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.     

Background Risks and the Value of a Statistical Life

We examine the effects of background mortality and financial risks on an individual's willingness to pay to reduce his mortality risk (the value of statistical life or VSL). Under reasonable assumptions about risk aversion and prudence with respect to wealth in the event of survival and with respect to bequests in the event of death, background mortality and financial risks decrease VSL. The effects of large mortality or financial risks on VSL can be substantial, but the effects of small background risks are negligible. These results suggest that the commonplace failure to account for background risk in evaluating VSL is unlikely to produce substantial bias in most applications.     

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     

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.     

Uncertainty aversion and aversion to increasing uncertainty

According to the original Ellsberg (1961) examples there is uncertainty version if the decision maker prefers to bet on an urn of known composition rather than on an urn of unknown composition. According to another definition (Schmeidler, 1989), there is uncertainty aversion if any convex combination of two acts is preferred to the least favorable of these acts. We show that these two definitions differ: while the first one truly refers to uncertainty aversion, the second one refers to aversion to increasing uncertainty. Besides, with reference to Choquet Expected Utility theory, uncertainty aversion means that there exists the core of a capacity, while aversion to increasing uncertainty means that the capacity is convex. Consequently, aversion to increasing uncertainty implies uncertainty aversion, but the opposite does not hold. We also show that a completely analogous situation holds for the case of risk and we define a set of risk and uncertainty premiums according to the previous analysis.     

On the intensity of downside risk aversion

The degree of downside risk aversion (or equivalently prudence) is so far usually measured by . We propose here another measure, , which has specific and interesting local and global properties. Some of these properties are to a wide extent similar to those of the classical measure of absolute risk aversion, which is not always the case for . It also appears that the two measures are not mutually exclusive. Instead, they seem to be rather complementary as shown through an economic application dealing with a simple general equilibrium model of savings.

Endogenous risks and the risk premium

This note tries to correct a deficiency of the microeconomic literature on decision making under uncertainty. Indeed, when considering meaningful comparative statics results in situations where risks are at least partially controllable (endogenous), this literature has mostly relied upon the traditional Arrow-Pratt risk aversion functions and has paid very little attention to the definition of the risk premium. However when they defined the risk premium and the risk aversion functions, Arrow and Pratt considered only roulette gambles, i.e. risks totally exogenous to the individual. This note highlights the fact that several definitions of the risk premium may be proposed for endogenous risks. Two of them, already used in the literature, do not preserve the intuitively-appealing properties of the Arrow-Pratt risk premium. An alternative definition is then proposed. It is shown that this new definition of the risk premium applied to endogenous risks exhibits the properties generally admitted for roulette gambles.The three authors have benefitted from Ph. Caperaa's advice and from a referee's comments.     

On the definition of risk aversion

Two definitions of risk aversion have recently been proposed for non-expected utility theories of choice under uncertainty: the former refers the measure of risk aversion (Montesano 1985, 1986 and 1988) directly to the risk premium (i.e. to the difference between the expected value of the action under consideration and its certainty equivalent); the latter defines risk aversion as a decreasing preference for an increasing risk (introduced as mean preserving spreads) (Chew, Karni and Safra 1987, Machina 1987, Röell 1987, Yaari 1987).When the von Neumann-Morgenstern utility function exists both these definitions indicate an agent as a risk averter if his or her utility function is concave. Consequently, the two definitions are equivalent. However, they are no longer equivalent when the von Neumann-Morgenstern utility function does not exist and a non-expected utility theory is assumed. Examples can be given which show how the risk aversion of the one definition can coexist with the risk attraction of the other. Indeed the two definitions consider two different questions: the risk premium definition specifically concerns risk aversion, while the mean preserving spreads definition concerns the increasing (with risk) risk aversion.The mean preserving spreads definition of risk aversion, i.e. the increasing (with risk) risk aversion, requires a special kind of concavity for the preference function (that the derivatives with respect to probabilities are concave in the respective consequences). The risk premium definition of local risk aversion requires that the probability distribution dominates on the average the distribution of the derivatives of the preference function with respect to consequences. Besides, when the local measure of the first order is zero, there is risk aversion according to the measure of the second order if the preference function is concave with respect to consequences.Yaari's (1969) measure of risk aversion is closely linked to the r.p. measure of the second order. Its sign does not indicate risk aversion (if positive) or attraction (if negative) when the measure of the first order is not zero (i.e., in Yaari's language, when subjective odds differ from the market odds).     

On the Effect of Risk Aversion in Bimatrix Games

Nash equilibria with identical supports are compared for bimatrix games that are different with respect to the risk aversion of player 2. For equilibria in 2× 2-bimatrix games and for equilibria with efficient supports in coordination games it is established for which cases increased risk aversion of player 2 benefits or hurts player 2.     

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.     

Benchmark values for higher order coefficients of relative risk aversion

The existing literature on savings, insurance, and portfolio choices under risk has revealed that quite often comparative statics results depend, among other things, upon the values of the coefficients of relative risk aversion and relative prudence. More specifically the benchmark values for these coefficients are, respectively, one and two. Recently, several papers investigated constraints on the higher degree extensions of the coefficients of relative risk aversion and of relative prudence. The present work provides a unified approach to this question based on the concept of elementary correlation increasing transformations, allowing for a better understanding of changes in risk in the multiplicative case.     

Loss Averse Behavior

A behavioral condition of loss aversion is proposed and tested. Forty-nine students participated in experiments on binary choices among lotteries involving small scale real gains and losses. At the aggregate level, a significant proportion of the choices are in the direction predicted by loss aversion. Individuals can be classified as loss averse (28 participants), gain seeking (12), and unclassified (9). A comparison with risk behavior for binary choices on lotteries involving only gains shows that risk attitudes vary across these domains of lotteries. A gender effect is also observed: proportionally more women are loss averse. In contrast to the predictions of comonotonic independence, the size of common outcomes has systematic influence on choice behavior. JEL Classification: D81, C91     

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.     

More on insurance as a Giffen good

In this article, we generalize the Hoy and Robson (1981) analysis and provide a necessary and sufficient condition for insurance not to be a Giffen good. The condition gives a bound for the variation of absolute risk aversion that permits the wealth effect to be always dominated by the substitution effect.     

Effects of Risk and Time Preference and Expected Longevity on Demand for Medical Tests

Despite their conceptual importance, the effects of time preference, expected longevity, uncertainty, and risk aversion on behavior have not been analyzed empirically. We use data from the Health and Retirement Study (HRS) to assess the role of risk and time preference, expected longevity, and education on demand for three measures used for early detection of breast and cervical cancer—regular breast self-exams, mammograms, and Pap smears. We find that individuals with a higher life expectancy and lower time preference are more likely to undergo cancer screening. Less risk averse individuals tend to be more likely to undergo testing.     

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.     

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.     

Comparative Risk Sensitivity with Reference-Dependent Preferences

Experimental evidence suggests that individuals are risk averse over gains and risk seeking over losses (i.e., they have S-shaped utility functions in an expected utility setting) and that they are loss averse. Furthermore, the evidence leads to a single definition of S-shaped utility, but it has led to several alternative specifications of loss aversion. This paper characterizes the relations more S-shaped than and more loss averse than for a utility function, and in so doing arrives at a new definition of loss aversion based on average instead of marginal utility.     

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.     

Choice under risk and the security factor: An axiomatic model

The particular attention paid by decision makers to the security level ensured by each decision under risk, which is responsible for the certainty effect, can be taken into account by weakening the independence and continuity axioms of expected utility theory. In the resulting model, preferences depend on: (i) the security level, (ii) the expected utility, offered by each decision. Choices are partially determined by security level comparison and completed by the maximization of a function, which express the existing tradeoffs between expected utility and security level, and is, at a given security level, an affine function of the expected utility. In the model, risk neutrality at a given security level implies risk aversion.