A new framework for portfolio diversification is introduced which goes beyond the classical mean-variance theory and other known portfolio allocation strategies such as risk parity. It is based on a novel concept called portfolio dimensionality and ultimately relies on the minimization of ratios of convex functions . The latter arises naturally due to our requirements that diversification measures should be leverage invariant and related to the tail properties of the distribution of portfolio returns. This paper introduces this new framework and its relationship to standardized higher order moments of portfolio returns. Moreover, it addresses the main drawbacks of standard diversification methodologies which are based primarily on estimates of covariance matrices. Maximizing portfolio dimensionality leads to highly non-trivial optimization problems with objective functions which are typically non-convex with potentially multiple local optima. Two complementary global optimization algorithms are thus presented. For problems of moderate size  , 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  through numerical experiments that the introduced diversification measures possess desired properties as introduced in the portfolio diversification literature.
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.
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