# Dividend discount model

The dividend discount model (DDM) is a method of valuing a company's stock price based on the theory that its stock is worth the sum of all of its future dividend payments, discounted back to their present value. [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 Gordon Model is as important to valuation theory as Bayes Theorem is to decision theory.

The variables are: ${\displaystyle P}$ is the current stock price. ${\displaystyle g}$ is the constant growth rate in perpetuity expected for the dividends. ${\displaystyle r}$ is the constant cost of equity capital for that company. ${\displaystyle D_{1}}$ is the value of the next year's dividends.

${\displaystyle P={\frac {D_{1}}{r-g}}}$

## Derivation of equation

The model uses the fact that the current value of the the dividend payment ${\displaystyle D_{0}\times (1+g)^{t}}$ at (discrete ) time ${\displaystyle t}$ is ${\displaystyle {\frac {D_{0}\times (1+g)^{t}}{{(1+r)}^{t}}}}$, and so the current value of all the future dividend payments, which is the current price ${\displaystyle P}$, is the sum of the infinite series

${\displaystyle P=\sum _{t=1}^{\infty }{D_{0}}\times {\frac {(1+g)^{t}}{(1+r)^{t}}}}$

This summation can be rewritten as

${\displaystyle P={D_{0}}\times r'\times (1+r'+{r'}^{2}+{r'}^{3}+....)}$

where

${\displaystyle r'={\frac {(1+g)}{(1+r)}}.}$

Clearly, the series in parenthesis is the geometric series with common ratio ${\displaystyle r'}$ and it sums to ${\displaystyle {\frac {1}{1-r'}}}$ if ${\displaystyle r'<1}$.

Thus.

${\displaystyle P={\frac {a}{1-r'}}}$, where a is the present value of next year's dividend i.e. ${\displaystyle {\frac {{D_{0}}\times (1+g)}{(1+r)}}}$ and r' is the common ratio

Substitution for ${\displaystyle r'}$ leads to

${\displaystyle P={\frac {\frac {{D_{0}}\times (1+g)}{(1+r)}}{1-{\frac {1+g}{1+r}}}}}$, which is simplified by multiplying by ${\displaystyle {\frac {1+r}{1+r}}}$, so that
${\displaystyle P={\frac {{\frac {{D_{0}}\times (1+g)}{(1+r)}}\times (r+1)}{r-g}}}$, or, finally,
${\displaystyle P={\frac {D_{0}(1+g)}{r-g}}}$

## Income plus capital gains equals total return

{{ safesubst:#invoke:Unsubst||$N=Confusing |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} The equation can also be understood to compute 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]

${\displaystyle {\text{Income}}+{\text{Capital Gain}}={\text{Total Return}}}$
${\displaystyle {\text{Dividend Yield}}+{\text{Growth}}={\text{Cost Of Equity}}}$
${\displaystyle {\frac {D}{P}}+g=r}$
${\displaystyle {\frac {D}{P}}=r-g}$
${\displaystyle {\frac {D}{r-g}}=P}$

## Growth cannot exceed cost of equity

From the first equation, one might notice that ${\displaystyle r-g}$ 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:

${\displaystyle P=\sum _{t=1}^{N}{\frac {D_{0}\left(1+g\right)^{t}}{\left(1+r\right)^{t}}}+{\frac {P_{N}}{\left(1+r\right)^{N}}}}$

Therefore,

${\displaystyle P={\frac {D_{0}\left(1+g\right)}{r-g}}\left[1-{\frac {\left(1+g\right)^{N}}{\left(1+r\right)^{N}}}\right]+{\frac {D_{0}\left(1+g\right)^{N}\left(1+g_{\infty }\right)}{\left(1+r\right)^{N}\left(r-g_{\infty }\right)}},}$

where ${\displaystyle g}$ denotes the short-run expected growth rate, ${\displaystyle g_{\infty }}$ denotes the long-run growth rate, and ${\displaystyle N}$ 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.

${\displaystyle P_{0}={\frac {D_{1}}{r}}}$.

b) This equation is also used to estimate cost of capital by solving for ${\displaystyle r}$.

${\displaystyle r={\frac {D_{1}}{P_{0}}}+g.}$

## 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. This approach is especially useful for computing a residual value of future periods.

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

## Related methods

The dividend discount model is closely related to both discounted earnings and discounted cashflow models. In either of the latter two, the value of a company is based on how much money is made by the company. For example, if a company consistently paid out 50% of earnings as dividends, then the discounted dividends would be worth 50% of the discounted earnings. Also, in the dividend discount model, a company that does not pay dividends is worth nothing.

## References

1. Investopedia – Digging Into The Dividend Discount Model
2. Gordon, M.J and Eli Shapiro (1956) "Capital Equipment Analysis: The Required Rate of Profit," Management Science, 3,(1) (October 1956) 102-110. Reprinted in Management of Corporate Capital, Glencoe, Ill.: Free Press of, 1959.
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4. Spreadsheet for variable inputs to Gordon Model

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