Assume the following: the investor\'s required rate of return is 15 percent, the

Question

Assume the following: the investor's required rate of return is 15 percent, the expected level of earnings at the end of this year (E1) is $5.00, the retention ratio is 50 percent, the return on equity (ROE) is 20 percent (that is, it can earn 20 percent vested earnings), and similar shares of stock sell at multiples of 10 times earnings per share. Determine the expected growth rate for dividends. Determine the price/earnings ratio (P/E1) using Equation (10-5a). What is the stock price using the P/E ratio valuation method? What is the stock price using the dividend discount model? What would happen to the P/E ratio (P/E1) and stock price if the firm could earn 25 percent on reinvested earnings (ROE)? What does this tell you about the relationship between the rate the firm can earn on reinvested earnings and P/E ratios?

Explanation / Answer

a. Growth rate can be calculated as follows
g = ROE * RR
ROE is return on investment and RR si retention ratio
g = 20% * 50% = 10.00%

b. Price to earnings ratio = price / earnings
Since it is given that shares sell at 10 multiple
Price / Earnings = 10x

c. Price using P/E valuation
Price = P/E * earnings = 10 * 5 = $50

d. Dividend discount model
Price = Dividend / (K-g)
Dividend = EPS * (1-RR) = 5 * (1-50%) = 2.5
Price = 2.5 / (15% - 10%) = 2.5 / 5% = $50

e. If ROE increases to 25%
g = 25% * 50% = 12.50%
Using dividend discount
Price = Dividend / (K-g)
Price = 2.5 / (15% - 12.50%) = 2.5 / 2.5% = $100
P/E = 100 / 5 = 20x

f. If ROE increases, P/E would increase accordingly
So ROE and P/E are directly proportional to each other

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