Portfolio Expected Return and Variance of Return

[Solutions] C

The correlation between two random variables R_i and R_j is defined as ρ(R_i,R_j) = Cov(R_i,R_j)/σ(R_i)σ(R_j). Using the subscript i to represent hedge funds and the subscript j to represent the market index, the standard deviations are σ(R_i) = 256^1/2 = 16 and σ(R_j) = 81^1/2 = 9. Thus, ρ(R_i,R_j) = Cov(R_i,R_j)/σ(R_i)σ(R_j) = 110/(16 × 9) = 0.764.

A.       26.39.

B.        26.56.

C.        28.12.

[Solutions] B

First, expected returns are

E(R_FI) = (0.25 × 25) + (0.50 × 15) + (0.25 × 10)

            = 6.25 + 7.50 + 2.50 = 16.25 and

E(R_DI) = (0.25 × 30) + (0.50 × 25) + (0.25 × 15)

            = 7.50 + 12.50 + 3.75 = 23.75.

Independence for Random Variables:

Two random variables X and Y are independent if and only if P(X, Y) = P(X)P(Y).

Independence is a stronger property than uncorrelatedness because correlation addresses only linear relationships.

Multiplication Rule for expected value of the product of uncorrelated random variables.

If X and Y are uncorrelated: E(XY) = E(X)E(Y)