The first question to which the investment theorist addresses himself is, Given a number of assets with known expectations, risks, and correlations, what is an optimum portfolio, that is, a portfolio with the maximum expectation for a given risk? This question has been dealt with exhaustively in a number of publications. The second question is, Given the value and returns of a number of portfolios managed by different managers or managed along different lines, which is the most successful? The question is complicated, because it involves an element of prediction. The statement that A is the most successful manager implies not that he has made the most money in the past -he may or may not have-but that he will in future make the most money for a given risk. This problem has been treated less widely and, in my view, less satisfactorily than the first. Quite elaborate analytical schemes have been described,' but their practical success has not been well demonstrated. In this paper I hold that, before attempting to select the best-managed portfolio from a group, it is necessary to establish that the members of the group do in fact differ from each other. No manager can be omniscient, and factors of which he is ignorant will always cause random fluctuations in the results. I wish to be reasonably certain that I do not ascribe the effects of chance to the deliberate actions of management. The consequences of such misattribution are not trivial, as may be seen from the following illustration. Imagine a penny tossed 40 times. The most probable number of heads is 20. However, in a particular experiment, we might find 18 heads, and if we were to attribute this to a real bias, our best estimate for the next 40 throws would be 18. Further, we might find that in the first 20 throws we had 10 heads and in the second, 8 heads. If we were to attribute this to a real trend, our best estimate for the next 40 throws would be 6 + 4 = 10 heads. If we bear in mind the predictive element in the evaluation of investment performance, the importance of avoiding the attribution of real causes to statistical fluctuations is obvious. In this work I examine several papers that have put forward schemes for the evaluation of investment performance. In each case I try to base the