Question

Suppose that the index model for stocks A and B is estimated from excess returns with the following results:

R_A = 2.8% + 1.00R_M + e_A

R_B = –1.0% + 1.30R_M + e_B

σ_M = 18%; R-square_A = 0.27; R-square_B = 0.13

Assume you create a portfolio Q, with investment proportions of 0.40 in a risky portfolio P, 0.35 in the market index, and 0.25 in T-bill. Portfolio P is composed of 70% Stock A and 30% Stock B.

a. What is the standard deviation of portfolio Q? (Calculate using numbers in decimal form, not percentages. Do not round intermediate calculations. Round your answer to 2 decimal places.)

b. What is the beta of portfolio Q? (Do not round intermediate calculations. Round your answer to 2 decimal places.

c. What is the "firm-specific" risk of portfolio Q? (Calculate using numbers in decimal form, not percentages. Do not round intermediate calculations. Round your answer to 4 decimal places.)

d. What is the covariance between the portfolio and the market index? (Calculate using numbers in decimal form, not percentages. Do not round intermediate calculations. Round your answer to 2 decimal places.

Answer 1

a). Variance = (beta*SDmarket)^2/Rsquare

Variance(stock A) = (1*18%)^2/0.27 = 0.120; SD(A) = 0.120^0.5 = 34.64%

Variance(stock B) = (1.30*18%)^2/0.13 = 0.421; SD(B) = 0.421^0.5 = 64.90%

Covariance(A,B) = BetaA*BetaB*Variance(market) = 1*1.30*18%^2 = 0.0421

SD(Portfolio P) =  [(wA*SDA)^2 + (wB*SDB)^2 + (2*wA*wB*Cov(A,B)]^0.5 = [(0.7*34.64%)^2 + (0.3*64.90%)^2 + (2*0.7*0.3*0.0421)]^0.5 = 33.82%

Beta for portfolio P = (wA*BetaA) + (wB*BetaB) = (0.7*1) + (0.3*1.30) = 1.09

Cov(P, Market) = BetaP*Variance(market) = 1.09*18%^2 = 0.035

SD for portfolio Q = [(wP*SDP)^2 + (wM*SDM)^2 + (2*wP*wM*Cov(P,Market)]^0.5

= [(0.40*33.82%)^2 + (0.35*18%)^2 + (2*0.40*0.35*0.035)]^0.5 = 17.93%

b). Beta of portfolio Q = (wP*BetaP) + (wM*BetaM) = (0.40*1.09) + (0.35*1) = 0.79

c). Firm-specific risk = Total risk - systematic risk

Systematic risk = (BetaQ*SDM)^2 = (0.79*18%)^2 = 0.02002; Total risk = variance of Q = 17.93%^2 = 0.0322

Firm-specific risk = 0.0322 - 0.02002 = 0.0121 (In terms of standard deviation, it is 11.02%)

d). Covariance (Q, Market) = BetaQ*Market variance = 0.79*18%^2 = 0.03