What is the equation for value?
Warren Buffett defined it in his 1992 annual shareholder letter. He referenced the formulas in the classic, The Theory of Investment Value by John Burr Williams.
This post will present and explain this exact formula published by John Burr Williams and help modernize it in today’s terms. Nobody has done it better than Warren Buffett himself, and so I recommend reading his 1992 letter for yourself for additional insight.
The equation for (intrinsic) value that eventually came to form the basis for modern DCF valuation is admittedly difficult to model even when reading the John Burr Williams classic. We’ll have to decode some of the Greek symbols presented throughout the book, but it will help make DCFs more clear.
You’ll find the equation for value in its final form on p. 94, as so:
Keep in mind that some of these Greek terms won’t necessarily correlate to the general terms used in regular algebra.
- Capital Vo (on the left of the equation) is defined as the investment value per share (page 76), which is what we are trying to solve here
- π (pi) is used to by Williams to signal dividends paid (also p.76)
- The subscript is used to indicate the year a dividend is paid
- The lower case “o” in this case indicates a dividend paid in the initial year examined
- ω (omega) is defined by Williams as lower case “u” times lower case “v”
- ω = uv
- Found on page 88
Breaking down ω = uv
- u = 1 + g (sourced from page 87)
- g = annual growth of dividend paying power
- v is another John Burr Williams special, defined (on page 76) in this picture below:
- where lower case “i” is equal to the interest rate
So, this fancy notation for ω (omega) comes down to a relationship between dividend growth and the interest rate.
Value to the investor comes from dividend growth, which when equated to modern DCF terms, refers to the growth of free cash flows. It is usually modeled by a mathematical summation in most finance textbooks (the upper case Greek symbol sigma: Σ).
ω (omega) here is scattered throughout the final equation, and will be summed to represent each year of cash flows being discounted.
As lower cases u and v are defined already in the notations of ω (omega), we now have a definition for all of the terms used by John Burr Williams in the equation for intrinsic value.
Last word of note here:
- lower case “n” is number of years dividends are paid
You’ll notice that John Burr Williams uses dividends in all of his formulas in the book, to assist in determining intrinsic value. Modern analysts have replaced most intrinsic value formulas to use free cash flows instead of dividends, partly because:
- Dividends come from free cash flows
- Free cash flows reinvested at higher rates of return can create exponentially higher future FCF and dividends
- Dividend yields have gradually decreased for decades now, adding to the perception that returns now come mostly from capital appreciation rather than dividends
- Among other factors…
Models such as the Dividend Discount Model are still used to evaluate some businesses such as banks and financials, whose dividends are generally more stable. However, this distinction is one of the few divergences between what Williams wrote his book and what is widely used in finance today.
“n” is found in the superscript for ω (omega) twice, as seen in the upper right corner of the symbol. A superscript essentially takes each ω (omega) to the power of n, which means it is multiplying each ω (omega) against itself by the number of years forecasted. The effect of doing this, rather than just multiplying by the number of years, is that it implements the force of compounding into the math.
You can see this clearly by using an example like this: say that you have 11% returns per year. After 3 years, 11% * 3 = 33% return, but that’s not really what happened with the money. In reality, the 11% in the second year is earned after the 11% return of the first year, and so on. So the correct function is to take a return by the power of the number of years, in this case to the power of 3, which equates to:
(11%) * (11%) * (11%) = 1.11 * 1.11 * 1.11 = 1.368 = 36.8%, which is the correct return if 11% is compounded over 3 years.
Simplifying the John Burr Williams Formula
By plugging these definitions in and making some algebraic simplifications, we can hopefully manipulate the equation to understand it in modern terms. It should then look more familiar, like the Discounted Cash Flow Models (DCFs) used today.
I did this on paper to make it a bit easier:
From this simplification, we can see an obvious parallel to the modern DCF. The “g” in growth of dividends is similar to a growth in free cash flow, and the “i” in the denominator represents the WACC, or discount rate, that an analyst is using to discount cash flows.
…allows for the compounding of growth to take effect, as represented by the “n” in the superscript.
The second group (below) attempts to formulate a figure for the continuing value, or the discounted value of cash flows once the company reaches a steady state of growth.
Burr Williams also graphically defined this phenomenon, where a company/industry experiences a rapid growth stage followed by a slowing in a mature state, with growth eventually approaching an upper limit.
He also explicitly defines an important caveat with the entire discounted cash flow process on page 89, which nicely summarizes the theory behind how the relationship between ROIC and WACC can increase value or destroy value.
By going back to that ω (omega) simplification again, you can see that if i > g, then ω (omega) returns a negative value, and leads to the ending Vo to be finite, and if continued, to become zero.
If ω (omega) is equal to one, or greater than 1, the mathematical value of Vo will be infinite, which of course is impossible in real life. This is why a continuing, or “terminal”, value is established as the final stage of the equation.
The Best Summary by the Author Himself
I realize that this is all a lot to take in, and I recommend learning the basics of a DCF valuation and working through some examples before you try and decipher the John Burr Williams equation. It’s really a bit more complex and hardly used directly today.
However, the lessons behind this book and its teachings are truly timeless, as the principles established throughout are still largely in practice and widely observed today.
I’ll leave you with a fantastic quote from the book that Williams wrote, which really summarizes how anyone’s approach to finding intrinsic value should be:
Updated for clarity: 10/5/2022