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CHAPTER 8 THE CAPITAL ASSET PRICING MODEL 1. E(rP) = rf + P[E(rM) – rf] 18 = 6 + (14 – 6) P = 12/8 = 1.5 2. If the covariance of the security doubles, then so will its beta and its risk premium. The current risk premium is 14 – 6 = 8%, so the new risk premium would be 16%, and the new discount rate for the security would be 16 + 6 = 22%. If the stock pays a constant perpetual dividend, then we know from the original data that the dividend, D, must satisfy the equation for the present value of a perpetuity: Price = Dividend / Discount rate 50 = D /.14 D = 50 .14 = $7.00 At the new discount rate of 22%, the stock would be worth only $7/.22 = $31.82. The increase in stock risk has lowered its value by 36.36%. 3. The appropriate discount rate for the project is: rf + [E(rM) – rf ] = 8 + 1.7(16 – 8) = 21.6% Using this discount rate, NPV = –40 + 1 (10/1.216t) + 20/1.2610 = –20 + 10 Annuity factor(26%, 9 years) + 20x Present value factor (26%, 10years) = 19.62. The internal rate of return (IRR) on the project is 35.73%. Recall from your introductory finance class that NPV is positive if IRR > discount rate (equivalently, hurdle rate). The highest value that beta can take before the hurdle rate exceeds the IRR is determined by 35.73 = 9 + (16 – 8) = 35.73 – 9 / 8 = 3.34 4. a. False. = 0 implies E(r) = rf, not zero. 9

b. False. Investors require a risk premium only for bearing systematic (undiversifiable or market) risk. Total volatility includes diversifiable risk.

8-1

c.

False. 75% of your portfolio should be in the market, and 25% in bills. Then, P = .75

1 + .25 0 = .75

5.

a. Call the aggressive stock A and the defensive stock D. Beta is the sensitivity of the stock’s return to the market return, i.e., the change in the stock return per unit change in the market return. Therefore, we compute each stock’s beta by calculating the difference in its return across the two scenarios divided by the difference in the market return: A

D

2 38 2.00 5 25 6 12 0.30 5 25

b. With the two scenarios equally likely, the expected return is an average of the two possible outcomes: E(rA ) = 0.5 (–2 + 38) = 18% E(rD ) = 0.5 (6 + 12) = 9% c, The SML is determined by the market expected return of [0.5(25 + 5)] = 15%, with a beta of 1, and the T-bill return of 6% with a beta of zero. See the following graph.

Expected Return - Beta Relationship
40 35 30 Expected Return 25 20 15 10 5 0 0 0.5 1 1.5 Beta 2 2.5 3 A

SML

D

M

A

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The equation for the security market line is: E(r) = 6 + (15 – 6) d. Based on its risk, the aggressive stock has a required expected return of: E(rA ) = 6 + 2.0(15 – 6) = 24% The analyst’s forecast of expected return is only 18%. Thus the stock’s alpha is: A = actually expected return – required return (given risk) = 18% – 24% = –6% Similarly, the required return for the defensive stock is: E(rD) = 6 + 0.3(15 – 6) = 8.7% The analyst’s forecast of expected return for D is 9%, and hence, the stock has a positive alpha:

= actually expected return – required return (given risk) = 9 – 8.7 = +0.3% The points for each stock plot on the graph as indicated above. D

e. The hurdle rate is determined by the project beta (0.3), not the firm’s beta. The correct discount rate is 8.7%, the fair rate of return for stock D. 6. Not possible. Portfolio A has a higher beta than B, but its expected return is lower. Thus, these two portfolios cannot exist in equilibrium. Possible. If the CAPM is valid, the expected rate of return compensates only for systematic (market) risk represented by beta rather than for the standard deviation which includes nonsystematic risk. Thus, A's lower rate of return can be paired with a higher standard deviation, as long as A's beta is lower than B's. Not possible. The SML for this situation is: E(r) = 10 + (18 – 10) Portfolios with beta of 1.5 have an expected return of E(r) = 10 + 1.5 (18 – 10) = 22%. A's expected return is 16%; that is, A plots below the SML ( A =...
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