Time-consistent investment and reinsurance strategies for mean-variance insurers with relative performance concerns under the Heston model

Highlights

• The non-zero sum stochastic differential game between two insurers is established.

• Relative performance concerns between two competitive insurers are considered.

• The volatility risk of finance market is considered.

• Explicit solutions for time-consistent Nash equilibrium strategies are obtained.

Abstract

This paper considers the optimal time-consistent investment and reinsurance strategies for two mean-variance insurers subject to the relative performance concerns. Each insurer can purchase a reinsurance protection and invest in a financial market consisted of one risk-free asset and one risky asset. We assume that the price process of risky asset is driven by the Heston model. The main objective of each insurer is to choose a investment and reinsurance strategy such that the mean and variance of his relative terminal wealth with respect to that of his competitor is maximized and minimized, simultaneously. By applying the stochastic control theory, closed-form expressions for the equilibrium investment-reinsurance strategies and corresponding value functions are derived. Finally, we provide some numerical studies and draw some economic interpretations.

Introduction

Investment and reinsurance problem has attracted much attention and became a popular research topic in insurance literature. In the early days, scholars only paid attention to investment problem, for example, Browne (1995) obtained the optimal investment strategies for an insurer who maximizes the expected utility of the terminal wealth or minimizes the ruin probability, where the surplus process of the insurer is modeled by a drifted Brownian motion. Yang and Zhang (2005) studied the optimal investment policies for an insurer who maximizes the expected exponential utility of the terminal wealth or maximizes the survival probability, where surplus process is driven by a jump-diffusion process. Elliott and Siu (2011) discussed a backward stochastic differential equation approach to a risk-based, optimal investment problem of an insurer who minimize the risk described by a convex risk measure of his/her terminal wealth, where the insurer’s risk process is modeled by a diffusion approximation to a compound Poisson risk process. Lin et al. (2012) investigated an optimal portfolio selection problem of an insurer who faces model uncertainty in a jump-diffusion risk model using a game theoretic approach. Liu et al. (2014) investigated an optimal investment problem of an insurance company in the presence of risk constraint and regime-switching using a game theoretic approach. In this topic, the investment problem for an insurer can be seen as an asset allocation problem. But the asset allocation problem for an insurer is different from the classic asset allocation problem, the key difference lies in the presence of insurance liabilities, which are mainly due to insurance claims (Zheng et al., 2016).

Due to the fact that reinsurance is an effective way to shift risk in the insurance business, optimal reinsurance problems for insurers with various stochastic investment opportunities have drawn great attention in recent years. Promislow and Young (2005) obtained the optimal reinsurance and investment strategies for an insurer to minimize the ruin probability. Bai and Guo (2008) studied the optimal investment and reinsurance problem with multiple risky assets, who aimed to maximizes the expected exponential utility of the terminal wealth or minimizes the ruin probability. Zhang and Siu (2009), Yi et al. (2015), Gu et al. (2018), and Li et al. (2018) investigated the robust investment-reinsurance strategies for insurers in different situations.

Recently, many scholars consider the optimal investment and reinsurance strategies for insurers under the mean-variance criterion proposed by Markowitz (1952). Delong and Gerrard (2007) considered two optimal investment problems for an insurer: one is the classical mean-variance portfolio selection and the other is the mean-variance terminal objective involving a running cost penalizing deviation of the insurer’s wealth from a specified profit-solvency target. They assume that the claim process is a compound Cox process with the intensity described by a drifted Brownian motion and the insurer invests in a financial market consisting of a risk-free asset and a risky asset whose price is driven by a Lévy process. Bai and Zhang (2008) studied the optimal investment-reinsurance policies for an insurer under the mean-variance criterion by the linear quadratic method and the dual method, where they assume that the surplus of the insurer is described by a Cramer–Lundberg model and a diffusion approximation model respectively. Zeng et al. (2010) assumed that the surplus of an insurer is modeled by a jump-diffusion process, and derived the optimal investment policies explicitly under the benchmark and mean-variance criteria by the stochastic maximum principle. More related studies include those of Zeng and Li (2011), Bi et al. (2016) and Li et al. (2017), etc.

In the above-mentioned literatures, they only focus on single-agent optimization problems, i.e., they usually take the investment and reinsurance problem as an optimal control problem, thus, they do not consider the strategic interaction among insurers. However, in a competitive economy, firms tend to compare themselves with one another, and relative performance concerns thus play a key role in decision-making (Garcia, Strobl, 2010, Basak, Makarov, 2014). With considering the relative performance concerns, Bensoussan et al. (2014) formulated a nonzero-sum stochastic differential investment and reinsurance game between two insurance companies whose surplus processes were modulated by continuous-time Markov chains. Meng et al. (2015) investigated an optimal reinsurance problem when the two insurers surpluses are subject to quadratic risk controls. Pun and Wong (2016) considered a reinsurance game problem for two ambiguity-averse insurers and obtain equilibrium under a worst-case scenario framework for the exponential utility functions. More studies on reinsurance and/or investment strategies under relative performance concerns can be found in (Pun et al., 2016), (Kwok et al., 2016), (Yan et al., 2017), (Siu et al., 2017), (Hu and Wang, 2018), (Deng et al., 2018) and (Chen et al., 2018).

In additional, it is well-known that the prices of risky assets may have different features in the real world and numerous studies have shown that the volatilities of risky assets prices are not deterministic. It is clear that a model with stochastic volatility will be more practical. Heston (1993) used a Cox-Ingersoll–Ross process to characterize the volatility of the risky asset. Since then, the Heston model has been widely used in the field of insurance. Li et al. (2012) applied the Heston model to study the reinsurance and investment problem under the meanvariance criterion. Zhao et al. (2013) discussed the optimal excess-of-loss reinsurance and investment problem for an insurer with jump-diffusion risk process under the Heston model. Yi et al. (2013) investigated the robust optimal reinsurance and investment strategies for an insurer with individual preferences when facing model uncertainty. Actually, the Heston model is classical and very popular for option pricing, and has been recognized as an important feature for asset price models. Meanwhile, the Heston model can be seen as an explanation of many well-known empirical findings, such as the volatility smile, the volatility clustering, and the heavy-tailed nature of return distributions. Therefore, we consider the optimal reinsurance and investment problem for two mean-variance insurers with relative performance concerns under the Heston model.

Moreover, it is apparent to all that the mean-variance criterion lacks the iterated-expectation property, which results in that continuous-time mean-variance problems are time-inconsistent. While time-consistence is an essential requirement in decision making, and the question of optimal investment and reinsurance for two mean-variance insurers subject to the relative performance concerns under Heston model has not been investigated, thus, time-consistent investment and reinsurance strategies for mean-variance insurers subject to the relative performance concerns under Heston model worthy to be further explored.

In this paper, we extend the work of Li et al. (2012) from single insurer to two competitive insurers. Each insurer aims to maximize his mean-variance utility of his terminal absolute and relative wealth by purchasing reinsurance and investing in a financial market. The financial market consists of one risk-free asset and one risky asset whose price follows the Heston model. By applying stochastic control theory, we establish the extended Hamilton–Jacobi–Bellman (HJB) equation, provide the corresponding verification theorem. Closed-form expressions for the equilibrium investment-reinsurance strategies and corresponding value functions are obtained via complicated analysis. Furthermore, we analyze the properties of the optimal strategy and present a numerical simulation to illustrate our results.

The rest of the paper is organized as follows. Section 2 introduces the basic model setup of the two competitive insurers. In Section 3, we derive the extended HJB equation for the case of mean-variance utility function; then, the closed-form expressions for equilibrium time-consistent strategies and the corresponding value functions are obtained. Section 4 provides detailed numerical studies to discuss the impact of model parameters on the equilibrium strategies. Section 5 concludes the paper with some suggestions for future research.