John Williams was one of the first to study mathematically various models of the intrinsic value of a stock in his now classic 1938 text, “The Theory of Investment Value.”  He credited the definition of intrinsic value to Robert F. Wiese, “Investing for True Values,” Barron’s, September 8, 1930 p. 5 who defined the intrinsic value of a stock as:

The proper price of any security, whether a stock or a bond, is the sum of all future income payments discounted at the current rate of interest in order to arrive at the present value.

Williams (1938) refined this to:

Let us define the investment value of a stock as the present worth of all dividends.

If we write this mathematically, let d_t be the dividend to be paid t years from now, and let k_e be the discount rate to calculate the present value.  Then, the intrinsic value is:

Williams’ contribution was acknowledged by Markowitz, who developed modern portfolio theory:

“The basic principles of portfolio theory came to me one day while I was reading John Burr Williams, The Theory of Investment Value. Williams proposed that the value of a stock should equal the present value of its future dividend stream. But clearly dividends are uncertain, so I took Williams’ recommendation to be to value a stock as the expected value of its discounted future dividend stream.” FOUNDATIONS OF PORTFOLIO THEORY, Nobel Lecture, December 7, 1990 by HARRY M. MARKOWITZ Baruch College, The City University of New York, New York, USA

In Markowitz’ formulation, we would replace d_t in each equation by E(d_t), the expected dividend. 

So two questions arise: how do we forecast dividends, and what is the discount rate (k_e)?  Williams provided two answers.  The simplest was that the current dividend grows at a constant rate for ever.  This model is sometimes called the constant dividend growth model or the Gordon growth model (following Gordon 1959).  Another is to assume there are two stages, an “abnormal” or “short run” stage, and then a “normal” or “long run” stage, and dividends grow at different rates in the two stages.  The simpler model is sometimes called the one-stage model (where the first stage lasts forever). Williams even discussed the application of the model to non-dividend paying stocks, which we will take up in the next chapter. For the discount rate, we have to use a risk adjusted discount rate because future dividends are unknown.  We will say more about this later in the chapter, but we can now define:

The intrinsic value is the present value of future dividends discounted at the appropriate risk adjusted interest rate. 

These two methods of modeling future dividends are the most popular implementations of the model even today.   However, Williams’ contribution did not stop here.   He identified two major advantages of having a formal model:

·   It identifies a consistent set of economic assumptions that bound an assessment of a stock's intrinsic value.

·   You can test the plausibility of the economic assumptions, and also infer whether market prices are consistent with these assumptions.

For example, Williams (page 522) in the section titled “The Degree of Overvaluation in the 1930 Price” wrote in relation to American Telephone:

“Even a change in the assumptions so as to make the company’s period of uninterrupted growth twenty years instead of ten, which would require it to grow to six and three quarters its 1929 size, would not have sufficed to give its stock an investment value in excess of the all-time high of 310 ¼ ex-rights price in 1920.  Twenty years of steady growth would only have warranted a price rights-on of V₀= 300 or a price ex-rights 0f V₀ = 283.”

So he is using the dividend model to argue that the market price of American Telephone cannot be justified.

In this chapter, we will describe the two implementations of the dividend model, and then also look at various ways to forecast the growth of dividends.  For the discount rate, we will use the Capital Asset Pricing Model (CAPM).  The CAPM provides an estimate of the risk-adjusted discount rate for a stock.

The simplest application of William’s theory is the constant dividend growth model.  In this model, d_t = d*(1+g)^t so the current dividend, d, grows for every at rate g.  Note that d_t is a forecast of the dividend at time t.  Mathematically this has a nice implied simplification which can be expressed as follows:

This simplifies to:

In the two stage model, dividends grow at rage g₁ for T years and then grow at rate g₂.  The dividends are discounted at rate k₁ in stage 1 and at rate k₂ thereafter.  This yields the following formula for the intrinsic value:

Where