Gordon model

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Gordon growth model is a variant of the Discounted cash flow model, a method for valuing a stock or business. Often used to provide difficult-to-resolve valuation issues for litigation, tax planning, and business transactions that are currently off market. It is named after Myron Gordon, who was a professor at the University of Toronto.

It assumes that the company issues a dividend that has a current value of D that grows at a constant rate g. It also assumes that the required rate of return for the stock remains constant at k which is equal to the cost of equity for that company. It involves summing the infinite series which gives the value of price current P.

P= \sum_{t=1}^{\infty} D\times\frac{(1+g)^t}{(1+k)^t}.
Summing the infinite series we get,

P = D\times\frac{1+g}{k-g}， In practice this P is then adjusted by various factors e.g. the size of the company.
k=\frac{D\times\left(1+g\right)}{P}+g， k denotes expected return = yield + expected growth.

It is common to use the next value of D given by D1 = D0(1 + g), thus the Gordon's model can be stated as [1]

P_0 = \frac{D_1}{k-g}.

Note that the model assumes that the earnings growth is constant for perpetuity. In practice a very high growth rate cannot be sustained for a long time. Often it is assumed that the high growth rate can be sustained for only a limited number of years. After that only a sustainable growth rate will be experienced. This corresponds to the terminal case of the Discounted cash flow model. Gordon's model is thus applicable to the terminal case.


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[edit] Problems with the model

a) The model requires one perpetual growth rate

But for many growth stocks, the current growth rate can vary with the cost of capital significantly year by year. In this case this model should not be used.

b) If the stock does not currently pay a dividend, like many growth stocks, more general versions of the discounted dividend model must be used to value the stock. One common technique is to assume that the Miller-Modigliani hypothesis of dividend irrelevance is true, and therefore replace the stocks's dividend D with E earnings per share.

But this has the effect of double counting the earnings. The model's equation recognizes the trade off between paying dividends and the growth realized by reinvested earnings. It incorporates both factors. By replacing the (lack of) dividend with earnings, and multiplying by the growth from those earnings, you double count.

c) Gordon's model is sensitive if k is close to g. For example, if

  • dividend = $1.00
  • cost of capital = 8%

Say the

  • growth rate = 1% - 2%

So the price of the stock

  • assuming 1% growth= $14.43 = 1.00(1.01/.07)
  • assuming 2% growth= $17.00 = 1.00(1.02/.06)

The difference determined in valuation is relatively small.

Now say the

  • growth rate = 6% - 7%

So the price of the stock

  • assuming 6% growth= $53 = 1.00(1.06/.02)
  • assuming 7% growth= $107 = 1.00(1.07/.01)

The difference determined in valuation is large.


[edit] Derivation

We want to find out the value of Pn as n \rightarrow \infty , where

P_n = \sum_{t=1}^{n} D\times \frac{(1+g)^t}{(1+k)^t} = D\times \sum_{t=1}^{n} \frac{(1+g)^t}{(1+k)^t} = D\times \sum_{t=1}^{n} \left(\frac{1+g}{1+k}\right)^t

Let

a = \frac{1+g}{1+k} .

Then

P_n = D\times \sum_{t=1}^{n} a^t .

Since

1-a^{(n+1)} = (1-a) \times (1 + a + a^2 + ... + a^n) = (1-a) \times (1 + \sum_{t=1}^{n} a^t) ,

we get

\frac{1-a^{(n+1)}}{1-a} - 1 = \sum_{t=1}^{n} a^t .

Therefore,

P_n = D \times \left[ \frac{1-a^{(n+1)}}{1-a} - 1 \right] .

If g < k, then a < 1 and

a^{(n+1)} \rightarrow 0 as n \rightarrow \infty .

Thus, we get

P_\infty = D \times \left( \frac{1}{1-a} - 1 \right) = D \times \left( \frac{a}{1-a} \right) = D \times \left(\frac{1+g}{1+k}\right) / \left[ 1 - \left(\frac{1+g}{1+k}\right) \right] = D \times \left( \frac{1+g}{k-g} \right) .


[edit] See also

[edit] References

  1. ^ http://www.dfaus.com/library/articles/earning_growth_stock/ Earnings Growth and Stock Returns, By Truman A. Clark, August 2000
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