Question

2. Portfolio Choice Suppose we have assets A and B with the following distribution of returns: Probability Return for A .01 Return for B -.14 .00 .03 TO .05 .07 14 .30 .09 .50 a. Compute the expected returns for assets A and B, rA and rg. b. Compute the variances of A and B, oả and oß. c. Compute the covariance of A and B, CAR- d. Use the formulas for portfolio returns and risk to write the expected portfolio return and standard deviation for an equally weighted portfolio of A and B, rp and Op. e. Use the formula to find the portfolio weights, a* and 1 - a* for the minimum variance portfolio of A and B. f. Find the portfolio return and standard deviation of the minimum variance portfolio.

Answer 1

a). Expected return is the sum of the probability weighted returns.

Expected return for A (ERa) = (0.1*0.01) +(0.2*0.03) +(0.4*0.05) + (0.2*0.07) +(0.1*0.09) = 0.05 (or 5%)

Expected return for A (ERb) = (0.1*-0.14) +(0.2*0.00) +(0.4*0.14) + (0.2*0.30) +(0.1*0.50) = 0.1520 (or 15.20%)

b). Variance = sum of [probability*(return - expected return)^2]

Variance for A (Va) = 0.1*(0.01-0.05)^2 + 0.2*(0.03-0.05)^2 + 0.4*(0.05-0.05)^2 + 0.2*(0.07-0.05)^2 + 0.1*(0.09-0.05)^2 = 0.00048

Variance for B (Vb) = 0.1*(-0.14-0.152)^2 + 0.2*(0.0-0.152)^2 + 0.4*(0.14-0.152)^2 + 0.2*(0.3-0.152)^2 + 0.1*(0.5-0.152)^2 = 0.029696

c). Covariance Cov(a,b) = sum of [probability*(return for A - ERa)*(return for B - ERb)]

= 0.1*(0.01-0.05)*(-0.14-0.152) + 0.2*(0.03-0.05)*(0.0-0.152) + 0.4*(0.05-0.05)*(0.14-0.152) + 0.2*(0.07-0.05)*(0.3-0.152) + 0.1*(0.09-0.05)*(0.5-0.152) = 0.00376

d). Equally weighted portfolio means that weights of A and B will be 50% (or 0.5) each.

Portfolio return is the sum of weighted expected returns.

Portfolio return (Rp) = (50%*5%) + (50%*15.20%) = 0.1010 (or 10.10%)

Portfolio standard deviation = [(weight of A*standard deviation of A)^2 + (weight of B*standard deviation of B)^2 + (2*weight of A*weight of B*Covariance)]^2

Portfolio standard deviation (SDp) = [(0.5^2*0.00048) + (0.5^2*0.29696) + (2*0.5*0.5*0.00376)]^0.5 = 0.0971

e). Portfolio weight for a stock in a minimum variance portfolio is given by:

Weight of A = [Vb - Cov(a,b)]/(Va + Vb - 2*Cov(a,b)) = (0.029696 - 0.00376)/(0.00048+0.029696-(2*0.00376)) = 1.145

Weight of B = 1 - weight of A = 1-1.145 = -0.145 (Negative weight implies that stock B is shorted)

f). Rp = (1.145*5%) + (-0.145*15.2%) = 0.0352 (or 3.52%)

SDp = ((1.145%^2*0.00048) + (-0.145*0.029696) + (2*1.145*-0.145*0.00376))^0.5 = 0.00227