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

Harrie Hendriks, Zinoviy Landsman

Research output: Contribution to journalArticlepeer-review

Abstract

We consider a multivariate distribution of the form ℙ (X1 > x1,...,Xn > xn), where the survival function h is a multiply monotonic function of order (n - 1) such that h(0) = 1, λi > 0 for all i and λi ≠ λj for i ≠ 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.

Original languageEnglish
Pages (from-to)785-803
Number of pages19
JournalScandinavian Actuarial Journal
Volume2017
Issue number9
DOIs
StatePublished - 21 Oct 2017

Bibliographical note

Publisher Copyright:
© 2016 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.

Keywords

  • divided differences
  • Exponential distribution
  • Pareto distribution
  • rapid variation
  • regular variation
  • tail conditional correlation
  • tail conditional expectation

ASJC Scopus subject areas

  • Statistics and Probability
  • Economics and Econometrics
  • Statistics, Probability and Uncertainty

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