Multivariate range Value-at-Risk and covariance risk measures for elliptical and log-elliptical distributions

In this article, we propose the multivariate range Value-at-Risk (MRVaR) and the multivariate range covariance (MRCov) as two risk measures and explore their desirable properties in risk management. To facilitate analytical analyses, we derive explicit expressions of the MRVaR and the MRCov in the context of the multivariate (log-)elliptical distribution family. Frequently-used cases in industry, such as normal, student-t, logistic, Laplace, and Pearson type VII distributions, are presented with numerical examples. As an application, we propose a range-based mean-variance framework for optimal portfolio selection.