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.

翻译：本文提出多元范围风险价值（MRVaR）与多元范围协方差（MRCov）两种风险度量，并探索其在风险管理中的优良性质。特别地，我们阐明此类基于范围的风险度量适用于监管与投资目的的风险管理。据此引入多元范围相关系数矩阵（MRCorr）。为便于解析分析，我们在多元（对数）椭圆分布族框架下推导了MRVaR与MRCov的显式表达式。针对行业常用情形，如正态分布、学生-t分布、逻辑分布、拉普拉斯分布及皮尔逊VII型分布，辅以数值算例进行说明。作为应用，我们提出基于范围的最优投资组合均值-方差选择框架。基于股票实际收益率数据，计算了最优投资组合的范围有效前沿。数值算例与有效前沿均验证了此类范围风险度量优良性质的一致性。