Conditional Tail Risk Measures for Skewed Generalised Hyperbolic Family

This paper deals with the estimation of loss severity distribution arising from the historical data on univariate and multivariate losses. We present an innovative theoretical framework where the closed-form expression for the tail conditional expectation (TCE) is derived for the skewed general hyperbolic (GH) family of distributions. Skewed GH family is especially suitable for right-skewed positive random variables such as insurance losses. It allows to capture the asymmetry in the distribution of losses that tends to have a heavy right tail. As opposed to a 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 general cases: Student-t, variance gamma, normal inverse gaussian and hyperbolic distributions. GH family and its special cases turn out to provide excellent fit to univariate and multivariate data on Danish fire losses. TCE risk measure computed for the skewed family of GH distributions provides conservative estimators of risk, addressing the main challenge faced by insurance companies on how to reliably quantify the risk arising from the loss distribution. We extend our analysis to the multivariate framework when modelling portfolio of losses, allowing multivariate GH distribution to capture the combination of correlated risks and demonstrate how TCE of the portfolio can be decomposed into individual components, representing individual risk in the aggregate (portfolio) loss