Efficient Frontier✓ Mathematical
◆ The PatternMarkowitz efficient frontier — balance return against portfolio variance
The 1952 Markowitz framework maximizes expected return for a given level of risk (or minimizes variance for a target return). The efficient frontier traces optimal portfolios.
min ½ w′ Σ w s.t. w′ μ ≥ rtarget, w′ 1 = 1
Estimation error in μ and Σ makes raw MVO unstable — shrinkage estimators (Ledoit-Wolf) and resampling improve robustness. The tangency portfolio maximizes the Sharpe ratio.
// Interactive — risk aversion and efficient frontier
Risk aversion
| Approach | Fix |
|---|---|
| Shrinkage | Reduce estimation noise in Σ |
| Resampling | Average across bootstrapped frontiers |
| Black-Litterman | Blend views with equilibrium priors |
Pattern bridge: Quadratic programming appears in ML Math — Optimization.