A dynamic pricing game for general insurance market

https://doi.org/10.1016/j.cam.2020.113349Get rights and content

Highlights

  • A novel dynamic pricing game model with multiple insurers is considered.

  • The risk exposure of an insurer is affected by all insurers in the market.

  • The insurers have different levels of ambiguity to the aggregate claim process.

  • Closed-form expression for premium strategies are obtained.

  • Comparative statics analysis for the model parameters is implemented.

Abstract

Insurance contracts pricing, that is determining the risk loading added to the expected loss, plays a fundamental role in insurance business. It covers the loss from adverse claim experience and generates a profit. As market competition is a key component in the pricing exercise, this paper proposes a novel dynamic pricing game model with multiple insurers who are competing with each other to sell insurance contracts by controlling their insurance premium. Different with the existing works assuming deterministic surplus/loss, we consider stochastic surplus and adopt the linear Brownian motion model, i.e., a diffusion approximation to the classical Cramér–Lundberg model, for the aggregate claim amount. The risk exposure of an insurer is assumed to be affected by all insurers in the market. By solving a system of Hamilton–Jacobi–Bellman (HJB) equations, Nash equilibrium premium strategies are explicitly obtained for the insurers who are aiming to maximize their expected terminal exponential utilities. The representation form of the equilibrium strategies relates to the so-called M-matrix, which appears in many economic models. To investigate the robustness of equilibrium pricing strategies under model uncertainty, we further extend the model by allowing insurers to perceive ambiguity towards the aggregate claim loss. Closed-form expression for the robust premium strategies are obtained and comparative statics are carried out for model parameters.

Introduction

In an insurance market, arguably the two most important decisions insurers need to make are to price insurance contracts and manage the associated risks. There is a considerable body of literature on the second topic, and numerous research studies are conducted to investigate how insurers could mitigate their risks via reinsurance, financial investment, and other methods. Nevertheless, the first topic of insurance contracts pricing, that is the calculation of insurance premium, is also of paramount importance but unfortunately has been much overlooked in the recent actuarial literature.

By the specialty of insurance contracts, their pricing approaches are a lot different from the well-developed no-arbitrage pricing theory for financial derivatives or the standard supply–demand principle in microeconomics. Insurance practice suggests the use of expectation, standard deviation, or quantiles of the underlying risk to determine a suitable premium (e.g., [1], [2]). Traditional premium principles include the net premium principle, the expected value premium principle, the variance premium principle, and the standard deviation premium principle, among others. Except the net premium principle which stipulates the insurance premium is simply the expectation of the potential claims on the policy over its duration, insurers usually add a risk loading to the expected loss so as to cover the loss from adverse claim experience and generate a profit. With the premium principle selected, the main challenge remained is to determine the value of risk loading, which shall depend on not only the risk aversion of insurers but also the competitive nature of insurance market.

In the scarce literature of general insurance pricing, Taylor [3] pioneered the work and used a simple discrete-time deterministic model to study how competition might affect an insurer’s premium strategy. In Taylor’s model, the premium was priced based on the relation between the market’s behavior and optimal response of an individual insurer. The law of demand was embedded in the modeling process to analyze the change of exposure volume through a comparison between the insurer’s and market average premiums. Over the past decade, there are a few extensions to Taylor’s work. Emms and Haberman [4] introduced an accrued premium paying scheme which assumes the continuously varying premium is paid at a rate fixed at the start of the policy. Emms et al. [5] extended Taylor’s model to the case that the market average premium is modeled by a geometric Brownian motion. Pantelous and Passalidou [6] proposed a stochastic demand function for the volume of business. The aforementioned works intend to solve the optimal pricing strategy for a single insurer by assuming the strategy adopted by this insurer does not affect the actions of other market participants who are competing to sell insurance products. Since insurance markets are often dominated by only a few major insurance companies and each of them can monitor the strategies of others, recent studies start to adopt game-theoretic approach to determine competitive premium; see, e.g., [7], [8], [9], [10], and [11].

The present paper is also devoted to studying competitive premium of general insurance pricing and our main contributions are summarized as follows.

First of all, we propose and analyze a novel dynamic pricing game model with multiple insurers. Specifically, we consider an insurance market with multiple insurers who are competing to sell insurance contracts. The aggregate claim amount process per exposure unit is modeled by a linear Brownian motion, i.e., the diffusion approximation model, which is a commonly used approximation to the classical Cramér–Lundberg model (see [12], [13], and the references therein). The risk exposure, that is the volume of insurance contracts, of an insurer is assumed to be affected by not only the premium of this insurer, but also the premium difference to that of other insurers. Therefore, a premium strategy adopted by a single insurer has influence on the surplus processes of other insurers and herself. Note that the previous mentioned works follow the development of retail pricing models in management science, but fail to consider the essential randomness of payoffs by selling insurance contracts. Unlike holding a certain good, the surplus of an insurance contract is stochastic and can take negative value. Our model thus fills in the gap with the large body of literature on insurance surplus process modeling.

Second, by solving a system of Hamilton–Jacobi–Bellman (HJB) equations, Nash equilibrium premium strategies are explicitly obtained for the insurers who are aiming to maximize their expected terminal exponential utilities. We find that the Nash equilibrium premium strategy is a kind of variance-related premium principle, that is, insurers’ risk loading factors depend on the variance of the underlying risk. It is also worth to mention that the aforementioned exiting works only involve deterministic optimal control because surplus processes are assumed to be deterministic (e.g., [7], [8], [9]). Moreover, the representation form of our equilibrium premium strategies relate to the so-called M-matrix, which appears in many economic models (see [14]). This also justifies the economic meaning of our novel pricing game model.

Third, to investigate the robustness of equilibrium pricing strategies under model uncertainty, we further extend the model by allowing insurers to perceive ambiguity towards the aggregate claim loss. Closed-form expression for the robust premium strategies are obtained and comparative statics are carried out for model parameters.

The remainder of this paper is organized as follows. Section 2 proposes the original dynamic pricing game model, presents and analyzes the Nash equilibrium premium strategies. Section 3 extends the previous section by incorporating ambiguity to the model formulation and studying the robust equilibrium premium strategies. Section 4 concludes this paper. All technical proofs, unless provided in the text, are placed in Appendix.

Section snippets

Perfect premium competition

Fix a time horizon T>0, and let (Ω,F,F={F(t)}t[0,T],P) be a filtered probability space satisfying the usual conditions of right continuity and completeness, where F is the augmented filtration generated by a standard Brownian motion {W(t)}t[0,T] and F=F(T). The expectation under P is denoted by E.

We consider a competitive insurance market with N insurers who are competing with each other by controlling their premium strategies. The aggregate claim amount process C(t)t[0,T] for per individual

Robust premium competition

In this section, we aim to extend the previous framework by allowing the N insurers to perceive ambiguity in the distribution of the aggregate claim amount process per unit C(t)t[0,T], and solve the so-called robust equilibrium premium strategies. Model uncertainty is one of the most influential extensions to the benchmark subjective expected utility framework, which has also been recently introduced to a few topics in insurance studies (e.g., [17], [18], and [19]). But to the best of our

Conclusion

In this paper, we investigate a novel dynamic pricing game model with multiple insurers who are competing with each other to sell insurance contracts by controlling their insurance premium. We consider stochastic surplus and adopt the linear Brownian motion model for the aggregate claim amount. The risk exposure of an insurer is assumed to be affected by all insurers in the market. By solving a system of Hamilton–Jacobi–Bellman (HJB) equations, Nash equilibrium premium strategies are explicitly

Acknowledgments

We are grateful for comments from the participants at the 10th Australasian Actuarial Education and Research Symposium (AAERS) in November 2019, and we would like to particularly thank Greg Taylor for helpful discussions. Danping Li gratefully acknowledges financial support from the Fundamental Research Funds for the Central Universities (No. 2019ECNU-HWFW028), the National Natural Science Foundation of China (No. 11801179, 71771220, 11971172, 12071147, 71931004), the “Chenguang Program”

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