Ranking Distributions when only Means and Variances are Known

Abstract

Consider a choice between two random variables, for which only means and variances are known. Is it possible to rank them by putting some constraints on risk preferences? We provide such a ranking by bounding how much marginal utility can change. Such bounds enable us to rank all distributions with given means and variances by first-order almost stochastic dominance. We show how our results can be used to compare a risky project and a sure payoff and also provide a new connection between the Sharpe and Omega ratios from finance.

Keywords: Decision analysis: stochastic dominance, utility, risk. Probability: distribution comparisons