90% of the purchase price, the term is less than six years, and it is a fair market value lease, this is an operating lease. d. A five-year fair market value lease with payments of $1000 per month and an option to cancel after three years with a $9000 cancellation penalty. Without the cancellation option, the PV of the lease payments would exceed 90% of the purchase price. With the cancellation option, PV(Min. Lease Pmts)  $    1 1 9000 1000   1  1   $38,560  35   36  0.09 / 12  (1  0.09 / 12)   (1  0.09 / 12)

As this is less than 90% of the purchase price, the lease qualifies as an operating lease. Evaluate: If the PV of the lease payments exceeds 90% of the purchase price or the lease term is more than 75% of the economic life of the asset, the lease must be classified as a capital lease. Features, such as a cancellation option, can change the classification of a lease. 6.

Plan: For a purchase, the purchase is a capital expenditure and then the depreciation will create tax shields. The lease payments for a true tax lease are tax deductible. Execute: a. FCF0

 Capital Expenditure  $756,000

FCF1-7  Depreciation Tax Shield  35%  756,000/7  $37,800

= –Lease Payment = –$130,000

b. FCF0

FCF1-6 = –Lease Payment + Income Tax Saving = –130,000 + 130,000 × 35% = – $84,500 FCF7

= + Income Tax Saving = +130,000 × 35% = +$45,500 = –130,000 – (–756,000) = +$626,000

c. FCF0

FCF1-6 = –84,500 – (37,800) = –$122,300 = +45,500 – (37,800) = $7700

FCF7 Evaluate:

The lease saves cash flow upfront in exchange for lower cash flows over the life of the lease. 7.

Excel Solution Plan: a. Compute the FCF from buying and from leasing and then compute the incremental FCF of leasing versus buying. The initial amount of the lease-equivalent loan must be the PV of the incremental FCF. b. Compare the upfront costs of the leasing versus the lease-equivalent loan. The one with the lower upfront cost is more attractive because the two approaches have the same future liabilities. c. You can compute the effective after-tax lease borrowing rate as the IRR of the incremental FCF calculated in part (a). Execute: If Riverton buys the equipment, it will pay $220,000 up front and have depreciation expenses of 220,000/5  $44,000 per year, generating a depreciation tax shield of 35%  44,000  $15,400 per year for years 1–5. If it leases, the lease payments are $55,000 at the beginning of each year and the tax shields of the lease payments are ($55,000 × 0.35)  $19,250 at the end of each year. Thus, the FCF of leasing versus buying is $  (220,000)  $165,000 in year 0; $$19,250  $15,400  $51,150 in years 1–4; and $19,250  $15,400  $3850 in year 5. a. The initial amount of the lease-equivalent loan is the PV of the incremental free cash flows in years 1–5 at Riverton’s after-tax borrowing rate of 8%(1 – 0.35) = 5.2%: 51,150 51,150 51,150 51,150 3,850     1.052 1.0522 1.0523 1.0524 1.0525  177,547.88

Loan Amt 

That is, leasing leads to the same future cash flows as buying the equipment and borrowing $177,547.88 initially.

b. Riverton is better off buying the equipment. If it buys, it could borrow $177,547.88 (the lease-equivalent loan amount) and this is more than the incremental Year 0 amount of $165,000 it would have to pay by buying instead of leasing. Thus, by buying instead of leasing, Riverton would save the difference equal to $12,547.88 in PV terms. c. We compute the effective after-tax lease borrowing rate as the IRR of the incremental FCF calculated in part (a): 165,000; $51,150; $51,150; $51,150; $51,150; $3850. Using Excel, we find the IRR is 8.496191%, which is higher than Riverton’s actual after-tax borrowing rate of 8%  (1  0.35)  5.2%. Thus, the lease is not attractive. Evaluate: The lease is not as attractive as borrowing and buying. Finding the lease-equivalent loan allows us to see this by comparing either the upfront costs or the borrowing rates. 8.

Excel Solution Plan: Calculate the FCF associated with buying and with leasing and then compute the NPV of lease versus buy. For part (b), use CCA and note the change the depreciation tax shield.

Execute: a. (See spreadsheet below for parts [a] and [b].) If Clorox buys the equipment, it will pay $4.25 million up front and have depreciation expenses of 4.25/5  $850,000 per year, generating a depreciation tax shield of 35%  850,000  $297,500 per year for years 1–5. If it leases, the lease payments at the beginning of each year are $975,000 and the tax shield of the lease payment at the end of each year is $975,000  35% = $341,250. Thus, the FCF of leasing versus buying is –$975,000  (4,250,000)  $3,275,000 in year 0; –$975,000 + 341,250 – (–850,000 + 297,500)  $931,250 in years 1–4; and $341,250297,500  $43,750 in year 5. We can determine the gain from leasing by discounting the incremental cash flows at Clorox’s after-tax borrowing rate of 7%(1  0.35)  4.55%: NPV (Lease – Buy)  $3,275,000 

931,250 931,250 931,250 931,250 43,750     1.0455 1.04552 1.04553 1.04554 1.04555

 $26,955.27

Under these assumptions, the lease is less attractive than financing a purchase of the computer. b. (See spreadsheet below for parts [a] and [b].) The depreciation tax shield if the Clorox Canada division buys is now $956,250 in year 1, and varies throughout the remaining years. The initial savings of leasing versus buying is still $3,275,000, as was shown in part (a) above. The FCF of leasing versus buying in years 1–4 have changed, however. Additionally, the tax effects of the terminal loss in year 6 must also be included. The NPV of leasing over buying, based on the spreadsheet data shown below, is therefore $48,346.80. And so the lease is still not attractive. Evaluate: The ability to accelerate the depreciation tax shields by using CCA increases the PV of the buying option and makes it even more attractive than leasing as compared to part (a).

Excel Solution Plan: Calculate the incremental FCF of leasing versus buying and then compute the NPV of the leasing minus buying. For part (b), the amount by which the lease payment must be increased or decreased is equal to the NPV of the Lease – Buy from part (a). Execute: a. (See the spreadsheet table below.) If P&G buys the equipment, it will pay $15 million up front and have depreciation expenses of 15/5  $3 million per year, generating a depreciation tax shield of 35%  3  $1.05 million per year for years 1–5. It will also have after-tax maintenance expenses of $1 million  (1  0.35)  $0.65 million at the end

of each year. Thus, the annual FCF from buying is $1.05  0.65  $0.4 million in years 1–5. If it leases, the after-tax lease payments are $4.2 million at the beginning of each year and the tax shield from the lease payments are $4.2 million  0.35  $1.47 million at the end of each year. Thus, the FCF of leasing versus buying is $  (15)  $10.8 million in year 0; $ +1.47  (1.05  .65)  $3.13 million in years 1–4; and $1.47 – (1.05 .65)  $1.07 million in year 5. We can determine the gain from leasing by discounting the incremental cash flows at P&G’s after-tax borrowing rate of 7%(1  0.35)  4.55%: 3.13 3.13 3.13 3.13 1.07     2 3 4 1.0455 1.0455 1.0455 1.0455 1.04555  $440,739.90

NPV (Lease  Buy)  10.8 

Under these assumptions, the lease is more attractive than financing a purchase of the computer. b. We can use Solver in Microsoft Excel to change the lease payment for year 0 (as other lease payments and income tax savings of leasing depend on it in the spreadsheet). Solver changes the lease payment until the NPV of leasing instead of buying equals 0. The break-even lease payment is $4,344,553.26. Evaluate: Leasing is more attractive than buying unless the lease rate is $4.344553 million or more.

10.

Excel Solution Plan: (See spreadsheet solutions below.) First, compute the FCF from buying the machine. Then, set the PV of the lease payments equal to the PV of the purchase FCF to find the lease payment that will make the NPV of the transaction (buying the equipment and then leasing it at that rate) equal to 0. For part (b), compute Netflix Canada’s lease versus buy NPV based on the lease rate found in part (a). Execute: a. Because the CCA is calculated using 100% rate, the half-year rule implies half the CapEx may be claimed as CCA in the first year and the other half in the second year. The breakeven lease rate for the lessor is $10,669,300.80 as shown in the spreadsheet at the end of the question. b. At a lease rate of $11.6693 and a tax rate of 10%, Netflix has a gain of $1,378,147.760. See spreadsheet below for details. Evaluate c. The source of the gain is the difference in tax rates. By leasing instead of buying, the government collects less tax revenue; this benefit accrues to the lessee as the lessor breaks even (which would be the same as if the lessor did nothing).

Chapter 24 Mergers and Acquisitions Note: All problems in this chapter are available in MyLab Finance. An asterisk (*) indicates problems with a higher level of difficulty. For a breakdown of the difficulty ratings of all of the problems in this Solutions Manual, please see the Preface – Question Difficulty Ratings. 1.

Parts a. and b. Plan: Compute the exchange ratio and the number of shares you need to issue. Calculate the total earnings from the combined firm and then divide by the total number of shares that will be outstanding. Execute:

a. TargetCo’s shares are worth $25, and your shares are worth $40. You will have to issue 25/40 ( 5/8) shares per share of Target Co to buy it. So, in aggregate, you have to issue (5/8)  1 million  625,000 new shares. After the merger, you will have a total of 1,625,000 shares outstanding (the original 1 million plus the 625,000 new shares). Your total earnings will be $6 million. This comes from the $4 per share  1 million shares  $4 million you were earning before the merger and the $2 per share  1 million shares  $2 million that TargetCo was earning. Thus, your new EPS will be $6 million/1.625 million shares  $3.69. b. A 20% premium means that you will have to pay $30 per share to buy TargetCo ( $25  1.20). Thus, you will have to issue $30/$40  0.75 of your shares per share of TargetCo, or a total of 750,000 new shares. With total earnings of $6 million and total shares outstanding after the merger of 1,750,000, you will have EPS of $6 million/1.75 million shares  $3.43. Evaluate: At no premium, you must issue 625,000 new shares to exchange for TargetCo’s shares. That means that the combined earnings of the new firm will be divided by a total of 1.625 million shares. Without synergies or a premium, the earnings of the two firms are combined, as are the shareholders. The larger earnings are spread among a larger number of shares. With a premium, you are paying more to the TargetCo shareholders. They are getting a larger fraction of the new firm (750,000/1,000,000 vs. 625,000/1,000,000). The EPS goes down to the higher number of shares, and the bidder shareholders are worse off than in part (a) and the TargetCo shareholders are better off than in part (a) (because of the premium, they have more shares, so they benefit even though the EPS has decreased).

c. In part (a), the change in the EPS simply came from combining the two companies, one of which was earning $4 per share and the other was earning $2 per share. However, you will notice that even though TargetCo has half your EPS, it is trading for more than half your value. That is possible if TargetCo’s earnings are less risky or if they are expected to grow more in the future. Thus, although your shareholders end up with lower EPS after the transaction, they have paid a fair price, exchanging their $4 per share before the transaction for either lower but safer EPS after the transaction, or lower EPS that are expected to grow more in the future. Either way, focusing on EPS alone cannot tell you whether shareholders are better or worse off. d. Plan: Compute the total value of the combined company and divide it by the total earnings. Execute: If you simply combine the two companies without any indicated synergies, then the total value of the company will be $40 million  $25 million  $65 million. You will have earnings totaling $6 million, so your P/E ratio (calculated with the total share value divided by total earnings) is $65 million/$6 million = 10.83. Your P/E ratio before the merger was $40/$4  10, and TargetCo’s was $25/$2 = 12.5. Evaluate: You can see that by buying TargetCo for its market price and creating no synergies, the transaction simply ends up with a company whose P/E ratio is between the P/E ratios of the two companies going into the transaction. Again, simply focusing on metrics like P/E does not tell you whether you are better or worse off. (Your P/E went up from 10 to 10.83, but your shareholders are no better or worse off.) 2.

Plan: Value TargetCo by multiplying its earnings per share by the P/E ratio of companies in its industry. Execute: Target Co has $2 in earnings, so if other companies in its industry are trading at 14 times earnings, then a starting point for a valuation of TargetCo in this transaction might be $28 per share, implying a 12% premium ($28/$25). Evaluate: Estimating Target Co’s value by comparing it to other companies in the same industry is one starting point for a purchase price.

Plan: Calculate the CEO’s portion of the loss: ownership stake multiplied by dollar loss and compare it to her compensation gain. Execute: Her portion of the $50 million loss in firm value is 3%, or $1.5 million. If her compensation increases by $5 million, even for only one year, she will be better off by $3.5 million. The CEO will be better off.

Evaluate: The CEO’s personal incentives are to go through with the transaction even though it is bad for shareholders. 4.

Plan: Use the premium to calculate the total price that must be offered and then base the exchange ratio on that price. Execute: The premium is 40%, so the compensation to Thor shareholders must be 1.4($40), or $56. Loki’s shares are worth $50, so it will need to offer $56/$50 = 1.12 shares of Loki for every share of Thor. Evaluate: The premium directly affects the exchange ratio: the higher the premium, the higher the exchange ratio.

Plan: Calculate the number of shares of LE by dividing the total value by price per share. Then, calculate the total value with synergies and the implied price per share. Because that implied price per share fully assigns the synergies to LE, it is the most NFF could pay without overpaying. Execute: First, calculate the number of shares of LE: Number of shares 

$4,000,000,000  1,600,000,000 $25

Including synergies, LE will be worth $4 billion  $1 billion  $5 billion, or $31.25 per share ( $5 billion/1.6 billion). Hence, the maximum exchange ratio that NFF can offer is Exchange Ratio 

$31.25  0.893 $35

Thus, NFF can offer a maximum exchange ratio of 0.893 of its share in exchange of each share of LE. Evaluate: If NFF offers a higher exchange ratio, it will be paying LE more than the total value of LE, including all synergies created by the merger. Thus, it will be overpaying and transferring value from NFF shareholders to LE shareholders.

Plan: Calculate the price per share of the new company by dividing the total value by the total number of shares that will be outstanding. Because bidder shares will survive and TargetCo shares will be exchanged, the bidder shares will have this value immediately after the announcement. TargetCo shares will be valued based on the exchange ratio. Finally, comparing the new price of TargetCo to the old price of TargetCo, you can calculate the actual premium. Execute: a. Since 0.75 million new shares will be issued, the share price will be ($40  $25)/1.75  $37.143. b. This is the same as the price after the merger, $37.143. c. Since TargetCo shareholders will receive 0.75  $37.143  $27.86 million, and there are 1 million shareholders, the share price will be $27.86. d. The premium paid will be 27.86/25  1  11.44%. Evaluate: Taking into account the changes in stock prices after the announcement, the true premium will be 11.44%.

*7.

Plan: Since there are no synergies, any premium is a pure transfer from ABC shareholders to XYZ shareholders, so the value of ABC must decrease by the total value of the premium. Execute: a. Upon the announcement, the price of XYZ will rise to $3. This is a premium of 20% over its pre-announcement price ($3/$2.5  1.20). The total premium paid will be $0.50 per share for 1 million shares, or $500,000. So, the value of ABC must decline by $500,000. ABC is worth $20 per share and has 1 million shares, so its value is $20 million, but it is $19.5 million after accounting for the $500,000 premium transferred to XYZ. The price per share is then $19.50. b. ABC will issue 0.15 million shares (0.15  1 million XYZ shares) so that the total number of shares for the combined company is 1.15 million shares. Since ABC will survive, each ABC share will have the value of a share of the combined company. The combined company is worth $20 million (ABC) plus $2.5 million (XYZ). ABC price  price of combined company  $22.50 million/1.15 million shares  $19.5652. The market knows that each XYZ share will become 0.15 shares in the combined entity, so XYZ price  amount shareholders will receive  0.15  $19.5652  $2.9345. Premium  2.9345/2.5 = 17.4% premium c. No, the premium in the stock offer is lower because market prices change to reflect the fact that ABC shareholders are giving XYZ shareholders money because they are paying a premium. The part (b) announcement means XYZ stock goes up and ABC stock goes down, which lowers the premium relative to the cash offer.

Evaluate: In a cash offer, the premium is fixed in dollar terms, but in a stock offer, the premium is defined by the exchange ratio, such that the actual premium paid fluctuates with the stock prices. *8.

Plan: A new share will be issued at a 50% discount for every share that you do not own. Your percentage ownership will then decrease due to the flood of new shares. The price will also decrease because new shares have been sold at a discount. You can calculate the effect on your holdings by comparing the prices from before to after the pill trigger. Execute: a. If you trigger the poison pill, then you own 20% of the company, or 400,000 shares ( 20%  2,000,000 shares). When you trigger the poison pill, every other shareholder will buy a new share for every share they hold, so 1,600,000 shares ( 2,000,000  400,000) will be issued. These shares will be issued at $10, which is 50% of the price immediately before triggering the poison pill (which we assume stays constant at $20). b. After the new 1,600,000 shares are issued, there will be a total of 3,600,000 shares ( 2,000,000  1,600,000). You will own 400,000 of them, so your participation will be 11.11% ( 400,000/3,600,000). c. When the poison pill is triggered, the market value of the firm will increase to $56 million [ ($20  2,000,000)  ($10  1,600,000)]. The new stock price will be $15.56 ( $56 million/3,600,000). d. You lose from triggering the poison pill (you bought shares at $20 that are now worth $15.56). Every other shareholder in the target firm gains—(they end with $31.12 ( $15.56  2) worth of shares for which they only paid $30 (= $20 + $10). Evaluate: Poison pills are very effective because they simultaneously transfer money from the bidder to the other shareholders while reducing the bidder’s voting stake in the company.

*9.

Plan: The post-takeover value of the firm will be the $40 million it is currently worth ($20  2 million shares) plus 40%. If you get control, you can assign the debt to the company. The equity value will be the total value minus the debt and the price per share can then be calculated. Shareholders will tender if the post-takeover price is expected to be less than the $25 you are offering. Execute: a. The value should reflect the expected improvement that you will make by replacing the management, so the value of the company will be $40 million plus 40%  $56 million. If you buy 50% of the shares for $25 apiece, you will buy 1 million shares, paying $25 million. However, you will borrow this money, pledging the shares as collateral, and then assign the loan to the company once you have control. This means that the new value of the equity will be $56 million  $25 million in debt  $31 million. With 2 million shares outstanding, the price of the equity will drop to $15.50.

b. Since the price of the shares will drop from $20 to $15.50 after the tender offer, everyone will want to tender their shares for $25. c. Assuming that everyone tenders their shares and you buy them all at $25 apiece, you will pay $50 million to acquire the company and it will be worth $56 million. You will own 100% of the equity, which will be $56 million  $50 million loan to buy the shares  $6 million. Evaluate: You can take over the company, increase the value, and capture $6 million of the value change.

Chapter 25 Corporate Governance Note: All problems in this chapter are available in MyLab Finance. An asterisk (*) indicates problems with a higher level of difficulty. For a breakdown of the difficulty ratings of all of the problems in this Solutions Manual, please see the Preface – Question Difficulty Ratings. 1.

The corporation allows for the separation of management and ownership. Thus, those who control the operations of the corporation and how its money is spent are not the same ones as those who have invested in the corporation. This creates a clear conflict of interest, and this conflict between the investors and managers creates the need for investors to devise a system of checks on managers—the system of corporate governance.

Examples of principal-agent problems are excessive perquisite consumption (more company jets/company jet travel than needed, nicer office than necessary, etc.) and shirking (management putting in less effort than desirable by shareholders). Another example is value-destroying acquisitions that nonetheless increase the pecuniary or non-pecuniary benefits to the CEO on net.

The corporate organizational form allows those who have the capital to fund an enterprise to be different from those who have the expertise to manage the enterprise. This critical separation allows a wide class of investors to share the risk of the enterprise. However, as mentioned in the answer to Question 1, this separation comes at a cost—the managers may act in their own best interests, not in the best interests of the shareholders who own the firm.

The board of directors is the primary internal control mechanism and the first line of defence to prevent fraud, agency conflicts, and mismanagement. The board is empowered to hire and fire managers, set compensation contracts, approve major investment decisions, etc.

Over time, a long-standing CEO can maneuver the nomination process so that his or her associates and friends are nominated to the board. Additionally, board members representing customers, suppliers, or others who have the potential for business relationships with the firm will sometimes compromise their fiduciary duty in order to keep the management of the firm happy. This desire to keep the CEO happy or a reluctance to challenge him or her interferes with the board’s primary function of monitoring the management.

By knowing a company and its industry as well as possible, security analysts are in a position to uncover irregularities. They also participate in earnings calls with the CEO and CFO, and sometimes ask difficult and probing questions.

Lenders are exposed to the firm as creditors and so are motivated to carefully monitor the firm. They often include covenants in their loans that require the company to maintain certain profitability and liquidity levels. Breaking these covenants can be a warning sign of deeper trouble.

Whistle-blowers can be anyone but are typically employees who uncover outright wrongdoing and ―blow the whistle‖ on the fraud by reporting it to the authorities.

The advantages are that, since options increase in value when the firm’s stock price increases, the CEO’s wealth and incentives will be more closely tied to the shareholders’ wealth. The disadvantage is that option grants can increase a CEO’s incentives to game the system by timing the release of information to fit the option granting schedule or to artificially smooth earnings.

10.

No. There are two counterarguments here. First, as Demsetz and Lehn (1985) argue, there is no reason to expect a simple relation between ownership and value or performance. There are many dimensions to the corporate governance system and a one-size-fits-all approach is too simplistic; the correct ownership level for one firm may not be the correct level for another. Second, some studies have shown a non-linear relationship between firm valuation and ownership—specifically, that increasing ownership is good at first, but that, in a certain range, managers can use their ownership level to partially block efforts to constrain them, even though they still own a minority of the shares. In this ―entrenching‖ range, increasing ownership could reduce performance.

11.

Proxy contests are simply contested elections for directors. In a proxy contest, two competing slates of directors rather than just one slate are proposed for the company. If a board has become captured or is unresponsive to shareholder demands, shareholders can put their own slate of new directors up for election in competition to the slate put up by incumbent management. If the dissident slate wins, then shareholders will have succeeded in placing new directors, presumably not beholden to the CEO, on the board.

12.

A say-on-pay vote is a non-binding vote whereby the shareholders indicate whether they approve of an executive’s pay package or not.

13.

When confronted with a dissident shareholder, a board can do either of the following:  Ignore the shareholder, which will result in the shareholder either going away or launching a proxy fight, in which case the board will need to expend resources in an attempt to convince shareholders not to side with the dissident; or  Negotiate with the dissident shareholder to come to a solution on which the board and the shareholder can agree.

14.

The government should be trying to maximize societal welfare. Thus, in designing regulation, it must trade off the effects of direct and indirect enforcement, compliance, and other costs associated with regulation against the aggregate benefits that accrue to shareholders and the economy as a whole.

15.

Auditors are important to corporate governance. Auditors ensure that the financial picture of the firm presented to outside investors is clear and accurate. Part of the role of auditors is to detect financial fraud before it threatens the viability of the firm. Sarbanes-Oxley included measures designed to reduce conflicts of interest among auditors and to increase the penalties for fraud.

16.

Trading is how prices come to reflect all material information about a company’s prospects. By restricting a set of investors from trading, we decrease the efficiency of the prices because it will take longer for the prices to reflect that private information. We rely on efficient prices to make sure that capital is allocated to its best use. While that is a cost of prohibiting insider trading, there is also a benefit. In order for a capital market to fulfill its function, uninformed investors must be willing to invest their money—providing liquidity and lowering the cost of capital. If investors thought that the stock market was just a fool’s game where they lost to insiders, they would be unwilling to invest or would price their expected loss into their required return. This increases the cost of capital for companies and slows economic growth.

17.

The laws are much stricter for merger-related trading. Anyone who has information about a pending merger is restricted from trading. Non-merger restrictions depend on the source of the material non-public information. If the source violated a fiduciary duty to the shareholders, then the trading is prohibited.

18.

They are better protected in the United States. The U.S. legal system is based on British common law, which offers considerably more protection to minority shareholders than French civil law does.

19.

Because pyramidal structures allow a controlling family to control firms in which they have few actual cash flow rights, the family can use their control to move profits away from firms where they get a small percentage of the cash flows to firms in which they can claim a larger fraction of the cash flows. For example, they can have one firm sell to another at a reduced price.