This study examined the RWH using the run tests, Ljung-Box statistics, and the unit root tests. The data covered the period starting January 1994 to June 2007 for the daily closing prices of the Ksh/UD dollar spot rate. The main finding of this study is that the RWH is strongly rejected at the 5% significance level. The results indicate that the rejections are due to autocorrelation in currency returns. The exchange rate tends to appreciate most of the time over the sample period. Therefore failure of the EMH could be due to exchange rate undershooting and overshooting phenomena. The unit root tests showed that the exchange rate data is non-stationary while returns are stationary. Therefore the evidence strongly suggested that the foreign exchange market is not efficient.