Tail Conditional Variance for Elliptically Contoured Distributions

Abstract

Abstract. The tail conditional expectation, TCE for short, provides a measure of the riskiness of the tail of a distribution and is an index that has gained popularity over the years. On the other hand, the tail conditional variance, TCV for short, is lesser known but provides a measure of the variability of the risk along the tail of its distribution. Landsman and Valdez (2003) derive explicit formulas for computing tail conditional expectations for elliptical distributions, a family of symmet-ric distributions which includes the more familiar Normal and Student-t distributions. In this paper, we are able to similarly exploit the properties of the elliptical distributions to derive similar explicit forms in computing the tail conditional vari-ance. In particular, the tail generator defined in the paper plays an important role in the process of developing these ex-plicit forms. We further investigate these results in the mul-tivariate case especially when the addition of several risks is concerned.

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