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.     

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.     

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 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.     

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.     

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     

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.     

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.     

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.     

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.     

The Design of an Optimal Area Yield Crop Insurance Contract

This article focuses on the design of a crop insurance contract when the indemnity is based on the aggregate yield of a surrounding geographical area. Coinsurance under a critical yield often provides an efficient sharing of systemic risk. Under a linear relationship between individual yield and aggregate yield, the optimal form depends on the individual beta coefficient, which measures the sensitivity of individual yield to aggregate yield. The optimal hedging position of the producer on the yield options market is to buy put options or call options depending upon whether his beta coefficient is positive or negative.     

Wealth effects and committed cost allocation: An agency theory perspective

In this article we examine the effects of committed costs (CC) on compensation and effort (production) decisions in a principal-agent (P-A) setting. In the case where moral hazard is present, the compensation and effort (production) decisions are independent of CC whenever P has constant absolute risk aversion. When P has decreasing absolute risk aversion, he demands as increased risk premium, therefore increases the spread of the compensation schedule, and induces A to increase effort (production) and vice versa. The optimal compensation scheme can be decomposed to conform to incentive schemes that are generally observed in practice. In particular, we decompose the optimal compensation scheme that depends on CC into two parts—a part that is based on cash flows and a part that is based on income (after allocating committed costs). In the case where effort is observable, CC does not effect the compensation scheme and effort decisions, when P has constant absolute risk aversion. In contrast to earlier studies that examine the owner-manager case, when P has decreasing absolute risk aversion, effort (production) could either increase or decrease. The presence of moral hazard affects the effort (production) decision differently than when risk-sharing considerations alone exist. The reduction in A's compensation induced by increased CC never exceeds the amount of CC.     

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.
     

Optimal Insurance Design Under a Value-at-Risk Framework

This study designs an optimal insurance policy form endogenously, assuming the objective of the insured is to maximize expected final wealth under the Value-at-Risk (VaR) constraint. The optimal insurance policy can be replicated using three options, including a long call option with a small strike price, a short call option with a large strike price, and a short cash-or-nothing call option. Additionally, this study also calculates the optimal insurance levels for these models when we restrict the indemnity to be one of three common forms: a deductible policy, an upper-limit policy, or a policy with proportional coinsurance. JEL Classification No: G22     

The Design of Optimal Insurance Contracts: A Topological Approach

This article deals with the optimal design of insurance contracts when the insurer faces administrative costs. If the literature provides many analyses of risk sharing with such costs, it is often assumed that these costs are linear. Furthermore, mathematical tools or initial conditions differ from one paper to another. We propose here a unified framework in which the problem is presented and solved as an infinite dimensional optimization program on a functional vector space equipped with an original norm. This general approach leads to the optimality of contracts lying on the frontier of the indemnity functions set. This frontier includes, in particular, contracts with a deductible, with total insurance and the null vector. Hence, we unify the existing results and point out some extensions.     

Bank Funding Risks,Risk Aversion,and the Choice of Futures Hedging Instrument

Currently, theories of financial futures hedging are based on either a portfolio-choice approach or a duration approach. This article presents an alternative: a firm-theoretic model of bank behavior with financial futures. Assuming the bank is uncertain about cash CD interest rates and the quantity of CDs it needs in the future, expressions for the optimal futures hedge are derived under constant absolute risk aversion and constant relative risk aversion. The performance of these two strategies is estimated from 1981–1983 using either the recently developed CD futures contract or the T-Bill futures contract. These results are also compared with the performance of a portfolio-choice strategy and a routine hedging strategy. The analysis indicates that the CD futures market can serve a hedging purpose that is not served by the previously established T-Bill futures market.     

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.     

Background Risk and the Performance of Insurance Markets under Adverse Selection

Background risk can influence the performance of insurance markets that must deal with adverse selection when applicants are risk vulnerable, since they are more averse to bearing the insurable risk as a result of their exposures to background risk. We show that background risk always results in a lower deductible for the incentive constrained contract, and that a broader range of markets attains the stable sequential equilibrium cross-subsidized pair of separating contracts. We conclude that background risk always improves the performance of markets for coverage against (insurable) foreground risks that must deal with adverse selection. We also find, however, that these improvements are never sufficient to offset the cost to insureds of bearing the background risk.     

Asymmetric information in insurance: general testable implications

Several recent articles on empirical contract theory and insurance have tested for a positive correlation between coverage and ex post risk, as predicted by standard models of pure adverse selection or pure moral hazard. We show here that the positive correlationproperty can be extended to general setups: competitive insurance markets and cases where risk aversion is public. We test our results on a French dataset. Our tests confirm that the estimated correlation is positive; they also suggest the presence of market power.     

Background Risk in the Principal-Agent Model

We examine the effect of background risk in the standard two-state, two-action principal-agent model. We analyse situations where the background risk is environmental (always present) and where the background risk is contractual (only present if the contract is accepted). With contractual background risk, expected wages always rise and the incentive scheme is flatter if the agent's preferences satisfy weak decreasing absolute risk aversion. With environmental background risk, the optimal incentive scheme becomes flatter if the agent is weakly prudent. We provide conditions under which the environmental background risk decreases the agent's expected wage.