Estimation Error in Optimal Portfolio Allocation Problems

Abstract

Markowitz showed that an investor who cares only about the mean and variance of portfolio returns should hold a portfolio on the efficient frontier. The application of this investment strategy proceeds in two steps. First, the statistical moments of asset returns are estimated from historical time series, and second, the mean-variance portfolio selection problem is solved separately, as if the estimates were the true parameters. The literature on portfolio decision acknowledges the difficulty in estimating means and covariances in many instances. This is particularly the case in high-dimensional settings. Merton notes that it is more difficult to estimate means than covariances and that errors in estimates of means have a larger impact on portfolio weights than errors in covariance estimates. Recent developments in high-dimensional settings have stressed the importance of correcting the estimation error of traditional sample covariance estimators for portfolio allocation. The literature has proposed shrinkage estimators of the sample covariance matrix and regularization methods founded on the principle of sparsity. Both methodologies are nested in a more general framework that constructs optimal portfolios under constraints on different norms of the portfolio weights including short-sale restrictions. On the one hand, shrinkage methods use a target covariance matrix and trade off bias and variance between the standard sample covariance matrix and the target. More prominence has been given to low-dimensional factor models that incorporate theoretical insights from asset pricing models. In these cases, one has to trade off estimation risk for model risk. Alternatively, the literature on regularization of the sample covariance matrix uses different penalty functions for reducing the number of parameters to be estimated. Recent methods extend the idea of regularization to a conditional setting based on factor models, which increase with the number of assets, and apply regularization methods to the residual covariance matrix.