The economics of adding and subdividing independent risks: Some comparative statics results

In this paper we address the problem of determining whether adding independent risks or subdividing them is a good substitute for insurance. Despite the fact that accepting more i.i.d. risks increases total risk, it is shown that some risk-averse decision makers can rationally reduce their demand for insurance by doing so. Similarly, a better diversified portfolio of i.i.d. risky assets can rationally be more insured, even if diversification is a risk-reduction scheme. We derive conditions sufficient to obtain unambiguous comparative statics results. Assuming that absolute risk aversion is decreasing and that the fourth derivative of the utility function is positive, we show that diversification is an exceptionally good substitute for insurance. Under the same conditions, adding independent risks to wealth reduces the demand for insurance on each unit.     

Joint measurement of risk aversion,prudence, and temperance

Risk aversion—but also the higher-order risk preferences of prudence and temperance—are fundamental concepts in the study of economic decision making. We propose a method to jointly measure the intensity of risk aversion, prudence, and temperance. Our theoretical approach is to define risk compensations of different orders, and in an experiment we elicit these compensations with a price list technique. We find evidence for risk aversion, prudence, and temperance. These traits correlate within subjects. The compensations elicited for prudence are significantly larger than those for risk aversion and temperance. In contrast to commonly used utility functions, prospect theory can predict this behavioral pattern. In our experiment, risk-averse, risk-loving, and risk-neutral subjects are prudent. This supports a recent theoretical observation that prudence may be a more universal trait than previously realized.     

Dynamic awareness and zero probability beliefs

We study behavioral patterns of insurance demand for low-probability large-loss events (catastrophic losses). Individual patterns of belief formation and risk attitude that were suggested in the behavioral decisions literature emerge robustly in the current set of insurance choices. However, social comparison effects are less robust. We do not find any evidence for peer effects (through social-loss aversion or imitation) on insurance take-up. In contrast, we find support for the prediction that people underweight others’ relevant information in their own decision making.     

Estimating the demand for risky assets via the indirect expected utility function

This article obtains demand functions for risky assets without making a priori assumptions about the form of the utility function. In a simple portfolio model, the envelope theorem is applied to the indirect expected utility function to derive estimating equations. Tests for the existence of constant absolute or constant relative risk aversion are also developed. Empirical estimation of the demand for financial assets held by U.S. households for the period 1946–1985 indicates that aggregate household behavior is consistent with the existence of constant relative risk aversion, with the coefficient of risk aversion having a value of approximately 1.3.The authors gratefully acknowledge helpful comments from the editor and an anonymous referee.     

Positivity of bid-ask spreads and symmetrical monotone risk aversion*

A usual argument in finance refers to no arbitrage opportunities for the positivity of the bid-ask spread. Here we follow the decision theory approach and show that if positivity of the bid-ask spread is identified with strong risk aversion for an expected utility market-maker, this is no longer true for a rank-dependent expected utility one. For such a decision-maker only a very weak form of risk aversion is required, a result which seems more in accordance with his actual behavior. We conclude by showing that the no-trade interval result of Dow and Werlang (1992a) remains valid for a rank-dependent expected utility market-maker merely exhibiting this weak form of risk aversion.     

Pareto utility

In searching for an appropriate utility function in the expected utility framework, we formulate four properties that we want the utility function to satisfy. We conduct a search for such a function, and we identify Pareto utility as a function satisfying all four desired properties. Pareto utility is a flexible yet simple and parsimonious two-parameter family. It exhibits decreasing absolute risk aversion and increasing but bounded relative risk aversion. It is applicable irrespective of the probability distribution relevant to the prospect to be evaluated. Pareto utility is therefore particularly suited for catastrophic risk analysis. A new and related class of generalized exponential (gexpo) utility functions is also studied. This class is particularly relevant in situations where absolute risk tolerance is thought to be concave rather than linear.     

Education and anomalies in decision making: Experimental evidence from Chinese adult twins

We estimate the effects of education on two dimensions of decision making behavior—risk and time—beyond those considered to be normal-ranged to encompass behavioral anomalies with respect to expected utility as well as time consistency. We conduct a number of incentivized choice experiments on Chinese adult twins to measure decision making behavior, and use a within-twin-pair fixed-effects estimator to deal with unobservable family-specific effects. The estimation results show that a higher education level tends to reduce the degree of risk aversion towards moderate prospects, moderate hazards, and longshot prospects. For anomalies under risk and uncertainty, college graduates exhibit significantly more Allais-type behavior compared to high school dropouts, while high school graduates exhibit more ambiguity aversion as well as a familiarity preference relative to high school dropouts. For decision making involving time, a higher education level tends to reduce the degree of impatience, and to reduce behavioral anomalies including hyperbolic discounting, dread, and hopefulness. The experimental observations suggest that people with a higher education level tend to exhibit more behavioral anomalies in risk attitudes but fewer behavioral anomalies involving time, hence implying that education has multi-functions in preference formation and human capability building. This study contributes to the understanding of the nature of these behavioral anomalies and the roles of education in human decision making.     

Uncertain indemnity and the demand for insurance

This paper considers the demand for insurance in a model with uncertain indemnity. Uncertain indemnity tends to increase the demand for insurance for precautionary reasons, but it also tends to decrease the demand due to the risk created by indemnity uncertainty. When the coefficient of relative prudence is not too large, uncertain indemnity reduces the demand for insurance and partial coverage is optimal even at actuarially fair premiums. In addition, insurance may be an inferior good or a normal good, depending on the behavior of absolute risk aversion and the magnitude of the coefficient of relative risk aversion.     

Risk and risk aversion for state-dependent utility

This paper analyses risk and risk aversion in the state-dependent utility model, which is useful for modelling health or life insurance purchase. We use Karni's (1983) definition of risk aversion, and extend the class of risks to which it can be applied.Research supported by the ESRC postdoctoral fellowship scheme. I would like to thank Jerry Nordquist for arousing my interest in this subject. For helpful comments on an earlier draft I am grateful to an anonymous referee and the editor of this journal.     

Risk and choice: A research saga

The economic theory of decision making under risk has seen remarkable advances over the 50 years since Pratt’s (1964) characterization of risk aversion under expected utility. We review developments in three key areas to which Louis Eeckhoudt has made significant contributions: (1) increases in risk and risk taking; (2) self-protection and risk aversion; and (3) higher (and lower) order derivatives of utility. For each, we identify seminal papers, puzzles, and recent developments. The saga of research on these topics reveals that important contributions were made long ago and yet significant gains in understanding continue to be made. Recent advances often have roots in early results and researchers can profit by examining the old as well as the new papers.     

On risk aversion,classical demand theory,and KM preferences

Building on Kihlstrom and Mirman (Journal of Economic Theory, 8(3), 361–388, 1974)’s formulation of risk aversion in the case of multidimensional utility functions, we study the effect of risk aversion on optimal behavior in a general consumer’s maximization problem under uncertainty. We completely characterize the relationship between changes in risk aversion and classical demand theory. We show that the effect of risk aversion on optimal behavior depends on the income and substitution effects. Moreover, the effect of risk aversion is determined not by the riskiness of the risky good, but rather the riskiness of the utility gamble associated with each decision.     

The risk aversion measure without the independence axiom

The risk premium (conveniently normalized) is defined as the measure of risk aversion. This measure does not require any relevant assumption in the theory of choice under uncertainty except the existence of a certainty equivalent. In particular, the independence axiom is not required. The measure of risk aversion of an action is provided not only for the case with one commodity and two consequences but also for the case with many commodities and consequences. The measure of mean risk aversion of all actions with given consequences is introduced and the local measure of risk aversion is obtained by making all these consequences approach the consequence under consideration. This measure is demonstrated to be zero when the von Neumann-Morgenstern utility function exists. In this case a measure of risk aversion of the second order is introduced, which turns out to be equal to the Arrow-Pratt absolute index when there is only one commodity and similar to the generalized measures proposed by several authors when there are many commodities and two consequences.Helpful comments by I. Gilboa and suggestions by the referee are gratefully acknowledged.     

Risk aversion in expected intertemporal discounted utilities bandit problems

We consider a situation where an individual is facing an uncertain situation, but may costly alter his knowledge of the uncertainties. We study in this context how risk aversion may modify the individual search behavior. We consider a one-armed bandit problem (where one arm is safe and the other is risky) and study how the agent risk aversion can change the sequence of arms selected. The main result is that when the utility function is more concave, the agent has more chances to select the safe arm. We also discuss how search is affected by risk aversion.     

Relative Risk Aversion: What Do We Know?

The relative risk aversion measure that represents the risk preferences of a decision maker depends on the outcome variable that is used as the argument of the utility function, and on the way that outcome variable is defined or measured. In addition, the relationship between any two such relative risk aversion measures is determined by the relationship between the corresponding outcome variables. These well-known facts are used to adjust several reported estimates of relative risk aversion so that those estimates can be directly compared with one another. After adjustment, the significant variation in the reported relative risk aversion measures for representative decision makers is substantially reduced. JEL Classification: D81     

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.

     

Estimating Risk Attitudes using Lotteries: A Large Sample Approach

Attitudes towards risk play a major role in many economic decisions. In empirical studies it is quite often assumed that attitudes towards risk do not vary across individuals. This paper questions this assumption and analyses which factors influence an individual's risk attitude. Based on questions on lotteries in a large household survey we first semiparametrically estimate an index for risk aversion. We only make weak assumptions about the underlying decision process and our estimation method allows for generalisations of expected utility. We then estimate a structural model based on Cumulative Prospect Theory. Expected utility is strongly rejected and both the value function and the probability weighting function vary significantly with (among other things) age, income, and wealth of the individual.     

Greater Downside Risk Aversion

Although investors are concerned foremost with mean and variance, they are also sensitive to downside risk. In this paper, we introduce an index of downside risk aversion to distinguish risk aversion from higher-order aspects of risk preference, including prudence. We show that the index of downside risk aversion S increases with monotonic downside risk averse transformations of utility, thereby directly linking S to the definition of downside risk aversion introduced by Menezes et al. (American Economic Review, 70, 921–932, 1980). Although the index S applies equally to risk averse and risk loving decision makers, for a given positive degree of risk aversion, S is greater when the index of prudence is greater and vice versa.     

Individual and couple decision behavior under risk: evidence on the dynamics of power balance

This article reports results of an experiment designed to analyze the link between risky decisions made by couples and risky decisions made separately by each spouse. We estimate both the spouses and the couples’ degrees of risk aversion, we assess how the risk preferences of the two spouses aggregate when they make risky decisions, and we shed light on the dynamics of the decision process that takes place when couples make risky decisions. We find that, far from being fixed, the balance of power within the household is malleable. In most couples, men have, initially, more decision-making power than women but women who ultimately implement the joint decisions gain more and more power over the course of decision making.     

Risk vulnerability: a graphical interpretation

The article gives a graphical interpretation of the concept of risk vulnerability. It shows that in a specific context of binary lotteries the assumption of risk vulnerability adds to prudence what the assumption of decreasing absolute risk aversion adds to risk aversion. We end the presentation showing that results can be extended to the concept of multiplicative risk vulnerability.     

Loss aversion leading to advantageous selection

Some insurance markets are characterized by “advantageous selection”, that is, ex-post risk and coverage are negatively correlated. We show that expectation-based loss aversion as in K?szegi and Rabin (Quarterly Journal of Economics, 121(4), 1133–1165, 2006; The American Economic Review, 97(4), 1047–1073, 2007) provides a natural explanation for this phenomenon in environments in which risk aversion models do not, e.g., when agents face modest-scale risks and/or in absence of moral hazard. More exposure to risk has two competing effects on an agent’s willingness to pay for insurance: a positive effect, as in standard expected utility models; and a negative one, due to a reference effect. We determine conditions under which an insurance provider optimally sets a high price at which only low risk agents buy.