Expected Utility and the Truncated Normal Distribution

This article demonstrates that: (1) When a normally distributed decision variable is combined with an analytic utility function (one with derivatives of all orders and a power series expansion involving those derivatives), the expected utility can be expressed in powers of μ and σ 2 . (2) In the case of the normal model, when the tails of the distribution do not reflect reality in the mind of a decision-maker, a truncated normal model is a possible alternative. (3) If the appropriate model is the truncated normal distribution, then the expected utility is approximately a linear function of μ and σ for several important classes of risk averse utility functions. (4) The negative exponential is an especially useful utility function since it has a simple closed form for both the truncated and nontruncated models, and since it gives an ordering similar to those of the log, arctangent or power utility functions.