One rational individual may be willing to pay less than another to insure a risk ~ when another risk ri, is present even though he would pay more to insure any isolated risk, and even though E(~ [~i,) = 0 for all w. Noticing this, Ross (1981) proposed excluding such reversals and gave equivalent analytical conditions. Reconsidering, we explain why some reversals are natural and show that prohibiting them has peculiar and undesirable properties. Although we also simplify the conditions and prove them necessary for partial-risk portfolio results, we conclude that they represent revealing restrictions on comparative statics rather than natural implications of increased aversion to risk.
Concepts of greater and monotonic risk aversion have proved convenient and powerful in comparing rational attitudes and behavior of different individuals, or one individual at different levels of wealth, when all risks are chosen simultaneously. In reality, of course, some risks are unavoidable and others are decided upon seriatim. Might the concepts apply to new risks ~ separately from ongoing risks if? This is clearly not the case if new and old risks are correlated, since correlation profoundly affects combined risk. If new and old risks are independent, the concepts carry over almost intact (Kihlstrom, Romer, and Williams, 1981;Pratt, 1988). In the intermediate case E(~ [if) = 0 without independence, where ~ might be considered purely added risk, Ross (1981) noticed that striking reversals in comparisons can occur. He suggested excluding them, and gave analytical conditions that are necessary and sufficient to do so. We reexamine this case here. Section 1 explains why such reversals, though perhaps surprising at a cursory glance, should sometimes be expected. Section 2 defines terms and gives simplified analytical conditions. Section 3 displays some counterintuitive and cantankerous properties that arise if reversals are excluded willy-nilly. Section 4 shows that this exclusion is unfortunately necessary for simple partial-risk portfolio results. Section 5 summarizes the intuitive lessons and conclusions to be drawn.1