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Calculation of Exchange Ratio from the perspective of the acquired and the acquiring firm

Whenever a firm A acquires another firm B, the compensation to the shareholders of the acquired firm is usually paid in the form of shares of the acquiring firm. In other words, shares of firm A will be given in exchange for shares of firm B. Thus, the exchange ratio is a very important factor in any kind of merger. Firm A will want to keep this ratio as low as possible, while firm B will want it to be as high as possible. In any case, both firms would ensure that post merger, their equivalent price per share will at least equal their pre-merger price per share.  Given below is the model developed by Conn and Nielson for determining the exchange ratio. The symbols used in this model are: -

ER = Exchange ratio
P = Price per share
EPS = Earning per share
PE = Price earning multiple
E = Earnings
S = Number of outstanding equity shares
AER = Actual exchange ratio

In addition, the acquiring, acquired and combined firms will be referred to by subscripts A, B and AB respectively.

Firm A would ensure that the wealth of its shareholders is preserved. This implies that the price per share of the combined firm is at least equal to the price per share of firm A before merger:

PAB >= PA

For the sake of simplicity consider that P AB =P A

Price earnings ratio of the combined firm x Earnings per share of the combined firm gives the Market Price per share.

P AB =PE AB x EPS AB = P A     -------  (1)        

Earnings per share of the combined firm can be expressed as:

EPS AB = (E A + EB ) / [S A  + S B (ER A )]   ------- (2)

ER A  = number of shares of firm A given in lieu of one share of firm B.

Substituting formula of EPS AB in equation 1 we get

P A  = PE AB (E A  +EB)/[S A +S B (ER)]   

From the above equation, we may solve for the value of ER A as follows

ER A = -(S A /S B ) + [(E A +E B) PE AB] / P A S B 

After discussing the maximum exchange ratio acceptable to the shareholders of firm A above, we will now calculate the minimum exchange ratio acceptable to the firm B(ER B)

The basic condition is

P AB (ER B) >= P B  ---------- (3)

Using the equality form of above equation and substituting P AB from equation 1 in equation 3 we get

PE AB x EPS AB   x ER B = P B

Substituting the value of  EPS AB  from equation 2 in the above equation, and solving the equation for  ER B we get

ER B = P B S A / [(PE AB)(E A + E B) P B S B]

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