The diversification quotient (DQ) is recently introduced for quantifying the degree of diversification of a stochastic portfolio model. It has an axiomatic foundation and can be defined through a parametric class of risk measures. Since the Value-at-Risk (VaR) and the Expected Shortfall (ES) are the most prominent risk measures widely used in both banking and insurance, we investigate DQ constructed from VaR and ES in this paper. In particular, for the popular models of elliptical and multivariate regular varying (MRV) distributions, explicit formulas are available. The portfolio optimization problems for the elliptical and MRV models are also studied. Our results further reveal favorable features of DQ, both theoretically and practically, compared to traditional diversification indices based on a single risk measure.

最近引入了多元化商数（DQ）来量化随机投资组合模型的多元化程度。它具有公理基础，可以通过参数类风险度量来定义。由于风险价值（VaR）和预期缺口（ES）是银行业和保险业广泛使用的最重要的风险衡量指标，因此我们在本文中研究了由 VaR 和 ES 构建的 DQ。特别是，对于流行的椭圆和多元正则变化 (MRV) 分布模型，可以使用显式公式。还研究了椭圆模型和 MRV 模型的投资组合优化问题。与基于单一风险度量的传统多元化指数相比，我们的结果进一步揭示了 DQ 在理论上和实践上的有利特征。