Decreasing absolute risk aversion, prudence and increased downside risk aversion

Downside risk increases have previously been characterized as changes preferred by all decision makers u(x) with u′′′(x) > 0. For risk averse decision makers, u′′′(x) > 0 also defines prudence. This paper finds that downside risk increases can also be characterized as changes preferred by all decision makers displaying decreasing absolute risk aversion (DARA) since those changes involve random variables that have equal means. Building on these findings, the paper proposes using “more decreasingly absolute risk averse” or “more prudent” as alternative definitions of increased downside risk aversion. These alternative definitions generate a transitive ordering, while the existing definition based on a transformation function with a positive third derivative does not. Other properties of the new definitions of increased downside risk aversion are also presented.