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.     

Insurance Contract Design When the Insurer Has Private Information on Loss Size

This article examines the optimal indemnity contract in an insurance market, when the insurer has private information about the size of an insurable loss. Both parties know whether or not a loss occurred, but only the insurer knows the true value of the loss and/or to what extent the losses are covered under the policy. The insured may verify the insurer's loss estimate for a fixed auditing cost. The optimal contract reimburses the auditing costs in addition to full insurance for losses less than some endogenous limit. For losses exceeding this limit, the contract pays a fixed indemnity and requires no monitoring. The optimal contract is compared with the contracts obtained in cases where it is only the insured who can observe the loss size.
     

The benefit of life insurance contracts with capped index participation when stock prices are subject to jump risk

We analyze the benefit to the insured of newly traded, innovative life insurance contracts. On a sequence of yearly reference days, the insured can choose between a guaranteed return (linked to the insurer’s asset result) and a capped index participation. The cap is adjusted at the beginning of each year such that both alternatives have the same value and the option to select is costless (product structuring condition). We point out that this condition cannot always be met. If the guaranteed return exceeds the upper bound of the capped index participation, the insurer can make a side profit. We show that a rather low insurance result also implies a rather low stock exposure, even if the insured opts for the index participation. Concerning the impact of the index dynamics, we emphasize that it is important to distinguish between jump and diffusion risk because the pricing of jump risk has an impact on cap rates that can be offered to an insured. Finally, we show that the optimal decision strategy of a CRRA investor implies an index selection even if it is unfairly priced such that the insurer indeed makes a side profit.     

Efficient Insurance Contracts under Epsilon-Contaminated Utilities

This paper analyses the qualitative properties of optimal contracts when agents have multiple priors and are uncertainty averse in an infinite state space framework. The case of the epsilon-contamination of a given prior, a basic tool in robustness theory is fully developped. It is shown that if both agents have strictly concave utility index, then if the insurer is less uncertainty averse than the insured, he provides a full insurance contract above a deductible for high values of the loss.     

Automobile Insurance Contracts and Risk of Accident: An Empirical Test Using French Individual Data

Insurance has for a long time been perceived as a way of transferring responsibility from insured agents to insurers and thus as potentially influencing insured agents' behavior. Two particular opportunistic behaviors have been analyzed. First, the theory of adverse selection predicts that high-risk agents are likely to demand more insurance than are low-risk agents. Second, the theory of moral hazard predicts that the wider the insurance coverage, the less agents will try to prevent accidents. Both theories thus conclude that agents who are totally insured should have a higher probability of accident than those with only partial insurance, ceteris paribus. Nevertheless, one of the aims of insurance rating systems is to control for these opportunistic behaviors. In this article, we use individual data to see if the French automobile insurance rating system has achieved this aim. We do this using a two-step maximum-likelihood method. First, we compute a probit model to estimate the probability of taking out comprehensive versus third-party insurance. We then calculate the generalized residual, which is included as an independent variable in a negative binomial model estimating the probability of having an accident. The coefficient of this variable is argued to represent adverse selection and ex-ante moral-hazard behavior.     

Insurance Market Effects of Risk Management Metrics

We extend the classical analysis on optimal insurance design to the case when the insurer implements regulatory requirements (Value-at-Risk). Presumably, regulators impose some risk management requirement such as VaR to reduce the insurers’ insolvency risk, as well as to improve the insurance market stability. We show that VaR requirements may better protect the insured and improve economic efficiency, but have stringent negative effects on the insurance market. Our analysis reveals that the insured are better protected in the event of greater loss irrespective of the optimal design from either the insured or the insurer perspective. However, in the presence of the VaR requirement on the insurer, the insurer's insolvency risk might be increased and there are moral hazard issues in the insurance market because the optimal contract is discontinuous.     

Deposit Insurance, Moral Hazard and Market Monitoring

The paper analyses the relationship between deposit insurance, debt-holder monitoring, and risk taking. In a stylised banking model we show that deposit insurance may reduce moral hazard, if deposit insurance credibly leaves out non-deposit creditors. Testing the model using EU bank level data yields evidence consistent with the model, suggesting that explicit deposit insurance may serve as a commitment device to limit the safety net and permit monitoring by uninsured subordinated debt holders. We further find that credible limits to the safety net reduce risk taking of smaller banks with low charter values and sizeable subordinated debt shares only. However, we also find that the introduction of explicit deposit insurance tends to increase the share of insured deposits in banks' liabilities.     

Corporate pension funding policy

What policy should a corporation adopt concerning the funding of a defined-benefit pension plan and the investment of the assets held in trust for the plan? Until recently, pension plans did not have to be insured, and some risk could be borne by intended beneficiaries. Federal legislation has now mandated such coverage. This paper analyzes corporate policy under three conditions which correspond, roughly, to the earlier situation (‘uninsure’ loans), the current situation (‘partially insured’ loans), and the situation required by law to be implemented in the future (‘completely insured’ plans). We show that if insurance premiums are set correctly, corporate policy in this area may not matter; otherwise the optimal policy may simply be that which maximizes the difference between the value of the insurance and its cost.     

Insurance law and incomplete contracts

We analyze how insurance law can mitigate moral hazard by allowing insurers to reduce or cancel coverage in some circumstances. We consider an incomplete contract setting in which the insurer may obtain information related to the policyholder's behavior through a costly audit of the circumstances of the loss. Court decisions are based on a standard of proof such as the balance of probabilities. We show that an optimal insurance law brings efficiency gains compared to the no-audit case. We also highlight the conditions under which the burden of proof should be on the insured, provided that insurers are threatened with sanctions for bad faith.     

Deposit Insurance, Capital Adequacy Requirements and Interest Rate Dynamics

This paper examines the long run interaction among deposit insurance, bank deposit rates and capital adequacy requirements. Using analysis similar to the price discrimination model of Lott and Roberts (1991) we find that a competitive environment among banks would link the spread between insured and uninsured deposit rates to the size of the insurance premium. We also find that banks that choose to operate at the regulatory minimum capital level, would increase asset risk with increased capital requirements if (1) the implicit interest paid to insured and uninsured depositors is equally sensitive to changes in risk and capital adequacy and (2) the insurance premium is independent of the level of risk and capital adequacy. Under the present risk-based premium structure, asset risk has the potential to decline when the regulatory agency raises capital requirements. Finally, we examine the time series behavior of insured and uninsured interest rates to see if it is consistent with our theoretical model. We find that insured and uninsured rates, along with deposit insurance premiums, are cointegrated series as suggested by our model.     

Who Obtains Greater Discounts on Automobile Insurance Premiums?

Insurance purchasers obtain varied discounts for insurance. This paper examines what drives these differences, specifically whether the loss probability and the wealth of the insured affect the size of the premium discount in automobile insurance. To describe a bargain between a client and an insurer over premiums and coverage, we first develop a sequential insurance bargaining game where the client has an outside option to bargain with another insurer. We find that the equilibrium involves full coverage and, based on the results of comparative statics, we propose hypotheses regarding the effects of the loss probability and the wealth of the insured on the size of the premium discount. We then use a unique data set of 85,806 observations of Taiwanese automobile liability insurance for property damage to empirically test the predictions. After controlling for underwriting and macroeconomic variables, we find that both (1) the insured with a lower claim probability (as a proxy for the insured with a lower loss probability) and (2) the insured with a higher salvage value car (as a proxy for the wealthier insured) receive a greater premium discount. These results support our theoretical results.     

Das deutsche und Schweizer Krankenversicherungssystem: Kosten,Leistungen und Anreizwirkungen aus Sicht der Versicherten

In the last years changes in the financing and in the service catalogue of the German health insurance system have been intensively discussed. The problematic incentive structures regarding the behaviour of the insured are frequently mentioned in these discussions. In particular, German insured have only little interest in a cost-efficient treatment of illness. At the same time the Swiss health insurance system is an example in which the insured show a higher cost consciousness. In the following article, the health insurance systems of Germany and Switzerland are compared from the perspective of the insured on the basis of the information-economic terms “moral hazard” and “adverse selection.” We find that both systems can learn from each other in reducing incentive problems.     

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.     

Gendiagnostikgesetz und Versicherung: Anspruch und Wirklichkeit

With the new German Gene Diagnostic Act (Gendiagnostikgesetz) the legislator aims at improving the protection of insurance applicants by prohibiting private insurers from collecting and using genetic information. However, the analysis of the new provisions shows that the provisions pertaining to insurance neither provide a comprehensive protection against genetic discrimination of insurance applicants and insured nor do they protect their right of gene-informational self-determination. Cuts of insurance benefits of the insured in private health insurance as well as incoherent disclosure obligations for insurance applicants unconstitutionally limit the rights of affected people as compared to the time before the Gene Diagnostic Act came into force. In summary, the new Gene Diagnostic Act does not only fail to meet its claims and thus falls short of the expectations, but also, due to numerous unclear provisions, poses a series of grave problems for insurance applicants, privately insured and private insurers.     

Welfare effects of insurance contract non-performance

Non-performance lies at the heart of much of the regulation that insurance companies face. Consumers’ concerns about non-performance of the insurance provider have also been cited as a possible explanation for low demand of microinsurance. We provide a behavioral evaluation of the welfare effects of non-performance risk. We test the hypothesis that the presence of non-performance risk negatively impacts not just take-up of insurance but more importantly the welfare of the insured. We also test if violations of the reduction of compound lotteries axiom could drive this decrease in take-up and welfare. The results show that the compound risk characteristic of non-performance risk does not significantly decrease the welfare of insurance choices made by individuals. This counter-intuitive result is sensitive to the structural modeling of risk preferences. If one assumes the reduction of compound lotteries axiom does characterize behavior towards risk, one finds evidence that non-performance risk reduces welfare for the insured. But if one correctly allows for violations in that axiom in the representation of risk preferences, which is appropriate if one is going to test for the effect of compound risk from non-performance, then the counter-intuitive result is obtained. Take-up is not a reliable proxy for welfare, and the behavioral drivers of take-up are again not the same drivers of welfare. These results provide structural behavioral insight to inform normative policy design with respect to insurance regulation.     

The design of an optimal insurance contract for irreplaceable commodities

This paper discusses optimal insurance contract for irreplaceable commodities. To describe the dual impacts on individuals when a loss occurs to the insured irreplaceable commodities, we use a state-dependent and bivariate utility function, which includes both the monetary wealth and sentimental value as two arguments. We show that over (full, partial) insurance is optimal when a decrease in sentimental value will increase (not change, decrease, respectively) the marginal utility of monetary wealth. Moreover, a non-zero deductible exists even without administration costs. Furthermore, we demonstrate that a positive fixed reimbursement is optimal if (1) the premium is actuarially fair, (2) the monetary loss is a constant, and (3) the utility function is additively separable and the marginal utility of money is higher in the loss state than in the no-loss state. We also characterize comparative statics of fixed-reimbursement insurance under an additively separable preference assumption. JEL Classification G22 · D86 The author acknowledge funding from National Science Council in Taiwan (NSC93-2416-H-130-020).     

从博弈论看保险市场信息不对称的本质及规范举措

博弈论是研究信息不对称问题的一种有效方法。保险业作为信息不对称最为集中的行业，保险人与被保险人之间存在着严重的信息不对称，特别是被保险人的逆向选择、道德风险和保险人的违规操作等现象普遍存在，长期困扰着保险界。     

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     

金融创新如何缓解信任品市场失灵?——中国食品安全责任强制保险的自然实验

信任品市场(如食品、医药等)存在的问题一直困扰着中国和世界很多国家。关于产品和服务质量的信息不对称(道德风险和逆向选择)会导致信任品市场失灵。缓解信任品市场失灵,一种常见的解决方案是政府监管;而经济学家认为更加基于市场的解决方案(例如强制责任保险等金融创新)可能更为有效。在理论上,强制责任保险有两种相反的效应:保险公司的监督减少了道德风险 vs. 保险加剧了企业的道德风险,但一直亟待实证检验。幸运的是,中国食品安全责任强制保险的改革实验走在了世界的前列。本文利用了中国在不同地区、不同时间推行的这个自然实验,通过双重差分的方法识别出责任保险对于信任品市场的因果效应。本文的研究发现,强制责任保险能显著降低食品安全事故发生概率。这表明政府强制推行的金融创新可以成为信任品市场失灵的一种有效的替代性解决方案。