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

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