Gordon Growth Conversion Factor

This article discusses the Gordon Growth  Conversion Factor. Many well trained and experienced appraisers who use the income method may easily overlook the fact that the standard discounting and capitalization of future cash flows assume that these cash flows will grow and continue forever at the same growth rate year and year out. It is an even surer bet that users of appraisal reports will not be aware of this implication when the income method is applied in the appraisal of a firm. It is worthwhile then to briefly review how the standard income method assumes the impossible.

The first assumption is that the present value ought to be the discounted value of the expected cash flows:

Where

V = the present value of the cash flow                                                                     

Ci = the expected cash flow in each period i                                                               

k = discount rate

Now this form of analysis is problematic in that we do not always know with any precision the specific expected cash flows in any given period in the future. To circumvent this problem appraisers generally make an assumption that cash flow will grow at a constant rate. If we assume that cash flow grows at a constant rate g, then starting with cash flow in the current period, call this C0 , we project that in the next year cash flow, C1, will grow to C0 [1+g] , C2 will grow to C0 [1+g]2 , C3 becomes C0 [1+g]3 and so on. So substituting these terms in (1.1) we get

Relabeling the term [1+g]/(1+k) as λ, (1.2) can be rewritten as

The terms in brackets form an infinite geometric series. Now somewhat counter intuitively a formula can be devised to compute the sum in brackets. Let:

represent the sum of the bracketed terms. Now multiply both sides of this equation by λ:

Subtract (1.4)–(1.5):

rearranging terms leaves us with:

Now we can substitute (1.6) back into (1.3) to derive the following expression:

Recall that λ stands for (1+g)/(1+k) where “g” is the expected growth rate and “k” is the discount rate. Now in most forecasts the expected growth rate can reasonably be expected to be less than the discount rate so the fraction, λ is less than one.

Critically if we assume that n, the number of expected periods that the cash flow will be available to the investor becomes very, very large then the λn term will become smaller and smaller because raising any fraction to higher and higher powers forces the fraction closer and closer to zero. In the language of real analysis as n + , λn 0.

Let’s take an example just to insure understanding and clarity. Suppose that we expect that the constant growth rate, g, will be 2%. Then 1+g will equal 1.02. Suppose that the discount rate, k, is 8%. Then 1+k will equal 1.08 and now we have:

Now start taking powers of this ratio, λn , for various values of n. Here is a table illustrating some sample results:

It can be seen that as n gets larger, λn gets smaller. Note however that n has to get well past fifty before the term gets really close to zero.

Making the critical assumption that n is allowed to get very, very large, the λn term goes to zero and equation for (1.7) becomes:

Retranslating λ we arrive at:

In the application of the income method the denominator, (k-g), is referred to as the capitalization rate. In actual practice a slight modification to (1.9) is made. The numerator of the formula is simply the projected next year’s cash flow:

This follows from the fact that (1+g) is the projected constant growth rate of cash flow. So whatever base period cash flow we take, the next period cash flow can be computed by multiplying it by one plus the growth rate. Now substituting (1.10) into (1.9) we derive the familiar capitalization formula of value:

Now I say this formula is familiar because it appears in virtually every application of the income method in one of two forms. It is applied in either a single stage or two stage approach. In a single stage approach the value of an equity is determined by utilization of (1.11) where the C term reflects the best estimate of future cash flow. The two stage approach involves a series of specific cash flow projections usually for no more than five years with a terminal value. The terminal value has the expression (1.11) in the numerator:

This terminal value is almost always based on a capitalization of the previous year’s earnings. Since this last expression presumes that earnings will continue forever it should probably be renamed the interminable value.

For more on infinite series check out this web page.

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