Shrink with Purpose: Optimal Covariance Matrix Estimation for Portfolio Selection

We introduce analytical linear and nonlinear shrinkage estimators of the sample covariance matrix that are optimal for mean-variance portfolio choice. Unlike the classical estimators based on statistical loss functions like the mean squared error, our shrinkage covariance matrices optimize the expected out-of-sample portfolio utility and account for estimation errors in mean returns. Our estimators shrink the sample eigenvalues more intensively than conventional methods, and they especially diminish the contribution of principal components with small squared Sharpe ratios. By jointly estimating the covariance matrix and the optimal portfolio in one step, our method delivers significant empirical performance gains relative to the usual two-step shrinkage approach. Our portfolios also help reduce turnover and outperform recent regularized mean-variance portfolio strategies.