Optimal Reinsurance Strategy Research Articles

Optimal reinsurance with multivariate risks and dependence uncertainty

In this paper, we study the optimal reinsurance design from the perspective of an insurer with multiple lines of business, where the reinsurance is purchased by the insurer for each line of business, respectively. For the risk vector generated by the multiple lines of business, we suppose that the marginal distributions are fixed, but the dependence structure between these risks is unknown. Due to the unknown dependence structure, the optimal strategy is investigated for the worst-case scenario. We consider two types of risk measures: Value-at-Risk (VaR) and Range-Value-at-Risk (RVaR) including Expected Shortfall (ES) as a special case, and general premium principles satisfying certain conditions. To be more practical, the minimization of the total risk is conducted under some budget constraint. For the VaR-based model with only two risks, it turns out that the limited stop-loss reinsurance treaty is optimal for each line of business. For the model with more than two risks, we obtain two types of optimal reinsurance strategies if the marginals have convex or concave distributions on their tail parts by constraining the ceded loss functions to be convex or concave. Moreover, as a special case, the optimal quota-share reinsurance with dependence uncertainty has been studied. Finally, after applying our findings to two risks, some studies have been implemented to obtain both the analytical and numerical optimal reinsurance policies.

The optimal reinsurance strategy with price-competition between two reinsurers

We study optimal reinsurance for an insurer and two reinsurers in the market through stochastic game theory. The relationship between the insurer and reinsurers is described by a Stackelberg model, where reinsurers, as market leaders, set prices for reinsurance treaties, and the insurer, as a price taker, determines reinsurance demand. Furthermore, we employ a Nash game to model the price competition between the two reinsurers who adopt different premium principles: the variance premium principle and the expected value premium principle. Both the insurer and reinsurers aim to maximize their respective mean-variance cost functions, leading to a time inconsistency control problem. This issue is resolved using a corresponding extended Hamilton-Jacobi-Bellman equation in the game-theoretic framework. We find that the insurer will adopt propositional and excess-of-loss reinsurance strategies with two reinsurers, respectively. Moreover, under an exponential claim size distribution, there exists a unique equilibrium reinsurance premium strategy. Our numerical analysis illuminates the effects of claim size, risk aversion, and the interest rates of the insurer and reinsurers on the equilibrium reinsurance and premium strategies, enhancing the understanding of competition in the reinsurance market.

Stackelberg differential reinsurance and investment game for a dependent risk model with Ornstein–Uhlenbeck process

This paper considers a reinsurance and investment problem under the Stackelberg differential game. It assumes that the insurer can purchase reinsurance and the claim businesses between the insurer and the reinsurer are correlated through common shock dependence, and both of them are allowed to invest in a common risky asset whose price follows an Ornstein–Uhlenbeck process. By the stochastic control theory, explicit expressions of the optimal controls and the value functions are obtained for both of the insurer and reinsurer. We show that the optimal reinsurance strategy is a constant, which is independent of the time and risk-free interest rate. We also show that compared with the independent model, the insurer will purchase less reinsurance and the reinsurer will increase the premium price under the dependent risk model.

Non-zero-sum reinsurance and investment game under thinning dependence structure: mean–variance premium principle

This paper considers a non-zero-sum stochastic differential reinsurance and investment game between two competitive insurers under the expected exponential utility. It is assumed that the surplus process of each insurer is characterized by a compound Poisson model and their claim businesses are correlated through thinning dependence structure. Each insurer can purchase proportional reinsurance and is allowed to invest in a risk-free asset and a risky asset. The insurers compete with each other and are concerned with the relative performance. By using the stochastic control technique, not only the existence and uniqueness of the optimal strategies are proved, but also the specific analyses for the constraint reinsurance strategies are provided. We find that (i) the thinning dependence structure has significant impact on optimal reinsurance strategies for both insurers and can help them reduce risks; (ii) the behavior of non-zero-sum game will reduce the proportion of purchasing reinsurance for each insurer.

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.

Research on the Estimation of The Scale of Catastrophe Insurance Funds in Various Provinces in China

The catastrophe insurance fund, as an important means for the steady and robust development of catastrophe risk management, has attracted significant attention from various sectors of society. However, natural disasters in different regions of China vary significantly in terms of frequency and severity, the centralized management of a uniform catastrophe insurance fund severely affects the efficiency and fairness of catastrophe risk management. This study selects the geological disaster loss data from various provinces of mainland China from 2004 to 2021 as the research sample. Based on the characteristics of relative losses caused by geological disasters, the geological disaster losses in various provinces of China are divided into four categories using cluster analysis, and the Generalized Pareto Distribution (GPD) is utilized to describe the loss distributions of each category; The scale of geological disaster insurance funds in each province is estimated through the optimal reinsurance strategy based on the Conditional Value-at-Risk (CVaR) - Expected Premium principle; Provinces with similar characteristics of relative losses from geological disasters have significantly different requirements for the scale of catastrophe insurance funds. Compared to traditional catastrophe models, this study introduces cluster analysis to consider the heterogeneity of geological disaster occurrences. In this paper, a new approach to enhance the management of catastrophe risks by fitting the distribution of geological disaster losses more reasonably is provided.

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.

Multiple per-claim reinsurance based on maximizing the Lundberg exponent

In this paper, we consider the optimal per-claim reinsurance problem for an insurer who designs a reinsurance contract with multiple reinsurance participants. In contrast to using the value-at-risk as a short-term risk measure, we take the Lundberg exponent in risk theory as a risk measure for the insurer over a long-term horizon because the Lundberg upper bound performs better in measuring the infinite-time ruin probability. To reflect various risk preferences of the reinsurance participants, we adopt a type of combined premium principle in which the expected premium principle, variance premium principle, and exponential premium principle are all special cases. Based on maximization of the insurer's Lundberg exponent, the optimal reinsurance is formulated within a static setting, and we derive optimal multiple reinsurance strategies within a general admissible policies set. In general, these optimal strategies are shown to have non-piecewise linear structures, differing from conventional reinsurance strategies such as quota-share, excess-of-loss, or linear layer reinsurance arrangements. In some special cases, the optimal reinsurance strategies reduce to classical results.

Optimal proportional reinsurance to maximize an insurer’s exponential utility under unobservable drift

AbstractWe study an optimal reinsurance problem for a diffusion model, in which the drift of the claim follows an Ornstein–Uhlenbeck process. The aim of the insurer is to maximize the expected exponential utility of its terminal wealth. We consider two cases: full information and partial information. Full information occurs when the insurer directly observes the drift; partial information occurs when the insurer observes only its claims. By applying stochastic control and by solving the corresponding Hamilton–Jacobi–Bellman equations, we find the value function and the optimal reinsurance strategy under both full and partial information. We determine a relationship between the value function and reinsurance strategy under full information with the value function and reinsurance strategy under partial information.

Robust optimal reinsurance strategy with correlated claims and competition

<abstract><p>This paper investigates the robust optimal reinsurance strategy, which simultaneously takes into account the ambiguity aversion, the correlated claims and the joint interests of an insurer and a reinsurer. The correlated claims mean that future claims are correlated with historical claims, which are measured by an extrapolative bias. The joint interests of the insurer and the reinsurer are reflected by the competition between them. To better reflect competition, we assume that the insurer and the reinsurer are engaged in related insurance business. The insurer is allowed to purchase proportional reinsurance or acquire a new business. Under ambiguity aversion and the criterion of maximizing the expected utility of terminal wealth, we obtain explicit solutions for the robust optimal reinsurance strategy and the corresponding value function by using the stochastic dynamic programming approach. Furthermore, we obtain the optimal reinsurance strategy under four typical cases. A series of numerical experiments were carried out to illustrate how the robust optimal reinsurance strategy varies with model parameters, and the result analyses reveal some interesting phenomena and provide useful guidance for reinsurance in reality.</p></abstract>

Optimal investment and reinsurance for the insurer and reinsurer with the joint exponential utility under the CEV model

<abstract><p>This paper considers the problem of optimal investment-reinsurance for the insurer and reinsurer under the constant elasticity of variance (CEV) model. It is assumed that the net claims process is approximated by a diffusion process, both the insurer and reinsurer can invest in risk-free assets and risky assets. We use the variance premium principle to calculate the premiums of the insurer and reinsurer, and the reinsurance proportion is constrained by the net profit condition. Our objective is to maximize the joint exponential utility of the insurer and reinsurer's terminal wealth for a fixed time. By solving the HJB equation, we obtain the explicit expressions of the optimal investment-reinsurance strategy and value function. We find that the optimal reinsurance strategy can be divided into many cases and is related to the risk aversion coefficient of the insurer and reinsurer, but independent of the price of risky assets. Furthermore, we give the proof of the verification theorem. Finally, we demonstrate a numerical analysis to explain the results.</p></abstract>

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>

Non-zero-sum reinsurance and investment game with non-trivial curved strategy structure under Ornstein–Uhlenbeck process

ABSTRACT This paper investigates a non-zero-sum stochastic differential game between two competitive CARA insurers, where we adopt the different classes of premium principles (including the expected value premium principle, the variance premium principle and the exponential premium principle) and each insurer aims to maximize the expected exponential utility of his terminal wealth relative to that of his competitor. Moreover, both insurers are allowed to purchase reinsurance treaty to mitigate individual claim risks and can invest in a financial market consisting of a risk-free asset, a risky asset where the instantaneous rate of investment return follows an Ornstein-Uhlenbeck process which can reflect the changes of bull market and bear market. The optimal reinsurance strategy has a non-trivial structure which is distinguished from the conventional proportional and excess-of-loss reinsurance strategies. Furthermore, we derive the optimal reinsurance and investment strategies under the variance premium principle and expected value principle. In addition, we give another model which considers the correlation between risk model and financial market under the expected value principle. Finally, numerical analyses are provided to analyze the effects of model parameters on the optimal strategies under different cases.

Robust optimal reinsurance in minimizing the penalized expected time to reach a goal

This paper studies a robust optimal reinsurance problem for an ambiguity-averse insurer, who does not have perfect information in the drift term of insurance process. The objective is to minimize the robust value involving the expected time to reach a given capital level before ruin and a penalization of model ambiguity. By using the techniques of stochastic control theory and exponential transformation, we derive the closed-form expressions of the optimal reinsurance strategy and the associated value function for the risk model with cheap reinsurance. For the non-cheap reinsurance, we prove that there exists a “safe level” such that the optimization problem becomes a trivial one when the initial surplus is below this safe level. Therefore, for this case, we focus on solving the corresponding boundary-value problems when the initial surplus is greater than the safe level, and the value function is obtained explicitly as well. Furthermore, we investigate the influence of model ambiguity in theory. Some properties and numerical examples are also presented to show the impact of model parameters on the optimal results.

Robust optimal proportional reinsurance and investment problem for an insurer with delay and dependent risks

In this article, we consider a robust optimal proportional reinsurance and investment problem in a model with delay and dependent risks, in which the insurer’s surplus process is assumed to follow a risk model with two dependent classes of insurance business. The insurer is allowed to purchase proportional reinsurance and invest his surplus in a financial market, which contains one risk-free asset and one risky asset whose price process satisfies a jump-diffusion model. Under the consideration of the performance-related capital inflow or outflow, the insurer’s wealth process is modeled by a stochastic differential delay equation. The insurer’s aim is to develop the robust optimal reinsurance and investment strategy for the insurer by maximizing the expected exponential utility of the combination of the average historical performance and terminal wealth under the worst-case scenario of the alternative measures. By adopting the stochastic dynamic programming technique, the expressions of the robust optimal strategy and value function are explicitly obtained. Finally, we present some numerical examples to illustrate the effects of some model parameters on the optimal strategy.

Optimal discounted drawdowns in a diffusion approximation under proportional reinsurance

AbstractA diffusion approximation to a risk process under dynamic proportional reinsurance is considered. The goal is to minimise the discounted time in drawdown; that is, the time where the distance of the present surplus to the running maximum is larger than a given level $d > 0$ . We calculate the value function and determine the optimal reinsurance strategy. We conclude that the drawdown measure stabilises process paths but has a drawback as it also prevents surpassing the initial maximum. That is, the insurer is, under the optimal strategy, not interested in any more profits. We therefore suggest using optimisation criteria that do not avoid future profits.

Optimal reinsurance pricing with ambiguity aversion and relative performance concerns in the principal-agent model

This paper first studies the optimal reinsurance problems for two competitive insurers and then studies the optimal reinsurance premium pricing problem for their common reinsurer by using the dynamic programming technique. The two insurers are subject to common insurance systematic risk. Each purchases proportional or excess-of-loss reinsurance for risk control. They aim to maximize the expected utilities of their relative terminal wealth. With the insurers' optimal reinsurance strategies, the reinsurer decides the reinsurance premiums for each insurer, also aiming to maximize the expected utility of her terminal wealth. Thus, the optimal reinsurance pricing problem is formulated as a Stackelberg game between two competitive insurers and a reinsurer, where the reinsurer is the leader, and the insurers are followers. Besides, all three players take model ambiguity into account. We characterize the optimal strategies for the insurers and the reinsurer and provide some numerical examples to show the impact of competition and model ambiguity on the pricing of reinsurance contracts.

Optimal per-loss reinsurance and investment to minimize the probability of drawdown

<p style='text-indent:20px;'>In this paper, we study an optimal reinsurance-investment problem in a risk model with two dependent classes of insurance business, where the two claim number processes are correlated through a common shock component. We assume that the insurer can purchase per-loss reinsurance for each line of business and invest its surplus in a financial market consisting of a risk-free asset and a risky asset. Under the criterion of minimizing the probability of drawdown, the closed-form expressions for the optimal reinsurance-investment strategy and the corresponding value function are obtained. We show that the optimal reinsurance strategy is in the form of pure excess-of-loss reinsurance strategy under the expected value principle, and under the variance premium principle, the optimal reinsurance strategy is in the form of pure quota-share reinsurance. Furthermore, we extend our model to the case where the insurance company involves <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M2">\begin{document}$ (n\geq3) $\end{document}</tex-math></inline-formula> dependent classes of insurance business and the optimal results are derived explicitly as well.</p>

Equilibrium reinsurance-investment strategy with a common shock under two kinds of premium principles

This paper investigates the optimal mean-variance reinsurance-investment problem for an insurer with a common shock dependence under two kinds of popular premium principles: the variance premium principle and the expected value premium principle. We formulate the optimization problem within a game theoretic framework and derive the closed-form expressions of the equilibrium reinsurance-investment strategy and equilibrium value function under the two different premium principles by solving the extended Hamilton–Jacobi–Bellman system of equations. We find that under the variance premium principle, the proportional reinsurance is the optimal reinsurance strategy for the optimal reinsurance-investment problem with a common shock, while under the expected value premium principle, the excess-of-loss reinsurance is the optimal reinsurance strategy. In addition, we illustrate the equilibrium reinsurance-investment strategy by numerical examples and discuss the impacts of model parameters on the equilibrium strategy.

Robust equilibrium investment and reinsurance strategy with bounded memory and common shock dependence

In this paper, we consider the robust investment and reinsurance problem with bounded memory and risk co-shocks under a jump-diffusion risk model. The insurer is assumed to be ambiguity-averse and make the optimal decision under the mean-variance criterion. The insurance market is described by two-dimensional dependent claims while the risky asset is depicted by the jump-diffusion model. By introducing the performance in the past, we derive the wealth process depicted by a stochastic delay differential equation (SDDE). Applying the stochastic control theory under the game-theoretic framework, together with stochastic control theory with delay, the robust equilibrium investment-reinsurance strategy and the corresponding robust equilibrium value function are derived. Furthermore, some numerical examples are provided to illustrate the effect of market parameters on the optimal investment and reinsurance strategy.