Question

Consider two risky assets, A and B, with expected returns and standard deviations (μA, σA) = (6%, 10%) and (μB , σB ) = (8%, 20%). Their correlation coefficient is de-noted by rho. You are a mean-variance investor with risk aversion parameter γ = 1. Suppose that you invest $1 between these assets to maximize your utility (no borrow-ing and saving; short position is prohibited; use the MV utility U = μ − γ 2 σ2).

(a) When rho = 1, what is the optimal investment (in dollar value) in asset A?
(b) When rho = 0.2, what is the optimal investment (in dollar value) in asset A?

Answer 1

When the correlation coefficient is 1, the optimal investment in asset A is $0.21. When the correlation coefficient is 0.2, the optimal investment in asset A is $0.025.

Step-by-step explanation:

a. When rho = 1, the formula to calculate the optimal investment in asset A can be derived from a mean-variance optimization. The formula is:

Optimal Investment in Asset A = (μB − μA + γ(σB^2 − σA^2)) / (2ρσAσB)

In this case, substituting the given values into the formula, we have:

Optimal Investment in Asset A = (0.08 - 0.06 + 1(0.2^2 - 0.1^2)) / (2 * 1 * 0.1 * 0.2) = 0.21

Therefore, the optimal investment in asset A when rho = 1 is $0.21 (or 21% of the total investment).

b. When rho = 0.2, we can use the same formula to calculate the optimal investment in asset A. Substituting the given values, we have:

Optimal Investment in Asset A = (0.08 - 0.06 + 1(0.2^2 - 0.1^2)) / (2 * 0.2 * 0.1 * 0.2) = 0.025

Therefore, the optimal investment in asset A when rho = 0.2 is $0.025 (or 2.5% of the total investment).