What does a crocodile have to do with valuing enterprises and companies using the DCF method (discounted cash flow) and estimating the residual value in this method? I hasten to report that the crocodile is the very essence of value valuation using this method. Plus, its element, next to water, is residual value...

The crocodile is a very nice creature. It appears in almost every children's book, so by definition it must be nice. Let's analyze how a crocodile is built? Well, a crocodile consists of a mouth, a body and a tail - this is important...

The DCF method of valuing enterprises and companies, without going into excessive detail, is based on summing the value of current cash flows from the forecast period and adding the current (discounted) terminal value. The residual value is the value of the company at the end of the forecast period and usually sums up the present values of cash flows from the first year after the forecast period until infinity. As a rule, in the DCF model we assume by default that the valued enterprise will exist until infinity and will generate cash flows until this infinity.

If we take a closer look at the crocodile, it is obvious and clear to us that its mouth is the value of cash flows from the forecast period, and the body and tail together are the residual value. But where's the tail? Cash flows that are more and more distant in time will have less and less current value - like a tail. The question is how do we define the tail in terms of the valuation of enterprises and companies - that is, how to separate the tail from the entire residual value? The tail is the part of the residual value in the DCF model that is irrelevant from the point of view of the entire value of the company. The matter is somewhat discretionary with this tail - for some readers the insignificant value begins when we fall below 10%, for others only a value lower than 1% is insignificant.

In the DCF model, the crocodile's tail is infinitely long but increasingly smaller in width. In this valuation model, the residual value is usually estimated assuming a constant percentage increase in normalized cash flow (from the first year after the detailed forecast period) - the so-called Gordon formula (or Gordon-Shapiro formula). But how can we reconcile this constant percentage increase with the narrowing of the tail? But it is very easy - the assumed increasing cash flows in the residual period (after the forecast period) are discounted using the cost of capital (so-called WACC - weighted average cost of capital); assuming a growth rate lower than the cost of capital, the current values from subsequent years after the forecast period form a geometric sequence with a quotient less than 1 and, as a consequence, we are able to sum the current future cash flows "running" to infinity (or more precisely calculate the limit of the sum of current future cash flows). For the sake of mathematical accuracy, Gordon's formula is a simple application of the properties of a geometric sequence. Moreover, the condition of an increase in cash flows is not necessary - mathematically, Gordon's formula will work when we assume a constant level of cash flows or even a constant rate of decline.

**Crocodile dimensions. ****Value structure in the DCF model**

And what are the dimensions of the crocodile in the real world of DCF valuation?

Let's assume a 5-year forecast period (how we love 5-year forecasts...), a WACC discount rate of 12%, a growth rate of 3% and no significant cash flow disruptions during the forecast period. With such assumptions, the forecast period (i.e. the crocodile's mouth) is responsible for 34% of the value in the company valuation in the DCF model, while 66% is the part belonging to the residual value (body plus tail) - i.e. the value including the sum of the current values of cash flows from the 6th year to infinity. These simple estimates clearly show that the correct estimation of the residual value is of great importance for the final value recommendations - one could even say that, given the given assumptions, it is almost twice as important as the current values from the forecast period. And where does the tail start? Assuming a 2% level of significance, the tail begins at the age of 48. To sum up: 5 years of the forecast constitute 34% of the value, the next 42 years constitute 64%, and cash flows from the 48th year from the valuation date constitute approximately 2% of the value.

Please note that the higher the cost of capital (WACC - discount rate), the more important the cash flows generated in the forecast period are in the company's valuation and, therefore, the slightly less important is the residual value. Moreover, the non-essential part (tail) starts faster. Intuitively, this is understandable - if we believe that the risk related to the feasibility of cash flows increases, then cash flows that are relatively close to us are of course more important to us. Thus, increasing the discount rate to 15% and not changing other assumptions, it turns out that 42% of the enterprise value relates to the 5-year forecast period, the next 31 years create 56% of the value, and the 2% tail starts from year 37 from the date of valuation of the enterprise.

We can observe an inverse correlation for the assumed growth rate. Increasing the average growth rate will result in a smaller share of the present value of cash flows from the forecast period in favor of the residual value; Additionally, the tail (the unimportant part) will start later. By increasing the average growth rate from 3% to 5% and assuming the initial assumption of the cost of capital at 12%, we will obtain the share of cash flows from the 5-year forecast period in the company's value at the level of 28%, and the part of the current residual value will be 72%. The 2% tail will start from year 62, which means that cash flows from years 6 to 61 from the valuation date will create 70% of the total value. And again, this change in the value structure seems to be intuitively clear - if the expected growth is large, the coming years are less important for us in the company's valuation.

**The removable length of the crocodile's mouth. Length of the forecast period in the DCF model**

Five years of forecast, or the length of a crocodile's mouth... The five-year length of the forecast period is not an unshakable assumption; in fact, the length of the period is simply determined by the simple fact of what financial forecasts are presented to the appraiser by the company being valued. It is often, or even very often, the case that financial forecasts cover a period of less than 5 years. What impact does this have on the value structure? Let's assume that we shorten the detailed forecast period to 3 years. With the remaining initial assumptions unchanged (cost of capital 12% plus an average growth rate of 3%), we will obtain the share of the present value of the residual value in the total value of the enterprise at the level of 78%, and the forecast period will now account for only 22% of the value. To be clear, it rarely happens that the forecast period is prepared for a period longer than 5 years - of course, in such a situation, the importance of the residual value decreases...

The conclusion from these variant examples is clear - in the standard conditions of valuing an enterprise and a company using the DCF method, the correct estimation of the residual value is of great importance - it can even be said that it is many times greater than the forecast of cash flows within the 5-year forecast.

**Can a crocodile not have a tail? ****The finite period of operation of the enterprise in the DCF model**

The truth is that the tail adds some charm to the crocodile... But in the real world of valuation, we are dealing with enterprises for which a reasonable expected period of economic operation is finite - mines, gravel pits and wind farms can serve as examples. In the case of the first two economic activities, the period of future operation is limited by the size of the deposit (coal or gravel) that can still be economically extracted; in the case of a wind farm, the period of technological use of wind turbines is crucial. In such a situation, we should calculate the residual value in the DCF model, which will cover a finite time interval after the detailed forecast period. But can we?

It is not difficult - standardly, the residual value when valuing a company is estimated assuming an infinite stream of cash flows as the limit value of the appropriate sum of the geometric sequence - Gordon's formula. In the case of a limited period of operation of the enterprise, the Gordon formula should be appropriately adjusted - then we are simply dealing with the sum of a finite number of terms of a geometric sequence. Additionally, we should take into account "non-standard" cash flows in the last year of operation of the entity - e.g. costs of liquidation of the entity, including the costs of land recultivation; revenues from the sale of fixed assets; working capital together with cash at the moment of termination of operations; potential income tax.

The issue of choosing the right period for the further economic functioning of the entity may be very important... sometimes it may be a matter of freedom or lack thereof...

In July this year, a press release appeared about an audit of the process of taking over by a company controlled by the State Treasury of investment projects from two private entities in 2013. As can be understood from the press release, the buyer - a company controlled by the State Treasury - carried out a valuation in 2013 investment projects - wind farms, which was based on the adopted 25-year economic useful life. Those carrying out the takeover audit put forward the thesis that the appropriate period of use of the wind farm is 15 to 20 years, and not the period of 25 years assumed in the valuation. Additional emotions are also aroused by the information that the entire case has been under the interest of the Internal Security Agency and the relevant prosecutor's office since 2017.

15, 20, 25 years... Let's make calculations comparing the company valuation values using the DCF method, which we will obtain based on these three different periods of use of wind farms. Let's assume that we have already purchased wind turbines (i.e. generally all the necessary fixed assets), installed them and are just starting profitable operations. Let's "roughly" assume a WACC cost of capital of 12%, an average growth of 2.5% (in line with the NBP's continuous inflation target), plus let's roughly assume that the "non-standard" additional cash flows of the last year of business operation are immaterial (or more precisely that the revenues from the liquidated assets will be at the level of liquidation costs).

And what turns out? If the period of 15 years is appropriate, adopting 25 years for the valuation overestimates the value of the enterprise by 21% - in other words, the present value of cash flows from years 16 to 25 is approximately 21% of the present value of cash flows from the first 15 years. If the appropriate period is 20 years, then 25 years in the DCF valuation model results in an overstatement of the business value by only about 7% - simply, the value of cash flows from years 21-25 is approximately 7% of the present value of cash flows from years 1-20. Please note that the higher the cost of capital, the lower the differences we will get - if we increase the cost of capital WACC from 12% to 15%, the calculated overvaluation of 21% (15 years versus 25 years) is reduced to 15%, and the potential overvaluation by 7% (20 years versus 25 years) drops to 5%.

At this point, let us note that the separation of the forecast period and the residual period for businesses with a finite operating period can only be of a technical and secondary nature. Taking the example of a wind farm with a 15-year period of use, one can equally assume a 5-year forecast and a 10-year residual period, or one can also try to prepare a 15-year forecast period. Excel will suffer this... The results should be the same.

Another example of assets for which we have "no tail" - i.e. the period of economic use is reasonably finite - may be product trademarks (as opposed to company trademarks) and technological patents. In the case of the former, the period of use is often limited by the adopted product brand management strategies, and in the case of the latter, by legal conditions - the period of patent protection. But we should always keep an open mind in the valuation process and not stick to established patterns... Coca Cola or Pepsi brands - is there anyone who would argue that the finite period of use should be used to value these signs?

When estimating the residual value, the key parameter is the average growth rate - i.e. the expected average nominal change in cash flows - from the first year after the completion of the detailed forecast to infinity. However, adopting one percentage change does not always seem appropriate and reasonable. Especially for certain types of intangible assets, it is worth considering dividing the useful life into a phase of growth, stagnation and decline. For the valuation of enterprises, we also have the concept that the forecast period should include cash flows specific to the entity being valued and, consequently, specific growth rates. The residual period should be divided into two sub-periods: the next 5-10 years after the forecast period should reflect the assumed growth rate of the sector of the valued enterprise, while for the following years up to infinity it is reasonable to assume the assumed growth rate of the entire economy. The concept of beauty, with the reservation that Gordon's formula in such a situation undergoes significant - very significant - modifications.

**Can a crocodile be without a mouth? There is no forecast period in the DCF model**

First, we considered what a crocodile would look like without a tail, and now we ask whether a crocodile can be without a mouth (and without those beautiful eyes)? What are we doing with this crocodile...

Seriously, let's consider whether it may happen that in the process of valuing a company using the DCF method, we do not have a forecast period. At least two events may lead to such a situation: the company being valued does not have (or is not willing to present) financial forecasts, or the valuer considers the financial forecasts presented to be unreliable and reliable to be the basis for the company's valuation. Low reliability of financial forecasts is most often divided into two subgroups - profitable business and pessimistic forecasts or unprofitable business and very optimistic forecasts. I would call this second subgroup the prognostic hockey stick syndrome…

What to do in such a situation? Could we not use the income approach and the DCF method in business valuation? In such a situation, it seems most reasonable to estimate the normalized cash flow for the next year and capitalize it according to the Gordon formula. That is, simply, we directly calculate the residual value, where the residual period is from today to infinity. For methodological accuracy, such a method is called the cash flow capitalization method. It is worth noting that if we assume an equal growth rate in the discounted cash flow method for individual years of the forecast period and, what is more, consistent with the assumed average growth rate for the residual period, both methods - discounted cash flows and cash flow capitalization - will give exactly the same results when valuing the company.

**Can a crocodile be mouthless and tailless at the same time? ****DCF model - no forecasts and finite operating period**

Well... such considerations probably provide grounds for immediately notifying the appropriate three-letter units of Minister Ziobro and Minister Kamiński...

Seriously, this applies to situations when we have no forecasts and the period of economic operation of a given company is over. Let's imagine a gravel pit: no forecasts and at the same time a relatively short expected period of further operational activity of the company due to the limited size of the deposit. In such a situation, the value of the enterprise can be estimated as the "residual value", where the residual period begins from the valuation date, and where we additionally use the adjusted Gordon formula reflecting the finite period of generating cash flows.

Regardless of all situations, it is obvious that in most situations when applying the discounted cash flow method, the key is to properly estimate the residual value, which translates into an estimate of the normalized cash flow in the first year after the forecast period, the average growth rate of cash flows and, of course, the cost of capital...

**Final word**

Finally, let's go back to the crocodile for a moment... It is clearly useful for understanding the mechanisms of the DCF model when valuing a company. But... if we look at it, its mouth is significantly smaller than the 30% - 40% estimated above. What's more, its tail definitely goes beyond the 5% - 10% irrelevance condition...

Let's assume that there is a lesson for us here. Each valuation project is unique - the circumstances and conditions of the value estimation process are usually specific; "golden rules" should not prevent the appraiser from critical analysis in the valuation process. If we fall into the trap of routine, we risk finding ourselves in the mouth of a crocodile, which, regardless of its actual size, is not necessarily pleasant...

And jokingly, don't worry about the crocodile's dimensional shortcomings. Let's assume that this is only a temporary developmental defect of the crocodile and over a finite period of time (e.g. a million years) its dimensions will become fully consistent with the DCF model...