Time-Consistent Investment-Reinsurance Strategies for the Insurer and the Reinsurer under the Generalized Mean-Variance Criteria

Since the insurer and the reinsurer can be allowed to invest their wealth in the securities market, they can obtain profits not only by collecting premiums but also by investing in securities. Different from the other institutional investors, the insurer and the reinsurer will face the double risks that exist in both the insurance market and the securities market. In order to reduce the risk of claims, the insurer can purchase reinsurance contracts from the reinsurer and transfer part of the risk of claims to the reinsurer, because the reinsurer is more risk-seeking than the insurer. Therefore, how to design a suitable reinsurance contract is also the concern of the insurer and the reinsurer. Obviously, the setting of the reinsurance contract depends on the mutual agreement between the insurer and the reinsurer. However, most of the existing literature mainly focuses on the optimal investment-reinsurance problems only from the perspective of the insurer, while the interest of the reinsurer is generally ignored (e.g., Schmidli [1], Zeng and Li [2], Zhu et al. [3], Huang et al. [4], Hu and Wang [5], Deng et al. [6] and so on). Actually, the optimal reinsurance contract for the insurer may not be optimal or even unacceptable for the reinsurer. That is, the reinsurance contract should be designed to take into account the interests of both the insurer and the reinsurer. To address this problem, we will propose an investment-reinsurance optimization problem considering the joint interests of the insurer and the reinsurer, and the corresponding investment-reinsurance strategy will be investigated.

As far as we know, some researchers have paid attention to the joint interests of the insurer and the reinsurer. Li et al. [7] considered the weighted sum of an insurer’s and a reinsurer’s mean-variance objectives and aimed to find the corresponding time-consistent reinsurance-investment strategy. Li et al. [8] discussed the optimal investment-reinsurance strategy by maximizing the expected exponential utility of the weighted sum of the insurer’s and the reinsurer’s terminal wealth. Under the mean-variance criterion, Zhao et al. [9] also discussed the time-consistent investment-reinsurance strategy by maximizing the utility of a weighted sum of the insurer’s and the reinsurer’s surplus processes. Zhou et al. [10] derived the optimal investment-reinsurance strategy with consideration of the joint interests of the insurer and the reinsurer, and also assumed that the decision-maker was an ambiguity-averse manager. Huang et al. [11] investigated a robust optimal investment and reinsurance problem on considering the product of the insurer’s and the reinsurer’s utilities. Obviously, the above optimization models on the joint interests of the insurer and the reinsurer can be classified into the following two categories. The first kind of model is built by maximizing the utility of a weighted sum of the insurer’s and the reinsurer’s surplus processes, while the second one is constructed by maximizing the weighted sum/product of the insurer’s and the reinsurer’s utilities. The former assumes that the insurer and the reinsurer have the same risk aversion coefficients, while the latter considers that they have different risk aversion coefficients. Actually, the latter is more compatible with reality, since the reinsurer is more risk-seeking compared to the insurer. In this paper, we mainly focus on deriving the corresponding time-consistent strategies by maximizing the weighted sum of the insurer’s and the reinsurer’s mean-variance objectives.

Additionally, the above literature is limited to the study of continuous-time problems, while the discrete-time problems are always ignored by researchers. In fact, the discrete-time setting is more realistic to decision-makers, because they cannot trade continuously since it will generate a lot of transaction costs. Brandt [12] also pointed out that the continuous-time strategies are often inadmissible in discrete time because they may cause negative wealth. Especially for insurers and reinsurers, their surplus processes are more likely to be negative values, because they bear the double risks from the insurance market and the securities market. More importantly, Zhu et al. [13] also presented that the bankruptcies occurring in the earlier stages of investments were greater than those in the later stages. The main reason is that the classical multi-period mean-variance optimizations only consider the performance of the terminal wealth. To deal with this problem, Costa and Nabholz [14] proposed a generalized mean-variance model considering the intertemporal restrictions (i.e., the investors will maximize the weighted sum of the mean-variance objectives over all the periods). Under this framework, the terminal and intermediate performance of the portfolio will be included in the decision-making. There are still many studies on this subject, such as Costa and Araujo [15], Costa and de Oliveira [16], Cui et al. [17], He et al. [18], Zhou et al. [19], Xiao et al. [20] and so on. Inspired by the works mentioned above, in this paper, we will build a generalized multiperiod mean-variance investment-reinsurance optimization model considering the joint interests of the insurer and the reinsurer.

However, the proposed model cannot be directly solved by using the dynamic programming approach because the variance measure does not satisfy the expected iterated property. To the best of our knowledge, there are two methods to solve this problem. The first method is the embedding scheme proposed by Li and Ng [21], and the derived optimal strategy is called the pre-commitment strategy. However, some researchers point out that the precommitment strategy does not satisfy time consistency since the future changes are not taken into account. The second method is provided by Basak and Chabakauri [22] and Björk and Murgoci [23], called the game method, which can provide a time-consistent strategy for decision-makers. Under the game framework, the decision-makers treat this optimization problem as a noncooperative game, and its Nash equilibrium solution is defined as the time-consistent strategy. Since then, many researchers have applied the game method to derive the time-consistent solutions of the various multiperiod mean-variance optimization problems, such as Björk and Murgoci [24], Bensoussan et al. [25], Wu and Zeng [26], Zhou et al. [19], Xiao et al. [20] and so on. Based on the game method shown above, in this paper, we will investigate the time-consistent investment-reinsurance strategies for the generalized mutiperiod mean-variance optimization problem considering the joint interests of the insurer and the reinsurer. In this framework, the intermediate and terminal performance of the insurer and the reinsurer can be both considered.

Motivated by the above studies, we assume the insurer and the reinsurer can invest their wealth in one risky asset and one risk-free asset, as well as the insurer can purchase a proportional reinsurance contract from the reinsurer. We use two generalized mean-variance criteria to measure the performance of the insurer and the reinsurer, and further propose a generalized multi-period investment-reinsurance optimization considering the joint interests of the insurer and the reinsurer by maximizing the weighted sum of the insurer’s and the reinsurer’s mean-variance objectives. We apply the game method to derive the time-consistent investment-reinsurance strategies for the proposed model and also discuss the time-consistent strategies under the expected and variance value principles. Finally, we provide some numerical simulations to show the impact of the intertemporal restrictions on the time-consistent strategies.

Different from the existing literature, this paper has four contributions. (i) We first propose a generalized multi-period investment-reinsurance optimization problem under the discrete-time framework, and the joint interests of the insurer and the reinsurer are also considered. Actually, it is more realistic to consider the joint interests of the insurer and the reinsurer in the discrete-time framework. On the one hand, it avoids the negative wealth that may be caused by the accumulation of transaction costs in the case of continuous transactions. On the other hand, it ensures that the reinsurance contract is optimal for both the insurer and the reinsurer. (ii) We consider the impact of intertemporal restrictions on the decision-making, that is, the insurer and the reinsurer not only consider the terminal performance but are also concerned with the intermediate performance of their portfolios, which is absent from the existing continuous-time literature (e.g., Li et al. [7], Zhao et al. [9], Huang et al. [11] and so on). In this framework, the insurer and the reinsurer can adjust the intertemporal restrictions dynamically according to their own risk appetites and the market environment. (iii) We first derive the time-consistent investment-reinsurance strategies rather than the traditional precommitment strategies. Compared with the precommitment strategies, the time-consistent strategies might be more suitable for the decision-makers who are more rational and sophisticated, since they take possible future revisions into account. (iv) We first investigate the impact of the intertemporal restrictions on the time-consistent strategies. The interesting finding is that the intertemporal restrictions will urge the insurer and the reinsurer to shrink the position invested in the risky asset. However, for the time-consistent reinsurance strategy, the role of the intertemporal restrictions depends on who is the leader in the proposed model. When the insurer is the leader, the intertemporal restrictions will reduce the retention level of claims, while for the case that the reinsurer is the leader, only when the impact of the reinsurer’s leading role is higher than that of the intertemporal restrictions will the intertemporal restrictions shrink the retention level of claims.

The remainder of this paper is organized as follows. In Section 2, we construct a generalized multi-period mean-variance optimization model considering the joint interests of the insurer and the reinsurer. In Section 3, we derive the time-consistent strategies by using the game method. In Section 4, we give some numerical examples to show the differences of the time-consistent strategies under different settings. Finally, we summarize the conclusions of this paper.

In this paper, we assume that both an insurer and a reinsurer will simultaneously enter the financial market at time 0 to carry out investment-reinsurance activities. Suppose that the insurer and the reinsurer have the initial wealth of $w_0^1$ and $w_0^2$, respectively, and they plan to take all the wealth into the capital market within a time horizon T. We suppose that the insurer and the reinsurer are both allowed to invest their wealth in a risk-free asset and a risky asset (note that they invest in the risky asset with different random returns), where the risk-free asset takes a determinate return $s_t$ and the random return on the risky asset invested by the insurer is $e_t^1$, while the risky asset invested by the reinsurer has the random return $e_t^2$. Let $u_t^1$ and $u_t^2$ represent the amount that the insurer and the reinsurer invest in the risky asset at the beginning of the time period t, respectively. Assume that $w_t^1$ and $w_t^2$ denote the insurer’s and the reinsurer’s wealth at time period t, then the wealth allocated in the risk-free asset can be expressed as $w_t^1-u_t^1$ and $w_t^2-u_t^2$, respectively. In addition to the investment, we also assume that a proportional reinsurance contract is applied between the insurer and the reinsurer. The proportion covered by the insurer at the time period t is denoted by $q_t$, where $q_t\in [0,1]$, $t=0,1,...,T-1$. Under this reinsurance contract, the insurer only requires to bear the claim amount $q_t{z}_t$ when facing a claim $z_t$ at the time period t. In this case, $q_t$ can also be treated as the retention level of the claim $z_t$. Meanwhile, the reinsurer undertakes the rest of the claim amount $(1-q_t)z_t$, as well as obtains a premium ${\delta }_t\left(q_t\right)$ from the insurer. Let $c_t$ denote the premium income of the insurer at the time period t (note that $c_t$ is assumed to be a determinate value), then the remaining premium of the insurer can be expressed as $c_t-{\delta }_t\left(q_t\right)$, $t=0,1,...,T-1$. Therefore, the wealth process of the insurer and the reinsurer can be shown as follows. and: where, $P_t^1=e_t^1-s_t$ and $P_t^2=e_t^2-s_t$ denote the excess return of risky assets 1 and 2, respectively, and $t=0,1,...,T-1$.

Apparently, the insurer and the reinsurer have conflicts of interest because of the reinsurance contract. Therefore, the formulation of the investment-reinsurance strategy should consider both the insurer and the reinsurer. In addition, since the insurer and the reinsurer not only face the risk of claims but also bear the investment risk of the securities market, they are more risk-averse compared to other institutional investors. More importantly, Zhu et al. [13] and Zhou et al. [19] showed that the precommitment and time-consistent strategies derived from the classical mean-variance model will lead to higher bankruptcy probabilities in the earlier periods of an investment. The cause is that the classical mean-variance model only considers the terminal performance of a portfolio and its intermediate performance is ignored. Therefore, the classical multiperiod mean-variance model may not be the best choice for the insurer and the reinsurer. To address this problem, we use two generalized mean-variance criteria to measure the performance of the insurer and the reinsurer. In this setting, the intermediate and terminal performance can be both considered. Simultaneously, we take the weighted sum of the insurer’s and the reinsurer’s mean-variance criteria into account, so as to measure the joint interests of the insurer and the reinsurer. Under this generalized mean-variance framework, the corresponding multiperiod investment-reinsurance optimization problem can be formulated as follows.

Here, we let $\pi =({\pi }_0,{\pi }_1,...,{\pi }_{T-1})$, ${\pi }_k=(u_k^1,u_k^2,q_k),k=0,1,...,T-1$, and the weight $\alpha$ satisfy the condition that $\alpha \in [0,1]$. Note that $\alpha$ can be regarded as the weighing coefficient between the benefit of the insurer and that of the reinsurer. Intuitively, $\alpha >0.5$ indicates that the benefit of the insurer is more concerned (i.e., the insurer is the leader); $\alpha <0.5$ means that the benefit of the reinsurer is paid more attention (i.e., the reinsurer is the leader); while for the case that $\alpha =0.5$, we consider that the interest of the insurer and that of the reinsurer are equally important. Additionally, we also assume that ${\xi }_t^1$ and ${\xi }_t^2$ are both the 0-1 variables, among which ${\xi }_t^1=1$ (${\xi }_t^2=1$) means that the insurer (reinsurer) will consider the intertemporal restriction at the time period t, otherwise, the intertemporal restriction is ignored here, $t=1,2,...,T$. Further, parameters ${\eta }_t^1$ and ${\eta }_t^2$ denote the risk aversion coefficients of the insurer and the reinsurer at time period t, respectively, $t=1,2,...,T$.

Similar to the existing literature, we further assume that $e_t={[e_t^1,e_t^2]}^{\prime }$ and $z_t$ are statistically independent, $t=0,1,...,T-1$. In other words, the covariance $Cov(e_{t_1},e_{t_2})=0$ for $t_1\ne t_2$, and the covariance $Cov(e_{t_1},z_{t_2})=0$, where $t_1,t_2=0,1,...,T-1$. Let ${\mu }_t^i=E\left(P_t^i\right)$, ${\sigma }_t^i=Var\left(P_t^i\right)$, ${\theta }_t=Cov(P_t^1,P_t^2)$, ${\tilde{\mu }}_t=E\left(z_t\right)$ and ${\tilde{\sigma }}_t=Var\left(z_t\right)$, where $i=1,2$ and $t=0,1,...,T-1$. In addition, for convenience, we define the notations that ${\sum }_{t=k}^l(·)=0$ and ${\prod }_{t=k}^l(·)=1$ for $k>l$. Under this framework, we aim to provide some suitable investment-reinsurance strategies for both the insurer and the reinsurer.

To the best of our knowledge, the expected iterated property is absent from the variance measure, thus Equation (3) is also a time-inconsistent one, that is, it cannot be directly solved by using the dynamic programming approach. Similar to the classical multiperiod mean-variance model, we have two approaches to solve Equation (3). The first approach is the embedding scheme provided by Li and Ng [21], and the derived strategy is called the precommitment strategy. However, some researchers point out the precommitment strategy does not satisfy time consistency since it does not take the future modifications into account. The second one is proposed by Basak and Chabakauri [22] and Björk and Murgoci [23], named as the game approach, and it can provide a time-consistent strategy for decision-makers. In this paper, we mainly focus on providing a time-consistent investment-reinsurance strategy for the rational decision-makers who will consider the decision modifications in the future. Similar to Björk and Murgoci [23], we define the following time-varying mean-variance optimization sub-objective. where $k=0,1,...,T-1$. Then, the time-consistent solution of Equation (3) can be defined similarly as follows.

According the definition of time-consistent strategy, we have the following proposition.

According to Proposition 1, we can derive the following theorem.

Theorem 1 only shows the time-consistent strategies when the weight coefficient $\alpha$ satisfies the condition that $\alpha \in (0,1)$. While for the case that $\alpha =1$ or $\alpha =0$, that is, Equation (3) only considers the benefit of the insurer or the reinsurer, then the corresponding time-consistent investment-reinsurance strategies can be derived similarly. The details are shown as follows.

As shown in Remark 1, we can find that the function $V_t(w_t^1,w_t^2)$ and $f_{t,\tau }(w_t^1,w_t^2)$ only depend on the insurer’s wealth at the time period t, i.e., $w_t^1$. In this situation, decision-makers only consider the benefit of the insurer, this is also the traditional approach to deal with the optimal investment-reinsurance problem. However, compared with the existing literature on the study of optimal investment and reinsurance in the continue-time setting (e.g., Schmidli [1], Zeng and Li [2], Zhu et al. [3], and Deng et al. [6] and so on), the proposed strategies in Remark 1 also consider the intermediate performance of the insurer.

Remark 2 shows the time-consistent strategies when Equation (3) only considers the interest of the reinsurer. In this situation, the value function $V_t(w_t^1,w_t^2)$ and the function $f_{t,\tau }\left(w_t^1\right)$ only depend on the wealth value $w_t^2$ (i.e., the reinsurer’s wealth at time period t).

Additionally, from Theorem 1 and Remarks 1 and 2, we can find that the reinsurance strategy and the investment strategies are independent of each other, that is, some changes in the reinsurance premium ${\delta }_t\left(q_t\right)$ will not affect the form of the investment strategies shown in Theorem 1. In the following, we will discuss the time-consistent strategies under some classical premium principles (e.g., the expected value principle and the variance value principle). The detailed results are presented in Section 3.1 and Section 3.2, respectively.