The Optimal Time to Merge Two First-Line Insurers with Proportional Reinsurance Policies

We examine the optimal time tomerge two first-line insurers with proportional reinsurance policies.*e problem is considered in a diffusion approximation model. *e objective is to maximize the survival probability of the two insurers. First, the verification theorem is verified.*en, 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. *en, they merge and apply the merged company’s optimal reinsurance strategy.


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. e topic has attracted more and more attention from scholars in recent years. e authors in [1] listed a number of advantages from mergers. e 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. e authors in [3] examined the effect of mergers on the wealth of firms' shareholders. To learn more about companies' mergers, see [4][5][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. eir objective was to maximize the sum of the two companies' expected discounted value. ey 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. e 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. e problem is a mixed regular control/two-dimensional optimal stopping problem (for optimal stopping problems, see [9][10][11]).
In eorem 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. e calculations in case 2 are more complex. First, we construct a function M(x). In Lemma 2, we analyze the property of this function. en, the constructed function is shown to satisfy the conditions in eorem 2. Finally, we prove that the constructed function is exactly the value function. e optimal policy can be obtained as a by product. e 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. is 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. e 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.

Problem Formulation
In this section, we set up the mathematical model of the problem. e problem is considered on a probability space (Ω, F, P). Suppose there are two insurers labeled 1 and 2.
eir safety loadings are η 1 and η 2 , 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 are where N 1 (t), N 2 (t), and N(t) are three independent Poisson processes defined on (Ω, F, P). eir intensities are λ 1 > 0, λ 2 > 0, and λ > 0, respectively. e claim sizes u i , i � 1, 2, . . . and u j , j � 1, 2, . . . are i.i.d. positive random variables with expectation μ and variance σ 2 . Let F t t≥0 be the underlying filtration.
Let θ be the reinsurance safety loading, where θ > η i , i � 1, 2. With self-retention rate b 1 , insurer 1's reserve process becomes With self-retention rate b 2 , insurer 2's reserve process becomes If the two insurers merge, the merged company's reserve process satisfies where I is the cost of the merger, x 1 is the reserve of insurer 1 at the time to merge, x 2 is the reserve of insurer 2 at the time to merge, and η m is the safety loading of the merged company. Here, we assume that η m ≤ θ.
With self-retention rate b m , the merged company's reserve process becomes e 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. erefore, from now on, we consider the problem under the diffusion approximation model. According to [16], the approximated diffusion process of Mathematical Problems in Engineering where B(t) is a standard Brownian motion on (Ω, F, F t t≥0 , P), and e approximated diffusion process of X b m m (t) satisfies the following: Considering a policy π � (T π , b π 1 , b π 2 , b π m ), where the control component T π represents the time of the merger, b π i,t (i � 1, 2) represent the proportions of risks undertaken by insurer i before the merger, and b π m represents the proportion of risk undertaken by the merged company after the merger. Denote the total surplus of the two companies at where x 1 and x 2 are the initial values of the two insurers. Let are progressively measurable (ii) T π is an F t -stopping time and T π ≤ τ π 0 (iii) ere exists a unique nonnegative solution of equation (9) under the policy π We denote the set of all admissible controls by Π. e 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. e 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.

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 b by δ b m (x). en, the value function is According to [12], we know that δ m (x) satisfies where By some simple calculations, we can obtain that the optimal proportional reinsurance policy is and the optimal survival probability is where Here, (18) Next, let us analyze the optimal proportional reinsurance policies of the two insurers if they do not merge. Define Define Here, b 1 is the proportional reinsurance policy of insurer 1 and b 2 is the proportional reinsurance policy of insurer 2. Let δ be the survival probability of the two insurers with policy (b 1 , b 2 ) if the merger does not occur. at is, e same methods used in [12] show that where Because considering proportional reinsurance policy 1 makes no sense, we can make b * 1 and b * 2 less than 1 by taking appropriate parameters. en, we can obtain that the optimal reinsurance policy is (b * 1 , b * 2 ) and the optimal value function is where In Section 4, we will consider two cases: We will show in case 1 that the two insurers do not merge; in case 2, the two insurers follow the reinsurance policy (b * 1 , b * 2 ) until the sum of their reserve processes reaches a boundary c, and then they merge and follow reinsurance policy b * m . In the following, we give two basic equations that are critical to find the value function. If the two insurers apply policy π m � (0, 0, 0, b * m ), then If the two insurers apply policy

The HJB Equation and the Verification Theorem
In this section, we give a verification result about δ(x). is result will help us find the optimal strategy and the value Proof. First, since for any π ∈ Π, we have Taking supremums on both sides of equation (31) with respect to π, we can get On the other hand, ∀π ∈ Π, construct a new policy π � (T π , b π 1 , b π 2 , b * m ; t ≥ 0), and we can easily get Since taking supermums on both sides and combining with equations (33) and (34), we can obtain en, the proof is finished. Next, we give a verification result about δ(x). □ Theorem 2. Suppose that we can find a nonnegative function w(x), piecewise twice continuously differentiable on [0, ∞) with bounded derivative and satisfying the following:

Proof.
For any control policy π ∈ Π, suppose X π 1 (0) + X π 2 (0) � x and consider w(X π 1 (t∧τ π 0 ) + X π 2 (t∧τ π 0 )). Using a generalized It o's formula from 0 to T π , we can get Since w ′ (x) is bounded, taking expectations on both sides and using the two conditions in this theorem, we can get Taking supremums with respect to π on both sides and referring to eorem 1, we can obtain the result.

The Value Function and the Optimal Strategy
e following theorem tells us that if k 1,2 ≥ k m , the two insurers never merge and follow reinsurance policy (b * 1 , b * 2 ).
Proof. Using equations (16) and (26), we can see that if k 1,2 ≥ k m , then Mathematical Problems in Engineering On the other hand, erefore, δ 1,2 (x) satisfies the conditions in eorem 2; thus, Since the proof is completed. e following lemma defines a function M(x). For k 1,2 < k m , we will prove that M(x) is the value function in eorem 4.

Lemma 1. Let
en, where c satisfies Proof. Using the optimal stopping theorem, we can obtain that Furthermore, there exists a c ≥ 0, for x < c, and for x ≥ c, Solving equation (48), we can obtain where k is the undetermined coefficient. Using the smooth fit principle, we know that k and c are determined by By simple calculations, we can get (53) Lemma 2 is used to prove that M(x) satisfies condition 2 in eorem 2.
combining with equation (52), we can obtain According to equations (16) and (26), define is implies that Furthermore, and then we have Taking supermums on both sides, we complete the proof. □ Theorem 4. If k 1,2 < k m , then δ(x) � M(x). 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 M(x), we know that 6 Mathematical Problems in Engineering (63) which implies that Combining with Lemma 2, we have for x ≥ 0, us, the two conditions in eorem 2 are satisfied, and we can obtain On the other hand, en, we have Clearly, by the definition of M(x), 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: □

Illustration of the Results
In this section, we discuss the effects of all the parameters on the optimal policy. k m − k 1,2 determines whether or not to merge, so in Section 6.1, let us show the effects of the parameters on the symbol of k m − k 1,2 . Figures 1-7 give the effects of all the parameters on k m − k 1,2 . Figure 1 shows that η m has a positive effect on k m but has no effect on k 1,2 . So, k m − k 1,2 increases as η m increases. At the beginning, k m < k 1,2 ; they are equal near η m � 0.24; for η m > 0.24, k m > k 1,2 . Figures 2 and 3 show that η 1 and η 2 have positive effects on k 1,2 but have no effect on k m . So, k m − k 1,2 decreases as η 1 or η 2 increases. At the beginning, k 1,2 < k m ; they are equal near η 1 � 0.26 and η 2 � 0.2, respectively. e two figures also indicate that the merged company has greater survival probability with a smaller safety loading. Figure 4 shows that θ has a negative effect on both k m and k 1,2 . Furthermore, k 1,2 decreases more quickly than k m as θ increases. At the beginning, k m < k 1,2 ; they are equal near θ � 0.49; for θ > 0.49, k m > k 1,2 . is indicates the following:

Effects of all the Parameters on
(i) e merger has more and more advantages as θ increases (ii) e merged company is better at resisting reinsurance rate risk Figure 5 shows that λ has negative effects on both k m and k 1,2 . Furthermore, k m decreases more quickly than k 1,2 as λ increases. is indicates the following: (i) e stronger the risk correlation (λ), the smaller the survival probability (refer to catastrophic insurance). (ii) e merged company's survival probability is more sensitive to the risk correlation (λ). So, the merger has more and more disadvantages as λ increases. (iii) k m − k 1,2 is sensitive to η m . A small increase in η m results in a change in the symbol of k m − k 1,2 . So, for different λ, we can set different η m to get a better merged result.
Let η 1 � 0.4 and η 2 � 0.35; 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 λ 1 has positive effects on both k m and k 1,2 . Furthermore, k 1,2 increases more quickly than k m as λ 1 increases. So, k m − k 1,2 decreases as λ 1 increases. At the beginning, k 1,2 < k m ; they are equal near λ 1 � 3; for λ 1 > 3, k m < k 1,2 . is indicates the following: (i) e 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) e merger has more and more disadvantages with the business expansion of insurer 1 Figure 7 shows that λ 2 has a positive effect on k m and has a negative effect on k 1,2 . At the beginning, k m < k 1,2 ; they are equal near λ 2 � 3; for λ 2 > 3, k m > k 1,2 . is 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 k m − k 1,2 , we can decide whether to merge. us, in this section, we have determined whether or not to merge for different situations. In the next section, we consider, for k m ≥ k 1,2 , the effects of k m , k 1,2 , and I on the time to merge. is is equivalent to analyzing the effects of k m , k 1,2 , and I on c.  Figure 8 shows that k m has a negative effect on c. is indicates the following: (i) As k m increases, the gap between k 1,2 and k m becomes larger and larger, synergy becomes more and       more obvious, and the boundary of the merger c becomes lower and lower (ii) As k m increases, the gap between k 1,2 and k m 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 k 1,2 has a positive effect on c. is indicates the following: (i) As k 1,2 increases, the gap between k 1,2 and k m becomes smaller and smaller and the boundary of the merger c becomes higher and higher (ii) As k 1,2 decreases, the gap between k 1,2 and k m 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 I has a positive effect on c (the larger the cost of the merger, the higher the boundary of the merger).

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. ese 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.
is 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
e authors declare that they have no conflicts of interest.