Bias–Variance Propagation from Precision Matrix Estimation to Minimum-Variance Portfolios

Abstract

This paper studies how estimation error in covariance and precision matrices propagates to the out-of-sample risk of the global minimum-variance portfolio. Using a Delta-method expansion, we derive a second-order bias–variance decomposition of the portfolio’s out-of-sample variance and show how the variance and bias of covariance and precision estimators map into excess portfolio risk. Within this framework we obtain an oracle intensity for linear covariance shrinkage that is tailored to minimizing out-of-sample variance and generally differs from the intensity that is optimal under standard mean-squared-error criteria for covariance estimation. Then, we propose a simple variance-targeted rule that selects the shrinkage intensity by minimizing a proxy of the portfolio’s out-of-sample variance on an internal validation block and rescales it to account for the larger outer estimation window.

A Monte Carlo study based on factor-structured data-generating processes, together with an empirical study on three large equity universes (S&P 100, S&P 500 and STOXX Europe 600), compares this procedure with the Ledoit–Wolf and Schäfer–Strimmer covariance estimators, and shows that the variance-targeted rule uses higher shrinkage intensities and typically achieves lower out-of-sample volatility and turnover.

[1]Alessandro Fulci

[1]Department of Economics and Management, University of Trento, Via Inama, 5, Trento, 38122, Italy