“Probability of risk aversion” and other applications of derivatives of the certainty equivalent

Abstract

“Probability of risk” aversion is principally concerned with reactions to scaling up of probabilities of non-zero values of a non-positive random variable by a common factor. Decreasing probability-of-risk aversion is defined and shown to be equivalent to ordinary risk aversion. Implications of this for insurance are pointed out. The sort of scaling involved is the same as that involved in “self-protection,” and it is shown that, for any expenditure on self-protection, say x, a concave utility function will prefer a coinsurance policy, costing x, which leaves probabilities unchanged, but scales down loss amounts by the same proportion as probabilities are scaled under self-protection. Properties of several comparative concepts of decreasing risk aversion are established. Derivatives of the certainty equivalent (CE) are used to elucidate well-known comparative static results in models of expected utility maximization. Finally, the study proves that concavity of the CE implies convexity of the coefficient of absolute risk aversion and examines the role of curvature of the CE in exploring relationships between properties of risk vulnerability, properness, and standardness.