Question

1. Derive the optimal portfolio weights, {w₁, w₂, w₃} for 3-asset case.

Hint: Solve the following constraint optimization problem:

        min σ²_p = [w²₁σ²₁ + w²₂σ²₂ + w²₃σ²₃] + 2w₁w₂σ₁₂ + 2w₁w₃σ₁₃ + 2w₂w₃σ₂₃

      w₁,w₂,w₃               

                                        (l) w₁E(r˜₁) + w₂E(r˜₂) + w₃E(r˜₃) = E(r˜_p)

                        s.t           (g) w₁ + w₂ + w₃=1

2. Derive the optimal portfolio variance, σ*_p²

Answer 1

Portfolio variance formula = w₁²* ơ₁² + w₂²* ơ₂² + w₃²* ơ₃² + 2 * ρ_1,2 * w₁ * w₂ * ơ₁ * ơ₂ + 2 * ρ_2,3 * w₂ * w₃ * ơ₂ * ơ₃ + 2 * ρ_3,1 * w₃ * w₁ * ơ₃ * ơ₁

The optimal portfolio may have more risk than the minimal variance portfolio. The efficient frontier shows different combinations of asset classes with different risk return profiles. Some clients may be able and want to take more risk. A minimum variance portfolio in contrast should exhibit much lower volatility than a higher risk portfolio. The only time the optimal portfolio and the minimum variance portfolio would be the same would be if you were talking about the minimum variance portfolio along each point on the efficient frontier. So you could have the minimum variance portfolio for the high risk portfolio.