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How can you measure and interpret the market risk, or beta, of a security?

The contribution of a security to the risk of a diversified portfolio depends on its market risk. But not all securities are equally affected by fluctuations in the market. The sensitivity of a stock to market movement is known as beta. Stocks with a beta greater than 1.0 are particularly sensitive to market fluctuations. Those with a beta of less than 1.0 are not so sensitive to such movements. The average beta of all stocks is 1.0.

What is the relationship between the market risk of a security and the rate of return that investors demand of that security?

The extra return that investors require for taking risk is known as the risk premium. The market risk premium—that is, the risk premium on the market portfolio—averaged almost 9.4 percent between 1926 and 1998. The capital asset pricing model states that the expected risk premium of an investment should be proportional to both its beta and the market risk premium. The expected rate of return from any investment is equal to the risk-free interest rate plus the risk premium, so the CAPM boils down to r = rf + P(rm - rf)

The security market line is the graphical representation of the CAPM equation. The security market line relates the expected return investors demand of a security to the beta.

How can a manager calculate the opportunity cost of capital for a project?

The opportunity cost of capital is the return that investors give up by investing in the project rather than in securities of equivalent risk. Financial managers use the capital asset pricing model to estimate the opportunity cost of capital. The company cost of capital is the expected rate of return demanded by investors in a company, determined by the average risk of the company's assets and operations.

The opportunity cost of capital depends on the use to which the capital is put. Therefore, required rates of return are determined by the risk of the project, not by the risk of the firm's existing business. The project cost of capital is the minimum acceptable expected rate of return on a project given its risk.

Your cash-flow forecasts should already factor in the chances of pleasant and unpleasant surprises. Potential bad outcomes should be reflected in the discount rate only to the extent that they affect beta.

Related Web Links

www.stanford.edu/~wfsharpe/ws/wksheets.htm William Sharpe's site contains "portfolio optimizers," spreadsheets that can be used to construct efficiently diversified portfolios www.riskmetrics.com RiskMetrics® Group maintains this site, which uses modern portfolio theory to help manage risk; some of the content at this site, including educational and demonstration materials, is free. www.riskview.com A nice site with historical risk and return data as well as software to manage and measure portfolio risk www.finance.yahoo.com You can find stock betas as well as other risk measures and company profiles here

KeY Terms market portfolio purity mariirt line beta company cost of capital market risk premium project cost °f capiM capital asset pricing model (CAPM)

Risk and Return. True or false? Explain or qualify as necessary.

Investors demand higher expected rates of return on stocks with more variable rates of return.

The capital asset pricing model predicts that a security with a beta of zero will provide an expected return of zero.

An investor who puts $10,000 in Treasury bills and $20,000 in the market portfolio will have a portfolio beta of 2.0.

Investors demand higher expected rates of return from stocks with returns that are highly exposed to macroeconomic changes.

Investors demand higher expected rates of return from stocks with returns that are very sensitive to fluctuations in the stock market.

2. Diversifiable Risk. In light of what you've learned about market versus diversifiable (unique) risks, explain why an insurance company has no problem in selling life insurance to individuals but is reluctant to issue policies insuring against flood damage to residents of coastal areas. Why don't the insurance companies simply charge coastal residents a premium that reflects the actuarial probability of damage from hurricanes and other storms?

3. Unique vs. Market Risk. Figure 4.14 plots monthly rates of return from 1993 to 1999 for the Snake Oil mutual fund. Was this fund well-diversified? Explain.

4. Risk and Return. Suppose that the risk premium on stocks and other securities did in fact

Quiz b

FIGURE 4.14

Monthly rates of return for the Snake Oil mutual fund and the Standard & Poor's Composite Index. See problem 3.

Snake Oil return, 5 -percent • • - . \ • * * |

1 2 3 Market return, percent |

rise with total risk (that is, the variability of returns) rather than just market risk. Explain how investors could exploit the situation to create portfolios with high expected rates of return but low levels of risk. 5. CAPM and Hurdle Rates. A project under consideration has an internal rate of return of 14 percent and a beta of .6. The risk-free rate is 5 percent and the expected rate of return on the market portfolio is 14 percent.

a. Should the project be accepted?

b. Should the project be accepted if its beta is 1.6?

c. Why does your answer change?

6. CAPM and Valuation. You are considering acquiring a firm that you believe can generate expected cash flows of $10,000 a year forever. However, you recognize that those cash flows are uncertain.

a. Suppose you believe that the beta of the firm is .4. How much is the firm worth if the risk-free rate is 5 percent and the expected rate of return on the market portfolio is 15 percent?

b. By how much will you overvalue the firm if its beta is actually .6?

7. CAPM and Expected Return. If the risk-free rate is 6 percent and the expected rate of return on the market portfolio is 14 percent, is a security with a beta of 1.25 and an expected rate of return of 16 percent overpriced or underpriced?

8. Using Beta. Investors expect the market rate of return this year to be 14 percent. A stock with a beta of .8 has an expected rate of return of 12 percent. If the market return this year turns out to be 10 percent, what is your best guess as to the rate of return on the stock?

9. Unique vs. Market Risk. Figure 4.15 shows plots of monthly rates of return on three stocks versus the stock market index. The beta and standard deviation of each stock is given beside its plot.

a. Which stock is riskiest to a diversified investor?

b. Which stock is riskiest to an undiversified investor who puts all her funds in one of these stocks?

FIGURE 4.15

These plots show monthly rates of return for (a) Exxon, (b) Polaroid, (c) Nike, and the market portfolio. See problem 9.

10 15 20 25

10 15 20 25

• • | ||

• |
V v — # • • | |

• | ||

Beta = .61 | ||

1 1 1 1 |
i i i i i i |

Market return, percent

Market return, percent

• | ||||

• |
• • |
• • •• • • • • • • • • • • | ||

• | ||||

Beta = .53 | ||||

1 1 1 1 |
i i i |
i i |

Market return, percent

Market return, percent

c. Consider a portfolio with equal investments in each stock. What would this portfolio's beta have been?

d. Consider a well-diversified portfolio made up of stocks with the same beta as Exxon. What are the beta and standard deviation of this portfolio's return? The standard deviation of the market portfolio's return is 20 percent.

e. What is the expected rate of return on each stock? Use the capital asset pricing model with a market risk premium of 8 percent. The risk-free rate of interest is 4 percent.

10. Calculating Beta. Following are several months' rates of return for Tumblehome Canoe Company. Prepare a plot like Figure 4.7. What is Tumblehome's beta?

FIGURE 4.15

(Continued)

Market return, percent

Market return, percent

1 |
0 |
+1 |

2 |
0 |
-1 |

3 |
-1 |
-2.5 |

4 |
-1 |
-0.5 |

5 |
+1 |
+2 |

6 |
+1 |
+1 |

7 |
+2 |
+4 |

8 |
+2 |
+2 |

9 |
-2 |
-2 |

10 |
-2 |
-4 |

11. Expected Returns. Consider the following two scenarios for the economy, and the returns in each scenario for the market portfolio, an aggressive stock A, and a defensive stock D.

Rate of Return

Scenario Market Aggressive Stock A Defensive Stock D

a. Find the beta of each stock. In what way is stock D defensive?

b. If each scenario is equally likely, find the expected rate of return on the market portfolio and on each stock.

c. If the T-bill rate is 4 percent, what does the CAPM say about the fair expected rate of return on the two stocks?

d. Which stock seems to be a better buy based on your answers to (a) through (c)?

12. CAPM and Cost of Capital. Draw the security market line when the Treasury bill rate is 10 percent and the market risk premium is 8 percent. What are the project costs of capital for new ventures with betas of .75 and 1.75? Which of the following capital investments have positive NPVs?

Project |
Beta |
Internal Rate of Return, % |

P |
1.0 |
20 |

Q |
0 |
10 |

R |
2.0 |
25 |

S |
0.4 |
16 |

T |
1.6 |
25 |

13. CAPM and Valuation. You are a consultant to a firm evaluating an expansion of its current business. The cash-flow forecasts (in millions of dollars) for the project are:

Years Cash Flow

Based on the behavior of the firm's stock, you believe that the beta of the firm is 1.4. Assuming that the rate of return available on risk-free investments is 5 percent and that the expected rate of return on the market portfolio is 15 percent, what is the net present value of the project?

14. CAPM and Cost of Capital. Reconsider the project in the preceding problem. What is the project IRR? What is the cost of capital for the project? Does the accept-reject decision using IRR agree with the decision using NPV?

15. CAPM and Valuation. A share of stock with a beta of .75 now sells for $50. Investors expect the stock to pay a year-end dividend of $2. The T-bill rate is 4 percent, and the market risk premium is 8 percent. If the stock is perceived to be fairly priced today, what must be investors' expectation of the price of the stock at the end of the year?

16. CAPM and Expected Return. Reconsider the stock in the preceding problem. Suppose investors actually believe the stock will sell for $54 at year-end. Is the stock a good or bad buy? What will investors do? At what point will the stock reach an "equilibrium" at which it again is perceived as fairly priced?

17. Portfolio Risk and Return. Suppose that the S&P 500, with a beta of 1.0, has an expected return of 13 percent and T-bills provide a risk-free return of 5 percent.

a. What would be the expected return and beta of portfolios constructed from these two assets with weights in the S&P 500 of (i) 0; (ii) .25; (iii) .5; (iv) .75; (v) 1.0?

b. Based on your answer to (a), what is the trade-off between risk and return, that is, how does expected return vary with beta?

c. What does your answer to (b) have to do with the security market line relationship?

18. Portfolio Risk and Return. Suppose that the S&P 500, with a beta of 1.0, has an expected return of 15 percent and T-bills provide a risk-free return of 5 percent.

a. How would you construct a portfolio from these two assets with an expected return of 12 percent?

b. How would you construct a portfolio from these two assets with a beta of .4?

c. Show that the risk premiums of the portfolios in (a) and (b) are proportional to their betas.

19. CAPM and Valuation. You are considering the purchase of real estate which will provide perpetual income that should average $50,000 per year. How much will you pay for the property if you believe its market risk is the same as the market portfolio's? The T-bill rate is 5 percent, and the expected market return is 12.5 percent.

20. Risk and Return. According to the CAPM, would the expected rate of return on a security with a beta less than zero be more or less than the risk-free interest rate? Why would in vestors be willing to invest in such a security? Hint: Look back to the auto and gold example.

21. CAPM and Expected Return. The following table shows betas for several companies. Calculate each stock's expected rate of return using the CAPM. Assume the risk-free rate of interest is 5 percent. Use a 9 percent risk premium for the market portfolio.

Company Beta

Bristol-Myers Squibb 1.13

General Mills 0.70

McGraw-Hill 0.92

22. CAPM and Expected Return. Stock A has a beta of .5 and investors expect it to return 5 percent. Stock B has a beta of 1.5 and investors expect it to return 13 percent. Use the CAPM to find the market risk premium and the expected rate of return on the market.

23. CAPM and Expected Return. If the expected rate of return on the market portfolio is 14 percent and T-bills yield 6 percent, what must be the beta of a stock that investors expect to return 10 percent?

24. Project Cost of Capital. Suppose General Mills is considering a new investment in the common stock of a publishing company. Which of the betas shown in the table in problem 21 is most relevant in determining the required rate of return for this venture? Explain why the expected return to General Mills stock is not the appropriate required return.

25. Risk and Return. True or false? Explain or qualify as necessary.

a. The expected rate of return on an investment with a beta of 2 is twice as high as the expected rate of return of the market portfolio.

b. The contribution of a stock to the risk of a diversified portfolio depends on the market risk of the stock.

c. If a stock's expected rate of return plots below the security market line, it is underpriced.

d. A diversified portfolio with a beta of 2 is twice as volatile as the market portfolio.

e. An undiversified portfolio with a beta of 2 is twice as volatile as the market portfolio.

26. CAPM and Expected Return. A mutual fund manager expects her portfolio to earn a rate of return of 12 percent this year. The beta of her portfolio is .8. If the rate of return available on risk-free assets is 5 percent and you expect the rate of return on the market portfolio to be 15 percent, should you invest in this mutual fund?

27. Required Rate of Return. Reconsider the mutual fund manager in the previous problem. Explain how you would use a stock index mutual fund and a risk-free position in Treasury bills (or a money market mutual fund) to create a portfolio with the same risk as the manager's but with a higher expected rate of return. What is the rate of return on that portfolio?

28. Required Rate of Return. In view of your answer to the preceding problem, explain why a mutual fund must be able to provide an expected rate of return in excess of that predicted by the security market line for investors to consider the fund an attractive investment opportunity.

29. CAPM. We Do Bankruptcies is a law firm that specializes in providing advice to firms in financial distress. It prospers in recessions when other firms are struggling. Consequently, its beta is negative, -.2.

a. If the interest rate on Treasury bills is 5 percent and the expected return on the market portfolio is 15 percent, what is the expected return on the shares of the law firm according to the CAPM?

b. Suppose you invested 90 percent of your wealth in the market portfolio and the remainder of your wealth in the shares in the law firm. What would be the beta of your portfolio?

CHALLENGE 30. Leverage and Portfolio Risk. Footnote 4 in the material asks you to consider a borrow-and-

_ invest strategy in which you use $1 million of your own money and borrow another $1 mil-

rROBLEM lion to invest $2 million in a market index fund. If the risk-free interest rate is 4 percent and the expected rate of return on the market index fund is 12 percent, what is the risk premium and expected rate of return on the borrow-and-invest strategy? Why is the risk of this strategy twice that of simply investing your $1 million in the market index fund?

Solutions to

Self-Test

Questions

1 See Figure 4.16. Anchovy Queen's beta is 1.0.

2 A portfolio's beta is just a weighted average of the betas of the securities in the portfolio. In this case the weights are equal, since an equal amount is assumed invested in each of the stocks in Table 4.9. The average beta of these stocks is (1.07 + 1.14 + .88 + .61 + .97 + 1.30 + .92 + 1.33 + 1.33 + 1.20)/10 = 1.07.

3 The standard deviation of a fully diversified portfolio's return is proportional to its beta. The standard deviation in this case is .5 x 20 = 10 percent.

5 Put 25 percent of your money in the market portfolio and the rest in Treasury bills. The portfolio's beta is .25 and its expected return is

This portfolio's beta is .6, since $600,000, which is 60 percent of the investment, is in the market portfolio. Investors in a stock with a beta of .6 would not buy it unless it also offered a rate of return of 11.4 percent and would rush to buy if it offered more. The stock price would adjust until the stock's expected rate of return was 11.4 percent.

Each point shows the performance of Anchovy Queen stock when the market is up or down by 1 percent. On average, Anchovy Queen stock follows the market; it has a beta of 1.0.

Anchovy Queen return, 2 -percent -.4 -.2 o |
• |
.2 .4 .6 .8 1.0 Market return, percent |

7 Present value = $50 million X 10-year annuity factor at 13.1%

= $270.23 million If P = .7, then the cost of capital falls to r = 4.8% + (.7 X 9%) = 11.1%

and the value of the 10-year annuity increases to $293.23 million.

8 Merck should use Compaq's cost of capital. Merck's company cost of capital tells us what expected rate of return investors demand from the pharmaceutical business. This is not the appropriate project cost of capital for Merck's venture into computer hardware.

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