No Country for Old Means: Optimal Input Shrinkage for Optimal Portfolios?

Estimation error costs mean-variance investors in the optimizer’s own currency, risk units, and shrinking the inputs is the standard way to pay less. We price the shrinkage of the mean when the covariance matrix it relies on is itself estimated. The estimated matrix enters the corrected mean through a single number with a known distribution, so the exact loss can be written down and the best shrinkage intensity follows from the number of assets and the length of the sample alone. No tuning, no validation window. We show that the corrected mean dominates the sample mean, the input of the most commonly used portfolio rules, whenever the sample is longer than the number of assets, and that the result extends James and Stein’s classic theorem to shrinkage toward the average return across assets, a target that is not an orthogonal projection in risk units. The corresponding analysis of the covariance matrix, where we correct in the operator norm of the optimizer rather than in the Frobenius norm of the shrinkage literature, is under development; this version contains a first result on the eigenvalue floor.

SSRN Scholarly Paper No. 7548058