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.     

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 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.     

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.     

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 Impact of Regret on the Demand for Insurance

We examine optimal insurance purchase decisions of individuals that exhibit behavior consistent with Regret Theory. Our model incorporates a utility function that assigns a disutility to outcomes that are ex post suboptimal, and predicts that individuals with regret‐theoretical preferences adjust away from the extremes of full insurance and no insurance coverage. This prediction holds for both coinsurance and deductible contracts, and can explain the frequently observed preferences for low deductibles in markets for personal insurance.     

Optimal asset allocation for participating contracts under the VaR and PI constraint

ABSTRACT

Participating contracts provide a maturity guarantee for the policyholder. However, the terminal payoff to the policyholder should be related to financial risks of participating insurance contracts. We investigate an optimal investment problem under a joint value-at-risk and portfolio insurance constraint faced by the insurer who offers participating contracts. The insurer aims to maximize the expected utility of the terminal payoff to the insurer. We adopt a concavification technique and a Lagrange dual method to solve the problem and derive the representations of the optimal wealth process and trading strategies. We also carry out some numerical analysis to show how the joint value-at-risk and the portfolio insurance constraint impacts the optimal terminal wealth.     

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.     

Time-consistent reinsurance and investment strategies for an AAI under smooth ambiguity utility

This paper investigates time-consistent reinsurance(excess-of-loss, proportional) and investment strategies for an ambiguity averse insurer(abbr. AAI). The AAI is ambiguous towards the insurance and financial markets. In the AAI's attitude, the intensity of the insurance claims' number and the market price of risk of a stock can not be estimated accurately. This formulation of ambiguity is similar to the uncertainty of different equivalent probability measures. The AAI can purchase excess-of-loss or proportional reinsurance to hedge the insurance risk and invest in a financial market with cash and an ambiguous stock. We investigate the optimization goal under smooth ambiguity given in Klibanoff, P., Marinacci, M., & Mukerji, S. [(2005). A smooth model of decision making under ambiguity. Econometrica 73, 1849–1892], which aims to search the optimal strategies under average case. The utility function does not satisfy the Bellman's principle and we employ the extended HJB equation proposed in Björk, T. & Murgoci, A. [(2014). A theory of Markovian time-inconsistent stochastic control in discrete time. Finance and Stochastics 18(3), 545–592] to solve this problem. In the end of this paper, we derive the equilibrium reinsurance and investment strategies under smooth ambiguity and present the sensitivity analysis to show the AAI's economic behaviors.     

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.     

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 in the presence of reinsurance

This paper studies an optimal insurance and reinsurance design problem among three agents: policyholder, insurer, and reinsurer. We assume that the preferences of the parties are given by distortion risk measures, which are equivalent to dual utilities. By maximizing the dual utility of the insurer and jointly solving the optimal insurance and reinsurance contracts, it is found that a layering insurance is optimal, with every layer being borne by one of the three agents. We also show that reinsurance encourages more insurance, and is welfare improving for the economy. Furthermore, it is optimal for the insurer to charge the maximum acceptable insurance premium to the policyholder. This paper also considers three other variants of the optimal insurance/reinsurance models. The first two variants impose a limit on the reinsurance premium so as to prevent insurer to reinsure all its risk. An optimal solution is still layering insurance, though the insurer will have to retain higher risk. Finally, we study the effect of competition by permitting the policyholder to insure its risk with an insurer, a reinsurer, or both. The competition from the reinsurer dampens the price at which an insurer could charge to the policyholder, although the optimal indemnities remain the same as the baseline model. The reinsurer will however not trade with the policyholder in this optimal solution.     

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.     

Risk-adjusted credibility premiums using distorted probabilities

Abstract

Denneberg (1990) and Wang (1996a) propose that one calculate risk-adjusted insurance premiums as the expectation with respect to a distorted probability measure, a non-additive set function. This premium principle is supported by the theories of decision making of Yaari (1987) and of Schmeidler (1989). Denneberg (1994a) presents three conditioning rules for updating non-additive set functions in light of available information. In this work, we show how to apply these three update rules to calculate a risk-adjusted credibility premium and, thereby, combine credibility theory with this relatively new premium principle. Our main result is that, for some pairs of distortion function and update rule, one gets the same risk-adjusted credibility premium by distorting the predictive probability distribution, as required by the theory of Yaari, or by updating the distorted probability, as required by the theory of Schmeidler.     

Optimal Reinsurance and Investment for a Jump Diffusion Risk Process under the CEV Model

Abstract

We consider an optimal reinsurance-investment problem of an insurer whose surplus process follows a jump-diffusion model. In our model the insurer transfers part of the risk due to insurance claims via a proportional reinsurance and invests the surplus in a “simplified” financial market consisting of a risk-free asset and a risky asset. The dynamics of the risky asset are governed by a constant elasticity of variance model to incorporate conditional heteroscedasticity. The objective of the insurer is to choose an optimal reinsurance-investment strategy so as to maximize the expected exponential utility of terminal wealth. We investigate the problem using the Hamilton-Jacobi-Bellman dynamic programming approach. Explicit forms for the optimal reinsuranceinvestment strategy and the corresponding value function are obtained. Numerical examples are provided to illustrate how the optimal investment-reinsurance policy changes when the model parameters vary.     

Optimal investment and risk control for an insurer with partial information in an anticipating environment

Abstract

This paper considers an optimal investment and risk control problem under the criterion of logarithm utility maximization. The risky asset process and the insurance risk process are described by stochastic differential equations with jumps and anticipating coefficients. The insurer invests in the financial assets and controls the number of policies based on some partial information about the financial market and the insurance claims. The forward integral and Malliavin calculus for Lévy processes are used to obtain a characterization of the optimal strategy. Some special cases are discussed and the closed-form expressions for the optimal strategies are derived.     

Insurer's insolvency risk and tax deductions for the individual's net losses

Using the representative agent approach as in Kaplow (Am Econ Rev 82:1013–1017, 1992b), this paper shows that providing tax deductions for the individual’s net losses is socially optimal when the insurer faces the risk of insolvency. We further show that the government should adopt a higher tax deduction rate for net losses when the insurer is insolvent than when the insurer is solvent. Thus, tax deductions for net losses could be used to provide an insurance for individuals against the insurer’s risk of insolvency. These findings could also be used to explain why a government provides supplementary public insurance or government relief. Finally, we discuss that, if the individuals are heterogeneous in terms of loss severity, loss probability, or income level, providing a tax deduction for the individual’s net losses may not always achieve a Pareto improvement, and cross subsidization should be taken into consideration.

Duality theory for robust utility maximisation

In this paper, we present a duality theory for the robust utility maximisation problem in continuous time for utility functions defined on the positive real line. Our results are inspired by – and can be seen as the robust analogues of – the seminal work of Kramkov and Schachermayer (Ann. Appl. Probab. 9:904–950, 1999). Namely, we show that if the set of attainable trading outcomes and the set of pricing measures satisfy a bipolar relation, then the utility maximisation problem is in duality with a conjugate problem. We further discuss the existence of optimal trading strategies. In particular, our general results include the case of logarithmic and power utility, and they apply to drift and volatility uncertainty.

     

Robust optimal excess-of-loss reinsurance and investment strategy for an insurer in a model with jumps

This paper considers a robust optimal excess-of-loss reinsurance-investment problem in a model with jumps for an ambiguity-averse insurer (AAI), who worries about ambiguity and aims to develop a robust optimal reinsurance-investment strategy. The AAI’s surplus process is assumed to follow a diffusion model, which is an approximation of the classical risk model. The AAI is allowed to purchase excess-of-loss reinsurance and invest her surplus in a risk-free asset and a risky asset whose price is described by a jump-diffusion model. Under the criterion for maximizing the expected exponential utility of terminal wealth, optimal strategy and optimal value function are derived by applying the stochastic dynamic programming approach. Our model and results extend some of the existing results in the literature, and the economic implications of our findings are illustrated. Numerical examples show that considering ambiguity and reinsurance brings utility enhancements.