SOLUTIONS MANUAL For Investment Analysis and Portfolio Management 12 Edition. Frank Reilly, Keith Br by welldoneassistant

Investment Analysis and Por�olio Management 12e Frank Reilly, Keith Brown, Sanford Leeds (Solu�ons Manual All Chapters, 100% Original Veriﬁed, A+ Grade)

Solution and Answer Guide

TABLE OF CONTENTS Answers to Questions..........................................................................................................1 Answers to Problems ......................................................................................................... 5 Appendix 1: Answers to Problems ..................................................................................... 9

ANSWERS TO QUESTIONS 1. When an individual’s current money income exceeds his or her current consumption desires, he or she saves the excess. Rather than keeping these savings in his or her possession, the individual may consider it worthwhile to forego immediate possession of the money for a larger future amount of consumption. This trade-off of present consumption for a higher level of future consumption is the essence of investment. An investment is the current commitment of funds for a period of time in order to derive a future flow of funds that will compensate the investor for the time value of money and the expected rate of inflation over the life of the investment, as well as provide a premium for the uncertainty associated with this future flow of funds. 2. Students in general tend to be borrowers because they are typically not employed and thus have no income (or they are employed with limited income), but they obviously consume and have expenses. The usual intent is to invest the money borrowed in order to increase their future income stream from employment. In other words, students expect to receive a better job and higher income due to their investment in education. 3. In the 20–30-year-old segment, an individual would tend to be a net borrower because s/he is in a relatively low-income bracket and has several expenditures, including automobile(s) and durable goods. In the 30–40-year-old segment, an individual would likely dissave, or borrow, as his or her expenditures would increase with the advent of family life, and conceivably, the purchase of a house. In the 40–50-year-old segment, the individual would probably be a saver because income would increase substantially with no increase in expenditures. Between the ages of 50 and 60, the individual would typically be a strong saver because income would continue to increase and by now the couple would be “emptynesters.” After this, depending upon when the individual retires, the individual

would probably be a dissaver as income decreases (transition from a regular income to income from a pension). Of course, the earlier that this individual can start saving, the better off s/he will be. Your goal should be to start saving once you finish school. 4. The saving–borrowing pattern would vary by profession to the extent that compensation patterns vary by profession and time spent in school also varies. For most white-collar professions (for example, lawyers), income would tend to increase with age. Thus, lawyers would tend to be borrowers in the early segments (when income is low) and savers later in life. Alternatively, for bluecollar professions (for example, plumbers), in which skill is often physical, compensation tends to remain constant or decline with age. Thus, plumbers would tend to be savers in the early segments and dissavers in the later segments (when their income declines). 5. The difference is because of the definition and the measurement of return. In the case of the Wall Street Journal, they only refer to the current dividend yield on common stocks, whereas in the case of the University of Chicago studies, they talk about the total rate of return on common stocks, which is the dividend yield plus the capital gain or loss yield during the period. In the long run, the dividend yield has been 4–5 percent, and the capital gain yield has averaged about the same. In recent years, the dividend yield has been closer to 2 percent (and the amount of share repurchases has increased). Therefore, it is important to compare alternative investments based on total return. 6. The variance of expected returns represents a measure of the dispersion of actual returns around the expected value. Everything else remaining constant, the larger the variance is, the greater the dispersion of expectations and the greater the uncertainty, or risk, of the investment. The purpose of the variance is to help measure and analyze the risk associated with a particular investment. A greater variance implies a greater possibility of returns that are very different from your expected return—and that is risk. 7. An investor’s required rate of return is a function of the economy’s risk-free rate (RFR), an inflation premium that compensates the investor for the loss of purchasing power, and a risk premium that compensates the investor for taking the risk. The RFR is the pure time value of money and is the compensation an individual demands for deferring consumption. More objectively, the RFR can be measured in terms of the long-run real growth rate in the economy because the investment opportunities available in the economy influence the RFR. We think of Treasury yields (i.e., the cost of government borrowing) as the RFR. The inflation premium is the additional protection an individual requires to compensate for the erosion in purchasing power resulting from increasing prices. Because the return on all investments is not certain as it is with T-bills, the investor requires a premium for taking on additional risk. The risk premium can be examined in terms of business risk, financial risk, liquidity risk, exchange rate risk, and country risk.

8. The three main factors that influence the nominal RFR are the real growth rate of the economy, the expected rate of inflation, and liquidity (i.e., supply and demand for capital in the economy). The real growth rate and inflationary expectations have positive relationships with the nominal RFR. In other words, the higher the real growth rate, the higher the nominal RFR, and the higher the expected level of inflation, the higher the nominal RFR. Liquidity has an inverse relationship with the nominal RFR, meaning that lower liquidity results in higher yields. It is unlikely that the economy’s long-run real growth rate will change dramatically during a business cycle. However, liquidity depends upon the government’s monetary policy and would change depending upon what the government considers to be the appropriate stimulus. Besides, the demand for business loans would be greatest during the early and middle parts of the business cycle. Inflation can also change significantly during a business cycle. 9. The five factors that influence the risk premium on an investment are business risk, financial risk, liquidity risk, exchange rate risk, and country risk. Business risk is a function of sales volatility and operating leverage, and the combined effect of the two variables can be quantified in terms of the coefficient of variation of operating earnings. Financial risk is a function of the uncertainty introduced by the financing mix. The inherent risk involved is the inability to meet future contractual payments (interest on bonds, etc.) or the threat of bankruptcy. Financial risk is measured in terms of a debt ratio (for example, debt/equity ratio) and/or the interest coverage ratio. Liquidity risk is the uncertainty an individual faces when he or she decides to buy or sell an investment. The two uncertainties involved are: (1) how long it will take to buy or sell this asset and (2) what price will be received. The liquidity risk on different investments can vary substantially (for example, real estate versus T-bills). Exchange rate risk is the uncertainty of returns on securities acquired in a different currency. The risk applies to the global investor or multinational corporate manager who must anticipate returns on securities in light of uncertain future exchange rates. A good measure of this uncertainty would be the absolute volatility of the exchange rate or its beta with a composite exchange rate. Country risk is the uncertainty of returns caused by the possibility of a major change in the political or economic environment of a country. The analysis of country risk is much more subjective and must be based on the history and current environment in the country. 10. The increased use of debt increases the fixed interest payment. Since this fixed contractual payment will increase, the residual earnings (net income) will become more variable. The required rate of return on the stock will increase since the financial risk (as measured by the debt/equity ratio) has increased. 11. According to the Capital Asset Pricing Model (which will be discussed in later chapters), all securities are located on the Security Market Line, with securities’ risk on the horizontal axis and securities’ expected return on the vertical axis. As to the locations of the five types of investments on the line, the U.S. government

bonds should be located to the left of the other four, followed by the U.K. government bonds, low-grade corporate bonds, common stock of large firms, and common stocks of Japanese firms. The U.S. government bonds have the lowest risk and the required rate of return simply because they virtually have no default risk at all. The U.K. government bonds are perceived to be default risk-free but expose the U.S. investor to exchange rate risk. Low-grade corporates contain business, financial, and liquidity risks but should be lower in risk than equities. Japanese stocks are riskier than U.S. stocks due to exchange rate risk.

12. An investor seeks a return that gives him or her a real rate of return plus compensation for inflation. If a market’s real RFR is, say, 3 percent, then the investor will require a 3 percent return on an investment because this will compensate him or her for deferring consumption. If there is no expected inflation, both the real and the nominal RFR would be 3 percent. However, if the expected inflation rate is 4 percent, the investor would be worse off in real terms if he or she invests at a rate of return of 3 percent. For example, you would receive $103, but the cost of $100 worth of goods at the beginning of the year would be $104 at the end of the year, which means that you could consume less real goods. Thus, for an investment to be desirable, it should have a return of 7.12 percent (1.03  1.04 ) − 1 or an approximate return of 7 percent (3% + 4%). In other words, you must receive both a real rate of return (3 percent) and compensation for inflation (4 percent). 13. Both changes cause an increase in the required return on all investments. Specifically, an increase in the real growth rate will cause an increase in the economy’s RFR because of a higher level of investment opportunities. In addition, the increase in the rate of inflation will result in an increase in the nominal RFR. Because both changes affect the nominal RFR, they will cause an equal increase in the required return on all investments of 5 percent. The following graph shows a parallel shift upward in the capital market line of 5 percent.

14. Such a change in the yield spread would imply a change in the market risk premium because, although the risk levels of bonds remain relatively constant, investors have changed the spreads they demand to accept this risk. In this case, because the yield spread (risk premium) declined, it implies a decline in the slope of the SML, as shown in the following graph. This also implies that there would be a lower risk premium for stocks. With a lower required rate of return, you would expect stock prices to increase.

15. The ability to buy or sell an investment quickly without a substantial price concession is known as liquidity. An example of a liquid investment asset would be a United States Government Treasury Bill. A T-bill can be bought or sold in minutes at a price almost identical to the quoted price. In contrast, an example of an illiquid asset would be a specialized machine or a parcel of real estate in a remote area. In both cases, it might take a considerable period of time to find a potential seller or buyer, and the actual selling price could vary substantially from expectations.

ANSWERS TO PROBLEMS Ending Value of Investment (Including Cash Flows) Beginning Value of Investment 39 + 1.50 40.50 = = = 1.191 1. 34 34 HPY = HPR − 1 = 1.191 − 1 = 0.191 = 19.1% HPR =

61 + 3 64 = = −0.985 2. 65 65 HPY = HPR − 1 = 0.985 − 1 = −0.015 = −1.5% HPR =

3. $4,000 used to purchase 80 shares = $50 per share (59  80) + (5  80) 4,720 + 400 5,120 = = = 1.280 4,000 4,000 4,000 HPY = HPR − 1 = 1.280 − 1 = 0.280 = 28%

HPR =

59  80 4,720 = = 1.180 4,000 4,000 HPY (Price Increase Alone) = 1.180 − 1 = 0.180 = 18%

HPR (Price Increase Alone) =

Therefore, HPY ( Total ) = HPY (Price Increase ) + HPY (Div ) 0.280 = 0.180 + HPY (Div ) 0.10 = HPY (Dividends )

4. “Real” Rate of Return =

Holding Period Return −1 1 + Rate of Inflation

For Problem #1:HPR = 1.191

1.191 1.191 −1 = − 1 = 1.145 − 1 = 0.145 = 14.5% 1 + .04 1.04 1.191 1.191 at 8% inflation: −1 = − 1 = 1.103 − 1 = 0.103 = 10.3% 1 + .08 1.08

at 4% inflation:

For Problem # 2:HPR = 0.985

0.985 − 1 = 0.947 − 1 = −0.053 = −5.3% 1.04 0.985 at 8% inflation: − 1 = 0.912 − 1 = −0.088 = −8.8% 1.08

at 4% inflation:

For Problem # 3: HPR = 1.280

1.280 − 1 = 1.231 − 1 = 0.231 = 23.1% 1.04 1.280 at 8% inflation: − 1 = 1.185 − 1 = 0.185 = 18.5% 1.08

at 4% inflation:

HPYi n i =1 (0.19) + (0.08) + (−0.12) + (−0.03) + (0.15) AMT = 5 0.27 = = 0.054 5 (0.08) + (0.03) + (−0.09) + (0.02) + (0.04) AMB = 5 0.08 = = 0.016 5

Arithemetic Mean (AM) = 

5(a).

Stock T is more desirable because the arithmetic mean annual rate of return is higher.

5(b). Standard Deviation ( ) =

 [R − E(R )] / n i =1

VarianceT = (0.19 − 0.054)2 + (0.08 − 0.054)2 + (−0.12 − 0.054)2 + (−0.03 − 0.054)2 + (0.15 − 0.054)2 = 0.01850 + 0.00068 + 0.03028 + 0.00706 + 0.00922 = 0.06574

 = 0.06574 / 5 = 0.01315 2

 T = 0.01314 = 0.11467 B = (0.08 − 0.016)2 + (0.03 − 0.016)2 + (−0.09 − 0.016)2 + (0.02 − 0.016)2 + (0.04 − 0.016)2 = 0.00410 + 0.00020 + 0.01124 + 0.00002 + 0.00058 = 0.01614

 = 0.01614 / 5 = 0.00323 2

 B = 0.00323 = 0.05681 By this measure, B would be preferable. Standard Deviation of Returns Expected Rate of Return 0.11466 CVT = = 2.123 0.054 0.05682 CVB = = 3.5513 0.016

Coefficient of Variation (CV) =

5(c).

By this measure, T would be preferable.

Geometric Mean ( GM) = π1/ n – 1 where π = Product of the HRs

5(d).

GMT = (1.19 ) (1.08 ) ( 0.88 ) ( 0.97 ) (1.15 ) 

1/5

−1

= [1.26160]1/5 – 1 = 1.04757 – 1 = 0.04757 GMB = (1.08 ) (1.03 ) ( 0.91 ) (1.02 ) (1.04 ) 

1/5

−1

= [1.07383]1/5 – 1 = 1.01435 – 1 = 0.01435

Stock T has more variability than Stock B. The greater the variability of returns, the greater the difference between the arithmetic and geometric mean returns.

7. 8.

E ( RMBC ) = ( 0.30 ) ( −0.10 ) + ( 0.10 ) ( 0.00 ) + ( 0.30 ) ( 0.10 ) + ( 0.30 ) ( 0.25) = ( −0.03) + 0.000 + 0.03 + 0.075 = 0.075

E ( RLCC ) = ( 0.05) ( −0.60 ) + ( 0.20 ) ( −0.30 ) + ( 0.10 ) ( −0.10 ) + ( 0.30 ) ( 0.20 ) + ( 0.20 ) ( 0.40 ) + ( 0.15) ( 0.80 ) = ( −0.03) + ( −0.06 ) + ( −0.01) + 0.06 + 0.08 + 0.12 = 0.16 Lauren’s range of possible returns is much wider ranging from −0.60 to 0.80 than that of Madison (from −0.10 to 0.25). The expected return is also higher for Lauren at 0.16 than that of Madison at 0.075. It presents a greater risk than the Madison Beer Company as an investment.

CPIn+1 − CPIn CPIn where CPI = the Consumer Price Index

Rate of Inflation =

172 − 160 12 = = 0.075 160 160 HPR Real Rate of Return = −1 1 + Rate of Inflation 1.055 U.S. Government T-Bills = − 1 = 0.9814 − 1 = −0.0186 1.075 1.075 U.S. Government LT Bonds = −1 = 0 1.075 1.1160 U.S. Common Stocks = − 1 = 1.0381 − 1 = 0.0381 1.075

Rate of Inflation =

10. NRFR = (1 + 0.03) (1 + 0.04 ) – 1 = 1.0712 – 1 = 0.0712 (An approximation would be the growth rate plus the inflation rate or 0.03 + 0.04 = 0.07.)

11. Return on Common Stock = 7.12% + 5% = 12.12% (An approximation would be 0.03 + 0.04 + 0.05 = 0.12 or 12%.) As an investor becomes more risk averse, the investor will require a larger risk premium to own common stock. As risk premium increases, so too will the required rate of return. In order to achieve a higher rate of return, stock prices should decline. Nominal Rate on T-Bills (or Risk-Free Rate) = (1 + 0.03) (1 + 0.05) – 1

12.

= 1.0815 – 1 = 0.0815 or 8.15%

(An approximation would be 0.03 + 0.05 = 0.08.) The required rate of return on common stock is equal to the RFR plus a risk premium. Therefore, the approximate risk premium for common stocks implied by these data is 0.14 − 0.0815 = 0.0585 or 5.85%. (An approximation would be 0.14 − 0.08 = 0.06.)

APPENDIX 1: ANSWERS TO PROBLEMS 1(a).

Expected Return = (Probability of Return)(Possible Return) n

E (RGDC ) =  Pi [Ri ] i =1

= (0.25)(−0.10) + (0.15)(0.00) + (0.35)(0.10) + (0.25)(0.25) = (−0.025) + (0.000) + (0.035) + (0.0625) = (0.0725) n

 2 =  Pi [Ri − E (Ri )]2 i =1

= (0.25)(−0.100 − 0.0725)2 + (0.15)(0.00 − 0.0725)2 + (0.35)(0.10 − 0.0725)2 + (0.25)(0.25 − 0.0725)2 = (0.25)(0.02976) + (0.15)(0.0053) + (0.35)(0.0008) + (0.25)(0.0315) = 0.0074 + 0.0008 + 0.0003 + 0.0079 = 0.0164

 GDC = 0.0164 = 0.128 1(b).

Standard deviation can be used as a good measure of relative risk between two investments that have the same expected rate of return.

1(c).

The coefficient of variation must be used to measure the relative variability of two investments if there are major differences in the expected rates of return.

2(a)

E ( RKCC ) = ( 0.15 )( −0.60 ) + ( 0.10 )( −0.30 ) + ( 0.05 )( −0.10 ) + ( 0.40 )( 0.20 ) + ( 0.20 )( 0.40 ) + ( 0.10 )( 0.80 ) = ( −0.09 ) + ( −0.03 ) + ( −0.005 ) + 0.08 + 0.08 + 0.08 = 0.115

 = ( 0.15 )( −0.60 − 0.115 ) + ( 0.10 )( −0.30 − 0.115 ) 2

+ ( 0.05 )( −0.10 − 0.115 ) + ( 0.40 ) ( 0.20 − 0.115 ) 2

+ ( 0.20 )( 0.40 − 0.115 ) + ( 0.10 )( 0.80 − 0.115 ) 2

= ( 0.15 )( −0.715 ) + ( 0.10 )( −0.415 ) + ( 0.05 )( −0.215 ) 2

+ ( 0.40 )( 0.085 ) + ( 0.20 )( 0.285 ) + ( 0.10 )( 0.685 ) 2

= ( 0.15 )( 0.5112 ) + ( 0.10 )( 0.1722 ) + ( 0.05 )( 0.0462 ) + ( 0.40 )( 0.0072 ) + ( 0.20 )( 0.0812 ) + ( 0.10 )( 0.4692 ) = 0.007668 + 0.01722 + 0.00231 + 0.00288 + 0.01624 + 0.04692 = 0.16225

 KCC = 0.16225 = 0.403

2(b).

Based on [E(Ri)] alone, Kayleigh Computer Company’s stock is preferable because of the higher return available.

2(c).

Based on standard deviation alone, the Gray Disc Company’s stock is preferable because of the likelihood of obtaining the expected return because it has a lower standard deviation. Standard Deviation Expected Return 0.128 CVGDC = = 1.77 0.0725 0.403 CVKCC = = 3.50 0.115

CV =

2(d).

Based on CV, Kayleigh Computer Company’s stock return has approximately twice the relative dispersion of Gray Disc Company’s stock return. A lower CV is better. 0.063 + 0.081 + 0.076 + 0.090 + 0.085 0.395 = = 0.079 5 5 0.150 + 0.043 + 0.374 + 0.192 + 0.106 0.865 AMU.K. = = = 0.173 5 5 AMU.S. =

3(a).

Standard deviation of U.S. T-bills: 0.92% or 0.0092. Standard deviation of U.K. common stock: 11.2% or 0.112. 3(b).

The average return of U.S. government T-bills is lower than the average return

of U.S. common stocks because U.S. government T-bills are riskless; therefore, their risk premium would equal 0. The U.S. common stocks are subject to the following types of risks: business risk, financial risk, liquidity risk, exchange rate risk, and, to a limited extent, country risk. The standard deviation of the T-bills and their range (9% to −6.3%) is much less than the standard deviation and range (37.4% to −4.3%) of the U.S. common stocks.

3(c).

GM = π1/ n – 1

πU.S. = (1.063) (1.081) (1.076 ) (1.090 ) (1.085) = 1.462 GMU.S. = (1.462 )

1/5

– 1 = 1.079 – 1 = 0.079

πU.K. = (1.150 ) (1.043) (1.374 ) (1.192 ) (1.106 ) = 2.1727 GMU.K. = ( 2.1727 )

1/5

– 1 = 1.1679 – 1 = 0.1679

In the case of the U.S. government T-bills, the arithmetic and geometric means are approximately equal (0.079), which indicates a small standard deviation (which, as we saw in 3(a), equals 0.0092). The geometric mean (0.1679) of the U.S. common stocks is lower than the arithmetic mean (0.173); this is always the case when the standard deviation is nonzero. A larger standard deviation means that the difference between the arithmetic and geometric means will be larger.

Solution and Answer Guide: Frank K. Reilly, Keith, C. Brown, Sanford J. Leeds, Investment Analysis & Portfolio Management, 12th Edition, © 2025, 9780357988176; Chapter 2: Asset Allocation and Secutiry Selection

Solution and Answer Guide

TABLE OF CONTENTS Answers to Questions..........................................................................................................1 Answers to Problems ......................................................................................................... 8 Appendix 2: Answers to Problems.................................................................................... 11

ANSWERS TO QUESTIONS 1. In answering this question, one assumes that the young person has a steady job, adequate insurance coverage, and sufficient cash reserves. The young individual is in the accumulation phase of the investment life cycle. During this phase, an individual should consider moderately high-risk investments, such as common stocks, because he or she has a long investment horizon and much earning ability over time. 2. In answering this question, one assumes that the 63-year-old individual has adequate insurance coverage and a cash reserve. Depending on her income from social security, she may need some current income from her retirement portfolio to meet living expenses. At the same time, she will need to protect herself against inflation. Removing money from her company’s retirement plan and investing it in money market funds and bond funds would satisfy the investor’s short-term and income needs. However, some long-term investments, such as common stock mutual funds, are needed to provide the investor with needed inflation protection. 3. Typically, investment strategies change during an individual’s lifetime. In the accumulating phase, the individual is accumulating net worth to satisfy shortterm needs (for example, house and car purchases) and long-term goals (for example, retirement and children’s college needs). In this phase, the individual is willing to invest in moderately high-risk investments in order to achieve aboveaverage rates of return. In the consolidating phase, an investor has paid off many outstanding debts and typically has earnings that exceed expenses. In this phase, the investor is becoming more concerned with the long-term needs of retirement or estate planning. Although the investor is willing to accept moderate portfolio risk, he or she is not willing to jeopardize the “nest egg.” In the spending phase, the typical investor is retired or semiretired. This investor wishes to protect the nominal value of his/her savings, but at the same time must

make some investments for inflation protection. The gifting phase is often concurrent with the spending phase. The individual believes that the portfolio will provide sufficient income to meet expenses, plus a reserve for uncertainties. If an investor believes there are excess amounts available in the portfolio, he/she may decide to make “gifts” to family or friends, institute charitable trusts, or establish trusts to minimize estate taxes. 4. A policy statement is important for both the investor and the investment advisor. A policy statement assists the investor in establishing realistic investment goals, as well as providing a benchmark by which a portfolio manager’s performance may be measured. 5. The 45-year-old uncle and 35-year-old sister differ in terms of time horizon. However, each has some time before retirement (20 versus 30 years). Each should have a substantial proportion of his/her portfolio invested in equities, with the 35year-old sister possibly having more equity investments in small firms or international firms (i.e., can tolerate greater portfolio risk). These investors could also differ in current liquidity needs (such as children and education expenses), tax concerns, and/or other unique needs or preferences. 6. Before constructing an investment policy statement, the financial planner needs to clarify the client’s investment objectives (for example, capital preservation, capital appreciation, current income, or total return) and constraints (for example, liquidity needs, time horizon, tax factors, legal and regulatory constraints, and unique needs and preferences). Data on current investments, portfolio returns, and savings plans (future additions to the portfolio) are helpful as well. 7. Student Exercise 8. CFA Examination III (1993) 8(a). At this point, we know (or can reasonably infer) that Mr. Franklin is •

unmarried (a recent widower)

•

childless

•

70 years of age

•

in good health

•

possessed a large amount of (relatively) liquid wealth intending to leave his estate to a tax-exempt medical research foundation, to whom he is also giving a large current cash gift

•

free of debt (not explicitly stated, but neither is the opposite)

•

in the highest tax brackets (not explicitly stated, but apparent)

•

not skilled in the management of a large investment portfolio, but also not a

complete novice because he owned significant assets of his own prior to his wife’s death •

not burdened by large or specific needs for current income

•

not in need of large or specific amounts of current liquidity

Taking this knowledge into account, his Investment Policy Statement will reflect these specifics: Objectives: Return requirements: The incidental throw-off of income from Mr. Franklin’s large asset pool should provide a more than sufficient flow of net spendable income. If not, such a need can easily be met by minor portfolio adjustments. Thus, an inflation-adjusted enhancement of the capital base for the benefit of the foundation will be the primary return goal (i.e., real growth of capital). Tax minimization will be a continuing collateral goal. Risk tolerance: Account circumstances and the long-term return goal suggest that the portfolio can take somewhat above-average risk. Mr. Franklin is acquainted with the nature of investment risk from his prior ownership of stocks and bonds, he has a still long actuarial life expectancy and is in good current health, and his heir—the foundation, thanks to his generosity—is already possessed of a large asset base. Constraints: Time horizon: Even disregarding Mr. Franklin’s still-long actuarial life expectancy, the horizon is long term because the remainder of his estate, the foundation, has a virtually perpetual life span. Liquidity requirement: Given what we know and the expectation of an ongoing income stream of considerable size, no liquidity needs that would require specific funding appear to exist. Taxes: Mr. Franklin is no doubt in the highest tax brackets, and investment actions should take that fact into account on a continuing basis. Appropriate tax-sheltered investments (standing on their own merits as investments) should be considered. Tax minimization will be a specific investment goal. Legal and regulatory: Investments, if under the supervision of an investment management firm (i.e., not managed by Mr. Franklin himself) will be governed by state law and the prudent person rule. Unique circumstances: The large asset total, the foundation as their ultimate recipient, and the great freedom of action enjoyed in this situation (i.e., freedom from confining considerations) are important in this situation, if not necessarily unique. 8(b). Given that stocks have provided (and are expected to continue to provide) higher risk-adjusted returns than either bonds or cash, and considering that the return goal is for long-term, inflation-protected growth of the capital base, stocks will be allotted the majority position in the portfolio. This is also consistent with Mr. Franklin’s absence of either specific current income needs (the ongoing cash flow should provide an adequate level for current spending) or specific liquidity needs. It is likely that income will accumulate to some extent and, if so, will automatically build a liquid emergency fund for Mr. Franklin as time passes. © 2025 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Because the inherited warehouse and the personal residence are significant (15 percent) real estate assets already owned by Mr. Franklin, no further allocation to this asset class is made. It should be noted that the warehouse is a source of cash flow, a diversifying asset and, probably, a modest inflation hedge. For tax reasons, Mr. Franklin may wish to consider putting some debt on this asset, freeing additional cash for alternative investment use. Given the long-term orientation and the above-average risk tolerance in this situation, about 70 percent of total assets can be allocated to equities (including real estate) and about 30 percent to fixed-income assets. International securities will be included in both areas, primarily for their diversification benefits. Municipal bonds will be included in the fixed-income area to minimize income taxes. There is no need to press for yield in this situation, nor any need to deliberately downgrade the quality of the issues utilized. Venture capital investment can be considered, but any commitment to this (or other “alternative” assets) should be kept small. The following is one example of an appropriate allocation that is consistent with the Investment Policy Statement and the historical and expected return and other characteristics of the various available asset classes: Current Range (%) Target (%) Cash/money market 0–5 0 U.S. fixed income 10–20 15 Non-U.S. fixed income 5–15 10 U.S. stocks (large cap) 30–45 30 (small cap) 15–25 15 Non-U.S. stocks 15–25 15 Real estate 10–15 15* Other 0–5 0 100 *Includes the Franklin residence and warehouse that together comprise the proportion of the total assets shown. 9. The major advantage of investing in common stocks is that generally, an investor would earn a higher rate of return than on corporate bonds over the long term. Also, while the return on bonds is prespecified and fixed, the return on common stocks can be substantially higher if the investor can pick a “winner,” that is, if the company’s performance turns out to be better than current market expectations. The main disadvantage of common stock ownership is the higher risk. While the income on bonds is certain (except in the extreme case of bankruptcy), the return on stocks will vary depending upon the future performance of the company and could well be negative. The coupon income from bonds will be taxed each year. The shareholder will be taxed for dividends (at a lower rate if the investor holds the stock for a period long enough to be considered a “qualified dividend”) and will not be taxed for any gains until the stock is sold. A line graph of returns over time should indicate a lower average level of return and lower variability of returns over time for bonds than for common stock. 10. The three factors are: (1) Limiting oneself to the U.S. securities market would imply effectively ignoring more than 50 percent of the world securities market. While U.S. markets are © 2025 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

still the largest single sector, foreign markets have been growing in absolute and relative size since 1969. (2) The rates of return available on non-U.S. securities often have substantially exceeded those of U.S. securities. (3) Diversification with foreign securities reduces portfolio risk. 11. International diversification reduces portfolio risk because of the low correlation of returns among the securities from different countries. This is due to differing international trade patterns, economic growth, fiscal policies, and monetary policies among countries. 12. There are different correlations of returns between securities from the U.S. and alternate countries because there are substantial differences in the economies of the various countries (at a given time) in terms of inflation, international trade, monetary and fiscal policies, and economic growth. 13. The correlations between U.S. stocks and stocks for different countries should change over time because each country has a fairly independent set of economic policies. Factors influencing the correlations include international trade, economic growth, fiscal policy, and monetary policy. A change in any of these variables will cause a change in how the economies are related. For example, the correlation between U.S. and Japanese stocks will change as the balance of trade shifts between the two countries. Closer economic ties and increased trade will likely result in higher correlations between financial markets. For example, we expect larger correlations between the U.S. and Canada; Canada is the largest trading partner of the U.S. 14. The major risks that an investor must consider when investing in any bond issue are business risk, financial risk, and liquidity risk. Additional risks associated with foreign bonds, such as Japanese or German bonds, are exchange rate risk and country risk. Country risk is not a major concern for Japanese or German securities. Exchange rate risk is the uncertainty that arises from floating exchange rates between the U.S. dollar and the Japanese yen or euro. 15. The additional risks that some investors believe international investing introduces include foreign exchange risk and country risk. As an example, if you invested in Japan in 2021, you would have been hurt if you had sold that investment during much of 2022 because the yen weakened significantly during that period. In addition, if you invested in Ukraine prior to Russia’s invasion, you would have lost 65 percent for the year ended November 30, 2022. 16. There are four alternatives to direct investment in foreign stocks available to investors: (1) purchase of American Depository Receipts (ADRs) (2) purchase of American shares

(3) direct purchase of foreign shares listed on a foreign or U.S. exchange (4) purchase of international mutual funds 17. Unlike corporate bonds, interest on municipal bonds is exempt from taxation by the federal government and by the state that issued the bond, provided the investor is a resident of that state. For instance, a marginal tax rate of 35 percent means that a regular bond with an interest rate of 8 percent yields a net return after taxes of only 5.20 percent [0.08  (1 − 0.35)] . A tax-free bond with a 6 percent yield would be preferable. 18. The convertible bond of the growth company would have a lower yield. This is intuitive because there is a greater potential for the price of the growth company stock to increase, which would make the conversion feature of the bond extremely attractive. Thus, the investor would be willing to trade-off the higher upside potential resulting from conversion for the lower yield. 19. Liquidity is the ability to buy or sell an asset quickly at a price similar to the prior price assuming no new information has entered the market. Common stocks have the advantage of liquidity because it is very easy to buy or sell a small position (there being a large number of potential buyers) at a price not substantially different from the current market price. Raw land is relatively illiquid because it is often difficult to find a buyer immediately and often the prospective buyer will offer a price that is substantially different from what the owner considers to be the true market value. A reason for this difference is that while common stock data are regularly reported in a large number of daily newspapers and several magazines and closely watched by a large number of individuals, raw land simply lacks this kind of interest. Further, the speculative nature of raw land investment calls for high risk and longer maturity before profits can be realized. Finally, the initial investment on a plot of raw land would be substantially greater than a round lot in most securities. As a result, the small investor is generally precluded from this kind of investment. 20. Art and antiques are considered illiquid investments because in most cases they are sold at auctions. The implication of being traded at auctions rather than on a developed exchange is that there is tremendous uncertainty regarding the price to be received and it takes a long time to contact a buyer who offers the “right” price. Besides, many buyers of art and antiques are accumulators rather than traders, and this further reduces trading. Coins and stamps are more liquid than art and antiques because an investor can determine the “correct” market price from several weekly or monthly publications. There is no such publication of current market prices of the numerous unique pieces of art and antiques and owners are forced to rely on dealer estimates. Further, while a coin or stamp can be readily disposed of to a dealer at a commission of about 10– 15 percent, the commissions on paintings range from 30 to 50 percent. To sell a portfolio of stocks that are listed on the New York Stock Exchange, an investor simply contacts his/her broker to sell the shares. The cost of trading stocks

varies depending on whether the trade is handled by a full-service broker or a discount broker. 21. Emerging market stocks will not be perfectly correlated with U.S. stocks. As a result, if these stocks are fairly priced, they could reduce the volatility of your portfolio. It is possible, however, that these stocks will be less liquid. 22. International stocks versus U.S. stocks—problems: (1) Information about foreign firms is often difficult to obtain on a timely basis and once obtained can be difficult to interpret and analyze due to language and presentation differences. (2) Financial statements are not comparable from country to country. Different countries use different accounting principles. Even when similar accounting methods are used, cultural, institutional, political, and tax differences can make cross-country comparisons hazardous and misleading. Stock valuation techniques useful in the United States may be less useful in other countries. Stock markets in different countries value different attributes. (3) Currency and political risk must be considered when selecting non-U.S. stocks for a portfolio. (4) Increased costs: custody, management fees, and transaction expenses are usually higher outside the United States. 23. Arguments in favor of adding international securities include the following: (1) Benefits gained from broader diversification, including economic, political, and/or geographic sources. (2) Expected higher returns at the same or lower (if properly diversified) level of portfolio risk. (3) Advantages accruing from improved correlation and covariance relationships across the portfolio’s exposures. (4) Improved asset allocation flexibility, including the ability to match or hedge non-U.S. liabilities. (5) A wider range of industry and company choices for portfolio construction purposes. (6) A wider range of managers through whom to implement investment decisions. (7) Diversification benefits are realizable despite the absence of non-U.S. pension liabilities. At the same time, there are a number of potential problems associated with moving away from a domestic-securities-only orientation: (1) Possible higher costs, including those for custody, transactions, and management fees. (2) Possibly reduced liquidity, especially when transacting in size. (3) Possible unsatisfactory levels of information availability, reliability, scope, timeliness, and understandability. (4) Risks associated with currency management, convertibility, and regulations/controls. (5) Risks associated with possible instability/volatility in both markets and governments. (6) Possible tax consequences or complications. (7) Recognition that overseas investments may underperform U.S. investments.

ANSWERS TO PROBLEMS 1.

Most experts recommend that about six months’ worth of living expenses be held in cash reserves. Although these funds are identified as “cash,” it is recommended that they be invested in instruments that can easily be converted to cash with little chance of loss in value (for example, money market mutual funds, etc.). Most experts recommend that an individual should carry life insurance equal to 7–10 times an individual’s annual salary, but the final determination needs to include the expected expenses and needs facing one’s dependents over their lifetime. An unmarried individual may not need coverage but should consider purchasing some insurance while they are “insurable,” meaning that they are young and have fewer health issues that may make obtaining insurance more difficult or expensive. A married individual with two children should definitely have coverage (possibly 9–10 times their salary as a starting point, to be refined after considering the living expenses of loved ones, desire to provide for the college education of children, and so on).

2(a). $10,000 invested in a 9 percent tax-exempt IRA (assuming annual compounding) in 5 years : $10,000 (FVIF @ 9% ) = $10,000 (1.5386 ) = $15,386

in 10 years : $10,000 (FVIF @ 9% ) = $10,000 (2.3674 ) = $23,674 in 20 years : $10,000 (FVIF @ 9% ) = $10,000 ( 5.6044 ) = $56,044 After-tax yield = Before-tax yield (1 − Tax rate)

2(b).

= 9%(1 − 0.36) = 5.76% $10,000 invested at 5.76 percent (assuming annual compounding) in 5 years : $10,000(FVIF @ 5.76%) =$13,231 in10 years : $10,000(FVIF @ 5.76%) =$17,507

in 20 years : $10,000(FVIF @ 5.76%) =$30,650

3(a).

$10,000 invested in a 10 percent tax-exempt compounding) in 5 years : $10,000(FVIF @10%) =$10,000(1.6105) =$16,105 in10 years : $10,000(FVIF @10%) =$10,000(2.5937) =$25,937

IRA

(assuming

annual

in 20 years : $10,000(FVIF @10%) =$10,000(6.7275) =$67,275

After-tax yield = Before-tax yield (1 − Tax rate)

3(b).

= 10%(1 − 0.15) = 8.50%

$10,000 invested at 8.50 percent (assuming annual compounding) in 5 years : $10,000(FVIF @8.50%) =$15,037

in10 years : $10,000(FVIF @8.50%) =$22,610 in 20 years : $10,000(FVIF @8.50%) =$51,120 .

With inflation growing at 3 percent annually, the above figures need to be deflated by the following factors: in 5 years : (1.03)5 = 1.1593 in10 years : (1.03)5 = 1.3439 in 20 years : (1.03)5 = 1.8061

The real values of the answers from 4(a) are: $10,000 invested in a 9 percent taxexempt IRA (assuming annual compounding). in 5 years : $15,386/1.1593 =$13,271.80 in10 years : $23,674/ 1.3439 =$17,615.89 in 20 years : $56,044/ 1.8061 =$31,030.40

The real values of the answers from 5(a) are: $10,000 invested in a 10 percent taxexempt IRA (assuming annual compounding). in 5 years: $16,105/ 1.1593=$13,892.00 in10 years: $25,937/1.3439 = $19,299.80 in 20 years: $67,275/1.8061 =$37,248.77

5. 6. 7. 8(a).

Student Exercise Student Exercise Student Exercise The arithmetic average assumes the presence of simple interest, while the geometric average assumes compounding or interest-on-interest. As long as there is variation in returns, the geometric mean will always be smaller than the arithmetic mean. In addition, the greater the standard deviation, the greater the disparity between the arithmetic and geometric means. The arithmetic mean is our best prediction for a single-period return. The geometric mean is our best prediction of how money will compound. The geometric mean internal rate of return is a critical concept in security and portfolio selection as well as performance measurement in a multiperiod framework. 8(b). Ranking is best accomplished by using the coefficient of variation (standard deviation/arithmetic mean, multiplied by 100): Real estate 47.42 Treasury bills 90.40 Long govt. bonds 125.49 Long corp. bonds 161.34 Common stocks 164.40

Expected mean plus or minus two standard deviations: Arithmetic:10.28%  16.9%(2) = − 23.52% to + 44.08% 9.

If inflation is 3 percent, Realrate of return = (1 + return)/(1+ inflationrate) − 1 T-bills:realreturn = 1.035/1.03 − 1 = 0.0049 Large-cap common stock:realreturn = 1.1175/1.03 − 1 = 0.0850 Long-term corporate bond:realreturn = 1.0550/1.03 − 1 = 0.0243 Long-term government bond:realreturn = 1.0490/1.03 − 1 = 0.0184 Small-cap common stock:realreturn = 1.1310/1.03 − 1 = 0.0981

APPENDIX 2: ANSWERS TO PROBLEMS Lauren’s average return

(5 + 12 − 11 + 10 + 12) 5 = 28 / 5 = 5.6

Kayleigh’s average return (5 + 15 + 5 + 7 − 10) 5 = 22 / 5 = 4.4

L−L

K −K

5 − 5.6 = − 0.6

5 − 4.4 =0.6

12 − 5.6 = 6.4

15 − 4.4 = 10.6

−11 − 5.6 = − 16.6

5 − 4.4 = 0.6

10 − 5.6 = 4.4

7 − 4.4 = 2.6

12 − 5.6 = 6.4

−10 − 4.4 = − 14.4

(L − L) (K − K) N (−0.6)(0.6) + (6.4)(10.6) + (−16.6)(0.6) + (4.4)(2.6) + (6.4)(−14.4) = 5 −23.2 = = −4.64 5

CovLK =

3. Calculation of correlation coefficient (L − L ) 1 −0.6 2 6.4 3 −16.6 4 4.4 5 6.4 377.2 = 75.44 5

(K − K )2 0.36 112.36 6.76 0.36 207.36 327.20 327.2  K2 = = 65.44 5

 L = 75.44 = 8.69

 K = 65.44 = 8.09

 L2 =

rLK =

Cov LK

 L K

(L − L )2 0.36 40.96 275.56 19.36 40.96 377.20

(K − K ) 0.6 10.6 2.6 0.6 −14.4

−4.64 = −0.066 (8.69)(8.09)

While there is a slight negative correlation, the two securities are essentially uncorrelated. Thus, even though the two companies produce similar products, their historical returns suggest that holding both of these securities would help reduce risk through diversification.

Solution and Answer Guide: Frank K. Reilly, Keith, C. Brown, Sanford J. Leeds, Investment Analysis & Portfolio Management, 12th Edition, © 2025, 9780357988176; Chapter 3: Organization And Functioning Of Securities Markets

Solution and Answer Guide

TABLE OF CONTENTS Answers to Questions...........................................................Error! Bookmark not defined. Answers to Problems ......................................................................................................... 4

ANSWERS TO QUESTIONS 1. A market is a means whereby buyers and sellers are brought together to aid in the transfer of goods and/or services. While it generally has a physical location, it need not necessarily have one. Secondly, there is no requirement of ownership by those who establish and administer the market—they need only to provide a cheap, smooth transfer of goods and/or services for a diverse clientele. A good market should provide accurate information on the price and volume of past transactions and current supply and demand. Clearly, there should be rapid dissemination of this information. Adequate liquidity is desirable so that participants may buy and sell their goods and/or services rapidly at a price that reflects the supply and demand. The costs of transferring ownership and middleman commissions should be low. Finally, the prevailing price should reflect all available information. 2. This is a good discussion question for class because you could explore with students what are some of the alternatives that are used by investors with regard to other assets, such as art and antiques. One primary concern is that you as a seller may not know what a fair price is for your stock. In order to try to sell the shares, one possibility is to put an ad in the paper of your local community or in large cities. Another obvious alternative is an auction or the use of a website such as eBay. With an ad, you would have to specify a price or be ready to negotiate with a buyer. With an auction (internet-based or otherwise), you would be very uncertain of what you would receive. In all cases, there would be a substantial time problem—it may take days, weeks, or longer before you obtain an acceptable price for your shares. 3. Liquidity is the ability to sell an asset quickly at a price not substantially different from the current market, assuming no new information is available. A share of AT&T is very liquid, while an antique would be a fairly illiquid asset. A share of AT&T is highly liquid because an investor could convert it into cash within a few cents of the current market price. An antique is illiquid because it is relatively difficult to find a buyer, and you are uncertain as what price the prospective buyer will offer. U.S. Treasury securities are usually considered to be the most liquid assets in the world.

4. The primary market in securities is where new issues are sold by corporations to acquire new capital via the sale of bonds, preferred stock, or common stock. The sale typically takes place through an investment banker. The secondary market is simply trading in outstanding securities. It involves transactions between owners after the issue has been sold to the public by the company. Consequently, the proceeds from the sale do not go to the company, as is the case with a primary offering. Thus, the price of the security is important to the buyer and the seller. The functioning of the primary market would be seriously hampered in the absence of a good secondary market. A good secondary market provides liquidity to an investor if he or she wants to alter the composition of his or her portfolio from securities to other assets (i.e., house, etc.). Thus, investors would be reluctant to acquire securities in the primary market if they felt they would not subsequently be able to sell the securities quickly at a known price. 5. An example of an initial public offering (IPO) would be a small company selling company stock to the public for the first time. By contrast, a seasoned equity refers to an established company, such as Google, offering a new issue of common stock to an existing market for the stock. The IPO involves greater risk for the buyer because there is not an established secondary market for the small firm. Without an established secondary market, the buyer incurs additional liquidity risk associated with the IPO. 6. Student Exercise 7. In competitive-bid underwriting, the issuer is responsible for specifying the type of security to be offered, the timing, and so on, and then soliciting competitive bids from investment banking firms wishing to act as an underwriter. The high bids will be awarded the contracts. Negotiated underwritings are contractual arrangements between an underwriter and the issuer wherein the underwriter helps the issuer prepare the security issue with the understanding that they have the exclusive right to sell the issue. 8. NASDAQ is the largest U.S. secondary market in terms of the number of issues traded and in the amount of trading. NYSE stocks, however, have a larger aggregate market capitalization. In 2022, there were 3,200 stocks trading on the NYSE and 4,800 stocks trading on the NASDAQ. 9(a). A market order is an order to buy/sell a stock at the most profitable ask/bid prices prevailing at the time the order hits the exchange floor or the quote system on the trading platform. A market order implies the investor wants the transaction completed quickly at the prevailing price. Example: I read good reports about AT&T and I’m certain the stock will go up in value. When I call my broker and submit a market buy order for 100 shares of AT&T, the prevailing asking price is 60. The total cost for my shares will be $6,000 plus commission. 9(b). A limit order specifies a maximum price that the individual will pay to purchase the stock or the minimum he will accept to sell it. Example: If AT&T is selling for $60, then I would put in a limit buy order for one week to buy 100 shares at $59. Or, I could put in a limit order to sell at $61. Whether you are placing a limit order to buy or a limit order to sell, you are effectively saying that you only want to enter into a transaction if you can obtain a more attractive price than currently is quoted (i.e., you can buy for less than the current price or sell for more than the

current price). 9(c).

A short sale is the sale of stock that is not currently owned by the seller with the intent of purchasing it later at a lower price. This is done by borrowing the stock from another investor through a broker. Example: If I expect AT&T to go to $48, then I would sell it short at $60 and hope to replace it when it gets to $55.

9(d).

A stop-loss order is a conditional order whereby the investor indicates that he wants to sell the stock if the price drops to a specified price, thus protecting himself from a large and rapid decline in price. Example: If I buy AT&T at $60 and put in a stop loss at $57, then that (hopefully) protects me from a major loss if it starts to decline. It is always possible that a stock will “gap down” and start trading at a much lower level after bad news is released. Imagine, for example, bad news comes out and the stock drops to $40. Since the stock went below $57, the stop loss order has effectively told the broker to sell the stock at the current market price ($40).

10.

The designated market maker (or “specialist”) acts as a broker in handling limit orders placed with member brokers. Being constantly in touch with current prices, he is in a better position to execute limit orders because they are entered into his books and executed as soon as appropriate. Second, he maintains a fair and orderly market by trading on his own account when there is inadequate supply or demand. If the spread between the bid and ask is substantial, he can place his own bid or ask in order to narrow the spread. This helps provide a continuous market with orderly price changes. On a call market exchange, a designated market maker would call the roll of stocks and ask for interest in one stock at a time. After determining the available buy and sell orders, exchange officials would specify a single price that will satisfy most of the orders, and all orders are transacted at this designated price. The NYSE, which is a continuous market, also employs a call-market mechanism at the open and during trading suspensions. The specialist obtains income from both his functions: commissions as a broker and outperforming the market in his dealer function using the monopolistic information he has on limit orders.

11(a). Dark Pools—Orders put into a dark pool are not displayed to other market participants in order to reduce information leakage and minimize market impact costs and are sold to anonymous buyers. The participants on both sides of the trade are generally in the pool by invitation. The advantage to participants beyond anonymity is better pricing (at the midpoint of the bid–ask spread) and lower transaction fees. In terms of regulation, dark pools are registered as ATSs. Finally, they report their transactions on the composite tape, and it is estimated that they are responsible for about 25 percent of trading volume. 11(b). Broker/Dealer Internalization—Internalization is when retail brokers/dealers internally transact an order by buying or selling the stock against their own account on a consistent basis. Put another way, the firm is the counterparty to all transactions and uses its own capital. It is considered “dark liquidity” because the brokers are acting as OTC market makers and are not required to display quotes prior to execution. These firms are also allowed to report all trades to the consolidated tape, so there is post-trade information and account for about 18 percent of total trading volume and almost 100 percent of all retail marketable order flow.

11(c). High-Frequency Traders—HFTs are professionals and institutions who use AT to create programs that are traded thousands of times a day for small profits. They bring significant liquidity to the market, smaller bid–ask spreads, and substantially lower transaction costs. It is estimated that about 50 percent of all trading volume is attributable to HFTs. They are reviled because they bring added volatility to the market because their algorithms can cause significant shifts in the volume of trading and prices. Also, they contribute to a short-term attitude toward investing when capital markets are meant to determine intrinsic value based on cash flows over very long horizons. 11(d). Algorithmic Trading—Algorithmic trading is basically creating computer programs to make trading decisions. The decisions have become more sophisticated and complex, including simultaneously buying in one market and selling in another for a small profit or programming that would trade based on important company news (for example, earning surprises or merger announcements) or macroeconomic events, such as Federal Reserve decisions or domestic or international political news.

ANSWERS TO PROBLEMS 1(a).

Assume you pay cash for the stock: Number of shares you could purchase = $40,000/$80 = 500 shares. (1)

If the stock is later sold at $100 a share, then the total share proceeds would be

$100  500 shares = $50,000 . Therefore, the rate of return from investing in the stock is as

follows: $50,000 − $40,000 = 25% $40,000

(2)

If the stock is later sold at $40 a share, then the total share proceeds would be $40  $500 shares = $20,000 . Therefore, the rate of return from investing in the stock

would be: =

$20,000 − $40,000 = −50% $40,000

1(b). Assuming you use the maximum amount of leverage in buying the stock, the leverage factor for a 60 percent margin requirement is = 1/percentage margin requirement = 1/0.60 = 5/3. Thus, the rate of return on the stock if it is later sold at $100 a share = 25%  5 / 3 = 41.67% . In contrast, the rate of return on the stock if it is sold for $40 a share: = −50%  5 / 3 = −83.33%.

2(a).

Because the margin is 40 percent and Lauren currently has $50,000 on deposit in her margin account, if Lauren uses the maximum allowable margin, then her $50,000 deposit must

represent 40 percent of her total investment. Thus, $50,000 = 0.4x and x = $125,000. This sum represents $50,000 of her own funds (equity) and $75,000 of borrowed funds. Because the shares are priced at $35 each, Lauren can purchase $125,000 / $35 = 3,571 shares (rounded).

Total Profit = Total Return − Total Investment

2(b).

If the stock rises to $45 / share, Lauren’s total return is: 3,571 shares  $45 = $160,695. Total profit = $160,695 − $125,000 = $35,695.

(1)

Lauren’s profit is computed as : $160,695 − $75,000 borrowing = $85,695; because her initial equity was $50,000, her profit is $85,695 − $50,000 = $35,695, which is the same as computed above. If the stock falls to $25 / share, Lauren’s total return is: 3,571 shares  $25 = $89,275.

(2)

Total loss = $89,275 − $125,000 = −$35,725.

2(c)

Margin =

Market Value − Debit Balance , Market Value

where Market Value = Price per Share  Number of Shares.

Initial Loan Value = Total Investment − Initial Margin = $125,000 − $50,000 = $75,000

Therefore, if the maintenance margin is 30 percent, 0.30 =

(3,571 shares  Price) − $75,000 (3,571 shares  Price)

0.30 ( 3,571  Price ) = ( 3,571  Price ) − $75,000 1,071.3  Price = ( 3,571  Price ) − $75,000 − 2,499.7  Price = −$75,000 Price = $30.00

3. Profit = Ending Value − Beginning Value + Dividends − Transaction Costs − Interest Beginning Value of Investment = $20  100 shares = $2,000

Your Investment = Margin Requirement = ( 0.55  $2,000 ) = $1,100 Ending Value of Investment = $27  100 shares = $2,700 Dividends = $0.50  100 shares = $50.00 Transaction Costs = $0

Interest = 0.05  ( 0.45  $2,000 ) = $45.00 Therefore, Profit = $2,700 − $2,000 + $50 −$45 = $705

The rate of return on your investment of $1,100 is: $706 / $1,100 = 64.1%

Profit on a Short Sale = Begin Value − Ending Value − Dividends − Trans. Costs − Interest

Beginning Value of Investment = $56.00  100 shares = $5,600

( sold under a short sale arrangement) Your Investment = Margin Requirement = ( 0.45  $5,600 ) = $2,520

Ending Value of Investment = $45.00  100 = $4,500

( Cost of closing out position) Dividends = $2.50  100 shares = $250.00 Transaction Costs = $0

Interest = 0.08  $5,600) = $448 ( an investor pays interest on the total value of the shares borrowed) Therefore, Profit = $5,600 − $4,500 − $250 − $448 = $402

The rate of return on your investment of $2,520 is: $402 / $2,520 = 15.95%

5(a).

I want to protect some of the profit I have; should prices drop I will still have a profit of $15/share. This is assuming that the stock is sold at $40. It is always possible that a stock can “gap down” and trade much lower. There is nothing guaranteeing you the ability to sell at $40.

5(b).

With the stop loss : ( $40 − $25 ) / $25 = 60%. Again, this is assuming that the stock is sold at $40. It is always possible that a stock can “gap down” and trade much lower. There is nothing guaranteeing you the ability to sell at $40.

6(a).

Assuming that you pay cash for the stock:

Rate of Return =

($45  300) − ($30  300) 13,500 − 9,000 = = 50% ($30  300) 9,000

This is the return earned over two years, so the annualized return is (1 + 0.50 )

1/2

6(b).

− 1 = 22.47% .

Assuming that you used the maximum leverage in buying the stock, the leverage factor for a 60 percent margin requirement is = 1/margin requirement = 1/.60 = 1.67. Thus, the rate of return on the stock if it is later sold at $45 a share = 50%  1.67 = 83.33% .

The annualized return is (1 + 0.8333 )

1/2

– 1 = 35.4%

Limit order @ $24: Assuming that the stock traded continuously from $28 down to $20, your order was executed at $24. Then, the price went to $36.

Rate of Return = ( $36 − $24 ) / $24 = 50% Assuming market order @ $28: Buy at $28, the price goes to $36.

Rate of Return = ( $36 − $28 ) / $28 = 28.57% Limit order @ $18: Because the market did not decline to $18 (the lowest price was $20), the limit order was never executed.

Solution and Answer Guide: Frank K. Reilly, Keith, C. Brown, Sanford J. Leeds, Investment Analysis & Portfolio Management, 12th Edition, © 2025, 9780357988176; Chapter 4: Security Market Indexes And Index Funds

Solution and Answer Guide

ANSWERS TO QUESTIONS 1. The purpose of security market indexes is to provide a general indication of the aggregate market changes or market movements. More specifically, the indexes are used to derive market returns for a period of interest and then used as a benchmark for evaluating the performance of alternative portfolios. A second use is in examining the factors that influence aggregate stock price movements by forming relationships between market (series) movements and changes in the relevant variables in order to illustrate how these variables influence market movements. A third use is by technicians who use past aggregate market movements to predict future price patterns. A fourth use is to provide the basis for an index fund (or an ETF) that tracks the index and gives investors a low-cost opportunity to earn the returns that an index earned during a period. Finally, a very important use is in portfolio theory, where the systematic risk of an individual security is determined by the relationship between the rates of return for the individual security and the rates of return for a market portfolio of risky assets. Here, a representative index is used as a proxy for the market portfolio of risky assets. 2. A characteristic that differentiates alternative market indexes is the sample—the size of the sample (how representative of the total market it is) and the source (whether securities are of a particular type or a given segment of the population [NYSE, TSE]). The weight given to each member plays a discriminatory role—with diverse members in a sample, it would make a difference whether the index is price-weighted, value-weighted, or unweighted. Finally, the computational procedure is used for calculating return, that is, whether arithmetic mean, geometric mean, etc. 3. A price-weighted series is an arithmetic average of the current prices of the securities included in the sample, that is, closing prices of all securities are summed and divided by the number of securities in the sample. Over time, as a result of stock splits, the divisor will move away from the number of stocks that are in the index. The divisor will be adjusted so that a stock split does not change the value of an index. For example, we wouldn’t want the Dow Jones Industrial Average to decrease in value simply because a member stock did a stock split. A $100 security will have a greater influence on the series than a $25 security because a 10 percent increase in the former increases the numerator by $10, while it takes a 40 percent increase in the price of the latter to have the same effect. Said differently, a 10 percent increase

in the higher-priced stock results in a 10-point increase in the numerator, while a 10 percent increase in the lower-priced stock results in a $2.50 increase in the numerator. 4. A value-weighted index begins by deriving the initial total market value of all stocks used in the series (market value equals number of shares outstanding multiplied by current market price). The initial value is typically established as the base value and assigned an index value of 100. Subsequently, a new market value is computed for all securities in the sample, and this new value is compared to the initial value to derive the percent change, which is then applied to the beginning index value of 100. 5. Given a four-security series and a 2-for-1 split for security A and a 3-for-1 split for security B, the divisor would change from 4 to 2.8 for a price-weighted series. Stock

Before Split Price

After Split Prices

$20

$10

Total

100/4 = 25

70/x = 25 x = 2.8

The price-weighted series adjusts for a stock split by deriving a new divisor that will ensure that the new value for the series is the same as it would have been without the split. The adjustment for a value-weighted series due to a stock split is automatic. The decrease in stock price is offset by an increase in the number of shares outstanding. Before Split Stock

Price/Share

# of Shares

Market Value

$20

1,000,000

$20,000,000

500,000

15,000,000

2,000,000

40,000,000

3,500,000

105,000,000

Total

$180,000,000

The $180,000,000 base value is set equal to an index value of 100. After Split Stock

Price/Share

# of Shares

Market Value

$10

2,000,000

$20,000,000

1,500,000

15,000,000

2,000,000

40,000,000

3,500,000

105,000,000

Total

$180,000,000

Current Market Value  Beginning Index Value Base Value 180,000,000 =  100 180,000,000 = 100 which is precisely what one would expect because there has been no change in prices other than the split. 6. In an unweighted price index series (perhaps more appropriately called an “unweighted” or “equally weighted” index), all stocks carry equal weight irrespective of their price and/or value. One way to visualize an unweighted series is to assume that equal dollar amounts are invested in each stock in the portfolio, for example, an equal amount of $1,000 is assumed to be invested in each stock. Therefore, the investor would own 25 shares of GM ($40/share) and 40 shares of Coors Brewing ($25/share). An unweighted price index that consists of these two stocks would be constructed as follows: New Index Value =

Stock

Price/Share

# of Shares

Market Value

$40

$1,000

Coors

1,000

Total

$2,000

A 20% price increase in GM: Stock

Price/Share

# of Shares

Market Value

$48

$1,200

Coors

1,000

Total

$2,200

A 20% price increase in Coors: Stock

Price/Share

# of Shares

Market Value

$40

$1,000

Coors

1,200

Total

$2,200

Therefore, a 20 percent increase in either stock would have the same impact on the total value of the index (i.e., in all cases the index increases by 10 percent). An alternative treatment is to compute percentage changes for each stock and derive the average of these percentage changes. In this case, the average would be 10 percent [(20% + 0%) / 2 = 10%]. So in the case of an unweighted price-index series, a 20 percent price increase in GM would have the same impact on the index as a 20 percent price increase in Coors Brewing. 7. All three of these indexes are value-based (i.e., based on market capitalization). The Dow Jones Total Stock Market Index includes all U.S.-listed public stocks. The NYSE composite includes all stocks listed on the New York Stock Exchange. Finally, the NASDAQ composite index includes all stocks listed on the NASDAQ. In March 2023, the market capitalization of the NYSE composite

index was ~$25 trillion and the market cap of the NASDAQ composite was ~$19 trillion. Since the NYSE and NASDAQ composites are each subset of the Dow Jones Total Stock Market Index, we would expect that the larger subset (the NYSE composite index) would be more highly correlated with the Dow Jones Total Stock Market Index. 8. The high correlations between returns for alternative NYSE price index series can be attributed to the source of the sample (i.e., stock traded on the NYSE). The four series differ in sample size, that is, the DJIA has 30 securities, the S&P 400 has 400 securities, the S&P 500 has 500 securities, and the NYSE composite has over 2,800 stocks. The DJIA differs in computation from the other series, that is, the DJIA is a price-weighted series, whereas the other three series are value-weighted. Even so, there is a strong correlation between the series because of the similarity of types of companies. 9. Because the equal-weighted series implies that all stocks carry the same weight, irrespective of price or value, the results indicate that on average all stocks in the index increased by 23 percent. On the other hand, the percentage change in the value of a large company has a greater impact than the same percentage change for a small company in the value-weighted index. Therefore, the difference in results indicates that for this given period, the smaller companies in the index outperformed the larger companies. 10. The bond-market series are more difficult to construct due to the wide diversity of bonds available. Also, bonds are hard to standardize because their maturities and market yields are constantly changing. In order to better segment the market, you could construct five possible subindexes based on coupon, quality, industry, maturity, and special features (such as call features, warrants, convertibility, etc.). 11. The Russell 1000 and Russell 2000 represent two different samples of stocks, segmented by size. The fact that the Russell 2000 (which is composed of the smallest 2,000 stocks in the Russell 3000) increased more than the Russell 1000 (composed of the 1,000 largest capitalization U.S. stocks) indicates that small stocks performed better during this time period. 12. We should expect a high-yield index to be more highly correlated with the S&P 500. Junk bonds are more “equity-like.” Junk bonds are closer to equity in the capital structure. 13. Indexes with the broadest representation of U.S. stocks include the Wilshire 5000, the NYSE composite, and the Dow Jones Total Stock Market Index. These indexes would be appropriate benchmarks for portfolio managers wishing to construct a broadly diversified portfolio. In addition, the S&P 500 and Russell 1000 are large-cap indexes, but they will include approximately 80 percent of the market cap of U.S. stocks (although they ignore small-cap stocks). 14. Two investment products that managers may use to track the S&P 500 index include index mutual funds, such as Vanguard’s 500 Index Fund (VFINX) and SPY, an ETF that tracks the S&P 500. Per the textbook, the more accurate means of tracking the S&P 500 index has been VFINX; the SPDR “shares do not track the index quite as closely as did the VFINX fund.”

ANSWERS TO PROBLEMS 1(a).

Given a three-security series and a price change from period t to t + 1, the percentage change in the series would be 42.85 percent. Period t

Period t + 1

$60

$80

Sum

$98

$140

Divisor

Average

32.67

46.67

Percentage change = 1(b).

Period t

46.67 − 32.67 14.00 = = 42.85% 32.67 32.67

Stock

Price/Share

# of Shares

Market Value

$60

1,000,000

$60,000,000

10,000,000

200,000,000

30,000,000

540,000,000

Total

$800,000,000

Period t + 1 Stock Price/Share

# of Shares

Market Value

$80

1,000,000

$80,000,000

10,000,000

350,000,000

30,000,000

750,000,000

Total

$1,180,000,000

1,180 − 800 380 = = 47.50% 800 800 The percentage change for the price-weighted series is a simple average of the differences Percentage change =

1(c).

in price from one period to the next. Equal weights are applied to each price change. The percentage change for the value-weighted series is a weighted average of the differences in price from one period t to t + 1. These weights are the relative market values for each stock. Thus, Stock C carries the greatest weight followed by B and then A. Because Stock B had the greatest percentage increase (75 percent) and the second largest weight, the percentage change would be larger for this series than the price-weighted series.

2(a).

Period t

Stock

Price/Share

# of Shares

Market Value

$60

16.67

$1,000,000

50.00

1,000,000

55.56

1,000,000

Total Stock

$3,000,000 Period t + 1 Price/Share

# of Shares

Market Value

$80

16.67

$1,333.60

50.00

1,750.00

55.56

1,389.00

Total

$4,472.60

Percentage change =

4,472.60 − 3,000 1,472.60 = = 49.09% 3,000 3,000

2(b). 80 − 60 20 = = 33.33% 60 60 35 − 20 15 B= = = 75% 20 20 25 − 18 7 C= = = 38.89% 18 18 33.33% + 75% + 38.89% Arithmetic average = 3 147.22% = = 49.07% 3 The answers are the same (slight difference due to rounding). This is what you would expect because Part A represents the percentage change of an equal-weighted series and Part B applies an equal weight to the separate stocks in calculating the arithmetic average. The geometric average is the nth root of the product of n items. A=

2(c).

Geometric average = [(1.3333) (1.75) (1.3889)]1/3 − 1 = [3.2407]1/3 − 1 = 1.4798 − 1 = 0.4798 or 47.98% The geometric average is less than the arithmetic average. This is because variability of return has a greater affect on the arithmetic average than the geometric average. It is important to realize that we don’t normally think about the geometric return for a single period of time. The geometric return helps investors to understand how their money compounds over multiple periods. It does not give us insight as to how a portfolio performed with three different stocks in a single period.

Student Exercise

4(a). 30

DJIA =  Pit /Dadj i =1

Company

Price/Share

Day 1

DJIA =

12 + 23 + 52 87 = = 29 3 3

(Before Split)

(After Split)

Company

Price/Share

DJIA =

10 + 22 + 55 3 87 = 29 3

10 + 44 + 55 X 109 29 = X X = 3.7586 (new divisor)

DJIA =

(Before Split)

(After Split)

Company

Price/Share

DJIA =

14 + 46 + 52 = 29.798 3.7586 112 3.7586

Day 2

Day 3

14 + 46 + 26 Y 86 29.798 = Y Y = 2.8861 (new divisor) DJIA =

Company

Price/Share

Day 4 DJIA =

Company

Price/Share

85 = 29.452 2.8861

Day 5 DJIA =

4(b).

13 + 47 + 25 2.8861

12 + 45 + 26 2.8861 83 = 28.759 2.8861

Because the index is a price-weighted average, the higher-priced stocks carry more weight. But when a split occurs, the new divisor ensures that the new value for the series is the same as it would have been without the split. Hence, the main effect of a split is just a repositioning of the relative weight that a particular stock carries in determining the index. For example, a 10 percent price change for company B would carry more weight in determining the percent change in the index on Day 3 after the reverse split that increased its price, than its weight on Day 2. In sum, the reverse (1:2) split increased the importance of Stock B. The 2:1 stock split reduced the importance of Stock C.

4(c). 5(a).

Student Exercise

Base = ( $12  500 ) + ( $23  350 ) + ( $52  250 ) = $6,000 + $8,050 + $13,000 = $27,050 Day 1 = ( $12  500 ) + ( $23  350 ) + ( $52  250 ) = $6,000 + $8,050 + $13,000 = $27,050 Index1 = ( $27,050 / $27,050 )  10 = 10

Day 2 = ( $10  500 ) + ( $22  350 ) + ( $55  250 ) = $5,000 + $7,700 + $13,750 = $26,450 Index 2 = ( $26,450 / $27,050 )  10 = 9.778

Day 3 = ( $14  500 ) + ( $46  175) + ( $52  250 ) = $7,000 + $8,050 + $13,000 = $28,050 Index 3 = ( $28,050 / $27,050 )  10 = 10.370

Day 4 = ( $13  500 ) + ( $47  175 ) + ( $25  500 ) = $6,500 + $8,225 + $12,500 = $27,225 Index 4 = ( $27,225 / $27,050 )  10 = 10.065

Day 5 = ( $12  500 ) + ( $45  175) + ( $26  500 )

= $6,000 + $7,875 + $13,000 = $26,875 Index 5 = ( $26,875 / $27,050 )  10 = 9.935

5(b).

The market values are unchanged due to splits and thus stock splits have no effect. The index, however, is weighted by the relative market values.

Price-weighted index (PWI)2011 = (20 + 80 + 40 ) / 3 = 46.67 To account for a stock split, a new divisor must be calculated: (20 + 40 + 40 ) / X = 46.67

X = 2.143(new divisor after stock split) Price-weighted index 2012 = ( 32 + 45 + 42 ) / 2.143 = 55.53

VWI2011 = 20 (100,000,000 ) + 80 ( 2,000,000 ) + 40 ( 25,000,000 ) = 2,000,000,000 + 160,000,000 + 1,000,000,000

= 3,160,000,000 Assuming a base value of 100 and 2011 as the base period, ( 3,160,000,000 / 3,160,000,000 ) 100 = 100 VWI2012 = 32 (100,000,000 ) + 45 ( 4,000,000 ) + 42 ( 25,000,000 ) = 3,200,000,000 + 180,000,000 + 1,050,000,000 = 4,430,000,000 Assuming a base value of 100 and 2011 as the period, ( 4,430,000,000 / 3,160,000,000 )  100 = 1.4019  100 = 140.19

6(a).

Percentage change in PWI = ( 55.53 − 46.67 ) / 46.67 = 18.99% Percentage change in VWI = (140.19 − 100 ) / 100 = 40.19%

6(b).

The percentage change in VWI was much greater than the change in the PWI because the stock with the largest market value (K) had the greater percentage gain in price (60 percent increase).

6(c). December 31, 2022 Stock

Price/Share

# of Shares

Market Value

$20

50.0

$1,000.00

12.5

1,000.00

25.0

1,000.00

Total

$3,000.00

December 31, 2023 Stock

Price/Share

# of Shares

Market Value

$32

50.0

$1,600.00

25.0*

1,125.00

25.0

1,050.00

Total

$3,775.00 (*Stock split 2-for-1 during the year.) 3,775.00 − 3,000 775.00 Percentage change = = = 25.83% 3,000 3,000 Geometric average = [(1.60) (1.125) (1.05)]1/3 − 1 = [1.89]1/3 − 1 = 1.2364 − 1

= 0.2364 or 23.64% Unweighted averages are not impacted by large changes in stock prices (i.e., price-weighted series) or in market values (i.e., value-weighted series). Using a spreadsheet and its functions, we obtain the following values:

R2: 0.98 Alpha or intercept term: 0.08 Beta or slope: 0.96 Average return difference (with signs): 0.08 Average return difference (without signs) 0.28 Portfolio Return

S&P

Difference

Return

Returns

in Returns

−0.20

0.20

0.70

0.9834

in Absolute Difference

Jan

5.0

5.2

Feb

−2.3

−3

Mar

−1.8

−1.6

Intercept 0.0822

−0.20

0.20

Apr

2.2

1.9

Slope

0.30

May 0.4

0.1

0.30

Jun

−0.8

−0.5

−0.30

0.30

Jul

0.2

−0.20

0.20

Aug

1.5

1.6

−0.10

0.10

Sep

−0.3

−0.1

−0.20

0.20

Oct

−3.7

−4

0.30

0.9571

Nov

2.4

0.40

Dec

0.3

0.2

0.10

0.08

0.28

Average

Solution and Answer Guide: Frank K. Reilly, Keith, C. Brown, Sanford J. Leeds, Investment Analysis & Portfolio Management, 12th Edition, © 2025, 9780357988176; Chapter 5: Efficient Capital Markets And Technical Analysis

Solution and Answer Guide

TABLE OF CONTENTS Answers to Questions ......................................................................................................... 1 Answers to Problems ..........................................................................................................7

ANSWERS TO QUESTIONS 1. There are several reasons why one would expect capital markets to be efficient, the foremost being that there are a large number of independent, profit-maximizing investors engaged in the analysis and valuation of securities. A second assumption is that new information comes to the market in a random fashion. The third assumption is that the numerous profit-maximizing investors will adjust security prices rapidly to reflect this new information. Thus, price changes would be independent and random. Finally, because stock prices reflect all information, one would expect prevailing prices to reflect “true” current value. Capital markets as a whole are generally expected to be efficient, but the markets for some securities may not be as efficient as others. Recall that markets are expected to be efficient because there are a large number of investors who receive new information and analyze its effect on security values. If there is a difference in the number of analysts following a stock and the volume of trading, one could conceive of differences in the efficiency of the markets. For example, new information regarding actively traded stocks such as IBM and Exxon is well publicized, and numerous analysts evaluate the effect. Therefore, one should expect the prices for these stocks to adjust rapidly and fully reflect the new information. On the other hand, new information regarding a stock with a small number of stockholders and low trading volume will not be as well publicized, and few analysts follow such firms. Therefore, prices may not adjust as rapidly to new information and the possibility of finding a temporarily undervalued stock is also greater. Some also argue that the size of the firms is another factor to differentiate the efficiency of stocks. Specifically, it is believed that the markets for stocks of small firms are less efficient than those of large firms. 2. The weak-form efficient market hypothesis contends that current stock prices reflect all available security-market information, including the historical sequence of prices, price changes, and any volume information. The implication is that there should be no relationship between past and future price changes. Therefore, any trading rule that uses past market data alone should be of little value. The two groups of tests of the weak-form EMH are (1) statistical tests of independence and (2) tests of trading rules. Statistical tests of independence can be divided further into two groups: the autocorrelation tests and the runs tests. The autocorrelation tests are used to test the existence of significant correlation, whether positive or negative, of price changes on a particular day with a series of consecutive previous days. The runs tests examine the sequence of positive

and negative changes in a series and attempt to determine the existence of a pattern. For a random series, one would expect 1/3(2n − 1) runs, where n is the number of observations. If there are too few runs (i.e., long sequences of positive changes or long sequences of negative changes), the series is not random; in other words, you would not expect a positive change to consistently follow a positive change and a negative change consistently after a negative change. Alternatively, if there are too many runs (+-+-+-+-, etc.), again the series is not random because you would not expect a negative change to consistently follow a positive change. In the trading rule studies, the second major set of tests, investigators attempted to examine alternative technical trading rules through simulation. The trading rule studies compared the risk-return results derived from the simulations, including transaction costs, to results obtained from a simple buy-and-hold policy. The semistrong form efficient market hypothesis contends that security prices adjust rapidly to the release of all new public information and that stock prices reflect all public information. The semistrong form goes beyond the weak form because it includes all market and nonmarket public information such as stock splits, economic news, and political news. Using the organization developed by Fama, studies of the semistrong form EMH can be divided into two groups: (1) studies that attempt to predict future rates of return using publicly available information (goes beyond weak-form EMH); these studies involve either time-series analysis of returns or the cross-section distribution of returns and (2) event studies that examine abnormal rates of return surrounding specific event or item of public information; these studies determine whether it is possible to make average risk-adjusted profits by acting after the information is made public. First, only use information or data that are publicly available at the time of the decision. As an example, if you use information that is typically not available until six weeks after a period and you assume you have it four weeks after, your investment results should be superior because you implicitly have prior information. Second, account all transaction costs for the trading rule. This is important because almost all trading rules involve more transactions than a buy-and-hold policy, and if you don’t consider this, then it will bias the results against buy-and-hold policy. Third, be sure to adjust all results for the risk involved because many trading rules will tend to select high-risk stocks that will have higher returns. The strong-form efficient market hypothesis asserts that stock prices fully reflect all information, whether public or private. It goes beyond the semistrong form because it requires that no group of investors have monopolistic access to any information. Thus, the strong-form efficient market hypothesis calls for perfect markets in which all information is available to everyone at the same time. The strong-form efficient market hypothesis goes beyond the semistrong form in that it calls for perfect markets, meaning that no group of investors has monopolistic access to information. Thus, tests for the strong-form efficient market hypothesis would center around examining whether any group has monopolistic access to information and can consistently obtain aboveaverage profits by using it. Four groups of investors have been featured in these tests: corporate insiders, stock exchange specialists, security analysts, and professional money managers.

7. Behavioral finance deals with individual investor psychology and how it affects individuals’ actions as investors, analysts, and portfolio managers. The goal of behavioral finance is to understand how psychological decisions affect markets and to be able to predict those effects. Behavioral finance looks to explain anomalies that can arise in markets due to psychological factors. 8. The proponents of behavioral finance contend that, although standard finance theory is acceptable in that it focuses on aggregate market behavior, it is incomplete because it fails to account for individual behavior. 9. The basic premise of technical analysis is that the information dissemination process is slow— thus, the adjustment of prices is not immediate but forms a pattern. This view is diametrically opposed to the concept of efficient capital markets, which contends that there is a rapid dissemination process and, therefore, prices reflect all information. Thus, there would be no value to technical analysis because technicians act after the news is made public, which would negate its value in an efficient market. 10. The proponents of fundamental analysis advocate that at one point in time, there is a basic intrinsic value for the aggregate stock market, alternative industries, and individual securities, and if this intrinsic value is substantially different from the prevailing market value, then the investor should make the appropriate investment decision. In the context of the efficient market hypothesis, however, if the determination of the basic intrinsic value is based solely on historical data, then it will be of little value in providing above-average returns. Alternatively, if the fundamental analyst makes superior projections of the relevant variables influencing stock prices, then, in accordance with the efficient market hypothesis, he could expect to outperform the market. The implication is that even with an excellent valuation model, if you rely solely on past data, you cannot expect to do better than a buy-and-hold policy. 11. To be superior in an efficient market, the analyst must be aware of the relevant variables influencing stock prices and be able to consistently project these accurately. If the analyst does not have access to inside information and lacks superior analytical ability, then there is little likelihood of obtaining above-average returns consistently. To establish the superiority of an analyst, it is appropriate to examine the performance of numerous buy-and-sell recommendations by the analyst over a period of time relative to a randomly selected sample of stocks in the same risk class. To be superior, the analyst must consistently perform better than the random selection. Consistency is emphasized because, on average, you would expect random selection to outperform the market about half the time. 12. The major efforts of the portfolio manager should be directed toward determining the risk preferences of his clients and offering, accordingly, a portfolio approximating the risk and return desires of the clientele. Further, the level of risk can be controlled by committing a portion of the portfolio to a risk-free asset and changing this proportion from time to time in accordance with the client’s risk preferences. Second, the portfolio manager should attempt to achieve complete diversification by eliminating all unsystematic risks. Thus, the portfolio should be highly correlated with the market portfolio of risky assets. Finally, it is important to minimize transaction costs, which include minimizing taxes for the client, minimizing commissions by reducing trading turnover, and minimizing liquidity costs by only trading currently liquid stocks. 13. Index funds are security portfolios specially designed to duplicate the performance of the overall security market, as represented by some selected market index series. The first group of index funds was created in the early 1970s because people started realizing that capital markets are efficient and that it is extremely difficult to be a superior analyst. Thus, instead of trying to

outperform the market, a large amount of money should be managed “passively” so that the investment performance simply matches that achieved by the aggregate market and costs are minimized so as not to drop returns below the market. An abundance of research has revealed that the performance of professional money managers is not superior to the market and often has been inferior. This is precisely what one would expect in an efficient capital market. Thus, rather than expending a lot of effort in selecting a portfolio, the performance of which may turn out to be inferior to the market, it is contended by some that portfolios should be designed to simply match the market. If you match the market and minimize transaction costs, then you will beat two-thirds of the institutional portfolio managers on average. The index funds are intended to match the market and minimize costs, as suggested above. Thus, they are consistent with the EMH. 14 (a). The efficient market hypothesis (EMH) states that a market is efficient if security prices immediately and fully reflect all available relevant information. Efficient means informationally efficient, not operationally efficient. Operational efficiency deals with the cost of transferring funds. If the market fully reflects information, the knowledge of that information would not allow anyone to profit from it because stock prices already incorporate the information. 1. Weak form asserts that stock prices already reflect all information that can be derived by examining market trading data, such as the history of past prices and trading volume. Empirical evidence supports the weak form. A strong body of evidence supports weak-form efficiency in the major U.S. securities markets. For example, test results suggest that technical trading rules do not produce superior returns after adjusting for transaction costs and taxes. 2. Semistrong form states that a firm’s stock price already reflects all publicly available information about a firm’s prospects. Examples of publicly available information are annual reports of companies and investment data. Empirical evidence mostly supports the semistrong form. Evidence strongly supports the notion of semi-strong efficiency, but occasional studies (for example, those identifying market anomalies, including the small-firm effect and the January effect) and events (for example, the stock market crash of October 1987) are inconsistent with this form of market efficiency. Black suggests that most so-called anomalies result from data mining. 3. The strong form of EMH holds that current market prices reflect all information, whether publicly available or privately held, that is relevant to the firm. Empirical evidence does not support the strong form. Empirical evidence suggests that strong-form efficiency does not hold. If this form were correct, then prices would fully reflect all information, although a corporate insider might exclusively hold such information. Therefore, insiders could not earn excess returns. Research evidence shows that corporate officers have access to pertinent information long enough before public release to enable them to profit from trading on this information. 14(b). Technical analysis in the form of charting involves the search for recurrent and predictable patterns in stock prices to enhance returns. The EMH implies that this type of technical analysis is without value. If past prices contain no useful information for predicting future prices, then there is no point in following any technical trading rule for timing the purchases and sales of securities. According to weak-form efficiency, no investor can earn excess returns by developing trading rules based on historical price and return information. A simple policy of buying and holding will be at least as good as any technical procedure. Tests generally show that technical trading rules do not produce superior returns after making adjustments for transaction costs and taxes.

Fundamental analysis uses earnings and dividend prospects of the firm, expectations of future interest rates, and risk evaluation of the firm to determine proper stock prices. The EMH predicts that most fundamental analysis is doomed to failure. According to semistrong form efficiency, no investor can earn excess returns from trading rules based on any publicly available information. Only analysts with unique insight receive superior returns. Fundamental analysis is no better than technical analysis in enabling investors to capture above-average returns. However, the presence of many analysts contributes to market efficiency. In summary, the EMH holds that the market appears to adjust so quickly to information about individual stocks and the economy as a whole that no technique of selecting a portfolio—using either technical or fundamental analysis—can consistently outperform a strategy of simply buying and holding a diversified group of securities, such as those making up the popular market averages. 14(c). Portfolio managers have several roles and responsibilities even in perfectly efficient markets. The most important responsibility is as follows: 1. Identify the risk/return objectives for the portfolio given the investor’s constraints. In an efficient market, portfolio managers are responsible for tailoring the portfolio to meet the investor’s needs rather than requirements and risk tolerance. Rational portfolio management also requires examining the investor’s constraints, such as liquidity, time horizon, laws and regulations, taxes, and unique preferences and circumstances, such as age and employment. Other roles and responsibilities include the following: 2. Develop a well-diversified portfolio with the selected risk level. Although an efficient market prices securities fairly, each security still has firm-specific risks that portfolio managers can eliminate through diversification. Therefore, rational security selection requires selecting a well-diversified portfolio that provides the level of systematic risk that matches the investor’s risk tolerance. 3. Reduce transaction costs with a buy-and-hold strategy. Proponents of the EMH advocate a passive investment strategy that does not try to find under-or-overvalued stocks. A buy-andhold strategy is consistent with passive management. Because the efficient market theory suggests that securities are fairly priced, frequently buying and selling securities, which generate large brokerage fees without increasing expected performance, makes little sense. One common strategy for passive management is to create an index fund that is designed to replicate the performance of a broad-based index of stocks. 4. Developing capital market expectations. As part of the asset-allocation decision, portfolio managers need to consider their expectations for the relative returns of the various capital markets to choose an appropriate asset allocation. 5. Implement the chosen investment strategy and review it regularly for any needed adjustments. Under the EMH, portfolio managers have the responsibility of implementing and updating the previously determined investment strategy of each client. 15. The principal contention of technicians is that stock prices move in trends that persist for long periods of time. Because these trends persist, they can be detected by analyzing past prices. 16. Technicians expect trends in stock price behavior because they believe that new information that causes a change in the relationship between supply and demand does not come to the market at one point in time; in other words, they contend that some investors get the information before others. Also, they believe that investors react gradually over time to new information. The result is a gradual adjustment of stock prices. 17. The problems encountered when doing a fundamental analysis of financial statements are

18.

19. 20.

21.

22. 23.

24.

25. 26.

the following: (1) much of the information in financial statements is not useful; (2) there are comparability problems for firms using alternative accounting practices; and (3) there are important psychological factors not included in financial statements. Also, with technical analysis, it is not necessary to invest until the move to a new equilibrium begins. The disadvantages of technical analysis are as follows: (1) past price patterns may not be repeated in the future; (2) the intense competition of those using the trading rules will render the technique useless; (3) the trading rules require a great deal of subjective judgment; and (4) the values that signal action are constantly changing. If mutual funds held a lot of cash, this would be bullish because (1) we think that extreme positions are normally contrarian indicators (for example, too much fear in this case) and (2) the large cash positions of mutual funds represent potential buying power. A significant increase in short selling would indicate extreme pessimism. Normally, we would consider this to be a contrarian indicator. In other words, if everyone is already pessimistic, sentiment can change and push stocks higher. For example, if some good news comes out, short sellers can start to cover their positions by buying stock. This will push stock prices higher. The Dow Theory contends that stock prices move in waves. Specifically, these waves may be grouped into three categories based upon the period of the wave: (1) major trends for long periods (tides); (2) intermediate trends (waves); and (3) short-run movements for very short periods (ripples). The major trend (the tide) is most important to investors. An intermediate reversal occurs when some investors decide to take profits. Technical analysts would prefer to see stock prices increase on high volume. They want to see high demand for the stock as opposed to a stock price increase due to few sellers being in the market (in which case the stock could rise on low volume). Technicians following the breadth of market rules may interpret the event as indicative of a possible market peak. Because the DJIA is a price-weighted series confined to 30 large wellknown stocks, the past trading period indicates that while the high-priced stocks are still advancing, the majority of individual issues are declining. Due to the technical analyst’s belief that information is disseminated on an unequal basis, the technicians would interpret a net cumulative decline as a sign of a market turn that has not been interpreted yet by small investors who hold funds of the large, well-known stocks. A support level is a price range in which considerable demand is expected, while a resistance level is a price range in which a large supply is expected. Support and resistance levels exist due to the behavior of a number of investors who are closely monitoring the market and will trade quickly at attractive price levels. Specifically, a support level occurs after a stock has increased in price followed by a brief period of profit-taking at which time some investors who did not get in on the first round decide to take the opportunity to get in. A resistance level occurs after a stock has declined, and when it experiences a recovery, some investors who missed selling at a price peak take the opportunity to sell. A price break through a resistance level on strong volume would be considered very bullish. This is because technical analysts believe that selling pressure is exhausted. Thus, a price breakthrough on strong volume would be bullish because it would mean the excess supply is gone. A moving average line indicates the major trend of a security’s price. When daily prices break through the long-term trend from below on heavy volume, it is considered a bullish action. The move above the trend line may indicate a new upward change in the trend. If current prices break through the 50-day MA line from above on heavy volume, technicians would expect a reversal of a rising trend and become bearish. If the 50-day MA line crosses

27.

28.

below the 200-day MA line from above on heavy volume, technicians would again expect a reversal in rising prices and become bearish. Relative strength is the ratio of a firm’s stock price to a market price series. If the relative strength ratio is increasing during a bear market, then the stock is not declining as much as the aggregate market. Under such favorable conditions, the technician would expect the stock to likewise outperform the market in the ensuing bull market. Technicians recognize that there is no single technical trading rule that is correct all the time—even the best ones miss certain turns or give false signals. Also, various indicators provide different information for alternative segments of the market. Therefore, you don’t want to depend on any one technique but look at several and derive a consensus.

ANSWERS TO PROBLEMS 1.

ARit = Rit − Rmt ARBt = 11.5 − 4.0 = 7.5

ARFt = 10.0 − 8.5 = 1.5 AR Tt = 14.0 − 9.6 = 4.4

ARCt = 12.0 − 15.3 = −3.3 AREt = 15.9 − 12.4 = 3.5

ARit = Rit −  i ( Rmt ) ARBt = 11.5 − 0.95 ( 4.0 ) =7.7

ARFt = 10.0 − 1.25 ( 8.5 ) = − 0.625 ARTt = 14.0 − 1.45 ( 9.6 ) =0.08 ARCt =12.0 − 0.70 (15.3 ) =1.29 AREt = 15.9 − (−0.3)(12.4)=19.62

The reason for the difference in each case is due to the implications of beta. Beta determines how the stock will move in relation to movements in the market. Considering stock C, a 1 percent change in the market return will result in a 0.70 percent change in stock C’s return. Therefore, comparing the abnormal return for stock C, the value becomes positive in Problem 2. Conversely, the 1.25 percent change expected by stock F, for every 1 percent change in the market, resulted in the abnormal return moving from positive

to negative. Stock E should move opposite the market because of the negative beta value. Thus, stock E has a very large abnormal return. For stocks B and T, the positive abnormal returns remain positive but change in value. 4.

Student Exercise

Student Exercise 40 39 38

X The price of the stock went to 38 1/2 before declining. Therefore, at the current price of around 30, the chart could imply a buying opportunity because the price has risen recently from 26 to 30 without a reversal.

Day 4 = (14,010 + 14,100 + 14,165 + 14,080 ) /4=56,355/4=14,088.75 Day 5 = (14,100 +14,165+14,080+14,070 ) /4 = 56,415/4 = 14,103.75 Day 6 = (14,165+14,080+14,070+14,150 ) /4 = 56,465/4 = 14,116.25 Day 7 = (14,080+14,070+14,150+14,220 ) /4 = 56,520/4 = 14,130.00

8.a.

Day 8 = (14,070+14,150+14,220+14,130 ) /4 = 56,570/4 = 14,142.50 Day 9 = (14,150+14,220+14,130+14,250 ) /4 = 56,750/4 = 14,187.50 Day 10 = (14,220+14,130+14,250+14,315 ) /4 = 56,915/4 = 14,228.75 Day 11= (14,130+14,250+14,315+14,240 ) /4 = 56,935/4 = 14,233.75 Day 12 = (14,250+14,315+14,240+14,310 ) /4 = 57,115/4 = 14,278.75

8.b.

With

this

value

for

Day

13,

the

four-day

moving

average

would

(14,315 + 14,240 + 14,310 + 14,300 ) / 4 = 57,165 / 4 = 14,291.25 Because the index closed above its four-day moving average, this would indicate the continuation of a bullish trend. 9.

Day 1: Start 21,240 + 448 net advances = 21,688 Day 2: 21,688 + 95 net advances = 21,783 Day 3: 21,783 + 519 net advance = 22,302 Day 4: 22,302 − 499 net declines = 21,803 Day 5: 21,803 − 193 net declines = 21,610

NOTE that the two problems that we are inserting have answers of “Student Exercise.”

Solution and Answer Guide: Frank K. Reilly, Keith, C. Brown, Sanford J. Leeds, Investment Analysis & Portfolio Management, 12th Edition, © 2025, 9780357988176; Chapter 6: An Introduction To Portfolio Management

Solution and Answer Guide

TABLE OF CONTENTS Answers to Questions..........................................................................................................1 Answers to Problems ......................................................................................................... 4 Appendix 6: Answers to Problems................................................................................... 10

ANSWERS TO QUESTIONS 1. Investors hold diversified portfolios in order to reduce risk, that is, to lower the variance of the portfolio, which is considered a measure of portfolio risk. A diversified portfolio should accomplish this because the returns for the different assets are typically not perfectly positively correlated, which leads to a reduction in the variance of the total portfolio.

(

)

2. The covariance is equal to  E ( Ri − E ( Ri ) ) R j − E ( R j )  and captures the absolute amount of   comovement in the return series for two different assets. If those returns tend to move in the same direction, covariance will be a large positive value and vice versa. Covariance is important in portfolio theory because the overall variance of a portfolio is a combination of individual variances and the covariances among all assets in the portfolio. For a portfolio with a large number of security holdings, the variance of the portfolio becomes the average of all the covariances. 3. Similar assets like common stock or stock for companies in the same industry will have high positive covariances because the sales and profits for the firms are affected by common factors, as their customers and suppliers are the same. Because their profits and risk factors move together, you should expect the stock returns to also move together as well and thus have high covariance. The returns from different assets will not have as much covariance because the returns will not be as closely connected. This is even more so for investments in different countries where the returns and risk factors can be very unique. 4. The covariance between the returns of assets i and j is affected by the variability of these two return series. Therefore, it is difficult to interpret the covariance figures without considering the variability of each return series. In contrast, the correlation coefficient is obtained by standardizing the covariance for the individual variability of the two return series, that is: rij = Cov ij /( i j ) . Thus, the correlation coefficient can only vary in the range of −1 to +1. A value of +1 would

indicate a perfect positive relationship between Ri and Rj. 5. The efficient frontier shows the trade-off in the expected return and risk (i.e., standard deviation) of the set of optimal portfolios available to an investor. This trade-off is a curved shape rather than a straight line whenever the assets contained in those optimal portfolio are not perfectly correlated with one another. One consequence of this relationship is that investors seeking higher levels of expected returns will need to assume increasingly more risk as they move to portfolio positions farther out on the efficient frontier. 6.

A portfolio dominates another portfolio if it has (1) a higher expected return than another portfolio with the same (or lower) level of risk or (2) a lower level of risk than another portfolio with equal (or higher) expected. For example, portfolio B dominates D by the first criterion. A dominates D by the second, and C dominates D by a combination of the two. The Markowitz efficient frontier is then simply the collection of portfolios that is not dominated by any other portfolio, namely those lying along the curved segment E–F. 7. The necessary information for the program would be the following: 1. 2. 3.

The expected rate of return of each asset The variance (or standard deviation) of each asset The covariance (or correlation) of all pairs of assets under consideration

8. Investors’ utility curves are important because they indicate the desired trade-off by investors between risk and return. Given the efficient frontier, utility curves indicate which portfolio is preferable for a given investor. Notably, because utility curves differ from one person to another, you should expect different investors to select different portfolios on the efficient frontier. Two investors will not choose the same portfolio from the efficient set unless their utility curves are identical. 9. The optimal portfolio for an investor is the point of tangency between his or her set of utility curves and the efficient frontier. This will most likely be a diversified portfolio because almost all the portfolios on the frontier are diversified except for the two end points: the minimum variance portfolio and the maximum return portfolio. 10. The hypothetical graph of an efficient frontier of U.S. common stocks only will have a curved shape, expressing the trade-off between risk and expected return for the optimal combination of those assets (see the graph in the answer to Question 6). Adding U.S. bonds to the portfolio will most likely generate a new efficient frontier that is shifted up (or to the left) of the original stock-only frontier. The reason for the shift is that we expect bonds to be less correlated with © 2025 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

stocks, thereby creating additional diversification potential and resulting in portfolios that can deliver the same expected returns with less risk than before. The third frontier (which includes global securities) will likely display another shift upward or to the left for similar reasons—lower asset correlations resulting in additional risk diversification potential. 11. The portfolio constructed containing stocks L and M, which have the lowest (i.e., most negative) correlation coefficient, would have the lowest standard deviation. As demonstrated, combining assets with equal risk and return but with low positive or negative correlations will reduce the risk level of the portfolio. 12. Standard deviation would be expected to decrease with an increase in stocks in the portfolio because an increase in number will increase the probability of having lower and inversely correlated stocks. There will be a major decline from 4 to 10 stocks and a continued decline from 10 to 20 but at a slower rate. Finally, from 50 to 100 stocks, there is a further decline but at a very slow rate because almost all unsystematic risk is likely to be eliminated after the inclusion of about 20–30 stocks. 13.

The existence of a risk-free asset excludes the E–A segment of the Markowitz efficient frontier because any point below A is dominated by the RFR. In fact, the entire efficient frontier below M is dominated by points on the RFR–M Line (combinations obtained by investing a part of the portfolio in the riskfree asset and the remainder in M); for example, the point P dominates the previously efficient B because it has lower risk for the same level of return. As shown, M is at the point where the line from RFR is tangent to the old efficient frontier. The new efficient frontier thus becomes RFR–M–F, which is called the Capital Market Line (CML). 14. The CML leads all investors to invest in the same risky collection of assets, Portfolio M. The investment prescription of the CML is that investors cannot do better, on average, than when they divide their investment funds between (1) the riskless asset and (2) the market portfolio rather than 100 percent allocation to Portfolio M. If an investor desires an expected return greater than that for Portfolio M, he or she should borrow money at RFR and buy more of Portfolio M (i.e., use leverage to increase his or her allocation to the best combination of risky assets available).

ANSWERS TO PROBLEMS 1.

[E(Ri)] for Lauren Labs Possible Probability Returns 0.10 −0.20 0.15 −0.05 0.20 0.10 0.25 0.15 0.20 0.20 0.10 0.40

Expected Return −0.0200 −0.0075 0.0200 0.0375 0.0400 0.0400 E(Ri) = 0.1100, or 11.00%

Market Stock Disney Starbucks Harley Davidson Intel Walgreens TOTAL

Value $15,000 17,000 32,000 23,000 7,000 94,000

Weight 0.160 0.181 0.340 0.245 0.074 1.0000

Security Return (Ri) 0.14 −0.14 0.18 0.16 0.12

Madison Cookies (Ri)

Portfolio Return Wi  Ri 0.022 0.025 0.061 0.039 0.009 E(Rp) = 0.1064, or 10.64%

Sophie Month Electric (Rj) Ri − E(Ri) Rj − E(Rj) 1 0.07 0.06 −0.04 −0.057 2 0.06 0.043 −0.02 −0.03 3 −0.07 −0.10 −0.087 −0.11 4 0.12 0.15 0.103 0.14 5 −0.02 −0.06 −0.037 −0.07 6 0.05 0.02 0.033 0.01 Sum 0.10 0.06 3(a). E ( RMadison ) = 0.10/6 = 0.0167 E ( RSophie ) = 0.06/6 = 0.01

[Ri − E(Ri)]  [Rj − E(Rj)] −0.0034 −0.0013 0.0096 0.0144 0.0026 0.0003 0.0222

 Madison = 0.0257 / 5 = 0.0051 = 0.0717 3(b).

 Sophie = 0.04120 / 5 = 0.0082 = 0.0908 3(c).

Cov ij = 1/5 ( 0.0222 ) = 0.0044

rij =

3(d).

0.0044 (0.0717) (0.0908)

0.0044 0.006510 = 0.6758 One should have expected a positive correlation between any two stocks, as their returns are influenced by the same macroeconomic factors (e.g., inflation) and so tend to move in the same =

direction. Risk can be reduced substantially by combining assets that have low positive or negative correlations, which is not the case for Madison Cookies and Sophie Electric. E ( R1 ) = 0.15E ( 1 ) =0.10w1 = 0.5 E ( R2 ) = 0.2E ( 2 ) = 0.20w2 = 0.5 E ( Rport ) = 0.5 ( 0.15) + 0.5 ( 0.20 ) = 0.175

If r1,2 = 0.40

 p = (0.5)2 (0.10)2 + (0.5)2 (0.20)2 + 2(0.5)(0.5)(0.10)(0.20)(0.40) = 0.0025 + 0.01 + 0.004 = 0.0165 = 0.12845 If r1,2 = −0.60

 p = (0.5)2 (0.10)2 + (0.5)2 (0.20)2 + 2(0.5)(0.5)(0.10)(0.20)(−0.60) = 0.0025 + 0.01 + (−0.006) = 0.0065 = 0.08062

The portfolio with the negative correlation coefficient reduces risk by a greater amount without sacrificing return. For all values of r1,2: E ( Rport ) = ( 0.6  0.10 ) + ( 0.4  0.15 ) = 0.12, or 12.0%

and:  port = (0.6)2 (0.03)2 + (0.4)2 (0.05)2 + 2(0.6)(0.4)(0.03)(0.05)(r1,2 ) = 0.000324 + 0.0004 + 0.00072(r1,2 ) = 0.000724 + 0.00072(r1,2 )

5(a).

so, 0.000724 + 0.00072(1.0) = 0.001444 = 0.0380

5(b).

0.000724 + 0.00072(0.75) = 0.001264 = 0.0356

5(c).

0.000724 + 0.00072(0.25) = 0.000904 = 0.0301

5(d).

0.000724 + 0.00072(0.00) = 0.000724 = 0.0269

5(e).

0.000724 + 0.00072(−0.25) = 0.000544 = 0.0233

5(f).

0.000724 + 0.00072(−0.75) = 0.000184 = 0.0136

5(g).

0.000724 + 0.00072(−1.0) = 0.000004 = 0.0020

6(a).

E ( Rp ) = (1.00  0.12 ) + ( 0.00  0.16 ) = 0.12

 p = (1.00)2 (0.04)2 + (0.00)2 (0.06)2 + 2(1.00)(0.00)(0.04)(0.06)(0.70) 6(b).

= 0.0016 + 0 + 0 = 0.0016 = 0.04 E ( Rp ) = ( 0.75  0.12 ) + ( 0.25  0.16 ) = 0.13

 p = (0.75)2 (0.04)2 + (0.25)2 (0.06)2 + 2(0.75)(0.25)(0.04)(0.06)(0.70) 6(c).

= 0.0009 + 0.000225 + 0.00063 = 0.001755 = 0.0419 E ( Rp ) = ( 0.50  0.12 ) + ( 0.50  0.16 ) = 0.14

 p = (0.50)2 (0.04)2 + (0.50)2 (0.06)2 + 2(0.50)(0.50)(0.04)(0.06)(0.70) 6(d).

= 0.0004 + 0.0009 + 0.00084 = 0.00214 = 0.0463 E ( Rp ) = ( 0.25  0.12 ) + ( 0.75  0.16 ) = 0.15

 p = (0.25)2 (0.04)2 + (0.75)2 (0.06)2 + 2(0.25)(0.75)(0.04)(0.06)(0.70) 6(e).

= 0.0001 + 0.002025 + 0.00063 = 0.002755 = 0.0525 E ( Rp ) = ( 0.05  0.12 ) + ( 0.95  0.16 ) = 0.158

 p = (0.50)2 (0.04)2 + (0.95)2 (0.06)2 + 2(0.05)(0.95)(0.04)(0.06)(0.70) = 0.000004 + 0.003249 + 0.00015960 = 0.0034126 = 0.0584

Changing the correlation coefficient does not affect the expected return of any portfolio. Also,

the standard deviation of a portfolio with a 100% allocation to a single stock would not change with a different correlation value. In all other cases, the portfolio standard deviation will decline as the correlation coefficient becomes smaller (i.e., less positive or more negative) due to increased benefits from diversification. Month

DJIA (R1)

S&P (R2)

Russell (R3)

Nikkei (R4)

R1 − E(R1)

R2 − E(R2)

R3 − E(R3)

R4 − E(R4)

0.03

0.02

0.04

0.01667

0.00333

0.01333

0.00833

0.07

0.06

0.10

−0.02

0.05667

0.04333

0.07333

−0.05167

−0.02

−0.01

−0.04

0.07

−0.0333 7

−0.0266 7

−0.0666

0.03883

−0.0033

0.01333

0.00333

−0.01167

3 4

0.01

0.03

0.02 3

7(a).

0.05

0.04

0.11

0.02

0.03667

0.02333

0.08333

−0.01167

−0.06 4

−0.0

−0.08

0.06

−0.0733 7

−0.0566

−0.10667

0.02833

0.08

0.10

Sum

3 0.16

0.19

0.08 0.10 = 0.01333 E (R2 ) = = 0.01667 6 6 0.16 0.19 E (R3 ) = = 0.02667 E (R4 ) = = 0.03167 6 6 2 2 2 2 2 2 1 = ( 0.01667 ) + ( 0.05667 ) + ( −0.03333 ) + ( −0.00333 ) + ( 0.03667 ) + ( −0.07333)

E (R1 ) =

7(b).

= 0.00028 + 0.00321 + 0.00111 + 0.00001 + 0.00134 + 0.00538 = 0.01133

 12 = 0.01133/5 = 0.00226  1 = ( 0.00226 )

1/2

= 0.0476

2 = ( −0.00333) + ( 0.04333) + ( −0.02667) + ( 0.01333) + ( 0.02333) + ( −0.05667 ) 2

= 0.00001 + 0.00188 + 0.00071 + 0.00018 + 0.00054 + 0.00321 = 0.00653

 22 = 0.00653/5 = 0.01306  2 = ( 0.01306 )

1/2

= 0.0361

3 = ( 0.01333) + ( 0.07333) + ( −0.06667) + ( 0.00333) + ( 0.08333) + ( −0.106672 ) 2

= 0.00018 + 0.00538 + 0.00444 + 0.00001 + 0.00694 + 0.01138 = 0.02833

 32 = 0.02833/5 = 0.00567  3 = ( 0.00567 )

1/2

= 0.0753

4 = ( 0.00833) + ( −0.05167 ) + ( 0.03833) + ( −0.01167) + ( −0.01167) + ( 0.02833) 2

= 0.00007 + 0.00267 + 0.00147 + 0.00014 + 0.00014 + 0.00080 = 0.000529 = 0.00007 + 0.00267 + 0.00147 + 0.00014 + 0.00014 + 0.00080 = 0.000529

 42 = 0.00529/5 = 0.001058  4 = ( 0.001058 )

1/2

= 0.0325

0.00006 + 0.00246 + 0.00089 − 0.00004 + 0.00086 + 0.00416 5 = 0.00839/5 = 0.001678

Cov1,2 =

0.00004 + 0.00318 + 0.00178 + 0.00004 + 0.00194 + 0.00604 5 = 0.01302/5 = 0.002604

Cov 2,3 =

7(c).

0.00003 − 0.00224 − 0.00102 − 0.00016 − 0.00027 − 0.00161 5 = −0.00527/5 = −0.001054

Cov 2,4 =

0.00011 − 0.00379 − 0.00256 − 0.00004 − 0.00097 − 0.00302 5 = −0.01027/5 = −0.002054 Correlation equals the covariance divided by each standard deviation.

Cov 3,4 =

7(d).

Correlation (DJIA, S&P ) = 0.001678/ ( 0.0476 )( 0.0361)  = 0.9765 Correlation ( S&P, R2000 ) = 0.002604/ ( 0.0361 )( 0.0753)  = 0.9579 Correlation ( S&P, Nikkei) = − 0.001054/ ( 0.0361)( 0.0325)  = −0.8984 Correlation (R2000, Nikkei) = −0.002054/ ( 0.0753)( 0.0325)  = − 0.8393

7(e).

 2,3 = (0.5)2 (0.0361)2 + (0.5)2 (0.0753)2 + 2 ( 0.5)( 0.5)( 0.002604 ) = 0.05518 E ( R )2,3 = ( 0.5)( 0.01667 ) + ( 0.5 )( 0.02667 ) = 0.02167

 2,4 = (0.5)2 (0.0361)2 + (0.5)2 (0.0325)2 + 2 ( 0.5)( 0.5)( −0.001054 ) = 0.00794 E ( R )2,4 = ( 0.5)( 0.01667 ) + ( 0.5 )( 0.03167 ) = 0.02417

The resulting correlation coefficients suggest a strong positive correlation in returns for the

8. 9a.

S&P 500 and the Russell 2000 combinations (0.96), which prevents any meaningful reduction in risk (0.05518) when they are combined. Because the S&P 500 and Nikkei have a negative correlation (−0.90), their combination results in a lower standard deviation (0.00794). Cov i, j 100 100 ri, j = = = = 0.3759  i j 19  14 266 E ( Rproposed ) = ( 0.5)( 0.086 ) + ( 0.3)( 0.056 ) + ( 0.2 )( 0.071) = 0.0598 = 7.40%

9b.

 2proposed = ( 0.5 ) ( 0.152 ) + ( 0.3 ) ( 0.0086 ) + ( 0.2 ) ( 0.117 ) 







+ 2 ( 0.5 )( 0.3 )( 0.152 )( 0.086 )( 0.2 )  + 2 ( 0.5 )( 0.2 )( 0.152 )( 0.117 )( 0.6 ) 



+ 2 ( 0.3 )( 0.2 )( 0.086 )( 0.117 )( 0.1 ) 

 proposed = ( 0.01002834 )

1/2

9c.

= 0.10014162 = 10.01%

Risk premium for current allocation = 7.40 – 3.1 /10.37 = 0.415

Risk premium for proposed allocation = 7.40 − 3.1 /10.01 = 0.430

9d.

The proposed allocation portfolio most likely falls on the Markowitz efficient frontier because it offers investors the best combination of risk and return. The current allocation does not match the 0.430 units of expected risk premium per unit of risk.

10a.

10b.

Q: 4.8%/10.5% = 0.4571 R: 7%/14% = 0.5000 S: 1.6%/5% = 0.3200 T: 8.7%/18.5% = 0.4703 U: 3.2%/7.5% = 0.4267 The CML slope, E ( RMKT ) − RFR  / MKT , is the ratio of risk premium per unit of risk. Portfolio

10c.

R has the highest ratio, 0.5000, of these five portfolios, so it is most likely the market portfolio. Thus, the slope of the CML is 0.5 and its intercept is 3%, which is the risk-free rate. The CML will be tangent to the Markowitz efficient frontier at Portfolio M. The CML equation, based on the above analysis, is E ( Rportfolio ) = 3%+ ( 0.50 )  portfolio . If the desired standard deviation is 7.0%, then the expected portfolio return is 6.5%: E ( Rportfolio ) = 3%+ ( 0.50 )( 7% ) = 6.5% . So, it is not possible to earn an expected return of 7% with a

10d.

portfolio whose standard deviation is only 7%. Using the CML equation, we set the expected portfolio return equal to 7% and solve for the standard deviation: E ( Rportfolio ) = 7% = 3%+ ( 0.50 ) portfolio → 4% = ( 0.50 )  portfolio →  = 4%/0.50 = 8% .

Thus, 8% is the standard deviation consistent with an expected return of 7%. To find the portfolio weights that result in a risk of 8% and an expected return of 7%, recall that the covariance between the risk-free asset and the market portfolio is zero. Thus, the portfolio standard deviation calculation simplifies to  portfolio = wMKT ( MKT ) , and the weight of the risk-free asset is 1 − wMKT . Doing

10e.

this,

have  portfolio = 8% = wMKT (14.0% ) ,

so wMKT = 8%/14.0% = 0.5714

and

wrisk-free asset = 1 – 0.5714 = 0.4286 . As a check, the weighted average expected return should equal 7%: 0.5714 (10% ) + 0.4286 ( 3% ) = 7.0% , which it does. Remember to use the expected return of the market portfolio, 10%, in this calculation. To find the portfolio weights that result in a risk of 18.2%, recall that the covariance

between the risk-free asset and the market portfolio is zero. Thus, the portfolio standard deviation calculation simplifies to  portfolio = wMKT ( MKT ) , and the weight of the risk-free asset is 1 − wMKT . Doing

this,

have  portfolio = 18.2% = wMKT (14.0% ) ,

so wMKT = 18.2%/14.0% = 1.30 ;

wrisk-free asset =1 – (1.3) = − 0.30 . This portfolio is a borrowing portfolio; 30% of the funds will be

borrowed (we will use margin), and 130% of the initial funds are invested in the market portfolio. The expected return will be the weighted average of the risk-free and market portfolio returns: 1.30 (10% ) + ( −0.30 )( 3% ) = 12.1%

We can also use the CML equation to find the expected return: E ( Rportfolio ) = 3% + ( 0.50 )  portfolio = 3% + ( 0.50 )(18.2% ) = 12.1%. Thus, both methods generate the same value for the expected portfolio return.

APPENDIX 6: ANSWERS TO PROBLEMS Appendix A 1(a). When ( 1 ) = ( 2 ) , the problem can be solved by substitution,

( 1 ) − ( r1,2 ) ( 1 )( 1 ) w1 = 2 2 ( 1 ) + ( 1 ) − 2 ( r1,2 ) ( 1 )( 1 ) 2

so that:

( 1 ) 1 − ( r1,2 )  w1 = 2 2 2 ( 1 ) − 2 ( r1,2 ) ( 1 ) 2 ( 1 ) 1 − ( r1,2 ) = 2 2 ( 1 ) 1 − ( r1,2 )  2

= ½ = 0.5

Appendix B If r12 = −1.0, the variance of a portfolio is zero when:

w1 =

( 2 ) = 0.06 = 0.6 ( 1 ) + ( 1 ) 0.04 + 0.06

so, w2 = 1 – w1 = 1 – 0.6 = 0.4

Solution and Answer Guide

ANSWERS TO QUESTIONS 1. In a capital asset pricing model (CAPM) world, the relevant risk variable is the security’s systematic risk, which is based on its covariance of returns with all other risky assets in the market. This risk cannot be eliminated through diversification. The unsystematic risk is not relevant because it can be eliminated through diversification—for instance, when you hold a large number of securities, the poor performance of some companies will be offset by the good performance of others. 2. Similarities: They both establish a linear relationship between risk and expected return, with the zero-risk expected return being the risk-free rate. Differences: First, the CML measures risk by the standard deviation (i.e., total risk) of the investment, while the SML explicitly considers only the systematic component of an investment’s volatility (i.e., beta). Second, as a consequence of the first point, the CML can only be applied to portfolio holdings that are already fully diversified, whereas the SML can be applied to any individual asset or collection of assets. On the CML, the lowest risk portfolio is 100 percent invested in the risk-free asset, with a standard deviation of zero. The SML deals primarily with security or asset risk. Security risk is measured by the asset’s systematic risk, or beta. Beta can be negative (if the asset’s returns and market returns are negatively correlated) such that the SML extends to the left of the vertical (expected return) axis. 3. The following are criticisms of beta as used in CAPM: a. Theory does not measure up to practice. In theory, a security with a zero beta should give a return exactly equal to the risk-free rate. But actual results do not come out that way, which implies that the market values something besides a beta measure of risk. b. A positive relationship between beta and realized returns does not always hold, particularly over short time horizons. Empirical evidence has shown that (1) in some short periods, investors may be penalized for taking on more risk; (2) in the long run, investors are not rewarded enough for high risk and are overcompensated for buying securities with low risk; and (3) in all periods, some unsystematic risk is being valued by the market. c. Estimated betas can be unstable. Major changes in a company that affect the character of the stock or some unforeseen event not reflected in past returns may decisively affect the

security’s future returns. d. Beta can be altered by changing the market index against which it is measured. That is, one can obtain quite different measures of the risk level of individual stocks and portfolios by selecting a different proxy for the market portfolio. As a result, one would make different predictions about the expected returns, and by changing indexes, one could change the riskadjusted performance ranking of a manager. 4(a).

The concepts are explained as follows: The Foundation’s portfolio currently holds a number of securities from two asset classes. Each of the individual securities has its own risk (and return) characteristics, described as unsystematic (or specific) risk. By including a sufficiently large number of holdings, the unsystematic risk of the individual holdings offset each other, diversifying away much of the overall unsystematic risk and leaving mostly nondiversifiable or market-related risk. Systematic risk is a market-related risk that cannot be diversified away. Because systematic risk cannot be diversified away, investors expect to be rewarded for assuming this risk. The variance of an individual security is the sum of the probability-weighted average of the squared differences between the security’s expected return and its possible returns. The standard deviation is the square root of the variance. Both variance and standard deviation measure total risk, including both systematic and unsystematic risk. Assuming the rates of return are normally distributed, the likelihood for a range of rates may be expressed using standard deviations. For example, 68 percent of potential returns can be expected to fall within + or −1 standard deviation of the mean, and 95 percent of returns can be expected to fall within 2 standard deviations of the mean. Covariance measures the extent to which two securities tend to move, or not move, together. The level of covariance is expressed by the degree of correlation between the securities (the correlation coefficient) as well as by each security’s standard deviation. As long as the correlation coefficient is less than 1, the portfolio standard deviation will be less than the weighted average of the individual securities’ standard deviations. The lower the correlation, the lower the covariance and the greater the diversification benefits (negative correlations provide more diversification benefits than positive correlations). The capital asset pricing model (CAPM) asserts that investors will hold only fully diversified portfolios and so beta measures the systematic risk of an individual security or portfolio relative to the market portfolio. By definition, the market itself has a beta of 1.0. Portfolios with betas greater than 1.0 have systematic risk higher than that of the market; portfolios with betas less than 1.0 have lower systematic risk. By adding securities with betas that are higher (lower), the systematic risk (beta) of the portfolio can be increased (decreased) as desired.

4(b).

Without performing the calculations, one can see that the portfolio return would increase because (1) real estate has an expected return equal to that of stocks and (2) its expected return is higher than the return on bonds. The addition of real estate would result in a reduction of overall portfolio risk because (1) the standard deviation of real estate is less than that of both stocks and bonds and (2) the correlation of real estate with both stocks and bonds is negative. The addition of an asset class that is not perfectly correlated with existing assets will reduce variance. The fact that real estate has a negative correlation with the existing asset classes will reduce risk even more.

4(c).

Capital market theory holds that in efficient markets, mispricing of assets is unlikely to occur meaning that the expected return will be proportional to the level of risk taken. In this instance, real estate is forecast to provide the same return as stocks and a higher return than bonds. Yet, it is projected to have a lower level of risk than both bonds and stocks. If these expectations were realistic, investors would sell the other asset classes and buy real estate to push down its return until it was proportionate to the level of risk. Appraised values differ from transaction prices, which reduces the accuracy of return and volatility measures for real estate. Capital market theory was developed and applied to the stock market, which is a very liquid market with relatively small transaction costs. In contrast to the stock market, real estate markets have high transaction costs and lack liquidity.

The hypothetical market portfolio contains all risky assets available and positively priced by investors. The weights for all risky assets are equal to their relative market value. The typical proxy for the market portfolio is stock market indexes such as the S&P 500 or Russell 3000. A misspecified proxy for the market portfolio can have two effects. First, the beta computed for alternative proxies could differ substantially, leading to an incorrect assessment of the asset’s true risk level. Second, the SML derived would be wrong because it goes from the RFR through the improperly specified market portfolio. In general, when comparing the performance of a portfolio manager to the “benchmark” portfolio, these errors will tend to misstate the performance of portfolio managers.

Both the capital asset pricing model and the arbitrage pricing model rest on the assumption that investors are rewarded with nonzero return for undertaking two activities: (1) committing capital (nonzero investment) and (2) taking risk. The main difference between the two theories is the way systematic risk is defined: a single market-wide beta factor for the CAPM versus several, unspecified risk exposures for the APT. If an investor could earn a positive return for no investment and no risk, then it should be possible for all investors to do the same. The subsequent trading that would result (i.e., buying undervalued securities and selling overvalued ones) would cause prices to adjust to their correct levels, thereby eliminating the source of the arbitrage (or “something for nothing”) return. In either model, superior performance could be established by actual returns that are in excess of the expected return predicted by the model, or alpha. This would be the return not explained by the risk factors in the respective model.

7a.

The capital asset pricing model (CAPM) is an equilibrium asset pricing theory showing that equilibrium rates of expected return on all risky assets are a function of their covariance with the market portfolio. The CAPM is a single-index model that defines systematic risk in relation to a broad-based market portfolio (i.e., the market index). This single factor (beta) is unchanging:

R j = R f + j ( Rm – R f ) where

Rj Rf Rm

j

= expected return on an asset or portfolio = risk-free rate of return = expected return on the market = volatility of the asset or portfolio to that of the market m.

Arbitrage pricing theory (APT) is an equilibrium asset pricing theory derived from a factor model by using diversification and arbitrage. The APT shows that the expected return on any risky asset is a linear combination of various factors. That is, the APT asserts that an asset’s riskiness and, hence, its average long-term return, is directly related to its sensitivities to certain factors. Thus, the APT is a multifactor model that allows for as many factors as are important in the pricing of assets. However, the model itself does not define these variables. Unlike the CAPM, which recognizes only one unchanging factor, the key factors in APT can change over time. R j = R f + j 1 ( RF1 − R f ) + … + jk ( RFk − R f )

where Rj = expected return on an asset Rf = risk-free rate of return jk = sensitivity of asset j to a particular factor k RFk = expected return on a portfolio with an average (1.0) sensitivity to a factor k Research suggests that several macroeconomic factors may be significant in explaining expected stock returns (i.e., these factors are systematically priced): (1) Inflation (2) Industrial production (3) Risk premia as measured by the spread between low- and high-grade bonds (4) Yield curve (i.e., slope of the term structure of interest rates) (5) Real GNP growth (6) Rate of growth of real oil prices (i.e., an energy factor) (7) Real defense spending (8) Market index 7b.

Because of APT’s more general formulation, it is more robust and intuitively appealing than the CAPM. Many factors, not just the market portfolio, may explain asset returns. This permits the stock selection process to take into account as many economic variables believed to be significant in a valuation context—not only to individual issues but also to groups, sectors, or even the market as a whole. For example, given a forecast of a sudden spurt in the inflation rate, and of the resulting effect on interest rates, the analyst or portfolio manager can, via APT, arrive at an estimate of valuation changes across the investable universe and adjust portfolio exposures accordingly.

The small firm effect refers to the tendency of small capitalization stocks to outperform large capitalization stocks. In and of itself, such evidence would not necessarily constitute evidence of market inefficiency because most tests of such “anomalies” are in fact joint tests of two elements: (1) an asset pricing model (for example, the CAPM or APT), which reflects the risk-return trade-off and (2) market efficiency, in the sense of prices reflecting some set of information. If a test fails to account for such anomalies, it could be due to (1) a misspecified asset pricing model, (2) market inefficiency, or (3) both.

A market factor of 1.2 means the mutual fund has a systematic risk level 20 percent greater than that of the market portfolio, all other factors held equal. The SMB (“small minus big”) factor is the return of a portfolio of small capitalization stocks minus the return to a portfolio of large capitalization stocks. A value of −0.3 indicates the fund tends to react negatively to small cap factors; thus, the fund is primarily invested in large cap stocks. The HML (high minus low) factor is the return to a portfolio of stocks with high book-to-market

ratios (i.e., value stocks) less the return to a portfolio of low book-to-market ratios (i.e., growth stocks). A value of 1.4 indicates the fund reacts positively to this factor; thus, the fund is weighted toward value stocks. 10.

The value of a stock can be viewed as the present value of its expected future cash flows discounted at some discount rate reflecting risk. Anticipated economic conditions are already incorporated in the valuation process. Unanticipated economic conditions can therefore affect valuations by altering expected future cash flows or changing risk levels (i.e., discount rates). Industrial production. Industrial production is related to cash flows in the traditional valuation process. The relative performance of a portfolio sensitive to unanticipated changes in industrial production should move in the same direction as the change in this factor. Portfolios sensitive to unanticipated changes in industrial production should be compensated for the exposure to this economic factor. Inflation. Inflation is reflected primarily in the discount rate, which generally is as follows: k = real return + hedge for inflation + risk premium. Unexpected inflation will quickly be factored into k by the market as investors attempt to hedge the loss of purchasing power. Thus, the discount rate would move in the same direction as the change in inflation. High inflation could also affect cash flows, but the effect will probably be more than offset by higher discount rates. Nominal cash flow growth rates do not always match expected inflation rates. If the growth in nominal cash flows lags the inflation rate, the relative performance of a portfolio sensitive to rising inflation should decline over time. Hence, higher unanticipated inflation will negatively affect portfolio values. Risk premia or quality spreads. Risk premia affect the magnitude of the discount rate in the valuation process. The risk premium measure represents investor attitudes toward riskbearing and perceptions about the general level of uncertainty. When the return on lowversus high-quality bonds widens, there is likely to be a negative impact on the values of stock, particularly for lower-quality companies. Riskier cash flows require higher discount rates, and widening quality spreads often signal greater uncertainty about profit levels and debt service requirements. Term structure. Changes in the term structure affect the discount rate. For example, if a parallel upward shift occurs, this means investors are requiring higher returns to hold all assets. Hence, with high discount rates, portfolio values would fall. If the curve becomes steeper, longer-duration assets, such as growth-oriented stocks, would be negatively affected more than shorter-term assets. Aggregate consumption. Aggregate consumption will likely affect cash flows in the valuation model. Consumer spending comprises about two-thirds of GDP in the US economy and similar levels overseas. Increased (decreased) levels of spending by consumers will likely lead to increases (decreases) in economic activity and corporate earnings. Oil prices. Oil prices will likely affect cash flows in the valuation formula. Higher oil prices will lead to higher inflation, as oil is a cost component for many items, including the cost of transportation of most goods we purchase, and thus may indirectly affect interest rates. However, a larger effect is that consumers seeing higher oil prices may reduce spending and adjust their budgets, thus cutting back on other types of consumption spending.

11.

The macroeconomic approach to identifying the factors in a multifactor asset pricing model tries to find variables that explain the underlying reasons for variations in the cash flows and investment returns over time (for example, unanticipated changes in industrial

production, inflation, yield spreads). The microeconomic approach concentrates on relevant characteristics of the securities themselves (for example, firm size and ratio of book-tomarket value) that are thought to be left out of the single, market-wide beta risk factor of the CAPM. Conceptually, it is possible for the two approaches to lead to the same estimate for expected returns. Practically speaking, it is not likely that they will. The two sets of factors, while probably significantly correlated, are not likely to be perfectly correlated, leading to differences in estimates.

ANSWERS TO PROBLEMS 1.

First, note that the Nominal RFR = Real RFR + E (Inflation ) . So, the SML in (a) would be a straight line intersecting the vertical (E(Ri)) axis at 0.06 and connecting through the point representing E ( Rm ) = 0.12 for  m = 1.0 . In (b), a change in expected inflation would lead to a change in the risk-free rate that, with other things being equal, would result in a new SMLb, which would intercept with the vertical axis at the new risk-free rate (0.09) and would be parallel in the original SMLa. In (c), this indicates that not only did the risk-free rate change from 0.06 to 0.09, but the market risk premium E ( Rm ) − R f  also changed from 0.06 (0.12−0.06) to 0.08 (0.17−0.09). Therefore, the new SMLc will have an intercept at 0.09 and a different slope, so it will no longer be parallel to SMLa. E ( Ri ) = RFR +  i ( E ( RM ) RFR ) = 0.10 +  i ( 0.14 − 0.10 )

= 0.10 + 0.04  i

2a.

Stock

Beta

(Required Return)

E ( Ri ) = 0.10 + 0.04βi

0.85

0.10 + 0.04 ( 0.85 )

= 0.10 + 0.034 = 0.134

1.25

0.10 + 0.04 (1.25 )

= 0.10 + 0.05 = 0.150

−0.20

0.10 + 0.04 ( −0.20 )

= 0.10 − 0.008 = 0.092

Stock

Current Price

2b. Expected Price

Expected Dividend

Forecasted Return

0.75

24 − 22 + 0.75 = 0.1250 22

2.00

51 − 48 + 2.00 = 0.1042 48

1.25

40 − 37 + 1.25 = 0.1149 37

Stock Beta Required Forecasted Evaluation U 0.85 0.134 0.1250 Overvalued N 1.25 0.150 0.1042 Overvalued D 0.092 0.1149 Undervalued −0.20 If you believe the appropriateness of these forecasted returns, you would buy stock D and sell stocks U and N.

3(a).

With a risk premium of 5% and the risk-free rate of 4.5%, the security market line is: E (return) = 4.5% + ( 5% )  . Information about the level of diversification of the portfolios is not given, nor is information about the market portfolio. But a portfolio’s beta is the weighted average of the betas of the securities held in the portfolio, so the SML can be used to evaluate managers Y and Z. Expected return ( Y ) = 4.5% + ( 5% )  = 4.5% + ( 5% )(1.20 ) = 10.50% . Expected return ( Z ) = 4.5% + ( 5% )  = 4.5% + ( 5% )( 0.80 ) = 8.50% .

3(b).

Alpha is the difference between the actual return and the expected return based on portfolio risk: Alpha of manager Y = actual return – expected return = 10.20% − 10.50% = −0.30%

3(c).

Alpha of manager Z = actual return – expected return = 8.80% − 8.50% = 0.30% A positive alpha means the portfolio outperformed the market on a risk-adjusted basis; it would plot above the SML. A negative alpha means the opposite, which is that the portfolio underperformed the market on a risk-adjusted basis; it would plot below the SML. In this case, manager Z outperformed the market portfolio on a risk-adjusted basis by 30 basis points (0.30%). Manager Y underperformed, returning 30 basis points less (−0.30%) than expected based on the risk of Y’s portfolio. Since both managers’ performances are being compared to the same model of expected returns, we can conclude that manager Z outperformed manager Y on a risk-adjusted basis even though the actual level of the return was lower.

4(a). Bi =

Cov i,m



2 m

and ri,m =

Cov i,m

( i )( m )

then Cov i,m = ( ri,m ) ( i )( m ) For Intel:

Cov i,m = ( 0.72 )( 0.1210 )( 0.0550 ) = 0.00479

0.00479 0.00479 = = 1.583 (0.055)2 0.003025 For Ford: Beta =

Cov i,m = ( 0.33)( 0.1460 )( 0.0550 ) = 0.00265 0.00265 = 0.876 0.003025 For Coca-Cola: Beta =

Cov i,m = ( 0.55)( 0.0760 )( 0.0550 ) = 0.00230 Beta =

0.00230 = 0.760 0.003025

For Merck: Cov i,m = ( 0.60 )( 0.1020 )( 0.0550 ) = 0.00337 0.00337 = 1.114 0.003025 E ( Ri ) = RFR + Bi ( RM − RFR )

Beta =

= 0.08 + Bi ( 0.15 − 0.08 )

4(b).

4(c).

= 0.08 + 0.07Bi Stock

Beta

E ( Ri ) = 0.08 + 0.07Bi

Intel Ford Anheuser-Busch Merck

1.583 0.876 0.760 1.114

0.1908 0.1413 0.1332 0.1580

Intel, Ford, and Coca-Cola all have forecasted returns (given in part c) exceeding their expected returns (computed in part b); they are undervalued and are potential “buy” candidates. Merck is overvalued, as its estimated return (10%) is less than the return required by the SML (15.8%); it is a potential candidate for selling. Year

Chelle Rtn

Index Rtn

Ch—Avg

(Ch— Avg)^2

Ind—Avg

(Ind— Avg)^2

1 2 3 4 5 6 Sum Average:

37 9 −11 8 11 4 58 9.67

15 13 14 −9 12 9 54 9.00

27.33 −0.67 −20.67 −1.67 1.33 −5.67

747.11 0.44 427.11 2.78 1.78 32.11 1211.33

6.00 4.00 5.00 −18.00 3.00 0.00

36.00 16.00 25.00 324.00 9.00 0.00 410

Var1 =

1211.33 = 242.267 5

VarM =

(Ch-Avg)  (InxAvg) 164.00 −2.67 −103.33 30.00 4.00 0.00 92

410 = 82.00 5

 1 = 242.267 = 15.5649

 M = 82 = 9.0554

92.00 = 18.40 5 The correlation coefficient can be computed as follows: Cov1,M =

5(a).

r1,M =

5(b). 5(c).

Cov1,M

18.40

= 0.13

Cov1,M

18.40 = 0.2244 VarM 82.00 (Note: The variance and covariance calculations in this problem were performed using the (N − 1) convention for sample-based statistics.) Security Market Line i Fair-value plot. The following template shows, using the CAPM, the expected return, ER, of Fund T and Fund U on the SML. The points are consistent with the following equations: Beta1 =

6(a).

18.40

 1 M (15.5649)(9.90554) 140.9464 The standard deviations are 15.5649% for Chelle Computer and 9.0554% for Index, respectively. Beta for Chelle Computer is computed as follows: =

ER on stock = Risk-free rate + Beta  (Market return − Risk-free rate ) ER for Fund T = 3.9% + 1.2 ( 6.1%) = 11.22%

ER for Fund U = 3.9% + 0.8 ( 6.1% )

ii 6(b).

6(c).

= 8.78% Analyst estimate plot. Using the analyst’s estimates, Fund T plots below the SML and Fund U, above the SML.

Over versus Undervalued Fund T is overvalued (a potential “sell” candidate) because it should provide a 11.22% return according to the CAPM, whereas the analyst has estimated only a 9.0% return. Fund U is undervalued (a potential “buy” candidate) because it should provide an 8.8% return according to the CAPM, whereas the analyst has estimated a 10% return.

7(b).

 = Cov i,m /( m )2 From a spreadsheet program, we find for Radar Tire and the Proxy,

Cov i,m = 187.4 m2 = 190.4 Using the proxy:

using proxy = 187.4/190.4 = 0.984 Using the true index, the covariance (Radar, true index) is 176.4, so

using true = 176.4/168 = 1.050 7(c).

Using the proxy: E ( RR ) = 0.08 + 0.984 ( 0.12 − 0.08 ) = 0.08 + 0.0394 = 0.1194 or 11.94% Using the true market: E ( RR ) = 0.06 +1.05 ( 0.15 – 0.06 ) = 0.06 + 0.0945

8(a).

= 0.1545 or 15.45% Radar’s performance of 13 percent would be inferior compared to expectations based on the true market portfolio, but would outperform expectations based on the market portfolio proxy. In general, for the APT, E ( Rq ) = 0 +1 bq1 +2 bq 2

For security J: E ( RJ ) = 0.05 + 0.02  0.80 + 0.04  1.40 = 0.05 + 0.016 + 0.056 = 0.1220 or 12.20% For Security L: E ( RL ) = 0.05 + 0.02  1.60 + 0.04  2.25 = 0.05 + 0.032 + 0.09

8(b).

= 0.172 or 17.20% Total return = dividend yield + capital gain yield

For security J, the dividend yield is $0.75/$22.50 = 0.033 or 3.33% . For security J, the expected capital gain is therefore 12.20% – 3.33% = 8.87% . Therefore: The expected price for security J is $22.50  (1.0887 ) = $24.50 . For security L, the dividend yield is $0.75/$15.00 = 0.05 or 5% . For security L, the expected capital gain is therefore 17.20% − 5.00% = 12.20% . Therefore: The expected price for security L is $15.00  (1.1220 ) = $16.83 . The answer can be found using the holding period return: For security J: ( P1 − $22.50 ) + 0.75 /$22.50 = 0.1220 ; solving for P1, we obtain $24.50. For security l: ( P1 − $15) +0.75 /$15 = 0.1720 ; solving for P1, we obtain $16.83. 9 (a). Three-factor model expected excess returns:

BCD ( 0.966 )( 7.23% ) + ( −0.018 )( 2.0% ) + ( −0.388 )( 4.41% ) = 5.24% FGH (1.042 )( 7.23% ) + ( −0.043)( 2.0% ) + ( 0.37 )( 4.41% ) = 9.08% JKL (1.178 )( 7.23% ) + ( 0.526 )( 2.0% ) + ( 0.517 )( 4.41% ) = 11.85% Four-factor model expected excess returns:

BCD (1.001)( 7.23% ) + ( −0.012 )( 2.0% ) + ( −0.341)( 4.41% ) + ( 0.073)( 4.91) = 6.07% FGH (1.122 )( 7.23% ) + ( −0.031)( 2.0% ) + ( 0.478 )( 4.41% ) + ( 0.166 )( 4.91) = 10.97% JKL (1.041)( 7.23% ) + ( 0.505)( 2.0% ) + ( 0.335)( 4.41% ) + ( −0.283)( 4.91) = 8.62% 9(b). 3-factor BCD FGH JKL

Coefficients MKT 0.966 1.042 1.178

SMB −0.018 −0.043 0.526

HML −0.388 0.370 0.517

MKT SMB HML

Factor Risk Premia 30-year 80-year 7.11% 7.92% 1.50% 3.61% 5.28% 5.02%

Expected excess returns: Using factor premium from: 30-year 80-year BCD 4.79% 5.64% FGH 9.30% 9.95% JKL 11.89% 13.82% Sample calculation: BCD, using 30-year premia:

( 0.966 )( 7.11% ) + ( −0.018 )(1.50% ) + ( −0.388 )( 5.28%) = 4.79%

9(c). With the addition of the Momentum risk factor, the previous factor loadings for the four-factor model generate the following expected excess returns: 30-year 80-year BCD 5.88% 6.89% FGH 11.78% 12.80% JKL 7.67% 8.98% 9(d). The excess returns for all the stocks for both sets of factor premia appear to be reasonable, although the estimates for FGH do seem moderately large. This is partly due to the fact that we are using out-of-sample numbers to do the estimating. That is, the factor betas were estimated by regression using a five-year sample period, but the factor risk premia were estimated using data from much longer periods. 10(a) RQRS = 4.5 + 7.5  1.24 = 4.5 + 6.825 = 13.8%

RTUV = 4.5 + 7.5  0.91 = 4.5 + 6.825 = 11.325%

RWXY = 4.5 + 7.5  1.03 = 4.5 + 7.725 = 12.225%

10(b)

RQRS = 4.5 + 7.5  1.24 + ( −0.3)  ( −0.42 ) + 0.6  0.00 = 4.5 + 9.30 + 0.126 + 0.00 = 13.926% RTUV = 4.5 + 7.5  0.91+ ( −0.3)  ( 0.54 ) + 0.6  0.23 = 4.5 + 6.825 – 0.162 + 0.138 = 11.301% RWXY = 4.5 + 7.5  1.03 + ( −0.3)  ( −0.09 ) + 0.6  0.00 = 4.5 + 7.725 + 0.027 + 0.00

= 12.252% 10(c). Assuming that the factor loadings are significant, the three-factor model should be more useful to the extent that the non-market factors pick up movements in returns not captured by the market return. To be practical, however, the differences in the expected returns are small. 10(d). Because the factor loadings on MACRO2 are zero for two of the stocks, it appears that MACRO2 is not a systematic factor; in other words, it is one that generally affects all stocks. It may represent an industry- or firm-specific risk exposure.

11(a)

E ( RD ) = 5.0 +1.21 +3.42 = 13.1% E ( RE ) = 5.0 +2.61 +2.62 = 15.4%

Solving the second equation for 1 in terms of 2, we get:

1 = (10.4 – 2.62 )/2.6 = (4.0 − 1.02 ) Substituting that into the first equation:

1.2(4 – 2 ) + 3.42 = 8.1 Solving for 2, we find 2 = 1.5. Using this value, we can determine from either equation that 1 is equal to 2.5. 11(b). Because neither stock pays a dividend, the total return is all due to price appreciation. Therefore, for stock D: P0  (1.131) = $55 P0 = $55/1.131 = $48.63 And for stock E: P0  (1.154 ) = $36 P0 = $36/1.154 = $31.20 11(c). From part (a), the risk premium for factor 1 was 2.5%. The new risk factor is thus 2.5% + 0.25%, or 2.75%. The new expected returns are as follows: E ( RD ) = 5.0 + (1.2  2.75 ) + ( 3.4  1.5 ) = 5.0 + 3.3 + 5.1 = 13.4%

E ( RE ) = 5.0 + ( 2.6  2.75 ) + ( 2.6  1.5 ) = 5.0 + 7.15 + 3.9 = 16.05% D: PD0 (1 + 0.134 ) = $55

11(d).

PD0 = $55/1.134 PD0 = $48.50

E: PE0 (1 + 0.1605 ) = $36 PE0 = $36/ (1.1605) PE0 = $31.02 12(a). Because no stock pays a dividend, all return is due to price appreciation. E ( RA ) = 1.1  0.04 + 0.8  0.02 = 0.044 + 0.016 = 0.06 or 6%

E (Price A ) = $30 (1.06 ) = $31.80

E ( RB ) = 0.7  0.04 + 0.6  0.02 = 0.28 + 0.012 = 0.04 or 4%

E (Price B ) = $30 (1.04 ) = $31.20 E ( RC ) = 0.3  0.04 + 0.4  0.02 = 0.12 + 0.008 = 0.02 or 2%

E (Price C ) = $30 (1.02 ) = $30.60 12(b). In order to create a riskless arbitrage investment, an investor would short one share of A and one share of C and buy two shares of B. The weights of this portfolio are WA = −0.5, WB = +1.0, and WC = −0.5 . The net investment is: Short one share of A = +$30 Buy two shares of B = −$60 Short one share of C = +$30 Net investment = $0 The risk exposure is as follows: Risk Exposure Factor 1 Factor 2 A ( −0.5)  0.8 ( −0.5)  1.1 C

( +1.0 )  0.7 ( −0.5)  0.3

( +1.0 )  0.6 ( −0.5)  0.4

Net Risk Exposure

At the end of the period, the profit is given by Profit = ( $30 − $31.50 ) +2  ( $35 – 30 ) + ( $30 − $30.50 ) = −$1.50+$10 − $0.50 = $8

13. 13(a). Using a spreadsheet program, we compute the following: Monthly: Average Std Dev

Portfolio A

Portfolio B

Factor 1

Factor 2

Factor 3

1.871 5.486

1.389 4.642

1.148 5.305

0.035 6.867

−1.287 4.970

Annual: Average 22.448 16.664 13.780 0.420 −15.440 Std Dev 19.006 16.079 18.376 23.789 17.217 13(b). No, it is not clear. Portfolio A earned a higher return than Portfolio B but also had higher risk (as measured by the standard deviation). 13(c). Using a spreadsheet program, we obtain correlation between 1&2: 0.2207

correlation between 1&3: −0.5505 correlation between 2&3: −0.7531 13(d). In theory, the correlations should be zero because we want the factors to be independent of each other. 13(e). Factors 1 and 2 are not highly correlated, but it appears that there is a significant negative correlation between factors 1 and 3 and factors 2 and 3. Statistically speaking, this leads to problems of multicollinearity, which could affect regression estimates. 14(a). Using a basic regression program, we get the following results (t-statistics given in parentheses): RA = 0.568+0.991  (Factor 1) − 0.201  (Factor 2) − 0.133  ( Factor 3) R2 = 0.971

(2.92*) (22.27*)

( −4.61*)

( −1.90 )

RB = 0.693+0.962  (Factor 1) +0.046  (Factor 2) +0.319  ( Factor 3) R2 = 0.915

(2.47*) (14.99*)

( 0.73)

( 3.14*)

*Significant at 5% level. 14(b). The R2’s in both regressions are very high (0.971 and 0.915, respectively.) This indicates that over 90% of the variation of each portfolio’s returns is correlated with variation in the three systematic risk factors. This, in turn, suggests that both funds are highly diversified. 14(c). Factor 1 is the most likely candidate for the market factor because it has a large, significant, and positive effect on both portfolios. Factors 2 and 3 have different signs in the two equations. 14(d). A positive HML factor loading indicates a value stock or a value-oriented portfolio. Portfolio B has a positive loading on this factor and therefore is the more likely candidate for the value-oriented portfolio. A negative HML factor loading indicates a growth stock or growthoriented portfolio. Portfolio A has a negative loading on this factor and therefore is more likely to be a growth-oriented portfolio.

Solution and Answer Guide

ANSWERS TO QUESTIONS 1. A fairly priced investment is one that gives a return that matches the risk. An overvalued investment is one that is so expensive that we will not receive a fair return if we buy it. An undervalued investment is so cheap that it offers a rate of return that is a greater reward than the risk that the investor has taken. 2. The top-down valuation process begins by examining the influence of the general economy on all firms and the security markets. The next step is to analyze the various industries in light of the economic environment. The final step is to select and analyze the individual firms within the superior industries and the common stocks of these firms. The top-down approach thus assumes that the first two steps (economy-market and industry) have a significant influence on the individual firm and its stock (the third step). In contrast, the bottom-up approach assumes that it is possible to select investments (i.e., firms) without considering the aggregate market and industry influences. 3. The FCFE model forecasts cash flows that belong to the equity holders. These cash flows are discounted using the cost of equity. The resulting value is the value of the equity. The free cash flow in the FCFF model is the cash flow that is available to distribute to all providers of capital (shareholders and debtholders) after estimating the cash flow generated from running the business and reinvesting in the business. The key difference is that the FCFE uses cash flow after the debtholders are already paid. With respect to the appropriate discount rate, the FCFE model uses the cost of equity. The idea is that you are discounting cash flows that belong to the equityholder and you discount them at the cost of equity and the result is the value of the equity. With the FCFF model, you use the weighted average cost of capital (WACC) for the discount rate. The idea is that the cash flow belongs to all of the providers of capital, so you must incorporate all of the costs of capital in the discount rate. When you do this, the result will be the value of the entire firm (the value of the equity plus the debt plus the preferred stock). 4. There is an important distinction between free cash flow to equity and net income. Net income is an attempt to determine how profitable a company was (or will be) for a particular period. FCFE is an attempt to determine how much cash was generated (or will be generated) for a particular period. There are many differences between the two. First, depreciation is subtracted to arrive at net income. Depreciation is not a cash flow. So, to move from net income to FCFE,

we must add the depreciation back. Second, the FCFE considers what the firm must invest in the long-term assets and short-term assets to continue in the business. This means that, after adding back depreciation, we must subtract capital expenditures (cash used to finance longterm assets) or add any cash that was received due to the sale of property, plant, and equipment. In addition, we must consider any changes to net working capital (short-term assets). Finally, we must consider any cash that is used to repay principal to debtholders or cash generated by taking on additional debt. (Remember, interest has already been subtracted in order to arrive at net income, which was our starting point to calculate FCFE.) 5. We discount nominal cash flows with nominal discount rates. Normally, models use nominal cash flows. Nominal cash flows include inflation. In other words, you forecast revenue based on how many units will be sold in the future and your estimate of the future price. Similarly, you forecast expenses that include future inflation. In the rare situation that real cash flows are used in a model, you would use a real discount rate. (This is extremely unusual.). If done correctly (and using the same assumptions), if you discount nominal cash flows with a nominal discount rate, you will receive the same answer as using real cash flows and a real discount rate. 6. The present value of the growth opportunity can be negative if the value is destroyed by earning less on capital than the capital costs. Growth is only valuable if the return on equity is greater than the cost of equity. 7. If a stock’s price is the same as its intrinsic value, then the investor’s rate of return will be the cost of equity. In other words, the stock investor will be earning a fair return. 8. The FCFE model and the FCFF model would result in the same value if certain assumptions are made. One of the assumptions is the basis of the weighted average cost of capital. For example, the weights used to calculate the weighted average cost of capital may differ from the actual amount of debt. But in order for the value of the equity to come out the same in the two models, you must take the value of the firm from the FCFF model and assume that the percentage of that value that belongs to the shareholders is the same as the percentage of the equity that is used in the WACC. 9. Differences in P/E ratios can be attributed to differences in earnings growth, the percentage of earnings that can be distributed to the shareholders, the equityholders, and the risk of the cash flows. Differences in price–sales ratios can be attributed to differences in profit margins, the percentage of earnings that can be distributed to the shareholders, the risk of the cash flows, and growth rates. Differences in price–book ratios can be attributed to differences in return on equity, risk (cost of equity), the percentage of earnings that can be distributed to the shareholders, and expectations of the growth rates of book value. 10. Advantages of relative valuation ratios: - Easy to use in that you don’t have to build a model or make a series of assumptions. - For a money manager with a mandate to be fully invested, relative value and not absolute value (intrinsic value) is what matters. - Relative value also incorporates sentiment—something that is not valued in a DCF. - Multiples are a great way to confirm the value that you have calculated with your discounted cash flow analysis. © 2025 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Disadvantages of relative valuation ratios: - Multiples are difficult to use correctly because of difficulties finding comparable companies and adjustments that need to be made. Companies use different accounting methodologies, and they often have different growth opportunities and risk profiles. - Even if multiples are used correctly, a significant assumption is the conclusion that if one stock is incorrectly priced, then it is possible that the other stocks may be mispriced.

ANSWERS TO PROBLEMS 1. Earnings per share: last year

$10.00

Dividends per share: last year

$7.00

Estimated earnings per share: this year

$11.00

Required rate of return

12%

Expected sales price at the end of the

$132.00

year 1(a).

Because the last dividend payout ratio = $7.00/$10.00 = 70%, and assuming you maintain the same payout ratio, then dividends per share at the end of the year is EPS  Payout = $11.00  70% = $7.70 .

Therefore, the present value of BBC’s share is Value =

$7.70 $132.00 + = $6.88 + $117.86 = $124.74 (1 + 0.12) (1 + 0.12)

Thus, $124.74 is the maximum price you would be willing to pay for BBC’s stock. 1(b). Earnings per share: last year

$10.00

Dividends per share: last year

$7.00

Required rate of return Expected sell price Value =

8% $110.00

$7.70 $110.00 + = $7.13 + $101.85 = $108.98 (1 + 0.08 ) (1 + 0.08 )

Thus, $108.98 is the maximum price you would be willing to pay for BBC’s stock. 2. Earnings per share: next year

$3.00

Return on equity

15%

Growth rate

Cost of equity

12%

2(a). Growth = ROE  RR, so 5% = 15%  RR, thus RR = 33.33% Intrinsic value:

3.00 (1 − 0.3333) 2.00 = = $28.57 0.12 − 0.05 0.07

2(b). No-growth value:

3.00 = $25.00 0.12

Present value: PVGO = $28.57 − $25.00 = $3.57

2(c). Growth = ROE  RR , so 3% = 15%  RR , thus RR = 20.0% and Payout ratio = 80% Forward P/E Multiple:

P /E1 = 0.80/ ( 0.12 − 0.03) = 8.89  3.

Dividend payout ratio

40%

Return on equity

16%

3(a). Growth rate = (Retention rate )  (Return on equity ) = (1 − payout ratio )  (Return on equity ) = (1 − 0.40 )  ( 0.16 ) = 0.60  0.16 = 9.6%

3(b). Sustainable growth rate = (Retention rate )  (Return on equity ) 2% = RR  16% RR = 12.5%

The company can afford to payout 1 − 0.125 = 0.875 or 87.5% 4. Required rate of return (k)

14%

Return on equity (ROE)

30%

Retention rate (RR)

90%

Earnings per share (EPS)

$5.00

Then, growth rate = RR  ROE = 0.90  0.30 = 0.27

P /E =

D /E 0.10 = k − g 0.14 − 0.27

Because the required rate of return (k) is less than the growth rate (g), the earnings multiplier cannot be used. (The answer is meaningless.) However, if ROE = 0.19 and RR = 0.60 , then growth rate = 0.60  0.19 = 0.114

𝑃/𝐸 =

0.40 0.40 = = 15.38 0.14 − 0.114 0.026

If next year’s earnings are expected to be: $5.57 = $5.00  (1 + 0.114 )

Applying the P /E: Price = (15.38 )  ( $5.57 ) = $85.69 Thus, you would be willing to pay up to $85.69 for Madison Computer Company stock. 5(a).

Projected dividends for next three years:

Year 1 ( $1.25  1.08 ) = $1.35 Year 2 ( $1.35  1.08 ) = $1.46 Year 3 ( $1.46  1.08 ) = $1.58 Required rate of return

12%

Growth rate of dividends

The present value of the stock is 1.35 1.46 1.58 40 + + + 2 3 1.12 (1.12) (1.12) (1.12)3 1.35 1.46 1.58 40 = + + + 1.12 1.2544 1.4049 1.4049 = 1.21 + 1.16 + 1.12 + 28.47 = $31.96

5(b). Growth rate

Required rate of return

12%

1.35 1.35 = = $33.75 0.12 − 0.08 0.04

5(c).

Assuming all the above assumptions remain the same, the price at the end of year 3 will be P3 =

D4 1.25  (1.08)4 1.25  1.3605 = = = $42.52 k −g 0.12 − 0.08 0.04

6. Earnings per share: last year

$5.00

Return on equity

12%

Cost of equity

12%

No-growth value:

V= 6(a).

5.00 = $41.67 0.12

No-growth value:

5.50 = $45.83 0.12

7. RFR

Market risk premium

Equity beta

0.9

Earnings per share

$2.50

Growth rate

ROE

12%

7(a). Cost of equity = k = RFR + beta (Risk premium) = 3%+ ( 0.9 )( 5% ) = 7.5% Growth = ROE  RR, so 4% = 12%  RR, thus RR = 33.33%

Intrinsic value: 2 $2.50 (1.04 )    3  = $49.52 Value = 0.075 − 0.04

7(b). No-growth value: No-growth value =

$2.50 (1.04 ) 0.075

= $34.67

NOTE: The no-growth model is based on the assumption that there will be no growth AFTER period 1. As a result, we still had to grow last year’s earnings to next year’s earnings. Then, we assumed a 100% payout. Present value: PVGO = $49.52 − $34.67 = $14.85

8. Sales per share

$18

Growth for 5 years

6.5%

Growth after 5 years

3.5%

0.9

(perpetuity) Net margins

Return on equity

14%

Cost of equity

11%

RR for 5 years

46.43%

Payout for 5 years

53.57%

RR after 5 years

25%

Payout after 5 years

75%

Year

Sales ($)

19.17

20.42

21.74

23.16

24.66

25.52

EPS

1.15

1.22

1.30

1.39

1.48

1.53

FCFE

0.62

0.66

0.70

0.74

0.79

1.15

Terminal

15.31

Value ($) PV of Cash Flows ($)

9.56 0.56

0.53

0.51

0.49

Value of Stock

$11.65

9. Assets

$300

Growth for 5 years = (450/300)1/5

8.45%

6.5%

Growth after 5 years (perpetuity)

0.9

ROA

1.2%

Return on equity

12.5%

Cost of equity

11.5%

RR for 5 years

67.60%

Payout for 5 years

32.40%

RR after 5 years

24%

Payout after 5 years

76%

Year

Assets ($)

325.35

352.84

382.66

414.99

450.00

463.50

ROA

3.90

4.23

4.59

4.98

5.40

5.56

FCFE

1.26

1.37

1.49

1.61

1.75

4.23

Terminal

49.73

Value ($) PV of Cash Flows ($)

1.13

Value of

$34.23

1.10

1.07

1.04

29.87

Stock 10. Sales (millions)

$1000

Growth for 5 years = (2,000/1,000)1/5

8.45%

6.5%

Operating margins

10%

0.9

Tax rate

28%

WACC

PP&E/Sales ($300/$1,000)

30%

11.5%

NWC/Sales ($100/$1,000)

10%

67.60%

($ millions) 1

Revenue

$1,148,698.35 $1,319,507.91 $1,515,716.57 $1,741,101.13 $2,000,000.00 $2,060,000.00

EBIT

$114,869.84

$131,950.79

$151,571.66

$174,110.11

$200,000.00

$206,000.00

NOPAT

$82,706.28

$95,004.57

$109,131.59

$125,359.28

$144,000.00

$148,320.00

Depreciation

$40,000.00

$45,947.93

$52,780.32

$60,628.66

$69,644.05

$71,733.37

OCF

$122,706.28

$140,952.50

$161,911.91

$185,987.94

$213,644.05

$220,053.37

Cap Ex + NWC

($59,479.34)

($68,323.82)

($78,483.46)

($90,153.82)

($103,559.55) ($24,000.00)

FCF

$63,226.94

$72,628.68

$83,428.45

$95,834.12

$110,084.50

Terminal Value

$196,053.37

$3,267,556.11

Cash Flow

$63,226.94

$72,628.68

$83,428.45

$95,834.12

$3,377,640.60

Discount Rate

1.09

1.1881

1.295029

1.41158161

1.538623955

$58,006.37

$61,130.11

$64,422.07

$67,891.31

$2,195,234.64

Value of Firm

$2,446,684.49

Debt

$(300,000.00)

Equity

$2,146,684.49

100,000

Per share

$21.47

Explanation: A. The growth rate is ~14.87% (which is from (2B/1B)(1/5) – 1. B. The EBIT is 10% times the revenue. C. The NOPAT is what remains from the EBIT after the 28% taxes are deducted. D. The Operating Cash Flow is the NOPAT + Depreciation. E. The NWC was 10% of revenue and the PP&E was 30%. So, 40% of incremental revenue will be used to invest in NWC and PPE. F. The FCF = OCF – Investment in NWC and Cap Ex G. The terminal Value = Yr 6 FCF/(WACC – perpetual growth rate) H. The equity value = the value of the firm minus the debt I.

The value per share = equity value/shares outstanding

11. 11a. Forward P/S Multiple: P /S1 =

Profit margin  Payout ratio 0.04  0.45 = = 0.24 k −g 0.10 − 0.025

11a. Trailing P/S Multiple: P /S0 = (1 + growth rate )  Profit margin  Payout ratio k −g

1.025  0.04  0.45 = 0.246 0.10 − 0.025

12.

12.a. The forward P/B Multiple is not forecast by analysts. 12b. Trailing P/B Multiple: P ROE  Payout = BV0 k −g P 0.16  0.75 = = 1.33 BV0 0.13 − 0.04

13. Free Cash Flow to the Firm—CSCO 2017

2018

2019

2020

Sales

$50.86

$52.89

$55.01

$57.21

$59.49

$61.73

EBIT

$16.78

$17.45

$18.15

$18.88

$19.63

$20.37

Taxes

−$3.69

−$3.84

−$3.99

−$4.15

−$4.32

−$4.48

NOPAT

$13.09

$13.61

$14.16

$14.72

$15.31

$15.89

Reinvestment

−$3.44

−$3.58

−$3.72

−$3.87

−$4.03

−$3.92

FCFF

$9.65

$10.03

$10.44

$10.85

$11.29

$11.97

Terminal Value

2021

2022

$244.38

$8.88

$202.17

Less: Debt

−$25.00

Equity Value

−$177.17

Shares Outs.

2 billion

Intrinsic Value

$35.43

$8.50

$8.14

$7.79

$168.87

Free Cash Flow to the Firm—Improved Methodology 2017

2018

2019

2020

2021

2022

Sales

$50.86

$52.89

$55.01

$57.21

$59.49

$61.73

EBIT

$16.78

$17.45

$18.15

$18.88

$19.63

$20.37

Taxes

−$3.69

−$3.84

−$3.99

−$4.15

−$4.32

−$4.48

NOPAT

$13.09

$13.61

$14.16

$14.72

$15.31

$15.89

Reinvestment

$3.44

-$3.58

−$3.72

$3.87

-$4.03

-$3.92

FCFF

$9.65

$10.03

$10.44

$10.85

$11.29

$11.97

Terminal value PV

$244.38 $8.88

$8.50

$8.14

$7.79

$168.87

$202.17

Less: Debt

−$28.88

Equity Value

$173.29

Shares Outst.

2 billion

Intrinsic Value

$34.66

Solution and Answer Guide: Frank K. Reilly, Keith, C. Brown, Sanford J. Leeds, Investment Analysis & Portfolio Management, 12th Edition, © 2025, 9780357988176; Chapter 9: The Top-Down Approach To Market, Industry, And Company Analysis

Solution and Answer Guide

FRANK K. REILLY, KEITH, C. BROWN, SANFORD J. LEEDS, INVESTMENT ANALYSIS & PORTFOLIO MANAGEMENT, 12TH EDITION, © 2025, 9780357988176; CHAPTER 9: THE TOP-DOWN APPROACH TO MARKET, INDUSTRY, AND COMPANY ANALYSIS

ANSWERS TO QUESTIONS 1. There are three problems with simply tracking the GDP data that has been released in order to make investment decisions: 1. Preliminary GDP data is released approximately one month after each quarter ends. This means that it is not timely. 2. The preliminary GDP data will be revised. Oftentimes, the revisions are meaningful. 3. The stock market moves ahead of the economy. In other words, investors are anticipating future cash flows. As a result, it is hard to identify stocks (which move ahead of the economy) by examining old economic data that may need to be revised significantly. 2. Analysts want to know the level of the federal funds rate and whether it is intended to stimulate the economy or restrict the economy (i.e., whether it is below the real neutral rate or above it). If a low federal funds rate results in lower long-term rates, this can make stocks more attractive. In addition to stocks being more attractive relative to low-yielding bonds, the value of a stock is the present value of future cash flows. Low rates mean that the present value of those distant cash flows is greater. In addition, if companies have a lower cost of capital, more projects will have a positive NPV (assuming cash flows don’t change). 3. An inverted yield curve occurs when the long-term yields are lower than short-term yields. The yield curve has inverted prior to every recession since 1970, and evidence shows that the likelihood of a recession increases with the severity of the inversion. Aside from its historical record (predicting recessions), the fundamental reasons why an inverted yield curve predicts recession are the following: (1) The yield curve usually inverts because the Fed has raised short-term rates in an attempt to slow the economy (to reduce inflation), and the longer-term yields drop due to the expectation that the Fed will eventually have to lower rates significantly (due to an eventual recession), and even the term premium that longer-term investors require will not be enough to keep the yield curve normal. (2) It is no longer profitable to lend because banks have to pay more for short-term funds, and they can charge less for longer-term loans. As bank lending decreases, the decrease in the availability of credit slows economic growth.

4. The four factors that impact operating margins for the overall market are as follows: (1) Capacity utilization (2) Unit labor costs (3) Rate of inflation (4) Foreign competition 5. When calculating the cost of equity, a required risk premium should be used, the compensation that will justify the risk of investing in equities as opposed to investing in risk-free bonds. 6. The four categories of structural (noncyclical) economic changes that an analyst should review are the following: (1) Demographics (2) Lifestyles (3) Technology (4) Politics and regulations 7. The five forces driving industry competition are as follows: (1) Threat of new entrants (2) Bargaining power of buyers (3) Threat of substitute products or services (4) Bargaining power of suppliers (5) Rivalry among existing firms 8. Growth companies are those that consistently grow sales and earnings at a rate that is faster than the overall economy. A growth stock is a stock with a higher expected rate of return than other stocks in the market with similar risk characteristics. A defensive stock’s rate of return is not expected to decline during an overall market decline, or it is expected to decline less than the overall market. A stock with low or negative systematic risk (a small positive or negative beta) may be considered a defensive stock according to this theory because its returns are unlikely to be harmed significantly in a bear market. A cyclical stock will experience changes in its rates of return greater than changes in overall market rates of return. In terms of the CAPM, these would be stocks that have high betas. 9. The firm that pursues the low-cost strategy is determined to become the low-cost producer and, hence, the cost leader in its industry. Cost advantages vary by industry and may include economies of scale, proprietary technology, or preferential access to raw materials. In order to benefit from cost leadership, the firm must command prices near the industry average, which means it must differentiate itself as well as other firms. If the firm discounts the price too much, it could erode the superior rates of return available because of its low cost. With the differentiation strategy, a firm seeks to identify itself as unique in its industry in an area that is important to buyers. Again, the possibilities for differentiation vary widely by industry. A company can attempt to differentiate itself based on its distribution system (selling in stores, by mail order, over the Internet, or door to door) or some unique marketing approach. A firm employing the differentiation strategy will enjoy above-average rates of return only if the price premium attributable to its differentiation exceeds the extra cost of being unique. 10. The growth duration model is based on two assumptions:

(1) Equal risk between the firms analyzed (2) No significant differences in the payout ratios

ANSWERS TO PROBLEMS 1. Growth rates and the implied inflation rate: a. ( $24.349 / $14.651 )

(1/12)

− 1 = 4.324%

b. $20.006 / $15.379)(1/12) − 1 = 2.216% c.

(1.04324 / 1.02216 ) − 1 = 2.06%

Link to chart: https://fred.stlouisfed.org/graph/?g=Y4bC 2. 2(a).1. The National Bureau of Economic Research has conducted extensive analysis of leading, coincident, and lagging indicators of general economic activity. Business Conditions Digest classifies economic indicators by their participation in the stage of the economic process and their relationship to business cycle movements. The leading indicators include those economic time series that usually reach peaks or troughs before the corresponding points in aggregate economic activity. The group includes 10 series. One of the 10 leading series is common stock prices, which turn lower before the economy turns down and move higher before the economy recovers. Coincident indicators turn higher and lower at close to the same time that the economy turns higher and lower. An example of this would be employees on payrolls. Lagging indicators tend to turn lower after the economy has already turned lower and they turn higher after the economy has already turned higher. These indicators confirm the evidence that we already have. An example would be the consumer price index for services (service inflation).

2(a).2. Leading indicators have historically been a good tool for anticipating the economy. Investment managers should be aware of this information and, where possible, investment decisions may reflect projected trends. However, these indicators are by no means infallible. They often generate false signals. A downturn in leading indicators may precede only a slowing of growth rather than a full-blown recession if the downturn is shallow or brief. One of the most consistent leading indicators is stock prices represented by the S&P 500 Stock Composite Index. Since stocks are a leading economic indicator, it is hard to use the leading economic indicators to predict stock movement. If stocks were a lagging indicator, it would be easier. 2(b).

Interest rate forecasts are usually important in investment management for the following reasons: Interest rates help determine the relative competitiveness of stocks versus bonds. They have an effect on the stock returns from interest rate-sensitive industries. They help determine the maturity structure of bond portfolios. They significantly affect investment in interest futures. They affect the discount rate used in various equity valuation models.

2(c).

Three economic time series, indicators, or data items that may be of key relevance to an auto analyst would be disposable personal income, consumer interest rate series, and consumer confidence survey results. An increase in disposable personal income may mean more discretionary income is available with which to purchase consumer durables such as automobiles. A decline in consumer interest rates may make the effective price of a car, including borrowing charges, more affordable. Survey results that showed a high and growing level of consumer confidence about the future of the economy would likely have a positive psychological effect on consumer willingness to commit to new, large-ticket purchases. In summary, the following variables constitute likely indicators of profits and performance in the automobile industry: Real disposable personal income per capita—to indicate affordability of automobiles Installment debt as a percent of disposable income—to indicate the ability of consumers to finance new car purchases The replacement cycle—indicating the average age of cars on the road, which suggests the inclination of consumers to consider new car purchases Trends in the cost of major components of auto production, including the prices charged by major suppliers 55 3(a). 𝑃/𝐸 = 0.09−0.04 = 0.55/0.05 = 11 3(b).

0.40

𝑃/𝐸 = 0.09−0.04 = 0.40/0.05 = 8

4. 4(a).

Growth = 0.60  0.12 = 0.072 = 7.2%

4(b).

𝑃/𝐸 = 0.10−0.072 = 14.29

0.40

4(c). The market price will rise to: Price = 14.29  $115 = $1,643.35 (or $1,642.86 if you used exact numbers from part (b).

In doing this problem, it is important to realize that you do not need to subtract depreciation. Depreciation has already been included in the operating margin. Think of the operating margin as the EBIT as a percentage of revenue. To get to EBIT, we already subtracted depreciation. $1,950  0.12 = $234.00 ( operating profit )

5(a). $234 − $58

= $176.00 (less interest )

$176  (1 − 0.32 ) = $119.68 (EPS after deducting taxes )

5(b).

Optimistic: $1,950  0.13

= $253.50 ( operating profit )

$253.50 − $58

= $195.50 (less interest )

$195.50  (1 − 0.32 ) = $132.94 (EPS after deducting taxes )

Pessimistic: $1,950  0.11

= $214.50 ( operating profit )

$214.50 − $58

= $156.50 (less interest )

$156.50  (1 − 0.32 ) = $106.42 (EPS after deducting taxes )

6. 6(a). Growth Rates:

Required Return:

Optimistic = (1 − 0.45 )  0.15 = 0.0825

Optimistic = 0.08 + 0.03 = 0.11

Pessimistic = (1 − 0.65)  0.11 = 0.0385

Pessimistic = 0.10 + 0.05 = 0.15

Consensus = (1 − 0.55 )  0.13 = 0.0585

Consensus = 0.09 + 0.04 = 0.13

P/E ratios: Optimistic = 0.45/ ( 0.11 – 0.0825) = 16.36 Pessimistic = 0.65/ ( 0.15 – 0.0385) = 5.83 Consensus = 0.55/ ( 0.13 – 0.0585) = 7.69

Price: (Note that all answers are based on P/E ratios taken out more decimal places than shown, so if you use the numbers shown, you may have some rounding error.) Optimistic = $132.94  16.36 = 2,175.38 Pessimistic = $106.42  5.83 = 620.39 Consensus = $119.68  7.69 = 920.62

Optimistic = ( 2,175.38/1000 ) − 1 = 117.54%

6(b).

Pessimistic = ( 620.39/1000 ) – 1 = −37.96% Consensus = ( 920.62/1000 ) –1 = −7.94%

Given that the consensus view is for a loss (and that probably has the highest likelihood), we may decide to slightly underweight stocks. 7. 7(a). FCFE

$80

Growth

years Growth years 4−6

Growth years 7+ =

1−3 = 9%

= 8%

Terminal value

PV of cash flows

k = 9%

Year

FCFE

$87.20

$80.00

$95.05

$80.00

$103.60

$80.00

$111.89

$79.27

$120.84

$78.54

$130.51

$139.64

$6982.24

$4,241.10 $4,638.90

Price

Sum

discounted

of cash

flows If the current value of the index were 4,000, one would overweight the U.S. equity market in the portfolio. 7(b).

A one percent increase in the rate of inflation would have two possible effects: (1) the required return would increase from 9% to 10%, decreasing the value and (2) the nominal cash flow growth rates would increase for all time periods by one percentage point. If both effects are at work, they would work to cancel each other out, as the increase in free cash flow is discounted back at an equivalently higher discount rate. In reality, compounding effects would lead to a slight increase in the estimated Index; it would rise from the above estimate to 4,678.91.

8(a).

RCara = 0.07+1.75 ( 0.15 − 0.07 ) = 0.07+0.14 = 0.21 or 21%

8(b).

RCara = 0.07+1.75 ( 0.10 − 0.07 ) = 0.07+0.0525 = 0.1225 or 12.25%

24 1 + 0.14 + 0.02 = T ln = 16 1 + 0.06 + 0.04 ln (1.50 ) = T ln (1.16/1.10 )

ln (1.50 ) = T ln (1.05455 ) T = ln (1.50 ) /ln (1.05455 ) T = 0.40547/0.05311 T = 7.635 years

10(a). Assuming the dividend yield is zero, we have the following growth duration formula: ln ( x ) = 10 ln (1.18/1.08 ) ln ( x ) = 10 ln (1.09259 ) ln ( x ) = 10 ( 0.08855) n ( x ) = 0.8855 x = 2.424

Thus, the P/E ratio would be 43.64  ( 2.424  18 ) . ln ( x ) = 5 ln (1.18/1.08 ) ln ( x ) = 5 ln (1.09259 )

10(b). ln ( x ) = 5 ( 0.08855) ln ( x ) = 0.44275 x = 1.55698

Thus, the P/E ratio would be 28.03  (1.55698  18 ) . 11(a). Company A 30 1 + 0.18 + 0.00 = T ln = 18 1 + 0.07 + 0.02 ln (1.6667 ) = T ln (1.18/1.09 )

ln (1.6667 ) = T ln (1.08257 ) T = ln (1.667 ) /ln (1.08257 ) T = 0.51083/0.07934 T = 6.44 years

Company B

27 1 + 0.15 + 0.01 = T ln = 18 1 + 0.07 + 0.02 ln (1.50 ) = T ln (1.16/1.09 )

ln (1.50 ) = T ln (1.06422 ) T = ln (1.50 ) /ln (1.06422 ) T = 0.40547/0.06224 T = 6.51 years

11(b). 30 1 + 0.18 + 0.00 = T ln = 27 1 + 0.15 + 0.01 ln (1.11) = T ln (1.18/1.16 )

ln (1.11) = T ln (1.01724 ) T = ln (1.11) /ln (1.01724 ) T = 0.10536/0.01709 T = 6.16 years

11(c). After computing the implied growth durations, the analyst must decide whether the projected growth rates can be sustained over the periods implied in the model. If the implied growth duration is greater than you believe is reasonable, you would likely advise against buying the respective stock. 12. 12(a). The dividend discount model is P=

D1 k −g

where P = value of the stock today D1 = next year’s dividend k = discount rate g = constant dividend growth rate Solving for k: ( k − g ) = D1 /P k = ( D1 /P ) + g

So k becomes the estimate for the long-term return of the stock. k = ( $0.50 / $20.00 ) +8% = 2.5%+8% = 10.5%.

12(b). Many professional investors shy away from the dividend discount framework analysis due to its many inherent complexities.

1) The model cannot be used where companies pay very small or no dividends and speculation

on the level of future dividends could be futile. (Dividend policy may be arbitrary.) In reality, we get around this by thinking about the FCFE. 2) The model presumes one can accurately forecast the long-term growth of earnings (dividends) of a company. Such forecasts become quite tenuous beyond two years. (A short-term valuation may be more pertinent.) 3) For variable growth firms, small differences in g for the first several years produce large differences in the valuations. 4) The correct k or the discount rate is difficult to estimate for a specific company as an infinite number of factors affect it, and these factors are difficult to forecast; for example, inflation, riskless rate of return, risk premium on stocks, and other uncertainties. 5) The model is not definable when g > k as with growth companies, so it is not applicable to a large number of companies. 6) Where a company has low or negative earnings per share or has a poor balance sheet, the ability to continue the dividend is questionable. 7) The components of income can differ substantially, thus reducing comparability. 12(c). Three alternative methods of valuation would include the following: 1) Price/earnings ratios 2) Price/asset value ratios (including market and book asset values) 3) Price/sales ratios 4) Liquidation or breakup value 5) Price/cash flow ratios 13. Fed Model: S&P 500 =

Next year’s earnings $130 = = 2,888.89 10-year yield 0.045

If the S&P is trading at 2,200, then the market is undervalued by

2,888.89 − 2,200 = 0.3131 or 31.1% 2,200

14. Student Exercise

15. Student Exercise

16. Student Exercise

Solution and Answer Guide: Frank K. Reilly, Keith, C. Brown, Sanford J. Leeds, Investment Analysis & Portfolio Management, 12th Edition, © 2025, 9780357988176; Chapter 10: The practice of fundamental investing

Solution and Answer Guide

ANSWERS TO QUESTIONS 1. Explanations and theories for underpricing include the following: 1. It is a way to compensate institutional investors for providing pricing information. 2. To the extent that the investor trusts the banker to systematically underprice the security based on the best available information, the investor will provide an accurate assessment of the value. 3. Bankers believe that the issuing company’s managers estimate their personal wealth based on the midpoint of the initial range. As a result, the initial range is set at a low level, knowing that the managers will feel much more pain if the range is adjusted down than they would feel joy if it were adjusted higher. 4. Bankers set prices low in order to minimize their litigation risk and/or to protect their reputational capital. 5. A bank may have an incentive to underprice the security in order to create a larger pool of profits, some of which the bank will receive back in the form of later commissions. 2. In most IPOs, there is an overallotment option. This provides an option to the investment bank to purchase 15% more shares than originally planned. As a result of this option, the bank will sell 15% more shares than planned. In effect, the bank is short 15% of the shares. If the stock increases in value, the bank will exercise the option and use those shares to cover the short position. If the stock decreases in value, the bank will not exercise the option. The bank will buy shares in the aftermarket. This aftermarket is considered to be stabilized. It is permitted by the SEC. It helps the bank to stabilize the falling stock price. 3. In a dirty auction, the issuing company could agree to sell the shares at a price below the clearing price. For example, even if $14 is the price at which 10 million shares could be sold, the stock may be issued at $13. There are bids for 18 million shares at $13 or higher. Because there are 10 million shares that have to be split between investors who placed orders for 18 million shares, every bid could be filled at 10/18 or 5/9. In other words, if you bid for 900 shares, you will receive 500 shares. This means that there should be continued demand for the stock in the aftermarket. 4. If an analyst is seen as influential within an industry, it is easier for the investment bankers in the firm to win the banking business. This means that influential analysts can demand more compensation because they help bring in more investment banking business. In addition, if an

analyst has superior access to management, this could be seen as very valuable to investors and could lead to more trading business through the firm. This would also make the analyst more valuable. 5. Management should repurchase shares when they are trading below intrinsic value and pay a dividend when the shares are trading above intrinsic value. When the stock is trading at intrinsic value, management should be indifferent between a repurchase and a dividend (although taxable investors may prefer a share repurchase). Investors should also be aware that earnings per share increases that result from share repurchases are different from increases that occur because companies have increased revenue or cut expenses. 6. Four important dates surrounding a dividend are as follows: 1. The declaration date: the date that a dividend is announced by the board of directors. 2. The date of record: the date on which the transfer agent “closes the book” and looks at who is the shareholder. 3. The ex-dividend date: the day on which the stock starts trading without the right to the dividend. 4. The payment date: the day the dividend is distributed. 7. To a large extent, investors don’t worry about capital expenditures in the same way that they worry when companies do acquisitions. The reason for this is that companies are not paying a premium when they do capital expenditures. In addition, investors tend to assume that these investments will earn returns similar to historic returns. Most importantly, management understands the business, and these expenditures do not disrupt the business as an acquisition can. 8. Poison pills effectively dilute a hostile takeover by issuing more shares (at a minimal price) to all shareholders other than the hostile acquirer. The result is that they discourage hostile takeovers. A poison pill may be good for shareholders if it allows management to negotiate a larger premium if the company is acquired. On the other hand, a poison pill may deter acquisitions and entrench management. It is hard to know if a poison pill is beneficial or detrimental to shareholders. 9. Many investors disagree about whether top managers are overpaid and whether compensation is related to performance. Arguments to say that CEOs are overpaid include the following: - Managers are paid over 200 times more than the average employee. - Managers often benefit when all companies do well, so it does not reflect their unique talents.

- Compensation is set by auditing committees that often rely on compensation consultants. No board believes that they have a bottom −50% CEO, so salaries spiral up (since anyone in the bottom 50% has to be increased to the top 50%). - Management compensation often has an upside if things go well and a limited downside if things go poorly. Arguments to say that CEO compensation is fair include the following: - Running a company has great complexity and requires a unique skill set. - CEO compensation should be compared to the compensation of the average worker. It should be compared to hedge fund managers, investment bankers, and partners at law firms. - CEO tenures are short and high risk. 10. Student Exercise.

ANSWERS TO PROBLEMS 1. 1 (a).

Market cap = 60 million shares  $19 = $1.14 billion

1 (b).

Underwriting fee = 0.07  $16  20 million = $22.4 million Total costs = $22.4+$60 million+$1 million+$2 million+$0.5 million = $85.9 million

Pre-cost equity value = market cap at the end of first day = $1.14 billion Total costs of going public = $85.9 million / $1,140 million = 7.535%

15% overallotment of 20 million shares = 3 million shares

3 (a).

7% discount = $16  (1 − 0.07 ) = $14.88 Underwriter profit = ( $16 − $14.88 )  3 million shares = $3.36 million

The underwriter has the right to buy 3 million shares from the issuing company for $14.88. This is the $16 offer price, but the underwriter is able to buy the shares at a 7 percent discount. In doing this, the underwriter makes an extra $1.12 per share profit on 3 million shares. The underwriter normally has this option for 30 days. Note that the underwriter has already sold those shares for $16 (so the underwriter completes the transaction by buying the shares from the issuing company for $14.88).

3 (b). If the stock price immediately drops to $11.50 in the secondary market, the underwriter will not exercise its option. There is no reason for the underwriter to pay $14.88 for shares that it could buy in the open market for $11.50. The bank will pay $11.50 for 3 million shares that it sold for $16 for a profit on the short sale of

( $16 − $11.50 )  3 million = $13.5 million 4.

In a traditional auction, the clearing price would be $18. In other words, this is how low the price has to go in order to sell 10 million shares. The people who bid $20 and $19 would have their orders completely filled. That would leave 0.5 million shares for the people who bid $18 for 4 million shares. Those bidders would receive 25% of the shares that they bid for.

In a dirty auction, the issuing company could agree to sell the shares at a price below the natural clearing price of $18. In this case, they choose to sell the shares for $17. There are bids for 18.5 million shares at $17 or higher. Because there are 10 million shares that have to be split between investors who placed orders for 18.5 million shares, every bid could be filled at 10/18.5. In other words, if you put in a bid for 185 shares, you’ll receive 100 shares at a price of $17.

a. The declaration date is May 18. b. The date of record is June 13. c. The ex-dividend date is June 9. This is because trades settle two days after the trade date (T + 2). Since we are looking at who is an owner as of the 13th, you would have to buy on the 11th—but that is a Sunday (because you were told that the 13th is a Tuesday). So you have to buy it by Friday, the 9th. d. The payment date is June 28.

7. 7 (a). If the company uses $300,000,000 to repurchase $35 shares, it will repurchase 8,571,429 shares. According to the model, the $4,500,000,000 ($45 × 100,000,000 shares) company will become a $4,200,000,000 company because it used $300 million to repurchase shares. Now, there are only 91,428,571 shares outstanding, and the intrinsic value should increase to $45.9375 ($4,200,000,000/891,428,571 shares). 7(b). PE multiple was 10, so EPS was (100,000,000 × $35)/10 = $3.50 per share or $350,000,000 in earnings.

After the repurchase, earnings per share increase from $3.50 to $3.828 per share ($350,000,000 divided by 91,428,571 shares). 7(c). If the company pays a $300 million cash dividend, the value of the company should drop from $4.5 billion to $4.2 billion. The value of the stock according to the model is now $42 per share ($4.2 billion/100 million shares). 7(d). It is safe to assume that a $3 dividend would result in the stock price dropping from $35 to $32. Using a PE multiple of 10 earnings before the dividend is 35/10 = $3.50 per share. The PE multiple after the dividend is $32/$3.50 = 9.14 . 8. 8 (a). If the company uses $300,000,000 to repurchase $35 shares, it will repurchase 8,571,429 shares. According to the model, the $2,500,000,000 ($25 × 100,000,000 shares) company will become a $2,200,000,000 company. Now, there are only 91,428,571 shares outstanding, and the intrinsic value should decrease to $24.0625 ($2,200,000,000/91,428,571 shares). 8(b). PE multiple was 10, so EPS was (100,000,000 × $35)/10 = $3.50 per share or $350,000,000 in earnings. After the repurchase, the earnings per share increase from $3.50 to $3.828 per share ($350,000,000 divided by 91,428,571 shares). 8 (c). If the company pays a $300 million cash dividend, the value of the company should drop from $2.5 billion to $2.2 billion. The value of the stock is now $2.2 billion/100 million shares = $22. 8 (d). It is safe to assume that a $3 dividend would result in the stock price dropping from $35 to $32. Using a PE multiple of 10 earnings before the dividend is 35/10 = $3.50 per share. The PE multiple after the dividend is $32/$3.50 = 9.14. 9.

If the shares are sold for $20 per share, the issuing firm will receive: 10 million shares  (1 − 0.07 )  20 = $186,000,000

10.

15% overallotment of 25 million shares = 3.75 million shares

7% discount = $18  (1 − 0.07 ) = $16.74 . This is the cost of covering the short position. Underwriter profit = ( $18 − $16.74 )  3.75 million shares = $4.725 million

If the underwriter used all its profits from the short position to purchase shares, after purchasing the 3.75MM shares for $16.74 per share, they could use the $4.725 million profit to buy shares for $20 per share (the market price). This would mean that they could buy $4.725MM / $20 = 236,250 shares . This would bring the total shares purchased to

3.75MM+.23625MM = 3,986,250 shares .

11.

Student Exercise.

Solution and Answer Guide: Frank K. Reilly, Keith, C. Brown, Sanford J. Leeds, Investment Analysis & Portfolio Management, 12th Edition, © 2025, 9780357988176; Chapter 11: Equity Portfolio Management Strategies

Solution and Answer Guide

ANSWERS TO QUESTIONS 1. Passive portfolio management strategies have grown in popularity because investors are recognizing that the stock market is fairly efficient, meaning that there is a smaller chance that an active manager can outperform her benchmark. Given that the costs of an actively managed portfolio are substantial, this leads many investors to adopt a passive strategy. 2. Numerous studies have shown that the majority of portfolio managers have been unable to match the risk-return performance of stock indexes. Following an indexing portfolio strategy, the portfolio manager builds a portfolio that matches both the contents and the performance of an index, thereby reducing the costs of research and trading. The portfolio manager’s evaluation is based upon how closely the portfolio tracks the index or tracking error, rather than a riskreturn performance evaluation. Another passive portfolio strategy, buy-and-hold, has the investor purchase a collection of securities and then not trade them—in other words, hold them—for a period of time. It differs from an indexing strategy in that the composition of the buy-and-hold portfolio does not need to mimic that of the index. Also, indexing does require some limited trading, such as when the composition of the index changes as firms merge or are added and deleted from the index. 3. There are a number of active management strategies discussed in the book, including sector rotation, the use of factor models, quantitative screens, and linear programming methods. All of these active strategies are an attempt by the manager to outperform his or her benchmark (that is, add alpha). Following a sector rotation strategy, the manager overweights certain economic sectors, industries, or other stock attributes in anticipation of an upcoming economic period or the recognition that the shares are undervalued. Using a factor model, portfolio managers examine the sensitivity of stocks to various economic variables or security characteristics, such as a value orientation, low volatility, and high momentum. The managers then “tilt” the portfolios by trading those shares most sensitive to the analyst’s economic forecast. Through the use of computer databases and quantitative screens, portfolio managers are able to identify groups of stocks based upon a set of characteristics. One characteristic that is often emphasized is the relationship between the stock’s market price and its intrinsic value, where

undervalued securities are purchased and overvalued securities are sold. Using linear programming techniques, portfolio managers are able to develop portfolios that maximize objectives while satisfying linear constraints. Any active management technique incorporates fundamental analysis, technical analysis, or the use of anomalies and attributes. For example, based upon the top-down fundamental approach, a factor model may be used to tilt a portfolio’s sensitivity toward those firms most likely to benefit from the economic forecast. Anomalies and attributes can be used as quantitative screens (for example, seek small stocks with low P/E ratios) to identify potential portfolio candidates. 4. Three basic techniques exist for constructing a passive portfolio: (1) full replication of an index, in which all securities in the index are purchased proportionally to their weight in the index; (2) sampling, in which a portfolio manager purchases only a sample of the stocks in the benchmark index; and (3) quadratic optimization or programming techniques, which utilize computer programs that analyze historical security information in order to develop a portfolio that minimizes tracking error. These represent trade-offs between the most accurate tracking of index returns versus cost. Two investment products that managers may use to track the S&P 500 index include index mutual funds, such as Vanguard’s 500 Index Fund (VFINX) and SPDRs, an ETF that tracks the S&P 500. Both of these products use a full replication approach to mimic the index. Per the textbook, the more accurate means of tracking the S&P 500 index has been VFINX; the SPDR shares do not track the index quite as closely as the VFINX fund. 5. Managers attempt to add value to their portfolio by (1) timing their investments in the various markets in light of market forecasts and estimated risk premiums; (2) shifting funds between various equity sectors, industries, or investment styles in order to catch the next “hot” concept; and (3) stock-picking of individual issues (buy low, sell high). A fundamental approach to investing includes “top-down” and “bottom-up” processes such as asset class rotation, sector rotation, and stock undervaluation/overvaluation. A factor-based approach forms portfolios that emphasize certain characteristics of a collection of securities that are believed to produce higher risk-adjusted returns than those in a traditional benchmark that is weighted by the market capitalization of the stocks in the index. 6. The job of an active portfolio manager is not easy. In order to succeed, the manager should maintain his or her investment philosophy, which is difficult to do in severe or volatile market conditions. Because the transaction costs of an actively managed portfolio typically amount to 1 percent or more of the portfolio assets, the portfolio must earn at least 1 percent above the passive benchmark just to keep even; this is very difficult to do in stock markets that are increasingly efficient. Therefore, it is recommended that a portfolio manager attempt to minimize the amount of portfolio trading activity. A high portfolio turnover rate will result in diminishing portfolio profits due to growing commission costs. 7. The four asset allocation strategies are as follows: (1) integrated asset allocation strategy, which separately examines capital market conditions and the investor’s objectives and constraints to establish an optimal portfolio mix; (2) strategic asset allocation strategy, which utilizes long-run projections to establish a set of target allocation weights; (3) tactical asset allocation strategy, which adjusts the portfolio mix as capital market expectations and relative asset valuations change while assuming that the investor’s objectives and constraints remain

constant over the planning horizon; and (4) insured asset allocation strategy, which presumes changes in investor’s objectives and constraints as his or her wealth changes as a result of rising or falling market asset values. 8. Value-oriented investors (1) focus on the current price per share, specifically, the price of the stock is valued as “inexpensive”; (2) should be less concerned about current earnings or the fundamentals that drive earnings growth; and/or (3) implicitly assume that the P/E ratio is below its natural level and that the (efficient) market will soon recognize the low P/E ratio and therefore drive the stock price upward (with little or no change in earnings). Growth-oriented investors (1) focus on earnings per share (EPS) and what drives that value; (2) look for companies that expect to exhibit rapid EPS growth in the future; and/or (3) implicitly assume that the P/E ratio will remain constant over the near term, that is, the stock price (in an efficient market) will rise as forecasted earnings growth is realized. Another perspective is that a single market-wide beta is not the only risk factor that is priced by the efficient market; other risk factors explain the difference in risk-adjusted returns between value and growth portfolios. Perhaps value stock returns reflect additional bankruptcy risk that growth stocks do not have. Some argue that the market may not be completely efficient; investor behaviors push down the price of stocks that become “value” stocks too far while being too optimistic about the future growth potential in “growth” stocks. 9. A price momentum strategy is based on the assumption that a stock’s recent price behavior will continue to follow its recent trend. Thus, an investor would buy a stock whose price has recently been rising and sell (or short) a stock whose price has been falling. An earnings momentum strategy rests on the idea that a firm’s stock price will ultimately follow the direction in which its earnings are moving. The measurement of earnings momentum is usually based on a comparison to expected earnings, with positive “surprises” occurring when actual earnings exceed those forecasted by professional stock analysts. Thus, an investor would buy a stock that has accelerating earnings relative to expectations and sell (or short) a stock whose earnings fall below expectations. These two approaches may produce similar portfolios if the company’s P/E ratios remain stable as their earnings (or price) exhibit momentum characteristics. 10. An investor can create a portfolio that includes his or her ESG beliefs by either excluding stocks of companies in business lines that do not comply with his or her views or screening companies by ESG scores to include those companies that do comply. This screening process—which can take place both initially and during an ongoing process—is what makes an ESG-oriented approach to investing an active strategy, even if the resulting positions in the portfolio match an existing stock index or are traded infrequently. Even though an ESG investor is mixing social impact objectives with traditional investment criteria, empirical evidence suggests that an ESG portfolio need not be expected to underperform an otherwise comparable “unconstrained” fund. 11. The manager would want to overweight natural resource-oriented investments in his or her stock portfolio, either in absolute terms or relative to his or her benchmark index. This could be accomplished by purchasing more individual stocks in the natural resources sector—that is, those specific companies that he or she thinks will particularly benefit from a higher inflationary environment—or by overweighting the entire sector by purchasing a sector-

oriented ETF.

ANSWERS TO PROBLEMS 1.

Using a spreadsheet, we obtain the following values: R2: 0.98 Alpha or intercept term: 0.08 Beta or slope: 0.96 Average return difference: 0.08 Portfolio

S&P

Difference

Return

in Returns

Jan

5.0

5.2

Feb

−2.3

−3

Mar

−1.8

−1.6

Intercept 0.0822

−0.20

Apr

2.2

1.9

Slope

0.30

May

0.4

0.1

0.30

Jun

−0.8

−0.5

−0.30

Jul

0.2

−0.20

Aug

1.5

1.6

−0.10

Sep

−0.3

−0.1

−0.20

Oct

−3.7

−4

0.30

Nov

2.4

0.40

Dec

0.3

0.2

0.10

0.9834

−0.20 0.70

0.9571

Average 0.08 Notice that in this case, the regression intercept and the average return differential—which are two different ways to measure the alpha for the portfolio relative to its benchmark index—are both equal to 0.08 per month. 2.

Foreman R-squared: The R-squared could be low if the S&P 500 is not the proper index to use given the goals of the fund manager. Although Foreman may emphasize large cap stocks, it may invest some of its assets in preferred stocks or bonds. Another possibility is that the fund has a value focus, whereas at times the S&P 500 has more of a growth orientation. Thus, the average P/E, dividend yield, price/book, or earnings growth estimates for the stocks held by Foreman may differ markedly from the S&P 500 stocks. Copeland may be a structured active fund that purposely keeps its tracking error low (1–3

percent) relative to the benchmark index while still trying to outperform. Another possibility is that it is a “closet” indexer that claims to be an active manager while remaining very similar in portfolio composition to the index. By charging fees for active management, but not straying far from the index, the fund can achieve gross returns close to the index and have a high R-squared. Its net fees, however, will fall below the index (as the average equity fund has fees of 1 percent or more). Risk-adjusted performance: No information is presented on risk, so it is possible that the total risk (standard deviation) for Copeland may be much less than the total risk of Foreman; we also have no information on systematic risk (beta) for the two funds. We can only tell that on the basis of gross returns, Foreman outperformed Copeland. We can make no judgment, as no data is presented, on net return and risk-adjusted return performance. 3(a).

Portfolio turnover is the dollar value of securities sold in a year divided by the average value of the assets: Fund W: 37.2/289.4 = 0.1285 or 12.85%

Fund X: 569.3/653.7 = 0.8709 or 87.09% Fund Y: 1,453.8/1,298.4 = 1.1197 or 111.97% 3(b)

3(c)

Fund Z: 437.1/5,567.3 = 0.0785 or 7.85% Passively managed funds will have low portfolio turnover ratios and should have low expense ratios. On this basis, Funds W and Z are the most likely passively managed portfolios; X and Y are most likely to be actively managed. The tax cost ratio is computed as 1 − (1+TAR ) / (1+PTR )   100 , where TAR represents taxadjusted return and PTR is the pre-tax return. Our calculations are as follows: Fund W: 1 − (1+0.0943 ) / (1+0.0998 )   100 = 0.50% Fund X: 1 − (1+0.0887 ) / (1+0.1065 )   100 = 1.61% Fund Y: 1 − (1+0.0934 ) / (1+0.1012 )   100 = 0.71% Fund Z: 1 − (1+0.0954 ) / (1+0.0983 )   100 = 0.26%

The tax cost ratio represents the percentage of an investor’s assets that are lost to taxes on a yearly basis due to the trading strategy employed by the fund manager. The two passive portfolios, Funds W and Z, are the most tax-efficient (least assets lost to taxes), and the actively managed Funds X and Y were the least tax-efficient. 4(a).

The following arguments could be made in favor of active management. Economic diversity—the diversity of the Otunian economy across various sectors may offer the opportunity for the active investor to employ “top-down,” sector rotation strategies. High transaction costs—very high transaction costs may discourage trading activity by international investors and lead to inefficiencies that may be exploited successfully by active investors. Good financial disclosure and detailed accounting standards—good financial disclosure and detailed accounting standards may provide the well-trained analyst an opportunity to perform fundamental research analysis to identify inefficiently priced securities. Capital restrictions—restrictions on capital flows may discourage foreign investor participation and serve to segment the Otunian market, thus creating exploitable market inefficiencies for the active investor. Developing economy and securities market—developing economies and markets are often

4(b).

characterized by inefficiently priced securities and rapid economic change and growth; these characteristics may be exploited by the active investor, especially one using a “topdown,” longer-term approach. Settlement problems—long delays in settling trades by nonresidents may serve to discourage international investors, leading to inefficiently priced securities that may be exploited by active management. Potential deregulation of capital restrictions—the potential deregulation of key industries in Otunia (media and transportation, for example) may be anticipated by active management. Poorly constructed index—a common characteristic of emerging country indices is domination by very large capitalization stocks, which may not capture the diversity of the markets or economy; to the extent that the Otunian index exhibits this poor construction, active management would be favored. Marketing appeal—it would be consistent for GAC, and appealing to GAC’s clients, if GAC were to develop local Otunian expertise in a regional office (similar to GAC’s other regional offices) to conduct active management of Otunian stocks. The following arguments could be made in favor of indexation. Economic diversity—economic diversity across a broad sector of industries implies that indexing may provide a diverse representative portfolio that is not subject to the risks associated with concentrated sectors. High transaction costs—indexation would be favored by the implied lower levels of trading activity and thus costs. Settlement problems—indexation might be favored by the implied lower levels of trading activity and thus settlement activity. Financial disclosure and accounting standards—wide public availability of reliable financial information presumably leads to greater market efficiency, reducing the value of both fundamental analysis and active management and favoring indexation. Restrictions of capital flows—indexation would be favored by the implied lower levels of trading activity and thus smaller opportunities for regulatory interference. Lower management fees—clients would receive the benefit of GAC’s cost savings by paying lower management fees for indexation than for regulatory interference. A recommendation for active management would focus on short-term inefficiencies in and long-term prospects for the developing Otunian markets and economy; these inefficiencies and prospects would not be easily found in more developed markets. A recommendation for indexation would focus on the factors of economic diversity, high transaction costs, settlement delays, capital flow restrictions, and lower management fees.

Reasons NewSoft shares may not be overvalued compared with shares of Capital Corporation although NewSoft’s higher ratios of price to earnings (P/E) and ratios of price to book (P/B) include the following: Higher P/E 1. Prices reflect expected future earnings. If NewSoft’s earnings are growing faster than those of Capital Corp., a higher P/E would be justified. 2. Different accounting practices. Accounting practices affect P/Es based on accounting profits because companies often have different accounting practices (for example, LIFO/FIFO, depreciation policy, and expense recognition). Historical accounting decisions (for example, write-offs) may also affect P/Es. Adjustments for such differences may be required before relative valuations can be properly assessed.

3. Cyclical behavior. Given the cyclical nature of the capital goods sector, the P/E ratio for Capital Corp. may have declined below its five-year average only because recent earnings have risen to a level that analysis indicates is unsustainable. 4. Difference between accounting and economic profits. Accounting profits used to calculate P/Es can differ from economic profits and may or may not be sustainable. NewSoft’s earnings performance may be understated (for example, by expensing noncash items such as goodwill). Capital Corp.’s earnings performance may be overstated (for example, by using low-cost tiers of LIFO inventory). 5. Differences in industries. Different industries have different valuation levels given by the market. NewSoft is a technology company, but Capital Corp. is a capital goods company. 6. Young versus mature industries. Industries at different points in their life cycles will have different valuations. Young industries often have higher valuations. Higher P/B 1. Off-balance-sheet assets. NewSoft may have significant off-balance-sheet assets that are reflected in its stock price but not in its book value. P/Bs have little interpretive value in situations in which intellectual property and human capital are key aspects of company success (for example, the software industry). Physical plant and equipment are typically small, resulting in high P/Bs. 2. Nature of assets. Although physical plant and equipment presumably are a larger portion of Capital Corp.’s total value, the nature of its assets will also affect the validity of using book value as a measure of economic value. Assets such as goodwill and plant and equipment may vary greatly in their balance-sheet cost versus economic value. For example, Capital Corp.’s balance sheet may include significant goodwill, which raises the company’s book value but not its stock price. 3. Efficient use of assets. The P/B does not show how efficiently either company is using its assets. To the extent that NewSoft is generating a higher margin of profit with its assets, a higher P/B may be justified. 4. Obsolete assets. Some of Capital Corp.’s assets may be functional but obsolete. The company’s lower P/B may merely reflect the limited market value of its plant and equipment and the prospective costs of replacement. 6(a).

EUpk = ER p – (sp2 /RTk ) Portfolios

6(b).

6(c).

Ms. A

Mr. B

8 – (5/8) = 7.38

8 – (5/27) = 7.81

9 – (10/8) = 7.75

9 – (10/27) = 8.63

10 – (16/8) = 8.00

10 – (16/27) = 9.41

11 – (25/8) = 7.88

11 – (25/27) = 10.07

The optimal portfolio is the one with the highest expected utility. Thus, Portfolio 3 represents the optimal strategic allocation for Ms. A, while Portfolio 4 is the optimal allocation for Mr. B. Because Mr. B has a higher risk tolerance, he is able to pursue more volatile portfolios with higher expected returns.

For Ms. A: Portfolio 1 = Portfolio 2 8 – ( 5 / RT ) = 9 – (10 / RT ) RT = 5

In other words, a risk tolerance factor of 5 would leave Ms. A indifferent between having Portfolio 1 or Portfolio 2 as her strategic allocation. This would imply a more risk-averse position than her current risk tolerance level of 8. 7.

a) The table below shows that Manager A’s average return is less than the index (−0.11%), while Manager B’s average exceeded that of the index (0.38%). However, the statistical significance of these return differentials has not been assessed, so it is not possible to conclude that these are meaningful distinctions. b) The table below shows the difference between Manager A’s performance and the index, as well as the difference between Manager B’s performance and the index. Manager A did a better job of limiting the client’s exposure to unsystematic risk, as the difference between A’s returns and those of the index has a smaller tracking error (that is, standard deviation of the return differential) than that of the difference between B’s returns and those of the index. Period Manager Manager Index A minus B minus A

Index

12.80%

13.90%

11.80%

1.00%

2.10%

−2.10%

−4.20%

−2.20%

0.10%

−2.00%

15.60%

13.50%

18.90%

−3.30%

−5.40%

0.80%

2.90%

−0.50%

1.30%

3.40%

−7.90%

−5.90%

−3.90%

−4.00%

−2.00%

23.20%

26.30%

21.70%

1.50%

4.60%

−10.40%

−11.20%

−13.20%

2.80%

2.00%

5.60%

5.50%

5.30%

0.30%

0.20%

2.30%

4.20%

2.40%

−0.10%

1.80%

19.00%

18.80%

19.70%

−0.70%

−0.90%

Average

5.89%

6.38%

6.00%

−0.11%

0.38%

Std Dev

11.41%

11.77%

11.66%

2.11%

3.00%

Std dev = tracking error

8a. Security 1

Benchmark

Fund X

Index (%)

(%) 10

w p,i − wb,ij

Fund Y

w p,i − wb,ij

(%) 40

Fund Z (%)

w p,i − wb,ij

 w −w p,i

b,ij

130

65%

Active Share 8b.

14%

Fund X is considered a closet (or enhanced) index fund because, although close to matching the benchmark, it differs in the investment weights of those holdings. Fund X could also be a structured portfolio. Fund Y is more concentrated than the benchmark and has fewer holdings and bigger investment positions and is therefore more likely to be considered an actively managed, concentrated stock-picking fund. Fund Z is considered a passive index fund because its security holdings come closer to matching the benchmark, both in number of holdings and the investment weights of those holdings.

9. Value Index Annualized Return Growth Index Annualized Return Month

(%)

Rvalue − Rgrowth

8.97

9.50

−0.53

17.38

17.48

−0.10

28.11

26.81

1.30

16.19

12.92

3.27

16.66

14.60

2.06

16.72

14.90

1.82

19.43

13.21

6.22

16.88

9.68

7.20

17.88

10.14

7.74

20.41

12.42

7.99

28.86

20.69

8.17

23.02

16.09

6.93

27.38

20.05

7.33

21.43

15.65

5.78

20.76

18.25

2.51

25.62

25.59

0.03

28.46

26.74

1.72

28.92

29.62

−0.70

18.87

22.49

−3.62

21.75

26.38

−4.63

20.18

21.63

−1.45

19.62

19.51

0.11

18.48

20.76

−2.28

21.97

24.59

−2.62

21.00

18.74

2.26

Arithmetic Mean 9a.

9b.

The arithmetic mean annual return for the Value Index is 21.00%, and the arithmetic mean annual return for the Growth Index is 18.74%. The Value Index appears to have outperformed the Growth Index over the 24-month period by 21,00% − 18.74% = 2.26% . As expected, the average of the return in the differential series is 2.26%, which is the same as the difference in the arithmetic means.

9c.

9d.

9e.

10a.

As the average return differential between the two indexes was 2.26%, a hedge fund that was following a strategy to go long in value stocks and short in growth stocks would have been quite profitable. The hedge fund would have earned the average annualized return in the Value Index of 21.00% while only losing 18.74% on the short position in growth stocks, thus earning the additional risk premium of 2.26%. The rolling average annualized return for the growth index was larger than that for the value index for eight out of the 24 months, or 33 percent of the time. This means that the value risk premium of 2.26% was a reliable indicator of value outperforming growth for two-thirds of the time in the period studied. However, it is important to understand that there will be frequent instances when the opposite outcome occurs. The mean and median characteristic levels of the 10 largest holdings for each manager are as follows: Market Cap ($ Est. EPS Dividend Company P/E P/BV Beta Bil) Growth (%) Yield (%) Manager 1:

Mean:

277.7

19.05

3.56

7.72

2.50

0.92

Median:

232.5

17.63

2.87

6.91

2.42

0.87

Manager 2:

10b.

Mean:

349.8

65.44

9.29

26.20

0.52

1.18

Median:

389.6

36.03

7.36

18.80

0.00

1.18

Growth stocks typically have higher valuation levels (higher P/E and P/BV ratios), lower dividend payout rates, higher EPS growth rates, and are often riskier (higher beta) than value stocks. From the characteristic averages reported above, it is clear that Manager 2 holds a growth-oriented portfolio and Manager 1 is a value-oriented manager. Examining the identical characteristic levels they possess, it is reasonable to conclude that Stock W (Manager 1) and Stock A (Manager 2) are the same company. Also, Stock S (Manager 1) and Stock D (Manager 2) are the same company. Based on the relative value measures, EPS growth rate, and dividend yield, Stock W/A appears to be better situated in the value portfolio. On the other hand, Stock S/D has elements of both a growth stock (high relative valuation levels) and a value stock (high dividend yield, mid-range EPS growth), but would most likely be situated in a growth portfolio. Stock S/D underscores the oftensubjective nature of the value–growth classification process.

Solution and Answer Guide: Frank K. Reilly, Keith, C. Brown, Sanford J. Leeds, Investment Analysis & Portfolio Management, 12th Edition, © 2025, 9780357988176; Chapter 12: Bond Fundamentals And Valuation

Solution and Answer Guide

ANSWERS TO QUESTIONS 1.

The three factors affecting the price (or value) of a bond are coupon, yield, and term to maturity. The relationship between price and coupon is a direct one—the higher the coupon, the higher the price. The relationship between price and yield is an inverse one— the higher the yield the lower the price, if all other factors are held constant. The relationship between price and maturity is not so clearly evident. Price changes resulting from changes in yields will be more pronounced the longer the term to maturity.

Interest income from municipal bonds is normally not taxable by the federal government or by the state or city in which it is issued. Interest income on U.S. Treasury bonds is taxable at the federal level, but not by state or local governments. Corporate bond interest is taxable at all levels, as are capital gains from any of the bonds.

A call option attached to a bond allows the issuer to retire the bond prior to its maturity. Callable bonds have a call premium, which is the dollar amount in excess of the bond’s par value that the issuer must pay to the bondholder for the possibility of prematurely retiring the bond. Refunding a bond would involve an issuer retiring an existing security at its maturity by using the proceeds of a newly issued bond, typically at a lower funding cost.

The purpose of bond ratings is to assess the possibility that the issuer will not be able to honor its commitment to pay the promised stream of cash flows to the investor, thereby triggering an act of default. The primary risk that a bondholder faces is that the borrower will not be able to pay these promised coupons and principal refund. The rating agencies analyze the issuing organization and the specific issue to determine the probability of default.

The differences between the Japanese and U.S. bond markets are clear from Exhibit 12.2. The Japanese bond market is dominated by government bonds, which accounted for over 90 percent of their market in the year shown. The corporate sector accounted for only about 2

percent. In contrast, in the U.S., federal government issues accounted for about 38 percent of the bond market, with federal agency issues taking up an additional 7 percent. Next to government issues, the largest sectors were securitized/collateralized bonds and corporate bonds, each with about 25 percent of the total. There are several reasons for the differences. In Japan, corporations tend to get long-term funding from banks. This is not the typical practice in the U.S. Also, U.S. federal agencies and government-sponsored organizations finance several legislative mandates, such as those involving funding for housing. 6.

The difference between a foreign bond and a Eurobond can be broken down as a difference in the issuer and the market in which they are issued. For example, a foreign bond in Japan (for example, a Samurai) is denominated in the domestic currency (yen) and is sold in the domestic market (Japan), but it is sold by non-Japanese issuers. On the other hand, a Eurobond is denominated in the domestic currency (yen), but it is sold outside the domestic country in a number of national markets. These bonds are typically underwritten by international syndicates. The relative size of these two markets varies by country.

The discounted cash flow valuation equation is more useful for the bond investor, largely because the bond investor has fewer uncertainties regarding future cash flows than does the common stock investor and, unlike stock, bonds have a predetermined maturity, meaning there are a finite number of cash flows to consider. By investing in bonds with relatively no default risk (that is, government securities), the investor can value a bond based primarily on expected cash flows (coupon rate and par value), required return (market yield), and the number of periods in the investment horizon (maturity date). Even with corporate bonds, default (credit) risk is incorporated into bond ratings, which makes the assignment of the appropriate credit spread (that is, risk premium) a more streamlined process. Examining market data based on time to maturity and credit rating allows the determination of the market interest rate on similar bonds. Each of these factors can be incorporated into the present value equation. By contrast, common stocks have no stated maturity date, and the valuation process is predominantly an estimate of future earnings. Although the present value method can be used for common stock analysis by estimating future cash flows and changes in price over a given time frame, the uncertainties involved are much greater.

The most crucial assumption in both cases that the investor makes is that cash flows will be received in full and reinvested at the promised yield. This assumption is crucial because it is implicit in the mathematical equation that solves for promised yield. If the assumption is not valid, an alternative method must be used, or the calculations will yield invalid solutions. The yield-to-maturity calculation assumes that the bond is held until it matures whereas the yield-to-call calculation requires an assumption about when in the future the bond issuer might exercise its option to call the bond prior to maturity.

9(a).

RFR is the real (that is, net of inflation) riskless rate of interest, I is the factor for expected inflation, and RP is the risk premium for the individual firm. The model considers the firm’s business conditions. The risk of not generating positive earnings at the firm level would be reflected in the model through changes in the variable RP. This uncertainty would change the nature of the frequency distribution for earnings.

9(b).

10(a). The term structure of interest rates refers to the relationship between yields and maturities for fixed-income securities that are alike in all other respects, including credit risk, liquidity, and taxation. Expectations regarding future interest rate levels give rise to different supply and demand pressures in the various maturity sectors of the bond market. These pressures are reflected in differences in the yield movements of bonds of different maturity. The term “structure of interest rates,” or “yield curve,” will normally be upward sloping in a period of relatively stable expectations. The theoretical basis for the upward sloping curve is the fact that investors generally demand a premium the longer the maturity of the issue to cover the risk through time and also to compensate for the greater price volatility of longer maturity bonds. 10(b). According to the expectations theory of yield curve determination, if borrowers prefer to issue shorter maturity bonds at the time lenders prefer to invest in longer-term issues, which happens when interest rates are expected to fall, longer maturity issues will tend to yield less than shorter maturity issues. The yield curve will be downward sloping. This generally occurs in periods when restrictive monetary policy by the Federal Reserve System, in an attempt to control inflation and inflation expectations, causes very high shortterm interest rates. In these circumstances, demand for short-term maturities is severely dampened. 10(c). The “real” rate of interest is simply the difference between nominal interest rates and some measure of inflation, such as the current consumer price index or GNP deflator. In other words, it is an inflation-adjusted interest rate. 10(d). A possible first reason is that investor preference for Treasuries stems from several factors, such as the fact that Treasury securities typically are extremely liquid and provide investors with more flexibility in the portfolio formation process. Second, Treasury securities typically do not have restrictive call features generally encountered with high-grade corporates. Thus, in a period of high interest rates, investors purchasing long-term securities anticipate an eventual decline in inflation and interest rates and thus prefer to lock in higher long-term yields. Third, in times of economic uncertainty or uncertainty about the values of other assets, a “flight to quality” may increase demand for Treasuries, thus increasing their prices and lowering their yields. 11.

The asset-backed securities (ABS) market involves pooling a specific type of debt holding (car loans, student loans, etc.) and then issuing new securities backed by the cash flows generated from the loan pool. Typically, ABS structures focus on a single type of loan, which can be anything other than residential or commercial mortgages. The credit rating of an ABS issue can be higher than the credit quality of the assets in the underlying portfolio through the use of overcollateralization (that is, holding assets in the loan pool that are worth more than the amount of new securities issued against the pool), or by the use of special investment vehicles and credit enhancements. Also, the development of a series of tranches with different credit ratings allows investors to select their preferred level of risk exposure.

ANSWERS TO PROBLEMS 1(a). For a zero-coupon bond, the valuation formulation simplifies to Present Value = Future Value / (1+discount rate )

PV = $1,000 / (1.06 ) = 174.11 30

The discount rate is 12% annually/2 interest payments per year, which equals 6% for 30 semiannual periods (15 years × 2 interest payments per year). 1(b).

PV = $1,000 / (1.05) = 142.05 40

The discount rate is 10% annually/2 interest payments per year, which equals 5% for 40 semiannual periods (20 years × 2 interest payments per year). 2.

ETY = 2(a).

i 1−t

For the 15% tax bracket

ETY = 2(b).

For the 25% tax bracket

ETY = 2(c).

0.084 0.084 = = 0.0988 or 9.88% 1 − 0.15 0.85

0.084 0.084 = = 0.1120 or 11.20% 1 − 0.25 0.75

For the 35% tax bracket

ETY =

0.084 0.084 = = 0.1292 or 12.92% 1 − 0.35 0.65

ETY =

i 0.055 0.055 = = = 0.07639 1 − t 1 − 0.28 0.72

Assuming all other relevant factors are equal, the corporate bond carrying an 8 percent coupon and selling at par offers a better tax-adjusted return than a 5.5 percent municipal bond (with an equivalent tax yield of 7.639 percent). 4(a).

Present Value = Future Value / (1+discount rate )

PV = $1,000 / (1.04 ) = 308.32 30

The discount rate is 8% annually/2 interest payments per year, which equals 4% for 30 semiannual periods (15 years × 2 interest payments per year).

4(b).

PV = $1,000 / (1.04 ) = 390.12 24

The discount rate is 8% annually/2 interest payments per year, which equals 4% for 24 semiannual periods (12 remaining years × 2 interest payments per year). 4(c).

PV = $1,000 / (1.05) = 310.07 24

The discount rate is 10% annually/2 interest payments per year, which equals 5% for 24 semiannual periods (12 remaining years × 2 interest payments per year). 5(a).

PV = $1,000 / (1.06 ) = 97.22 40

The discount rate is 12% annually/2 interest payments per year, which equals 6% for 40 semiannual periods (20 years × 2 interest payments per year). 5(b).

PV = $1,000 / (1+r ) inserting the known items, n

$624.60 = $1,000 / (1.04 )

Solving algebraically or with a financial calculator, (PV = −624.60; FV = 1000; i = 4; CPT n ) we find that the number of periods equals 12.00 semiannual periods or 6.0 years for maturity. 5(c).

PV = $1,000 / (1+r ) inserting the known items, n

$350.34 = $1,000 / (1+r )

(use 18 as the number of periods is 2 × 9)

Solving algebraically or with a financial calculator (PV = −350; FV = 1000; n = 18; CPT i ) , we find that the periodic yield equals 6.0% or an annual yield of 12%. 6(a).

Using a financial calculator: PV = −1000, FV = 1100, PMT = 35, n = 8 , as the bonds were purchased four years ago. The resulting periodic return is 4.564% or 9.13% annually.

6(b).

Using a financial calculator: FV = 1000, PMT = 35, n = 42 , as the bonds mature in 21 years. If the periodic interest rate is 5%/2 or 2.5%, the price of the bonds is $1,258.21.

7(a).

Using a financial calculator: PV= − 01012.50, FV = 1000, PMT = 40, n = 24 , as the bonds have 12 years until maturity. The resulting periodic yield is 3.919% or 7.84% annually.

7(b).

Using a financial calculator: PV = −1012.50, FV = 1080, PMT = 40, n = 6 , as the bond is callable in three years. The resulting periodic return is 4.932% or 9.86% annually.

8(a).

Current yield = Annual dollar coupon interest / Price = 70/960 = 7.29%

8(b).

The annual yield to maturity (YTM), using a financial calculator, is found using these inputs: PV = −960; FV = 1000; PMT = 35; n = 10 . The resulting periodic yield to maturity is 3.993%

semiannually or 7.99% annually.

8(c).

Horizon yield (also called total return) accounts for coupon interest, interest on interest, and proceeds from the sale of the bond. The first step is to find the future value of the reinvested coupons: PMT = 35, n = 6, reinvestment rate = 3% ; using these inputs, the future value is $226.39.

Next, we determine the projected sale price at the end of three years. In this case, it will equal $1,000 because the required return of 7% is the same as the bond’s coupon rate. Adding these values, the value of the cash flows at the end of year 3 is $1,226.39. Finally, we find the semiannual total return = ($1,226.39/$960)(1/6) − 1 = 0.04166 . Doubling this to annualize the periodic return, we have a horizon yield of 8.33%. 9. The price of Bond A using spot discount rates would be

10 10 110 + + = 9.524+8.573+80.431 = 98.528 2 1.05 (1.08 ) (1.11)3 The market price of Bond A is 98.40 based on the single yield to maturity discount rate, which is about 13 cents (12.8 basis points of market price) less than the spot rate-based price. The price of Bond B based on spot discount rates would be

6 6 106 + + = 5.714+5.144+77.506 = 88.364 2 1.05 (1.08 ) (1.11)3 The market price of Bond B is 88.34, which is only about 2 cents (2.4 basis points of market price) less than the spot rate-based price. So, despite having a lower yield to maturity (10.65 percent versus 10.75 percent), Bond A is the better value because the difference between its market price and theoretical value is greater than that for Bond B. 10(a). Bond Valuation Step #1: On December 4, 2023: P0 = PV of Coupons + PV of Face Value Using a financial calculator: Payments = Semiannual, FV = 100, PMT = 2.3, n = 11, YTM = 1.9% . The resulting PV = 103.2805. Step #2: On April 2, 2018:

Pn +AIn = (103.2805)( 1+0.038/2)

(118/180)

= 104.5628 = Total Invoice Price

10(b). Step #3:

(i)

AIn = ( 2.3)(118/180 ) = 1.5078

(ii)

“Clean” or “Flat” Price = 104.5628 − 1.5078 = 103.0550

Solution and Answer Guide: Frank K. Reilly, Keith, C. Brown, Sanford J. Leeds, Investment Analysis & Portfolio Management, 12th Edition, © 2025, 9780357988176; Chapter 13: Bond Analysis And Portfolio Management Strategies

Solution and Answer Guide

ANSWERS TO QUESTIONS 1(a). 1(b).

Given that you expect interest rates to decline during the next six months, you should choose bonds that will have the largest price increase, that is, bonds with long durations. Set 1: Given a choice between bonds A and B, you should select bond B, which will have the largest duration because of its lower coupon and lower yield to maturity. Set 2: Given a choice between bonds C and D, you should select bond C, which will have the largest duration because of its longer maturity and lower coupon. Set 3: Given a choice between bonds E and F, you should select bond F, which will have the largest duration because of its longer maturity and lower yield to maturity.

You should select portfolio A because it has a longer modified duration (5.7 versus 4.9 years) and greater convexity (125.18 versus 40.30), thereby offering greater price appreciation in a market where rates are expected to decline. Portfolio A is also noncallable; therefore, there is no danger of the bonds being refunded by the issuer when interest rates decline, as you expect they will.

3(a).

Call-adjusted duration takes into account the probability of the bond being called and its impact on the actual duration of a bond. If a bond is noncallable, duration is based on all cash flows up to and including maturity. If interest rates drop and a call becomes likely, then duration should be calculated based on the time to call, which can be substantially sooner than maturity. The range for duration now is 8.2 to 2.1 years. Because the bond is trading at par, its duration should be near 8.2 years. If rates increase, then duration will decrease (recall that duration is inversely related to yield to maturity.) However, the call is now very unlikely, so call-adjusted duration will stay near the high end of the range. If rates fall to 4 percent, then call becomes highly probable, and so call-adjusted duration will drop to the low end. (Remember here that the duration to call is now greater than 2.1 years because of the inverse relationship between duration and yield to maturity). Negative convexity refers to slow price increases of callable bonds as interest rates fall. Indeed, at some point, the price change will be zero. This is due to the call price placing a ceiling on the price of a callable bond.

3(b). 3(c). 3(d).

4(a). Duration Adding a call feature to a proposed bond issue will shorten the duration of the bond. Duration may be lowered because the call feature can potentially accelerate the payments, depending on what happens to interest rates. When interest rates are high, a call is unlikely, and the call-adjusted duration of the callable bond will be slightly lower than the duration of the noncallable bond. With decreasing rates, the likelihood of call increases, and the callable bond’s duration decreases, relative to the duration of the noncallable bonds. The resulting duration measure will be greater than the duration to call and less than the duration to maturity. An additional impact is that the call feature increases the uncertainty of duration relative to a noncallable bond. Convexity Relative to a noncallable bond, the call feature reduces the convexity (call-adjusted convexity) of the callable bond. The call feature reduces the convexity across all interest rates, but the reduction is minimal at high rates when calling the bond is less likely and is greatest as the market price approaches the strike price and the call becomes more likely. In fact, the call feature may induce negative convexity as the market price approaches and exceeds the strike price as interest rates in the market decrease. When the call is more likely to be exercised, the market price won’t appreciate much above the call price. 4(b). To use callable bonds and still maintain duration and convexity targets, the value and price sensitivity of the call feature must be taken into account. The managers will need to calculate call-adjusted duration and convexity for callable bonds and modified duration and convexity for option-free (noncallable) bonds to use them both in the same strategy. 4(c). Some advantages of including callable bonds are as follows: • Callable bonds have higher offering yields (or more income or more cash flow) than comparable noncallable bonds. • The larger supply of callable bonds makes it easier to rebalance the portfolio to maintain duration and convexity targets. Some disadvantages are as follows: • The reinvestment risk of having the bonds called away and having to invest at lower rates. • Management time needed to rebalance when securities called or when rates change. • Uncertainty about cash flows. 5(a).

5(b).

As interest rates decline, the price of both bonds would increase. However, the price appreciation of Celeste will be limited by the call price of 102.00. As interest rates decline, the probability of the issuer calling the bonds increases, as the company will consider issuing new bonds at lower interest rates. On the other hand, the price appreciation of Sophie bond would not be limited in this manner, as the bond is not callable. Therefore, Sophie bond would be the preferred investment if you expect interest rates to decrease by more than 100 basis points. Burke would prefer the Celeste bond in either a rising or a stable interest rate scenario. The Celeste bond has an embedded option, which is sold by the investor to the issuer of the bond. The higher yield compensates Burke for the risk of the embedded call option. If rates are stable or increase, the investor earns the extra income without having to worry about having the bond called from them. Additionally, if rates increase, Celeste bond price should

5(c).

6(a).

6(b).

decrease less relative to Sophie bond because of Celeste’s shorter effective duration due to the embedded call option. (1) Because the Sophie bond is noncallable, increased interest rate volatility would not impact its directional price change. (2) Callable bond value = Noncallable bond value − Call option value The level and volatility of interest rates are key factors in determining the value of a bond with an embedded call option. The greater the variance or uncertainty of interest rates, the greater the value of the embedded call option. As the embedded option value increases, it causes the value of Celeste’s callable bond to decrease. You might expect it to be easier to match the performance of a bond index than that of a stock index because of the more homogenous nature of the bond market, which depends on matching fewer broad characteristics (credit quality, interest rate sensitivity, or duration) than in the stock market. As a result, you might be able to match the performance of the bond index with substantially fewer issues. As an example, in order to match the performance of the S&P 500 Stock Index, it generally requires anywhere from 300 to 450 issues. In contrast, one could do a fairly good job of tracking a bond index that would include thousands of issues with less than l00 bonds simply because bonds are so heavily influenced by the general movements in interest rates. While it might be possible to match the bond index with fewer issues, the selection and operational process of running the bond index fund would be more difficult. First, it is going to require more characteristics to derive the desired diversification. While the equity market only requires serious consideration of capitalization and risk decile, bonds have many characteristics that can effect return including maturity, duration, credit quality, capitalization, coupon, industrial classification, sinking fund, and call features. Thus, it will be necessary to determine the makeup for each of these characteristics and attempt to match it in the portfolio. A second factor would be the difficulty of obtaining bonds as opposed to stocks. In the case of a stock index, you are typically dealing with very large capitalization stocks traded on an exchange or involved in an active over-the-counter market. In contrast, the secondary corporate bond market is not nearly as liquid, and so it is difficult to buy and sell for the bond index fund. Finally, there is greater difficulty in reinvestment of the cash flows from a bond index fund rather than a stock index. Because of larger cash flows from a bond index fund, you are going to have to establish more frequent buying programs. While this additional reinvestment might allow you to change the makeup of the fund, it could be difficult to avoid changing the fund with small reinvestment programs. Balancing all relevant bond portfolio characteristics when doing these reinvestments will be challenging. It is likely to be easier to maintain an equity index ETF for many of the previous reasons. Also, a significant difference between investing in bonds versus stocks is that bonds mature whereas stock does not. So, a bond index manager will generally have more cash flow reinvesting to do as bonds in the portfolio mature and need to be replaced with new issues. a. “The Trinity Fund is being managed well.” Disagree. The fund is not managed properly because a pure bond indexing strategy should not deviate significantly from its benchmark by more than the amount of its expenses. Thus, by outperforming the index by almost 50 basis points (net of fees), it is clear that the Trinity manager was executing a strategy different than trying to match the index performance.

b. “I expected that, as an active manager, Montego would outperform the index; therefore, the fund should be sold.” Disagree. While is true that you should expect an active manager to deliver a positive alpha (that is, risk-adjusted return in excess of the benchmark) to justify their higher management fee, this may not happen in every investment period. Six months is too short a time frame to evaluate any active portfolio manager, so essentially firing the Montego manager (that is selling the fund) is premature. 8(a).

8(b).

8(c).

In normal market conditions, we expect safer assets to be more liquid than riskier assets. Given the size of the Treasury market, it will be most liquid. High-grade corporate bonds will be more liquid than lower-grade corporate securities. In times of a credit crisis, a “flight to quality” will maintain liquidity in Treasuries, and reduce liquidity for all corporate bonds, possibly all but eliminating liquidity for low-rated bonds. Market liquidity should be a major consideration when investing in high-yield bonds, which generally are substantially less widely traded than Treasury securities and high-grade corporate bonds. In fact, there have been times when there was almost no trading at all in the high-yield bond market. Even though high-yield bonds are less liquid than Treasury and high-grade corporate bonds, it is important to recognize that they are still more liquid than private market debt transactions, such as corporate bank loans or private placement bond issues. Because high-yield bonds are generally less liquid, they are also generally more difficult to price, which may lead to an investor requiring a liquidity/pricing premium to hold these issues. Also, because of the relative illiquidity, when adding high-yield bonds to a portfolio, it must be recognized that they cannot be used for short-term trading. Alternatively, a portfolio manager may decide that she wants to invest in high-yield bonds but will limit investments to the set of high-yield bonds that enjoy fairly active, liquid markets with relatively frequent pricing. The reason that Treasury bills and high-grade corporate bonds are so highly correlated is that the major factor causing changes in their rates of return is aggregate market interest rates (that is, the yield curve for Treasury securities). In terms of the capital asset pricing model (CAPM), the fact that these rates of return move together means that they possess very high systematic risk (that is, all rates of return are being driven by one major aggregate market variable which in this case is market interest rates). The rates of return for high-yield bonds are likewise influenced by market interest rates, but they also have a very large unique component that is the performance of the company and the credit quality of the specific issue. In terms of the CAPM, this unique risk possessed by high-yield bonds would be described as unsystematic risk. Notably, this unsystematic risk can be diversified away in a large portfolio. This implies that while it is essential to diversify a portfolio of high-yield bonds to eliminate the unsystematic risk, it also means that such bonds are good additions to a portfolio with high systematic risk, as described above. Because the rates of return for high-yield bonds are not very highly correlated with rates of return for Treasuries and high-grade corporate bonds, they would be an excellent addition to such a portfolio because they would add diversification and thereby help reduce the portfolio’s overall risk. This would be like adding common stocks to such a bond portfolio. In addition, since the expected rates of return on these securities are likely higher than those on treasuries and high-grade corporates, you could not only reduce the overall risk of the portfolio but also increase its average rate of return. The duration for high-yield bonds would be much lower than the duration for high-grade corporates because of the impact of all three of the major factors that influence duration: maturity, coupon, and market yield. The average maturity of high-yield bonds is shorter

than high-grade corporates simply because of the inherent risk, which causes investors to demand shorter maturities (for example, l0- to 15-year maturities for high-yield bonds compared to 15–30 years for high-grade corporates). The average maturity for high-yield bonds is lower than for high-grade corporates, and duration is positively related to maturity. Second, because of the risk of high-yield bonds, they have a higher required rate of return that is reflected in a higher coupon for these securities, and there is an inverse relationship between the coupon and the duration of the security. Third, not only is the coupon high, but the market yield to maturity on these securities would likewise be high. Again, there is an inverse relationship between the discount rate used to compute duration (that is, the yield to maturity) and the duration of the bond. 9(a).

9(b).

10.

The bond portfolio manager should be generally concerned about the ability of the new portfolio structure to outperform the existing one in a falling interest rate environment. He should ask about the degree of negative convexity being added through the use of the corporate bonds. Also, for bonds with the same maturity, new Treasury issues will have longer durations than new corporate bonds because of their lower coupon rates. Because the move forecasted is relatively large, both these duration and convexity factors will become quite important. Given the outlook for significantly lower rates, it is important to minimize the negative convexity effect and duration differential. The recommendation can be modified by using noncallable corporates and lowering the coupon level of the purchases to accomplish this. The briefing note also should take a close look at the option-adjusted spreads received on any corporate to see if it is worth reducing convexity in a declining rate environment. Investment horizon a year later = 3.0 years Duration of portfolio a year later = 3.2 years While the term to maturity has declined by a year, the bond’s duration will decline by less than one year because duration declines more slowly than term to maturity (see, for example, Exhibit 13.2). This means that, assuming no changes in market rates, as time passes, a portfolio that was originally designed to have its duration match its planning horizon (that is, the duration of the liabilities) will become out of balance. In this case, the portfolio manager must rebalance the portfolio to reduce its duration to three years.

11(a). With an immunized portfolio, the goal is to provide a minimum dollar amount of assets at a single horizon. Contingent immunization is primarily an active strategy. It is based on the premise that if the manager is willing to accept a lower guaranteed rate than is available in the current market, she can actively manage the bond portfolio in an effort to produce positive risk-adjusted returns. However, should the market value of the portfolio deteriorate to the point where this minimum required return is threatened, a switch to full immunization of the portfolio would be necessary. The purpose of a cash-matched dedicated portfolio is to have a portfolio that will generate cash flows that specifically match the required stream of cash outflows that represent the fund’s liabilities. Therefore, it is necessary to match maturities and amounts over a time period, not a single time period. This is accomplished by planning maturities and interim cash flows from the portfolio. The purpose of a duration-matched dedication portfolio is to match the interest rate sensitives of the cash flows from the portfolio to the required cash outflows over time. The

major difference from the cash-matched dedication is that you recognize that you do this by matching the weighted average duration of the liabilities with the duration of your investment portfolio. 11(b). When managing an immunized portfolio, it is necessary to maintain the duration of the portfolio equal to the investment horizon (that is, the duration of the fund’s liabilities). This will require periodic rebalancing because (1) duration declines slower than term to maturity and (2) duration is affected by changes in market yields; in other words, there is an inverse relationship between yield and duration. 11(c). With a cash-matched dedication portfolio, it is necessary to make several major decisions: (1) Predictability of liabilities: The most important thing for any matched portfolio strategy to consider is how predictable the required cash payments are. To implement a pure cash-matched portfolio, the required cash flows should be highly predictable in terms of amount, timing, and finite in number. (2) Timing and amount of investments: Once the required cash flows are established, a portfolio of bonds will have to be established. Ideally, this could be accomplished with a series of zero-coupon bonds with a face value and maturity that match the required stream of future payments. This may be difficult to achieve, meaning several decisions will need to be made regarding alternative bond issues to hold in the portfolio. (3) Managing reinvestment risk: Assuming required cash flows cannot be matched precisely, the important thing will be to design a portfolio that generates at least as much cash as will be needed for each future payment. This means that the manager will need to develop a plan for reinvesting the excess cash generated by the portfolio that is not needed to pay current liabilities. These excess cash flow reinvestments will make the portfolio subject to interest rate risk as yields change in the market. 11(d). There are several basic components that should be specified to implement a contingent immunization strategy. The most important ones are as follows: (1) Current market return: This is the return that could be earned if the portfolio is immunized at today’s rates. (2) Potential return: This is the return that the manager is willing to accept as a lower outcome than the current market return. (3) Safety margin and trigger point: The amount by which the value of the actively managed portfolio exceeds the value required to satisfy the potential return. The trigger point is the market yield at which the safety margin becomes zero. In addition to the above, the client and manager should agree on the flexibility to allow the manager in an active strategy. The agreement should specify the time horizon and level of active risk the manager is permitted to take. 11(e). Once it is established, the cash-matched dedicated portfolio will require the least supervision over time. A contingent immunization strategy requires continual active management in the portfolio and a duration-matched portfolio will require periodic rebalancing to account for imbalances in how the planning horizon and portfolio duration change as time passes. 12(a). Three issues that Devlin and Parish should address are the following: (1) The starting yield level relative to the U.S.: If the spread is positive, this provides a cushion against unfavorable moves in either interest rates in the foreign market or in the value of the foreign currency. If the spread is negative, the foreign market must make up the difference by outperforming in local currency terms or by experiencing an appreciation in its currency (or both).

(2)

The prospects for internal price movements relative to the U.S. bond market: What is the likely trend in yield spreads between the foreign market and the U.S.? Unlike in the U.S., where yields in different sectors will generally move in the same direction, albeit at different rates, yields in foreign markets may move in opposite directions to the U.S., due to differences in economic, social, and political factors in those foreign markets. (3) The prospects for currency gain or loss versus the dollar: The factors Devlin and Parish should look at to assess prospects for the Euro and the Australian dollar include the following: (a) Trends in the balance of payments (b) Inflation and interest rate differentials (c) The social and political atmosphere, particularly as it relates to foreign investment (d) The extent of central bank intervention in the currency markets 12(b). The two reasons for investing in a mixture of international bonds are (1) the opportunity for superior rates of return and (2) diversification. With respect to return, economic and interest rate cycles tend not to move in perfect concert worldwide. As a result, being able to invest in a host of different markets presents opportunities for above-average returns in comparison to having access to only one individual and relatively homogeneous market. With regard to diversification, foreign bond markets are not perfectly correlated with the U.S. bond market. This means that the volatility of return for a portfolio of global bonds will be less than for a portfolio comprised only of U.S. bonds. The ERISA account does have a 10% position in Canadian bonds, but the close interrelationship of the Canadian economy and its capital markets makes Canada highly correlated with the U.S. In that sense, the Canadian position does not afford the return and diversification opportunities that other foreign bond markets would offer.

ANSWERS TO PROBLEMS 1(a). (1)

(2)

(3)

(4)

(5)

Cash

PV as %

Flow

at 5%

of Flow

of Price

(1)  ( 5)

$40 0.9524

$38.10

0.04014

40 0.9070

36.28

0.03822

0.07644

40 0.8638

34.55

0.03640

0.10920

40 0.8227

32.91

0.03467

0.13868

40 0.7835

31.34

0.03302

0.16510

1,040 0.7462

776.05

81756

4.90536

$949.23

1.00000

5.43492

Period

(6)

The duration equals 5.43492 semiannual periods or 2.71746 years.

Percentage change in price = −Dmod  Δi

= − ( 2.588 )  ( −50/100 )

1(b).

= 1.294 percent

The bond price should increase by 1.294% in response to a drop in the bonds YTM from 10% to 9.5%. If the price of the bond before the decline was $949.23, the price after the decline in the YTM should be approximately $949.23  1.01294 = $961.51 . 2(a).

With semiannual interest payments: 8.8 8.8 = = 8.565years 1 + (0.055 / 2) 1.0275 Percentage change in price = −Dmod  Di Modifiedduration

= − ( 8.565)  ( +150/100 ) = −12.85%

This approximates the price change because the modified-duration line is a linear estimate of a curvilinear function. That is, convexity measures the rate of change in modified duration as yields change. The effect of convexity on price should be added to the effect of duration on price to obtain an improved approximation of the change in price given a change in yield. In this case, adding the convexity effect will result in a smaller decrease in the bond’s price when interest rates rise. 2(b).

Percentage change in price = −8.565  −3% = +25.70%

2(c).

Percentage change in price = −8.565  −3% + 0.5  92.8  ( −3% ) = +29.87%





2 (d). The 3% decline in rates may not elevate the bond price if the bond’s call price is reached and the issuer decides to refund the issue. 3(a).

Modified duration is Macaulay duration divided by 1 plus the yield to maturity divided by the number of coupons per year: Modified duration =

Macaulay duration Yield 1+ k

where k is the number of coupons per year. If the Macaulay duration is 10 years and the yield to maturity is 8 percent, then modified duration equals 9/ (1+ ( 0.08/2 ) ) = 8.65 . 3(b).

3(c). 3(d).

For noncallable coupon bonds, modified duration is a better measure of the bond’s sensitivity to changes in interest rates. Maturity considers only the final cash flow (that is, the principal repayment and the last coupon), while modified duration considers all of the bond’s cash flows, as well as other factors. These factors are the size of coupon payments, the timing of coupon payments, and the level of interest rates (yield-to-maturity). Modified duration increases as the coupon rate decreases. Modified duration decreases as maturity decreases. Convexity measures the rate of change in modified duration as yields change. Convexity

refers to the shape of the price–yield relationship and can be used to refine the modified duration approximation of the sensitivity of prices to interest rate changes. Convexity shows the extent to which bond prices rise at a greater rate (as yields fall) than they fall (as yields rise). The effect of duration on price and the effect of convexity on price should be added together to obtain an improved approximation of the change in price for a given change in the market yield. 4(a).

Calculation of One-Year Forward Rate for January 1, 2027:

(1+ z4 ) = (1+z3 ) (1+4 r3 ) 4 3 (1+0.055) = (1.05) (1+4 r3 ) 4 3 4 r3 = ( 1+0.055 ) / (1.05 ) − 1 4

r = 1.2388/1.1576 − 1

4 3

r = 0.701 or 7.01%

4 3

OR Date

Calculation of Forward Rate (Linking Method)

1/1/24

3.5%

1/1/25

(1.045) = (1.035)(1+2r1 ) 2

r = 0.0551 or 5.51%

2 1

1/1/26

(1.05) = (1.035)  (1.0551)  (1+3r2 ) 3

r = 0.0601 or 6.01%

3 2

1/1/27

(1.055) = (1.035)  (1.0551)  (1.0601)  (1+4 r3 ) 4

r = 0.0701 or 7.01%

4 3

4(b).

4(c).

4(d).

The conditions would be those that underlie the pure expectations theory of the term structure: market participants who are willing to substitute among maturities solely on the basis of yield differentials. This behavior would rule out liquidity or term premia relating to risk as well as market segmentation based on maturity preferences. Under the expectations hypothesis, lower implied forward rates would indicate lower expected future spot rates for the corresponding period. Because the lower expected future rates embodied in the term structure are nominal rates, either lower expected future real rates or lower expected future inflation rates would be consistent with the specified change in the observed (implied) forward rate. Multiple scenario forecasting is a technique for developing a set of expectations that reflects the effects that different realizations of key economic variables will have on capital market returns. By providing a mechanism that reveals information about market participants’ expectations for future interest rate levels (and about changes in these expectations), the term structure can be a useful source of input in the process of assembling scenario forecasts. The decline in anticipated future short-term nominal rates could be depicted either by revising the scenario outcome (the states) to reflect the possibility of

lower future rates or by reweighting the probabilities attached to the scenarios to increase the odds associated with lower future rates. 5. n +t n r = (1+zn +t ) / (1+zn )   

t +n t

1/n

−1

5 3 r = (1 + 0.0651) / (1+0.0619 )   

5 3

1/2

− 1 = (1.37073/1.197432 )

1/2

− 1 = 6.99%

The assumption underlying the calculation of the implied forward rate is that, regardless of the expected holding period, the expected return would be the same for all maturity strategies. That is, it would not matter whether an investor would purchase a one-year investment or a two-year investment and sell it at the end of one year. The return would be the same for any strategy. 6.

(i) The manager purchased a longer maturity, lower coupon bond; by purchasing a longer duration bond, the manager must expect market interest rates to fall. (ii) The manager will benefit if the shape of the yield curve either stays flat or becomes downward sloping because he has replaced a shorter-term bond that makes payments every six months with one that has no interim cash flows. Thus, beyond predicting a general decline in rates, the manager may also be anticipating a shift in the shape of the yield curve. Notice that a forecast of an inverted yield curve often precedes a recessionary economic environment, which is consistent with the “flight to safety” trade indicated here. (iii) By selling a corporate and purchasing a Treasury bond, the manager evidently is concerned that the credit spread (risk premium) for corporate bonds will widen; if this indeed comes to pass, the Treasury will outperform as the corporate yields rise relative to Treasuries and as corporate bond prices fall relative to Treasuries.

7(a).

Computation of Duration (assuming 8% market yield) (1)

(2)

(3)

(4)

Year

Cash Flow

PV@8%

(5)

(6)

(1)  ( 5)

of PV as % of Price

Flow 1

100

0.9259

92.59

0.0868

100

0.8573

85.73

0.0804

0.1608

100

0.7938

79.38

0.0745

0.2234

1100

0.7350

808.50

0.7583

3.0332

1066.24

1.0000

3.5042

Duration = 3.50 years 7(b).

Computation of Duration (assuming 12% market yield) (1)

(2)

(3)

(4)

Year

Cash Flow

PV@12%

PV of Flow

(5) PV as % of Price

(6)

(1)  ( 5)

100

0.8929

89.29

0.0951

100

0.7972

79.72

0.0849

0.1698

100

0.7118

71.18

0.0758

0.2274

1100

0.6355

699.05

0.7442

2.9768

939.24

1.0000

3.4691

Duration = 3.47 years 7(c).

A portfolio of bonds is immunized from interest rate risk if the duration of the portfolio is always equal to the desired investment horizon. In this example, although nothing changes regarding the bond, there is a change in market rates, which causes a change in duration in the opposite direction. This could mean that, assuming the portfolio duration matched the planning horizon (duration of liabilities) when the market yield was 8%, it would no longer be perfectly immunized when rates increased to 12%.

Assuming semiannual coupons. Current and one-year-later prices can easily be established using a financial calculator or spreadsheet model: CURRENT CANDIDATE BOND

BOND

Dollar Investment

839.5388

961.1675

Coupon

45.00

55.00

i on One Coupon

2.5875

3.1625

Principal Value at Year End

841.9526

961.7171

Total Accrued

934.5401

1,074.8796

Realized Compound Yield

11.0127

11.5000

where



Realized Compound Yield = ( Total Accrued) / (Dollar Investment ) 

0.5



– 1 2

so, Value of swap: 48.73 basis points in one year 9(a).

The portfolio’s modified duration will be a weighted average of those of the component bonds. Because there are five equally weighted bonds, the weight of each is 0.20: Portfolio duration = 0.20 (2.727+6.404+3.704+4.868+10.909 ) = 5.722 years

9(b).

The cash coming into the portfolio (via interest payments and redemptions) needs to be reinvested. As the liability’s duration (6.50 years) is greater than the portfolio’s duration, the endowment is subject to the net reinvestment risk. That is, the assets are invested for a shorter average period of time than when the money is needed to pay for the liabilities, so those funds must be reinvested until the horizon date. Kritzman has shown that a portfolio’s convexity can be increased by spreading out the cash flows—that is, use a bond ladder strategy and/or seek higher coupon bonds so that cash flows will occur both earlier and later than the duration time frame. In this case, greater

9(c).

9(d).

convexity could be generated by increasing the investment weights to the most extreme positions (Bonds A and E) and reducing the weights to the other bonds, with the specific adjustments being managed so that the modified duration of the portfolio remains at 5.722. If Treasury yields are expected to decline and corporate credit spreads will decline, a bond portfolio manager should (i) extend portfolio duration to capture more price appreciation as yields decline; this can be done by purchasing longer maturity bonds and bonds with lower coupon rates (such as zeros) and (ii) purchase corporate bonds because if their yields follow Treasuries and decline—and the corporate credit spreads decline, too—they will benefit from price appreciation. In the current portfolio, this would lead to increased allocations for Bonds B and E.

10(a). $300 million  (1.05) = $488.6684 million 10

PV of $488.6684 million for 4 years @ 10 percent: $488.76 / (1.05)

10(b). PV of $488.6684 million for 3 years @ 8 percent: $488.6684 / (1.04 )

= $330.7500 million 6

= $386.2017 million

PV of $488.6684 million for 2 years @ 12 percent: $488.6684 / (1.06 ) = $387.0711 million 4

340.9 − 330.7500 = $10.1500 million margin of error

10(c). 405.5 − 386.2017 = $19.2983 million margin of error 395.2 − 387.0711 = $8.1289 million margin of error 10(d). If the margin of error at the end of any year had been zero or negative, the portfolio manager would discontinue active management of the portfolio and immunize it, thereby locking in as closely as possible the original floor rate that the client agreed to (10 percent, in this case). 11(a) (i) Adjusted Interest Coverage = EBITDA/Interest Expense = 2.81. This is a reduction of 1.91 from the pre-adjustment ratio of 4.72. Off-Balance Sheet

EBITDA Impact

Item

Interest

Expense Interest

Impact

Coverage Impact

Pre-adjustment

$4,450,000

$942,000

+$40,000

+$614,400

Net adjustment

+$40,000

+$654,400

Post-adjustment

$4,490,000

$1,596,400

2.81

- Guarantee of debt

4.72

(n/a) - Sale of receivables (1) - Operating lease (2)

(1)

Sale of receivables = Interest income (EBITDA ) & interest expense $500,000 @ 8% = $40,000

(2) Operating lease = Interest expense: $6,144,000 @ 10% = $614,400

(ii)

In adjusting for the accounts receivable that were sold and treating it as a secured loan, operating income (EBITDA) should be increased by the interest income ($40,000). This is because, presumably, the “loan proceeds” from the financed receivables would be invested to generate interest income (for simplicity’s sake, the same rate of interest is assumed—an alternative rate could be justified). The financing costs of the loan ($40,000) would be added to interest expense and will not impact EBITDA. In adjusting for the operating lease and treating it as a capital lease, the interest expense for the first year of the lease ($614,400) should be added to the adjusted interest expense. [Note that this is a new operating lease and therefore there is no adjustment necessary for operating lease expenses because they have not yet been incurred and included in the financials. Had the lease been in effect for the past year, the present value of the lease would now be less than the full amount ($6,144,000), and the principal and interest adjustments would be less. In addition, there would need to be other adjustments made: operating income (EBITDA) would be increased by the amount of the annual lease payment— $1,000,000 for operating lease expense eliminated and, correspondingly, depreciation expense would be increased to cover the first year’s write-off of the lease asset.] Adjusted Leverage = Long-term Debt/Equity = 0.50. This is an increase of 0.20 from the pre-adjustment ratio of 0.30. Off-Balance Sheet

Long-term Debt

Item

Impact

Pre-adjustment

Equity Impact

Leverage Impact

$10,000,000

$33,460,000

+$995,000

- Operating lease (2)

+$5,758,400

Net adjustment

+$6,753,400

Post-adjustment

$16,753,400

$33,460,000

- Guarantee of debt

0.30

(n/a) - Sale of receivables (1)

0.50

(1) Guarantee of debt = Long-term debt: $995,000 . (2) Operating lease = Long-term debt: $6,144,000 (PV of lease ) − $385,600 ( current portion) = $5,758,400. In adjusting for the guarantee of the affiliate’s debt and treating it as internal long-term debt, long-term debt should be increased by the amount of the guarantee ($995,000). In adjusting for the operating lease and treating it as a capital lease, long-term debt should be increased by the present value of the lease ($6,144,000) less the current or short-term portion—the principal due in the next 12 months ($1,000,000–$614,400 = $385,600).

(iii).

Adjusted Current Ratio = Current Assets/Current Liabilities = 0.97. This is a reduction of 0.08 from the pre-adjustment ratio of 1.05. Off-Balance Sheet

Current Assets

Current Liabilities

Current

Item

Impact

Ratio Impact

Pre-adjustment

$4,735,000

$4,500,000

+$500,000

+$385,600

Net adjustment

+$500,000

+$885,600

Post-adjustment

$5,235,000

$5,385,600

- Guarantee of debt

1.05

(n/a) - Sale of receivables (1) - Operating lease (2)

(1)

0.97

Sale of receivables = Accounts receivable ( current assets ) & notes payable

( current liabilities ): $500,000 Operating lease = Current portion of lease obligation ( current liabilities ):

(2)

$1,000,000 ( annual payment ) − $614,400 (interest expense ) = $385,600 ( principal payment ) .

In adjusting for the accounts receivable that were sold and treating it as a secured loan, accounts receivable (current assets) and notes payable (current liabilities) should both be increased by the amount of the sale ($500,000). In adjusting for the operating lease and treating it as a capital lease, leases payable (current liabilities) should be increased by the principal due in the next 12 months on the lease. This is equal to the annual lease payment less the first year’s interest expense ( $1,000,000 − $614,400 = $385,600 ) . 11(b). The current “A” rating of the Montrose bond does not incorporate the effect of the offbalance-sheet items, and the current credit spread of 55 basis points is not sufficient to compensate Smith for the true credit risk of the bond. After adjusting for the three offbalance sheet items, all three internal bond-rating criteria indicate that the Montrose bond should have a lower credit rating: • The lower interest coverage ratio (2.81) indicates that the bond is riskier, should have a lower (BB) rating, and should have a higher credit spread (+125 basis points). • The higher leverage ratio (0.50) indicates that the bond is riskier, should have a lower (BBB) rating, and should have a higher credit spread (+100 basis points).

•

The lower current ratio (0.97) indicates that the bond is riskier, should have a lower (BBB) rating, and should have a higher credit (+100 basis points). Montrose should be rated BBB instead of A and should be paying a yield risk premium of at least 100 basis points. The Montrose bond is riskier than it appears, and Smith should not purchase the bond at its current price.

Solution and Answer Guide: Frank K. Reilly, Keith, C. Brown, Sanford J. Leeds, Investment Analysis & Portfolio Management, 12th Edition, © 2025, 9780357988176; Chapter 14: An Introduction To Derivative Markets And Securities

Solution and Answer Guide

ANSWERS TO QUESTIONS 1.

It is generally true that futures contracts are traded on exchanges, whereas forward contracts are transacted directly with a financial institution. Consequently, there is a liquid market for most exchange-traded futures, whereas there is no guarantee of closing out a forward position quickly or cheaply. However, the liquidity of futures comes at a price. Because the futures contracts are exchange traded, they need to be standardized with set delivery dates, contract sizes, and underlying assets. This standardization leads to tracking error in a hedge position to the extent that any of the prespecified terms are not consistent with the counterparty’s actual portfolio position. If having a specialized delivery date, contract size, or underlying asset that is not available with a futures contract is important to a counterparty, then the forward may be more appealing despite the greater difficulty getting out of the contract before the expiration date. If liquidity is an important factor, then the user may prefer the futures contract. Another consideration is the mark-to-market property of futures. If a firm is hedging an exposure that is not marked to market, it may prefer to not have any intervening cash flows associated with the exchange-traded derivative; hence, it will prefer forwards.

For forwards, calls, and puts, what the long position gains, the short position loses, and vice versa. However, while payoffs to forward positions are symmetric, payoffs to call and put positions are asymmetric. That is, long and short forwards can gain as much as they can lose, whereas long calls and puts have a gain potential dramatically greater than their loss potential. Conversely, short calls and puts have gains limited to the option premium received upfront, but have unlimited or extremely large liabilities. This can be summarized as follows: assuming the underlying asset is an equity position, which itself has limited downside liability and unlimited upside potential and X represents the exercise price for the option contract: Position

Loss Potential

Gain Potential

Long Forward

100%

Unlimited

Symmetric

Short Forward

Unlimited

100%

Symmetric

Call premium

Unlimited

Asymmetric

Long Call

Payoff Symmetry

Short Call

Unlimited

Call premium

Asymmetric

Long Put

Put premium

X—premium

Asymmetric

Short Put

X—premium

Put premium

Asymmetric

3(a). Derivatives can be used in an attempt to bridge the 90-day time gap in the following ways: (1) The hedge fund could buy (long) calls on an equity index, such as the S&P 500 Index, and on Treasury bonds, notes, or bills. This strategy would require the hedge fund to make an immediate cash outlay for the premiums on the calls. If the hedge fund were to buy calls in the amount of the anticipated future cash inflows, the cost of these calls could be substantial, particularly if their strike prices were close to current stock and bond prices (that is, the calls were close to being “in the money”). (2) The hedge fund could write or sell (short) puts on an equity index and on Treasury bonds, notes, or bills. By writing puts, the hedge fund would receive an immediate cash inflow equal to the premiums on the puts (less brokerage commissions). If stock and bond prices rise as the manager expects, the puts would expire out of the money, and the hedge fund would keep the premiums, thus realizing at least part of the market increase. If the prices fall, however, the hedge fund loses the difference between the strike price and the current market price, which may be greater than the value of the premiums. (3). The hedge fund could buy (long) equity and fixed-income futures. This is probably the most practical way for the hedge fund to hedge its expected gift. Futures are available on the S&P 500 Index and on Treasury bonds, notes, and bills. No cash outlay would be required. Instead, the hedge fund could use some of its current portfolio as a good faith deposit, or margin, to take the long positions. The market value of the futures contracts will, in general, mirror changes in the underlying market values of the S&P 500 Index and Treasuries. Although no immediate cash outlay is required, any gains (losses) in the value of the contracts will be added (subtracted) from the margin deposit on a daily basis (that is, marked to market). Hence, if stock and bond markets advance as the manager expects, the balances in the hedge fund’s futures account will reflect the market increase. 3(b). There are both positive and negative factors to be considered in using derivatives to bridge the time gap before the expected cash inflows arrive in three months. Positive factors (1) The hedge fund could establish its position in stock and bond markets using derivatives today and benefit from any subsequent increases in market values in the S&P Index and Treasury instruments in the time period. In effect, the hedge fund would have a synthetic position in those markets beginning today. (2) The cost of establishing the synthetic position is relatively low, depending on the derivative strategy used. If calls are used, the cost is limited to the premiums paid. If futures are used, the losses on the futures contracts would be similar to the amounts that would be lost if the hedge fund invested cash today. Writing the puts is the riskiest strategy because there is an open-ended loss if the market declines, but here again the losses would be similar if the hedge fund invested today and stock and bond markets declined. (3) Relative to holding the underlying asset directly, doing so indirectly using derivatives effectively creates a leveraged investment; it can be equivalent to investing in the underlying stock or bond position with borrowed money. This leverage can dramatically increase the potential return, assuming market prices move up as expected.

(4) Derivative markets (for the types of contracts under consideration here) are liquid, which makes it relatively straightforward to trade out of futures or option positions if necessary. Negative factors (1) The expected cash flows could be delayed or not received at all. This would create a situation in which the hedge fund would have to unwind its position and could experience losses, depending on market movements in the underlying assets. (2) The manager may be wrong in his or her expectation that stock and bond prices will rise in the time period. If prices decline on stocks and bonds, the leverage inherent in the derivative contracts would work against the manager and the losses could become extreme for even a relatively modest change in stock and bond prices. (3) Because there is a limited choice of option and futures derivative contract compared to the universe that the manager may wish to invest in, there could be a mismatch between the specific equities and bonds the hedge fund wishes to invest in and the contracts available in size for expected cash flows. Unless the time period exactly matches, say, the 90-day period before expiration dates on the contracts, there may be a timing mismatch. (4) The cost of the derivatives is potentially high. For example, if the market in general shares the manager’s optimistic outlook, the premiums paid for calls would be expensive and the premiums received on puts would be lean. The opportunity cost of all derivative strategies discussed would be large if the manager is wrong on the outlook for one or both markets. The negative factors appear to outweigh the positive factors if the outlook for the market is neutral; therefore, the manager’s decision to use derivatives to bridge the gap for three months will depend on the strength of his or her conviction that stock and bond prices will rise in that period. The manager should certainly be aware that there is a cost to establish the derivative positions, especially if his or her expectations do not work out. The manager may want to consider a partial hedge of the expected cash flows. 4.

Because the manager is considering adding either short index futures or long index put options (a form of protective put) to an existing well-diversified equity portfolio, he or she evidently intends to create a hedged position for the existing portfolio. Both the short futures and the long put positions will reduce the risk of the resulting combined portfolios but in different ways. Assuming that the short futures contract is perfectly negatively correlated with the existing equity portfolio and that the size of the futures position is sufficient to hedge the risk of the entire equity portfolio, any movement up or down in the level of stock market prices will result in offsetting gains and losses in the combined portfolio’s two segments (the equity portfolio itself and the short futures position). Thus, the manager is effectively removing the portfolio from exposure to market movements by eliminating all risk, thereby becoming a synthetic T-bill position. Once the equity portfolio has been perfectly hedged in this manner, the manager can expect to receive a risk-free rate of return on the combined portfolio. If the hedge is less than perfect, some risk and some potential for return beyond the risk-free rate are present, but only in proportion to the completeness of the hedge. Also, since this strategy involves futures contracts, there is no upfront cost to the hedged position. If, on the other hand, the manager hedges the portfolio by purchasing stock index puts, he will be placing a floor price on the equity portfolio. If the market declines and the index value drops below the strike price of the puts, the value of the puts increases, thus offsetting the loss in the equity portfolio. Conversely, if the stock market rises, the value of the put

options will decline, and they may expire worthless; however, the potential return to the combined portfolio is unlimited, reduced only by the cost of the puts. As with the short futures, if the long options hedge is less than perfect, downside risk remains in the combined portfolio in proportion to the amount not covered by the puts. However, in exchange for keeping the upside potential in the underlying stock portfolio, the manager will have to pay an upfront price for this “insurance,” which is the premium on the put option. In summary, either short futures or long puts can be used to reduce or eliminate risk in the equity portfolio. The use of the protective put strategy, however, permits unlimited potential returns to be realized (less the cost of the options), while the use of the short futures strategy effectively guarantees the risk-free rate but reduces or eliminates potential returns above that level. Neither strategy dominates the other; each offers a different risk/return profile and involves different costs. Arbitrage ensures that, on a risk-adjusted basis, neither approach is superior. 5.

When used as stand-alone investments (that is, not combined with the underlying asset), options represent leveraged ways to participate in the upward (calls) or downward (puts) price movement in the underlying stock positions. This is because those derivatives participate in the full price movement of the stock (relative to the contract’s exercise price) but for a relatively small fraction of the price of the stock itself. This leverage feature, which is embedded in the design of the option contract, has the effect of substantially increasing the risk-return profile of the underlying asset. By contrast, when options are used in conjunction with a position in the underlying asset, they can substantially reduce the overall level of risk in the portfolio. An example of this is the protective put position, which combines a long position in the stock with a long position in a put option on that stock. The put effectively provides price insurance to the portfolio by adding back any stock price declines below the exercise price of the contract. That is, the protective put eliminates some or all of the stock’s downside risk at the expense of the initial premium on the option.

A long position in a forward contract gives the holder the right and the full obligation to buy the underlying asset at a prespecified price (forward contract price) at a prespecified date in the future (expiration date). If the expiration date spot price for the asset is above (below) the forward contract price, this will result in a positive (negative) payoff to the long forward position since the forward contract holder is fully obligated to purchase the asset regardless of the terminal spot price. By contrast, a long call position allows the holder to buy the asset at the expiration—which she will do if the spot price exceeds the exercise price—but does not require that purchase to happen. This is equivalent to owning the “good half” of the forward contract (that is, being able to purchase the asset in a favorable market condition). The “bad half” of the forward would be having to purchase the asset when you do not want to, which occurs when the spot price at expiration is below the exercise price. This would be equivalent to having a short position in a put option at the same exercise price and expiration date as the long position in the call.

The put–call–spot parity relationship can be expressed as follows:

(Long Stock ) + (Long Put ) + ( Short Call) = (Long T-Bill) or

S0 + P0,T − C 0,T =

X (1 + RFR)T

One of the four assets represented in this equation is always redundant because its cash flow profile can always be replicated by a properly designed portfolio that combines the other three securities. For instance, the above equation summarizes how a synthetic T-bill position could be constructed by combining the stock, the put, and the call. The transactions that would be necessary to make the call option the redundant asset would be (1) long in the stock, (2) long in the put option, and (3)short in the T-bill, or X C 0,T = S0 + P0,T − (1 + RFR)T 8.

If the underlying security pays a dividend during the life of an option contract, the current stock price must be adjusted downward by the present value of the dividend. The payment of the dividend reduces the price of the call relative to the put by the discounted amount of the cash distribution. Expressed as an extension of the put–call–spot parity model, suppose the stock pays a dividend of DT immediately prior to the expiration of the options at Date T and that the amount of this distribution is known when the investment is initiated. With this adjustment, the terminal value of the long stock position will be ( ST +DT ) , while the terminal payoffs to the put and call options remain max  0, X − ST  and max  0, ST – X  , respectively, as the holders of the two derivative contracts will not participate directly in the payment of dividends to the stockholder. Thus, the net Date T value of the portfolio acquired originally for ( S0 +P0 ,T − C 0 ,T ) is ( X +DT ) . Because the dividend payment is known at Date 0, the portfolio long in the stock, long in the put, and short in the call once again can be viewed as equivalent to a T-bill, now having a face value of ( X +DT ) . This allows the put–call–spot parity condition to be adapted as follows: X + DT DT X S0 + P0,T − C 0,T = + + T T (1 + RFR) (1 + RFR) (1 + RFR)T which can be interpreted as

(Long Stock ) + (Long Put ) + ( Short Call) = (Long T-Bill) + (Long Present Value of Dividends ) This can be rearranged as follows: DT X   + P0,T − C0,T =  S0 − T  (1 + RFR)  (1 + RFR)T 

9. (i) Short a three-month forward contract. This is a hedge position, where the price risk of the underlying asset is fully offset by a supplementary derivative transaction. To neutralize the risk of falling stock prices, the fund manager requires a hedge position with payoffs that are negatively correlated with the existing exposure. The expiration date value of the hedged stock position will be as follows: (1) the value of the unhedged stock portfolio, plus (2) the value of the short forward position less (3) the initial cost of the derivative position, which is zero in the case of a

forward contract. This position is equivalent to holding a synthetic T-bill because the short forward positions offset all of the future price risk of the underlying stock. (ii) Buy an at-the-money three-month put option for cash. The purchase of a put option to hedge the downside risk of an underlying security holding is called a protective put position. The put contract exactly offsets any expiration date share price decline below the exercise price while allowing the portfolio to increase in value as stock prices increase. Thus, the put provides the manager with insurance against falling prices with no deductible. The initial cost of this insurance is the upfront premium or the cost of the put option. The expiration date payoff potential of this protective put position generates the same expiration date payoff as a long position in a call option with equivalent characteristics and a long position in a T-bill with a face value equal to the options’ exercise price: X S0 + P0,T = C 0,T + (1 + RFR)T (iii) Buy an out-of-the-money three-month put option and sell an out-of-the-money threemonth call option. The simultaneous purchase of an out-of-the-money put and sale of an out-of-the-money call on the same underlying asset and with the same expiration date and market price is a strategy known as a collar agreement. Like the short forward hedge, there is no initial outof-pocket expense associated with this derivative combination. Instead, the manager effectively pays for portfolio insurance (that is, the put option) by surrendering an equivalent amount of the portfolio’s future upside potential (that is, selling the call option). As with the protective put (and unlike with the short forward position), the manager retains some of the benefits of a rising stock market. However, this upside gain potential stops at the exercise price of the call option. 10.

Options have asymmetric payoff structures, so any portfolio that contains options will also see their risk-return profiles altered in a way that changes their inherent symmetry profile. When this occurs, symmetric risk measures such as the standard deviation, which capture stock price volatility both above and below the expected return, may be misleading. For example, a portfolio that is put-protected may not move down much if the market declines by 10 percent but may move up nearly 10 percent if the market rises by 10 percent. Consequently, the returns are asymmetric or skewed. This makes standard deviation a less informative statistic because it reveals information only about the degree of variation and not the “direction” of variation. Because investors usually care the most about downside risk, they will likely not be concerned if all of the portfolio’s price variation is on the upside. The standard deviation statistic could be modified to only measure the variation in negative returns so that it is a measure of downside risk only.

ANSWERS TO PROBLEMS l(a). (i). A long position in a forward with a contract price of $50. Expiration Date Sophia

Long Forward

Initial Long

Stock Price (S)

(X = $50) Payoff = S−50

Forward Premium

Net Profit

($25.00)

$0.00

($25.00)

($20.00)

$0.00

($20.00)

($15.00)

$0.00

($15.00)

($10.00)

$0.00

($10.00)

($5.00)

$0.00

($5.00)

$0.00

$5.00

$0.00

$5.00

$10.00

$0.00

$10.00

$15.00

$0.00

$15.00

$20.00

$0.00

$20.00

$25.00

$0.00

$25.00

(ii). A long position in a call option with an exercise price of $50 and a front-end premium expense of $5.20. Expiration Date Sophia

Long Call (K = $50)

Initial Long

Stock Price (S)

Payoff = max (0,S−50)

Call Premium

$0.00

($5.20)

$0.00

($5.20)

$0 00

($5.20)

$0.00

($5.20)

$0 00

($5.20)

$0.00

($5 20)

$5.00

($5 20)

($0.20)

$10.00

($5.20)

$4.80

$15.00

($5.20)

$9.80

$20.00

($5.20)

$14.80

$25.00

($5.20)

$19.80

Net Profit

(iii). A short position in a call option with an exercise price of $50 and a front-end premium receipt of $5.20. Expiration Date Sophia

Short Call (K = $50)

Initial Short

Stock Price (S)

Payoff = -max (0,S−50)

Call Premium

Net Profit

$0.00

$5.20

$0.00

$5.20

$0.00

$5.20

$0.00

$5.20

$0.00

$5.20

$0.00

$5.20

($5.00)

$5.20

$0.20

($10.00)

$5.20

($4.80)

($15.00)

$5.20

($9.80)

($20.00)

$5.20

($14.80)

($25.00)

$5.20

($19.80)

l(b). (i). A long position in a forward with a contract price of $50.

(ii.) A long position in a call option with an exercise price of $50 and a front-end premium expense of $5.20.

(iii.) A short position in a call option with an exercise price of $50 and a front-end premium receipt of $5.20.

THE BREAKEVEN POINT FOR THE CALL OPTIONS IS $55.20. l(c). The long position in a forward with a contract price of $50: The purchaser believes that the price of Sophia Enterprises stock will be above $50. The long position in a call option with an exercise price of $50 and a front-end premium expense of $5.20: The purchaser believes the price will be above $55.20. The short position in a call option with an exercise price of $50 and a front-end premium receipt of $5.20: The seller believes the price of Sophia Enterprises stock will be below $55.20.

2(a). (i). A short position in a forward with a contract price of $50. Expiration Date Sophia

Short Forward

Stock Price (S)

(K = $50) Payoff = S−50

Initial Short Forward Premium

Net Profit

$25.00

$0.00

$25 00

$20.00

$0.00

$20.00

$15.00

$0.00

$15.00

$10.00

$0.00

$10.00

$5.00

$0.00

$5.00

$0.00

($5.00)

$0.00

($5.00)

($10.00)

$0.00

($10.00)

($15.00)

$0.00

($15.00)

($20.00)

$0.00

($20.00)

($25.00)

$0.00

($25.00)

(ii). A long position in a put option with an exercise price of $50 and a front-end premium expense of $3.23. Expiration Date Sophia

Long Put (K = $50)

Initial Long

Stock Price (S)

Payoff = max (0,50−S)

Put Premium

$25.00

($3.23)

$21.77

$20.00

($3.23)

$16.77

$15.00

($3.23)

$11.77

$10.00

($3.23)

$6.77

$5.00

($3.23)

$1.77

$0.00

($3.23)

$0.00

($3.23)

$0.00

($3.23)

$0.00

($3.23)

$0.00

($3.23)

$0.00

($3.23)

Net Profit

(iii.) A short position in a put option with an exercise price of $50 and a front-end premium receipt of $3.23. Expiration Date Sophia

Short Put (K = $50)

Initial Short

Stock Price (S)

Payoff = −max (0,50-S)

Put Premium

($25.00)

$3.23

($21.77)

($20.00)

$3.23

($16.77)

($15.00)

$3.23

($11.77)

($10.00)

$3.23

($6.77)

($5.00)

$3.23

($1.77)

$0.00

$3.23

$0.00

$3.23

$0.00

$3.23

$0.00

$3.23

$0.00

$3.23

$0.00

$3.23

Net Profit

2(b). (i). A short position in a forward with a contract price of $50.

(ii). A long position in put option with an exercise price of $50 and front-end premium expenses of $3.23.

(iii). A short position in a put option with an exercise price of $50 and a front-end premium receipt of $3.23.

THE BREAKEVEN POINT FOR BOTH PUT OPTIONS IS $46.77. 2(c).

A short position in a forward with a contract price of $50: The seller believes the price of Sophia Enterprises will be below $50. A long position in a put option with an exercise price of $50 and front-end premium expense of $3.23: The buyer of the put believes the price will be below $46.77. A short position in a put option with an exercise price of $50 and a front-end premium receipt of $3.23: The seller of the put believes the price will be above $46.77.

3(a). (i). A short position in a forward option with an exercise price of $50. Expiration Date Sophia

Short Forward (X = $50)

Initial Short

Stock Price (S)

Payoff = max (0,S−50)

Forward Premium

Net Profit

$25.00

$0.00

$50.00

$20.00

$0.00

$50.00

$15.00

$0.00

$50.00

$10.00

$0.00

$50.00

$5.00

$0.00

$50.00

$0.00

$50.00

($5.00)

$0.00

$50.00

($10.00)

$0.00

$50.00

($15.00)

$0.00

$50.00

($20.00)

$0.00

$50.00

($25.00)

$0.00

$50.00

(ii). A long position in a put option with an exercise price of $50 and a front-end premium expense of $3.23. Expiration Date Sophia

Long Put (K = $50)

Initial Long Put

Stock Price (S)

Payoff = max (0,50−S)

Put Premium

Net Profit

$25.00

($3.23)

$46.77

$20.00

($3.23)

$46.77

$15.00

($3.23)

$46.77

$10.00

($3.23)

$46.77

$5.00

($3.23)

$46.77

$0.00

($3.23)

$46.77

$0.00

($3.23)

$51.77

$0.00

($3.23)

$56.77

$0.00

($3.23)

$61.77

$0.00

($3.23)

$66.77

$0.00

($3.23)

$71.77

(iii). A short position in a call option with an exercise price of $50 and a front-end premium receipt of $5.20. Expiration Date Sophia

Short Call (K = $50)

Initial Short

Stock Price (S)

Payoff = −max (0,S−50)

Call Premium

Net Profit

$0.00

$5.20

$30.20

$0.00

$5.20

$35.20

$0.00

$5.20

$40.20

$0.00

$5.20

$15.20

$0.00

$5.20

$50.20

$0.00

$5.20

$55.20

($5.00)

$5.20

$55.20

($10.00)

$5.20

$55.20

($15.00)

$5.20

$55.20

($20.00)

$5.20

$55.20

($25.00)

$5.20

$55.20

3(b). (i). A short position in a forward with a contract price of $50.

(ii). A long position in a put option with an exercise price of $50 and a front-end premium expense of $3.23.

(iii). A short position in a call option with an exercise price of $50 and a front-end premium expense of $5.20.

3(c). F0,T = Call − Put + PV ( Strike ) $50.00 = 5.20 − 3.23 + PV ( $50 ) $48.03 = PV ( $50 )

so, PV Factor = $48.03/$50 = 0.9606 S = F * (PV Factor ) $50 = F *0.9606 $52.05 = F

The zero-value contract price, $52.05, differs from the $50 contract price because the put and the call prices are not the same. If they were, the combination of the two would yield a zero-value forward price. 4(a).

With $13,700 to spend, one could: (1) purchase 100 shares of Breener Inc. stock (@ $137 per share) or

(2) purchase call options with an exercise price of $140 based on 100 shares of the underlying Breener stock. The potential payoff is unlimited in both cases; however, the leverage that options provide will translate into a higher percentage gain than purely purchasing stock. However, leverage works both ways and the call option position will experience a 100% loss (that is, be out of the money) if Breener stock price is below $140 on the expiration date. 4(b). (1) Stock price increases to $155. a.

Stock return = ( $155 − $137 ) /$137 = 13.14%

Option Payoff: Max ( 0, 155 − 140 ) = $15

Option expense = $10 Option return = (15 − 10 ) /10 = 50%

(2) Stock price decreases to $135. Stock return = ( $135 − $137 ) /$137 = −1.46%

Option return: Option would not be exercised, lose entire option purchase price

b. 4(c).

or, ( 0 − 10 ) /10 = −100%

The breakeven on this call option is $150, which is equal to the exercise price ($140) plus the call premium ($10). In other words, the writer of the call option will receive the initial premium of $10, which is all he or she will ever receive from the transaction. So, once the stock price rises to $150, the short position in the call will have exhausted the premium he or she received. If the seller does not currently own the stock, his or her loss is potentially unlimited as there is no limit to how high Breener’s stock price might rise.

5(a).

Given: Current Price of XYZ = $42 Put Premium ( X = $40 ) = $1.45 Call Premium ( X = $40 ) = $3.90 RFR = 8% ( annual ) ; 4% ( semiannual )

(i). Buy one call option Expiration Date

Long Call (K = $40)

Initial Long

XYZ Stock Price (S)

Payoff = max (0,S−40)

Call Premium

$0.00

($3.90)

Net Profit

($3.90)

$0.00

($3.90)

$0.00

($3.90)

$0.00

($3.90)

$0.00

($3.90)

$5.00

($3.90)

$1.10

$10.00

($3 90)

$6.10

$15.00

($3.90)

$11.10

$20.00

($3.90)

$16.10

Expiration Date

Short Call (K = $40)

Initial Short

XYZ Stock Price (S)

Payoff = −max (0,S−40)

Call Premium

$0.00

$3.90

$0.00

$3.90

$0.00

$3.90

$0.00

$3.90

$0.00

$3.90

($5.00)

$3.90

($1.10)

($10.00)

$3.90

($6.10)

($15.00)

$3.90

($11.10)

($20.00)

$3.90

($16.10)

(ii). Short one call option

Net Profit

Both call positions will break even at a stock price of $43.90.

5(b). (i). Buy one put option Expiration Date

Long Put (K = $40)

Initial Long

XYZ Stock Price (S)

Payoff = max (0,40−S)

Put Premium

$20.00

$1.45

$18.55

$15.00

$1.45

$13.55

$10.00

$1.45

$8.55

$5.00

$1.45

$3.55

$0.00

$1.45

($1.45)

$0.00

$1.45

($1.45)

$0.00

$1.45

($1.45)

$0.00

$1.45

($1.45)

$0.00

$1.45

($1.45)

Net Profit

(ii). Short one put option Expiration Date

Long Put (K = $40)

Initial Long

Payoff = −max (0,40−S)

Put Premium

Net Profit

($20.00)

$1.45

($18.55)

($15.00)

$1.45

($13.55)

($10.00)

$1.45

($8.55)

($5.00)

$1.45

($3.55)

$0.00

$1.45

$0.00

$1.45

$0.00

$1.45

$0.00

$1.45

$0.00

$1.45

XYZ Stock Price (S)

Both put positions will break even at a stock price of $38.55.

Does Call − Put = S − PV ( exercise price ) ?

5(c). $3.90 − $1.45  $42 − 40/ ( 1.04 ) $2.45  $3.54

So, put–call parity does not hold in this situation. The prevailing call price of $3.90 is too low relative to the prevailing put price of $1.45. 6(a).

The put–call–forward parity relationship can be summarized as P − C = PV ( X ) − PV ( F )

where X is the exercise price for the two options and F is the forward contract price. So, when X = 50, the “no arbitrage price of the put option is P = C +PV ( X ) − PV ( F ) = 2.47+ ( 50/1.03 ) – ( 48/1.03 ) = $4.41

Similarly, the “no arbitrage” price of a call option with an exercise price of $45 can be expressed as C = P – PV ( X ) +PV ( F ) = 1.93 – ( 45/1.03 ) + ( 48/1.03 ) = $4.84

6(b).

The “no arbitrage” difference between the call and the put both having the same exercise price is given as C – P = PV ( F ) − PV ( X )

So, for X = 40 and F = 48: C – P = ( 48/1.03 ) – ( 40/1.03 ) = $7.77

However, the actual price difference between the call and the put with an exercise price of $40 is ( 8.73 − 0.59 ) = $8.14 . This means that the current market price of the call option is too high relative to the price of the put (or, equivalently, the put price is too low relative to the call price). Thus, the arbitrage trade to correct this pricing “mistake” would require buying the call option and selling the put option. Specifically, the following combination of positions would capture this $0.37 ( = 8.14 − 7.77 ) price differential: (1) short call option, (2) long put option, (3) long T-bill with a face value of $40, (4) short futures with a contract price of $48. Notice that transactions (1) and (2) are equivalent to “selling” the actual [C−P], which is overpriced, and transactions (3) and (4) are equivalent to “buying” the theoretical [C−P], which is the cheaper alternative. 7(a).

Alternative 1 (Buy Protective Put Options)

•

Buy S&P 500 put options with market exposure equal to equity holdings to protect these holdings.

•

Buy Government bond put options with market exposure equal to the bond holdings to protect these holdings.

This could be done by (1) buying $350 million of S&P 500 put options. Because each option is equivalent to $45,000 market exposure, this would require $350,000,000/ ( $45,000/option ) = 7,778 S&P 500 put options

and (2) buying $350 million of Government bond put options. Because each option is equivalent to $100,000 market exposure, this would require $350,000,000/ ( $100,000/option) = 3,500 Government bond put options

By buying 7,778 S&P 500 put options and 3,500 Government bond put options, the portfolio is protected against both stock market losses (below the current stock index price of $1,000) and bond market losses (below the current bond price of $100). Also, this strategy retains the upside potential of rising stock and bond prices, should either of those occur over the next three months. However, this “insurance” comes at a considerable upfront cost: $140,004 ( = 7,778  $18.00 ) for the stock options and $22,750 ( = 3,500  $6.50 ) for the bond

options. Alternative 2 (Short Futures) •

Sell S&P stock index futures equal to the equity exposure in the portfolio.

•

Sell Government bond futures equal to the bond exposure in the portfolio.

This could be done by the following: (1) Shorting $350 million of S&P futures. Because each future is equivalent to $250,000 of equity exposure, this would require $350,000,000/ ( $250,000/future ) = 1,400 S&P futures

(2) Shorting $350 million of Government bond futures contracts. Because each bond future is equivalent to $100,000 of bond exposure, this would require $350,000,000/ ( $100,000/future ) = 3,500 bond futures

By shorting 1,400 S&P futures and 3,500 bond futures, the manager would be exactly offsetting all of the price volatility in the stock and bond markets over the next three months, effectively converting the endowment fund into a risk-free position (that is, a synthetic T-bill). This would require no upfront premium payment, but the strategy

surrenders all of the upside potential in the stock and bond markets over the next three months. 7(b).

Given the put–call parity relationship, the put options appear misvalued compared to the call options. Given the S&P 500 call price, the put should be priced at Put = 21.00 − index price + present value of ( strike + income )

Since the dividend yield is 3% and the time frame is 3 months, the expected income yield is 0.75% of the S&P: = 21.00 – 1,000+1000/1.012+ (1000*0.0075 ) /1.012 = 21.00 – 1,000+988.14+7.41 = $16.55

whereas the current price is $18.00. Given the bond call price, the bond put option should be priced at Put = 6.00 − bond price+present value of ( strike+income ) = 6.00 − 100+100/1.012+ (100* 0.01) /1.012 = 6.00 − 100+98.81+0.99 = $5.80

whereas the current price is $6.50. For both the S&P 500 options and the Government bond options, the puts appear overvalued compared to the prices of the calls. The prices of the futures also appear high. A fair price for the S&P 500 future would be

S& P 500 future = Index price+ ( T-bill income − dividend income ) = 1,000+ ( 0.012 − 0.0075)  (1,000 )  = $1,004.50 compared to the actual futures price of $1,007. A fair price for the bond future would be

Bond future price = T-bond price+ ( T-bill income – T-bond income ) = 100+ ( 0.012 − 0.01)  (100 )  = $100.20 compared to the actual futures price of $103.00. From this analysis, all of the derivatives being considered (stock and bond put options, stock and bond futures contracts) have prices that are too high relative to their theoretical “no arbitrage” levels. Alternative 1 involves buying relatively overvalued assets (the put options). Alternative 2 involves shorting overvalued assets (the futures contracts).

Alternative 2 where protection is gained by shorting futures is recommended, providing the manager is comfortable with surrendering the upside potential in the stock and bond markets in exchange for an above-market risk-free return over the next three months. 8(a).

8(b).

With a price of $97 for six-month T-bills, the six-month risk-free rate is 100/97 − 1 = 3.09% . Using put–call parity, the “no arbitrage price” is S = PV ( exercise price ) +C − P S = $60/1.0309+3.18 − 3.38 S = $58.00

8(c).

Because put–call parity indicates the “no arbitrage” price of the stock is $58 and the stock selling at $60, the stock is currently overvalued relative to the other three securities (that is, the T-bill, the put option, and the call option). So, the arbitrage trade would require the following four positions: (1) short the stock and use the proceeds to (2) buy the T-bill, (3) buy the call option, and (4) short the put option. This set of trades would generate the net of $2 in arbitrage profits. Because by put–call parity we know that the portfolios consisting of Trade (1) and Trades (2)–(4) will have exactly offsetting terminal payoffs, the entire transaction is riskless and requires no upfront capital investment, but will still make a profit.

9(a) Expiration Date

Long Put (X = $55)

Initial Long

Stock Price (S)

Payoff = max (0,55−S)

Put Premium

Net Profit

$20.00

($1.32)

$53.68

$15.00

($1.32)

$53.68

$10.00

($1.32)

$53.68

$5.00

($1.32)

$53.68

$0.00

($1.32)

$53.68

$0.00

($1.32)

$58.68

$0.00

($1.32)

$63.68

$0.00

($1.32)

$68.68

$0.00

($1.32)

$73.68

9(b)

The basis put–call parity relationship can be expressed as S +P − C = PV ( X ) , or (Long Stock ) + (Long Put ) + ( Short Call ) = (Long T-bill )

Rearranging this relationship, the protective put position can be synthetically reconstructed as S +P = PV ( X ) +C , or (Long Stock ) + (Long Put ) = (Long T-bill ) + (Long Call)

In this case, the purchase price of the T-bill position would be the option exercise price of $55 discounted at the risk-free rate of 7 percent. Also, the payoff diagram is essentially that of a long call position that has been “elevated” by the T-bill holding. 9(c). Expiration Date

Short Call (K = $55)

Initial Short

Stock Price (S)

Payoff = max (0,S−55)

Call Premium

Net Profit

$0.00

$2.55

$37.55

$0.00

$2.55

$42.55

$0.00

$2.55

$47.55

$0.00

$2.55

$52.55

$0.00

$2.55

$57.55

($5.00)

$2.55

$57.55

($10.00)

$2.55

$57.55

($15.00)

$2.55

$57.55

($20.00)

$2.55

$57.55

9(d).

Using put–call parity, the covered call strategy (that is, long stock and short calls) can be synthetically recreated as S − C = PV ( X ) − C , or (Long Stock ) + ( Short Call ) = (Long T-bill ) + ( Short Put )

So, the payoff diagram is essentially that of a short put position that has been “elevated” by the T-bill holding. 10(a). By put–call parity, we have S + P – C = PV ( X ) , or (Long Stock ) + (Long Put ) + ( Short Call ) = (Long T-bill)

So, the transactions needed to construct the synthetic T-bill would be (1) long the stock, (2) long the put, and (3)short the call. 10(b). Using the put–call parity formula, the “no arbitrage” present value of a T-bill with a face value of $21 would be 21.50+4.50 − 5.50 = $20.50

So, the three-return would be

21.00/20.50–1 = 2.44% , which would annualize to 9.76% ( = 2.44  4 ) 10(c). The difference between the actual T-bill yield (3.00 percent) and the synthetic T-bill yield

(9.76%) indicates that the actual T-bill is overvalued relative to the other three securities. So, the arbitrage trade would involve (1) shorting the actual T-bills and (2) purchasing the portfolio representing the synthetic T-bill. Specifically, the strategy would be to short 21 actual T-bills and to long 100 synthetic T-bills. Assume the actual T-bill was quoted on a bond equivalent basis and yields a 0.75% (= 3.00/4) quarterly return. The present value of the actual T-bill position would be $210,000/1.0075 = $208,437

At the time of initiation, the long position in the synthetic T-bill would be worth the following: Long stock

$215,000

Long put

$45,000

Short call

−$55,000 $205,500

Therefore, the net initial cash flow is $208,437 − $205,500 = $2,937

which represents the arbitrage profit since both the actual and the synthetic T-bills will be worth $210,000 (the option exercise price, scaled to the investment level) at the expiration date. 10(d). At the three-month expiration, the value of the long synthetic position (on a per-share basis) would be X = P + S –C

or = $0+$23 − $2 = $21

However, the maturity value of the short T-bill position would also be $21, which was the original face value, so that at the expiration date Net cash flow = $21 − $21 = $0

This can be seen a little differently looking at the expiration date values of all four related positions when the stock price in three months is $23: Long stock position

= $23

Short call position = $21 − $23

= $−2

Long put position

= $0

Short Treasury bill position

= −$21

Net position

Finally, notice that it makes no difference what the final stock price ends up being because the combination of being short the call and long the put insures that the stock will be sold for the common exercise price of $21 no matter what.

Solution and Answer Guide

ANSWERS TO QUESTIONS 1.

There are many different reasons why some futures contracts succeed and some fail, but the most important is demand. If people need a particular contract to expose themselves to or hedge price risk, then the contract will succeed. Most people use Treasury bond futures to gain exposure to or hedge general long-term interest rate risk. The only additional advantage of futures on corporate bonds would be that the investors could gain exposure to changes in the credit spread for a specific company. Apparently, there is little demand for this, either because investors do not want to hedge or gain exposure to this risk or because the underlying market is not liquid enough to support a widely traded futures contract. Either way, the lack of futures is motivated by a lack of demand in the asset or futures contract. The lack of chicken contracts most likely derives from a similar lack of demand. It could be that chicken prices are not extremely volatile or they are highly correlated with other existing contract prices, so investors do not need the additional chicken futures contract.

At any point in time, the basis is the difference between the prevailing spot price for the underlying asset and the futures contract price for a contract that expires at a later date. If the underlying asset for the futures contract is the same as the one being held, at the expiration of the futures contract, the basis will be zero as the spot and futures contract prices converge at maturity. In that scenario, an investor could remove all of the price risks of holding a risky asset with a short position in the futures contract. Expanding on this notion, before entering into a futures or forward contract, investors have exposure to price changes in the underlying asset. To hedge this risk, hedgers enter into those contracts that most closely offset this price risk. The problem is that for most hedgers, there is not a contract that exactly matches their exposure. Perhaps, the commodity they use is a different grade or needed in a different location than specified in the contract, so differences in prices between the actual asset the company is exposed to and the asset in the contract may exist and change over time. Likewise, a portfolio manager hedging a stock portfolio may hold a portfolio that is not perfectly correlated with the index future he or she is using. To minimize basis risk, it is necessary to find the contract whose price is most highly correlated with the price of the asset to be hedged.

3(a).

To hedge price risk, you could enter into a long position in a futures contract for 100,000

3(b).

gallons worth of unleaded gasoline, expiring in three months. That transaction would lock in the purchase price of the gasoline you have agreed to deliver to your client. In this case, by using futures, you will not be able to match the quantity or the timing of the delivery in the forward contract you sold. This gives rise to two sources of basis risk. First, because the position size for each futures contract does not divide evenly into the amount you have promised to deliver, you will be forced to over- or underhedge, which will expose you to the general price movements in gas one way or the other. Second, the timing mismatch between when you have committed to delivering gas and the future expiration dates will also create basis risk. If you synthetically create a three-month expiration using two-month and four-month contracts, then you will be exposed to relative price changes in the two futures contracts in the near term. Later, after the two-month contract expires, you will be forced to hedge a shorter-term position with a longer-term contract, so again relative near-term and longer-term price differentials will lead to basis risk.

There are two types of basis risk that this hedge is exposed to. The first is from changes in the shape of the yield curve. Because the company wishes to hedge a seven-year issue’s cost of funds with a ten-year contract, the hedge is exposed to changes in the relative level of interest rates between the seven- and ten-year maturities. Specifically, if the seven-year treasury rate rises relative to the 10-year rate—perhaps because a yield curve that was originally upward sloping begins to flatten—then the hedge will not completely neutralize the position and may even result in a higher funding cost for the firm. The second source of basis risk is from the credit spread over Treasuries in the Eurobond market. Because the company will have to sell its bonds with a spread over the Treasury rate, changes in this spread will also affect the quality of the hedge. Specifically, increases in the spread will lead to a higher fund cost for the firm that is not hedged by the futures contract.

5(a).

The hedging strategy using the 10-year Treasury note futures contract that would provide the best protection against this possible decline in yields is taking a long position in $50 million worth of U.S. Treasury 10-year-note futures contracts. The specific 10-year-note futures contracts employed should be the one with an expiration date closest to the actual date the $50 million will be invested. The long position in the futures contract essentially locks in the CFO’s purchase price for 10year notes in six months’ time, which in turn locks in the yield to maturity on this eventual bond holding. Again, this hedge was done to guard against the possibility that the actual 10year yield available in the market six months from now was lower than what could be locked in using the futures contract. So, if the yield curves move up instead of down over the next six months, the yield on the hedged position will be lower than what the CFO would have realized by being unhedged over this period. Thus, with perfect hindsight, it would have been better for the CFA not to hedge. However, the reason for hedging is to reduce or manage one’s risk exposure. By entering the hedge, there was protection against an opportunity cost loss if interest rates fell before the expected cash inflow arrived. The entire point of the hedge was to offset this risk altogether by locking in today the investment rate available six months from now.

5(b).

It is most likely that a single position in an index futures market would be the best hedge. There are several reasons for this. Perhaps the most important is that using 50 individual equity contracts instead of a single index derivative would result in a significant amount of over-hedging. To understand why, remember that the total risk exposure for an individual

stock comprises both systematic (that is, marketwide) and unsystematic (that is, companyspecific) elements. When an individual stock position is hedged separately, both sources of risk will have to be considered. However, when the stock is included in a well-diversified portfolio (that is, index), only the systematic portion of risk will need to be hedged. There are other advantages to hedging the index instead of the individual equities. The overall cost of the hedge. Because there are a limited number of exchange-traded futures for individual stocks, entering 50 different positions might have to be done through an over-the-counter derivative dealer. This typically would mean that the higher the transaction costs (that is, bid– ask spreads) to cover the fees from setting up the one-off deal, the lower the liquidity of individual stocks, and the increased commissions for the dealer. Another disadvantage is the liquidity of the position. Index options are very liquid and can be closed out quickly with little trading cost. Closing the 50 different positions would entail paying many of the start-up costs twice. Finally, it is easy to go short an index future but rather hard and more expensive to short the underlying stocks, which the OTC dealer would have to do to hedge the position in the 50 different stocks. The only advantage to the 50 different positions is that they would provide a nearly perfect hedge, whereas there would be some basis risk in the index futures position to the extent that the underlying index is not fully diversified. 7.

The fourth factor affecting the price of a stock index futures contract is the risk-free interest rate, which is usually measured by the Treasury bill yield. By the cost of carry model, futures contract prices will increase directly with increases in the risk-free interest rate, all other factors held constant. For stock index futures, the cost of carry is the difference between the risk-free rate and the dividend yield on the underlying stock index, multiplied by the spot price of the index. In a stock index arbitrage transaction, this cost of carry represents the implicit net cost of financing the purchase of the underlying stock position (that is, paying a borrowing cost of the risk-free rate less the dividends received) against which an overvalued futures contract might be shorted. For this reason, there is always a direct and positive relationship between the risk-free interest rate and futures contract prices.

8(a).

If the pension fund invests in bonds, they can use the current spot exchange rate between yen and dollars to calculate how many bonds they will receive today for $900 million. We can assume the pension fund is satisfied with the bond interest rate and motivated primarily by a desire to diversify and reduce its overall interest rate exposure. Even if it could guarantee the value of the bonds held, however, the fund would still face exchange rate risk. A currency futures contract would lock in the rate for future exchange of yen for dollars. Specifically, the fund would need to structure the currency derivative transaction to pay the yen-denominated coupons it receives on the Japanese bonds and receive a fixed number of dollars in exchange. Short futures positions will incur losses as the exchange rate rises (that is, each yen becomes worth more dollars than in the original futures contract) and gains as the exchange rate falls. The dollar value of the bonds will change in the opposite direction. A perfect hedge would have pension funds buy just enough futures (face value in yen) to cover the dollars they will repatriate. So, if the yen appreciated relative to the dollar, the pension fund would have been better off (that is, acquire more dollars in six for the same yen-denominated coupons) without the currency futures hedge. However, the ultimate point of the hedge is that it offsets the pension fund’s FX exposure, which in this case amounted to a fear that the yen would depreciate relative to the dollar. That is what it

8(b).

wanted to accomplish in the first place. Another factor worth noting is that futures positions require posting a margin account with the exchange. Marked-to-market losses must be settled on a daily basis. So, if the pension fund does not liquidate its gains on the bonds to fund losses on the futures, it may need to hold additional cash balances during the next six months to satisfy its margin requirements. 9.

Because the currency from one country could be converted to the currency of another in the spot market and then, through the use of forward contracts, converted back to the original currency at the same rate, an arbitrage opportunity is available. Investors from Country B could borrow at a lower rate in their country, convert it into the currency of Country A, earn a higher rate of return on that foreign investment, and then pay back their loan, thereby pocketing the difference between the higher investment return and the lower borrowing cost without losing (or gaining) anything from the round-trip currency exchange. Thus, the only market force that would prevent all of the currency in Country B from flowing to Country A is if investors from Country B (that is, the low-interest rate country) lose enough on the round-trip currency translation to offset the interest rate differential in the two countries. This would require investors in Country B to receive less of Currency A (per unit of Currency B) upfront than they will have to pay to get back a unit of Currency B with a forward contract. Said differently, this is equivalent to the currency of Country A trading at a forward discount to that of Country B.

l0(a). An interest rate swap is a customized risk-management vehicle. It constitutes an agreement between two parties to exchange a series of future cash flows for a certain period of time (term) based on a stated (notional) amount of principal. For example, one party will agree to make a series of floating-rate coupon payments to another party in exchange for receipt of a series of fixed-rate coupon payments (or vice versa, in which case the swap would work in reverse). No exchange of principal payments is made. The fixed rate in the agreement is specified upfront and does not change during the life of the swap. The variable rate is determined on a periodic basis for each future settlement date according to the fluctuating level of an underlying reference rate (for example, SOFR). l0(b). Strategies using interest rate swaps to affect the duration or improve the return in a domestic fixed-income portfolio can be divided into two categories: Duration modification. Swapping floating- for fixed-rate interest payments effectively converts a long position in a floating-rate note (FRN) into a long position in a fixed-rate bond over the term of the swap. This increases portfolio duration (and, vice versa, decreases duration for the floating-rate recipient). This method of modifying duration can be used either to control risk (for example, keep it within policy guidelines/ranges) or to enhance return (for example, to profit from a rate anticipation bet while remaining within an allowed maturity trading range). Seeking profit opportunities in the swap market. Opportunities occur in the swap market, as in the cash markets, to profit from temporary disequilibrium between demand and supply. If, in the process of exploiting such opportunities, portfolio duration would be moved beyond a policy guideline/range, it can be controlled by using bond futures contracts or by making appropriate rebalancing transactions in the cash market. Finally, it is worth mentioning that converting a bond holding synthetically using a swap agreement is almost always more cost-effective and more efficiently executed than physically transforming the portfolio (for example, selling an FRN holding and then buying a fixed-rate bond).

11.

An interest rate swap is an agreement to exchange a series of cash flows based on the difference between a fixed interest rate and a variable (or floating) interest rate, based on a prespecified notional principal amount. The fixed-rate receiver in the swap agreement would get the difference between the fixed rate and the prevailing floating rate if the fixed rate was above the floating rate, and pay the difference if the floating was above the fixed. The fixed rate is set so that no cash changes hands upon initiation of the deal. This can be thought of as follows: i. A series of forward contracts on the floating rate because forward contracts also have no initial cash flow and will net the difference between the floating rate and the forward rate (which acts like a fixed rate). To make the analogy precise, only one forward rate is chosen, but it is chosen such that the sum of the values of all the contracts is zero. The fixed-rate receiver to the swap is like having the short position in a series of interest rate forwards with different maturity dates because if interest rates go up, he or she loses. ii. A pair of bonds, one with a floating-rate coupon and the other with a fixed-rate coupon (both selling at par with the same face value and maturity). Being the fixed-rate receiver in the swap is the same as being long the fixed-rate bond and short in the floating-rate bond. As long as the floating rate is less than the fixed rate, the coupon payment from the fixed-rate bond will cover the interest due on the floating-rate bond, and the difference is positive. If the floating rate is above the fixed rate, then the fixed-rate receiver must make up the difference. Because the bonds are of the same present and face values, there is no net cash flow at the beginning or end of the agreement.

ANSWERS TO PROBLEMS 1(a). Price

Adjustment

Margin

Maintenance

March 9

$173.00

3,000

April 9

$179.75

−675

2,325

May 9

$189.00

−925

1,400

1,600 ( = 100+1,500 )

June 9

$182.50

650

3,650

July 9

$174.25

825

4,475

Net loss of ( $173.00 – 174.25)  100 = −$125 or = $4,475 – ( $3,000+$1,600 )  (including maintenance) Total return ( as % of initial margin only ) = −125/3,000 = −4.17%

1(b).

Futures (Forwards) unwind without (with) discounting the net price differential, so Short Futures

Long Forward

Net

May 9

− (189 − 173)  100 = ( $1,600 )

(189 − 173)  100 /1.010 = $1,584.16

−15.84

June 9

− (182.5 − 173)  100 = ( $950.00 )

(182.5 − 173)  100 / 1.005 = $945.27

−4.73

This implies that the forwards underhedge with equal notional amounts of forwards and futures. 2(a).

Purchase coffee in the spot market in Month 3: 82,000  $0.5856 = $48,019.20 Settlement from long futures hedge position: 2  ( 0.5920 – 0.5856 )  37,500 = $2,437.50 Net coffee purchase in Month 3: 48,019.20 − 2,437.50 = $45,581.70 so, Effective purchase price (per pound): 45,581.70/82,000 = $0.5559

2(b).

3(a).

3(b).

Notice that this effective price is lower than the spot price in Month 3, but not equal to the futures contract price at the initiation of the hedge positions. That means that while the long futures hedge worked to reduce your exposure to rising coffee prices over the three-month period, there was still some basis risk present in the transaction. There are at least two sources of basis risk in this transaction. First, the amount of coffee to purchase (82,000 lb) is not evenly divisible by the standardized delivery amount in each futures contract (37,500 lb). This means that you will be underhedged with two contracts or overhedged with three contracts. Second, given that the Month 3 futures price and spot price have not converged to the same level, it is likely that the expiration date of the futures contract occurs sometime after the date of the actual coffee purchase. Finally, it is also possible that the futures contract is based on a different type or grade of coffee than that purchased in the spot market, which could result in a price differential as well. Futures are an efficient, low-cost tool that can be used to alter the risk and return characteristics of an entire portfolio with less disruption than using conventional methods. There may also be both institutional constraints and unfavorable tax consequences that prevent a portfolio manager such as Klein from liquidating the entire portfolio. Because Treasury bonds and Treasury bond futures have a very high correlation, the futures approach allows one to effectively create a temporary fully liquidated position without having to physically rebalance the underlying portfolio. Futures can be sold against the portfolio to replicate the price response of the portfolio with the desired duration. In addition, there are cost advantages of using futures contracts, including lower execution costs (bid–ask spread), speed and ease of execution (time required), and the higher marketability and liquidity of futures contracts. Consequently, the bond sale strategy may well be disadvantageous on all counts. Shortening the duration by attempting to sell the actual bond portfolio would be more costly, time-consuming, and disruptive, with possible adverse tax implications as well. In Klein’s case, there may be more bonds to sell than futures contracts because many bonds in the portfolio could be in denominations as low as $1,000. Also, the bond sales would invoke liquidity problems not encountered by the bond futures strategy. The value of the futures contract is 94–05 (i.e., 94 5/32% of $100,000), which translates into 0.9415625  $100,000 = $94,156.25 . Using the information given, there are at least two ways, modified duration (ModD) and basis point value (BPV), to calculate the number of contracts. (i) Using modified duration: From the hedge ratio formula in Equation 15.7, the optimal number (N*) of futures contract to short is N* = (10/8 )  (1.0 )  (100,000,000/94,156.25) = 1,327.58 or 1,328 contracts short. Note here that the yield beta in the hedge ratio calculation is assumed to be 1.0 since the futures contract is

traded in parity with the underlying bond holding (that is, has a conversion ratio of 1.0). (ii) Using basis point value: BPVtarget = BPVportfolio – BPVhedge

BPVtarget = BPVportfolio – ( N*  BPVper futures )

And N* = (BPVtarget – BPVportfolio ) /BPVper futures

Because the target BPV is zero, then: N = ( $0 − $100,000 ) /$75.32 = –1327.67 or short 1328 contracts

3(c).

Klein is selling the contracts, as indicated by the negative value of the contracts. The difference in the two exact answers is due to rounding the BPV number to the nearest cent. Because the newly modified portfolio has approximately a zero modified duration and basis point value, the value of this portfolio would remain relatively constant for small parallel changes in rates. With an interest rate increase, the bond portfolio’s immediate market value would decline, but the positive cash flow from the Treasury bond futures contracts would offset this loss. As shown in part B, either modified duration or basis point value can be used to compute the change in value. Change in value using MD = MD  change in yield  value, or

Change in value using BPV = BPV  BP change 3(c)i.

The $100,000 BPV for the portfolio means that the portfolio value will decrease (increase) by $100,000 for each basis point increase (decrease). A 10-basis point increase in interest rates would mean a $1,000,000 decline (or loss in the market value of the original portfolio) Change in value = ModD  change in yield  change in value = 10  0.001  $100,000,000 = $1,000,000 OR Change in value = BPV  BP change = 10  $100,000 = $1,000,000

3(c)ii. A $75.32 BPV for the futures contract represents a $75.32 change in value per basis point per contract. When rates increase by 1 basis point, each futures contract will decrease by $75.32. However, because Klein is short contracts, she will receive a cash flow of $75.32 per contract for each basis point increase. Using ModD, the total cash inflow from the futures position is $94,156  8  0.0001  1,328 = $1,000,316

Using BPV, the total cash inflow from the futures position is 10  $75.32  1,328 = $1,000,249

Differences from exactly $1,000,000 are due to rounding the number of contracts. 3(c)iii. The change in the value of the hedged portfolio is the sum of the change in the value of the

original portfolio and the cash flow from the hedge (futures) position, or: Newly hedged portfolio change = −$1,000,000+$1,000,316 = $316 (using ModD ) = −$1,000,000+$1,000,249 = $249 (using BPV )

3(d).

4(a).

These values differ from zero only slightly due to having to round off the number of contracts used in the hedge position, which is the only source of basis risk in this case. Klein’s hedging strategy may not fully protect the portfolio against interest rate risk for several reasons. First, immunization risk would remain even after the execution of the strategy because of the possibility of nonparallel shifts in the yield curve. If the yield curve shifts in a nonparallel fashion, the modified portfolio is not immunized against interest rate risk because the original bond portfolio and T-bond futures exist at different points on the yield curve and hence face different interest rate changes. If the curve became steeper, for example, then the market value loss on the original bond portfolio would be accompanied by a less-than-compensating value gain on the futures position. Second, the volatility of the yield between the T-bond futures and the government bond portfolio may not be one-toone. Hence, a different yield beta adjustment than the value of 1.0 assumed here may be needed. Third, this may still be a cross-hedge because the government bonds in the portfolio may not be the same as the cheapest-to-deliver bond. Fourth, the durations for the actual bonds and futures contract bonds will change at different rates as time passes, so risk will arise unless continual rebalancing takes place. Finally, because fractional futures contracts cannot be sold, the duration may not be able to be set exactly to zero. Both bonds in portfolio 1 are zero coupon bonds, so D1 = ( 4/10 )  14+ ( 6/10 )  3 = 7.4 ModD1 =7.4/1.0731= 6.896 years

1.0731 1.0731+9  ( 0.046 − 0.0731) = 7.4 0.0731 0.046 (1.0731)9 − 1 + 0.0731   ModD2 = 7.4/1.0731 = 6.896 years D=

For both portfolios: DP/P@ − 6.896  0.006 = 4.137%

4(b).

To hedge these bond positions, the number of futures contracts required for each portfolio position (using Equation 15.7) is calculated as HR1 = ( 6.896/11.769 )  0.6  1.13+0.4  1.03  ( −10,000,000/104,750 ) = −60.97 HR2 = ( 6.896/11.769 )  1.01  ( −11,500,000/104,750 ) = +64.97

Combining these two gives a net hedge ratio of +4.00 contracts. Consequently, the hedge would be to enter into a long position in four futures contracts. 5(a).

The potential for an arbitrage transaction exists any time two positions that have otherwise identical patterns of cash flows and risks sell for different prices in the current market. In this case, an arbitrage trade is possible because the actual futures contract price ($112.15) is different than its theoretical level using the cost of carry model:

Theoretical Level of Futures Price = Spot Price+ Cost of Carry  = 102.30+ 1.10 – 2.20 = $101.20

So, the arbitrage transaction would involve (1) shorting the actual futures contract and (2) taking an offsetting long position in the theoretical futures contract. This theoretical long futures position could be achieved by (i) borrowing $102.30 to buy the bond position in the spot market today, (ii) holding the bond until the futures contract expiration date in three months, collecting the income of $2.20 along the way, and (iii) selling the bond in three months using the settle price from the short position in the actual futures contract in order to repay the borrowed funds, plus the financing costs. 5(b). The cash flows from the arbitrage trade can be illustrated as follows, where B is the (unknown) bond price in three months’ time: Transaction Date 0 3 Months 1. Short Actual F

—

112.15—B

—Borrow Spot Price

+102.30

−(102.30 + 1.10)

—Buy Spot Index

−102.30

—Collect Income

—

+2.20

Net Amount:

0.00

10.95

2. Long Theoretical F

5(c).

6(a).

Notice that this arbitrage transaction involved none of Hershey’s own wealth (that is, she used borrowed funds, not her own capital) and involved no price risk (that is, she had already locked in the future sales price of the bond and she borrowed money to purchase today). Nevertheless, Hershey still made a net profit of $10.95 on the trade, which represents the difference in prices between the actual futures contract she sold and the theoretical long futures contract positions she assembled on her own (that is, 10.95 = 112.15–101.20). As noted above, the spot price of the bond in three months’ time does not matter since Hershey locked in her eventual selling price by shorting the actual futures contract at the same time she borrowed the money to buy the bond at the inception of the trade. This futures position removed all of the price risks inherent in holding an unhedged security position over any future time horizon. Recalling the traditional money market convention that the interest expense on floatingrate debt is determined in advance and paid in arrears, the quarterly loan receipts will be based on the following 90-day VREF rates:

Loan Receipt due in 3 months: 90-day VREF today = 4.60% Loan Receipt due in 6 months: 90-day VREF today = 4.96% Loan Receipt due in 9 months: 90-day VREF today = 5.63% Loan Receipt due in 12 months: 90-day VREF today = 6.44%

Based on a non-amortizing loan balance of $1,000,000 and 90-day quarters, these percentages imply the following sequence of quarterly cash payments: $1,000,000  ( 0.0460 )  ( 90/360 ) = $11,500 $1,000,000  ( 0.0496 )  ( 90/360 ) = $12,400 $1,000,000  ( 0.0563 )  ( 90/360 ) = $14,075 $1,000,000  ( 0.0644 )  ( 90/360 ) = $16,100

6(b).

6(c).

Although there are four interest payments due, the convention of setting VREF at the front end of a borrowing period means that there are only three uncertain cash flows at the time the funding originated. This means that the customer will have to “lock in” a 90-day VREF on settlement dates 90, 180, and 270 days from now. This can be done by going long in the following strip of VREF futures contracts, each having a notional principal equal to the loan amount of $1,000,000: long one 90-day contract, long one 180-day contract, and long one 270-day contract. Notice that this problem would require a long hedge on the part of the bank (that is, the lender receiving VREF payments) because the floating-rate lender would need a hedge as compensation when VREF rates fall (that is, when VREF futures prices rise). The annuity that would be equivalent to locking in the preceding series of quarterly cash receipts with the long futures strip is calculated as the solution to: $11,500 $12,400 $14,075 $16,100 + + + 1 + ( 0.046 )( 90 ) / 360  1 + ( 0.0475)( 180 ) / 360  1 + ( 0.050 )( 270 ) / 360  1 + ( 0.053)( 360 ) / 360  =

Annuity Annuity Annuity Annuity + + + 1 + ( 0.046 )( 90 ) / 360  1 + ( 0.0475)( 180 ) / 360  1 + ( 0.050 )( 270 ) / 360  1 + ( 0.053)( 360 ) / 360 

or Annuity = (52,337.50/3.878955) = $13,492.68 Expressing this dollar amount on a percentage basis on terms comparable to VREF leaves ( $13,492.68/$1,000,000 )( 360/90 ) = 5.40% 7.

An arbitrage trade is possible in this situation if the actual futures price of $614.75 differs substantially from the theoretical futures price using the cost of carry model. In this case, we have Theoretical Level of Futures Price = Spot Price+ Cost of Carry  = 614.75+ 614.75  ( ( 0.08 − 0.03)  0.25)  = $609.78

So, the arbitrage transaction would involve (1) shorting the actual futures contract and (2) taking an offsetting long position in the theoretical futures contract. This theoretical long futures position could be achieved by (i) borrowing $602.25 to buy the stock index portfolio in the spot market today, (ii) holding the stock until the futures contract expiration date in three months, collecting the dividends of $4.52 (=602.25  0.03  0.25) along the way, and (iii) selling the stock index in three months using the settle price from the short position in © 2025 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

the actual futures contract in order to repay the borrowed funds, plus the financing costs of $12.05 (=602.25  0.08  0.25). The overall cash flows from this combination of transactions can be shown as follows: Transaction

Date 0

3 Months

1. Short Actual F

—

614.75—S3

+602.25

614.30

2. Long Theoretical F —Borrow Spot Price

(= 602.25[1+(0.08)(0.25)]) —Buy Spot Index

−602.25

—Collect Dividends

—

4.52 (= 602.25(0.03*0.25))

Net Amount:

+4.97

Notice that the amount of the arbitrage profit received in three months is equal to the difference between the actual and theoretical futures prices (that is, 4.97 = 614.75 – 609.78). This underscores the important point that it is this price differential that creates the arbitrage opportunity in the first place. 8(a). The number of futures contracts required is determined by the hedge ratio formula: N* = (Market Value Portfolio / Market Value Index Futures )  Beta of Portfolio = $15,000,000/ (1,000  250 )   0.88 = $15,000,000/250,000  0.88 = 60  0.88 = 52.8 contracts

8(b).

Selling (going short) 52 or 53 contracts will hedge $15,000,000 of equity exposure. Alternative methods that replicate the futures strategy in Part a include the following: 1. Short selling an exchange-traded index fund, such as SPDRs. SPDRs would be more expensive than futures to trade in terms of liquidity and transaction costs. Tracking error, in theory, would be higher for futures than for SPDRs because S&P 500 futures can close under and over fair value. SPDRs do not incur the cost of rolling over, which a position in futures might incur if held longer than one expiration date. 2. Creating a synthetic short futures position using a combination of calls and puts. Either options on the underlying index or options on the futures could be used. Selling an index call option and purchasing an index put option with the same contract specifications (that is, exercise price and time to expiration) would create a synthetic short futures position. This strategy would likely be more costly than futures because two transactions are required. Longer-term options tend to be less liquid than futures and SPDRs. If the call and put are traded separately, bid/ask spreads and market impact may increase the cost of the strategy.

3. Entering into an equity swap contract in which Andrew pays the capital appreciation and dividends on the portfolio and receives a fixed interest rate. The price of this transaction is negotiated between the two parties, but in general, the swap would be more costly than the futures hedge. Also, equity swaps are not liquid and may prove difficult to reverse once entered. Unlike futures, which are standardized contracts with no customization possible, this alternative has the advantage of customization; negotiable terms include the length of the contract, margin requirements, cost of closing position early, and timing of payments. 4. Shorting a forward contract on the S&P 500 index. The price of this transaction is negotiated between the two parties. Forwards are not liquid, may prove difficult to reverse once entered, and may involve counterparty risk. Unlike futures, which are standardized contracts with no customization possible, this alternative has the advantage of customization; negotiable terms include the length of the contract, margin requirements, cost of closing the position early, and timing of payments. 9(a).

The basic premise of interest rate parity is that an investor from Country 1 should receive the same return from the following two investment schemes: (1) investing directly in Country 1’s sovereign debt issue versus (2) an alternative involving (i) converting his or her local currency to Country 2’s currency in the spot market, (ii) investing those proceeds in Country 2’s sovereign bond, and (iii) convert the terminal value of this position back into Country 1’s currency using the futures market to lock in the exchange rate. For this to be the case, any interest rate differential in the sovereign bonds in the two countries must be neutralized by differences in the spot and futures exchange rates. In this case, the client that starts with $1 to invest would have the following outcomes at the end of the year: Investment Scheme #1: $1.0000  (1+0.0425) = $1.0425 Investment Scheme #2: ( CHF1.5035)  (1+Rchf )  ( $0.6586/CHF )

So, interest rate parity would require the Swiss investment rate to be such that

(1.5035)  (1+Rchf )  ( 0.6586 ) = 1.0425 or Rchf = 5.281%

9(b).

If the actual rate is 5.50 percent for a one-year Swiss government bond, then the return on investing in CHF with the second scheme would generate the highest return. This can be seen by plugging in Rchf = 0.055 into the left-hand side of the preceding parity equation to get (1.5035)  (1+0.055)  ( 0.6586 ) = 1.0447

9(c).

A U.S.-based arbitrageur could borrow $250,000 domestically at 4.25 percent, convert those proceeds into CHF375,875 ( = 250,000  1.5035) , buy one-year Swiss government bonds paying 5.50 percent, and enter into a forward contract to reconvert the proceeds after a year. After a year invested at 5.50 percent, the arbitrageur would have CHF396,548.125, which could be converted back into dollars at the forward rate of $0.6586/CHF. This would result in $261,166.60, of which $260,625 would be needed to repay the loan plus interest. Thus, the arbitrageur would be left with a $541.60 profit before transaction costs are considered.

10(a). According to the cost-of-carry model, the theoretical futures contract settlement price must equal the spot price plus the cost of carrying the spot commodity forward to the delivery date of the futures contract. So, Theoretical F0,t = S0 + S0  ( Ann. RF – Ann. Div Yld)  (Holding Period)  = 1100+1100  ( 0.032–0.018 )  ( 0.5)  = 1,107.70

10(b). If the futures contract price were substantially higher than the theoretically correct level of 1,107.70, Singer should • Short the stock index future contract • Borrow funds at the risk-free rate to purchase the stock index • Hold the position until maturity, collecting dividends and then sell the stock index to repay the loan with interest Note that this is buying low and selling high: the higher-price futures contract is sold and the lower-priced stocks are purchased, locking in a profit (considering as well the dividend yield and borrowing cost). If the futures contract price were substantially lower than 1,107.70, Singer should reverse the above transactions, again implementing the basic strategy of buy low, sell high: • Go long the stock index future contract • Lend funds at the risk-free rate and short-sell the stock index, paying the dividends during the holding period • Cover the short index position at maturity, using the stock index futures to cover the purchase price of the short index position 10(c). Arbitrage will not be profitable when the profit, including the transaction costs, is not greater than zero. If the actual futures contract price in this example is equal to the theoretical level of 1,107.70, either of the two arbitrage trades just described would generate a profit of exactly zero, before accounting for any transaction costs. Thus, with round-trip trading costs of $20, to justify the first trading scheme (that is, shorting the stock index futures), the actual futures price would have to be greater than Upper Bound F: 1,107.70+20 = 1,127.70 To justify the second trading scheme (that is, going long the stock index futures), the actual future price would have to be less than Lower Bound F: 1,107.70 − 20 = 1,087.70 11.

From Exhibit 15.17, we find that the fixed rate receiver will get the bid rate for the two-year interest rate swap, or 4.487 percent. So, for a receive-fixed swap agreement with a notional principal of $22,500,000, we have the following:

Notional principal:

$22,500,000 Number

Net Payment

(Receipt)

Settlement

Fixed

Number of

“30/360

Current

Fixed-Rate

Floating Rate

Date (yrs)

Rate

Actual Days

” Days

VREF

Receipt

Payment

4.487%

—

4.25%

—

0.5

4.487%

183

180

4.40%

$504,787.50

$486,093.75

($18,693.75)

1.0

4.487%

183

180

4.90%

$504,787.50

$503.250.00

($1,537.50)

1.5

4.487%

182

180

5.05%

$504,787.50

$557,375.00

$52,587.50

2.0

4.487%

183

180

4.60%

$504,787.50

$577,593.75

$72,806.25

The values for the receipts and payments are found using the following: Fixed-rate receipt = Swap fixed rate  (180/360 )  Notional principal = 0.04487  (180/360 )  $22,500,000 = $504,787.50 Floating-rate payment = VREF-1  ( #days/360 )  Notional principal Example for first settlement ( 0.5 yrs ) : 0.0425  (183/360 )  $22,500,000 = $486,093.75

The number of actual days for each settlement period is listed in Exhibit 15.19. 12(a). With these estimates, the settlement payments can be calculated as follows: March 2: Floating-rate payment = ( 0.0350 − 0.0010 )  ( $50,000,000 )  ( 90/360 ) = $425,000 Equity-index receipt = [ ( 477.51 − 463 .11) /463.111  ( $50,000,000 ) = $1,554,706

So the net receipt the fund expects would be ( $1,554,706 − $425,000 ) = $1,129,706 June 2:

Floating-rate payment = ( 0.0325 − 0.0010 )  ( $50,000,000 )  ( 92/360 ) = $402,500 Equity-index receipt = ( 464.74 − 477.51) /477.51  ( $50,000,000 ) = −$1,337,145 So the net payment the fund owes would be ( $1,337,145+$402,500 ) = $1,739,645 September 2:

Floating-rate payment = ( 0.0375 − 0.0010 )  ( $50,000,000 )  ( 92/360 ) = $466.389 Equity-index receipt = ( 480.86 − 464.74 ) /464.74   ( $50,000,000 ) = $1,734,303 So the net receipt the fund expects would be ( $1,734,303 − $466,389 ) = $1,267,914

December 2:

Floatingnrate payment = ( 0.0400 − 0.0010 )  ( $50,000,000 )  ( 91/360 ) = $492,917 Equitynindex receipt = ( 482.59 − 480.86 ) /480.86   ( $50,000,000 ) = $179,886 So the net payment the fund owes would be ( $492,917 − $179,886 ) = $313,031 . 12(b). It is also quite common for equity index swaps to be based on a notional principal amount that varies directly with the level of the underlying index. If, for instance, the swap participants had agreed to let the initial notional principal of $50 million vary over time, it would have been adjusted up on March 2 to $51.555 million. This adjustment is calculated as $50 million (1+ ( 477.51 − 463.11 ) /463.11]) . That is, on each settlement date, the notional principal is adjusted up (down) by the percentage of appreciation (decline) in the starting level of the index. This adjustment process, which is equivalent to adding the gross equity settlement payment to the initial notional principal, simulates the return that investors with direct stock positions would obtain as their actual equity exposure would rise or fall with market conditions. In contrast, a fixed notional principal in an equity index swap is equivalent to an asset allocation strategy in which the equity exposure is kept constant.

Solution and Answer Guide

ANSWERS TO QUESTIONS 1.

A long straddle consists of a long call and a long put on the same stock with the same exercise price. The position profits from dramatic price movement by the stock, whether up or down. A short straddle involves the sale of a call and a put on the same stock with the same exercise price and profits from little or no stock price change. Investors going long would anticipate volatility in excess of that incorporated in the options’ prices, while investors going short would expect volatility below that already discounted. Because volatility enhances option prices, long-straddle investors would tend to pay higher premiums for more volatile options, whereas short-straddle investors must accept lower premiums for less volatile options.

A range forward is actually an option strategy that combines a long call and a short put (or vice versa) through a costless transaction (that is, the call and the put have the same upfront premium). Because the options will not have the same exercise price, the combination is classified as a range forward as opposed to an actual forward, created by combining long and short options with the same exercise price that has the same upfront premium. It is fair to view actual forwards as a special case (or zero-cost version) of range forwards. There will always be one combination of a long call/short put transaction for which the options have the same exercise price and the same initial premium; this common exercise price is the actual forward contract price.

Call options give the owner the right, but not the obligation, to purchase yen for a prespecified amount of domestic currency. Purchasing an at-the-money call option would guarantee the current exchange rate over the life of the option. If the yen declines in value, the call will not be exercised because the yen can be purchased more cheaply in the open market, and redeeming the bond issue will be less costly. Contrasting characteristics: (1) Currency options are traded worldwide and enjoy a generally liquid market; (2) exchange-traded currency option contracts have standard amounts, maturities, etc.; (3) over-the-counter options could be tailored to meet TechnoLogical’s needs; (4) the initial cash outflow would be the premium (that is, the cost of the “insurance”); (5) the use of options preserves the ability to profit from favorable future FX rate movements; (6) there may be counterparty credit risk depending on how the contract is collateralized; and (7) must roll to match annual debt obligation.

Currency forward contracts commit the short position to deliver the specified amount of currency to the long position on a specified future date at a fixed price. Both the short position and the long position in a forward contract are fully obligated to complete the agreed-upon transaction at the expiration date of the contract. Contrasting characteristics: (1) The market for forward contracts is over-the-counter and sometimes may not be as liquid as option or futures markets; (2) forward contracts may be custom-designed for specific applications; (3) cash does not change hands until a forward contract is settled; (4) there may be counterparty credit risk depending on how the contract is collateralized; and (5) can be tailored to best match the five-year debt obligation. Currency futures are like forward contracts except the gain or loss on the contract is marked to market daily and settled under the supervision of an organized exchange. A short position in the futures requires either offset or delivery at expiration. Contrasting characteristics: (1) Futures are traded in standardized contracts and are highly liquid; (2) cash is required for daily settlement; (3) a margin (that is, collateral) account is required; (4) management and administration costs are higher than with a forward or option contract; (5) no counterparty credit risk due to margin accounts; and (6) must roll to match annual debt obligation. 4.

The other three factors affecting the value of call options and the ways that changes in them affect value are as follows: (1). Return volatility of the underlying asset. A call cannot be worth less than zero no matter how far the stock price falls, but rising stock prices can increase the call’s value without limit. Therefore, the wider the range within which a stock’s price can fluctuate (that is, the greater its volatility), the greater the chance that the option will expire in the money, the higher the expected payoff from owning it, and the higher its initial value. A wider range of probable future prices on the underlying stock increases the probability of higher payoffs in general but because the call’s value cannot decline below zero (that is, does not have to be exercised under unfavorable circumstances), this volatility does not symmetrically increase the probability of lower payoffs. (2). The risk-free interest rate. The call value will increase with increases in interest rates (given constant stock prices) because higher interest rates make the ownership of call options more attractive. The call owner does not pay for the stock until the option is exercised, so a higher interest rate (that is, discount factor) will reduce the present value of the strike price that the investor will eventually have to pay, which makes the current call premium higher. (3). The exercise price of the option. Call values decrease with increases in the exercise price. When a call option is exercised, the payoff is the difference between the stock price at the time of exercise and the exercise (or strike) price. A higher exercise price decreases the expected payoff from the call, thus reducing the option’s value.

Considered as stand-alone investments, options are at least as risky as the underlying asset they are based on and are usually much riskier than that due to the embedded leverage built into the contract terms. However, an option (or combination of options) can also be used to offset (that is, hedge) the price volatility of the underlying asset altogether, effectively creating a portfolio position that is risk free (that is, a synthetic T-bill). The three-step process underlying the valuation of an option contract is as follows:

(1) Create a portfolio consisting of the underlying asset and option contracts that delivers a certain payoff at the expiration date, regardless of future price movements for the underlying asset. This is the riskless hedge portfolio. (2) Assume that any riskless investment should be priced to earn the risk-free rate during its holding period. This is the “no arbitrage” (that is, efficient markets) assumption. (3) Solve for the option value that is consistent with the previous two steps. The first of these steps—forming the riskless hedge portfolio—is the key to the valuation process. In the put–call parity model, the riskless hedge portfolio is straightforward: one stock long, one put long, one call short. In the two-state or Black-Scholes model, the risk hedge portfolio consists of one stock long and a multiple number of calls short (that is, the hedge ratio). This hedge ratio has to be adjusted as the price for the underlying stock changes over time to keep the hedge portfolio risk free. 6.

A covered call position is a portfolio consisting of the following holdings: one stock long, one call short. By selling the call option—usually one that is at or near the money—the stockholder receives the upfront premium, which serves as a supplement to any dividends that the stock pays. For this reason, the covered call strategy is sometimes referred to as a yield enhancement strategy. The risk of the strategy is twofold: downside movements in the stock price are only protected to the extent of the call premium received, and upside movements in the stock price beyond the exercise price of the call will be surrendered when the contract is exercised to the disadvantage of the stockholder. Thus, this is a strategy that works best when the stock price is expected to trade in a very narrow range around its original position.

If there are no transaction costs, it is only rational to exercise a call option early immediately before a dividend payment. When a firm pays a dividend, the price of the stock usually declines by approximately the amount of the dividend. This has two impacts on the option holder. First, the decrease in the stock price decreases the value of his or her option. Second, the investor does not get the dividend payment unless he or she actually has exercised the option and owns the stock. Consequently, if the value of the dividend is greater than C + X − S, it will be beneficial to exercise early. Note that this is most likely to happen for an in-the-money option. For put options, because the underlying stock price can only fall to zero, there is a limit on how high the benefit of exercising the contract can go. If a stock is already trading near zero (that is, almost bankrupt) and the stock still has three months to expiration, the present value of what an investor would get by waiting to exercise may be lower than what he or she would get immediately, in which case the put should be exercised early.

In the Black-Scholes model, the expected future value of a stock is a function of the risk-free interest rate and the dividend yield. As long as the risk-free rate is greater than the dividend yield, the future expected value will be greater than today’s price. The longer the time period, the higher the expected price. So, as time to expiration increases, there are two opposing forces on the value of a European put. First, the increased time to expiration increases the chances of the option being more in the money. This increases put value. Second, the higher expected price at expiration decreases the present value of the exercise price, which adversely affects the expected value of the put’s payoff at expiration and, therefore, decreases the put value. Depending on which of these two effects is larger, the put may increase or decrease in price with an increase in time to expiration.

For a European call option, these two effects work in the same direction because an increase in expected future price increases the value of a call. Hence, an increase in the time to expiration always increases the value of a European call. 9.

For any option, volatility—which effectively contains the investor’s forecast of future price movements in the underlying asset—is the only factor that is not known in advance during the valuation process. Increasing volatility will increase all option values, both puts and calls, because this increases the possibility that the option will finish in the money. So, an option contract that is viewed as trading at too high of a price is consistent with one that has a volatility level that is too high; an option priced too low is consistent with a volatility level that is too low. Consequently, “buy low vol, sell high vol” is simply a trader’s way of saying “buy low, sell high,” treating volatility as the “asset” that is being bought or sold.

10.

On October 19, 1987, implied volatilities in the stock market sky-rocketed, from the normal level of around 20 percent to almost 120 percent, due to the unprecedented events of that day. At certain points, this jump in implied volatility increased the value of call options more than enough to offset the negative impact of the steep declines in the index level itself, which is the situation that created the possibility of buying call options on the stock index and then selling them shortly thereafter at a profit.

11.

Owning a convertible bond can be viewed as holding a portfolio of two separate positions: long in a “regular” bond from the issuing company, and long in a call option that allows for the exchange of the bond into a prespecified number of stock shares and within a predetermined time horizon. So, the convertible investor is effectively participating in the upward stock price movements, with the safety of not having to own the stock outright in case its price moves downward in the future. In addition, the bond position will typically deliver semiannual coupon payments during the life of the security, which provides additional downside risk protection for the investor. Of course, owning this conversion feature will increase the price of the convertible bond relative to the regular bond, thereby lowering the yield to maturity that the investor can expect to receive.

ANSWERS TO PROBLEMS 1(a)–(b). Here u = ( 70/50 ) = 1.4 and d = ( 35/50 ) = 0.7 , so

Implied Probability ( p ) = Call Value ( C 0 ) =

r − d (1.04 − 0.7) = = 0.486 u − d (1.4 − 0.7)

pCu + (1 − p)Cd 0.486(20) + 0.514(0) = = $9.34 r 1.04

OR Set up a binomial stock and call option price tree and calculate the option values at expiration for each ending stock price. Current price = $50 which equals the exercise price. If the stock price rises to $70, the call premium is $20; if the stock price falls to $35, the call’s value is $0. Step 1 Solve for the amount to invest in the stock and the amount to borrow in order to replicate the option given its value in the up state and its value in the down state. Solve these equations simultaneously. Value of stock+options if price rises: $70+h ( $20 ) = $35+h ( $0 ) = value of stock+options if price rises (riskless arbitrage )

Solving for h, the number of call options to purchase: $70+h ( $20 ) = $35, so h = −1.75 ; sell 1.75 call options for every share purchased. Steps 2 and 3 Use the values derived in Step 2 to solve for the value of the option at the beginning of the period. Beginning portfolio of stock + 1.75 written calls will earn the RFR of 4 percent; this must equal the certain value of $35:

$50 – 1.75C0  (10+RFR ) = $35 = $50 – 1.75C0  (1+0.04 ) giving C0 = $9.34 1

2(a).

Critique of Belief Joel Franklin’s belief is incorrect. There are two fundamental kinds of options: American style and European style. An American option permits the owner to exercise the option at any time before or at expiration. The owner of a European option may exercise it only at expiration. If an option is at expiration, it will have the same value whether it is American or European.

The owner of an American option can treat the option as a European option simply by postponing the decision to exercise until expiration. Therefore, the American option cannot be worth less than the European option. However, the American option can be worth more. The American option will be worth more if circumstances make an exercise of the option before its expiration desirable. So, it may have a higher premium than the European contract, but never a lower one. 2(b).

European-Style Option’s Value The formula to calculate a call option using put–call parity is c = S +p − PV ( X ) where c =the price of a European call option S = the price of the underlying stock p =the price of a European put X = the exercise price for both options t = time to expiration for both options Therefore, from the information given,

Call option premium = 43+4.00 − 45/ ( 1.055) = $4.346 1

2(c).

Effect of Variables Effect on Call Option’s Value i. An increase in short-term interest

Increase

rate ii. An increase in stock price volatility

Increase

iii. A decrease in time to option

Decrease

expiration 3(a) Calculate the following parameters, option values, and hedge ratios at each node:

u = 33.00/30.00 = 36.00/33.00 = 1.10 d = 27.00/30.00 = 24.30/27.00 = 0.90 r = 1.05 Implied probability ( p ) = ( r – d ) / ( u – d ) = (1.05 – 0.9 ) / (1.1 – 0.9 ) = 0.75 Cuu = Max  0, 36.30 – 28  = 8.30 Cud = Max  0, 29.70 – 28  = 1.70 C dd = Max  0, 24.30 – 28  = 0

(i). Up State in S1 (S = 33): If the stock moves up, the option will be worth $8.30 (Cuu); if the stock moves down, the option will be worth $1.70 (Cud). The value of the option and hedge ratio at this state are

Cu =

0.75(8.30) + 0.25(1.70) = 6.333 1.05

hu =

(36.30 − 29.70) = −1.000 (1.70 − 8.30)

(ii). Down State in S1 (S = 27) If the stock moves up, the option will be worth $1.70 (Cu); if the stock moves down, the option will be worth $0. The value of the option and hedge ratio at this state are

Cd =

0.75(1.70) + 0.25(0) = 1.214 1.05

hd =

(29.70 − 24.3) = −3.176 (0 − 1.70)

(iii). Initial State at Time 0 (S = 30) With previous values of Cu and Cd, the initial value of the option and hedge ratio are

Co =

0.75(6.333) + 0.25(1.124) = 4.813 1.05

ho =

(33.00 − 27.00) = −1.172 (1.214 − 6.333)

So the riskless portfolio will initially contain one share of stock and be short 1.172 calls. Because after an initial up-move to 33.00 the option can only finish in the money, the hedge ratio is 1.00. To summarize, the call option tree is given as follows:

3(b). Ending Price

Number of Paths

Path Probability

Total Probability

$36.30

(0.75)(0.75) = 0.5625

0.5625

$29.70

(0.75)(0.25) = 0.1875

0.3750

$24.30

(0.25)(0.25) = 0.0625

0.0625 1.000

3(c).

From above

u = 33.00/30.00 = 36.00/33.00 = 1.10 d = 27.00/30.00 = 24.30/27.00 = 0.90 r = 1.05 so, p = (1.05 – 0.9 ) / (1.1 – 0.9 ) = 0.75

Cuu = Max  0, 36.30 – 28  = 8.30

Cud = Max  0, 29.70 – 28  = 1.70 C dd = Max  0, 24.30 – 28  = 0

Using the full binomial option valuation formula, we have Co =

1(0.75)2 (8.30) + 2(0.75)(0.25)(1.70) + 1(0.25)2 (0) = 4.813 (1.05)2

which is the same value as shown above for the two-state approach. 3(d).

The only path where the put option has a positive intrinsic value at expiration is after two consecutive downward stock price movements. Its intrinsic value in that state would be: $28.00 – $24.30 = $3.70. So, Po =

1(0.25)2 (3.70) = 0.210 (1.05)2

We can check this valuation result with put–call parity for consistency: P = C +PV ( X ) – S = 4.813+28/ (1.05) – 30 = 0.210 2

Put–call parity does hold. 4(a).

Using the dividend yield-adjusted version of the Black-Scholes model, we have the following inputs: S = 75, X = 70, T = 0.25, RF = 0.09,  = 0.20, Annual Div Yld = ( 4  $2 ) /75 = 0.1067

so,

d1 = 0.6983 and N ( d1) = 0.7575; d 2 = 0.5983 and N ( d 2 ) = 0.7252

and Black-Scholes C = e −( 0.1067)(0.25) ( 75 )( 0.7575 ) − 70 e −( 0.09 )(0.25) ( 0.7252 ) = $5.685

An alternative approach would be to use a stock price adjusted for the present value of the dividend payment during the life of the option and set the dividend yield to zero: S ’ = 75 − 2e −( 0.09)(0.25) = 73.044, X = 70, T = 0.25, RF = 0.09,  = 0.20, Div Yld = 0

so, d1 = 0.7007 and N ( d1) = 0.7582; d 2 = 0.6007 and N ( d 2 ) = 0.7260

and Black-Scholes C = ( 73.044 )( 0.7582 ) − 70 e −( 0.09 )(0.25) ( 0.7260 ) = $5.698

4(b).

Using put–call parity with the dividend-adjusted stock price: P = C − S +PV ( D ) +PV ( X ) P = 5.698 − 75+ ( 70+2 ) *exp ( −0.09*0.25) P = $1.096

Alternatively, the dividend yield-adjusted Black-Scholes formula gives P = 70 e−( 0.09 )(0.25) (1 − 0.7252 ) − 75 e −( 0.1067)(0.25) (1 – 0.7575) = $1.101

4(c).

With no dividend payment and the original stock price, S = 75, X = 70, T = 0.25, RF = 0.09,  = 0.20, Div Yld = 0

so, d1 = 0.9649 and N ( d1) = 0.8327; d 2 = 0.8649 and N ( d 2 ) = 0.8065

and Black-Scholes C = ( 75)( 0.8327 ) − 70 e −( 0.09 )(0.25) ( 0.8065) = $7.257

So, the call option value increases by $1.572 ( = 7.257 – 5.685 ) , which is less than the $2 dividend foregone. 4(d).

All else held constant, an increase in the volatility to 30 percent would increase the call’s value. A decrease in the risk-free rate to 8 percent would decrease the call’s value.

5(a). Hedge Price

Call

Ratio

$25

0.0146

0.0103

$30

0.2324

0.0995

$35

1.2750

0.3400

$40

3.7239

0.6342

$45

7.4672

0.8436

$50

11.9775 0.9458

$55

16.8191 0.9839

X = $40; r = 0.09; T = 6 months ( 0.5) ;  = 0.25

d1 =

ln ( S /X ) + [r + ( 2 / 2)]T

d2 = d1 −  (T )1/2

 (T )

1/2

For example, if S = 25, d1 =

ln ( 25/40 ) + 0.09+ ( 0.25 2 /2 )  ( 0.5)

( 0.25)( 0.5) −0.47+  0.09+0.03125 ( 0.5) −0.47+0.060625 = = = −0.409375/0.17678 = − 2.3157 0.17678 ( 0.25)( 0.7071) 1/2

d2 = −2.3157 – ( 0.25)( 0.5) = −2.31 − 0.17678 = −2.4926 ½

N ( d1 ) = 1 − 0.9897 = 0.0103

N ( d2 ) = 1 − 0.9937 = 0.0063

C = S ( N ( d1 ) ) − Xe − rT ( N ( d2 ) ) = 25 ( 0.0103) – 40e (

− 0.09 )(0.5)

5(b).

( 0.0063) = 0.26 – 0.24 = $0.0146

The call value for each level of the current stock price is lower than those shown in Exhibit 16.8 because: (1)

A decrease in time to expiration (one year to six months) causes a decrease in the call value.

(2) A decrease in security volatility () causes a decrease in the call value. 5(c).

For S = 40 Using put–call parity, P = C − S +PV ( X ) P = 3.7239 − 40+ ( 40 ) *exp ( −0.09*0.5) P = 1.9638

6(a).

For this index call option, we have S = 653.5, X = 670, T = 0.25, RF = 0.055,  = 0.16, Div Yld = 0.028

so,

d1 = −0.1873 and N ( d1 ) = 0.4257; d2 = −0.2673 and N ( d2 ) = 0.3946

and Black-Scholes C = e −( 0.028 )(0.25) ( 653.5 )( 0.4257 ) − 670 e −( 0.055)(0.25) ( 0.3946 ) = $15.4784

6(b).

At a market price of $17.40, the call is trading higher than its theoretical value of $15.48. The market price would be consistent with a Black-Scholes value using a higher volatility estimate than 16 percent. The implied volatility value is the volatility forecast that would set the Black-Scholes model value for the call equal to its market price of $17.40. In this case, the implied volatility would have to be larger than 16 percent. (Although no calculation was required in this problem, the implied volatility consistent with the current market price of this index call option is 17.51 percent.)

6(c).

The market price of an option may differ from the Black-Scholes model value for various reasons. Two of the most likely reasons are the following: (1) the assumption of continuous dividend payments in the formula does not match the reality of how and when dividends are actually paid and (2) the formula itself may be incorrect—for instance, the BlackScholes model may not capture accurately the stock price forecast process or it may handle deep in-the-money or out-of-the money contracts differently. The implied volatility statistic will capture not only investors’ composite beliefs about stock volatility but also every other way in which the formula may be incorrect.

7(a).

The volatility estimates are calculated as follows: Period A Price

Price Relative A

Period B Price

Price Relative B

168.375

—

122.5

—

162.875

−0.0332

124.5

0.0162

162.5

−0.0023

121.875

−0.0213

161.625

−0.0054

120.625

−0.0103

160.75

−0.0054

119.5

−0.0094

157.75

−0.0188

118.125

−0.0116

157.25

−0.0032

117.75

−0.0032

157.75

0.0032

119.25

0.0127

161.125

0.0212

122.25

0.0248

162.5

0.0085

121.625

−0.0051

157.5

−0.0313

120

−0.0135

156.625

−0.0056

117.75

−0.0189

157.875

0.0079

118.375

0.0053

155.375

−0.0160

115.625

−0.0235

150.5

−0.0319

117.75

0.0182

155.75

0.0343

117.5

−0.0021

154.25

−0.0097

118.5

0.0085

155.875

0.0105

117.625

−0.0074

156

0.0008

114.625

−0.0258

152.75

−0.0211

110.75

−0.0344

150.5

−0.0148

150.75

0.0017

Std. Deviation:

1.71%

1.63%

Annual Std. Dev.:

27.08%

25.70%

7(b). In this case, we have the following inputs and option call value: S = 120.625, X = 115, T = 62/365 = 0.17, RF = 0.0742,  = 0.2570, Div Yld = 0.0365

so, d1 = 0.5642 and N ( d1 ) = 0.7137; d 2 = 0.4583 and N ( d 2 ) = 0.6766

and Black-Scholes C = e −( 0.0365)(0.17) (120.625 )( 0.7137 ) − 115 e −( 0.0742)(0.17) ( 0.6766 ) = $8.720

The market price of $12.25 is much higher than the calculated Black-Scholes value. This implies that if all of the other parameters of the model are correct, the implied volatility is much higher than the historical volatility. This could be because recent developments in the company make it a riskier investment than it has been in the past, so the historical volatility statistic is not a good predictor of future events. 8(a). Cost/Contract

x 31,250

Long 1.44 Call

−0.0422

−$1,318.75

Short 1.48 Call

0.0255

$796.88

Long 1.40 Put

−0.0260

−$ 812.50

Short 1.44 Put

0.0422

$1,318.75

June

Net Initial

Long Call

Short Call

Long Put

Short Put

Total Net

USD/GBP

Cost

1.44 Payoff

1.48 Payoff

1.40 Payoff

1.44 Payoff

Profit

1.36

($15.625)

1,250

−2,500

($1,265.625)

1.40

($15.625)

−1,250

($1,265.625)

1.44

($15.625)

1.48

($15.625)

1,250

$1,234.375

1.52

($15.652)

2,500

−1,250

$1,234.375

Example: Long Call 1.44 profit = (1.48 – 1.44)  31,250 = $1,250 8(b).

This position resembles a bull vertical spread. The purchaser of this portfolio is probably moderately bullish on the exchange rate. We see this from the willingness to give up the extreme upside in exchange for limiting the downside. 8(c).

This is a simple application of put-call parity. C − P = R * exp ( −rf * t ) − X *exp ( nrd *t )

where R is the exchange rate (1.385), X is the exercise price (1.44), rd (rf) is the domestic (foreign) risk-free rate, and t is the time to expiration (2/12 = 0.1667). C − P = 1.385*exp ( −0.07*0.1667 ) − 1.44*exp ( −0.05*0.1667 ) = ( $0.0591)

9(a).

The purchase of both a call and a put on the same underlying stock with the same exercise price and expiration date is called a straddle. In this case, it is an at-the-money straddle since the current stock price and common exercise price are both $100. The initial cost of

this position is $17 (= 9 + 8). The chart below shows the net profit of the straddle strategy for a range of potential Friendwork stock prices at the expiration date in one year. (Note that the call payoff is Max(0, S − 100) and the put payoff is Max(0, 100 – S)): Expiration Date

Call Payoff

Put Payoff

Net Straddle

Stock Price

(X = 100)

Option Cost

Profit

−17

−7

100

−17

110

−17

−7

117

−17

121

−17

130

−17

This shows that the maximum loss would occur at S = 100 when both options would expire out of the money. The breakeven points are 83 (= 100 – 17) and 117 (= 100 + 17) as the future stock price will have to move either up or down by enough to cover the cost of the initial option purchase. This is shown in the following diagram:

9(b).

The purchase of both a call and a put on the same underlying stock with the same expiration date but different exercise prices is called a strangle. In this case, both options are out of the money, so the initial cost of the position is cheaper at $11 (= 6 + 5). The chart below shows the net profit of the strangle strategy for a range of potential Friendwork stock prices at the

expiration date in one year. (Note that the call payoff is Max(0, S − 110) and the put payoff is Max(0, 90 – S)): Expiration Date

Call Payoff

Put Payoff

Net Strangle

Stock Price

(X = 110)

(X = 90)

Option Cost

Profit

−11

−4

−11

100

−11

110

−11

117

−11

−4

121

−11

130

−11

This shows that the maximum loss would occur at a range of S between 90 and 110 when both options would expire out of the money. The breakeven points are 79 (= 90 – 11) and 121 (= 110 + 11) as the future stock price will have to move either up or down by enough to cover the out-of-the-money strike price as well as the cost of the initial option purchase. This is shown in the previous diagram. 10(a). Price of ARB

Profit on

Net Profit on

Stock at Expiration

Initial Cost

Call #1 Position

Call #2 Position

Total Position

($1.72)

$0.00

($1.72)

$0.00

($1.72)

$0.00

($1.72)

$5.00

$0.00

$3.28

($1.72)

$10.00

$0.00

$8.28

($1.72)

$15.00

($10.00)

$3.28

($1.72)

$20.00

($20.00)

($1.72)

$25.00

($30.00)

($6.72)

10 (b).

Breakeven points are $51.72 and $68.28. Maximum profit occurs at $60. Maximum profit occurs when the short calls are at the money; at prices above $60, the losses on the short calls reduce profit. Breakeven on the low side occurs at the long call strike price ($50) plus sufficient stock price increase to cover the position’s net initial cost ($1.72). This equals $51.72. Breakeven on the high side occurs with the position’s net initial cost ($1.72), profit on the call (P − $50), and loss of the short calls 2($60 − P) equal zero:

( P − $50 ) +2 ( $60 − P ) − $1.72 = 0 which occurs when price = $68.28. 10(c). The user of this position is betting on low volatility (that prices will stay between breakeven points). The holder has limited liability for substantial price declines and unlimited liability for substantial price increases. 11(a). Conversion value = 48.852 shares  12.125 = $592.33 The conversion option embedded in this bond is currently out of the money because the conversion value is below the current market price of the bond. 11(b). Conversion parity price = Bond price/conversion ratio = 965/48.852 = $19.754 11(c). Payback =

Bond price – Conversion value Bond income – Income from equal investment in common stock

Sold the bond for $965.00 and used the proceeds to purchase 79.588 shares (=$965.00/$12.125) of Bildon Enterprises stocks, the payback period would be 4.887 years; this is viable as the bond matures in 7 years.

38.125 1000 + t (1 + i / 2)14 t =1 (1 + i / 2)

11(d). $965.00 = 

Solving for i = 4.147% (semiannual) or 8.294% (annual). Using an annual (APR) rate of 9.25 percent, we compute the “straight” price as 14

38.125 1000 + t (1 + 0.04625)14 t =1 (1 + 0.04625)

$917.61 = 

This

means

that

the

net

value

the

combined

option

positions

$965.00 − $917.61 or $47.39 .

Solution and Answer Guide: Frank K. Reilly, Keith, C. Brown, Sanford J. Leeds, Investment Analysis & Portfolio Management, 12th Edition, © 2025, 9780357988176; Chapter 17: Professional Portfolio Management, Alternative Assets And Industry Ethics

Solution and Answer Guide

FRANK K. REILLY, KEITH, C. BROWN, SANFORD J. LEEDS, INVESTMENT ANALYSIS & PORTFOLIO MANAGEMENT, 12TH EDITION, © 2025, 9780357988176; CHAPTER 17: PROFESSIONAL PORTFOLIO MANAGEMENT, ALTERNATIVE ASSETS AND INDUSTRY ETHICS

ANSWERS TO QUESTIONS 1.

Private wealth management and advisory firms typically develop a personal relationship with their clients, getting to know the specific investment objectives and constraints of each. The collection of assets held can then be tailored to the special needs of the client. Conversely, a mutual fund (investment company) offers a general solution to an investment problem (for example, a large cap, growth-oriented equity portfolio) and then markets the portfolio to investors who may fit that profile. Special attention comes at a cost and for that reason, private management firms are used mainly by investors with substantial levels of capital, such as pension funds and high-networth individuals. Conversely, individual investors with relatively small pools of capital are typically the primary clients of investment companies. Management and advisory firms hold the assets of both individual and institutional investors in separate accounts, which allows for the possibility of managing each client’s portfolio in a unique manner. Conversely, investment companies are pools of assets that are managed collectively. Investors in these funds receive shares representing their proportional ownership in the underlying portfolio of stocks, bonds, or other securities. Investment companies usually offer investors several different mutual funds based on different asset classes or investment strategies to appeal to the widest clientele possible.

Based upon Exhibit 17.2, there has been a rapid increase in the number of large asset management firms. Much of this asset growth can be explained by the strong performance of the U.S. equity markets during this period, but there has been a trend toward consolidating assets under management in large, multiproduct firms. As of December 2022, all of the 50 largest firms had an AUM of more than $220 billion and the 10 largest firms had an AUM of over $1 trillion.

After the initial public sale of shares in the investment company, the open-end fund will continue to sell new shares to the public at the NAV with or without a sales charge and will redeem (buy back) shares of the fund at the NAV. In contrast, the closed-end fund does not buy or sell shares once the original issue is sold. Therefore, the purchase price or sales price for a closed-end fund is determined in the secondary market. Because it may have to redeem investor shares on a daily basis, a mutual fund company needs to have a portfolio of securities that is relatively liquid (that is, can be converted to cash fairly quickly without a

substantial price discount). Conversely, a closed-end fund does not face that shareholder redemption pressure, which allows it to hold assets in the underlying portfolio that also do not need to be sold quickly. 4.

A load fund charges a fee for the sale of shares (front-end load) and/or redeeming shares (back-end load). It will sell shares at its net asset value plus the sales charge. The purpose of this load fee is to compensate the salesperson assisting the investor with the transaction; it is not a fee associated with superior portfolio management skills. A no-load fund has no initial sales charge, so it will sell or repurchase its shares at its net asset value. No-load funds are permitted to impose on investors an annual expense for marketing their services, which are known as 12b-1 fees.

You definitely should care about how well a mutual fund is diversified. One of the major advantages of a mutual fund is that it provides investors with instant diversification in a cost-effective manner, so it truly is important. Given the CAPM, it is well established that the capital market only rewards investors for their systematic risk exposure, meaning that it is important to eliminate unsystematic risk. An undiversified fund is making unsystematic “bets” that the investor may not be aware of unless they carefully review the fund’s statements and statistical analyses.

As an investor, it is the net return that is important because these are the returns that you ultimately receive. The net return for a fund is the return after all research and management costs are deducted. The gross return is the return that the fund produces before these expenses are deducted, meaning that it represents the performance of the actual security selection decisions that the fund manager has made. On the other hand, the net return is the relevant number for the investor because it accounts for both the manager’s investment skill and what the manager charges for providing those services. Typically, the average difference between gross and net returns is about one percent a year, but this varies by fund.

In a limited partnership, one or more general partners are responsible for running the organization and assuming its legal obligations, while the remaining limited partners are liable only to the extent of their investments. For example, in a private equity partnership, the general partner develops, implements, and maintains the investment portfolio around an initial strategy, while the limited partners (high-net-worth individuals, pension funds, endowment funds) provide most of the capital but have no direct involvement in the actual investment process of the underlying portfolio of assets.

1. The alpha from the long–short strategy can be transported to other asset classes by combining a long–short portfolio with other derivatives, such as swaps and futures. The appropriate derivatives to overlay on the long–short position would be determined by the type of return that the investor seeks to replicate. This class of investment schemes are sometimes referred to as “portable alpha” strategies. 2. The three major quantifiable sources of risk are the following: i) Market risk ii) Credit risk iii) Liquidity risk Supplemental information (not required):

i) Market risk refers to losses that could arise as a result of changes in market factors. ii) Credit risk refers to losses that could arise as a result of declines in the creditworthiness of the fund’s investments or counterparties to derivative positions. iii)Liquidity refers to losses that could arise as a result of changes in a fund’s investment strategies, the liquidity of the portfolio assets, and the rights of investors to redeem their shares. 3(a). Risk-based leverage uses a measure of market risk (such as downside risk measures, for example, VAR) relative to a measure of the resources available to absorb risk (such as cash or equity). Accounting-based leverage is based on the traditional concept of measuring asset values relative to equity capital. 3(b). The hedge fund investment strategies best characterized by each of the three strategy components reviewed by Marco are as follows: 1. Quantitative and technical strategies 2. Risk arbitrage and relative value strategies 3. Quantitative and technical strategies Supplemental information (not required) 1. The first component—describing a quantitative strategy—involves purchasing stocks after positive earnings surprise announcements, anticipating that stock prices will rise in the short term. This strategy employs a tested, historically profitable pattern. 2. The second component—describing a risk arbitrage/relative value strategy— involves buying and selling stocks of companies that have announced or are rumored to be considering a merger or acquisition transaction. In this strategy, investors predict the likelihood that the announced or rumored transactions will be closed. 3. The third component—describing a quantitative strategy—involves using neural networks to detect patterns in historical data; computer-generated analyses of past trading patterns are used to determine future trading strategies. 3(c). Marco’s

Five Correct

or If Incorrect, Give One Reason Why the

Conclusions

Incorrect?

Conclusion Is Incorrect

1. Increase

Correct

Low correlations between hedge funds

diversification

and traditional asset classes Evidence of positive alpha (Exhibit 17.17).

2. Lack of

Incorrect

Lack of transparency is often a concern

transparency/inability

with hedge fund investments but

to add value

managers in different fund categories have positive alphas, indicating they are adding value (Exhibit 24.19).

3. Directional hedge

Incorrect

Client has exposure to stock and bond

fund is more

market “beta” through their current

appropriate for this

portfolio. Sources of “pure” (that is,

client

zero-beta exposure) alpha and low correlation need to be identified, possibly via market neutral or arbitrage-based strategies.

4. Macro hedge fund is

Incorrect

Risk level is moderate (Exhibit 17.17);

appropriate for the

other strategies tend to exhibit far less

client

volatility than those that adopt directional positions (such as macro).

5. Equitized long–

Correct

short strategy

Nondirectional strategy provides lowcorrelated alpha. Exhibit 17.17 indicates equity market neutral strategy has the lowest standard deviation of the funds presented.

Managers are often compensated with a base salary and a bonus that depends on the performance of their portfolios relative to those of their peers. Therefore, a manager with a relatively poor performance midway through a compensation period could be more likely to increase the risk of the portfolio in an effort to increase his or her final standing in the peer group. Of course, altering fund risk to enhance compensation suggests that some managers may not always act in their client’s best interest.

10.

Soft dollars are generated when a manager commits the investor to pay a brokerage commission that is higher than the simple cost of executing a stock trade in exchange for the manager receiving additional bundled services from the broker. One example would be for a manager to route his or her trades through a non-discount broker in order to receive security research reports that the brokerage firm produces. Academic studies have debated the merits and shortcomings of these soft dollar arrangements, reaching the conclusion that capital markets might be more efficient without them.

ANSWERS TO PROBLEMS 1. 2.

$50,000 = 6,250 $8.00 Current NAV = ( $75,800 − $0 ) /6,250 shares = $12.13

Initial number of shares =

Load Fund:

Initial Investable Capital: $1,000 − $80 = $920 Capital after One Year: $920  1.15 = $1,058.00 Net Return on Initial Contribution: ( $1,058/$1,000 − 1) = 5.80% or 5.80 percent growth No-Load Fund: Initial Investable Capital: $1,000 Capital after One Year: $920  1.12 = $1,200.00

Net of Back-End Redemption: $1,200  (1 − 0.01) = $1,108.80 Net return on Initial Contribution: ( $1,108.80/$1,000 − 1) = 5.80% or 10.88 percent growth The no-load fund offers an extra $50.80 over the load fund for a $1,000 initial contribution held over a one-year time period, net of purchase and redemption fees. The difference in percent growth is 5.08 percent.

3. Period

NAV

Premium/Discount

Market Price

Annual Return

3(a).

$10.00

0.0

$10.000

11.25

−5.0

10.688

6.88%

9.85

+2.3

10.077

−5.72

10.50

−3.2

10.164

0.87

12.30

−7.0

11.439

12.54

Using the above data, the arithmetic average return per year is 3.64 percent. On an annual

compounded (geometric average) basis, the average annual return is 3.42 percent. This latter answer is the same as if the annual return is computed using only the end points; shares were worth $11.439 at the end of year 4 and were purchased for $10, giving a compounded return of 0.25 ( $11.439 / $10 ) − 1 = 3.42% 3(b).

(12.30 / 10.00 )

3(c).

Ignoring commission, shares were purchased for $10.69 and sold at 10.08 and thus a return

(1/4)

− 1 = 5.31%

of −5.72% ( = 10.08/10.69 − 1 ) . 3(d).

Change in NAV is $9.85 − $11.25 = $ − 1.40 ; the percentage change is $ − 1.40 / 11.25 = −12.44%

4(a). Client 1

Client 2

0.0100  5,000,000 = 50,000

0.0075  5,000,000 = 37,500

0.0060  10,000,000 = 60,000

0.0040 

4(b). 4(c)

7,000,000 28,000 = 27,000,000 175,500

0.0040 

77,000,000 308,000 = 97,000,000 455,500

175,500/27,000,000 = 0.0065 or 0.65% 455,000/97,000,000 = 0.004696 = 0.47%

Costs of management (such as research expenses, office costs, and support personnel

salaries) do not increase at the same rate as the managed assets because substantial economies of scale exist in managing assets. 5. Conduct

Potential Conflict

a. Compensation

A fee structure of this type can cause a conflict because the portfolio manager

based on

has an incentive to turn over (that is, “churn”) security positions in client

commissions

accounts simply to increase fees, rather than for valid investment strategy

from clients’

purposes. A high volume of trading may conflict with a client’s investment

trades.

objectives.

b. Use of client

A conflict may be created when client brokerage is used to generate soft dollars

brokerage (soft

for the purchase of goods or services that benefit the firm (such as those that

dollars).

are used in the management of the firm) rather than benefit the clients whose trades generated the soft dollars. An investment manager may pay higher commissions to obtain “soft dollar” credits to buy goods and services that do not necessarily provide direct benefits to the client whose commissions are being used; such actions, however, must be justifiable on the basis that the goods and services aid the manager in its investment decision-making process. Absent such justification, the manager is putting its own interest ahead of the client’s interest.

c. Purchase of

There is a potential conflict of interest when a portfolio manager trades, for the

stock in a

account he or she manages, in the shares of a company in which he or she

company whose

personally owns warrants that can be used to purchase shares of that same

warrants are

company. The conflict arises whether the manager purchases or holds such

owned by the

shares for the fund because that transaction could be viewed as an attempt to

portfolio manager

increase the value of the warrants.

Accepting reimbursement for such expenses as meals, lodging, or plane fare

Reimbursement

from the issuer of the stock about which the analyst is writing a research

of analyst

report creates an obvious conflict of interest. Such conduct gives rise to the

expenses.

perception that the analyst’s independence and objectivity may have been compromised, causing the validity of the opinions expressed in the report to be in doubt.

Beginning value

6(a).

= $27.15  257.876 = $7,001.33

Capital gain & dividends = $1.12  257.876 = 288.82 Ending value

Return =

= $30.34  257.876 = 7,823.96

( $7,823.96 − $7,001.33) +288.82

= 15.87% $7,001.33 which can be computed on a per-share basis: ( $30.34 – 27.15 ) +1.12  / $27.15 = 0.1587 or 15.87%

6(b).

Only the dividend and capital gain distribution are taxable; the shares are not yet sold, so

the change in NAV does not represent a taxable (realized) gain or loss:

6(c).

( $30.34 – 27.15) + (1.12 )(1 − 0.30 )  / $27.15 = 3.974 / $27.15 = 0.1464 or 14.64% The investor received a distribution of $1.12 per share which, at the year-end NAV,

purchases $1.12/$30.34 = 0.036915 additional shares for each original share held. Because the investor owned 257.876 shares, he or she can purchase 9.519 shares. Notice that the same number of shares would be purchased for both the taxable and the non-taxable investor, but the taxable investor would still be responsible for paying taxes on the $1.12 distribution, regardless of whether that amount was paid in cash or reinvested into additional shares. 7. Year 1:

Year 2:

Stock

Price

100,000

$45.25

$4,525,000

100,000

$48.75

$4,875,000

225,000

25.38

$5,710,500

225,000

24.75

$5,568,750

375,000

14.5

$5,437,500

375,000

12.38

$4,642,500

115,000

87.13

$10,019,950

115,000

98.5

$11,327,500

154,000

56.5

$8,701,000

154,000

62.5

$9,625,000

175,000

$11,025,000

175,000

$13,475,000

212,000

$6,784,000

212,000

38.63

$8,189,560

275,000

15.25

$4,193,750

275,000

8.75

$2,406,250

450,000

9.63

$4,333,500

450,000

27.45

$12,352,500

90,000

71.25

$6,412,500

90,000

75.38

$6,784,200

87,000

42.13

$3,665,310

87,000

49.63

$4,317,810

137,000

19.88

$2,723,560

27.88

17.75

150,000

19.75

$2,962,500

Ttl MV of

$73,531,570

$86,526,570

Cash:

$3,542,000

$2,873,000

Expenses:

$730,000

$830,000

Ttl Net

$76,343,570

Fund Value: 7(a)

$88,569,570

Year 1 NAV = $76,343,570/5,430,000 = $14.06

7(b) Year 2 NAV = $88,569,570/5,430,000 = $16.31, so one-year return ( growth) = 16.31/14.06 − 1 = 16.01%

7(c)

Shares redeemable with cash on hand = $2,873,000/$16.31 = 176,137.132

7(d). Net shares to be redeemed ( after liquidating cash balances ) = 500,000 – 176,137.132 = 323,862.868 Net dollars of shares to be liquidated = 323,862.868  16.31 = $5,282,577.35 Proportion of year 2 portfolio MV to be sold = $5,282,577.35/$86,256,570 = 6.11% So, every portfolio holding must be reduced by slightly more than 6 percent. For example,

Stock A will require a sale of $297,626.09 ( = $4,875,000  0.0611 ) , which translates into 6,105.2 ( =$297,626.09/48.75) shares sold.

Stock

Price

Position Reduce

Shares Sold

$48.75

$4,875,000

$297,626.09

6105.2

24.75

$5,568,750

$339,980.57

13736.6

12.38

$4,642,500

$283,431.61

22894.3

98.5

$11,327,500

$691,560.93

7020.9

62.5

$9,625,000

$587,620.74

9401.9

$13,475,000

$822,669.03

10684.0

38.63

$8,189,560

$499,984.97

12942.9

8.75

$2,406,250

$146,905.18

16789.2

27.45

$12,352,500

$754,138.72

27473.2

75.38

$6,784,200

$414,185.62

5494.6

49.63

$4,317,810

$263,608.80

5311.5

27.88

$0.00

0.0

19.75

$2,962,500

$180,865.08

9157.7

Total: 8(a).

(1)

$5,282,577.35

3 percent front-end load = $100,000 (1 − 0.03) = $97,000 $97,000 (1+0.12 ) = $97,000 (1.4049 ) = $136,278.02 3

(2) a 0.50 percent annual deduction (assumed to be deducted at year end) Year 1: $100,000 ( 1+0.12 ) = $112,000 ( 1 − 0.005) = $111,440 Year 2: $111,440 ( 1+0.12 ) = $124,812.80 ( 1 − 0.005 ) = $124,188.74 Year 3: $124,188.74 ( 1+0.12) = $139,091.38 ( 1 − 0.005) = $138,395.93, or 100,000  ( 1+0.12 )  ( 1 − 0.005) 3

(3) a 2 percent back-end load

$100,000 (1+0.12) = $100,000 (1.4049 ) = $140,492.80 3

8(b).

$140,492.80 (1 − 0.02) = $137,682.94 Choice (2) with ending wealth of $138,395.93

(1)

3 percent front-end load = $100,000 (1 − 0.03) = $97,000 $97,000 (1+0.12 ) = $97,000 ( 3.10585) = $301,267.28 10

(2) a 0.50 percent annual deduction $100,000 (1 + 0.12 )

(1 − 0.005) = $100,000 ( 3.10585)( 0.9511) 10

= $295,400.37 (3) a 2 percent back-end load

$100,000 (1+0.12 ) = $100,000 ( 3.10585) = $310,584.82 10

8(c).

$310,584.82 (1 − 0.02 ) = $304,373.12 The answer would change, now choice (3) with an ending wealth of $304,373.12. A front-end load takes the money out right away, thus reducing your initial investment contribution. The annual fee is usually less than one percent, which is a small amount and based on the example the preferred choice for a holding period of three years. A back-end load is usually a smaller percentage than a front-end load, though the dollar amount has typically grown during the holding period. However, back-end loads are not due

until the fund is liquidated, which allows the full initial contribution to experience compounded growth for a longer period of time. 9.

You should refer to the CFA Institute’s Code of Ethics and Standards of Professional Conduct in Appendix B at the back of the book. 9(a). The following CFA Institute Standards of Professional Conduct apply to Clark if C&K provides all three functions on a combined basis. 1. According to CFA Institute Standard VI.A, Disclosure of Conflicts, C&K must disclose to its clients any material conflict of interest relating to the firm that may be perceived by clients to influence C&K’s objectivity. If the Europension Group is hired as a pension consultant and it recommends to the client that C&K International be hired to manage the pension portfolio, C&K needs to disclose to the client C&K’s interest in the affiliate. If C&K International executes pension portfolio securities transactions through Alps Securities, C&K needs to disclose to clients C&K’s interest in the affiliate so that they can decide whether C&K’s self-interests are compromising its interests. 2. According to CFA Institute Standard IV.B, Disclosure of Additional Compensation Arrangements Compensation, C&K must inform its customers and clients of compensation or other benefit arrangements concerning its services to them that are in addition to compensation from them for such services. Therefore, C&K needs to disclose to clients that C&K earns through its affiliates additional compensation in the management of portfolios and in brokerage transactions. 3. According to CFA Institute Standard VI. C, Disclosure of Referral Fees, C&K must inform prospective customers or clients of any considerations paid or other benefits delivered to any of its affiliates for recommending to them (the clients or customers) services of a sister organization. Such disclosure should help the customer or client evaluate any possible partiality shown in any recommendation of services or evaluate the full cost of such services. 4. According to CFA Institute Standard III. B, Fair Dealing with Customers and Clients, C&K must deal fairly with all customers and clients when taking investment actions. Alps Securities needs to deal fairly with all customers when executing security orders and not favor the pension clients of C&K International over the other customers of Alps Securities. 9(b). The following CFA Institute Standards of Professional Conduct apply to this situation: 1. According to CFA Institute Standard III. C1, Portfolio Investment Recommendations and Actions, C&K must, when taking an investment action for a specific portfolio or client, consider its appropriateness and suitability for such a portfolio or client. In considering such matters, C&K must consider the needs and circumstances of the client. C&K must also use reasonable judgment to determine the applicable relevant factors. According to these requirements, C&K should consider differences in relevant factors in various countries. If the duties, practices, and customs in that country stipulate a particular pension fund asset allocation, then C&K may reasonably be expected to respect this practice. Because his clients and pension beneficiaries specifically want to conform to conservative asset allocation practices in their country, C&K may reasonably be expected to respect its clients’ investment policies. 2. According to CFA Institute Standard III. C and Standard V.A, Fiduciary Duties, Clark should determine applicable fiduciary duties in that country and comply with them. Clark must also determine to whom the duties are owed and what asset allocation is best suited to the investment objectives of the respective clients together with the investment conditions and circumstances in the host country. If the prevailing fiduciary practice

stipulates a particular asset allocation, C&K may have to respect that practice. Therefore, the practice of allocating at least 80 percent of pension fund assets to fixed-income securities may be appropriate within the context of this European country’s practices, as opposed to a North American context. 9(c). The following CFA Institute Standards of Professional Conduct apply in this situation. 1. According to CFA Institute Standard I. A, Required Knowledge and Compliance, Clark must comply with the CFA Institute Standards of Professional Conduct and the accompanying Code of Ethics. Although local laws, rules, and regulations may not relate specifically to the use of material nonpublic information, any violation of another standard would be considered a violation of Standard II. A. Therefore, the latter standard applies. 2. According to CFA Institute Standard I.D, Prohibition Against Assisting Legal and Ethical Violations, Clark must not knowingly participate in any act that would violate the CFA Institute Standards of Professional Conduct. Lacking specific local or other regulatory requirements in a country, or when the CFA Institute Standards impose a higher degree of responsibility or higher duty than that required by local or other law or custom, Clark is held to the CFA Institute Standards. 3. According to CFA Institute Standard II. A, Prohibition Against Use of Material Nonpublic Information, Clark cannot use insider information in his investment actions. Specifically, Clark must not take investment actions based on material nonpublic information if (1) such actions violate a special or confidential relationship with an issuer or with others or (2) such information was disclosed to him in a breach of duty or was misappropriated. If a breach of duty exists, Clark should try to achieve public distribution of the information. 10.

The Muellers’ portfolio can be evaluated in terms of the following criteria: i. Preference for “Minimal Volatility.” The volatility of the Muellers’ portfolio is likely to be much greater than minimal. The asset allocation of 95 percent stocks and 5 percent bonds indicates that substantial fluctuations in asset value will likely occur over time. The asset allocation’s volatility is exacerbated by the fact that the beta coefficient of 90 percent of the portfolio (that is, the four growth stock allocations) is substantially greater than 1.0. Thus, the allocation to stocks should be reduced, as should the proportion of growth stocks or higher beta issues. Furthermore, the 5 percent allocation to bonds is in a long-term, zerocoupon bond fund that will be highly volatile (that is, high duration) in response to longterm interest rate changes; this bond allocation should be exchanged for one with lower volatility (perhaps shorter duration, higher credit grade issues.) ii. Equity Diversification. The most obvious equity diversification issue is the concentration of 35 percent of the portfolio in the high beta small cap stock of Andrea’s company, a company that may have a highly uncertain future. A substantial portion of the stock should be sold, with care to tax implications. Another issue is the 90 percent concentration in high beta growth stocks, which contradicts the Muellers’ preference for minimal volatility investments. The same is true of the portfolio’s 45 percent allocation to higher volatility small cap stocks. Finally, the entire portfolio is concentrated in the domestic market. Diversification away from Andrea’s company’s stock, into more value-oriented stocks, into more large cap stocks, and into at least some global (that is, non-U.S.) stocks is justified. iii. Asset Allocation (Including Cash Flow Needs). The portfolio has a large equity weighting that appears to be very aggressive. If, for example, they have below-average risk tolerance and limited growth objectives, a more conservative, balanced allocation is more appropriate. The Muellers are not invested in any asset class other than stocks and the small bond fund holding. A reduction in equity investments, especially growth and small cap

equities, and an increase in debt investments—as well as other asset classes, such as real estate and natural resources—are justified to produce more consistent and desired results over a complete market cycle. 11(a). Implied probability that the takeover bid will be successful is ( 42 − 30 ) / ( 45 − 30 ) = 0.80 or 80% meaning that the market price already captures $12 ( = 42 – 30 ) of the $15 ( = 45 − 30 ) tender offer premium in the proposed takeover deal. 11(b). According to this market-implied probability, a risk arbitrage investor will need to believe that the deal has better than a four-in-five chance of being completed in order to justify purchasing XYZ stock for $42 in the hope of selling at the tender offer price of $45; if the deal falls apart, XYZ’s shares will return to $30. Other factors: - The shares have already risen to $42. - The target firm benefits from the arbitrage, not the acquiring firm, and it is normal to see a temporary drop in the share price of the acquiring firm. - If interest rates have a negative impact, XYZ’s shares could actually fall below $30 if the deal falls apart. 12(a). Market Down 15%

Fund A Return:

Fund B Result:

Long Positions Down 10%

( +150%  −10% ) +

( +75%  −10% ) +

Short Positions Down 20%

( −100%  −20% )

( −25%  −20% )

Up 5.0%

Down 2.5%

Notice that each manager added alpha on both their long stock selections (that is, their portfolios only lost 10 percent in a market that was down 15 percent) and their short positions (their portfolios were down 20 percent, which benefits short holdings). 12(b). Although both hedge funds had the same Net Exposure in the portfolios, the manager with the smaller Long–Short Ratio (that is, Fund A) did better because short positions outperformed long positions in this falling market environment. The exact opposite outcome would have occurred if the overall market return had been positive since Fund B (higher Long–Short Ratio) is positioned with a larger proportion of long positions in the fund.

Solution and Answer Guide: Frank K. Reilly, Keith, C. Brown, Sanford J. Leeds, Investment Analysis & Portfolio Management, 12th Edition, © 2025, 9780357988176; Chapter 18: Evaluation of portfolio performance

Solution and Answer Guide

ANSWERS TO QUESTIONS 1.

The two main questions an investor should ask regarding the performance of her portfolio manager are: (1) How did the manager actually perform? and (2) Why did the manager perform as he did? An answer to the first question is best established by assessing a combination of different risk-adjusted performance measures (Sharpe ratio, Treynor ratio, and Jensen’s alpha) that compare the manager’s actual return performance to an expected return that captures what the portfolio should have produced given the level of risk involved in the investment. An answer to the second question involves a decomposition of investment returns using attribution analysis, which deconstructs the manager’s valueadded return (that is, alpha) relative to the relevant benchmark, into components representing his security selection skills and asset allocation (market timing) skills.

a) Treynor ratio and Jensen’s alpha measure risk by systematic risk (beta). The Sharpe ratio measure the total risk of the portfolio by using the standard deviation of returns over time. The information ratio (IR) uses the standard deviation of excess returns where the excess return is the difference between the return on a portfolio and its benchmark. The Sortino ratio uses a semi-deviation measure of downside risk, including only those returns that fall below a specified minimum acceptable return. b) Treynor divided a fund’s average excess return (nominal return less risk-free rate) by its beta. For a fund not completely diversified, Treynor’s “T” value will understate risk and overstate performance. Sharpe divided a fund’s average excess return by its standard deviation. Sharpe’s “S” value will produce evaluations very similar to Treynor’s “T” value for funds that are well diversified. The Jensen alpha measures performance as the estimated intercept of a regression between a fund’s actual excess return and the excess return on the market portfolio proxy. Because the latter return is based on the capital asset pricing model (CAPM) and a fund’s beta, Jensen makes the same implicit assumptions as Treynor—namely that funds are completely diversified. The IR measures a portfolio’s average return in excess of that of a benchmark, divided by the standard deviation of this excess return. Like the Sharpe measure, it can be used when the fund is not necessarily well diversified. The Sortino ratio defines the excess return as the difference between the portfolio return and a minimum acceptable return defined by the investor; the average of this excess return value is divided by a semi-deviation measure that is computed using only returns less than the minimum acceptable return.

For portfolios with R2 values, noticeably less than 1.0, it would make sense to compute both measures. Differences in the rankings generated by the two measures would suggest lessthan-complete diversification by some funds—specifically those that were ranked higher by Treynor than by Sharpe. For portfolios that are completely diversified (R 2 = 1.0), the Sharpe and Treynor measures will produce identical rankings of investment performance.

Jensen’s alpha ( ) is found from the equation R jt – RFRt =  j +  j Rmt – RFRt  + e jt . The  j indicates whether a manager has superior ( j  0) or inferior ( j  0) ability in market timing or stock selection, or both. So, Jensen defines superior (inferior) performance as a positive (negative) difference between a manager’s actual return and his CAPM-based required return. For poorly diversified funds, Jensen’s rankings would more closely resemble that of Treynor. For well-diversified funds, Jensen’s rankings would follow those of Treynor and Sharpe. By replacing the CAPM with a multifactor risk model (such as the Fama–French three-factor model) to measure the expected return of the managed fund, differences between funds’ actual and required returns (or alpha) would produce a different evaluation of the manager’s ability to add value beyond merely taking on risk in the investment process.

The IR is calculated by dividing the average return on the portfolio less a benchmark return by the standard deviation of that excess return. Thus, the numerator of the IR can be viewed as the fund’s alpha relative to the benchmark (the benefit), and the denominator is viewed as the fund’s tracking error (the additional “cost” in terms of incremental risk). Said differently, the IR can be viewed as a benefit–cost ratio in that the standard deviation of return can be viewed as a cost associated in the sense that it measures the unsystematic risk taken on by active management; therefore, the IR is a cost–benefit ratio that assesses the quality of the investor’s information deflated by unsystematic risk generated by the investment process.

Returns-based measures compare the actual returns on a portfolio to a benchmark. The Sharpe and Treynor measures are straightforward measures that compare portfolio returns to a risk-free rate. The IR and Sortino measures can use the returns of the benchmark portfolio as the standard of comparison. Another type of returns-based analysis is returnsbased style analysis, where portfolio returns are regressed over time against various style benchmarks to determine the portfolio’s underlying risk exposures and its performance against the regression-determined benchmark. Returns-based measures are easier to compute, as they use available historical returns. The biggest disadvantage of returns-based performance measures is that they do not directly observe the actions (security sales and purchases) of the manager but infers what those actions might have been by looking at the returns the portfolio produces. Holding-based measures, such as the Grinblatt–Titman statistic, review performance by examining how the portfolio holdings are actually adjusted over time. By reviewing the securities in the portfolio, it can be determined if the portfolio’s style is changing as well as precisely by analyzing why it underperformed or outperformed the benchmark. The advantage of this approach is that it provides a direct assessment of the quality of the decisions the manager actually made. The disadvantage is that it is frequently difficult or impossible for investors to obtain information about the specific security trades that a fund manager has made.

The difference by which a manager’s overall actual return beats her overall benchmark return is termed the total value-added return, and it decomposes into an allocation effect and a selection effect. The former effect measures differences in weights assigned by the actual and benchmark portfolios to various asset classes (stocks, bonds, and cash) times the respective differences between market-specific benchmark returns and the overall benchmark return. The latter effect focuses on the asset class-specific actual returns less the corresponding asset class-specific benchmark returns times the weights assigned to each market by the actual portfolio. Of course, the foregoing analysis implicitly assumes that the actual and benchmark market-specific portfolios (for example, stocks) are risk equivalent. If this is not true, then the analysis would not be valid.

8(a). Benchmark

Market Index

Explain two different weaknesses of using each of the benchmarks to measure the performance of the portfolio. •

• • • • • • Benchmark Normal Portfolio

• • •

Median of the Manager Universe

• •

A market index may exhibit a survivorship bias; firms that have gone out of business are removed from the index, resulting in a performance measure that overstates the actual performance had the failed firms been included. A market index may exhibit double counting that arises because of companies owning other companies and both being represented in the index. It is often difficult to exactly and continually replicate the holdings in the market index without incurring substantial trading costs. The chosen index may not be an appropriate proxy for the management style of the managers. The chosen index may not represent the entire universe of securities (for example, S&P 500 Index represents 65–70 percent of U.S. equity market capitalization). The chosen index may have a large capitalization bias (for example, S&P 500 has a large capitalization bias). The chosen index may not be investable. There may be securities in the index that cannot be held in the portfolio. This is the most difficult performance measurement method to develop and calculate. The normal portfolio must be continually updated, requiring substantial resources. Consultants and clients are concerned that managers who are involved in developing and calculating their benchmark portfolio may produce an easily beaten normal portfolio, making their performance appear better than it actually is. It can be difficult to identify a universe of managers appropriate for the investment style of the plan’s managers. Selection of a manager universe for comparison involves some, perhaps much, subjective judgment.

•

Comparison with a manager universe does not take into account the risk taken in the portfolio. • The median of a manager universe does not represent an “investable” portfolio, meaning a portfolio manager may not be able to invest in the median manager portfolio. • Such a benchmark may be ambiguous. The names and weights of the securities constituting the benchmark are not clearly delineated. • The benchmark is not constructed prior to the start of an evaluation period; it is not specified in advance. • A manager universe may exhibit the survivorship bias; managers that have gone out of business are removed from the universe resulting in a performance measure that overstates the actual performance had those managers been included. 8(b)i. The Sharpe ratio is calculated by dividing the portfolio risk premium (that is, actual portfolio return minus risk-free return) by the portfolio standard deviation of return: Sharpe ratio = ( Rp – R f ) /  p Where: R p = Actual portfolio return R f = Risk-free return

 p = Standard deviation of portfolio return The Treynor measure is calculated by dividing the portfolio risk premium (that is, actual portfolio return minus risk-free return) by the portfolio beta ( p ): Treynor measure = ( Rp – R f ) /  p

Jensen’s alpha is calculated by subtracting the market premium, adjusted for risk by the portfolio’s beta, from the actual portfolio’s excess return (risk premium). It is usually computed as the estimated intercept term in a regression equation of the portfolio return in excess of the risk-free rate and the excess return on a proxy for the market portfolio:  p = Rp – R f –  p ( Rm − R f ) or  p = Rp – [R f +  p ( Rm – R f ) 

The IR is computed as the average excess return of the portfolio relative to the benchmark return (or alpha) divided by the standard deviation of that excess return (or tracking error): IR = ( Rp – Rb ) /TE The Sortino ratio is the average excess return of a portfolio over a predetermined return threshold level ( ) divided by a measure of downside risk in the portfolio, such as the return semi-deviation statistic: ST = (Rp –  )/DR 8(b)ii. •

The Sharpe ratio assumes that the relevant risk is total risk and measures excess return per unit of total risk.

• • • •

9(a).

The Treynor measure assumes that the relevant risk is systematic risk and measures excess return per unit of systematic risk. Jensen’s alpha assumes that the relevant risk is systematic risk and measures excess return at a given level of systematic risk. The IR views both return and risk of a portfolio relative to that fund’s benchmark portfolio, so it measures the manager’s value-added return per unit of incremental risk (tracking error). The Sortino ratio looks only a measure of downside (underperformance) risk and measures excess returns relative a return threshold definition supplied by the investor.

The basic procedure in portfolio evaluation is to compare the return on a managed portfolio to the return expected on an unmanaged portfolio having the same risk, via use of the CAPM. That is, expected return (Rp ) is calculated from: Rp = R f +  p ( Rm − R f )

where: R f = risk-free rate

Rm = the unmanaged portfolio or the market return

 p = the beta coefficient or systematic risk of the managed portfolio

9(b).

9(c).

Then the benchmark of performance is the unmanaged portfolio. The typical proxy for this unmanaged portfolio is some aggregate stock market indexes, such as the S&P 500 Index. The benchmark error often occurs because the unmanaged portfolio used in the evaluation process is not “optimized.” That is, market indexes, such as the S&P 500, chosen as benchmarks are not on the evaluator’s ex ante mean/variance efficient frontier. The benchmark error may also occur because of an error in the estimation of the risk-free return. Together, these two sources of error will cause the implied security market line (SML) to be mispositioned. The main concept can be shown assuming that the true risk-free rate is lower than the measured risk-free rate and the true market is above the measured market. The result is underperformance relative to the true SML rather than superior performance relative to the measured SML.

9(d).

9(e).

10.

The response depends upon one’s beliefs about whether these portfolios represent the true market portfolio. The Dow Jones Industrial Average comprises only 30 industrial stocks; S&P 500 includes more firms and broader economic sector exposure but has a large-cap bias. The NYSE Composite includes only NYSE-traded stocks. All three measures exclude asset classes such as bonds, cash, real estate, and non-U.S. equities. Defense of CAPM: It is valid as a normative theory as it describes the proper measure of risk (systematic risk, not total risk or another risk definition), and it shows what the relationship should be between risk and expected return; it is linear and affected by the market risk premium and the asset’s level of systematic risk. Scrap CAPM: Opponents may argue that a theory is not worth much if variables cannot be adequately measured. We cannot do empirical testing to prove or disprove it because (a) benchmark error may or may not exist and (b) if they do exist, we cannot adequately measure benchmark returns. Simple (that is, non-risk-adjusted) measures of portfolio performance, such as peer group comparisons or drawdown measures, have the advantage of being relatively easy to compute and they also provide a direct comparison between funds that may be substitutes (opportunity costs) for one another in the mind of the investor. A peer group comparison, for example, ranks all funds of a comparable investment style, making it easy to judge “winners” and “losers” among the competitors. Similarly, comparing drawdown measures across funds indicates which managers were better able to protect their investors against severe market declines. The drawback to these simple measures is that they do not explicitly account for the risk of the funds involved in the comparison, which could make a simple contrasting of returns an invalid exercise since funds with higher risk should produce higher returns in a way that has nothing to do with a manager’s value-added skills.

ANSWERS TO PROBLEMS 1(a). 0.15 − 0.07 0.08 = = 1.60 0.05 0.05 0.20 − 0.07 0.13 SQ = = = 1.30 0.10 0.10 0.10 − 0.07 0.03 SR = = = 1.00 0.03 0.03 0.17 − 0.07 0.10 SS = = = 1.67 0.06 0.06 0.13 − 0.07 0.06 Market = = = 1.50 0.04 0.04 SP =

1(b). 0.15 − 0.07 0.08 = = 0.0800 1.00 1.00 0.20 − 0.07 0.13 TQ = = = 0.0867 1.50 1.50 0.10 − 0.07 0.03 TR = = = 0.0500 0.60 0.60 0.17 − 0.07 0.10 TS = = = 0.0909 1.10 1.10 0.13 − 0.07 0.06 Market = = = 0.0600 1.00 1.00 TP =

Sharpe

Treynor

Market

1(c).

It is apparent from the rankings above that Portfolio Q was poorly diversified because Treynor ranked it #2 and Sharpe ranked it #4. Otherwise, the rankings are similar.

2(a).

The Treynor measure (T) relates the rate of return earned above the risk-free rate to the portfolio beta during the period under consideration. Therefore, the Treynor measure shows the risk premium (excess return) earned per unit of systematic risk, ignoring any remaining unsystematic (undiversified) risk in the portfolio:

Ti =

Ri − R f

i

where: Ri = average rate of return for portfolio i during the specified period R f = average rate of return on a risk-free investment during the specified period

 i = beta of portfolio i during the specified period. Treynor Measure Performance Relative to the Market (S&P 500) T=

10% − 6% = 6.7% 0.60

Outperformed

Market (S&P 500)

TM =

12% − 6% = 6.0% 1.00

The Treynor measure examines portfolio performance in relation to the SML. Because the portfolio would plot above the SML, it outperformed the S&P 500 Index. Because T was greater than TM , 6.7 percent versus 6.0 percent, respectively, the portfolio clearly outperformed the market index. The Sharpe measure (S) relates the rate of return earned above the risk-free rate to the total risk of a portfolio by using the standard deviation of returns. Therefore, the Sharpe measure indicates the risk premium (excess return) per unit of total risk: R − Rf S= i

i

where: Ri = average rate of return for portfolio i during the specified period R f = average rate of return on a risk-free investment during the specified period

 i = standard deviation of the rate of return for portfolio i during the specified period. Sharpe Measure Performance Relative to the Market (S&P 500) S=

10% − 6% = 0.222% 18%

Underperformed

Market (S&P 500)

SM =

2(b).

12% − 6% = 0.462% 13%

The Sharpe measure uses total risk to compare portfolios with the capital market line (CML). The portfolio would plot below the CML, indicating that it underperformed the market. Because S was less than SM , 0.222 versus 0.462, respectively, the portfolio underperformed the market. The Treynor measure assumes that the appropriate risk measure for a portfolio is its systematic risk (or beta). Hence, the Treynor measure implicitly assumes that the portfolio being measured is fully diversified. The Sharpe measure is similar to the Treynor measure except that the excess return on a portfolio is divided by the standard deviation of the portfolio. For perfectly diversified portfolios (that is, those without any unsystematic or specific risk), the Treynor and Sharpe measures would give consistent results relative to the

market index because the total variance of the portfolio would be the same as its systematic variance (beta). A poorly diversified portfolio could show better performance than the market if the Treynor measure is used but lower performance than the market if the Sharpe measure is used. Any difference between the two measures relative to the markets would come directly from a difference in diversification. In particular, Portfolio X outperformed the market if measured by the Treynor measure but did not perform as well as the market using the Sharpe measure. The reason is that Portfolio X has a large amount of unsystematic risk. Although not stated in the problem, it is likely that Portfolio X has an R 2 of 0.6 or less. Such risk is not a factor in determining the value of the Treynor measure for the portfolio because the Treynor measure considers only systematic risk. The Sharpe measure, however, considers total risk (that is, both systematic and unsystematic risks). Portfolio X, which has a low amount of systematic risk, could have a high amount of total risk because of its lack of diversification. Hence, Portfolio X would have a high Treynor measure (because of low systematic risk) and a low Sharpe measure (because of high total risk). 3(a).

Portfolio MNO enjoyed the highest degree of diversification because it had the highest R 2 (94.8 percent). The statistical logic behind this conclusion comes from the CAPM, which says that all fully diversified portfolios should be priced along the SML. R 2 is a measure of how well assets conform to the SML, so R 2 is also a measure of diversification. A completely diversified portfolio would have an R 2 of 100 percent. 3(b). Note that the mean returns are net of the risk-free rate. Doing the calculations we obtain: Fund Sharpe Rank Treynor Rank Jensen Rank ABC

1.022/1.193 = 0.857

1.022/1.048 = 0.975

0.192

DEF

0.473/0.764 = 0.619

0.473/0.662 = 0.715

−0.053

GHI

0.935/0.793 = 1.179

0.935/0.594 = 1.574

0.463

JKL

0.955/1.044 = 0.915

0.955/0.757 = 1.262

0.355

MNO

0.890/0.890 = 1.000

0.890/0.785 = 1.134

0.296

3(c). Fund

t(alpha)

ABC

0.192/0.11 = 1.7455

DEF

−0.053/0.19 = -0.2789

GHI

0.463/0.19 = 2.4368

JKL

0.355/0.22 = 1.6136

MNO

0.296/0.14 = 2.1143

With a t-statistic greater than 2 being the relevant criterion, only GHI and MNO have significantly positive alphas at a 95 percent of confidence interval.

Overall performance (Fund 1) = 26.40% − 6.20% = 20.20%

4(a).

4(b).

Overall performance (Fund 2 ) = 13.22% − 6.20% = 7.02% E ( Ri ) = 6.20 +  (15.71 – 6.20 ) = 6.20 +  ( 9.51) Total return (Fund 1 ) = 6.20 + (1.351 )( 9.51 ) = 6.20 + 12.85 = 19.05% where 12.85 percent is the required return for risk

Total return (Fund 2 ) = 6.20 + ( 0.905 )( 9.51 ) = 6.20 + 8.61 = 14.81% where 8.61% is the required return for risk

Selectivity1 = 20.2% − 12.85% = 7.35%

4(c)(i).

Selectivity2 = 7.02% − 8.61% = −1.59%

4(c)(ii).

Ratio of total risk1 = 1/ m = 20.67/13.25 = 1.56 Ratio of total risk2 = 2/ m = 14.20/13.25 = 1.07

R1 = 6.20 + 1.56 ( 9.51) = 6.20 + 14.8356 = 21.04% R2 = 6.20 + 1.07 ( 9.51) = 6.20 + 10.1757 = 16.39% Diversification1 = 21.04% – 19.05% = 1.99% Diversification2 = 16.39% – 14.81% = 1.58% Net Selectivity = Selectivity − Diversification

4(c)(iii). Net Selectivity1 = 7.35% − 1.99% = 5.36% Net Selectivity 2 = −1.59% − 1.58% = −3.17% 4(d).

Even accounting for the added cost of incomplete diversification, Fund 1’s performance was above the market line (best performance), while Fund 2’s performance fell below the line, before and after adjusting for the lack of complete diversification.

5(a). Manager 1: Average return = ( −1.5 + −1.5 =



+ 13.5 + 17.5) / 10 = 4.50%

Standard deviation = ( −1.5 − 4.5) + ( −1.5 − 4.5) +  2



 = 6.90%

+ ( 17.5 − 4.5)  / 10  2

0.5

 = 5.63%

Semi-deviation = ( −1.5 − 4.5) + ( −1.5 − 4.5) + ( −1.5 − 4.5) + ( −1.0 − 4.5) + (17.5 − 4.5)  / 5   2

0.5

Manager 2: Average return = ( −6.5 + −3.5 =



+ 12.5 + 13.5 ) / 10 = 4.50%

 = 6.63% Semi-deviation = ( −6.5 − 4.5 ) + ( −3.5 − 4.5 ) + ( −1.5 − 4.5 ) + ( 3.5 − 4.5 )  / 4 = 7.45%  

Standard deviation = ( −6.5 − 4.5 ) + ( −3.5 − 4.5 ) +  2

+ (13.5 − 4.5 )  / 10  2

0.5

Sharpe ratio: (average return minus risk-free rate) / standard 5(b).

deviation Mgr X:

0.435

Mgr Y:

0.452

Best performer

Sortino ratio: (average return − minimum acceptable return)/semi5(c). deviation Mgr X:

0.533

Mgr Y:

0.403

Best performer

5(d). The Sharpe and Sortino measures should provide the same performance ranking when the return distributions are symmetrical for the funds or managers under consideration. Asymmetric distributions, such as when one manager is hedging risk exposure or using a portfolio insurance strategy, the performance rankings should differ. 6(a)(i).

6 ( −5) + 0.3 ( −3.5) + 0.1 ( 0.3) = −4.02%

6(a)(iii). 0.5 ( −4 ) + 0.2 ( −2.5) + 0.3 ( 0.3) = −2.41% 6(a)(iii). 0.3 ( −5) + 0.4 ( −3.5) + 0.3 ( 0.3) = −2.81% Manager A outperformed the benchmark fund by 161 basis points, while Manager B beat the benchmark fund by 121 basis points. In this case, both managers outperformed by losing less value in their portfolios than the benchmark. 6(b). Manager A:

Selection: 0.5 ( −4 + 5) + 0.2 ( −2.5 + 3.5) + 0.3 ( 0.3 − 0.3)  = 0.70% Allocation: ( 0.5 − 0.6 ) ( −5 + 4.02 ) + ( 0.2 − 0.3 ) ( −3.5 + 4.02 ) + ( 0.3 − 0.1)( 0.3 + 4.02 )  = 0.91% 6(b). Manager B:

Selection: 0.5 ( −5 + 5) + 0.2 ( −3.5 + 3.5) + 0.3 ( 0.3 − 0.3)  = 0.00% Allocation: ( 0.3 − 0.6 ) ( −5 + 4.02 ) + ( 0.4 − 0.3) ( −3.5 + 4.02 ) + ( 0.3 − 0.1)( 0.3 + 4.02 )  = 1.21% Notice that the outperformance for both managers can be decomposed into the selection and allocation components (A: 1.61 = 0.70 + 0.91, B: 1.21 = 0.00 + 1.21). Manager A added value with a mixture of skills (security selection and market timing), while Manager B added value exclusively with market timing skills.

7(a).

Overall, both managers added value by mitigating the currency effects present in the index. Both exhibited an ability to pick superior stocks in the markets they chose to be in (Manager B in particular). Manager B used her opportunities not to be in stocks quite effectively (via the cash/bond contribution to return), but neither of the managers matched the passive index in picking the country markets in which to be invested (Manager B in particular). Manager A Manager B Strengths

Currency management

Stock selection

Stock selection Use of cash/bond flexibility

Weaknesses

Country selection

(to a limited degree) 7(b). The “Currency” column reveals the effect on performance in local currency terms after adjustment for movements in the U.S. dollar and, therefore, the effect on the portfolio. Currency gains/losses arise from translating changes in currency exchange rates versus the U.S. dollar over the measuring period (three years in this case) into U.S. dollars for the U.S. pension plan. The index mix lost 12.9 percent to the dollar reducing what would otherwise have been a very favorable return from the various country markets of 19.9 percent to a net return of only 7.0 percent. As discussed above, the findings in the column indicate that both managers did a better job at mitigating currency losses for their dollar-based portfolios.

8. Evaluation begins with the selection of the appropriate benchmark against which to measure the firms’ results: Firm A. The Aggregate Index and “Managers using the Aggregate Index” benchmark are appropriate here because Firm A maintains market-like sector exposures. Performance has been strong; Firm A outperformed the Aggregate Index by 50 basis points and placed in the first (top) quartile of managers’ results. Based on the description of their investment strategy, this alpha of 50 basis points is probably due to superior security selection skills, since the manager intends to keep portfolio duration and industry sector weights relatively neutral. Firm B. Firm B does not use mortgages; therefore, the government/corporate for both index and universe comparisons would be the appropriate benchmark. Although Firm B produced the highest absolute performance (9.3 percent), it did not perform as well as either the index (9.5 percent) or the “typical” (median) other peer managers investing only in the government/corporate sectors (third quartile). This fund’s negative alpha (−20 basis points to the benchmark, −10 basis points to the median manager in their category) is likely due to inferior security selection, but it could also be from a mismatch in portfolio duration. Firm C. Like Firm A, this firm maintains broad market exposures and should be compared with the Aggregate Index for both index and universe comparisons. Performance has been good (30 basis points ahead of the index and in the second quartile of the manager peer group results) but not as good as Firm A’s show during this relatively short measurement period. Assuming this manager followed the indicated strategy, the source of her outperformance is due to her ability to anticipate shifts in the yield curve and trade in and out of portfolio holdings accordingly.

9(a). Money-weighted return Manager L:

500,000 = −12,000 / (1 + r ) − 7,500 / (1 + r ) − 13,500 / (1 + r ) − 6,500 / (1 + r ) − 10,000 / (1 + r ) 2

+ 625,000 / (1 + r )

Solving for r, the internal rate of return or MWR is 2.75% Manager M:

700,000 = 35,000 / (1 + r ) + 35,000 / (1 + r ) + 35,000 / (1 + r ) + 35,000 / (1 + r ) + 35,000 / (1 + r ) 2

+ 625,000 / (1 + r )

Solving for r, the internal rate of return or MWR is 2.98%. 9(b).

Time-weighted return Manager L: Periods

HPR

( 527,000 – 500,000 ) – 12,000  / 500,000 = 0.0300

( 530,000 – 527,000 ) – 7,500  / 527,000 = − 0.0085

( 555,000 – 530,000 ) – 13,500  / 530,000 = 0.0217

( 580,000 – 555,000 ) – 6,500  / 555,000 = 0.0333

( 625,000 – 580,000 ) – 10,000  / 580,000 = 0.0603

TWR = (1 + .03 )(1 − .0085 )(1 + .0217 )(1 + .0333 )(1 + .0603 )  = (1.143 )

1/5

−1

– 1 = 1.02712 – 1 = .02712 = 2.71%

Manager M: Periods

HPR

( 692,000 – 700,000 ) + 35,000  / 700,000 = 0.03857

( 663,000 – 692,000 ) + 35,000  / 692,000 = 0.00867

( 621,000 – 663,000 ) + 35,000  / 663,000 = − 0.01056

( 612,000 – 621,000 ) + 35,000  / 621,000 = 0.04187

( 625,000 – 612,000 ) + 35,000  / 612,000 = 0.0784

TWR = (1 + 0.03857 )(1 + 0.00867 )(1 − 0.01056 )(1 + 0.04187 )(1 + 0.0784 )  = (1.1646 )

1/5

−1

– 1 = 1.03094 – 1 = 0.03094 = 3.094%

9(c). Dietz Approximation =

(Ending Value ) − (Beginning Value ) − ( Total Contributions ) ( (Beginning Value) + i ( Contribution)i DWi )

In this case, DW = ( 91 − 45.5) /91 = 0.50 Manager L: Periods

HPY

( 527,000 – 5000,000 − (12,000 )  /  500,000 + ( 0.50 )(12,000 )     = 15,000 / 506,000 = 0.0296

( 530,000 – 527,000 − (7,500 ) / 527,000 + (0.50 )(7,500 ) = −4,500 / 530,750 = −0.0085

( 555,000 − 530,000 − (13,500) / 530,000 + ( 0.50)(13,500) = 11,500 / 536,750 = 0.0214

( 580,000 – 555,000 − (6,500 ) / 555,000 + (0.50 )(6,500 ) = 18,500 / 558,250 = 0.0331

( 625,000 – 580,000 − (10,000 ) / 580,000 + ( 0.50 )(10,000 ) = 35,000 / 585,000 = 0.0598

Manager M: Periods

HPY

( 692,000 – 700,00 − ( −35,000 )  /  700,000 + ( 0.50 )( −35,000 )     = 27,000 / 682,500 = 0.0396

(663,000 – 692,000 − ( −35,000 ) / 692,000 + ( 0.50 )( −35,000 ) = 6,000 / 674,500 = 0.0089

( 621,000 – 663,000 − ( −35,000 ) / 663,000 + ( 0.50 )( −35,000 ) = −7,000 / 645,5000 = −0.0108

( 612,000 – 621,000 − ( −35,000 ) / 621,000 + ( 0.50 )( −35,000 )

= 26,000 / 603,500 = 0.0431

( 625,000 – 612,000 − ( −35,000 ) / 612,000 + ( 0.50 )( −35,000 )

= 48,000 / 594,500 = 0.0807 10(a). Average Return AFNDX − RF = 1.627 Average Return SPX − RF = 1.531 Average Return RF = 0.405 Standard Deviation AFNDX − RF = 5.729 Standard Deviation SPX − RF = 4.824

Sharpe Ratio AFNDX − RF = 1.627/5.729 = 0.284 Sharpe Ratio SPX − RF = 1.531/4.824 = 0.317

10(b). SUMMARY OUTPUT Regression Statistics Multiple R

0.7727885

0.5972021

Adjusted R2

0.5868739

Standard error

3.681994

Observations

ANOVA df

Significance F

Regression

783.90748

783.9075

57.8227

3.22017E-09

Residual

39 528.72612

13.55708

Total

40 1312.6336

Coefficients

Standard Error

t-Stat

P-value

Lower 95%

Upper 95%

0.2226204

0.6039757

0.368592

0.71443

−0.9990358

1.444277

Rm − R f 0.9176815

0.120682

7.604126

3.2E-09

0.673579002

1.161784

Alpha

(1) One-factor Jensen’s alpha coefficient

0.2226

The value indicates that the manager generated a return of 22.3 basis points per month more than what was expected given the portfolio’s risk level. However, this intercept term is statistically insignificant. (2) Beta coefficient

0.9177

The risk variable beta measures systematic risk or volatility. The fund’s beta is less than 1 at 0.9177, just slightly below the market’s beta, so the fund is only slightly less volatile than the market. (3) R2 measure

0.5972

The R2 of 0.5972 indicates the correlation between the fund and the benchmark. It is higher than 0.50, which means that the fund’s performance is statistically related to the benchmark but that there remains a considerable amount of undiversified volatility in the portfolio. 10(c). Treynor’s ratio performance measure AFNDX − RF: 1.627 / 0.9177 = 1.773 SPX − RF: 1.531 / 1.0000 = 1.531 10(d). Although the portfolio has a low risk premium per unit of total risk as indicated by the Sharpe ratio of 0.284 (lower than the market portfolio of 0.317), it has a higher T value of 1.773 than the market portfolio of 1.531. It plots above the SML, indicating slightly superior systematic risk-adjusted performance. 10(e). The tracking error (TE) for AFNDX on a monthly basis is 3.657, and on an annualized basis, 0.5 it is 3.657  (12 ) = 12.669. 10(f). The monthly IR = 0.097 / 3.657 = 0.026. This represents the manager’s average alpha per unit of incremental risk. The annualized IR is 0.026  (12)0.5 = 0.092. As the IR is positive but considerably less than 0.50, the active manager has added the value relative to the benchmark but not enough to be considered a superior performer. 10(g). SUMMARY OUTPUT Regression Statistics Multiple R

0.8345862

0.6965341

Adjusted R2

0.6719288

Standard error

3.2811482

Observations

ANOVA

Regression

Significance F

914.29406

304.7647

28.3082

1.0964E-09

Residual

398.33954

Total

1312.6336

10.76593

Coefficients

Standard Error

t-Stat

P-value Lower 95%

Alpha

0.0724799

0.540428

0.134116 0.89404 −1.02253008

1.16749

Rm − R f

0.894159

0.1321262

6.767462 5.8E-08 0.626446175

1.161872

SMB

0.0439858

0.1298708

0.338689 0.73676 −0.21915725

0.307129

HML

−0.179041

0.1989471

−0.89994 0.37397 −0.58214549

0.224064

(1)

Alpha (intercept) is 0.0725 but is statistically insignificant.

(2)

Beta coefficients:

Upper 95%

Excess market 0.8942 SMB 0.0440 HML −0.1790 (3)

R2 measure 0.6965 which is statistically significant.

10(h). The majority of the evidence on the active manager’s performance supports the view that the fund has done at least as well as it should have done on a risk-adjusted basis. So, it is recommended that senior management includes AFNDX on its premier recommended list of approved managers.