Luck versus skill

300 active funds have ten years of monthly returns. Treat their alpha $t$-statistics as independent standard normal under the null of zero alpha. (a) If no fund has skill, how many funds are expected to show $t > 2$? What is the probability that at least five do? (b) Now suppose 45 of the funds are skilled, with expected $t$-statistic 3, and the other 255 have zero alpha. Compute the expected numbers of true and false discoveries at $t > 2$ and the false discovery rate. (c) A website lists the "top 10 funds by 5-year alpha". Using (a) and (b), say what you would need to know before investing in any of them. (d) Explain how Fama and French (2010) use bootstrap simulation instead of the normal approximation, and why.