In 1991 letter to shareholders, Buffett explained his thought process in valuing a business. Given below is the excerpt from the letter.
A few years ago the conventional wisdom held that a newspaper, television or magazine property would forever increase its earnings at 6% or so annually and would do so without the employment of additional capital, for the reason that depreciation charges would roughly match capital expenditures and working capital requirements would be minor. Therefore, reported earnings (before amortization of intangibles) were also freely-distributable earnings, which meant that ownership of a media property could be construed as akin to owning a perpetual annuity set to grow at 6% a year. Say, next, that a discount rate of 10% was used to determine the present value of that earnings stream. One could then calculate that it was appropriate to pay a whopping $25 million for a property with current after-tax earnings of $1 million. – 1991 shareholders letter
If we translate Buffett’s writings into a mathematical equation, we would get an equation as shown below. This equation is called as Gordon Growth Model. It is named after Myron J. Gordon who originally published it in 1956. His work borrowed heavily from the writings of John Burr Williams 1938 book The Theory of Investment Value.
Intrinsic Business Value = [Owners earnings / (Discount rate - Growth rate)]
I was using this equation for a very long time without understanding how it was derived. This is akin to learning about swimming by reading a book without getting into the water. I decided to learn how Gordon arrived at this equation. And it is the subject of this post. Before venturing into the proof, I wanted to make sure that Gordon’s method produced the same result as the traditional DCF valuation model. The calculation assumes that the growth rate is 6% and the discount rate is 10%. From the calculations given below we can clearly see that the intrinsic business value for the DCF model converges to the value of $25 million produced by Gordon’s method. Spend sometime to make sure that you really understand the calculations.
The next step is to prove that Gordon’s method is going to work for every possible values of discount rate and growth rate. Let E be the owners earnings, D be the discount rate, and G be the growth rate. So we need to prove what is shown in the image.
At this point, I have no idea how to proceed with this proof. What should we do? Let’s ask George Polya, the Hungarian Mathematician, who wrote the fantastic book How to solve it. What would he say? Look at the unknown! And try to think of a familiar problem having the same or a similar unknown. Just a simple cue from Polya helped me to come up with a similar problem; summation of infinite geometric series. In mathematics, a geometric series is a series with a constant ratio between successive terms. Using a simple example, I have explained how to solve a geometric series.
Using the geometric series formula derived above I proved that Gordon’s method is same as traditional DCF model. The proof given below is very easy to understand.
Upon closer inspection we can see that Gordon’s growth model is used in calculating the terminal value of DCF. Also Penmann uses the same formula for figuring out the intrinsic value. Gordon’s model breaks down when growth (G) is equal to or greater than the discount rate (D). Will this happen? Yes it does for businesses with durable competitive advantage; moats. And moat enables businesses to grow their owners earnings above the discount rate for a long time. The period in which owners earnings grow abnormally is called as competitive advantage period.
In the example given below, the owners earnings is expected to grow at 20% for the first 7 years due to moat and thereafter it mean reverts to 3% due to competition. In that case the business at 10% discount rate will be worth around $43 million after 7 years. If the business is selling today for $15 million then your expected return after 7 years will be around 16.25%. Thus by splitting the calculations in two parts we have addressed the limitations of Gordon growth model. This should also explain why DCF has two pieces of calculations.
Few weeks back Prof Bakshi wrote an excellent post on valuation which can be found here. The excerpt from the post caught my attention. I wanted to know if he uses Gordon’s growth model. And the answer is yes. In his calculations he uses exit multiple which is nothing but an inverse of earnings yield. If his discount rate is 10% then at 10x multiple his G will be zero. At 15x multiple his G will be around 3.4%. And at 20x multiple his G will be around 5%. The table given below shows the relationship between exit multiple and growth rates.
Also when I said 20x multiple ten years from now as maximum I will value the firm at, I mean it. Many of them are valued at 15x and some as low as 10x – Prof Bakshi