  The diversification quotient (DQ) is recently introduced for quantifying the
degree of diversification of a stochastic portfolio model. It has an axiomatic
foundation and can be defined through a parametric class of risk measures.
Since the Value-at-Risk (VaR) and the Expected Shortfall (ES) are the most
prominent risk measures widely used in both banking and insurance, we
investigate DQ constructed from VaR and ES in this paper. In particular, for
the popular models of elliptical and multivariate regular varying (MRV)
distributions, explicit formulas are available. The portfolio optimization
problems for the elliptical and MRV models are also studied. Our results
further reveal favourable features of DQ, both theoretically and practically,
compared to traditional diversification indices based on a single risk measure.
