The diversification quotient (DQ) is a recently introduced tool 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 multivariate 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.