Proving the Gordon Growth Model: Geometric Series and Their Applications

Financial theory (and our own intuition) tells us the value of any asset is the sum of its future cash flows, discounted at the firm’s cost of capital to arrive at their present value, with cost of capital being a blend of the systematic risk inherent in the firm (beta, if you are using the CAPM), and the future cost of borrowing (yield to maturity).

When modeling, we forecast out the cash flows of the company for the next two to five years (depending on your prophetic capability), and then assign a terminal value to find the value of the firm from the last year into perpetuity. How exactly does a model sum up an infinite series of cash flows? The answer lies in the Gordon Growth Model.

The formula for the Gordon Growth Model is as follows:

g = terminal growth rate

r = Weighted Average Cost of Capital (WACC)

D0 = Cash flow in year 5 (or 3, or whatever)

Where does this come from? If we assume the cash flow in year five will grow at a constant rate, we can write the sum as:

This calculation will take the initial value, D0, and grow it using the growth rate g, discounting each cash flow to the value in year five. Although n is going from one to infinity, it is actually going from year six to infinity. We will still have to discount the sum back another five years at the end.

Our goal is to prove that:

How do we do this? Let’s begin by replacing “(1+g)/(1+r)” with “a”. Algebra tells us we can also move D0 outside of the Riemann Sum.

We then have:

We recall from calculus that the sum of the geometric series,

as long as |a|<1

When r = 1, the equation can be simplified to:

In our case, r does in fact equal one. You will notice that in our equation, n starts at one, but in the example geometric series, n begins at zero. In order to begin n at zero, we will factor out an a from all terms in the series:

Notice that on the right hand side of the equation, the series portion (which is in parentheses), has the first term of one. It has been written it this way instead of writing it as “a to the power of zero.”

We now have a series that looks exactly like the geometric series we discussed earlier.

Because r = 1, it simplifies to:

Simplifying again, we get:

Recall that:

So, replacing a:

Using a bit of algebra, if we multiply the numerator and denominator by (1+r), this results in:

Simplifying the denominator:

This is exactly what we were trying to prove initially. So, we have shown that:

Remember, this will only hold true when the absolute value of a is less than one. Because a= (1+g)/(1+r), the denominator must always be higher than the numerator in order for that inequality to hold true. In essence, the growth rate you choose must always be less than the Weighted Average Cost of Capital (WACC). This also should make sense intuitively. If the company is growing at a higher rate than its total risk and future cost of borrowing, its value is infinite.