Performance-based regularization in mean-CVaR portfolio optimization

Regularization is a technique widely used to improve the stability of solutions to statistical problems. We propose a new regularization concept, performance-based regularization (PBR), for data-driven stochastic optimization. The goal is to improve upon Sample Average Approximation (SAA) in finite-sample performance while maintaining minimal assumptions about the data. We apply PBR to mean-CVaR portfolio optimization, where we penalize portfolios with large variability in the constraint and objective estimations, which constrains the probabilities that the estimations deviate from the respective true values. This results in a combinatorial optimization problem, but we prove its convex relaxation is tight. We prove PBR is asymptotically optimal, and derive its first-order behavior by extending the theory of M-estimators. To calibrate the constraint right-hand side, we develop a new, performance-based k-fold cross-validation algorithm. An extensive empirical investigation demonstrates that PBR can improve upon SAA and standard regularization methods in the out-of-sample Sharpe ratio with statistical significance.