Marginal Indemnification Function formulation for optimal reinsurance

In this paper, we propose to combine the Marginal Indemnification Function (MIF) formulation and the Lagrangian dual method to solve optimal reinsurance model with distortion risk measure and distortion reinsurance premium principle. The MIF method exploits the absolute continuity of admissible indemnification functions and formulates optimal reinsurance model into a functional linear programming of determining an optimal measurable function valued over a bounded interval. The MIF method was recently introduced to analyze the reinsurance model but without premium budget constraint. In this paper, a Lagrangian dual method is applied to combine with MIF to solve for optimal reinsurance solutions under premium budget constraint. Compared with the existing literature, the proposed integrated MIF-based Lagrangian dual method provides a more technically convenient and transparent solution to the optimal reinsurance design. To demonstrate the practicality of the proposed method, analytical solution is derived on a particular reinsurance model that involves minimizing Conditional Value at Risk (a special case of distortion function) and with the reinsurance premium being determined by the inverse-S shaped distortion principle.     

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??Motivated by[1] and [2], we study in this paper the optimal (from the insurer's point of view) reinsurance problem when risk is measured by a general risk measure, namely the GlueVaR distortion risk measures which is firstly proposed by [3].Suppose an insurer is exposed to the risk and decides to buy a reinsurance contract written on the total claim amounts basis, i.e. the reinsurer covers and the cedent covers . In addition, the insurer is obligated to compensate the reinsurer for undertaking the risk by paying the reinsurance premium, ( is the safety loading), under the expectation premium principle. Based on a technique used in [2], this paper derives the optimal ceded loss functions in a class of increasing convex ceded loss functions. It turns out that the optimal ceded loss function is of stop-loss type.     

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Based on the default risk effect of reinsurance company for reinsurer, this paper studies the optimal reinsurance strategy by VaR optimality criterion. In a reinsurance contract, reinsurance company will charge the number of premium to undertake part of the insurer's loss. However, if the reinsurance company's commitment exceeds its solvency, the default risk will occur. In order to avoid the default risk and minimize the total risk of the insurance company, the paper introduces Wang's premium principle to obtain the optimal reinsurance policy under VaR risk measure. Some numerical examples are given to illustrate these results.     

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.     

Optimality of general reinsurance contracts under CTE risk measure

By formulating a constrained optimization model, we address the problem of optimal reinsurance design using the criterion of minimizing the conditional tail expectation (CTE) risk measure of the insurer’s total risk. For completeness, we analyze the optimal reinsurance model under both binding and unbinding reinsurance premium constraints. By resorting to the Lagrangian approach based on the concept of directional derivative, explicit and analytical optimal solutions are obtained in each case under some mild conditions. We show that pure stop-loss ceded loss function is always optimal. More interestingly, we demonstrate that ceded loss functions, that are not always non-decreasing, could be optimal. We also show that, in some cases, it is optimal to exhaust the entire reinsurance premium budget to determine the optimal reinsurance, while in other cases, it is rational to spend less than the prescribed reinsurance premium budget.     

Risk-adjusted Bowley reinsurance under distorted probabilities

In the seminal work of Chan and Gerber (1985), one of the earliest game theoretical approaches was proposed to model the interaction between the reinsurer and insurer; in particular, the optimal pricing density for the reinsurer and optimal ceded loss for the insurer were determined so that their corresponding expected utilities could be maximized. Over decades, their advocated Bowley solution (could be understood as Stackelberg equilibria) concept of equilibrium reinsurance strategy has not been revisited in the modern risk management framework. In this article, we attempt to fill this gap by extending their work to the setting of general premium principle for the reinsurer and distortion risk measure for the insurer.     

Expected Shortfall风险度量下的最优再保险（英文）

为了避免由高理赔额造成的违约,保险公司通常通过签订再保合约将一部分风险转移给再保险公司.近年来对最优再保策略的研究着眼于最小化自留损失的方差,保险公司总风险的value-at-risk或conditional tail expectation.本文研究了在expected shortfall准则下的再保策略.我们给出了最优的增凸转移损失函数,并分别讨论了有无保费限制的情形.     

带干扰的相依双险种最优再保险

研究一类带干扰的理赔相依的双险种风险模型,其中两险种分别采取成数再保险和超额损失再保险.在期望保费计算原理下,利用调节系数最大化得到成数再保险及超额损失再保险的最优自留水平.     

Risk Measure and Premium Distribution on Catastrophe Reinsurance

In this paper, we propose a new risk measure which is based on the Orlicz premium principle to characterize catastrophe risk premium. The intention is to develop a formulation strategy for Catastrophe Fund. The logarithm equivalent form of reinsurance premium is regarded as the retention of reinsurer, and the differential earnings between the reinsurance premium and the reinsurer's retention is accumulated as a part of Catastrophe Fund. We demonstrate that the aforementioned risk measure has some good properties, which are further confirmed by numerical simulations in R environment.     

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.     

保险公司和再保险公司之间的停止损失再保险策略选择博弈

本文在扩散逼近风险模型下考虑保险公司和再保险公司之间的停止损失再保险策略选择博弈问题.假设保险公司和再保险公司都以期望终端盈余效用增加作为购买停止损失再保险和接受承保的条件.在保险公司和再保险公司都具有指数效用函数条件下,运用动态规划原理,通过求解其对应的Hamilton-Jacobi-Bellman方程,得到了三种博弈情形下保险公司和再保险公司之间的停止损失再保险策略和值函数的显示解,以及再保险合约能够成交时再保费满足的条件.结果显示,在适当的条件下,保险公司和再保险公司之间的停止再保险合约是可以成交的.最后,通过灵敏性分析给出了最优停止损失再保险策略和再保费,以及效用损益与模型主要参数之间的关系,并给出相应的经济分析.     

最优成数再保险决策模型研究

根据期望值计算原理下,给出了由n种索赔次数相关的相关险种构成的总体最优成数再保险函数.     

基于风险调整资本收益率下的最优再保险策略

风险调整资本收益率是一个用来描述赚取收益所承担风险的重要指标,是衡量风险调整后的财务绩效的一个有效工具,在银行业中得到广泛采用。近年来,在保险公司的再保险业务中,也越来越多地采用风险调整资本收益率这一指标来衡量收益和风险。本文重新考虑了有再保险控制下的风险调整资本收益率,并得到:在一般再保险自留函数下,我们证明了分层再保险策略是最优再保险形式;进一步,我们获得了最优再保险分层水平及最优风险调整资本收益率;最后,我们也给出了算例及数值分析。     

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.     

Stochastic Stackelberg differential reinsurance games under time-inconsistent mean–variance framework

We study optimal reinsurance in the framework of stochastic Stackelberg differential game, in which an insurer and a reinsurer are the two players, and more specifically are considered as the follower and the leader of the Stackelberg game, respectively. An optimal reinsurance policy is determined by the Stackelberg equilibrium of the game, consisting of an optimal reinsurance strategy chosen by the insurer and an optimal reinsurance premium strategy by the reinsurer. Both the insurer and the reinsurer aim to maximize their respective mean–variance cost functionals. To overcome the time-inconsistency issue in the game, we formulate the optimization problem of each player as an embedded game and solve it via a corresponding extended Hamilton–Jacobi–Bellman equation. It is found that the Stackelberg equilibrium can be achieved by the pair of a variance reinsurance premium principle and a proportional reinsurance treaty, or that of an expected value reinsurance premium principle and an excess-of-loss reinsurance treaty. Moreover, the former optimal reinsurance policy is determined by a unique, model-free Stackelberg equilibrium; the latter one, though exists, may be non-unique and model-dependent, and depend on the tail behavior of the claim-size distribution to be more specific. Our numerical analysis provides further support for necessity of integrating the insurer and the reinsurer into a unified framework. In this regard, the stochastic Stackelberg differential reinsurance game proposed in this paper is a good candidate to achieve this goal.     

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 reinsurance with regulatory initial capital and default risk

In a reinsurance contract, a reinsurer promises to pay the part of the loss faced by an insurer in exchange for receiving a reinsurance premium from the insurer. However, the reinsurer may fail to pay the promised amount when the promised amount exceeds the reinsurer’s solvency. As a seller of a reinsurance contract, the initial capital or reserve of a reinsurer should meet some regulatory requirements. We assume that the initial capital or reserve of a reinsurer is regulated by the value-at-risk (VaR) of its promised indemnity. When the promised indemnity exceeds the total of the reinsurer’s initial capital and the reinsurance premium, the reinsurer may fail to pay the promised amount or default may occur. In the presence of the regulatory initial capital and the counterparty default risk, we investigate optimal reinsurance designs from an insurer’s point of view and derive optimal reinsurance strategies that maximize the expected utility of an insurer’s terminal wealth or minimize the VaR of an insurer’s total retained risk. It turns out that optimal reinsurance strategies in the presence of the regulatory initial capital and the counterparty default risk are different both from optimal reinsurance strategies in the absence of the counterparty default risk and from optimal reinsurance strategies in the presence of the counterparty default risk but without the regulatory initial capital.     

Optimal reinsurance under the mean–variance premium principle to minimize the probability of ruin

We consider the problem of minimizing the probability of ruin by purchasing reinsurance whose premium is computed according to the mean–variance premium principle, a combination of the expected-value and variance premium principles. We derive closed-form expressions of the optimal reinsurance strategy and the corresponding minimum probability of ruin under the diffusion approximation of the classical Cramér–Lundberg risk process perturbed by a diffusion. We find an explicit expression for the reinsurance strategy that maximizes the adjustment coefficient for the classical risk process perturbed by a diffusion. Also, for this risk process, we use stochastic Perron’s method to prove that the minimum probability of ruin is the unique viscosity solution of its Hamilton–Jacobi–Bellman equation with appropriate boundary conditions. Finally, we prove that, under an appropriate scaling of the classical risk process, the minimum probability of ruin converges to the minimum probability of ruin under the diffusion approximation.     

双险种最优再保险策略

本文对双险种风险模型,在一险种采取比例再保险,另一险种采取超出损失再保险策略下,得到调节系数与再保险自留水平之间的函数关系式,在理赔额为指数分布和Erlang(2)分布的条件下,得到最优比例再保险和超出损失再保险的自留水平,以及调节系数最大值。     

关于再保险效应的注记

本文在Sparre Anderson模型中采用超额损失再保险与成数分保混合的策略,其中成数分保再保险费按照原始条款计算,超额损失再保险费按Esscher保费原则计算。通过调整系数来研究再保险的效应,将调整系数看作自留额水平的函数,证明了在M充分大时保险人的调整系数关于自留额水平M单调增加,在一定程度上有利于保险公司确定更合理的自留额水平M。