Mean-Variance Optimization

# E aw b numerical solution using quadratic programming

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This document was uploaded on 02/17/2014 for the course COURANT G63.2751.0 at NYU.

Ask a homework question - tutors are online