A new framework for portfolio diversification is introduced which goes beyond
the classical mean-variance approach and portfolio allocation strategies such
as risk parity. It is based on a novel concept called portfolio dimensionality
that connects diversification to the non-Gaussianity of portfolio returns and
can typically be defined in terms of the ratio of risk measures which are
homogenous functions of equal degree. The latter arises naturally due to our
requirement that diversification measures should be leverage invariant. We
introduce this new framework and argue the benefits relative to existing
measures of diversification in the literature, before addressing the question
of optimizing diversification or, equivalently, dimensionality. Maximising
portfolio dimensionality leads to highly non-trivial optimization problems with
objective functions which are typically non-convex and potentially have
multiple local optima. Two complementary global optimization algorithms are
thus presented. For problems of moderate size and more akin to asset allocation
problems, a deterministic Branch and Bound algorithm is developed, whereas for
problems of larger size a stochastic global optimization algorithm based on
Gradient Langevin Dynamics is given. We demonstrate analytically and through
numerical experiments that the framework reflects the desired properties often
discussed in the literature
