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     

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.     

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.     

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).     

Risk preferences of Australian academics: where retirement funds are invested tells the story

Risk preferences of Australian academics are elicited by analyzing the aggregate distribution of their retirement funds (superannuation) across available investment options. Not more than 10 % of retirement funds are invested as if their owners maximize expected utility under the assumption of constant relative risk aversion with an empirically plausible level of risk aversion. An implausibly high level of risk aversion is required to rationalize any investment into bonds when stocks are available. Not more than 36.54 % of all investments can be rationalized by a model of loss averse preferences. Moreover, the levels of loss aversion typically reported in the experimental studies imply overinvestment in bonds, which is not observed in the data. Up to 67.18 % of all investments can be rationalized by rank-dependent utility or Yaari’s (Econometrica 55:95–115 1987) dual model with empirically plausible parameters. A median Australian academic behaves as if maximizing rank-dependent utility with parameter \(\gamma \in [0.76, 0.79]\) in a Tversky and Kahneman (J Risk Uncertain 5:297–323 1992) probability weighting function.     

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.     

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.     

Rationality on the rise: Why relative risk aversion increases with stake size

How does risk tolerance vary with stake size? This important question cannot be adequately answered if framing effects, nonlinear probability weighting, and heterogeneity of preference types are neglected. We show that the observed increase in relative risk aversion over gains cannot be captured by the curvature of the value function. Rather, it is predominantly driven by a change in probability weighting of a majority group of individuals who weight probabilities of high gains more conservatively. Contrary to gains, no coherent change in relative risk aversion is observed for losses. These results not only challenge expected utility theory, but also prospect theory.     

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.     

Horizon Length and Portfolio Risk

In this paper, we compare the attitude towards current risk of two expected-utility-maximizing investors who are identical except that the first investor will live longer than the second one. It is often suggested that the young investor should take more risks than the old investor. We consider as a benchmark the case of complete markets with a zero risk-free rate. We show that a necessary and sufficient condition to assure that younger is riskier is that the Arrow-Pratt index of absolute tolerance (T) be convex. If we allow for a positive risk-free rate, the necessary and sufficient condition is T convex, plus T(0) = 0. It extends the well-known result that rational investors can behave myopically if and only if the utility function exhibits constant relative risk aversion.     

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.     

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.     

Risk Premiums and Benefit Measures for Generalized-Expected-Utility Theories

Standard tools for the analysis of economic problems involving uncertainty, including risk premiums, certainty equivalents and the notions of absolute and relative risk aversion, are developed without making specific assumptions on functional form beyond the basic requirements of monotonicity, transitivity, continuity, and the presumption that individuals prefer certainty to risk. Individuals are not required to display probabilistic sophistication. The approach relies on the distance and benefit functions to characterize preferences relative to a given state-contingent vector of outcomes. The distance and benefit functions are used to derive absolute and relative risk premiums and to characterize preferences exhibiting constant absolute risk aversion (CARA) and constant relative risk aversion (CRRA). A generalization of the notion of Schur-concavity is presented. If preferences are generalized Schur concave, the absolute and relative risk premiums are generalized Schur convex, and the certainty equivalents are generalized Schur concave.     

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.     

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.

     

Mean utility preserving increases in risk for state dependent utility functions

This paper defines the concept of a mean utility preserving spread across states (MUPSAS) for state dependent utility functions and analyzes the behavioural impact of shifts in the probability distribution of wealth across states such that overall mean utility is preserved. The main result provides an alternative way of ranking state dependent utility functions according to their degree of risk aversion (thus extending Kami's theorem of comparative risk aversion) and establishes a link between increases in risk and risk aversion for state dependent preferences. In a portfolio problem where preferences and the rate of return of the risky venture are state dependent, we find sufficient conditions to determine the impact of a MUPSAS on the optimal share of the portfolio invested in the risky asset.

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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.     

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.     

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.