A new coherent multivariate average-value-at-risk

Abstract

A new operator for handling the joint risk of different sources has been presented and its various properties are investigated. The problem of risk evaluation of multivariate risk sources has been studied, and a multivariate risk measure, so-called multivariate average-value-at-risk, ${\mathrm{m}\mathrm{A}\mathrm{V}\mathrm{a}\mathrm{R}}_{\mathit{\alpha }}$, is proposed to quantify the total risk. It is shown that the proposed operator satisfies the four axioms of a coherent risk measure while reducing to one variable average-value-at-risk, ${\mathsf{A}\mathsf{V}\mathsf{a}\mathsf{R}}_{\alpha }$, in case N = 1. In that respect, it is shown that ${\mathrm{m}\mathrm{A}\mathrm{V}\mathrm{a}\mathrm{R}}_{\mathit{\alpha }}$ is the natural extension of ${\mathsf{A}\mathsf{V}\mathsf{a}\mathsf{R}}_{\alpha }$ to N-dimensional case maintaining its axiomatic properties. We further show ${\mathrm{m}\mathrm{A}\mathrm{V}\mathrm{a}\mathrm{R}}_{\mathit{\alpha }}$ is flexible by giving the investor the option to choose the risk level ${\alpha }_i$ of each random loss i differently. This flexibility is novel and can not be achieved applying univariate ${\mathsf{A}\mathsf{V}\mathsf{a}\mathsf{R}}_{\alpha }$ with corresponding risk level α to the sum of the risk marginals. The framework is applicable for Gaussian mixture models with dependent risk factors that are naturally used in financial and actuarial modelling. A multivariate tail variance and its connection with ${\mathrm{m}\mathrm{A}\mathrm{V}\mathrm{a}\mathrm{R}}_{\mathit{\alpha }}$ is also presented via Chebyshev inequality for tail events. Examples with numerical simulations are also illustrated throughout.

Keywords:

• Financial risk management
• multivariate coherent risk measures
• average-value-at-risk

2010 Mathematics Subject Classifications:

• 91G10
• 91G40

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported by Nazarbayev University [grant number SSH2020016].