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Unformatted text preview: (β B + .25)(MRP) .134 = Rf + β B(MRP) We can rewrite the CAPM equation for Stock A as: .151 = Rf + β B(MRP) + .25(MRP) Subtracting the CAPM equation for Stock B from this equation yields: .017 = .25MRP MRP = .068 or 6.8% which is the same answer as our previous result. 38. a. A typical, risk-averse investor seeks high returns and low risks. For a risk-averse investor holding a well-diversified portfolio, beta is the appropriate measure of the risk of an individual security. To assess the two stocks, we need to find the expected return and beta of each of the two securities. Stock A: Since Stock A pays no dividends, the return on Stock A is simply: (P 1 – P0) / P0. So, the return for each state of the economy is: RRecession = ($63 – 75) / $75 = –.160 or –16.0% RNormal = ($83 – 75) / $75 = .107 or 10.7% 291 RExpanding = ($96 – 75) / $75 = .280 or 28.0% 292 The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring. So, the expected return of the stock is: E(RA) = .20(–.160) + .60(.107) + .20(.280) = .0880 or 8.80% And the variance of the stock is: σ 2 = .20(–0.160 – 0.088)2 + .60(.107 – .088)2 + .20(.280 – .088)2 A σ 2 = 0.0199 A Which means the standard deviation is: σ A = (0.0199)1/2 σ A = .1410 or 14.10% Now we can calculate the stock’s beta, which is: β A = (ρ A,M)(σ A) / σ M β A = (.80)(.1410) / .18 β A = .627 For Stock B, we can directly calculate the beta from the information provided. So, the beta for Stock B is: Stock B: β B = (ρ B,M)(σ B) / σ M β B = (.25)(.34) / .18 β B = .472 The expected return on Stock B is higher than the expected return on Stock A. The risk of Stock B, as measured by its beta, is lower than the risk of Stock A. Thus, a typical risk-averse investor holding a well-diversified portfolio will prefer Stock B. Note, this situation implies that at least one of the stocks is mispriced since the higher risk (beta) stock has a lower return than the lower risk (beta) stock. b. The expected return of the portfolio is the sum of the weight of each asset times the expected return of each asset, so: E(RP) = wAE(RA) + wBE(RB) E(RP) = .70(.088) + .30(.13) E(RP) = .1006 or 10.06% 293 To find the standard deviation of the portfolio, we first need to calculate the variance. The variance of the portfolio is: σ 2 = w 2 σ 2 + w 2 σ 2 + 2wAwBσ Aσ Bρ A,B P A A B B σ 2 = (.70)2(.141)2 + (.30)2(.34)2 + 2(.70)(.30)(.141)(.34)(.48) P σ 2 = .02981 P And the standard deviation of the portfolio is: σ P = (0.02981)1/2 σ P = .1727 or 17.27% c. The beta of a portfolio is the weighted average of the betas of its individual securities. So the beta of the portfolio is: β P = .70(.627) + .30(0.472) β P = .580 39. a. The variance of a portfolio of two assets equals: σ 2 = w 2 σ 2 + w 2 σ 2 + 2wAwBσ Aσ BCov(A,B) P A A B B Since the weights of the assets must sum to one, we can write the variance of the portfolio as: σ 2 = w 2 σ 2 + (1 – wA)σ P A A
2 B + 2wA(1 – wA)σ Aσ BCov(A,B) To find the minimum for any function, we find the derivative and set the derivative equal to zero. Finding the derivative of the variance function with respect to the weight of Asset A, setting the derivative equal to zero, and solving for the weight of Asset A, we find: wA = [σ 2 – Cov(A,B)] / [σ 2 + σ 2 – 2Cov(A,B)] B A B Using this expression, we find the weight of Asset A must be: wA = (.452 – .001) / [.222 + .452 – 2(.001)] wA = .8096 This implies the weight of Stock B is: wB = 1 – wA wB = 1 – .8096 wB = .1904 b. Using the weights calculated in part a, determine the expected return of the portfolio, we find: E(RP) = wAE(RA) + wBE(RB) E(RP) = .8096(.09) + .1904(0.15) 294 E(RP) = 0.1014 or 10.14% c. Using the derivative from part a, with the new covariance, the weight of each stock in the minimum variance portfolio is: wA = [σ 2 + Cov(A,B)] / [σ 2 + σ 2 – 2Cov(A,B)] B A B wA = (.452 + –.05) / [.222 + .452 – 2(–.05)] wA = .7196 This implies the weight of Stock B is: wB = 1 – wA wB = 1 – .7196 wB = .2804 d. The variance of the portfolio with the weights on part c is: σ 2 = w 2 σ 2 + w 2 σ 2 + 2wAwBσ Aσ BCov(A,B) P A A B B σ 2 = (.7196)2(.22)2 + (.2804)2(.45)2 + 2(.7196)(.2804)(.22)(.45)(–.05) P σ 2 = .0208 P And the standard deviation of the portfolio is: σ P = (0.0208)1/2 σ P = .1442 or 14.42% 295 CHAPTER 12 AN ALTERNATIVE VIEW OF RISK AND RETURN: THE ARBITRAGE PRICING THEORY
Answers to Concept Questions 1. Systematic risk is risk that cannot be diversified away through formation of a portfolio. Generally, systematic risk factors are those factors that affect a large number of firms in the market, however, those factors will not necessarily affect all firms equally. Unsystematic risk is the type of risk that can be diversified away through portfolio formation. Unsystematic risk factors are specific to the firm or industry. Surprises in these factors will affec...
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