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     

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     

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.     

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.     

Multiple Losses, EX ANTE Moral Hazard, and the Implications for Umbrella Policies

Under certain cost conditions the optimal insurance policy offers full coverage above a deductible, as Arrow and others have shown. However, many insurance policies currently provide coverage against several losses although the possibilities for the insured to affect the loss probabilities by several prevention activities (multiple moral hazard) are substantially different. This article shows that optimal contracts under multiple moral hazard generally call for complex reimbursement schedules. It also examines the conditions under which different types of risks can optimally be covered by a single insurance policy and argues that the case for umbrella policies under multiple moral hazard is limited in practice.     

Remarks on the Mossin Theorem

We establish several new results regarding the Mossin Theorem under both nonrandom initial wealth and random initial wealth. For the nonrandom initial wealth case, we show that the Mossin Theorem holds for any constrained class of insurance contracts whose maximum coverage provides full coverage of the potential loss. This result not only settles an open conjecture, but also provides a unified treatment for extant varieties of the Mossin Theorem. For the random initial wealth case, we give a thorough study of the upper-limit insurance. In particular, we show that (1) for a fair premium, the Generalized Mossin Theorem for coinsurance does not hold for upper-limit insurance, and (2) for an unfair premium, partial insurance will always be optimal, regardless of the risk preference of the individual and the dependence structure between the random loss and the random initial wealth.     

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 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.     

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 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).     

On Compulsory Per-Claim Deductibles in Automobile Insurance

The purposes of this paper are to analyze the theoretical characteristics of the compulsory deductible system and to verify the rationality of an increasing per-claim deductible in automobile insurance. We derive the optimal variable per-claim deductible by assuming the insurers are financially balanced and the expected utility of the insured is maximized in the absence of moral hazard. Our result suggests that a variable per-claim deductible increasing with the number of claims per year is not optimal. Instead, deductibles should be charged in a decreasing rate forming a second-best solution.     

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.     

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.     

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.     

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.     

A note on Mossin’s theorem for deductible insurance given random initial wealth

Mossin’s theorem for deductible insurance given random initial wealth is re-examined. For a fair premium, it is shown that a necessary and sufficient condition, in the spirit of the Generalized Mossin Theorem for coinsurance, is impossible using the notion of expectation dependence. Next, it is established that for a fair premium, full insurance will be optimal for a risk-averse individual if the random loss and the random initial wealth are negative quadrant dependent, improving upon an extant result in the literature. In view of a set of examples given in this paper, such a sufficient condition cannot be obtained using the notion of expectation dependence. Finally, for an unfair premium, it is shown that partial insurance will always be optimal, irrespective of the risk preference of the individual as well as the dependence structure between the random loss and the random initial wealth.     

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.     

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.     

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.     

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.