Sparse Precision Matrices for Minimum Variance Portfolios

Abstract

Financial crises are typically characterized by highly positively correlated asset returns due to the simultaneous distress on almost all securities, high volatilities and the presence of extreme returns. In the aftermath of the 2008 crisis, investors were prompted even further to look for portfolios that minimize risk and can better deal with estimation error in the inputs of the asset allocation models. The minimum variance portfolio à la Markowitz is considered the reference model for risk minimization, due to its simplicity in the optimization as well as its need for just one input estimate: the inverse of the covariance estimate, or the so-called precision matrix. In this paper, we propose a data-driven portfolio framework that relies on two regularization methods, glasso and tlasso. They provide sparse estimates of the inverse of the covariance matrix by penalizing the 1-norm of the precision matrix relying on asset returns normality or t-Student assumptions, respectively. Simulation and actual data results support the proposed methods compared to state-of-art methods, such as random matrix and Ledoit-Wolf shrinkage.

Keywords: minimum variance, precision matrix, graphical lasso, tlasso

JEL Classification: G11, C58