Tail conditional expectations for elliptical distributions

Significant changes in the insurance and financial markets are giving in-creasing attention to the need for developing a standard framework for risk measurement. Recently, there has been growing interest among insurance and investment experts to focus on the use of a tail conditional expectation be-cause it shares properties that are considered desireable and applicable in a variety of situations. In particular, it satisfies requirements of a “coherent” risk measure in the spirit developed by Artzner, et al. (1999). In this paper, we derive explicit formulas for computing tail conditional expectations for el-liptical distributions, a family of symmetric distributions which includes the more familiar normal and student-t distributions. We extend this investiga-tion to multivariate elliptical distributions allowing us to model combinations of correlated risks. We are able to exploit properties of these distributions nat-urally permitting us to decompose the conditional expectation so that we are able to allocate contribution of individual risks to the aggregated risks. This is meaningful in practice particularly in the case of computing capital require-ments for an institution who may have several lines of correlated business and is concerned of fairly allocating the total capital to these constituents. ∗Keywords: elliptical distributions, tail VAR, tail conditional expectations, coherent risk measure. 1