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35 Pages

### Chapter 10 Solutions V2

Course: FIN FIN/554, Summer 2006
School: Phoenix
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10: Chapter Return and Risk: The Capital Asset Pricing Model (CAPM) 10.1 a. Expected Return = (0.1)(-0.045) + (.2)(0.044) + (0.5)(0.12) + (0.2)(0.207) = 0.1057 = 10.57% The expected return on Q-marts stock is 10.57%. b. Variance (2) = (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 = 0.005187 Standard Deviation () = (0.005187)1/2 = 0.0720 = 7.20% The standard...

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10: Chapter Return and Risk: The Capital Asset Pricing Model (CAPM) 10.1 a. Expected Return = (0.1)(-0.045) + (.2)(0.044) + (0.5)(0.12) + (0.2)(0.207) = 0.1057 = 10.57% The expected return on Q-marts stock is 10.57%. b. Variance (2) = (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 = 0.005187 Standard Deviation () = (0.005187)1/2 = 0.0720 = 7.20% The standard deviation of Q-marts returns is 7.20%. 10.2 a. Expected ReturnA = (1/3)(0.063) + (1/3)(0.105) + (1/3)(0.156) = 0.1080 = 10.80% The expected return on Stock A is 10.80%. Expected ReturnB = (1/3)(-0.037) + (1/3)(0.064) + (1/3)(0.253) = 0.933 = 9.33% The expected return on Stock B is 9.33%. b. VarianceA (A2) = (1/3)(0.063 0.108)2 + (1/3)(0.105 0.108)2 + (1/3)(0.156 0.108)2 = 0.001446 Standard DeviationA (A) = (0.001446)1/2 = 0.0380 = 3.80% The standard deviation of Stock As returns is 3.80%. VarianceB (B2) = (1/3)(-0.037 0.0933)2 + (1/3)(0.064 0.0933)2 + (1/3)(0.253 0.0933)2 = 0.014447 Standard DeviationB (B) = (0.014447)1/2 = 0.1202 = 12.02% The standard deviation of Stock Bs returns is 12.02%. c. Covariance(RA, RB) = (1/3)(0.063 0.108)(-0.037 0.0933) + (1/3)(0.105 0.108)(0.064 0.933) + (1/3)(0.156 0.108)(0.253 0.0933) = 0.004539 The covariance between the returns of Stock A and Stock B is 0.004539. Correlation(RA,RB) = Covariance(RA, RB) / (A * B) = 0.004539 / (0.0380 * 0.1202) = 0.9937 The correlation between the returns on Stock A and Stock B is 0.9937. 10.3 a. Expected ReturnHB = (0.25)(-0.02) + (0.60)(0.092) + (0.15)(0.154) = 0.0733 = 7.33% The expected return on Highbulls stock is 7.33%. Expected ReturnSB = (0.25)(0.05) + (0.60)(0.062) + (0.15)(0.074) = 0.0608 = 6.08% The expected return on Slowbears stock is 6.08%. b. VarianceA (HB2) = (0.25)(-0.02 0.0733)2 + (0.60)(0.092 0.0733)2 + (0.15)(0.154 0.0733)2 = 0.003363 Standard DeviationA (HB) = (0.003363)1/2 = 0.0580 = 5.80% The standard deviation of Highbears stock returns is 5.80%. VarianceB (SB2) = (0.25)(0.05 0.0608)2 + (0.60)(0.062 0.0608)2 + (0.15)(0.074 0.0608)2 = 0.000056 Standard DeviationB (B) = (0.000056)1/2 = 0.0075 = 0.75% The standard deviation of Slowbears stock returns is 0.75%. c. Covariance(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) = 0.000425 The covariance between the returns on Highbulls stock and Slowbears stock is 0.000425. Correlation(RA,RB) = Covariance(RA, RB) / (A * B) = 0.000425 / (0.0580 * 0.0075) = 0.9770 The correlation between the returns on Highbulls stock and Slowbears stock is 0.9770. 10.4 Value of Atlas stock in the portfolio = (120 shares)(\$50 per share) = \$6,000 Value of Babcock stock in the portfolio = (150 shares)(\$20 per share) = \$3,000 Total Value in the portfolio = \$6,000 + \$3000 = \$9,000 Weight of Atlas stock = \$6,000 / \$9,000 = 2/3 The weight of Atlas stock in the portfolio is 2/3. Weight of Babcock stock = \$3,000 / \$9,000 = 1/3 The weight of Babcock stock in the portfolio is 1/3. 10.5 a. The expected return on the portfolio equals: E(RP) = (WF)[E(RF)] + (WG)[E(RG)] where E(RP) E(RF) E(RG) WF WG = the expected return on the portfolio = the expected return on Security F = the expected return on Security G = the weight of Security F in the portfolio = the weight of Security G in the portfolio E(RP) = (WF)[E(RF)] + (WG)[E(RG)] = (0.30)(0.12) + (0.70)(0.18) = 0.1620 = 16.20% The expected return on a portfolio composed of 30% of Security F and 70% of Security G is 16.20%. b. The variance of the portfolio equals: 2P = (WF)2(F)2 + (WG)2(G)2 + (2)(WF)(WG)(F)(G)[Correlation(RF, RG)] where 2P WF WG F G RF RG = the variance of the portfolio = the weight of Security F in the portfolio = the weight of Security G in portfolio = the standard deviation of Security F = the standard deviation of Security G = the return on Security F = the return on Security G 2P = (WF)2(F)2 + (WG)2(G)2 + (2)(WF)(WG)(F)(G)[Correlation(RF, RG)] = (0.30)2(0.09)2 + (0.70)2(0.25)2 + (2)(0.30)(0.70)(0.09)(0.25)(0.2) = 0.033244 The standard deviation of the portfolio equals: P = (2P)1/2 where P 2P = the standard deviation of the portfolio = the variance of the portfolio P = (2P)1/2 = (0.033244)1/2 = 0.1823 =18.23% If the correlation between the returns of Security F and Security G is 0.2, the standard deviation of the portfolio is 18.23%. 10.6 a. The expected return on the portfolio equals: E(RP) = (WA)[E(RA)] + (WB)[E(RB)] where E(RP) E(RA) E(RB) WA WB = the expected return on the portfolio = the expected return on Stock A = the expected return on Stock B = the weight of Stock A in the portfolio = the weight of Stock B in the portfolio E(RP) = (WA)[E(RA)] + (WB)[E(RB)] = (0.40)(0.15) + (0.60)(0.25) = 0.21 = 21% The expected return on a portfolio composed of 40% stock A and 60% stock B is 21%. The variance of the portfolio equals: 2P = (WA)2(A)2 + (WB)2(B)2 + (2)(WA)(WB)(A)(B)[Correlation(RA, RB)] where 2P WA WB A B RA RB = the variance of the portfolio = the weight of Stock A in the portfolio = the weight of Stock B in the portfolio = the standard deviation of Stock A = the standard deviation of Stock B = the return on Stock A = the return on Stock B 2P = (WA)2(A)2 + (WB)2(B)2 + (2)(WA)(WB)(A)(B)[Correlation(RA, RB)] = (0.40)2(0.10)2 + (0.60)2(0.20)2 + (2)(0.40)(0.60)(0.10)(0.20)(0.5) = 0.0208 The standard deviation of the portfolio equals: P = (2P)1/2 where P = the standard deviation of the portfolio 2P = the variance of the portfolio P = (0.0208)1/2 = 0.1442 =14.42% If the correlation between the returns on Stock A and Stock B is 0.5, the standard deviation of the portfolio is 14.42%. b. 2P = (WA)2(A)2 + (WB)2(B)2 + (2)(WA)(WB)(A)(B)[Correlation(RA, RB)] = (0.40)2(0.10)2 + (0.60)2(0.20)2 + (2)(0.40)(0.60)(0.10)(0.20)(-0.5) = 0.0112 P = (0.0112)1/2 = 0.1058 =10.58% If the correlation between the returns on Stock A and Stock B is -0.5, the standard deviation of the portfolio is 10.58%. c. 10.7 a. As Stock A and Stock B become more negatively correlated, the standard deviation of the portfolio decreases. Value of Macrosoft stock in the portfolio Value of Intelligence stock in the portfolio = (100 shares)(\$80 per share) = \$8,000 = (300 shares)(\$40 per share) = \$12,000 Total Value in the portfolio = \$8,000 + \$12,000 = \$20,000 Weight of Macrosoft stock = \$8,000 / \$20,000 = 0.40 Weight of Intelligence stock = \$12,000 / \$20,000 = 0.60 The expected return on the portfolio equals: E(RP) = (WMAC)[E(RMAC)] + (WI)[E(RI)] where E(RP) E(RMAC) E(RI) WMAC WI = the expected return on the portfolio = the expected return on Macrosoft stock = the expected return on Intelligence Stock = the weight of Macrosoft stock in the portfolio = the weight of Intelligence stock in the portfolio E(RP) = (WMAC)[E(RMAC)] + (WI)[E(RM)] = (0.40)(0.15) + (0.60)(0.20) = 0.18 = 18% The expected return on her portfolio is 18%. The variance of the portfolio equals: 2P = (WMAC)2(MAC)2 + (WI)2(I)2 + (2)(WMAC)(WI)(MAC)(I)[Correlation(RMAC, RI)] where 2P WMAC = the variance of the portfolio = the weight of Macrosoft stock in the portfolio WI MAC I RMAC RI = the weight of Intelligence stock in the portfolio = the standard deviation of Macrosoft stock = the standard deviation of Intelligence stock = the return on Macrosoft stock = the return on Intelligence stock 2P = (WMAC)2(MAC)2 + (WI)2(I)2 + (2)(WMAC)(WI)(MAC)(I)[Correlation(RMAC, RI)] = (0.40)2(0.08)2 + (0.60)2(0.20)2 + (2)(0.40)(0.60)(0.08)(0.20)(0.38) = 0.018342 The standard deviation of the portfolio equals: P = (2P)1/2 where P = the standard deviation of the portfolio 2P = the variance of the portfolio P = (0.018342)1/2 = 0.1354 =13.54% The standard deviation of her portfolio is 13.54%. b. Janet started with 300 shares of Intelligence stock. After selling 200 shares, she has 100 shares left. Value of Macrosoft stock in the portfolio Value of Intelligence stock in the portfolio = (100 shares)(\$80 per share) = \$8,000 = (100 shares)(\$40 per share) = \$4,000 Total Value in the portfolio = \$8,000 + \$4,000 = \$12,000 Weight of Macrosoft stock = \$8,000 / \$12,000 = 2/3 Weight of Intelligence stock = \$4,000 / \$12,000 = 1/3 E(RP) = (WMAC)[E(RMAC)] + (WI)[E(RI)] = (2/3)(0.15) + (1/3)(0.20) = 0.1667 = 16.67% The expected return on her portfolio is 16.67%. 2P = (WMAC)2(MAC)2 + (WI)2(I)2 + (2)(WMAC)(WI)(MAC)(I)[Correlation(RMAC, RI)] = (2/3)2(0.08)2 + (1/3)2(0.20)2 + (2)(2/3)(1/3)(0.08)(0.20)(0.38) = 0.009991 P = (0.009991)1/2 = 0.1000 =10.00% The standard deviation of her portfolio is 10.00%. 10.8 a. Expected ReturnA = (0.20)(0.07) + (0.50)(0.07) + (0.30)(0.07) = 0.07 = 7% The expected return on Stock A is 7%. VarianceA (A2) = (0.20)(0.07 0.07)2 + (0.50)(0.07 0.07)2 + (0.30)(0.07 0.07)2 =0 The variance of the returns on Stock A is 0. Standard DeviationA (A) = (0)1/2 = 0.00 = 0% The standard deviation of the returns on Stock A is 0%. Expected ReturnB = (0.20)(-0.05) + (0.50)(0.10) + (0.30)(0.25) = 0.1150 = 11.50% The expected return on Stock B is 11.50%. VarianceB (B2) = (0.20)(-0.05 0.1150)2 + (0.50)(0.10 0.1150)2 + (0.30)(0.25 0.1150)2 = 0.011025 The variance of the returns on Stock B is 0.011025. Standard DeviationB (B) = (0.011025)1/2 = 0.1050 =10.50% The standard deviation of the returns on Stock B is 10.50%. b. Covariance(RA, RB) = (0.20)(0.07 0.07)(-0.05 0.1150) + (0.50)(0.07 0.07)(0.10 0.1150) (0.30)(0.07 0.07)(0.25 0.1150) =0 The covariance between the returns on Stock A and Stock B is 0. Correlation(RA,RB) = Covariance(RA, RB) / (A * B) = 0 / (0 * 0.1050) =0 The correlation between the returns on Stock A and Stock B is 0. c. The expected return on the portfolio equals: E(RP) = (WA)[E(RA)] + (WB)[E(RB)] where E(RP) E(RA) E(RB) WA WB = the expected return on the portfolio = the expected return on Stock A = the expected return on Stock B = the weight of Stock A in the portfolio = the weight of Stock B in the portfolio E(RP) = (WA)[E(RA)] + (WB)[E(RB)] = (1/2)(0.07) + (1/2)(0.115) = 0.0925 = 9.25% The expected return of an equally weighted portfolio is 9.25%. 2P = (WA)2(A)2 + (WB)2(B)2 + (2)(WA)(WB)(A)(B)[Correlation(RA, RB)] where 2P WA WB A B RA RB = the variance of the portfolio = the weight of Stock A in the portfolio = the weight of Stock B in the portfolio = the standard deviation of Stock A = the standard deviation of Stock B = the return on Stock A = the return Stock B 2P = (WA)2(A)2 + (WB)2(B)2 + (2)(WA)(WB)(A)(B)[Correlation(RA, RB)] = (1/2)2(0)2 + (1/2)2(0.105)2 + (2)(1/2)(1/2)(0)(0.105)(0) = 0.002756 The standard deviation of the portfolio equals: P = (2P)1/2 where P = the standard deviation of the portfolio 2P = the variance of the portfolio P = (0.002756)1/2 = 0.0525 =5.25% The standard deviation of the returns on an equally weighted portfolio is 5.25%. 10.9 a. The expected return on the portfolio equals: E(RP) = (WA)[E(RA)] + (WB)[E(RB)] where E(RP) E(RA) E(RB) WA WB = the expected return on the portfolio = the expected return on Stock A = the expected return on Stock B = the weight of Stock A in the portfolio = the weight of Stock B in the portfolio E(RP) = (WA)[E(RA)] + (WB)[E(RB)] = (0.30)(0.10) + (0.70)(0.20) = 0.17 = 17% The expected return on the portfolio is 17%. The variance of a portfolio equals: 2P = (WA)2(A)2 + (WB)2(B)2 + (2)(WA)(WB)(A)(B)[Correlation(RA, RB)] where 2P WA WB A B RA RB = the variance of the portfolio = the weight of Stock A in the portfolio = the weight of Stock B in the portfolio = the standard deviation of Stock A = the standard deviation of Stock B = the return on Stock A = the return on Stock B 2P = (WA)2(A)2 + (WB)2(B)2 + (2)(WA)(WB)(A)(B)[Correlation(RA, RB)] = (0.30)2(0.05)2 + (0.70)2(0.15)2 + (2)(0.30)(0.70)(0.05)(0.15)(0) = 0.01125 The standard deviation of the portfolio equals: P = (2P)1/2 where P = the standard deviation of the portfolio 2P = the variance of the portfolio P = (0.01125)1/2 = 0.1061 = 10.61% The standard deviation of the portfolio is 10.61%. b. E(RP) = (WA)[E(RA)] + (WB)[E(RB)] = (0.90)(0.10) + (0.10)(0.20) = 0.11 = 11% The expected return on the portfolio is 11%. 2P = (WA)2(A)2 + (WB)2(B)2 + (2)(WA)(WB)(A)(B)[Correlation(RA, RB)] = (0.90)2(0.05)2 + (0.10)2(0.15)2 + (2)(0.90)(0.10)(0.05)(0.15)(0) = 0.00225 P = (0.00225)1/2 = 0.0474 = 4.74% The standard deviation of the portfolio is 4.74%. c. No, you would not hold 100% of Stock A because the portfolio in part b has a higher expected return and lower standard deviation than Stock A. You may or may not hold 100% of Stock B, depending on your risk preference. If you have a low level of risk-aversion, you may prefer to hold 100% Stock B because of its higher expected return. If you have a high level of risk-aversion, however, you may prefer to hold a portfolio containing both Stock A and Stock B since the portfolio will have a lower standard deviation, and hence, less risk, than holding Stock B alone. 10.10 The expected return on the portfolio must be less than or equal to the expected return on the asset with the highest expected return. It cannot be greater than this assets expected return because all assets with lower expected returns will pull down the value of the weighted average expected return. Similarly, the expected return on any portfolio must be greater than or equal to the expected return on the asset with the lowest expected return. The portfolios expected return cannot be below the lowest expected return among all the assets in the portfolio because assets with higher expected returns will pull up the value of the weighted average expected return. 10.11 a. Expected ReturnA = (0.40)(0.03) + (0.60)(0.15) = 0.1020 = 10.20% The expected return on Security A is 10.20%. VarianceA (A2) = (0.40)(0.03 0.102)2 + (0.60)(0.15 0.102)2 = 0.003456 Standard DeviationA (A) = (0.003456)1/2 = 0.0588 = 5.88% The standard deviation of the returns on Security A is 5.88%. Expected ReturnB = (0.40)(0.065) + (0.60)(0.065) = 0.0650 = 6.50% The expected return on Security B is 6.50%. VarianceB (B2) = (0.40)(0.065 0.065)2 + (0.60)(0.065 0.065)2 =0 Standard DeviationB (B) = (0)1/2 = 0.00 = 0% The standard deviation of the returns on Security B is 0%. b. Total Value of her portfolio = \$2,500 + \$3,500 = \$6,000 Weight of Security A Weight of Security B = \$2,500 / \$6,000 = 5/12 = \$3,500 / \$6,000 = 7/12 E(RP) = (WA)[E(RA)] + (WB)[E(RB)] where E(RP) E(RA) E(RB) WA WB = the expected return on the portfolio = the expected return on Security A = the expected return on Security B = the weight of Security A in the portfolio = the weight of Security B in the portfolio E(RP) = (WA)[E(RA)] + (WB)[E(RB)] = (5/12)(0.102) + (7/12)(0.065) = 0.0804 = 8.04% The expected return of her portfolio is 8.04%. The variance of a portfolio equals: 2P = (WA)2(A)2 + (WB)2(B)2 + (2)(WA)(WB)(A)(B)[Correlation(RA, RB)] where 2P WA WB A B RA RB = the variance of the portfolio = the weight of Security A in the portfolio = the weight of Security B in the portfolio = the standard deviation of Security A = the standard deviation of Security B = the return on Security A = the return on Security B 2P = (WA)2(A)2 + (WB)2(B)2 + (2)(WA)(WB)(A)(B)[Correlation(RA, RB)] = (5/12)2(0.0588)2 + (7/12)2(0)2 + (2)(5/12)(7/12)(0.0588)(0)(0) = 0.000600 The standard deviation of the portfolio equals: P = (2P)1/2 where P = the standard deviation of the portfolio 2P = the variance of the portfolio P = (0.00600)1/2 = 0.0245 =2.45% The standard deviation of her portfolio is 2.45%. 10.12 The wide fluctuations in the price of oil stocks do not indicate that these stocks are a poor investment. If an oil stock is purchased as part of a well-diversified portfolio, only its contribution to the risk of the entire portfolio matters. This contribution is measured by systematic risk or beta. Since price fluctuations in oil stocks reflect diversifiable plus non-diversifiable risk, observing the standard deviation of price movements is not an adequate measure of the appropriateness of adding oil stocks to a portfolio. a. Expected Return1 = (0.10)(0.25) + (0.40)(0.20) + (0.40)(0.15) + (0.10)(0.10) = 0.1750 = 0.1750 10.13 The expected return on Security 1 is 17.50%. Variance1 (12) = (0.10)(0.25 0.175)2 + (0.40)(0.20 0.175)2 + (0.40)(0.15 0.175)2 + (0.10)(0.10 0.175)2 = 0.001625 Standard Deviation1 (1)= (0.001625)1/2 = 0.0403 = 4.03% The standard deviation of the returns on Security 1 is 4.03%. Expected Return2 = (0.10)(0.25) + (0.40)(0.15) + (0.40)(0.20) + (0.10)(0.10) = 0.1750 = 0.1750 The expected return on Security 2 is 17.50%. Variance2 (22) = (0.10)(0.25 0.175)2 + (0.40)(0.15 0.175)2 + (0.40)(0.20 0.175)2 + (0.10)(0.10 0.175)2 = 0.001625 Standard Deviation2 (2)= (0.001625)1/2 = 0.0403 = 4.03% The standard deviation of the returns on Security 2 is 4.03%. Expected Return3 = (0.10)(0.10) + (0.40)(0.15) + (0.40)(0.20) + (0.10)(0.25) = 0.1750 = 0.1750 The expected return on Security 3 is 17.50%. Variance3(32) = (0.10)(0.10 0.175)2 + (0.40)(0.15 0.175)2 + (0.40)(0.20 0.175)2 + (0.25)(0.10 0.175)2 = 0.001625 Standard Deviation3 (3)= (0.001625)1/2 = 0.0403 = 4.03% The standard deviation of the returns on Security 3 is 4.03%. b. Covariance(R1, R2) = (0.10)(0.25 0.175)(0.25 0.175) + (0.40)(0.20 0.175)(0.15 0.175) + + (0.40)(0.15 0.175)(0.20 0.175) + (0.10)(0.10 0.175)(0.10 0.175) = 0.000625 The covariance between the returns on Security 1 and Security 2 is 0.000625. Correlation(R1,R2) = Covariance(R1, R2) / (1 * 2) = 0.000625 / (0.0403 * 0.0403) = 0.3848 The correlation between the returns on Security 1 and Security 2 is 0.3848. Covariance(R1, R3) = (0.10)(0.25 0.175)(0.10 0.175) + (0.40)(0.20 0.175)(0.15 0.175) + + (0.40)(0.15 0.175)(0.20 0.175) + (0.10)(0.10 0.175)(0.25 0.175) = -0.001625 The covariance between the returns on Security 1 and Security 3 is -0.001625. Correlation(R1,R3) = Covariance(R1, R3) / (1 * 3) = -0.001625 / (0.0403 * 0.0403) = -1 The correlation between the returns on Security 1 and Security 3 is -1. Covariance(R2, R3) = (0.10)(0.25 0.175)(0.10 0.175) + (0.40)(0.15 0.175)(0.15 0.175) + + (0.40)(0.20 0.175)(0.20 0.175) + (0.10)(0.10 0.175)(0.25 0.175) = -0.000625 The covariance between the returns on Security 2 and Security 3 is -0.000625. Correlation(R2,R3) = Covariance(R2, R3) / (2 * 3) = -0.000625 / (0.0403 * 0.0403) = -0.3848 The correlation between the returns on Security 2 and Security 3 is 0.3848. c. The expected return on the portfolio equals: E(RP) = (W1)[E(R1)] + (W2)[E(R2)] where E(RP) E(R1) E(R2) W1 W2 = the expected return on the portfolio = the expected return on Security 1 = the expected return on Security 2 = the weight of Security 1 in the portfolio = the weight of Security 2 in the portfolio E(RP) = (W1)[E(R1)] + (W2)[E(R2)] = (1/2)(0.175) + (1/2)(0.175) = 0.175 = 17.50% The expected return of the portfolio is 17.50%. The variance of a portfolio equals: 2P = (W1)2(1)2 + (W2)2(2)2 + (2)(W1)(W2)(1)(2)[Correlation(R1, R2)] where 2P W1 W2 1 2 R1 R2 = the variance of the portfolio = the weight of Security 1 in the portfolio = the weight of Security 2 in the portfolio = the standard deviation of Security 1 = the standard deviation of Security 2 = the return on Security 1 = the return on Security 2 2P = (W1)2(1)2 + (W2)2(2)2 + (2)(W1)(W2)(1)(2) [Correlation(R1, R2)] = (1/2)2(0.0403)2 + (1/2)2(0.0403)2 + (2)(1/2)(1/2)(0.0403)(0.0403)(0.3848) = 0.001125 The standard deviation of the portfolio equals: P = (2P)1/2 where P = the standard deviation of the portfolio 2P = the variance of the portfolio P = (0.001125)1/2 = 0.0335 = 3.35% The standard deviation of the returns on the portfolio is 3.35%. d. E(RP) = (W1)[E(R1)] + (W3)[E(R3)] = (1/2)(0.175) + (1/2)(0.175) = 0.175 = 17.50% The expected return on the portfolio is 17.50%. 2P = (W1)2(1)2 + (W3)2(3)2 + (2)(W1)(W3)(1)(3) [Correlation(R1, R3)] = (1/2)2(0.0403)2 + (1/2)2(0.0403)2 + (2)(1/2)(1/2)(0.0403)(0.0403)(-1) =0 P = (0)1/2 =0 = 0% The standard deviation of the returns on the portfolio is 0%. e. E(RP) = (W2)[E(R2)] + (W2)[E(R3)] = (1/2)(0.175) + (1/2)(0.175) = 0.175 = 17.50% The expected return of the portfolio is 17.50%. 2P = (W2)2(2)2 + (W3)2(3)2 + (2)(W2)(W3)(2)(3) [Correlation(R2, R3)] = (1/2)2(0.0403)2 + (1/2)2(0.0403)2 + (2)(1/2)(1/2)(0.0403)(0.0403)(-0.3848) = 0.000500 P = (0.000500)1/2 = 0.0224 = 2.24% The standard deviation of the returns on the portfolio is 2.24%. f. As long as the correlation between the returns on two securities is below 1, there is a benefit to diversification. A portfolio with negatively correlated stocks can achieve greater risk reduction than a portfolio with positively correlated stocks, holding the expected return on each stock constant. Applying proper weights on perfectly negatively correlated stocks can reduce portfolio variance to 0. 10.14 a. State 1 2 3 4 b. Return on A 15% 15% 10% 10% Return on B 35% -5% 35% -5% Probability (0.40)(0.50) = 0.2 (0.40)(0.50) = 0.2 (0.60)(0.50) = 0.3 (0.60)(0.50) = 0.3 E(RP) = (0.20)[(0.50)(0.15) + (0.50)(0.35)] + (0.20)[(0.50)(0.15) + (0.50)(-0.05)] + (0.30)[(0.50)(0.10) + (0.50)(0.35)] + (0.30)[(0.50)(0.10) + (0.50)(-0.05)] = 0.135 = 13.5% The expected return on the portfolio is 13.5%. 10.15 a. The expected return on a portfolio equals: E(RP) = E(Ri) / N where E(RP) = the expected return on the portfolio E(Ri) = the expected return on Security i N = the number of securities in the portfolio E(RP) = E(Ri) / N = [(0.10)(N)] / N = 0.10 = 10% The expected return on an equally weighted portfolio containing all N securities is 10%. The variance of a portfolio equals: P2 = [Covariance(Ri, Rj) / N2] + i2 / N2 where P2 Ri Rj N i2 = the variance of the portfolio = the returns on security i = the return on security j = the number of securities in the portfolio = the variance of security i P2 = [Covariance(Ri, Rj) / N2] + i2 / N2 Since there are N securities, there are (N)(N-1) different pairs of covariances between the returns on these securities. P2 = (N)(N-1)(0.0064) / N2 + [N(0.0144)] / N2 = (0.0064)(N-1) / N + (0.0144)/(N) The variance of an equally weighted portfolio containing all N securities can be represented by the following expression: (0.0064)(N-1) / N + (0.0144)/(N) b. As N approaches infinity, the expression (N-1)/N approaches 1 and the expression (1/N) approaches 0. It follows that, as N approaches infinity, the variance of the portfolio approaches 0.0064 [= (0.0064)(1) + (0.0144)(0)], which equals the covariance between any two individual securities in the portfolio. The covariance of the returns on the securities is the most important factor to consider when placing securities into a well-diversified portfolio. c. 10.16 The statement is false. Once the stock is part of a well-diversified portfolio, the important factor is the contribution of the stock to the variance of the portfolio. In a well-diversified portfolio, this contribution is the covariance of the stock with the rest of the portfolio. The covariance is a more appropriate measure of a securitys risk in a well-diversified portfolio because the covariance 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 securities. Since covariance measures the impact of an individual security on the variance of the portfolio, covariance is the appropriate measure of risk. If we assume that the market has not stayed constant during the past three years, then the lack in movement of Southern Co.s stock price only indicates that the stock either has a standard deviation or a beta that is very near to zero. The large amount of movement in Texas Instrument stock price does not imply that the firms beta is high. Total volatility (the fluctuation) price is a function of both systematic and unsystematic risk. The beta only reflects the systematic risk. Observing the standard deviation of price movements does not indicate whether the price changes were due to systematic factors or firm specific factors. Thus, if you observe large stock price movements like that of TI, you cannot claim that the beta of the stock is high. All you know is that the total risk of TI is high. Because a well-diversified portfolio has no unsystematic risk, this portfolio should like on the Capital Market Line (CML). The slope of the CML equals: SlopeCML = [E(RM) rf] / M where E(RM) rf M = the expected return on the market portfolio = the risk-free rate = the standard deviation of the market portfolio 10.17 10.18 10.19 SlopeCML = [E(RM) rf] / M = (0.12 0.05) / 0.10 = 0.70 a. The expected return on the portfolio equals: E(RP) = rf + SlopeCML(P) where E(RP) rf P = the expected return on the portfolio = the risk-free rate = the standard deviation of the portfolio E(RP) = rf + SlopeCML(P) = 0.05 + (0.70)(0.07) = 0.99 = 9.9% A portfolio with a standard deviation of 7% has an expected return of 9.9%. b. E(RP) = rf + SlopeCML(P) 0.20 = 0.05 + (0.70)(P) P = (0.20 0.05) / 0.70 = 0.2143 = 21.43% A portfolio with an expected return of 20% has a standard deviation of 21.43%. 10.20 a. The slope of the Characteristic Line (CL) of Fuji equals: SlopeCL = [E(RFUJI)BULL E(RFUJI)BEAR] / [(RM)BULL (RM)BEAR] where E(RFUJI)BULL E(RFUJI)BEAR (RM)BULL (RM)BEAR = the expected return on Fuji in a bull market = the expected return on Fuji in a bear market = the return on the market portfolio in a bull market = the return on the market portfolio in a bear market Characteristic Line of Fuji Expected Return on Fuji 0.15 0.1 0.05 0 0 0.05 0.1 0.15 0.2 Return on the Market SlopeCL = [ E(RFUJI)BULL E(RFUJI)BEAR ] / [(RM)BULL (RM)BEAR] = (0.128 0.034) / (0.163 0.025) = 0.68 Beta, by definition, equals the slope of the characteristic line. Therefore, Fujis beta is 0.68. 10.21 10.22 Polonius portfolio will be the market portfolio. He will have no borrowing or lending in his portfolio. a. E(RP) = (1/3)(0.10) + (1/3)(0.14) + (1/3)(0.20) = 0.1467 = 14.67% The expected return on an equally weighted portfolio is 14.67%. b. The beta of a portfolio equals the weighted average of the betas of the individual securities within the portfolio. P = (1/3)(0.7) + (1/3)(1.2) + (1/3)(1.8) = 1.23 The beta of an equally weighted portfolio is 1.23. c. If the Capital Asset Pricing Model holds, the three securities should be located on a straight line (the Security Market Line). For this to be true, the slopes between each of the points must be equal. Security Market Line 0.25 0.2 0.1 5 0.1 0.05 0 0 0.5 1 B eta 1 .5 2 Slope between A and B = (0.14 0.10) / (1.2 0.7) = 0.08 Slope between A and C = (0.20 0.10) / (1.8 0.7) = 0.091 Slope between B and C = (0.20 0.14) / (1.8 1.2) = 0.10 Since the slopes between the three points are different, the securities are not correctly priced according to the Capital Asset Pricing Model. 10.23 According to the Capital Asset Pricing Model: E(r) = rf + (EMRP) where E(r) rf EMRP = the expected return on the stock = the risk-free rate = the stocks beta = the expected market risk premium In this problem: rf = 0.06 = 1.2 EMRP = 0.085 The expected return on Holups stock is: E(r) = rf + (EMRP) = 0.06 + 1.2(0.085) = 0.162 The expected return on Holups stock is 16.2%. 10.24 According to the Capital Asset Pricing Model: E(r) = rf + (EMRP) where E(r) rf EMRP = the expected return on the stock = the risk-free rate = the stocks beta = the expected market risk premium In this problem: rf = 0.06 = 0.80 EMRP = 0.085 The expected return on Stock A equals: E(r) = rf + (EMRP) = 0.06 + 0.80(0.085) = 0.128 The expected return on Stock A is 12.8%. 10.25 According to the Capital Asset Pricing Model: E(r) = rf + [E(rm) rf] where E(r) rf E(rm) = the expected return on the stock = the risk-free rate = the stocks beta = the expected return on the market portfolio In this problem: rf E(rm) = 0.08 = 1.5 = 0.15 The expected return on Stock B equals: E(r) = rf + [E(rm) rf] = 0.08 + 1.5(0.15 0.08) = 0.185 The expected return on Stock B is 18.5%. 10.26 According to the Capital Asset Pricing Model: E(r) = rf + (EMRP) where E(r) rf EMRP = the expected return on the stock = the risk-free rate = the stocks beta = the expected market risk premium In this problem: E(r) = 0.142 rf = 0.037 EMRP = 0.075 E(r) = rf + (EMRP) 0.142 = 0.037 + (0.075) = (0.142 0.037) / 0.075 = 1.4 The beta of Tristars stock is 1.4. 10.27 Because the Capital Asset Pricing Model holds, both securities must lie on the Security Market Line (SML). Given the betas and expected returns on the two stocks, solve for the slope of the SML. Slope of SML = [E(rMP) E(rPSD)] / (MP - PSD) where E(rMP) E(rPSD) MP PSD = the expected return on Murck Pharmaceutical = the expected return on Pizer Drug Corp = the beta of Murck Phamraceutical = the beta of Pizer Drug Corp Security Market Line 0.3 0.25 0.2 0.15 0.1 0.05 0 0 0.5 Beta 1 1.5 Slope of SML = [E(rMP) E(rPSD)] / (MP - PSD) = (0.25 0.14) / (1.4 0.7) = 0.1571 A security with a beta of 0.7 has an expected return of 0.14. As you move along the SML from a beta of 0.7 to a beta of 1, beta increases by 0.3 (= 1 0.7). Since the slope of the security market line is 0.1571, as beta increases by 0.3, expected return increases by 0.0471 (= 0.3 * 0.1571). Therefore, the expected return on a security with a beta of one equals 18.71% (= 0.14 + 0.0471). Since the market portfolio has a beta of one, the expected return on the market portfolio is 18.71%. Expected Return According to the Capital Asset Pricing Model: E(r) where = rf + [E(rm) rf] E(r) rf E(rm) = the expected return on the security = the risk-free rate = the securitys beta = the expected return on the market portfolio Since Murck Pharmaceutical has a beta of 1.4 and an expected return of 0.25, we know that: 0.25 = rf + 1.4(0.1871 rf) rf = 0.03 The risk-free rate is 3%. Thus, the entire SML looks like: Security Market Line 0.3 Expected Return 0.25 0.2 0.15 0.1 0.05 0 0 0.5 Beta 1 1.5 10.28 a. E(rA) = (0.25)(-0.10) + (0.50)(0.10) + (0.25)(0.20) = 0.075 The expected return on Stock A is 7.5%. E(rB) = (0.25)(-0.30) + (0.50)(0.05) + (0.25)(0.40) = 0.05 The expected return on Stock B is 5%. b. From part a, we know that: E(rA) E(rB) = 0.075 = 0.05 We also know that the beta of A is 0.25 greater than the beta of B. Therefore, as beta increases by 0.25, the expected return on a security increases by 0.025 (= 0.075 0.5). Consider the following graph: Security Market Line 0.08 Expected Return 0.06 0.04 0.02 0 Beta The slope of the security market line (SML) equals: SlopeSML = Rise / Run = Increase in Expected Return / Increase in Beta = (0.075 0.05) / 0.25 = 0.10 The slope of the Security Market Line equals 10%. The expected market risk premium is the difference between the expected return on the market and the risk-free rate. Since the markets beta is 1 and the risk-free rate has a beta of zero, the slope of the Security Market Line equals the expected market risk premium. The expected market risk premium is 10%. 10.29 a. Security Market Line 0.2 Expected Return 0.15 0.1 0.05 0 0 0.5 1 Beta 1.5 2 2.5 b. According to the security market line drawn in part a, a security with a beta of 0.80 should have an expected return of: E(r) = rf + (EMRP) = 0.07 + 0.8(0.05) = 0.11 = 11% Since this asset has an expected return of only 9%, it lies below the security market line. Because the asset lies below the security market line, it is overpriced. Investors will sell the overpriced security until its price falls sufficiently so that its expected return rises to 11%. c. According to the security market line drawn in part a, a security with a beta of 3 should have an expected return of: E(r) = rf + (EMRP) = 0.07 + 3(0.05) = 0.22 = 22% Since this asset has an expected return of 25%, it lies above the security market line. Because the asset lies above the security market line, it is underpriced. Investors will buy the underpriced security until its price rises sufficiently so that its expected return falls to 22%. 10.30 According to the Capital Asset Pricing Model (CAPM), the expected return on the stock should be: E(r) = rf + (EMRP) where E(r) rf EMRP = the expected return on the stock = the risk-free rate = the stocks beta = the expected market risk premium E(r) = rf + (EMRP) = 0.05 + 1.8(0.08) = 0.194 According to the CAPM, the expected return on the stock should be 19.4%. However, the security analyst expects the return to be only 18%. Therefore, the analyst is pessimistic about this stock relative to the markets expectations. 10.31 a. According to the Capital Asset Pricing Model: E(r) = rf + [E(rm) rf] where E(r) rf E(rm) = the expected return on the stock = the risk-free rate = the stocks beta = the expected return on the market portfolio In this problem: rf E(rm) = 0.064 = 1.2 = 0.138 The expected return on Solomons stock is: E(r) = rf + [E(rm) rf] = 0.064 + 1.2(0.138 0.064) = 0.1528 The expected return on Solomons stock is 15.28%. b. If the risk-free rate decreases to 3.5%, the expected return on Solomons stock is: E(r) = rf + [E(rm) rf] = 0.035 + 1.2(0.138 0.035) = 0.1586 The expected return on Solomons stock is 15.86%. 10.32 First, calculate the standard deviation of the market portfolio using the Capital Market Line (CML). We know that the risk-free rate asset has a return of 5% and a standard deviation of zero and the portfolio has an expected return of 25% and a standard deviation of 4%. These two points must lie on the Capital Market Line. Capital Market Line 0.3 Expected Return 0.25 0.2 0.15 0.1 0.05 0 0 0.01 0.02 0.03 0.04 0.05 Standard Deviation The slope of the Capital Market Line equals: SlopeCML = Rise / Run = Increase in Expected Return / Increase in Standard Deviation = (0.25 0.05) / (0.04 - 0) =5 According to the Capital Market Line: E(ri) = rf + SlopeCML(i) where E(r) rf SlopeCML i = the expected return on security i = the risk-free rate = the slope of the Capital Market Line = the standard deviation of security i Since we know the expected return on the market portfolio is 20%, the risk-free rate is 5%, and the slope of the Capital Market Line is 5, we can solve for the standard deviation of the market portfolio (m). E(rm) = rf + SlopeCML(m) 0.20 = 0.05 + (5)(m) m = (0.20 0.05) / 5 = 0.03 The standard deviation of the market portfolio is 3%. Next, use the standard deviation of the market portfolio to solve for the beta of a security that has a correlation with the market portfolio of 0.5 and a standard deviation of 2%. Security = [Correlation(RSecurity, RMarket)*(Security)] / Market = (0.5*0.02) / 0.03 = 1/3 The beta of the security equals 1/3. According to the Capital Asset Pricing Model: E(r) = rf + [E(rm) - rf] where E(r) rf E(rm) = the expected return on the security = the risk-free rate = the securitys beta = the expected return on the market portfolio In this problem: rf E(rm) = 0.05 = 1/3 = 0.20 E(r) = rf + [E(rm) - rf] = 0.05 + 1/3(0.20 - 0.05) = 0.10 A security with a correlation of 0.5 with the market portfolio and a standard deviation of 2% has an expected return of 10%. 10.33 a. According to the Capital Asset Pricing Model: E(r) = rf + (EMRP) where E(r) rf EMRP = the expected return on the stock = the risk-free rate = the stocks beta = the expected market risk premium In this problem: E(r) rf = 0.167 = 0.076 = 1.7 E(r) = rf + (EMRP) 0.167 = 0.076 + 1.7(EMRP) EMRP = (0.167 0.076) / 1.7 = 0.0535 The expected market risk premium is 5.35%. b. According to the Capital Asset Pricing Model: E(r) where = rf + (EMRP) E(r) rf EMRP = the expected return on the stock = the risk-free rate = the stocks beta = the expected market risk premium In this problem: rf = 0.076 = 0.8 EMRP = 0.0535 E(r) = rf + (EMRP) = 0.076 + 0.8(0.0535) = 0.1188 The expected return on Magnolia stock is 11.88%. c. The beta of a portfolio is the weighted average of the betas of the individual securities in the portfolio. The beta of Potpourri is 1.7, the beta of Magnolia is 0.8, and the beta of a portfolio consisting of both Potpourri and Magnolia is 1.07. Therefore: 1.7 where = (WP)(1.7) + (WM)(0.8) WP = the weight of Potpourri stock in the portfolio WM = the weight of Magnolia stock in the portfolio Because your total investment must equal 100%: WP = 1 - WM 1.7 = (1 WM)(1.7) + (WM)(0.8) 1.07 = 1.7 1.7WM + 0.8WM -0.63 = -0.90WM WM WP = 0.70 = 1 - WM = 1 0.70 = 0.30 You have 70% of your portfolio (\$7,000) invested in Magnolia stock and 30% of your portfolio (\$3,000) invested in Potpourri stock. E(r) = (0.70)(0.1188) + (0.30)(0.167) = 0.1333 The expected return of the portfolio is 13.33%. 10.34 According to the Capital Asset Pricing Model: E(rP) = rf + P[E(rm) rf] where E(rP) = the expected return on the portfolio rf = the risk-free rate P = the beta of the portfolio E(rm) = the expected return on the market portfolio The beta of a portfolio equals: P = [Correlation(RP, Rm) * P] / m where RP Rm P m = the return on the portfolio = the return on the market portfolio = the standard deviation of the portfolio = the standard deviation of the market portfolio Since the market portfolio has a variance of 0.0121, it has a standard deviation of 11% [= (0.0121)1/2]. Since the portfolio has a variance of 0.0169, it has a standard deviation of 13% [= (0.0169)1/2]. Therefore, the beta of the portfolio equals: P = [Correlation(RP, Rm) * P] / m = (0.45*0.13) / 0.11 = 0.5318 The beta of the portfolio is 0.5318. The expected return on the portfolio is: E(rP) = rf + P[E(rm) rf] = 0.063 + 0.5318(0.148 0.063) = 0.1082 The expected return on portfolio Z is 10.82%. 10.35 a. The equation for the Security Market Line is: E(r) = rf + (EMRP) Since the risk-free rate equals 4.9% and the expected market risk premium is 9.4%, the CAPM implies: E(r) = 0.049 + (0.094) b. First, calculate the beta of Durham Companys stock. = Covariance(RDurham, RMarket) / (Market)2 = 0.0635 / 0.04326 = 1.467 Use the Capital Asset Pricing Model to determine the required return on Durhams stock. According to the Capital Asset Pricing Model: E(r) where = rf + (EMRP) E(r) rf EMRP = the expected return = the risk-free rate = the stocks beta = the expected market risk premium In this problem: rf EMRP E(r) = 0.049 = 1.467 = 0.094 = rf + (EMRP) = 0.049 + 1.467(0.094) = 0.1869 The required return on Durhams stock is 18.69%. 10.36 Because the Capital Asset Pricing Model holds, both securities must lie on the Security Market Line (SML). Given the betas and expected returns of the two stocks, solve for the slope of the SML. Slope of SML = [E(rJ) E(rW)] / (J - W) where E(rJ) E(rW) J W = the expected return on Johnsons stock = the expected return on Williamsons Stock = the beta of Johnsons stock = the beta of Williamsons stock Security Market Line 0.2 Expected Return 0.15 0.1 0.05 0 0 0.5 1 Beta 1.5 2 Slope of SML = [E(rJ) E(rW)] / (J - W) = (0.19 0.14) / (1.7 1.2) = 0.10 A security with a beta of 1.2 has an expected return of 0.14. As you move along the SML from a beta of 1.2 to a beta of 1, beta decreases by 0.2 (= 1.2 1). Since the slope of the security market line is 0.10, as beta decreases by 0.2, expected return decreases by 0.02 (= 0.2 * 0.10). Therefore, the expected return on a security with a beta of one equals 12% (= 0.14 - 0.02). Since the market portfolio has a beta of one, the expected return on the market portfolio is 12%. According to the Capital Asset Pricing Model: E(r) where = rf + [E(rm) rf] E(r) rf E(rm) = the expected return on the security = the risk-free rate = the securitys beta = the expected return on the market portfolio Since Williamson has a beta of 1.2 and an expected return of 0.14, we know that: 0.14 = rf + 1.2(0.12 rf) rf = 0.02 The risk-free rate is 2%. Thus, the entire SML looks like: Security Market Line 0.2 Expected Return 0.15 0.1 0.05 0 0 0.5 1 Beta 1.5 2 10.37 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 Capital Asset Pricing Model. E(r) = rf + [E(rm) rf] where E(r) rf = the expected return on a security = the risk-free rate = the securitys beta E(rm) = the expected return on the market portfolio If < 0, E(r) < rf 10.38 First, determine the beta of the portfolio. Total Amount Invested = \$5,000 + \$10,000 + \$8,000 + \$7,000 = \$30,000 Weight of Stock A = \$5,000 / \$30,000 = 1/6 Weight of Stock B = \$10,000 / \$30,000 = 1/3 Weight of Stock C = \$8,000 / \$30,000 = 4/15 Weight of Stock D = \$7,000 / \$30,000 = 7/30 The beta of a portfolio is the weighted average of the betas of its individual securities. Portfolio = (1/6)(0.75) + (1/3)(1.1) + (4/15)(1.36) + (7/30)(1.88) = 1.293 Use the Capital Asset Pricing Model (CAPM) to find the expected return on the portfolio. According to the CAPM: E(r) = rf + [E(rm) rf] where E(r) rf E(rm) = the expected return on the portfolio = the risk-free rate = the portfolios beta = the expected return on the market portfolio In this problem: rf E(rm) = 0.04 = 1.293 = 0.15 E(r) = rf + [E(rm) rf] = 0.04+ 1.293(0.15 0.04) = 0.1822 The expected return on the portfolio is 18.22%. 10.39 a. Let i i m i,m = the beta of Security i = the standard deviation of Security i = the standard deviation of the market = the correlation between returns on Security i and the market (i) i = (i,m)(i) / m 0.9 = (i,m)(0.12) / 0.10 i,m = 0.75 (ii) i = (i,m)(i) / m 1.1 = (0.4)(i) / 0.10 i (iii) i = 0.275 = (i,m)(i) / m = (0.75)(0.24) / 0.10 = 1.8 The market has a correlation of 1 with itself. The beta of the market is 1. The risk-free asset has 0 standard deviation. (iv) (v) (vi) (vii) The risk-free asset has 0 correlation with the market portfolio. (viii) The beta of the risk-free asset is 0. b. According to the Capital Asset Pricing Model: E(r) = rf + [E(rm) rf] E(r) rf E(rm) = the expected return on the stock = the risk-free rate = the stocks beta = the expected return on the market portfolio where Firm A rf = 0.05 = 0.9 E(rm) = 0.15 E(r) = rf + [E(rm) rf] = 0.05 + 0.9(0.15 0.05) = 0.14 According to the CAPM, the expected return on Firm As stock should be 14%. However, the expected return on Firm As stock given in the table is only13%. Therefore, Firm As stock is overpriced, and you should sell it. Firm B rf = 0.05 = 1.1 E(rm) = 0.15 E(r) = rf + [E(rm) rf] = 0.05 + 1.1(0.15 0.05) = 0.16 According to the CAPM, the expected return on Firm Bs stock should be 16%. The expected return on Firm Bs stock given in the table is also 16%. Therefore, Firm As stock is correctly priced. Firm C rf = 0.05 = 1.8 E(rm) = 0.15 E(r) = rf + [E(rm) rf] = 0.05 + 1.8(0.15 0.05) = 0.23 According to the CAPM, the expected return on Firm Cs stock should be 23%. However, the expected return on Firm Cs stock given in the table is 25%. Therefore, Firm As stock is underpriced, and you should buy it. 10.40 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, 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 [(P1 P0) 1]. where P0 = the price of the stock today P1 = the price of the stock one period from today E(rA) = (0.10)(40/50 -1) + (.80)(55/50 -1) + (0.10)(60/50 1) = (0.10)(-0.20) + (0.80)(0.10) + (0.10)(0.20) = 0.08 A2 A A = (0.10)(-0.20 0.08)2 + (0.80)(0.10 0.08)2 + (0.10)(0.20 0.08)2 = 0.0096 = (0.0096)1/2 = 0.098 = Correlation(RA, RM)(A) / M = (0.8)(0.098) / 0.10 = 0.784 Stock A has an expected return of 8% and a beta of 0.784. Stock B E(r) = 0.09 (Given) B = Correlation(RB, RM)(B) / M = (0.2)(0.12) / 0.10 = 0.24 Stock B has an expected return of 9% and a beta of 0.24. 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. b. E(rP) = (0.70)E(rA) + (0.30)E(rB) = (0.70)(0.08) + (0.30)(0.09) = 0.083 The expected return on a portfolio consisting of 70% of Stock A and 30% of Stock B is 8.3%. P2 P = (0.70)2(0.098)2 + (0.30)2(0.12)2 + 2(0.70)(0.30)(0.60)(0.098)(0.12) = 0.008965 = (0.008965)1/2 = 0.0947 The standard deviation of a portfolio consisting of 70% of Stock A and 30% of Stock B is 9.47%. c. The beta of a portfolio is the weighted average of the betas of its individual securities. P = (0.70)(0.784) + (0.30)(0.24) = 0.6208 The beta of a portfolio consisting of 70% of Stock A and 30% of Stock B is 0.6208. 10.41 a. The variance of a portfolio of two assets equals: P2 where = (XA)2(A)2 + (XB)2(B)2 + 2(XA)(XB)Covariance(RA, RB) XA XB A B = the weight of Stock A in the portfolio = the weight of Stock B in the portfolio = the standard deviation of Stock A = the standard deviation of Stock B Let XA = 1 XB. Then, P2 = (1-XB)2(A)2 + (XB)2(B)2 + 2(1-XB)(XB)Covariance(RA, RB) From the problem, we know: A B Covariance(RA, RB) Therefore, P2 = (1-XB)2(0.10)2 + (XB)2(0.20)2 + 2(1-XB)(XB)(0.001) = (0.01)(1-XB)2+ (0.04)(XB)2 + (0.002)(1-XB)(XB) = (0.01)[1 2XB + (XB)2] + (0.04)(XB)2 + (0.002)[XB (XB)2] = 0.01 0.02XB + 0.01(XB)2 + (0.04)(XB)2 + 0.002XB 0.002(XB)2 = 0.10 = 0.20 = 0.001 Minimize this function (the portfolio variance). We do this by differentiating the function with respect to XB. (P2) / (XB) = -0.02 + 0.02XB + 0.08XB + 0.002 0.004XB = -0.018 + 0.096XB Set this expression equal to zero. Then solve for XB. 0 = -0.018 + 0.096XB XB XA = (0.018) / (0.096) = 0.1875 = 1 - XB = 1 0.1875 = 0.8125 In order to minimize the variance of the portfolio, the weight of Stock A in the portfolio should be 81.25% and the weight of Stock B in the portfolio should be 18.75%. b. Using the weights calculated in part a, determine the expected return of the portfolio. E(rP) = (XA)[E(rA)] + (XB)[E(rB)] = (0.8125)(0.05) + (0.1875)(0.10) = 0.0594 The expected return on the minimum variance portfolio is 5.94%. c. From the problem, we know: A B Covariance(RA, RB) Therefore, P2 = (1-XB)2(0.10)2 + (XB)2(0.20)2 + 2(1-XB)(XB)(-0.02) = (0.01)(1-XB)2+ (0.04)(XB)2 + (-0.04)(1-XB)(XB) = (0.01)[1 2XB + (XB)2] + (0.04)(XB)2 + (-0.04)[XB (XB)2] = 0.01 0.02XB + 0.01(XB)2 + (0.04)(XB)2 - 0.04XB + 0.04(XB)2 = 0.10 = 0.20 = -0.02 Differentiate this function with respect to XB. (P2) / (XB) = -0.02 + 0.02XB + 0.08XB - 0.04 + 0.08XB) = -0.06 + 0.18XB Set this expression equal to zero. Then solve for XB. 0 = -0.06 + 0.18XB XB XA = (0.06) / (0.18) = 1/3 = 1 - XB = 1 (1/3) = 2/3 In order to minimize the variance of the portfolio, the weight of Stock A in the portfolio should be 1/3 and the weight of Stock B in the portfolio should be 2/3. d. The variance of a portfolio of two assets equals: P2 = (XA)2(A)2 + (XB)2(B)2 + 2(XA)(XB)Covariance(RA, RB) where XA XB A B = the weight of Stock A in the portfolio = the weight of Stock B in the portfolio = the standard deviation of Stock A = the standard deviation of Stock B In this problem: XA = 1/3 XB = 2/3 A = 0.10 B = 0.20 Covariance(RA, RB) = -0.02 Therefore, the variance of the portfolio is: P2 = (XA)2(A)2 + (XB)2(B)2 + 2(XA)(XB)Covariance(RA, RB) = (1/3)2(0.10)2 + (2/3)2(0.20)2 + (2)(1/3)(2/3)(-0.02) = 0.01 The variance of the portfolio is 0.01.
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