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First Order Versus Second Order Risk Aversion
Abstract
This paper defines a new concept of attitude towards risk. For an actuarially fair random variable is the risk premium the decisionmaker is willing to pay to avoid . In expected utility, and as it turns out, in the case of smooth Freéchet differentiability of the representation functional, π′(0) = 0. There are models (e.g., rank dependent probabilities) in which . We call the latter attitude as being of order 1, and we call the first one attitude of order 2. These concepts are then applied to analyze the problem of full insurance.
... We say that a DM displays attitude towards probability of order 1 and order 2 if, for risk changes with small probability mass, the associated probability premium is linear in the probability mass and proportional to the square of the probability mass, respectively. These orders of attitude towards probability can be seen to represent the "probability counterparts" of the well-known orders of attitude towards risk introduced by Segal and Spivak [37]. ...
... The orders of probability aversion can be viewed as the probability equivalents of the orders of risk aversion. Indeed, in important work, Segal and Spivak [37] introduce firstand second-order risk aversion. They consider the risk premium of Pratt [28] and Arrow [2,3] and analyze its limiting behavior for a zero-mean risk with payoff tending to zero. ...
... We thus find that RDU and EU maximizers usually exhibit first-order probability aversion, while DT DMs with smooth probability weighting function are second-order probability averters. From Segal and Spivak [37] we already know that RDU and DT maximizers usually exhibit first-order risk aversion, while EU DMs with smooth utility function are risk averse of order 2. That is, under the RDU model, both probability aversion and risk aversion are usually first-order phenomena. ...
Employing a generalized definition of Pratt (1964) and Arrow's (1965, 1971) probability premium, we introduce a new concept of attitude towards probability. We illustrate in a problem of risk sharing that whether attitude towards probability is a first-order or second-order phenomenon has important economic applications. By developing a local approximation to the probability premium, we show that the canonical rank-dependent utility model usually exhibits attitude towards probability of first order, whereas under the dual theory with smooth probability weighting functions attitude towards probability is a second-order trait.
... Several authors have explained a preference for full insurance at actuarially unfair premiums. One explanation is based on first-order risk aversion (Schmidt 1998;Segal and Spivak 1990;Schlesinger 1997), which can be accommodated by EU (Dionne and Li 2014), but arises more prominently in RDU with a concave or convex probability weighting function, see Segal and Spivak's (1990) Proposition 4. In a similar vein, Doherty and Eeckhoudt (1995) use Yaari's (1987) dual theory with a concave probability weighting function to explain full insurance at unfair premiums. By assumption, these papers rule out the commonly found pattern of inverse S-shaped probability weighting (e.g., Abdellaoui et al. 2011). ...
... Several authors have explained a preference for full insurance at actuarially unfair premiums. One explanation is based on first-order risk aversion (Schmidt 1998;Segal and Spivak 1990;Schlesinger 1997), which can be accommodated by EU (Dionne and Li 2014), but arises more prominently in RDU with a concave or convex probability weighting function, see Segal and Spivak's (1990) Proposition 4. In a similar vein, Doherty and Eeckhoudt (1995) use Yaari's (1987) dual theory with a concave probability weighting function to explain full insurance at unfair premiums. By assumption, these papers rule out the commonly found pattern of inverse S-shaped probability weighting (e.g., Abdellaoui et al. 2011). ...
We provide a comprehensive analysis of the impact of probability weighting on optimal insurance demand in a unified framework. We identify decreasing relative overweighting as a new local condition on the probability weighting function that is useful for comparative static analysis. We discuss the effects of probability weighting on coinsurance, deductible choice, insurance demand for low-probability, high-impact risks versus high-probability, low-impact risks, and insurance demand in the presence of nonperformance risk. Probability weighting can make better or worse predictions than expected utility depending on the insurance demand problem at hand.
... This dichotomy can be linked to the notions of first-and second-order risk aversion introduced by Segal and Spivak [26]; see also Lang [22] and Eeckhoudt and Laeven [12]. Indeed, the insightful analysis of Segal and Spivak [26] shows that the risk premia have distinct limiting behavior for "small" risks under RDU and under expected utility: under the RDU model risk aversion is a first-order phenomenon, while under the expected utility model risk aversion is only a second-order phenomenon; see also Eeckhoudt and Laeven [12], Section 5.3. ...
... This dichotomy can be linked to the notions of first-and second-order risk aversion introduced by Segal and Spivak [26]; see also Lang [22] and Eeckhoudt and Laeven [12]. Indeed, the insightful analysis of Segal and Spivak [26] shows that the risk premia have distinct limiting behavior for "small" risks under RDU and under expected utility: under the RDU model risk aversion is a first-order phenomenon, while under the expected utility model risk aversion is only a second-order phenomenon; see also Eeckhoudt and Laeven [12], Section 5.3. ...
We analyze the limiting behavior of the risk premium associated with the Pareto optimal risk sharing contract in an infinitely expanding pool of risks under a general class of law-invariant risk measures encompassing rank-dependent utility preferences. We show that the corresponding convergence rate is typically only $n^{1/2}$ instead of the conventional $n$, with $n$ the multiplicity of risks in the pool, depending upon the precise risk preferences.
... Thus, our main results in Section 3 remain valid. This specification of function g(·) also implies that the policyholder is first-order risk averse in the net payoff of insurance, as defined in Segal and Spivak (1990), at the break-even point. In the following proposition, we present our result on the optimal insurance contract under narrow framing with loss aversion. ...
... This may be because the reference point is shifted when moving from one decision situation to another. A recent work by Eeckhoudt et al. (2018) studied insurance purchase from the policyholder who is first-order risk averse, as defined by Segal and Spivak (1990), at the full insurance wealth level which plays a role of reference point or target. Not reaching this target is painful for the policyholder. ...
In this paper, we study insurance decisions when the policyholder evaluates insurance with narrow framing. We show that due to aversion to risk on the net insurance payoff, i.e., insurance indemnity minus insurance premium, narrow framing reduces insurance demand. This helps explaining the observed low insurance demand in many insurance markets. We also show that the optimal insurance contract involves a deductible and the coinsurance of losses above the deductible when transaction costs depend on the actuarial value of the policy. Moreover, when the policyholder is loss averse over the net insurance payoff, a fixed indemnity equal to insurance premium should be paid for a range of intermediate losses.
... If the expected gross return of the risky asset exceeds unity, E k > 1, a secondorder risk-averse individual invests a positive amount in the risky asset (see Segal and Spivak, 1990). We denote by δ * A the share of the endowment the individual invests in the risky asset. ...
We conduct the first incentivized laboratory experiment to test the effects of multiplicative background risk on risk-taking behavior. Using a within-subject design, subjects make two investment decisions, one in the absence and one in the presence of multiplicative background risk. We consider a symmetric and a left-skewed background risk. Based on the first three moments of the final wealth distribution, we predict that multiplicative background risk can increase or decrease risk taking, with the increase being more plausible when the background risk is left-skewed. We find that multiplicative background risk increases risk taking. Consistent with our prediction , this effect is driven by the left-skewed background risk and leads to an estimated increase in risk taking of approximately 12%. We do not find a significant change in risk taking in response to the symmetric background risk. In the experiment, risk taking decreases for 34%, stays the same for 27%, and increases for 39% of the subjects.
... W 1991 r. Kahneman i Tversky ustalili i przebadali empirycznie, że najbardziej pożądanym stosunkiem straty do zysku jest 1:2 (loss -aversion to gain -attraction) [13]. Wielu autorów, takich jak Rabin, Thaler (2001), Segal, Spivak (1990), Epstein (1992) prowadziło rozważania dotyczące uczestnictwa w grach o zróżnicowanych stawkach, a także rozważało stosunek do ryzyka uczestników takich gier w różnych kontekstach, przy różnych funkcjach użyteczności. W 2000 r. ...
Many important economic decisions involve an element of risk. Risk aversion is a concept in economic, game theory, finance and psychology related to the behavior of consumers, players and investors under uncertainty. Loss aversion is an important component of a phenomenon that has been discussed a lot in recent years. Loss aversion is a tendency to feel the pain of a loss more acutely than pleasure of the equal – sized gain. Many scientists have analyzed the problem of profitability in the game. Some authors presented certain features, by which “safe” games played once should be characterized. Kahneman and Tversky (1991) showed that loss – aversion – to – gain – attraction ratio should amount to 1:2. The aim of this paper is to show an asymptotically effective strategy which enables the risk – aversive player to establish boundary variables loss and gain at each stage of the repeated game.
... The larger expected rates of return of riskier assets translate into a larger Weighted Average 4 Of course, prudence and ambiguity aversion matter too but we will focus on risk aversion for the purpose of this paper. 5 This result requires that risk aversion has only second-order effects, as is the case under Expected Utility (Segal and Spivak, 1990). This is illustrated by the Arrow-Pratt approximation for the risk premium, which is proportional to the variance (i.e., the square of the size of the risk) of final consumption. ...
... In 1991 they, as well as others, estimated the loss aversion to gain attraction ratio to be about 2:1. Many authors such as Rabin and Thaler (2001), Segal and Spivak (1990), and Epstein (1992) have discussed the problem of participation in games of different stakes, and examined the participants' attitude to risk in different contexts and with different utility functions. ...
... In 1991 they, as well as others, estimated the loss aversion to gain attraction ratio to be about 2:1. Many authors such as Rabin and Thaler (2001), Segal and Spivak (1990), and Epstein (1992) have discussed the problem of participation in games of different stakes, and examined the participants' attitude to risk in different contexts and with different utility functions. ...
... In 1991 they, as well as others, estimated the loss aversion to gain attraction ratio to be about 2:1. Many authors such as Rabin and Thaler (2001), Segal and Spivak (1990), and Epstein (1992) have discussed the problem of participation in games of different stakes, and examined the participants' attitude to risk in different contexts and with different utility functions. ...
... 4 This result requires that risk aversion has only second-order effects, as is the case under Expected Utility (Segal and Spivak, 1990). This is illustrated by the Arrow-Pratt approximation for the risk premium, which is proportional to the variance (i.e., the square of the size of the risk) of final consumption. ...
We surveyed economists' attitudes toward adjusting discount rates to the risk profile of public programs. Three-quarters of respondents recommend to use project-specific discount rates. For example, on average, respondents discount railway infrastructures more than hospitals and climate mitigation. But the degree of discount discrimination between distinct risk profiles of different projects is fairly limited in our sample given the differences in beta's for these projects. Economic experts thus penalize risky public projects far less than financial markets penalize private investments. We call this the "discount premium puzzle". Climate-economy models typically have a beta for global warming damages close to one, which sits uneasy with the low discount rate recommended to assess climate mitigation measures. Finally, among experts in favor of a single discount rate, there is no consensus on whether it should be based on the average cost of capital in the economy, the sovereign borrowing cost, or the Ramsey rule, which gives rise to disagreement over the level of the recommended discount rate.
... Rank-dependent expected utility theory sparked a great interest in the research community ( [61]). Axiomatizations were presented ( [77,92,1,97,93]). Generalizations have also been proposed ( [18,32]), and extensions were discussed ( [80,59,84]). ...
This thesis is an interdisciplinary work at the intersection of applied mathematics, economics, and machine learning. Our leitmotiv is to study models for Internet problematics: large interacting populations, games in social networks, decision making when facing risky choices, online click learning algorithms, and digital advertising auctions. We develop models for each of these thematics and study their properties. The problems we solve all have advertising, commercial, political, and sociological applications. The first part of the thesis studies controlled large populations with mean-field interactions, with natural interpretation to targeted advertising problems in a population under social influence. The second part studies games and economic behaviors, with the general goal to predict people's choices in complex situations, the first study being a social game in a large population, and the second one a decision under risk model parametrizing the well-established Cumulative Prospect theory, with explicit gamble valuation for gaussian rewards. We also discuss commercial and political applications. Finally, the third part formulates concrete strategic problems for digital advertising. We provide an online classification learning algorithm designed for click prediction in digital advertising, and in a second study, we develop an optimal control problem for targeted advertising auctions in a dynamic population.
... For example, as demonstrated by Rabin (2000), an expected-utility maximizer who always turns down 50-50 lose $10/gain $11 bets will always turn down ones where you lose $100 or can gain any sum of money with equal probability, simply because under expected utility theory risk aversion increases with the lottery outcomes. More generally, Hansson's (1988) and Rabin's (2000) critique concerns "first order risk aversion" models which allow for risk-averse behavior even if the stakes are small (Segal & Spivak, 1990; and see also emergence of loss aversion for smaller losses results in baffling predictions for a simple expected utility model. ...
Loss aversion, the argument that losses are given more weight than gains, has been recently shown to be absent in small losses. However, a series of studies by Mrkva et al. (2020) appear to demonstrate the existence of loss aversion even for smaller losses. We re-ran Mrkva et al.’s decision tasks after removing features of the task that differentiated losses from the gains, particularly asymmetries in sizes of gains and losses, an increasing order of losses, and status quo effects. The results show that we replicate Mrkva et al.’s (2020) findings in their original paradigm with online participants, yet in five studies where gains and losses were symmetrically presented in random order (n = 2,001), we find no loss aversion for small amounts, with loss aversion surfacing very weakly only for average losses of $40 (mean λ = 1.16). We do find loss aversion for higher amounts such as $100 (mean λ = 1.54) though it is not as extreme as previously reported. Furthermore, we find weak correlation between the endowment effect and loss aversion, with the former effect existing simultaneously with no loss aversion. Thus, when items are presented symmetrically, significant loss aversion emerges only for large losses, suggesting that it cannot be argued that (all) “losses loom larger than gains.”
... Because m (x) = 1 when m(x) is a nonlinear-form function. 10 It could be related to the work ofSegal and Spivak (1990), since the first-order risk aversion arises when the support of reference levels includes the risk-free outcome. ...
This article provides some analytical proofs for the effects of loss aversion and diminishing sensitivity on portfolio choice. We use the performance index (Omega ratio) of return distribution to characterize the threshold value that determines non-zero investment in the risky asset and show the impact of stochastic improvement of risky return on the threshold. We propose the measure of greater diminishing sensitivity and examine the loss aversion characterizations for large and small stakes. Moreover, we demonstrate that diminishing sensitivity and loss aversion may make opposite predictions on willingness to invest in the risky asset. Specifically, although loss aversion decreases the investment in the risky asset, diminishing sensitivity in the loss domain will predict the opposite.
... It presents the utility framework and how investors' responses to risk are shaped by investment outcomes. It is argued that the prospect theory's central feature is that value is carried by changes in wealth rather not absolute levels or final outcomes (Segal & Spivak, 1990). ...
Purpose: As the global community continues to grapple with the effect of the COVID-19 pandemic, there is a prospect of negative/undesirable business outcomes in Nigeria. The scourge has been a regular topic of discussion in businesses, policy-making, and the academic community in many countries around the world. The pandemic has introduced a novel systematic risk class in the real estate sector. This study investigated how real estate investors in Lagos State perceive the emerging business climate following the pandemic. Methodology: The study had a population of 2,400 while a sample size of 240 was drawn using a 10% rate. The study adopted a descriptive survey design. The statistical Package for Social Sciences (SPSS) was used to analyze data collected through questionnaires. Findings: The findings showed that COVID-19 had a significant relationship with the level of investment in real estate in Lagos State though at a weak linear associative level. Secondly, while risk managers in these sectors have adopted different strategies to minimize their risk exposures, the study revealed that funds providers are placing a higher risk premium on investible capital as a result of uncertainties in the economic outlook occasioned by the COVID-19 pandemic. However, the study shows that the demand for real estate property in Lagos State was not negatively affected to a significant extent as a result of the pandemic, but rather a shift in preferences by property owners and occupiers. The study has implications for taxation and housing policy as well as business planning decisions. Recommendation: The study recommends that government should create a favorable investment climate for real estate investors with appropriate risk mitigation measures that can reactivate the real estate sector while aiding its recovery from the COVID-19 debilitations.
... This feature of EU is the driving force in deriving R AS as a representation of WU . With the preferences in (1), rejection ofg ∈ G at all wealth levels is not possible unless η > 0, which in turn implies deviation from "second-order risk aversion" at all wealth levels -something ruled out by EU (see, for example, the discussion in Segal and Spivak 1990). Deviations from second-order risk aversion can be instrumental in addressing the Rabin (2000) critique of EU as well as the equity premium puzzle (e.g., Epstein and Zin 1990). ...
Hart (J Polit Econ, 119(4):617–638, 2011) argues that the Aumann and Serrano (J Polit Econ, 116(5): 810–836, 2008) and Foster and Hart (J Polit Econ, 117(5):785–814, 2009) measures of riskiness have an objective and universal appeal with respect to a subset of expected utility preferences, UH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {U}}}_H$$\end{document}. We show that mean-riskiness decision-making criteria using either measure violate expected utility and are generally inconsistent with optimal portfolio choices made by investors with preferences in UH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {U}}}_H$$\end{document}. We also demonstrate that riskiness measures satisfying Hart’s other behavioral requirements do not generally exist when his argument is generalized to incorporate non-expected utility preferences. Finally, we identify other attributes of the Aumann-Serrano and Foster-Hart measures that raise concerns over their operationalizability and usefulness in various decision making, risk management, and risk assessment settings.
... Although CT-preferences that satisfy strong risk aversion imply that u is strictly concave and also that weak risk aversion holds, the indifference sets of lotteries have convex kinks at certainty. Therefore, such CT-preferences exhibit first-order risk aversion as defined by Segal and Spivak (1990). This means that a CT-investor requires a strictly positive premium in order to invest in a risky asset. ...
In a temporal context, sure outcomes may yield higher utility than risky ones as they are available for the execution of plans before the resolution of uncertainty. By observing a disproportionate preference for certainty, empirical research points to a fundamental difference between riskless and risky utility. Chance Theory (CT) accounts for this difference and, in contrast to earlier approaches to separate risky and riskless utility, does not violate basic rationality principles like first-order stochastic dominance or transitivity. CT evaluates the lowest outcome of an act with the riskless utility v and the increments over that outcome, called chances, by subjective expected utility (EU) with a risky utility u . As a consequence of treating sure outcomes differently to risky ones, CT is able to explain the EU-paradoxes of Allais ( Econometrica, 21 (4): 503–546, 1953) that rely on the certainty effect, and also the critique to EU put forward by Rabin ( Econometrica, 68 (5): 1281–1292, 2000). Moreover, CT separates risk attitudes in the strong sense, captured entirely by u , from attitude towards wealth reflected solely through the curvature of v .
... Typically, nonexpected utility theories have indifference curves that are either smooth or have a kink at the point of full insurance. Accordingly, the latter predicts that consumers will demand either just as much or more insurance than does expected utility theory (c.f., Segal and Spivak, 1990). By contrast, when individuals frame their insurance purchases narrowly, they will buy less insurance than under expected utility. ...
This paper tests the claim that insurers often engage in risk‐shifting years before the materialization of a failure. It compares the mechanisms of insurance insolvency across different jurisdictions, using a first‐of‐its‐kind international database assembled by the authors, merging individual financial data together with information on impairments over the last 30 years in four of the largest insurance markets in the world (France, Japan, the UK, and the United States). Results show evidence that low profitability is a leading indicator of failures. Further, there is an asymmetry between life insurance, where bond investment is highly significant, and nonlife insurance sectors, where operating inefficiency plays a larger role. Moreover, this paper highlights differences across countries: a stronger reaction to operating inefficiency in nonlife insurance in France and a less positive impact of bond investment in life insurance in Japan. Both results are linked to differences in the functioning of insurance markets.
... The value function may change depending on the individual. For example, individuals may be risk-averse (Christopoulos et al., 2009;Dohmen et al., 2010;Albert and Duffy, 2012) and the Expected-Utility theory considers this possibility (Hershey and Schoemaker, 1980;Segal and Spivak, 1990;Karni and Schmeidler, 1991). Risk aversion occurs when the value function is concave, that is, it shows diminishing marginal utility (Hanoch and Levy, 1975). ...
Obtaining information from the world is important for survival. The brain, therefore, has special mechanisms to extract as much information as possible from sensory stimuli. Hence, given its importance, the amount of available information may underlie aesthetic values. Such information-based aesthetic values would be significant because they would compete with others to drive decision-making. In this article, we ask, “What is the evidence that amount of information support aesthetic values?” An important concept in the measurement of informational volume is entropy. Research on aesthetic values has thus used Shannon entropy to evaluate the contribution of quantity of information. We review here the concepts of information and aesthetic values, and research on the visual and auditory systems to probe whether the brain uses entropy or other relevant measures, specially, Fisher information, in aesthetic decisions. We conclude that information measures contribute to these decisions in two ways: first, the absolute quantity of information can modulate aesthetic preferences for certain sensory patterns. However, the preference for volume of information is highly individualized, with information-measures competing with organizing principles, such as rhythm and symmetry. In addition, people tend to be resistant to too much entropy, but not necessarily, high amounts of Fisher information. We show that this resistance may stem in part from the distribution of amount of information in natural sensory stimuli. Second, the measurement of entropic-like quantities over time reveal that they can modulate aesthetic decisions by varying degrees of surprise given temporally integrated expectations. We propose that amount of information underpins complex aesthetic values, possibly informing the brain on the allocation of resources or the situational appropriateness of some cognitive models.
... 9 Such utility functions exhibit first-order risk aversion at the kink, which can explain a preference for full insurance even when the premium is actuarially unfair (see Segal and Spivak, 1990). This behavior is often observed in the laboratory and in the field (e.g., Shapira and Venezia, 2008;Sydnor, 2010;Corcos et al., 2017;Jaspersen et al., 2022). ...
We compile, generalize and extend the results about the comparative static effects of risk changes on optimal risk-reduction and saving behavior. We use the time-separable discounted expected-utility model and consider income risk, inflation risk, and interest rate risk. For each type of risk, we trace precautionary risk reduction and precautionary saving to the individual's underlying risk apportionment preference. Risk reduction and saving are shown to be Edgeworth-Pareto substitutes for (mixed) risk-averters and Edgeworth-Pareto complements for (mixed) risk-lovers. This introduces feedback effects, and we identify conditions for jointly optimal risk-reduction and saving decisions to be N th-degree risk substitutes or N th-degree risk complements for the different types of risks. Keywords: Changes in risk · precautionary effects · risk apportionment · comparative statics · income risk · inflation risk · interest rate risk JEL-Classification: D81 · D90 · D91 · E21 · G22 * An earlier version of this paper was circulated under the title "Risk management and saving: On income effects and background risk". We are indebted to Cassandra Cole, Roland Eisen and Sebastian Schlütter for helpful comments and suggestions. We also would like to thank conference participants at the 39th Seminar
... For example, given any risk-averse expected utility preference, if a gamble where one loses $100 or wins $110 with equal probability is rejected at all wealth levels below $300,000, then a gamble where one loses $2000 and wins $12,000,000 with equal probability must also be rejected at wealth levels below $290,000. Ang, Bekaert, and Liu (2005) show that this shortcoming of expected utility may be addressed by alternative preference theories that feature first-order risk aversion (Segal and Spivak, 1990), such as loss aversion, rankdependent utility or disappointment aversion (Kahneman and Tversky, 1979;Quiggin, 1982;Gul, 1991). 4 Nonetheless, Barberis, Huang, and Thaler (2006) highlight that in standard dynamic models of recursive preferences, background risk can smooth out first-order risk aversion. ...
Building on Pomatto, Strack, and Tamuz (2020), we identify a tight condition for when background risk can induce first-order stochastic dominance. Using this condition, we show that under plausible levels of background risk, no theory of choice under risk can simultaneously satisfy the following three economic postulates: (i) decision-makers are risk averse over small gambles, (ii) their preferences respect stochastic dominance, and (iii) they account for background risk. This impossibility result applies to expected utility theory, prospect theory, rank-dependent utility, and many other models. (JEL D81, D91)
... However, in the 1980s, two definitions emerged, with the first one established by Abderrezak (1985): non-expected utility theories are used to define risk aversion as the difference between certainty equivalence and the expected value under consideration. According to Segal and Spivak (1990), the second definition was proposed by Chew, Karni and Safra (1987), which asserts that risk aversion exists when preferences decrease as risk increases. Even though the definitions look different, when the von Neumann-Morgenstern theorem is applied to the two definitions, it 2.Approximately $0.33. ...
Orientation: Poverty is deeply entrenched in South Africa, and various initiatives to reduce it have not been successful. Behavioural economics may help explain this by testing whether individuals exhibit path dependence when making decisions under uncertainty. Research purpose: The aim of this article was to test the presence of path dependence and the determinants of risk aversion. Motivation for the study: This study was conducted in Eastern Cape province of South Africa, which is largely rural, with a high level of poverty, where 36% of households depend on social security grants, and unemployment is higher than the national average. Research approach/design and method: An experimental design approach was applied, eliciting risk under different prospects. Descriptive statistics was used to determine path dependency, with random effects regression used to investigate the determinants of risk aversion. Main findings: The results show the existence of path dependence, with individuals who are more risk averse unwilling to change their choices even if they stand to win or lose more than in the previous period. Practical/managerial implications: Policymakers need to consider these behavioural characteristics in formulating policies to reduce poverty. Information and how that information is presented (framing) are central for impactful policy formulation. Contribution/value-add: The results here have policy implications on addressing triple challenges bedeviling South Africa and many developing countries.
... For example, given any risk-averse expected utility preference, if a gamble where one loses $100 or wins $110 with equal probability is rejected at all wealth levels below $300,000, then a gamble where one loses $2000 and wins $12,000,000 with equal probability must also be rejected at wealth levels below $290,000. Ang, Bekaert, and Liu (2005) show that this shortcoming of expected utility may be addressed by alternative preference theories that feature first-order risk aversion (Segal and Spivak, 1990), such as loss aversion, rankdependent utility or disappointment aversion (Kahneman and Tversky, 1979;Quiggin, 1982;Gul, 1991). 4 Nonetheless, Barberis, Huang, and Thaler (2006) highlight that in standard dynamic models of recursive preferences, background risk can smooth out first-order risk aversion. ...
We show that under plausible levels of background risk, no theory of choice under risk---such as expected utility theory, prospect theory, or rank dependent utility---can simultaneously satisfy the following three economic postulates: (i) Decision makers are risk-averse over small gambles, (ii) they respect stochastic dominance, and (iii) they account for background risk.
... Comme le font remarquer Eeclchoudt et ail.(1995), ceci est une illustration du concept de "first order risk aversion"(Segal_& Spivak 1990). Si x • est une solution intérieure du problème de maximisation (29), on obtient le corollaire suivant.Soient C = O, r > 0 et une jonction de production stochastique y = 8yx avec ro = 8yw et .Ero=µ. ...
Dans cet article, nous montrons comment l'introduction d'une contrainte financière dans le problème de maximisation de l'utilité espérée d'un agriculteur modifie ses décisions d'investissement. Bien que supposé intrinsèquement neutre au risque, l'agriculteur va se comporter comme un agent risquophobe ou risquophile, selon que ses liquidités initiales sont suffisantes ou insuffisantes pour honorer ses engagements de fin de période. De tels choix sont dus à la discontinuité de la fonction d'utilité apparente. Nous analysons ses choix d'investissement face à un projet risqué indivisible, puis lorsqu'il dispose d'une fonction de production stochastique à un intrant. Les modifications des conditions d'accès aux marchés financiers peuvent s'avérer être un outil de politique économique intéressant.
... It is easy to check that the slightly stronger Assumption 2 used in Section 4 holds in B)). Finally, many other families of preferences, involving, for instance, first-order risk aversion (Segal and Spivak (1990)), also fit within our general framework. ...
This paper studies competitive allocations under adverse selection. We first provide a general necessary and sufficient condition for entry on an inactive market to be unprofitable. We then use this result to characterize, in an active market, a unique budget-balanced allocation implemented by a market tariff making additional trades with an entrant unprofitable. Motivated by the recursive structure of this allocation, we finally show that it emerges as the essentially unique equilibrium outcome of a discriminatory ascending auction. These results yield sharp predictions for competitive nonexclusive markets.
We study a generalization of the classical monopoly insurance problem under adverse selection (see Stiglitz 1977) where we allow for a random distribution of losses, possibly correlated with the agent’s risk parameter that is private information. Our model explains patterns of observed customer behavior and predicts insurance contracts most often observed in practice: these consist of menus of several deductible-premium pairs or menus of insurance with coverage limits–premium pairs. A main departure from the classical insurance literature is obtained here by endowing the agents with risk-averse preferences that can be represented by a dual utility functional (Yaari 1987). (JEL D81, D82, D86, D91, G22)
Most Western countries use a single discount rate to evaluate public investments and policies. This ignores the cost of risk, in a world where most risk markets exhibit surprisingly large prices of risk. The current discounting guidelines generate a misallocation of capital that entails a large welfare cost. We claim that the well-established asset pricing literature provides a strong normative justification in favor of risk-adjusting discount rates. More specifically, project-specific discount rates should be increasing in the income elasticity of the project's net benefit. This will favor projects whose net benefit materializes preferentially in low-income states, thereby recognizing their insurance benefit ex ante. The intuition is simple, the welfare benefit of the reform is large, and the methodology only requires evaluators to estimate an income elasticity on top of what is required in the current approach. It is time to fix our public discounting systems.
Expected final online publication date for the Annual Review of Financial Economics, Volume 15 is November 2023. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.
Gul and Pesendorfer (2015) propose a promising theory of decision under uncertainty, they dub Hurwicz expected utility (HEU). HEU is a special case of α ‐maxmin EU that allows for preferences over sources of uncertainty. It is consistent with most of the available empirical evidence on decision under risk and uncertainty. We show that HEU is also tractable and can readily be measured and tested. We do this by deriving a new two‐parameter functional form for the probability weighting function, which fits our data well and which offers a clean separation between ambiguity perception and ambiguity aversion. In two experiments, we find support for HEU's predictions that ambiguity aversion is constant across sources of uncertainty and that ambiguity aversion and first order risk aversion are positively correlated.
This article compares the classic liability rules, negligence, and strict liability, under the hypothesis that injurers and victims formulate subjective beliefs about the probabilities of harm. Parties may reasonably disagree in their assessment of the precautionary measures available: a measure regarded as safe by one party may be regarded as not safe by the other. By relying on the notions of Pareto efficiency and “No Betting” Pareto efficiency, the article shows that negligence is the optimal liability rule when injurers believe that the probability of harm is always higher than the victims do, while strict liability with overcompensatory damages is the optimal rule in the opposite case. The same results apply to bilateral accidents and, specifically, to product-related harms in competitive markets. Overcompensatory (“punitive”) damages provide consumers with insurance against their own pessimism. (JEL K13, D83, D62)
When an investor is ambiguous about the asset returns' correlation and evaluates the portfolio in a multi-prior framework, we show that the optimal portfolio may contain only a fraction of risky assets and may be independent of feasible correlation matrices. In particular, if the level of ambiguity is high enough, the optimal portfolio contains only the one with the highest Sharpe ratio. Moreover, we demonstrate that the optimal portfolio may not change when the Sharpe ratios of some assets change. Ambiguity-aversion on correlation uncertainty explains portfolio concentration and portfolio inertia in household portfolios and retirement accounts, and the model can explain the growth of indexing and ETFs from an optimal portfolio choice perspective. We further show that these properties do not hold in an alternative smooth ambiguity model, suggesting that the smooth ambiguity model does not depart from the standard model sufficiently to explain portfolio concentration and portfolio inertia.
We study auction design for bidders equipped with non-expected utility preferences that exhibit constant risk aversion (CRA). The CRA class is large and includes loss-averse, disappointment-averse, mean-dispersion and Yaari’s dual preferences as well as coherent and convex risk measures. Any preference in this class displays first-order risk aversion, contrasting the standard expected utility case which displays second-order risk aversion. The optimal mechanism offers “full-insurance” in the sense that each agent’s utility is independent of other agents’ reports. The seller excludes less types than under risk neutrality, and awards the object randomly to intermediate types. Subjecting intermediate types to a risky allocation while compensating them when losing allows the seller to collect larger payments from higher types. Relatively high types are willing to pay more, and their allocation is efficient.
This work examines optimal demand for insurance coverage when the insured has nonadditive subjective probability beliefs about loss uncertainty. By showing the equivalent conditions of Jensen's inequality under the Choquet expected utility framework, we not only provide the threshold on the insurance premium for Mossin's theorem to hold but also explore the joint effect of both risk aversion and uncertainty aversion on optimal insurance demand.
We survey 2,484 U.S. individuals with at least $1 million of investable assets about how well leading academic theories describe their financial beliefs and personal investment decisions. The wealthy's beliefs about financial markets and the economy are surprisingly similar to those of the average U.S. household, but the wealthy are less driven by discomfort with the market, financial constraints, and labor income considerations. Portfolio equity share is most affected by professional advice, time until retirement, personal experiences, rare disaster risk, and health risk. Concentrated equity holding is most often motivated by belief that the stock has superior risk-adjusted returns. Beliefs about how expected returns vary with stock characteristics frequently differ from historical relationships, and more risk is not always associated with higher expected returns. Active equity fund investment is most motivated by professional advice and the expectation of higher average returns. Berk and Green (2004) rationalize return chasing in the absence of fund performance persistence by positing that past returns reveal managerial skill but there are diminishing returns to scale in active management. Forty-two percent of respondents agree with the first assumption, 33% with the second, and 19% with both.
The standard problem of portfolio choice between one risky and one riskless asset is analyzed in the model of expected utility with a safety-first component that is represented by the probability of final wealth exceeding a “safety” wealth level. It finds that a positive expected excess return remains sufficient for investing a positive amount in the risky asset except in the special situation where the safety wealth level coincides with the wealth obtained when the entire initial wealth is invested in the riskless asset. In this situation, the optimal amount invested in the risky asset is zero if the weight on the safety-first component is sufficiently large. Comparative statics analysis reveals that whether the optimal amount invested in the risky asset becomes smaller as the weight on the safety-first component increases depends on whether the safety wealth level is below the wealth obtained when the entire initial wealth is invested in the riskless asset. Further comparative statics analyses with respect to the safety wealth level and the degree of risk aversion in the expected utility component are also conducted.
We propose a general family of piecewise hyperbolic absolute risk aversion (PHARA) utility, including many non-standard utilities as examples. A typical application is the composition of an HARA preference and a piecewise linear payoff in hedge fund management. We derive a unified closed-form formula of the optimal portfolio, which is a four-term division. The formula has clear economic meanings, reflecting the behavior of risk aversion, risk seeking, loss aversion and first-order risk aversion. One main finding is that risk-taking behaviors are greatly increased by non-concavity and reduced by non-differentiability.
Many experiments and field studies indicate that individuals have an asymmetric attitude towards gains versus losses. In this paper, we extend the canonic tournament model by assuming the workers' preferences exhibit disappointment aversion. First, we find the winning prize is first increasing and then decreasing in volatility and the losing prize shows the opposite. Furthermore, when the volatility exceeds a threshold, both the winning and losing prizes are reduced to zero. By contrast, there is no such kink for the risk aversion case. Finally, we find the piece rates always dominate rank‐order tournaments when the workers are disappointment averse.
We introduce the concept of forward rank‐dependent performance criteria, extending the original notion to forward criteria that incorporate probability distortions. A fundamental challenge is how to reconcile the time‐consistent nature of forward performance criteria with the time‐inconsistency stemming from probability distortions. For this, we first propose two distinct definitions, one based on the preservation of performance value and the other on the time‐consistency of policies and, in turn, establish their equivalence. We then fully characterize the viable class of probability distortion processes and provide the following dichotomy: it is either the case that the probability distortions are degenerate in the sense that the investor would never invest in the risky assets, or the marginal probability distortion equals to a normalized power of the quantile function of the pricing kernel. We also characterize the optimal wealth process, whose structure motivates the introduction of a new, ‘distorted’ measure and a related market. We then build a striking correspondence between the forward rank‐dependent criteria in the original market and forward criteria without probability distortions in the auxiliary market. This connection also provides a direct construction method for forward rank‐dependent criteria. Finally, our results on forward rank‐dependent performance criteria motivate us to revisit the classical (backward) setting. We follow the so‐called dynamic utility approach and derive conditions for existence and a construction of dynamic rank‐dependent utility processes.
Expected utility theory with a smooth utility function predicts that, when allocating wealth between a risky and a riskless asset, investors allocate a positive amount to the risky asset whenever its expected return exceeds the riskless rate of return. A large number of people invest none of their wealth in risky assets, though, leading to the ”participation puzzle.” This paper explores whether the participation puzzle can be addressed when the utility function has a kink at the reference wealth level. It shows that when the reference wealth level is initial wealth increased by the riskless rate of return, there exists a range of expected excess returns for the risky asset for which the investor takes no position. Moreover, this range of expected excess returns is described by comparing a common performance measure of stock returns, the Omega Function, to a function of preference parameters. However, if the reference wealth level is any other constant, the usual expected utility prediction holds and investors allocate at least some of their wealth to the risky asset whenever it has a positive expected excess return.
This paper characterizes the stochastic deterioration resulting from taking a zero-mean financial risk in the presence of correlated non-financial background risk. We show in particular that it has an equivalent stochastic order as well as a necessary and sufficient “integral condition” that implies and is implied by a particular sense in which the stochastic deterioration can be decomposed into a “correlation increase” and a “marginal risk increase”. We further characterize a measure of aversion to the stochastic deterioration. These characterizations provide for a more general framework for formulating concepts of increases in risk and correlation and for better understanding risk management decisions governed by individuals’ attitudes to them.
Experimental studies have shown that the key behavioral assumption of expected utility theory, the so-called "independence axiom," tends to be systematically violated in practice. Such findings would lead us to question the empirical relevance of the large body of literature on the behavior of economic agents under uncertainty which uses expected utility analysis. The first purpose of this paper is to demonstrate that the basic concepts, tools, and results of expected utility analysis do not depend on the independence axiom, but may be derived from the much weaker assumption of smoothness of preferences over alternative probability distributions. The second purpose of the paper is to show that this approach may be used to construct a simple model of preferences which ties together a wide body of observed behavior toward risk, including the Friedman-Savage and Markowitz observations, and both the Allais and St. Petersburg Paradoxes.
The main result of this paper is a generalization of the quasilinear mean of Nagumo [29], Kolmogorov [26], and de Finetti [17]. We prove that the most general class of mean values, denoted by $M_{\alpha \phi}$, satisfying Consistency with Certainty, Betweenness, Substitution-independence, Continuity, and Extension, is characterized by a continuous, nonvanishing weight function α and a continuous, strictly monotone value-like function φ. The quasilinear mean $M_{\phi}$ results whenever the weight function α is constant. Existence conditions and consistency conditions with first and higher degree stochastic dominance are derived and an extension of a well known inequality among quasilinear means, which is related to Pratt's [31] condition for comparative risk aversion, is obtained. Under the interpretation of mean value as a certainty equivalent for a lottery, the $M_{\alpha \phi}$ mean gives rise to a generalization of the expected utility hypothesis which has testable implications, one of which is the resolution of the Allais "paradox." The $M_{\alpha \phi}$ mean can also be used to model the equally-distributed-equivalent or representative income corresponding to an income distribution. This generates a family of relative and absolute inequality measures and a related family of weighted utilitarian social welfare functions.
This paper evaluates the benefits to consumers from price stabilization in terms of the convexity-concavity properties of the consumer's indirect utility function. It is shown that in the case where only a single commodity price is stabilized, the consumer's preference for price instability depends upon four parameters: the income elasticity of demand for the commodity, the price elasticity of demand, the share of the budget spent on the commodity, and the coefficient of relative risk aversion. All of these parameters enter in an intuitive way and the analysis includes the conventional consumer's surplus approach as a special case. The analysis is extended to consider the benefits of stabilizing an arbitrary number of commodity prices. Finally, some issues related to the choice of numeraire and certainty price in this context are discussed.
This paper concerns utility functions for money. A measure of risk aversion in the small, the risk premium or insurance premium for an arbitrary risk, and a natural concept of decreasing risk aversion are discussed and related to one another. Risks are also considered as a proportion of total assets.
Expected utility with rank dependent probability theory is a model of decision-making under risk where the preference relations on the set of probability distributions is represented by the mathematical expectation of a utility function with respect to a transformation of the probability distributions on the set of outcomes. This paper defines, based on Gâteaux differentiability, measures of risk aversion for such preferences which characterize the relation “more risk averse” and applies these measures to the analysis of unconditional and conditional portfolio choice problems.
Almost every phase of economic behavior is affected by uncertainty. The economic system has adapted to uncertainty by developing methods that facilitate the reallocation of risk among individuals and firms. The most apparent and familiar form for shifting risks is the ordinary insurance policy. Previous insurance decision analyses can be divided into those in which the insurance policy was exogenously specified (see John Gould, Jan Mossin, and Vernon Smith), and those in which it was not (see Karl Borch, 1960, and Kenneth Arrow, 1971, 1973). In this paper, the pioneering work of Borch and Arrow—the derivation of the optimal insurance contract form from the model—is synthesized and extended.
We develop a Savage-type model of choice under uncertainty in which agents identify uncertain prospects with subjective compound lotteries. Our theory permits issue preference; that is, agents may not be indifferent among gambles that yield the same probability distribution if they depend on different issues. Hence, we establish subjective foundations for the Anscombe-Aumann framework and other models with two different types of probabilities. We define second-order risk as risk that resolves in the first stage of the compound lottery and show that uncertainty aversion implies aversion to second-order risk which implies issue preference and behavior consistent with the Ellsberg paradox.
This remark proves that Quiggin's anticipated utility function may solve the Allais paradox and the common ratio effect. For some generalizations of these it is needed to assume that the decision-weight function is concave.
A new theory of cardinal utility, with an associated set of axioms, is presented. It is a generalization of the von Neumann-Morgenstern expected utility theory, which permits the analysis of phenomena associated with the distortion of subjective probability.
This paper studies the rank-dependent model of choice under uncertainty proposed by J. Quiggin in 1982 and elaborated by M. E. Yaari in 1984. First, a rigorous axiomatic foundation for the model is provided. A very close analogy with expected utility theory is drawn permitting a considerably simplified treatment. Risk aversion and its measurement are then studied; two characterizations, one weaker and one stronger, are presented in addition to the one considered by Yaari. Lastly, risk aversion and other properties of th model are related to empirically observed departures from expected utility maximizing behavior. Copyright 1987 by Royal Economic Society.
This paper investigates the consequences of the following modification of Expected Utility theory: instead of requiring independence with respect to probability mixtures of risky prospects, require independence with respect to direct mixing of payments o f risky prospects. A new theory of choice under risk- a so-called Dual theory-is obtained. Within this new theory, the following questions are considered: (1) numerical representation of preferences; (2) properties of the utility function; ( 3) the possibility for resolving the "paradoxes" of Expected Utilit y theory; ( 4) the characterization of risk aversion; and (5) comparative statics. The paper ends with a discussion of other non-Expected Utility theories proposed recently. Copyright 1987 by The Econometric Society.
A theory of choice under uncertainty is proposed which removes the completeness assumption from the Anscombe-Aumann formulation of Savage's theory and introduces an inertia assumption. The inertia assumption is that there is such a thing as the status quo and an alternative is accepted only if it is preferred to the status quo. This theory is one way of giving rigorous expression to Frank Knight's distinction between risk and uncertainty.
Knightian decision theory: Part I, Cowles Foundation DP #807
- T F Bewley
T. F. BEWLEY, Knightian decision theory: Part I, Cowles Foundation DP #807, Yale University, 1986.