Question

WEIGHT E(r) Sigma

ASSET A: 0.3 1.487% 6.344%

ASSET B: 0.5 2.078% 6.353%

ASSET C: 0.2 1.66% 7.616%

ρAB=0.313 ρBC=0.374 ρAC=0.321

1. What is the standard deviation and expected return of the portfolio?

Answer 1

Expected return of three-asset portfolio R_p = w₁R₁ + w₂R₂ + w₃R₃

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

w₃ = weight of Asset 3

R₃ = expected return of Asset 3

standard deviation for a three-asset portfolio σ_p = (w₁²σ₁² + w₂²σ₂² + w₃²σ₃²+ 2w₁w₂Cov_1,2 + 2w₂w₃Cov_2,3 + 2w₁w₃Cov_1,3 )^1/2

where σ_p = standard deviation of the portfolio

w₁ = weight of Asset 1

w₂ = weight of Asset 2

w₃ = weight of Asset 3

σ₁ = standard deviation of Asset 1

σ₂ = standard deviation of Asset 2

σ₃ = standard deviation of Asset 3

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

Cov_2,3 = covariance of returns between Asset 2 and Asset 3

Cov_1,3 = covariance of returns between Asset 1 and Asset 3

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

Cov_2,3 = ρ_2,3 * σ₂ * σ_3, where ρ_2,3 = correlation of returns between Asset 2 and Asset 3

Cov_1,3 = ρ_1,3 * σ₁ * σ_3, where ρ_1,3 = correlation of returns between Asset 1 and Asset 3

Expected return = (0.3 * 1.487%) + (0.5 * 2.078%) + (0.2 * 1.66%)

Expected return = 1.817%

standard deviation = ((0.3² * 0.06344²) + (0.5² * 0.06353²) + (0.2² * 0.07616²) + (2 * 0.3 * 0.5 * 0.313 * 0.06344 * 0.06353) + (2 * 0.5 * 0.2 * 0.374 * 0.06353 * 0.07616) + (2 * 0.3 * 0.2 * 0.321 * 0.06344 * 0.07616))^1/2

standard deviation = 5.030%