Robustifying Markowitz

TL;DR

This paper introduces a robust optimization method for Markowitz portfolios that reduces estimation error and transaction costs by stabilizing weights using a robust gradient descent approach and median-of-means estimators, confirmed through empirical equity market studies.

Contribution

It presents a novel robustification toolbox for Markowitz portfolios employing projected gradient descent and median-of-means estimators to improve stability and reduce transaction costs.

Findings

Robustified portfolios have lower turnover than shrinkage and constrained portfolios.

The approach maintains or slightly improves out-of-sample performance.

Empirical studies confirm the method's effectiveness on equity markets.

Abstract

Markowitz mean-variance portfolios with sample mean and covariance as input parameters feature numerous issues in practice. They perform poorly out of sample due to estimation error, they experience extreme weights together with high sensitivity to change in input parameters. The heavy-tail characteristics of financial time series are in fact the cause for these erratic fluctuations of weights that consequently create substantial transaction costs. In robustifying the weights we present a toolbox for stabilizing costs and weights for global minimum Markowitz portfolios. Utilizing a projected gradient descent (PGD) technique, we avoid the estimation and inversion of the covariance operator as a whole and concentrate on robust estimation of the gradient descent increment. Using modern tools of robust statistics we construct a computationally efficient estimator with almost Gaussian properties based on median-of-means uniformly over weights. This robustified Markowitz approach is confirmed by empirical studies on equity markets. We demonstrate that robustified portfolios reach the lowest turnover compared to shrinkage-based and constrained portfolios while preserving or slightly improving out-of-sample performance.