Question

Assume you manage a risky portfolio with an expected rate of return of 20% and a standard deviation of 30%. The T-bill rate is 5%. Your client invests 60% of his portfolio in your fund and 40% in a T-bill money market fund. What is the expected return and standard deviation of your client’s portfolio?  What is the Sharpe ratio of your client’s portfolio?

Answer 1

Expected return of two-asset portfolio R_p = w₁R₁ + w₂R_2,

where R_p = expected return

w₁ = weight of Asset 1

R₁ = expected return of Asset 1

w₂ = weight of Asset 2

R₂ = expected return of Asset 2

Expected return of client portfolio = (60% * 20%) + (40% * 5%)

Expected return of client portfolio = 14.00%

standard deviation for a two-asset portfolio σ_p = (w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov_1,2)^1/2

where σ_p = standard deviation of the portfolio

w₁ = weight of Asset 1

w₂ = weight of Asset 2

σ₁ = standard deviation of Asset 1

σ₂ = standard deviation of Asset 2

Cov_1,2 = covariance of returns between Asset 1 and Asset 2

Cov_1,2 = ρ_1,2 * σ₁ * σ_2, where ρ_1,2 = correlation of returns between Asset 1 and Asset 2

standard deviation of client portfolio = ((0.60² * 0.30²) + (0.40 * 0²) + (2 * 0.60 * 0.40 * 0))^1/2

standard deviation of client portfolio = 18.00%

Sharpe ratio = (portfolio return - risk free rate) / portfolio standard deviation

Sharpe ratio = (14% - 5%) / 18%

Sharpe ratio = 0.50