Mean-Variance Optimization

10232012 p kolm 9 summary the solution to the mean

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Unformatted text preview: linear system above in matrix form ⎛A B ⎞⎛λ ⎞ ⎛ 1 ⎞ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎟ ⎟ ⎜ ⎜B C ⎟⎜γ ⎟ = ⎜μ ⎟ ⎟⎜ ⎠ ⎝ 0 ⎠ ⎟⎜⎟ ⎟⎜ ⎜ ⎝ ⎠⎝ ⎟ ⎜ ⎟ where A = ι ′Σ−1ι B = ι ′Σ−1μ C = μ′Σ−1μ Solving, we obtain (For you: derive this) C − μ0B Δ μ A−B γ= 0 Δ λ= with Δ ≡ AC − B 2 We call w ∗ the optimal portfolio (with λ and γ chosen as above) RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 9 Summary: The solution to the mean-variance problem is w ∗ = λΣ−1ι + γΣ−1μ with C − μ0B λ= , Δ μ0A − B γ= , and Δ Δ ≡ AC − B 2 where A = ι ′Σ−1ι, B = ι ′Σ−1μ, and C = μ′Σ−1μ We call w ∗ a mean-variance efficient portfolio RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 10 Understanding the Solution We can compute the variance at the optimal point 2 ∗ ∗ σ * = w ′Σw ′ −1 −1 −1 −1 = λΣ ι + γΣ μ Σ λΣ ι + γΣ μ ( )( ( −1 −1 ′ = (λι + γμ) λΣ ι + γΣ μ ) ) −1 −1 −1 −1 2 2 = λ ι ′Σ ι + λγι ′Σ μ + λγμ ′Σ...
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This document was uploaded on 02/17/2014 for the course COURANT G63.2751.0 at NYU.

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