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