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 approach introducing a total-order multivariate Value at Risk, referred to as the Kendall Value at Risk, which links 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 thanks to the Kendall function so as to get a multivariate VaR with some advantageous properties compared to the standard orthant VaR: it is based on a total order and, for a non-atomic and Rd-supported density function, there is no distinction anymore between the d-dimensional VaRs based on the distribution function or on the survival function. We quantify the differences between this new kendall VaR and orthant VaRs. In particular, we show that the Kendall VaR is less (respectively more) conservative than the lower-orthant (resp. upper-orthant) VaR. 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.