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.     

Measures of risk aversion with expected and nonexpected utility

In the expected utility case, the risk-aversion measure is given by the Arrow-Pratt index. Three proposals of a risk-aversion measure for the nonexpected utility case are examined. The first one sets “the second derivative of the acceptance frontier as a measure of local risk aversion.” The second one takes into account the concavity in the consequences of the partial derivatives of the preference function with respect to probabilities. The third one measures risk aversion through the ratio between the risk premium and the standard deviation of the lottery. The third proposal catches the main feature of risk aversion, while the other two proposals are not always in accordance with the same crude definition of risk aversion, by which there is risk aversion when an agent prefers to get the expected value of a lottery rather than to participate in it.     

Risk seeking with diminishing marginal utility in a non-expected utility model

The present work takes place in the framework of a non-expected utility model under risk: the RDEU theory (Rank Dependent Expected Utility, first initiated by Quiggin under the denomination of Anticipated Utility), where the decision maker's behavior is characterized by two functionsu andf. Our first result gives a condition under which the functionu characterizes the decision maker's attitude towards wealth. Then, defining a decision maker as risk averter (respectively risk seeker) when he always prefers to any random variable its expected value (weak definition of risk aversion), the second result states that a decision maker who has an increasing marginal utility of wealth (a convex functionu) can be risk averse, if his functionf issufficiently below his functionu, hence if he is sufficientlypessimistic. Obviously, he can also be risk seeking with a diminishing marginal utility of wealth. This result is noteworthy because with a stronger definition of risk aversion/risk seeking, based on mean-preserving spreads, Chew, Karni, and Safra have shown that the only way to be risk averse (in their sense) in RDEU theory is to have, simultaneously, a concave functionu and a convex functionf.     

A strong (Ross) characterization of multivariate risk aversion

Using the addition of uncorrelated noise as a natural definition of increasing risk for multivariate lotteries, I interpret risk aversion as the willingness to pay a (possibly random) vector premium in exchange for a reduction in multivariate risk. If no restriction is placed on the sign of any co-ordinate of the vector premium then (as was the case in Kihlstrom and Mirman's (1974) analysis) only pairs of expected utility maximizers with thesame ordinal preferences for outcomes can be ranked in terms of their aversion to increasing risk. However, if we restrict the premium to be a non-negative random variable then comparisons of aversion to increasing risk may be possible between expected utility maximizers withdistinct ordinal preferences for outcomes. The relationship between their utility functions is precisely the multi-dimensional analog of Ross's (1981)global condition forstrongly more risk averse.     

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.     

Another Tale of Two Tails: On Characterizations of Comparative Risk

We characterize the classes of utility functions that are consistent with different notions of mean preserving spreads introduced in the literature. This gives rise to a unified approach and extension of some definitions of increasing risk, including the concepts of Rothschild and Stiglitz (1970) and Landsberger and Meilijson (1990a,b). The main idea is to restrict the centers of the mean preserving spreads to an arbitrary subset.     

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.

     

A foundation of Bayesian statistics (How to deal with fears)

A rational statistical decision maker whose preferences satisfy Savage's axioms will minimize a Bayesian risk function: the expectation with respect to a revealed (or subjective) probability distribution of a loss (or negative utility) function over the consequences of the statistical decision problem. However, the nice expected utility form of the Bayesian risk criterion is nothing but a representation of special preferences. The subjective probability is defined together with the utility (or loss) function and it is not possible, in general, to use a given loss function - say a quadratic loss - and to elicit independently a subjective distribution.I construct the Bayesian risk criterion with a set of five axioms, each with a simple mathematical implication. This construction clearly shows that the subjective probability that is revealed by a decider's preferences is nothing but a (Radon) measure equivalent to a linear functional (the criterion). The functions on which the criterion operates are expected utilities in the von Neumann-Morgenstern sense. It then becomes clear that the subjective distribution cannot be eliciteda priori, independently of the utility function on consequences.However, if one considers a statistical decision problem by itself, losses, defined by a given loss function, become the consequences of the decisions. It can be imagined that experienced statisticians are used to dealing with different losses and are able to compare them (i.e. have preferences, or fears over a set of possible losses). Using suitable axioms over these preferences, one can represent them by a (linear) criterion: this criterion is the expectation of losses with respect to a (revealed) distribution. It must be noted that such a distribution is a measure and need not be a probability distribution.     

Ross' measure of risk aversion and portfolio selection

This article shows that if Ross' definition of riskier is replaced by a more traditional definition, such as a mean-preserving spread or second-degree stochastic dominance, then the application of Ross's stronger measure of risk aversion to the portfolio problem may no longer produce the desired result. It is also shown that the stronger measure may not perform satisfactorily when applied to exponential utility functions.The authors are grateful to John Pratt for his helpful comments.     

Endogenous risk and protection premiums

Endogenous risk implies an individual perceives he can influence the likelihood that a state of nature will occur. To add structure to endogenous risk models, I define a protection premium for reduced uncertainty about protection efficiency when a stochastic variable enters the probability functionp(x) rather than the utility function. For a binary lottery, a measure of aversion of uncertain protection efficiency(x) =-p(x)/p(x) is defined to unambiguously determine the effects of increased risk on an individual's voluntary contribution to public good supply earmarked to reduce the probability of an undesirable state. Finally, I examine the protection premium in ann-state discrete lottery and when uncertainty exists in both the probability and utility function.     

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.     

An Experiment on Rational Insurance Decisions

We describe the results of an experiment on decision making in an insurance context. The experiment was designed to test for the underlying rationality of insurance consumers, where rationality is understood in usual economic terms. In particular, using expected utility as the preference function, we test for positive marginal utility, risk aversion, and decreasing absolute risk aversion, all of which are normal postulates for any microeconomic decision context under uncertainty or risk. We find that there the discrepancy from rational decision making increases with the sophistication of the rationality criteria, that irrationality concerning fair premium contracts is uncharacteristically high, and that the slope of absolute risk aversion seems to depend on the format of the insurance contract. This revised version was published online in June 2006 with corrections to the Cover Date.     

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.     

Ambiguity aversion in the small and in the large for weighted linear utility

The widely observed preference for lotteries involving precise rather than vague of ambiguous probabilities is called ambiguity aversion. Ambiguity aversion cannot be predicted or explained by conventional expected utility models. For the subjectively weighted linear utility (SWLU) model, we define both probability and payoff premiums for ambiguity, and introduce alocal ambiguity aversion function a(u) that is proportional to these ambiguity premiums for small uncertainties. We show that one individual's ambiguity premiums areglobally larger than another's if and only if hisa(u) function is everywhere larger. Ambiguity aversion has been observed to increase 1) when the mean probability of gain increases and 2) when the mean probability of loss decreases. We show that such behavior is equivalent toa(u) increasing in both the gain and loss domains. Increasing ambiguity aversion also explains the observed excess of sellers' over buyers' prices for insurance against an ambiguous probability of loss.     

Total and partial Bivariate Risk Premia: An Extension

This paper investigates the link between the total bivariate risk premium and the sum of partial bivariate risk premia. Whereas in the case of small risks, the non interaction between risks is a sufficient condition to obtain the equality between the total risk premium and the sum of partial risk premia, the paper shows that this condition is not sufficient for large risks. The non interaction between risks occurs in two cases: if risks are independent or if individual's marginal utility of one good is independent of the endowment in the other. Without restriction on the utility function, none of these two conditions is sufficient for large risks. If attention is restricted to preferences that exhibit constant absolute risk aversion, the non variability of the marginal utility of good one with respect to variations in endowment in the other remains a sufficient condition, while the independence between risks does not.     

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     

On probabilities and loss aversion

This paper reviews the most common approaches that have been adopted to analyze and describe loss aversion under prospect theory. Subsequently, it is argued that loss aversion is a property of observable choice behavior and two new definitions of loss averse behavior are advocated. Under prospect theory, the new properties hold if the commonly used utility based measures of loss aversion are corrected by a probability based measure of loss aversion and their product exceeds 1. It is shown that prominent parametric families of weighting functions, while successful in accommodating empirical findings on probabilistic risk attitudes, may not fit well with the theoretical implications of the new loss averse behavior conditions.     

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.     

Normalized measures of concavity and Ross’s strongly more risk averse order

The Arrow-Pratt (A-P) definitions of absolute and relative risk aversion dominate the discussion of risk aversion and defining “more risk averse”. Ross (Econometrica 49:621–663, 1981) notes, however, that being A-P more risk averse is not sufficient for addressing many important comparative static questions. Consequently he introduces “a new and stronger measure for comparing two agents’ attitudes towards risk…”. Ross does not provide a corresponding measure of risk aversion. This paper uses a normalized measure of concavity to characterize the Ross definition of strongly more risk averse on bounded intervals. Other properties and uses of these normalized measures of concavity are also presented.     

Decreasing absolute risk aversion, prudence and increased downside risk aversion

Downside risk increases have previously been characterized as changes preferred by all decision makers u(x) with u????(x) > 0. For risk averse decision makers, u????(x) > 0 also defines prudence. This paper finds that downside risk increases can also be characterized as changes preferred by all decision makers displaying decreasing absolute risk aversion (DARA) since those changes involve random variables that have equal means. Building on these findings, the paper proposes using ??more decreasingly absolute risk averse?? or ??more prudent?? as alternative definitions of increased downside risk aversion. These alternative definitions generate a transitive ordering, while the existing definition based on a transformation function with a positive third derivative does not. Other properties of the new definitions of increased downside risk aversion are also presented.