Mean-Variance Optimization

E aw b numerical solution using quadratic programming

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Unformatted text preview: an analytical solution • With linear inequality constraints (i.e. Aw ≤ b ) → numerical solution using quadratic programming (QP) • With non-linear inequality constraints (i.e. g(w ) ≤ b ) → numerical solution using nonlinear programming (NP) RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 6 Solution of the Mean-Variance Using Lagrange Multipliers 1 min w ′Σw w2 s.t. w ′μ = μ0 w ′ι = 1 ′ ι = ⎡⎢⎣1,...,1⎤⎥⎦ The Lagrangian becomes 1 L ≡ L(w, λ, γ ) ≡ w ′Σw + λ (1 − w ′ι ) + γ (μ0 − w ′μ) 2 Differentiating with respect to w, we have the FOC ∂L = Σw − λι − γμ = 0 ∂w RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 7 ∂L = Σw − λι − γμ = 0 ∂w Solving for the optimal w, we obtain w = λΣ−1ι + γΣ−1μ (For you: Is this the minimum? How so?) How do we determine λ and γ ? Substituting the “parametrized” solution for w above into the constraints, we obtain a linear system for λ and γ ι ′w = λι ′Σ−1ι + γι ′Σ−1μ ≡ 1 μ′w = λμ ′Σ−1ι + γμ′Σ−1μ ≡ μ0 RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 8 We can write the...
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