A Note on Optimal Insurance in the presence of a Nonpecuniary Background Risk

This note examines the theory of optimal insurance purchasing in the presence of a nonpecuniary background risk. The occurrence of the qualitative uninsurable background loss can increase, decrease or can leave the marginal utility of wealth unchanged, whereas a financial background loss (as in Doherty and Schlesinger, 1883a) always increases it. Existing theorems on the optimal level of insurance and the optimal form of insurance contracts are shown to hold only under restrictive assumptions on the correlation level between risks. The paper shows that sufficient conditions for the validity of the theorems depend not only on the correlation between the insurable and noninsurable losses, but also on the variation of the marginal utility of wealth with respect to the nonpecuniary variable.     

The Value of a Statistical Life and the Coefficient of Relative Risk Aversion

Individuals’ risk preferences are estimated and employed in a variety of settings, notably including choices in financial, labor, and product markets. Recent work, especially in financial economics, provides estimates of individuals’ coefficients of relative risk aversion (R’s) in excess of one, and often significantly higher. However, it can be shown that high R’s imply equally high values for the income elasticity of the value of a statistical life. Yet estimates of this elasticity, derived from labor and product markets, are in the range of 0.5 to 0.6. Furthermore, it turns out that even an R below one is difficult to reconcile with these elasticity estimates. Thus, there appears to be an important (additional) anomaly involving individuals’ risk-taking behavior in different market settings.JEL Classification: D80, G11, G12, I10, J17     

Endogenous risks and the risk premium

This note tries to correct a deficiency of the microeconomic literature on decision making under uncertainty. Indeed, when considering meaningful comparative statics results in situations where risks are at least partially controllable (endogenous), this literature has mostly relied upon the traditional Arrow-Pratt risk aversion functions and has paid very little attention to the definition of the risk premium. However when they defined the risk premium and the risk aversion functions, Arrow and Pratt considered only roulette gambles, i.e. risks totally exogenous to the individual. This note highlights the fact that several definitions of the risk premium may be proposed for endogenous risks. Two of them, already used in the literature, do not preserve the intuitively-appealing properties of the Arrow-Pratt risk premium. An alternative definition is then proposed. It is shown that this new definition of the risk premium applied to endogenous risks exhibits the properties generally admitted for roulette gambles.The three authors have benefitted from Ph. Caperaa's advice and from a referee's comments.     

On the definition of risk aversion

Two definitions of risk aversion have recently been proposed for non-expected utility theories of choice under uncertainty: the former refers the measure of risk aversion (Montesano 1985, 1986 and 1988) directly to the risk premium (i.e. to the difference between the expected value of the action under consideration and its certainty equivalent); the latter defines risk aversion as a decreasing preference for an increasing risk (introduced as mean preserving spreads) (Chew, Karni and Safra 1987, Machina 1987, Röell 1987, Yaari 1987).When the von Neumann-Morgenstern utility function exists both these definitions indicate an agent as a risk averter if his or her utility function is concave. Consequently, the two definitions are equivalent. However, they are no longer equivalent when the von Neumann-Morgenstern utility function does not exist and a non-expected utility theory is assumed. Examples can be given which show how the risk aversion of the one definition can coexist with the risk attraction of the other. Indeed the two definitions consider two different questions: the risk premium definition specifically concerns risk aversion, while the mean preserving spreads definition concerns the increasing (with risk) risk aversion.The mean preserving spreads definition of risk aversion, i.e. the increasing (with risk) risk aversion, requires a special kind of concavity for the preference function (that the derivatives with respect to probabilities are concave in the respective consequences). The risk premium definition of local risk aversion requires that the probability distribution dominates on the average the distribution of the derivatives of the preference function with respect to consequences. Besides, when the local measure of the first order is zero, there is risk aversion according to the measure of the second order if the preference function is concave with respect to consequences.Yaari's (1969) measure of risk aversion is closely linked to the r.p. measure of the second order. Its sign does not indicate risk aversion (if positive) or attraction (if negative) when the measure of the first order is not zero (i.e., in Yaari's language, when subjective odds differ from the market odds).     

On the Effect of Risk Aversion in Bimatrix Games

Nash equilibria with identical supports are compared for bimatrix games that are different with respect to the risk aversion of player 2. For equilibria in 2× 2-bimatrix games and for equilibria with efficient supports in coordination games it is established for which cases increased risk aversion of player 2 benefits or hurts player 2.     

Proper and Standard Risk Aversion in Two-Moment Decision Models

For linear distribution classes, mean-variance and expected utility specifications have been shown in the literature to be fully compatible when studying the concepts of risk aversion, prudence, risk vulnerability and temperance. This paper shows that such compatibility does hold for the concept of standard risk aversion but not for the concepts of proper risk aversion and proper prudence.Jel Classification: D81     

The risk aversion measure without the independence axiom

The risk premium (conveniently normalized) is defined as the measure of risk aversion. This measure does not require any relevant assumption in the theory of choice under uncertainty except the existence of a certainty equivalent. In particular, the independence axiom is not required. The measure of risk aversion of an action is provided not only for the case with one commodity and two consequences but also for the case with many commodities and consequences. The measure of mean risk aversion of all actions with given consequences is introduced and the local measure of risk aversion is obtained by making all these consequences approach the consequence under consideration. This measure is demonstrated to be zero when the von Neumann-Morgenstern utility function exists. In this case a measure of risk aversion of the second order is introduced, which turns out to be equal to the Arrow-Pratt absolute index when there is only one commodity and similar to the generalized measures proposed by several authors when there are many commodities and two consequences.Helpful comments by I. Gilboa and suggestions by the referee are gratefully acknowledged.     

Arrow-Pratt Risk Aversion, Risk Premium and Decision Weights

This paper analyzes two issues: (a) the effect of decision-weights on risk premium, and (b) whether risk-aversion characterizes most investors. We theoretically show that cumulative prospect theory decision-weights systematically increase Arrow's risk premium, and may induce a positive risk premium even in the absence of risk-aversion. However, decision-weights may either increase or reduce Pratt's risk premium. We present three experiments revealing a striking result: a large proportion of the subjects' choices contradicts risk-aversion. This may be due to non-concave preferences, or to decision-weights. This result may have a dramatic impact on equilibrium models in economics and finance.     

Comparative Risk Sensitivity with Reference-Dependent Preferences

Experimental evidence suggests that individuals are risk averse over gains and risk seeking over losses (i.e., they have S-shaped utility functions in an expected utility setting) and that they are loss averse. Furthermore, the evidence leads to a single definition of S-shaped utility, but it has led to several alternative specifications of loss aversion. This paper characterizes the relations more S-shaped than and more loss averse than for a utility function, and in so doing arrives at a new definition of loss aversion based on average instead of marginal utility.     

The logic of partial-risk aversion: Paradox lost

One rational individual may be willing to pay less than another to insure a risk % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacuWF1oG8gaacaaaa!41B4!\[\tilde \varepsilon \] when another risk % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia% aaaa!36F7!\[\tilde w\] is present even though he would pay more to insure any isolated risk, and even though % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraiGacI% cadaabcaqaamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLbac% faGaf8xTdiVbaGaaaiaawIa7aiqadEhagaacaiGacMcaciGG9aGaai% imaaaa!47F3!\[E(\left. {\tilde \varepsilon } \right|\tilde w) = 0\] for all w. Noticing this, Ross (1981) proposed excluding such reversals and gave equivalent analytical conditions. Reconsidering, we explain why some reversals are natural and show that prohibiting them has peculiar and undesirable properties. Although we also simplify the conditions and prove them necessary for partial-risk portfolio results, we conclude that they represent revealing restrictions on comparative statics rather than natural implications of increased aversion to risk.     

Estimating Risk Attitudes using Lotteries: A Large Sample Approach

Attitudes towards risk play a major role in many economic decisions. In empirical studies it is quite often assumed that attitudes towards risk do not vary across individuals. This paper questions this assumption and analyses which factors influence an individual's risk attitude. Based on questions on lotteries in a large household survey we first semiparametrically estimate an index for risk aversion. We only make weak assumptions about the underlying decision process and our estimation method allows for generalisations of expected utility. We then estimate a structural model based on Cumulative Prospect Theory. Expected utility is strongly rejected and both the value function and the probability weighting function vary significantly with (among other things) age, income, and wealth of the individual.     

Premium Risk and Managed Care

This paper analyzes how individuals can insure premium risk and obtain high quality health insurance in a managed care environment. Insurers choose health care providers. Only a fraction of high risk individuals is unambiguously identifiable in front of a court. Premium insurance is not able to reach a first-best risk allocation while health insurers have an incentive to stint on quality under guaranteed renewable contracts. It is shown that a contract exists which can implement the first-best. This contract specifies payments to individuals and a third party upon switching to create a credible self-commitment by the insurer to provide high quality.     

Risk Aversion when Gains are Likely and Unlikely: Evidence from a Natural Experiment with Large Stakes

In the television show Deal or No Deal a contestant is endowed with a sealed box, which potentially contains a large monetary prize. In the course of the show the contestant learns more information about the distribution of possible monetary prizes inside her box. Consider two groups of contestants, who learned that the chances of their boxes containing a large prize are 20% and 80% correspondingly. Contestants in both groups receive qualitatively similar price offers for selling the content of their boxes. If contestants are less risk averse when facing unlikely gains, the price offer is likely to be more frequently rejected in the first group than in the second group. However, the fraction of rejections is virtually identical across two groups. Thus, contestants appear to have identical risk attitudes over (large) gains of low and high probability.      

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

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

Effects of Risk and Time Preference and Expected Longevity on Demand for Medical Tests

Despite their conceptual importance, the effects of time preference, expected longevity, uncertainty, and risk aversion on behavior have not been analyzed empirically. We use data from the Health and Retirement Study (HRS) to assess the role of risk and time preference, expected longevity, and education on demand for three measures used for early detection of breast and cervical cancer—regular breast self-exams, mammograms, and Pap smears. We find that individuals with a higher life expectancy and lower time preference are more likely to undergo cancer screening. Less risk averse individuals tend to be more likely to undergo testing.     

审计风险补偿金理论分析

事务所对于风险偏好不同 ,而采用收取高于收费水平的风险补偿金的方法来回避高风险。在SimunicandStein提出的审计收费的单期简单模型的前提下 ,进行实证研究。是对实际收费或者实验性案例分析中的期望收费进行各影响因素的回归分析 ,找出解释能力较强的显著因素。PrattandStice的结果显示 ,在存在审计风险范畴以外的经营风险的条件下 ,审计师的收费中的确含有风险补偿的部分。     

On the intensity of downside risk aversion

The degree of downside risk aversion (or equivalently prudence) is so far usually measured by . We propose here another measure, , which has specific and interesting local and global properties. Some of these properties are to a wide extent similar to those of the classical measure of absolute risk aversion, which is not always the case for . It also appears that the two measures are not mutually exclusive. Instead, they seem to be rather complementary as shown through an economic application dealing with a simple general equilibrium model of savings.

党内基层民主建设风险管理研究

党内基层民主的风险如何研判,是目前民主建设急需研究的一项基础内容.本文以层次分析法为基本路径,分析了党内基层民主建设的风险体系,对风险进行了认定与分类;对风险体系中最为切要、最具联系性的系统性风险进行了研究,并对风险的发展趋势进行了分析.     

Willingness to Pay for Risk Reduction and Risk Aversion without the Expected Utility Assumption

By means of minimal assumptions on the individual preferences, I show that the Willingness To Pay (WTP) for both a FSD and SSD reduction of risk is the sum of a mean effect, a pure risk effect and a wealth effect. As a result, the WTP of a risk-averse decision maker may be lower than the WTP of a risk-neutral one, for a large class of individual preferences’ representation and a large class of risks.     

Comparative Mixed Risk Aversion: Definition and Application to Self-Protection and Willingness to Pay

We analyze the optimal choices of agents with utility functions whose derivatives alternate in sign, an important class that includes most of the functions commonly used in economics and finance (Mixed Risk Aversion, MRA, Caballé and Pomansky, 1996). We propose a comparative mixed risk aversion definition for this class of utility functions, namely, More Risk Averse MRA, and provide a sufficient condition to compare individuals. We apply the model to optimal prevention and willingness to pay. More risk averse MRA agents spend less to reduce accident probabilities that are above 1/2. They spend more only when accident probabilities are below 1/2. Explanations in terms of risk premiums are provided. The results presented also allow for the presence of background risk.