A Generalized Precision Matrix for Multivariate T-Student and Skew Distributions in Portfolio Optimization

Abstract

The Markowitz model is still the cornerstone of modern portfolio theory. In particular, when focusing on the minimum-variance portfolio, the covariance matrix or better its inverse, the so-called precision matrix, is the only input required. So far, most scholars worked on improving the estimation of the inverse of the covariance matrix, however little attention has been given to its limitations in capturing the dependence structure in the data in a non-Gaussian setting. In this paper, exploiting a local dependence function, a definitions of a generalized precision matrix (GPM), which holds for a general class of distributions, is introduced. Applications are provided for the multivariate t, multivariate skew-normal and multivariate skew-t distributions. We test then the performance of the proposed GPM on simulated and real-world financial data. As expected, the multivariate skew-t model seems a better fit to crisis periods.

Keywords: Generalized Precision Matrix, heavy tails, multivariate t distribution, multivariate skew-normal and skew-t distributions, minimum-variance portfolio

JEL Classification: C46, C58, G11