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

Abstract

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.