Mean-variance optimization
Quantitative Finance glossary · Numerical Methods & Optimization
Definition
Markowitz framework selecting weights boldsymbolw to minimise boldsymbolw^T boldsymbolΣ boldsymbolw subject to boldsymbolw^T boldsymbolμ = R^*, boldsymbolw^T boldsymbol1 = 1. Closed form boldsymbolw^* propto boldsymbolΣ^-1(boldsymbolμ - r_fboldsymbol1) for the tangency portfolio. The infamous flaw: boldsymbolw^* is extraordinarily sensitive to noise in hatboldsymbolμ (1% input change to corner-solution weights), motivating shrinkage covariance (Ledoit-Wolf), Black-Litterman, risk-parity, and resampled-efficiency alternatives.
Mean-variance optimization in plain English
This is the original math recipe for building an investment portfolio, and its big idea is that you shouldn't judge an investment alone — you judge how it mixes with everything else you own. It searches for the exact percentages to put in each asset (the "weights") that give you a target return R^* while making the bundle's overall bounciness — its risk — as small as possible, since some investments zig while others zag and partly cancel out. The catch that made it famous: it's dangerously touchy about your guess for future returns — nudge that guess by 1% and it may suddenly dump everything into one stock. It's like a GPS so sensitive that mistyping your destination by a block reroutes you to another city, which is why people invented steadier versions (Ledoit-Wolf shrinkage, Black-Litterman, risk-parity).
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