  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.
