Essentials Of Investments 8th Ed Bodie 投资学精要（第八版）

Chapter 07 - Capital Asset Pricing and Arbitrage Pricing Theory

CHAPTER 07

CAPITAL ASSET PRICING AND ARBITRAGE PRICING

THEORY

1. The required rate of return on a stock is related to the required rate of return on the

stock market via beta. Assuming the beta of Google remains constant, the increase in the risk of the market will increase the required rate of return on the market, and thus increase the required rate of return on Google.

2. An example of this scenario would be an investment in the SMB and HML. As of yet,

there are no vehicles (index funds or ETFs) to directly invest in SMB and HML. While they may prove superior to the single index model, they are not yet practical, even for professional investors.

3. The APT may exist without the CAPM, but not the other way. Thus, statement a is

possible, but not b. The reason being, that the APT accepts the principle of risk and return, which is central to CAPM, without making any assumptions regarding

individual investors and their portfolios. These assumptions are necessary to CAPM.

4. E(rP) = rf + ?[E(rM) – rf]

20% = 5% + ?(15% – 5%) ? ? = 15/10 = 1.5

5. If the beta of the security doubles, then so will its risk premium. The current risk

premium for the stock is: (13% - 7%) = 6%, so the new risk premium would be 12%, and the new discount rate for the security would be: 12% + 7% = 19%

If the stock pays a constant dividend in perpetuity, then we know from the original data that the dividend (D) must satisfy the equation for a perpetuity:

Price = Dividend/Discount rate 40 = D/0.13 ? D = 40 ? 0.13 = $5.20 At the new discount rate of 19%, the stock would be worth: $5.20/0.19 = $27.37

The increase in stock risk has lowered the value of the stock by 31.58%.

6. The cash flows for the project comprise a 10-year annuity of $10 million per year plus an

additional payment in the tenth year of $10 million (so that the total payment in the tenth year is $20 million). The appropriate discount rate for the project is:

rf + ?[E(rM) – rf ] = 9% + 1.7(19% – 9%) = 26% Using this discount rate:

10

NPV = –20 + ?t?1101.26t?101.2610

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Chapter 07 - Capital Asset Pricing and Arbitrage Pricing Theory

7.

c. False. You should invest 0.75 of your portfolio in the market portfolio, and the

remainder in T-bills. Then: 8.

a. The beta is the sensitivity of the stock's return to the market return. Call the

aggressive stock A and the defensive stock D. Then beta is the change in the stock return per unit change in the market return. We compute each stock's beta by calculating the difference in its return across the two scenarios divided by the difference in market return.

?A?2?325?20?2.00

= –20 + [10 ? Annuity factor (26%, 10 years)] + [10 ? PV factor (26%, 10 years)] = 15.64

The internal rate of return on the project is 49.55%. The highest value that beta can take before the hurdle rate exceeds the IRR is determined by:

49.55% = 9% + ?(19% – 9%) ? ? = 40.55/10 = 4.055 a. False. ? = 0 implies E(r) = rf , not zero.

b. False. Investors require a risk premium for bearing systematic (i.e., market or

undiversifiable) risk.

?P = (0.75 ? 1) + (0.25 ? 0) = 0.75

?D?3.5?145?20?0.70b. With the two scenarios equal likely, the expected rate of return is an average of

the two possible outcomes:

E(rA) = 0.5 ? (2% + 32%) = 17% E(rB) = 0.5 ? (3.5% + 14%) = 8.75%

c. The SML is determined by the following: T-bill rate = 8% with a beta equal to

zero, beta for the market is 1.0, and the expected rate of return for the market is:

0.5 ? (20% + 5%) = 12.5% See the following graph.

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Chapter 07 - Capital Asset Pricing and Arbitrage Pricing Theory

E(r) SML A M 12.5% ?D 8% D .7 1.0 2.0 ??

The equation for the security market line is: E(r) = 8% + ?(12.5% – 8%) d. The aggressive stock has a fair expected rate of return of:

E(rA) = 8% + 2.0(12.5% – 8%) = 17%

The security analyst’s estimate of the expected rate of return is also 17%.

Thus the alpha for the aggressive stock is zero. Similarly, the required return for the defensive stock is:

E(rD) = 8% + 0.7(12.5% – 8%) = 11.15%

The security analyst’s estimate of the expected return for D is only 8.75%, and hence:

?

?D = actual expected return – required return predicted by CAPM = 8.75% – 11.15% = –2.4%

The points for each stock are plotted on the graph above.

e. The hurdle rate is determined by the project beta (i.e., 0.7), not by the firm’s

beta. The correct discount rate is therefore 11.15%, the fair rate of return on stock D.

9. Not possible. Portfolio A has a higher beta than Portfolio B, but the expected return

for Portfolio A is lower.

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Chapter 07 - Capital Asset Pricing and Arbitrage Pricing Theory

10. Possible. If the CAPM is valid, the expected rate of return compensates only for

systematic (market) risk as measured by beta, rather than the standard deviation, which includes nonsystematic risk. Thus, Portfolio A's lower expected rate of return can be paired with a higher standard deviation, as long as Portfolio A's beta is lower than that of Portfolio B.

11. Not possible. The reward-to-variability ratio for Portfolio A is better than that of the

market, which is not possible according to the CAPM, since the CAPM predicts that the market portfolio is the most efficient portfolio. Using the numbers supplied:

SA =SM =

16?101218?1024?0.5

?0.33These figures imply that Portfolio A provides a better risk-reward tradeoff than the market portfolio.

12. Not possible. Portfolio A clearly dominates the market portfolio. It has a lower

standard deviation with a higher expected return.

13. Not possible. Given these data, the SML is: E(r) = 10% + ?(18% – 10%)

A portfolio with beta of 1.5 should have an expected return of: E(r) = 10% + 1.5 ? (18% – 10%) = 22%

The expected return for Portfolio A is 16% so that Portfolio A plots below the SML (i.e., has an alpha of –6%), and hence is an overpriced portfolio. This is inconsistent with the CAPM.

14. Not possible. The SML is the same as in Problem 12. Here, the required expected

return for Portfolio A is: 10% + (0.9 ? 8%) = 17.2%

This is still higher than 16%. Portfolio A is overpriced, with alpha equal to: –1.2%

15. Possible. Portfolio A's ratio of risk premium to standard deviation is less attractive

than the market's. This situation is consistent with the CAPM. The market portfolio should provide the highest reward-to-variability ratio.

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Chapter 07 - Capital Asset Pricing and Arbitrage Pricing Theory

16.

a. FordBeta 5 yearsBeta first twp yearsBeta last two yearsSE of residualSE beta 5 yearsIntercept 5 yearsIntercept first two yearsIntercept last two years1.812.011.9712.010.42-0.93-2.370.81GM 0.861.050.698.340.29-1.44-1.82-3.41ToyotaS&P0.711.000.473.78SD0.495.140.180.451.80-1.91 b.

As a first pass we note that large standard deviation of the beta estimates. None of the subperiod estimates deviate from the overall period estimate by more than two standard deviations. That is, the t-statistic of the deviation from the overall period is not significant for any of the subperiod beta estimates. Looking beyond the aforementioned observation, the differences can be attributed to different alpha values during the subperiods. The case of Toyota is most revealing: The alpha estimate for the first two years is positive and for the last two years negative (both large). Following a good performance in the \Toyota surprised investors with a negative performance, beyond what could be expected from the index. This suggests that a beta of around 0.5 is more reliable. The shift of the intercepts from positive to negative when the index moved to largely negative returns, explains why the line is steeper when estimated for the overall period. Draw a line in the positive quadrant for the index with a slope of 0.5 and positive intercept. Then draw a line with similar slope in the negative quadrant of the index with a negative intercept. You can see that a line that reconciles the observations for both quadrants will be steeper. The same logic explains part of the behavior of subperiod betas for Ford and GM.

17. Since the stock's beta is equal to 1.0, its expected rate of return should be equal to that

of the market, that is, 18%.

18. If beta is zero, the cash flow should be discounted at the risk-free rate, 8%:

PV = $1,000/0.08 = $12,500

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E(r) =0.18 =

D?P1P0?P0

??P1 = $109

9?P1?100100

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