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

arXiv: https://arxiv.org/abs/2112.07521 · HTML: https://mlsanity.com/paper/2112.07521/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Dec 2021.

## Built on by (most cited first, 5 of 5)

- [End-to-End Large Portfolio Optimization for Variance Minimization with Neural Networks through Covariance Cleaning](https://mlsanity.com/paper/2507.01918/) (2025)
- [Estimating Covariance for Global Minimum Variance Portfolio: A Decision-Focused Learning Approach](https://mlsanity.com/paper/2508.10776/) (2025)
- [The Decision Geometry of Covariance Estimation for the Global Minimum-Variance Portfolio under Heavy Tails](https://mlsanity.com/paper/2606.27462/) (2026)
- [Covariance-Aware Simplex Projection for Cardinality-Constrained Portfolio Optimization](https://mlsanity.com/paper/2512.19986/) (2025)
- [Physics-Informed Singular-Value Learning for Cross-Covariances Forecasting in Financial Markets](https://mlsanity.com/paper/2601.07687/) (2026)

## Abstract (arXiv)

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.