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

Paper: arXiv 2305.09097

Authors: Baishuai Zuo, Chuancun Yin, Jing Yao

Abstract

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.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 5.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides heavy mathematical derivations for new multivariate risk measures (MRVaR and MRCov) within the elliptical distribution family. While it includes an application to optimal portfolio selection using real stock return data, the empirical testing serves primarily as a demonstration of theoretical properties rather than an exhaustive empirical backtest.

Research Flowchart

  flowchart TD
    A[Research Goal: Develop Multivariate Range-Based Risk Measures MRVaR & MRCov for Regulatory & Investment Risk] --> B[Derive Explicit Analytical Formulas under Multivariate Elliptical & Log-Elliptical Distributions]
    B --> C[Evaluate Specific Distributions & Empirical Data<br><i>Normal, Student-t, Logistic, Laplace, Stock Returns</i>]
    C --> D[Construct Range-Based Mean-Variance Framework & Calculate Efficient Frontiers]
    D --> E[Key Findings: Range-based measures ensure robust risk management & consistent optimal portfolio selection]