Diversification and the Distribution of Portfolio Variance, Part 3: Polynomial Optimisation for Asset Allocation

Abstract

Diversification is a fundamental topic for all investors but there remains little agreement on how to measure it. Often it is defined ambiguously through risk-based portfolio construction techniques. Recently it has been suggested to connect maximising diversification with minimising risk instability, via kurtosis, which presents practical optimisation challenges. In particular, minimising kurtosis is a non-convex problem that is typically solved using deterministic Branch-and-Bound methods, that do not scale well, or stochastic methods that provide limited guarantees on finding minima. We thus apply a deterministic hierarchical polynomial optimization framework that allows realistic asset allocation problems to be readily solved and also provides a numerical certificate of optimality.

Keywords: Portfolio Diversification, Higher-Order Moments, Variance of Variance, Kurtosis, Dimensionality, Minimum Variance, Diversification Ratio, Risk Parity, Polynomial Optimization

JEL Classification: C10, C40, C49, C61, G11