Equity Valuation Formulas William L. Silber and Jessica Wachter
I. The Dividend Discount Model Suppose a stock with price P0 pays dividend D1 one year from now, D2 two years from now, and so on, forthe rest of time. P0 is then equal to the discounted value of the future dividends: (1)
D1 D2 D3 + + +L 2 1 + k (1+ k ) (1 + k )3
The discount factor, k, is the firm’s cost of equity capitaland is given by the CAPM’s required rate of return for holding the stock:
k = Rf + β ( RM − Rf ) .
k is sometimes called the firm’s capitalization rate.
II. Some Simplifications and Extensions We cansimplify equation (1) by assuming that the company pays the same expected dividend forever. For some companies, this is not a bad approximation. Then:
P0 = D D D + + +L 2 1 + k (1+ k ) (1 + k )3
P0is simply a perpetuity with cash payment D and discount rate k. Using the formula for perpetuities: (2)
P0 = D k
This formula can be related to the price-earnings ratio because dividends are paidout of earnings. Let b denote the plowback ratio, i.e., the fraction of earnings that are “plowed back” into the company. The rest are paid out as dividends. Then:
D = (1− b)E .
Substituting thisexpression into equation (2) above, we have:
P0 = E(1− b ) k
which implies: (3)
P0 1 − b = . E k
This simple model implies that the price-earnings ratio is inversely related to the firm’s cost of equitycapital, k. The lower is k the higher is the firm’s price-earnings ratio. Note that when b=0 the price-earnings ratio becomes 1/k. More on this special case below. III. Dividend Growth Model Thesimplified dividend discount model does not capture a feature that is important for many companies: dividends are expected to grow over time. We need to modify the model to account for dividend growth. Asimple assumption is that dividends are expected to grow at a constant rate, g, forever. This means:
D1 = (1 + g )D0 D 2 = (1+ g ) D0
D3 = (1+ g ) 3 D0
and so forth. When we substitute these...
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