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

Authors: Yanan Li; Chuanzheng Li
Journal: Mathematical Problems in Engineering
 (2021)

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## 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.


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## Body

## 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 Theorem2, 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 Mx. 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. Section2 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 spaceΩ,ℱ,P. Suppose there are two insurers labeled 1 and 2. Their safety loadings are η1andη2, 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(1)X1t=x1+λ+λ11+η1μt−∑i=1N1t+Ntui,X2t=x2+λ+λ21+η2μt−∑j=1N2t+Ntuj,where N1t,N2t, and Nt are three independent Poisson processes defined on Ω,ℱ,P. Their intensities are λ1>0,λ2>0, and λ>0, respectively. The claim sizes ui,i=1,2,… and uj,j=1,2,… are i.i.d. positive random variables with expectation μ and variance σ2. Let ℱtt≥0 be the underlying filtration.Letθ be the reinsurance safety loading, where θ>ηi,i=1,2. With self-retention rate b1, insurer 1’s reserve process becomes(2)X1b1t=x1+λ1+λμ1+θb1−θ−η1t−b1∑i=1N1t+Ntui.With self-retention rateb2, insurer 2’s reserve process becomes(3)X2b2t=x2+λ2+λμ1+θb2−θ−η2t−b2∑j=1N2t+Ntuj.If the two insurers merge, the merged company’s reserve process satisfies(4)Xmt=x1+x2−I+λ1+λ2+2λ1+ηmμt−∑i=1N1t+Ntui−∑j=1N2t+Ntuj,where I is the cost of the merger, x1 is the reserve of insurer 1 at the time to merge, x2 is the reserve of insurer 2 at the time to merge, and ηm is the safety loading of the merged company. Here, we assume that ηm≤θ.With self-retention ratebm, the merged company’s reserve process becomes(5)Xmbmt=x1+x2−I+λ1+λ2+2λμ1+θbm−θ−ηmt−bm∑i=1N1t+Ntui+∑i=1N2t+Ntui.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 X1b1t+X2b2t satisfies the following:(6)X1b1t+X2b2t=x1+x2+μ1+μ2t+γ12+γ22+2ργ1γ2Bt,where Bt is a standard Brownian motion on Ω,ℱ,ℱtt≥0,P, and(7)μ1=λ1+λμθb1−θ−η1,μ2=λ2+λμθb2−θ−η2,γ1=λ1+λμ2+σ2b1,γ2=λ2+λμ2+σ2b2,ρ=λγ1γ2b1b2μ2.The approximated diffusion process ofXmbmt satisfies the following:(8)Xmbmt=x1+x2−I+λ1+λ2+2λμθbm−θ−ηmt+λ1+λ2+2λμ2+σ2+2λμ2bmBt.Considering a policyπ=Tπ,b1π,b2π,bmπ, where the control component Tπ represents the time of the merger, bi,tπi=1,2 represent the proportions of risks undertaken by insurer i before the merger, and bmπ represents the proportion of risk undertaken by the merged company after the merger. Denote the total surplus of the two companies at time t with policy π=Tπ,b1π,b2π,bmπ;t≥0 by Xπt. Then, we can get(9)Xπt=x1+x2−I1t≥Tπ+∫0t∧TπdX1πs+X2πs+∫t∧TπtdXmπs,where x1 and x2 are the initial values of the two insurers. Let τ0π=inft≥0:Xπt≤0. A control policy π=Tπ,b1π,b2π,bmπ;t≥0 is said to be admissible if(i)
bi,tπ∈0,1i=1,2,m are progressively measurable(ii)
Tπ is an ℱt-stopping time and Tπ≤τ0π(iii)
There exists a unique nonnegative solution of equation (9) under the policy πWe denote the set of all admissible controls byΠ.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(10)δπx=Pτ0π=∞|X1π0+X2π0=x.Denote the value function by(11)δx=supπ∈Πδπx.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 insurerm. Denote the survival probability of insurer m with reinsurance policy b by δmbx. Then, the value function is(12)δmx=supb∈0,1δmbx.According to [12], we know that δmx satisfies(13)supb∈0,1ℒmbδmx=0,where(14)ℒmbδmx=λ1+λ2+2λμθbm−θ−ηmδm′x+12λ1+λ2+2λμ2+σ2+2λμ2bm2δm′′x.By some simple calculations, we can obtain that the optimal proportional reinsurance policy is(15)bm∗=21−ηmθ∧1,and the optimal survival probability is(16)δmx=1−e−kmx,where(17)km=Aθ22θ−ηm,ηm≤θ≤2ηm,2Aηm,θ≥2ηm.Here,(18)A=λ1+λ2+2λμλ1+λ2+2λμ2+σ2+2λμ2.Next, let us analyze the optimal proportional reinsurance policies of the two insurers if they do not merge. Define(19)ℒ1,2b1,b2gx=μλ1+λθb1−θ−η1+λ2+λθb2−θ−η2g′x+12λ1+λμ2+σ2b12+λ2+λμ2+σ2b22+2λb1b2μ2g″x.Define(20)τb1,b2=inft≥0|X1b1t+X2b2t=0.Here,b1 is the proportional reinsurance policy of insurer 1 and b2 is the proportional reinsurance policy of insurer 2. Let δ1,2b1,b2x be the survival probability of the two insurers with policy b1,b2 if the merger does not occur. That is,(21)δ1,2b1,b2x=Pτb1,b2=∞|X1b10+X2b20=x.Define(22)δ1,2x=sup0≤bi≤1i=1,2δ1,2b1,b2x.The same methods used in [12] show that(23)supbi∈0,1,i=1,2ℒ1,2b1,b2δ1,2x=0.Let(24)b1∗=λ+λ1μ2+σ2−λμ2λ+λ1B,b2∗=λ+λ2μ2+σ2−λμ2λ+λ2B,where(25)B=2θ−η1λ1+λ+θ−η2λ2+λθσ2+μ2λ1+λ2+2λσ2.Because considering proportional reinsurance policy 1 makes no sense, we can makeb1∗ and b2∗ less than 1 by taking appropriate parameters. Then, we can obtain that the optimal reinsurance policy is b1∗,b2∗ and the optimal value function is(26)δ1,2x=1−e−k1,2x,where(27)k1,2=θμλ2+λ2b2∗μ2+σ2λ2+λ+2b1∗λμ2.In Section4, we will consider two cases:(i)
k1,2≥km(ii)
k1,2<kmWe will show in case 1 that the two insurers do not merge; in case 2, the two insurers follow the reinsurance policyb1∗,b2∗ until the sum of their reserve processes reaches a boundary c, and then they merge and follow reinsurance policy bm∗.In the following, we give two basic equations that are critical to find the value function. If the two insurers apply policyπm=0,0,0,bm∗, then(28)δx≥δπmx=δmx−I.If the two insurers apply policyπ0=∞,b1∗,b2∗,0, then(29)δx≥δπ0x=δ1,2x.


## 4. The HJB Equation and the Verification Theorem

In this section, we give a verification result aboutδx. 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 functionδx satisfies(30)δx=supπ∈ΠExδmX1πTπ+X2πTπ−I.Proof.
First, since for anyπ∈Π, we have(31)δπx=Ex1τ0π=∞=ExE1τ0π=∞|X1πTπ+X2πTπ≤ExδmX1πTπ+X2πTπ−I.
Taking supremums on both sides of equation (31) with respect to π, we can get(32)δx≤supπ∈ΠExδmX1πTπ+X2πTπ−I.
On the other hand,∀π∈Π, construct a new policy π¯=Tπ,b1π,b2π,bm∗;t≥0, and we can easily get(33)δπ¯x=ExδmX1πTπ+X2πTπ−I.
LetΠ¯=π¯=Tπ,b1π,b2π,bm∗;t≥0:π∈Π, then(34)supπ∈ΠExδmX1πTπ+X2πTπ−I=supπ∈Π¯ExδmX1πTπ+X2πTπ−I.
Since(35)δx≥δπ¯x,taking supermums on both sides and combining with equations (33) and (34), we can obtain(36)δx≥supπ∈ΠExδmX1πTπ+X2πTπ−I.
Then, the proof is finished.
Next, we give a verification result aboutδx.Theorem 2.
Suppose that we can find a nonnegative functionwx, piecewise twice continuously differentiable on 0,∞ with bounded derivative and satisfying the following:(1)
supbi∈0,1,i=1,2ℒ1,2b1,b2wx≤0(2)
wx≥δmx−I
With the initial condition,w0=0. Then, wx≥δx for all x≥0.Proof.
For any control policyπ∈Π, suppose X1π0+X2π0=x and consider wX1πt∧τ0π+X2πt∧τ0π. Using a generalized It o^’s formula from 0 to Tπ, we can get(37)wX1πTπ+X2πTπ=wx+∫0Tπℒ1,2πwX1πt+X2πtdt+∫0Tπγ12+γ22+2ργ1γ2w′X1πt+X2πtdBt.
Sincew′x is bounded, taking expectations on both sides and using the two conditions in this theorem, we can get(38)wx≥ExwX1πTπ+X2πTπ≥ExδmX1πTπ+X2πTπ−I.
Taking supremums with respect toπ 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 ifk1,2≥km, the two insurers never merge and follow reinsurance policy b1∗,b2∗.Theorem 3.
Ifk1,2≥km, then(39)δx=δ1,2x.Proof.
Using equations (16) and (26), we can see that if k1,2≥km, then(40)δ1,2x≥δmx−I.
On the other hand,(41)supbi∈0,1,i=1,2ℒ1,2b1,b2δ1,2x=ℒ1,2b1∗,b2∗δ1,2x=0.
Therefore,δ1,2x satisfies the conditions in Theorem 2; thus,(42)δ1,2x≥δx.
Since(43)δ1,2x≤δx,the proof is completed.
The following lemma defines a functionMx. For k1,2<km, we will prove that Mx is the value function in Theorem 4.Lemma 1.
Let(44)Mx=supτ∈TEδmX1π0τ+X2π0τ−I.Then,(45)Mx=kδ1,2x,x<c,δmx−I,x≥c,where c satisfies(46)kmek1,2−kmc+k12−kme−kmc=k1,2e−kmI,k=kmexpck1,2−km+kmIk1,2.Proof.
Using the optimal stopping theorem, we can obtain that(47)maxℒ1,2b1∗,b2∗Mx,δmx−I−Mx=0.
Furthermore, there exists ac≥0, for x<c,(48)ℒ1,2b1∗,b2∗Mx=0,and for x≥c,(49)Mx=δmx−I.
Solving equation (48), we can obtain(50)Mx=kδ1,2x,x<c,where k is the undetermined coefficient. Using the smooth fit principle, we know that kandc are determined by(51)kδ1,2c=δmc−I,(52)kδ1,2′c=δm′c−I.
By simple calculations, we can get(53)kmek1,2−kmc+k12−kme−kmc=k1,2e−kmI,k=kmexpck1,2−km+kmIk1,2.
Lemma2 is used to prove that Mx satisfies condition 2 in Theorem 2.Lemma 2.
Ifk1,2<km, for x>c, we have(54)supbi∈0,1,i=1,2ℒ1,2b1,b2Mx=supbi∈0,1,i=1,2ℒ1,2b1,b2δmx−I≤0.Proof.
Since(55)ℒ1,2b1∗,b2∗δmc−I≤kℒ1,2b1∗,b2∗δ1,2c=0,combining with equation (52), we can obtain(56)δm′′c−I≤kδ1,2′′c.
According to equations (16) and (26), define(57)Gx=δm′′x−Ikδ1,2′′x=ekmIkm2kk1,22e−km−k1,2x.
Clearly, ifkm>k1,2, Gx is strictly decreasing. For x>c, we have(58)δm′′x−Ikδ1,2′′x<δm′′c−Ikδ1,2′′c≤1.
This implies that(59)δm′′x−I<kδ1,2′′x,x>c.
Furthermore,(60)δm′c−I=kδ1,2′c, and then we have(61)δm′x−I<kδ1,2′x,x>c.
Thus,∀b1,b2,(62)ℒ1,2b1,b2δmx−I<kℒ1,2b1,b2δ1,2x≤0,x>c.
Taking supermums on both sides, we complete the proof.Theorem 4.
Ifk1,2<km, then δx=Mx. 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 ofMx, we know that(63)Mx≥δmx−I.
Forx≤c,(64)Mx=kδ1,2x,which implies that(65)supbi∈0,1,i=1,2ℒ1,2b1,b2Mx=ksupbi∈0,1,i=1,2ℒ1,2b1,b2δ1,2x=kℒ1,2b1∗,b2∗δ1,2x=0.
Combining with Lemma2, we have for x≥0,(66)supbi∈0,1,i=1,2ℒ1,2b1,b2Mx≤0.
Thus, the two conditions in Theorem2 are satisfied, and we can obtain(67)δx≤Mx.
On the other hand,(68)δx≥Mx.
Then, we have(69)δx=Mx.
Clearly, by the definition ofMx, 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:(70)T∗=inft|X1b1∗t+X2b2∗t=c.


## 6. Illustration of the Results

In this section, we discuss the effects of all the parameters on the optimal policy.km−k1,2 determines whether or not to merge, so in Section 6.1, let us show the effects of the parameters on the symbol of km−k1,2.


### 6.1. Effects of all the Parameters onkm−k1,2

Figures1–7 give the effects of all the parameters on km−k1,2.Figure 1
The effect ofηm on k12 and km.Figure 2
The effect ofη1 on k12 and km.Figure 3
The effect ofη2 on k12 and km.Figure 4
The effect ofθ on k12 and km.Figure 5
The effect ofλ on k12 and km.Figure 6
The effect ofλ1 on k12 and km.Figure 7
The effect ofλ2 on k12 and km.Figure1 shows that ηm has a positive effect on km but has no effect on k1,2. So, km−k1,2 increases as ηm increases. At the beginning, km<k1,2; they are equal near ηm=0.24; for ηm>0.24, km>k1,2.Figures2 and 3 show that η1 and η2 have positive effects on k1,2 but have no effect on km. So, km−k1,2 decreases as η1 or η2 increases. At the beginning, k1,2<km; they are equal near η1=0.26 and η2=0.2, respectively. The two figures also indicate that the merged company has greater survival probability with a smaller safety loading.Figure4 shows that θ has a negative effect on both km and k1,2. Furthermore, k1,2 decreases more quickly than km as θ increases. At the beginning, km<k1,2; they are equal near θ=0.49; for θ>0.49, km>k1,2. This indicates the following:(i)
The merger has more and more advantages asθ increases(ii)
The merged company is better at resisting reinsurance rate riskFigure5 shows that λ has negative effects on both km and k1,2. Furthermore, km decreases more quickly than k1,2 as λ increases. This indicates the following:(i)
The stronger the risk correlation (λ), the smaller the survival probability (refer to catastrophic insurance).(ii)
The 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)
km−k1,2 is sensitive to ηm. A small increase in ηm results in a change in the symbol of km−k1,2. So, for different λ, we can set different ηm to get a better merged result.Letη1=0.4 and η2=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.Figure6 shows that λ1 has positive effects on both km and k1,2. Furthermore, k1,2 increases more quickly than km as λ1 increases. So, km−k1,2 decreases as λ1 increases. At the beginning, k1,2<km; they are equal near λ1=3; for λ1>3, km<k1,2. 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 1Figure7 shows that λ2 has a positive effect on km and has a negative effect on k1,2. At the beginning, km<k1,2; they are equal near λ2=3; for λ2>3, km>k1,2. 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 knowkm−k1,2, 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 km≥k1,2, the effects of km,k1,2, and I on the time to merge. This is equivalent to analyzing the effects of km,k1,2, and I on c.


### 6.2. Effects ofk1,2,km,andI on c

Figures8–10 present the results of the problem when km≥k1,2.Figure 8
The effect ofkm on c.Figure 9
The effect ofk12 on c.Figure 10
The effect ofI on c.Figure8 shows that km has a negative effect on c. This indicates the following:(i)
Askm increases, the gap between k1,2 and km becomes larger and larger, synergy becomes more and more obvious, and the boundary of the merger c becomes lower and lower(ii)
Askm increases, the gap between k1,2 and km becomes larger and larger, and the slope of the line approaches zero (this result is consistent with the diminishing marginal effect)Figure9 shows that k1,2 has a positive effect on c. This indicates the following:(i)
Ask1,2 increases, the gap between k1,2 and km becomes smaller and smaller and the boundary of the merger c becomes higher and higher(ii)
Ask1,2 decreases, the gap between k1,2 and km becomes larger and larger and the slope of the line approaches zero (this result is consistent with the diminishing marginal effect)Figure10 shows that I has a positive effect on c (the larger the cost of the merger, the higher the boundary of the merger).


## 6.1. Effects of all the Parameters onkm−k1,2

Figures1–7 give the effects of all the parameters on km−k1,2.Figure 1
The effect ofηm on k12 and km.Figure 2
The effect ofη1 on k12 and km.Figure 3
The effect ofη2 on k12 and km.Figure 4
The effect ofθ on k12 and km.Figure 5
The effect ofλ on k12 and km.Figure 6
The effect ofλ1 on k12 and km.Figure 7
The effect ofλ2 on k12 and km.Figure1 shows that ηm has a positive effect on km but has no effect on k1,2. So, km−k1,2 increases as ηm increases. At the beginning, km<k1,2; they are equal near ηm=0.24; for ηm>0.24, km>k1,2.Figures2 and 3 show that η1 and η2 have positive effects on k1,2 but have no effect on km. So, km−k1,2 decreases as η1 or η2 increases. At the beginning, k1,2<km; they are equal near η1=0.26 and η2=0.2, respectively. The two figures also indicate that the merged company has greater survival probability with a smaller safety loading.Figure4 shows that θ has a negative effect on both km and k1,2. Furthermore, k1,2 decreases more quickly than km as θ increases. At the beginning, km<k1,2; they are equal near θ=0.49; for θ>0.49, km>k1,2. This indicates the following:(i)
The merger has more and more advantages asθ increases(ii)
The merged company is better at resisting reinsurance rate riskFigure5 shows that λ has negative effects on both km and k1,2. Furthermore, km decreases more quickly than k1,2 as λ increases. This indicates the following:(i)
The stronger the risk correlation (λ), the smaller the survival probability (refer to catastrophic insurance).(ii)
The 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)
km−k1,2 is sensitive to ηm. A small increase in ηm results in a change in the symbol of km−k1,2. So, for different λ, we can set different ηm to get a better merged result.Letη1=0.4 and η2=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.Figure6 shows that λ1 has positive effects on both km and k1,2. Furthermore, k1,2 increases more quickly than km as λ1 increases. So, km−k1,2 decreases as λ1 increases. At the beginning, k1,2<km; they are equal near λ1=3; for λ1>3, km<k1,2. 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 1Figure7 shows that λ2 has a positive effect on km and has a negative effect on k1,2. At the beginning, km<k1,2; they are equal near λ2=3; for λ2>3, km>k1,2. 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 knowkm−k1,2, 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 km≥k1,2, the effects of km,k1,2, and I on the time to merge. This is equivalent to analyzing the effects of km,k1,2, and I on c.


## 6.2. Effects ofk1,2,km,andI on c

Figures8–10 present the results of the problem when km≥k1,2.Figure 8
The effect ofkm on c.Figure 9
The effect ofk12 on c.Figure 10
The effect ofI on c.Figure8 shows that km has a negative effect on c. This indicates the following:(i)
Askm increases, the gap between k1,2 and km becomes larger and larger, synergy becomes more and more obvious, and the boundary of the merger c becomes lower and lower(ii)
Askm increases, the gap between k1,2 and km becomes larger and larger, and the slope of the line approaches zero (this result is consistent with the diminishing marginal effect)Figure9 shows that k1,2 has a positive effect on c. This indicates the following:(i)
Ask1,2 increases, the gap between k1,2 and km becomes smaller and smaller and the boundary of the merger c becomes higher and higher(ii)
Ask1,2 decreases, the gap between k1,2 and km becomes larger and larger and the slope of the line approaches zero (this result is consistent with the diminishing marginal effect)Figure10 shows that I has a positive effect on c (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.



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*Source: 1423325-2021-08-16.xml*