Greater parametric downside risk aversion

We show that, just as an expected utility maximizer with utility function u responds to a compensated increase in risk by adjusting a control variable to reduce the degree of risk aversion measured by the Arrow-Pratt index Ru=−u′′∕u′ (Diamond & Stiglitz, 1974), so the response to a compensated increase in downside risk entails adjusting the control to reduce the degree of downside risk aversion measured by the Schwarzian Su=u′′′∕u′−(3∕2)Ru2. We also show that, ceteris paribus, increases in Su and in Ru result in reduced exposure to downside risk and, therefore, greater demand for self-protection activities that reduce downside risk to future income. An increase from Su to Sv is characterized by downside risk-averse transformations of utility everywhere along a path from u to v, which together constitute what we define to be a parametric increase in downside risk aversion. These parametric increases yield comparative statics predictions not true if v is simply a downside risk-averse transformation of u, and predictions for incremental changes in risk preferences can be extended immediately to global changes.