Risk measures: a generalization from the univariate to the matrix-variate

María A. Arias-Serna; Francisco J. Caro-Lopera; Jean-Michel Loubes

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This paper proposes a method to calculate matrix-variate value-at-risk.
This paper develops a method for estimating the value-at-risk and the conditional value-at-risk when the underlying risk factors follow a beta distribution in a univariate and matrix-variate setting.
Analytical expressions of the risk measures are developed.
A numerical solution for the risk measures for any parameterization of beta distributed loss variables is presented.
Of fundamental importance is the application of computer-based algorithms for solving classically analytic problems in financial risk management. The data we acquired from Colombian financial institutions are considered using both algorithmic and analytic methods. Our results demonstrate a correspondence between the two. Although our results are motivated by problems in finance, we believe that our methods may well more general applications as well.

Abstract

This paper develops a method for estimating value-at-risk and conditional value-at-risk when the underlying risk factors follow a beta distribution in a univariate and a matrix-variate setting. For this purpose, we connect the theory of the Gaussian hypergeometric function of matrix argument and integration over positive definite matrixes. For certain choices of the shape parameters, a and b, analytical expressions of the risk measures are developed. More generally, a numerical solution for the risk measures for any parameterization of beta-distributed loss variables is presented. The proposed risk measures are finally used for quantifying the potential risk of economic loss in credit risk.

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