A generalization of multivariate Pareto distributions: tail risk measures, divided differences and asymptotics

Harrie Hendriks and Zinoviy Landsman

Scandinavian Actuarial Journal, 2017, vol. 2017, issue 9, 785-803

Abstract: We consider a multivariate distribution of the form P(X1>x1,…,Xn>xn)=h∑i=1nλixi$ { \mathbb P }(X_{1}>x_{1},\ldots ,X_{n}>x_{n})=h\left( \sum _{i=1}^{n}\lambda _{i}x_{i}\right) $, where the survival function h is a multiply monotonic function of order (n-1)$ (n-1) $ such that h(0)=1$ h(0)=1 $, λi>0$ \lambda _{i}>0 $ for all i and λi≠λj$ \lambda _{i}\ne \lambda _{j} $ for i≠j$ i\ne j $. This generalizes work by Chiragiev and Landsman on completely monotonic survival functions. We show that the considered dependence structure is more flexible in the sense that the correlation coefficient between two components may attain negative values. We demonstrate that the tool of divided difference is very convenient for evaluation of tail risk measures and their allocations. In terms of divided differences, formulas for tail conditional expectation (tce), tail conditional variance and tce-based capital allocation are obtained. We obtain a closed form for the capital allocation of aggregate risk. Special attention is paid to survival functions h that are regularly or rapidly varying.

Date: 2017
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DOI: 10.1080/03461238.2016.1255249

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