Rankings of utility functions generated by simple n-th order risk-averse transformations are not partial orders, and therefore, do not yield reliable comparative statics predictions, except at the second order. Restrictions have been identified that rectify this deficiency at the third order...

It is shown that the downside risk aversion measure du is a special case of the more general Schwarzian downside risk aversion measure Su for appropriate preferences, in which case the downside measure du then inherits the desirable properties of the Schwarzian measure Su