###### The Dividend (Gordon) Growth Model

Finance theory views the price of an asset as the sum of its future discounted cash flows. In general, the cash flows can arrive at any future date. They need not be spaced evenly, say, at annual periods.

For example, the first cash flow could arrive in one year, the second cash flow could arrive six months after that, the third cash flow could arrive one day later, and the next cash flow (which could be the last cash flow) could arrive five years after that.

Also, in general each of those cash flows can have its own discount rate. For example, the risk of the cash flow to come in one month could be very different than the risk of the cash flow to be generated in seven months, and both of these might be different than the risk of the cash flow to come in two years, and so on.

A general model for the price of an asset is then the sum of its future cash flows discounted at the required rate of return appropriate for that cash flow's risk and timing. That is,

Notice that the price at time t (e.g., today), Pt, is the sum of the cash flow in each future period discounted at the rate appropriate to the risk of that cash flow and its timing.

The important thing to keep in mind is that a general pricing formula can assume different cash flows at different periods of time and different appropriate discount rates for those cash flows, which in general could be at many different times and need not be spaced out equally (e.g., annually).

Step 1 to the DGM: Assume a constant time interval for cash flows

We can start on the way to the dividend-growth model ("DGM", also known as the Gordon Growth Model) by first assuming that the future cash flows occur annually at the end of every year. That is, we will no longer allow cash flows to come in an arbitrary time intervals. That means that

where t is today and each increment of 1 is a year, so that t+2 is, for example, two years from today.

Step 2 to the DGM: Assume a constant growth rate in the cash flows

We continue toward the DGM by assuming that cash flows are related to each other by a constant growth rate, g. For example, CFt+2 cannot be any amount; it must be CFt+2 x (1+g), while

and so on.

Step 3 to the DGM: Assume a constant required rate of return

The next step is to assume that the annual required rate of return is constant at r to be earned each period, so that

etc.

Step 4 to the DGM: Assume that cash flows continue forever

The next step is to assume that the cash flow stream is infinite. This is why put that cool image of the mathematical symbol for infinity at the top of this page.

Step 5 to the DGM: Solve the infinite series of cash flows to get the DGM formula

We can now get to the DGM. We started with

This last equation is an infinite series, a sum of an infinite number of terms, with each term being a cash flow and its discount factor. As long as g is less than r (g < r), each term (each year's additional discounted cash flow) is getting smaller the farther out we go, so the series "converges" to (using a mathematical result for the sum of this kind of infinite series)

Games People Play (with this model)

Note that the DGM formula is accurate only if the assumptions that we made hold in the real world. That is, (1) assuming a constant time interval; (2) assuming a constant growth rate; (3) assuming a constant rate of return; and (4) assuming that the cash flows go on forever. You get the idea. This model is never going to be very satisfactory, because no (interesting) firm can satisfy these assumptions.

People play a variety of games - that is, push bad valuations - with the DGM. One bad practice is assume a growth rate of cash flows that is impossible for eternity.

We cannot comprehend the idea of eternal growth but we can all agree that a company in an infinite economy better not grow at a larger rate than the economy as whole or it will eventually dwarf the economy.

The growth rate used in the model must be a perpetual growth rate. It is very, very wrong to use stock analyst forecasts of future growth since these are not estimates of perpetual growth rates.

To the contrary, analysts' long-term forecasts are typically just 3-5 years ahead.

A firm might grow more than 2% for 3-5 years, but not forever.