Forecasting Methods of Key Ratios and Their Impact in Company’s Value

Abstract

This paper aims to develop a comprehensive procedure for calculating the fair value of a company by predicting its future values using historical data of key ratios and applying dynamic algorithms to improve the selection of forecasting methods. The most important business valuation methodologies are based on discounting a firm’s future variables, and there are many ways to predict them through financial and quantitative methodologies. This paper provides the most important and commonly used time series forecasting methodologies that can be used for variables, such as financial ratios, and proposes three different algorithms to help and improve the selection of the best-fit method for each of the model’s variables. Another, more indirect way of predicting values is using operational research methodologies, such as Monte Carlo simulation, where the output of the sensitivity analysis gives the most likely firm value, taking into account the distribution of each variable. This paper includes a complete example of using the above procedures in a real Greek company to calculate its fair value. It offers alternative approaches to the problem that exists around the process of predicting variables, with the help of technology. We hope this will be a useful tool for future use.

1. Introduction

Analysts often struggle with the challenge of accurately predicting future prices when valuing a company. This task is highly subjective and can vary from one valuation to another. Is there a way to automate this process? This is the question we seek to answer in this paper.

There are various business valuation methodologies, some of which are based on the current state of the company without taking into account the future. Others, such as discounted free cash flow, calculate the value of the company by discounting projections of its future performance. This paper explores this methodology and proposes an automated process for direct or indirect price forecasting that analysts can use when constructing their models. Two dynamic algorithms are proposed, as follows: one that exists in the literature, namely AIC (Akaike information criterion), and one that was constructed by us, which is simple and easy to implement. These algorithms accept different time series forecasting methods as input, and their output is the preferable methodology to use for the respective model variable. Additionally, an indirect forecasting methodology is proposed that uses the probability distributions of the variables and simulates the model to provide a confidence interval of the model’s output.

In the third section, we summarize the valuation process of a company using the discounted free cash flow method, providing the necessary characteristics and formulas. The fourth section outlines the theoretical elements of the most well-known time series forecasting methods, such as exponential smoothing, autoregressive, moving average and ARCH/GARCH. Additionally, two dynamic algorithms are described in this section, one based on error scoring and the other on the AIC criterion. The fourth section also describes the indirect price forecasting methodology of Monte Carlo simulation. Finally, we present the whole process of valuing a firm using real data from the Greek market, applying all the above-mentioned methodologies. We then record our conclusions about our findings.

2. Literature Review

Forecasting the future values of a variable has been a challenge since the dawn of civilization. People have tried to predict the weather, the wind, and other aspects of their lives for centuries. As statistical and mathematical science progressed, models were created to explain past values, which could then be used to estimate what the variable might be in the future. Although errors can occur, the right tools can help minimize them. If we were able to accurately predict the future, many undesirable events might have been avoided in all areas.

In today’s world, forecasting future prices in accounting and finance has become more imperative than ever (Jansen van Vuuren 2016). The constant introduction of new standards from the FASB and IASB to prevent past problems in the proper recording of a company’s assets and liabilities have created a close relationship with the science of statistics and econometrics. In 2013, the International Accounting Standards Board (the Board) introduced the fair value measurement. Its target is to improve disclosures for fair value (FV) in order that users could better assess the valuation techniques and inputs that are used to measure the real value of a company’s property (Lawrence et al. 2022). This new implementation recommended three valuation methods depending on the type of asset under consideration, namely the market approach, the cost approach, and the most widely used, the income approach (Hodder et al. 2014). The first method evaluates the price using the market, while the second method evaluates the price through estimation of the replacement cost. The income statement method evaluates future cash flows that the asset is assumed to generate in the future (Palea 2013). Thus, the present value of the asset is equal to the sum of discounted cash flows that it will generate in the future. In order to calculate and evaluate future cash flows, certain assumptions regarding coming years’ flows should be applied. Present value techniques are usually used for measurement of cash generating units (CGU) and businesses (based on estimated revenue, expenses, working capital changes, etc.), financial assets/liabilities when prices are not available for identical or similar items (based on contractual and/or estimated cash flows), investment properties (based estimated rental revenue and operating expenses), or project management models (Song et al. 2010). Forecasting technics can be used to predict these assumptions. Forecasting is based on the reasoning that by observing past trends combined with experience and knowledge, one can estimate the future outcome. So far, quantitative and statistical methods are widely used in the field of finance and economics in order to explain past events and evaluate future prospects. In macroeconomics, prediction are used to forecast agent theories and their role in forming trends and prospects in business units (Behavior et al. 2019; Bordalo et al. 2018; Fuhrer 2018). Forecasting applications are used to predict GDP and inflation rates in order to allow for creating new strategies (Kolasa and Rubaszek 2015; Del Negro et al. 2015). In the investment sector, surveys are carried out in order to predict the interest rate of the markets, whether it is the rate free rate or the extra risk premium that applies. That way investors can protect themselves from unwanted market movements and interest-sensitive securities. These studies usually apply autoregression models and a mixture of ARCH and GARCH in order to examine volatility persistence (Barclay et al. 2003; Brüggemann et al. 2006). In addition, the investment industry is using forecasting to predict future returns from portfolios. Investors want to know their risk exposure and the possible movement of their invested money in the near future (Haugen and Baker 1996; Arnott et al. 2017; Asness 2016). Complicated regression models are used to predict certain estimators that have a good fit in market (Huang et al. 2015; Kelly and Pruitt 2013). On the other hand, the autoregressive conditional heteroskedasticity (ARCH) model is used to capture high changes in volatility that can explain possible high future market changes (Engle 1982). Nevertheless prediction of market failure still remains very difficult, and (Engle and Granger 2003) used regression to predict duration among the negative returns and, thus, trace the negative market cycle.

The aforementioned is a drop in the ocean compared to the studies that have been carried out and are still going on today. In addition, in recent years, prediction through neural networks has begun to be applied more and more often. Knowledge has always existed, but now it has evolved and is being used in a variety of ways. Taking into consideration all the above applications of the different forecasting methodologies in this article we will apply a combination of these techniques in order to present another route for predicting assumptions in fair value measurement with the DFCF method.

3. Value in Use Method

Valuation of a business is a process of great importance to all those involved, ranging from shareholders and creditors to investors and analysts. This is why there are numerous reasons to use a valuation, such as for mergers and acquisitions (M&A), increasing share capital, initial public offerings (IPO), impairment testing, estimating potential investments in company shares, bankruptcy tests, and helping shareholders make informed decisions.

To calculate the value in use (VIU), we focused on methodologies that rely on Level 3 inputs. Level 3 inputs are those that are unobservable in regard to the asset or liability being measured. [IFRS 13:86] Unobservable inputs are utilized when relevant observable inputs are not accessible, thus, allowing for situations in which there is limited, if any, market activity for the asset or liability at the measurement date. An entity develops unobservable inputs by using the best available data in the given circumstances, which may include the entity’s own data, considering all information about market participant assumptions that is reasonably available. [IFRS 13:87-89] The IFRS 13 does not include a hierarchy of valuation techniques, nor does it suggest the utilization of a specific valuation technique for the purpose of determining fair value.

Income-based valuation (Level 3) is one of the classifications of business valuation techniques. According to (Adair et al. 2010), it is possible to perform the valuation of a firm, asset, or CGU based on income, utilizing various methods. Generally, when evaluating either a company or CGU that has cash flows and outflows, the cash flow discounting method, also known as DCF, can be employed. Additionally, when financing information of the underlying asset is available and costs can be estimated, free cash flow discounting techniques can be used.

Discounted Free Cash Flow

The formula for calculating free cash flow involves subtracting any necessary investments in fixed assets and other investments from the cash received. Loan repayments are not taken into account when calculating this metric, as it is intended to represent the funds available to all interested parties, including creditors. To arrive at the final figure, a discounted rate, such as the weighted average cost of capital (WACC), is used (Damodaran 1999).

The corresponding formula is as follows:

$$\begin{gathered} FC{F}_{to the firm}=EBITDA\ast \left(1-Tax Rate\right)-Capital Expenditure \\ -DWorking Capital \end{gathered}$$

(1)

$$Firm Value={\displaystyle \sum }_{i=1}^n\frac{FCF{F}_i}{{\left(1+WACC\right)}^i}+\frac{\frac{FCF{F}_{\left(n\right)}\ast \left(1+g\right)}{WACC-g}}{{\left(1+WACC\right)}^n}$$

(2)

where the second part of the above formula is often called the terminal value of the company.

4. Direct Forecasting Methods

The discounted free cash flow (DFCF) business valuation method is based solely on the future free cash flow of the firm, as discussed in the previous section. To calculate these future free cash flow values, analysts must use financial ratios from the firm’s financial statements. However, this process is complicated by the need to forecast these financial variables for at least five years into the future. While there are many algorithms that can be used to make these predictions, there is no guarantee that they will be accurate.

In this section, we are attempting to compile the most important and well-known forecasting methods. These methods are used to predict the inputted variables for the specified periods. Financial variables are typically time series data, as they are collected over a period of time. Therefore, we are mainly discussing time series forecasting methods. The challenge here is that each variable has different characteristics, so there is a high chance that a forecasting method may be more suitable for one variable and, thus, predict its future values more accurately, but may not be as effective for another. Taking this into consideration, we are attempting to provide a dynamic algorithm that can select the best forecasting methods from a pool of methods for each of the inputted variables.

In this unit, we will begin by conducting a theoretical analysis of several time series forecasting methods, including exponential smoothing, moving average, autoregressive and autoregressive conditional heteroskedasticity/generalized autoregressive conditional heteroskedasticity.

4.1. Time Series Forecasting Methods

A time series is a set of data collected over time that expresses the evolution of the values of a variable over equal consecutive time periods (Hamilton 2020). It is essentially a sequence of numbers that expresses the situation at a given point in time and evolves in a random way. The main purpose of time series analysis is to investigate models that can describe the mechanism of the stochastic process from which they originate, as well as to identify the characteristics that contribute to understanding the historical behavior of a variable and allow for the prediction of future values. Time series analysis is a key tool in a variety of applications, such as predicting future values of financial ratios used to calculate the future free cash flow of a firm and, thus, to calculate its fair value. The main characteristics of a time series are trend and seasonality. A trend is a long-term change in the average level of the values of a time series, while seasonality refers to specific periods related to natural seasons of the year (month, quarter, etc.). It is assumed that every observation at time t is a function of the behavior of the time series in previous times. The main prerequisite for a correct prediction of a time series is its stationarity, which is affected by the existence of trend and seasonality. For a stationary time series, the mean, variance and covariance must be time-independent.

A typical form of a time series method is like the Equation (3).

$$Y_t={\mu }_t+s_t+X_t$$

(3)

where

• ${\mu }_t$ is the component of trend;
• $s_t$ is the component of seasonality;
• $X_t$ is the time series of the residuals, if we subtract the trend and seasonality from the observed time series.

The two most popular methods for calculating the trend are the least squares method and the first or second differences method. Trend is a low-degree function of time, which can be described by a known or estimated function of time known as a deterministic trend. Generally, the trend is linear, but it can also be a polynomial of degree p. For seasonality, the moving average method is typically used. The least squares method is used to determine the trend in a time series.

4.1.1. Exponential Smoothing

This methodology is designed to perform forecasting by giving more weight to the most recent values and decreasing the weight as the time goes backwards. This method also takes into account the trend and seasonality of the variables, which are adjusted to minimize the error of the forecasted values. Nowadays, this process is usually performed automatically with statistical packages. Below, we provide two versions of exponential smoothing method, namely Simple and Holt’s.

Simple Exponential Smoothing

This method, which is also called the single exponential method, is mainly used for short-term forecasts. This is because the data fluctuate relatively close to a constant average with no indication of trend or seasonality.

The formula of simple exponential smoothing is as follows:

$$F_t=A_t\ast w+F_{t-1}\ast \left(1-w\right)$$

(4)

where

• $F_t$ = is the forecast of the current period;
• $A_t$ = the actual value of the current period;
• w = the weight we give to the actual value and 0 ≤ w ≤ 1.

We can calculate the forecast for a given period by taking a weighted average of the actual value of the current period and the immediate past forecast period. The weight (w) determines how much importance we give to the prior values; if w is 1, then we do not give any importance to past values, and the forecasts are equal to the actual values. If w is close to 0, then we pay more attention to past predictions. The first value Ft plays a very crucial role, and often, in practice, the average of the first four true values is used. After forecasting the first value, w takes the value of 0, as we do not have real data to use.

Holt’s Exponential Smoothing

Holt’s method, otherwise known as double exponential smoothing, is used when the time series data follow a trend. It is very similar to the simple exponential method in that it considers two additional variables for each period, level and trend.

The formula for double exponential smoothing is as follows:

$$F_t=A_t\ast w+\left(1-w\right)\ast \left(F_{t-1}+T_{t-1}\right)$$

(5)

$$T_t=r\ast \left(F_t-F_{t-1}\right)+\left(1-r\right)\ast T_{t-1}$$

(6)

where

• $T_t$ = trend;
• r = trend rate.

Of particular importance, however, is how we find the initial values of the above variables. Most often, for the trend value we usually take the following value:

$$T_1=\frac{\left(A_t-A_1\right)}{t-1}$$

(7)

or

$$T_1=\left(A_2-A_1\right)$$

(8)

or

$$T_1=\frac{\left(A_2-A_1\right)+\left(A_3-A_2\right)+\left(A_4-A_3\right)}{3}$$

(9)

For the initial value of the case, we take the actual value of the first period.

4.1.2. Moving Average

The moving average (MA) model is a popular technique for analyzing univariate time series data. This model assumes that the dependent variable is correlated with a random variable that is not identical to itself. Estimating MA models is more complex than autoregressive models, as the lagged error terms are not visible. The autocorrelation function (ACF) of an MA(q) process is zero at lags q + 1 and higher. To determine the maximum lag for estimation, we examine the sample autocorrelation function to identify where it becomes insignificantly different from zero for all lags beyond a certain lag, which is designated as the maximum lag q.

The Moving Average function is as follows:

$$Y_t=\mu +{\epsilon }_t+{\theta }_1{\epsilon }_{t-1}+{\theta }_2{\epsilon }_{t-2}+\dots {\theta }_p{\epsilon }_{t-p}$$

(10)

where degree p refers to the length of the lag of the variable ε, which represents historical values. The term moving average refers to the fact that $Y_t$ appears as a weighted sum of ${\epsilon }_t$ values.

4.1.3. Autoregressive

The autoregressive model specifies that the dependent variable depends linearly on its own previous values and on a stochastic term; thus, the model is in the form of a stochastic difference equation (or recurrence relation which should not be confused with differential equation).

$$Y_t=a_0+a_1{Y}_{t-1}+a_2{Y}_{t-2}+\dots a_p{Y}_{t-p}+{\epsilon }_t$$

(11)

Degree p refers to the length of the lag, while the term autoregression comes from the fact that the above relationship is in essence a regression model, when the interpretive variables are the values of the dependent variable $Y_t$ with a time lag.

4.1.4. Autoregressive—Generalized Conditional Heteroskedasticity

When testing for heteroskedasticity in econometric models, the White test is generally the best option. However, when dealing with time series data that exhibit time-varying volatility and volatility clustering, the ARCH test is more appropriate. This test describes the variance of the current error term as a function of the error terms from previous time periods. Additionally, the GARCH test can also be used to assess ARCH errors.

The ARCH model is as follows:

$${\sigma }_t{}^2=a_0+{\displaystyle \sum }_{i=1}^q{a}_i{\epsilon }_{t-i}{}^2\hspace{1em}{a}_0,i,a_i>0$$

(12)

An ARCH(q) model can be estimated using ordinary least squares.

The GARH model is as follows:

$${\sigma }_t{}^2=\omega +{\displaystyle \sum }_{i=1}^q{a}_i{\epsilon }_{t-i}{}^2+{\displaystyle \sum }_{i=1}^p{\beta }_i{\sigma }_{t-i}{}^2\hspace{1em}{a}_0,i,a_i>0$$

(13)

Sometimes the ACF and partial autocorrelation function (PACF) will suggest that an MA model would be a better model choice, and sometimes both AR and MA terms should be used in the same model.

4.2. Dynamic Algorithms

The goal of this paper is to present algorithms that can enhance the selection of forecasting methods. As mentioned before, each variable has its own unique characteristics, so not all forecasting techniques are suitable. These characteristics can be identified from a chart of the historical data or by using mathematical and statistical methods.

In this section, we present two dynamic algorithms that can be used to determine the best-fit forecasting method for any given variable by inputting its historical data. The first algorithm is straightforward to use and is based on error scoring of each method. The second algorithm is called AIC (Akaike information criterion) and is usually used with the help of statistical packages such as Palisade.

4.2.1. Error Scoring

To evaluate the differences between different price forecasting methodologies, we created a process to calculate the error between the actual values and the values generated by each model. This error is used to determine which method is most accurate. We also take into account the trend and seasonality of each variable, as some methodologies are designed to account for these characteristics while others are not. For example, ice cream sales during summer are usually higher than in other periods. Thus, the model with the smallest error is considered to be the most correct statistical methodology.

The steps which we intend to follow in this paper are as follows:

• We need to understand the nature of our data (time series, panel data, etc.). However, we are mainly referring to time series data, as this will be the data for these models.
• Then, we should choose two or more forecasting methodologies depending on the nature of the data we found in step 1.
• For each historical time in our data sample, we generate equations (regressions) for all the forecasting methodologies in step 2. By applying these equations, we can calculate the corresponding predicted value. This means that for each historical time, we will have three or more values, namely the actual value and the expected values from each of the time series forecasting methods we have chosen.
• For each method we have chosen and for each time, we calculate the squared difference between the actual value and expected value and sum these differences. We choose to square the differences because otherwise we are very likely to miss information, as it is fortuitous that some differences are either positive or negative and their sum may lead to over-estimation of the differences.
• We end up with one value for each method we have chosen. The most statistically correct prediction method is the one that has the smallest value, as it is the one that had the smallest error from the actual values.

Although the algorithm mentioned above cannot guarantee absolute accuracy, we believe it is a great tool as it will select the most suitable methodology for the variable based on the past data. We recommend that the sample size of historical values should be sufficiently large.

4.2.2. AIC Algorithm

The Akaike information criterion (AIC) is a measure of the quality of statistical models for a given set of data, formulated by Ozcicek and McMilli (1999). It compares the interpretive ability of different models, which may differ in the number of parameters and/or sample size, by using maximum likelihood estimation (log-likelihood) as a measure of fit. According to Billah et al. (2005), the model with the lowest AIC value is the one that the algorithm proposes to use.

The mathematical form of AIC is as follows:

$$\mathrm{AIC}=2\mathrm{k}-2\log \mathrm{l}$$

(14)

where

• k represents the number of independent variables that have used in the model;
• l represents the likelihood that the model could have produced the dependent variable.

Thus, since the best model is the one with the smallest AIC criterion and regarding the above mathematical form, it is obvious that the best model to use is the one that has relatively few parameters (2k) and high likelihood (−2logl).

The AIC criterion is an algorithm that is used for all the types of models. In this paper, we are using it to show us the best time series forecasting method for each individual financial variable. We also use it in order to choose the distribution of each variable, and this is something that is presented in the next units.

5. Indirect Forecasting Methods

Another approach to the forecasting problem is by using more “indirect” methods to calculate future values. In the previous section we provided algorithms and forecasting methods that give as an output exact variable’s value for future periods. In this section we are trying to present an idea of forecasting without predicting the exact values of the variables, which is why this idea called indirect forecasting.

The idea behind the indirect approach is that each variable, regarding its historical values, follows a distribution. There are many tests that analyst can follow in order to find the distribution of the variables using, of course, statistical packages, such as Palisade. Knowing the distribution of each variable, we can predict a confidence interval of a firm’s value by using a simulation method. As such, without calculating and forecasting the values of each variable, we can have an assumption of the firm value. This paper focuses only on Monte Carlo simulation in order to calculate the confidence interval of the model.

Monte Carlo Simulation

Monte Carlo simulation, also known as probability simulation, is a mathematical technique used to estimate the impact of uncertainty and risk in financial forecasting models, such as business plans and investment valuations. It involves generating random variables to model the risk or uncertainty of a system. A computer is used to generate random numbers and simulate the outcome using different values as inputs. This process is repeated a large number of times, and the occurrence rate of each result is calculated. Monte Carlo simulation is now considered one of the most effective ways of capturing risk, especially in complex models with an element of uncertainty. It is a probabilistic method for modelling risk in a system and is never deterministic. The Monte Carlo method is better at predicting possible outcomes and estimating the likelihood of each one. This is useful when modelling variables that are related, such as in gambling or risk taking. The Monte Carlo simulation produces the result of a model using multiple simulations and values that follow the defined probability distribution. However, there is no guarantee that the most expected outcome will occur, or that actual movements will not exceed the wildest projections.

6. Model

The purpose of this model is to calculate the value of a firm using the valuation technique proposed by IFRS 13. In particular, in order for our model to represent a real case scenario, the data were taken from a real Greek company whose main activity is coffee trading. At this point it should be declared that the above actions, although in most cases are similar for other categories of companies, may differ in other circumstances. Furthermore, our historical data are very limited for reflecting the real world problems where in most cases analysts won’t have historical data for more than 5 years. This because company’s historical data exceeding 5 years can be biased due to its difficulty in maintaining stability over that period, as well as a range of macroeconomic variables. Having more than 5 years data would be great from a statistical view, but in real cases would raise doubts. This paper aims to provide the whole process of predicting future values using technology, in order to calculate the fair value of the company, with limited data available.

As mentioned in a previous unit, in order to evaluate the value of the company, we used the discounted free cash flow to the firm technique. More specifically, we rely on the historical published data of the company for the years 2016–2021 and calculate the future values for the next 5 years. For this method, we need to calculate the free cash flows values that will be available to all stake holders e.g., shareholders, creditors, etc.

$$\begin{array}{ll}{\mathrm{FCF}}_{to the firm}=& \mathrm{EBITDA}\ast \left(1-\mathrm{Tax}\mathrm{Rate}\right)\\ & -\mathrm{Capital}\mathrm{Expenditure}\mathsf{\Delta }\mathrm{Working}\mathrm{Capital}\end{array}$$

(15)

In order to predict the values, we will use the methods mentioned in the previous unit. Specifically, after cleaning and organized our historical data, we will apply the time series forecasting methods followed by our algorithm for choosing the best method regarding the error scoring. Furthermore, we will implement algorithm AIC to have a different perspective on which forecasting method to follow. At the end we will use PERT distribution for each variable, and we will simulate our model using the Monte Carlo technique.

In total, there will be a chance to see the different results that each one of the mentioned methods calculate. In the real world, there is no such thing as a method that can predict with 100% probability the future values. For this reason, analyst often choose the the average of these values as the value of the company, or otherwise they estimate a valuation range in which the correct value is placed.

According to the authors, the most appropriate method is differentiated by the life cycle of a firm. For those that are at a relatively early stage, the linear method is the best fit, while at mature stages where saturation and diseconomies of scale occur, all other methods can be applied. The above hypothesis will be the subject of future research for the authors of this paper.

6.1. Organizing Historical Data

We are calculating the free cash flow (FCF) for the firm, so we do not need to consider interest expenses. The relevant data can be found in

Table 1. Income statement.

Income Statement
€k	2016 A	2017 A	2018 A	2019 A	2020 A	2021 A
Sales	8795	8808	10,399	11,692	10,126	13,324
Cost of sales	(4928)	(5022)	(6281)	(6986)	(6017)	(7886)
Total Revenue	3867	3786	4118	4706	4109	5438
Operating Expenses	(2249)	(2519)	(2666)	(2858)	(2659)	(2872)
Other Expenses	(342)	(193)	(206)	(217)	(90)	(94)
Total Expenses	(2592)	(2712)	(2871)	(3075)	(2749)	(2966)
EBITDA	1275	1074	1247	1631	1360	2472
D&A	(253)	(326)	(386)	(416)	(506)	(742)
EBIT	1022	748	861	1215	854	1730

Source—author’s calculations.

, which is the income statement, and

Table 2. Balance sheet.

Balance Sheet
€k	2016 A	2017 A	2018 A	2019 A	2020 A	2021 A
Property, plant, and equipment	1020	1298	1643	2216	2885	3973
Intangible assets	26	21	18	15	12	11
Other non-current financial assets	10	10	10	10	10	10
Non-current assets	1055	1329	1671	2241	2907	3993
Inventories	1637	1879	2122	2383	2333	2661
Trade receivables	2982	3033	2883	2747	2613	2647
Other receivables	480	413	382	507	327	417
Prepayment expenses	3	48	41	27	39	8
Cash and cash equivalents	179	26	115	275	744	181
Current assets	5282	5400	5544	5939	6056	5914
Total assets	6338	6728	7214	8180	8963	9906
Share capital	59	59	59	59	59	59
Owners’ deposits	223	223	223	223	223	223
Reserves	182	182	182	182	182	182
Retained earnings	1593	1160	1439	1633	1365	1734
Equity	2056	1623	1902	2096	1828	2197
Provisions	-	-	-	-	-	183
Repayable advance payment	-	-	-	-	700	750
Long-term loans	-	-	-	428	800	943
Other long-term liabilities	339	274	294	336	293	184
Long-term liabilities	339	274	294	764	1793	1877
Short-term loans	1538	1631	2077	2311	2476	2215
Short-term or long-term loans	270	358	384	416	372	571
Trade payables	972	861	919	1207	755	1080
Income tax payable	150	300	256	314	259	413
Other taxes	836	1077	1109	743	953	794
Social security organizations	76	104	79	86	176	216
Other short-term liabilities	100	498	194	244	351	360
Short-term liabilities	3942	4830	5019	5320	5342	5649
Total equity and liabilities	6338	6728	7214	8180	8963	9906

Source—author’s calculations.

, which is the balance sheet.

Using the above historical values, we can calculate the working capital for each historical period as it is presented in

Table 3. Working capital.

Working Capital
€k	2016 A	2017 A	2018 A	2019 A	2020 A	2021 A
Current Assets
Receivables	3463	3446	3265	3254	2940	3064
Other current assets	1641	1927	2163	2410	2372	2668
Current Liabilities
Suppliers	972	861	919	1207	755	1080
Other current liabilities	100	498	194	244	351	360
Net working Capital	4032	4014	4316	4213	4205	4292
(Increase) / Decrease in WC	(829)	18	(302)	103	7	(86)

Source—author’s calculations.

.

The mathematical form of working capital is as follows:

$$\begin{array}{ll}\mathrm{Working}\mathrm{capital}& =\mathrm{Receivables}+\mathrm{Other}\mathrm{current}\mathrm{assets}\\ & -\left(\mathrm{Payables}\mathrm{other}\mathrm{current}\mathrm{liabilities}\right)\end{array}$$

(16)

After gathering and organizing the historical data, the future values of the model must be calculated in order to create a 5-year forecast. In this example, we have calculated nine key variables that are essential for forecasting these accounts. These are as follows:

$$\mathrm{Sales}\mathrm{Growth}=\frac{{\mathrm{Sales}}_{t+1}}{{\mathrm{Sales}}_t}-1$$

(17)

This variable shows the percentage change in the company’s revenue.

$$\mathrm{Margin}\mathrm{of}\mathrm{Cos}\mathrm{t}\mathrm{of}\mathrm{sales}=\frac{{\mathrm{Cos}\mathrm{t}\mathrm{Of}\mathrm{Sales}}_t}{{\mathrm{Sales}}_t}$$

(18)

The variable expresses the ratio of cost of sales to revenue.

$$\mathrm{Expenses}\mathrm{Margin}=\frac{{\mathrm{Total}\mathrm{Expenses}}_{\mathrm{t}}}{{\mathrm{Sales}}_{\mathrm{t}}}$$

(19)

This variable shows the ratio of total expenditure to revenue.

$$\mathrm{Tax}\mathrm{Rate}=\frac{{\mathrm{Tax}\mathrm{Expenses}}_{\mathrm{t}}}{{\mathrm{Ebit}}_{\mathrm{t}}}$$

(20)

This variable expresses the tax rate.

$$\mathrm{Receivable}\mathrm{Days}\mathrm{Ratio}=\frac{365\ast \mathrm{Average}{\left(\mathrm{Receivables}\right)}_{}}{{\mathrm{Sales}}_{\mathrm{t}}}$$

(21)

This variable expresses how many days it takes to collect the receivables.

$$\mathrm{Payables}\mathrm{Days}\mathrm{Ratio}=\frac{365\ast \mathrm{Average}{\left(\mathrm{Payables}\right)}_{}}{{\mathrm{Cos}\mathrm{t}\mathrm{of}\mathrm{Sales}}_{\mathrm{t}}}$$

(22)

This variable expresses how many days the liabilities are repaid.

$$\mathrm{Other}\mathrm{Current}\mathrm{Assets}=\frac{{\mathrm{Other}\mathrm{Current}\mathrm{Asstes}}_{\mathrm{t}}}{{\mathrm{Sales}}_{\mathrm{t}}}$$

(23)

This variable expresses other current assets in units of income.

$$\mathrm{Other}\mathrm{Current}\mathrm{Liabilities}=\frac{{\mathrm{Other}\mathrm{Current}\mathrm{Liabilities}}_{\mathrm{t}}}{{\mathrm{Cos}\mathrm{t}\mathrm{of}\mathrm{Sales}}_{\mathrm{t}}}$$

(24)

This variable expresses other liabilities in units of income.

$$\mathrm{Net}\mathrm{Capital}\mathrm{Asset}\mathrm{Expenditure}=\frac{{\mathrm{CAPEX}}_{\mathrm{t}}}{{\mathrm{Sales}}_{\mathrm{t}}}$$

(25)

The

Table 4. Assumptions.

Assumptions
Variables	2016A	2017 A	2018 A	2019 A	2020 A	2021 A
Increase in sales	12%	0%	18%	12%	−13%	32%
Margin of cost of sales	56%	57%	60%	60%	59%	59%
Expenses margin	29%	31%	28%	26%	27%	22%
Income tax	24%	31%	24%	22%	36%	20%
Days on receivables	144	143	118	102	112	82
Days on suppliers	72	67	52	56	60	42
Other current Assets	19%	22%	21%	21%	23%	20%
Other current Liabilities	2%	10%	3%	3%	6%	5%
Net Capital assets exp	5%	7%	7%	8%	12%	14%

Source—author’s calculations.

below summarizes the calculations for the above variables, named as “assumptions” from now on.

6.2. Valuation Process

In this chapter we will use all available information about the company in order to develop a forecasted model and, consequently, predict the next 5 years’ future free cash flows.

We will provide various estimates for above values using two different methods, as follows:

• Using dynamic algorithms with time series forecasting methods.
• Using the derived probability distributions.

Both aforementioned dynamic algorithms can be used to develop a consistent forecasting approach for each individual financial variable.

6.2.1. Using Dynamic Algorithms

Error scoring

In order to implement the dynamic algorithm of error scoring, it is necessary to select forecasting methods which will use the forecasted values as inputs. As we only have time series data, we chose time series forecasting methods as they are the most widely used. These methods are as follows:

• Linear trendline forecasting with independent variable the time.
• Logarithmic trendline forecasting with independent variable the time.
• Simple exponential smoothing.
• Holt’s exponential smoothing.

It is not necessary to limit oneself to just these forecasting methods; the algorithm can be employed with as many forecasting methods as the analyst desires. Furthermore, it is important to note that we have taken for granted that all of the regression’s assumptions to be followed.

Therefore, we initially followed the steps outlined in the fourth section above for each of the variables mentioned above, except for the tax rate, which tends to remain constant over time or can be accurately predicted from external information. In this way we can predict the future values of the variables with as little error as possible.

Table 5. Error scoring.

Error scoring
Variables	Linear Frcst	Logar Frcst	Exp sm Single	Exp Sm Holt’s
Increase in sales	11.57%	11.81%	12.11%	16.19%
Margin of cost of sales	0.07%	0.05%	0.13%	0.11%
Expenses margin	0.10%	0.17%	0.39%	0.16%
Days on receivables	387	579	1934	563
Days on suppliers	170	161	564	212
Other current assets	0.12%	0.10%	0.23%	0.20%
Other current liabilities	0.40%	0.39%	0.68%	0.58%
Net capital assets exp	0.03%	0.11%	0.20%	0.07%

Source—author’s calculations.

shows the error scoring for each variable with each method.

As can be seen, for all assumptions, the method that was selected was either linear trendline forecasting with the independent variable of time, or logarithmic trendline forecasting with the independent variable of time.

Table 6. Assumptions with forecasted values.

Assumptions
Variables	2016A	2017 A	2018 A	2019 A	2020 A	2021 A	2022 E	2023 E	2024 E	2025 E	2026 E
Increase in sales	12%	0%	18%	12%	−13%	32%	6%	6%	6%	6%	6%
Margin of cost of sales	56%	57%	60%	60%	59%	59%	61%	61%	62%	62%	62%
Expenses margin	29%	31%	28%	26%	27%	22%	23%	21%	20%	18%	17%
Income tax	24%	31%	24%	22%	36%	20%	20%	20%	20%	20%	20%
Days on receivables	144	143	118	102	112	82	76	63	51	39	27
Days on suppliers	72	67	52	56	60	42	47	45	43	42	40
Other current assets	19%	22%	21%	21%	23%	20%	22%	22%	22%	23%	23%
Other current liabilities	2%	10%	3%	3%	6%	5%	5%	5%	5%	5%	5%
Net capital assets exp	5%	7%	7%	8%	12%	14%	15%	17%	18%	20%	22%

Source—author’s calculations.

presents the assumptions’ forecasted values using the chosen forecasting method.

For sales growth, we thought it would be more appropriate to consider only the constant term of the regression, for practical reasons and to be realistic. This was a move based on the authors’ experience. It does not, however, invalidate the algorithms that have been presented in the above unit. It is important that, in practice, we must not consider only historical data but also experience and economic and market conditions. Using the forecasted assumption, it is easy to forecast the actual values of the financial statements as follows.

More specifically, having the value of sales growth for each one of the predicted years, it is easy to find sales account, as follows:

$$Sale{s}_t=Sale{s}_{t-1}\ast sales growt{h}_t$$

(26)

After calculating sales for each future period, it is easy to calculate the corresponding cost of sales, as follows:

$$Cost of sale{s}_t=Sale{s}_t\ast Margin of Cost of sale{s}_t$$

(27)

Thus, after we have sales and the cost of sales, we use the same technique above in order to calculate the rest of the variables - as it is presented in the

Table 7. Income statement (actual and forecasted).

Income Statement
€k	2016 A	2017 A	2018 A	2019 A	2020 A	2021 A	2022 E	2023 E	2024 E	2025 E	2026 E
Sales	8795	8808	10,399	11,692	10,126	13,324	14,124	14,971	15,869	16,822	17,831
Cost of sales	−4928	−5022	−6281	−6986	−6017	−7886	−8626	−9183	−9772	−10,393	−11,051
Total Revenue	3867	3786	4118	4706	4109	5438	5498	5788	6098	6428	6780
Operating expenses	−2249	−2519	−2666	−2858	−2659	−2872	−2972	−2953	−2921	−2874	−2811
Other expenses	−342	−193	−206	−217	−90	−94	−224	−222	−220	−216	−212
Total Expenses	−2592	−2712	−2871	−3075	−2749	−2966	−3196	−3175	−3141	−3090	−3022
EBITDA	1275	1074	1247	1631	1360	2472	2302	2613	2957	3338	3758
D&A	−253	−326	−386	−416	−506	−742	−852	−994	−1176	−1411	−1718
EBIT	1022	748	861	1215	854	1730	1449	1619	1782	1927	2040

Source—author’s calculations.

and

Table 8. Working capital (actual and forecasted).

Working Capital
Current Assets
Receivables	3463	3446	3265	3254	2940	3064	2925	2600	2226	1798	1311
Other Current Assets	1641	1927	2163	2410	2372	2668	3133	3345	3568	3803	4052
Current Liabilities
Suppliers	972	861	919	1207	755	1080	1100	1126	1155	1188	1225
Other Current Liabilities	100	498	194	244	351	360	448	483	519	556	596
Net working Capital	4032	4014	4316	4213	4205	4292	4510	4337	4121	3857	3542
(Increase)/Decrease in WC	(829)	18	(302)	103	7	(86)	(218)	173	217	264	315

Source—author’s calculations.

as follows:

$$\mathrm{e}.\mathrm{g}.,Total expense{s}_t=Sale{s}_t\ast Expenses Margi{n}_t$$

(28)

Having predicted the actual values of the financial statements, we can proceed to calculate the forecasted free cash flow from operations, as it is presented in

Table 9. Free cash flow.

Cash Flow from Operations
	2016A	2017 A	2018 A	2019 A	2020 A	2021 A	2022 E	2023 E	2024 E	2025 E	2026 E
EBITDA	1.275	1.074	1.247	1.631	1.36	2.472	2.302	2.613	2.957	3.338	3.758
Less: Capex (Replacement capital expenditure)	−441	−603	−731	−989	−1.175	−1.83	−2.102	−2.484	−2.904	−3.366	−3.873
Less: (Increase)/decrease in WC	−829	18	−302	103	7	−86	−218	173	217	264	315
Unlevered pre-tax CFO	6	489	214	745	192	557	−18	302	270	236	200
Less: income tax	−247	−230	−210	−272	−305	−347	290	324	356	385	408
Cash flow from operations—after tax	−242	258	4	473	−113	210	272	625	626	621	608

Source—author’s calculations: https://pages.stern.nyu.edu/~adamodar/New_Home_Page/datafile/wacc.html (accessed on 10 December 2022).

.

To calculate the discount rate, we use the formula of the weighted average cost of capital (WACC) as it is presented in

Table 10. WACC.

€k	2016A	2017 A	2018 A	2019 A	2020 A	2021 A	2022 E	2023 E	2024 E	2025 E	2026 E
Ke	10%	9%	9%	10%	8%	6%	10%	10%	10%	10%	10%
KD	6%	6%	6%	6%	6%	6%	6%	6%	6%	6%	6%
%EQUITY	0.4	0.4	0.4	0.4	0.4	0.4	0.4	0.4	0.4	0.4	0.4
%DEBT	0.6	0.6	0.6	0.6	0.6	0.6	0.6	0.6	0.6	0.6	0.6
WACC	8%	7%	7%	8%	7%	6%	7.4%	7.4%	7.4%	7.4%	7.4%

Source—author’s calculations.

. We use this discount rate because free cash flow values are reported for all stake holders. In this example, we have calculated and assume a constant cost of capital for debt. Regarding the cost of equity, we have considered the annual survey of Professor Damodaran from NYU. For the future values of ke, we used the average method. Alternatively, it is possible to use the above techniques in order to predict future value of ke.

$$\mathrm{WACC}=\frac{D}{D+E}\ast K_d+\frac{E}{D+E}\ast K_e$$

(29)

Then, the value of the company is calculated as it presented in

Table 11. VIU.

VIU
€k		2022 E	2023 E	2024 E	2025 E	2026 E
FCF		272	625	626	621	608
Discount rate	VIU	0.931	0.867	0.807	0.752	0.700
DFCF	2193 €	253	542	505	467	426

Source—author’s calculations.

, using all of the above, and the formula of DFCF method without terminal value is as follows:

$$\mathrm{DFCF}={{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}\frac{{\mathrm{FCF}}_{\mathrm{i}}}{{\left(1+\mathrm{wacc}\right)}^{\mathrm{i}}}$$

(30)

Finally, we come to the result of the value of the company. However, much of the above is based on the authors’ experience and it is very likely that much of the above will vary from appraiser to appraiser. However, the main interest here is the algorithm for finding the best time series forecasting methodology.

AIC Criterion

In order to select the most suitable forecasting method, we can employ the Akaike information criterion (AIC) to evaluate the different options. By following the output of the AIC, we can determine which method is best suited for our needs. Thus, we follow the same process as above to calculate the fair value of the firm. The time series that we use for our data are as follows:

• Autoregressive;
• Moving average;
• ARMA;
• ARCH;
• GARCH.

As like the previous dynamic algorithm, analysts can use as many algorithms as they want as input for this dynamic algorithm. Using the historical values of each variable, we implemented the AIC criterion using Palisade’s @RISK module, and we obtained the output as in the

Table 12. AIC.

AIC
Variables	Forecasting Method
Increase in sales	AR
Margin of cost of sales	ARCH
Expenses margin	MA
Days on receivables	GARCH
Days on suppliers	ARCH
Other current assets	AR
Other current liabilities	GARCH
Net capital assets exp	AR

Source—author’s calculations and Palisade.

.

After utilizing the AIC to determine which forecasting technique to use for each variable, the same process should be followed as before. It is important to keep in mind that this method does not guarantee the best estimated value of the company. The analyst’s expertise and the format of the model also play a role in the method that should be employed.

6.2.2. Using Monte Carlo Simulation

It is a general assumption that the values of the hypothesis variables must follow some probability distribution. On the other hand, the constant term in a time series estimation function gives us the constant value for the variable in addition to the effect of the trend that the values take.

The PERT distribution was used in this example due to its ability to generate estimates for parameters where exact values are not available. This makes it an ideal tool for creating probabilistic models, such as Monte Carlo simulations. The three scenarios that PERT relies on (minimum, maximum, and most likely value) are chosen to generate the estimates. These three values were taken from the historical data. The PERT distribution is widely supported by most statistical software packages, and when combined with Monte Carlo simulations, it is a powerful tool for estimating risk in complex situations. Using PERT distribution for each variable we can determine a confidence interval for the firm’s value by utilizing 100,000 iterations of a Monte Carlo simulation via the @Risk software.

The above simulation method is primarily used to calculate the risk of valuation, or to determine its sensitivity. However, it can also be employed as a forecasting tool, as the methodology of this simulation involves making assumptions about the independent variables in the future in order to generate a distribution of the dependent variable.

In conclusion, the estimated value of the firm based on the PERT distribution of each variable is the mean of the confidence interval that the Monte Carlo simulation produced. An alternative is to use the Akaike information criterion (AIC) to determine which distribution each variable follows. Once this is established, a Monte Carlo simulation can be run to obtain the corresponding results.

7. Conclusions

This paper focuses solely on the process of valuing firms using the DFCF method. As outlined in Section 3, this method involves discounting future free cash flows, which are derived from financial statement accounts. Forecasting these accounts is notoriously difficult due to their subjective nature. Nevertheless, we have attempted to create an automated process that analysts can use as a starting point and further refine their model, even if they do not choose to rely on it exclusively.

At the outset, our research was focused on predicting financial ratios based on their past values. However, the small amount of data available has limited the accuracy of our findings. This is a common issue that analysts face when attempting to value a company, so it is essential to consider the analyst’s expertise in such situations.

Dynamic stochastic trend analysis models provide us with an accurate and optimal prediction of future values of variables based on error scoring and the AIC criterion. These methodologies can help us identify the most suitable forecasting technique based on the past behavior of the variable. In the example above, linear and log linear regression were found to be the best predictors for most of the variables using the first methodology. The second methodology revealed that moving average, autoregressive, GARCH, and ARCH were the most suitable forecasting techniques. It is important to note that this example is not a model and that no definitive conclusion can be drawn as to which variable is best predicted by which methodology. The example is only meant to illustrate the procedure, and the results may vary in reality; different forecasting techniques may also be used. Furthermore, by utilizing probability distributions of each variable, the Monte Carlo simulation can be employed to indirectly forecast the variables in the model. This simulation will generate a confidence interval of the company’s value.

Further research could involve applying the methods and estimates discussed above to more companies at different stages of their life cycle. Additionally, analyzing the external factors that influence a company’s valuation and utilizing more accounting data could be beneficial. Our research has contributed to the discussion on the best way to calculate a firm’s value and has provided alternative approaches for industry professionals to use when determining a firm’s value in uncertain circumstances.