Covariance Versus Precision Matrix Estimation for Efficient Asset Allocation

Asset allocation constitutes one of the most crucial and most challenging tasks in financial engineering, which often requires the estimation of large covariance or precision matrices, from short time-span multivariate observations, a mandatory yet difficult step. The present contribution reviews and compares a large selection of estimators for covariance and precision matrices, organized into classes of estimation principles (direct, factor, shrinkage, sparsity). This includes the theoretical derivation of several additional estimators not available in the literature. Rather than assessing estimation performance from synthetic data based on a priori selected models of questionable practical interest, it is chosen here to evaluate practically the quality of these estimators directly from portfolio selection performance, quantified by financial criteria. Portfolio selection is conducted over two datasets of different natures: a 15-year large subset (within Stoxx Europe 600) of 244 European stock returns, and a 50-year benchmark dataset of 90 US equity portfolios. This large scale comparative study addresses issues such as the relative benefits and difficulties of using robust versus direct estimates, of choosing precision or covariance estimates, of quantifying the impacts of constraints.