Non-linear shrinkage of the price return covariance matrix is far from optimal for portfolio optimisation

TL;DR

This paper challenges the effectiveness of non-linear shrinkage for covariance matrix estimation in portfolio optimization, proposing a new optimal target that better accounts for non-stationary asset dependencies, leading to potential improvements.

Contribution

It introduces a new optimal covariance estimator tailored for non-stationary conditions, surpassing the traditional non-linear shrinkage method in portfolio optimization.

Findings

Non-linear shrinkage is not optimal for non-stationary asset dependencies.

The proposed optimal target improves covariance estimation accuracy.

Reopens questions on best covariance estimators for realistic portfolio conditions.

Abstract

Portfolio optimization requires sophisticated covariance estimators that are able to filter out estimation noise. Non-linear shrinkage is a popular estimator based on how the Oracle eigenvalues can be computed using only data from the calibration window. Contrary to common belief, NLS is not optimal for portfolio optimization because it does not minimize the right cost function when the asset dependence structure is non-stationary. We instead derive the optimal target. Using historical data, we quantify by how much NLS can be improved. Our findings reopen the question of how to build the covariance matrix estimator for portfolio optimization in realistic conditions.