Sparse Portfolio Selection via Quasi-Norm Regularization

Abstract

In this paper, we propose ${\ell }_p$-norm regularized models to seek near-optimal sparse portfolios. These sparse solutions reduce the complexity of portfolio implementation and management. Theoretical results are established to guarantee the sparsity of the second-order KKT points of the ${\ell }_p$-norm regularized models. More interestingly, we present a theory that relates sparsity of the KKT points with Projected correlation and Projected Sharpe ratio. We also design an interior point algorithm to obtain an approximate second-order KKT solution of the ${\ell }_p$-norm models in polynomial time with a fixed error tolerance, and then test our ${\ell }_p$-norm modes on S&P 500 (2008-2012) data and international market data.\ The computational results illustrate that the ${\ell }_p$-norm regularized models can generate portfolios of any desired sparsity with portfolio variance and portfolio return comparable to those of the unregularized Markowitz model with cardinality constraint. Our analysis of a combined model lead us to conclude that sparsity is not directly related to overfitting at all. Instead, we find that sparsity moderates overfitting only indirectly. A combined ${\ell }_1$-${\ell }_p$ model shows that the proper choose of leverage, which is the amount of additional buying-power generated by selling short can mitigate overfitting; A combined ${\ell }_2$-${\ell }_p$ model is able to produce extremely high performing portfolios that exceeded the 1/N strategy and all ${\ell }_1$ and ${\ell }_2$ regularized portfolios.

34 pages,7 figures

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