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Risk Aversion and Prudence in the Large
Abstract
The familiar measures of absolute and relative risk aversion constructed by Pratt and Arrow, along with the measures of absolute and relative prudence inspired by Leland and later developed by Kimball, are local instruments based on the first and second derivatives of utility at a specific level of wealth. As such, they are applicable only to infinitesimal risks—those for which differential calculus is a suitable analytical tool. Consequently, they may not accurately gauge preferences regarding the larger risks typically encountered in practice. To address this problem, the present paper develops more general, closed-form index measures of risk aversion and prudence that are applicable to either large or small risks. The new measures are exact in that they do not rely on approximations, they can be implemented empirically without knowledge of the functional form of utility, and they do not require information regarding pre-existing wealth.
... It is also possible, however, to conceive of relative risk aversion as an arc elasticity, whose computation does not rely on differential calculus (Eisenhauer, 2010). Specifically, Eisenhauer (2006) develops a discrete approach for use with the expected utility model, and shows that it is the more appropriate method for evaluating aversion to large-scale risks. We now proceed to demonstrate that this approach also resolves the conflicts outlined above. ...
... Following Eisenhauer (2006), let us use the slopes of the chords between these points to define the following discrete analogues of marginal utility (MU): ...
... In much the same manner, the overreliance on variance in the calculus-based Pratt-Arrow measures of risk aversion induces a counterintuitive result for the lotteries described above. 15 In presenting the discrete alternative, Eisenhauer (2006) demonstrates that, like a(w) or r(w), R can also be represented in terms of the mean and variance of wealth, though these terms enter the calculation of R in a more complex manner than they enter the 13 As noted above, the increase in g reduces a(w) provided that is not too large. 14 Indeed, an increase in g alone does not constitute a mean-preserving spread, and thus does not represent a risk in the sense of Rothschild and Stiglitz (1970). ...
When the functional form of utility is unknown, conventional measures of risk aversion are often approximated by applying a Taylor series expansion to expected utility. This is shown to produce counterintuitive rank-orderings of risk preferences for individuals who are willing to pay equal reservation prices in lotteries with different prizes. Moreover, individuals who are unwilling to participate in favorable lotteries may be incorrectly identified as having a finite aversion to risk. Correct orderings are obtained by applying a discrete measure of relative risk aversion. The contrast between the conventional and discrete measures is illustrated with data from three Dutch surveys.
... To convert a(X) into an index number, Pratt (1964) and Arrow (1965) multiply a(X) by X to define the coefficient of relative risk aversion, r(X) = Xa(X), which represents the point elasticity of marginal utility with respect to X. Pratt's approximation then gives . Unfortunately, both the exact and approximate coefficients are based on differential calculus, and are therefore applicable exclusively to infinitesimal changes in the argument of the utility function; that is, they are only suitable for use with trivial risks (Eisenhauer, 2006(Eisenhauer, , 2008. 5 Thus, Pratt (1964) described them as "risk aversion in the small". For large risks, the Pratt-Arrow coefficient often yields perverse rankings of preferences (Eisenhauer, 2010a), and frequently fails to capture statistically significant differences between population subgroups (Eisenhauer and Ventura, 2011). ...
... For large risks, the Pratt-Arrow coefficient often yields perverse rankings of preferences (Eisenhauer, 2010a), and frequently fails to capture statistically significant differences between population subgroups (Eisenhauer and Ventura, 2011). 6 To address these problems, an arc elasticity measure of risk aversion has been developed for use with large, discrete changes (Eisenhauer, 2006); the new metric is especially suited to standard gambles involving two risky states (Eisenhauer, 2010b). From (1), define the following discrete utility changes, or discrete marginal utilities: ...
Researchers have commonly used financial gambles to assess risk preferences, though attitudes regarding monetary gambles may differ from those concerning health risks. Moreover, the conventional measure of risk preference--the Pratt-Arrow coefficient of relative risk aversion--is a point-elasticity suitable only for trivial risks, whereas the risks pertaining to medical treatment--often involving life-or-death decisions--are quite substantial. Our objective is to examine attitudes toward health risks and the variables that influence those attitudes. In addition to the conventional metric, we employ an alternative measure of risk aversion designed for large-scale risks; the latter is calculated as an arc-elasticity of marginal utility. Both measures are applied to responses from a health preference survey, which incorporates a standard gamble over longevity. Measured risk aversion is then related to health and demographic variables using multiple linear regressions. Our research suggests that tolerance of health risks may be influenced by an endowment effect reflecting the asymmetry of medical information between patients and providers. We also find that the arc-elasticity measure captures more of the significant differences and has a stronger correlation to demographic and socio-economic variables than does the more traditional point-elasticity measure, indicating that the conventional measures of risk aversion are indeed ill suited to large-scale risks such as those involving health. In addition, our results suggest that health-related risk preferences may differ from financial risk attitudes. Although our sample is relatively small and largely homogeneous, these results collectively provide a basis for replication and a justification for additional study on the measurement of risk aversion with regard to health.
... We cannot observe high levels of risk aversion in this study because there are no bonus payments in the Portuguese version of the game, and the value of the final showcase is lower than in versions from other countries. Individuals exhibit aversion only towards more significant risks, and when risks are small, behaviour is roughly risk-neutral [53]. Another reason is that individuals who dare to appear on TV are likely to be more self-confident, outgoing and risk-loving or risk-neutral individuals in the general population [54]. ...
The theoretical literature on risk aversion and Expected Utility Theory is extensive; however, the analysis of this behaviour with natural experiments could be more comprehensive. In this paper, we use data from 120 episodes of the Portuguese version of the TV game show The Price is Right, namely from The Wheel game, to explore risk aversion as well as the impact of gender in decision-making. The Wheel game has straightforward rules and huge expected payoffs. All contestants have access to the same information and distributions of uncertainty, making it a unique field laboratory to conduct an experimental test of rational decision theory. The objective is to infer the risk aversion levels of decision-makers from their choice to turn the wheel and the influence of gender on risk attitudes. There is a widespread view that women are more risk-averse than men. However, we could not reject the hypothesis that women and men have the same level of risk aversion. Nevertheless, we have evidence that contestants are more risk-averse than risk-seeking. The omission bias, loss aversion and regret can explain that behaviour.
... In the second direction, since Kimball's index of absolute prudence (like Pratt's index of absolute risk aversion) is a local measure, Eisenhauer (2006) proposes an extended version of this measure that is applicable to both small and large risks. A different extension of Kimball's analysis is provided by Hau (2002), who derives a version of Kimball's precautionary premium for the case of a non-additively separable utility function and shows that the comparison between the precautionary saving chosen by different households depends on both their absolute prudence and their intertemporal substitution rates. ...
This work reviews recent developments in the literature analyzing precautionary saving. After a description of traditional precautionary saving theory, which considers labor income risk and interest rate risk, we present different research lines which introduce a wide range of extensions and generalizations of the classical model: the contemporaneous presence of multiple risks, changes in risks of different types, multiple variables affecting household utility, preferences non-featuring risk aversion and joint decisions on many choice variables. For each of these issues, we provide specific highlights which summarize the main results obtained in the literature. Lastly, we briefly discuss the analyzes beyond the classical model.
... In 1960s Pratt-Arrow measure of absolute and relative risk aversion were the important developments for understanding risk taking behavior (Pratt, 1964). Eisenhauer (2006) criticizes Pratt-Arrow risk measure because it requires wealth levels to be known and size of the stakes. If stakes are infinitesimal, theory is working. ...
We analyze the relationship between the value of information (VOI) and the decision-maker’s risk aversion in the lake monitoring where one needs to decide about whether to implement costly management actions or not. We calculate the value of perfect as well as imperfect information for risk neutral and risk averse decision-makers. The risk aversion is measured using the certain equivalent and the Arrow-Pratt risk aversion measures, which are defined using the derivatives of the utility function. We consider two utility functions for a risk averse decision-maker, an exponential utility and a power utility function, and demonstrate their use with lake management data from Finland. The results show that, in this context, a risk averse decision-maker’s VOI may be lower or higher than a risk neutral decision-maker’s VOI, depending on the prior probability of the lake being in the need of management actions and the cost of the actions. The risk aversion seems to have a clear impact on decisions. This may encourage the decision-makers to contemplate their risk preferences instead of hastily assuming the risk neutrality.
The relation between the leader and alliance partners is viewed as the principal-agent relationship in the technology alliance. The leading enterprise is a principal, and alliance partner is an agent. On this basis, a game mode of knowledge transfer between them is established based on the principalagent theory. The incentive behaviors of the leader's and partners' knowledge transfer are analyzed in order to explore how the leading enterprise designs the superior incentive mechanism for encouraging partners transfer knowledge actively in the alliance, which will maintain the stability of operations, thereby enhancing the knowledge transfer effectiveness of the technology alliance.
The conventional measures of risk aversion developed by Pratt and Arrow, as well as Kimball's measures of prudence, assume differential utility functions and infinitesimal changes in wealth, and consequently yield strictly local assessments of preferences. Most of the meaningful risks encountered in practice, however, occur on a large scale, making it necessary to measure preferences over broad intervals of wealth. to that end, the present paper uses discrete rather than continuous changes in utility to develop reduced-form index measures of risk aversion and prudence suitable for use with risks of any magnitude. Survey data from the Center for Economic Research (CentER) at Tilburg University in the Netherlands are then applied to the model to determine construct validity and illustrate mean and marginal demographic differences in attitudes toward large-scale risks.
Multicriteria decision analysis (MCDA) has been applied to various energy problems to incorporate a variety of qualitative and quantitative criteria, usually spanning environmental, social, engineering, and economic fields. MCDA and associated methods such as life-cycle assessments and cost-benefit analysis can also include risk analysis to address uncertainties in criteria estimates. One technology now being assessed to help mitigate climate change is carbon capture and storage (CCS). CCS is a new process that captures CO2 emissions from fossil-fueled power plants and injects them into geological reservoirs for storage. It presents a unique challenge to decisionmakers (DMs) due to its technical complexity, range of environmental, social, and economic impacts, variety of stakeholders, and long time spans. The authors have developed a risk assessment model using a MCDA approach for CCS decisions such as selecting between CO2 storage locations and choosing among different mitigation actions for reducing risks. The model includes uncertainty measures for several factors, utility curve representations of all variables, Monte Carlo simulation, and sensitivity analysis. This article uses a CCS scenario example to demonstrate the development and application of the model based on data derived from published articles and publicly available sources. The model allows high-level DMs to better understand project risks and the tradeoffs inherent in modern, complex energy decisions.
Risk aversion is the central reason why individuals purchase insurance and undertake other forms of risk management. But deriving the Pratt–Arrow coefficient of relative risk aversion from a utility function requires familiarity with differential calculus—a level of mathematics beyond the prerequisites for most introductory risk management courses. Thus, students are not exposed to one of the most important and fundamental concepts in the field unless and until they take more advanced courses. The present article demonstrates that relative risk aversion can be obtained as an arc elasticity using only elementary mathematics. This approach highlights the relationship between risk aversion and the demand for insurance, and integrates concepts from the principles of economics course, helping to unify the business curriculum. Numerical examples are easily computed and graphed using electronic spreadsheets, providing students with a hands-on learning experience. For sufficiently small risks, the arc elasticity measure reduces to the Pratt–Arrow coefficient, providing a platform for discussing the difference between large-scale and small-scale risk aversion in upper-level courses.
The conventional measures of absolute and relative risk aversion are appropriate for measuring preferences locally, but because they rely on differential calculus, they cannot accurately capture attitudes towards high-stakes risks involving potentially large changes in wealth. Eisenhauer (200613.
Eisenhauer , JG . 2006. Risk aversion and prudence in the large. Research in Economics, 60: 179–87. View all references) has recently proposed an alternative approach which avoids the use of calculus. The present article extends that work in two ways. First, the Pratt–Arrow coefficient of absolute risk aversion is generalized into a measure of interval risk aversion, suitable for analysing preferences over risks of any magnitude, and a corresponding interval measure of relative risk aversion is constructed from preference and risk parameters, without explicit reference to initial wealth or income. Second, the new measures are applied to survey data from the Bank of Italy, to illustrate their empirical applicability to large-scale risk.
I. Introduction, 465. — II. A two period model of consumption, 466. — III. The effect of uncertainty on saving, 467. — IV.
Decreasing risk aversion and precautionary saving, 468. — V. Speculations on the consumption function, 470. — VI. Conclusion,
472.
In this paper, we propose the infimum of the Arrow–Pratt index of absolute risk aversion as a measure of global risk aversion of a utility function. We show that, for any given arbitrary pair of distributions, there exists a threshold level of global risk aversion such that all increasing concave utility functions with at least as much global risk aversion would rank the two distributions in the same way. Furthermore, this threshold level is sharp in the sense that, for any lower level of global risk aversion, we can find two utility functions in this class yielding opposite preference relations for the two distributions.
We use information from the television game show with the highest guaranteed average payoff in the United States, Hoosier Millionaire, to analyze risktaking in a high-stakes experiment. We characterize gambling decisions under alternative assumptions about contestant behavior and preferences, and derive testable restrictions on individual risk attitudes based on this characterization. We then use an extensive sample of gambling decisions to estimate distributions of risk-aversion parameters consistent with the theoretical restrictions and revealed preferences. We find that although most contestants display risk-averse preferences, the extent of the risk aversion implied by our estimates varies substantially with the stakes involved in the different decisions. Copyright (c) 2003 President and Fellows of Harvard College and the Massachusetts Institute of Technology.
We introduce a simple measure of risk aversion in the large. Besides satisfying properties which are conceptually analogous to the usual properties of the Arrow-Pratt measure, the index of risk aversion in the large leads to a stronger concept of decreasing risk aversion, which necessarily implies decreasing absolute risk aversion but not necessarily non-increasing relative risk aversion. The index also leads to a recursive procedure for refining the set of vN - M utility functions. We show that the majority of refinements considered in the theory of multiple risk bearing including that of mixed risk aversion can be obtained from this procedure. Finally, as an illustration, we apply the measure to characterize individual behaviour under uncertainty in the principal-agent model of optimal income tax enforcement in which the risks involved are indeed large.
This paper proposes an extension of the Pratt-Arrow analysis of decision making under uncertainty, based on a three-term Taylor approximation of the utility function. A two-parameter measure of risk aversion is defined? which incorporates both the risk’s variance and its skewness. The present approach gives better numerical approximations of the risk premium for risks which are not necessarily very small, provides new theoretical results, and also predicts unexpected behavior under certain conditions.
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.
The modern literature on risk aversion and risk tolerance almost
universally posits a monotonically concave, linear, or convex utility
function. More than 50 years ago, however, Friedman and Savage
(1948) theorized that individuals who both gamble and buy insurance
may exhibit utility functions with both concave and convex segments.
Despite its intuitive appeal, the Friedman-Savage hypothesis has not
been widely tested on an empirical basis. Using a unique survey data
set, this paper obtains evidence indicating that at least 18 to 20 percent
of individuals evaluate risks in a manner consistent with Friedman-
Savage preferences.
This paper provides additional evidence on life‐cycle patterns of relative risk aversion, using spline functions generated on Consumer Expenditure Survey data. Human capital is hypothesized to affect relative risk aversion; age has been used in previous work as a proxy for human capital. The objective of this study is to determine whether there is a life‐cycle pattern that is independent of the effect of human capital. The results suggest an affirmative answer. Moreover, this independent life‐cycle pattern is the opposite of that estimated in a previous study that used age as a proxy. © The Southern Finance Association and the Southwestern Finance Association
* General Introduction * Tools from Logic * Differential and Integral Calculus * General Topology * Functions of a Real Variable * Functions of a Complex Variable * Linear Spaces * Topological Groups and Lie Groups * Selected Topics * Concerning the History of the Calculus
Within the expected-utility framework, the only explanation for risk aversion is that the utility function for wealth is concave: A person has lower marginal utility for additional wealth when she is wealthy than when she is poor. This paper provides a theorem showing that expected-utility theory is an utterly implausible explanation for appreciable risk aversion over modest stakes: Within expected-utility theory, for any concave utility function, even very little risk aversion over modest stakes implies an absurd degree of risk aversion over large stakes. Illustrative calibrations are provided. Keywords: Diminishing Marginal Utility, Expected Utility, Risk Aversion JEL Classifications: B49, D11, D81 Acknowledgments: Many people, including David Bowman, Colin Camerer, Eddie Dekel, Larry Epstein, Erik Eyster, Mitch Polinsky, Drazen Prelec, Richard Thaler, and Roberto Weber, as well as Andy Postlewaite and two anonymous referees, have provided useful feedback on this paper. I th...
Purpose
Conventional measures of risk aversion based on first and second derivatives of utility are strictly local instruments, valid only for infinitesimally small changes in wealth. This paper to develop a global index suitable for assessing attitudes toward large‐scale risks.
Design/methodology/approach
Integral calculus is used to measure the geometric area between an individual's actual utility function and a linear function displaying risk neutrality, over the entire range of potential wealth outcomes for a given risk. The area is then converted to an index number.
Findings
Local and global measures of risk aversion yield similar interpersonal comparisons only for small risks; with larger risks, local measures distort interpersonal differences. The analysis also shows that individuals having exponential utility functions evaluate risk exclusively on the basis of wealth dispersion, whereas those with logarithmic or square‐root utilities consider both the mean and variance of wealth.
Research/limitations/implications
The global index is quantifiable if the functional form of utility is known; further research is needed to approximate the index when information about utility is limited.
Practical implications
The most important risks encountered in practice, such as the possibility of unemployment or disability, involve variations in wealth far larger than differential calculus is designed to accommodate. The integral index therefore provides a more appropriate basis for measuring and comparing risk preferences.
Originality/value
The paper provides an innovative geometric interpretation of global risk aversion, and in contrast to local measures, the integral index captures differences in the intensity of an individual's aversion toward risks of various magnitudes.
A wide range of empirical applications rely on linear approximations to dynamic Euler equations. Among the most notable of these is the large and growing literature on precautionary saving that examines how consumption growth and saving behavior are affected by uncertainty and prudence. Linear approximations to Euler equations imply a linear relationship between expected consumption growth and uncertainty in consumption growth, with a slope coefficient that is a function of the coefficient of relative prudence. This literature has produced puzzling results: estimates of the coefficient of relative prudence (and the coefficient of relative risk aversion) from linear regressions of consumption growth on uncertainty in consumption growth imply estimates of prudence and risk aversion that are unrealistically low. Using numerical solutions to a fairly standard intertemporal optimization problem, our results show that the actual relationship between expected consumption growth and uncertainty in consumption growth differs substantially from the relationship implied by a linear approximation. We also present Monte Carlo evidence that shows that the instrumental-variables methods that are commonly used to estimate the parameters correct some, but not all, of the approximation bias. © 2001 by the President and Fellows of Harvard College and the Massachusetts Institute of Technolog
This paper contributes to an important recent debate around expected utility and risk aversion. Rejecting a gamble over a given range of wealth levels imposes a lower bound on risk aversion. Using this lower bound and empirical evidence on the range of the risk aversion coefficient, we calibrate the relationship between risk attitudes over small-stakes and large-stakes gambles. We find that rejecting small gambles is consistent with expected utility, contrary to a recent literature that concludes that expected utility is fundamentally unfit to explain decisions under uncertainty. Paradoxical behavior is only obtained when calibrations are made in a region of the parameter space that is not empirically relevant.
The principle that it is better to let some guilty individuals be set free than to mistakenly convict an innocent person is generally shared by legal scholars, judges and lawmakers of modern societies. The paper shows why this common trait of criminal procedure is also efficient. It extends the standard Polinsky and Shavell (2007) model of deterrence and shows that when the costs of convictions are positive, and guilty individuals are more likely to be convicted than innocent individuals it is always efficient to minimize the number of wrongful convictions, while a more than minimal amount of wrongful acquittals may be optimal.
This paper attempts to answer the question: When is a random variable Y “more variable” than another random variable X?
The theory of precautionary saving is shown to be isomorphic to the Arrow-Pratt theory of risk aversion, making possible the application of a large body of knowledge about risk aversion to precautionary saving--and more generally, to the theory of optimal choice under risk. In particular, a measure of the strength of the precautionary saving motive analogous to the Arrow-Pratt measure of risk aversion is used to establish a number of new propositions about precautionary saving and to give a new interpretation of the Dreze-Modigliani substitution effect. Copyright 1990 by The Econometric Society.
This paper develops a matrix-measure of multivariate risk aversion which is related to a notion of risk premium and states the restrictions that must be imposed upon the matrix-measures of two utility functions in order that one require a higher risk premium than another for every small multivariate risk. A necessary and sufficient condition for comparability of global attitude towards risk is that the local restrictions hold over the entire domain. The usefulness of the measures of risk aversion is discussed within the context of a multivariate risk-sharing problem.
In this work, the notion of risk premium is applied to a situation for which the individual is confronted with a risk of illness. This health risk can breed a risk of wealth via the recourse to health care. Following the nature of health care undertaken, different premiums are defined which verify the usual properties of the Arrow-Pratt risk premiums. Finally, the comparison of these premiums informs us of the individual preferences in terms of health care. Dans ce travail, nous appliquons la notion de prime de risque à une situation pour laquelle l’individu est confronté à un risque de maladie qui peut engendrer un risque de richesse via le recours aux soins de santé. Suivant la nature des soins entrepris, différentes primes sont définies qui vérifient pour la plupart les propriétés usuelles des primes à la Arrow-Pratt. Enfin, la comparaison de ces primes nous renseigne sur les préférences individuelles en matière de soins de santé.
The asymmetric approximation originally employed by Pratt (1964) to construct reduced-form measures of risk aversion s a downward bias when used for empirical estimation. Calculations based on recent survey data indicate that estimates from a symmetric approximation are generally three times larger than their asymmetric counterparts, a finding that may help to explain the equity premium puzzle.
This note proposes a decomposition of the familiar scalar multivariate risk premium into components which can be easily interpreted in the context of consumer theory. The premium under consideration is the standard one used to ascertain the impact of price and income risk on consumer welfare. This proposed premium decomposition allows for a more intuitive identification of the detrimental and beneficial effects that arise from income and price risk. As an illustrative example, this decomposition is used to ascertain the welfare effects arising from the price fluctuations experienced by UK households over the period 1963-97. Copyright Blackwell Publishers Ltd and the Board of Trustees of the Bulletin of Economic Research, 2004.
A simple measure of risk aversion in the large and an application
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