Should smart investors buy funds with high past returns?

We fully characterize the equilibria in a game between a fund manager of unknown ability who control the riskiness of his portfolio and investors who only observe realized returns. We derive two types of equilibria. The first one is such that (i) investors invest in the fund if the realized return falls within some interval, i.e., is neither too low nor too high, (ii) a good manager picks a portfolio of minimal riskiness and (iii) a bad manager picks a portfolio with higher risk, "gambling" on a lucky outcome. The second type of equilibrium is more traditional: (i) investors invest in the fund if the observed return is larger than some threshold, and (ii) good and bad managers choose the same risk level. [PUBLICATION ABSTRACT]