The Gordon Growth Model
There are some things that are at risk of growing out of control in the future. Dividends aren't one of them.
When forecasting dividends of a stock for a dividend discount model, one common assumption is that there is some growth rate that dividends can grow at each year until the end of time. But there are limits.
Suppose a dividend of 1.00 was paid today. You might have reason to believe that this dividend will grow at a 3% growth rate, so that next year the dividend will be 1.03. And perhaps you assume that this will go on forever. In the distant future, those dividends will be enormous. What discount rate do you think is necessary to keep the present value of these dividends finite?
Not quite. This would mean that the present value of the cash flows is an infinite series of 1.00 added up. That's infinite.
No. If it were lower than 3%, then the PV of each dividend would be getting larger, and the sum of this series of PVs would be infinite.
Exactly!
As long as the required rate of return _r_ is larger than the growth rate _g_ of 3%, then next year's dividend of 1.03 still isn't as good as today's dividend of 1.00. Each dividend after that would be smaller and smaller in present value, moving toward zero.
| Time | Dividend | Present Value, r=6% |
|---|---|---|
| 1 | 1.0300 | 0.9717 |
| 2 | 1.0609 | 0.9442 |
| 3 | 1.0927 | 0.9175 |
| 10 | 1.3439 | 0.7504 |
| 100 | 19.2186 | 0.0566 |
| 1,000 | 6.9 trillion | 0.0000 |
So this would be a finite sum that could be added up. And in fact, it's a geometric series that collapses to this convenient form.
$$\displaystyle V_0 = \frac{D_1}{r - g} $$
So to complete this example with a 6% discount rate,
$$\displaystyle V_0 = \frac{1.03}{0.06 - 0.03} = 34.33 $$.
Suppose the required rate of return rose from 6% to 7%. What would happen to the estimated value of the stock?
Absolutely!
No. Actually it would fall.
Logically, this makes sense. If you require a higher rate of return, then any given set of cash flows won't look as good. That's always the case. But just by plugging in this new value to the Gordon growth model, you can see that the value falls from 34.33 to 25.75.
$$\displaystyle V_0 = \frac{D_1}{r - g} = \frac{1.03}{0.07 - 0.03} = 25.75 $$
This also works if the growth rate is zero or negative. Just plug in whatever you'd like for _g_, and you'll get a value. This is helpful for __fixed-rate perpetual preferred stock__. If a preferred stock pays a set dividend of 6.50 each year, then you just need a discount rate like 5% to obtain a price.
$$\displaystyle V_0 = \frac{D_1}{r - 0} = \frac{D}{r} = \frac{6.50}{0.05} = 130 $$
How could you rearrange this to clearly state what the rate of return is on a share of preferred stock?
No. Just the opposite: dividend divided by the price.
That's right!
Either way, at least in estimating a terminal price of a stock in a different valuation form, the Gordon growth model and its variations aren't likely going away any time soon. In fact, don't discount the chance that the number of analysts using it could grow at a constant rate... forever.
To summarize:
[[summary]]
This rate is called a __capitalization rate__ when it's used with a perpetuity like this.
This is some really convenient math, but it's also only as accurate as the inputs, of course. If dividends can really be forecast that accurately, great. It's usually not that easy, though. Also, the growth rate can't be too big. Infinity is a long time, and it's not going to grow faster than GDP forever, which is often 1-3% in developed nations.
Exactly 3%
Something lower than 3%
Something larger than 3%
It would fall
It would rise
Dividend divided by price
Price divided by dividend
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