Risk aversion over finite domains

We investigate risk attitudes when the underlying domain of payoffs is finite and the payoffs are, in general, not numerical. In such cases, the traditional notions of absolute risk attitudes, that are designed for convex domains of numerical payoffs, are not applicable. We introduce comparative notions of weak and strong risk attitudes that remain applicable. We examine how they are characterized within the rank-dependent utility model, thus including expected utility as a special case. In particular, we characterize strong comparative risk aversion under rank-dependent utility. This is our main result. From this and other findings, we draw two novel conclusions. First, under expected utility, weak and strong comparative risk aversion are characterized by the same condition over finite domains. By contrast, such is not the case under non-expected utility. Second, under expected utility, weak (respectively: strong) comparative risk aversion is characterized by the same condition when the utility functions have finite range and when they have convex range (alternatively, when the payoffs are numerical and their domain is finite or convex, respectively). By contrast, such is not the case under non-expected utility. Thus, considering comparative risk aversion over finite domains leads to a better understanding of the divide between expected and non-expected utility, more generally, the structural properties of the main models of decision-making under risk.

Dual Moments and Risk Attitudes

The well-known Pratt–Arrow approximation, developed independently by John W. Pratt and Kenneth Arrow, provides an insightful dissection of the risk premium under the expected utility (EU) model. It is given by one-half the product of the variance of the risk and the local index of absolute risk aversion of the decision maker. Quite surprisingly, despite many important developments on “global” risk aversion in non-EU models, the “local” approach to risk aversion has received little attention outside EU. By considering the first two dual moments, mean and maxiance, on equal footing with the first two primal moments, mean and variance, the authors develop a dissection of the risk premium under the popular rank-dependent utility (RDU) model. This yields a simple approximation to the risk premium and a local index of absolute risk aversion under the RDU model.

Revealed Preferences over Risk and Uncertainty

We develop a nonparametric method, called Generalized Restriction of Infinite Domains (GRID), for testing the consistency of budgetary choice data with models of choice under risk and under uncertainty. Our test can allow for risk-loving and elation-seeking attitudes, or it can require risk aversion. It can also be used to calculate, via Afriat’s efficiency index, the magnitude of violations from a particular model. We evaluate the performance of various models under risk (expected utility, disappointment aversion, rank-dependent utility, and stochastically monotone utility) using data collected from several recent portfolio choice experiments. (JEL C14, D11, D12, D81)

BILATERAL RISK SHARING WITH HETEROGENEOUS BELIEFS AND EXPOSURE CONSTRAINTS

This paper studies bilateral risk sharing under no aggregate uncertainty, where one agent has Expected-Utility preferences and the other agent has Rank-dependent utility preferences with a general probability distortion function. We impose exogenous constraints on the risk exposure for both agents, and we allow for any type or level of belief heterogeneity. We show that Pareto-optimal risk-sharing contracts can be obtained via a constrained utility maximization under a participation constraint of the other agent. This allows us to give an explicit characterization of optimal risk-sharing contracts. In particular, we show that an optimal risk-sharing contract contains allocations that are monotone functions of the likelihood ratio, where the latter is obtained from Lebesgue’s Decomposition Theorem.

On comparison of the principles of equivalent utility and its applications

An insurance premium principle is a way of assigning to every risk, represented by a non-negative bounded random variable on a given probability space, a non-negative real number. Such a number is interpreted as a premium for the insuring risk. In this paper the implicitly defined principle of equivalent utility is investigated. Using the properties of the quasideviation means, we characterize a comparison in the class of principles of equivalent utility under Rank-Dependent Utility, one of the important behavioral models of decision making under risk. Then we apply this result to establish characterizations of equality and positive homogeneity of the principle. Some further applications are discussed as well.