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.     

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     

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.     

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.     

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 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.     

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.     

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.

     

An Examination of Adverse Selection in the Public Provision of Insurance

Using a unique data set from Florida's residual property insurer, we test for adverse selection in the public provision of homeowners’ insurance in Florida. We find a significant relationship between the losses and deductible choices of insureds in Florida's residual homeowners’ insurance market. This relationship provides strong evidence of the existence of an adverse selection problem in Florida's residual property insurance market. While this relationship is important to Florida regulators (and taxpayers) specifically, a finding of an adverse selection problem in residual markets in general has implications more broadly for government providers of insurance as an adverse selection problem in these settings will impact the public policy debates and decisions involving these markets.     

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.     

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.     

不真正连带债务理论在保险活动中的应用——兼评“无责免赔”问题

保险活动中,保险人与其他责任人对同一损失同时负有补偿义务的情形时常发生,保险界对其如何处理至今尚未找到明确统一的、具有说服力的理论或者法律依据,以致于虽然《保险法》第60条对相关问题进行了规定,但保险实务中仍然出现了包括机动车保险中"无责免赔"这类被法院认定为无效条款的约定,交通事故人身损害赔偿纠纷中保险人也被不当地判决承担连带责任。将不真正连带债务理论应用于保险活动中,能够为保险竞合和包括保险人补偿义务在内的补偿义务竞合情形提供广为接受的处理方案,能够为保险条款和《保险法》的完善提供理论指导,能够为"无责免赔"争议和交通事故人身损害赔偿纠纷中保险人权利义务合理确定等现实问题的解决提供思路。     

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 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.     

Optimal excess-of-loss reinsurance contract with ambiguity aversion in the principal-agent model

ABSTRACT

We discuss an optimal excess-of-loss reinsurance contract in a continuous-time principal-agent framework where the surplus of the insurer (agent/he) is described by a classical Cramér-Lundberg (C-L) model. In addition to reinsurance, the insurer and the reinsurer (principal/she) are both allowed to invest their surpluses into a financial market containing one risk-free asset (e.g. a short-rate account) and one risky asset (e.g. a market index). In this paper, the insurer and the reinsurer are ambiguity averse and have specific modeling risk aversion preferences for the insurance claims (this relates to the jump term in the stochastic models) and the financial market's risk (this encompasses the models' diffusion term). The reinsurer designs a reinsurance contract that maximizes the exponential utility of her terminal wealth under a worst-case scenario which depends on the retention level of the insurer. By employing the dynamic programming approach, we derive the optimal robust reinsurance contract, and the value functions for the reinsurer and the insurer under this contract. In order to provide a more explicit reinsurance contract and to facilitate our quantitative analysis, we discuss the case when the claims follow an exponential distribution; it is then possible to show explicitly the impact of ambiguity aversion on the optimal reinsurance.     

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.     

On the Theory of Rational Insurance Purchasing: A Note

The authors consider the optimal amount of insurance purchased by an individual who behaves according to the Hurwicz criterion of choice under uncertainty. Their results are compared with earlier results obtained in alternative frameworks (expected utility maximization and Savage's regret criterion). It is shown that a positive amount deductible is often suboptimal.     

Creating Customer Value in Participating Life Insurance

The value of a life insurance contract may differ depending on whether it is looked at from the customer's point of view or that of the insurance company. We assume that the insurer is able to replicate the life insurance contract's cash flows via assets traded on the capital market and can hence apply risk‐neutral valuation techniques. The policyholder, on the other hand, will take risk preferences and diversification opportunities into account when placing a value on that same contract. Customer value is represented by policyholder willingness to pay and depends on the contract parameters, that is, the guaranteed interest rate and the annual and terminal surplus participation rate. The aim of this article is to analyze and compare these two perspectives. In particular, we identify contract parameter combinations that—while keeping the contract value fixed for the insurer—maximize customer value. In addition, we derive explicit expressions for a selection of specific cases. Our results suggest that a customer segmentation in this sense, that is, based on the different ways customers evaluate life insurance contracts and embedded investment guarantees while ensuring fair values, is worthwhile for insurance companies as doing so can result in substantial increases in policyholder willingness to pay.     

Pareto-Optimal Insurance Policies in the Models with a Premium Based on the Actuarial Value

This article analyzes the problem of designing Pareto‐optimal insurance policies when both the insurer and the insured are risk averse and the premium is calculated as a function of the actuarial value of the insurer's risk. Two models are considered: in the first, the set of admissible policies is constrained by a given size of the premium; in the second, the premium size is not constrained so that it varies with the actuarial value of a policy chosen by the agents. For both cases a characterization of the Pareto‐optimal policies is derived. The corresponding optimality equations for the Pareto‐optimal policies are obtained and compared with the results on the classical risk exchange model.