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THE DIVIDEND DISCOUNT MODEL

We have managed to explain today s stock price P0 in terms of the dividend DIV1 and the expected stock price next year P1. But future stock prices are not easy to forecast directly, though you may encounter individuals who claim to be able to do so. A formula that requires tomorrow s stock price to explain today s stock price is not generally helpful.

As it turns out, we can express a stock s value as the present value of all the forecast future dividends paid by the company to its shareholders without referring to the future stock price. This is the dividend discount model:

How far out in the future could we look? In principle, 40, 60, or 100 years or more corporations are potentially immortal. However, far-distant dividends will not have significant present values. For example, the present value of $1 received in 30 years using a 10 percent discount rate is only $.057. Most of the value of established companies comes from dividends to be paid within a person s working lifetime.

How do we get from the one-period formula P0 = (DIV1 + P1)/(1 + r) to the dividend discount model? We look at increasingly long investment horizons.

Let s consider investors with different investment horizons. Each investor will value the share of stock as the present value of the dividends that she expects to receive plus the present value of the price at which the stock is eventually sold. Unlike bonds, however, the final horizon date for stocks is not specified stocks do not mature. More over, both dividends and final sales price can only be estimated. But the general valuation approach is the same. For a one-period investor, the valuation formula looks like this:

In fact we can look as far out into the future as we like. Suppose we call our horizon date H. Then the stock valuation formula would be

In words, the value of a stock is the present value of the dividends it will pay over the investor s horizon plus the present value of the expected stock price at the end of that horizon.

Does this mean that investors of different horizons will all come to different conclusions about the value of the stock? No! Regardless of the investment horizon, the stock value will be the same. This is because the stock price at the horizon date is determined by expectations of dividends from that date forward. Therefore, as long as the investors are consistent in their assessment of the prospects of the firm, they will arrive at the same present value. Let s confirm this with an example.

DIVIDEND DISCOUNT MODEL Discounted cashflow model of today s stock price which states that share value equals the present value of all expected future dividends.

Valuing Blue Skies Stock

Take Blue Skies. The firm is growing steadily and investors expect both the stock price and the dividend to increase at 8 percent per year. Now consider three investors, Erste, Zweiter, and Dritter. Erste plans to hold Blue Skies for 1 year, Zweiter for 2, and Dritter for 3. Compare their payoffs:

Remember, we assumed that dividends and stock prices for Blue Skies are expected to grow at a steady 8 percent. Thus DIV2 = $3 АГАз 1.08 = $3.24, DIV3 = $3.24 АГАз 1.08 = $3.50, and so on.

Erste, Zweiter, and Dritter all require the same 12 percent expected return. So we can calculate present value over Erste s 1-year horizon:

All agree the stock is worth $75 per share. This illustrates our basic principle: the value of a common stock equals the present value of dividends received out to the investment horizon plus the present value of the forecast stock price at the horizon. Moreover, when you move the horizon date, the stock s present value should not change. The principle holds for horizons of 1, 3, 10, 20, and 50 years or more.

Look at Table 3.5, which continues the Blue Skies example for various time horizons, still assuming that the dividends are expected to increase at a steady 8 percent compound rate. The expected price increases at the same 8 percent rate. Each row in the table represents a present value calculation for a different horizon year. Note that present value does not depend on the investment horizon. Figure 3.12 presents the same data in a graph. Each column shows the present value of the dividends up to the horizon and the present value of the price at the horizon. As the horizon recedes, the dividend stream accounts for an increasing proportion of present value but the total present value of dividends plus terminal price always equals $75.

If the horizon is infinitely far away, then we can forget about the final horizon price it has almost no present value and simply say Stock price = PV (all future dividends per share) This is the dividend discount model.