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Unformatted text preview: : σ 2 =.33(.082 – .0800)2 + .33(.095 – .0800)2 + .33(.063 – .0800)2 = .00017 σ = (.00017)1/2 = .0131 or 1.31% And the standard deviation of Stock B is: σ 2 =.33(–.065 – .0813)2 + .33(.124 – .0813)2 + .33(.185 – .0813)2 = .01133 σ = (.01133)1/2 = .1064 or 1064% To find the covariance, we multiply each possible state times the product of each assets’ deviation from the mean in that state. The sum of these products is the covariance. So, the covariance is: Cov(A,B) = .33(.092 – .0800)(–.065 – .0813) + .33(.095 – .0800)(.124 – .0813) + .33(.063 – .0800)(.185 – .0813) Cov(A,B) = –.000472 And the correlation is: ρ A,B = Cov(A,B) / σ A σ B ρ A,B = –.000472 / (.0131)(.1064) ρ A,B = –.3373 27. 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 each stock is: E(RA) = .30(–.020) + .50(.138) + .20(.218) = .1066 or 10.66% E(RB) = .30(.034) + .50(.062) + .20(.092) = .0596 or 5.96% To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, and then add all of these up. The result is the variance. So, the variance and standard deviation of Stock A are: σ 2 =.30(–.020 – .1066)2 + .50(.138 – .1066)2 + .20(.218 – .1066)2 = .00778 A σ A = (.00778)1/2 = .0882 or 8.82% And the standard deviation of Stock B is: σ 2 =.30(.034 – .0596)2 + .50(.062 – .0596)2 + .20(.092 – .0596)2 = .00041 B σ B = (.00041)1/2 = .0202 or 2.02% 282 To find the covariance, we multiply each possible state times the product of each assets’ deviation from the mean in that state. The sum of these products is the covariance. So, the covariance is: Cov(A,B) = .30(–.020 – .1066)(.034 – .0596) + .50(.138 – .1066)(.062 – .0596) + .20(.218 – .1066)(.092 – .0596) Cov(A,B) = .001732 And the correlation is: ρ A,B = Cov(A,B) / σ A σ B ρ A,B = .001732 / (.0882)(.0202) ρ A,B = .9701 28. a. 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) = wFE(RF) + wGE(RG) E(RP) = .30(.10) + .70(.17) E(RP) = .1490 or 14.90% b. The variance of a portfolio of two assets can be expressed as:
2 2 σ 2 = w 2 σ 2 + w G σ G + 2wFwG σ Fσ Gρ F,G P F F σ 2 = .302(.262) + .702(.582) + 2(.30)(.70)(.26)(.58)(.25) P σ 2 = .18675 P So, the standard deviation is: σ P = (.18675)1/2 = .4322 or 43.22% 29. a. 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) = .45(.13) + .55(.19) E(RP) = .1630 or 16.30% The variance of a portfolio of two assets can be expressed as: σ 2 = w 2 σ 2 + w 2 σ 2 + 2wAwBσ Aσ Bρ A,B P A A B B σ 2 = .452(.382) + .552(.622) + 2(.45)(.55)(.38)(.62)(.50) P σ 2 = .20383 P So, the standard deviation is: σ P = (.20383)1/2 = .4515 or 45.15% 283 b. σ 2 = w 2 σ 2 + w 2 σ 2 + 2wAwBσ Aσ Bρ A,B P A A B B σ 2 = .452(.382) + .552(.622) + 2(.45)(.55)(.38)(.62)(–.50) P σ 2 = .08721 P So, the standard deviation is: σ = (.08721)1/2 = .2953 or 29.53% c. 30. a. As Stock A and Stock B become less correlated, or more negatively correlated, the standard deviation of the portfolio decreases. (i) We can use the equation to calculate beta, we find: β A = (ρ A,M)(σ A) / σ M 0.85 = (ρ A,M)(0.27) / 0.20 ρ A,M = 0.63 (ii) Using the equation to calculate beta, we find: β B = (ρ B,M)(σ B) / σ M 1.50 = (.50)(σ B) / 0.20 σ B = 0.60 (iii) Using the equation to calculate beta, we find: β C = (ρ C,M)(σ C) / σ M β C = (.35)(.70) / 0.20 β C = 1.23 (iv) The market has a correlation of 1 with itself. (v) The beta of the market is 1. (vi) The riskfree asset has zero standard deviation. (vii) The riskfree asset has zero correlation with the market portfolio. (viii) The beta of the riskfree asset is 0. b. Using the CAPM to find the expected return of the stock, we find: Firm A: E(RA) = Rf + β A[E(RM) – Rf] E(RA) = 0.05 + 0.85(0.12 – 0.05) E(RA) = .1095 or 10.95% 284 According to the CAPM, the expected return on Firm A’s stock should be 10.95 percent. However, the expected return on Firm A’s stock given in the table is only 10 percent. Therefore, Firm A’s stock is overpriced, and you should sell it. Firm B: E(RB) = Rf + β B[E(RM) – Rf] E(RB) = 0.05 + 1.5(0.12 – 0.05) E(RB) = .1550 or 15.50% According to the CAPM, the expected return on Firm B’s stock should be 15.50 percent. However, the expected return on Firm B’s stock given in the table is 14 percent. Therefore, Firm B’s stock is overpriced, and you should sell it. Firm C: E(RC) = Rf + β C[E(RM) – Rf] E(RC) = 0.05 + 1.23(0.12 – 0.05) E(RC) = .1358 or 13.58% According to the CAPM, the expected return on Firm C’s stock should be 13.58 percent. However, the expected return on Firm C’s stock given in the table is 17 percent. Therefore,...
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 Spring '10
 eshmalwi
 Finance, Corporate Finance

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