Optimal Insurance Design Research Articles

Optimal Insurance Design under Different Dependency Structures of Insurable Risk and Background Risk

Optimal Insurance Design under Different Dependency Structures of Insurable Risk and Background Risk

Optimal insurance design under belief-dependent utility and ambiguity

Optimal insurance design under belief-dependent utility and ambiguity

Optimal insurance design under mean-variance preference with narrow framing

In this paper, we study an optimal insurance design problem under mean-variance criterion by considering the local gain-loss utility of the net payoff of insurance, namely, narrow framing. We extend the existing results in the literature to the case where the decision maker has mean-variance preference with a constraint on the expected utility of the net payoff of insurance, where the premium is determined by the mean-variance premium principle. We first show the existence and uniqueness of the optimal solution to the main problem studied in the paper. We find that the optimal indemnity function involves a deductible provided that the safety loading imposed on the “mean part” of the premium principle is strictly positive. Our main result shows that narrow framing indeed reduces the demand for insurance. The explicit optimal indemnity functions are derived under two special local gain-loss utility functions – the quadratic utility function and the piecewise linear utility function. As a spin-off result, the Bowley solution is also derived for a Stackelberg game between the decision maker and the insurer under the quadratic local gain-loss utility function. Several numerical examples are presented to further analyze the effects of narrow framing on the optimal indemnity function as well as the interests of both parties.

Revisiting the optimal insurance design under adverse selection: Distortion risk measures and tail-risk overestimation

This paper studies the design of optimal insurance from an insurer's perspective when it is subject to adverse selection issue. Different from the literature, the insureds who are exposed to different types of risks are allowed to apply different preference measures. By assuming that the insureds' preferences are dictated by some distortion risk measures that always over-estimate the tail risk, we figure out the optimal policy menu without assuming the parametric form of indemnity functions. We also find that the insureds who deem their losses riskier than those of others will always purchase full insurance, which is consistent with the results in past studies. Furthermore, we show that in the presence of adverse selection the optimal policy menu always outperforms the optimal single policy in the sense that the former can yield a larger expected profit for the insurer. This outcome also echoes some existing results in the literature.

MEAN–VARIANCE INSURANCE DESIGN WITH COUNTERPARTY RISK AND INCENTIVE COMPATIBILITY

AbstractThis paper studies the optimal insurance design from the perspective of an insured when there is possibility for the insurer to default on its promised indemnity. Default of the insurer leads to limited liability, and the promised indemnity is only partially recovered in case of a default. To alleviate the potential ex post moral hazard, an incentive compatibility condition is added to restrict the permissible indemnity function. Assuming that the premium is determined as a function of the expected coverage and under the mean–variance preference of the insured, we derive the explicit structure of the optimal indemnity function through the marginal indemnity function formulation of the problem. It is shown that the optimal indemnity function depends on the first and second order expectations of the random recovery rate conditioned on the realized insurable loss. The methodology and results in this article complement the literature regarding the optimal insurance subject to the default risk and provide new insights on problems of similar types.

Regret-based optimal insurance design

In this paper, we investigate an optimal insurance design problem in which the insured's risk preference is of the regret-theoretical expected utility type and admissible insurance contracts are set to satisfy both the principle of indemnity and the incentive-compatible condition. When the insurance premium is calculated by the expected value principle, we show that the optimal solution for a regret-averse insured can be in the form of no insurance or partial insurance above a deductible. We obtain a necessary and sufficient condition for the optimality of no insurance and provide a numerical algorithm to derive the optimal partial insurance above a deductible. Notably, the deductible disappears if and only if the insurance premium is actuarially fair. Finally, we carry out a comparative statics analysis to examine the effect of the insured's regret coefficient on the insured's final wealth and insurance demand. When the insurance pricing is actuarially fair, we find that the increase of the regret coefficient incentivizes the insured to purchase less insurance coverage and leads to an increase in risk of the insured's final wealth. However, our numerical analysis indicates that this finding is not always true for unfair insurance pricing.

OPTIMAL INCENTIVE-COMPATIBLE INSURANCE WITH BACKGROUND RISK

ABSTRACTIn this paper, the optimal insurance design is studied from the perspective of an insured, who faces an insurable risk and a background risk. For the reduction of ex post moral hazard, alternative insurance contracts are asked to satisfy the principle of indemnity and the incentive-compatible condition. As in the literature, it is assumed that the insurer calculates the insurance premium solely on the basis of the expected indemnity. When the insured has a general mean-variance preference, an explicit form of optimal insurance is derived explicitly. It is found that the stochastic dependence between the background risk and the insurable risk plays a critical role in the insured’s risk transfer decision. In addition, the optimal insurance policy can often change significantly once the incentive-compatible constraint is removed.

Mean-variance insurance design with counterparty risk and incentive compatibility

This paper studies the optimal insurance design from the perspective of an insured when there is possibility for the insurer to default on its promised indemnity. Default of the insurer leads to limited liability, and the promised indemnity is only partially recovered in case of a default. To alleviate the potential ex post moral hazard, an incentive compatibility condition is added to restrict the permissible indemnity function. Under the actuarial premium principle and mean-variance preferences of the insured, we derive the explicit structure of the optimal indemnity function through the marginal indemnity function formulation of the problem. It is shown that the optimal indemnity function depends on the first and second order conditional expectations of the random recovery rate. The methodology and results in this paper complement the literature regarding the optimal insurance subject to the default risk and provide new insights on problems of similar types. Moreover, we also apply the techniques in this paper to the case of additive background risk, which yields an alternative proof of the main result of Chi and Tan (2021).

Optimal insurance design under Vajda condition and exclusion clauses

In this paper, we explore the optimal insurance problem where the exclusion clause is taken into account. Assume that the insurable loss is mutually exclusive from another loss that is denied in the insurance coverage. Our objective is to characterize the optimal insurance strategy by minimizing the risk-adjusted value of a policyholder’s liability, where the unexpected loss is calculated by either the value at risk (VaR) or the tail value at risk (TVaR). To prevent moral hazard and to reflect the spirit of insurance, we analyze the optimal solutions over the class of ceded loss functions such that the policyholder’s retained loss and the proportion paid by an insurer are both increasing. We show that every admissible insurance contract is suboptimal to a ceded loss function composed of three interconnected line segments if the insurance premium principles satisfy risk loading and convex order preserving. The form of optimal insurance can be further simplified if the premium principles satisfies an additional weak property. Finally, we derive the optimal insurance explicitly for the expected value principle and Wang’s principle.

Optimal insurance design under narrow framing

In this paper, we study insurance decisions when the policyholder evaluates insurance with narrow framing. We show that due to aversion to risk on the net insurance payoff, i.e., insurance indemnity minus insurance premium, narrow framing reduces insurance demand. This helps explaining the observed low insurance demand in many insurance markets. We also show that the optimal insurance contract involves a deductible and the coinsurance of losses above the deductible when transaction costs depend on the actuarial value of the policy. Moreover, when the policyholder is loss averse over the net insurance payoff, a fixed indemnity equal to insurance premium should be paid for a range of intermediate losses.

On Belief Singularity and the Optimality of Deductible Indemnity Schedules

In Ghossoub (2019), we studied a problem of optimal insurance design under belief heterogeneity. We examined the effect of the divergence in beliefs, and the singularity between the beliefs, on optimal indemnities; and we provided a closed-form characterization of optimal indemnity schedules. In this paper, we build upon the work of Ghossoub (2019) and introduce a belief singularity metric to examine the effect of the level of singularity between the beliefs on the shape of optimal indemnity schedules. Our main result shows that for any level x of belief singularity with respect to the insurer's belief Q, there exists a probability measure P exhibiting a level x of belief singularity with respect to Q, such that an optimal indemnity for a decision maker with belief P is a linear deductible with Q-probability 1, and full insurance with Q-probability 0. In this sense, we show that from the perspective of the insurer, linear deductible indemnity schedules are robust to belief singularity, which gives a possible explanation of the popularity of these contracts in practice.

Optimal insurance under rank-dependent expected utility

We re-visit the problem of optimal insurance design under Rank-Dependent Expected Utility (RDEU) examined by Bernard et al. (2015), Xu (2018), and Xu et al. (2018). Unlike the latter, we do not impose the no-sabotage condition on admissible indemnities, that is, that indemnity and retention functions be nondecreasing functions of the loss. Rather, in a departure from the aforementioned work, we impose a state-verification cost that the insurer can incur in order to verify the loss severity, hence automatically ruling out any ex post moral hazard that could otherwise arise from possible misreporting of the loss by the insured. We fully characterize the optimal indemnity schedule and discuss how our results relate to those of Bernard et al. (2015) and Xu et al. (2018). We then extend the setting by allowing for a distortion premium principle, with a distortion function that differs from that of the insured, and we provide a characterization of the optimal retention in that case.

Optimal Insurance Under Rank-Dependent Expected Utility

We re-visit the problem of optimal insurance design under Rank-Dependent Expected Utility (RDEU) examined by Bernard et al. (2015), Xu (2018), and Xu et al. (2015). Unlike the latter, we do not impose the no sabotage condition on admissible indemnities, that is, the comonotonicity of indemnity functions and retention functions with the loss. Rather, in a departure from the aforementioned work, we impose a state-verification cost that the insurer can incur in order to verify the loss severity, hence automatically ruling out any ex post moral hazard that could otherwise arise from possible misreporting of the loss by the insured. Hence, monotonicity properties of indemnification schedules become of second-order concern. We fully characterize the optimal indemnity schedule and discuss how our results relate to those of Bernard et al. (2015) and Xu et al. (2015). We then extend the setting by allowing for a distortion premium principle, with a distortion function that differs from that of the insured, and we provide a characterization of the optimal retention in that case.

Optimal Insurance Without the Nonnegativity Constraint on Indemnities: Ambiguity and Belief Heterogeneity

In Arrow's (1971) classical problem of optimal insurance design, a linear deductible schedule is optimal for an Expected-Utility (EU) maximizing decision maker (DM), if the premium depends on the indemnity's actuarial value, if the DM and the insurer share the same probabilistic beliefs about the realizations of the random loss, and under the classical constraints that, in each state of the world, the indemnity is nonnegative and does not exceed the value of the loss. Raviv (1979) re-examined Arrow's problem and concluded that the presence of a deductible is due to both the nonnegativity constraint on the indemnity and the variability in the cost of insurance. In an effort to test this statement, Gollier (1987) relaxes the nonnegativity constraint and argues that the existence of a deductible is only due to the variability in the cost of insurance. In this paper, we test whether the intuition behind Gollier's result still holds under more general preferences for the DM and the insurer. We consider a setting of ambiguity (one-sided and then two-sided) and a setting of belief heterogeneity. We drop the nonnegativity constraint and assume no cost (or a fixed cost) to the insurer, and we derive closed-form analytical solutions to the problems that we formulate. In particular, we show that an optimal indemnity (resp., retention) no longer includes a deductible provision (resp., is not equal to the realized loss). Moreover, the optimal indemnity (resp., retention) can be negative (resp., higher than the realized loss) for small values of the loss, or in case of no loss. This is in line with the intuition behind Gollier's finding in the case of belief homogeneity and no ambiguity.

Optimal insurance under rank‐dependent utility and incentive compatibility

AbstractBernard, He, Yan, and Zhou (Mathematical Finance, 25(1), 154–186) studied an optimal insurance design problem where an individual's preference is of the rank‐dependent utility (RDU) type, and show that in general an optimal contract covers both large and small losses. However, their results suffer from the unrealistic assumption that the random loss has no atom, as well as a problem of moral hazard that provides incentives for the insured to falsely report the actual loss. This paper addresses these setbacks by removing the nonatomic assumption, and by exogenously imposing the “incentive compatibility” constraint that both indemnity function and insured's retention function are increasing with respect to the loss. We characterize the optimal solutions via calculus of variations, and then apply the result to obtain explicitly expressed contracts for problems with Yaari's dual criterion and general RDU. Finally, we use numerical examples to compare the results between ours and Bernard et al.

Optimal insurance design under background risk with dependence

In this paper, we revisit the problem of optimal insurance under a general criterion that preserves stop-loss order when the insured faces two mutually dependent risks: background risk and insurable risk. According to the local monotonicity of conditional survival function, we derive the optimal contract forms in different types of interval. Because the conditional survival function reflects the dependence between background risk and insurable risk, the dependence structure between the two risks plays a critical role in the insured’s optimal insurance design. Furthermore, we obtain the optimal insurance forms explicitly under some special dependence structures. It is shown that deductible insurance is optimal and the Mossin’s Theorem is still valid when background risk is stochastically increasing in insurable risk, which generalizes the corresponding results in Lu et al. (2012). Moreover, we show that an individual will purchase no insurance when the sum of the two risks is stochastically decreasing in insurable risk.

Optimal Insurance Design Under Narrow Framing

In this paper, we incorporate a psychological factor called narrow framing into standard vNM expected utility model to explain low insurance purchase rate observed in many insurance markets which are sometimes heavily subsidized by governments. In this psychological model, insurance purchase is not only viewed as a way of transferring wealth from the states of low loss to the states of high loss, but also as a pure gamble against insurance companies. When the received indemnity is lower than the paid insurance premium, the economic agent feels like losing the gamble and vice-versa. We study the insurance demand when coinsurance policy is offered and show that partial insurance is strictly preferred even in case of zero transaction cost. We also discuss the optimal insurance design under narrow framing. It turns out that a (non-linear) coinsurance contract is optimal for losses above a certain threshold. We further allow for possible loss aversion and derive the optimal insurance scheme under both narrow framing and loss aversion. As a result, a flat payment of insurance premium should be applied to some intermediate losses.

Optimal insurance design with a bonus

This paper investigates an insurance design problem, in which a bonus will be given to the insured if no claim has been made during the whole lifetime of the contract, for an expected utility insured. In this problem, the insured has to consider the so-called optimal action rather than the contracted compensation (or indemnity) due to the existence of the bonus. For any pre-agreed bonus, the optimal insurance contract is given explicitly and shown to be either the full coverage contract when the insured pays high enough premium, or a deductible one otherwise. The optimal contract and bonus are also derived explicitly if the insured is allowed to choose both of them. The contract turns out to be of either zero reward or zero deductible. In all cases, the optimal contracts are universal, that is, they do not depend on the specific form of the utility of the insured. A numerical example is also provided to illustrate the main theoretical results of the paper.

Optimal insurance design in the presence of exclusion clauses

The present work studies the design of an optimal insurance policy from the perspective of an insured, where the insurable loss is mutually exclusive from another loss that is denied in the insurance coverage. To reduce ex post moral hazard, we assume that both the insured and the insurer would pay more for a larger realization of the insurable loss. When the insurance premium principle preserves the convex order, we show that any admissible insurance contract is suboptimal to a two-layer insurance policy under the criterion of minimizing the insured’s total risk exposure quantified by value at risk, tail value at risk or an expectile. The form of optimal insurance can be further simplified to be one-layer by imposing an additional weak condition on the premium principle. Finally, we use Wang’s premium principle and the expected value premium principle to illustrate the applicability of our results, and find that optimal insurance solutions are affected not only by the size of the excluded loss but also by the risk measure chosen to quantify the insured’s risk exposure.

Optimal Insurance with Heterogeneous Beliefs and Disagreement about Zero-Probability Events

In problems of optimal insurance design, Arrow’s classical result on the optimality of the deductible indemnity schedule holds in a situation where the insurer is a risk-neutral Expected-Utility (EU) maximizer, the insured is a risk-averse EU-maximizer, and the two parties share the same probabilistic beliefs about the realizations of the underlying insurable loss. Recently, Ghossoub re-examined Arrow’s problem in a setting where the two parties have different subjective beliefs about the realizations of the insurable random loss, and he showed that if these beliefs satisfy a certain compatibility condition that is weaker than the Monotone Likelihood Ratio (MLR) condition, then optimal indemnity schedules exist and are nondecreasing in the loss. However, Ghossoub only gave a characterization of these optimal indemnity schedules in the special case of an MLR. In this paper, we consider the general case, allowing for disagreement about zero-probability events. We fully characterize the class of all optimal indemnity schedules that are nondecreasing in the loss, in terms of their distribution under the insured’s probability measure, and we obtain Arrow’s classical result, as well as one of the results of Ghossoub as corollaries. Finally, we formalize Marshall’s argument that, in a setting of belief heterogeneity, an optimal indemnity schedule may take “any”shape.