A tutorial on discounted cashflow (DCF) analysis - Part 2
From a statistician and data scientist perspective
In Part 1 of this tutorial, we went through the concepts behind discounted cash and how I think about discount rates. In Part 2 of this tutorial, I’d like to go into the numbers and mathematics of DCF analysis. The goal at the end of this tutorial is to develop an understanding of what really drives the valuation output of a DCF analysis and why this number can look so varied when run by different analysts.
To Infinity and Beyond!
In Part 1, we provided an example of how cash is valued looking 10 years into the future. But what happens after 10 years? Assuming the company didn’t go bankrupt, it should still be around right? Therefore, when valuing a company, we generally project generating cash forever to infinity. As an example, suppose we invest in a company and that first year, they generate $10 dollars in free cashflow. On top of that, every year, this company gets a little bigger so they generate 5% more FCF than the year before. On Year 2, you get $10*1.05 = $10.50. On Year 3, you get $10.50*1.05 = $11.03, so on and so forth…
Remembering from Part 1 that cash today is worth more than cash in the future, we assign a discount rate of 9% to money that comes in the future. This actually means that even though we’re making more and more money nominally each year, the value of that money is constantly decreasing. In Year 2, we made $10.50 but the present day value is actually $10.50/1.09 = $9.63. In Year 3, the present day value is actually only $9.28. This number keeps going down until it eventually approaches 0. In fact, the present day value of any given Year X for this company can be nicely expressed as:
If we now add up all the years of future cash flow and adjust each year to what it’s worth today based on our chosen discount rate, we get the fair value of this investment today. This is known in mathematics as a geometric series and has a nice closed form solution. Specifically the total present day value of the investment can be expressed as:
In other words, the total amount of money this company will generate from now until the universe explodes measured in today’s dollars is $272.50. This is also why we sometimes refer to this as the fair value.
Mathematically, the above estimate only works when the growth rate of the FCF is smaller than the discount rate. If the growth rate is bigger, then the entire calculation doesn’t make sense as you would make more money each year than the previous year even as measured by today’s dollars and the Total Present Day Value would be infinite. In reality, we shouldn’t need to worry about this as all companies are restricted in growth as they reach maturity. Eventually, you run out of places to growth and customers to sell to. This is why this low rate of growth is also referred to as the Terminal Rate. This is the rate of growth that the company is expected to be able to sustain forever.
Fleshing out the DCF model
What we laid out in the previous section could already be considered a DCF analysis. However, most of the time, companies we invest in are still in their growth phase. They’re continuing to develop into new countries, products, or customer basis. Typically in a DCF model, we try to model these years differently with a higher FCF growth rate. Note that because this doesn’t extend out to infinity, it can be higher than the discount rate.
In keeping with the previous example, we could assume the growth rate in the first 3 years are 20% before returning back to the terminal rate of 5%. In that case, the Year 2, the present day value of what was made is $10 * 1.2 / 1.09 = $11.01. In Year 3, it would be $12.12, and so on. Running through all the calculations would get a Total Present Day Value of $410.08 instead.
DCF analysis can also be used to easily capture multiple scenarios of the future. Suppose for instance we thought there was an 80% chance this growth plan would succeed and achieve 20% growth rate for 3 years but if it fails, it would just continue to grow at the slow 5% rate we previously assumed. In that case, we can simply apply the probabilities to the Total Present Day Values in each respective scenario as follows:
Ultimately, under this framework, various adjustments can be made including varying the discount rate, limiting the extension of the terminal rate to a set number of years into the future, etc.
Terminal Rate - the real weakness of the DCF model
Up to this point, the DCF analysis as described may seem perfectly logical. It seems like as long we can accurately predict the cash flow, it would give us an exact fair value by which to measure investments. Unfortunately, DCF analysis is not a magic bullet and there are some clear issues that became evident to me from diving into the details and they mostly derive from trying to predict too far into the future.
While researching companies, I spent a fair bit of effort trying to accurately project the growth rates in the next few years but the terminal rate is usually an afterthought. However, as we can clearly see in the formula, this rate has an outsized contribution to the fair value. For example, in the first simple example above with a constant 5% FCF growth rate, we estimated a total present day value of $272.50. Just changing this terminal rate slightly to 4% would change this value to $218, almost a 20% difference! Worse yet, this terminal growth rate rarely has much justification behind it. In fact, I’ve even observed some investment influencers increasing this number during their valuations as the market has gotten more expensive to show companies at a discount.
Other faulty assumptions such as assuming the company will exist forever and the duration of the growth period can both significantly impact the final valuation estimate as well.
To wrap up, I am reminded of a quote from a famous statistician, George Box, “All models are wrong, but some are useful.” I believe it nicely describes the DCF analysis. Despite it’s shortcomings, I believe it has tremendous value in helping better understand the intrinsic value of an investment. The key is however, not to just take a single output at face value but to understand the inputs required for the model as a whole.
Missed part one of this series? Find it here.
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Disclaimer: Any information contained here is not intended as, and shall not be understood or construed as, financial advice. I am not a financial advisor and this is only a documentation of my personal investment journey and decisions. You should always do your own research before making any final decision on investments.