Question 15.3: If P gives 25,000 shares having a market price of $80 per sh...

It is incorrect to argue that the P/E of P+S will be the same as the P/E of P, leading to a price of $11.20 × 8 or $89.60. The new price/earnings ratio can be expected to be a weighted average of the old P/E ratios, where the weights are the total earnings of each of the companies divided by the total earnings of both companies.

New   P/E = \frac{E_{P} }{E_{P}+E_{S} } (P/E  of  P)+\frac{E_{S} }{E_{P}+E_{S} } (P/E  of   S)

=\frac{\$1,000,000}{\$1,400,000}(8)+\frac{\$400,000}{\$1,400,000}(5)

= 5.714 + 1.429 = 7.143.

Table 15.1. Financial Information for Firms P and S.

This leads to a market value of a share of P +S equal to 7.143($11.20) = $80. The total market value is

$80(125,000) = $10,000,000,

which is equal to the sum of the values of firms P plus S. Value is being neither created nor destroyed by the acquisition.

With no change in operations, expectations, or payouts, the post-acquisition P/E ratio should be a weighted average of the pre-acquisition P/Es, where the weights are the relative amounts of earnings of each component.

To understand better the logic of why the P/E of firm P + S is less than that of firm P, we will determine the implied growth rates for P and for S. Table 15.2 shows the current dividend, the cost of equity, and the retention rate. For example, P earned $10 and paid a $6 dividend; therefore, the retention rate is 0.4.

Using the one-stage growth model,

P=\frac{D}{k_{e}-g }

and, if D = (1 − b)E, then

P=\frac{(1-b)E}{k_{e}-g }

Solving this relationship for g, we obtain the implicit growth rate:

g=k_{e} -(1-b)\frac{E}{P} .

For firm P, we have

g=0.15-\frac{1}{8} (1-0.4)=0.15-0.075=0.075.

Table 15.2. Additional Information.

With zero debt, and a return on investment of r for new investments, then

g = rb
0.075 = 0.4r
r = 0.1875.

For firm S, we have a lower implied growth rate:

g=k_{e} -\frac{E}{P} (1-b)

0.15 -\frac{1}{5} (1-0.5)=0.05.

The implicit return on new investments is

g = rb
0.05 = 0.5r
r = 0.10.

Firm S has a lower growth rate than does P; we also see that S is expected to earn only 0.10 on new investments, whereas P is expected to earn 0.1875. P’s higher P/E ratio implies that the market is more optimistic about its future growth and earnings opportunities.

Each P/E ratio and dividend retention percentage implies a different growth rate (and thus a different return on new investment). Table 15.3 gives a few illustrative values for a retention rate of 0.6 and a 0.15 cost of equity capital.
The relationship g = rb assumes that there is zero debt. If we changed that assumption, the growth rate formulation would be somewhat more complex.

Table 15.3. Implied Growth Rates and Returns

Note: b = 0.4;  k_{e} = 0.15;   g=k_{e}-(1-b)\frac{E}{P} ;   g=rb.

Thus, assuming that a firm can be acquired for its current market value, it is not important if the current P/E of the candidate for acquisition is above or below the acquirer’s P/E. It is important whether the acquisition can be expected to result in an increase in the acquirer’s P/E as a result of a change in operations (changing retention rates or debt utilization), a decrease in financing costs (k_{i} and k_{e}), or real synergistic effects such as better rates of return on reinvestment or direct reduction of cost (higher current earnings). If the weighted average post-acquisition P/E ratio exceeds the weighted average pre-acquisition P/E ratio, or if earnings improve, the acquisition will tend to be beneficial to stockholders of both the acquired and acquiring firms.