Insights into robust optimization: decomposing into mean–variance and risk-based portfolios

Thomas Heckel and Raul Leote de Carvalho

Abstract

ABSTRACT

For a number of different formulations of robust portfolio optimization, quadratic and absolute, we show that

(a) in the limit of low uncertainty in estimated asset mean returns, the robust portfolio converges toward the mean-variance portfolio obtained with the same inputs, and

(b) in the limit of high uncertainty, the robust portfolio converges toward a risk-based portfolio, which is a function of how the uncertainty in estimated asset mean returns is defined.

We give examples in which the robust portfolio converges toward the minimum variance, inverse variance, equal-risk budget and equally weighted portfolios in the limit of sufficiently large uncertainty in asset mean returns. At intermediate levels of uncertainty, we find that a weighted average of the mean-variance portfolio and the respective limiting risk-based portfolio offers a good representation of the robust portfolio, particularly in the case of the quadratic formulation. The results remain valid even in the presence of portfolio constraints, in which case the limiting portfolios are the corresponding constrained mean-variance and constrained risk-based portfolios. We believe our results are important, particularly for risk-based investors who wish to take expected returns into account in order to gently tilt away from their current allocations, eg, risk parity or minimum variance.