Diversification Quotients: Quantifying Diversification via Risk Measures

We establish the first axiomatic theory for diversification indices using six intuitive axioms: nonnegativity, location invariance, scale invariance, rationality, normalization, and continuity. The unique class of indices satisfying these axioms, called the diversification quotients (DQs), are defined based on a parametric family of risk measures. A further axiom of portfolio convexity pins down DQs based on coherent risk measures. The DQ has many attractive properties, and it can address several theoretical and practical limitations of existing indices. In particular, for the popular risk measures value at risk and expected shortfall, the corresponding DQ admits simple formulas, and it is efficient to optimize in portfolio selection. Moreover, it can properly capture tail heaviness and common shocks, which are neglected by traditional diversification indices. When illustrated with financial data, the DQ is intuitive to interpret, and its performance is competitive against other diversification indices. This paper was accepted by Manel Baucells, behavioral economics and decision analysis. Funding: X. Han is supported by the National Natural Science Foundation of China [Grants 12301604, 12371471, and 12471449). L. Lin is supported by the Hickman Scholarship from the Society of Actuaries. R. Wang is supported by the Natural Sciences and Engineering Research Council of Canada [Grants CRC-2022-00141 and RGPIN-2024-03728] and the Sun Life Research Fellowship. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.00513 .