Overcoming Markowitz's Instability with the Help of the Hierarchical Risk Parity (HRP): Theoretical Evidence

Abstract

In this paper we compare two methods of portfolio allocation: the classical Markowitz one and the hierarchical risk parity (HRP) approach. We derive analytical values for the noise of allocation weights coming from the estimated covariance. We demonstrate that the HRP is indeed less noisy (and thus more robust) w.r.t. the classical Markowitz.

The second part of the paper is devoted to a detailed analysis of the optimal portfolio variance for which we derive analytical formulas and theoretically demonstrate the superiority of the HRP w.r.t to the Markowitz optimization.

We also address practical outcomes of our analytics. The first one is a fast estimation of the confidence level of the optimization weights calculated for a single (real-life) scenario. The second practical usefulness of the analytics is an HRP portfolio construction criterion which selects assets and clusters minimizing the analytical portfolio variance. We confirm our theoretical results with numerous numerical experiments. Our calculation technique can be also used in other areas of portfolio optimization.

Keywords: Trading strategy, portfolio allocation, Markowitz optimisation, Hierarchical Risk Parity, Clustered Optimisation, optimisation weights noise

JEL Classification: G0, G1, G2, G11, E44, E47