Optimal insurance contracts without the non-negativity constraint on indemnities: revisited

In the literature on optimal indemnity schedules, indemnities are usually restricted to be non-negative. Keeler [1974] and Gollier [1987] show that this constraint might well bind: insured could get higher expected utility if insurance contracts would allow payments from the insured to the insurer at some losses. This paper extends Collier’s findings by allowing for negative indemnity payments for a broader class of insurers’ cost functions and argues that the indemnity schedule derived here is more appropriate for practical applications (e.g. in health insurance). JEL Classification D80 · D81 · D89     

保险法损失补偿原则规范功能之重塑——以“禁止得利”向“充分补偿”之学说演进为中心

损失补偿原则的规范功能具有"二元性",即禁止被保险人不当得利和充分填补被保险人的损失。但在传统保险法理论和实践中,为了达到防范道德风险的功能,损失补偿原则完全被禁止不当得利原则所取代。随着保险技术与保险观念的进步,传统理论下的保险制度难以满足被保险人充分补偿需求并容易造成保险合同效率的低下,因此损失补偿原则规范之重心应完成从"禁止得利"向"充分补偿"的转变。在制度设计及保险合同中,应恰当缩减合同条款对补偿金额的限制、重视发展重置成本保险、重复保险中应完成分摊中按份责任向连带责任之转变、优先实现被保险人的剩余损害赔偿请求权,以更全面保障被保险人的利益。     

Arrow's Theorem of the Deductible with Heterogeneous Beliefs

In Arrow's classical problem of demand for insurance indemnity schedules, it is well-known that the optimal insurance indemnification for an insurance buyer—or decision maker (DM)—is a deductible contract when the insurer is a risk-neutral Expected-Utility (EU) maximizer and when the DM is a risk-averse EU maximizer. In Arrow's framework, however, both parties share the same probabilistic beliefs about the realizations of the underlying insurable loss. This article reexamines Arrow's problem in a setting where the DM and the insurer have different subjective beliefs. Under a requirement of compatibility between the insurer's and the DM's subjective beliefs, we show the existence and monotonicity of optimal indemnity schedules for the DM. The belief compatibility condition is shown to be a weakening of the assumption of a monotone likelihood ratio. In the latter case, we show that the optimal indemnity schedule is a variable deductible schedule, with a state-contingent deductible that depends on the state of the world only through the likelihood ratio. Arrow's classical result is then obtained as a special case.     

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.     

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.     

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.     

The Demand for Insurance With an Upper Limit on Coverage

The demand for insurance is examined when the indemnity schedule is subject to an upper limit. The optimal contract is shown to display full insurance above a deductible up to the cap. Some results derived in the standard model with no upper limit on coverage turn out to be invalid; the optimal deductible of an actuarially fair policy is positive and insurance may be a normal good under decreasing absolute risk aversion. An increase in the upper limit would induce the policyholder with constant absolute risk aversion to reduce his or her optimal deductible and therefore this would increase the demand for insurance against small losses.     

论投保人缴纳保费或保险公司签发保单附条件条款的法律效力

我国保险实践中普遍将投保人缴纳保费或者保险公司签发保单作为保险合同的成立、生效或者保险责任开始的条件。但该条件不是新《保险法》规定的附条件保险合同所指的条件。同时该附条件条款赋予了保险公司对要约进行承诺时或者履行合同义务时不受时间限制的权利,该附条件条款因不具有合理性应为无效合同条款。     

Equilibrium in Competitive Insurance Markets Under Adverse Selection and Yaari's Dual Theory of Risk

Under Yaari's dual theory of risk, we determine the equilibrium separating contracts for high and low risks in a competitive insurance market, in which risks are defined only by their expected losses, that is, a high risk is a risk that has a greater expected loss than a low risk. Also, we determine the pooling equilibrium contract when insurers are assumed non-myopic. Expected utility theory generally predicts that optimal insurance indemnity payments are nonlinear functions of the underlying loss due to the nonlinearity of agents' utility functions. Under Yaari's dual theory, we show that under mild technical conditions the indemnity payment is a piecewise linear function of the loss, a common property of insurance coverages.     

Optimales Vertragsdesign bei moralischem Risiko in der Rückversicherung

This paper addresses the optimal design of risk sharing arrangements in reinsurance contracts with asymmetric information concerning the primary insurer’s behavior. The latter usually has significant unobservable discretions, for instance with respect to risk selection, implying a moral hazard problem. We show that the existence of moral hazard strongly affects the characteristics of the reinsurance indemnification rule, i. e. the connection between the level of losses and the indemnity, which is specified in the contract. For this analysis, a standard model framework from the theory of optimal reinsurance with perfect information is modified by the assumption that the primary insurer has unobservable control of the probability distribution of the extent of losses. In particular, the solution indicates that for this situation, a Pareto-optimal indemnity rule is less steep, and therefore the primary insurer’s share in a marginal increase of the loss is greater, compared to the case of complete information. A deductible, however, turns out not to be a suitable approach in this context.     

Ex ante loss control by insurers: Public interest for higher profit

This article examines the incentives of an insurer to modify loss distributions prior to the sale of insurance. While actions such as lobbying Congress for mandatory airbags in automobiles are undertaken by insurers for the stated purpose of reducing the aggregate loss in society, they also change the nature of the risk being insured and, hence, affect the profitability of insurance sales. For the case of loss prevention (reducing the probabilty of a loss), insurers do not always have an incentive to invest in loss control. For loss reduction (reducing the severity of any loss that does occur), the incentive is to reduce the size of small losses while simultaneously increasing the size of large losses. Venezian Associates     

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.     

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.     

Die Haftpflichtversicherung nach der VVG-Reform

The reform of the German Insurance Contract Act (Versicherungsvertragsgesetz, ?VVG“) also targets key aspects of third-party liability insurance. The changes go beyond the findings made by both the courts and legal authorities to date.Compulsory insurance aside, the law still provides that an injured third party has no standing to assert a claim directly against the tortfeasor’s liability insurer. The tortfeasor may assign its indemnity claim against the insurer solely to the injured third party and may no longer be precluded from doing so under the General Insurance Conditions (AVB). Consequently, the tortfeasor’s indemnity claim against the insurer effectively becomes a pecuniary claim. This is criticised by the insurance industry particularly with regard to eliminating the prohibition against acknowledgment and satisfaction of claims.In the future, third parties will be able to assert claims directly against the tortfeasor’s insurer and this will be the case for compulsory insurance across the board. Provisions currently in effect in the motor vehicle liability insurance industry will be carried over to the entire compulsory insurance sector. Compulsory insurance does permit agreements involving self-deductibles. However, such agreements are generally effective only as between the insurer and the tortfeasor inter se, i.e. they are not effective as against third parties — in contrast to valid disclaimers of risk.Another change in compulsory insurance is the hierarchy of claims for compensatory damages and relief in the event the insured amount is inadequate. Specifically, the hierarchy gives preference to individual claims of injured parties which are not otherwise covered, such as claims for pain and suffering.The prohibition against the retroactive loss of provisional coverage for failure to pay the first premium, which had been criticised primarily by motor vehicle liability insurers, has been omitted in the Government bill.     

A Continuous-Time Optimal Insurance Design with Costly Monitoring

We provide a theoretical and numerical framework to study optimal insurance design under asymmetric information. We consider a continuous-time model where neither the efforts nor the outcome of an insured firm are observable to an insurer. The insured may then cause two interconnected information problems: moral hazard and fraudulent claims. We show that, when costly monitoring is available, an optimal insurance contract distinguishes the one problem from the other. Furthermore, if the insured’s downward-risk aversion is weak and if the participation constraint is not too tight, then a higher level of the monitoring technology can mitigate both problems.     

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.     

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.     

临时保险制度：赔付理由、期间及其限制

我国《保险法》未规定临时保险制度,本文建议,我国的临时保险制度应当分为强制临时保险与自愿临时保险两种,在保险人预收保险费的情况下,根据履行提前、对价平衡、合理期待等理论,保险人应当对临时保险事故予以赔付,此种临时保险的期间自投保人交付保险费之时起算,于正式保险生效之时或投保人收到拒保通知书并由保险人退还保险费之时终止;...     

A note on increased probability of loss and the demand for insurance

It is shown that the effect of increased probability of loss on the demand for insurance depends on whether both insured and insurer are aware of the change. When both insurer and insured share the same beliefs about the probability of loss (symmetric information), an increase in the loss probability may lead risk-averse agents to demandless insurance.     

Optimal insurance under maxmin expected utility

We examine a problem of demand for insurance indemnification, when the insured is sensitive to ambiguity and behaves according to the maxmin expected utility model of Gilboa and Schmeidler (J. Math. Econ. 18:141–153, 1989), whereas the insurer is a (risk-averse or risk-neutral) expected-utility maximiser. We characterise optimal indemnity functions both with and without the customary ex ante no-sabotage requirement on feasible indemnities, and for both concave and linear utility functions for the two agents. This allows us to provide a unifying framework in which we examine the effects of the no-sabotage condition, of marginal utility of wealth, of belief heterogeneity, as well as of ambiguity (multiplicity of priors) on the structure of optimal indemnity functions. In particular, we show how a singularity in beliefs leads to an optimal indemnity function that involves full insurance on an event to which the insurer assigns zero probability, while the decision maker assigns a positive probability. We examine several illustrative examples, and we provide numerical studies for the case of a Wasserstein and a Rényi ambiguity set.