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