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Unformatted text preview: Chapter 10: Return and Risk: The CapitalAssetPricing Model (CAPM) 10.1 a. b. R 2 10.2 a. b. RA RB A2 A B2 B c. = 0.1 ( 4.5%) + 0.2 (4.4%) + 0.5 (12.0%) + 0.2 (20.7%) = 10.57% = 0.1 (0.045 0.1057)2 + 0.2 (0.044 0.1057)2 + 0.5 (0.12 0.1057)2 + 0.2 (0.207 0.1057)2 = 0.0052 = (0.0052)1/2 = 0.072 = 7.20% = (6.3 + 10.5 + 15.6) / 3 = 10.8% = (3.7 + 6.4 + 25.3) / 3 = 9.3% = {(0.063 0.108)2 + (0.105 0.108)2 +{(0.156 0.108)2} / 3 = 0.001446 = (0.001446)1/2 = 0.0380 = 3.80% = {( 0.037 0.093)2 +(0.064 0.093)2 +(0.253 0.093)2} / 3 = 0.014447 = (0.014447)1/2 = 0.1202 = 12.02% Cov(RA,RB) = [(.063 .108) ( .037 .093) + (.105 .108) (.064 .093) + (.156 .108) (.253 .093) ] / 3 = .013617 / 3 = .004539 Corr(RA,RB) = .004539 / (.0380 x .1202) = .9937 = 0.25 (2.0) + 0.60 (9.2) + 0.15 (15.4) = 7.33% = 0.25 (5.0) + 0.60 (6.2) + 0.15 (7.4) R SB = 6.08% HB2 = 0.25 ( 0.02 0.0733)2 + 0.60 (0.092 0.0733)2 + 0.15 (0.154 0.0733)2 = 0.003363 HB = (0.003363)1/2 = 0.05799 = 5.80% 2 SB = 0.25 (0.05 0.0608)2 + 0.60 (0.062 0.0608)2 + 0.15 (0.074 0.0608)2 = 0.000056 SB = (0.000056)1/2 = 0.00749 = 0.75% Cov (RHB, RSB) = 0.25 ( 0.02 0.0733) (0.05 0.0608) + 0.60 (0.092 0.0733) (0.062 0.0608) + 0.15 (0.154 0.0733) (0.074 0.0608) = 0.000425286 Corr (RHB, RSB) = 0.000425286 / (0.05799 0.00749) = 0.9791 R HB 10.3 a. b. c. 10.4 Holdings of Atlas stock = 120 $50 = $6,000 Holdings of Babcock stock = 150 $20 = $3,000 Weight of Atlas stock = $6,000 / $9,000 = 2 / 3 Answers to EndofChapter Problems B103 Weight of Babcock stock = $3,000 / $9,000 = 1 / 3 10.5 a. b. RP P 2 P 10.6 a. RP P 2 = 0.3 (0.12) + 0.7 (0.18) = 0.162 = 16.2% = 0.32 (0.09)2 + 0.72 (0.25)2 + 2 (0.3) (0.7) (0.09) (0.25) (0.2) = 0.033244 = (0.033244)1/2 = 0.1823 = 18.23% b. c. 10.7 = 0.4 (0.15) + 0.6 (0.25) = 0.21 = 21% = 0.42 (0.1)2 + 0.62 (0.2)2 + 2 (0.4) (0.6) (0.1) (0.2) (0.5) = 0.0208 P = (0.0208)1/2 = 0.1442 = 14.42% P 2 = 0.42 (0.1)2 + 0.62 (0.2)2 + 2 (0.4) (0.6) (0.1) (0.2) (0.5) = 0.0112 P = (0.0112)1/2 = 0.1058 = 10.58% As the stocks are more negatively correlated, the standard deviation of the portfolio decreases. Macrosoft: 100 $80 = $8,000 Intelligent: 300 $40 = $12,000 Weight:Macrosoft: $8,000 / $20,000 = 0.4 Intelligent: $12,000 / $20,000 = 0.6 a. = 0.4 (0.15) + 0.6 (0.20) = 0.18 = 18% RP 2 P = 0.42 (0.08)2 + 0.62 (0.2)2 + 2 (0.4) (0.6) (0.38) (0.08) (0.20) = 0.0183424 P = (0.0183424)1/2 = 0.1354 = 13.54% b. New weight: Macrosoft: $8,000 / $12,000 = 0.667 Intelligent: $4,000 / $12,000 = 0.333 = 0.667 (0.15) + 0.333 (0.20) = 0.1666 = 16.66% RP 2 P = 0.6672 (0.08)2 + 0.3332 (0.2)2 + 2 (0.667) (0.333) (0.38) (0.08) (0.20) = 0.009984 P = (0.009984)1/2 = 0.09992 = 9.99% 10.8 a. b. c. = 7% = 0.2 (0.05) + 0.5 (0.10) + 0.3 (0.25) = 0.115 = 11.5% = U = 0 = 0.2 (0.05  0.115)2 + 0.5 (0.10  0.115)2 + 0.3 (0.25  0.115)2 = 0.0110 V = (0.0110)1/2 = 0.105 = 10.5% Cov (RU, RV) = 0.2 (0.05  0.115) (0.07  0.07) + 0.5 (0.10  0.115) (0.07  0.07) + 0.3 (0.25  0.115) (0.07  0.07) =0 Corr (RU, RV) = 0 = 0.5 (0.115) + 0.5 (0.07) = 0.0925 = 9.25% RP RU RV U2 V2 B104 Answers to EndofChapter Problems P 2 P 10.9 a. = 0.52 (0.0110) = 0.00275 = (0.00275)1/2 = 0.0524 = 5.24% b. c. = 0.3 (0.10) + 0.7 (0.20) = 0.17 = 17.0% RP P 2 = 0.32 (0.05)2 + 0.72 (0.15)2 = 0.01125 P = (0.01125)1/2 = 0.10607 = 10.61% = 0.9 (0.10) + 0.1 (0.20) = 0.11 = 11.0% RP 2 P = 0.92 (0.05)2 + 0.12 (0.15)2 = 0.00225 P = (0.00225)1/2 = 0.04743 = 4.74% No, I would not hold 100% of stock A because the portfolio in b has higher expected return but less standard deviation than stock A. I may or may not hold 100% of stock B, depending on my preference. 10.10 The expected return on any portfolio must be less than or equal to the return on the stock with the highest return. It cannot be greater than this stock's return because all stocks with lower returns will pull down the value of the weighted average return. Similarly, the expected return on any portfolio must be greater than or equal to the return of the asset with the lowest return. The portfolio return cannot be less than the lowest return in the portfolio because all higher earning stocks will pull up the value of the weighted average. 10.11 a. b. c. = 0.4 (0.03) + 0.6 (0.15) = 0.102 = 10.2% = 0.4 (0.065) + 0.6 (0.065) = 0.065 = 6.5% = 0.4 (0.03  0.102)2 + 0.6 (0.15  0.102)2 = 0.003456 A = (0.003456)1/2 = 0.05878 = 5.88% 2 B = B = 0 XA = $2,500 / $6,000 = 0.417 XB = 1  0.417 = 0.583 = 0.417 (0.102) + 0.583 (0.065) = 0.0804 = 8.04% RP 2 P = XA2 A2 = 0.0006 P = (0.0006)1/2 = 0.0245 = 2.45% Amount borrowed = 40 $50 = $2,000 XA = $8,000 / $6,000 = 4 / 3 XB = 1  XA = 1 / 3 = (4 / 3) (0.102) + (1 / 3) (0.065) = 0.1143 = 11.43% RP 2 P = (4 / 3)2 (0.003456) = 0.006144 P = (0.006144)1/2 = 0.07838 = 7.84% RA RB A2 10.12 The wide fluctuations in the price of oil stocks do not indicate that oil is a poor investment. If oil is purchased as part of a portfolio, what matters is only its beta. Since the price captures beta plus idiosyncratic risks, observing price volatility is not an adequate measure of the appropriateness of adding oil to a portfolio. Remember that total variability should not be used when deciding whether or not to put an asset into a large portfolio. Answers to EndofChapter Problems B105 10.13 a. R1 R2 R3 = 0.1 (0.25) + 0.4 (0.20) + 0.4 (0.15) + 0.1 (0.10) = 0.175 = 17.5% = 0.1 (0.25) + 0.4 (0.15) + 0.4 (0.20) + 0.1 (0.10) = 0.175 = 17.5% = 0.1 (0.10) + 0.4 (0.15) + 0.4 (0.20) + 0.1 (0.25) = 0.175 = 17.5% R1  R 1 0.075 0.025 0.025 0.075 (R1  R 1 )2 0.005625 0.000625 0.000625 0.005625 Variance = 0.001625 = 0.0403 (R2  R 2 )2 0.005625 0.000625 0.000625 0.005625 Variance = 0.001625 = 0.0403 (R3  R 3 )2 0.005625 0.000625 0.000625 0.005625 Variance = 0.001625 = 0.0403 P (R1  R 1 )2 0.0005625 0.0002500 0.0002500 0.0005625 0.0016250 R1 if State occurs 0.25 0.20 0.15 0.10 Standard deviation R2 if State occurs 0.25 0.15 0.20 0.10 R2  R 2 0.075 0.025 0.025 0.075 P (R2  R 2 )2 0.0005625 0.0002500 0.0002500 0.0005625 0.0016250 Standard deviation R3 if State occurs 0.10 0.15 0.20 0.25 R3  R 3 0.075 0.025 0.025 0.075 P (R3  R 3 )2 0.0005625 0.0002500 0.0002500 0.0005625 0.0016250 Standard deviation b. Cov(1,2) = .10 (.25  .175) (.25  .175) + .40 (.20  .175) (.15  .175) + .40 (.15  .175) (.20  .175) + .10 (.10  .175) (.10  .175) = 0.000625 Cov(1,3) = .10 (.25  .175) (.10  .175) + .40 (.20  .175) (.15  .175) + .40 (.15  .175) (.20  .175) + .10 (.10  .175) (.25  .175) =  0.001625 Cov(2,3) = .10 (.25  .175) (.10  .175) + .40 (.15  .175) (.15  .175) + .40 (.20  .175) (.20  .175) + .10 (.10  .175) (.25  .175) =  0.000625 Corr(1,2) = 0.000625 / (0.0403 x 0.0403) = 0.385 Corr(1,3) =  0.001625 / (0.0403 x 0.0403) =  1 Corr(2,3) =  0.000625 / (0.0403 x 0.0403) =  0.385 B106 Answers to EndofChapter Problems c. E(R) = .5 x .175 + .5 x .175 = .175 Var = .5 x .5 x .0403 x .0403 + .5 x .5 x .0403 x .0403 + 2 x .5 x .5 x .000625 = 0.0011245 = 0.0335 E(R) = .5 x .175 + .5 x .175 = .175 Var = .5 x .5 x .0403 x .0403 + .5 x .5 x .0403 x .0403 + 2 x .5 x .5 x (.001625) =0 =0 E(R) = .5 x .175 + .5 x .175 = .175 Var = .5 x .5 x .0403 x .0403 + .5 x .5 x .0403 x .0403 + 2 x .5 x .5 x (.000625) = 0.0004995 = 0.0224 Portfolio with negatively correlated stocks can achieve higher degree of diversification than portfolio with positively correlated stocks, holding expected return for each stock constant. Applying proper weights on perfectly negatively correlated stocks can reduce portfolio variance to 0. As long as the correlation is not 1, there is benefit of diversification. State Return on A Return on B Probability 1 15% 35% 0.4 0.5 = 0.2 2 15% 5% 0.4 0.5 = 0.2 3 10% 35% 0.6 0.5 = 0.3 4 10% 5% 0.6 0.5 = 0.3 = 0.2 [0.5 (0.15) + 0.5 (0.35)] + 0.2[0.5 (0.15) + 0.5 (0.05)] RP + 0.3 [0.5 (0.10) + 0.5 (0.35)] + 0.3 [0.5 (0.10) + 0.5 (0.05)] = 0.135 = 13.5% = R i / N = {N (0.10)} / N = 0.10 = 10% = Cov (Ri, Rj) / N2 + i2 / N2 = N (N  1) (0.0064) / N2 + N (0.0144) / N2 = (0.0064) (N  1) / N + (0.0144) / N As N , P 2 0.0064 = Cov (Ri, Rj) The covariance of the returns of the securities is the most important factor to consider when placing securities in a welldiversified portfolio. RP P 2 d. e. f. 10.14 a. b. 10.15 a. b. c. 10.16 The statement is false. Once the stock is part of a welldiversified portfolio, the important factor is the contribution of the stock to the variance of the portfolio. In a welldiversified portfolio, this contribution is the covariance of the stock with the rest of the portfolio. The covariance is a more appropriate measure of risk in a welldiversified portfolio because it reflects the effect of the security on the variance of the portfolio. Investors are concerned with the variance of their portfolios and not the variance of the individual 10.17 Answers to EndofChapter Problems B107 securities. Since covariance measures the impact of an individual security on the variance of the portfolio, covariance is the appropriate measure of risk. 10.18 If we assume that the market has not stayed constant during the past three years, then the low volatility of Southern Co.'s stock price only indicates that the stock has a beta that is very near to zero. The high volatility of Texas Instruments' stock price does not imply that the firm's beta is high. Total volatility (the price fluctuation) is a function of both systematic and unsystematic risk. The beta only reflects the systematic risk. Observing price volatility does not indicate whether it was due to systematic factors, or firm specific factors. Thus, if you observe a high price volatility like that of TI, you cannot claim that the beta of TI's stock is high. All you know is that the total risk of TI is high. Note: The solution to this problem requires calculus. Specifically, the solution is found by minimizing a function subject to a constraint. Calculus ability is not necessary to understand the principles behind a minimum variance portfolio. Min { XA2 A2 + XB2 B2 + 2 XA XB Cov(RA , RB)} subject to XA + XB = 1 Let XA = 1  XB. Then, Min {(1  XB)2 A2 + XB2 B2+ 2(1  XB) XB Cov (RA, RB)} Take a derivative with respect to XB. d{} / dXB = (2 XB  2) A2 + 2 XB B2 + 2 Cov(RA, RB)  4 XB Cov(RA, RB) Set the derivative equal to zero, cancel the common 2 and solve for XB. XB A2  A2 + XB B2 + Cov(RA, RB)  2 XB Cov(RA, RB) = 0 XB = { A2  Cov(RA, RB)} / { A2 + B2  2 Cov(RA, RB)} and XA = { B2  Cov(RA, RB)} / { A2 + B2  2 Cov(RA, RB)} Using the data from the problem yields, XA = 0.8125 and XB = 0.1875. a. Using the weights calculated above, the expected return on the minimum variance portfolio is E(RP) = 0.8125 E(RA) + 0.1875 E(RB) = 0.8125 (5%) + 0.1875 (10%) = 5.9375% Using the formula derived above, the weights are XA = 2 / 3 and XB = 1 / 3 The variance of this portfolio is zero. P 2 = XA2 A2 + XB2 B2 + 2 XA XB Cov(RA , RB) = (4 / 9) (0.01) + (1 / 9) (0.04) + 2 (2 / 3) (1 / 3) (0.02) =0 This demonstrates that assets can be combined to form a riskfree portfolio. 10.19 b. c. 10.20 The slope of the capital market line is ( R M  Rf) / M = (12  5) / 10 = 0.7 B108 Answers to EndofChapter Problems a. b. 10.21 R P = 5 + 0.7 (7) = 9.9% P = ( R P  Rf) / 0.7 = (20  5) / 0.7 = 21.4% The slope of the characteristic line of Fuji is R Fuji (Bull)  R Fuji ( Bear) = (12.8  3.4) / (16.3  2.5) R M (Bull)  R M (Bear)
a. b. = 0.68 Beta = slope of the characteristic line = 0.68 The responsiveness to the market = 0.68 Slope = 0.68 = {12.8  R Fuji ( Bear ) } / {16.3  (4.0)} Thus, R Fuji ( Bear ) = 12.8  0.68 (16.3 + 4.0) = 1.00% 10.22 Polonius' portfolio will be the market portfolio. He will have no borrowing or lending in his portfolio. 10.23 a. b. c. = (0.10 + 0.14 + 0.20) / 3 = 0.1467 = 14.67% RP P = (0.7 + 1.2 + 1.8) / 3 = 1.23 To be in equilibrium, three securities should be located on a straight line (the Security Market Line). Check the slopes. Slope between A & B = (0.14  0.10) / (1.2  0.7) = 0.08 Slope between B & C = (0.20  0.14) / (1.8  1.2) = 0.10 Since the slopes are different, these securities are not in equilibrium. 10.24 10.25 10.26 10.27 10.28 Expected Return For Alpha = 6% + 1.2 8.5% =16.2% Expected return for Ross = 6% + (0.8 8.5%) = 12.8% Expected Return in Jordan = 8% + (1.5 7%) =18.5% 14.2%= 3.7%+(7.5%) = 1.4 (I) (II) (III) 0.25 = Rf + 1.4 [RM Rf] 0.14 = Rf + 0.7 [RM Rf] (I) (II)=0.11 = 0.7 [RM Rf] [RM Rf ]= 0.1571 Put (III) into (I) [RM Rf ]= 0.1571 RM = 0.1571 + 0.03 = 18.71% 0.25 = Rf + 1.4[0.1571] Rf = 3% Answers to EndofChapter Problems B109 10.29 a. b. E(RA) = (0.25)(0.1) + (0.5)(0.1) + (0.25)(0.2) = 0.075 E(RB) = (0.25)(0.3) + (0.5)(0.05) + (0.25)(0.4)= 0.05 (I) E(RA) = Rf + A [E(RM) Rf] = 0.075 (II) E(RB) = Rf + B [E(RM) Rf] = 0.05 (I) (II) = 0.025 = ( A  B)[E(RM) Rf] 0.025 = 0.25[E(RM) Rf], so the market risk premium = [E(RM) Rf ]= 10% b c 10.30 a. E(R) E(Ri) = 7% + i (5%) 22% 20% 12% 7% B 4% 2% 1 0 1 3 C 1.4 i b. i. ii. c. i. ii. See point B on the graph in part a. There does exist a mispricing of the security. According to the SML, this asset should have a return of 2% [= 7% + (1) (5%)]. Since the return is too high, the price of this asset must be too low. (Remember, asset prices and rates of return are inversely related!) Since the asset is underpriced, you should buy it. See point C on the graph in part a. There does exist a mispricing of the security. According to the SML, this asset should have a return of 22% [= 7% + (3) (5%)]. Since the return is too low, the price of this asset must be too high. Since the asset is overpriced, you should sell it. 10.31 10.32 Expected return = 0.05 + 1.8 (0.08) = 0.194 = 19.4% The analyst expects only 18%, so he is pessimistic. a. R = 6.4 + 1.2 (13.8  6.4) = 15.28% b. R = 3.5 + 1.2 (13.8  3.5) = 15.86% B110 Answers to EndofChapter Problems 10.33 Market excess return = E(RM)  Rf = 20%  5% = 15% Portfolio excess return = E(RE)  Rf = 25%  5% = 20% Portfolio beta = E = 20% / 15% = 4 / 3 E = {Corr(RE, RM) (RE)} / (RM) Therefore, 4 / 3 = (1 4%) / (RM) (RM) = 3% Note: Corr(RE, RM) = 1 because this portfolio is a combination of the riskless asset and the market portfolio. For the security with Corr(RS, RM) = 0.5, S = {Corr(RS, RM) (RS)} / (RM) = (0.5 2%) / 3% = 0.3333 Thus, E(RS) = 5% + S (15%) = 5% + (0.3333) (15%) = 10% 10.34 a. The risk premium = R M  Rf Potpourri stock return: 16.7 = 7.6 + 1.7 ( R M  Rf) R M  Rf = (16.7  7.6) / 1.7 = 5.353% R Mag = 7.6 + 0.8 (5.353) = 11.88% XPot Pot + XMag Mag = 1.07 1.7 XPot + 0.8 (1  XPot) = 1.07 0.9 XPot = 0.27 XPot = 0.3 XMag = 0.7 Thus invest $3,000 in Potpourri stock and $7,000 in Magnolia. R P = 7.6 + 1.07 (5.353) = 13.33% Note: The other way to calculate R P is R P = 0.3 (16.7) + 0.7 (11.88) = 13.33% b. c. 10.35 RZ Z = Rf + Z ( R M  Rf) = Cov(RZ, RM) / M 2 = Corr(RZ, RM) Z M / M 2 = Corr(RZ, RM) Z / M Z = (0.0169)1/2 = 0.13 M = (0.0121)1/2 = 0.11 Thus, Z = 0.45 (0.13) / 0.11 = 0.5318 = 0.063 + 0.5318 (0.148  0.063) RZ = 0.1082 = 10.82% Answers to EndofChapter Problems B111 10.36 a. b. R i = 4.9% + i (9.4%) D = Cov(RD, RM) / M 2 = 0.0635 / 0.04326 = 1.468 R D = 4.9 + 1.468 (9.4) = 18.70% 10.37 CAPM: Johnson Williamson 19 = Rf + 1.7 ( R M  Rf) 14 = Rf + 1.2 ( R M  Rf) 5 = 0.5 ( R M  Rf) Thus ( R M  Rf) = 10% = Rf + 1.7 (10) = 2% = 12% 19 Rf RM 10.38 The statement is false. If a security has a negative beta, investors would want to hold the asset to reduce the variability of their portfolios. Those assets will have expected returns that are lower than the risk free rate. To see this, examine the SML equation. E(Ri) = Rf + i {E(RM)  Rf)} If i < 0, E(Ri) < Rf. Weights: XA = 5 / 30 = 0.1667 XB = 10 / 30 = 0.3333 XC = 8 / 30 = 0.2667 XD = 1  XA  XB  XC = 0.2333 Beta of portfolio = 0.1667 (0.75) + 0.3333 (1.10) + 0.2667 (1.36) + 0.2333 (1.88) = 1.293 R P = 4 + 1.293 (15  4) = 18.22% a. (i) A A,M B C M,M M f f,M f E(Ri) = A,M A / M = A M / A = (0.9) (0.10) / 0.12 = 0.75 = B M / B,M = (1.10) (0.10) / 0.40 = 0.275 = C,M C / M = (0.75) (0.24) / 0.10 = 1.80 =1 =1 =0 =0 =0 = Rf + i {E(RM)  Rf} 10.39 10.40 (ii) (iii) (iv) (v) (vi) (vii) (viii) b. SML: B112 Answers to EndofChapter Problems = 0.05 + (0.10) i Security A B C Ri 0.13 0.16 0.25 i 0.90 1.10 1.80 E(Ri) 0.14 0.16 0.23 Security A performed worse than the market, while security C performed better than the market. Security B is fairly priced. c. According to the SML, security A is overpriced while security C is underpriced. Thus, you could invest in security C while sell security A (if you currently hold it). The typical riskaverse investor seeks high returns and low risks. To assess the two stocks, find the risk and return profiles for each stock. Returns: State of economy Recession Normal Expansion Probability 0.1 0.8 0.1 Return on A* 0.20 0.10 0.20 10.41 a. * Since security A pays no dividend, the return on A is simply (P1 / P0)  1. RA RB = 0.1 (0.20) + 0.8 (0.10) + 0.1 (0.20) = 0.08 = 0.09 This was given in the problem. Risk: RA  R A 0.28 0.02 0.12 (RA  R A )2 0.0784 0.0004 0.0144 Variance P (RA  R A )2 0.00784 0.00032 0.00144 0.00960 Standard deviation (RA) = 0.0980 A = {Corr(RA, RM) (RA)} / (RM) = 0.8 (0.0980) / 0.10 = 0.784 B = {Corr(RB, RM) (RB)} / (RM) = 0.2 (0.12) / 0.10 = 0.24 Answers to EndofChapter Problems B113 The return on stock B is higher than the return on stock A. The risk of stock B, as measured by its beta, is lower than the risk of A. Thus, a typical riskaverse investor will prefer stock B. b. RP P 2 P = (0.7) R A + (0.3) R B = (0.7) (0.8) + (0.3) (0.09) = 0.083 = 0.72 A2 + 0.32 B2 + 2 (0.7) (0.3) Corr (RA , RB) A B = (0.49) (0.0096) + (0.09) (0.0144) + (0.42) (0.6) (0.0980) (0.12) = 0.0089635 = 0.0089635 = 0.0947 c. The beta of a portfolio is the weighted average of the betas of the components of the portfolio. P = (0.7) A + (0.3) B = (0.7) (0.784) + (0.3) (0.240) = 0.621 B114 Answers to EndofChapter Problems ...
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