One-Switch Utility Functions and a Measure of Risk

Consider the relative attractiveness to a decision maker of two financial gambles as the wealth of that individual varies. It may seem reasonable that either one alternative should be preferred for all wealth levels or that there exists a unique critical wealth level at which the decision maker switches from preferring one alternative to the other. Decreasing risk aversion is not sufficient for this property to hold: we identify the small class of utility functions for which it does. We show how the property leads naturally to a measure of risk. The results of this paper apply equally well to discounting functions for cash flows: one-switch discount functions permit at most one change in preference between cash flows as all payoffs are deferred in time.