Asymptotic Analysis of Multivariate Tail Conditional Expectations

Tail conditional expectations refer to the expected values of random variables conditioning on some tail events and are closely related to various coherent risk measures. In the univariate case, the tail conditional expectation is asymptotically proportional to the value-at-risk, a popular risk measure. The focus of this paper is on asymptotic relations between the multivariate tail conditional expectation and value-at-risk for heavy-tailed scale mixtures of multivariate distributions. Explicit tail estimates of multivariate tail conditional expectations are obtained using the method of regular variation. Examples involving multivariate Pareto and elliptical distributions, as well as application to risk allocation are also discussed.