Submitted to International Journal of Real Options and Strategy
Theoretical Paper

Mergers and acquisitions strategy under imperfect information: A
mixed payment model
Wenjun Chen∗

Graduate School of Economics and Business Administration, Hokkaido University

Abstract: This paper develops a dynamic model of a joint takeover to determine the timing, acquisition premiums, and terms. The model incorporates imperfect information and the strategy bysolving a Markov perfect Nash equilibrium. The results predict that the bidder will make a high cash
payment to the target to gain high post-merger management control. The abnormal return to theparticipating firm can be positive or negative due to asymmetric information. In addition, the model
relates the acquisition premium payment and the merger threshold to the growth rate, volatility, andcorrelation coeﬃcient of the bidder and target.
Keywords: Sharing-rule, Real option, Takeover, Acquisition premium, Synergy, Bargain power∗

Received: July 18, 2017; Accepted: October 18, 2017.
Corresponding author. Address: Kita 9, Nishi 7, Kita-ku, Sapporo 060-0809, Japan; Phone: +81-11-706-3183;
E-mail: chenwenjun@eis.hokudai.ac.jp†

1Submitted to International Journal of Real Options and Strategy

1

Introduction

Generally, enterprises have two ways to grow: one is by expanding production, which is internal
reinvestment; and another is by acquisition and reorganization. Acquisitions will create synergies
from cost savings, customer expansion, or other financial benefits resulting from the cooperation of
two firms.
The optimal timing and post-merger ownership structure are two keys in the analysis of merger
and acquisition strategies. Several famous studies analyzed the timing and terms of mergers andacquisitions. Lambrecht [11] studies the timing and terms of mergers motivated by economies of scaleand shows that merger activities are positively correlated with markets, meaning that firms are willingto merge during economic expansions. Thijssen [21] extends Lambrecht [11] into a two uncertain
conditions model and optimizes the timing for considering both eﬃciency gains and diversification
benefits. The results show that mergers and acquisitions will during both economic upswings anddownswings. Bernile, Lyandres and Zhdanov [4], Hackbarth and Morellec [10], and Lambrecht andMyers [12] also develop models of optimal timing. Hackbarth and Miao [9] develop a joint model ofoligopolistic industries that determines the industry’s product equilibrium and the timing and termsof takeovers. The result shows the relationship between the return from mergers and company size.
In terms of firm-size analysis, Moeller, Schlingemann, and Stulz [16] show the firm size eﬀect inacquisition announcement returns.
There are two general types of mergers and acquisitions: pure-cash and pure-share. The studies
described above assume pure stock-for-stock (pure-share) mergers conditions in their models, withno cash-payment in the process. In stock-for-stock mergers, shareholders of both participating firm
remain shareholders in the continuing combined enterprise. They negotiate the post-merger structure
based on the pre-transaction ratio of their company market value. In the pure-cash type, the acquirer
pays an equivalent cash amount to the target and receives all of the shares of the new combined firm.
Shareholders of the acquiring firm are remained the only shareholders of the new company. However,Goergen and Renneboog [8] analyze 156 takeover bid samples in Europe during the 1990s; of thesample, 93 and 37 of 156 were pure-cash and pure-share, respectively, with 18 cases of the mixed
cash-share type. In the mixed cash-share type, the acquirer pays both cash and shares to the target,and shareholders of both companies maintain ownership in the new company. A study by Faccio
and Masulis [7] shows that 11.3% of a sample of European 3,667 mergers or acquisitions are mixed
cash-share types, and these deals always have a larger transaction size. In Martynova and Renneboog[15], the proposition of the mixed cash-share bids increased to 19% in a sample of 1,361 European
acquisitions.
This study develops a model of mixed cash-share mergers in which both the bidder and target
remain shareholders of the new combined enterprise and negotiate over the post-merger sharing-rule.
The bidder pays a cash premium to the target. Acquirers have a lot of motivations to pay a cash
premium. In this paper, we assume that the bidder pays a cash premium to negotiate a higher
post-merger share in the new company. This study aims to establish a joint merger model of timing,
acquisition premium, and terms in markets with both perfect and imperfect information. Furthermore,
the model will examine the eﬀect of information on the decision-making.
In this study, we base our analysis on Morellec and Zhdanov [18]’s model of a joint determination2

Submitted to International Journal of Real Options and Strategy

of timing and terms of takeovers under competition and imperfect information. We extend the model
to study the returns generated from asymmetric information. This study also diﬀers from previous
studies in several important dimensions. First, we consider a cash premium, which Morellec andZhdanov [18] do not include. We establish the model using a non-cooperative game in which thebidder provides a tender oﬀer, and the target can accept the oﬀer or wait. The study also closely
relates to Lukas and Welling [13], who develop a two-stage model analyzing the pricing and timing ofmergers and acquisitions.
Second, we analyze the terms, assuming that the bidder and target will negotiate the terms of
the post-merger combined enterprise, which we solve via a Nash barging solution (Nash [19]). Several
studies combine game theory and real option theory. Azevedo and Paxson [2] discuss the discrete-timeand continuous-time frameworks of a standard real option game and review two decades of academic
research on standard and non-standard real option games. Lukas, Reuer and Welling [14] use a gametheoretic option approach to model the value of contingent earn-outs, finding that the firm will tend
to postpone the investment under larger transaction costs, greater uncertainty in cash flows, a longer
earn-out period, and higher performance targets.
Finally, we assume that both the bidder and target will probably mis-estimate the price, of which
the managers of both firms can take advantage. In Morellec and Zhdanov [18], asymmetric information occurs between the participating firms and investors. The paper analyzes the abnormal returns
from the announcement. We extend the model by assuming asymmetric information between two
participating firms.
This study examines two scenarios. The first is a single bidder case in which we develop a basicmodel of the target acquisition in a market with perfect information without competition. We thenextend the model to a second scenario with asymmetric information. Both the bidder and target
may be mis-estimated in the market, which means the market price may be under- or over-estimated.
However, both participating firms have complete information about their own company. Therefore,
they will generate an optimal strategy using the estimated parameters based on the available information. The merger and acquisition activities can increase information disclosure and push the market
price towards the real value. Therefore, the model generates abnormal returns to both the bidder andtarget around the merger.
The paper is organized as follows. In section 2, we introduce the model’s framework and provide thebasic model with full information. We then extend the basic model to one with imperfect informationin Section 2.3. In section 3, we analyze the impact of the information on the results in Section 2. In
Section 4, we give several numerical examples. Finally, Section 5 concludes.

2
2.1

The model

Basic framework

We construct the framework based on Morellec and Zhdanov [18]’s model. Consider two firms: abidder and a target, which operate in the same market. We assume capital stocks of KX for thebidder and KY for the target. The stock market valuation of each firm, denoted by SX (t) and SY (t),respectively, without takeover is
SX (t) = KX X(t), SY (t) = KY Y (t),(1)

3Submitted to International Journal of Real Options and Strategy

where X(t) and Y (t) denote the per-unit value of capital, which follows a geometric Brownian motion,dX(t) = µX X(t) dt + σX X(t) dWX ,(2)

dY (t) = µY Y (t) dt + σY Y (t) dWY ,where drift µX , µY > 0, and diﬀusion σX , σY > 0 are constant parameters. WX and WY are standard
Brownian motions. The correlation coeﬃcient between WX and WY is constant, represented as ρ ∈(−1, 1).
We assume that all participants are risk neural, and the risk-less interest rate is r, r ≥ µi (i = X, Y ).
Suppose the bidder aims to hold a certain fraction of the post-merger company to gain at least
management rights in the company. To achieve this purpose, the bidder has to pay an acquisition
premium to the target. We use parameter λ to denote the acquisition premium meaning that thebidder will pay λSY (t) to the target, where SY (t) is the market value of the acquired company before
the acquisition, given by equation (1). The bidder will receive a ξ(ξ > 0) fraction of the post-merger
combined entity in return. Consequently, the target gets a cash payment of λSY (t) and receives (1−ξ)
as the friction of the new combined company. The parameter λ can even be negative, in which case,the target shareholders bargain a higher post-merger ownership than the friction they could without
cash payment. The bidder asks the target to pay for the higher bargain share. There is also the needto pay the transaction costs, denoted as cSY , when merging to the two companies.
From the target’s standpoint, the higher the sale price λSY (t) is, the higher the cash payment they
will receive upon selling. On the other hand, the remaining part of ownership will decrease, and thetarget will therefore receive a lower fraction (1 − ξ). From the bidder’s perspective, if they provide
a higher payment to the target, the cost will increase, though they will accordingly gain a higher
fraction ξ of the new entity. Therefore, there is an optimal payment strategy and sharing-rule ξ forboth the bidder and target.
The acquisition process proceeds in two steps. At time t0 , the bidder oﬀers a certain λ to the target,
which can accept or reject the oﬀer. The target firm will not decide immediately upon receiving theoﬀer, and they can postpone the decision. Accepting the oﬀer leads to an immediate merger of thetwo firms. The target firm will receive a payment of λSY (t) from the bidder and a (1 − ξ) friction of
the new combined entity, and has to give up a claim of SY (t) instead.
Following Morellec and Zhdanov [18] and Shleifer and Vishny [20], we assume a linear combinationof pre-takeover values in terms of per-unit value of capital. Hence, the post-merger value of the firm
is
(
)
(
)
SM X(t), Y (t) = SX (t) + SY (t) + G X(t), Y (t) ,()
where G X(t), Y (t) is net synergy gains generated from the merger, given by
( (
)
(
)
)
G X(t), Y (t) = KY α X(t) − Y (t) − cY (t) , (α, c) ∈ R2++ ,

(3)

(4)

where α is the synergy parameter, which all participants can observe. c denotes the per-unit merger
cost of the capital value of the target firm. This assumption indicates that the synergy will be positive
only when the bidder outperforms the target, which has the same meaning as X(t) > Y (t).

4Submitted to International Journal of Real Options and Strategy

2.2

A perfect information model

In this section, we consider a scenario with perfect information and no competition in the process.
As in the framework, there are two stages in the merger process. We use the Markov Perfect Nash
Equilibrium to analyze the strategy, in which the bidder will provide an optimized cash payment of
λSY (t) ( λ > 0) at stage one. The target receives this oﬀer in stage one with a given λ and will accept
the oﬀer at the threshold of τ that maximizes their profit. The bidder and target will bargain theterms according to the optimized price λ and the threshold.
The target firm reacts to receiving a price oﬀer of λ in stage one. They hold the option of accepting
the oﬀer. Conditional on the oﬀered price λ, the target will choose a threshold τ in stage two andaccept the oﬀer. The acquisition premium to accept the oﬀer is λSY (τ ). Therefore, the target will
give up their claim, worth SY (τ ), and receive both the payment value λSY (τ ) and a fraction (1 − ξ)()
of the new combined firm, worth (1 − ξ)SM X(τ ), Y (τ ) . The optimization function of the reacting
party (target firm) at stage two is
{
]}
[
(
)
(
)
f sale X(t), Y (t) = max E e−rτ (1 − ξ)SM X(τ ), Y (τ ) + λSY (τ ) − SY (τ ) .
(5)
τLemma 1 (Optimal threshold for the target in stage two) Based on the value-maximizing strategy, the target firm will accept the oﬀer and merge with the bidder when the ratio of capital price,
denoted by R(t) = X(t)/Y (t), reaches the levelR∗ (λ, ξ) =[() 1 − λ ] ϑ1
KY.
α+c−1 +
1 − ξ ϑ1 − 1 KX + αKY(6)

The value of merger option for the target is

f

(
sale

)
R(t) =( R(t) )ϑ1(1 − ξ)(α + c − 1) + (1 − λ)KY Y (t) ∗,R(t) < R∗ (λ)
ϑ
−
1
R
(λ)
1
[
]

 (1 − ξ)(KX + αKY )R(τ ) + [(1 − ξ)(1 − α − c) + (λ − 1)]KY Y (t), R(t) ≥ R∗ (λ),(7)

and the first passing time isτ ∗ = inf{t|R(t) = R∗ (λ, ξ)}.

(8)

ϑ1 > 1 is the positive root of the quadratic equations
1
1 2+ σY2 − ρσX σY )ϑ(ϑ − 1) + (µX − µY )ϑ − (r − µY ) = 0.
( σX
2
2

(9)

(Appendix A provides the proofs)
The threshold (6) is a reaction function of the cash payment parameter λ and the bargain sharingrule (1 − ξ). Holding (1 − ξ) constant, a higher λ will decrease R∗ and accelerate the merger. The
threshold also largely relies on the ratio of firm size, represented by KX /KY . A higher ratio of firm
size (KX /KY ) will also accelerate the merger.
At stage two, the bidder will give up their claims, worth SX (τ ∗ ), and also pay the payment value of()λSY (τ ∗ ). In return, they will receive a fraction ξ of the new combined firm, worth ξSM X(τ ∗ ), Y (τ ∗ ) .

5

Submitted to International Journal of Real Options and Strategy

The bidder will choose an optimal λ to maximize their benefit in stage one. The optimization function
for the bidder is
{
[
]}
(
)
(
)
∗
f M A X(t), Y (t) = max E e−rτ ξSM X(τ ∗ ), Y (τ ∗ ) − λSY (τ ∗ ) − SX (τ ∗ )λ(10)Proposition 1 (Optimal tender oﬀer in stage one) Maximizing the payoﬀ function (10) yields
the optimal oﬀered portion λ∗ from the bidder, given asλ∗ (ξ) =[()]ϑ1 (α + c)(1 − ξ)(KX + αKY )α + c (1 − ξ) + ξ +(1 − ξ)KX − α(ϑ1 + ξ − 1)KY

(11)

Substituting result (11) into (6) yields
R∗ (ξ) =( ϑ2 )(α + c)KY
1
ϑ1 − 1 α(ϑ1 + ξ − 1)KY − (1 − ξ)KX

(12)Receiving the optimal oﬀer of λ∗ , the target firm will accept the oﬀer when the ratio of capital priceR(t) satisfies (12). The bidder will immediately merge with the target after the target accepts the oﬀer.
(Appendix B contains the proofs)
According the result from (11), dλ∗ (ξ)/ d(1 − ξ) < 0 (equivalently, dλ∗ (ξ)/ dξ > 0), the cash
payment parameter λ∗ positively relates to the sharing-rule ξ. Hence, the bidder is willing to pay ahigher cash premium to the target for a higher post-merger share ξ. On the other hand, a target’s
higher (1 − ξ) requirement will decrease λ∗ and increase the R∗ , slowing the merging process.
We suppose the bidder and target will negotiate the post-merger sharing-rule at stage one. As in
Section 2.1, the post-merger sharing-rule is ξ for the bidder and (1 − ξ) for the target. We obtain the
optimal ξ using the Nash bargaining solution. The optimization function is
[
)]1−β
(
)
(
)]β [ sale (,(13)fX(t), Y (t); ξΠ X(t), Y (t) = max f M A X(t), Y (t); ξξwhere the bargain power parameter β is subject to β ∈ (0, 1). Solving the maximization problem (13)yields the following.
Proposition 2 (Optimal merger terms) The bidder will pay the optimal oﬀered portion λ∗ andexpect to receive the optimal ξ ∗ fraction of the combined entity, where ξ ∗ satisfiesξ∗ = 1 −(1 − β)αKY.
KX + αKY(14)

Receiving the optimal oﬀered portion λ∗ , the optimal strategy for the target is to require (1 − ξ ∗ ) as afraction of the combined entity and choose to accept the oﬀer at τ ∗ . The terms for the target are
1 − ξ∗ =(1 − β)αKY.
KX + αKY(15)(Appendix C provides the proofs)
The optimal sharing-rule (equation (14)) under a Nash bargaining solution positively relates to theratio of the firm size (KX /KY ). With a higher firm size (KX /KY ), the bidder is willing to require ahigher ξ ∗ and therefore pay a higher cash premium.

6Submitted to International Journal of Real Options and Strategy

2.3

An imperfect information model

In this section, we analyze the strategy in a market with asymmetric information. In this scenario, it
is possible for the market price to be misjudged. In contrast to assumption (4), the net synergy gains
generated from the merger here are
( (
)
(
)
)
(16)
G X(t), Y (t) = KY α ωB X(t) − ωT Y (t) − cY (t) , (α, c) ∈ R2++ ,where ωB > 0 and ωT > 0 are information parameters. The market price is underestimated if
ωi ∈ (1, ∞), (i = B, T ), and vice versa. We assume that ωB is only observable to the managers of thebidder and ωT is only observable to the managers of the target. Thus, managers know only the realvalue of their own company. Additionally, managers cannot trade on their inside information due tolegal prohibitions.
For the bidder, the net synergy gains generated from the merger are
( (
)
(
)
)
GB X(t), Y (t) = KY α ωB X(t) − Y (t) − cY (t) ;(17)
ωT is not observable to the bidder, who believes that the target firm’s market price is the real price.
Based on the bidder’s information, the post-merger value of the combined firm should be
(
)
(
)
B
SM
X(t), Y (t) = SX (t) + SY (t) + GB X(t), Y (t) .
For the target, the net synergy gains generated from the merger are
( (
)
)
(
)
GT X(t), Y (t) = KY α X(t) − ωT Y (t) − cY (t) ,

(18)

(19)

Based on the target firm’s information, the post-merger value of the combined firm should be
(
)
(
)
T
SM
X(t), Y (t) = SX (t) + SY (t) + GT X(t), Y (t) .

(20)

Asymmetric information will be disclosed after the merger. Therefore, the merger of the two firms
will generate an abnormal return.
Similar to the model in the perfect information market, the optimization function of the reacting
party (target firm) at stage two is
{
[
]}
(
)
(
)
T
f˜sale X(t), Y (t) = max E e−rτ (1 − ξ)SM
X(τ ), Y (τ ) + λSY (τ ) − SY (τ ) .
(21)
τLemma 2 (Optimal threshold in stage two in a market with imperfect information) Basedon the value-maximizing strategy, the target firm will accept the oﬀer and merge with the bidder when
the capital price ratio R(t) reachesR̃∗ (λ, ξ) =[() 1 − λ ] ϑ1
KY
αωT + c − 1 +.
1 − ξ ϑ1 − 1 KX + αKY(22)The value of merger option for the target is()
f˜sale R(t) =( R(t) )ϑ1(1 − ξ)(αωT − c − 1) + (1 − λ)KY Y (t) ∗,R(t) < R∗ (λ)
ϑ
−
1
R
(λ)
1
[
]

 (1 − ξ)(KX + αKY )R(t) + [(1 − ξ)(1 − αωT − c) + (λ − 1)]KY Y (t), R(t) ≥ R∗ (λ)

7Submitted to International Journal of Real Options and Strategy(23)
The first passing time isτ̃ ∗ = inf{t|R(t) = R̃∗ (λ, ξ)},(24)

where ϑ1 > 1 is the positive root of equation (9). (Appendix D provides the proofs)
At stage one, the bidder’s optimization function is
{
[
]}
(
)
(
)
∗
B
f˜M A X(t), Y (t) = max E e−rτ ξSM
X(τ ∗ ), Y (τ ∗ ) − λ̃SY (τ ∗ ) − SX (τ ∗ ) .

(25)

λ̃

With the available information, the bidder will estimate the target firm’s threshold as R∗ (λ̃), which
is given by equation (6).
Proposition 3 (Optimal tender oﬀer in stage one in a market with imperfect information)
The information parameter ωT is not observable before τ̃ ; the bidder then optimizes the acquisition
premium under a distorted estimation. The optimal oﬀered portion λ̃∗ the bidder chooses is()λ̃∗ (ξ) = (1 − ξ) α + c + ξ +ϑ1 (α + c)(1 − ξ)(KX + αKY ).
(1 − ξ)KX − α[(ϑ1 + ξ − 1) + ϑ1 ξ(ωB − 1)]KY(26)

Given λ̃∗ (ξ), the threshold finally is equal to R̃∗ (λ̃∗ , ξ). According to the result of (22), the optimal
threshold to accept the oﬀer is]ϑ1 [ α(ωT − 1)KY
ϑ1 (α + c)KYR̃∗ (ξ) =+(27)
ϑ1 − 1 KX + αKYα[(ϑ1 + ξ − 1) + ϑ1 ξ(ωB − 1)]KY − (1 − ξ)KX
(Appendix E provides the proofs)
Similar to the model in the perfect information market, the optimization function of the sharingrule is[()()]β [ sale ()]1−βΠ̃ X(t), Y (t) = max f˜M A X(t), Y (t); ξ˜(28)f˜X(t), Y (t); ξ˜ξ̃where the bargaining power parameter β is subject to β ∈ (0, 1). Solving the maximization problem(28) yields the following result.
Proposition 4 (Optimal merger terms in a market with imperfect information) The bidder
will pay the optimal oﬀered portion λ̃∗ and expect to receive the optimal ξ˜∗ fraction of the combined
entity. Receiving the optimal oﬀered portion λ̃∗ , the optimal strategy for the target is to require (1− ξ˜∗ )as a fraction of the combined entity and choose to accept the oﬀer at τ̃ ∗ . The optimal merger terms
in a market with imperfect information satisfies
KY
ωB
γ1
α
(1 − β)γ1= 2+
α + c KX + αKY 1 − ξ˜∗γ2βϑ1 γ22 − (1 − β)(1 − ϑ1 )γ1 γ2

(29)

where
γ1 = (1 − ξ˜∗ )(α + c − 1) + (1 − λ̃∗ (ξ˜∗ ))γ2 = (1 − ξ˜∗ )(αωT + c − 1) + (1 − λ̃∗ (ξ˜∗ ))

(30)
(31)

(Appendix F provides the proofs.)8Submitted to International Journal of Real Options and Strategy

The entire strategy will be decided in stage one; therefore, both participating firms have no chance toadjust the strategy after the decision. The optimal sharing-rule under a Nash bargaining solution in
the imperfect information market is the solution to equation (29). For the bidder, the strategy will be(λ̃∗ (ξ˜∗ ), ξ˜∗ ). For the target, the strategy will be (R̃∗ (λ̃, ξ˜∗ ), 1 − ξ˜∗ ). In the next section, we analyze
the impact of the information on the strategy.

3

Impact of information

All information is revealed after the entire merging process is complete. After merging, the market
price reflects the real value of the enterprise. Before merging, the participating firms know only theirown real value and estimate their counterpart’s value assuming that their value is equal to their market
value. Because the information parameter ωT is unknown to the bidder until after the merger, ωT has
no impact on the decision, which is λ̃∗ for the bidder. Comparing the results of (11) and (26), thehigher the information parameter ωB is, the higher the acquisition premium the bidder will provide.
In the range ωB ∈ (0, 1), λ̃∗ < λ∗ and the reversal relationship is in the range ωB ∈ (1, ∞).
Proposition 5 (Impact of information on the acquisition premium) The bidder is willing to
pay a higher acquisition premium when their firm is underestimated, and vice versa. A higher acquisition premium therefore accelerates the merger process and the disclosure of underestimated information.
The target will choose their strategy following the reaction function (22), the eﬀect of informationon their function is as follows.
Proposition 6 (Impact of information on the threshold) If the parameters satisfy(1 − ξ)α(ωT − 1) > λ̃∗ − λ∗ ,(32)

or equivalently,
ωT − 1ξϑ21 (α + c)(KX + αKY )KY>, (33)
ωB − 1[(1 − ξ)KX − α(ϑ1 + ξ − 1)KY )][(1 − ξ)KX − α[(ϑ1 + ξ − 1) + ϑ1 ξ(ωB − 1)KY ]the threshold in the situation with imperfect information is higher than that with perfect information,
which leads toR̃∗ > R∗ ,(34)

and vice versa.
If ωB ∈ (0, 1) and ωT ∈ (1, ∞), the left side of (32) is positive and the right side is negative. The
inequality (32) will always be satisfied. If ωB ∈ (1, ∞) and ωT ∈ (0, 1), the inequality (32) will neverbe satisfied because of the diﬀerent signs of the both sides. Therefore, if the target is overvalued
and the bidder is undervalued, inequality (32) will never be satisfied, and vice versa. If inequality(32) cannot be satisfied, the condition R̃∗ < R∗ always holds. In this situation, the symmetric
information will always accelerate the merger. On the other hand, if both of the target and bidder
are overestimated, which leads to ωT ∈ (0, 1) and ωB ∈ (0, 1), inequality (33) means that the higher

9Submitted to International Journal of Real Options and Strategy

the target is overestimated, the higher the threshold is. Additionally, if both of the target and bidder
are underestimated, which leads to ωT ∈ (1, ∞) and ωB ∈ (1, ∞), inequality (33) means that thehigher the target is underestimated, the higher the threshold is. Therefore, if the mis-estimations of
per-unit market value of capital are in the same direction, the higher the distortion is and the slower
the merger process will be.
When the bidder and target merge, they will receive a certain fraction of the post-merging firm
under the real net synergy gain from merging, given by (16). We can write the value of the bidder
pre-merger as()
SX (τ̃−∗ ) = KX X(τ̃−∗ ) + f˜M A X(τ̃−∗ ), Y (τ̃−∗ ); (ωB , 1) .

(35)

In addition, the value of the target pre-merger is()
SY (τ̃−∗ ) = KY Y (τ̃−∗ ) + f˜sale X(τ̃−∗ ), Y (τ̃−∗ ); (1, ωT ) .
The abnormal returns satisfy()()
˜M A X(τ̃ ∗ ), Y (τ̃ ∗ ); (ωB , ωT ) − f˜M A X(τ̃ ∗ ), Y (τ̃ ∗ ); (ωB , 1)f;
ARX (τ̃ ∗ ) =SX (τ̃−∗ )ξα(1 − ωT )KY=.
ξ(KX + αωB KY )R̃∗ + [ξ(1 − α − c) − λ̃∗ ]KY()()
f˜sale X(τ̃ ∗ ), Y (τ̃ ∗ ); (ωB , ωT ) − f˜sale X(τ̃ ∗ ), Y (τ̃ ∗ ); (1, ωT )∗ARY (τ̃ ) =;SY (τ̃−∗ )(1 − ξ)α(ωB − 1)KY R∗=.
(1 − ξ)(KX + αKY )R̃∗ + [(1 − ξ)(1 − ωT α − c) + λ̃∗ ]KY

(36)

(37)

(38)

In the next section, we examine the abnormal returns and compare the results with previous studies.

4

Numerical examples

This section provides several numerical tests of the results. Hackbarth and Morellec [10] find that
acquiring firms earn low or negative abnormal returns, while target firms earn substantially positive
abnormal returns around the announcement date of the takeover. We first study the abnormal returns
in this model and compare the results with those reported in the literature. Second, we examine theacquisition premium and the reaction threshold of merger changes with endogenous parameters. Third,
we provide insights into eﬀect of information on the results. Table 1 summarizes the basic parametervalues.

10

Submitted to International Journal of Real Options and Strategy

Table 1: Parameter values.
Parameter
Drift of the bidderµX
Drift of the targetµY
Volatility of the bidderσX
σY
Volatility of the target
Ratio of the firm size
KX /KY
Correlation coeﬃcientρ
Synergy parameterα
Per unit merger costcRisk-free rate
r

Value
0.05
0.02
0.20
0.30
3.00
0.50
0.30
0.01
0.06

Figure 1: The eﬀects of information on abnormal returns. The black and blue lines represent thebidder and target’s abnormal returns, respectively. We first test the abnormal return change with
the information parameter ωT , assuming ωB = 1.5 and ωB = 0.5, meaning that the bidder is underor over-estimated. We then test the abnormal return change with the information parameter ωB ,assuming ωT = 1.5 and ωT = 0.5.
Figure 1 plots the abnormal returns as a function of the information parameters ωB and ωT . In
the figures, the participating firms’ abnormal returns can be positive or negative according to therelationship between ωB and ωT . Figure 1 shows a diﬀerent result than that reported in Hackbarthand Morellec [10], in which the abnormal returns to the target shareholders are not always higher
than those of the bidder are. The abnormal return to the target shareholders is largely influenced by11Submitted to International Journal of Real Options and Strategy

ωB and increases with ωB . The more the bidder is underestimated, the more the abnormal returnto the target shareholders is. The zero abnormal return point to the target shareholders is aroundωB = 1. The impact of ωT was observable prior to the merger, and hence has less impact on thetarget’s abnormal return. The abnormal return to the bidder shareholders is largely influenced by
ωT and decrease with ωT . For the bidder, ωB was observable before the merger and thus has lessinfluence.

Figure 2: Parameter impacts on the acquisition premium λ∗ under ξ = 0.5. We assume that thebidder and target will negotiate a bisection sharing-rule and test the acquisition premium with thegrowth rate, volatility, correlation coeﬃcient, and synergy parameter. The black, red, and blue lines
represent the relationship when ωB = 1, ωB = 0.5, and ωB = 1.5, respectively. ωT does not aﬀect thebidder’s strategy. Therefore, we set ωT = 1.
.

The merger’s timing and acquisition premium depend on the growth rate and volatility of thefirms’ core business valuations. We release the basic parameter values, which we want to test, and12Submitted to International Journal of Real Options and Strategy

Figure 3: Parameter impacts on the threshold R∗ under ξ = 0.5. We again assume that the bidder
and target will negotiate a bisection sharing-rule and test the acquisition premium with the growthrate, volatility, correlation coeﬃcient, and synergy parameter. The black, red, and blue lines represent
the relationship when ωT = 1, ωT = 0.5, and ωT = 1.5, respectively. ωB does not aﬀect the bidder’sstrategy. Therefore, we set ωB = 1.

13

Submitted to International Journal of Real Options and Strategy

fix the others, as in Table 1. In Figure 2, the bidder will pay a lower acquisition premium when thebidder’s growth rate is high, and pay more when the target’s growth rate is high. As in the reaction
function (22), the lower the acquisition premium is, the higher the threshold will be. Therefore,
in Figure 3, a bidder’s higher growth rate will decelerate the merger process, and a target’s higher
growth rate will accelerate the process. The parameter ϑ1 has a positive relationship with volatility
σX when σX < ρσY , and vice versa. Thus, the acquisition premium will increase with volatility σXand then decrease. When the volatility both firms is high, the bidder will pay a lower acquisition
premium. Hence, it represent a reverse relationship in Figure 3. The correlation coeﬃcient andsynergy parameter will also increase the acquisition premium. In Figure 2, information ωB increases
the acquisition premium. The blue line is always above the other two lines in both Figure 2 andFigure 3.

Figure 4: Eﬀect of the firm size on strategy decision. Firm size equals KX /KY . We assume ωT = 1
and ωB = 1 in this figure and test the impacts of firm size of the bidder and target. Hence, the figure
represents the relationship in a prefect information market.

The ratio of the two participating firms always decreases the acquisition premium. A small target
will generate a small synergy according to the assumption in (4). Therefore, the bidder will pay a lower
acquisition premium. Although the threshold R∗ is a reaction function of the acquisition premiumλ∗ , a lower λ∗ will increase the threshold and decelerate the process. On the other hand, the ratio
KX /KY has a direct negative eﬀect on the threshold if given a fixed λ∗ . The target has two source ofreturns: the cash premium payment λSY (τ−∗ ), which decreases with KX /KY ; and the share payment(1 − ξ). When KX /KY is large, the target shareholders will gain a relatively high post-merger sharepayment (the (1 − ξ)KX X(τ−∗ ) part is higher), even under lower acquisition premium. And therefore
accelerates the merging process. Thus, the threshold will have a maximum when the merger process
is slowest.

14Submitted to International Journal of Real Options and Strategy

Figure 5: Relationship between the terms ξ, acquisition premium, and merger threshold. The left
side represents the bidder’s strategy and we set ωT = 1. The left side represents the target’s strategy.
Therefore, ωB = 1.

In Figure 5, if the bidder negotiates a higher post-merger share ξ, they need to pay a higher
acquisition premium. The bidder may expect more management control after the merger and will
provide the target with an addition cash payment. The target firm is willing to receive a higher cash
payment; therefore, the merger threshold is also lower when ξ is high. In terms of the information
parameter, an underestimated bidder will increase the acquisition premium, and an underestimatedtarget decelerates the merger process.

5

Conclusion

This paper develops a dynamic model of a joint takeover to determine the timing, acquisition premiums, and terms of a takeover. The model considers a mixed cash-share payment, with both acquisition
premiums and a sharing-rule between the bidder and target. We extend the model to a market withimperfect information and show how the strategy interacts with asymmetric information. The merger
generates an abnormal return. The model also tests the relationship between the abnormal returnand information parameters.
The timing, acquisition premiums, and terms of the takeover interact. The model predicts that (i)the abnormal returns to the bidder’s shareholders are positive when the target is overestimated andnegative when the target is underestimated, (ii) the abnormal returns to the target’s shareholders arenegative when the bidder is overestimated and positive when the bidder is underestimated, (iii) the
abnormal return to the target’s shareholders can be higher or lower than those to the bidder’s shareholders, (iv) a higher acquisition premium will accelerate the merger process, and (v) an undervalued
bidder will accelerate the merger process and an underestimated target will decelerate the merger
process.

15

Submitted to International Journal of Real Options and Strategy

A

Proof of Lemma 1

Substituting the firm’s post-merger value given by equation (3) into the value of the option for thetarget to accept the oﬀer, given by (5), yields
[
(
]}
{
(
))
(
)
f sale X(t), Y (t) = max E e−rτ (1 − ξ) SX (τ ) + SY (τ ) + GM X(τ ), Y (τ ) + (λ − 1)KY Y (τ )τ

(A.1)
which is in the region for the two state variables. The payoﬀ function (A.1) satisfies the partial
diﬀerential equation()
1 2 2 sale 1 2 2 sale
sale
sale
rf sale X(t), Y (t) = µX XfX+ µY Y fYsale + σX
X fXX + σY Y fY Y + ρσX σY XY fXY
.
2
2

(A.2)

The value function (A.1) is linearly homogeneous in X(t) and Y (t). We set the ratio of the twocapital prices as
R(t) =

X(t)
Y (t)

(A.3)

We can rewrite the synergy equation (4) as
(
)
(
)
G
GM X(t), Y (t) = Y (t)gR
R(t) ,

(A.4)

where
(
)
[ (
)
]
G
gR
R(t) = KY α R(t) − 1 − c .

(A.5)

Substituting equations (A.4) and (A.5) into (A.1) yields the payoﬀ function
f

sale

(

{
[
]}
)
[
]
−rτ
X(t), Y (t) = max E e Y (τ ) (1 − ξ)(KX + αKY )R(τ ) + (1 − ξ)(1 − α − c) + (λ − 1) KY
τ(A.6)
The payoﬀ function above increases with the ratio R(t). We can assume that when the ratio reaches
a certain threshold R∗ , the target firm will obtain the oﬀer and merge with the bidder. Suppose()()
f sale X(t), Y (t) = Y (t)g sale R(t) .

(A.7)

Successive diﬀerentiation equations of (A.7) with respect to R(t) give
(
)
(
)
sale
sale
fX
X(t), Y (t) = gR
R(t) ,
(
)
(
)
(
)
sale
fYsale X(t), Y (t) = g sale R(t) − R(t)gR
R(t) ,
(
)
(
)
sale
sale
fXX
X(t), Y (t) = gRR
R(t) /Y (t),()()
2 salefYsaleY X(t), Y (t) = R(t) gRR R(t) /Y (t),
(
)
(
)
sale
sale
fXY
X(t), Y (t) = −R(t)gRR
R(t) /Y (t).

(A.8)
(A.9)
(A.10)(A.11)(A.12)

Substituting equations (A.9) - (A.12) into (A.2) yields the ordinary diﬀerential equation
(
)
(
)
(
)
1 2
1
sale
sale
( σX
+ σY2 − ρσX σY )R2 gRR
R(t) + (µX − µY )RgR
R(t) − (r − µY )g sale R(t) = 0. (A.13)2216

Submitted to International Journal of Real Options and Strategy()
Using the technique for solving O.D.E., the general solution of (A.13) can be written as g sale R(t) =AR(t)ϑ1 + CR(t)ϑ2 , where ϑ1 > 1 and ϑ2 < 0 are roots of the quadratic equation
1
1 2+ σY2 − ρσX σY )ϑ(ϑ − 1) + (µX − µY )ϑ − (r − µY ) = 0.
( σX
2
2
(
)
As the no-bubble condition lim g sale R(t) = 0, we have the solution(A.14)

R(t)→0()
g sale R(t) = AR(t)ϑ1 , ϑ1 > 1.

(A.15)()
We can solve g sale R(t) subject to the value-matching and smoothing-pasting conditions()()
g sale R(t) |t=τ ∗ = g G R(t) |t=τ ∗ ,
(
)
(
)
sale
G
gR
R(t) |t=τ ∗ = gR
R(t) |t=τ ∗ .

(A.16)The target will accept the oﬀer at τ ∗ , which leads to R(t) = R∗ at τ ∗ . Substituting equations (A.6)
and (A.15) into (A.16) yields[]
A(R∗ )ϑ1 = (1 − ξ)(KX + αKY )R∗ + (1 − ξ)(1 − α − c) + (λ − 1) KY ,(A.17)
ϑ1 A(R∗ )ϑ1 −1 = (1 − ξ)(KX + αKY ).
Solving the equations above yields) 1 − λ ] ϑ1
KY,
α+c−1 +
1 − ξ ϑ1 − 1 KX + αKY(1 − ξ)(α + c − 1) + (1 − λ)A=
KY (R∗ (λ))−ϑ1
ϑ1 − 1R∗ (λ) =[((A.18)(A.19)

Substituting the results above into (A.15) yields( R(t) )ϑ1() (1 − ξ)(α + c − 1) + (1 − λ)g sale R(t) =KY.
ϑ1 − 1R∗ (λ)(A.20)

Therefore,( R(t) )ϑ1()(1 − ξ)(α + c − 1) + (1 − λ)f sale X(t), Y (t) = Y (t)KY,
ϑ1 − 1R∗ (λ)(A.21)

where R(t) = X(t)/Y (t).

B

Proof of proposition 1

Submitting the result of Lemma 1 into the optimization function (10), we have{[(]( R(t) )ϑ1 }
(
)
)
(
)
f M A X(t), Y (t) = maxY (t)(ξ − 1)KX + ξαKY R∗ (λ) + ξ(1 − α − c) − λ KY
λR∗ (λ)(B.1)
The value function (B.1) is linearly homogeneous in X(t) and Y (t); thus, we assume()()
f M A X(t), Y (t) = Y (t)g M A R(t) ,

(B.2)

17

Submitted to International Journal of Real Options and Strategy

where
{[(
]( R(t) )ϑ1 }
(
)
(
)
)
g M A R(t) = max(ξ − 1)KX + ξαKY R∗ (λ) + ξ(1 − α − c) − λ KY
λR∗ (λ)
We can solve the optimization problem when()
dg M A R(t)
=0
dλ

(B.3)

(B.4)

The result of condition (B.4) isλ∗ = (α + c)(1 − ξ) + ξ +ϑ1 (α + c)(1 − ξ)(KX + αKY )(1 − ξ)KX − α(ϑ1 + ξ − 1)KY

(B.5)

Therefore,R∗ =( ϑ2 )(α + c)KY
1
ϑ1 − 1 α(ϑ1 + ξ − 1)KY − (1 − ξ)KX

C

(B.6)

Proof of proposition 2

We can solve the maximization problem (13) when the sharing-rule satisfies
(
)
(
)
(
)
df M A X(t), Y (t); ξf M A X(t), Y (t); ξ df sale X(t), Y (t); ξ)β+ (1 − β) sale (
=0
dξ
dξ
f
X(t), Y (t); ξAccording to equations (A.7) and (B.2), we can rewrite equation (C.1) as
(
)
(
)
(
)
dg M A R(t); ξg M A R(t); ξ dg sale R(t); ξ)β+ (1 − β) sale (
=0
dξ
dξ
g
R(t); ξ

(C.1)

(C.2)

Substituting the optimal acquisition premium λ∗ , given by (11), and the threshold (12) into the payoﬀ
function (B.3), yields() (α + c)KY ( R(t) )ϑ1
g M A R(t); ξ =ϑ1 − 1R∗ (λ)(C.3)

The first-order diﬀerentiation of equation (C.3) with respect to ξ is()
( R(t) )ϑ1
dg M A R(t); ξϑ1(α + c)KY (αKY + KX )=dξϑ1 − 1 α(ϑ1 + ξ − 1)KY − (1 − ξ)KX R∗ (ξ)(C.4)

We simplify the payoﬀ function of (A.20) as()
g sale R(t); ξ =ϑ1 (1 − ξ)(α + c)(KX + αKY )KY ( R(t) )ϑ1
ϑ1 − 1 αϑ1 KY − (1 − ξ)(KX + αKY ) R∗ (λ)(C.5)

The first-order diﬀerentiation of equation (C.5) with respect to ξ is
(
)
(
) (
dg sale R(t); ξR(t) )ϑ1 ϑ21 (α + c) (1 − ξ)(KX + αKY ) − αKY (KX + αKY )KY
=
(C.6)
dξ
R∗ (λ)
ϑ1 − 1[αϑ1 KY − (1 − ξ)(KX + αKY )]2
Substituting equations (C.3) - (C.6) into (C.2) gives1−ξ =(1 − β)αKY
KX + αKY

(C.7)

18

Submitted to International Journal of Real Options and Strategy

D

Proof of Lemma 2

We can simplify the target firm’s value of the option to accept the oﬀer, given by (21), as
{
[
]}
(
)
[
]
f˜sale X(t), Y (t) = max E e−rτ (1−ξ)(KX +αKY )X(τ )+ (1−ξ)(1−αωT −c)+(λ−1) KY Y (τ )
τ

(D.1)
(
)
(
)
Assume f˜sale X(t), Y (t) = Y (t)g̃ sale R(t) , which satisfies the ordinary diﬀerential equation
(
)
(
)
(
)
1
1 2
sale
sale
+ σY2 − ρσX σY )R2 g̃RR
( σX
R(t) + (µX − µY )Rg̃R
R(t) − (r − µY )g̃ sale R(t) = 0. (D.2)
22()
Suppose g̃ sale R(t) = DR(t)ϑ1 , where ϑ1 is the positive root of equation (9). We can solve()
g sale R(t) subject to the value-matching and smoothing-pasting conditions[]
D(R̃∗ )ϑ1 = (1 − ξ)(KX + αKY )R̃∗ + (1 − ξ)(1 − αωT − c) + (λ − 1) KY ,(D.3)
ϑ1 D(R̃∗ )ϑ1 −1 = (1 − ξ)(KX + αKY ).
Solving the equations above yields[() 1 − λ ] ϑ1
KY,R̃∗ (λ) = αωT + c − 1 +1 − ξ ϑ1 − 1 KX + αKY(1 − ξ)(αωT + c − 1) + (1 − λ)D=
KY (R∗ (λ))−ϑ1
ϑ1 − 1(D.4)

Hence,( R(t) )ϑ1() (1 − ξ)(αωT + c − 1) + (1 − λ)KYg̃ sale R(t) =,ϑ1 − 1R̃∗ (λ)(D.5)

where R(t) = X(t)/Y (t).

E

Proof of proposition 3

We can rewrite the optimization function (25) as{[]( R(t) )ϑ1 }()[]
f˜M A X(t), Y (t) = max Y (t) [(ξ−1)KX +ξαωB KY ]R∗ (λ̃)+ ξ(1−α−c)− λ̃ KY(E.1)
R∗ (λ̃)λ̃Suppose()()
f˜M A X(t), Y (t) = Y (t)g̃ M A R(t) ,

(E.2)

(
)
where g̃ M A R(t) is given by{[]( R(t) )ϑ1 }()[]
g̃ M A R(t) = max [(ξ − 1)KX + ξαωB KY ]R∗ (λ̃) + ξ(1 − α − c) − λ̃ KY(E.3)
R∗ (λ̃)λ̃
where the threshold function R∗ (λ̃) given λ̃ is (6).
We can solve the optimization problem if the following equation is satisfied:()
dg̃ M A R(t)
= 0.
dλ̃

19

(E.4)

Submitted to International Journal of Real Options and Strategy

Therefore, the optimal oﬀered portion satisfies(ϑ1 − 1)[(ξ − 1)KX + ξαωB KY ][] 1 dR∗ (λ̃)
dR∗ (λ̃)+ ϑ1 ξ(1 − α − c)KY − λ̃KY
+ KY = 0 (E.5)
dλ̃R∗ (λ̃) dλ̃

Solving the equation above yields()
λ̃∗ = (1 − ξ) α + c + ξ +ϑ1 (α + c)(1 − ξ)(KX + αKY )(1 − ξ)KX − α[(ϑ1 + ξ − 1) + ϑ1 ξ(ωB − 1)]KY(E.6)

Substituting the result above into the equation (22) leads toR̃∗ =]ϑ1 [ α(ωT − 1)KY
ϑ1 (α + c)KY+.
ϑ1 − 1 KX + αKYα[(ϑ1 + ξ − 1) + ϑ1 ξ(ωB − 1)]KY − (1 − ξ)KX

F

(E.7)

Proof of proposition 4

The sharing rule is determined by negotiations between the bidder and target. According to the
optimization function (28), the sharing-rule satisfies
(
)
(
)
(
)
dg̃ M A R(t); ξ˜g̃ M A R(t); ξ˜ dg̃ sale R(t); ξ˜β+ (1 − β)
=0
(F.1)
(
)
dξ˜
g̃ sale R(t); ξ˜dξ˜For the bidder firm, ωT is unknown until the merger succeeds. Therefore, they will provide an
optimal acquisition premium λ̃ and estimate the threshold at which the target will decide in a market
with perfect information. We can then rewrite the bidder’s payoﬀ function as
}(
(
) {(
)
(
)
g̃ M A R(t); ξ = ξ(KX + αωB KY ) − KX R̃∗ (ξ) + ξ(1 − α − c) − λ̃∗ (ξ) KY(α + c)KY ( R(t) )ϑ1=()
ϑ1 − 1R∗ λ̃∗ (ξ)R(t) )ϑ1()
R∗ λ̃∗ (ξ)

According to the results from (6), this satisfies() [dR∗ λ̃∗ (ξ)
11dλ̃∗ (ξ) 1 − λ̃∗ (ξ) ]
= −
+
(
)
(
)
∗
∗
dξ
dξ
1−ξ
R λ̃ (ξ)(1 − ξ) α + c − 1 + 1 − λ̃∗ (ξ)

(F.2)

(F.3)

The first-order diﬀerentiation of equation (F.2) with respect to ξ is()()
dg̃ M A R(t); ξdR∗ λ̃∗ (ξ)ϑ1 ( R(t) )ϑ1
1= − (α + c)KY
(
)
(
)
dξ
ϑ1 − 1 R∗ λ̃∗ (ξ)
dξR∗ λ̃∗ (ξ)(F.4)( R(t) )ϑ1 ϑ [ dλ̃∗ (ξ) 1 − λ̃∗ (ξ) ](α + c)KY
1
=
−
(
)
(
)
ϑ1 − 1dξ
1−ξR∗ λ̃∗ (ξ)(1 − ξ) α + c − 1 + 1 − λ̃∗ (ξ)
Substituting (F.2) - (F.4) into the optimization function (F.1), we have
)2
KY
ωB (
α
(1 − ξ)(αωT + c − 1) + (1 − λ̃∗ (ξ))α + c KX + αKY 1 − ξ(1 − β)= (1 − ξ)(α + c − 1) + (1 − λ̃∗ (ξ)) +βϑ1(1 − ξ)(α + c − 1) + (1 − λ̃∗ (ξ))

20−

(1 − β)(1 − ϑ1 )(1 − ξ)(αωT + c − 1) + (1 − λ̃∗ (ξ))

Submitted to International Journal of Real Options and Strategy(F.5)
Supposeγ1 = (1 − ξ)(α + c − 1) + (1 − λ̃∗ (ξ))(F.6)

γ2 = (1 − ξ)(αωT + c − 1) + (1 − λ̃∗ (ξ))(F.7)

Thus, the optimal sharing-rule satisfies
KYωB 2(1 − β)αγ2 = γ1 + βϑ1
α + c KX + αKY 1 − ξ− (1−β)(1−ϑ1 )γ1γ2(1 − β)γ1
α
KY
ωB
γ1
⇒
= 2+
2
α + c KX + αKY 1 − ξγ2βϑ1 γ2 − (1 − β)(1 − ϑ1 )γ1 γ2(F.8)

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