The Synergy and Timing of Multi-Firms’ Merges under Uncertainty
Kuo-Jung Lee
Shih Chien Univeristy, Kaohsiung Campus
David S. Shyu*
National Sun Yat-sen University
Abstract
By using real options approach, this paper constructs a model to examine the
multi-firms’ merges strategies under stochastic demand shock. The strategies include
the benefits and timing of merges. This study derives the synergy effect; timing of
multi-firms’ merges and shares between each other under the different market
structure. In addition, we find that increasing merger costs and product market
uncertainty tend to delay mergers. On the contrary, larger synergies or economy of
scale speed up mergers.
Keywords: Mergers, Real options, Threshold, Takeover game
*
Corresponding author. Department of Finance, National Sun Yat-sen University, 70 Lien-hai Rd.
Kaohsiung, Taiwan; Tel: +886-7-525-2000 ext. 4814; Fax: +886-7-525-4899.
E-mail addresses: leekj@pchome.com.tw (K.J. Lee) dshyu@cm.nsysu.edu.tw (D.S. Shyu),
1
The Synergy and Timing of Multi-Firms’ Merges under Uncertainty
1. Introduction
Most successful, large businesses have developed into large, interconnected
organizations either through self-development or mergers and acquisitions. Mergers
and acquisitions, hence, have been the subject of considerable research in financial
economics. The transactions of mergers and acquisitions are larger and strategic.
Many mergers are integrated by multi-firms simultaneously. These merging deals
consist of tree, four, or even more firms. In Taiwan, for example, Lite-On Electronics,
Silitek, Lite-On Technology, and GVC merge and become collectively known as the
Lite-On Technology Corporation in2002. In addition, Taiwan International Securities
Corporation attempts to merge with First Securities and Far Eastern Securities. This
paper structures the model to examine the profits and timing of multi-firms mergers.
That is, we study the problem of whether and when to merge when the case is
multi-firms merger and not just two firms merger.
The traditional approach used to value merger include discounted cash flow model,
asset appraisal model, and market comparative approach（Ernst and Young, 1994）.
These methods, however, ignores the value of timing and operation flexibility（Dixit
and Pindyck, 1994; Trigeorgis, 1996）. As emphasized by Trigeorgis (1993) and Dixit
and Pindyck (1994), among others, the options approach helps explain why actual
investment decisions by the business sector cannot be explained by conventional
wisdom such as the net present value (NPV) approach. In reality, firms invest only in
projects that are expected to yield a return well in excess of the required rate of return.
Baba (2001) thank that firms should consider the option to wait until economic
conditions improve sufficiently under the assumption of uncertainty and irreversibility.
This is a typical setting of a so-called real options approach first applied by McDonald
and Siegel (1984) and later extensively reviewed by Dixit and Pindyck (1994). Real
options analysis is more focusing on describing uncertainty and in particular the
managerial flexibility inherited in many investments.
In recent year, some studies applied real options approach to value the project of
mergers. The advantage is considering the flexibility value of mergers strategy under
uncertainty. Margrabe (1978) is the first to analyze takeovers as exchange options. In
his model, the takeover involves a zero-sum game and its timing is exogenous.
Shleifer and Vishny (2003) also consider a zero-sum game with exogenous timing. In
their framework, the market does not reflect the potential surplus associated with the
takeover in the stock market valuations of participating firms. Managers of the firms
2
understand stock market inefficiencies, and take advantage of them, in part through
merger decisions. Their model is consistent with available empirical findings about
characteristics and returns of merging firms. Herath and Jahera (2002) develop a
theoretical model based on option pricing theory to value managerial flexibility
arising in stock for stock exchanges. They show how a mergers and acquisition
(M&A) deal may be optimally structured as a real options swap by including
managerial flexibility of both the acquiring and target firms when stock prices are
volatile.
As argued by Huisman, Kort, Pawlina, and Thijssen (2004), the real option theory
mainly considers single decision maker problems of firms operating in monopoly or
perfect competition markets. But capital budgeting decisions can be strongly
influenced by existing as well as potential competitors. They conclude this brand new
topic requires a merge between game theory and real options. Grenadier (2000) finds
that only a few contributions, at present, deal with the effects of strategic interactions
on the option value of waiting associated with investments under uncertainty. One of
the main reasons is that the application of game theory to continuous-time models is
not well developed and often quite tricky. However, due to the importance of studying
the topic of investment under uncertainty in an oligopolistic setting, it can be expected
that more publications will appear in the immediate future.
Some articles use game theory to investigate the firms’ interaction of mergers. For
example, Kwoka (1989) studies the effect of market competition on a merger
production and profit. He finds that a merger occurs more easily in more competitive
intensity and in which the firm of post-merger makes higher profit. Rodrigues (2001)
assumes that merger occurrence is determined by the interplay of the initial number of
firms in the industry, the expected competitive intensity, and the possibility of
economizing on fixed costs through merger. He shows that the equilibrium market
concentration is decreasing in the first of these factors and increasing in the other two.
However, these studies are not within the dynamic economic environment and cannot
value the flexibility from takeovers.
The game-theoretic real options approach is employed to analyze the effects of
uncertainty on mergers decisions. These are used to model corporate restructuring as a
strategic investment decision under uncertainty. Related papers include Smith and
Triantis (1995), Anderson and Sundaresan (1996), Grenadier (1996), Grenadier
(1999), Mella-Barral (1999), and Grenadier (2000). There are a number of theoretical
papers that have modeled the terms of takeovers under alternative assumptions, but
usually not within the dynamic economic environment that is the setting in this paper.
A good review of the literature is given by Weston, Chung, and, Hoag (1990) and
Weston, Chung, and Siu (1998). A related paper by Smith and Triantis (1995)
3
examines the valuation of strategic options associated with acquisitions. They
illustrate how option-pricing theory can be used to value growth, flexibility and
divestiture options arising from takeovers.
Besides, Morellec (2001) develops an equilibrium framework for the joint
determination of the timing and the terms of takeovers. His article analyzes takeovers
as exchange options and derives equilibrium restructuring strategies by solving option
exercise games between bidding and target shareholders. Lambrecht (2004) thinks
that a merger is a cooperative decision where the shares of both firms are determined
according to Pareto optimality. So, firms decide on an optimal time first and then
share the profits accordingly. In a (hostile) takeover the target first chooses its profit
share upon which the acquirer determines the optimal time of takeover. Lambrecht
(2004) concludes that M&A activity only takes place during periods of economic
expansion.
The starting point of the paper is that firms merge to generate a surplus to their
shareholders. In the model, the surplus results from synergies arising from increased
returns to scale and a reduction in the unit production cost. The surplus is defined as
the difference between the market capitalization of the joint (merged) entity and the
value of the individual companies. Since the market capitalization of the merged
entity and the individual companies are influenced by economic uncertainty, it follows
that the surplus is likely to be influenced by economic uncertainty too. In other words
the gains from merging depend on the state of the industry or economic environment.
In this paper the source of uncertainty considered is the demand shock for the firms'
output.
Mergers not only generate benefits, however, but also bring with them costs such as
legal fees, fees to investment banks and other merger promoters, costs of restructuring
and integrating the two companies. These are typically one-off costs that, once
incurred, are sunk and irreversible. When merging, both companies therefore have to
trade off the stochastic benefit of merging against the cost of merging (Lambrecht,
2004).
The companies of mergers have the right (but not the obligation) to exchange their
current firm for a share in a new firm with higher returns to scale and lower unit costs.
The higher profits they are forgoing by not merging act as an incentive to exercise the
merger option. Given that corporate restructurings are (at least partially) irreversible,
the merging companies have to decide when it is optimal for them to exercise their
option to merge. A key issue in the paper is therefore to examine the optimal timing of
corporate restructurings under demand shock uncertainty and irreversibility. We
assume that all firms and investors have complete information. The assumption of
complete information is an important one as almost all theories on the timing or
4
returns of takeovers are based on asymmetric information arguments. We show that
takeovers that increase the firms' returns to scale occur in a rising product market.
The research presented in this paper is motivated by the fact that present takeover
projects are more blooming and complicate. A merger project maybe includes tree,
four, or possible more firms. The above literatures only analyze the merger case of
two firms. To include this kind of mechanism, this paper extends the model of
Lambrecht (2004) by incorporating more than two firms in mergers. The main aim of
this paper is to provide a theoretical strategy in multi-firms mergers. We, hence,
derive closed form solutions for the takeover options, optimal threshold, and
acquirer's and acquiree's ownership shares. We also find that increasing merger costs
and product market uncertainty tend to delay mergers.
From a methodological point of view, this paper also uses continuous-time real
options techniques and game theoretical concepts, which enable us to model corporate
restructuring as a strategic investment decision under uncertainty. The market
structure we assume is an oligopolistic market in which the firms will extend their
size, increase market power or improve their efficient by merging. Even by acquiring
numerous firms simultaneously the acquirer will be a monopoly in the industry.
The remainder of the paper is organized as follows. Section 2 describes the basic
theoretical framework of the real options approach. We show that this surplus is a
convex increasing function of the output price and that it has option-like
characteristics. Section 3 derives the optimal merger timing from a global optimizer’s
viewpoint. We show that mergers are optimal in a rising market, causing merger
activity to be pro-cyclical. We provide a closed form solution for the merger timing. A
closed form solution for the post-merger ownership shares is provided. Section 5
concludes the study.
2. The basic models
Consider an oligopolistic industry comprised of n identical firms producing a single,
homogeneous good. Firm i produces qi units of output. Output is infinitely divisible.
Assume that the market inverse demand function is of a constant-elasticity form:
P  XQ
 1
(1)
where P is the price of a unit of output, X is an exogenous shock process to demand,
and Q is the industry supply process. We assume that  > 1/n to ensure that marginal
profits (for any assumed value of n) are increasing in X.1 X represents a multiplicative
1
Concavity of the profit function with respect to each firm’s output is guaranteed for all n > 1.
5
demand shock, and evolves as a geometric Brownian motion:
dX  X dt   X dz
(2)
where  denotes the expected growth rate parameter,  the volatility parameter, and dz
the increment of the standard Wiener process. Assume that r   in order to ensure
convergence.
In addition, following the setting of Rodrigues (2001), the cost functions of firms
are:
TCi = cqi + F
(3)
where F and c are, respectively, fixed and marginal costs, which we assume are
identical for all firms, and qi is the quantity i supplied by firm i. Of course, Q = qi + q-i,
with q-i representing the quantity supplied by the ith firm (n - 1) competitors. Hence,
firms’ profits are:
 i ( X t )  pi ( X t )qi  cqi  F
(4)
Rodrigues (2001) let i  dqi / dqi represent firm i’s conjectural variation, i.e. its
expectation regarding the change in its competitors production resulting from a
change in its own production level, and assume that this conjecture is identical for all
firms ( i   ). We also use the definition and refer to  as the competitive intensity
of the industry, with lower values of  corresponding to more intense competition.
The range of  from -1 to n-1 represent, respectively, competitive equilibrium and
perfectly collusive one.
From the equation (4), the profit maximization requires that  i / qi  0 . It is
then straightforward to show that, in equilibrium:
X
qi  1 
(5)
nc
In order to ensure the quality is positive, it requires   (  1) / n . Substituting the
equilibrium production into  i , each firm’s profit can then be written:


 1 
n
   1  1    1   X 
  1 
 
 i  1 
F
n 
n   nc 1


(6)
One can easy find that the fixed and marginal costs will reduce each firm’s profit. As
for the inference of firms’ number among firms in the industry on firm’s profit, we
can see from the following equations.
 1

X
  z (1  z ) 2 2  z   z  2  1  0 , z 
1
n
n
n c
where the third term can be rewritten by [(1  z )  (1  n1 )] , which is positive.
6

1  z    1 
 2  1 1 
X  0
 n c 
n 
The increasing firms’ number and competition among firms in the industry will
decrease each firm’s profit. This result is the same with that of Kwoka (1989) and
Rodrigues (2001).
Applying Ito’s lemma, we can get dynamic program of profit function:
d i ( X t )
 [i  12  i ( i  1) 2 ]dt   i dz
i (Xt )
(7)
The value of the firm, Vi, can be seen as the discounted value of all future profits and
is given by:
Vi ( X )   i (,n,c)
X F

n
r
(8)
   1  1    1  
1
  1 
 
 i  1 
n 
n   [r  g ( )]c 1


where r is the riskless rate, g () 1 [(r   )  12  2 ( 1  1)] , and   r   2.
Suppose that m + 1 of the n firms comprising an industry merger, i.e. m + 1 firms
merger. The new number of firms in the industry will be n’ = n – m. Assume, further,
that the merger has no impact on firms’ conjectural variations and marginal costs. The
post-merger equilibrium will still be characterized by identical production for each of
the n – m firms, with:
q
m 1
i
 1 
 1
 ( nm)


X
( n  m) c 
(9)
where the superscript, m + 1, denotes that m + 1 firms comprise an industry merger.
Similarly, this equation requires (  1) /  (n  m)  1 ensuring that the post-merger
quality is positive. From the equilibrium production before and after merger, it can be
shown that the firm resulting from the merger will produce less than the sum of the
pre-merger production of the merged firms, but more than any of them produced
separately. Each of the non-merging firms will increase its production, and the overall
production level will decrease. This will lead to a price increase that substantiates the
market power motive for merger.
Using the setting of Rodrigues (2001), the efficiency motive is incorporated in the
model through fixed cost economies. Let  be a coefficient measuring the degree in
which the merger allows such economies, with 0    m . With   0 , fixed costs
 is usually referred to as the convenience yield. We assume it to be positive, since the problem
would otherwise not have a solution. This condition is interpreted in Dixit and Pindyck (1994).
2
7
have a sunk nature, and are not adjustable post-merger: the firm resulting from the
merger has the same fixed costs as the whole of the merged firms. At the other
extreme, with   m , the firm resulting from the merger has the same fixed costs as
one of the merged firms. This can happen if those fixed costs result from an
indivisibility in the production factors or if the merger is purely anti-competitive,
resulting in the closure of all but one of the merged firms. Intermediate cases are, of
course, conceivable.
When the firm i merges m firms in the industry, the multi-firms merger profit
will be:

m 1
i
 1



 1 
 1  
X
m 1





 1 
 1 

F
 1


 1
  (n  m) 
 (n  m)   (n  m)c
(10)
where the superscript of  im 1 denotes that there are m + 1 firms comprising an
industry merger. The effect of m on merger profit can be found from the
differentiation:
 im1
X
F
 1
, z' 
 z ' (1  z ' ) 2
(2  z 'z ' ) 
2  1
m
 1
 ( n  m)
( n  m) c
We can find that the result includes two parts. The second part indicates the effect of
fixed cost economies. The increasing  will increase a benefit from the number of a
merger increasing. If the first part is larger than the second part, the differentiation is
positive, increasing number of a merger will increase profit. The sign of this result,
however, depends on the third term (i.e. 2  z 'z ' ) of first part. Since the
requirement of 1 > z’, the third term is positive when m  (n    1) . That is, when
the competition intensity is high (i.e.  is small) or number of mergers is low,
increasing number of a merger will increase profit.
Considering these multi-firms that have the option to merge into a new company
with value Vm+1. By using the same approach above, we can get the new firm i’s
post-merger value:
Vi m1 ( X t )   im1 (,n, m,c)
where 
m 1
i
X t
m 1 F

n  m (  1) r
(11)
 1



 1 
 1  
1
  1 
 
 1 
 1
  (n  m) 
 (n  m)   [r  g ( )]c
The synergy assumed that comes from scale economy in this paper is the main
reason to merge. Considering the synergy of m + 1 firms merger, which can be
8
denoted by the difference between the post- and pre-mergers value:
S i  Vi m 1 ( X t )  (m  1)Vi ( X t )
  m1

 (m  1) F

 (m  1)  X  
n
 1 r
n  m
(12)
We can check whether the merger of any subset of the n firms in the industry is
privately profitable. When   1, equivalent to a competitive industry, the first term
in equation (12) is zero. Then, a merger among a subset of the firms in the industry
will only be profitable if it allows for fixed cost economies. In fact, even without any
merger, the industry profit is already maximized, there being room for no
improvement. The merger of all the n firms, creating a multi-plant monopoly, would
only maintain the pre-merger industry profit, while the merger of any smaller subset
of the firms would imply a loss for the merged firms because, transforming them in a
single firm, it would reduce their share in the same industry profit pie.
3. The timing and terms of multi-firms’ mergers
3.1 The timing of multi-firms’ mergers
When merging, each firm incurs a fixed cost denoted by I. In considering a merger,
firms make a tradeoff between the stochastic benefit of merging and the cost of
merging. Since firms have the option but not the obligation to merge, they will do so
only when it is in their interest. We use the approach of Lambrecht (2004) and show
that firm i’s option value in m + 1 firms merger is given by:

Oi ( X t )  max V m1 ( X t )  (m  1)V ( X t )  (m  1) I , 0

  m1

 (m  1)
 m a x
 (m  1)  X  
F  (m  1) I ,
n 
r (  1)
 n  m

0

According to the assumption of model, the demand shock follows geometric
Brownian motion. The partial differential equation of mergers option is（see Dixit and
Pindyck, 1994）:
1
2
 2 X t2 O' ' ( X t )  (r   ) X t O' ( X t )  rO ( X t )  0
(13)
As suggested by Dixit and Pindyck (1994), in such a case, the solution has a form3:
3
Strictly speaking, one must write the solution as
9
O( X t )  AX  ,   0
(14)
where A and β are constants to be determined. The expression for β > 0 can be found
by solving the following fundamental quadratic equation:
Q  12  2  (   1)  ( r   )   r  0
(15)
We consider the boundary conditions4 that must be satisfied at a threshold value
for entry to close the model:
(16)
(m  1)V ( X )  O( X )  V m ( X )
'
(m  1)V ' ( X )  O ' ( X )  V m ( X )
(17)
where X is the threshold to merge. Equation (16) is referred to as the
value-matching condition. The second boundary condition, equation (17), is called the
high-order contact or smooth-pasting condition. According to the boundary condition
above, the parameter A can be solved as:
  m1


 (m  1)
A  
 (m  1)  X  
F  (m  1) I  X 
n 
r (  1)
 n  m

(18)
Concerning existence and uniqueness of X t , we were able to show the following
results. The merger takes place the first time when the state variable Xt hits the
threshold X from below. The value of firm's option to merge is given by:
  m1
 X  
    (m  1)

O( X t )  
 (m  1)
X 
F  (m  1) I  t  , X t  X (19)


n

m
n
r
(


1
)

 X 

Under the condition of O( X ) / X  0 , we can get the threshold X to merge:
1
  (m  1) I  ((m11) ) Fr  
X 

 m 1

    n  m  (m  1) n 
(20)
Using the threshold, a firm’s manager has the principal to deal with the problem of
mergers. The optimal timing, T, to multi-firms merge is: T  inf[ t  0 : X (t )  X ] .
The optimal merger threshold without concern as to how the merger option is
divided between the firms of two parts. The above expressions for the value of the
merger options, Equation (19), have a simple and intuitive interpretation. The first
factor represents the surplus, S(Xt), generated at the time of the merger. The factor
O( X t )  A1 X 1  A2 X 2 , 1  0;  2  0
4
Generally speaking, the value-matching and smooth-pasting conditions consist of so-called boundary
conditions.
10
( X t / X )  is a discount factor that discounts the surplus back to the present.
Lambrecht (2004) thinks that the discount factor can also be interpreted as the
probability of the state variable ever hitting the merger threshold X given the
current level, Xt, of the state variable.
From the merger threshold in equation (20), we can find that the larger merger costs,
I, and hysteresis factor,  /(    ) , delay merger. This is in line with the predictions of
real options theory. On the contrary, larger synergies including
 m 1
nm
 ( m 1) F
 1 r
and
 (m  1) n speed up mergers.
For each firm of multi-firms merger, the cumulative return, R, at X is given by:
O( X )
R

(m  1)V ( X )
 m 1
X   m11 Fr  (m  1) I
nm
m 1
n
 X   (m  1) Fr
1
(21)
From the cumulative return of multi-firms merger, we can see the return of merger
will increase when merger effect,  m1 , increases and merger cost, I, decreases.
Besides, the return of merger will increase through parameter of fixed cost economies,
 , increasing.
3.2 The terms of multi-firms’ mergers
In a merger the equity of the individual merging companies is exchanged for a
fraction of the equity in the newly created firm. We assume that the acquirer consists
of m firms in the merger with m + 1 firms. This acquire is denoted by V am and the
acquiree in the merger with m + 1 firms is denoted by Ve1 . In a stock offer the
acquirer buys the shares of the target and gives a fraction of its own shares in
exchange. So similar to a merger both the acquirer and the acquiree hold shares in the
restructured company. We model mergers as the outcome of a Cournot-Nash game in
which both firms determine simultaneously the timing and the terms of the merger.
We study further the options value from individual firm’s merging. The payoff to each
firm has the option to exchange its own shares for shares in a new company.
Oa ( X t , s a )  max[ s aV m 1 ( X t )  Vam ( X t )  I a , 0] ,
11
Xt  X
Oe ( X t , s e )  max[ s eV m 1 ( X t )  Ve1 ( X t )  I e , 0] ,
Xt  X
(22)
where Vm+1 is a new company value after merging. Besides, Ij (j = a, e) is firm j’s cost
incurred in merger and Ia + Ie = I. Besides, sj denotes the fraction of the new equity
capital owned by firm j and sa + se = 1. Hence, at the time of the stock offer, the
payoff of acquirer and acquiree can be written as: s jV m1 ( X t )  V j ( X t ) , j = a, e. The
acquirer constructs a new company with value Vm+1, but has to offer a fraction 1-sa.
The firm’s option of merger is:

O j ( X t , s j )  s jV
m 1


X 
Vj  I j  t  ,
 X 
j = a, e and X t  X
(23)
Let X a  X e  X and the conditions:
O j ( X t , X j , s j )
 0 (j = a, e) and sa + se = 1
X j
X j X
Hence, the ownership structure of post-merger firm (sa, se) ensures that both firms
want to execute the merger at the efficient threshold (i.e. X a  X e  X ) and that the
total merger surplus gets divided (i.e. sa + se = 1). The ownership structures of the
post-merger firm are:
sa 
Va   n X   I a
m 1
V m 1   n  m X 
,
(24)
Ve   n m1 X   I e
m
and se 
(25)
m 1
V m1   n m X 
The optimal merger threshold is:
 
X 
  



 
 nm1  n 
I

m 1
nm
 F
1 r
m
1

(26)
The implication of threshold is similar to that in equation (20). The main difference is
synergy, the second term of threshold.
3.3 The terms of hostile takeovers
In acquisition, we assume the first firm in the industry is acquiring firm and the
12
mth firm is the target firm. We follow Lambrecht (2004) to assume that the process of
acquisition is described by Stackelberg leader-follower game. The target first decides
on the terms, s, at which it is prepared to offer its shares, and the acquirer
subsequently decides on the timing (i.e. when to make the offer). That is, the timing
and terms decisions are not made simultaneously. This situation is similar to a hostile
takeover where the target management dictates the minimum terms required for
relinquishing control. Lambrecht (2004) thinks that the costs of renegotiating the
terms demanded by the target are prohibitively high and the target can make a
take-it-or-leave-it offer with respect to the takeover terms. The target's and the
acquirer's optimization problem can be formulated respectively as:
max Oe ( X t , X ( s a ), s a ) and max Oa ( X t , X , s a )
sa
X
By using similar method above one can show that takeover options value and
threshold. The acquirer’s takeover option value is:

Oa ( X t , X , s a )  s aV
m 1


X 
( X )  Va ( X )  I a  t  ,
 X 
Xt  X
 s aV m 1 ( X )  Va ( X )  I a ,
Xt  X
(27)
Besides, the target’s takeover option value is:

Oe ( X t , X , s a )  (1  s a )V
m 1


X 
( X )  Ve ( X )  I e  t  ，
 X 
 (1  s a )V m 1 ( X )  Ve ( X )  I e ，
The post-takeover ownership shares are given by:
(    )Va   m1 Fr   I a
sa 
and se  1  s a
(    )V m 1   m11 Fr
Xt  X
Xt  X
(28)
(29)
However, we cannot find the closed form of takeover threshold. For simply, we
assume a firm’s fixed cost F = 0. The post-takeover ownership shares then are given
by:
(    )Va   I a
and se  1  s a
(30)
sa 
(    )V m1
The optimal takeover threshold is given by
 
X 
  


1


m
 nm1  n 
  I a  I e
 m 1
nm
(31)
13
From the threshold above, we can also find that the increasing takeover costs, Ia and Ie,
and hysteresis factor,  /(    ) , will delay mergers. On the contrary, larger synergies
 m 1
nm
 nm1  n speed up mergers.
m
4. Extension: merger to monopoly
Mergers within an industry, horizontal mergers, can increase concentration within
an industry. According to Demsetz (1973), high market concentration helps firms in a
particular industry to reach collusive agreement, thus increasing the degree of
monopolization within that industry and creating monopolistic profits.
Assuming there is an industry with n firms originally. The acquirer plants to merge
the remainder (i.e. n –1 firms) in the industry. The acquirer’s value is denoted by V1
and the value of remainder entirely is denoted by Vn-1. The merger of all the n firms
creates a multi-plant monopoly with the new value Vn. When merging to monopoly,
the option to merge is given by:
O( X t )  max[ V n ( X t )  V 1 ( X t )  V n 1 ( X t )  I , 0]
(32)
where I is the cost incurred in merger. For all the n firms, the optimal threshold X for
merger to monopoly is:
 
I
X  
n
n 1
1
1
     2  n
 F
1 r




1

(33)
Similarly, from the threshold above, we find that the increasing takeover costs, I and
hysteresis factor,  /(    ) , will delay mergers. On the contrary, larger synergies
including
 F
 1 r
n 1
and  n   2  n speed up mergers.
For merger to monopoly, the firm’s cumulative return, R, at X is given by:
R
 n X    n1 Fr  nI
1
 X   n Fr
(34)
From this cumulative return of multi-firms merger above, we can find that the return
of merger will increase when synergy effect,  n , increases and merger cost, I,
decreases. Besides, the return of merger will increase through parameter of fixed cost
economies,  , increasing. If the fixed cost F = 0, we can see the return more clearly.
The cumulative return in this time is:
14
Ri 
n
I

1
 Vi ( X )
Then the firm’s cumulative return is synergy effect minus merger cost rate and firm’s
value.
5. Conclusion
Owing to there are multiple firms participating in a merger sometimes and they
want to merge together. This study therefore develops a model for the timing and
terms of multi-firms mergers in a dynamic economic environment where the firm’s
demand shock is stochastic. We model mergers as the outcome of a Cournot Nash
game where the timing and terms of the takeover are simultaneously determined,
whereas tender offers result from a Stackelberg leader follower game where the target
decides first on the terms and the acquirer subsequently on the timing. This paper,
hence, obtain closed form solutions for the takeover synergy, options, optimal
threshold, and acquirer's and acquiree's ownership shares. We find that the expected
time to merge is inversely related to the magnitude of the synergies generated. That is,
increasing merger costs, product market uncertainty and decreasing scale economy
tend to delay mergers. As future research, we plan to test empirically some of the
resulting hypotheses. Besides, the extensions can be aimed at incorporating the
market of perfect competition into the model.
15
Reference
Anderson, R., and S. Sundaresan, 1996, “Design and Valuation of Debt Contracts,”
Review of Financial Studies, 9(1), 37-68.
Baba, N., 2001, “Uncertainty, Monitoring Cost, and Private Banks’ Lending Decisions
in a Duopolistic Loan Market: A Game-Theoretic Real Options Approach,”
Monetary and Economic Studies, 19(2), 21-47.
Demsetz, H., 1973, “The Market Concentration Doctrine,” Washington: American
Enterprise Institute- Hoover Policy Studies.
Dixit, A. K. and R. S. Pindyck, 1994, Investment Under Uncertainty, NJ: Princeton
University Press.
Ernst & Young, 1994, Mergers and Acquisitions: Back-to-Basics Techniques for the
1990s, 2nd edition, New York: John Wiley and Sons.
Grenadier, S. R., 1996, “The Strategic Exercise of Options: Development Cascades
and Overbuilding in Real Estate Markets,” Journal of Finance, 51(5), 1653-1679.
Grenadier, S. R., 1999, “Information Revelation through Option Exercise,” Review of
Financial Studies, 12, 95–130.
Grenadier, S. R. 2000, "Option Exercise Games: The Intersection of Real Options and
Game Theory" Journal of Applied Corporate Finance, 13, 99-107.
Herath, H.S.B. and J. S. Jr. Jahera, 2002, “Real Options: Valuing Flexibility in
Strategic Mergers & Acquisitions As an Exchange Ratio Swap,” Working paper.
Huisman, K.J.M., P.M. Kort, G. Pawlina, and J.J.J. Thijssen (2004), “Strategic
Investment under Uncertainty: Merging Real Options with Game Theory”,
Zeitschrift für Betriebswissenschaft, 67, 97-123.
Kwoka, Jr. J.E., 1989, “The private profitability of horizontal mergers with
non-Cournot and maverick behavior,” International Journal of Industrial
Organization, 7, 403–411.
Lambrecht, B., 2004, “The Timing and Terms of Mergers Motivated by Economies of
Scale,” Journal of Financial Economics, 72, 41-62.
Margrabe, W., 1978, “The Value of the Option to Exchange One Asset for Another,”
Journal of Finance, 33, 177-186.
McDonald, R. and D. Siegel, 1984, “Option Pricing When the Underlying Asset Earns
a Below-Equilibrium Rate of Return: A Note,” Journal of Finance, 39, 261-265.
Mella-Barral, P., 1999, “The Dynamics of Default and Debt Reorganization,” Review
of Financial Studies, 12, 535-578.
Morellec, E., 2001, “The Dynamics of Mergers and Acquisitions,” Working paper,
William E. Simon Graduate School of Business Administration, Rochester.
Shleifer, A. and R. Vishny, 2003, “Stock Market Driven Acquisitions,” Journal of
16
Financial Economics, 70, 295-311.
Smith, K. W. and A. J. Triantis, 1995, “The Value of Options in Strategic
Acquisitions,” in Real Options in Capital Investment: Models, Strategies, and
Applications, ed. by L. Trigeorgis, 135-149. Praeger, London.
Weston, J. F., K. S. Chung, and S. E. Hoag, 1990, Mergers, Restructuring, and
Corporate Control. Prentice Hall International, Englewood Cliffs, New Jersey.
Rodrigues, V., 2001, “Endogenous mergers and market structure,” International
Journal of Industrial Organization, 19, 1245-1261.
Trigeorgis, L., 1996, Real Options, The MIT Press.
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