A class of generalised hyper-elliptical distributions and their applications in computing conditional tail risk measures

Katja Ignatieva, Zinoviy Landsman

Research output: Contribution to journalArticlepeer-review


This paper introduces a new family of Generalised Hyper-Elliptical (GHE) distributions providing further generalisation of the generalised hyperbolic (GH) family of distributions, considered in Ignatieva and Landsman (2019). The GHE family is constructed by mixing an elliptical distribution with a Generalised Inverse Gaussian (GIG) distribution. We present an innovative theoretical framework where a closed form expression for the tail conditional expectation (TCE) is derived for this new family of distributions. We demonstrate that the GHE family is especially suitable for heavy-tailed insurance losses data. Our theoretical TCE results are verified for two special cases, Laplace - GIG and Student-t - GIG mixtures. Both mixtures are shown to outperform the GH distribution, providing excellent fit to univariate and multivariate insurance losses data. The TCE risk measure computed for the GHE family of distributions provides a more conservative estimator of risk in the extreme tail, addressing the main challenge faced by financial companies on how to reliably quantify the risk arising from extreme losses. Our multivariate analysis allows to quantify correlated risks by means of the GHE family: the TCE of the portfolio is decomposed into individual components, representing individual risks in the aggregate loss.

Original languageEnglish
Pages (from-to)437-465
Number of pages29
JournalInsurance: Mathematics and Economics
StatePublished - Nov 2021

Bibliographical note

Funding Information:
This research was supported by the Israel Science Foundation (Grant N 1686/17 ).

Publisher Copyright:
© 2021 Elsevier B.V.


  • Conditional tail risk measures
  • Generalised Hyper-Elliptical (GHE) distributions
  • Generalised Inverse Gaussian distribution
  • Portfolio allocation
  • Tail conditional expectation

ASJC Scopus subject areas

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


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