The document discusses pre-merger and post-merger price-to-earnings (P/E) ratios when firm A acquires firm B in an all-stock transaction. It shows that if markets rationally value the acquisition, the post-merger P/E ratio of firm A will equal a weighted average of the pre-merger P/E ratios of firms A and B, with the weights based on each firm's relative contribution to post-merger earnings. However, if investors mistakenly interpret the increase in earnings per share as a sign of high future growth, the post-merger P/E ratio may exceed the rational weighted average, temporarily inflating the stock price until the market recognizes its

1. PRE AND POST MERGER P/E RATIOS, AND THE BOOTSTRAPPING GAME
Suppose that A buys B in a stock-for-stock transaction. Let A, B and AB refer to firm A before
the merger, firm B before the merger, and firm A after it has acquired firm B. Following the
Brealey & Myers (B&M) discussion of bootstrapping on pages 935-936, and in order to focus on
the main point, we will assume that the merger is non-synergistic from an economic and
accounting perspective (the same general principle holds with synergies or diseconomies), and
that earnings perfectly reflect cash flows (so that market value can be computed using earnings).
For simplicity assume all-equity firms (this is not necessary but simplifies the discussion). We
will use the following definitions:
AV , BV , ABV = market values of firm A before the merger, firm B before the merger,
and firm A after it has acquired firm B, respectively
AE , BE , ABE = earnings of firm A, firm B, and firm A after the merger, respectively
A]e/p[ , B]e/p[ , AB]e/p[ = price-to-earnings ratio of firm A before the merger, firm B
before the merger, and firm A after the merger, respectively. The price-to-earnings ratio
depends on the future growth rate of earnings and on the risk of earnings. The price-to-
earnings ratio increases as expected future growth increases and as perceived risk decreases
(all else held constant, higher growth is good and greater risk is bad).
AV = A]e/p[ × AE , BV = B]e/p[ × BE , ABV = AB]e/p[ × ABE
PRICE-TO-EARNINGS RATIOS AND EQUITY VALUES IF MARKETS ARE EFFICIENT
Since the merger is of the conglomerate (non-synergistic) type, we know that:
ABE = AE + BE (1)
As shown in the Appendix, it follows that:
AB]e/p[ = A]e/p[ 





×
AB
A
E
E
+ B]e/p[ 





×
AB
B
E
E
(2)
Equation (2) says that the price-to-earnings ratio of AB must equal the weighted-average of the
price-to-earnings ratios of firms A and B, where the weights (in big brackets) equal the
percentage of ABE that is represented by the earnings of A ( AE ) and the earnings of B ( BE ).
This makes intuitive sense. Post-merger firm A will have a risk level and growth rate
somewhere between those of pre-merger firms A and B. Therefore, since price-to-earnings
ratios depend on earnings growth rate and risk, AB]e/p[ is an average of A]e/p[ and B]e/p[ .
Furthermore, the greater is AE relative to BE , the closer will be the post-merger firm A to pre-
merger firm A, and the closer will be AB]e/p[ to A]e/p[ .
Example: To illustrate equation (2), suppose that (all dollar amounts in $millions):
1

2. AE = $40, BE = $60 and ABE = $100 (3a)
A]e/p[ = 30 (3b)
B]e/p[ = 10 (3c)






AB
A
E
E
= .4 (4a)






AB
B
E
E
= .6 (4b)
Using (2), it follows that:
AB]e/p[ = A]e/p[ 





×
AB
A
E
E
+ B]e/p[ 





×
AB
B
E
E
= [30 × .4] + [10 × .6]
= 18 (5)
Thus:
AV = A]e/p[ × AE = 30 × $40 = $1,200 (6a)
BV = B]e/p[ × BE = 10 × $60 = $600 (6b)
ABV = AB]e/p[ × ABE = 18 × $100 = $1,800 (6c)
EQUITY VALUATION, MARKET EFFICIENCY AND “BOOTSTRAPPING”
2

3. Bootstrapping is the process under which the price-earnings ratio of an acquiring firm is
excessively high because the market misinterprets an increase in earnings per share due to
the acquisition as a sign of high earnings-per-share growth in the future. Exhibit 1 shows
the outcomes if markets correctly value acquiring firm A and acquired firm B both before and
after the merger. To focus on the principle issue, assume a non-synergistic merger of firms A
and B, where A and B are equally risky but acquiring firm A has a higher earnings per share
growth rate. Exhibit 1 uses the data in equations (3a) through (6c).
Exhibit 1. Valuation in Efficient Markets - No Bootstrapping (in $million)
Firm A before
merger
Firm B before
merger
Firm A after the
merger (AB)
1. Earnings per share $2 $1 $3.33
2. Price per share $60 $10 $60
3. Price-earnings ratio 30 10 18
4. Number of shares 20 mil. 60 mil. 30 mil.
5. Total earnings $40 mil. $60 mil. $100 mil.
6. Total market value $1,200 mil. $600 mil. $1,800 mil.
7. Earnings per dollar invested in
the stock (line 1 ÷ line 2) $.033 $.10 $.056
The firm A share price is $60 both before and after the merger even though firm A earnings per
share rises from $2 to $3.33. This is because the merger is non-synergistic (no value is created
by the merger) and a fair price is paid by firm A for firm B. The post-merger firm A price-to-
earnings ratio is 18 (rather than the pre-merger 30) because earnings-per-share growth is lower
for firm A after the merger than before. The post-merger firm A earnings are a blend of the pre-
merger firm A high-growth earnings and the firm B low-growth earnings; the growth rate of
post-merger firm A earnings will be between the growth rates of pre-merger firm A earnings and
firm B earnings. As a result, A]e/p[ > AB]e/p[ > B]e/p[ (30 > 18 > 10).
Exhibit 2. Valuation in Inefficient Markets with Bootstrapping (in $million)
Firm A before
merger
Firm B before
merger
Firm A after the
merger (AB)
1. Earnings per share $2 $1 $3.33
2. Price per share $60 $10 $100
3. Price-earnings ratio 30 10 30
4. Number of shares 20 mil. 60 mil. 30 mil.
5. Total earnings $40 mil. $60 mil. $100 mil.
6. Total market value $1,200 mil. $600 mil. $3,000 mil.
7. Earnings per dollar invested in
the stock (line 1 ÷ line 2) $.033 $.10 $.0333
Now suppose that, as illustrated in Exhibit 2 above, investors misinterpret the leap in firm A
earnings per share from $2 to $3.33 as a sign of rapid future earnings per share growth, and that
they keep the firm A price-to-earnings ratio at 30, i.e., AB]e/p[ = A]e/p[ = 30 (any AB]e/p[
above 18 will make the point). Although the merger has in reality produced no economic value
3

4. added (since we assume a non-synergistic merger), the market value of post-merger firm A
shares ($3 billion) exceeds the sum of the market values of firms A and B were they not to merge
($1.8 billion). This market value increase is due to investor misunderstanding of the
combination and its future consequences for earnings. As time passes, of course, the market will
come to realize the truth and firm A’s market price will reflect that realization. Initially, though,
there is a mispricing of firm A’s stock. As Brealey & Myers point out, the firm A management
may encourage this temporary illusion in order to produce personal benefits (for example,
because management plans to exercise stock options and then liquidate the stock before the
market perceives the mispricing).
4

5. APPENDIX
Assume markets in which assets are rationally valued. Assume the following definitions:
AV , BV , ABV = market values of firm A before the merger, firm B before the merger,
and firm A after it has acquired firm B, respectively
AE , BE , ABE = earnings of firm A, firm B, and the merged firm AB, respectively
A]e/p[ , B]e/p[ , AB]e/p[ = price-to-earnings ratio of firm A, firm B, and the merged
firm AB, respectively
The merger is non-synergistic (from an economic or accounting perspective); therefore:
ABE = AE + BE (A.1)
The merger produces no economies or diseconomies. In markets in which assets are consistently
and rationally valued, investors are not fooled and we have:
ABV = AV + BV (A.2)
The price-to-earnings must equal total equity value divided by total earnings. That is:
A]e/p[ = ]E/V[ AA (A.3a)
B]e/p[ = ]E/V[ BB (A.3b)
AB]e/p[ = ]E/V[ ABAB (A.3c)
Using (A.1) through (A.3c) we have:
AB]e/p[ =
AB
AB
E
V
=
BA
BA
EE
VV
+
+
=
AB
A
E
V
+
AB
B
E
V
= 





×





AB
A
A
A
E
E
E
V
+ 





×





AB
B
B
B
E
E
E
V
= A]e/p[ 





×
AB
A
E
E
+ B]e/p[ 





×
AB
B
E
E
(A.4)
5/25/2004
5

6. APPENDIX
Assume markets in which assets are rationally valued. Assume the following definitions:
AV , BV , ABV = market values of firm A before the merger, firm B before the merger,
and firm A after it has acquired firm B, respectively
AE , BE , ABE = earnings of firm A, firm B, and the merged firm AB, respectively
A]e/p[ , B]e/p[ , AB]e/p[ = price-to-earnings ratio of firm A, firm B, and the merged
firm AB, respectively
The merger is non-synergistic (from an economic or accounting perspective); therefore:
ABE = AE + BE (A.1)
The merger produces no economies or diseconomies. In markets in which assets are consistently
and rationally valued, investors are not fooled and we have:
ABV = AV + BV (A.2)
The price-to-earnings must equal total equity value divided by total earnings. That is:
A]e/p[ = ]E/V[ AA (A.3a)
B]e/p[ = ]E/V[ BB (A.3b)
AB]e/p[ = ]E/V[ ABAB (A.3c)
Using (A.1) through (A.3c) we have:
AB]e/p[ =
AB
AB
E
V
=
BA
BA
EE
VV
+
+
=
AB
A
E
V
+
AB
B
E
V
= 





×





AB
A
A
A
E
E
E
V
+ 





×





AB
B
B
B
E
E
E
V
= A]e/p[ 





×
AB
A
E
E
+ B]e/p[ 





×
AB
B
E
E
(A.4)
5/25/2004
5