Question

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

R_A = 5.0% + 1.30R_M + e_A

R_B = –2.0% + 1.60R_M + e_B

σ_M = 20%; R-square_A = 0.20; R-square_B = 0.12

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?

b. What is the beta of portfolio Q?

c. What is the "firm-specific" risk of portfolio Q?

d. What is the covariance between the portfolio and the market index?

Answer 1

a). Variance for stock A = (beta*market standard deviation)^2/R-squareA = (1.30*20%)^2/0.20 = 0.3380

Standard deviation (SD) for stock A = 0.3380^0.5 = 58.14%

Variance for stock B = (1.60*20%)^2/0.12 = 0.8533

SD for stock B = 0.8533^0.5 = 92.38%

Covariance(A,B) = BetaA*BetaB*market variance = 1.30*1.60*20%^2 = 0.0832

SD for portfolio P = [(wA*SDA)^2 + (wB*SDB)^2 + (2*wA*wB*Cov(A,B)]^0.5

= [(0.7*58.14%)^2 + (0.3*92.38%)^2 + (2*0.7*0.3*0.0832)]^0.5 = 52.67%

Beta for portfolio P = (wA*BetaA) + (wB*BetaB) = (0.7*1.3) + (0.3*1.6) = 1.39

Cov(P, Market) = 0.7*Cov(A, Market) + 0.3*Cov(B, Market)

= 0.7*(Correlation (A, Market)*SDA*SDM)) + 0.3*(Correlation (B, Market)*SDB*SDM))

= 0.7*(0.2^0.5*58.14%*20%) + 0.3*(0.12^0.5*92.38%*20%) = 0.0556

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

= [(0.40*52.67%)^2 + (0.35*20%)^2 + (2*0.40*0.35*0.0556)]^0.5 = 25.46%

b). Beta for portfolio Q = (wP*BetaP) + (wM*BetaM) = (0.40*1.39) + (0.35*1) = 0.906

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

Systematic risk = (BetaQ*SDM)^2 = (0.906*20%)^2 = 0.0532; Total risk = variance of Q = 25.46%^2 = 0.0648

Firm-specific risk = 0.0648 - 0.0532 = 0.0116 (or, in terms of standard deviation 0.0116^0.5 = 10.78%)

d). Covariance (Q, Market) = BetaQ*Market variance = 0.906*20%^2 = 0.03624