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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. <ref>[http://www.investopedia.com/articles/fundamental/04/041404.asp Investopedia – Digging Into The Dividend Discount Model]</ref>  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;<ref>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.</ref><ref>{{cite journal |last=Gordon |first=Myron J. |authorlink= |coauthors= |year=1959 |month= |title=Dividends, Earnings and Stock Prices |journal=Review of Economics and Statistics |volume=41 |issue= 2|pages=99–105 |doi= 10.2307/1927792|jstor= 1927792|publisher=The MIT Press }}</ref> although the theoretical underpin was provided by [[John Burr Williams]] in his 1938 text "[[The Theory of Investment Value]]".
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The variables are: <math>P</math> is the current stock price.  <math>g</math> is the constant growth rate in perpetuity expected for the dividends. <math>r</math> is the constant cost of equity capital for that company. <math>D_1</math> 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.
:<math>P = \frac{D_1}{r-g}</math>
 
==Derivation of equation ==
The model sums the [[infinite series]] which gives the current price P.
: <math> P= \sum_{t=1}^{\infty}  D_1\times\frac{(1+g)^{t-1}}{(1+r)^t}</math>
: <math>P = D_1\times\frac{1}{1+r}\times\frac{1+r}{r-g}</math>
:<math>P = \frac{D_1}{r-g}</math>
 
==Income plus capital gains equals total return ==
{{Confusing|section|date=February 2013}}
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.<ref>[http://www.retailinvestor.org/perpetuity.xls Spreadsheet for variable inputs to Gordon Model]</ref>
 
:<math>\text{Income} + \text{Capital Gain} = \text{Total Return}</math>
 
:<math>\text{Dividend Yield} + \text{Growth} = \text{Cost Of Equity}</math>
 
:<math>\frac{D}{P} + g = r</math>
 
:<math>\frac{D}{P} = r - g</math>
 
:<math>\frac{D}{r -g} = P</math>
 
==Growth cannot exceed cost of equity ==
From the first equation, one might notice that <math>r-g</math> 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:
:<math>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}</math>
 
Therefore,
:<math>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)},</math>
 
where <math>g</math> denotes the short-run expected growth rate, <math>g_\infty</math> denotes the long-run growth rate, and <math>N</math> 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.
:<math>P_0 = \frac{D_1}{r}</math>.
 
''' b)'''
This equation is also used to estimate [[cost of capital]] by solving for <math>r</math>.
:<math>r = \frac{D_1}{P_0} + g.</math>
 
==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 stock]]s, more general versions of the discounted dividend model must be used to value the stock. One common technique is to assume that the [[Modigliani-Miller theorem|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 <math>g</math> chosen.
 
==References==
{{reflist}}
 
==Further reading==
*{{cite book |title=The Investment, Financing, and Valuation of the Corporation |last=Gordon |first=Myron J. |authorlink= |coauthors= |year=1962 |publisher=R. D. Irwin |location=Homewood, IL |isbn= |pages= |url= }}
*{{cite web|title=Equity Discounted Cash Flow Models|url=http://pages.stern.nyu.edu/~adamodar/pdfiles/damodaran2ed/ch5.pdf}}
 
==External links==
*[http://papers.ssrn.com/sol3/papers.cfm?abstract_id=996016/ Alternative derivations of the Gordon Model and its place in the context of other DCF-based shortcuts]
*[http://www.fightfinance.com/?t=DDM Mathematics of the DDM: questions and solutions.]
{{stock market}}
[[Category:Financial terminology]]
[[Category:Finance theories]]
[[Category:Valuation (finance)]]

Latest revision as of 02:29, 5 December 2014

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