Abstract
We examine the optimal time to merge two first-line insurers with proportional reinsurance policies. The problem is considered in a diffusion approximation model. The objective is to maximize the survival probability of the two insurers. First, the verification theorem is verified. Then, we divide the problem into two cases. In case 1, never merging is optimal and the two insurers follow the optimal reinsurance policies that maximize their survival probability. In case 2, the two insurers follow the same reinsurance policies as those in case 1 until the sum of their surplus processes reaches a boundary. Then, they merge and apply the merged company’s optimal reinsurance strategy.
1. Introduction
Mergers of companies bring a range of benefits, such as diversification, management and operational risk decentralization, elimination of competition, tax reduction, and optimization of resource allocation. The topic has attracted more and more attention from scholars in recent years. The authors in [1] listed a number of advantages from mergers. The authors in [2] deemed that, in contrast to acquisition, little cash is paid during a merger and the merger is realized through the exchange of shares. The authors in [3] examined the effect of mergers on the wealth of firms’ shareholders. To learn more about companies’ mergers, see [4–6] and so on.
However, the above analysis is all qualitative and only little quantitative work has been done. Only the authors in [7] considered the problem of a merger of two companies with dividend policies. Their objective was to maximize the sum of the two companies’ expected discounted value. They constructed a situation in which the merger of the two companies results in a gain and gave a useful guideline on corporate governance. An open problem of finding the optimal time to merge in a more realistic situation was raised at the end of this paper. The authors in [8] solved this problem with some additional conditions. In this paper, we also determine the optimal time to merge, but it is different from what was found in [7, 8]:(i)In this paper, we seek to find the optimal time to merge to maximize the survival probability of two first-line insurers. The problem is a mixed regular control/two-dimensional optimal stopping problem (for optimal stopping problems, see [9–11]).(ii)The problem is considered with proportional reinsurance (for optimal reinsurance problems, see [12–15]).
In Theorem 2, we give the verification theorem of this problem. To find the optimal strategy and the value function, we focus on two critical inequalities and consider the problem separately in two cases. In case 1, never merging is optimal and the two insurers apply the optimal reinsurance strategies that maximize their survival probability. The calculations in case 2 are more complex. First, we construct a function . In Lemma 2, we analyze the property of this function. Then, the constructed function is shown to satisfy the conditions in Theorem 2. Finally, we prove that the constructed function is exactly the value function. The optimal policy can be obtained as a by product. The two insurers follow the optimal reinsurance policies that maximize their survival probability until the sum of their surpluses reaches a boundary c, and then they merge and apply the merged company’s optimal reinsurance strategy.
This paper is organized as follows. Section 2 presents the formulation. In Section 3, we analyze the reinsurance problem of the two first-line insurers without a merger and the reinsurance problem of the merged company, respectively. In Section 4, the conditions for a function to be greater than the value function are given. The value function and the optimal policy are derived in Section 5. Section 6 reveals the effects of all parameters on the optimal strategy and shows that the results are consistent with economic phenomena. Conclusions are presented in Section 7.
2. Problem Formulation
In this section, we set up the mathematical model of the problem. The problem is considered on a probability space . Suppose there are two insurers labeled 1 and 2. Their safety loadings are , and their risk processes are governed by compound Poisson processes. Similar to the procedure in [16], we suppose that the reserve processes of the two insurers arewhere , and are three independent Poisson processes defined on . Their intensities are , and , respectively. The claim sizes and are i.i.d. positive random variables with expectation and variance . Let be the underlying filtration.
Let be the reinsurance safety loading, where . With self-retention rate , insurer 1’s reserve process becomes
With self-retention rate , insurer 2’s reserve process becomes
If the two insurers merge, the merged company’s reserve process satisfieswhere is the cost of the merger, is the reserve of insurer 1 at the time to merge, is the reserve of insurer 2 at the time to merge, and is the safety loading of the merged company. Here, we assume that .
With self-retention rate , the merged company’s reserve process becomes
The martingale central limit theorem tells us that the diffusion approximation is a good approximation of a compound Poisson process provided the number of insurance contracts is large enough. Therefore, from now on, we consider the problem under the diffusion approximation model. According to [16], the approximated diffusion process of satisfies the following:where is a standard Brownian motion on , and
The approximated diffusion process of satisfies the following:
Considering a policy , where the control component represents the time of the merger, represent the proportions of risks undertaken by insurer before the merger, and represents the proportion of risk undertaken by the merged company after the merger. Denote the total surplus of the two companies at time with policy by . Then, we can getwhere and are the initial values of the two insurers. Let . A control policy is said to be admissible if(i) are progressively measurable(ii) is an -stopping time and (iii)There exists a unique nonnegative solution of equation (9) under the policy
We denote the set of all admissible controls by .
The two insurers want to determine an admissible control policy to maximize their survival probability (i.e., if the merger occurs, they want to maximize the survival probability of the merged company); that is, they want to maximize
Denote the value function by
Remark 1. The bankruptcy occurs if and only if the sum of the two insurers’ values reaches zero. So, in reality, the two insurers can be regarded as two subsidiaries of a company.
3. Preliminaries
First, let us analyze the optimal proportional reinsurance problem of the merged insurer m. Denote the survival probability of insurer m with reinsurance policy by . Then, the value function is
According to [12], we know that satisfieswhere
By some simple calculations, we can obtain that the optimal proportional reinsurance policy isand the optimal survival probability iswhere
Here,
Next, let us analyze the optimal proportional reinsurance policies of the two insurers if they do not merge. Define
Define
Here, is the proportional reinsurance policy of insurer 1 and is the proportional reinsurance policy of insurer 2. Let be the survival probability of the two insurers with policy if the merger does not occur. That is,
Define
The same methods used in [12] show that
Letwhere
Because considering proportional reinsurance policy 1 makes no sense, we can make and less than 1 by taking appropriate parameters. Then, we can obtain that the optimal reinsurance policy is and the optimal value function iswhere
In Section 4, we will consider two cases:(i)(ii)
We will show in case 1 that the two insurers do not merge; in case 2, the two insurers follow the reinsurance policy until the sum of their reserve processes reaches a boundary , and then they merge and follow reinsurance policy .
In the following, we give two basic equations that are critical to find the value function. If the two insurers apply policy , then
If the two insurers apply policy , then
4. The HJB Equation and the Verification Theorem
In this section, we give a verification result about . This result will help us find the optimal strategy and the value function of our problem. The following theorem gives a crucial equation to prove the verification result.
Theorem 1. The value function satisfies
Proof. First, since for any , we haveTaking supremums on both sides of equation (31) with respect to , we can getOn the other hand, , construct a new policy , and we can easily getLet , thenSincetaking supermums on both sides and combining with equations (33) and (34), we can obtainThen, the proof is finished.
Next, we give a verification result about .
Theorem 2. Suppose that we can find a nonnegative function , piecewise twice continuously differentiable on with bounded derivative and satisfying the following:(1)(2)With the initial condition, . Then, for all .
Proof. For any control policy , suppose and consider . Using a generalized It ’s formula from 0 to , we can getSince is bounded, taking expectations on both sides and using the two conditions in this theorem, we can getTaking supremums with respect to on both sides and referring to Theorem 1, we can obtain the result.
5. The Value Function and the Optimal Strategy
The following theorem tells us that if , the two insurers never merge and follow reinsurance policy .
Theorem 3. If , then
Proof. Using equations (16) and (26), we can see that if , thenOn the other hand,Therefore, satisfies the conditions in Theorem 2; thus,Sincethe proof is completed.
The following lemma defines a function . For , we will prove that is the value function in Theorem 4.
Lemma 1. Let
Then,where satisfies
Proof. Using the optimal stopping theorem, we can obtain thatFurthermore, there exists a , for ,and for ,Solving equation (48), we can obtainwhere k is the undetermined coefficient. Using the smooth fit principle, we know that are determined byBy simple calculations, we can getLemma 2 is used to prove that satisfies condition 2 in Theorem 2.
Lemma 2. If , for , we have
Proof. Sincecombining with equation (52), we can obtainAccording to equations (16) and (26), defineClearly, if , is strictly decreasing. For , we haveThis implies thatFurthermore, and then we haveThus, ,Taking supermums on both sides, we complete the proof.
Theorem 4. If , then . The optimal strategy is that the two insurers follow the reinsurance policies that maximize their survival probability until the sum of their surplus processes reaches c, and then they merge and apply the merged company’s optimal reinsurance strategy.
Proof. First, by the definition of , we know thatFor ,which implies thatCombining with Lemma 2, we have for ,Thus, the two conditions in Theorem 2 are satisfied, and we can obtainOn the other hand,Then, we haveClearly, by the definition of , the optimal strategy is that the two insurers follow the reinsurance policies that maximize their survival probability until the sum of their surplus processes reaches c, and then they merge and apply the merged company’s optimal reinsurance strategy.
In this case, the optimal merge time is as follows:
6. Illustration of the Results
In this section, we discuss the effects of all the parameters on the optimal policy. determines whether or not to merge, so in Section 6.1, let us show the effects of the parameters on the symbol of .
6.1. Effects of all the Parameters on
Figures 1–7 give the effects of all the parameters on .
Figure 1 shows that has a positive effect on but has no effect on . So, increases as increases. At the beginning, ; they are equal near ; for , .
Figures 2 and 3 show that and have positive effects on but have no effect on . So, decreases as or increases. At the beginning, ; they are equal near and , respectively. The two figures also indicate that the merged company has greater survival probability with a smaller safety loading.
Figure 4 shows that has a negative effect on both and . Furthermore, decreases more quickly than as increases. At the beginning, ; they are equal near ; for , . This indicates the following:(i)The merger has more and more advantages as increases(ii)The merged company is better at resisting reinsurance rate risk
Figure 5 shows that has negative effects on both and . Furthermore, decreases more quickly than as increases. This indicates the following:(i)The stronger the risk correlation (), the smaller the survival probability (refer to catastrophic insurance).(ii)The merged company’s survival probability is more sensitive to the risk correlation (). So, the merger has more and more disadvantages as increases.(iii) is sensitive to . A small increase in results in a change in the symbol of . So, for different , we can set different to get a better merged result.
Let and ; that is, the safety loading of insurer 1 is greater than the safety loading of insurer 2. It implies that insurer 1 is an insurer with a better reputation and service. We plot Figures 6 and 7 to illustrate the effect of different insurers’ idiosyncratic claim intensities on their optimal policy.
Figure 6 shows that has positive effects on both and . Furthermore, increases more quickly than as increases. So, decreases as increases. At the beginning, ; they are equal near ; for , . This indicates the following:(i)The business expansion of insurer 1 results in greater survival probabilities regardless of whether merger occurs (this is clear because the business expansion of a better insurer will bring more profits than risks)(ii)The merger has more and more disadvantages with the business expansion of insurer 1
Figure 7 shows that has a positive effect on and has a negative effect on . At the beginning, ; they are equal near ; for , . This indicates that the business expansion of the bad insurer (insurer 2) decreases the survival probability, but if it is merged with some good insurer (insurer 1), the business expansion increases survival probability.
If we know , we can decide whether to merge. Thus, in this section, we have determined whether or not to merge for different situations. In the next section, we consider, for , the effects of , and on the time to merge. This is equivalent to analyzing the effects of , and on .
6.2. Effects of on
Figures 8–10 present the results of the problem when .
Figure 8 shows that has a negative effect on . This indicates the following:(i)As increases, the gap between and becomes larger and larger, synergy becomes more and more obvious, and the boundary of the merger becomes lower and lower(ii)As increases, the gap between and becomes larger and larger, and the slope of the line approaches zero (this result is consistent with the diminishing marginal effect)
Figure 9 shows that has a positive effect on . This indicates the following:(i)As increases, the gap between and becomes smaller and smaller and the boundary of the merger becomes higher and higher(ii)As decreases, the gap between and becomes larger and larger and the slope of the line approaches zero (this result is consistent with the diminishing marginal effect)
Figure 10 shows that has a positive effect on (the larger the cost of the merger, the higher the boundary of the merger).
7. Conclusion
Economies of scale, competitive advantage theory, and agency theory have led to the rapid development of enterprise merger and acquisition theory, making them one of the most active fields in Western economics. However, the existing research results are mainly address the motivation for mergers and acquisitions. From the perspective of enterprise management and financial analysis, those papers mainly focus on economies of scale, management efficiency, and enterprise pricing. Most of these research results are qualitative analysis and ignore the measurement of enterprise risk.
From the perspective of risk control, this paper gives the optimal merger time and reinsurance strategies of two insurance companies by means of optimal stopping theory and stochastic optimal control theory. By analyzing the influence of changes in parameters on the merger strategy, we obtain many meaningful conclusions. For example, the merged company is more competitive and more adaptable to changes in reinsurance rate. Expanding the business of the company with a better reputation and service will reduce the bankruptcy probability. These conclusions are in line with the theories of economies of scale and competitive advantage. We also find that the more obvious the advantages of the company’s merger, the earlier the merger time; the higher the merger cost, the later the merger time.
This paper gives the optimal strategy on the premise of equal bargaining between two companies. However, the merger of two companies with different bargaining power is a topic worthy of further discussion. As this problem is more complex, it requires more auxiliary tools such as game theory and so on.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
Chuanzheng Li was supported by National Natural Science Foundation of China (11901404).