Assume that both portfolios a and b are well diversified

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20. Go to Connect and link to Chapter 7 materials, where you will find a spreadsheet with monthly returns for GM, Ford, and Toyota, the S&P 500, and Treasury bills. (LO 7-1)

a. Estimate the index model for each firm over the full five-year period. Compare the betas of each firm. b. Now estimate the betas for each firm using only the first two years of the sample and then using only

the last two years. How stable are the beta estimates obtained from these shorter subperiods?

Templates and spreadsheets are available in Connect

In Problems 21–23 below, assume the risk-free rate is 8% and the expected rate of return on the market is 18%. 21. 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. What do investors expect the stock to sell for at the end of the year? (LO 7-2) 22. 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, how much more will I offer for the firm than it is truly worth? (LO 7-2)

23. A stock has an expected return of 6%. What is its beta? (LO 7-2) 24. 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. (LO 7-2) 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? c. What if the T-bill rate were 3% and the market return 15%?

25. Suppose the yield on short-term government securities (perceived to be risk-free) is about 4%. Suppose also that the expected return required by the market for a portfolio with a beta of 1 is 12%. According to the capital asset pricing model: (LO 7-2) 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 β = –.5. Is the stock overpriced or underpriced?

26. Based on current dividend yields and expected capital gains, the expected rates of return on portfolios A and B are 11% and 14%, respectively. The beta of A is .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%. (LO 7-2) 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?

27. 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? (LO 7-4)

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28. 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 βA = 1 while βB = 1.1, what must be the risk-free rate? (LO 7- 4)

29. Assume a market index represents the common factor and all stocks in the economy have a beta of 1. 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%, 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. (LO 7-4) a. What is the expected profit (in dollars), and what is the standard deviation of the analyst’s profit? b. How does your answer change if the analyst examines 50 stocks instead of 20? 100 stocks?

30. If the APT is to be a useful theory, the number of systematic factors in the economy must be small. Why? (LO 7-4)

31. 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? (LO 7-3)

32. 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 on IP and .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? (LO 7-3)

33. 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.

What is the expected return–beta relationship in this economy? (LO 7-5)

Challenge

34. As a finance intern at Pork Products, Jennifer Wainwright’s assignment is to come up with fresh insights concerning the firm’s cost of capital. She decides that this would be a good opportunity to try out the new material on the APT that she learned last semester. As such, she decides that three promising factors would be (i) the return on a broad-based index such as the S&P 500; (ii) the level of interest rates, as represented by the yield to maturity on 10-year Treasury bonds; and (iii) the price of hogs, which are particularly important to her firm. Her plan is to find the beta of Pork Products against each of these factors and to estimate the risk premium associated with exposure to each factor. Comment on Jennifer’s choice of factors. Which are most promising with respect to the likely impact on her firm’s cost of capital? Can you suggest improvements to her specification? (LO 7-3)

35. Suppose the market can be described by the following three sources of systematic risk. Each factor in the following table has a mean value of zero (so factor values represent realized surprises relative to prior expectations), and the risk premiums associated with each source of systematic risk are given in the last column.

Systematic Factor Risk Premium