High-dimensional Portfolio Selection via an $\ell _0$-Constrained Regression

Abstract

The Markowitz mean-variance model is a foundational tool for portfolio allocation, designed to minimize risk for a given return and budget constraint. However, traditional methods like the plug-in portfolio can be unstable, especially in high-dimensional settings where the number of assets significantly exceeds the sample size. To address this, we propose a new unconstrained regression model equivalent to the Markowitz mean-variance optimization problem but with an essential constraint: the sum of portfolio weights equals 1, incorporating the $\ell _0$ constraint to enhance sparsity. By estimating the mean and variance using the Fama-French Factor Model, we demonstrate the convergence of these parameters to their true values. Our sparse regression model asymptotically achieves minimum risk while satisfying maximum expected return and budget constraints under mild assumptions and effectively controls the number of selected assets. Simulations and empirical analyses highlight the efficacy of our approach in high-dimensional contexts, consistently selecting fewer stocks and often achieving higher returns and Sharpe ratios compared to traditional methods.

Keywords

High-dimensional portfolio selection / Mean-variance portfolio / Budget constraint / Fama-french factor model / $\ell _0$ constraint')">$\ell _0$ constraint / 62P05

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Funding

National Natural Science Foundation of China(72171216)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature