Tail Conditional Expectations for Elliptical Distributions

TLDR

Significant changes in insurance and financial markets have heightened the need for a standard risk‑measurement framework, and tail conditional expectation is increasingly favored as a coherent risk measure. The study derives explicit formulas for tail conditional expectations of elliptical distributions, including normal and student‑t, and extends the analysis to multivariate cases to model correlated risks. By exploiting properties of elliptical distributions, the authors decompose the conditional expectation and allocate individual risk contributions to aggregate risks. These results enable institutions with multiple correlated business lines to compute capital requirements and fairly allocate total capital among constituents. © 1999.

Abstract

Abstract Significant changes in the insurance and financial markets are giving increasing 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 because it shares properties that are considered desirable 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). This paper derives explicit formulas for computing tail conditional expectations for elliptical distributions, a family of symmetric distributions that includes the more familiar normal and student-t distributions. The authors extend this investigation to multivariate elliptical distributions allowing them to model combinations of correlated risks. They are able to exploit properties of these distributions, naturally permitting them to decompose the conditional expectation, and allocate the contribution of individual risks to the aggregated risks. This is meaningful in practice, particularly in the case of computing capital requirements for an institution that may have several lines of correlated business and is concerned about fairly allocating the total capital to these constituents.