  In this paper, 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. In particular, we explain that
such range-based risk measures are appropriate for risk management of
regulation and investment purposes. The multivariate range correlation matrix
(MRCorr) is introduced accordingly. 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 of optimal portfolio selection.
We calculate the range-based efficient frontiers of the optimal portfolios
based on real data of stocks' returns. Both the numerical examples and the
efficient frontiers demonstrate consistences with the desirable properties of
the range-based risk measures.
