Question

1. An investor can allocate wealth into risky assets, A and B. The returns of the assets RA and Rp are considered bivariate normal random variables with parameters as follows: HA = .25,0Ả = .09, MB = .15,03 = .04 and PAB = -1. What is the expected return of a portfolio that achieves op = 0?

Answer 1

Standard deviation for A =

Let the weight of A & B be x & (1-x)

Standard deviation of portfolio = (x^2)*(std deviation for A)^2 + (1-x)^2*(std deviation for B)^2 + 2* (co-variance for A & B)*(x)*(std deviation for A)* (1-x)*(std deviation for B)

= (x^2)* 0.09 + (1-x)^2*0.04+ 2* (-1)*(x)*(1-x)*(0.09)^(1/2)*(0.04)^(1/2)

= (x^2)* 0.09 + (1-x)^2*0.04 - 2*(x)*(1-x)*(0.09)^(1/2)*(0.04)^(1/2)

=  (x^2)* 0.09 + (1-x)^2*0.04 - 0.12*(x)*(1-x) which is to be set to zero.

Solving the quadratic equation,

(x^2)* 0.09 + (1-x)^2*0.04 - 0.12*(x)*(1-x) = 0

We obtain x = 0.40 & 1-x = 0.60

So, weight of A = 40% & weight of B = 60%

Expected return for portfolio = weight of A * Return on A + weight of B * Return on B

= 0.4*0.25 + 0.6*0.15

= 0.19

= 19% (Ans)