Reinsurance Contracts Research Articles

A two-layer stochastic game approach to reinsurance contracting and competition

We propose a two-layer stochastic game model to study reinsurance contracting and competition in a market with one insurer and two competing reinsurers. The insurer negotiates with both reinsurers simultaneously for proportional reinsurance contracts that are priced using the variance premium principle. The reinsurance contracting between the insurer and each reinsurer is modeled as a Stackelberg game. The two reinsurers compete for business from the insurer and optimize the so-called relative performance, instead of their own surplus, and their competition is settled by a noncooperative Nash game. We obtain a sufficient and necessary condition, related to the competition degrees of the two reinsurers, for the existence of an equilibrium. We show that the equilibrium, if exists, is unique, and the equilibrium strategy of each player is constant, fully characterized in semiclosed form. Furthermore, we obtain interesting sensitivity results for the equilibrium strategies through both analytical and numerical studies.

Static Risk Measures in a Frequency-Severity Framework with Systematic Risk: Application in Reinsurance

This article presents the concept of static risk measures as an approach to assessing risk by focusing on loss severity within a frequency-severity framework that encompasses systematic and common shocks. This article will discuss a set of important properties of static risk measures and compare them with the ruin-based risk measures. It will also give a robust representation of static risk measures and discuss the implications of such representation to designing optimal reinsurance contracts. By introducing a flexible framework, the model accommodates additional elements such as systematic and common shocks, enhancing its applicability in the field of reinsurance.

Reinsurance contracts under Stackelberg game and market equilibrium

In this paper, we investigate the robust reinsurance contracts under Stackelberg game and market equilibrium. Each reinsurance contract contains two decision makers, one insurer and one reinsurer. The insurer is ambiguity-neutral and adopts a loss-dependent premium principle to collect premium. The reinsurer is ambiguity-averse and is a Bayesian learner. By using the stochastic dynamic programming method and the inverse method, the analytical expressions of the optimal risk allocation proportion and reinsurance price are derived for the two types of reinsurance contracts. We shows that the loss-dependent premium principle has the penalty-reward nature. Both the reinsurance price and demand decrease as the extrapolative intensity increases. Learning has important significance and always puts down the reinsurance price and puts up the reinsurance demand. On the contrary, the reinsurer’s ambiguity aversion raises the reinsurance price and decreases the reinsurance demand. Finally, numerical analysis reveals that the reinsurance price is greater under the Stackelberg game than that under the market equilibrium.

Irreversible reinsurance: minimization of capital injections in presence of a fixed cost

AbstractWe propose a model in which, in exchange to the payment of a fixed transaction cost, an insurance company can choose the retention level as well as the time at which subscribing a perpetual reinsurance contract. The surplus process of the insurance company evolves according to the diffusive approximation of the Cramér-Lundberg model, claims arrive at a fixed constant rate, and the distribution of their sizes is general. Furthermore, we do not specify any particular functional form of the retention level. The aim of the company is to take actions in order to minimize the sum of the expected value of the total discounted flow of capital injections needed to avoid bankruptcy and of the fixed activation cost of the reinsurance contract. We provide an explicit solution to this problem, which involves the resolution of a static nonlinear optimization problem and of an optimal stopping problem for a reflected diffusion. We then illustrate the theoretical results in the case of proportional and excess-of-loss reinsurance, by providing a numerical study of the dependency of the optimal solution with respect to the model’s parameters.

재보험계약원칙(PRICL) version 1.0에 대한 약간의 검토

In 2010, the European Principle of Insurance Contracts (PEICL) stipulated that PEICL did not apply to reinsurance contracts. The reason was that the insurance contract law of most EU member states completely excluded the application of the insurance contract law to reinsurance contracts. Accordingly, the project group led by the private sector began to write the Reinsurance Contract Law Principles (PRICL). The goal of this work is to provide a unified soft law for contract law to the reinsurance market. This project has a similar purpose to the restatement promoted by the American Legal Association (ALI) and has a similar writing format. ALI's restatement is intended to enhance the clarity and simplification of the law, accommodate social needs, and encourage the performance of academic and scientific legal work useful in trials. Project Group published its achievements so far in 2019. It was named PRICL version 1.0. PRICL Project Group planned to complete the work until the end of 2023, but failed to complete it due to COVID-19 pandemic and other reasons. Korea occupies a significant portion of the global insurance(and reinsurance)market. Nevertheless, the author regret the lack of research on PRICL in Korean academia. This paper briefly introduces the basic matters of PRICL's preparation process and version 1.0. As well as the provisions to be added in the future, a detailed review of the entire article is required.

The Time-Consistent Optimal Reinsurance Strategy of Insurance Group under the CEV Model

The article introduces proportional reinsurance contracts under the mean-variance criterion, studying the time-consistence investment portfolio problem considering the interests of both insurance companies and reinsurance companies. The insurance claims process follows a jump-diffusion model, assuming that the risk asset prices of insurance companies and reinsurance companies follow CEV models different from each other. In the framework of game theory, the time-consistent equilibrium reinsurance strategy is obtained by solving the extended HJB equation analytically. Finally, numerical examples are used to illustrate the impact of model parameters on equilibrium strategies and provide economic explanations. The results indicate that the decision weights of insurance companies and reinsurance companies do have a significant impact on both the reinsurance ratio and the equilibrium reinsurance strategy.

Blockchain adoption and optimal reinsurance design

We study blockchain adoption in insurance–reinsurance markets. We consider operational costs related to claim verification and record-keeping. Traditionally, the majority of these costs scale linearly with the volume of claims. Instead, with a consortium blockchain these costs, per firm, become independent of claim volume and decrease with the adoption rate since they are distributed. In a consortium of insurance firms, we quantify how the equilibrium adoption decisions depend on the reinsurance contract characteristics, the risk aversion of insurance companies, the distributions of their potential losses and the blockchain cost structure. Under the optimal contract, the reinsurance firm internalizes the benefits of adoption on other insurance firms, thereby acting as a central planner. We then characterize the adoption gap between decentralized (Nash) and centralized blockchain consortia.

Analisis Perhitungan Retensi Optimal Reasuransi Stop Loss dengan Metode Value at Risk (VaR)

Insurance companies do not cover the entire risk of policyholders. The risk is generally transferred in part to the reinsurance company. Stop loss reinsurance is a form of reinsurance contract where there is a limit on the risk value that can be borne by the insurance company. This value is the retention value or retention limit in the reinsurance contract, which is the maximum risk or insurance value that can be borne by the insurance company. Determining the right retention value is very important. Optimizing the Value at Risk risk measure is one approach in calculating optimal retention. Optimal retention criteria were identified in this research so that optimal retention values ​​could be calculated. This research uses a large claim data sample with the Weibull distribution. Determining optimal retention depends on the distribution of claim sizes and loading factors (additional factors in the insurance policy). With a 95% confidence level, for loading factors of 10%, 15%, and 20%, the estimated optimal retention values ​​are $1,080.56, $1,393.54, and $1,567.20. This means that the risk transferred to the reinsurer is the remaining claim amount, if the claim size exceeds the optimal retention value.

Bowley solution under the reinsurer's default risk

In this paper, we examine how a monopolistic reinsurer designs a Bowley reinsurance contract, under the assumption that the reinsurer will default on payment if the compensated loss exceeds the sum of the initial capital and the premium charged from the contract. The problem is divided into two subproblems faced by the insurer and the reinsurer in turn. The optimal reinsurance contract is analyzed when both the insurer and the reinsurer minimize their retained risks, as quantified by the VaR measure, and the optimal ceded loss function and the optimal pricing function are provided. Explicit expressions are then derived when the reinsurer adopts either VaR- or TVaR-based regulation capital and charges premiums by the expected-value premium principle. Numerical examples using exponential and Pareto distributions are provided to illustrate the sensitivity effect generated by the confidence levels of the VaR for both parties, as well as those for the initial capitals on the set of Bowley reinsurance contracts.

Međunarodni institut za unifikaciju privatnog prava i ugovorno pravo reosiguranja

Within the International Institute for the Unification of Private Law (UNIDROIT), the idea of working on an instrument of reinsurance contract law existed since the 30’s in the twentieth century. However, the final realization of this project happened only in 2019 with the publication of the Principles of Reinsurance Contract Law and in cooperation with the so-called PRICL Project Group. This harmonization endeavor is special in many ways, considering that in addition to the application of a specific bottom-up method of law creation, there was also an innovative approach in the use of a successful soft law regulation - the UNIDROIT Principles for International Commercial Contracts as lex generalis in relation to the new adopted regulation for the field of reinsurance. The author analyzes the relationship between the two mentioned acts in general, but also regarding the crisis caused by the COVID-19 virus pandemic. The paper also analyzes the contribution of the Principles of Reinsurance Contract Law to legal certainty and the characteristic role that the International Institute for the Unification of Private Law plays in the process of legal harmonization, and more specifically in the process of reinsurance contract law harmonization.

FINANCIAL STABILITY OF THE INSURANCE COMPANY AND KEY CONDITIONS FOR ITS PROVIDING

In the article, we considered the evaluation of the financial reliability of insurance organiza­tions. The insurer’s financial reliability usually means the insurer’s ability to fulfill insur­ance obli­gations under insurance and reinsurance contracts in the event of adverse factors. We investigated that the financial stability of an insurance organization is a complex indicator that depends on both internal and external factors. We studied the main classifications of factors that affect the level of financial stability of an insurance organization: according to the possibility of management and depending on direct and indirect influence. We have analyzed the main ap­proaches to assessing the financial reliability of an insurance organization. We proposed ap­proaches using single indicators (liquidity indicator, solvency indicators, reinsurance indicators, etc.), systems of relative indicators and integral indicators. We identified the advantages and disad­vantages of such approaches, ana­lyzed the advantages and disadvantages of early warning tests for assessing the financial reliability of an insurance organization, and showed that the advantage of using integral indicators is the ease of application and the possibility of taking into account various factors. However, when using inte­gral indicators, certain difficulties arise, which are associated with the fact that, as a rule, partial indicators that have different dimensions are combined into an integral indicator, therefore, for their combination or convolution, each of the partial indicators should be transformed into a di­mensionless quantity using the appropriate scale. In order to build an integral indicator for as­sessing the financial reliability of an insurance organization, it is recom­mended to use the Harring­ton method and take into account the factors that characterize the ac­tivity of an insurance organi­zation: internal: (level of payments; provision of insurance reserves; reinsurer participation; sol­vency ratio; current liquidity ratio; reinsurance ratio) and external fac­tors that take into account the general economic situation in the country and the efficiency of the functioning of the state’s economy (inflation rate; GDP growth rates).

Robust reinsurance contract and investment with delay under mean-variance framework

We consider a (non) zero-sum differential game between two insurers and determine a robust reinsurance contract from joint interests of the insurers and the reinsurer under the framework of Stackelberg differential game, that is, a hybrid stochastic differential game framework is introduced in this article. We assume that each party can invest in a financial market consisting of a risk-free asset and a risky asset whose price evolution follows a stochastic volatility model, and the reinsurer charges the premium according to both the expected value premium principle and the variance premium principle. Since the performance-related capital inflow or outflow feature is introduced, the insurers and reinsurer all aim to maximize their own time-inconsistent mean-variance criterion with delay. Using the dynamic programming approach, we derive the equilibrium investment strategy, the robust equilibrium reinsurance contract, and the corresponding equilibrium value functions under two premium principles with certain conditions, respectively. Some interesting properties of the equilibrium investment strategy are proved explicitly, and several numerical examples and sensitivity analysis are presented to demonstrate the effects of model parameters on the equilibrium reinsurance contract.

Quantum Computing in Insurance Capital Modelling under Reinsurance Contracts

In this study, we design an algorithm to work on gate-based quantum computers. Based on the algorithm, we construct a quantum circuit that represents the surplus process of a cedant under a reinsurance agreement. This circuit takes into account a variety of factors: initial reserve, insurance premium, reinsurance premium, and specific amounts related to claims, retention, and deductibles for two different non-proportional reinsurance contracts. Additionally, we demonstrate how to perturb the actuarial stochastic process using Hadamard gates to account for unpredictable damage. We conclude by presenting graphs and numerical results to validate our capital modelling approach.

Risk sharing in equity-linked insurance products: Stackelberg equilibrium between an insurer and a reinsurer

AbstractWe study the optimal investment-reinsurance problem in the context of equity-linked insurance products. Such products often have a capital guarantee, which can motivate insurers to purchase reinsurance. Since a reinsurance contract implies an interaction between the insurer and the reinsurer, we model the optimization problem as a Stackelberg game. The reinsurer is the leader in the game and maximizes its expected utility by selecting its optimal investment strategy and a safety loading in the reinsurance contract it offers to the insurer. The reinsurer can assess how the insurer will rationally react on each action of the reinsurer. The insurance company is the follower and maximizes its expected utility by choosing its investment strategy and the amount of reinsurance the company purchases at the price offered by the reinsurer. In this game, we derive the Stackelberg equilibrium for general utility functions. For power utility functions, we calculate the equilibrium explicitly and find that the reinsurer selects the largest reinsurance premium such that the insurer may still buy the maximal amount of reinsurance. Since in the equilibrium the insurer is indifferent in the amount of reinsurance, in practice, the reinsurer should consider charging a smaller reinsurance premium than the equilibrium one. Therefore, we propose several criteria for choosing such a discount rate and investigate its wealth-equivalent impact on the expected utility of each party.

Optimal risk management with reinsurance and its counterparty risk hedging

In this paper, we revisit the study of an optimal risk management strategy for an insurer who wants to maximize the expected utility by purchasing reinsurance and managing reinsurance counterparty risk with a default-free hedging instrument, where the reinsurance premium is calculated by the expected value principle and the price of the hedging instrument equals the expected payoff plus a proportional loading. Different to previous studies, we exclude ex post moral hazard by imposing the no-sabotage condition on reinsurance contracts and derive the optimal strategy analytically. We find that the stop-loss reinsurance is always optimal, but the form of the optimal hedging payoff depends on the cost difference between reinsurance and hedging instrument. We further show that full risk transfer is optimal if and only if both reinsurance pricing and the hedging price are fair. Finally, numerical analyses are conducted to illustrate the effects of some interesting factors on the optimal risk management strategy.

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 contract in a Stackelberg game framework: a view of social planner

In this paper, we consider an optimal reinsurance contract under a mean-variance criterion in a Stackelberg game theoretical framework. The reinsurer is the leader of the game and decides on an optimal reinsurance premium to charge, while the insurer is the follower of the game and chooses an optimal per-loss reinsurance to purchase. The objective of the insurer is to maximize a given mean-variance criterion, while the reinsurer adopts the role of social planner balancing its own interests with those of the insurer. That is, we assume that the reinsurer determines the reinsurance premium by maximizing a weighted sum of the insurer's and reinsurer's mean-variance criteria. Under the general mean-variance premium principle, we derive the optimal reinsurance contract by solving the extended Hamilton–Jacobi–Bellman (HJB) systems. Moreover, we provide an intuitive way to set the weight of each party in the reinsurer's objective. Finally, we consider some special cases to illustrate our main results.

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.

An insurer's optimal strategy towards a new independent business

In this paper, we investigate the optimal decision making of an insurer towards a new insurable business, whose risk is independent of the existing risk faced by the insurer. We assume that the insurer, with the objective of maximizing the expected utility of its final wealth, together with the solvency constraint and the availability of reinsurance as a risk transfer mechanism, is deciding if it is viable to underwrite a new insurance business risk. If this new business is underwritten, it is shown that a stop-loss reinsurance contract is optimal when the solvency risk is quantified by the conditional value at risk. If the regulatory regime changes to the value at risk, the optimal reinsurance form becomes more complicated; it can be either stop-loss or two-layer under the assumption that the new risk has a strictly decreasing probability density function. Numerical examples are provided to illuminate the insurer's decision making and the optimal form of the reinsurance strategy.

An efficient algorithm for pricing reinsurance contract under the regime-switching model

This paper suggests a new and efficient variance reduction technique based on the Monte-Carlo simulation method for pricing the reinsurance contract in a regime-switching environment. The proposed approach extends the standard antithetic variables method to K>2 antithetic variables per replication of the Monte-Carlo simulation. Since the model under the regime-switching framework describes a market with arbitrage opportunities, a risk-neutral measure by Esscher transform is provided. Under this probability measure, the proposed estimator is unbiased for pricing the reinsurance contract and increasing K which leads to more variance reduction. Finally, the success of the introduced method is confirmed numerically, and the efficacy of contract parameters is evaluated on its price.