#08-41
A Martingale Test for Alpha
Dean P. Foster, Robert Stine and H. Peyton Young, February 2010

Abstract: This paper describes a new way of testing whether the returns from a financial asset are systematically higher than a perfectly efficient market would allow. Such an asset is said to have positive alpha. The standard way of estimating alpha is to regress the asset’s returns against the market returns over an extended period of time and to apply the t-test. The difficulty is that the residuals often fail to satisfy independence and normality. In fact, portfolio managers may have an incentive to employ strategies whose residuals depart by design from independence and normality. To address these problems we propose a robust test for alpha based on the martingale maximal inequality. Unlike the t-test, our test places no restrictions on the distribution of returns while retaining substantial statistical power. The method is illustrated for four assets: a stock, a mutual fund, a hedge fund, and a fabricated fund that is deliberately designed to fool standard tests of significance.

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