Optimal reinsurance policy: The adjustment coefficient and the expected utility criteria

This paper is concerned with the optimal form of reinsurance from the ceding company point of view, when the cedent seeks to maximize the adjustment coefficient of the retained risk. We deal with the problem by exploring the relationship between maximizing the adjustment coefficient and maximizing the expected utility of wealth for the exponential utility function, both with respect to the retained risk of the insurer.Assuming that the premium calculation principle is a convex functional and that some other quite general conditions are fulfilled, we prove the existence and uniqueness of solutions and provide a necessary optimal condition. These results are used to find the optimal reinsurance policy when the reinsurance premium calculation principle is the expected value principle or the reinsurance loading is an increasing function of the variance. In the expected value case the optimal form of reinsurance is a stop-loss contract. In the other cases, it is described by a nonlinear function.     

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 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.     

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 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.     

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．     

基于再保险和投资的随机微分博弈

本文研究了具有再保险和投资的随机微分博弈.应用线性-二次控制的理论,在指数效用和幂效用下,求得了最优再保险策略、最优投资策略、最优市场策略和值函数的显示解,推广了文[8]的结果.通过本文的研究,当市场出现最坏的情况时,可以指导保险公司选择恰当的再保险和投资策略使自身所获得的财富最大化.     

Optimal proportional reinsurance and investment with transaction costs, I: Maximizing the terminal wealth

We consider a problem of optimal reinsurance and investment with multiple risky assets for an insurance company whose surplus is governed by a linear diffusion. The insurance company’s risk can be reduced through reinsurance, while in addition the company invests its surplus in a financial market with one risk-free asset and n risky assets. In this paper, we consider the transaction costs when investing in the risky assets. Also, we use Conditional Value-at-Risk (CVaR) to control the whole risk. We consider the optimization problem of maximizing the expected exponential utility of terminal wealth and solve it by using the corresponding Hamilton-Jacobi-Bellman (HJB) equation. Explicit expression for the optimal value function and the corresponding optimal strategies are obtained.     

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.     

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.     

Optimal quota-share and stop-loss reinsurance from the perspectives of insurer and reinsurer

Reinsurance plays a vital role in the insurance activities. The insurer and the reinsurer, which have conflicting interests, compose the two parties of a reinsurance contract. In this paper, we extend the results achieved by Tan et al. (N Am Actuar J 13(4):459–482, 2009) to the case in which the perspectives of both the insurer and the reinsurer are considered. We study the optimal quota-share and stop-loss reinsurance models by minimizing the convex combination of the VaR risk measures of the insurer’s cost and the reinsurer’s cost. Furthermore, as many as 16 reinsurance premium principles are investigated. The results show that optimal quota-share and stop-loss reinsurance may or may not exist depending on the chosen principles. Moreover, we establish the sufficient and necessary conditions for the existence of the nontrivial optimal reinsurance.     

Optimal proportional reinsurance with common shock dependence

In this paper, we consider the optimal proportional reinsurance strategy in a risk model with multiple dependent classes of insurance business, which extends the work of Liang and Yuen (2014) to the case with the reinsurance premium calculated under the expected value principle and to the model with two or more classes of dependent risks. Under the criterion of maximizing the expected exponential utility, closed-form expressions for the optimal strategies and value function are derived not only for the compound Poisson risk model but also for the diffusion approximation risk model. In particular, we find that the optimal reinsurance strategies under the expected value premium principle are very different from those under the variance premium principle in the diffusion risk model. The former depends not only on the safety loading, time and interest rate, but also on the claim size distributions and the counting processes, while the latter depends only on the safety loading, time and interest rate. Finally, numerical examples are presented to show the impact of model parameters on the optimal strategies.     

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 investment and life insurance strategies under minimum and maximum constraints

We derive optimal strategies for an individual life insurance policyholder who can control the asset allocation as well as the sum insured (the amount to be paid out upon death) throughout the policy term. We first consider the problem in a pure form without constraints (except nonnegativity on the sum insured) and then in a more general form with minimum and/or maximum constraints on the sum insured. In both cases we also provide the optimal life insurance strategies in the case where risky-asset investments are not allowed (or not taken into consideration), as in basic life insurance mathematics. The optimal constrained strategies are somewhat more complex than the unconstrained ones, but the latter can serve to ease the understanding and implementation of the former.     

Stability of solutions in optimal reinsurance problem

A discrete-time insurance model with stop-loss reinsurance is considered. One-period insurance claims form a sequence of independent identically distributed nonnegative random variables with finite mean. The insurer maintains the company surplus above a chosen level a by capital injections. We study the stability of optimal capital injections to the variability of claims distribution. The term “optimal” means the minimal amount of injections that can be found from the corresponding Bellman equation.     

On the optimal design of insurance contracts with guarantees

The paper analyzes insurance contracts where the benefits of the insured depend on the performance of an investment strategy and which guarantee a certain interest rate on the contributions made by the insured. The insured has to decide simultaneously on the investment strategy and the guarantee scheme. For a CRRA insured and in a BS economy, the optimal combination is given by a constant mix strategy and the contribution guarantee scheme. In case the insured has a subsistence level, the CPPI strategy turns out to be optimal for arbitrary schemes. We illustrate our results by numerical examples and analyze the utility losses of a CRRA insured due to the use of a suboptimal combination of investment strategy and guarantee scheme.     

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.     

基于个体公平原则的寿险产品定价模型

在无套利框架的基础上,讨论基于个体公平原则下的寿险产品定价问题,即运用倒向随机微分方程理论,将投保人和保险人置于同一系统中进行考虑:首先,根据双方的随机投资决策目标分别建立无套利寿险定价模型和动态资产份额定价模型,得出两个特殊线性倒向随机微分方程的显式解;然后,建立基于个体公平原则的寿险定价模型,从投保人和保险人双方的角度对寿险产品进行公平定价,得出了从供需双方考虑的投资回报定价公式;最后,利用所建立的模型进行案例分析,计算出基于个体公平原则的保费及保险公司的投资策略.该寿险产品定价模型不仅考虑了保险人的意愿,还同时考虑了投保人的实际情况,因此,按此定价理念开发出的保险产品,不仅可以提高产品研发的成功率,而且使得研发出的新产品更能在竞争激烈的保险市场中站稳脚步.     

马尔科夫调节风险模型下的最优投资策略:最大化终端效用

本文用跳-扩散模型模拟保险公司的盈余过程,并允许该盈余在由1个无风险资产和N个风险资产组成的金融市场上进行投资.盈余过程和资产价格过程模型中的参数皆受到一个可观察的有限状态连续马尔科夫过程的影响.为了最大化终端效用,我们寻找最优的投资策略,借助HJB方程等工具问题得到解决.当公司的效用函数为指数型时,我们给出了最优投资策略与其对应的值函数的显示表达式,以及相关的经济解释.Browne (1995)和Yang和Zhang (2005)的一些结论得到推广.