Optimal reinsurance under VaR and CTE risk measures

Let X denote the loss initially assumed by an insurer. In a reinsurance design, the insurer cedes part of its loss, say f(X), to a reinsurer, and thus the insurer retains a loss I_f(X)=X−f(X). In return, the insurer is obligated to compensate the reinsurer for undertaking the risk by paying the reinsurance premium. Hence, the sum of the retained loss and the reinsurance premium can be interpreted as the total cost of managing the risk in the presence of reinsurance. Based on a technique used in [Müller, A., Stoyan, D., 2002. Comparison Methods for Stochastic Models and Risks. In: Willey Series in Probability and Statistics] and motivated by [Cai J., Tan K.S., 2007. Optimal retention for a stop-loss reinsurance under the VaR and CTE risk measure. Astin Bull. 37 (1), 93–112] on using the value-at-risk (VaR) and the conditional tail expectation (CTE) of an insurer’s total cost as the criteria for determining the optimal reinsurance, this paper derives the optimal ceded loss functions in a class of increasing convex ceded loss functions. The results indicate that depending on the risk measure’s level of confidence and the safety loading for the reinsurance premium, the optimal reinsurance can be in the forms of stop-loss, quota-share, or change-loss.     

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.     

���ϱ���׼���������ٱ��չ�˾ΥԼ���յ������ٱ������

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.     

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.     

On optimal reinsurance treaties in cooperative game under heterogeneous beliefs

In this paper, we study the optimal reinsurance policies as the result of a two-person cooperative game. We assume that both the insurer and the reinsurer are risk averse and expected-utility maximizers. In addition, we assume that they “agree to disagree” on the distribution of the underlying losses in the contract negotiation.In our analysis, we consider two scenarios. In the first one, the reinsurance premium is fully negotiable, whereas in the second one, the premium is determined by the reinsurer using the expected value premium principle. For both scenarios, we first derive the set of Pareto-optimal reinsurance contracts and then identify the reinsurance contract corresponding to the Nash bargaining solution as well as that corresponding to the Kalai–Smorodinsky bargaining solution.     

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.     

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.     

GlueVaRʧ����ն����µ������ٱ���

??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.     

Pareto-optimal reinsurance arrangements under general model settings

In this paper, we study Pareto optimality of reinsurance arrangements under general model settings. We give the necessary and sufficient conditions for a reinsurance contract to be Pareto-optimal and characterize all Pareto-optimal reinsurance contracts under more general model assumptions. We also obtain the sufficient conditions that guarantee the existence of the Pareto-optimal reinsurance contracts. When the losses of an insurer and a reinsurer are both measured by the Tail-Value-at-Risk (TVaR) risk measures, we obtain the explicit forms of the Pareto-optimal reinsurance contracts under the expected value premium principle. For the purpose of practice, we use numerical examples to show how to determine the mutually acceptable Pareto-optimal reinsurance contracts among the available Pareto-optimal reinsurance contracts such that both the insurer’s aim and the reinsurer’s goal can be met under the mutually acceptable Pareto-optimal reinsurance contracts.     

Measuring the effects of reinsurance by the adjustment coefficient

In this paper we consider the optimal levels of reinsurance in cases where the cedent has a choice between a pure quota-share treaty, a pure excess of loss treaty or any combination of the two. The optimality criterion that we use is that the insurer's adjustment coefficient should be maximized, subject or not to a constraint on the insurer's expected net profit. The solution is given, assuming that the claims have a compound Poisson distribution, that the quota-share premium is calculated on a proportional basis with a commission payment and that the excess of loss reinsurance premium is calculated according to the expected value principle.     

Optimality of excess-loss reinsurance under a mean–variance criterion

In this paper, we study an insurer’s reinsurance–investment problem under a mean–variance criterion. We show that excess-loss is the unique equilibrium reinsurance strategy under a spectrally negative Lévy insurance model when the reinsurance premium is computed according to the expected value premium principle. Furthermore, we obtain the explicit equilibrium reinsurance–investment strategy by solving the extended Hamilton–Jacobi–Bellman equation.     

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 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.     

A continuous-time theory of reinsurance chains

A chain of reinsurance is a hierarchical system formed by the subsequent interactions among multiple (re)insurance agents, which is quite often encountered in practice. This paper proposes a novel continuous-time framework for studying the optimal reinsurance strategies within a chain of reinsurance. The transactions between reinsurance buyers and sellers are formulated by means of Stackelberg games, in order to reflect the conflicting interests and unequal negotiation powers in the bargaining process. Assuming the variance premium principle and the mean–variance criterion on the surplus processes, we solve the time-consistent optimal reinsurance demands and pricing strategies in explicit forms, which are surprisingly plain.Based on the proposed reinsurance chain models, our in-depth theoretical analysis shows that: (a.) it is optimal to situate more (resp. less) risk averse reinsurers to the latter (resp. former) positions in a chain of reinsurance; (b.) adding new reinsurers will lower the reinsurance prices at all levels in a chain of reinsurance, promoting the existing agents to rationally control their respective risk exposures; and essentially (c.) alleviate the systemic risk in the chain structure.     

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.     

A Bowley solution with limited ceded risk for a monopolistic reinsurer

Borch (1969) advocated that the study of optimal reinsurance design should take into consideration the conflicting interests of both an insurer and a reinsurer. Motivated by this and exploiting a Bowley solution (or Stackelberg equilibrium game), this paper proposes a two-step model that tackles an optimal risk transfer problem between the insurer and the reinsurer. From the insurer’s perspective, the first step of the model provisionally derives an optimal reinsurance policy for a given reinsurance premium while reflecting the reinsurer’s risk appetite. The reinsurer’s risk appetite is controlled by imposing upper limits on the first two moments of the coverage. Through a comparative analysis, the effect of the insurer’s initial wealth on the demand for reinsurance is then examined, when the insurer’s risk aversion and prudence are taken into account. Based on the insurer’s provisional strategy, the second step of the model determines the monopoly premium that maximizes the reinsurer’s expected profit while still satisfying the insurer’s incentive condition. Numerical examples are provided to illustrate our Bowley solution.     

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。     

Optimal reinsurance in a compound Poisson risk model with dependence

This paper considers the problem of optimal reinsurance in a compound Poisson risk model with dependent classes of insurance business. It is assumed that the risk process in each class follows a compound Poisson process, and that all classes are correlated due to the so-called thinning-dependence structure. Under the criterion of maximizing the adjustment coefficient, methods for finding the optimal reinsurance strategies are discussed for both the expected value premium principle and the variance premium principle. Numerical examples are also provided to illustrate the impact of the model parameters on the optimal reinsurance strategies.     

Optimal Investment-Reinsurance Strategy with Exchange Rate Risk under Variance Premium Principl

It is assumed that both an insurance company and a reinsurance company adopt the variance premium principle to collect premiums. Specifically, an insurance company is allowed to investment not only in a domestic risk-free asset and a risky asset, but also in a foreign risky asset. Firstly, we use a geometry Brownian motion to model the exchange rate risk, and assume that the insurance company could control the insurance risk by transferring the insurance business into the reinsurance company. Secondly, the stochastic dynamic programming principle is used to study the optimal investment and reinsurance problems in two situations. The first is a diffusion approximation risk model and the second is a classical risk model. The optimal investment and reinsurance strategies are obtained under these two situations. We also show that the exchange rate risk has a great impact on the insurance company's investment strategies, but has no effect on the reinsurance strategies. Finally, a sensitivity analysis of some parameters is provided.