Ross5eChap12sm

# Ross5eChap12sm - Chapter 12 An Alternative View of Risk and...

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Chapter 12: An Alternative View of Risk and Return: The Arbitrage Pricing Theory 1. Since we have the expected return of the stock, the revised expected return can be determined using the innovation, or surprise, in the risk factors. So, the revised expected return is: R = 11% + 1.2(4.2% – 3%) – 0.8(4.6% – 4.5%) R = 12.36% 2. a. If m is the systematic risk portion of return, then: m = β GNP ΔGNP + β Inflation ΔInflation + β r ΔInterest rates m = .000586(\$5,436 – 5,396) – 1.40(3.80% – 3.10%) – .67(10.30% – 9.50%) m = 0.83% b. The unsystematic return is the return that occurs because of a firm specific factor such as the bad news about the company. So, the unsystematic return of the stock is –2.6 percent. The total return is the expected return, plus the two components of unexpected return: the systematic risk portion of return and the unsystematic portion. So, the total return of the stock is: ε + + = m R R R = 9.50% + 0.83% – 2.6% R = 7.73% 3. a. If m is the systematic risk portion of return, then: m = β GNP ΔGNP + β r ΔInterest rates m = 2.04(4.8% – 3.5%) – 1.90(7.80% – 7.10%) m = 1.32% b. The unsystematic is the return that occurs because of a firm specific factor such as the increase in market share. If ε is the unsystematic risk portion of the return, then: ε = 0.36(27% – 23%) ε = 1.44% c. The total return is the expected return, plus the two components of unexpected return: the systematic risk portion of return and the unsystematic portion. So, the total return of the stock is: ε + + = m R R R = 10.50% + 1.32% + 1.44% R = 13.26% 4. The beta for a particular risk factor in a portfolio is the weighted average of the betas of the assets. This is true whether the betas are from a single factor model or a multi–factor model. So, the betas of the portfolio are: F 1 = .20(1.20) + .20(0.80) + .60(0.95) Answers to End–of–Chapter Problems B–175

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F 1 = 0.97 F 2 = .20(0.90) + .20(1.40) + .60(–0.05) F 2 = 0.43 F 3 = .20(0.20) + .20(–0.30) + .60(1.50) F 3 = 0.88 So, the expression for the return of the portfolio is: R i = 5% + 0.97 F 1 + 0.43 F 2 + 0.88 F 3 Which means the return of the portfolio is: R i = 5% + 0.97(5.50%) + 0.43(4.20%) + 0.88(4.90%) R i = 16.45% 5. We can express the multifactor model for each portfolio as: E(R P ) = R F + β 1 F 1 + β 2 F 2 where F 1 and F 2 are the respective risk premiums for each factor. Expressing the return equation for each portfolio, we get: 18% = 6% + 0.75F 1 + 1.2F 2 14% = 6% + 1.60 F 1 – 0.2 F 2 We can solve the system of two equations with two unknowns. Multiplying each equation by the respective F 2 factor for the other equation, we get: 3.6% = 1.2% + .15 F 1 + 0.24 F 2 16.8% = 7.2% + 1.92 F 1 – 0.24 F 2 Summing the equations and solving F1 fr gives us: 20.40% = 8.40% + 2.07 F 1 F 1 = 5.80% And now, using the equation for portfolio A, we can solve for F 2 , which is: 18% = 6% + 0.75(5.80%) + 1.2F 2 F 2 = 6.38% 6. a. The market model is specified by: ε β + - + = ) ( M M R R R R so applying that to each Stock: Stock A: Answers to End–of–Chapter Problems B–176
A M M A A A R R R R ε β + - + = ) ( R A = 10.5% + 1.2(R M – 14.2%) + ε A Stock B: B M M B B B R R R R ε β + - + = ) ( R B = 13.0% + 0.98(R M – 14.2%) + ε B Stock C: C M M C C C R R R R ε β + - + = ) ( R C = 15.7% + 1.37(R M – 14.2%) + ε C b. Since we don't have the actual market return or unsystematic risk, we will get aformula with those values as unknowns: R P = .30R A + .45R B + .30R C R P = .30[10.5% + 1.2(R M – 14.2%) + ε A ] + .45[13.0% + 0.98(R M – 14.2%) + ε B ] + .25[15.7% + 1.37(R M – 14.2%) + ε C R P = .30(10.5%) + .45(13%) + .25(15.7%) + [.30(1.2) + .45(.98) + .25(1.37)] (R M – 14.2%)+ .30 ε A + .45 ε

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