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A Model of Horizontal Merger Based on Shareholder-Competition and Firm Debt

ZHANG Li-bing1, 2，WANG Chu-ming1

1 Department of Finance, Shanghai Lixin University of Commerce, P.R.China, 201620 2 Antai School of Economics and Management, Shanghai Jiaotong University, P.R.China, 200052

Abstract: Merger is a relatively rapid and

convenient means for firms to expand and enhance competitive ability. The existing literatures take no account of the fact that there’s debt in most firms’ capital structure, whereas take the object of M&A as to maximize the firm value change. However, the decision-makers should be the stockholders of both firms, thus the merger object is to maximize the equity value. Applying the analysis method of contingent claim pricing, this paper constructs a merger model based on shareholders maximizing equity value, and finds the necessary conditions of merger, containing some linear or non-linear functions. The numerical results are significant, showing that the debt scale of the both firms affects the merger threshold, and the threshold grows higher as the debt scale grows. Although our study is only on horizontal mergers, it contributes to better understanding the effect of debt.

Keywords: cash flow, merger, bankruptcy, perpetual debt 1 Introduction

Merger and acquisition, i.e. M&A, a relatively rapid and convenient means of firm expansion, has always been a hot subject of economic and financial study, for a firm can kill its competitor, enter a new industry, or obtain the resources up or down the industry chain, by horizontal, mixed and lengthways M&A respectively. Although many researchers have studied both theoretically and empirically the motives and performances of M&A in the world[1-7], the relation between the timing and terms of takeovers is still unclear. Vast existing papers of takeover models have set the timing exogenously.

Since to merge or not is an option of each firm respectively, many researchers have began to analyze the M&A using structural models, especially real option ones, which specialize in timing decision as McDonald and Siegel[8], Pindyck[9], Dixit and Pindyck[10] have presented. Although Magrabe[11] is often regarded as the first to analyze takeovers as exchange options in whose

Supported by the National Natural Science Foundation of China (70701023)

model the takeover involves a zero-sum game and timing is exogenous, Lambrecht[12] is the first researcher to study horizontal takeover using real option method obtaining endogenous timing, in whose model the underlying source of uncertainty is the same for both firms. Morellec and Zhdanov[13] extended Lambrecht’s model to a situation, where the two firms have different but related cash-flow, which means the uncertain factors of the two firms are correlative. Zhang and Wu[14], however, combined the product market and firm’s management ability into firm’s cash flow and studied their effect on takeover timing. Except for these mentioned above, there are still some related literatures[15-17].

All these literatures suffer some limits in spite of the fact that they have solved some problems. These limits include:

The effect of debt on merger is still unclear. All these papers mentioned above focus on the firm value, that is, they don’t think of debt and thus the aim of decision makers is to maximize firm value. As we know, firms in real economy bear some debt and have some relatively good capital structures[18,19], on this occasion, shareholders’ interest is not to maximize firm but equity value. From this perspective, the models of these existing papers are limited.

Competition between the two firms in M&A is seldom taken into account. In most of these models, such as Lambrecht[12], the decision makers focus on the maximization of merged firm value, thus they cooperate to some extend, in other words, they try to make the cake as big as possible firstly and then decide how to divide it. The models like Lambrecht care only the first step, not the division. Actually, the shareholders can’t be so rational, they only want to make the part of their own as big as possible, not the summation of the two parts, in that they think others would think so. In this way, the shareholders from two firms are competitive and each side of them wants to maximize their own share, which means at any moment, coming to an agreement on takeover depends on the consistence of interest shares of the merged firm

As for horizontal mergers, the relative size of the two firms must be considered. Firm size may affect the partition of the merged firm equity, for institution, those

2008 International Conference on Management Science & Engineering (15th) September 10-12, 2008 Long Beach, USA

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shareholders from a bigger firm should have a larger equity partition if the two firms have the same finance situation. However, if the two firms have different situation and size, how will the partition be?

To study the effect of debt on mergers, particularly the timing, is very important. As we all know, debt may cause firm bankruptcy. A firm with merger opportunity may wind up before merger because of the shortage of cash flow, thus the life of the option to merge depends on the bankruptcy trigger, and while on the contrary, a firm without any debt surely has no probability of bankruptcy and thus has a perpetual option of merging with another one. On the other hand, capital scale is a substitute description of firm size, thus the effect of capital scale is that of firm size too. If the firm sizes are small in some period, then merger causes benefit, for scale economy is positive.

Since nearly no existing papers have combined these aspects into a model, we try to do so. For simplicity, we consider only horizontal merger, and suppose the motive of merger is scale economy as that in Lambrecht[13]. However, our model is much more complicated: the shareholders of the two firms are competitive, which is quite different from that of Lambrecht[13], and the decision makers are to maximize the equity value because of the introduction of firm debt, which is different from that of Zhang and Wu[14], or of Morrelec and Zhdanov[13].

The rest of this paper is organized as follows: in section 2, the firm’s cash flow is analyzed, constructing the interaction between the firm’s product price and its capital scale, which, of course, is also representative of firm size. In section 3, we analyze the aim of merger decision-makers, to maximize their own equity value, not firm value, thus valuate the equity of both firms before and after merger exogenously; we obtain the merger threshold standing for the merger timing and share-division. In section 4, we summarize some numerical results of our model, and section 5, conclusions. 2 Cash flow

For research conveniences, we suppose as follows. Firstly, there are two firms that have a potential

opportunity to merge with the other and are in the same industry. This kind of supposing is similar to that of Lambrecht. Furthermore, we suppose that the assets of the two firms are tradable and the financial market is efficient, which means that no hidden information are not contained in asset value.

Secondly, no agency-conflict between manager and shareholders, or between shareholders themselves.

Thirdly, the merger is based on equity payment, which means that the shareholders of the two firms will be the those of the merged firm and

Lastly, the tax and cost of bankruptcy are not taken into account.

With these hypothesis, our study is in the

neoclassical framework, thus we can apply the contingent claim analysis method presented by Duffie[20] and others.

In this paper the merger is horizontal, thus the supposing is reasonable that both related firms have the same product market, where the product price follows the following process

/ ( )dP P dt dwμ δ σ= − + (1) where 0δ > , stands for the cost ratio of goods-holding, for instance, the storage fares[8]. Standard Brownian motion w describes the risk factor of the economy, whileσ is volatility and μ , expected return. Notice that σ and μ are both constant. As we think of the industry, P can be regarded as the main industry risk factor.

Suppose that firm i has its cash flow ix with the following formulation

i ix AK Pρ= (2)

where iK is firm i ’s capital scale, 0ρ > reflects the effect of capital scale on firm’s cash flow, while constant A contains other potential factors affecting cash flow of

the firm, such as management, technology, and so on. Formulation (2) is similar to Lambrecht[12], who supposed that the initial motive of merger is to obtain the economy of scale in his paper. Here in this paper, 1ρ > means economy, while 1ρ < , diseconomy.

Because the merged firm is still in the same industry as the two before merger, we can thus suppose the merged firm has the following cash flow

m mx AK Pρ= (3)

where 1 2mK K K= + is the capital scale of the merged firm. The formulation (2) and (3) means in this paper, that the main role of firm cash flow is capital scale and product price. Actually, there may be other factors contained in the parameter A , but they are not what this paper focuses on.

To take firm debt into account, suppose that each firm has perpetual debt with continuous coupon rate 0, 1,2ic i> = . Perpetual debt means that the firm can always issue new debt to pay the old one, which is a simplified way of asset pricing to consider the effect of capital structure. Although we can still consider the maturity date of debt, we won’t do that for simplicity and as we will soon find in the following sections, the effect of perpetual debt is significant enough. After merger, the merged firm takes over the debt of the two firms, thus the coupon paid by the merged firm is 1 2mc c c= + . 3 Merger decision 3.1 Equity before merger

After the introduction of capital structure, we have

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to say, the aim of the merger decision-maker, i.e. the share-holders, is to maximize the equity value, not firm value any more.

The shareholders of both firms have their own aim to maximize the equity-value iE , which is the present value of future cash flow under some risk-neutral probability and contains two parts: cash flow before either bankruptcy or merger, and the present value of equity value after merger. Thus the equity value before merger is

()

,1 ,2

,1 ,2 ,3

,2

,2 ,1

( )

, ,

( ){ } ,2

sup ( )

( ( ))

i i

i i i

i

i i

Q r z ti t i it

r t mi i i

E E x c e dz

e E x

τ τ

τ τ τ

ττ τ τ

∧ − −

− −<

= −

+

∫1

()

,1 ,2

,1 ,2 ,3

,2

,2 ,1

( )

, ,

( ){ } ,2

sup ( )

( ( ))

i i

i i i

i

i i

Q r z tt i it

r t mi i

E AK P c e dz

e E P

τ τ ρ

τ τ τ

ττ τ τ

∧ − −

− −<

= −

+

∫1

(4)

where the notation,1 ,2 ,3, ,sup

i i iτ τ τdenotes

choosing 1 2 3, ,i i iτ τ τ to maximize, and 1 2 3, ,i i iτ τ τ are the times at which firm i is bankrupted before merger, merges with the other one, and is bankrupted after merge respectively. Q

tE is conditional expectation at t under the neutral probability. The notation ∧ is to choose the smaller one, and { }1 i is an indicator, equaling to 1

when { }i stands and 0 otherwise. i iAK P cρ − is the

firm i ’s cash flow before merger and bankruptcy, and

,2( ( ))mi iE P τ is the firm i ’s equity value at merger

time. Risk free rate r is set as constant. With Ito’s formula and (4), we can

have 2, ,

1( ) ( )2i i i i P i PPdE AK P c dt E dP E dPρ= − + + ,

applying standard contingent claim analysis method, we can obtain the following differential function that iE follows

2 2, ,

1( )2i i i i P i PPrE AK P c r PE P Eρ δ σ= − + − + (5)

Notice that in function (5), there’s no term of ,i

i tEEt

∂=

∂,

it’s because there’s no debt maturity. Solve function(5)，we can have

1 2,1 ,2

i ii i i

AK P cE B P B Pr

ρα α

δ= − + + (6)

where , , 1, 2i jB j = are constant parameters to be

determined later, and 1 0α < and 2 0α > , are respectively the negative and positive root of the

quadratic equation 21( ) ( 1)2 yr r x x xδ σ= − + − , thus

2 2 2 2

1 2

1 1( ) ( ) 22 2

r r rσ δ δ σ σα

σ

+ − − − − +=

and

2 2 2 2

2 2

1 1( ) ( ) 22 2

r r rσ δ δ σ σα

σ

+ − + − − += .

Notice that ( )( )Q r z ti i i

t it

AK P c cE AK P e dzr r

ρρ

δ∞ − −− = −∫

is the present value of perpetual cash flow of firm i ’s shareholders and also the value of the equity without the opportunity of merger or bankruptcy, thus

1 2,1 ,2i iB P B Pα α+ is the combined value the options

both to merge and to be bankrupted. Before merger, shareholders of each firm face two

kinds of choice, to declare bankruptcy, or to merge with the other firm. These two kinds of potential behavior are related with some thresholds, or triggers, which we set as

*,1iP and *

,2iP , that is, *,1iP and *

,2iP are firm i ’s bankruptcy and merger trigger points respectively as the following analysis, and we also set

*, ,inf{ : ( ) }i j i jt P t Pτ = =

as the jth time of firm i ’s shareholders, which is also

first passage-time up or down to *,i jP of the

process P . To find out directly *,i jP is apparently easer

than to find out these first passage-times. (1) *

,1iP P=

When *,1iP P= , firm i ’s shareholders choose

bankruptcy, and equity value satisfies the following two conditions

*,1( ) 0i iE P = (7)

*,1,

0ii P

E = (8)

Formulation(7)is the value-matching condition of firm i ’s shareholders’ equity, at which the shareholders give up their claim, thus equity value is 0. Formulation(8)is smooth-pasting condition, the necessary condition of maximizing equity value. According to formulation (6), formulation (7) and (8) change into the following two functions

1 2

*,1 * *

,1 ,1 ,2 ,1 0i i ii i i i

AK P c B P B Pr

ρα α

δ− + + = (9)

1 21 1* *,1 1 ,1 ,2 2 ,1 0i

i i i iAK B P B P

ρα αα α

δ− −+ + = (10)

(2) *,2iP P=

When *,2iP P= , firm i ’s should merge with the

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other one, and equity value satisfies * *,2 ,2( ) ( )m

i i i i iE P I E P+ = (11)

*,2

*,2

, i

i

mi

i PP P

EEy

=

∂=

∂ (12)

Formulation(11)means when merging, equity value plus

iI equals equity value after merger. This is the value

matching condition, and iI is the constant cost that firm i pays at merger trigger and can be called as sinked cost. Formulation (12) is the smoothing condition. Equity value after merger, m

iE , will be determined in section3.2. From the analysis above, we can see that before

merger, firm i ’s equity value has the following form

1 2

*,1

* *,1 ,2 ,1 ,2

0,

,

i

i i ii i i i

P PE AK P c B P B P P P P

r

ρα α

δ

⎧ ≤⎪= ⎨

− + + < ≤⎪⎩

3.2 Equity after merger

If firm i ’s share iλ after merger is determined

aforehand, the equity value miE can be written as the

following formulation ,3

,3

,3

,3

( )

( )

( ) sup ( )

sup ( )

i

i

i

i

m Q r z ti t i m mt

Q r z tt i m mt

E P E x c e dz

E AK P c e dz

τ

τ

τ ρ

τ

λ

λ

− −

− −

= −

= −

∫

∫(13)

In (13), mAK Pρ is the merged firm’s cash flow, while

m mAK P cρ − means that of all share-holders, including those of the previous firms.

As in section 3.1, formulation(13)satisfies the following function as

,

2 2,

( ) ( )12

m mi i m m i P

my i PP

rE AK P c r PE

P E

ρλ δ

σ

= − + −

+(14)

Solve function(14),we have

1 2,3

m m mi i i

AK P cE B P CPr

ρα αλ

δ⎛ ⎞

= − + +⎜ ⎟⎝ ⎠

(15)

where the constant parameters ,3iB and C are to be determined. Notice the fact that after merger, share holders face only one potential behavior, when to declare bankruptcy, and they have no chance to merge with other firms for simplicity. Because when P → ∞ , the net cash flow of equity is ∞ , and thus the probability of bankruptcy is 0, so we can have

,3

,3 ,3

( )

lim ( )

lim sup ( )i

i i

miP

Q r z tt i m mt

m mi

E P

E AK P c e dz

AK P cr

τ ρ

τ τ

ρ

λ

λδ

→∞

− −

→∞= −

⎛ ⎞= −⎜ ⎟

⎝ ⎠

∫(16)

Compare formulation(15)and(16), we find that 2lim 0

yCyα

→∞= , and surely 0C = . Thus the

shareholders’ equity value of merged firm is

1,3

m m mi i i

AK P cE B Pr

ραλ

δ⎛ ⎞

= − +⎜ ⎟⎝ ⎠

(17)

Now we consider the bankruptcy decision of merged firm’s shareholders. At *

,3iτ , i.e. when *,3iP P= , shareholders declare bankruptcy, and like

formulation (7) and (8), we have *

,3( ) 0mi iE P = (18)

*,3,

0i

mi P

E = (19)

Formulation(18)is the value-matching condition, while (19), the smooth-pasting condition. Here we should notice that formulation (7)-(8) and (18)-(19) are consistent with Leland[18], who supposed a firm’s debt was perpetual to find an optimal capital structure, while this paper takes no account of tax effect as Leland.

From formulation(18)and(19), and equity valuation after merger of the previous firm i ,

1,3

m m mi i i

AK P cE B Pr

ραλ

δ⎛ ⎞

= − +⎜ ⎟⎝ ⎠

with a given iλ , we

can obtain the merger threshold of shareholders from firm i

* *1,3 3

1 1m

im

cP PrAK ρ

δαα

= =−

(20)

and

1

*,3

,3 *,3

1( )m imi i

i

AK PcBr P

ρ

αλδ

= − (21)

We can see from the formulation(20) that the bankruptcy threshold is related to the total debt mc only,

but not iλ , which is because after merger, all the shareholders, no matter which firm do they come from, have been united into one, and thus have the same interest of bankruptcy. 3.3 Equilibrium of merger

So far, we have obtained the equity value of shareholders from each firm, but when will the two groups of shareholders come to merge?

We don’t need to care the process of the two group of shareholders’ negotiation, but focus on the necessary condition under which both group of shareholders would

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arrive at an agreement of merger. It’s obvious that at any moment, if the two firms merge, each group of the shareholders must acquire share of the merged firm according to maximizing there own equity value. Simply, provided that P is the threshold, firm i ’s share is iλ ,

thus cause a curve ( , )i Pλ . Only 1ii

λ =∑ stands, or in

other words, they agree with each other how the partition of the merged firm should be, can the merger happen. This is, of course, a local equilibrium. The intersection point of the two curves, 1( , )Pλ and 2( , )Pλ , is the equilibrium we try to find. This idea is similar to that of Zhang and Wu[14]. The competition between these two firms is thus introduced in this paper’s model.

According to the analysis above, we can see now when the two firms come to merge, there must be the following two conditions

* * *1,2 2,2P P P= = (22)

1, 1, 2ii

iλ = =∑ (23)

Formulation(22) means the consistence of merger timing, while (23), of shares. Combine formulation(9)—(12) and (17)—(23), we can have the equilibrium problem of merger changed into solving the following equations

1 2

1 2

1 2

1

1

1 2

*,1 * *

,1 ,1 ,2 ,1

1 1* *,1 1 ,1 ,2 2 ,1

** *

,1 ,2

*

* *,3

*3

1* *,1 1 ,2 2

0, 1,2

0, 1,2

( )

( ) , 1,2

i i ii i i i

ii i i i

i ii i

m mi

m imi

ii i

AK P c B P B P ir

AK B P B P i

AK P c B P B Pr

AK P cr

AK Pc P ir P

AK B P B P

ρα α

ρα α

ρα α

ρ

ρ α

α

ρα α

δ

α αδ

δ

λδ

λδ

α αδ

− −

− −

− + + = =

+ + = =

− + +

= −

+ − =

+ +

1

1

1

* 1*,3 1

*3

2

1

( ) , 1,2

1

m im mi i

ii

AK PAK c P ir P

ρρ α

α

αλ λδ δ

λ

−

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪ = + − =⎪⎪⎪ =⎪⎩∑ (24)

The group of equations (24) contains nine equations,

most of which are no-linear, and its solution is a vector * * *

1 2 1,1 1,2 2,1 2,2 1,1 2,1( , , , , , , , , )P B B B B P Pλ λ ,

where *P is the threshold to merge, implying the merger time. Obviously the equation group (24) is too complicated for us to obtain a closed-form solution, but has to be solved numerically.

There’s still the possibility that (24) has no resolution under some unknown parameter conditions, just as a game model has no equilibrium, which means under these conditions the merger cannot happen.

From (24) we know that the parameters of merger are difficult to obtained, thus the pricing of equity is difficult too. So we can surely conclude that the ordinary shareholders cannot judge correctly, this may be a cause that the stock prices have abnormal return when two firms have opportunity to merge.

4 Numerical tests and results

We can see now from section 3 that the effect of debt and capital scale on M&A is not easy to find out, in spite of this, we can say that the time or threshold of this paper will be different from that of those papers focusing on maximizing firm value, for maximizing firm value conflict with maximizing equity value to some extent because of capital structure, in other words, shareholders own interest may make them choose the bankruptcy and merger threshold not maximizing firm value and thus damage both firm and debt value. To see the results of this research, give some parameters and simulate the equations of (24). The basic parameters are set as follows 0.05r = , 0.02η = , 0.3σ = , 1A = , 1.1ρ = .

For simplicity, we set 0iI = .The debt and capital scale will be adjusted in numerical computation process so as to show there affect. The computing program is written with Matlab 7.1, and the main results are as follows.

Fig.1shows us the effect of debt on merger threshold,

standing for the merger time. In Fig.1, 1

m

KkK

= is the

relative ratio of capital scale, and 1 1 / mC c c= , the relative debt ratio of firm 1. As for the merger timing stood for by the threshold *P , the results under the two

conditions of 1

m

KkK

= and 21m

KkK

− = , must be

symmetric. It is similar to calculate the results based on

1C or 2C .

We can see directly from Fig.1, where 100mK =

in (a) and 20mc = in (b), that under the condition of constant total capital scale of the two firms, the lager total debt scale causes higher merger threshold, which thus causes the delay of merger. In the meantime, given the total debt, the merger threshold will change with the debt ratio of each firm.

When firm 1 has little debt as shown in Fig.1 (a), that is 1c is small, the threshold is relatively high and firm 1 asks much share according to our computation. While the debt ratio of firm 1 grows, the merger threshold and its share of the merged firm get down. In

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any cases, there exists a critical value of 1c , under which the threshold is the lowest and firm 1 and firm2 are both near bankruptcy point. Over the critical value, the threshold goes up. On the other hand, if the debt scale is given as in Fig.1 (b), the effect of debt ratio on merger threshold is similar to Fig.1 (a).

As for the effect of capital scale, we cans see from Fig.1 that it’s very significant.

Firstly, given the total capital scale, the relatively firm size or the capital scale affects the critical value as shown in Fig.1 (a). Particularly, the critical debt ratio of firm 1 is mainly affected by the relatively capital scale, if firm 1is small, the critical value is small, otherwise, it’s large.

Relative debt ratio of firm 1(C1)

Mer

ger t

hres

hold

(P*)

0

5

10

15

0 0.2 0.4 0.6 0.8 1.0

20

k=0.1

Relative debt ratio of firm 1(C1)

Mer

ger t

hres

hold

(P*)

0

2

4

6

8

10

0 0.2 0.4 0.6 0.8 1.0

12 k=0.3

Mer

ger t

hres

hold

(P*)

(a)

Secondly, as in Fig.1 (b), the total capital scale operates too. In general, provided that the total debt is set as constant, the higher capital scale causes lower merger threshold, the reason of which is that a higher capital scale generates a larger cash flow, reducing the bankruptcy possibility and thus the firms can wait longer to merge.

Relative debt ratio of firm 1(C1)M

erge

r thr

esho

ld (P

*)

0 0.5 1.00

5

10

15 k=0.1

M

erge

r thr

esho

ld (P

*)

Relative debt ratio of firm 1(C1)

0 0.5 1.00

2

4

6

8

10 k=0.5

(b) Fig.1 Merger threshold

(a)

(b)

Except for those results above, there is still something unsurprising that sometimes the merger

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threshold is just one of the two firms’ bankruptcy points, the one of the firm with higher debt level. The reason is that the firm with lower debt level asks a share of merged firm higher than that the other one would agree on, thus the merger has to be hindered till the industry business recession when one comes to bankruptcy that the other takeovers without any cost. 5 Conclusion This paper constructs a merger model based on firm debt and capital scale, supposing that the shareholders from different firms are competitive, and studies the effect of debt and capital scale on merger timing. As the debt gives shareholders an option to declare bankruptcy, the merger timing is sensitive to debt. Capital scale, which affects the cash-flow and the possibility of bankruptcy, and thus the merger decision. Applying the analysis method of contingent claim pricing, we construct a model based on shareholders maximizing equity value, and find the necessary conditions of merger, containing some linear or non-linear functions and needing numerical computation to solve.

The numerical computation results show that, the debt scale of the both firms affects the merger threshold, the threshold grows higher as the debt scale grows, for the high debt level enlarges the bankruptcy threshold and thus reduces the possibility of merger. Although our study is only on horizontal mergers, it contributes to better understanding the effect of debt. Furthermore, with this framework and other constraint conditions, we can study more aspects of mergers. Moreover, our study presents a method of pricing under the condition of debt and merger opportunity.

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