## Example

Ross et al.: Fundamentals I III. Valuation of Future I 8. Stock Valuation I I © The McGraw-Hill of Corporate Finance, Sixth Cash Flows Companies, 2002

Edition, Alternate Edition

### 248 PART THREE Valuation of Future Cash Flows

The only tricky thing here is that the next dividend, D^ is given as \$4, so we won't multiply this by (1 + g). With this in mind, the price per share is given by:

Because we already have the dividend in one year, we know that the dividend in four years is equal to D1 X (1 + g)3 = \$4 X 1.063 = \$4.764. The price in four years is therefore:

P4 = D4 X (1 + g )/(R - g) = \$4.764 X 1.06/(.16 - .06) = \$5.05/.10 = \$50.50

Notice in this example that P4 is equal to P0 X (1 + g)4.

P4 = \$50.50 = \$40 X 1.064 = P0 X (1 + g)4 To see why this is so, notice first that:

P4 = D5/(R - g) However, D5 is just equal to ^ X (1 + g)4, so we can write P4 as:

This last example illustrates that the dividend growth model makes the implicit assumption that the stock price will grow at the same constant rate as the dividend. This really isn't too surprising. What it tells us is that if the cash flows on an investment grow at a constant rate through time, so does the value of that investment.

You might wonder what would happen with the dividend growth model if the growth rate, g, were greater than the discount rate, R. It looks like we would get a negative stock price because R - g would be less than zero. This is not what would happen.

Instead, if the constant growth rate exceeds the discount rate, then the stock price is infinitely large. Why? If the growth rate is bigger than the discount rate, then the present value of the dividends keeps on getting bigger and bigger. Essentially, the same is true if the growth rate and the discount rate are equal. In both cases, the simplification that allows us to replace the infinite stream of dividends with the dividend growth model is "illegal," so the answers we get from the dividend growth model are nonsense unless the growth rate is less than the discount rate.

Finally, the expression we came up with for the constant growth case will work for any growing perpetuity, not just dividends on common stock. If C1 is the next cash flow on a growing perpetuity, then the present value of the cash flows is given by:

Notice that this expression looks like the result for an ordinary perpetuity except that we have R - g on the bottom instead of just R.

Ross et al.: Fundamentals of Corporate Finance, Sixth Edition, Alternate Edition

III. Valuation of Future Cash Flows

8. Stock Valuation

### CHAPTER 8 Stock Valuation

Nonconstant Growth The last case we consider is nonconstant growth. The main reason to consider this case is to allow for "supernormal" growth rates over some finite length of time. As we discussed earlier, the growth rate cannot exceed the required return indefinitely, but it certainly could do so for some number of years. To avoid the problem of having to forecast and discount an infinite number of dividends, we will require that the dividends start growing at a constant rate sometime in the future.

For a simple example of nonconstant growth, consider the case of a company that is currently not paying dividends. You predict that, in five years, the company will pay a dividend for the first time. The dividend will be \$.50 per share. You expect that this dividend will then grow at a rate of 10 percent per year indefinitely. The required return on companies such as this one is 20 percent. What is the price of the stock today?

To see what the stock is worth today, we first find out what it will be worth once dividends are paid. We can then calculate the present value of that future price to get today's price. The first dividend will be paid in five years, and the dividend will grow steadily from then on. Using the dividend growth model, we can say that the price in four years will be:

P4 = D4 X (1 + g)/(R - g) = Ds/(R - g) = \$.50/(.20 - .10) = \$5

If the stock will be worth \$5 in four years, then we can get the current value by discounting this price back four years at 20 percent:

The stock is therefore worth \$2.41 today.

The problem of nonconstant growth is only slightly more complicated if the dividends are not zero for the first several years. For example, suppose that you have come up with the following dividend forecasts for the next three years:

 Year Expected Dividend
0 0