Portfolio Optimization

Portfolio theory promised one perfect blend of assets sitting on a curved efficient frontier. Actually computing that blend from real data is where the bodies are buried — a textbook-perfect optimizer can produce portfolios so unstable that practitioners spent decades learning to distrust their own machine. This topic teaches both halves: how the optimizer works, and why it betrays you.

What you will learn to do:

• Mean-variance mechanics — set up the constrained minimization over expected returns μ and the covariance matrix Σ, walk the Lagrangian intuition, and crank the small-matrix arithmetic for two and three assets.
• Minimum-variance & tangency portfolios — compute the calmest mix (no return forecast needed) and the maximum-Sharpe point the capital market line kisses, by hand.
• Error maximization — see why tiny errors in the estimated means get levered into +200%/−150% weight swings, making the naive optimizer an “error maximizer” that collapses out of sample.
• Shrinkage estimators — use Ledoit–Wolf to pull the noisy sample covariance toward a structured target for stable, better-conditioned weights, plus shrinking the means and the Black–Litterman view-blending idea.
• Risk parity — equalize each asset’s risk contribution instead of forecasting returns, because equal dollars are emphatically not equal risk.
• Real constraints — long-only and box limits, turnover penalties, and transaction costs that decide whether a strategy is profitable on paper or after the broker takes their cut.

By the end you will know when to trust an optimizer, how to tame it, and why the sharpest desks prefer a robust, shrunk, cost-aware portfolio to the razor-sharp “optimal” one — the math-plus-humility judgment that separates a quant from a calculator. A graded final exam closes the loop.