Conditional Tail Moments of the Exponential Family and Its Related Distributions

The risk measure is a central theme in the risk management literature. For good reasons, the conditional tail expectation (CTE) has received much interest in both insurance and finance applications. It provides for a measure of the expected riskiness in the tail of the loss distribution. In this article we derive explicit formulas of the CTE and higher moments for the univariate exponential family class, which extends the natural exponential family, using the canonical representation. In addition we show how to compute the conditional tail expectations of other related distributions using transformation and conditioning. Selected examples are presented for illustration, including the generalized Pareto and generalized hyperbolic distributions. We conclude that the conditional tail expectations of a wide range of loss distributions can be analytically obtained using the methods shown in this article.