Time-consistent non-zero-sum stochastic differential reinsurance and investment game under default and volatility risks

Highlights

• Study a time-consistent mean–variance reinsurance and investment game problem.

• Consider the insurance risk, volatility risk and default risk.

• Provide verification theorem at pre-default and post-default cases.

• Derive explicit solutions of equilibrium strategies.

• Present detailed sensitivity analysis to show the insurers’ economic behaviors.

Abstract

This paper investigates a non-zero-sum stochastic differential game between two mean–variance insurers. These two insurers are concerned about their terminal wealth and the relative performance compared with each other. We assume that they can buy proportional reinsurance and invest in a financial market consisting of a risk-free asset, a stock and a defaultable bond. The price process of stock is driven by the constant elasticity of variance (CEV) model and the defaultable bond recovers a proportion of value at default. So, these two insurers are faced with insurance risk, volatility risk and default risk. The non-zero-sum goal of these insurers is to maximize the mean–variance utility of a weighted value of their terminal and relative wealths. We solve the mean–variance problem in the time-consistent case and establish the extended Hamilton$-$Jacobi$-$Bellman systems for the post-default case and the pre-default case, respectively. Furthermore, we derive the closed form solutions of the Nash equilibrium reinsurance and investment strategies for these two insurers. In the end of this paper, we calibrate the parameters based on real data and several numerical examples are provided to illustrate the effects of economic parameters on the equilibrium strategies.

Introduction

Original from Browne (1995) [1], risk management for an insurer has attracted more and more attention. When selling insurance contracts and investing in the financial market, the insurer is faced with insolvency risk and financial risks. In reality, the insurer can take different ways (investment, reinsurance) to manage financial and insurance risks. As Eling et al. (2007) [2] noted, financial crises worldwide along the history prompt the development of risk regulation and risk-based capital standards are more effective for risk regulation. So, on one hand, in order to meet the regulation requirement, the insurer should take the risks into account comprehensively. On the other hand, to achieve a higher expected utility or lower the bankruptcy probability as in [1], effective financial and insurance risk models should be proposed.

Because reinsurance and investment are two efficient ways for an insurer to manage risks, there are numerous studies of optimal reinsurance and investment for insurer. In the reinsurance market, there exist different kinds of reinsurance contracts to meet the insurers’ different kinds of demand. So, reinsurance is a popular way for the insurer to manage insurance risk. The insurer can take different reinsurance policies to divide part of his insurance risks, such as proportional reinsurance in [3], [4], [5], etc., excess of loss reinsurance in [6], [7], [8] etc., and combinational reinsurance in [9], [10], etc.

However, the wealth of the insurer is also influenced by different financial risks. The stock price in [1] is characterized by a geometric Brownian motion. But empirical findings such as volatility clustering, volatility smile in the market address the stochastic volatility model for the stock. Lin and Li (2011) [11], Li et al. (2015) [12] both assume that the insurer can invest in a financial market with risky asset governed by a constant elasticity of variance model to incorporate conditional heteroscedasticity. Moreover, Heston’s volatility model which assumes a mean reverting volatility process is studied in [7], [13]. Although many scholars have recognized and analyzed the performance of an insurer faced with volatility risk, most of the above mentioned literature supposes that the bond market is risk free. However, bond market is also a very large market worldwide and default risk can largely affect the insurer’s wealth. Bo et al. (2013) [14] originally derive the optimal investment and consumption policies that maximize the infinite horizon expected discounted HARA utility of the consumption with a perpetual defaultable bond. Later, Zhao et al. (2016) [15] consider an optimal investment and reinsurance problem involving a defaultable security. They establish the extended Hamilton–Jacobi–Bellman systems of equations for the post-default and the pre-default cases and obtain the closed form solution for the insurer. Sun et al. (2017) [16] combine the ambiguity aversion and default risk for an insurer and calculate the insurance premium by variance premium principle. In [17], jump risk, ambiguity aversion and default risk are studied and optimal proportional reinsurance and investment strategies are obtained. Meanwhile, Wang et al. (2019) [18] consider a reinsurance–investment problem with delay for an insurer under the mean–variance criterion in a defaultable market and present numerical examples to show the effect of different risks on the insurer’s behavior.

Most of the works above consider the management for a single agent. However, cooperation or competition exists among different agents and we think two aspects ought to be explored further. On one hand, the insurer and reinsurer may cooperate to achieve a higher utility. Cai et al. (2013) [19] consider the interests of both insurer’s and reinsurer’s and study the joint survival and profitable probabilities of insurers and reinsurers. Zhao et al. (2017) [20] consider the time-consistent mean–variance criterion and formulate the optimal decision to maximize a weighted sum of the insurer’s and the reinsurer’s surplus processes. Later, Zhou et al. (2017) [21] and Huang et al. (2018) [22] incorporate ambiguity aversion and maximize the minimal expected utility of the weighted sum surplus process of the insurer and the reinsurer. More related works can refer to [23], [24], [25] etc. On the other hand, different insurers will also compete with each other to attract more clients. Bensoussan et al. (2014) [26] study non-zero-sum stochastic differential investment and reinsurance game between two insurance companies. Each insurer is concerned with the relative performance over his competitor and Nash equilibrium strategies are derived by dynamic programming method. Pun and Wong (2016) [27] incorporate ambiguity aversion and competition together for two competitive insurers. Recently, Deng et al. (2018) [28] consider default risk and study a non-zero-sum stochastic differential game between two insurers. They derive the pre-default and post-default Nash equilibrium strategies explicitly.

Inspired by these studies, we study the non-zero-sum stochastic differential reinsurance and investment game between two insurers. We suppose that these insurers invest in bond market and equity market simultaneously. In order to simplify the model, we assume that the financial market are the same for these two insurers: cash, defaultable bond and a stock with stochastic volatility. The stochastic volatility process follows a CEV model which captures stochastic volatility and the leverage effect as in [6], [20]. And the description of the defaultable bond is similar to Deng et al. (2018) [28], Zhang et al. (2019) [17]. The process of the defaultable bond is divided to pre-default case and post-default case. At default time, the defaultable bond only recovers a proportion of value to the investor. Besides, we assume that the accumulated insurance claims of two insurers are not independent while affected by a same compound Poisson process, which is more realist in the insurance market. Therefore, in this risk model, we include volatility risk and default risk for these insurers and also consider dependence between their claims, which provide an efficient and comprehensive basis for further risk management.

We suppose that these insurers can invest and purchase proportional reinsurance business continuously in the market. The goal of this paper is to design the optimal reinsurance and investment strategies for these two competitive insurers. The insurers are concerned with their relative performance. In [28], the insurers aim to maximize the expected utility of terminal relative wealth. However, the insurers may also study the mean–variance criterion in [29]. The profit of the wealth is judged by the mean of the terminal wealth and the risk is calculated by the variance. The agent needs to seek compromise between the profit and risk. The original mean–variance criterion searches the pre-commitment solution and many scholars have investigated the (mean–variance) efficient frontier and strategy for an insurer, see [30], [31], etc. However, the solution in the pre-commitment case is not time-consistent and many recent literatures search the time-consistent strategy defined in [32]. Zeng and Li (2011) [33] first explore this time-consistent goal for an insurer and derive the explicit solution by extended HJB equation. Later, many related works also appear, such as [12], [20] and the time-consistent strategy proves to be very efficient for an insurer. Because competition exists among these two insurers, they are concerned with a weighted value of the terminal wealth and relative wealth. In this paper, we suppose that the insurers hold the mean–variance criterion and search the time-consistent Nash equilibrium reinsurance and investment strategies. We present the extended HJB equations in pre-default and post-default cases. Then, competition exists among these two insurers and we derive the Nash equilibrium investment and reinsurance strategies explicitly. In the end of this paper, numerical results are presented to show the economic behaviors of these two insurers.

The remainder of this paper is organized as follows. The financial and insurance markets are described in Section 2. Section 3 shows the time-consistent non-zero-sum game between these insurers. The extended HJB equation is also presented. In Section 4 we derive the equilibrium strategies in post-default case and pre-default case explicitly. Numerical examples are showed in Section 5 and Section 6 is a conclusion.