Buying Insurance for Disaster-Type Risks: Experimental Evidence

This paper presents a series of experiments that confront subjects with low probability, high loss situations. A rich parameter set is examined and we find subjects respond to low probability, high loss risks in predictable ways. As loss events become more likely, or loss amounts get larger, or the cost of insurance falls, subjects are more likely to buy indemnifying insurance, even for the class of low probability risks that usually presents problems for standard expected utility theory. A novel application of Cameron's method to estimate willingness to pay from dichotomous choice responses allows us to estimate willingness to pay for insurance. We do not observe the bimodal distribution of bids found in other studies of similar risk situations.     

Optimal Insurance With Divergent Beliefs About Insurer Total Default Risk

This paper extends the classic expected utility theory analysis of optimal insurance contracting to the case where the insurer has a positive probability of total default and the buyer and insurer have divergent beliefs about this probability. The optimal marginal indemnity above the deductible is smaller (greater) than one if the buyer's assessment of default risk is more pessimistic (optimistic) than the insurer's. As an application of the model, we consider the market for reinsurance against catastrophic property loss and propose an expected utility theory explanation for the increasing and concave marginal indemnity schedule observed in this market.     

The impact of governmental assistance on insurance demand under ambiguity: a theoretical model and an experimental test

This article deals with the impact of governmental assistance on insurance demand under ambiguity, i.e., in situations where probabilities are uncertain. First, using a model of insurance demand under ambiguity, we derive theoretical predictions about the impact of several governmental assistance programmes on optimal insurance demand. For example, governmental assistance through a fixed public support scheme implies that partial insurance is always optimal under fair insurance with ambiguity. Second, we present the results of an experiment designed to test these predictions. We find support for several of our theoretical predictions. For example, the presence of governmental assistance through a fixed public support scheme decreases individuals’ willingness to pay to be fully insured. Finally, we compare these results with those obtained for a risk situation. We find that, regardless of the form of governmental assistance, participants in the ambiguity context are consistently willing to pay more to be fully insured than participants in the risk situation.     

Optimal insurance without expected utility: The dual theory and the linearity of insurance contracts

Models of the insurance markets and institutions are routinely based on expected utility. Since EU is being challenged by an increasing number of decision models, we examine whether EU-based models are robust in their predictions. To do so, we rework some basic models of optimal insurance contracts and equilibrium using the “dual” theory to EU of Yaari. When there is a single, insurable source of risk, dual theory permits only corner solutions if the contract itself is linear. This contrasts sharply with EU. Nonlinearity, and thereby the possibility of interior solutions, is introduced in two ways. First, the contract itself is nonlinear, i.e., a deductible insurance policy. Or second, the decision maker is subject to some background risk such as uninsurable risky assets or default of the insurer. When decision problems are subject to nonlinearity, the predictions on optimal insurance are more similar to, though not identical with, those generated with EU.     

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

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

On probabilities and loss aversion

This paper reviews the most common approaches that have been adopted to analyze and describe loss aversion under prospect theory. Subsequently, it is argued that loss aversion is a property of observable choice behavior and two new definitions of loss averse behavior are advocated. Under prospect theory, the new properties hold if the commonly used utility based measures of loss aversion are corrected by a probability based measure of loss aversion and their product exceeds 1. It is shown that prominent parametric families of weighting functions, while successful in accommodating empirical findings on probabilistic risk attitudes, may not fit well with the theoretical implications of the new loss averse behavior conditions.     

Loss aversion leading to advantageous selection

Some insurance markets are characterized by “advantageous selection”, that is, ex-post risk and coverage are negatively correlated. We show that expectation-based loss aversion as in K?szegi and Rabin (Quarterly Journal of Economics, 121(4), 1133–1165, 2006; The American Economic Review, 97(4), 1047–1073, 2007) provides a natural explanation for this phenomenon in environments in which risk aversion models do not, e.g., when agents face modest-scale risks and/or in absence of moral hazard. More exposure to risk has two competing effects on an agent’s willingness to pay for insurance: a positive effect, as in standard expected utility models; and a negative one, due to a reference effect. We determine conditions under which an insurance provider optimally sets a high price at which only low risk agents buy.     

Confusion of Relative and Absolute Risk in Valuation

Subjects were less willing to pay for government medical insurance for diseases when the number of people who could not be cured was higher, holding constant the number who could be cured. In a second experiment, willingness to pay (from a hypothetical government windfall) for risk reduction was unaffected by whether the risk was described in terms of percentage or number of lives saved, even though subjects knew that the risks in question differed in prevalence. These results are consistent with the findings of Fetherstonhaugh et al., Jenni and Loewenstein, and others. I suggest that these results can be explained in terms of a general tendency to confuse proportions and differences, a confusion that is analogous to other confusions of quantitative dimensions in children, adults, the news media, and perhaps even the epidemiological literature.     

Risk aversion elicitation: reconciling tractability and bias minimization

Risk attitude is known to be a key determinant of various economic and financial choices. Behavioral studies that aim to evaluate the role of risk attitudes in contexts of this type, therefore, require tools for measuring individual risk tolerance. Recent developments in decision theory provide such tools. However, the methods available can be time consuming. As a result, some practitioners might have an incentive to prefer “fast and frugal” methods to clean but more costly methods. In this article, we focus on a tractable procedure initially proposed by Holt and Laury (2002) to elicit risk attitude. We generalize this method to measure utility and risk aversion as follows. First, we allow measurement of probabilistic risk attitude through violations of expected utility due to probability weighting. Second, we use the outcome scale rather than the probability scale in the menu of choices. Third, we compare sure payoffs with lotteries instead of comparing non-degenerate lotteries. A within-subject experimental study illustrates the gains in tractability and bias minimization that can result from such an extension.     

How Much Internalization of Nuclear Risk Through Liability Insurance?

An important source of conflict surrounding nuclear energy is that with a very small probability, a large-scale nuclear accident may occur. One way to internalize the associated financial risks is through mandating nuclear operators to have liability insurance. This paper presents estimates of consumers' willingness to pay for increased financial security provided by an extension of coverage, based on the 'stated choice' approach. A Swiss citizen with median characteristics may be willing to pay 0.14 US cents per kwh to increase coverage beyond the current CHF 0.7 billion (bn.) (US$ 0.47 bn.). Marginal willingness to pay declines with higher coverage but exceeds marginal cost at least up to CHF 4 bn.(US$ 2.7 bn.). An extension of nuclear liability insurance coverage therefore may be efficiency-enhancing.     

A parametric analysis of prospect theory’s functionals for the general population

This article presents the results of an experiment that completely measures the utility function and probability weighting function for different positive and negative monetary outcomes, using a representative sample of N = 1,935 from the general public. The results confirm earlier findings in the lab, suggesting that utility is less pronounced than what is found in classical measurements where expected utility is assumed. Utility for losses is found to be convex, consistent with diminishing sensitivity, and the obtained loss-aversion coefficient of 1.6 is moderate but in agreement with contemporary evidence. The estimated probability weighting functions have an inverse-S shape and they imply pessimism in both domains. These results show that probability weighting is also an important phenomenon in the general population. Women and lower educated individuals are found to be more risk averse, in agreement with common findings. In contrast to previous studies that ascribed gender differences in risk attitudes solely to differences in the degree utility curvature, however, our results show that this finding is primarily driven by loss aversion and, for women, also by a more pessimistic psychological response toward the probability of obtaining the best possible outcome.     

Average willingness to pay for disease prevention with personalized health information

Personal health related information modifies individuals’ willingness to pay for disease prevention programs inasmuch as it allows health status assessment based on intrinsic (instead of average) characteristics. In this paper, we examine the effect that personalized information about the baseline probability of disease has on the average willingness to pay for programs reducing either the probability of disease (self-protection) or the severity of disease (self-insurance). We show that such information raises the average willingness to pay for self-protection while it increases the average willingness to pay for self-insurance if health and wealth are complements (i.e. the marginal utility of wealth rises with health).     

Insurance demand and social comparison: An experimental analysis

This paper analyzes whether social comparison can explain the low take-up of disaster insurance usually reported in field studies. We argue that risks in the case of disasters are highly correlated between subjects whereas risks for which high insurance take-up can be observed (e.g. extended warranties or cell phone insurance) are typically idiosyncratic. We set up a simple model with social reference points and show that in the presence of inequality aversion social comparison makes insurance indeed less attractive if risks are correlated. In addition we conducted a simple experiment which confirms these theoretical results. The average willingness to pay for insurance is significantly higher for idiosyncratic than for correlated risks.     

Admissible utility functions for health, longevity, and wealth: integrating monetary and life-year measures

The value of reducing health and mortality risks is often measured using value per statistical life (VSL) or one of several life-year measures (e.g., life years, quality-adjusted life years, disability-adjusted life years). I derive the utility function that is admissible when preferences for health and longevity, conditional on wealth, are consistent with any life-year measure (LYM) and examine the implications for marginal willingness to pay (WTP) for increases in health, longevity, and current-period survival probability. I conclude that marginal WTP for any LYM is decreasing and that VSL is increasing in the LYM. These results imply that cost-effectiveness analysis using a fixed monetary value per LYM is not consistent with economic welfare theory and that the benefit of a health improvement cannot be calculated by multiplying the change in a LYM by a constant.     

Avoiding the curves: Direct elicitation of time preferences

We propose and test a new method for eliciting curvature-controlled discount rates that are invariant to the form of the utility function. Our method uses a single elicitation task and obtains individual discount rates without knowledge of risk attitude or parametric assumptions about the form of the utility function. We compare our method to a double elicitation technique in which the utility function and discount rate are jointly estimated. Our experiment shows that these methods yield consistent estimates of the discount rate, which is reassuring given the wide range of estimates in the literature. We find little evidence of probability weighting, but in a second experiment, we observe that discount rates are sensitive to the length of the front-end delay, suggesting present bias. When the front-end delay is at least two weeks, we estimate average discount rates to be 11.3 and 12.2% in the two experiments.     

What is Loss Aversion?

A behavioral definition of loss aversion is proposed and its implications for original and cumulative prospect theory are analyzed. Original prospect theory is in agreement with the new loss aversion condition, and there utility is capturing all effects of loss aversion. In cumulative prospect theory loss aversion is captured by both the weighting functions and the utility function. Further, some restrictions apply for the weighting functions involved in the latter model.We are indebted to Michèle Cohen and Peter Wakker for helpful comments. The suggestions of an anonymous referee have improved the presentation of the paper.     

Comparing risk preferences over financial and environmental lotteries

This paper investigates whether preferences over environmental risks are best modeled using probability-weighted utility functions or can be reasonably approximated by expected utility (EU) or subjective EU models as is typically assumed. I elicit risk attitudes in the financial and environmental domains using multiple-price list experiment. I examine how subjects?? behavioral, attitudinal, and demographic characteristics affect their probability weighting functions first for financial risks, then for oil-spill risks. I find that most subjects tend to overweight extreme positive outcomes relative to expected utility in both the environmental and financial domains. Subjects are more likely to overemphasize low probability, extreme environmental outcomes than low probability, extreme financial outcomes, leading subjects to offer more support for mitigating environmental gambles than financial gambles with the same odds and equivalent outcomes. I conclude that EU models are likely to underestimate subjects?? willingness to pay for environmental cleanup programs or policies with uncertain outcomes.     

Imprecise preferences and the WTP-WTA disparity

This article reports the results of a study designed to elicit willingness to pay (WTP) and willingness to accept (WTA) values for changes in the risk of nonfatal road injuries. We examine the possibility that individuals' preferences over combinations of wealth, risk, and safety are imprecise, and that this imprecision might result in the observed disparity between WTP and WTA measures of value. The results confirm that individuals' preferences for safety are significantly imprecise, but that this alone is insufficient to explain more than part of the disparity. Indeed, respondents' estimates of the minimum that they would be prepared to accept for a risk increase frequently exceed the maximum that they would be prepared to pay for an equivalent risk reduction.     

Investigating Risky Choices Over Losses Using Experimental Data

We conduct a battery of experiments in which agents make choices from several pairs of all-loss-lotteries. Using these choices, we estimate a representation of individual preferences over lotteries. We find statistically and economically significant departures from expected utility maximization for many subjects. We also estimate a preference representation based on summary statistics for behavior in the population of subjects, and again find departures from expected utility maximization. Our results suggest that public policies based on an expected utility approach could significantly underestimate preferences and willingness to pay for risk reduction.JEL Classification: C91, D81