Sparse Precision Matrices for Minimum Variance Portfolios

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 `{a} la Markowitz is considered the reference model for risk minimization in equity markets, 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 based on two regularization methods, extit{glasso} and extit{tlasso}, that provide sparse estimates of the precision matrix by penalizing its $L_1$-norm. extit{Glasso} and extit{tlasso} rely on asset returns Gaussianity or t-Student assumptions, respectively. Simulation and real-world data results support the proposed methods compared to state-of-art approaches, such as random matrix and Ledoit-Wolf shrinkage.