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.     

Optimal excess-of-loss reinsurance and investment polices under the CEV model

This paper focuses on risk control problem of the insurance company in enterprise risk management. The insurer manages its financial risk through purchasing excess-of-loss reinsurance, and investing its wealth in the constant elasticity of variance stock market. We model risk process by Brownian motion with drift, and study the optimization problem of maximizing the exponential utility of terminal wealth under the controls of reinsurance and investment. Using stochastic control theory, we obtain explicit expressions for optimal polices and value function. We also show that the optimal excess-of-loss reinsurance is always better than optimal proportional reinsurance. And some numerical examples are given.     

Optimal excess-of-loss reinsurance and investment problem for an insurer with jump–diffusion risk process under the Heston model

In this paper, we study the optimal excess-of-loss reinsurance and investment problem for an insurer with jump–diffusion risk model. The insurer is allowed to purchase reinsurance and invest in one risk-free asset and one risky asset whose price process satisfies the Heston model. The objective of the insurer is to maximize the expected exponential utility of terminal wealth. By applying stochastic optimal control approach, we obtain the optimal strategy and value function explicitly. In addition, a verification theorem is provided and the properties of the optimal strategy are discussed. Finally, we present a numerical example to illustrate the effects of model parameters on the optimal investment–reinsurance strategy and the optimal value function.     

Optimal reinsurance under dynamic VaR constraint

This paper deals with the optimal reinsurance strategy from an insurer’s point of view. Our objective is to find the optimal policy that maximises the insurer’s survival probability. To meet the requirement of regulators and provide a tool to risk management, we introduce the dynamic version of Value-at-Risk (VaR), Conditional Value-at-Risk (CVaR) and worst-case CVaR (wcCVaR) constraints in diffusion model and the risk measure limit is proportional to company’s surplus in hand. In the dynamic setting, a CVaR/wcCVaR constraint is equivalent to a VaR constraint under a higher confidence level. Applying dynamic programming technique, we obtain closed form expressions of the optimal reinsurance strategies and corresponding survival probabilities under both proportional and excess-of-loss reinsurance. Several numerical examples are provided to illustrate the impact caused by dynamic VaR/CVaR/wcCVaR limit in both types of reinsurance policy.     

Optimal investment–reinsurance strategies with state dependent risk aversion and VaR constraints in correlated markets

In this paper, we investigate the optimal time-consistent investment–reinsurance strategies for an insurer with state dependent risk aversion and Value-at-Risk (VaR) constraints. The insurer can purchase proportional reinsurance to reduce its insurance risks and invest its wealth in a financial market consisting of one risk-free asset and one risky asset, whose price process follows a geometric Brownian motion. The surplus process of the insurer is approximated by a Brownian motion with drift. The two Brownian motions in the insurer’s surplus process and the risky asset’s price process are correlated, which describe the correlation or dependence between the insurance market and the financial market. We introduce the VaR control levels for the insurer to control its loss in investment–reinsurance strategies, which also represent the requirement of regulators on the insurer’s investment behavior. Under the mean–variance criterion, we formulate the optimal investment–reinsurance problem within a game theoretic framework. By using the technique of stochastic control theory and solving the corresponding extended Hamilton–Jacobi–Bellman (HJB) system of equations, we derive the closed-form expressions of the optimal investment–reinsurance strategies. In addition, we illustrate the optimal investment–reinsurance strategies by numerical examples and discuss the impact of the risk aversion, the correlation between the insurance market and the financial market, and the VaR control levels on the optimal strategies.     

Optimal reinsurance-investment strategy for a dynamic contagion claim model

We study the optimal reinsurance-investment problem for the compound dynamic contagion process introduced by Dassios and Zhao (2011). This model allows for self-exciting and externally-exciting clustering effect for the claim arrivals, and includes the well-known Cox process with shot noise intensity and the Hawkes process as special cases. For tractability, we assume that the insurer’s risk preference is the time-consistent mean–variance criterion. By utilizing the dynamic programming and extended HJB equation approach, a closed-form expression is obtained for the equilibrium reinsurance-investment strategy. An excess-of-loss reinsurance type is shown to be optimal even in the presence of self-exciting and externally-exciting contagion claims, and the strategy depends on both the claim size and claim arrivals assumptions. Further, we show that the self-exciting effect is of a more dangerous nature than the externally-exciting effect as the former requires more risk management controls than the latter. In addition, we find that the reinsurance strategy does not always become more conservative (i.e., transferring more risk to the reinsurer) when the claim arrivals are contagious. Indeed, the insurer can be better off retaining more risk if the claim severity is relatively light-tailed.     

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.     

VaR Criteria for optimal limited change-loss and truncated change-loss reinsurance

Reinsurance can provide an effective way for insurer to manage its risk exposure. In this paper, we further analyze the optimal reinsurance models recently proposed by J. Cai and K. S. Tan [Astin Bulletin, 2007, 37(1): 93-112]. With the criteria of minimizing the value-at-risk (VaR) risk measure of insurer’s total loss exposure, we derive the optimal values of sharing proportion a, retention d, and layer l of two reinsurance treaties: the limited changeloss f(x) = a{(x - d)+ - (x - l)+} and the truncated change-loss f(x) = a(x-d)₊I_(x≤l). Both of the reinsurance plans have been considered to be more realistic and practical in the real business. Our solutions have several appealing features: (i) there is only one condition to verify for the existence of optimal limited change-loss reinsurance while there always exists an optimal truncated change-loss reinsurance, (ii) the resulting optimal parameters have simple analytic forms which depend only on assumed loss distribution, reinsurer’s safety loading, and insurer’s risk tolerance, (iii) the optimal retention d for limited change-loss reinsurance is the same as that by Cai and Tan while the optimal value is smaller for truncated change-loss, (iv) the optimal sharing proportion and layer are always the same for both reinsurance plans, (v) minimized VaR are strictly lower than the values derived by Cai and Tan, (vi) the optimization results reveal possible drawbacks of VaR-based risk management that a heavy tail risk exposure may be expressed by lower VaR.     

Robust equilibrium excess-of-loss reinsurance and CDS investment strategies for a mean–variance insurer with ambiguity aversion

This paper considers the robust equilibrium reinsurance and investment strategies for an ambiguity-averse insurer under a dynamic mean–variance criterion. The insurer is allowed to purchase excess-of-loss reinsurance and invest in a financial market consisting of a risk-free asset and a credit default swap (CDS). Following a game theoretic approach, robust equilibrium strategies and equilibrium value functions for the pre-default case and the post-default case are derived, respectively. For the ambiguity-averse insurer, in general the equilibrium strategies can be characterized by unique solutions to some algebraic equations. For the degenerate case with an ambiguity-neutral insurer, closed-form expressions of equilibrium strategies and equilibrium value functions are obtained. Numerical examples demonstrate that the consideration of model uncertainty and CDS investment improves the insurer’s utility. In this regard, our paper establishes theoretical and numerical support for the importance of ambiguity aversion, credit risk and their interplay in insurance business.     

Optimal dynamic excess-of-loss reinsurance and multidimensional portfolio selection

In this paper, the surplus process of the insurance company is described by a Brownian motion with drift. In addition, the insurer is allowed to invest in a risk-free asset and n risky assets and purchase excess-of-loss reinsurance. Under short-selling prohibition, we consider two optimization problems: the problem of maximizing the expected exponential utility of terminal wealth and the problem of minimizing the probability of ruin. We first show that the excess-of-loss reinsurance strategy is always better than the proportional reinsurance under two objective functions. Then, by solving the corresponding Hamilton-Jacobi-Bellman equations, the closed-form solutions of their optimal value functions and the corresponding optimal strategies are obtained. In particular, when there is no risky-free interest rate, the results indicate that the optimal strategies, under maximizing the expected exponential utility and minimizing the probability of ruin, are equivalent for some special parameter. This validates Ferguson’s longstanding conjecture about the relation between the two problems.     

基于股票价格随机脉冲模型的保险人再保险和投资的最优动态组合选择

本文假设保险人可以进行再保险,并且允许其在金融市场中将资产投资于风险资产和无风险资产,其中风险资产价格采用随机脉冲模型来刻画.当目标是最大化在某一确定终止时刻所拥有财富的二次效用函数期望时,分别得到了超额损失再保险和比例再保险情况下保险人的再保险和投资最优动态选择的显式解和闭解.利用得到的显式解,考虑了金融风险和保险风险之间相关性对最优动态选择的影响,做了相关数值计算.     

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.     

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.     

状态相依效用下的超额损失再保险——投资策略

假设保险公司的盈余过程和金融市场的资产价格过程均由可观测的连续时间马尔科夫链所调节, 以最大化终端财富的状态相依的期望指数效用为目标, 研究了保险公司的超额损失再保险-投资问题. 运用动态规划方法, 得到最优再保险-投资策略的解析解以及最优值函数的半解析式. 最后, 通过数值例子, 分析了模型各参数对最优值函数和最优策略的影响.     

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.     

风险相依下再保险双方的联合最优再保险问题

结合保险人和再保险人的共同利益,研究了具有两类相依险种风险模型下的最优再保险问题.假定再保险公司采用方差保费原理收取保费,利用复合Poisson模型和扩散逼近模型两种方式去刻画保险公司和再保险公司的资本盈余过程,在期望效用最大准则下,证明了最优再保险策略的存在性和唯一性,通过求解Hamilton-Jacobi-Bellman（HJB）方程,得到了两种模型下相应的最优再保险策略及值函数的明晰解答,并给出了数值算例及分析.     

On randomized reinsurance contracts

In this paper we discuss the potential of randomizing reinsurance treaties for efficient risk management. While it may be considered counter-intuitive to introduce additional external randomness in the determination of the retention function for a given occurred loss, we indicate why and to what extent randomizing a treaty can be interesting for the insurer. We illustrate the approach with a detailed analysis of the effects of randomizing a stop-loss treaty on the expected profit after reinsurance in the framework of a one-year reinsurance model under regulatory solvency constraints and cost of capital considerations.     

Optimal Mean-Variance Problem with Constrained Controls in a Jump-Diffusion Financial Market for an Insurer

In this paper, we study the optimal investment and optimal reinsurance problem for an insurer under the criterion of mean-variance. The insurer’s risk process is modeled by a compound Poisson process and the insurer can invest in a risk-free asset and a risky asset whose price follows a jump-diffusion process. In addition, the insurer can purchase new business (such as reinsurance). The controls (investment and reinsurance strategies) are constrained to take nonnegative values due to nonnegative new business and no-shorting constraint of the risky asset. We use the stochastic linear-quadratic (LQ) control theory to derive the optimal value and the optimal strategy. The corresponding Hamilton–Jacobi–Bellman (HJB) equation no longer has a classical solution. With the framework of viscosity solution, we give a new verification theorem, and then the efficient strategy (optimal investment strategy and optimal reinsurance strategy) and the efficient frontier are derived explicitly.     

Optimal Reinsurance and Dividend Strategies Under the Markov-Modulated Insurance Risk Model

In this article, we consider the optimal reinsurance and dividend strategy for an insurer. We model the surplus process of the insurer by the classical compound Poisson risk model modulated by an observable continuous-time Markov chain. The object of the insurer is to select the reinsurance and dividend strategy that maximizes the expected total discounted dividend payments until ruin. We give the definition of viscosity solution in the presence of regime switching. The optimal value function is characterized as the unique viscosity solution of the associated Hamilton–Jacobi–Bellman equation and a verification theorem is also obtained.     

Robust optimal reinsurance–investment strategy with price jumps and correlated claims

This paper considers the robust optimal reinsurance–investment strategy selection problem with price jumps and correlated claims for an ambiguity-averse insurer (AAI). The correlated claims mean that future claims are correlated with historical claims, which is measured by an extrapolative bias. In our model, the AAI transfers part of the risk due to insurance claims via reinsurance and invests the surplus in a financial market consisting of a risk-free asset and a risky asset whose price is described by a jump–diffusion model. Under the criterion of maximizing the expected utility of terminal wealth, we obtain closed-form solutions for the robust optimal reinsurance–investment strategy and the corresponding value function by using the stochastic dynamic programming approach. In order to examine the influence of investment risk on the insurer’s investment behavior, we further study the time-consistent reinsurance–investment strategy under the mean–variance framework and also obtain the explicit solution. Furthermore, we examine the relationship among the optimal reinsurance–investment strategies of the AAI under three typical cases. A series of numerical experiments are carried out to illustrate how the robust optimal reinsurance–investment strategy varies with model parameters, and result analyses reveal some interesting phenomena and provide useful guidances for reinsurance and investment in reality.