Tail risk measures with application for mixtures of elliptical distributions

In this paper we derive explicit formulas of tail conditional expectation ($ \text{TCE} $) and tail variance ($ \text{TV} $) for the class of location-scale mixtures of elliptical distributions, which includes the generalized hyper-elliptical ($ \text{GHE} $) distribution. We also develop portfolio risk decomposition with $ \text{TCE} $ for multivariate location-scale mixtures of elliptical distributions. To illustrate our findings, we focus on the generalized hyperbolic ($ \text{GH} $) family which is a popular subclass of the $ \text{GHE} $ for stocks modelling.