A Comprehensive Methodology for Investment Project Assessment Based on Monte Carlo Simulation

Abstract

This article presents a methodology for assessing investment projects representing the sequence of steps necessary for a comprehensive assessment and optimization of an investment project. The assessment process starts with the quantitative forecasting of stochastic input factors, with the selection of risk factors and the definition of their uncertainty. That is followed by the design of a mathematical model for calculating the criterion of economic efficiency of investment, its calculation mathematically, and forecasting by Monte Carlo simulation. The simulation output is assessed from the point of view of risk, and in case of an unacceptable result, the possibilities for project optimization are proposed. Finally, the proposed methodology was applied to an investment project model, where individual principles are practically demonstrated.

1. Introduction

Investment decision-making is a process with long-term effects on a company’s economy. The investments in real capital are especially one of the essential activities in ensuring the competitiveness of the company in the long term. The profitability of investment projects is affected by many factors. Therefore, the project evaluation process should incorporate all factors that possibly influence its value [1,2]. Financial investments are one of the ways in which a company can invest its finances. Many research works have dealt with this issue. Reference [3] established a deterministic equivalent income model based on risk cost and risk aversion of investors. The model fully considered subjective and objective factors affecting risk investment and reasonably evaluated risk investment schemes. In [4], authors developed a general framework for analysing corporate investment risk management and financing policies. In [5], financial risks were divided into three types (environment, resource-allocation, stakeholder-cooperation risk), which created a three-dimensional financial risk system. A methodology for streamlining a company’s investment activities to solve financial risk using the Monte Carlo method was presented by [6].

On the other hand, real investments are crucial for manufacturing companies. In a competitive market, manufacturing companies must utilize their available capacity efficiently to obtain high profits. Ref. [7] presented efficient approaches to find an optimal product mix for the company to achieve optimal manufacturing. The most important decisions made in production systems concern the determination of the product mix in such a way as to obtain maximum throughput. Researchers [8] discussed the inefficiency of the traditional theory of constraints algorithm in handling the multiple bottleneck problem. Production plan optimization in connection with investment decision-making was presented by [9]. A framework for conducting the economic analysis and identifying the most suitable methodology before proceeding with the investment was proposed in [10].

In general, investment activity is characterized by high risk and uncertainty. Many investment project failures are due to scale-up issues and uncertain estimations of the capital and production costs. Decision-making strategies and risk assessment modelling are crucial to decrease the risks of losing large investment projects [11]. The literature sources introduce many review papers and case studies on corporate investments and investment risks in various industrial fields. In the oil and gas industry, ref. [12] presented background on the use of risk analysis methods and carried out Monte Carlo simulation to predict a cumulative distribution function for well drilling. In the paper by [13], a probabilistic hesitant fuzzy set was used for handling uncertainty in multiple attribute decision-making. Authors of [14] identified a link between cash flow risk, capital structure decisions, and operating cash flows. Ref. [15] presented a methodology that uses the Monte Carlo Method to estimate economic parameters, which may help a decision, considering the risk in project sustainability. Ref. [16] dealt with introducing a Monte Carlo Simulation approach to risk analysis, based on a life-cycle representation of renewable energy technology investment projects. Authors of [17] presented a computer-based model for cash flow forecasting and for studying the impact of risk factors on cash flow.

In investment decision-making, an important role is played not only by the analysis and assessment of investment risks but also by choosing the optimal investment variant with respect to the risk (in the case of financial investments) and the optimal use of production capacities taking the risk into account (in the real investment case). Optimization in investment decision-making is the subject of other research papers. We selected the following: authors of [18] who proposed an investment strategy for portfolio optimization problems that maximized the expected portfolio value bounded within a targeted range. Research in [19] surveyed the application of Monte Carlo sampling-based methods for stochastic optimization problems. Through economic analysis in [20], a working framework was proposed to generate a complex procedure for optimizing investment decisions in oil field development. The problem evaluated in [21] is related to optimizing an industrial enterprise budget using simulations of the production process. The authors of [22] advanced a methodology for optimizing intertemporal investment decisions through a consensus-oriented process. In addition, ref. [23] proposed a procedure for assessing financial risks and investment development, while the statistical modelling was performed using ANOVA analysis. The paper by [24] developed a mixed integer optimization model that aids designers in selecting building costs, considering the materials and risks involved in the selection process. In the case of investment in a regional distribution network, optimization was solved by [25]. The on-condition maintenance period of the equipment, for the lowest maintenance cost in unit time, was optimized by Monte-Carlo in the paper [26]. In ref. [27] the application of Monte Carlo simulations, using the example of optimising vehicle fleet capacity, was shown. A reliability-based design optimization enhancing sequential optimization and a reliability assessment method was proposed in [28]. Multi-objective optimization is an important problem in various areas. The paper [29] proposed a multi-objective flower pollination algorithm combined with Monte Carlo simulation. An improved SPEA2 algorithm to optimize the multi-objective decision-making of investment was introduced in [30]. The paper [31] presented the optimizing investment decision dilemma in a multiple variation production process.

The aim of this article was to present a complex methodology for assessing corporate investment projects. The methodology includes a sequence of steps necessary for a comprehensive assessment of the economic efficiency and risk of the investment project. The novelty and complexity of solving the investment problem lie in the fact that the assessment is carried out by a software tool using quantitative forecasting methods when defining input variables, Monte Carlo simulation when forecasting the economic efficiency of an investment project, and optimization when creating a production plan. We present the methodology connecting forecasting, simulation, and optimization in investment decision-making to fill the existing knowledge gap, as it is currently neither mentioned in the professional and scientific literature, nor is it applied in conditions of industrial practice.

2. Materials and Methods

2.1. Problem Description

The assessment of an investment project in terms of profitability and risk and the subsequent decision to approve it or to select the optimal investment variant from several options generally involves the solution of the following partial tasks:

• First, it concerns the selection of input variables and the definition of their relationship to overall economic efficiency. Many input variables are stochastic, and the numerical value we consider in the calculation has a direct and significant impact on the forecast of investment profitability. Here, if their development in the past is known, we can use time series forecasting methods to forecast their future evolution; it can be the development of demand, the prices of essential materials and energy inputs, the selling price of products, etc.
• The second step is the choice of a financial indicator for assessing profitability and its calculation. It is essentially a forecast variable that can be calculated deterministically or, if the input variables are stochastic, by Monte Carlo simulations.
• The third step is to assess the result of the simulations in terms of risk and optimize the production program concerning the requirement—maximum profitability at an acceptable level of risk.

To implement this sequence of steps in the investment decision-making process, we propose the following methodological procedure.

2.2. Methodology for Investment Assessment

The proposed methodology provides a structured guidance for a comprehensive assessment of an investment problem. It includes (see Figure 1) quantitative forecasting to obtain input data necessary for the creation of a financial model (to determine the economic efficiency of the investment); Monte Carlo simulations focused on the analysis of investment-related risks; an optimization process aimed at improving the economic efficiency of the investment under limiting conditions; finally, archiving the financial model including results from simulations and optimization of the investment problem. The individual steps of the methodology are defined in more detail in the following text.

1st step: Forecasting.

In many cases, while creating a financial model to evaluate the economic efficiency of the considered investment, it is necessary to predict the main parameters, such as product demand, product prices, product costs, and others, using the appropriate forecasting method. Forecasting methods can be applied via various software programs such as Crystal Ball, @Risk, etc., which include a forecasting module. As a rule, these programs allow forecasting methods such as Non-seasonal Methods, Seasonal Methods, ARIMA, and Multiple Linear Regression. The output of forecasting from time series is a pie chart and forecast values are presented in a table.

For forecasting itself, it is not necessary to define the uncertainties for the input variables in the form of distribution functions. However, if the uncertainties are defined, the program produces, in addition to the graph from the forecasting, a histogram and other related outputs described in the section of simulations. Forecast results obtained by selected methods are assessed based on the forecasting error. The forecasting results can be displayed for any method selected in the context menu at the beginning of forecasting.

2nd step: Monte Carlo simulation.

Creating a financial model. The starting point is to determine the simulated value, which represents the output financial criterion (for example, Net Present Value) and will be the subject of a risk analysis. A deterministic financial model is established based on input variables of a deterministic nature. In this case, the financial criterion can be calculated deterministically, without uncertainty. If at least one of the input variables is variable or stochastic in nature, the financial criterion can only be determined with a certain probability. Simulations are logically possible only in the case of such variables. For this reason, it is necessary first to create a register of input risk variables and examine their nature. In the identification, it is necessary to go into considerable depth, preferably in such a way that the root risk factors are identified.

In the next step, it is necessary to determine the risk criteria. They will be assessed based on their occurrence and severity. That is important if the output quantity depends on many inputs. Risk criteria relate to expressing the probability and impact of risks and how the risks are assessed in terms of severity (or acceptability/unacceptability). In the analysis of risk variables probability of occurrence is mainly assessed. Impacts are either not assessed at this stage or only as a preliminary estimate. The impact analysis itself is performed only on the basis of the simulation results. The analysis result is a register of risks, based on the risk criteria, that have been evaluated as unacceptable or of such a level that it is necessary to deal with them.

Subsequently, it is necessary to create a stochastic financial model for the output quantity. The financial model should go into such depth and detail so as to include all input variables that have been identified as significant risk factors. It is necessary to define probability distributions for them. If historical data are available, on the basis of these data the stochastic nature of the variables can be transformed into a suitable distribution function (via the FIT function), and it is then advisable to use this option. However, there is not always a sufficient amount of data, or they are of such a nature that the distribution function cannot be “fitted”. Then it is necessary to define the distribution function. It is based on history, trends, forecasts, etc. The basis is also a result of the analysis from the previous phase. There are different functions to choose for both discrete and continuous variables. The reliability of the simulation result directly depends on the type and accuracy of the probability distribution estimate. This activity should be performed on the basis of an expert assessment by a qualified employee.

In many cases, the input risk variables are statistically dependent on real conditions and their impact on the output is influenced by this. In such a case, it is necessary to specify the correlation between the inputs, usually in the form of a correlation matrix. In practice, this is, for example, the correlation of the variables product price and sales volume, which represent input variables with a negative correlation (an increase in price reduces demand). This establishes a stochastic financial model.

The simulation process. The simulation process consists of a large number of simulation steps that are repeated until the end of the simulation, determined by the number of repetitions or the specified accuracy of the results. Each simulation step generates random values of risk factors, and the average value of the output quantity—the financial criteria—is subsequently calculated from their probability distribution. Depending on the type of simulation program, other simulation parameters are also defined in this phase, such as the type of simulation technique, the simulation speed, the expected output target value, lower and upper tolerance, or the form of displaying the simulation results. Simulation outputs are presented graphically and tabularly.

Evaluation of simulation outputs. The simulation outputs, the average value of the simulated output quantity, including its probability profile and the impact of risk factors, are assessed against the established goals and risk criteria. If the value of the simulated variable or its reliability is out of tolerance, optimization should be considered.

If the simulation outputs are acceptable with respect to the value and reliability of the simulated variable, the impacts of the risk factors are assessed against the established risk criteria. If the risk factors are assessed as unacceptable, according to their severity, it is necessary to propose measures to reduce the risk of achieving the target.

From the output of the simulation, it is possible to determine the severity of individual risk factors in terms of their impact on the output, how they affect the value of the output (or the probability of achieving the goal), and how they affect the risk of the output (usually represented by the standard deviation).

3rd step: Optimization.

The goal of optimization is to find the optimal solution for the monitored problem using a simulation model. Most of the software uses the optimization tool OptQuest, which allows for the combination of classical linear programming and optimization under uncertainty. Before starting the optimization itself, in addition to the uncertain input variables, it is necessary to define the output quantity and the decision variables. Optimization follows at least one standard simulation cycle, and the reason for optimization is when the output does not meet expectations, requirements, or limitations.

Optimization starts by setting the optimization goal. A goal refers to an output quantity and is most often defined as a requirement to maximize or minimize its average value. However, it is not a rule. The goal can also be focused on other statistical characteristics of the output quantity, such as mode, median, standard deviation, coefficient of variation, skewness, spikiness, etc. This goal can subsequently be specified in more detail as a requirement not to exceed a certain value or the limits of the interval of the selected statistical characteristic of the target variable.

The following is a selection of decision variables, the minimum of which is one variable. These are variables over which it is possible to have control and regulate their value in such a way as to find the best solution with respect to the set goal. Constraints must be defined for each decision variable, i.e., the upper and lower bounds at which a solution is searched for the decision variable, the type of decision variable (continuous or discrete), and the step size of the discrete decision variable that is greater than zero. If some of the input variables represent capacity limitations (e.g., amount of available material resources, available time, space limitations, etc.), they are defined as constraints in the form of the maximum, minimum, or exact value of the variable. Before starting optimization, it is necessary to establish the conditions for its implementation, namely the number of repetitions or the simulation duration.

The outputs from the optimization are in graphic form in the form of a diagram, which shows the process of finding the optimal solution, and in numerical form, which provides data about the optimal solution. An integral part of the outputs is data of the target size, i.e., of the output quantity after optimization in the form of a probability distribution graph, statistical indicators, or other forms of outputs according to the evaluator’s requirements.

4th step: Archiving of the model, simulation and optimization outputs.

Records of individual simulations (records of input and output quantities) should be archived due to their possible repeated use, e.g., for the purpose of deeper analysis, comparison, monitoring of trends, assessment of the effectiveness of the measures taken, etc.

3. Application of the Methodology on a Virtual Investment Project

3.1. Investment Project Description

The proposed methodology is now verified on an investment project. It is an investment in a new production line, which will enable the production of three types of products (A, B, C). The production plan of the line should ensure the following:

• optimal use of the line time,
• maximization of the economic efficiency of the investment using the NPV financial criterion,
• minimization of the investment risk (i.e., achieving an acceptable level of risk while simultaneously maximizing the economic efficiency of the investment).

The production plan of the line is based on market requirements and the development of orders over the last annual period is known (

Table 1. The number of orders for the last year (in a monthly period).

Period (month)	Products (t/month)
	A	B	C
1	72.1	9.5	11.2
2	73.5	9.3	23.8
3	81.9	10.0	18.2
4	72.8	10.5	13.3
5	78.4	9.7	23.5
6	74.9	10.7	18.9
7	74.2	10.3	14.8
8	83.3	10.9	16.0
9	79.1	10.1	13.4
10	77.7	11.3	19.3
11	74.9	10.8	12.9
12	74.2	10.9	24.6
Total	91.0	124.0	210.0
Preliminary production plan	850	130	140

). The limiting factor is the company’s contract orders for 700 t/year of product B, which represent the minimum production volume of this product.

The investment represents production line purchase at an estimated cost of EUR 1,463,190. The other main production input parameters relating to time, material, and production inputs, as well as variable costs and product prices, are presented in the further text of the article.

3.2. Timeseries Forecasting of Demand for Products A, B, C

One of the stochastic input variables is the demand for products. Based on time series data (Table 1), quantitative forecasting of future demand is performed by software. Figure 2, Figure 3 and Figure 4 show the demand forecast in monthly periods for one year. The forecasting has also considered the characteristics of the time series in terms of seasonality and trend. According to the nature of the data, the software selected appropriate forecasting methods. Figure 2, Figure 3 and Figure 4 show the graphical progression of the forecasts for products A, B, and C.

In Figure 2 it can be seen that demand for product A is seasonal, and the most accurate forecast is by the SARIMA (1,0,0)(1,0,0) method. The forecast error, assessed according to MAPE (mean absolute percentage error), is 1.59%. Demand A is stable and forecastable with high accuracy. In the case of product B, it is a demand with an upward trend (Figure 3) and is relatively stable.

Forecast error (forecast by the Double Exponential Smoothing Method) is 4.15%, according to MAPE. Product C is characterized by high dynamic demand with no seasonality or trend (Figure 4). The higher demand variability may be due to the unstable price of the product, replicating input prices, and market behaviour. The resulting forecast values for the next 12 months are presented in

Table 2. Forecast of products A, B, C demand.

Month	Forecast: Products Demand (t/month)
	A	B	C
13	82.71	11.06	16.60
14	78.88	11.18	17.50
15	77.59	11.31	17.50
16	75.03	11.44	17.50
17	74.38	11.56	17.50
18	82.19	11.69	17.50
19	78.67	11.81	17.50
20	77.50	11.94	17.50
21	75.14	12.06	17.50
22	74.55	12.19	17.50
23	81.71	12.31	17.50
24	78.48	12.44	17.50
Total	936.84	140.99	209.10

. The forecast errors according to the MAPE, MAD, and RMSE methods are presented in

Table 3. Accuracy of demand forecasts.

Method	Forecast Accuracy
	A	B	C
MAPE	1.59%	4.15%	19.06%
RMSE	1.97	0.54	3.4
MAD	1.24	0.44	2.8

. The forecast errors should also be taken into account when drawing up the production plan, which must account for deviations in demand.

3.3. Creating a Financial Model and Calculating NPV Deterministically

The economic efficiency of the investment project is assessed using the NPV financial criterion, the value of which is calculated according to the relationship (1). The prediction of annual CF from an operational activity is determined according to relationship (2):

$$NPV={\displaystyle \sum }_{n=1}^{N}C{F}_n·\frac{1}{{\left(1+d_r\right)}^n}-IC,$$

(1)

$$C{F}_n=\left({\displaystyle \sum }_{j=1}^3{S}_{jn}-{\displaystyle \sum }_{j=1}^3{C}_{jn}\right)·\left(1-T_n\right)+\left(D_n·T_n\right)-\mathsf{\Delta }NCW{C}_n,$$

(2)

where CFn is cash flow in year n, n number of years of life of the investment, N life of the investment, d_r discount rate, IC investment costs, S sales, p price, C costs, j product range (A, B, C), D depreciation, T income tax rate, and NCWC non-cash working capital.

Table 4. Input variables and NPV calculation.

Input Variables	Unit	Value
		A	B	C
Planned production	t/year	850	130	140
Average production time	h/t	6.0	7.6	7.5
Price	EUR/t	1300	1600	1800
Variable costs	EUR/t	370	550	650
Nominal time of the line	h/year	8760
Loss times (repairs, cleaning, and others)	h/year	460
Operating time of the line	h/year	7300
Number of shifts	day	3
The length of a shift	h/shift	8
Fixed costs personal (2% annual increase from year 2)	EUR/year	120,000
Fixed costs other (3% from investment costs)	EUR/year	43,896
Income tax	%	21
Discount rate	%	3.5
Investment costs	EUR	1,463,190
NPV	EUR	3,778,179

presents the input variables necessary to calculate the financial criterion NPV deterministically and its calculated value of EUR 3,778,179.

3.4. Simulation of NPV by Monte Carlo Method

Distribution functions express the stochastic nature of the input variables. They can be defined based on data obtained from their registration for a certain period. In this case, there must be a sufficient number of data for the FIT function to be used. In the case of our investment project, the distribution function was chosen by estimation according to the stochastic behavior of the variables. The distribution functions, as they are defined, are presented in

Table 5. Distribution function of input variables.

Variable	Unit	Statistical Characteristics	Distribution Function
Revenue Variables
Price A	EUR/t	Likeliest 1300; Min. 1250; Max. 1330	Triangular
Price B	EUR/t	Likeliest 1600; 5% 1500; 95% 1700	Triangular
Price C	EUR/t	Likeliest 1800; Min. 1720; Max. 1880	BetaPERT
Cost Variables
Variable costs A	EUR/t	Likeliest 370; 5% 350; 95% 380	Triangular
Variable costs B	EUR/t	Likeliest 550; Min. 495; Max. 605	Triangular
Variable costs C	EUR/t	Likeliest 1800; Min. 1720; Max. 1880	BetaPERT
Personal costs	EUR	Likeliest 40,000; Min. 39,000; Max. 43,000	BetaPERT
Other fixed costs	EUR	Mean 123,500; 90% 135,000	Normal
Investment costs	EUR	Likeliest 1,339,690; 5% 1,248,086; 95% 1,431,294	Triangular
Time Variables
Average production time A	h/t	Mean 6; Std. Dev. 0.2	Normal
Average production time B	h/t	Mean 7.6; Std. Dev. 0.76	Normal
Average production time C	h/t	Mean 7.5; Std. Dev. 0.3	Normal

.

By assigning distribution functions to the stochastic variables and running the simulation, 10,000 attempts were made to calculate the NPV. The result is a forecast of the NPV financial indicator, including its statistical profile. The results of the simulation are presented in the form of a histogram (Figure 5) or in tabular form. These are mainly the statistical characteristics of the simulated variable. The simulation found that for the preliminary production plan, the mean NPV is EUR 3,761,324, and the standard deviation (σ) is EUR 123,008. One indicator of the level of risk is precisely the standard deviation, which indicates the reliability of the forecast. In practice, the predicted NPV is within 68% confidence in the interval of the mean NPV ± σ.

One of the risk analysis methods is a sensitivity analysis. The diagram in Figure 6 shows the seven most serious input stochastic transformations, which contribute the most to the overall uncertainty of the NPV forecast. If the column is to the right of the y-axis, it is a revenue item, if it is to the left, it is a cost item.

The price of product A contributes the most to the uncertainty of NPV—almost 40%. It is mainly due to the highest share of product A in the total produced volume and also because of the higher volatility of the price of this product. Then comes the value of the invested capital, i.e., the price of the line and the cost of its acquisition, while next in line are the variable costs of product A. The price and variable costs of products B and C are significantly lower contributors to the overall uncertainty, i.e., even weaker risk factors.

3.5. Optimization of the Production Plan

The preliminary production plan, partially based on market requirements and considering the contracted production volume for product A, is subsequently optimized for the target requirement of maximizing NPV.

In the software settings, it necessary to define the production volume A, B, and C as controlled variables. That means that the program will search for an optimal solution so that the produced amounts of products A, B, and C are such that we maximize the goal and meet all restrictions.

The optimization requirements are set as follows:

• The objective of the optimization is to maximize the mean NPV.
• Restriction for optimization:

  ○

  Constraints in the use of the time of the production line. Minimum use of 7100 h and maximum available line time capacity of 7300 h.

  ○

  The production volume of product A is limited to a minimum of 700 t/year due to the contracts.

  ○

  For the controlled variables (production volumes A, B, C), a minimum volume of 10 t/year was set (because at least a minimum production would be available and the product would not be dropped from the production program), and a maximum volume of 1000 t/year (which was only a theoretical limitation—such a volume would not be sold on the market due to the demand forecast).

In this optimization, the requirement for restrictions in terms of risk is not considered. As a risk, it is possible to understand, e.g., high dispersion of NPV values expressed by the standard deviation or the high coefficient of variation. It can also be the minimum value of the NPV indicator or another statistical characteristic of the output (skewness, kurtosis, etc.).

The output of the optimization is presented in Figure 7. It can be seen that the optimal solution was found very quickly during simulation, and it represents production in the volume of 999 t/year of product A, 10 t/year of product B, and 166 t/year of product C. These are extreme values within the intervals for the production volume limit. Although this production would represent the maximum NPV value (EUR 3,989,753), the standard deviation would rise to EUR 126,749, which is EUR 6,746 more than with the original production program (Figure 7 and Figure 8).

3.6. Optimization of the Production Plan Taking Risk into Account

The optimization result was assessed as an output with an unacceptable risk value. The standard deviation is taken as the risk criterion. The requirement for acceptable risk is set so that the standard deviation is no greater than that of the NPV forecast under the original production plan, i.e., EUR 123,000. The standard deviation of NPV must be less than or equal to EUR 123,000.00 (Figure 9). The other constraints remain unchanged. In the same way, the optimization could be performed by setting a different statistical parameter for the risk criterion.

The optimization adjusted the production plan as follows: product A at 895 t/year, product B at 88 t/year, and product C at 152 t/year. The mean NPV will be EUR 3,821,643, which is EUR 211,574 more than the original plan (Figure 10).

When comparing this production plan with the demand forecasts, we can conclude that the production plan is feasible and the expected demand will cover this supply from the company. The sensitivity analysis has shown that product A again has the highest impact on the NPV uncertainty (due to the high volume), followed by the investment cost and the price of product C (Figure 11).

Comparing this production plan with the forecasted demand, we can evaluate it as feasible, i.e., the expected demand will cover the supply from the company (compare Table 2 and Figure 9). The sensitivity analysis (Figure 11) showed that product A has the highest impact on NPV uncertainty (due to the high production volume), followed by the investment cost of acquiring the line and the price of product C.

4. Conclusions

Evaluating the economic efficiency of investments is the starting point for investment decision-making. Several methods are used to assess investment potential profitability or loss deterministically. When making investment decisions, it is very simplistic to work only with deterministic evaluation methods and not to include the risk in the assessment. This risk is present in investments due to the long-term time horizon and the dynamics of the market. This distortion can seriously affect the correctness of the decision. The use of modern software tools, such as Monte Carlo simulation-based software, enables the efficient processing of large amounts of data of a stochastic nature in investment decision-making. Software support, in the form of Monte Carlo simulations, can be applied to varying extents and at different phases of the project appraisal.

The proposed methodology was intended to cover, as comprehensively as possible, the investment project appraisal process. At the same time, the intention was to make the most effective use of the possibilities of such software tools, which are nowadays commonly available not only on the market but also affordable for business entities.

The methodology covers the data processing stage and its use when forecasting the development of stochastic variables. Simulations are an effective tool for evaluating an investment’s economic efficiency. Last but not least, the simulation approach offers significant benefits in planning the optimal utilization of the investment. At each stage, there is scope for identifying and assessing risks. In the process of forecasting stochastic inputs, the risk is in the form of forecast error, while the selection of the optimal forecasting method is based on the forecast error. When calculating the financial criterion at the forecasting stage, the risk is evaluated according to statistical characteristics such as standard deviation, coefficient of variation, etc. Finally, after optimization, the sensitivity analysis provides an overview of the riskiest variables that contribute the most to the uncertainty of the financial criterion.

The example of the methodology applied in the case of a virtual investment project has provided a practical guide on using simulation tools in investment decision-making. In the individual steps and the context of the solved problem, the principles of preparing the simulation model, setting the simulation parameters, and the possibilities of processing the outputs are explained. The methodology can be a supporting tool for the assessor in investment decision-making. However, the software tool itself is not sufficient for quality decision-making. Even in this process, the expertise and competence of the assessor are crucial.