Corporate valuation using stochastic variables

Posted on:Dec 5,2020


This study uses the free cash flow to equity to estimate the market value of the company. Valuation based on free cash flow may give a more accurate result than a valuation based on other business indicators. It is important to consider risk in all corporate evaluations. The authors use the Monte Carlo simulation to analyse the sensitivity of the calculations using the @RISK program of Palisade company. The @RISK program allows us to consider everything we know about the variables, including the full ranges of possible values and the probabilities of their occurrence. The solution of the presented company valuation model confirms that using stochastic simulation, we can get a more accurate picture of the value of the company.

Keywords: corporate valuation, free cash flow, business risk, simulation

1. Introduction

Company valuation methods allow such valuation estimations that can be a basis of the intelligent investment decisions since the value of securities is not equal to their price either in relatively efficient capital markets. The price can be observed, but the value is not, that must estimate. Therefore, estimating the value of a company is a common goal in the analysis of financial statements. The analysis and valuation process of financial statements allows investors, analysts, portfolio managers, investment bankers and corporate managers to estimate the value of equity in a reliable way.

It is possible to achieve a reliable estimate of the company value depends entirely on the prudent application of the method of analysis and evaluation, and the analysis and evaluation of the results of the financial statement. An essential point of view to understand the main economic characteristics of the sector in which the company operates, which is not always a simple task because nowadays, larger companies can even operate in several sectors. Once we have learned about the sector(s) in which the company evaluated operates, the next step is to evaluate the strategy of the company. Then there is a need to evaluate the accounting system used by the company (Hamad and Dékán Tamásné Orbán, 2018). There is also essential to assess the company’s profitability and risk. These analyses provide useful information for forecasting future financial statements. Forecasts can provide expectations about future income, assets and dividends, which are the essential characteristics on which it bases the valuation. Finally, different models can be applied to these expectations to determine the values ​​of the unique characteristics (Kovács et al., 2020). A critical step in the valuation is to determine the risk-adjusted discount rate, which requires an unbiased assessment of the risk inherent in the set of expected future cash flows. Based on the above, a reliable estimate of corporate value depends on unbiased expectations about future cash flows and an appropriate risk-adjusted discount rate (Fazzini, 2018).

The purpose of the valuation process is to determine the distribution of value estimated between the critical forecast assumptions and ranges of valuation parameters. Need to determine the most likely range of value by examining the sensitivity of value estimated, that can be used to make an intelligent investment decision. Experience shows that applying the simulation method to calculate the company value can provide a better insight into a company’s value than it can achieve without it.

In this study, we deal with the cash flow-based company valuation.

2. Risk analysis and company valuation

Discounted cash flow (DCF) valuation calculates the intrinsic value of assets as the sum of the present value of expected future cash flows (Takács, 2007; Takacs et al 2020) . In the study, DCF-based valuation is applied to free cash flow to equity (FCFE).

There are no easily available data to determine the FCFE. The professionals performing the calculations must calculate these values based on the available financial statements and information, which requires an understanding of free cash flows (FCFs) and the ability to interpret and use the information correctly. Forecasting future FCFs is also a task that requires serious knowledge and work. The persons performing the calculations must understand the company’s financial statements, and have knowledge of the company’s operations, financing, and industry position. Even so, many analysts believe that models based on FCFs may provide a more economically sound basis for company valuation (Massari et al., 2016). Consequently, controlling can also play an essential role in defining these characteristics more precisely (Lakatos, 2017, Böcskei-Hágen, 2017).

The FCFE is the cash flow available to the company’s ordinary shareholders after paying operating costs, interests and capital payments and they have performed all necessary investments in working capitals and fixed assets (Hamad – Szekeres, 2019).

The advantage of FCFs over other cash-flow concepts is that it can use directly in the DCF framework to measure total corporate assets or equity. Other cash flows – or incomes – such as operating cash flow, net income, EBIT and EBITDA – do not have this feature because they calculate doubly or omit cash flows. From a shareholder perspective, EBITDA and similar other values do not take into account different capital structures or the resources provided by bondholders to finance the investments in assets operated. These measures do not take into account the reinvestment of cash flows that the company invests into maintain or maximise its long-term value.

Using free cash flows for valuation is more challenging because when forecasting FCFs, the analyst must integrate the cash flows from the company’s operations with the cash flows from its investing and financing activities (Yaari et al., 2016). The value of equity can estimate directly by discounting of FCFEs with the expected rate of return on equity (since FCFE is cash flow to ordinary equity holders, the expected return on equity or cost of equity is the risk-adjusted rate for discounting of FCFE):

Formula (1) gives the estimated market value of the firm’s equity, which is equal to the firm’s estimated market value because the book value of the firm’s market value is equity. In both cases, there are usable models with a constant rate of return, or a rate of return changing with a constant value, or a rate of return changing with varying degrees. It is practical to use the third form, where the expected rates of return (or cost of capital) can change with a varying degree.

3. Az @RISK Excel bõvítmény bemutatása

@RISK („at risk”) is a program to analyse business and technical situations affected by risk. Risk analysis techniques are important tools to support decision-makers when managing situations of uncertainty. @RISK allows using these techniques in a Microsoft Excel spreadsheet. By linking @RISK and Excel, any risky situation can be modelled, and this software combination allows to create models that best meet the expectations. In the course of a decision or an analysis involving uncertainty to use @Risk to gain insight into future opportunities. That is the reason the stochastic estimation of model variables is necessary because future values ​​are not known with absolute certainty (Lehman – Groenendaal, 2019).

In real-world, many things do not turn out as planned. We may have been too conservative for some estimations and too optimistic for others. A decision based on the expected results can be irrational that would not make if we would have a complete picture of all possible outcomes. Some business, technical, and scientific decisions apply estimates and assumptions (Casey, 2001). With @RISK, the uncertainty of the estimations can take into account by generating the results using a wide range of possible outcomes.

@RISK applies the Monte Carlo simulation to combine all the uncertainties of several situations. We are no longer forced to reduce to a single value what we should know about a variable. Instead, we can incorporate everything we know about the variable, including the full range of its possible values and the probabilities of their occurrence. @RISK uses all the information about the model constructed in Excel to produce a range of possible outputs. It looks like simultaneously executing hundreds or thousands of „what if” scenarios. @RISK allows taking full account of what might happen in a situation (Seila et al., 2003).

@RISK can be used in all cases where we perform an analysis that includes uncertainty. That is why the @RISK was applied to solve a company valuation model, which can use not only to calculate the results but also to perform their risk analysis. In a broader sense, risk analysis is a quantitative and/or qualitative method to assess the effects of risks on the decision outcomes. These methods can help decision-makers to choose a course of action and allow them to understand better the possible outcomes occurred. @RISK risk analysis is a quantitative method to determine the probability distribution of the outcomes of decisions (Albright – Winston, 2017).

4. Corporate valuation using @RISK

4.1. Presentation of the company valuation model

The company valuation model was compiled using real company data (Tables 1 and 2). The data of the model representing the years „x-4 … x” were extracted from the annual financial statements (balance sheet and income statement) of the given company. Thus, the data of „x-4… x” years represent the data for the previous five years of the company, which data were used to estimate the data for the next five years („x + 1 … x + 5”). After that, the stochastic (probability) variables of the model were determined, and they are in the light grey rows of the model. These rows show the same value in each year in the base model because they contain different probability functions, provided by @RISK. In the model, we used two types of probability functions, one with a normal distribution (RiskNormal) and one with a triangular distribution (RiskTriang) (Kotz – van Dorp, 2004). These functions are used to generate the random numbers in @RISK that need for the Monte Carlo simulation. The program recalculates the values ​​of the functions with each simulation run and recalculates the model using them.

The normal distribution has two parameters, mean and standard deviation, so for these types of distribution functions, these values were estimated based on data from previous years. We used a normal distribution in the following cases: change in net sales, change in operating expenses, base return, inflation rate, risk premium. In other cases, the triangular distribution was used, which has three parameters, minimum, mode, and maximum. The mean was used as the mode.

We could not determine precisely the cost of capital for the firm, Formula (2) was used to determine the expected rate of return (discount rate) and considered all three components in the formula as probability variables.

expected rate of return = base expected return + inflation premium + risk premium (2)

In the model, the FCFE was used to determine the firm value, which was determined by the formula (3).

FCFE = Profit after tax – Net investment – Change in current assets – Net debt (3)

The FCFE values and their present values for the 5 years following the reference year were determined explicitly. The so-called residual value also had to be determined. Residual value is the sum of the present values of cash flows outside the explicit period. The perpetuity calculation was used to determine the residual value.

There has been added together the FCFE values determined explicitly and the present values of the residual value to determine the corporate value. There was set 5,000 runs (iterations) to solve the model using the Monte Carlo simulation. Once the model is finished running, the results can be displayed in several ways. The results can also be saved to an Excel file, which saves all model-specific input and output values for each run, and key statistical characteristics of the simulation result and key figures. Further analyses can be performed, and additional diagrams can be created using the saved results.

4.2. Results of the company valuation model

It can be seen from Table 2 that the company value calculated with the basic data and the average values ​​of the probability variables is 10,347,386 thousand HUF. However, it should also be seen that this is only a single value, which is uncertain because we used estimated values ​​to calculate it, which may change in the future. The application of the simulation allows us to produce several variants by randomly changing the selected variables using random numbers generated by a given distribution. During the calculation, 5,000 iterations were performed; that is the model was counted out 5,000 times. All this means we have 5,000 results of the model after performing the calculations. If only one value would be determined, it could not be decided whether it is a fair value or a result influenced by estimation errors. The large number of model solutions performed can ensure that we can evaluate the result more precisely.

It can be seen from Table 3 that the firm value can take both negative and positive values, which could cause a problem in the valuation, but the 95% confidence interval falls in a positive range. However, the range of the 95% confidence interval is quite large (17,175,813 thousand HUF). The probability that the value of a company is less than or equal to zero is 2.29%, which is a fairly small value. It also follows that the probability that the value of the company is greater than zero is 97.71%.

We can see from the table that the distribution of the results differs only slightly from the normal distribution, which is also supported by the minor difference between the mean and the median and the low value of the skewness index. Figure 1 shows the distribution of company values generated by the simulation. The figure also shows that the distribution of company values calculated by the simulation is very close to the normal distribution.

After performing the simulation, it can be seen from Table 3 and the first figure that the corporate value can take on a very wide range of values. So it is uncertain that the value of the company will be HUF 10,300,491 thousand as the basic table of the model shows (Table 2). Table 4 presents what probability value would belong to a condition. It can be read from Table 4 that the probability that the value of the company will be greater than 10,000,000 thousand HUF is 53.66%, and the probability that it will exceed HUF 15,000,000 thousand HUF is already less than 20%. However, it is essential to emphasise that the corporate value is with a 95% probability to fall within the range given in Table 3 and can take any of the values.

It may be raised a question of what means the risk analysis in the assessment presented. In risk analysis, we examine how the output value would change if the input values changed. The risk is well characterised by the distribution of the calculated company values and the 95% confidence interval, which relates to the distribution. The larger the confidence interval, the greater the risk because the final value can take different values. The fact that the estimation of input values contains errors is modelled using probability variables. Table 6 shows the change in operating expenses for 5 years. In the table, we can also see the distribution functions.

The trend in the data in Table 6 is very similar to that reported in Table 5.

Table 7 shows the order of the effect of each probability variable. The company value is the most sensitive to the first variable. This means that the Long term debts / Total assets ratio in year x + 5 has the strongest effect on the company value. The effects are examined annually by the program because the distribution functions are also given in this way.

A 2. ábra a tornádó diagramot mutatja, amely a vállalati érték érzékenységét vizs-
gálja, és eredményei nagyon hasonlítanak a 7. táblázathoz.

Tornado chart is beneficial for sensitivity analysis because we can compare the relative importance of variables. For each variable examined, one needs estimates what would be the low, base, and high outcomes. The sensitive variables are modelled with uncertain values while all other variables have stable values. This enables to test the sensitivity/risk associated with the uncertain variables.


One problem with discounted cash flow-based valuations is that we need to estimate the values to be discounted because we do not know their future values. If only one value is determined during the calculation, it cannot take the estimation errors into account. The other problem with this calculation is the determination of the discount rate, the future development of which can also only be estimated. Both problems can be bridged by calculating not constant but variable values and making this change dependent on chance. The probability variables and the distribution functions associated with them can provide a random effect. Our calculations prove that the uncertainty of the corporate value of HUF 10,300,491 thousand got with constant input values is high. It is highly probable (95%) that the fair value of the company is expected to be between 2,158,819 and 19,334,632 thousand HUF. Because the previous confidence interval has a relatively large range, it can also be concluded that the value of the firm is highly sensitive to input variables, so the risk is high. The latter finding is also supported by Table 7 and Figure 2. Using simulation and the statistical and sensitivity investigations performed on the results allow us to make a more informed decision.


The study was supported by the ROHU 217 (CIFIDE) Interreg V-A Romania-Hungary Program, which was funded by the European Regional Development Fund of the European Union.


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Dr. Veronika Fenyves
University of Debrecen, Faculty of Economics and Business

Dr. Tibor Tarnóczi
University of Debrecen, Faculty of Economics and Business