Reinsurance Game Research Articles

The equilibrium strategy of insurance companies’ dividends and reinsurance games

We investigated the optimal risk management strategies of an insurance company within a two-player non-cooperative differential game framework. The key feature of this model is considering the company’s surplus as a strategic variable to make the insurance company more attractive to shareholders. In this setup, shareholders are participants who demand a share of the surplus, while managers are participants concerned with the risk management of the surplus. The objective of the shareholders is to maximize the expected discounted dividends. We address this asymmetric game under two different assumptions about the managers’ objectives: in the first scenario, the managers aim to minimize the probability of bankruptcy; in the second scenario, the managers aim to maximize the expected discounted utility derived from the surplus.

Stochastic differential reinsurance game for two competitive insurers with ambiguity-aversion under mean-variance premium principle

Stochastic differential reinsurance game for two competitive insurers with ambiguity-aversion under mean-variance premium principle

Stochastic differential investment and reinsurance game between an insurer and a reinsurer under thinning dependence structure

This paper investigates a non-zero-sum stochastic differential investment and reinsurance game between an insurer and a reinsurer. It is assumed that the insurer can purchase proportional reinsurance and the claim businesses between the insurer and the reinsurer are correlated through thinning dependence structure. Besides, both the insurer and the reinsurer are allowed to invest in a risk-free asset and a risky asset, in which the two risky assets are supposed to be correlated. The objective of each is to maximize the mean–variance utility of the difference between its terminal wealth and that of its cooperator. By solving the extended Hamilton–Jacobi–Bellman systems within the game theoretic framework, explicit expressions of the optimal time-consistent strategies and value functions of the insurer and the reinsurer are derived, and some comparison results with and without game are obtained as well. Finally, several sensitivity analyses and numerical examples are presented to illustrate the effects of market parameters on the optimal strategies as well as the economic interpretation behind.

A non-zero-sum investment and reinsurance game between two mean–variance insurers with dynamic CVaR constraints

This paper is devoted to investigating a non-zero-sum game between two competing insurers. The insurers can diversify their insurance risks by purchasing proportional reinsurance and investing their collected premiums into a financial market composed of one risk-free asset and one stock. The reinsurance premiums charged by the reinsurer follow the generalized mean–variance premium principle. Moreover, the dynamic CVaR constraints are incorporated in the game problem to control risks. With the dynamic mean–variance objective, we introduce two forward deterministic auxiliary processes to represent the expectations of the insurers’ wealth processes and transform the original time inconsistent game problem into a standard time consistent game problem with two state variables for each insurer. By adopting the dynamic programming principle and the Lagrange duality method, we derive the Nash equilibrium investment–reinsurance strategies for the two insurers. Finally, the effects of several important model parameters on the optimal policies are analyzed by numerical examples.

The investment and reinsurance game of insurers and reinsurers with default risk under CEV model

On the premise of considering the interests of insurance companies and reinsurance companies at the same time, this paper studies the investment and reinsurance game between them. Suppose that the compensation process faced by an insurance company is described by Brownian motion with drift. Insurance companies can purchase proportional reinsurance from reinsurance companies, and both companies can invest in a risk-free asset, a risky asset whose price process follows the constant elasticity of variance (CEV) model, and a defaultable bond. With the goal of maximizing the expected utility of weighted terminal wealth, the corresponding Hamilton–Jacobi–Bellman (HJB) equations are established and solved by using the principle of dynamic programming, and the analytical expressions of the equilibrium investment-reinsurance strategies of insurers and reinsurers are derived respectively. Finally, the influence of each model parameter on the equilibrium strategy is analyzed by numerical examples.

Reinsurance games with variance-premium reinsurers: from tree to chain

AbstractThis paper studies dynamic reinsurance contracting and competition problems under model ambiguity in a reinsurance market with one primary insurer and n reinsurers, who apply the variance premium principle and who are distinguished by their levels of ambiguity aversion. The insurer negotiates reinsurance policies with all reinsurers simultaneously, which leads to a reinsurance tree structure with full competition among the reinsurers. We model the reinsurance contracting problems between the insurer and reinsurers by Stackelberg differential games and the competition among the reinsurers by a non-cooperative Nash game. We derive equilibrium strategies in semi-closed form for all the companies, whose objective is to maximize their expected surpluses penalized by a squared-error divergence term that measures their ambiguity. We find that, in equilibrium, the insurer purchases a positive amount of proportional reinsurance from each reinsurer. We further show that the insurer always prefers the tree structure to the chain structure, in which the risk of the insurer is shared sequentially among all reinsurers.

Reinsurance games with two reinsurers: Tree versus chain

This paper studies reinsurance contracting and competition in a continuous-time model with ambiguity. The market consists of one insurer and two reinsurers, who apply a generalized expected-value premium principle and a generalized variance premium principle to price reinsurance contracts, respectively. The reinsurance contracting problems between the insurer and reinsurers are resolved by Stackelberg differential games, and the reinsurance competition between two reinsurers is settled by a non-cooperative Nash game. We obtain the closed-form equilibrium strategies for all three players under both a tree structure and a chain structure. A detailed comparison study reveals that the tree structure is preferred to the chain structure from a social planner’s perspective, and the tree structure is generally preferred from the insurer’s perspective.

Reinsurance Games with Two Reinsurers: Tree Versus Chain

Reinsurance Games with Two Reinsurers: Tree Versus Chain

Non-zero-sum reinsurance and investment game with correlation between insurance market and financial market under CEV model

<p style='text-indent:20px;'>In this paper, we study a non-zero-sum investment and reinsurance game for two insurers. Each insurer's surplus process is described by a Brownian motion with drift. Both insurers are allowed to purchase proportional reinsurance and invest in a risk-free asset and a risky asset. The price process of the risky asset follows the constant elasticity of variance (CEV) model, and the correlation between the risky asset's price process and the claim process is considered. Each insurer aims to maximize the expected exponential utility of his terminal wealth relative to that of his competitor. By applying stochastic control theory, we establish the corresponding Hamilton-Jacobi-Bellman (HJB) equation and derive optimal investment-reinsurance strategies for two insurers under exponential utility function. Furthermore, we consider the insurer's optimal investment-reinsurance strategies without competition. Finally, numerical analyses are provided to analyze the effects of model parameters on the optimal strategies.</p>

Stochastic asset allocation and reinsurance game under contagious claims

In this paper, we consider a stochastic asset allocation and reinsurance game between two insurance companies with contagious claims, where the insurance claim of one insurer can simultaneously affect the claim intensities of itself and its competitor. This clustering feature of claims is modelled by a mutual-excitation Hawkes process with exponential decays. Furthermore, we assume that the management of the insurance company wants to maximise the expected utility of the relative difference between its terminal surplus and that of its competitor at a fixed time point. The Nash equilibrium strategies have been constructed by solving the Hamilton–Jacobi–Bellman equations, where the explicit formulas of the optimal allocation policies have been derived to be independent of the claim intensities. We also introduce an iterative scheme based on the Feynman–Kac formula to compute the optimal proportional reinsurance policies numerically, where the existence and uniqueness of the solution to the fixed point equation and the convergence of the iterative numerical algorithm are proved rigorously. Finally, numerical examples are presented to show the effect of claim intensities on the optimal controls.

A two‐layer stochastic differential investment and reinsurance game with default risk under the bi‐fractional Brownian motion environment

This paper is concerned with the investment and reinsurance problem between two insurance companies and a reinsurance company by constructing a two‐layer stochastic differential game. Insurance companies invest in a risk‐free asset, a defaultable bond, and a risky asset under the bi‐fractional Brownian motion environment; reinsurance companies invest in a risk‐free asset and a risky asset under the bi‐fractional Brownian motion environment. In order to maximize the expected utility of the insurance companies' relative wealth and the expected utility of the reinsurance company's wealth at the terminal time, we solve the Hamilton–Jacobi–Bellman (HJB) equations by using the differential game theory and stochastic optimal control theory and obtain the equilibrium investment–reinsurance and reinsurance premium strategies. Finally, we investigate the influence of the parameters on the equilibrium strategy through numerical examples and analyze its economic implications.

Equilibrium reinsurance strategies for n insurers under a unified competition and cooperation framework

We propose a unified competition and cooperation framework for n insurers and investigate the resulting reinsurance game problem. Each insurer's surplus is assumed to be a diffusion process. Each insurer can purchase the proportional reinsurance to reduce his claim risk, and the reinsurance premium is determined via the variance value principle. The objective of each insurer is to find a reinsurance strategy so as to maximize the expected utility of his terminal payoff. We establish a Hamilton–Jacobi–Bellman (HJB) equation and the corresponding verification theorem. Furthermore, we derive the explicit solutions for both the equilibrium reinsurance strategy and the value function by solving the HJB equation. Finally, numerical experiments are carried out to illustrate the influences of model parameters such as the size of a group, the number of groups on the equilibrium reinsurance strategy. The numerical results reveal some similarities and differences between the competition case and the cooperation case, and the detailed effects of different competition and cooperation patterns, which provide useful insights for reinsurance in reality.

The Role of Risk Aversion in Stochastic Differential Reinsurance Games

The Role of Risk Aversion in Stochastic Differential Reinsurance Games

Stochastic differential investment and reinsurance games with nonlinear risk processes and VaR constraints

This paper investigates a class of non-zero-sum stochastic differential investment and reinsurance games between two insurance companies. We allow both insurers to purchase a proportional reinsurance contract and invest in risky and risk-free assets. When applying the generalized mean–variance premium principle in determining reinsurance premium, the surplus process becomes quadratic in the retained proportion of the claims. The optimization criterion of each insurer is to maximize the expected utility of the insurer’s terminal performance relative to that of his competitor. In addition, we incorporate dynamic Value-at-Risk (VaR) constraints in the optimization problems of both insurers to satisfy the capital requirements from regulators. The results show that this game problem can be converted to solving a system of nonlinear equations by means of dynamic programming principle and Karush–Kuhn–Tucker (KKT) conditions. Specifically, when both insurers are constant absolute risk aversion (CARA) institutions and the reinsurance premium principle reduces to the expected value principle, we derive the simplified expressions for the Nash equilibrium strategies. Finally, we use some numerical examples to illustrate the effects of several model parameters on the Nash equilibrium strategies under three different scenarios.

Stochastic differential reinsurance games in diffusion approximation models

In this paper, we study some noncooperative and cooperative stochastic differential games between two insurers with reinsurance controls. The surplus processes are modeled by diffusion approximation processes and the insurers can purchase reinsurance dynamically (i.e. control the drift and diffusion terms continuously over time). We consider two types of reinsurance: quota-share (QS) reinsurance and excess-of-loss (XL) reinsurance. In the noncooperative game, one insurer tries to minimize the probability that the surplus difference of the two insurers reaches a low target before it hits a high target, while the other aims to maximize the probability. We consider two cases of the game: in the first case, both insurers purchase XL reinsurance; and in the second case, one insurer purchases QS reinsurance while the other purchases XL reinsurance. In some parameter cases, we solve the game by finding the value function and Nash equilibrium strategy explicitly. In some parameter cases, the value function and Nash equilibrium strategy do not exist and we find the sup-value and sub-value functions of the game. We also establish and solve a cooperative game in which both insurers make joint efforts to minimize the probability that the sum of surplus processes reaches a low target before it hits a high target. Numerical examples and economic implications are given to illustrate the results.

Nonzero-Sum Stochastic Differential Reinsurance Games with Jump–Diffusion Processes

In this paper, a nonzero-sum stochastic differential reinsurance game is studied. A model including controls for the market share (advertising), investment, and reinsurance policies is considered. A jump–diffusion process is used to represent insurance claims. Necessary conditions that would lead to the Nash equilibrium can be found in the duopoly game we consider. Cases with and without controls on the reinsurance are discussed separately. Closed-form solutions for optimal strategies are derived by applying the Hamilton–Jacobi–Bellman equation. Numerical examples are given to validate the correctness of our results.

A Reinsurance and Investment Game between Two Insurers under the CEV Model

In this paper, the problem of nonzero-sum stochastic differential game between two competing insurance companies is considered, i.e., the relative performance concerns. A certain proportion of reinsurance can be taken out by each insurer to control his own risk. Moreover, each insurer can invest in a risk-free asset and risk asset with the price dramatically following the constant elasticity of variance (CEV) model. Based on the principle of dynamic programming, a general framework regarding Nash equilibrium for nonzero-sum games is established. For the typical case of exponential utilization, we, respectively, give the explicit solutions of the equilibrium strategy as well as the equilibrium function. Some numerical studies are provided at last which assist in obtaining some economic explanations.

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.

Stochastic differential reinsurance games with capital injections

This paper investigates a class of reinsurance game problems between two insurance companies under the framework of non-zero-sum stochastic differential games. Both insurers can purchase proportional reinsurance contracts from reinsurance markets and have the option of conducting capital injections. We assume the reinsurance premium is calculated under the generalized variance premium principle. The objective of each insurer is to maximize the expected value that synthesizes the discounted utility of his surplus relative to a reference point, the penalties caused by his own capital injection interventions, and the gains brought by capital injections of his competitor. We prove the verification theorem and derive explicit expressions of the Nash equilibrium strategy by solving the corresponding quasi-variational inequalities. Numerical examples are also conducted to illustrate our results.

Approximation of a class of non-zero-sum investment and reinsurance games for regime-switching jump–diffusion models

This work develops an approximation procedure for a class of non-zero-sum stochastic differential investment and reinsurance games between two insurance companies. Both proportional reinsurance and excess-of loss reinsurance policies are considered. We develop numerical algorithms to obtain the approximation to the Nash equilibrium by adopting the Markov chain approximation methodology. We establish the convergence of the approximation sequences and the approximation to the value functions. Numerical examples are presented to illustrate the applicability of the algorithms.