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

Abstract

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.

Keywords:

• Multivariate elliptical distribution
• multivariate log-elliptical distribution
• multivariate range covariance
• multivariate range value-at-risk
• multivariate risk measure
• range variance

Mathematics Subject Classification 2020::

• 62H05
• 62E10

Acknowledgments

The authors are grateful for reviewers’ constructive comments and editor’s inspiring encouragement that helped them to improve their work to a significant extent.

Disclosure statement

The authors declare that they have no conflicts of interest.

Additional information

Funding

The research was supported by the National Natural Science Foundation of China (Nos. 12071251, 12371472, 12371474)