Conditional tail risk measures for the skewed generalised hyperbolic family

Katja Ignatieva, Zinoviy Landsman

Research output: Contribution to journalArticlepeer-review


This paper deals with the estimation of loss severity distributions arising from historical data on univariate and multivariate losses. We present an innovative theoretical framework where a closed-form expression for the tail conditional expectation (TCE) is derived for the skewed generalised hyperbolic (GH) family of distributions. The skewed GH family is especially suitable for equity losses because it allows to capture the asymmetry in the distribution of losses that tends to have a heavy right tail. As opposed to the widely used Value-at-Risk, TCE is a coherent risk measure, which takes into account the expected loss in the tail of the distribution. Our theoretical TCE results are verified for different distributions from the skewed GH family including its special cases: Student-t, variance gamma, normal inverse gaussian and hyperbolic distributions. The GH family and its special cases turn out to provide excellent fit to univariate and multivariate data on equity losses. The TCE risk measure computed for the skewed family of GH distributions provides a conservative estimator of risk, addressing the main challenge faced by financial companies on how to reliably quantify the risk arising from the loss distribution. We extend our analysis to the multivariate framework when modelling portfolios of losses, allowing the multivariate GH distribution to capture the combination of correlated risks and demonstrate how the TCE of the portfolio can be decomposed into individual components, representing individual risks in the aggregate (portfolio) loss.

Original languageEnglish
Pages (from-to)98-114
Number of pages17
JournalInsurance: Mathematics and Economics
StatePublished - May 2019

Bibliographical note

Publisher Copyright:
© 2019 Elsevier B.V.


  • Conditional tail risk measures
  • Generalized Inverse Gaussian distribution
  • Portfolio allocation
  • Skewed generalised hyperbolic distributions
  • Tail conditional expectation

ASJC Scopus subject areas

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


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