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In the present work I derive the risk functions of 5 standard estimators for expected asset returns which are frequently advocated in the literature, viz the sample mean vector, the James-Stein and Bayes-Stein estimator, the minimum-variance estimator, and the CAPM estimator. I resolve the question why it is meaningful to study the risk function in the context of optimal asset allocation. Further, I derive the quantities which determine the risks of the different expected return estimators and show which estimators are preferable with respect to optimal asset allocation. Finally, I discuss the question whether it pays to strive for the optimal portfolio by using time series information. It turns out that in many practical situations it is better to renounce parameter estimation altogether and pursue some trivial strategy such as the totally risk-free investment.

An Analytical Investigation of Estimators for Expected Asset Returns from the Perspective of Optimal Asset Allocation