## The Constantgrowth Dividend Discount Model

The dividend discount model requires a forecast of dividends for every year into the future, which poses a bit of a problem for stocks with potentially infinite lives. Unless we want to spend a lifetime forecasting dividends, we must use simplifying assumptions to reduce the number of estimates. The simplest simplification assumes a no-growth perpetuity which works for no-growth common shares.

Here's another simplification that finds a good deal of practical use. Suppose forecast dividends grow at a constant rate into the indefinite future. If dividends grow at a steady rate, then instead of forecasting an infinite number of dividends, we need to forecast only the next dividend and the dividend growth rate.

Recall Blue Skies Inc. It will pay a $3 dividend in 1 year. If the dividend grows at a constant rate of g = .08 (8 percent) thereafter, then dividends in future years will be

DIV2 = $3 x (1 + g) = $3 x 1.08 = $3.24 DIV3 = $3 x (1 + g)2 = $3 x 1.082 = $3.50

Plug these forecasts of future dividends into the dividend discount model:

CONSTANT-GROWTH DIVIDEND DISCOUNT MODEL Version of the dividend discount model in which dividends grow at a constant rate.

Although there is an infinite number of terms, each term is proportionately smaller than the preceding one as long as the dividend growth rate g is less than the discount rate r. Because the present value of far-distant dividends will be ever-closer to zero, the sum of all of these terms is finite despite the fact that an infinite number of dividends will be paid. The sum can be shown to equal

This equation is called the constant-growth dividend discount model, or the Gordon growth model after Myron Gordon, who did much to popularize it.7

V EXAMPLE 3 Blue Skies Valued by the Constant-Growth Model

Let's apply the constant-growth model to Blue Skies. Assume a dividend has just been paid. The next dividend, to be paid in a year, is forecast at DIVj = $3, the growth rate of dividends is g = 8 percent, and the discount rate is r = 12 percent. Therefore, we solve for the stock value as p DIV1 $3 $75 P°= r - g = .12 - .08 = $75

The constant-growth formula is close to the formula for the present value of a perpetuity. Suppose you forecast no growth in dividends (g = 0). Then the dividend stream is a simple perpetuity, and the valuation formula is P0 = DIVj/r. This is precisely the formula you used in Self-Test 5 to value Moonshine, a no-growth common stock.

The constant-growth model generalizes the perpetuity formula to allow for constant growth in dividends. Notice that as g increases, the stock price also rises. However, the constant-growth formula is valid only when g is less than r. If someone forecasts perpetual dividend growth at a rate greater than investors' required return r,then two things happen:

1. The formula explodes. It gives nutty answers. (Try a numerical example.)

2. You know the forecast is wrong, because far-distant dividends would have incredibly high present values. (Again, try a numerical example. Calculate the present value of a dividend paid after 100 years, assuming DIV1 = $3, r = .12, but g = .20.)

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