On the design of optimal health insurance contracts under ex post moral hazard

We analyze the design of optimal medical insurance under ex post moral hazard, i.e., when illness severity cannot be observed by insurers and policyholders decide for themselves on their health expenditures. The trade-off between ex ante risk sharing and ex post incentive compatibility is analyzed in an optimal revelation mechanism under hidden information and risk aversion. The optimal contract provides partial insurance at the margin, with a deductible when insurers’ rates are affected by a positive loading, and it may also include an upper limit on coverage. The potential to audit the health state leads to an upper limit on out-of-pocket expenses.     

Optimal insurance contract and coverage levels under loss aversion utility preference

This study develops an optimal insurance contract endogenously and determines the optimal coverage levels with respect to deductible insurance, upper-limit insurance, and proportional coinsurance, and, by assuming that the insured has an S-shaped loss aversion utility, the insured would retain the enormous losses entirely. The representative optimal insurance form is the truncated deductible insurance, where the insured retains all losses once losses exceed a critical level and adopts a particular deductible otherwise. Additionally, the effects of the optimal coverage levels are also examined with respect to benchmark wealth and loss aversion coefficient. Moreover, the efficiencies among various insurances are compared via numerical analysis by assuming that the loss obeys a uniform or log-normal distribution. In addition to optimal insurance, deductible insurance is the most efficient if the benchmark wealth is small and upper-limit insurance if large. In the case of a uniform distribution that has an upper bound, deductible insurance and optimal insurance coincide if benchmark wealth is small. Conversely, deductible insurance is never optimal for an unbounded loss such as a log-normal distribution.     

Optimal Insurance Under Random Auditing

We provide a characterization of an optimal insurance contract (coverage schedule and audit policy) when the monitoring procedure is random. When the policyholder exhibits constant absolute risk aversion, the optimal contract involves a positive indemnity payment with a deductible when the magnitude of damages exceeds a threshold. In such a case, marginal damages are fully covered if the claim is verified. Otherwise, there is an additional deductible that disappears when the damages become infinitely large. Under decreasing absolute risk aversion, providing a positive indemnity payment for small claims with a nonmonotonic coverage schedule may be optimal.     

A Note on Deductible Insurance and Production

This paper examines the optimal production decision of a firm facing revenue risk. We show that the purchase of actuarially fair deductible insurance unambiguously induces the firm to produce more if the firm is not only risk averse but also prudent. If the firm's perferences satisfy constant absolute risk aversion, buying actuarially unfair deductible insurance unambiguously enhances production should the positive loading factor be sufficiently small. When there are moral hazard problems in that the firm's output cannot be contracted upon, we show that the purchase of actuarially fair deductible insurance unambiguously induces the firm to produce more if the firm's utility function is quadratic.     

To Insure or Not to Insure?: An Insurance Puzzle

The selection of a deductible level in insurance is governed by the willingness to limit the risk borne by risk-averse agents at an acceptable cost, given the deadweight insurance loading. We examine the demand for insurance in a simple lifecycle model with a liquidity constraint and no serial correlation in the insurable risk. This allows for consumers to follow a time-diversification (self-insurance) strategy by accumulating buffer stock wealth. We conclude that insurance would only be demanded for catastrophic risks, or by people that are currently liquidity constrained. The added value of the insurance sector is thus surprisingly low in such an economy.     

Insurance demand in the presence of loss-dependent background risk

We analyze insurance demand when insurable losses come with an uninsurable zero-mean background risk that increases in the loss size. If the individual is risk vulnerable, loss-dependent background risk triggers a precautionary insurance motive and increases optimal insurance demand. Prudence alone is sufficient for insurance demand to increase in two cases: the case of fair insurance and the case where the smallest possible loss exceeds a certain threshold value (referred to as the large loss case). We derive conditions under which insurance demand increases or decreases in initial wealth. In the large loss case, prudence determines whether changes in the background risk lead to more insurance demand. We generalize this result to arbitrary loss distributions and find conditions based on decreasing third-degree Ross risk aversion, Arrow–Pratt risk aversion, and Arrow–Pratt temperance.     

Optimal insurance and reinsurance policies chosen jointly in the individual risk model

The paper studies the so-called individual risk model where both a policy of per-claim insurance and a policy of reinsurance are chosen jointly by the insurer in order to maximize his/her expected utility. The insurance and reinsurance premiums are defined by the expected value principle. The problem is solved under additional constraints on the reinsurer’s risk and the residual risk of the insured. It is shown that the solution to the problem is the following: The optimal reinsurance is a modification of stop-loss reinsurance policy, so-called stop-loss reinsurance with an upper limit; the optimal insurer’s indemnity is a combination of stop-loss- and deductible policies. The results are illustrated by a numerical example for the case of exponential utility function. The effects of changing model parameters on optimal insurance and reinsurance policies are considered.     

The Spillover effect of compulsory insurance

The assumption usually made in the insurance literature that risks are always insurable at the desired level does not hold in the real world: some risks are not—or are only partially—insurable, while others, such as civil liability or health and workers' injuries, must be fully insured or at least covered for a specific amount. We examine in this paper conditions under which a reduction in the constrained level of insurance for one risk increases the demand of insurance for another independent risk. We show that it is necessary to sign the fourth derivative of the utility function to obtain an unambiguous spillover effect. Three different sufficient conditions are derived if the expected value of the exogenous risk is zero. The first condition is that risk aversion be standard—that is, that absolute risk aversion and absolute prudence be decreasing. The second condition is that absolute risk aversion be decreasing and convex. The third condition is that both the third and the fourth derivatives of the utility function be negative. If the expected value of the exogenous risk is positive, a wealth effect is added to the picture, which goes in the opposite direction if absolute risk aversion is decreasing.     

Disappointment and the Optimal Insurance Contract

This paper studies the optimal insurance contract under disappointment theory. We show that, when the individuals anticipate disappointment, there are two types of optimal insurance contract. The first type contains a deductible and a coinsurance above the deductible. We find that zero marginal cost is just a sufficient but not a necessary condition for a zero deductible. The second type has no deductible and the optimal insurance starts with full coverage for small losses and includes a coinsurance above an upper value of the full coverage.     

The comparative statics of deductible insurance in expected- and non-expected-utility theories

This paper identifies comparative statics results for insurance contracts that distinguish between various models of decision making under risk—specifically, expected utility, rank-dependent expected utility, and weighted utility. Insurance contracts offer full coverage above a deductible. Firms offer premium schedules that give the premium charged as a function of the deductible; households choose both an insurance company and a deductible to maximize utility. A competitive equilibrium requires zero expected profit for firms. We identify changes in the distribution of losses such that the optimal deductible increases for utility representations in a particular class but decreases for some representations outside that class. We give results both for the demand for insurance, as well as for the equilibrium contract.     

A Value at Risk Approach to Background Risk

This paper studies the effects of an uninsurable background risk (BR) on the demand for insurance (proportional and with deductible). We study both the case of BR uncorrelated with the insurable one and the perfectly correlated one, in a Gaussian world. In order to perform our study, we exploit the new risk measure known as Value at Risk (VaR) and consider insurance contracts which are Mean-VaR efficient. We obtain results which depend on the parameters (moments) of both risks and on the magnitude of loadings charged by the insurance company, instead of depending on the risk attitudes of the insured, such as risk aversion and prudence.We demonstrate that, if loadings are not too high, the demand for insurance increases with positively correlated BR; it decreases with BR negatively correlated if the latter is less risky than the insurable one (in this case it can even go to zero, if loadings are too high); it goes to zero with BR which is negatively correlated and more risky than the insurable one.     

Equity Risk, Conversion Risk, and the Demand for Insurance

Existing insurance theory fails when applied to real property because it does not account for variations in the economic environment. The article studies optimal property insurance in the presence of two sources of variation: equity risk and conversion risk. Equity risk is randomness of the value of a property. It tends to raise demand for conventional insurance. In contrast, conversion risk is randomness in the value the property would have if, after severe damage, it were converted to the highest‐valued use. It is distinct from equity risk because the highest‐valued use is typically not the current one. Under independent conversion risk, the optimum upper limit is a compromise among underlying conversion thresholds. Absent independence, the optimum can be quite different. Conversion risk can raise or lower the demand for property insurance. Insurance contracts that fail to address conversion tend to undermine the orderly disposition of obligations and reduce the gains from reallocation of risks through insurance.     

我国农业保险市场失衡分析——基于效用理论视角

本文在分析农业保险发展现状的基础上,基于马斯洛需求层次理论、风险偏好理论及风险规避程度度量理论,建立农户风险态度呈动态变化的效用函数曲线,推导出政府对农业保险补贴的上限和下限,得出结论:政府补贴过低无法达到刺激农户购买欲望转化为有效需求;但政府对农业保险的补贴并不是越多越好,超过一定限度,反而引发更为严重的道德风险,并...     

Consumers’ collision insurance decisions: a mental models approach to theory evaluation

Using interviews with 74 drivers, we elicit and analyze how people think about collision insurance coverage and decide whether to buy coverage, and if so, what deductible level to carry. We compare respondents’ judgments and behaviors to predictions of three models: baseline expected utility (EU) theory, which predicts that insurance is an inferior good, meaning more wealthy people buy less; a modified EU model, which incorporates income constraints and suggests that property insurance is a normal good, meaning more wealthy people buy more; and a mental accounting model which predicts that consumers budget income across consumption categories. The results suggest they purchase insurance as a normal good, guided by a cognitive model that emphasizes budget constraints. Verbal reports reveal a desire to balance two conflicting goals in deductible decisions: keeping premiums ‘affordable’ and keeping deductible level ‘affordable.’ Thus, wealth does not distinguish people by risk aversion, but by ability to pay. In other words, the behavior of less wealthy people is not driven by greater risk aversion, but by their lesser ability to pay, both now and later. We find that a simple heuristic using only vehicle value accounts for most decisions of whether to purchase optional collision coverage: out of 45 respondents who did not have loans on their vehicles, 90% of those with vehicles worth more than $1000 carried collision coverage, while less than 30% of those with lower‐valued vehicles did.     

Optimal insurance contract under a value-at-risk constraint

This study develops an optimal insurance contract endogenously under a value-at-risk (VaR) constraint. Although Wang et al. [2005] had examined this problem, their assumption implied that the insured is risk neutral. Consequently, this study extends Wang et al. [2005] and further considers a more realistic situation where the insured is risk averse. The study derives the optimal insurance contract as a single deductible insurance when the VaR constraint is redundant or as a double deductible insurance when the VaR constraint is binding. Finally, this study discusses the optimal coverage level from common forms of insurances, including deductible insurance, upper-limit insurance, and proportional coinsurance. JEL Classification G22     

When Is a Coinsurance‐Type Insurance Policy Inferior or Even Giffen?

This article derives the necessary and sufficient conditions for a coinsurance‐type insurance policy covering a particular risk to be inferior and to be Giffen. Mossin's decreasing absolute risk aversion assumption for insurance to be inferior is avoided. The result generalizes Hoy and Robson and Briys, Dionne, and Eeckhoudt's results to the case with a continuum of states and relaxes their assumption of constant relative risk aversion. It is shown that knowledge about the distribution of risk can be used to relax assumptions on an utility function for a coinsurance‐type insurance policy to be inferior and to be Giffen.     

Dynamic Portfolio Choice with Stochastic Wage and Life Insurance

We study optimal insurance, consumption, and portfolio choice in a framework where a family purchases life insurance to protect the loss of the wage earner's human capital. Explicit solutions are obtained by employing constant absolute risk aversion utility functions. We show that the optimal life insurance purchase is not a monotonic function of the correlation between the wage and the financial market. Meanwhile, the life insurance decision is explicitly affected by the family's risk preferences in general. The model also predicts that a family uses life insurance and investment portfolio choice to hedge stochastic wage risk.     

A Gentle Introduction to Risk Aversion and Utility Theory

While the topics of risk aversion and utility theory have been discussed extensively in the academic literature on risk and insurance, this literature does not include a pedagogical discussion that is widely accessible for classroom use. This article provides a practical introduction to risk aversion that is designed for readers with little prerequisite course work in economics or statistics. We describe a simple model of insurance demand that can be applied to the property, liability, life, and health insurance markets. We also demonstrate how risk aversion affects a variety of real-life insurance decisions made under conditions of uncertainty, including how much the market will bear to pay for insurance administrative expenses and how demand varies for different types of auto insurance coverage. Exercises and practice problems are provided so that readers can test their mastery of the concepts presented in the article. An instructional note on using this article to teach risk aversion in the classroom is also provided.     

Mossin's Theorem for Upper-Limit Insurance Policies

Although Mossin's Theorem (“full insurance with a fair premium and less‐than‐full coverage with a proportional premium loading”) is well known for the classes of coinsurance contracts and for deductible‐insurance contracts, it has not been proven for the class of upper‐limit insurance contracts. This article provides a proof for this case.     

Optimal Insurance Under the Insurer's VaR Constraint

In this paper, we impose the insurer's Value at Risk (VaR) constraint on Arrow's optimal insurance model. The insured aims to maximize his expected utility of terminal wealth, under the constraint that the insurer wishes to control the VaR of his terminal wealth to be maintained below a prespecified level. It is shown that when the insurer's VaR constraint is binding, the solution to the problem is not linear, but piecewise linear deductible, and the insured's optimal expected utility will increase as the insurer becomes more risk-tolerant. Basak and Shapiro (2001) showed that VaR risk managers often choose larger risk exposures to risky assets. We draw a similar conclusion in this paper. It is shown that when the insured has an exponential utility function, optimal insurance based on VaR constraint causes the insurer to suffer larger losses than optimal insurance without insurer's risk constraint.