Sparse Markowitz Portfolio Selection by Penalty Methods

Abstract

We consider the framework of the classical Markowitz mean-variance model when multiple solutions exist, among which the sparse solutions are stable and cost-efficient. We study a least-$p$-norm sparse portfolio model with $p\in(0,1)$ solved by the penalty method. This model finds the least-$p$-norm sparse asset allocation in the solution set of the Markowitz problem, which saves the transaction cost and stabilizes the optimization problem. We apply the sample average approximation (SAA) method to the least-$p$-norm sparse portfolio model and give a detailed convergence analysis. We implement this method on the data sets of 20 A\&H stocks, Fama \& French 12 industry sectors (FF12), and Fama \& French 25 portfolios formed on size and book-to-market (FF25). Using portfolios constructed in the training sample, we test them in the out-of-sample data and find their Sharpe ratios outperform the $0$-norm sparse portfolio, $\ell_1$ penalty regularized portfolios, cardinality constrained portfolios, and $1/N$ investment strategy.