A simple discrete-time financial market model is introduced. The market participants consist of a collection of noise traders as well as a distinguished agent who uses the price information as it arrives to update her demand for the assets. It is shown that the distinguished agent's demand converges, both almost surely and in mean square, to a demand consistent with the rational expectations hypothesis, and the rate of convergence is calculated explicitly. Furthermore, the convergence of the standardised deviations from this limit is established. The rate of convergence, and hence the efficiency of this market, is an increasing function of both the risk-free interest rate and the relative number of noise traders in the market. An efficient market, therefore, measured in terms of a high proportion of informed traders, seems incompatible with the notion that efficient markets converge quickly.