Question

2. An investor can design a risky portfolio based on two stocks, S and B. Stock S has an expected return of 12% and a standard deviation of return of 25%. Stock B has an expected return of 10% and a standard deviation of return of 20%. The correlation coefficient between the returns of S and B is 0.4. The risk-free rate of return is 5%

a. The proportion of the optimal risky portfolio that should be invested in stock B is approximately __________.

b. The expected return on the optimal risky portfolio is __________.

Answer 1

(a) Let the proportion invested in Stock S be w₁. Hence, proportion invested in Stock B = 1 - w₁

Objective is to find the weights to reduce the risk of the portfolio i.e. minimize the variance

Standard Deviation of Stock S = σ_S = 0.25

Standard Deviation of Stock B = σ_B = 0.20

Correlation between stocks = Corr_SB = 0.4

covariance of portfolio C = w₁²σ_S² + (1-w₁)²σ_B² + 2w₁(1-w₁)*Corr_SB = w₁²0.25² + (1-w₁)²0.20² + 2w₁(1-w₁)*0.4

To minimize the covariance, dC/dw¹ = 0

=> 2*0.25²w₁ - 2*0.20²(1-w₁) + 0.8(1-w₁) - 0.8w₁ = 0

=> 0.125w₁ - 0.08w₁ + 0.8 - 0.8w₁ - 0.8w₁ = 0

=> w₁ = 0.5144

Hence, proportion invested in Stock B = 1- 0.5144 = 0.4856 or 48.56%

(b) Expected Return = w₁R_S + (1-w₁)R_B = 0.5144*0.12 + 0.4856*0.10 = 0.1103 or 11.03%