A novel multivariate risk measure: the Kendall VaR

The definition of multivariate Value at Risk is a challenging problem, whose most common solutions are given by the lower- and upper-orthant VaRs, which are based on copulas: the lower-orthant VaR is indeed the quantile of the multivariate distribution function, whereas the upper-orthant VaR is the quantile of the multivariate survival function. In this paper we introduce a new multivariate Value at Risk, referred to as the Kendall Value at Risk, which linkd the copula approach to an alternative definition of multivariate quantiles, known as the quantile surface, which is not used in finance, to our knowledge. We more precisely transform the notion of orthant VaR tanks to the Kendall function so as to get a multivariate VaR, that is to say a set of loss vectors, with some advantageous properties compared to the standard orthant VaR: i/ it is based on a total order, ii/ the probability level of the VaR is consistent with the probability measure of the set of the more severe loss vectors, iii/ the d-dimensional Vars based on the distribution function or on the survival function have vectors in common, which conciliate both upper- and lower-orthant approaches. We quantify the differences between this new Kendall VaR and orthant VaRs. In particular, we show that Kendall VaRs are less (respectively more) conservative than lower-orthant (resp. upper-orthant) VaRs. the definition and the properties of the Kendall VaR are illustrated using Gumbel and Clayton copulas with lognormal marginal distributions and several levels of risk