The standard approach, therefore, cannot properly measure the odds of observing a group of funds having genuine positive alphas. For example, suppose that 20 out of 200 funds have positive estimated alphas at a given significance level γ. Obviously, the true performance of these 20 funds depends on the proportion of lucky funds. For instance, if the latter is equal to 100%, all 20 funds produce, in reality, zero alphas. Another problem with the standard approach is that it assumes that an observed increase in the number of significant funds as γ rises is only due to the detection of new funds with differential performance
