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line (SCL), 231

PROBLEM

1. Which of the following statements about the security market line (SML) are true?

SETS

a. The SML provides a benchmark for evaluating expected investment performance.

b. The SML leads all investors to invest in the same portfolio of risky assets.

c. The SML is a graphic representation of the relationship between expected return

and beta.

d. Properly valued assets plot exactly on the SML.

2. Risk aversion has all of the following implications for the investment process except:

a. The security market line is upward sloping.

b. The promised yield on AAA-rated bonds is higher than on A-rated bonds.

c. Investors expect a positive relationship between expected return and risk.

d. Investors prefer portfolios that lie on the efficient frontier to other portfolios with

equal expected rates of return.

3. What is the beta of a portfolio with E(rP) 20%, if rf 5% and E(rM) 15%?

4. The market price of a security is $40. Its expected rate of return is 13%. The risk-free rate

is 7%, and the market risk premium is 8%. What will the market price of the security be

if its beta doubles (and all other variables remain unchanged)? Assume the stock is

expected to pay a constant dividend in perpetuity.

5. You are a consultant to a large manufacturing corporation considering a project with the

following net after-tax cash flows (in millions of dollars)

Years from Now After-Tax CF

0 20

1â€“9 10

10 20

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The projectâ€™s beta is 1.7. Assuming rf 9% and E(rM) 19%, what is the net present

value of the project? What is the highest possible beta estimate for the project before its

NPV becomes negative?

6. Are the following statements true or false? Explain.

a. Stocks with a beta of zero offer an expected rate of return of zero.

b. The CAPM implies that investors require a higher return to hold highly volatile

securities.

c. You can construct a portfolio with a beta of 0.75 by investing 0.75 of the budget in

T-bills and the remainder in the market portfolio.

Bodieâˆ’Kaneâˆ’Marcus: II. Portfolio Theory 7. Capital Asset Pricing Â© The McGrawâˆ’Hill

Essentials of Investments, and Arbitrage Pricing Companies, 2003

Fifth Edition Theory

252 Part TWO Portfolio Theory

7. Consider the following table, which gives a security analystâ€™s expected return on two

stocks for two particular market returns:

Market Return Aggressive Stock Defensive Stock

5% 2% 3.5%

20 32 14

a. What are the betas of the two stocks?

b. What is the expected rate of return on each stock if the market return is equally

likely to be 5% or 20%?

c. If the T-bill rate is 8%, and the market return is equally likely to be 5% or 20%, draw

the SML for this economy.

d. Plot the two securities on the SML graph. What are the alphas of each?

e. What hurdle rate should be used by the management of the aggressive firm for a

project with the risk characteristics of the defensive firmâ€™s stock?

If the simple CAPM is valid, which of the situations in problems 8â€“14 below are possible?

Explain. Consider each situation independently.

8.

Expected

Portfolio Return Beta

A 20% 1.4

B 25 1.2

9.

Expected Standard

Portfolio Return Deviation

A 30% 35%

B 40 25

10.

Expected Standard

Portfolio Return Deviation

Risk-free 10% 0%

Market 18 24

A 16 12

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11.

Expected Standard

Portfolio Return Deviation

Risk-free 10% 0%

Market 18 24

A 20 22

Bodieâˆ’Kaneâˆ’Marcus: II. Portfolio Theory 7. Capital Asset Pricing Â© The McGrawâˆ’Hill

Essentials of Investments, and Arbitrage Pricing Companies, 2003

Fifth Edition Theory

253

7 Capital Asset Pricing and Arbitrage Pricing Theory

12.

Expected

Portfolio Return Beta

Risk-free 10% 0

Market 18 1.0

A 16 1.5

13.

Expected

Portfolio Return Beta

Risk-free 10% 0

Market 18 1.0

A 16 .9

14.

Expected Standard

Portfolio Return Deviation

Risk-free 10% 0%

Market 18 24

A 16 22

In problems 15â€“17 below, assume the risk-free rate is 8% and the expected rate of

return on the market is 18%.

15. A share of stock is now selling for $100. It will pay a dividend of $9 per share at the

end of the year. Its beta is 1.0. What do investors expect the stock to sell for at the end

of the year?

16. I am buying a firm with an expected perpetual cash flow of $1,000 but am unsure of its

risk. If I think the beta of the firm is zero, when the beta is really 1.0, how much more

will I offer for the firm than it is truly worth?

17. A stock has an expected return of 6%. What is its beta?

18. Two investment advisers are comparing performance. One averaged a 19% return and

the other a 16% return. However, the beta of the first adviser was 1.5, while that of the

second was 1.0.

a. Can you tell which adviser was a better selector of individual stocks (aside from the

issue of general movements in the market)?

b. If the T-bill rate were 6%, and the market return during the period were 14%, which

adviser would be the superior stock selector?

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c. What if the T-bill rate were 3% and the market return 15%?

19. In 2002, the yield on short-term government securities (perceived to be risk-free) was

about 4%. Suppose the expected return required by the market for a portfolio with a

beta of 1.0 is 12%. According to the capital asset pricing model:

a. What is the expected return on the market portfolio?

b. What would be the expected return on a zero-beta stock?

c. Suppose you consider buying a share of stock at a price of $40. The stock is

expected to pay a dividend of $3 next year and to sell then for $41. The stock risk

has been evaluated at 0.5. Is the stock overpriced or underpriced?

Bodieâˆ’Kaneâˆ’Marcus: II. Portfolio Theory 7. Capital Asset Pricing Â© The McGrawâˆ’Hill

Essentials of Investments, and Arbitrage Pricing Companies, 2003

Fifth Edition Theory

254 Part TWO Portfolio Theory

20. Based on current dividend yields and expected capital gains, the expected rates of return

on portfolio A and B are 11% and 14%, respectively. The beta of A is 0.8 while that of B

is 1.5. The T-bill rate is currently 6%, while the expected rate of return of the S&P 500

Index is 12%. The standard deviation of portfolio A is 10% annually, while that of B is

31%, and that of the index is 20%.

a. If you currently hold a market index portfolio, would you choose to add either of

these portfolios to your holdings? Explain.

b. If instead you could invest only in bills and one of these portfolios, which would you

choose?

21. Consider the following data for a one-factor economy. All portfolios are well

diversified.

Portfolio E(r) Beta

A 10% 1.0

F 4 0

Suppose another portfolio E is well diversified with a beta of 2/3 and expected return of

9%. Would an arbitrage opportunity exist? If so, what would the arbitrage strategy be?

22. Following is a scenario for three stocks constructed by the security analysts of PF Inc.

Scenario Rate of Return (%)

Stock Price ($) Recession Average Boom

A 10 15 20 30

B 15 25 10 10

C 50 12 15 12

a. Construct an arbitrage portfolio using these stocks.

b. How might these prices change when equilibrium is restored? Give an example

where a change in stock Câ€™s price is sufficient to restore equilibrium, assuming the

dollar payoffs to stock C remain the same.

23. Assume both portfolios A and B are well diversified, that E(rA) 14% and

E(rB) 14.8%. If the economy has only one factor, and A 1.0 while B 1.1,

what must be the risk-free rate?

24. Assume a market index represents the common factor, and all stocks in the economy

have a beta of 1.0. Firm-specific returns all have a standard deviation of 30%.

Suppose an analyst studies 20 stocks and finds that one-half have an alpha of 3%,

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and one-half have an alpha of 3%. The analyst then buys $1 million of an equally

weighted portfolio of the positive alpha stocks and sells short $1 million of an equally

weighted portfolio of the negative alpha stocks.

a. What is the expected profit (in dollars), and what is the standard deviation of the

analystâ€™s profit?

b. How does your answer change if the analyst examines 50 stocks instead of 20?

100 stocks?

25. If the APT is to be a useful theory, the number of systematic factors in the economy

must be small. Why?

Bodieâˆ’Kaneâˆ’Marcus: II. Portfolio Theory 7. Capital Asset Pricing Â© The McGrawâˆ’Hill

Essentials of Investments, and Arbitrage Pricing Companies, 2003

Fifth Edition Theory

255

7 Capital Asset Pricing and Arbitrage Pricing Theory

26. The APT itself does not provide information on the factors that one might expect to

determine risk premiums. How should researchers decide which factors to investigate?

Is industrial production a reasonable factor to test for a risk premium? Why or why not?

27. Suppose two factors are identified for the U.S. economy: the growth rate of industrial

production, IP, and the inflation rate, IR. IP is expected to be 4% and IR 6%. A stock

with a beta of 1.0 on IP and 0.4 on IR currently is expected to provide a rate of return of

14%. If industrial production actually grows by 5%, while the inflation rate turns out to

be 7%, what is your best guess for the rate of return on the stock?

28. Suppose there are two independent economic factors, M1 and M2. The risk-free rate is

7%, and all stocks have independent firm-specific components with a standard deviation

of 50%. Portfolios A and B are both well diversified.

Portfolio Beta on M1 Beta on M2 Expected Return (%)

A 1.8 2.1 40

B 2.0 0.5 10

What is the expected returnâ€“beta relationship in this economy?

29. The security market line depicts:

a. A securityâ€™s expected return as a function of its systematic risk.

b. The market portfolio as the optimal portfolio of risky securities.

c. The relationship between a securityâ€™s return and the return on an index.

d. The complete portfolio as a combination of the market portfolio and the risk-free

asset.

30. Within the context of the capital asset pricing model (CAPM), assume:

â€¢ Expected return on the market 15%.

â€¢ Risk-free rate 8%.

â€¢ Expected rate of return on XYZ security 17%.

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