Sparse Portfolio Selection via Quasi-Norm Regularization

TL;DR

This paper introduces $ ext{l}_p$-norm regularized models for sparse portfolio selection, providing theoretical guarantees, an efficient algorithm, and empirical evidence that these models produce competitive, sparse portfolios with controlled overfitting.

Contribution

The paper develops a novel $ ext{l}_p$-norm regularization framework for sparse portfolios, along with a polynomial-time interior point algorithm and theoretical insights into sparsity and overfitting.

Findings

$ ext{l}_p$-norm models generate sparse portfolios with comparable performance to traditional models.

Sparsity is shown to indirectly moderate overfitting, not directly.

Combined $ ext{l}_1$-$ ext{l}_p$ and $ ext{l}_2$-$ ext{l}_p$ models outperform standard strategies.

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.