Approximation of Some Multivariate Risk Measures for Gaussian Risks

TL;DR

This paper derives precise approximations for key multivariate risk measures of Gaussian risks, revealing their connection to conditional limit theorems and extending results to elliptical risks.

Contribution

It introduces new approximation formulas for multivariate risk measures of Gaussian vectors, linking them to conditional limit theorems and extending to elliptical risks.

Findings

Accurate approximations for marginal mean excess and expected shortfall.

Links established between risk measures and conditional limit theorems.

Results applicable to elliptical and Gaussian-like risks.

Abstract

Gaussian random vectors exhibit the loss of dimension phenomena, which relate to their joint survival tail behaviour. Besides, the fact that the components of such vectors are light-tailed complicates the approximations of various multivariate risk measures significantly. In this contribution we derive precise approximations of marginal mean excess, marginal expected shortfall and multivariate conditional tail expectation of Gaussian random vectors and highlight links with conditional limit theorems. Our study indicates that similar results hold for elliptical and Gaussian like multivariate risks.