On multivariate extensions of value-at-risk (Q391656)

The authors introduce two alternative extensions of the univariate value-at-risk (VaR) in a multivariate setting -- lower-orthant VaR and upper-orthant VaR. These are vector-valued measures with the same dimension as the underlying risk portfolio. The lower-orthant VaR is constructed from the level sets of multivariate distribution function. The upper-orthant VaR is based on level sets of multivariate survival functions. Positive homogeneity and translation invariance properties of both measures are derived. A comparison between univariate risk measures and components of multivariate VaR is provided. The authors also track the impact on these measures of a change in marginal distribution, in dependence structure and in risk level. Illustrations of the considered properties are provided in the class of Archimedean copulas.