Sparsity and Performance Enhanced Markowitz Portfolios Using Second-Order Cone Programming

A mixed-integer second order cone program (MISOCP) formulation is proposed for solving Markowitz’s asset portfolio construction problem under a cardinality constraint. Compared with a standard alternative big-M linearly constrained formulation, our reformulation is solved significantly faster using state-of-the-art integer programming solvers. We consider learning methods that are based on the MISCOP formulation: cardinality-constrained Markowitz (CCM) solves the MISCOP for a given cardinality k and training set data of asset returns. We also find reinforcing evidence for factor model theory in the selection of factors to form optimal CCM portfolios. For large datasets in the absence of a hard-cardinality constraint, we propose a method (CCM-R) that is based on the continuous relaxation of our MISCOP, where k selected by rolling time window validation. In predictive performance experiments, based on historical stock exchange data, our learning methods usually outperform a competing extension of the Markowitz model that penalizes the L1 norm of asset weights.