Decreasing Downside Risk Aversion and Background Risk

To analyze the impact of background risks, decreasing absolute risk aversion (DARA) must be combined with other restrictions on the shape of the utility function in order to make preferences risk vulnerable. In this note, we indicate that risk vulnerability can also be associated with the sole assumption of decreasing downside risk aversion (DDRA). That is, no matter how absolute risk aversion changes with wealth, DDDRA in the Arrow-Pratt sense and DDRA in the Ross sense are shown to be respectively necessary and sufficient for a background risk to raise the aversion to other independent risks.