Relations between Kendall distributions and families of VaR curves in bivariate survival models

We consider the two concepts of Multivariate Value at Risk and Kendall distribution function. Attention will be restricted to the case of n = 2 non-negative, exchangeable random variables; furthermore, conditions are assumed under which, for any level v ∈ (0, 1), the bivariate Value at Risk (VaR) is a curve within the quadrant R2+. As main purposes, we study the relations between the two concepts; in particular, we focus attention on the survival models simultaneously admitting a specified Kendall distribution and a specified family of bivariate VaR’s. Furthermore we obtain some conditions of qualitative type in terms of the notions of IFRA and PKD dependence. For our results we combine two different methods. On the one hand we use a slight extension of a transformation result proved by Genest and Rivest (2001); on the other hand we resort to the bivariate aging functions introduced by Bassan and Spizzichino (2001) to represent the family of the level curves of a bivariate survival function. Different aspects of our results will be outlined by considering in some detail the cases of Archimedean and Cuadras-Auge ́ survival copulas