Question

Suppose you want to invest $ 1 million and you have two assets to invest in: Risk free asset with return of 12% per year and a risky asset with expected return of 30% and standard deviation of 40%. If you want a portfolio with standard deviation of 30% how much do you invest in each of the assets?

Answer 1

Let the weight allocated to Risk free asset be w₁ and risky asset be w₂ = 1 - w₁

Standard deviation of risk free asset = σ₁ = 0

Standard deviation of risky asset = σ₂ = 0.40

Covariance between risky asset and risk free asset = Cov(1, 2) = 0 (since the covariance/correlation with risk free security is always zero)

Standard deviation of the portfolio = P = 0.30

P = [ w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov(1, 2) ]^1/2 = [ w₁²0² + (1-w₁)²0.40² + 2w₁(1 - w₁)*0 ]^1/2 = 0.30

=> (1-w₁)*0.40 = 0.30

=> w₁ = 1 - 0.30/0.40 = 0.25 or 25%

=> w₂ = 1 - w₁ = 1 - 0.25 = 0.75 or 75%

Hence, Investment in risk free asset = 0.25*1000000 = $250000

Investment in risky asset = 0.74*1000000 = $750000