A new coherent multivariate average-value-at-risk

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, m AVaR(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, AVaR(alpha), in case N = 1. In that respect, it is shown that mAVaR(alpha) is the natural extension of AVaR(alpha) to N-dimensional case maintaining its axiomatic properties. We further show mAVaR(alpha) is flexible by giving the investor the option to choose the risk level ai of each random loss i differently. This flexibility is novel and can not be achieved applying univariate AVaR(alpha) with corresponding risk level a 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 m AVaR(alpha) is also presented via Chebyshev inequality for tail events. Examples with numerical simulations are also illustrated throughout.

Automated summary

A new operator for handling joint risk of different sources has been presented and investigated. It proposes a multivariate risk measure called multivariate average-value-at-risk, m AVaR(alpha), to quantify total risk. The operator satisfies the four axioms of a coherent risk measure and can be simplified to the conventional average-value-at-risk in certain cases. It also offers flexibility by allowing investors to choose different risk levels for each random loss. The research explores the connection between multivariate tail variance and m AVaR(alpha) using the Chebyshev inequality for tail events.