Downside risk aversion vs decreasing absolute risk aversion: an intuitive exposition

Downside risk aversion (downside RA) and decreasing absolute risk aversion (DARA) are different concepts that describe preferences for which the harm from bearing risk is lessened by an increase in wealth. This note presents some intuitive explanations of the difference between the two concepts using simple lotteries and graphical analysis. All risk-averse utility functions exhibit downside risk aversion, except those that exhibit sufficiently strong increasing absolute risk aversion. In a sense, downside RA is to be expected: adding downside risk to a baseline lottery is analogous to increasing risk while adding upside risk is analogous to decreasing risk. The difference between the two concepts can be attributed to the use of different measures of the harm from risk bearing: downside RA measures harm using the utility premium and DARA measures harm using the risk premium. The two premia can change at different rates and even in different directions as wealth increases.