Question

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

R_A = 3.00% + 1.05R_M + e_A

R_B = -1.20% + 1.20R_M + e_B

σ_M = 29%; R-square_A = 0.29; R-square_B = 0.14

Assume you create portfolio P with investment proportions of 0.60 in A and 0.40 in B.

a. What is the standard deviation of the portfolio? (Do not round your intermediate calculations. Round your answer to 2 decimal places.) Standard deviation 0 % b. What is the beta of your portfolio? (Do not round your Intermediate calculations. Round your answer to 2 decimal places.) Portfolio beta c. What is the firm-specific variance of your portfolio? (Do not round your Intermediate calculations. Round your answer to 4 decimal places.) Firm-specific d. What is the covarlance between the portfolio and the market Index? (Do not round your intermediate calculations. Round your answer to 3 decimal places.) Covariance

Answer 1

Given information:

R_A = 3.00% + 1.05R_M + e_A This implies $\beta_A$ = 1.05

R_B = -1.20% + 1.20R_M + e_B This implies $\beta_B$ = 1.20

σ_M = 29%; R-square_A = 0.29; R-square_B = 0.14

a.What is the standard deviation of the portfolio?

To calculate standard deviation of portfolio, first we need to determine standard deviation of A and B.

The standard deviation can be derived from R² equation

$R_A^2$ = Explained variance / Total variance = $\beta_A^2$ * $\sigma_M^2$ / $\sigma_A^2$

0.29 = 1.05² * 0.29² / $\sigma_A^2$

solving we get $\sigma_A$ = 0.5654

Similarly for stock B,

$R_B^2$ = Explained variance / Total variance = $\beta_B^2$ * $\sigma_M^2$ / $\sigma_B^2$

0.14 = 1.2² *0.29² / $\sigma_B^2$

solving we get $\sigma_B$ = 0.9301

Covariance between A and B is given by the formula:

Cov (RA, RB) = $\beta_A$ * $\beta_B$ * $\sigma_M^2$ = 1.05 * 1.2 * 0.29² = 0.106

Variance of portfolio =

w* 0+ W *OB+2 * WA* WB* Cou(A,B)

= 0.6²*0.5654²+0.4²*0.9301²+2*0.6*0.4*0.106

= 0.3044

Hence standard deviation of portfolio = square root (0.3044) = 0.5517

b. Calculate beta of portfoli0

R(Portfolio) = 0.6RA + 0.4RB

=0.6*(3+1.05RM +eA) +0.4 *(-1.20 + 1.20R_M + e_B)

= 0.18 + 0.630RM +0.6eA - 0.48 + 0.48RM + 0.4eB

= 0.66+1.11 RM + 0.6eA + 0.4eB

The co-efficient of RM is the beta of the portfolio. Hence beta of portfolio is $\beta_P$ = 1.11

c. What is the firm specific variance of the portfolio?

Firm specific risk = Total risk - Systematic risk

We have already calculate total risk of portfolio in part a. Now we need to calculate systematic risk.

Systematic risk of portfolio =

BOM

Beta of portfolio = 1.11 which we calculated in part b.

Hence systematic risk of portfolio = 1.11²*0.29² = 0.1036

substitute this value in the formula above:

Firm specific risk = 0.5517² - 0.1036 = 0.2001

Part d:

What is the covariance between portfolio and market index?

Cov (RP, RM) =

Bp * o

substitute the values

Cov (RP, RM) = 1.11 * 0.29² = 0.09335

Answer:

Covariance between portfolio and market is 0.09335