an exchange ratio determination model

8/9/2019 An Exchange Ratio Determination Model

1/6

An Exchange Ratio Determination Model1In any merger that is financed by the issue of stock, the determination of swap ratio (or exchange ratio) is

always very important. As we will see in this chapter, the exchange ratio determines the way the synergyis distributed between the shareholders of the merged and the merging company. Secondly, the swap ratio

also determines the control that each group of shareholders will have over the combined firm.

Most of the times mergers become controversial because of the inappropriate determination of the

exchange ratio. One can site the example of the TOMCO and the HLL merger.In this chapter we will see how the exchange ratios are determined, what are the properties of the

exchange ratio, how the synergy gets distributed for a given exchange ratio, and how considerations like

corporate control affect the decision affect the determination of the exchange ratio.

The Model

Suppose A and B are merging. A is the merged company and B is the merging company. A wants to pay

partly in cash and party in stock to the shareholder of B. Lets use the following notation.

A B

No of Shares n1 n2

Earnings per Share E1 E2Stock Price p1 P2

Suppose A gives x shares of A for each 1 shares of B to the shareholders of B. A can give some cashC also to the shareholders of B. We will look at these as two separate cases. Assume that the synergy is

S.

Case 1: The entire merger is financed by stock swap onlyHere, Shareholders of B swap complete ownership over B for a partial ownership over A+B. The

shareholders of A receive B and swap certain number of shares in return.

Analysis from the point of the shareholders of A:

The shareholders of A will agree to go for the merger only if the stock price of A increases as a result ofthe merger. This is given by the following condition:

p1 is less that p1+2.

The price of the merged entity after the merger will be:

2*1

2*21*1

nxn

Snpnp

+

++

Hence shareholders of A will accept the merger if the following condition is satisfied.

2*1

2*2

2*21*12**11*1

2*1

2*21*11

np

Snpx

Snpnpnxpnp

nxn

Snpnpp

+

+++

+

++

The above in-equation puts the upper limit for the exchange ratio that will be acceptable to the

shareholders of A. From the above in-equation, we can find one interesting point. When the shareholders

of A perceive no synergy in the merger, (i.e., S=0), the maximum swap ratio that is acceptable to theshareholders of A is given by p2/p1. As we will see shortly, if there is no synergy in the merger, then the

1 Prof. Pitabas Mohanty


2/6

minimum swap ratio that is acceptable to the shareholders of B is also given by p2/p1. Hence in a merger

when there is no synergy, the only exchange ratio that will be acceptable to the shareholders of both thecompanies is given by p2/p1.

In the above we only looked at the total wealth of the shareholders of A. It is also possible that the

promoter of A does not want to let his stake fall below a certain level after the merger. If both A and B aremanaged by the same promoter this will usually not be a serious problem. If however, B is managed by a

different management, then the promoter of A will also look at considerations other than value while

determining the swap ratio. Lets see how such considerations affect the predictions of our model.Lets use the following notations.

Promoters stake in A:

The minimum stake that the promoter wants in the combined entity:

(The promoter may want the value of to be at least 26%, or 51%, or 76%.)

If the promoter wants a stake of at least after the merger, then the following condition must be satisfied.

+

21

1

nxn

n

Here, the left hand side of the above equation shows the actual stake of the promoter after the merger for a

given swap ratio x. This must be greater than so that the promoter of A will ok the merger. From the

above we can derive the following value for x.

2

1)(

211

n

nx

nnn

+

Hence if the promoter also puts down a condition of certain minimum stake after the merger, then the

maximum swap ratio that will be acceptable to all the shareholders of A (including the promoter) will be

given by

+

21

22,

2

1)(min

np

Snp

n

n

There are two possibilities in this case. If the maximum swap ratio that is acceptable to the promoter islower than the maximum swap ratio that is acceptable to the other shareholders, then it is possible that the

promoter will reject some of the mergers, which are in the interests of most of the shareholders. It is also

possible that the maximum swap ratio acceptable to all the shareholders is less than that is acceptable tothe promoter. As we will discuss later, it is possible that in some cases, the promoter may go ahead with

such mergers.

Analysis from the point of shareholders of B:

The shareholders of B are actually getting x number of shares of A for each share of B that they have.The shareholders of B will accept the merger if the value of the x number of shares of A that they are

getting is higher than the value of each share of B. This will happen if the following condition is satisfied.


3/6

Snp

npx

xSnpnp

xSxnpxnpxnpnp

xnxn

Snpnpp

+

+

+++

+

++

1*1

1*2

*)1*1(1*2

**2*2*1*1*2*21*2

*2*1

2*21*12

The above in-equation gives us the minimum exchange ratio that will be acceptable for a given level ofsynergy. You can see from the above in-equation that when synergy is zero, the shareholders of B will

demand an exchange ratio of at least p2/p1.

We can plot the above two in-equations graphically.

If a promoter different from the promoter of A is managing B, then the promoter of B may put a different

constraint for the merger. The promoter of B may also demand that its stake does not fall below certainkey percentage.

Lets see how this affects our model.

Lets assume that the promoter of B has a stake in B given by . Suppose, also that the promoter of B

wants its stake to be at least after the merger. Then the promoter of B will accept the merger only if thefollowing condition gets satisfied.

+ nn

xn

*21

*2*

Here, the left hand side tells us what is the stake of the promoter of B in the combined entity for differentvalues of x. The above in-equation can be further reduced to

)1*2*(

1*

1*)1*2*(

*2*1**2*

nn

nx

nnnx

xnnxn

+

Synergy

E

x

c

h

a

n

g

e

R

a

t

i

o

Zone 1

Zone 2

Zone 4

Zone 3


4/6

Hence the promoter of B will accept the merger only if the actual exchange ratio is greater than or equal

to the term on the right side of the above equation.We can combine the constraints given in the above two in-equations as follows. The shareholders and the

promoter of B will accept the merger only if the exchange ratio is given by

+ Snp

np

nn

n

1*1

1*2,

1*2*

1*max

There are two possibilities. Either the first term in the bracket is lower than or equal to the second term, orit is higher than the second term. If the first term is higher than the second term, then it is possible that amerger which is beneficial to the shareholders of B (on considerations of value) will not be acceptable to

the promoter and hence will not take place at all.

The above analysis clearly shows that if the promoters are worried about their control in a company, thenthey will reject some of the value-maximizing mergers.

Case 2: Merger is being financed by a mix of cash and stock swap

Here, we will assume that A is financing the acquisition through an issue of stock and part payment of

cash. The share price of the merged entity after the merger will be equal to

21

22211

****

nxnSCnpnpn

+++

Here, the numerator stands for the total value of equity after the merger. The first two terms, namely,n1*p1+n2*p2 stand for the sum total of the value of the two companies. Since A is making a cash paymentof C for each share of B, the value of the merged entity is going to come down after the merger by an

amount equal to n2*C. Finally, since the management is expecting a synergy of S from the merger, the

value of the combined entity will increase by an amount equal to S.2

The numerator tells us that after the merger there will be n1+x*n2 shares outstanding in the merged

company. Initially, there are n1 number of shares in A and consequent to the terms of agreements of the

merger A has to issue x shares of A for each share of A to the shareholders of B.

Analysis from the point of view of the shareholders of A

The proposed merger will be acceptable to the shareholders only if the stock price of the merged entity

after the merger is greater than the pre-merger stock price. That is, the following condition needs to be

satisfied if the shareholders of A will accept the merger.

2*1

)2(*2

*22*21*12**11*1

2*1

*22*21*11

np

SCpnx

SCnpnpnnxpnp

nxn

SCnpnpnp

+

+++

+

++

The above in-equation tells us that if the proposed exchange ratio is greater than the right hand side of the

above equation, then the shareholders of A will not accept the merger. We can see from the above in-

equation that as C increases, the value of x comes down. This quite easy to understand. If A is payinga large sum to the shareholders of B in the form of cash, then the exchange ratio cannot be large. In the

extreme case, when C is set equal to p2, we obtain:

2 It is possible that the actual synergy will be less than what the management expects. However, we are going to ignore these

issues in this Chapter.


5/6

2*1 np

Sx

This tells us that if there is some synergy in the proposed merger, then the shareholders of A will not mind

swapping some shares for the shares of B even when C is set equal to the pre-merger price of B.A second interesting observation that we can find from the above in-equation is that x is a positive

function of the synergy, i.e., S. This is again easy to understand. If the shareholders of A perceive a

large amount of synergy from the merger, then they will not mind exchanging a larger number of shares

of A for every share of B.

Analysis from the point of view of the shareholders of B

The shareholders of B are exchanging their ownership over B for a partial ownership over the new

merged company and a part payment of cash. The shareholders of B will accept this merger only if the

following condition is satisfied:

Cxnxn

SCnpnpnp +

+

++ *

2*1

*22*21*12

Here, each share of B is being exchanged for x number of shares of A and a cash payment of C. The

above in-equation further reduces to:

( )( )

Spn

nCpx

SpnxnCp

CnpnSCnpnpnxnCp

xSCnpnpnnCpnCp

+

+

+++

+++

1*1

1*)2(

1*1*1*)2(

*22*2*22*21*1*1*)2(

*)*22*21*1(2**)2(1*)2(

The right hand side of the above in-equation gives the minimum exchange ratio that is acceptable to the

shareholders of B. We can see certain interesting points from the above in-equation.The value of x is negatively related to the cash payment made (i.e., C). This means that if the

shareholders of B receive a large cash payment, then they will not minding receiving a lower exchange

ratio in the merger. In the extreme case, if we set C equal to p2, then any exchange ratio will beacceptable to the shareholders of B.Secondly, there is also a negative relationship between x and the synergy. This is again very easy to

understand. If there is higher synergy, then the shareholders of B will not mind receiving a lower

exchange ratio.

Determination of Exchange Ratio in Consolidations

Lets use the following notation.

Company A Company B

No of shares n1 n2

Stock price p1 p2

Exchange Ratio x1 x2

Suppose the synergy in consolidation is S.

Shareholders of A will accept this merger if the following condition is satisfied.

p12 p1

This can be written as follows:


6/6

Snp

xnpx

xnpxnpSxnpxnpx

pxnxn

Snpnpx

+

+++

+

++

2*2

2*2*11

2*2*11*1*1*12*2*11*1*1

12*21*1

2*21*1*1

Here, we can see that

.01

,02

1

S

x

x

x

We can similarly derive the following condition for the shareholders of B.

Snp

xnpx

+

1*1

1*1*22

We can rewrite the above two in-equations in the following fashion:

1*2

1*1

2

1

np

Snp

x

x +

for the shareholders of the first company, and

Snp

np

x

x

+

2*2

2*1

2

1

for the shareholders of the second company.

We can see both the curves intersect at the point p1/p2.

an exchange ratio determination model

Documents