Tail variance for generalised hyper-elliptical models

This paper introduces a novel theoretical framework that offers a closed-form expression for the tail variance (TV) for the novel family of generalised hyper-elliptical (GHE) distributions. The GHE family combines an elliptical distribution with the generalised inverse Gaussian (GIG) distribution, resulting in a highly adaptable and powerful model. Expanding upon the findings of Ignatieva and Landsman ((2021) Insurance: Mathematics and Economics, 101, 437-465.) regarding the tail conditional expectation (TCE), this study demonstrates the significance of the TV as an additional risk measure that provides valuable insights into the tail risk and effectively captures the variability within the loss distribution's tail. To validate the theoretical results, we perform an empirical analysis on two specific cases: the Laplace - GIG and the Student-t - GIG mixtures. By incorporating the TV derived for the GHE family, we are able to quantify correlated risks in a multivariate portfolio more efficiently. This contribution is particularly relevant to the insurance and financial industries, as it offers a reliable method for accurately assessing the risks associated with extreme losses. Overall, this paper presents an innovative and rigorous approach that enhances our understanding of risk assessment within the financial and insurance sectors. The derived expressions for the TV in addition to TCE within the GHE family of distributions provide valuable insights and practical tools for effectively managing risk.