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This paper examines the hypothesis that the expected rate of return to speculation in the forward foreign exchange market is zero; that is, the logarithm of the forward exchange rate is the market's conditional expectation of the logarithm of the future spot rate. A new computationally tractable econometric methodology for examining restrictions on a k-step-ahead forecasting equation is employed. Using data sampled more finely than the forecast interval, we are able to reject the simple market efficiency hypothesis for exchange rates from the 1970s and the 1920s. For the modern experience, the tests are also inconsistent with several alternative hypotheses which typically characterize the relationship between spot and forward exchange rates.