Question

The following are estimates for two stocks. Stock Expected Return Beta Firm-Specific Standard Deviation A 10 % 0.70 28 % B 18 1.25 42 The market index has a standard deviation of 22% and the risk-free rate is 7%. a. What are the standard deviations of stocks A and B? (Do not round intermediate calculations. Round your answers to 2 decimal places.) b. Suppose that we were to construct a portfolio with proportions: Stock A 0.35 Stock B 0.35 T-bills 0.30 Compute the expected return, standard deviation, beta, and nonsystematic standard deviation of the portfolio

Answer 1

This is a fairly straight forward question.

If _A, _B are the standard deviations of stocks A and B respectively then,

Total Variance = ² = Beta² x _M² + Firm specific variance

where _M = standard deviation of returns from market = 22% = 0.22

Part (a)

Total Variance of returns from stock A= _A² = Beta_A² x _M² + Firm specific variance_A = 0.7² x 0.22² + 0.28² = 0.102116

Hence, _A = (0.102116)^1/2 = 0.3196 = 31.96%

Total Variance of returns from stock B = _B² = Beta_B² x _M² + Firm specific variance_B = 1.25² x 0.22² + 0.42² = 0.252025

Hence, _B = (0.252025)^1/2 = 0.5020 = 50.20%

Part (b)

Suppose that we were to construct a portfolio with proportions:

Stock A = W_A = 0.35 and return, R_A = 10% = 0.10

Stock B = W_B = 0.35 and return, R_B = 18% = 0.18

and T-bills = W_T = 0.30 and return = risk free return = R_F = 7% = 0.07

The expected return of the portfolio = W_A x R_A + W_B x R_B + W_T x R_F = 0.35 x 0.10 + 0.35 x 0.18 + 0.3 x 0.07 = 0.1190 = 11.90%

Let's first calculate the beta of the portfolio = Beta_P = W_A x Beta_A + W_B x Beta_B + W_T x Beta_T = 0.35 x 0.70 + 0.35 x 1.25 + 0.3 x 0 = 0.6825

Nonsystematic variance of the portfolio = (W_A x Firm Specific Standard deviation_A)² + (W_B x Firm Specific Standard deviation_B)² + (W_T x Standard deviation_T)² = (0.35 x 0.28)² + (0.35 x 0.42)² + (0.3 x 0)² = 0.031213

Hence, Nonsystematic standard deviation of the portfolio = (Nonsystematic variance of the portfolio)^1/2 = (0.031213)^1/2 = 0.176672012 = 17.67%

Variance of the portfolio = = _P² = Beta_P² x _M² + Nonsystematic variance of the portfolio = (0.6825 x 0.22)² + 0.031213 = 0.053758023

Hence, standard deviation of the portfolio = _P = (0.053758023)^1/2 = 0.231857764 = 23.19%