On reinsurance and investment for large insurance portfolios

We consider a problem of optimal reinsurance and investment for an insurance company whose surplus is governed by a linear diffusion. The company’s risk (and simultaneously its potential profit) is reduced through reinsurance, while in addition the company invests its surplus in a financial market. Our main goal is to find an optimal reinsurance-investment policy which minimizes the probability of ruin. More specifically, in this paper we consider the case of proportional reinsurance, and investment in a Black-Scholes market with one risk-free asset (bond, or bank account) and one risky asset (stock). We apply stochastic control theory to solve this problem. It transpires that the qualitative nature of the solution depends significantly on the interplay between the exogenous parameters and the constraints that we impose on the investment, such as the presence or absence of shortselling and/or borrowing. In each case we solve the corresponding Hamilton-Jacobi-Bellman equation and find a closed-form expression for the minimal ruin probability as well as the optimal reinsurance-investment policy.     

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.     

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 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.     

考虑交易费用和债务的再保险和投资问题

该文考虑了保险公司的再保险和投资在多种风险资产中的策略问题. 假设保险公司本身有着一定的债务, 债务的多少服从线性扩散方程. 保险公司可以通过再保险和将再保险之后的剩余资产投资在m种风险资产和一种无风险资产中降低其风险. 资产中风险资产的价格波动服从几何布朗运动, 其债务多少的演化也是依据布朗运动而上下波动. 该文考虑了风险资产与债务之间的相互关系, 考虑了在进行风险投资时的交易费用, 并且利用HJB方程求得保险公司的最大最终资产的预期指数效用, 给出了相应的最优价值函数和最优策略的数值解.     

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.     

Robust optimal investment and reinsurance problem for a general insurance company under Heston model

In this paper, we study a robust optimal investment and reinsurance problem for a general insurance company which contains an insurer and a reinsurer. Assume that the claim process described by a Brownian motion with drift, the insurer can purchase proportional reinsurance from the reinsurer. Both the insurer and the reinsurer can invest in a financial market consisting of one risk-free asset and one risky asset whose price process is described by the Heston model. Besides, the general insurance company’s manager will search for a robust optimal investment and reinsurance strategy, since the general insurance company faces model uncertainty and its manager is ambiguity-averse in our assumption. The optimal decision is to maximize the minimal expected exponential utility of the weighted sum of the insurer’s and the reinsurer’s surplus processes. By using techniques of stochastic control theory, we give sufficient conditions under which the closed-form expressions for the robust optimal investment and reinsurance strategies and the corresponding value function are obtained.     

Robust optimal reinsurance and investment strategies for an AAI with multiple risks

This paper studies the robust optimal reinsurance and investment problem for an ambiguity averse insurer (abbr. AAI). The AAI sells insurance contracts and has access to proportional reinsurance business. The AAI can invest in a financial market consisting of four assets: one risk-free asset, one bond, one inflation protected bond and one stock, and has different levels of ambiguity aversions towards the risks. The goal of the AAI is to seek the robust optimal reinsurance and investment strategies under the worst case scenario. Here, the nominal interest rate is characterized by the Vasicek model; the inflation index is introduced according to the Fisher’s equation; and the stock price is driven by the Heston’s stochastic volatility model. The explicit forms of the robust optimal strategies and value function are derived by introducing an auxiliary robust optimal control problem and stochastic dynamic programming method. In the end of this paper, a detailed sensitivity analysis is presented to show the effects of market parameters on the robust optimal reinsurance policy, the robust optimal investment strategy and the utility loss when ignoring ambiguity.     

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.     

Time-consistent reinsurance and investment strategies for mean–variance insurer under partial information

In this paper, based on equilibrium control law proposed by Björk and Murgoci (2010), we study an optimal investment and reinsurance problem under partial information for insurer with mean–variance utility, where insurer’s risk aversion varies over time. Instead of treating this time-inconsistent problem as pre-committed, we aim to find time-consistent equilibrium strategy within a game theoretic framework. In particular, proportional reinsurance, acquiring new business, investing in financial market are available in the market. The surplus process of insurer is depicted by classical Lundberg model, and the financial market consists of one risk free asset and one risky asset with unobservable Markov-modulated regime switching drift process. By using reduction technique and solving a generalized extended HJB equation, we derive closed-form time-consistent investment–reinsurance strategy and corresponding value function. Moreover, we compare results under partial information with optimal investment–reinsurance strategy when Markov chain is observable. Finally, some numerical illustrations and sensitivity analysis are provided.     

Optimal proportional reinsurance and investment based on Hamilton-Jacobi-Bellman equation

In the whole paper, the claim process is assumed to follow a Brownian motion with drift and the insurer is allowed to invest in a risk-free asset and a risky asset. In addition, the insurer can purchase the proportional reinsurance to reduce the risk. The paper concerns the optimal problem of maximizing the utility of terminal wealth. By solving the corresponding Hamilton-Jacobi-Bellman equations, the optimal strategies about how to purchase the proportional reinsurance and how to invest in the risk-free asset and risky asset are derived respectively.     

具有随机利率、随机变差的最优投资和联合比例-超额损失再保险

对跳-扩散风险模型,研究了最优投资和再保险问题.保险公司可以购买再保险减少理赔,保险公司还可以把盈余投资在一个无风险资产和一个风险资产上.假设再保险的方式为联合比例-超额损失再保险.还假设无风险资产和风险资产的利率是随机的,风险资产的方差也是随机的.通过解决相应的Hamilton-Jacobi-Bellman(HJB)方程,获得了最优值函数和最优投资、再保险策略的显示解.特别的,通过一个例子具体的解释了得到的结论.     

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.     

带交易费用的最优投资和比例再保险

研究了保险公司的最优投资和再保险问题.保险公司的盈余通过跳-扩散风险模型来模拟,可以把盈余的一部分投资到金融市场,金融市场由一个无风险资产和n个风险资产组成,并且保险公司还可以购买比例再保险;在买卖风险资产时,考虑了交易费用.通过随机控制的理论,获得了最优策略和值函数的显示解.     

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??In this paper, we investigate a robust optimal portfolio and reinsurance problem under inflation risk for an ambiguity-averse insurer (AAI), who worries about uncertainty in model parameters. We assume that the AAI is allowed to purchase proportional reinsurance and invest his/her wealth in a financial market which consists of a risk-free asset and a risky asset. The objective of the AAI is to maximize the minimal expected power utility of terminal wealth. By using techniques of stochastic control theory, closed-form expressions for the value function and optimal strategies are obtained.     

基于新巴塞尔协议监管下保险人的均值-方差最优投资-再保险问题

本文研究了均值-方差优化准则下,保险人的最优投资和最优再保险问题.我们用一个复合泊松过程模型来拟合保险人的风险过程,保险人可以投资无风险资产和价格服从跳跃-扩散过程的风险资产.此外保险人还可以购买新的业务(如再保险).本文的限制条件为投资和再保险策略均非负,即不允许卖空风险资产,且再保险的比例系数非负.除此之外,本文还引入了新巴塞尔协议对风险资产进行监管,使用随机二次线性(linear-quadratic,LQ)控制理论推导出最优值和最优策略.对应的哈密顿-雅克比-贝尔曼(Hamilton-Jacobi-Bellman,HJB)方程不再有古典解.在粘性解的框架下,我们给出了新的验证定理,并得到有效策略(最优投资策略和最优再保险策略)的显式解和有效前沿.     

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.     

Non-zero-sum stochastic differential reinsurance and investment games with default risk

This paper investigates the implications of strategic interaction (i.e., competition) between two CARA insurers on their reinsurance-investment policies. The two insurers are concerned about their terminal wealth and the relative performance measured by the difference in their terminal wealth. The problem of finding optimal policies for both insurers is modelled as a non-zero-sum stochastic differential game. The reinsurance premium is calculated using the variance premium principle and the insurers can invest in a risk-free asset, a risky asset with Heston’s stochastic volatility and a defaultable corporate bond. We derive the Nash equilibrium reinsurance policy and investment policy explicitly for the game and prove the corresponding verification theorem. The equilibrium strategy indicates that the best response of each insurer to the competition is to mimic the strategy of its opponent. Consequently, either the reinsurance strategy or the investment strategy of an insurer with the relative performance concern is riskier than that without the concern. Numerical examples are provided to demonstrate the findings of this study.     

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??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.     

Robust non-zero-sum investment and reinsurance game with default risk

This paper investigates a non-zero-sum stochastic differential game between two competitive CARA insurers, who are concerned about the potential model ambiguity and aim to seek the robust optimal reinsurance and investment strategies. The ambiguity-averse insurers are allowed to purchase reinsurance treaty to mitigate individual claim risks; and can invest in a financial market consisting of one risk-free asset, one risky asset and one defaultable corporate bond. The objective of each insurer is to maximize the expected exponential utility of his terminal surplus relative to that of his competitor under the worst-case scenario of the alternative measures. Applying the techniques of stochastic dynamic programming, we derive the robust Nash equilibrium reinsurance and investment policies explicitly and present the corresponding verification theorem. Finally, we perform some numerical examples to illustrate the influence of model parameters on the equilibrium reinsurance and investment strategies and draw some economic interpretations from these results.