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It is observed that the measure S u = u ′′′/ u ′ − (3/2)( u ′′/ u ′) 2 , previously shown to be a relevant measure of the degree of downside risk aversion, is known in the mathematics literature as the Schwarzian derivative. The Schwarzian derivative has invariance properties under composition of functions that make it particularly well-behaved as a ranking of downside risk aversion. Indeed, it has the same invariance properties as the measure R u = − u ′′/ u ′, familiar to economists as a ranking of utility functions by degree of Arrow-Pratt risk aversion.
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