# Consistent Cash Creators, Part 2: Linear vs. Exponential Growth

Stop using compound growth rates now! While using a percent growth rate for free cash flows might be conventional, mathematically convenient and easier to convey to others, it is not as accurate or conservative as using an absolute rate of change from a linear trend model. Value investing is all about being conservative and accurate, so think twice before projecting out future cash flows next time you value a company. My statistical analysis below will show you why using a linear growth trend is often the right way to go.

## Exponential Growth

When most of us think about growth rates, we automatically think of the percent growth rate. Is that company growing earnings at 5% or 10%? That’s the way we’ve all been taught to think. Our bank accounts grow at a compound interest rate, so we often like to think of how our other investments are also compounding.

Compound growth can be expressed as:

Yt = Y0(1+r)t

where Y0 is the initial amount, r is the growth rate, and t is time.

This exponential growth equation can be transformed into a linear form so it can be modeled using linear regression. To do this, all you have to do is take the natural logarithm of each side of the equation. You get the following equation:

lnYt = mt + b

where m is lnY0 and b = ln(1 + r)

Using this equation, you can take the linear regression of the natural log of free cash flow to model its exponential growth over time. This is what was suggested by a comment regarding exponential trends in earnings in Part 1 of this article. That comment was the inspiration for this post.

Exponential growth is seen in nature when reproduction is not limited by food, space, or disease. When bacteria colonies first form, they often grow exponentially. In the business world, when a radically new product catches on with the masses, or goes “viral”, its sales can grow exponentially for a time. Young innovative companies with low capital reinvestment requirements can sometimes grow their free cash flows exponentially. However, as time passes on, these exponential growers face resource scarcity, market saturation, and labor scarcity.  The law of diminishing marginal returns kicks in and it becomes increasingly more expensive to produce additional units of goods or services because each additional input added is less and less productive. Often while this is going on, competitors seeing these excess profits enter the market and start to drive down prices. It is exceptionally difficult to maintain exponential growth for long in competitive markets. This is the reasoning that has always made me hesitant in using exponential growth rates in my valuations.

## Linear Growth of Free Cash Flows

A simpler alternative is to model growth in a linear fashion.  This is simply the straight line trend of free cash flows over time. The linear trend model assumes growth occurs at an absolute level each, such as a \$10 million increase in free cash flows each year.

Linear growth can be expressed as:

Yt = mt + b

where m is the slope, t is time, and b is a constant.

This is the simplest type of model to fit to the data with regression. It can handle negative free cash flows without having to do anything special. The exponential model requires that the data is non negative, so you have to transform the data if you end up with some years of negative free cash flows. I’m sure some software does the transformation automatically, but Excel doesn’t. In addition, the regression of a linear trend model can produce a coefficient of determination, r². R² measures the proportion of the variation in Y explained by the variables. In the exponential function, R² doesn’t quite mean the same thing. It measure the proportion of the variation in lnY explained by the variables. I almost made the mistake of comparing the R² of the linear and exponential form, but then I realized it wasn’t that easy. I’ve suspected that mature companies with sustainable competitive advantages often grow free cash flows in a more linear fashion, at least over a period of about 10 years. I’ve actually seen some DCF models that gradually lower growth rates over time, which basically creates a linear trend but in a less direct way. I’m a big believer in the Occam’s razor principle that states that you should describe phenomenon as simply as possible using the least amount of assumptions. Therefore, unless there is sufficient evidence supporting that free cash flows grow exponentially, I prefer using a simpler linear trend instead.

## Linear versus Exponential FCF Growth Statistics

Let’s take a look at the data. I returned back to the same dataset I used in Part 1 of Consistent Cash Creators. However, this time I once again filtered out the financial sector but I instead only included companies that grew free cash flows over the past seven years above zero and also exhibited positive free cash flow in each of the seven years. I did this in order to avoid having to transform the data when running log linear regressions. This gave me a universe of 563 companies. I thought I would see a longer list of companies. Given that this list was pretty small, I didn’t filter for return on equity this time.

I ran both a linear regression and log linear regression for each of these 563 companies. For the exponential growth, I had to create an R² that was comparable to the R² of the linear functional form called a quasi r-squared. This is critical because the normal R² value for the log linear transformed exponential growth models often underestimates the prediction error in the most recent years of growth because it fails to capture the overestimate of growth in the most current years of data since FCF is still in a log scale. The results are summarized in the table below.

Goodness of Fit: Linear & Exponential Growth

 % Variation Explained Linear r2 Exponential quasi R2 > 95 12 12 95 – 90 27 26 90 – 80 64 50 80 – 70 51 39 70 – 50 90 87 < 50 319 349 Total 563 563

As you can see, there are an equal number of stocks with R² above 95%. However, there are more poorly fits for the exponential forum than the linear form.

A better way to compare the two functional forms is to look at the number of companies with higher quasi R² values than r². Only 132 of the 563 stocks analyzed had higher quasi R² values than the linear r² value. This means that most of the companies had free cash flows that were better described by linear growth than exponential growth. Of course there were exceptions to this rule, but these often occurred for high growth companies that value investors would often not be as attracted to. For example, Trimble Navigation (TRMB) displays an exponential growth curve. However, the more mature Yum! Brands (YUM) displays a very consistent linear growth in free cash flows.

The data clearly indicates that straight line growth in free cash flows is the way to go. While requiring seven years of positive free cash flows might introduce a bias, this bias is likely to be similar to take of most value investors. Value investors want positive steady histories of free cash flows. Some young high growth companies with less than 7 years of positive free cash flows might not be included in the data analyzed, but those are the types of companies that must be analyzed more carefully due to greater difficulty in predicting their future cash flows. For the vast majority of the situations you might encounter, using a linear growth pattern for free cash flows is the more sound method.

### 14 thoughts on “Consistent Cash Creators, Part 2: Linear vs. Exponential Growth”

• February 29, 2008 at 8:24 pm

Would you happen to have a simple excel spreadsheet to calculate yearly and total returns?

• February 29, 2008 at 9:40 pm

“For the exponential growth, I had to create an R² that was comparable to the R² of the linear functional form called a quasi r-squared. This is critical because the normal R² value for the log linear transformed exponential growth models often underestimates the prediction error in the most recent years of growth because it fails to capture the overestimate of growth in the most current years of data since FCF is still in a log scale. The results are summarized in the table below.”

I have never heard of this criticism before. Could you post the formula that you used so I may understand your point better? Thanks.

• February 29, 2008 at 10:10 pm

I used the process described at this University of Illinois at Chicago econometrics course excersize. A similar process is also described in my old Basic Econometrics text by Gujarati. Basically you have to manually calculate the TSS (total sum of squares) and RSS (residual sum of squares) using the non transformed exponential curve equation. I took the antilog of the slope times X plus the intercept and used that when calculating the residual sum of squares.

• February 29, 2008 at 11:07 pm

Is this really a value site, R², linear regression? The more formulaic you get the more you lose.

• March 12, 2008 at 8:27 am

Faro, I’m not sure how analyzing free cash flows is not value investing. All I’m really doing is saying that using a simple linear trend is not a bad way to analyze free cash flows. I used the statistics to test my theory. I might be using formulas but so far none of them are for making buy or sell decisions. I’ve only used this to find consistent cash generating firms for further research.

• March 24, 2008 at 5:03 am

What variables did you use as the dependent and independent variables?

• March 24, 2008 at 5:29 am

I used free cash flow and year as the dependent and independent variables, respectively.

• March 25, 2008 at 12:40 am

George,

Correct me if I am wrong here, but shouldn’t you be using an independent variable that has a greater predictive power for explaining the changes in FCF? Granted that an R^2 of 0.97 using number of years as the independent variable explains 97% of the variation in the FCF, but wouldnt you be better off using a variable like cash flow from operations?

• March 25, 2008 at 6:59 am

It all depends on your goal, Vivek. The goal of Consistent Cash Creators is to find the easiest companies to predict their future owner earnings. For the most part, I predicted that the resulting companies would be simple businesses with low capital requirements, just the kind of businesses Buffett loves. I use free cash flows as a proxy for owner’s earnings. While using cash flows from operations might be a “smoother” and more predictive variable, it would fail to let us know about businesses that require irregular infusions of cash. These types of businesses can drain free cash flows quickly, even though cash flow from operations is consistently positive and growing.

• March 25, 2008 at 2:37 pm

George,
May I suggest running a Fama-MacBeth regression on these companies: you could sort them into groups/portfolios based on R^2 and then regress the portfolios against the market and other factors?

• March 25, 2008 at 3:47 pm

Hi John,
Thank you for the suggestion. I’m not familiar with Fama-MacBeth regression (due to my lack of training in finance) but from a quick Google search it appears that this would require more historical market data than my database currently contains. Plus, I’m not really a fan of the CAPM and the Fama-MacBeth appears to be used to estimate CAPM parameters.

• March 26, 2008 at 10:57 am

I agree that there are substantial problems with the CAPM approach, but you wouldn’t be testing for CAPM if you did a Fama-MacBeth regression here. You would essentially be testing if there is a growth stability effect (like value or size effect). In others words, you would test the portfolios created by sorting based on R^2 to see if there is a risk premium associated with the stability of cash generation. This is similar to the work of Fama and French in 92 and 93 testing for the value and size effects. The only reason I mentioned other factors is because current work running Fama-MacBeth regressions control using multi-factor models (at least) instead of just CAPM.

Fama and French ran their regressions with portfolios formed once a year. I imagine you could probably get enough power if you had 20 years of data if yours is yearly. Without a long enough, or fine enough (yours couldn’t be more than quarterly) data, it wouldn’t be worth it. However, it would still be interesting.

• June 13, 2010 at 12:07 am