We are all familiar with the company valuation model using the discounted cash flow (DCF) method. This method is based on summing the current values of cash flows from the forecast period and adding the discounted value from the Gordon Growth Model (GGM) based on the cash flow from the last year of the forecast.

For the end-period discounting convention, it can be expressed by the formula [1]:

CF_{1} /(1+r)+CF_{2} /(1+r)^{2} +…+CF_{n} /(1+r)^{n} +CF_{n} (1+g)/[(rg)(1+r)^{n} ] , Where:

- CF
_{i}free cash flow in the ith period

- n length of the forecast period;

- r cost of capital;

- g assumed long-term growth rate after the forecast period.

One sentence of note - the end-of-period convention assumes by default that the flow is realized at the end of each period and from that point in time the value of the flow is discounted to the valuation date.

Are you sure this pattern is correct? Unfortunately not...

The correct formula is: CF_{1} /(1+r)+CF_{2} /(1+r) ^{2} +…+CF_{n} /(1+r)^{n} +CF_{n+1} /[(rg)(1+r)^{n} ], where CF_{n+1} means the normalized cash flow in the first year after the end of the forecast [2].

The difference between the first and second formula is small - in the second one, instead of CF _{n} (1+g), we have CF _{n+1} and, what's more, we demand that CF _{n+1} be normalized. It is the standardized nature that is crucial here. Of course, in both enterprise value formulas, the last component is supposed to mean the discounted residual value in accordance with the Gordon formula. In the first formula, the discounted residual value CF _{n} (1+g)/[(rg)(1+r) ^{n} ] was calculated assuming that in the first year after the end of the forecast period we have a cash flow CF _{n} (1+g) and in the following years, this flow will continue to increase at a rate g.

But CF_{n} means the cash flow in the last forecast period - if this flow assumes a lower level of investment outlays than depreciation, then, through the adopted mechanism of positive growth g, we will implicitly assume that the value of the difference between investment outlays and depreciation will increase at a rate of gi, Moreover, the net value of fixed assets will decrease. Similarly, with working capital - if we assume its decline in the last year of the forecast, we also assume by default its decline in the residual period (residual period - the period after the adopted forecast period).

Let's analyze a simple example.

Let us assume that at the end of the forecast period the net value of fixed assets and net working capital is estimated at 10,000 thousand PLN, respectively and 5,000 thousand PLN; in the last year of the forecast, net operating profit less adjusted tax (NOPLAT), investment outlays, depreciation and a decrease in working capital were assumed in the amount of 1,600 thousand PLN, respectively, 200 thousand PLN, 1,200 thousand PLN and 600 thousand PLN; Additionally, the average growth in the residual period was assumed to be 4%. If we use the first formula from "CF_{n} (1+g)", what do such assumptions actually mean?

Of course, they mean an increase in the negative difference between depreciation and investment outlays in the following years - at a rate of 4%. Plus means a decrease in the net value of fixed assets and working capital to... negative values. How do we know this? For working capital, the matter is obvious - the balance at the beginning of the year minus the decrease is the balance at the end of the year. For fixed assets, the relationship is also simple: the net value at the beginning of a given year plus investment outlays minus depreciation will give us the net value at the end of a given year.

Thus, in the fourth year of the residual period, the excess of depreciation over investment outlays will amount to 1,170,000 PLN (1,000(1+4%)^{4}), and at the end of this year the net value of fixed assets and working capital decreases to 5,584 thousand PLN, respectively. (10,000-1,040-1,082-1,125-1,170) and 2,350 thousand PLN (5.000-624-649-675-702). At the same time, NOPLAT in the fourth year of the residual period increases to 1,872 thousand PLN (1,600(1+4%)^{4}). Continuing, at the end of the eighth year of the residual period, the value of working capital will reach a negative level (minus 750,000 PLN), and at the end of the following year the negative level will be exceeded by the net value of fixed assets (minus 1,006,000 PLN). Just as working capital can theoretically assume negative values, there is no possibility, even theoretical, for net fixed assets to be below zero.... That is, an innocent assumption about the flow in the last year of the forecast leads by default to impossible things.

What is the return on invested capital ROIC=NOPLAT/IC (calculated in relation to the initial values)? Invested capital (IC) is, on the one hand, equity plus bank debt, and on the other hand, it is the sum of net fixed assets and working capital. In the first year after the end of the forecast period, ROIC is estimated at 11.1% (1,664/15,000) and increases significantly thereafter. For example, this indicator reaches 19.1% in the fourth year and then 117.9% in the eighth year of the residual period. From a purely mathematical point of view, I can add that in the next year we assume by default that the ROIC will be minus 685%. Where does this "cool" value come from? According to our forecast, NOPLAT will be equal to 2,227,000 PLN, and the capital invested at the beginning of the year minus 332,000 PLN (net fixed assets 417 thousand PLN; working capital minus 750 thousand PLN)…

Of course, such relationships make no sense and distort the value of the company. In our example, the residual value has actually been overstated, which in turn leads to an overstatement of the value of the entire submot. It is worth recalling here that under the standard conditions of the DCF model, the discounted residual value is at the level of 60%-80% of the entire value of the enterprise.

Of course, the relationship can also be skewed the other way - if we assume very significant investment outlays, far exceeding depreciation, and a high increase in working capital in the last year of the forecast, we will actually lead to an underestimation of the value in the DCF model in accordance with the first formula.

From a rational point of view, years after the detailed forecast period, we should assume stable operation of the enterprise - common sense would require the adoption of a constant and appropriate long-term ratio of sales and EBIT (NOPLAT) to net fixed assets and working capital and, consequently, an unchanged return on invested capital.

To sum up, the error in the first formula is the automatic assumption of the cash flow in the first year of the residual period as the forecast cash flow from the last year of the forecast adjusted for the assumed growth rate. The solution is simple - separately estimate the normalized cash flow for the first year of the residual period. This normalized flow (also called residual flow) is the basis for proper estimation of the residual value.

Is it possible for the first formula to be true? Exists. The condition is the normalized nature of cash flow in the last year of the forecast. If this flow is normalized, then after increasing it by the assumed growth rate g, we should obtain a normalized cash flow in the first year of the residual period...

It should be noted here that the appraiser very often prepares a valuation using the DCF method based on financial forecasts presented by the management board of the entity being valued. In such a situation, the possibilities of changing financial forecasts are usually limited. Let us add that it may be that the financial forecasts are prepared reliably and, at the same time, the cash flows of the last year of the forecast are not standardized. In such a situation, they cannot be the basis for estimating the residual value.

Similarly to the end-period discounting convention, for the mid-period discounting convention the formula CF_{1} /(1+r) ^{0.5} +CF_{2} /(1+r) ^{1.5} +…+CF_{n} /(1+r)^{n -0.5} +CF_{n} (1+g)/[(rg)(1+r)^{n-0.5} ] is not valid. The correct formula is: CF_{1} /(1+r)^{0.5} +C_{2} /(1+r)^{1.5} +…+ CF_{n} /(1+r)^{n-0.5} +CF_{n+1} /[ (rg)(1+r)^{n-0.5} ]

And how to estimate the normalized cash flow in the first year after the end of the forecast - i.e. the residual cash flow? And this is a separate issue...