Economic theorists have interpreted the "efficient markets hypothesis" to assert that equilibrium asset prices reveal all decision-relevant information in the market. This paper establishes conditions on investors' utility functions of future wealth which are necessary for the efficient markets hypothesis to be satisfied and be robust to slight perturbations of endowments and the joint distribution of current information and future asset values. The main result states that over the relevant range of future wealth values, there are three possible cases: (i) all investors are risk-neutral; (ii) modulo a change in the wealth origin, each investor has constant relative risk aversion with the same constant for all investors; or (iii) all investors have constant absolute risk aversion.