A generalized precision matrix for t-Student distributions in portfolio optimization

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 input, however little attention has been given to the limitations of the inverse covariance matrix when capturing the dependence structure in a non-Gaussian setting. While the precision matrix allows to correctly understand the conditional dependence structure of random vectors in a Gaussian setting, the inverse of the covariance matrix might not necessarily result in a reliable source of information when Gaussianity fails. In this paper, exploiting the local dependence function, different definitions of the generalized precision matrix (GPM), which holds for a general class of distributions, are provided. In particular, we focus on the multivariate t-Student distribution and point out that the interaction in random vectors does not depend only on the inverse of the covariance matrix, but also on additional elements. We test the performance of the proposed GPM using a minimum-variance portfolio set-up by considering S\&P 100 and Fama and French industry data. We show that portfolios relying on the GPM often generate statistically significant lower out-of-sample variances than state-of-art methods