Tail risk measures with application for mixtures of elliptical distributions

Pingyun Li; Chuancun Yin; Pingyun Li; Chuancun Yin

doi:10.3934/math.2022491

AIMS Mathematics

2022, Volume 7, Issue 5: 8802-8821. doi: 10.3934/math.2022491

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Research article

Tail risk measures with application for mixtures of elliptical distributions

Pingyun Li ,
Chuancun Yin ^,

School of Statistics and Data Science, Qufu Normal University, Qufu, Shandong 273165, China

Received: 10 August 2021 Revised: 22 February 2022 Accepted: 23 February 2022 Published: 04 March 2022
MSC : 62E10, 62H05

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.
- tail conditional expectations,
- tail variance,
- portfolio risk decomposition with $ \text{TCE} $,
- location-scale mixtures,
- elliptical distributions
Citation: Pingyun Li, Chuancun Yin. Tail risk measures with application for mixtures of elliptical distributions[J]. AIMS Mathematics, 2022, 7(5): 8802-8821. doi: 10.3934/math.2022491

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Abstract

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.

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