Equity Valuation Formulas. William L. Silber and Jessica Wachter
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1 Equity Valuation Formulas William L. Silber and Jessica Wachter I. The ividend iscount Model Suppose a stoc with price pays dividend one year from now, two years from now, and so on, for the rest of time. is then equal to the discounted value of the future dividends: () L + (+ ( + The discount factor,, is the firm s cost of equity capital and is given by the CAM s required rate of return for holding the stoc: R + β R R ). f ( M f is sometimes called the firm s capitalization rate. II. Some Simplifications and Extensions We can simplify equation () by assuming that the company pays the same expected dividend forever. For some companies, this is not a bad approximation. Then: (+ ( + +L is simply a perpetuity with cash payment and discount rate. Using the formula for perpetuities: ()
2 This formula can be related to the price-earnings ratio because dividends are paid out of earnings. Let b denote the plowbac ratio, i.e., the fraction of earnings that are plowed bac into the company. The rest are paid out as dividends. Then: ( b) E. Substituting this expression into equation () above, we have: E( b) which implies: () E b. This simple model implies that the price-earnings ratio is inversely related to the firm s cost of equity capital,. The lower is the higher is the firm s price-earnings ratio. Note that when b the price-earnings ratio becomes /. More on this special case below. III. ividend Growth Model The simplified 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. A simple assumption is that dividends are expected to grow at a constant rate, g, forever. This means: (+ (+ (+ and so forth. When we substitute these into equation () above we get: ( + (+ ( (+ ( + +L Factoring out (+ ), produces: g
3 + g ( + ( L + (+ (+ It turns out that when >g the term in bracets has a finite sum equal to /(. If <g the braceted sum goes to infinity. Thus, as long as >g, we have: (4) ( + g g Equation (4) gives the value of a stoc according to the dividend growth model. IV. Where oes Growth Come From? The dividend growth model wors better than the model with constant expected dividends, but it does require an estimate for g, the growth rate. A simple numerical example shows that if ROE is the return on the firm s equity, then: g ROE b In words, the growth rate equals the return on equity times the plowbac ratio, or growth is determined by how much of earnings is put bac into the firm, and how profitable those earnings are. Therefore, substituting g ROE b into equation (4), produces: (5) ROE b V. eterminants of the rice-earnings Ratio We can go one step further and derive the implications of the dividend growth model for a firm s price-earnings ratio. Because E( b) we can rewrite (5) as:
4 E b ( ) ROE b This gives us: (6) E b ROE b Thus, the price-earnings ratio is determined by the maret capitalization rate, the plowbac ratio b, and the return on equity ROE. When ROE, something interesting happens: E It doesn t matter how much is plowed bac into the firm in this case. Why does this happen? When ROE, the earnings that are ept in the firm provide the same return as earnings that are paid out. So the price of the firm is the same whether those earnings are left in the firm or returned to shareholders. When ROE the price-earnings ratio does depend on the plowbac ratio but the direction of the effect depends on whether derivative of /E in expression (6) with respect to b is, as follows: ROE > or ROE <. In particular, the E b ( ROE b)( ) ( b)( ROE) ( ROE b) This expression can be simplified to:
5 (7) E b ROE ( ) The sign of (7) is positive if ROE > and the sign is negative if ROE <. This is a comforting result because only if a company s ROE is greater than its cost of capital,, will an increase in the fraction plowed bac into the firm be rewarded with a high price-earnings ratio. Companies that have a low ROE relative to will be punished with a low price-earnings ratio when they plowbac a larger fraction of earnings.
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