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.     

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.     

Ambiguity framed

In his exposition of subjective expected utility theory, Savage (1954) proposed that the Allais paradox could be reduced if it were recast into a format which made the appeal of the independence axiom of expected utility theory more transparent. Recent studies consistently find support for this prediction. We consider a salience-based choice model which explains this frame-dependence of the Allais paradox. We then derive the novel prediction that the presentation format responsible for reductions in Allais-style violations of expected utility theory will also reduce Ellsberg-style violations of subjective expected utility theory. This format makes the appeal of Savage’s “sure thing principle” more transparent. We design an experiment to test this prediction and find strong support for such frame-dependence of ambiguity aversion in Ellsberg-style choices. In particular, we observe markedly less ambiguity-averse behavior in Savage’s matrix format than in a more standard “prospect” format. This finding poses a new challenge for the leading models of ambiguity aversion.     

Changes in multiplicative background risk and risk-taking behavior

This article analyzes the conditions under which any change in a multiplicative background risk induces a more cautious behavior. We give necessary and sufficient conditions under which any change in the multiplicative background risk with respect to the Nth-degree stochastic dominance raises local risk aversion. Surprisingly, decreasing relative risk aversion of any order up to N in the sense of Pratt (Econometrica 32:122–136, 1964) coupled with decreasing relative risk aversion in the sense of Ross (Econometrica 49:631–638, 1981) are sufficient to guarantee an increase in local risk aversion after any deterioration of the multiplicative background risk thanks to the Nth-degree stochastic dominance. We link our results concerning second-order stochastic dominance with the concept of multiplicative risk vulnerability.     

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.     

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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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.     

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.

     

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.     

Observing different orders of risk aversion

A decision maker's attitude towards risk is said to be of orderi, i=1, 2, if for every given riskē with expected value zero, the risk premium the decision maker is willing to pay to avoid the risktē goes witht to zero at the same order ast ⁱ. This article presents an experiment testing the order of decision makers' attitudes toward risk. Its major result is that both attitudes exist, each in significant proportions. Moreover, two classes of first-order behavior are defined. The rank-dependent model (Quiggin, 1982) belongs to one, the disappointment aversion model (Gul, 1991) to the other. We show that only the first of these two classes appears among our subjects.     

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.     

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.     

Risk behavior for gain,loss, and mixed prospects

This study extends experimental tests of (cumulative) prospect theory (PT) over prospects with more than three outcomes and tests second-order stochastic dominance principles (Levy and Levy, Management Science 48:1334–1349, 2002; Baucells and Heukamp, Management Science 52:1409–1423, 2006). It considers choice behavior of people facing prospects of three different types: gain prospects (losing is not possible), loss prospects (gaining is not possible), and mixed prospects (both gaining and losing are possible). The data supports the distinction of risk behavior into these three categories of prospects, Further, probability weighting and diminishing sensitivity of utility as predicted by PT are observed. Loss aversion is, however, less pronounced, except for choices where one prospect is degenerate. The data suggests that the probability of losing may be relevant for loss aversion.     

The ostrich effect: Selective attention to information

We develop and test a model which links information acquisition decisions to the hedonic utility of information. Acquiring and attending to information increases the psychological impact of information (an impact effect), increases the speed of adjustment for a utility reference-point (a reference-point updating effect), and affects the degree of risk aversion towards randomness in news (a risk aversion effect). Given plausible parameter values, the model predicts asymmetric preferences for the timing of resolution of uncertainty: Individuals should monitor and attend to information more actively given preliminary good news but “put their heads in the sand” by avoiding additional information given adverse prior news. We test for such an “ostrich effect” in a finance context, examining the account monitoring behavior of Scandinavian and American investors in two datasets. In both datasets, investors monitor their portfolios more frequently in rising markets than when markets are flat or falling.

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.     

Intentional time inconsistency

We propose a theoretical model to explain the usage of time-inconsistent behavior as a strategy to exploit others when reputation and trust have secondary effects on the economic outcome. We consider two agents with time-consistent preferences exploiting common resources. Supposing that an agent is believed to have time-inconsistent preferences with probability p,  we analyze whether she uses this misinformation when she has the opportunity to use it. Using the model originally provided by Levhari and Mirman (Bell J Econ 11(1):322–334, 1980), we determine the optimal degree of present bias that the agent would like to have while pretending to have time-inconsistent preferences and we provide the range of present-bias parameter under which deceiving is optimal. Moreover, by allowing the constant relative risk aversion class of utility form, we characterize the distinction between pretending to be naive and sophisticated.

     

Reflections on gains and losses: A 2 × 2 × 7 experiment

What determines risk attraction or aversion? We experimentally examine three factors: the gain-loss dichotomy, the probabilities (0.2 vs. 0.8), and the money at risk (7 amounts). We find that the majority display risk attraction for small amounts of money, and risk aversion for larger amounts. Yet the frequency of risk attraction varies according to the gain-loss dichotomy and the probabilities. Kahneman and Tversky studied gain-loss reflections. We submit that a reflection can be decomposed into a translation and a probability switch. We find significant translation and switch effects, which are of comparable magnitude, a result that is equidistant from the diverging two popular views inspired by Prospect Theory: the gain-loss asymmetry, and the fourfold pattern.     

Risk preferences and changes in background risk

We present two theorems that yield necessary and sufficient conditions for first- and second-degree stochastic dominance deteriorations of background risk to increase risk aversion with respect to foreground risk. We require that any change in a foreground risk that is undesirable remains so after a background risk changes in a way that is either unfair, undesirable in the sense of reducing expected utility, or undesirable in the sense of increasing expected marginal utility. Our results thus characterize utility functions that are, respectively, vulnerable, proper, or standard with respect to changes in background risk.

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