Reviewing the basics of mean–variance portfolio optimization and the capital asset pricing model, this paper discusses the plausibility of some of the underlying assumptions. It is pointed out that a positive in-sample relationship between the expected return of an asset and its covariance with the market portfolio can be a statistical artifact because it can be explained without using any economic arguments. In an empirical analysis of two sets of assets consisting of individual stocks and indices, respectively, no indication of any out-of-sample relationship is found. In the absence of such a relationship or any other additional information about the expected returns, simple averages of past returns must be used as input for portfolio-optimization procedures. Empirical evidence is presented which suggests that portfolio optimization is of little practical value in this case and, in addition, that the use of robust estimators can hardly make any difference.