Reversibly Greater Downside Risk Aversion

An intrinsic, but seldom recognized property of greater risk aversion in expected utility theory is its reversibility, viz., utility v=phi(u) is more risk averse than u if and only if the transformation function phi is concave, while equivalently, u=psi(v) is less risk averse than v if and only if psi is convex. Moreover, this reversibility is exactly mirrored in the ranking of utility functions by their Arrow-Pratt measures, which ensures that the rankings are transitive and so capable of yielding meaningful comparative statics predictions for greater and less risk aversion. We extend this reversibility property to the third order concerned with downside risk aversion by deriving a utility measure of third-order risk preference whose relative magnitude, along with that of the Arrow-Pratt measure of risk aversion, yields a strict partial ordering of utility functions by greater downside risk aversion, and when the ranking by these measures is reversed the result is an ordering by less downside risk aversion. A sufficient condition for using this reversible measure is developed in terms of the well-known prudence measure. We also provide applications, particularly one in the context of saving