Dividend Discount Model

The Dividend Discount Model (DDM) is a widely used approach to value common stocks. Financial theory states that the value of any securities is worth the present value of all future cash flow the owner will receive. If we assume that stock investor receive all their cash fow in the form of dividend, a DDM will give the intrinsic value for a stock.

A common stock can be tough as the right to receive future dividends. A stock's intrincic value can be defined as the value of all future dividends discounted at the appropriate discount rate. In its simpliest form, the DDM uses, as discount rate, the investor's required rate of return.

Mathematically, it can be expressed as: $V_{0}=\sum _{t=1}^{\infty }{\frac {D_{t}}{(1+k)^{n}}}$,

where $D_{t}$ is the expected dividend in period $t$ and $k$ is the required rate of return for the investor.

Historically, this formula is a special form of the discounted cash flow model proposed by the seminal book by John B. Williams, The Theory of Investment Value published in 1938.

From this formula, one can deduct that the most important components of the value of a stock are likely to be the size and the timing of the expected dividend. The larger it is, and the more quick the shareholder receive it, the higher the share value will be.

Assumptions of the model

• The future value of dividend is know by the investor.
• Dividends are expected to be distributed at the end of each year until infinity.
• Dividends are the only way investors get money back from the company. This implies that any share buyback would be ignored.
• The implication of the second assumption is that the investor is expected to hold the share for an infite period: he will not sell it, an any moment.

While the model may be of some (perhaps significant) theoretical value, these assumptions, at the same time, indicate the limitations of this model. In reality, dividends can vary considerably. Also, few investors have an indefinite holding period of a particular stock. In general, investors tend to sell a particular stock after a certain period, for a variety of reasons. These limitations mean that the practical value of the Dividend Discount Model is limited.

Inputs to the model

To estimate the value of a common share, one must know at least:

• $D_{1}$: the expected dividend to be received in one year;
• $k$: the required rate of return on the investment. There are many methods to estimate this required rate of return, the most common is the Nobel-prize rewarded Capital Asset Pricing Model;
• $g$: the expected growth rate in dividends.

Zero Growth dividend

In the case where the dividend is not expected to growth in the future ($g=0$), then the stock is also known as a perpetuity.

In that case, the price of the stock would be equal to:

$P_{0}={\frac {D}{1+k}}+{\frac {D}{(1+k)^{2}}}+{\frac {D}{(1+k)^{3}}}+...={\frac {D}{k}}$,

where $D$ is the expected constant dividend and $k$ is the required rate of return for the investor.

Constant Growth dividend

In the case where the dividend is expected to grow at a definite constant growth rate $g$, the value of the stock will be equal to

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_0=\frac{D}{1+k}+\frac{D(1+g)}{(1+k)^2}+\frac{D(1+g)^2}{(1+k)^3}+...}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_0=\frac{D_1}{(k-g)}} ,

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_1} is the expected constant dividend at period Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_1=D_0*(1+g)} ), Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} is the dividend growth rate and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} is the required rate of return for the investor.

It is also known as the Gordon Model for evaluating stocks.

Supernormal growth model

In many cases, companies do not growth at a constant rate during their life. They are expected to growth at a "supernormal" growth rate at the beginning of their activities and then, at maturity, the growth rate will be reduce to a constant "normal" growth rate. This model is also know as the Two-Stage Dividend Discount Model.

In that case, we will have to adjust the calculation to take in account those two different growth periods.

If we assume that a company will have its dividend growing at a rate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_1} during the first Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} periods and thereafter, until infinity, at a lower rate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_2} , the value of the company will be equal to:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_0=\sum_{t=1}^{N}\frac{D_0*(1+g_1)^t}{(1+k)^t}+\frac{\frac{D_N*(1+g_2)}{(k-g_2)}}{(1+k)^N}}

Where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_0} is the dividend distributed today, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} is the required rate of return or the investor.

This method is usually used by analysts in valuing companies as the short-term growth is,in most cases, higher than the long-term growth (generally set at the general economic growth rate).

References

• Damodaran A. (2002) , Investment Valuation, Tools and Techniques for Determining the Value of Any Asset, 2nd Edition, Wiley.
• Ross S., Westerfield R., Jaffe J.(2005) Corporate Finance, 6th Edition, Mc-Graw Hill
• Williams, John B. (1938), The Theory of Investment Value, Cambridge, MA., Harvard University Press

See Also

Gordon Model