Question

You invest $10,000 in portfolio XYZ. The portfolio XYZ is composed of a risky asset with an expected rate of return of 15% and a standard deviation of 20% over the one year time period. The risk free asset has a rate of return of 5% over the same time period. How much money should be invested in the risky asset so that the standard deviation of returns on XYZ portfolio is 10% over the one year time horizon

Answer 1

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

For a risk free asset, the standard deviation is zero, and the covariance with the risky asset is zero

σ_p = (w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov_1,2)^1/2

0.10 = (w₁²0.20² + w₂²0.00² + (2w₁w₂ * 0)/)^1/2

0.10 = (w₁²0.20²)^1/2

0.10 = w₁ * 0.20

w₁ = 0.50

Proportion of portfolio to invest in risky asset = 0.50, or 50%

Money to invest in risky asset = $10,000 * 50%

Money to invest in risky asset = $5,000