Machine Learning and Portfolio Optimization

We modify two popular methods in machine learning, regularization and cross-validation, for the portfolio optimization problem. First, we introduce performance-based regularization (PBR), where the idea is to constrain the sample variances of the estimated portfolio risk and return. The goal of PBR is to steer the solution towards one associated with less estimation error in the performance. We consider PBR for mean-variance and mean-CVaR portfolio optimization problems. For the mean-variance problem, PBR introduces a quartic polynomial constraint, from which we make two convex approximations; one based on rank-1 approximation and another based on the best convex quadratic approximation. We then analytically show that, rank-1 approximation PBR adds a bias to the optimal allocation, and the convex quadratic approximation PBR shrinks the sample covariance matrix. For the mean-CVaR problem, the PBR model is a combinatorial optimization problem, but we prove its convex relaxation is tight. We show that the PBR models can be cast as robust optimization problems and establish asymptotic optimality of both SAA and PBR solutions, and show this extends to the corresponding efficient frontiers. To calibrate the right hand sides of the PBR constraints, we develop a new, performance-based k-fold cross-validation algorithm. Using this algorithm, we carry out an extensive empirical investigation of PBR against SAA, as well as other methods, including L1, L2 regularization and the equally-weighted portfolio on three different data sets. We find that PBR dominates all other benchmarks in the literature for two widely-used data sets with statistical significance.