Parametric Characterizations of Risk Aversion and Prudence

We show that in order to determine whether one decision-maker is more risk averse than another, it is sufficient to consider their attitudes towards a given two-parameter family of risks. When all risks belong to this family, useful comparisons of risk aversion can be made even in situations of ‘background risk’. Since expected utility becomes a function of mean and standard deviation, risk aversion can be measured by the marginal rate of substitution between mean and standard deviation. A utility function exhibits decreasing risk aversion if, and only if, this slope is a decreasing function of the mean. Second, we use the concept of prudence to solve a long-standing problem in mean-variance analysis: what is the economic interpretation of the concavity of a utility function which is a function of mean and variance? We show that in the case of normal distributions, utility is concave as a function of variance and mean if, and only if, it exhibits decreasing prudence.