On Multivariate Extensions of Value-at-Risk

TL;DR

This paper introduces two new multivariate extensions of Value-at-Risk that are vector-valued, satisfy key properties, and are analyzed under various distributional changes with illustrations using Archimedean copulas.

Contribution

The paper proposes two novel multivariate VaR measures based on level sets of distribution and survival functions, extending univariate VaR to multivariate contexts.

Findings

Both measures satisfy positive homogeneity and translation invariance.

The measures' sensitivities to distributional changes are analyzed.

Illustrations demonstrate behavior under Archimedean copulas.

Abstract

In this paper, we introduce two alternative extensions of the classical univariate Value-at-Risk (VaR) in a multivariate setting. The two proposed multivariate VaR are vector-valued measures with the same dimension as the underlying risk portfolio. The lower-orthant VaR is constructed from level sets of multivariate distribution functions whereas the upper-orthant VaR is constructed from level sets of multivariate survival functions. Several properties have been derived. In particular, we show that these risk measures both satisfy the positive homogeneity and the translation invariance property. Comparison between univariate risk measures and components of multivariate VaR are provided. We also analyze how these measures are impacted by a change in marginal distributions, by a change in dependence structure and by a change in risk level. Illustrations are given in the class of Archimedean copulas.