Diversification quotients: Quantifying diversification via risk measures

TL;DR

This paper introduces diversification quotients (DQs), a new axiomatic framework for quantifying diversification using risk measures, addressing limitations of existing indices and applicable to popular risk metrics like VaR and Expected Shortfall.

Contribution

It establishes the first axiomatic theory for diversification indices, defining DQs based on risk measures and demonstrating their theoretical and practical advantages.

Findings

DQs satisfy six intuitive axioms and are uniquely characterized by them.

DQs based on VaR and Expected Shortfall have simple formulas and are computationally efficient.

DQs effectively capture tail risks and common shocks, outperforming traditional indices.

Abstract

We establish the first axiomatic theory for diversification indices using six intuitive axioms: non-negativity, 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 DQ based on coherent risk measures. 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, DQ is intuitive to interpret, and its performance is competitive against other diversification indices.