*2.2. Methods*

The methodological procedure was based on collection, analysis, and examination of relevant data concerning the economic efficiency assessment of individual investment projects. The outputs were aimed at answering research questions concerning the interconnectedness of individual analyses of future project uncertainties.

#### 2.2.1. Qualitative Analysis

The significance of project risks (R) was divided into four categories: very high (VH), high (H), medium (M), and low (L). This was determined on the basis of the product of the project risk impact intensity (I) and its occurrence probability (p), with a five-interval scale of both variables, according to the following relation:

$$\mathbf{R} = \mathbf{I} \times \mathbf{p} \tag{2}$$

The probability (value) and the impact intensity had the determined ranges presented in following Tables 3 and 4.


**Table 3.** Scale of risk occurrence probability (p).

Source: Departmental methodology of the Ministry of Transport [7].



Source: Departmental methodology of the Ministry of Transport [7].

Table 5 shows the occurrence frequency of very high, high, and medium risks in the researched sample of projects, according to the risk register (see Table 1). In addition to the risk frequency, the table also shows the dependent variable, which enters the economic CF of the projects as a basis for the calculation of economic efficiency indicators.

It is clear from the overview given in Table 5 that the most significant risks for transport infrastructure projects identified in the pre-investment phase lie in the estimation of future demand for new infrastructure use (R1), design and preparatory work (R2), (R3), delays in obtaining construction permits (R5), land purchase (R7), and excess of project costs (R8).

The R1 risk is related to the demand, which affects the income part of the projects in the operational phase of their life cycle by a possible reduction in their expected socio-economic benefits.

The influence of other risks has a direct impact on investment costs, which thus become a significant variable in the economic assessment.


**Table 5.** Risk frequency according to their significance, including the dependent variable identification.

Source: Feasibility Studies of Investment projects, SFDI, authors' own processing.

#### 2.2.2. Sensitivity Analysis

The outputs of the sensitivity analysis (elasticity coefficients and switching values of economic efficiency indicators) were investigated for individual projects in the following phase of the research in order to determine project resilience to changes in variables potentially affected by risks. The elasticity coefficients were determined both for investment costs and for all relevant socio-economic benefits, which as a total amount, form the income part of the economic CF (following the R1 risk).

It can be seen from the data in Table 6 that variables such as accident rate, externalities, and/or total operating costs generally have low elasticity coefficients, and are not in most cases identified as critical variables. Investment costs and the time savings of infrastructure users already showed that they very often become critical variables (EC > 1). For this reason, occurrences of switching values (i.e., ENPV = 0), which show the influence of these critical variables, were investigated in the following phase of the research. Outputs were divided into the interval of changes up to 10%, up to 30%, and over 30%. It can be clearly seen from Table 7 that the projects showed a relatively high efficiency robustness; about 70% of projects met a limit of efficiency when changing one of these critical variables up to 30%.


**Table 6.** Frequency of elasticity coefficient (EC) values.


**Table 7.** Switching values of project efficiency.

The outputs of the sensitivity analysis and qualitative risk analysis showed that the total investment costs and time savings of transport infrastructure users represented fundamental risk variables that affected the efficiency of the investment projects. For this reason, these independent variables were tested by subsequent quantitative analysis, which was carried out by the Monte Carlo method, using Crystal Ball software [15].

In the case of the quantitative analysis, a relative index BCR was chosen, because it allows comparing the efficiency of projects of different sizes (investment demanding), and it shows the benefit of one invested currency unit. The utilization of the BCR index as one of the criterial indicators for the evaluation of the economic efficiency of public projects is methodically described in references [6,7]. The authors focused on comparing two assumptions of the probability distribution of the investment costs critical variable. The simulations were therefore performed in two variants, in the first variant the beta-PERT probability distribution was chosen for the investment costs, in the second variant a triangular asymmetric probability distribution was used. In order to be able to correctly compare the impact of the use of partial probability distributions of investment costs on the overall project results, an equally normal distribution was used for the second critical variable "time savings of infrastructure users" for both simulation variants.

The parameters of the probability distribution of investment costs in the case of the beta-PERT probability distribution assumption were therefore chosen as follows:


The parameters of the probability distribution of investment costs in the case of the asymmetric triangular probability distribution assumption were, in accordance with the recommendations arising from the background source [9], set with parameters comparable with the beta-PERT probability distribution, i.e., as follows:


Probability distribution for the time savings of infrastructure users was chosen as a normal probability distribution, where the mean value corresponded to the project value of time savings and standard deviation 10%.

#### **3. Results**

The performance of the quantitative analysis can be demonstrated on one of the projects of the tested set. The D10 Prague-Kosmonosy project, with a total investment cost of CZK 9,272,678,497 (€361,367,050), was used as an example. Simulation results when the beta-PERT probability distribution of total investment costs and the normal probability distribution for time savings of the infrastructure users were chosen, are shown in Table 8 and Figure 1. The simulated quantity dependent variable was cost-effectiveness (BCR).


**Table 8.** Results of the simulation of a random cost-effectiveness variable. Investment costs beta-PERT probability distribution.

The resulting probability distribution for the random BCR variable is shown in the following chart.

**Figure 1.** Probability distribution for a random cost benefit ratio (BCR) variable. Investment costs beta-PERT probability distribution.

Simulation results, when an asymmetric triangular probability distribution for total investment costs and a normal probability distribution for time savings of the infrastructure users were chosen, are shown in Table 9 and Figure 2. The simulated quantity dependent variable was cost-effectiveness (BCR).

**Table 9.** Results of the simulation of a random cost-effectiveness variable. Investment costs: asymmetric triangular probability distribution.


The resulting probability distribution for the random BCR variable of the project D10 Prague-Kosmonosy is shown in the following chart.

**Figure 2.** Probability distribution for a random BCR variable. Investment costs: asymmetric triangular probability distribution.

It is evident from the probability distribution shown in Figures 1 and 2 that with a certain probability the random BCR variable will take values below the critical value, and the project will therefore be economically inefficient.

Table 10 shows the outputs of the quantitative analysis of all the researched projects for both variants of the considered probability distribution of the investment costs critical variable. The table for each project presented the following statistical characteristics indicators: BCR: mean, median, standard deviation (*σ*), and certainty level (CL).


The outputs of all projects showed a normal distribution of the BCR indicator. The research in [11] came to the same results, where an experiment which was identified as a pseudo-random number sequence as normally distributed was carried out.

In the interpretation of results it is necessary to respect certain limits connected with the elaborated analysis. As mentioned above, in this paper is presented the case study elaborated using projects being prepared for realization in the Czech Republic. Even if the original methodical steps used in this paper are generally accepted and used, it is necessary to respect certain national specificities in the evaluation of public investment projects. The next limit, which it is necessary to consider, is the definition of probability distributions for the simulation. In the presented analysis it was for the random variable "investment costs", and the triangle and beta-PERT probability distributions were alternatively used, which is in harmony with the present state in the references, and opinions of other experts. However, it is not possible to exclude that the real probability distribution of investment costs of partial projects will be different. However, for the correct evaluation, and the identification of the influence of the selected probability distribution on the results of the evaluated projects it was necessary to uniformly use the chosen probability distributions. In a similar limitation, it is necessary to also note the probability distributions of the random variable "time savings of infrastructure users". In this case it was uniformly selected for both variants of the simulation normal probability distribution, even if the real probability distribution of this variable can be, for partial projects, slightly different.

#### **4. Discussion**

It can be concluded from the above-stated calculations that one of the important settings of the input variables is their assumed probability distribution. From the available literature research and the authors' own expert opinion, it can be assumed that the investment costs variable tends to have a rather asymmetric probability distribution. This was also confirmed by the CBA guide [6], which considers an asymmetric triangular probability distribution in the range −5% to 20%. Makovšek [16], who dealt with a long-term analysis of cost over-runs of road constructions in Slovenia, addressed this issue in detail. Two fundamental conclusions emerged from his analysis: the fact that cost over-runs are systematic (not randomly distributed around zero) and that cost over-runs appear constantly over a time period of several decades and do not decrease (and thus do not show signs of improved forecasting tools and methods). A conclusion can also be drawn from these deductions, that the probability distribution of investment costs tends to be rather asymmetric.

An interesting comparison was published by Emhjellen [17], who dealt with the difference of values when setting different limits of normal distribution and their effect on the resulting values. Kumar [18] noted that the concessionaire aims to bear minimal cost, so maximum probability occurs at lower cost values, and hence it followed a lognormal probability distribution. Jakiukevicius [19,20] worked with normal and triangular distributions, for which he set theoretical parameters which he, based on simulations, converted to log logistics parameters. Kumar [18] adhered to a lognormal distribution of project costs. Gorecki [21] used a triangular distribution. The Czech author Hnilica [22] worked with the beta-PERT distribution, which he considered to be smoother, with possible values more concentrated around the most probable value, and the probability decreases towards the limit values faster than linearly. The authors of this article believe that the beta-PERT distribution best fits an expert estimate of the investment costs behaviour in comparing their values in the ex-ante and ex-post phases. The authors of this article carried out project simulations as mentioned above, assuming both a probability distribution of beta-PERT, and an asymmetric triangular one, and state that the results of the outputs in the expected value of "BCR-mean" ranged up to 7% for all of the projects. The outputs of all projects in both variants of solutions proved the normal distribution of the BCR indicator. The authors of the background research [4] reached the same results, where they stated that an experiment which identifies a pseudo-random number sequence as normally distributed

was carried out. The reading of the frequency distribution of the evaluation indicator provides information of extreme importance, as regards the riskiness of the investment project [23].

#### **5. Conclusions**

It is clear from the above-stated findings that attention must be paid to the setting of statistical characteristics of variables which enter into the calculations of economic efficiency indicators, and on the basis of which it is decided whether or not to accept projects for financing. At present, data on post-audits of major transport infrastructure projects are beginning to be collected and analysed in the Czech Republic, and it is expected that the analyses will make possible, among other things, reaching more precise assumptions.

Although the projects proved efficient, a combination of negative changes to both variables can already bring projects with a certain value of probability into negative results. Based on the analysis of the research sample, it is clear that it cannot be clearly established for projects that a certain value of the BCR ratio predicts 100% stability of the project under the action of several critical variables. It is obvious from the mean value simulations determining the expected BCR value that projects with BCR < 1.1 show, at a certain percentage of probability, and at the critical variable limits specified above, that they shall not be 100% effective. However, the variance of the results obtained was large. Project P10 also showed an interesting result; a relatively high mean BCR ratio showed with a 5% probability that it will not be effective.

The results of the research point to the fact that it is always necessary to perform a quantitative analysis, since the results of the combination of the interaction of critical variables cannot be derived from the partial results of the sensitivity and qualitative analyses. The result will always depend on the absolute values of the critical variables of each unique project.

**Author Contributions:** Conceptualization. J.K. and V.H.; methodology. J.K. and V.H.; validation. J.K. and V.H.; formal analysis. J.K. and V.H.; investigation. J.K. and V.H.; resources. J.K. and V.H.; data curation. J.K. and V.H.; writing—original draft preparation. J.K. and V.H.; writing—review and editing. J.K. and V.H.; visualization. J.K.; supervision. J.K.; project administration. J.K.; funding acquisition. J.K. Both authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by project Brno University of Technology No. FAST-S-20-6383 Selected Economic and Managerial Aspects in Construction Engineering.

**Acknowledgments:** This paper has been worked out under the project of Brno University of Technology no. FAST-S-20-6383 Selected Economic and Managerial Aspects in Construction Engineering.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

