Reinsurance–investment game between two mean–variance insurers under model uncertainty

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Abstract

This paper investigates a class of robust non-zero-sum reinsurance–investment stochastic differential games between two competing insurers under the time-consistent mean–variance criterion. We allow each insurer to purchase a proportional reinsurance treaty and invest his surplus into a financial market consisting of one risk-free asset and one risky asset to manage his insurance risk. The surplus processes of both insurers are governed by the classical Cramér–Lundberg model and each insurer is an ambiguity-averse insurer (AAI) who concerns about model uncertainty. The objective of each insurer is to maximize the expected terminal surplus relative to that of his competitor and minimize the variance of this relative terminal surplus under the worst-case scenario of alternative measures. Applying techniques in stochastic control theory, we obtain the extended Hamilton–Jacobi–Bellman (HJB) equations for both insurers. We establish the robust equilibrium reinsurance–investment strategies and the corresponding equilibrium value functions of both insurers by solving the extended HJB equations under both the compound Poisson risk model and its diffusion-approximated model. Finally, we conduct some numerical examples to illustrate the effects of several model parameters on the Nash equilibrium strategies.

Introduction

In practice, insurance companies can purchase reinsurance contracts to transfer parts of the underwriting risk to a reinsurer and invest in the financial markets to increase their profits. The investigation of the insurer’s optimal investment and reinsurance strategies has attracted considerable attention in the area of actuarial science. For example, Bai and Guo [1] considered the optimal investment–reinsurance problem by maximizing the expected exponential utility of the insurer’s terminal wealth under the no-shorting constraint. Shen and Zeng [2] applied the mean–variance criterion to derive the insurer’s optimal reinsurance and investment strategies. Zhang et al. [3] aimed to seek the optimal reinsurance and investment strategies by minimizing the probability of ruin, where the reinsurance premium was determined by the generalized mean–variance principle. For more literature, the readers may refer to [4], [5], [6] and [7], just to name a few.

As was shown in [8], decision makers were not only risk-averse but also ambiguity-averse. In many situations, the parameters, especially the drift parameters, are difficult to estimate with precision, and thus it is controversial to determine which model is completely true so as to represent the real-world probability and be used in the optimization problems. Therefore, it is reasonable to assume that the decision maker concerns about model misspecification. In the literature, one popular approach to describe model ambiguity was proposed in [9] which studied asset pricing problems in stochastic continuous-time settings by incorporating the investor’s consideration of model misspecification. Under their assumption, the investor regarded the specific probability measure as his reference measure and could then find robust strategies that worked over the nearby measures known as alternative measures. Since then, due to its analytical tractability, the formulation of the robust optimization procedures conducted in [9] has been adopted in portfolio selection, asset pricing and optimal reinsurance–investment problems. For example, Maenhout [10] obtained the robust optimal portfolio decision for an investor with ambiguity aversion attitudes. Pun and Wong [11] discussed a robust optimal reinsurance–investment problem for an ambiguity-averse insurer (AAI) under a general class of utility functions when the risky asset followed a multiscale stochastic volatility (SV) model. Zeng et al. [12] derived optimal time-consistent investment and proportional reinsurance strategies for an AAI under mean–variance criterion. They assumed that the surplus of the AAI followed Cramér–Lundberg model and the price of the risky asset could be characterized by a jump–diffusion process. Li et al. [13] articulated the optimal investment and excess-of-loss reinsurance problem for an AAI who was concerned about ambiguity with respect to the diffusion and jump components arising from the financial and insurance markets. Gu et al. [14] investigated a robust optimal investment and proportional reinsurance problem for an AAI who could invest his surplus into one risk-free asset, one market index and a pair of mispriced stocks.

The aforementioned literature is devoted to single-agent optimization problems. However, the real-world economy is a complex and interactive system where the financial institutions have to make optimal decisions by concerning the performance of their competitors. Hence, some scholars focus on dealing with the competition between two institutions. Some of them formulated the problem into a zero-sum stochastic differential game, for example, Browne [15] proposed a zero-sum stochastic differential portfolio game between two investors. Liu and Yiu [16] studied a zero-sum stochastic differential reinsurance and investment game for two competing insurance companies, and they imposed constant Value at Risk (VaR) constraints for the purpose of risk management. On the other hand, some works studied the relative performance concerns and formulated non-zero-sum game problems. Along this direction, Bensoussan et al. [17] applied dynamic programming techniques to consider a non-zero-sum reinsurance and investment game under the regime-switching framework. Siu et al. [18] extended the results obtained in [17] by allowing the insurers to purchase an excess-of-loss reinsurance contract and invest in the risky asset whose price was described by Heston SV model. Moreover, Siu et al. [18] studied the effects of systematic risks on the equilibrium reinsurance strategy. Subsequently, Deng et al. [19] studied a non-zero-sum stochastic differential reinsurance–investment game between two competitive constant absolute risk aversion (CARA) insurers, and their investment options included a risk-free bond, a risky asset with Heston’s SV model and a defaultable corporate zero-coupon bond. Hu and Wang [20] investigated a class of non-zero-sum reinsurance and investment games under time-consistent mean–variance criterion. Some attempts have also been made in addressing the robust game problems. For instance, Zhang and Siu [21] considered an optimal reinsurance–investment problem in the presence of model uncertainty and formulated the problem into a zero-sum stochastic differential game between the insurer and the market. Pun and Wong [22] explored the non-zero-sum stochastic differential game between two competitive AAIs who aimed to seek the robust optimal proportional reinsurance strategies by maximizing the expected utility of the terminal surplus relative to that of his competitor. Wang et al. [23] formulated a class of non-zero-sum reinsurance and investment games between two AAIs who faced default risk and explicit expressions for Nash equilibrium investment and reinsurance strategies were established. Wang and Siu [24] discussed a robust reinsurance contracting problem in the presence of model uncertainty and a risk constraint based on VaR under the principal–agent modelling framework.

To the best of our knowledge, there exists rare literature studying the robust non-zero-sum stochastic differential game between two AAIs under the mean–variance criterion. In this paper, we aim to fill this gap and investigate how ambiguity aversion and relative performance concerns impact the equilibrium reinsurance and investment strategies. More specifically, we suppose that each insurer has the choice to purchase a proportional reinsurance contract and invest in one risk-free asset and one risky asset. The surplus process of each insurer is assumed to follow the classical Cramér–Lundberg model. Inspired by [18], we assume that the insurers face both idiosyncratic and systematic jump risks. In regard to the objective function, we incorporate relative performance and model ambiguity concerns into mean–variance criterion. In the traditional mean–variance optimization problems, most of the literature obtained the pre-commitment strategies which were time-inconsistent and only optimal at the initial time. However, time-consistency of the optimal strategies is a basic requirement for a rational decision maker. On account of this opinion, in this paper we follow the approach proposed in [25] and [26] in order to obtain time-consistent reinsurance–investment strategy. In fact, this approach tackles the problem within a non-cooperative game framework, where the players are the future incarnations of the decision maker. In the contexts of insurance, this approach was applied by many researchers, see, for example, Li et al. [27], Lin and Qian [28] and Li et al. [29]. More recently, Pun [30] established a general and tractable framework for stochastic control problems when model uncertainty was incorporated with time-inconsistent preference. Therefore, the formulation in this paper induces each insurer not only to compete with himself but also with his competitor under the worst-case scenario of the alternative measures. The non-cooperative game with himself stems from the approach applied to obtain time-consistent reinsurance and investment strategies under the mean–variance criterion and the game with his competitor derives from relative performance consideration. Employing the Hamilton–Jacobi–Bellman (HJB) dynamic programming principle in stochastic optimal control theory, we obtain the robust equilibrium reinsurance and investment strategies as well as the corresponding equilibrium value functions.

Comparing with the existing literature, the main contributions of our paper are summarized as follows. First, we extend the robust optimal reinsurance–investment problem under the mean–variance criterion in [12], where only a single insurer was considered, to a continuous-time game framework by taking two insurers’ relative performance concerns into account. The key reason for considering strategic interaction between two insurers is that there always exist several competing insurers in the market in reality, and they often assess their performance against a relative benchmark of their competitors. Therefore, we derive the Nash equilibrium investment and reinsurance strategies of a non-zero-sum game in this paper. Numerical examples demonstrate that the competition makes each insurer more risk-seeking compared with the case without competition because they would increase their exposure on the risky asset and elevate their respective retention levels of the claims. Second, the effects of ambiguity aversion on the optimal reinsurance and investment strategies under a non-zero-sum stochastic differential game framework are investigated, which was not considered in [18] and [20], although these two papers formulated non-zero-sum games under expected utility maximization criterion and mean–variance criterion, respectively. Our numerical experiments show that an AAI would prefer more conservative investment and reinsurance strategies than an ambiguity-neutral insurer (ANI), which is reflected in the reduction of retention level of insurance risks and the amount invested in the risky asset.

The remainder of this paper is organized as follows. Section 2 formulates a non-zero-sum stochastic differential reinsurance and investment game between two AAIs under the mean–variance criterion. In Section 3, we derive the extended HJB equation and present the time-consistent equilibrium reinsurance and investment strategies under the classic Cramér–Lundberg model. Section 4 provides the results under the diffusion-approximated model. In Section 5, we carry out some numerical examples to illustrate the effects of some important parameters on the time-consistent equilibrium reinsurance and investment strategies. Finally, we provide some concluding remarks in Section 6.

Section snippets

Model formulation

Let (Ω,F,{Ft}t[0,T],P) be a complete filtered probability space, where the filtration {Ft}t[0,T] is right continuous and P-complete; [0,T] is a fixed time horizon for investment and reinsurance. In what follows, all stochastic processes are assumed to be adapted to {Ft}t[0,T].

We consider an insurance market consisting of two competing insurance companies, for simplicity, referred to as insurer 1 and insurer 2, whose surplus processes Sk(t),k{1,2}, are described by the classical

Nash equilibrium in compound Poisson risk model

In this section, we first present the verification theorem and then derive the robust Nash equilibrium reinsurance–investment strategy under the compound Poisson risk model. For notational convenience, we first define C1,2([0,T]×R){f(t,x)|f(t,x) is continuously differentiable for t[0,T] and twice continuously differentiable for xR}, Dp1,2([0,T]×R){f(t,x)|f(t,x)C1,2([0,T]×R) and all first order partial derivatives satisfy the polynomial growth condition on R}.

We suppress the arguments of

Nash equilibrium in diffusion approximated model

Having obtained the robust equilibrium reinsurance–investment strategies for both insurers in the case of classical compound Poisson risk process, in this section we investigate the robust reinsurance–investment game problem when the compound Poisson risk process of each insurer is approximated by a diffusion model. Specifically, according to [38], the aggregate claim process i=1Nk(t)+N(t)Zki in (2.1) could be approximated by a Brownian motion with drift, that is i=1Nk(t)+N(t)Zki(λk+λ)μktσkλ

Numerical examples

In this section, we conduct some numerical experiments to provide sensitivity analyses for the robust equilibrium reinsurance and investment strategies derived in Sections 3 Nash equilibrium in compound Poisson risk model, 4 Nash equilibrium in diffusion approximated model. The model parameters as our benchmark are shown in Table 1. The values of some parameters are borrowed from Bensoussan et al. [17], Zeng et al. [12], Siu et al. [18] and Deng et al. [19] to make our analyses reasonable.

Concluding remarks

In this paper, we study a robust non-zero-sum reinsurance–investment game between two competing AAIs who take into account model uncertainty and intend to seek robust equilibrium reinsurance and investment strategies. We formulate the competition between these two insurers by assuming that they concern about their relative performance and aim to outperform one another at terminal time. In addition, the mutual dependency between these two AAIs is described by a common Poisson process in their

Acknowledgements

We would like to thank two anonymous referees for their constructive comments and suggestions that resulted in an improved version. This work was supported in part by the National Social Science Foundation Key Program (17ZDA091), National Natural Science Foundation of China (11901201, 11771147, 11971172, 11871220), the State Key Program of National Natural Science Foundation of China (71931004), the 111 Project, China (B14019), Research Grants Council of the Hong Kong Special Administrative

References (42)

  • DengC. et al.

    Non-zero-sum stochastic differential reinsurance and investment games with default risk

    Eur. J. Oper. Res.

    (2018)
  • ZhangX. et al.

    Optimal investment and reinsurance of an insurer with model uncertainty

    Insurance Math. Econom.

    (2009)
  • WangN. et al.

    Robust non-zero-sum investment and reinsurance game with default risk

    Insurance Math. Econom.

    (2019)
  • LiD. et al.

    Time-consistent reinsurance-investment strategy for a mean–variance insurer under stochastic interest rate model and inflation risk

    Insurance Math. Econom.

    (2015)
  • LiD. et al.

    Optimality of excess-of-loss reinsurance under a mean–variance criterion

    Insurance Math. Econom.

    (2017)
  • PunC.S.

    Robust time-inconsistent stochastic control poblems

    Automatica

    (2018)
  • BrangerN. et al.

    Robust portfolio choice with uncertainty about jump and diffusion risk

    J. Bank. Financ.

    (2013)
  • LiB. et al.

    Alpha-robust mean–variance reinsurance-investment strategy

    J. Econom. Dynam. Control

    (2016)
  • SunZ. et al.

    Optimal mean–variance investment and reinsurance problem for an insurer with stochastic volatility

    Math. Methods Oper. Res.

    (2018)
  • EllsbergD.

    Risk, ambiguity and the savage axioms

    Q. J. Econ.

    (1961)
  • AndersonE.W. et al.

    A quartet of semigroups for model specification, robustness, prices of risk and model detection

    J. Eur. Econom. Assoc.

    (2003)
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