Diversification Quotients: Quantifying Diversification via Risk Measures

To overcome several limitations of existing diversification indices, we introduce the diversification quotient (DQ). Defined through a parametric family of risk measures, DQ satisfies three natural properties, namely, non-negativity, location invariance and scale invariance, which are shown to be conflicting for any traditional diversification index based on a single risk measure. We pay special attention to the two most important classes of risk measures in banking and insurance, the Value-at-Risk (VaR) and the Expected Shortfall (ES, also called CVaR). DQs based on VaR and ES enjoy many convenient technical properties, and they are efficient to optimize in portfolio selection. By analyzing the popular multivariate models of elliptical and regular varying distributions, we find that DQ 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 when contrasted with other diversification methods in portfolio optimization.