Risk Aversion for Nonsmooth Utility Functions

This paper generalizes the notion of risk aversion for functions which are not necessarily differentiable nor strictly concave. Using an approach based on superdifferentials, we define the notion of a risk aversion measure, from which the classical absolute as well as relative risk aversion follows as a RadonNikodym derivative if it exists. Using this notion, we are able to compare risk aversions for nonsmooth utility functions, and to extend a classical result of Pratt to the case of nonsmooth utility functions. We prove how relative risk aversion is connected to a super-power property of the function. Furthermore, we show how boundedness of the relative risk aversion translates to the one of the conjugate function. We propose also a weaker ordering of the risk aversion, denoted by essential bounds for the risk aversion, which requires only that bounds of the (absolute or relative) risk aversion have to hold up to a certain tolerance.