Suppose that the index model for stocks A and B is estimated

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

Break down the variance of each stock to the systematic and firm-specific components

Solution

Solution: corr(A, M) = sqrt(0.30) = 0.547722
cov(A,M) = corr(A, M) * M*A = sqrt(0.30)*0.15* A = .08*A
beta = cov(A, M)/M^2 = sqrt(0.30)*0.15* A/0.15^2 = sqrt(0.30)* A/0.15

and from regression equation above that beta = 0.4 so

Solve

0.4 = sqrt(0.30)*A/0.15

A = .1095 And Variance = 0.01199

Similar for B

corr(B, M) = sqrt(0.22)

cov(B,M) = corr(B, M) * M*B = sqrt(0.22)*0.15* B
beta = cov(B, M)/M^2 = sqrt(0.22)*0.15* B/0.15^2 = sqrt(0.22)* B/0.15

and from regression equation above that beta = 0.9 so

Solve

0.9 = sqrt(0.22)*B/0.15

B = .2878 and Variance = 0.2878^2 = 0.0828