Optimal Reinsurance and Derivative-Based Investment Decisions for Insurers with Mean-Variance Preference

Abstract

In our study, we investigate reinsurance issues and optimal investment related to derivatives trading for a mean-variance insurer, employing game theory. Our primary objective is to identify strategies that are time-consistent. In particular, the insurer has the flexibility to purchase insurance in proportion to its needs, explore new business, and engage in capital market investments. This is under the assumption that insurance companies’surplus capital adheres to the classical Cramér-Lundberg model. The capital market is made up of risk-free bonds, equities, and derivatives, with pricing dependent on the underlying stock’s basic price and volatility. To obtain the most profitable expressions and functions for the associated investment strategies and time guarantees, we solve a system of expanded Hamilton–Jacobi–Bellman equations. In addition, we delve into scenarios involving optimal investment and reinsurance issues with no derivatives trading. In the end, we present a few numerical instances to display our findings, demonstrating that the efficient frontier in the case of derivative trading surpasses that in scenarios where derivative trading is absent.

1. Introduction

In the realm of insurance and risk management, a fundamental problem revolves around the optimization of investment and reinsurance strategies. This task entails figuring out the most effective investment strategy and reinsurance policy for insurers, with the goal of enhancing the projected utility or reducing the likelihood of financial catastrophe. In recent years, this issue has garnered significant attention from researchers, resulting in a substantial body of literature on the topic. These studies explore various approaches to optimize the allocation of capital and manage risk, considering factors such as stochastic volatility, jump risks, and market dependencies. The integration of derivative-based investments into these strategies has been a particular focus, with research consistently demonstrating their potential to significantly enhance risk management and financial outcomes for insurance companies. Consequently, insurers are increasingly looking to leverage investment opportunities in the derivatives market to bolster their overall risk management frameworks. Some of the notable contributions in this field include Browne’s work on optimum policies for investing for a company subject to random risk see [1]. For example, Hipp and Plum’s study on the best investments for insurers, see [2]; Schmidli’s investigation into the best proportional reinsurance policies in an evolving environment, see [3]; and Luo, Taksar, and Tsoi’s study on reinsurance and investment are some of the notable articles in this field, as well as the study by Luo, Taksar, and Tsoi on reinsuring and investing in large portfolios of insurance companies, see [4]. Other significant works include those by Yang and Zhang on the best investments for insurers with jump-diffusion risk processes, see [5]; Wang on the best investments for insurers with exponential utility preferences, see [6]; Bai and Guo on the best proportional reinsurance as well as investments with multiple-risk assets and no short-selling constraints, see [7]; and Xu, Wang, and Yao on the best maximization of expected terminal utility, see [8]. The wide-ranging body of research in this field highlights its significance and the ongoing efforts to tackle the complex challenges of reinsurance strategies and optimal investment in the realm of insurance and risk management.

The works discussed above concentrate on either maximizing the expected wealth or reducing the likelihood of financial devastation. Within the domain of optimum reinsurance and investment issues, the mean-variance criterion emerges as a frequently utilized framework. As an example, ref. [9] employed this framework while taking into account an ideal proportional reinsurance issue for a complex Poisson risk model. Likewise, following this framework, ref. [10] deduced the viscosity methods for a reinsurance investment issue. It is worth noting that dynamic mean-variance optimization presents a significant challenge due to its inherent time inconsistency. These two works, however, merely disregard the time-inconsistency problem, while instead focusing on the related prior commitment issue. The quest for strategies that exhibit time consistency, particularly in the context of reinsurance and investment equilibrium, holds paramount importance. This is because the consistency of optimal strategies over time is an essential prerequisite for a well-informed decision-making process. In response to this need, refs. [11,12] applied a game theoretic approach to examine the optimal investment and reinsurance issues. More papers on mean-variance reinsurance and investment issues in various contexts include [13,14,15,16,17,18,19,20,21] and many others.

Despite the extensive examination of the optimal investment and reinsurance issues in various contexts, two aspects warrant further investigation, building upon the aforementioned literature. On one hand, the inclusion of derivatives as options for investment remains relatively uncommon in studies addressing optimal investment and reinsurance concerns. However, due to their success in mitigating financial risk, investments in derivatives have also drawn considerable attention as viable investment instruments. Financial institutions are heavily utilizing numerous financial instruments built on derivatives, such as options, swaps, credit derivatives, forwards, and futures exchanges, to fit within their comprehensive risk management strategies. For instance, ref. [22] applied derivatives to a portfolio-choosing issue and argued that derivatives serve as a crucial tool for improving investor prosperity. Ref. [23] delved into the dynamic allocation and consumption of assets problem involving derivative securities with an iterative utility function approach. Ref. [24] analyzed the most suitable investing approach to align with the preferences of risk-averse investors who have connections to both stock and derivative platforms. Ref. [25] applied the mean-variance criterion while considering derivative-based optimization of investment plans to solve an asset–liability management issue, accounting for stochastic volatility. Ref. [26] developed a derivative-based optimizing strategy for investment to cater to ambiguity-averse participants in a pension plan, who faced hazards associated with time-varying earnings and long-term economic unpredictability, as well as market return volatility.

In a related study, ref. [27] investigated the most efficient methods for a Constant Absolute Risk Aversion (CARA) insurer to control the risks associated with its company. They achieved this by incorporating equity derivatives trading in addition to equity investment and proportional reinsurance. Ref. [28] examined the optimal strategies for derivative-based investment and proportional reinsurance in the presence of stochastic volatility and jump risks. Their findings demonstrate that derivative trading consistently yields a significant positive value, particularly when compared with the magnitude of positions held in the financial market. Additionally, assuming that the risks in the financial market are independent from those in the insurance market, the optimal reinsurance strategy is found to be independent of the investment strategy.In this study, a general investment opportunity set that includes an asset without risk (i.e., a bond), an asset that is risky (i.e., a stock), and a form of derivative that has the risky asset as its fundamental component is examined.

On the other hand, the decision-making process regarding portfolio management, specifically concerning the use of derivatives, has seen increased interest over the past decade. While traditional approaches have primarily focused on direct investment and proportional reinsurance, the integration of derivatives offers a sophisticated means to hedge against market volatility and enhance returns. This growing trend underscores the need for a deeper exploration into how these instruments can be optimally employed alongside conventional investment and reinsurance strategies.

To address these gaps, we created an expanded system of Hamilton–Jacobi–Bellman (HJB) equations and acquired the explicit demonstration for a well-balanced strategy of investment and reinsurance for the insurer by employing a method akin to that utilized by [12,19]. In addition, we included numeric instances of sensitivity analyses to provide concrete examples that enhance the understanding of our findings. The study is a contribution to the evolving system of knowledge regarding risk management strategies and underscores the value of incorporating derivatives in the pursuit of efficient risk control.

The following is a summary of our paper’s key contributions. Firstly, a novel optimal reinsurance investment model containing derivatives trading according to the mean-variance criterion is developed. This is a model that includes the equilibrium value expression, which corresponds to the explicit derivation of the equilibrium strategy. Our research builds upon and extends the findings of prior studies in this field. Secondly, we investigate how derivatives trading impacts optimum reinsurance and investment strategies, an aspect that prior works by [12,19] did not explore. Although these two papers apply the mean-variance criterion and separately formulated optimal reinsurance-investment models, they did not consider derivatives trading. Our experiments provide compelling evidence that the inclusion of derivatives trading results in a more effective frontier compared with scenarios without derivatives trading.

The remainer of the paper is structured as follows: Section 2 presents the basic model and underlying hypotheses. This sets the foundation for the subsequent analysis. The formulation of the time-inconsistent issue is elaborated on in Section 3, presenting a generic framework. Additionally, it introduces the verification theorem associated with this framework. Section 4 is dedicated to solving the optimization issue, deriving explicit strategies for their investment and reinsurance within the consistent time using the developed model. Section 5 explores solutions and strategies that do not involve derivatives trading, offering insights into alternative approaches. Section 6 contains a few numeric instances to help provide concrete examples that elucidate and reinforce the research findings. Section 7 outlines recommendations and directions for future research, indicating potential areas for further exploration and study.

2. Model Setup

In the model setup, $(\Omega ,\mathbb{F}{\left\{{\mathcal{F}}_t\right\}}_{t\in [0,T]},\mathbb{P})$ is a given filtered complete probability space, and meets the general situation, where $T>0$ represents a finite constant planning horizon; ${\mathcal{F}}_t$ embodies the information obtainable up to time t; and $\mathbb{P}$ serves as the reference measure. It is important to note that every resolution attained at time t is contingent upon the information in $mathcal{F}_t$, and all random procedures below should be clearly defined and tailored for this probability space. In addition, our model operates under the assumption of a market for financial assets without trading taxes or costs, where transactions persist continuously, and short sales are allowed.

2.1. Surplus Process

Assuming that the surplus process of the insurer conforms to classical Cramér–Lundberg model, when there is neither reinsurance nor investment involved, the surplus process can be described as follows:

$$R\left(t\right)=x_0+ct-\sum _{i=1}^{N\left(t\right)}{Y}_i,$$

(1)

In this context, $x_0\ge 0$ represents the the insurer’s initial surplus, and the premium rate is represented by c. The term ${\sum }_{i=1}^{N\left(t\right)}{Y}_i$ corresponds to a compound Poisson process, which embodies the accumulated claims incurred by the insurer to time t. Here, ${\left\{N\left(t\right)\right\}}_{t\in [0,T]}$ represents a homogeneous Poisson process having an intensity $\lambda >0$, and the values of individual claims, denoted as $Y_1,Y_2,\cdots$, are not dependent on $N\left(t\right)$ and are regarded as independently and equally distributed (i.i.d.) positive random variables. These random variables possess a finite first moment, denoted as $\mathbb{E}\left[Y_i\right]=\mu$, and a second moment, denoted as $\mathbb{E}\left[Y_i^2\right]={\sigma }_y^2$. In addition, it is assumed that the insurer calculates the premium rate c based on the expected value principle, i.e., $c=(1+\alpha )\lambda \mu$, with $\alpha >0$ representing the insurer’s safety loading.

Furthermore, in our model, we make the assumption that an insurance company possesses the capability to mitigate its insurance risk. This risk mitigation can take various forms, including expanding its portfolio by acquiring new business or engaging in proportional reinsurance arrangements. This perspective is aligned with the insights provided by Bauerle et al, as documented in their research see [9]. To be more specific, at any given time point, denoted as each $t\in [0,T]$, the level of proportional reinsurance or the acquisition of a new business can be expressed through the risk exposure value, denoted as $q\left(t\right)\in [0,+\infty )$. When $q\left(t\right)\in [0,1]$, it signifies the adoption of proportional reinsurance coverage. Under this arrangement, the insurer transfers a fraction of the premiums to the reinsurer atthe rate of $(1+\beta )\left(1-q\left(t\right)\right)\lambda \mu$, Where $\beta$ stands for the reinsurer’s safety loading with $\beta >\alpha$. The insurer covers $100q\left(t\right)\%$ of any claim that occurs during time t, while the reinsurer takes responsibility for the remaining $100\left(1-q\left(t\right)\right)\%$. On the other hand, $q\left(t\right)\in [1,+\infty )$ corresponds to the pursuit of new business opportunities. In this scenario, ${\left\{q\left(t\right)\right\}}_{t\in [0,T]}$ can be referred to as a reinsurance strategy for simplicity. Through the implementation of such a reinsurance strategy, the surplus process of the insurer is governed in the following way:

$$\mathrm{d}{R}^q\left(t\right)=\lambda \mu \left[\vartheta +(1+\beta )q\left(t\right)\right]\mathrm{d}t-q\left(t\right)\mathrm{d}\sum _{i=1}^{N\left(t\right)}{Y}_i,$$

(2)

with $\vartheta \triangleq \alpha -\beta$.

2.2. Financial Market

Furthermore, the insurer may make financial investments to obtain reinsurance protection. In particular, the insurer holds assets such as risk-free bonds, shares of stock, and derivatives. The behavior of the risk-free bond over time can be expressed in terms of its dynamic:

$$\mathrm{d}{S}_0\left(t\right)=r{S}_0\left(t\right)\mathrm{d}t,S_0\left(0\right)=1,$$

(3)

and $r>0$ represents the risk-free interest rate. The stock is priced below

$$\mathrm{d}S\left(t\right)=S\left(t\right)\left[\left(\sqrt{V\left(t\right)}\mathrm{d}{B}_1\left(t\right)\right.+\eta V\left(t\right)\right.+r\left)\mathrm{d}t\right],S\left(0\right)=s_0,$$

(4)

while the value of the stock $V\left(t\right)$ is controlled by

$$\mathrm{d}V\left(t\right)=\kappa \left[\delta -V\left(t\right)\right]\mathrm{d}t+\sigma \sqrt{V\left(t\right)}\left[\rho \mathrm{d}{B}_1\left(t\right)+\sqrt{1-{\rho }^2}\mathrm{d}{B}_2\left(t\right)\right],V\left(0\right)=v_0,$$

(5)

where two standard Brownian motions that are independent are denoted as $B_1\left(t\right)$ and $B_2\left(t\right)$. The model assumes the $V\left(t\right)$ follows a $\sigma >0$ volatility coefficient, a $\kappa >0$ mean-reversion rate, and a stochastic process with a $\delta >0$ long-run mean. The correlation between volatility and price, denoted by the coefficient $\rho \in (-1,1)$, and is a crucial aspect of the real-world data. In addition, $\eta$ represents a constant that captures the price for the risk factor $B_1\left(t\right)$.

Furthmore, apart from investments in risk-free bonds and stocks, the insurer may also invest in derivatives with hazardous assets as their underlying basis. This concept, inspired by the methodology initially proposed by [22], helps us think of the derivative with price $O(t,S(t),V(t\left)\right)$, at time t. The value of the derivative is contingent with both the basic stock price, represented as $S\left(t\right)$, and its associated volatility, denoted as $V\left(t\right)$. At the expiration time $\tau$, the payoff structure is expressed as $O\left(\tau \right)=f\left(S\right(\tau ),V(\tau \left)\right)$ based on some function f. Building on the work conducted by [22], as well as the subsequent contributions of [24,26], we make an assumption about the price process of the $O(t,S(t),L(t\left)\right)$ as follows:

$$\left\{\begin{array}{c}\mathrm{d}O\left(t\right)=rO\left(t\right)\mathrm{d}t+\left(\sigma \rho O_v+O_{s}S\left(t\right)\right)\left[\sqrt{V\left(t\right)}\mathrm{d}{B}_1\left(t\right)+\eta V\left(t\right)\mathrm{d}t\right]\hfill \\ +\sigma \sqrt{1-{\rho }^2}{O}_v\left[\xi V\left(t\right)\mathrm{d}t+\sqrt{V\left(t\right)}\mathrm{d}{B}_2\left(t\right)\right],\tau <t,\hfill \\ O\left(\tau \right)=f\left(S\left(\tau \right),V\left(\tau \right)\right),\hfill \end{array}\right.$$

(6)

Here, $\xi$ represents the constant that captures the stochastic volatility risk’s market pricing associated with $B_2\left(t\right)$. Moreover, the partial derivatives of O about $S\left(t\right)$ and $V\left(t\right)$ are denoted by $O_s$ and $O_v$, respectively. It is evident that in this setting, the financial market is considered complete, and there exists only one pricing kernel that aligns with the given Equation (7)

$$\mathrm{d}\Pi \left(t\right)=-\Pi \left(t\right)\left[r\mathrm{d}t+\sqrt{V\left(t\right)}\left(\eta \mathrm{d}{B}_1\left(t\right)+\xi \mathrm{d}{B}_2\left(t\right)\right)\right].$$

(7)

2.3. Wealth Process

Using $u_1$ and $u_2$ to signify the amounts of money allocated in the stocks and derivatives of the insurer, respectively, the strategy of the reinsurance investment can be expressed using a stochastic process with three-dimensional, denoted as $\pi \left(t\right)\triangleq {\left(q\left(t\right),u_1\left(t\right),u_2\left(t\right)\right)}_{t\in [0,T]}$. The insurer’s wealth process, denoted as $X^{\pi }\left(t\right)$, can be described below with the adoption of a reinsurance-investment strategy $\pi \left(t\right)$:

$$\left\{\begin{array}{c}\mathrm{d}{X}^{\pi }\left(t\right)=\left[X^{\pi }\left(t\right)-u_1\left(t\right)-u_2\left(t\right)\right]\frac{S_0\left(t\right)}{\mathrm{d}{S}_0\left(t\right)}+u_1\left(t\right)\frac{S\left(t\right)}{\mathrm{d}S\left(t\right)}+u_2\left(t\right)\frac{O\left(t\right)}{\mathrm{d}O\left(t\right)}+\mathrm{d}{R}^q\left(t\right)\hfill \\ =\left\{r{X}^{\pi }\left(t\right)+\eta {\theta }_1\left(t\right)V\left(t\right)+\xi {\theta }_2\left(t\right)V\left(t\right)+\lambda \mu \left[\vartheta +(1+\beta )q\left(t\right)\right]\right\}\mathrm{d}t\hfill \\ +{\theta }_1\left(t\right)\sqrt{V\left(t\right)}\mathrm{d}{B}_1\left(t\right)+{\theta }_2\left(t\right)\sqrt{V\left(t\right)}\mathrm{d}{B}_2\left(t\right)-q\left(t\right)\mathrm{d}{\displaystyle \sum _{i=1}^{N\left(t\right)}}{Y}_i,\hfill \\ X^{\pi }\left(0\right)=x_0,\hfill \end{array}\right.$$

(8)

where

$$\theta \left(t\right)\triangleq \left(\begin{array}{c}{\theta }_1\left(t\right)\\ {\theta }_2\left(t\right)\end{array}\right)=\left(\begin{array}{cc}1& \frac{O_{s}S\left(t\right)+\sigma \rho O_v)}{O\left(t\right)}\\ 0& \frac{\sigma \sqrt{1-{\rho }^2}{O}_v}{O\left(t\right)}\end{array}\right)\left(\begin{array}{c}{u}_1\left(t\right)\\ u_2\left(t\right)\end{array}\right)$$

(9)

represent the insurer’s exposures to $B_1\left(t\right)$ and $B_2\left(t\right)$. The market return risk is represented by $B_1\left(t\right)$ and the additional volatility risk is described by $B_2\left(t\right)$. In the following parts of the study, firstly, we obtain the equilibrium risk exposures denoted as ${\theta }_i$, where $i=1,2$. Subsequently, we transform them back utilizing the relation (9) to determine the equilibrium positions in the risky assets $S\left(t\right)$ and $O\left(t\right)$.

Definition 1

(Admissible strategies). A strategy $\pi \left(t\right)=\left(q\left(t\right),{\theta }_1\left(t\right),{\theta }_2\left(t\right)\right)$ shall be deemed to be permissible for the insurer in the case of

(i) 

$\pi \left(t\right)$ is a ${\mathcal{F}}_t$—progressively measurable process such that $q\left(t\right)\ge 0$ and $\mathbb{E}\left[{\int }_0^T\left|\right|{\theta }_i\left(t\right)\sqrt{V\left(t\right)}{\left|\right|}^2\mathrm{d}t\right]<\infty$, $i=1,2$;

(ii) 

There is a pathwise-unique solution $X^{\pi }\left(t\right)$ satisfying $\mathbb{E}\left[\underset{0\le t\le T}{sup}\left|\right|X^{\pi }\left(t\right){\left|\right|}^2\right]<\infty$ in Equation (8), For any $(t,x,v)\in \Gamma \triangleq [0,T]\times \mathbb{R}\times \mathbb{R}$.

The set of all admissible strategies is denoted by $\Pi$.

3. Problem Formation and Verification Theorem

Much of the existing literature, see [10,29,30], concerns mean-variance problems that solely take into account the optimization of the result at the initial time. They employ formulations consistent with the wealth process $X^{\pi }$, as shown below:

$$\underset{\pi \in \Pi }{sup}\left\{{\mathbb{E}}_{0,x_0,v_0}\left[X^{\pi }\left(T\right)\right]-\frac{\gamma }{2}\mathbb{V}a{r}_{0,x_0,v_0}\left[X^{\pi }\left(T\right)\right]\right\},$$

(10)

where ${\mathbb{E}}_{t,x,v}\left[·\right]=\mathbb{E}\left[·|X^{\pi }\left(t\right)=x,V\left(t\right)=v\right]$, $\mathbb{V}a{r}_{t,x,v}\left[·\right]=\mathbb{V}ar\left[·|X^{\pi }\left(t\right)=x,V\left(t\right)=v\right]$, and $\gamma >0$ represent the insurer’s risk aversion coefficient. Formulations like (10) represent problems regarding static optimization, where the objectives are entirely determined by relying upon the knowledge available from the starting time 0. Consequently, resultant solutions should be considered pre-commitments, as they do not adapt to the accumulation of information over time. As highlighted by [11], decision-making in a real-world setting often involves changing objectives as time progresses. This characteristic renders the solutions derived from (10) time-inconsistent in the sense they are optimum only for the initial time point, i.e., time 0, but cease to be so as time advances. From a practical perspective, solutions based on (10) are not applicable, as rational insurers must adhere to the fundamental requirement of time consistency in decision-making processes.

The current paper examines the subsequent formulation using a time-varying objective instead. For each $(t,x,v)\in \Gamma$, the insurer aims to derive

$$\underset{\pi \in \Pi }{sup}\left\{{\mathbb{E}}_{t,x,v}\left[X^{\pi }\left(T\right)\right]-\frac{\gamma }{2}\mathbb{V}a{r}_{t,x,v}\left[X^{\pi }\left(T\right)\right]\right\}.$$

(11)

The value function for (11) fails to meet the Bellman principle of optimality, which is because of the absence of the iterated expectation property in the variance term of the objective. As a result, this circumstance results in an issue that is time-inconsistent, meaning that the solutions found at a given time point $t\in [0,T]$ are no longer the best at times $s>t$ in the future. To find a time-consistent solution for (11), let us turn to the concept of the equilibrium strategy, which will be explained by Definition 2 below. In broad terms, for an infinitesimal $\Delta t>0$, the decision taken within the equilibrium strategy on t remains consistent when compared with the decision obtained on $t+\Delta t$, thereby ensuring time consistency.

To facilitate this understanding, we introduce some notations. Consider $C^{1,2}(\Gamma )$, which denotes the space of function $\psi$ meeting condition $\psi$ and its partial derivatives ${\psi }_t$, ${\psi }_x$, ${\psi }_v$, ${\psi }_{xx}$, and ${\psi }_{xv}$, which are continuous on $\Gamma$. Similarly, $D^{1,2}(\Gamma )$ represents the space of functions $\psi$ meeting the condition $\psi \in C^{1,2}(\Gamma )$ and having first-order partial derivatives satisfying the polynomial growth conditions.

For any $(t,x,v)\in \Gamma$, we assume that

$$y^{\pi }(t,x,v)={\mathbb{E}}_{t,x,v}\left[X^{\pi }\left(T\right)\right],z^{\pi }(t,x,v)={\mathbb{E}}_{t,x,v}\left[{\left(X^{\pi }\left(T\right)\right)}^2\right].$$

(12)

Then, the mean-variance optimization issue (11) is rewritten as Equations (13) and (14)

$$\underset{\pi \in \Pi }{sup}f\left(t,x,v,y^{\pi }(t,x,v),z^{\pi }(t,x,v)\right),$$

(13)

$$f\left(t,x,v,y^{\pi },z^{\pi }\right)=y^{\pi }-\frac{\gamma }{2}\left(z^{\pi }-{\left(y^{\pi }\right)}^2\right),$$

(14)

where $f:\Gamma \to \mathbb{R}$ represents a function in $D^{1,2}(\Gamma )$ and $y^{\pi }$, $z^{\pi }$ are short for $y^{\pi }(t,x,v)$ and $z^{\pi }(t,x,v)$, respectively.

Below, you will find the definitions of the equilibrium value functions and equilibrium strategies.

Definition 2

(Equilibrium strategies). For a valid strategy ${\pi }^{*}=\left(q^{*}\left(t\right),{\theta }_1^{*}\left(t\right),{\theta }_2^{*}\left(t\right)\right)$ with a predetermined initial state $(t,x,v)\in \Gamma$, we can describe the following

$${\pi }_{ϵ}\left(s\right)=\left\{\begin{array}{c}(\tilde{q},{\tilde{\theta }}_1,{\tilde{\theta }}_2),t\le s<t+ϵ,\\ {\pi }^{*}\left(s\right),t+ϵ\le s<T,\end{array}\right.$$

with $\tilde{q}\in {\mathbb{R}}^{+}$, ${\tilde{\theta }}_1\in \mathbb{R}$, $ϵ\in {\mathbb{R}}^{+}$ and ${\tilde{\theta }}_2\in \mathbb{R}$. If $\forall (\tilde{q},{\tilde{\theta }}_1,{\tilde{\theta }}_2)\in \mathbb{R}\times \mathbb{R}\times {\mathbb{R}}^{+}$,

$$\underset{ϵ\to 0}{lim}inf\frac{f(t,x,v,y^{{\pi }^{ϵ}},z^{{\pi }^{ϵ}})-f(t,x,v,y^{{\pi }^{*}},z^{{\pi }^{*}})}{ϵ}\le 0,$$

herem ${\pi }^{*}$ is referred to as the equilibrium value function and the equilibrium strategies are described as

$$J(t,x,v)=f(t,x,v,y^{{\pi }^{*}},z^{{\pi }^{*}}).$$

To establish the equilibrium strategies for solving the mean-variance issue stated in (13) and obtain the extended Hamilton–Jacobi–Bellman (HJB) system, introducing a variational operator is essential.

$$\begin{array}{c}{\mathcal{A}}^{\pi }\psi \triangleq {\psi }_t+\kappa (\delta -v){\psi }_v+\frac{1}{2}{\sigma }^{2}v{\psi }_{vv}+\left[rx+\eta {\theta }_{1}v+\xi {\theta }_{2}v+\lambda \mu \vartheta +\lambda \mu (1+\beta )q\right]{\psi }_x+\frac{v}{2}\left({\theta }_1^2+{\theta }_2^2\right){\psi }_{xx}\hfill \\ +\sigma v\left({\theta }_1\rho +{\theta }_2\sqrt{1-{\rho }^2}\right){\psi }_{xv}+\lambda \mathbb{E}\left[\psi (t,x-q{Y}_i,v)-\psi (t,x,v)\right],\hfill \end{array}$$

for any $\psi (t,x,v)\in C^{1,2}(\Gamma )$.

The below proof theorem provides an extended HJB system to solve problem (13).

Theorem 1

(Verification theorem). Refer to problem (13), where two real value functions $W(t,x,v),g(t,x,v)\in C^{1,2}(\Gamma )$ exist which fulfill the subsequent extended HJB systems: $\forall (t,x,v)\in \Gamma$,

$$\underset{\pi \in \Pi }{sup}\left\{{\mathcal{A}}^{\pi }W(t,x,v)-{\mathcal{A}}^{\pi }\frac{\gamma }{2}{g}^2(t,x,v)+\gamma g(t,x,v){\mathcal{A}}^{\pi }g(t,x,v)\right\}=0,W(T,x,v)=x,$$

(15)

$${\mathcal{A}}^{{\pi }^{*}}g(t,x,v)=0,g(T,x,v)=x.$$

(16)

where

$${\pi }^{*}\triangleq arg\underset{\pi \in \Pi }{sup}\left\{{\mathcal{A}}^{\pi }W(t,x,v)-{\mathcal{A}}^{\pi }\frac{\gamma }{2}{g}^2(t,x,v)+\gamma g(t,x,v){\mathcal{A}}^{\pi }g(t,x,v)\right\},$$

(17)

then, $J(t,x,v)=W(t,x,v)$, ${\mathbb{E}}_{t,x,v}\left[X^{{\pi }^{*}}\left(T\right)\right]=g(t,x,v)$, and ${\pi }^{*}$ will be the optimally time-consistent strategies.

This proves this theorem bears a resemblance to that of Theorem 4.1 in [31], Theorem 1 in [32], and Theorem 1 in [11]. Therefore, it will be omitted from our analysis.

4. Solutions

In this section, we summarize the precise expressions of the optimal value function and the time-consistent reinsurance-investment strategies in Theorem 2.

Theorem 2.

Regarding problem (13), the optimal time-consistent strategies involve ${\pi }^{*}\left(t\right)=(q^{*}\left(t\right),{\theta }_1^{*}\left(t\right),{\theta }_2^{*}\left(t\right))$, where the time-consistent reinsurance strategies can be

$$q^{*}\left(t\right)=\frac{{\mathrm{e}}^{-r(T-t)}}{\gamma }·\frac{\beta \mu }{{\sigma }_y^2}$$

(18)

The time-consistent risk exposures to risk factors $B_1$ and $B_2$ are listed below:

$$\left\{\begin{array}{c}{\theta }_1^{*}\left(t\right)=\frac{{\mathrm{e}}^{-r(T-t)}}{\gamma }·\left[\eta -\sigma \rho \overline{B}\left(t\right)\right],\hfill \\ {\theta }_2^{*}\left(t\right)=\frac{{\mathrm{e}}^{-r(T-t)}}{\gamma }·\left[\xi -\sigma \sqrt{1-{\rho }^2}\overline{B}\left(t\right)\right].\hfill \end{array}\right.$$

(19)

Through the utilization of Equation (9), we transform the optimal risk exposures into the optimal allocation of investment in the risky stock, denoted as ${\theta }_1\left(t\right)$, and the derivative, denoted as ${\theta }_2\left(t\right)$. This process results in the formulation of time-consistent investment strategies, which can be expressed as follows:

$$\left\{\begin{array}{c}{u}_1^{*}\left(t\right)={\theta }_1^{*}\left(t\right)-\frac{O_{s}S\left(t\right)+\sigma \rho O_v}{O\left(t\right)}{u}_2^{*}\left(t\right),\hfill \\ u_2^{*}\left(t\right)=\frac{O\left(t\right){\theta }_2^{*}\left(t\right)}{\sigma \sqrt{1-{\rho }^2}{O}_v}.\hfill \end{array}\right.$$

(20)

Moreover, this value function is as follows:

$$J(t,x,v)={\mathrm{e}}^{r(T-t)}x+\frac{B\left(t\right)}{\gamma }v+\frac{C\left(t\right)}{\gamma },$$

(21)

and

$${\mathbb{E}}_{t,x,v}\left[X^{{\pi }^{*}}\left(T\right)\right]=g(t,x,v)={\mathrm{e}}^{r(T-t)}x+\frac{\overline{B}\left(t\right)}{\gamma }v+\frac{\overline{C}\left(t\right)}{\gamma },$$

(22)

where

$$B\left(t\right)=-\frac{({\eta }^2+{\xi }^2){\mathrm{e}}^{-(\kappa +\sigma \eta \rho +\sigma \xi \sqrt{1-{\rho }^2})(T-t)}}{\kappa +\sigma \eta \rho +\sigma \xi \sqrt{1-{\rho }^2}}+d_3{\mathrm{e}}^{-\kappa (T-t)}+\frac{d_2}{\kappa },$$

(23)

$$\overline{B}\left(t\right)=\frac{{\eta }^2+{\xi }^2}{\kappa +\sigma \eta \rho +\sigma \xi \sqrt{1-{\rho }^2}}\left[1-{\mathrm{e}}^{-(\kappa +\sigma \eta \rho +\sigma \xi \sqrt{1-{\rho }^2})(T-t)}\right],$$

(24)

$$C\left(t\right)=\kappa \delta {\int }_t^{T}B\left(s\right)\mathrm{d}s+\frac{\gamma \lambda \mu \vartheta }{r}\left[{\mathrm{e}}^{r(T-t)}-1\right]+\frac{\lambda {\beta }^2{\mu }^2}{2{\sigma }_y^2}(T-t).$$

(25)

$$\overline{C}\left(t\right)=\kappa \delta {\int }_t^T\overline{B}\left(s\right)\mathrm{d}s+\frac{\gamma \lambda \mu \vartheta }{r}\left[{\mathrm{e}}^{r(T-t)}-1\right]+\frac{\lambda {\beta }^2{\mu }^2}{{\sigma }_y^2}(T-t),$$

(26)

with $d_1=\frac{({\eta }^2+{\xi }^2)\left(\sigma \eta \rho +\sigma \xi \sqrt{1-{\rho }^2}\right)}{\kappa +\sigma \eta \rho +\sigma \xi \sqrt{1-{\rho }^2}}$, $d_2=\frac{{\eta }^2+{\xi }^2}{2}-d_1$, $d_3=\frac{d_1}{\sigma \eta \rho +\sigma \xi \sqrt{1-{\rho }^2}}-\frac{d_2}{\kappa }$.

Proof. 

To solve Equations (15) and (16), we aim to propose the following solutions.

$$\begin{array}{c}W(t,x,v)=A\left(t\right)x+\frac{B\left(t\right)}{\gamma }v+\frac{C\left(t\right)}{\gamma },A\left(T\right)=1,B\left(T\right)=0,C\left(T\right)=0,\hfill \\ g(t,x,v)=\overline{A}\left(t\right)x+\frac{\overline{B}\left(t\right)}{\gamma }v+\frac{\overline{C}\left(t\right)}{\gamma },\overline{A}\left(T\right)=1,\overline{B}\left(T\right)=0,\overline{C}\left(T\right)=0.\hfill \end{array}$$

(27)

We can obtain the fiollowing partial derivatives:

$$\left\{\begin{array}{c}{W}_t=A^{\prime }\left(t\right)x+\frac{B^{\prime }\left(t\right)}{\gamma }v+\frac{C^{\prime }\left(t\right)}{\gamma },W_x=A\left(t\right),W_v=\frac{B\left(t\right)}{\gamma },\hfill \\ g_t={\overline{A}}^{\prime }\left(t\right)x+\frac{{\overline{B}}^{\prime }\left(t\right)}{\gamma }v+\frac{{\overline{C}}^{\prime }\left(t\right)}{\gamma },g_x=\overline{A}\left(t\right),g_v=\frac{\overline{B}\left(t\right)}{\gamma }.\hfill \end{array}\right.$$

(28)

Substituting Equation (28) back into Equation (15), we have

$$\begin{array}{c}\underset{\pi \in \Pi }{sup}\left\{A^{\prime }\left(t\right)x+\frac{B^{\prime }\left(t\right)}{\gamma }v+\frac{C^{\prime }\left(t\right)}{\gamma }+\kappa (\delta -v)\frac{B\left(t\right)}{\gamma }-\frac{1}{2}{\sigma }^{2}v\frac{{\overline{B}}^2\left(t\right)}{\gamma }\right.\hfill \\ +\left[rx+{\theta }_1\eta v+{\theta }_2\xi v+\lambda \mu \vartheta +\lambda \mu \beta q\right]A\left(t\right)-\frac{v}{2}\left({\theta }_1^2+{\theta }_2^2\right)\gamma {\overline{A}}^2\left(t\right)\hfill \\ \left.-\sigma v\left({\theta }_1\rho +{\theta }_2\sqrt{1-{\rho }^2}\right)\overline{A}\left(t\right)\overline{B}\left(t\right)-\frac{\gamma }{2}\lambda q^2{\sigma }_y^2{\overline{A}}^2\left(t\right)\right\}=0.\hfill \end{array}$$

(29)

Differentiating Equation (29) with respect to $\pi$ implies

$$\left\{\begin{array}{c}{q}^{*}\left(t\right)=\frac{\beta \mu A\left(t\right)}{\gamma {\sigma }_y^2{\overline{A}}^2\left(t\right)},\hfill \\ {\theta }_1^{*}\left(t\right)=\frac{\eta A\left(t\right)-\sigma \rho \overline{A}\left(t\right)\overline{B}\left(t\right)}{\gamma {\overline{A}}^2\left(t\right)},\hfill \\ {\theta }_2^{*}\left(t\right)=\frac{\xi A\left(t\right)-\sigma \sqrt{1-{\rho }^2}\overline{A}\left(t\right)\overline{B}\left(t\right)}{\gamma {\overline{A}}^2\left(t\right)}.\hfill \end{array}\right.$$

(30)

Inserting Equation (30) into Equations (16) and (29) yields

$$\begin{array}{c}\left[A^{\prime }\left(t\right)+rA\left(t\right)\right]x+\frac{B^{\prime }\left(t\right)}{\gamma }v+\frac{C^{\prime }\left(t\right)}{\gamma }+\kappa (\delta -v)\frac{B\left(t\right)}{\gamma }-\frac{1}{2}{\sigma }^{2}v\frac{{\overline{B}}^2\left(t\right)}{\gamma }+\lambda \mu \vartheta A\left(t\right)\hfill \\ +\frac{\lambda {\beta }^2{\mu }^2{A}^2\left(t\right)}{2\gamma {\sigma }_y^2{\overline{A}}^2\left(t\right)}+\frac{{\left[\eta A\left(t\right)-\sigma \rho \overline{A}\left(t\right)\overline{B}\left(t\right)\right]}^2}{2\gamma {\overline{A}}^2\left(t\right)}v+\frac{{\left[\xi A\left(t\right)-\sigma \sqrt{1-{\rho }^2}\overline{A}\left(t\right)\overline{B}\left(t\right)\right]}^2}{2\gamma {\overline{A}}^2\left(t\right)}v=0\hfill \end{array}$$

(31)

and

$$\begin{array}{c}\left[{\overline{A}}^{\prime }\left(t\right)+r\overline{A}\left(t\right)\right]x+\frac{{\overline{B}}^{\prime }\left(t\right)}{\gamma }v+\frac{{\overline{C}}^{\prime }\left(t\right)}{\gamma }+\kappa (\delta -v)\frac{\overline{B}\left(t\right)}{\gamma }+\lambda \mu \vartheta \overline{A}\left(t\right)+\frac{\lambda {\beta }^2{\mu }^{2}A\left(t\right)}{\gamma {\sigma }_y^2\overline{A}\left(t\right)}\hfill \\ +\frac{{\eta }^{2}A\left(t\right)-\sigma \eta \rho \overline{A}\left(t\right)\overline{B}\left(t\right)}{\gamma \overline{A}\left(t\right)}v+\frac{{\xi }^{2}A\left(t\right)-\sigma \xi \sqrt{1-{\rho }^2}\overline{A}\left(t\right)\overline{B}\left(t\right)}{\gamma \overline{A}\left(t\right)}v=0.\hfill \end{array}$$

(32)

From the separation of variables with and without x and v, we can obtan the following equations:

$$A^{\prime }\left(t\right)+rA\left(t\right)=0,{\overline{A}}^{\prime }\left(t\right)+r\overline{A}\left(t\right)=0,$$

(33)

$$B^{\prime }\left(t\right)-\kappa B\left(t\right)-\frac{1}{2}{\sigma }^2{\overline{B}}^2\left(t\right)+\frac{{\left[\eta A\left(t\right)-\sigma \rho \overline{A}\left(t\right)\overline{B}\left(t\right)\right]}^2}{2{\overline{A}}^2\left(t\right)}+\frac{{\left[\xi A\left(t\right)-\sigma \sqrt{1-{\rho }^2}\overline{A}\left(t\right)\overline{B}\left(t\right)\right]}^2}{2{\overline{A}}^2\left(t\right)}=0,$$

(34)

$$C^{\prime }\left(t\right)+\kappa \delta B\left(t\right)+\lambda \mu \vartheta A\left(t\right)+\frac{\lambda {\beta }^2{\mu }^2{A}^2\left(t\right)}{2{\sigma }_y^2{\overline{A}}^2\left(t\right)}=0,$$

(35)

$${\overline{B}}^{\prime }\left(t\right)-\kappa \overline{B}\left(t\right)+\frac{{\eta }^{2}A\left(t\right)-\sigma \eta \rho \overline{A}\left(t\right)\overline{B}\left(t\right)}{\overline{A}\left(t\right)}+\frac{{\xi }^{2}A\left(t\right)-\sigma \xi \sqrt{1-{\rho }^2}\overline{A}\left(t\right)\overline{B}\left(t\right)}{\overline{A}\left(t\right)}=0,$$

(36)

$${\overline{C}}^{\prime }\left(t\right)+\kappa \delta \overline{B}\left(t\right)+\gamma \lambda \mu \vartheta \overline{A}\left(t\right)+\frac{\lambda {\beta }^2{\mu }^{2}A\left(t\right)}{{\sigma }_y^2\overline{A}\left(t\right)}=0.$$

(37)

Considering those boundary constraints, we have

$$A\left(t\right)=\overline{A}\left(t\right)={\mathrm{e}}^{r(T-t)},$$

(38)

$B\left(t\right)$, $\overline{B}\left(t\right)$, $C\left(t\right)$ and $\overline{C}\left(t\right)$ are given by Equations (23)–(26), respectively.

Therefore, Equation (30) becomes

$$\left\{\begin{array}{c}{q}^{*}\left(t\right)=\frac{{\mathrm{e}}^{-r(T-t)}}{\gamma }·\frac{\beta \mu }{{\sigma }_y^2},\hfill \\ {\theta }_1^{*}\left(t\right)=\frac{{\mathrm{e}}^{-r(T-t)}}{\gamma }·\left[\eta A\left(t\right)-\sigma \rho \overline{B}\left(t\right)\right],\hfill \\ {\theta }_2^{*}\left(t\right)=\frac{{\mathrm{e}}^{-r(T-t)}}{\gamma }·\left[\xi A\left(t\right)-\sigma \sqrt{1-{\rho }^2}\overline{B}\left(t\right)\right].\hfill \end{array}\right.$$

(39)

Equations (9) and (20) provide the desired expressions of the optimal investment strategies. □

Remark 1.

According to the definition of the optimal value function and Theorem 2, the following can be obtained:

$$\mathbb{V}a{r}_{t,x,v}\left[X^{{\pi }^{*}}\left(T\right)\right]=-\frac{2}{\gamma }\left(J(t,x,v)\right.-{\mathbb{E}}_{t,x,v}\left[X^{{\pi }^{*}}\left(T\right)\right])=-\frac{2}{{\gamma }^2}\left[\left(B\left(t\right)-\overline{B}\left(t\right)\right)v-\overline{C}\left(t\right)+C\left(t\right)\right],$$

we can obtain:

$$\frac{1}{\gamma }=\sqrt{\frac{\mathbb{V}a{r}_{t,x,v}\left[X^{{\pi }^{*}}\left(T\right)\right]}{2\left[\left(\overline{B}\left(t\right)-B\left(t\right)\right)v+\overline{C}\left(t\right)-C\left(t\right)\right]}}.$$

Subsequently, the substitution of the expression for $1/\gamma$ in Equation (22) yields

$${\mathbb{E}}_{t,x,v}\left[X^{{\pi }^{*}}\left(T\right)\right]={\mathrm{e}}^{r(T-t)}x+\frac{\overline{C}\left(t\right)+\overline{B}\left(t\right)v}{2\left[\overline{C}\left(t\right)-C\left(t\right)\right.+\left(\overline{B}\left(t\right)-B\left(t\right)\right)v]}\sqrt{\mathbb{V}a{r}_{t,x,v}\left[X^{{\pi }^{*}}\left(T\right)\right]}.$$

(40)

Therefore, for every $(t,x,v)\in \Gamma$, the relation between ${\mathbb{E}}_{t,x,v}\left[X^{{\pi }^{*}}\left(T\right)\right]$ and $\mathbb{V}a{r}_{t,x,v}\left[X^{{\pi }^{*}}\left(T\right)\right]$ is represented with (40). The relationship of (40) is also referred to as the effective frontier of problem (11) in the beginning situation $(t,x,v)$ incurrent investment theory.

5. Special Case of No Derivatives Trading

This subsection elaborates from the insurer’s viewpoint and illustrates the significant role of derivatives trading. We investigate an alternate scenario in which the insurer does not have access to derivative trading and only considers investment in stock and reinsurance. In this alternate situation, the financial market will be inadequate from the perspective of the insurer. We will proceed to formulate the value function for an insurer who refrains from participating in derivative trading. We will also establish the equilibrium investment-reinsurance strategy for this scenario. Following this, we will conduct a comparative analysis by contrasting its value function with that of an insurer who can engage in derivative trading within the market.

The insurer who refrains from participating in derivative trading shares the same objective as outlined in Equation (11), with the extra restrictive condition of being unable to engage in derivative trading, i.e., $u_2\left(t\right)\equiv 0$; therefore, the wealth process undergoes the following modification:

$$\left\{\begin{array}{c}\mathrm{d}{\mathcal{X}}^{\tilde{\pi }}\left(t\right)=\left\{r{\mathcal{X}}^{\tilde{\pi }}\left(t\right)+\eta {\tilde{u}}_1\left(t\right)V\left(t\right)+\lambda \mu \left[\vartheta +\left(1+\beta \right)\tilde{q}\left(t\right)\right]\right\}\mathrm{d}t\hfill \\ +{\tilde{u}}_1\left(t\right)\sqrt{V\left(t\right)}\mathrm{d}{B}_1\left(t\right)-\tilde{q}\left(t\right)\mathrm{d}{\displaystyle \sum _{i=1}^{N\left(t\right)}}{Y}_i,\hfill \\ {\mathcal{X}}^{\tilde{\pi }}\left(0\right)=x_0,\hfill \end{array}\right.$$

(41)

here, we define $\tilde{\pi }\left(t\right)\triangleq {\left(\tilde{q}\left(t\right),{\tilde{u}}_1\left(t\right)\right)}_{t\in [0,T]}$, where the risk exposure equals the investment strategy, i.e., ${\tilde{\theta }}_1\left(t\right)={\tilde{u}}_1\left(t\right)$. This leads to the following optimization problem:

$$\underset{\tilde{\pi }\in \Pi }{sup}\left\{{\mathbb{E}}_{t,x,v}\left[{\mathcal{X}}^{\tilde{\pi }}\left(T\right)\right]-\frac{\gamma }{2}\mathbb{V}a{r}_{t,x,v}\left[{\mathcal{X}}^{\tilde{\pi }}\left(T\right)\right]\right\},$$

(42)

so the extended HJB system becomes

$$\underset{\tilde{\pi }\in \Pi }{sup}\left\{{\mathcal{A}}^{\tilde{\pi }}\tilde{W}(t,x,v)-{\mathcal{A}}^{\tilde{\pi }}\frac{\gamma }{2}{\tilde{g}}^2(t,x,v)+\gamma \tilde{g}(t,x,v){\mathcal{A}}^{\tilde{\pi }}\tilde{g}(t,x,v)\right\}=0,\tilde{W}(T,x,v)=x,$$

(43)

$${\mathcal{A}}^{{\tilde{\pi }}^{*}}\tilde{g}(t,x,v)=0,\tilde{g}(T,x,v)=x.$$

(44)

with

$${\tilde{\pi }}^{*}\triangleq arg\underset{\tilde{\pi }\in \Pi }{sup}\left\{{\mathcal{A}}^{\tilde{\pi }}\tilde{W}(t,x,v)-{\mathcal{A}}^{\tilde{\pi }}\frac{\gamma }{2}{\tilde{g}}^2(t,x,v)+\gamma \tilde{g}(t,x,v){\mathcal{A}}^{\tilde{\pi }}\tilde{g}(t,x,v)\right\},$$

(45)

where

$$\begin{array}{c}{\mathcal{A}}^{\tilde{\pi }}\psi (t,x,v)\triangleq {\psi }_t+\kappa (\delta -v){\psi }_v+\frac{1}{2}{\sigma }^{2}v{\psi }_{vv}+\left[rx+\eta {\tilde{u}}_{1}v+\lambda \mu \vartheta +\lambda \mu (1+\beta )\tilde{q}\right]{\psi }_x+\frac{1}{2}{\tilde{u}}_1^{2}v{\psi }_{xx}\hfill \\ +\sigma {\tilde{u}}_1\rho v{\psi }_{xv}+\lambda \mathbb{E}\left[\psi (t,x-\tilde{q}{Y}_i,v)-\psi (t,x,v)\right].\hfill \end{array}$$

The theorem below describes the time-consistent investment-reinsurance strategies and the optimal value function of the insurer without derivatives trading.

Theorem 3.

For problem (42) without derivatives trading, the optimal time-consistent investment strategies and risk exposure will be

$${\tilde{u}}_1^{*}\left(t\right)={\tilde{\theta }}_1^{*}\left(t\right)=\frac{{\mathrm{e}}^{-r(T-t)}}{\gamma }·\left[\eta -\sigma \rho \widehat{B}\left(t\right)\right],$$

(46)

the optimal time-consistent reinsurance strategies remain unchanged as $q^{*}\left(t\right)$ defined by Equation (18), and the associated optimum value function follows

$$\tilde{J}(t,x,v)={\mathrm{e}}^{r(T-t)}x+\frac{\tilde{C}\left(t\right)}{\gamma }+\frac{\tilde{B}\left(t\right)}{\gamma }v,$$

(47)

and

$${\mathbb{E}}_{t,x,v}\left[{\mathcal{X}}^{{\tilde{\pi }}^{*}}\left(T\right)\right]=\tilde{g}(t,x,v)={\mathrm{e}}^{r(T-t)}x+\frac{\widehat{C}\left(t\right)}{\gamma }+\frac{\widehat{B}\left(t\right)}{\gamma }v,$$

(48)

where

$$\widehat{B}\left(t\right)=\frac{{\eta }^2}{\kappa +\sigma \eta \rho }\left[1-{\mathrm{e}}^{-(\kappa +\sigma \eta \rho )(T-t)}\right],$$

(49)

$$\tilde{B}\left(t\right)=\frac{M{\mathrm{e}}^{-2(\kappa +\sigma \eta \rho )(T-t)}}{\kappa +2\sigma \eta \rho }-\frac{(2M+N){\mathrm{e}}^{-(\kappa +\sigma \eta \rho )(T-t)}}{\sigma \eta \rho }+\chi {\mathrm{e}}^{-\kappa (T-t)}+\frac{Q}{\kappa },$$

(50)

$$\widehat{C}\left(t\right)=\kappa \delta {\int }_t^T\widehat{B}\left(s\right)\mathrm{d}s+\frac{\gamma \lambda \mu \vartheta }{r}\left[{\mathrm{e}}^{r(T-t)}-1\right]+\frac{\lambda {\beta }^2{\mu }^2}{{\sigma }_y^2}(T-t),$$

(51)

$$\tilde{C}\left(t\right)=\kappa \delta {\int }_t^T\tilde{B}\left(s\right)\mathrm{d}s+\frac{\gamma \lambda \mu \vartheta }{r}\left[{\mathrm{e}}^{r(T-t)}-1\right]+\frac{\lambda {\beta }^2{\mu }^2}{2{\sigma }_y^2}(T-t),$$

(52)

with $M=\frac{(1-{\rho }^2){\sigma }^2{\eta }^4}{2{(\kappa +\sigma \eta \rho )}^2}$, $N=\frac{\sigma \rho {\eta }^3}{\kappa +\sigma \eta \rho }$, $Q=\frac{{\eta }^2}{2}-(M+N)$, $\chi =\frac{2M+N}{\sigma \eta \rho }-\frac{M}{\kappa +2\sigma \eta \rho }-\frac{Q}{\kappa }$.

The proof process of Theorem 3 is comparable to that of Theorem 2 and is, therefore, omitted here.

6. Numerical Studies

In the following part of the study, we aim to shed light on how different model parameters affect both investment strategies and equilibrium reinsurance decisions. We will do this by presenting various numerical examples. In addition, we depict the effects of these parameters on the frontier of efficiency. It is important to note that, unless stated otherwise in this section, we will be using the market parameters outlined in

Table 1. Values of the model parameters.

$\mathit{\alpha }$	$\mathit{\beta }$	$\mathit{\lambda }$	$\mathit{\mu }$	${\mathit{\sigma }}_{\mathit{y}}^2$	r	$\mathit{\eta }$	$\mathit{\kappa }$	$\mathit{\delta }$
$0.2$	$0.3$	4	$0.6$	1	$0.05$	4	5	$0.{13}^2$
$\sigma$	$\rho$	$\xi$	$v_0$	$s_0$	$x_0$	t	T	$\gamma$
$0.5^2$	$-0.4$	$-6$	$0.{15}^2$	2	4	0	5	$0.1$

, following the conventions established by [26,27] and related references.

6.1. How the Parameters Affect the Equilibrium Reinsurance Strategies

We illustrate model parameters’ effects on equilibrium reinsurance strategies numerically In this section.

Figure 1 depicts the influence of the reinsurer’s safety loading parameter $\beta$ and the insurer’s risk-aversion parameter $\gamma$ on their equilibrium reinsurance strategy $q^{*}\left(0\right)$. It has been observed that the value of $q^{*}\left(0\right)$ increases when the parameter $\beta$ rises. Conversely, $q^{*}\left(0\right)$ decreases when parameter $\gamma$ decreases. This indicates a positive correlation between $q^{*}\left(0\right)$ and $\beta$, and a negative correlation between $q^{*}\left(0\right)$ and $\gamma$. This phenomenon can be explained by the fact that $\beta$ reflects the relative safety loading satisfying for the reinsurer, where an increase in $\beta$ implies a higher cost for acquiring new business or proportional reinsurance. To minimize these costly reinsurance payments for reinsurance, the insurer is inclined to accept additional insurance business and increase the reinsurance’s retention level. Alternatively, the insurer with a greater $\gamma$ tends to be more averse to risk and is more likely to give up greater risks to the reinsurer.

Figure 2 demonstrates the sensitivity of $q^{*}\left(0\right)$ to changes in $\mu$ and ${\sigma }_y$. As parameter $\mu$ increases, it is clear that the value of $q^{*}\left(0\right)$ also rises correspondingly. Conversely, when there is a decrease in ${\sigma }_y$, the value of $q^{*}\left(0\right)$ declines. This indicates a direct relationship between $\mu$ and $q^{*}\left(0\right)$, such that higher values of $\mu$ result in higher values of $q^{*}\left(0\right)$. Similarly, there is an inverse relationship between ${\sigma }_y$ and $q^{*}\left(0\right)$, meaning that lower values of ${\sigma }_y$ lead to lower values of $q^{*}\left(0\right)$. This behavior can be attributed to the fact that as $\mu$ rises, the expected claim size increases, thereby prompting the insurer to buy less insurance and take on a greater proportion of the risks. On the other hand, an increase in ${\sigma }_y$ introduces greater uncertainty among the insurer, prompting the insurer to seek increased reinsurance coverage to mitigate the associated risks.

6.2. How the Parameters Affect the Equilibrium Investment Strategies

In the upcoming portion, to demonstrate the impact that model parameters have on investment strategies, we utilize the European call option as a representative example. Our approach in this analysis closely aligns with the methodology introduced in the research conducted by Xue and his colleagues, as detailed in their study [27]. This choice of the European call option as a case study allows us to examine how various model parameters affect investment decisions in a concrete and practical context.

Figure 3 depicts the influence of $\gamma$ on the insurer’s equilibrium investment strategy. When the risk increases, $u_1$ decreases, which is consistent with the classical asset allocation theory. Conversely, $u_2$ is opposite of $u_1$. As the risk increases, since derivatives play a hedging role in trading, the insurer needs to manage risk, leading to increased investment in derivatives. The insurer holds opposing positions in derivatives and stocks. When the insurer becomes increasingly risk averse, signified by a higher value of $\gamma$, which represents the coefficient of absolute risk aversion, their investment approach consequently becomes less aggressive. As a result, we have noticed that the insurer has fewer short positions in derivatives and they have fewer long positions in stocks.

In Figure 4, we investigate how the optimal investment allocation corresponds to the change in the average mean-reversion rate $\kappa$, which reflects the volatility persistence in Heston’s stochastic volatility model. The insurer will choose increased investments in stocks and adopt bearish positions in derivatives as $\kappa$ rises. This also confirms that derivatives can serve a hedging function and aid in risk management.

Figure 5 illustrates the impact of $\sigma$. An increase in $\sigma$ rises, leads to reduced investments in stocks and shorting less derivatives by the insurer.

Figure 6 and Figure 7 illustrate the effect of the risk premiums $\eta$ and $\xi$, respectively. As $\eta$ and $\xi$ increase, the hazards of the financial market rise, and the insurer takes long positions in smaller stocks and short positions in smaller derivatives to mitigate portfolio risk.

6.3. How the Parameters Affect Equilibrium Efficient Frontier

In the following part of our study, we demonstrate the model parameters’ effect on the equilibrium efficient frontier. We consider both scenarios: one involving derivative trading and another where derivatives are unavailable.

Figure 8 and Figure 9 illustrate the impact of the mean-reversion rate $\kappa$ as well as the volatility coefficient $\sigma$ on the equilibrium efficient frontier. A greater mean-reversion rate $\kappa$ and lower stock return variance process volatility $\sigma$ reflect reduced uncertainties in the variance process. As a result, the insurer encounters a smaller risk of volatility. In both cases, the equilibrium efficient frontiers are observed to expand as $\kappa$ increases and contract as $\sigma$ decreases.

Figure 10 and Figure 11 illustrate the effect of the risk premiums $\eta$ and $\xi$, respectively. An increase in $\eta$ and $\xi$ causes the equilibrium efficient frontier to shift upward towards the axix y, indicating the growing value of derivatives trading.

7. Conclusions

The decision-making process of an insurer about investment policies and reinsurance is investigated. The insurer is presented with the choice of acquiring proportional reinsurance, while, at the same time, engaging in financial market investments that encompass risk-free assets, risky assets, and derivatives. The Heston stochastic volatility model is used to model the price dynamics for the risky assets. We explicitly derive the corresponding value function and establish the time-consistent reinsurance-investment strategy by resolving an extended HJB equation. The paper concludes with a numerical simulation and sensitivity analysis, which provide insights into how model parameters impact the time-consistent strategy. The results show the insurer is inclined to accept additional insurance business and increase the reinsurance’s retention level to minimize these costly reinsurance payments for reinsurance. Alternatively, the insurer with high risk-aversion tends to be more averse to risk and is more likely to give up greater risks to the reinsurer. The insurer has fewer short positions in derivatives and they have fewer long positions in stocks. One of the novel aspects of the study is the incorporation of derivatives trading into the realms of reinsurance and optimal investment problems. It is demonstrated that derivatives trading can have a substantial positive effect on investment performance. More specifically, the efficient frontier, which considers a combination of stocks alongside risk-free bonds and derivatives, significantly outperforms that of investing solely in risk-free bonds and stocks. This underscores the potential for derivatives to serve as a valuable tool for insurers who are aiming to enhance the effectiveness of their investment strategies. The above results indicate that derivative investments are effective; therefore, insurance companies can take advantage of investment opportunities in the derivatives market to improve their risk management.

This paper lays the foundation for several potential avenues of future research. Firstly, addressing the issue of model uncertainty is of paramount importance, as it is a crucial factor in every modeling endeavor. As a result, investigating the optimum reinsurance-investment issue for mean-variance insurers facing model uncertainty in both financial and insurance markets merits a promising direction for further research. Secondly, delving into the model of regime-switching jump diffusion, which offers a more realistic representation of market dynamics, could be an intriguing extension of this work. On the other hand, it is essential to acknowledge that such an approach may introduce a rather complex nonlinear equation, which will require the use of numerical solution methods. Such investigations could be reserved for future studies.These proposed extensions have the potential to enrich our understanding of insurer decision-making in the face of diverse market conditions and uncertainties. They open up avenues for further exploration and refinement in the fields of insurance and financial modeling.