A Martingale Test for Alpha

We present a new method for testing whether a fund manager's track record allows us to infer that he is able to beat the market with high probability or is just plain lucky. The test is based on the martingale maximal inequality. Unlike other standard approaches the test is robust to the assumed distribution of returns while retaining substantial statistical power. The method is illustrated using stock market data from 1926‐2007. 1. Estimating alpha Fund managers who deliver returns that are consistently higher than the returns from a market portfolio are said to generate 'alpha'. This is the extra return attributable to the manager's superior investment skill. Needless to say, such managers are in great demand and many of them earn large sums of money. But how can investors tell whether a given manager actually is able to generate positive alpha over an extended period of time? The usual method for estimating alpha empirically is to regress a given series of returns generated by a fund manager against the returns generated by a market portfolio, such as the S&P 500. Let 0,1, 2,3,... t = be discrete times at which returns are reported, say the end of each month. Consider first a fund that is invested entirely in the stock market, and let m t r be the return generated by the market portfolio in period t. Let f t r be the risk‐free rate of return in period t , that is, the return available on a safe asset such as US Treasury bills. The excess