# Gordon model

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The Gordon model, also called Gordon's model or the Gordon growth model is a variant of the discounted dividend model, a method for valuing a stock. It is named after Myron Gordon, who is currently 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.

${\displaystyle \sum _{t=1}^{\infty }D*{\frac {(1+g)^{t}}{(1+k)^{t}}}}$.

The current price of the above security should be

${\displaystyle P=D*{\frac {1+g}{k-g}}={\frac {D_{1}}{k-g}}}$.

The model requires a constant growth rate and that g<k. 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 dividendD with E earnings per share.

## Restrictions to the model

As a model, the discounted dividend model suffers from its hypothesises.

• Growth rate: the model requires a perpetual dividend growth rate that is
• constant: if dividends are expected to grow at a constant growth rate ${\displaystyle g}$, then earnings and the stock price are also expected to grow at that rate.
• between -1 and the cost of capital ${\displaystyle k}$. In the case where the growth rate exceeds the discount rate, the share price would be infinite. That can be easily understood as in that case the present value of the dividends keeps on getting bigger and bigger.
• Payment of a dividend: if the stock does not currently pay a dividend, like many growth stocks, one should use a more general form of the Gordon model (see dividend discount model).
• Sensitivity to the estimation of ${\displaystyle (k-g)}$. If we assume for example a stock with an expected dividend of $3 next period, a required of return of 15% and an expected growth of 5 percent forever. According to the model, the stock value is: ${\displaystyle V_{0}={\frac {3}{0.15-0.05}}=\30}$ If the analyst following this stock considered that the growth rate was expected to be 8% (3% more than the initial 5%), the stock value would be$42 (40% more than $30). If the projected growth rate is 14%, the stock value would skyrock to$300 !

## Proof

We can prove that the current price of a security ${\displaystyle P}$ is equal to ${\displaystyle {\frac {D_{1}}{k-g}}}$

By definition, ${\displaystyle V_{0}={\frac {D_{1}}{1+k}}+{\frac {D_{1}*(1+g)}{(1+k)^{2}}}+{\frac {D_{1}*(1+g)^{2}}{(1+g)^{3}}}+...}$ (1)

Multipliying both sides by ${\displaystyle (1+k)/(1+g)}$, we have

${\displaystyle {\frac {(1+k)}{(1+g)}}V_{0}={\frac {D_{1}}{1+g}}+{\frac {D_{1}}{(1+k)}}+{\frac {D_{1}*(1+g)}{(1+g)^{2}}}+...}$ (2)

Subtracting equation (1) from (2), we find that ${\displaystyle {\frac {1+k}{1+g}}V_{0}={\frac {D_{1}}{1+g}}}$

which implies that ${\displaystyle {\frac {(k-g)V_{0}}{(1+g)}}={\frac {D_{1}}{(1+g)}}}$, and then, ${\displaystyle V_{0}={\frac {D_{1}}{k-g}}}$