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Chap009 - Chapter 09 The Capital Asset Pricing Model...

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Chapter 09 - The Capital Asset Pricing Model CHAPTER 9: THE CAPITAL ASSET PRICING MODEL PROBLEM SETS 1. E(r P ) = r f + β P [E(r M ) – r f ] 18 = 6 + β P (14 – 6) β P = 12/8 = 1.5 2. If the security’s correlation coefficient with the market portfolio doubles (with all other variables such as variances unchanged), then beta, and therefore the risk premium, will also double. The current risk premium is: 14 – 6 = 8% 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/0.14 D = 50 × 0.14 = $7.00 At the new discount rate of 22%, the stock would be worth: $7/0.22 = $31.82 The increase in stock risk has lowered its value by 36.36%. 3. a.False. β = 0 implies E(r) = r f , not zero. b. False. Investors require a risk premium only for bearing systematic (undiversifiable or market) risk. Total volatility includes diversifiable risk. c. False. Your portfolio should be invested 75% in the market portfolio and 25% in T-bills. Then: β P = (0.75 × 1) + (0.25 × 0) = 0.75 4. The appropriate discount rate for the project is: r f + β [E(r M ) – r f ] = 8 + [1.8 × (16 – 8)] = 22.4% Using this discount rate: × + - = + - = = 15 [$ 40 $ 224 . 1 15 $ 40 $ NPV 10 1 t t Annuity factor (22.4%, 10 years)] = $18.09 The internal rate of return (IRR) for the project is 35.73%. Recall from your introductory finance class that NPV is positive if IRR > discount rate (or, equivalently, hurdle rate). The highest value that beta can take before the hurdle rate exceeds the IRR is determined by: 35.73 = 8 + β (16 – 8) β = 27.73/8 = 3.47 9-1
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Chapter 09 - The Capital Asset Pricing Model 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: 00 . 2 25 5 38 2 A = - - - = β 30 . 0 25 5 12 6 D = - - = β b. With the two scenarios equally likely, the expected return is an average of the two possible outcomes: E(r A ) = 0.5 × (–2 + 38) = 18% E(r D ) = 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 0 5 10 15 20 25 30 35 40 0 0.5 1 1.5 2 2.5 3 Beta SML D M A α A The equation for the security market line is: E(r) = 6 + β (15 – 6) 9-2
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Chapter 09 - The Capital Asset Pricing Model d. Based on its risk, the aggressive stock has a required expected return of: E(r A ) = 6 + 2.0(15 – 6) = 24% The analyst’s forecast of expected return is only 18%. Thus the stock’s alpha is: α A = actually expected return – required return (given risk) = 18% – 24% = –6% Similarly, the required return for the defensive stock is: E(r D ) = 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: α D = actually expected return – required return (given risk) = 9 – 8.7 = +0.3% The points for each stock plot on the graph as indicated above.
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