Behavioral optimal insurance

The present work studies the optimal insurance policy offered by an insurer adopting a proportional premium principle to an insured whose decision-making behavior is modeled by Kahneman and Tversky’s Cumulative Prospect Theory with convex probability distortions. We show that, under a fixed premium rate, the optimal insurance policy is a generalized insurance layer (that is, either an insurance layer or a stop–loss insurance). This optimal insurance decision problem is resolved by first converting it into three different sub-problems similar to those in Jin and Zhou (2008); however, as we now demand a more regular optimal solution, a completely different approach has been developed to tackle them. When the premium is regarded as a decision variable and there is no risk loading, the optimal indemnity schedule in this form has no deductibles but a cap; further results also suggests that the deductible amount will be reduced if the risk loading is decreased. As a whole, our paper provides a theoretical explanation for the popularity of limited coverage insurance policies in the market as observed by many socio-economists, which serves as a mathematical bridge between behavioral finance and actuarial science.     

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.     

A methodology for integrated critical spare parts and insurance management

Critical spare‐parts stock optimization has become a relevant topic for academy and industry. In most articles, the problem has been stated as a trade‐off between economic risks of shortages and financial costs. Risk optimization in this context has been mainly studied from a logistics point of view. The most common decision variables have been stock levels, stock location, and reorder points. In this context, buying insurance to cover shortage cost can be a complementary (or exclusive) measure for risk mitigation. Insurance optimization traditionally has been studied from a microeconomic and financial perspective. The main decision variable has been the indemnity function, and occasionally, the insurance premium. Its use in the context of physical asset management has not been observed to the best of our knowledge. This creates an opportunity to link inventory optimization techniques with insurance optimization for shortage losses. In this work, we present a novel approach to jointly manage the shortage risk of a critical non‐repairable component in a unique critical system. We develop an original model to integrate critical spare‐parts stock optimization with insurance optimization techniques. The result is a decision model to select the optimal stock and insurance policy that maximizes the decision maker's expected utility. This allows for a business‐centered integrated perspective in critical parts decisions. We present a case study representative of the mining industry, illustrating the complementary nature of selecting optimal stock levels and contracting an optimal insurance. Our results show that contracting an insurance can lead to policies preferred by a risk‐averse decision maker. The case study shows that this may even occur lowering stock levels and increasing profits. Copyright © 2015 John Wiley & Sons, Ltd.     

风险资产的最优保险

本文采用期望方差方法，引入无风险投资；建立多元风险模型，从投保人角度讨论了最优保险决策，分析了投资风险，无风险投资收益和保费政策等因素对最优决策的影响，为现代企业采取综合措施降低风险提供了理论依据．     

系统风险不同预期下的存款保险费率测算

将影响银行资产价值的风险因素分解为系统风险因素和银行特定风险因素,进而在系统风险因素点估计和区间估计的不同预期下测算银行存款保险费率水平,得到的费率能够反映银行资产风险随经济形势波动的变化情况。通过模拟测算了我国16家上市银行2008~2016年间特定经济形势情境下的存款保险费率水平,并在极端压力下与传统Merton费率进行了比较。得到的基本结论包括:不同年度不同银行费率对系统风险因素的敏感程度不同;经济形势尾部极端分布对费率的影响具有非对称性特点,风险极高区间对费率的贡献远大于风险极低区间;与传统的Merton费率相比,系统风险特定预期下测算的费率更契合经济形势的变化,这在存款保险制度运行初期,有利于增强基金的抗压能力。     

Deductibles in health insurance

This study is an extension to a simulation study that has been developed to determine ruin probabilities in health insurance. The study concentrates on inpatient and outpatient benefits for customers of varying age bands. Loss distributions are modelled through the Allianz tool pack for different classes of insureds. Premiums at different levels of deductibles are derived in the simulation and ruin probabilities are computed assuming a linear loading on the premium. The increase in the probability of ruin at high levels of the deductible clearly shows the insufficiency of proportional loading in deductible premiums. The PH-transform pricing rule developed by Wang is analyzed as an alternative pricing rule. A simple case, where an insured is assumed to be an exponential utility decision maker while the insurer’s pricing rule is a PH-transform is also treated.     

Optimal insurance with belief heterogeneity and incentive compatibility

People may evaluate risk differently in the insurance market. Motivated by this, we examine an optimal insurance problem allowing the insured and the insurer to have heterogeneous beliefs about loss distribution. To reduce ex post moral hazard, we follow Huberman et al. (1983) to assume that alternative insurance contracts satisfy the principle of indemnity and the incentive-compatible constraint. Under the assumption that the insurance premium is calculated by the expected value principle, we establish a necessary and sufficient condition for an optimal insurance solution and provide a practical scheme to improve any suboptimal insurance strategy under an arbitrary form of belief heterogeneity. By virtue of this condition, we explore qualitative properties of optimal solutions, and derive optimal insurance contracts explicitly for some interesting forms of belief heterogeneity. As a byproduct of this investigation, we find that Theorem 3.6 of Young (1999) is not completely true.     

An optimal insurance strategy for an individual under an intertemporal equilibrium

In this paper, we discuss how a risk-averse individual under an intertemporal equilibrium chooses his/her optimal insurance strategy to maximize his/her expected utility of terminal wealth. It is shown that the individual’s optimal insurance strategy actually is equivalent to buying a put option, which is written on his/her holding asset with a proper strike price. Since the cost of avoiding risk can be seen as a risk measure, the put option premium can be considered as a reasonable risk measure. Jarrow [Jarrow, R., 2002. Put option premiums and coherent risk measures. Math. Finance 12, 135-142] drew this conclusion with an axiomatic approach, and we verify it by solving the individual’s optimal insurance problem.     

Threshold control of mutual insurance with limited commitment

We study the optimal premium policy of mutual insurance when the charged premium cannot be higher than a preset rate. We provide a complete solution to the problem and use numerical simulations to illustrate how the optimal premium policy responds to changes of outside factors. The results are useful for mutual insurance firms to design premium policies and can be used to test the behavior of these firms in empirical studies.     

A participatory multi-criteria approach for power generation and transmission planning

It is well-known that reinsurance can be an effective risk management solution for financial institutions such as the insurance companies. The optimal reinsurance solution depends on a number of factors including the criterion of optimization and the premium principle adopted by the reinsurer. In this paper, we analyze the Value-at-Risk based optimal risk management solution using reinsurance under a class of premium principles that is monotonic and piecewise. The monotonic piecewise premium principles include not only those which preserve stop-loss ordering, but also the piecewise premium principles which are monotonic and constructed by concatenating a series of premium principles. By adopting the monotonic piecewise premium principle, our proposed optimal reinsurance model has a number of advantages. In particular, our model has the flexibility of allowing the reinsurer to use different risk loading factors for a given premium principle or use entirely different premium principles depending on the layers of risk. Our proposed model can also analyze the optimal reinsurance strategy in the context of multiple reinsurers that may use different premium principles (as attributed to the difference in risk attitude and/or imperfect information). Furthermore, by artfully imposing certain constraints on the ceded loss functions, the resulting model can be used to capture the reinsurer’s willingness and/or capacity to accept risk or to control counterparty risk from the perspective of the insurer. Under some technical assumptions, we derive explicitly the optimal form of the reinsurance strategies in all the above cases. In particular, we show that a truncated stop-loss reinsurance treaty or a limited stop-loss reinsurance treaty can be optimal depending on the constraint imposed on the retained and/or ceded loss functions. Some numerical examples are provided to further compare and contrast our proposed models to the existing models.     

Optimal insurance in the presence of insurer’s loss limit

In this paper we consider the optimal insurance problem when the insurer has a loss limit constraint. Under the assumptions that the insurance price depends only on the policy’s actuarial value, and the insured seeks to maximize the expected utility of his terminal wealth, we show that coverage above a deductible up to a cap is the optimal contract, and the relaxation of insurer’s loss limit will increase the insured’s expected utility.When the insurance price is given by the expected value principle, we show that a positive loading factor is a sufficient and necessary condition for the deductible to be positive. Moreover, with the expected value principle, we show that the optimal deductible derived in our model is not greater (lower) than that derived in Arrow’s model if the insured’s preference displays increasing (decreasing) absolute risk aversion. Therefore, when the insured has an IARA (DARA) utility function, compared to Arrow model, the insurance policy derived in our model provides more (less) coverage for small losses, and less coverage for large losses.Furthermore, we prove that the optimal insurance derived in our model is an inferior (normal) good for the insured with a DARA (IARA) utility function, consistent with the finding in the previous literature. Being inferior, the insurance can also be a Giffen good. Under the assumption that the insured’s initial wealth is greater than a certain level, we show that the insurance is not a Giffen good if the coefficient of the insured’s relative risk aversion is lower than 1.     

Vigilant measures of risk and the demand for contingent claims

We examine a class of utility maximization problems with a non-necessarily law-invariant utility, and with a non-necessarily law-invariant risk measure constraint. Under a consistency requirement on the risk measure that we call Vigilance, we show the existence of optimal contingent claims, and we show that such optimal contingent claims exhibit a desired monotonicity property. Vigilance is satisfied by a large class of risk measures, including all distortion risk measures and some classes of robust risk measures. As an illustration, we consider a problem of optimal insurance design where the premium principle satisfies the vigilance property, hence covering a large collection of commonly used premium principles, including premium principles that are not law-invariant. We show the existence of optimal indemnity schedules, and we show that optimal indemnity schedules are nondecreasing functions of the insurable loss.     

Competitive insurance pricing with complete information,loss-averse utility and finitely many policies

In a recent paper, Ramsay and Oguledo (2012) show that in a competitive insurance market with complete information about individuals’ accident probabilities and production costs, which are proportional to the amount of insurance purchased and to the premium charged, only individuals whose accident probability is in a medium range are insurable and desire insurance. The purpose of this paper is to complement the analysis of Ramsay and Oguledo by considering production costs which are proportional to the number of policies offered by an insurer. In addition to the result of Ramsay and Oguledo we show that the group of individuals who obtain insurance is partitioned into several subgroups, where each subgroup is offered the same insurance policy. To derive this result we introduce the concept of incentive compatibility which ensures that an individual has no incentive to buy another policy. Assuming that individuals have loss-averse utility, we fully characterize the boundaries of these subgroups as the result of an undercutting process in premiums between the insurers.     

Pricing general insurance in a reactive and competitive market

A simple parameterisation is introduced which represents the insurance market’s response to an insurer adopting a pricing strategy determined via optimal control theory. Claims are modelled using a lognormally distributed mean claim size rate, and the market average premium is determined via the expected value principle. If the insurer maximises its expected wealth then the resulting Bellman equation has a moving boundary in state space that determines when it is optimal to stop selling insurance. This stochastic optimisation problem is simplified by the introduction of a stopping time that prevents an insurer leaving and then re-entering the insurance market. Three finite difference schemes are used to verify the existence of a solution to the resulting Bellman equation when there is market reaction. All of the schemes use a front-fixing transformation. If the market reacts, then it is found that the optimal strategy is altered, in that premiums are raised if the strategy is of loss-leading type and lowered if it is optimal for the insurer to set a relatively high premium and sell little insurance.     

财政补贴政策与长期护理保险市场演化

财政补贴政策是政府矫正市场失灵,驱动长期护理保险市场高质量演化发展的重要手段。本文基于多主体仿真模型,考察不同补贴政策驱动长期护理保险市场演化的过程。研究发现:实行产品创新补贴政策时,可以形成健康有序的长期护理保险市场竞争格局;实行经营性补贴政策时,长期护理保险市场容易演化成垄断市场。保费补贴政策能够扩大需求规模,且差异化的保费补贴政策能够更加有效地激励需求空间的大幅增长。     

Purchasing life insurance to reach a bequest goal

We determine how an individual can use life insurance to meet a bequest goal. We assume that the individual’s consumption is met by an income from a job, pension, life annuity, or Social Security. Then, we consider the wealth that the individual wants to devote towards heirs (separate from any wealth related to the afore-mentioned income) and find the optimal strategy for buying life insurance to maximize the probability of reaching a given bequest goal. We consider life insurance purchased by a single premium, with and without cash value available. We also consider irreversible and reversible life insurance purchased by a continuously paid premium; one can view the latter as (instantaneous) term life insurance.     

Dynamic hybrid products in life insurance: Assessing the policyholders’ viewpoint

Dynamic hybrid life insurance products are intended to meet new consumer needs regarding stability in terms of guarantees as well as sufficient upside potential. In contrast to traditional participating or classical unit-linked life insurance products, the guarantee offered to the policyholders is achieved by a periodical rebalancing process between three funds: the policy reserves (i.e. the premium reserve stock, thus causing interaction effects with traditional participating life insurance contracts), a guarantee fund, and an equity fund. In this paper, we consider an insurer offering both, dynamic hybrid and traditional participating life insurance contracts and focus on the policyholders’ perspective. The results show that higher guarantees do not necessarily imply a higher willingness-to-pay, but that in case of dynamic hybrid contracts, a minimum guarantee level should be offered in order to ensure that the willingness-to-pay exceeds the minimum premium the insurer has to charge when selling the contract. In addition, strong interaction effects can be found between the two products, which particularly impact the willingness-to-pay of the dynamic hybrids.     

Pareto-optimal insurance contracts with premium budget and minimum charge constraints

In view of the fact that minimum charge and premium budget constraints are natural economic considerations in any risk-transfer between the insurance buyer and seller, this paper revisits the optimal insurance contract design problem in terms of Pareto optimality with imposing these practical constraints. Pareto optimal insurance contracts, with indemnity schedule and premium payment, are solved in the cases when the risk preferences of the buyer and seller are given by Value-at-Risk or Tail Value-at-Risk. The effect of our constraints and the relative bargaining powers of the buyer and seller on the Pareto optimal insurance contracts are highlighted. Numerical experiments are employed to further examine these effects for some given risk preferences.     

Multivariate reinsurance designs for minimizing an insurer’s capital requirement

This paper investigates optimal reinsurance strategies for an insurer with multiple lines of business under the criterion of minimizing its total capital requirement calculated based on the multivariate lower-orthant Value-at-Risk. The reinsurance is purchased by the insurer for each line of business separately. The premium principles used to compute the reinsurance premiums are allowed to differ from one line of business to another, but they all satisfy three mild conditions: distribution invariance, risk loading and preserving the convex order, which are satisfied by many popular premium principles. Our results show that an optimal strategy for the insurer is to buy a two-layer reinsurance policy for each line of business, and it reduces to be a one-layer reinsurance contract for premium principles satisfying some additional mild conditions, which are met by the expected value principle, standard deviation principle and Wang’s principle among many others. In the end of this paper, some numerical examples are presented to illustrate the effects of marginal distributions, risk dependence structure and reinsurance premium principles on the optimal layer reinsurance.     

Optimal inventory and insurance decisions for a supply chain financing system with downside risk control

This research considers a supply chain financing system consisting of a capital‐constrained retailer, a supplier and a risk‐averse bank. The retailer may be subject to credit limit because of the bank's downside risk control, and hence, credit insurance should be needed to enhance his financing ability. This paper develops a mathematical optimization model by incorporating insurance policy into the well‐known newsvendor financing model. The optimal inventory and insurance decisions under different scenarios, that is, no insurance, insurance with symmetric information and insurance with asymmetric information, are derived. This work also discusses how the retailer's capital level, the bank's risk aversion, and the insurer's loading factor affect the optimal inventory and insurance decisions. The results show that the retailer will use credit insurance if he is sufficiently capital‐constrained or the insurer's risk loading factor is low enough. Moreover, credit insurance can bring Pareto improvement to the supply chain financing system, which verifies the prevalence of credit insurance in practice. Several numerical experiments are presented to examine the sensitivities of key parameters. Copyright © 2016 John Wiley & Sons, Ltd.