### Gordon Growth Model

The **dividend discount model** (DDM) is a method of valuing a company based on the theory that a stock is worth the discounted sum of all of its future dividend payments.^{[1]} In other words, it is used to value stocks based on the net present value of the future dividends. The equation most widely used is called the **Gordon growth model**. It is named after Myron J. Gordon of the University of Toronto, who originally published it along with Eli Shapiro in 1956 and made reference to it in 1959;^{[2]}^{[3]} although the theoretical underpin was provided by John Burr Williams in his 1938 text "The Theory of Investment Value".

The variables are: $P$ is the current stock price. $g$ is the constant growth rate in perpetuity expected for the dividends. $r$ is the constant cost of equity capital for that company. $D\_1$ is the value of the next year's dividends. There is no reason to use a calculation of next year's dividend using the current dividend and the growth rate, when management commonly disclose the future year's dividend and websites post it.

- $P\; =\; \backslash frac\{D\_1\}\{r-g\}$

## Contents

## Derivation of equation

The model sums the infinite series which gives the current price P.

- $P=\; \backslash sum\_\{t=1\}^\{\backslash infty\}\; D\_1\backslash times\backslash frac\{(1+g)^\{t-1\}\}\{(1+r)^t\}$
- $P\; =\; D\_1\backslash times\backslash frac\{1\}\{1+r\}\backslash times\backslash frac\{1+r\}\{r-g\}$
- $P\; =\; \backslash frac\{D\_1\}\{r-g\}$

## Income plus capital gains equals total return

**
The equation can also be understood to generate the value of a stock such that the sum of its dividend yield (income) plus its growth (capital gains) equals the investor's required total return. Consider the dividend growth rate as a proxy for the growth of earnings and by extension the stock price and capital gains. Consider the company's cost of equity capital as a proxy for the investor's required total return.**^{[4]}

- $\backslash text\{Income\}\; +\; \backslash text\{Capital\; Gain\}\; =\; \backslash text\{Total\; Return\}$

- $\backslash text\{Dividend\; Yield\}\; +\; \backslash text\{Growth\}\; =\; \backslash text\{Cost\; Of\; Equity\}$

- $\backslash frac\{D\}\{P\}\; +\; g\; =\; r$

- $\backslash frac\{D\}\{P\}\; =\; r\; -\; g$

- $\backslash frac\{D\}\{r\; -g\}\; =\; P$

## Growth cannot exceed cost of equity

From the first equation, one might notice that $r-g$ cannot be negative. When growth is expected to exceed the cost of equity in the short run, then usually a two stage DDM is used:

- $P\; =\; \backslash sum\_\{t=1\}^N\; \backslash frac\{D\_0\; \backslash left(\; 1+g\; \backslash right)^t\}\{\backslash left(\; 1+r\backslash right)^t\}\; +\; \backslash frac\{P\_N\}\{\backslash left(\; 1\; +r\backslash right)^N\}$

Therefore,

- $P\; =\; \backslash frac\{D\_0\; \backslash left(\; 1\; +\; g\; \backslash right)\}\{r-g\}\; \backslash left[\; 1-\; \backslash frac\{\backslash left(\; 1+g\; \backslash right)^N\}\{\backslash left(\; 1\; +\; r\; \backslash right)^N\}\; \backslash right]$

+ \frac{D_0 \left( 1 + g \right)^N \left( 1 + g_\infty \right)}{\left( 1 + r \right)^N \left( r - g_\infty \right)},

where $g$ denotes the short-run expected growth rate, $g\_\backslash infty$ denotes the long-run growth rate, and $N$ is the period (number of years), over which the short-run growth rate is applied.

Even when *g* is very close to *r*, P approaches infinity, so the model becomes meaningless.

## Some properties of the model

** a)**
When the growth *g* is zero the dividend is capitalized.

- $P\_0\; =\; \backslash frac\{D\_1\}\{r\}$.

** b)**
This equation is also used to estimate cost of capital by solving for $r$.

- $r\; =\; \backslash frac\{D\_1\}\{P\_0\}\; +\; g.$

## Problems with the model

** a)**
The presumption of a steady and perpetual growth rate less than the cost of capital may not be reasonable.

**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. However, this requires the use of earnings growth rather than dividend growth, which might be different.

**c)**
The stock price resulting from the Gordon model is hyper-sensitive to the growth rate $g$ chosen.

## References

## Further reading

## External links

- Alternative derivations of the Gordon Model and its place in the context of other DCF-based shortcuts
- Mathematics of the DDM: questions and solutions.

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