Reinsurance-investment Strategy Research Articles

Non-zero-sum investment-reinsurance game with delay and ambiguity aversion

In this paper, we investigate a investment-reinsurance game problem with delay and ambiguity aversion for two insurers under mean–variance preference. Each insurer can purchase proportional reinsurance whose surplus process is a diffusion approximation process, and invest oneself’s wealth in a financial market consisting of a risk-free asset and a risky asset whose price evolution follows a jump–diffusion process. Based on the introduction of delay, we obtain wealth dynamics for each insurer depicted by a stochastic delay differential equation. By the stochastic control theory in the framework of game theory, the robust time-consistent Nash equilibrium investment-reinsurance strategies and the corresponding robust equilibrium value functions for each insurer are derived. Furthermore, some numerical examples are provided to illustrate the effect of market parameters on the optimal investment-reinsurance strategy.

Equilibrium investment–reinsurance strategy under information asymmetry and random horizon

This paper investigates an investment–reinsurance problem incorporating information asymmetry and random horizon. At the beginning of the transaction, the insurer owns some inside information related to the risky asset price and the insurance claims. Simultaneously, the return rate and the driving noise for the risky asset process cannot be observed directly. Moreover, the trading horizon is random due to some exogenous factors. Under the dynamic mean–variance criterion, the equilibrium feedback strategy and the associated equilibrium value function are derived in closed form by solving the extended Hamilton–Jacobi–Bellman (HJB) equations. Several special cases are further discussed. Finally, numerical simulations are conducted to analyze the influences of some important parameters on the equilibrium strategy and the efficient frontier.

Optimal robust reinsurance contracts with investment strategy under variance premium principle

This paper investigates the optimal robust proportional reinsurance contracts with investment in a liquid financial market under variance premium principle in a principal-agent framework. The surplus process of the insurer (agent) is assumed to follow an approximating compound Poisson risk process. Both the insurer and reinsurer (principal) are allowed to invest in a risk-free asset and a risky asset whose price process is governed by the constant elasticity of variance model. The insurer and reinsurer aim to maximize the expected exponential utility of terminal wealth. The reinsurer is ambiguity-averse and has deterministic ambiguity aversion preferences against the diffusion risk caused by the financial market and the approximated diffusion risk which comes from the claim process. The reinsurance price is described by the reinsurer's safety loading which can be decided by the optimal strategies of the insurer and the reinsurer. By utilizing stochastic optimal control principle and HJB (or HJBI) equations, the optimal (robust) proportional reinsurance-investment strategies and the corresponding value functions are obtained explicitly. Numerical examples are provided.

Optimal reinsurance-investment problem with default risk for an insurer under the constant elasticity of variance model

Abstract Accepted by: Giorgio Consigli In this paper we study the optimal reinsurance-investment problem for an insurer with constant absolute risk aversion (CARA) utility. Because the insurer needs to avoid large claims and make profits in the actual operation process, he/she is allowed to buy proportional reinsurance and invest in a risk-free bond, a stock and a default bond. Moreover, the price process of the stock follows the constant elasticity of variance (CEV) model and the correlation between the risk process and the stock’s price is considered. For the objective of expected utility maximization, we establish the Hamilton–Jacobi–Bellman equation and derive the optimal reinsurance-investment strategy explicitly for the pre-default case and the post-default case via the dynamic programming. Finally, numerical examples and sensitivity analyses are provided to illustrate the effects of model parameters on the optimal strategy. We find that (i) the default risk has no impact on the optimal strategy of the stock and the parameters of the stock’s price have no impact on the optimal money amount invested in the defaultable bond. (ii) The optimal reinsurance strategy depends on the volatility of the stock’s price under the CEV model. (iii) The correlation between risk model and the stock’s price does have effects on the insurer’s optimal reinsurance-investment strategy. These findings may provide guidance to the management of insurers in practice.

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.

Optimal Reinsurance–Investment Strategy Based on Stochastic Volatility and the Stochastic Interest Rate Model

This paper studies insurance companies’ optimal reinsurance–investment strategy under the stochastic interest rate and stochastic volatility model, taking the HARA utility function as the optimal criterion. It uses arithmetic Brownian motion as a diffusion approximation of the insurer’s surplus process and the variance premium principle to calculate premiums. In this paper, we assume that insurance companies can invest in risk-free assets, risky assets, and zero-coupon bonds, where the Cox–Ingersoll–Ross model describes the dynamic change in stochastic interest rates and the Heston model describes the price process of risky assets. The analytic solution of the optimal reinsurance–investment strategy is deduced by employing related methods from the stochastic optimal control theory, the stochastic analysis theory, and the dynamic programming principle. Finally, the influence of model parameters on the optimal reinsurance–investment strategy is illustrated using numerical examples.

Optimal reinsurance contract and investment strategy for multiple competitive-cooperative insurers and a reinsurer

Abstract Accepted by: Giorgio Consigli In this article, we consider a reinsurance contract design by taking into account the joint interests of multiple insurers and a reinsurer. The reinsurance contract consists of optimal reinsurance strategies and reinsurance prices. The former are chosen by the competitive–cooperative insurers and the latter are determined by a reinsurer. Both insurers and the reinsurer can invest in the common risk-free asset and one different risky asset. The optimal time-consistent reinsurance–investment strategies of the insurers and the optimal reinsurance prices and investment strategy of the reinsurer are derived analytically. Numerical experiments are carried out to illustrate the influences of model parameters on the optimal reinsurance contract and optimal investment strategies. We find that the establishment of the reinsurance contract is affected by the correlation between the claim sizes of insurers as well as that of claim numbers. The results reveal that competition and cooperation will lead to a decrease and increase of the reinsurance price, respectively, showing the importance of opting for cooperation among insurers. Both competition and cooperation are beneficial to insurers, especially for those with high risk aversion.

Optimal reinsurance-investment game for two insurers with SAHARA utilities under correlated markets

In this paper, we study the optimal reinsurance-investment game between two insurers with the same insurance business but different wealth and risk preferences. Assume that the insurers who have the symmetric asymptotic hyperbolic absolute risk aversion (SAHARA) utilities and the price of risky asset obeys the constant elasticity of variance (CEV) model. It is impossible to obtain closed-form solution of the optimal reinsurance-investment strategy due to the non-homothetic property and the complicity of SAHARA utilities. According to establish a strong duality relationship of the value function, we successfully propose an efficient dual control Monte Carlo method for computing the Nash equilibrium strategies. Finally, numerical analysis is given to illustrate the impact of model parameters to Nash equilibrium strategies.

Optimal reinsurance-investment strategy with thinning dependence and delay factors under mean-variance framework

In this paper, we study the optimal time-consistent reinsurance-investment problem for a risk model with the thinning-dependence structure. The insurer’s wealth process is described by a jump-diffusion risk model with two dependent classes of insurance business. We assume that the insurer is allowed to purchase per-loss reinsurance and invest its surplus in a financial market consisting of a risk-free asset and a risky asset. Also, the performance-related capital inflow or outflow feature is introduced, and the wealth process is modeled by a stochastic delay differential equation. Under the time-inconsistent mean-variance criterion, we derive the explicit optimal reinsurance-investment strategy and value function under the expected value premium principle as well as the variance premium principle by solving the extended Hamilton–Jacobi–Bellman (HJB) delay system. In particular, we prove the existence and uniqueness of the optimal strategy under the expected value premium principle. Finally, some numerical examples are provided to illustrate the influence of model parameters on the optimal strategy.

Robust optimal reinsurance–investment for [formula omitted]-maxmin mean–variance utility under Heston’s SV model

Most literatures about robust optimal reinsurance–investment problem aim to maximum the value function under the worst-case scenario, but some insurers are optimistic, so it is more reasonable to consider a more smooth criterion named α-maxmin criterion which seek a balance between the worst-case scenario and best-case scenario. Furthermore, the past performance of insurance company will heavily impact the reinsurance–investment strategy of insurer, by introducing the capital flow related to the historical performance of the insurer, the wealth process can be described by stochastic delay differential equation. In this paper, we consider the robust optimal reinsurance–investment strategy for an α-maxmin mean–variance insurer with delay under Heston’s stochastic volatility stock model, the verification theorem is given and the closed-form solutions of value function and optimal strategies are obtained, respectively. In the part of numerical analysis, we illustrate the influence of some important parameters on the optimal strategies.

Robust equilibrium investment-reinsurance strategy for n competitive insurers with square-root factor process

We investigate a robust equilibrium investment-reinsurance problem for n ambiguity-averse competitive insurers, . Each insurer is allowed to purchase proportional reinsurance and invest in a risk-free asset and a risky asset. Each insurer aims to maximize the expected utility of a weighted relative terminal wealth with respect to the other competitors. In this article, the risky asset is assumed to follow a general and flexible model: the square root factor process. Following the game theory approach, we derive the closed solutions of the robust equilibrium investment-reinsurance strategies. Moreover, the verification theorem is provided in this article. Finally, we demonstrate some numerical analyses and give the economic explanations as well.

Optimal reinsurance and investment with a common shock and a random exit time

Under the mean-variance framework, we study the continuous-time optimal reinsurance and investment problem with a common shock and a random exit time. To describe the influence of the common shock, we propose a new interdependence mechanism between the insurance market and the financial market. It can reflect both the impact of the occurrence of a common shock and its influence degree on the two markets. Both the termination times of reinsurance and investment are random, and the random exit time is affected simultaneously by exogenous and endogenous random events. The insurer’s objective is to minimize the variance of her terminal wealth under a given level of expected terminal wealth. We derive the explicit optimal reinsurance-investment strategy by employing stochastic optimal control and Lagrange duality techniques. The influences of the market interdependence and the random exit time on the optimal strategy are demonstrated through numerical experiments. The results reveal some meaningful phenomena and provide insightful guidance for reinsurance and investment practice in reality.

Optimal investment and reinsurance strategies for an insurer with stochastic economic factor

This work considers optimal investment and reinsurance strategies for an insurer with stochastic economic factor. In our mathematical model, a risk-free asset and a risky asset are assumed to rely on a stochastic economic factor which is described by a diffusion process. We generalize the claim process to a compound Poisson process with the stochastic economic factor. Using expected utility maximization, we characterize the optimal strategy of investment-reinsurance under the power utility function. We use dynamic programming principle to derive the Hamilton–Jacobi–Bellman (HJB) equation. Then, by analysing the solution of the HJB equation, the optimal investment-reinsurance strategy is obtained and given in the verification theorem. Finally, sensitivity analysis is given to show the economic behavior of the optimal investment and reinsurance strategies.

A Heston local-stochastic volatility model for optimal investment–reinsurance strategy with a defaultable bond in an ambiguous environment

A Heston local-stochastic volatility model for optimal investment–reinsurance strategy with a defaultable bond in an ambiguous environment

Mean field and n-insurers games for robust optimal reinsurance-investment in correlated markets

The paper investigates a family of robust reinsurance-investment strategies under relative performance criteria for the insurers who have exponential utility and trade in a common reinsurance-investment horizon in log-normal markets. Assume that two correlated Brownian motions used to characterize the insurance surplus risk and stock price risk. By applying stochastic control theories, we derive the closed-form solutions for the n-players game and the corresponding mean field game, furthermore the equilibria strategies are shown to be unique in the class of constant equilibria. Finally, we present the influence of model parameters on reinsurance and investment strategies with economic explanations by some numerical figures.

A Stackelberg reinsurance-investment game under Heston's stochastic volatility model

<p style='text-indent:20px;'>This paper studies the optimal reinsurance-investment problem under Heston's stochastic volatility (SV) in the framework of Stackelberg stochastic differential game, in which an insurer and a reinsurer are the two players. The aim of the insurer as the follower in the game is to find the optimal reinsurance strategy and investment strategy such that its mean-variance cost functional is maximized, while the aim of the reinsurer as the leader is to maximize its mean-variance cost functional through finding its optimal premium pricing strategy and investment strategy. To overcome the time-inconsistency problem in the game, we derive the optimization problems of the two players as embedded games and obtain the corresponding extended Hamilton–Jacobi–Bellman (HJB) equations. Then we present the verification theorem and get the equilibrium reinsurance-investment strategies and the corresponding value functions of both the insurer and the reinsurer. Finally, we provide some numerical examples to draw the findings.</p>

Optimal reinsurance-investment problem for a general insurance company under a generalized dynamic contagion claim model

<p style='text-indent:20px;'>In this paper, we study an optimal management problem for a general insurance company which holds shares of an insurance company and a reinsurance company. The general company aims to derive the equilibrium reinsurance-investment strategy under the mean-variance criterion. The claim process described by a generalized compound dynamic contagion process introduced by [<xref ref-type="bibr" rid="b18">18</xref>] which allows for self-exciting and externally-exciting clustering effect for the claim arrivals and the processes of the risky assets are described by the jump-diffusion models. Based on practical considerations, we suppose that the externally-exciting clustering effect will simultaneously affect both the price of risky assets and the intensity of claims. To overcome the inconsistency issue caused by the mean-variance criterion, we formulate the optimization problem as an embedded game and solve it via a corresponding extended Hamilton-Jacobi-Bellman equation. The equilibrium reinsurance-investment strategy is obtained, which depends on a solution to an ordinary differential equation. In addition, we demonstrate the derived equilibrium strategy and the economic implications behind it through a large number of mathematical analysis and numerical examples.</p>

Robust optimal reinsurance-investment problem for $ n $ competitive and cooperative insurers under ambiguity aversion

<abstract><p>We investigate a robust optimal reinsurance-investment problem for $ n $ insurers under multiple interactions, which arise from the insurance market, the financial market, the competition mechanism and the cooperation mechanism. Each insurer's surplus process is assumed to follow a diffusion model, which is an approximation of the classical Cramér-Lundberg model. Each insurer is allowed to purchase proportional reinsurance to reduce their claim risk. To reflect the first moment and second moment information on claims, we use the variance premium principle to calculate reinsurance premiums. To increase wealth, each insurer can invest in a financial market, which includes one risk-free asset and $ n $ correlated stocks. Each insurer wants to obtain the robust optimal reinsurance and investment strategy under the mean-variance criterion. By applying a stochastic control technique and dynamic programming approach, the extended Hamilton-Jacobi-Bellman (HJB) equation is established. Furthermore, we derive both the robust optimal reinsurance-investment strategy and the corresponding value function by solving the extended HJB equation. Finally, we present numerical experiments, which yield that competition and cooperation have an important influence on the insurer's decision-making.</p></abstract>

A non-zero-sum stochastic differential game between two mean-variance insurers with inside information

This paper is devoted to the study of a non-zero-sum investment-reinsurance game between two insurers with different opinions about some inside information. Each insurer is concerned about her terminal wealth level and the relative performance of wealth. We assume that the surplus processes of the insurers are correlated jump-diffusion processes. The insurers can invest in the financial market consisting of a risk-free asset and a risky asset. From the beginning of the transaction, the insurers have some inside information about the risky asset price in the future, which is disturbed by noise. However, one insurer trusts it while the other does not. We use the filtration expansion technique to transform the wealth processes for these two insurers with inside information. Then, under the dynamic mean-variance criterion, the explicit solutions for the equilibrium investment-reinsurance strategy and the equilibrium value function are derived by solving the extended HJB equations. Finally, the numerical examples are provided to analyze the effects of the inside information and the model parameters on the equilibrium strategy and the effective frontier.