Tail conditional risk measures for location-scale mixture of elliptical distributions

Abstract

We present general results on the univariate tail conditional expectation (TCE) and multivariate tail conditional expectation (MTCE) for location-scale mixture of elliptical distributions. Examples include the location-scale mixture of normal distributions, location-scale mixture of Student-$t$ distributions, location-scale mixture of logistic distributions and location-scale mixture of Laplace distributions. We also consider portfolio risk decomposition with TCE for location-scale mixture of elliptical distributions. More specifically, we give MTCEs of generalized hyperbolic and slash distributions, and discuss the difference of MTCEs for generalized hyperbolic and slash distributions. As an illustrative example, we discuss the MTCE of five stocks including Amazon, Goldman Sachs, IBM, Google and Apple.

Keywords:

• Tail conditional expectations
• portfolio allocations
• multivariate risk measures
• location-scale mixture
• elliptical distributions

Acknowledgements

The authors thank the anonymous referees and the Editor for their helpful comments and suggestions, which have led to the improvement of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research was supported by the National Natural Science Foundation of China [grant numbers 12071251, 11571198, 11701319].