EFFICIENCY INDICES OF INVESTMENT IN IT PROJECTS WITH EQUAL LIVES

. Theoretical results not always give an unambiguous answer regarding the preference of using the indices of efficiency of investment in IT projects with equal lives. To complement some of such results, the Net Present Value (NPV), Profitability (PI) and Internal Rate of Return (IRR) indices are researched by computer simulation. In this aim, a model of comparative analysis of projects with equal lives is defined and the SIMINV application is made up. Using SIMINV, the percentage of cases when the solutions, obtained according to indices of each of the pairs {NPV, PI}, {NPV, IRR}, {PI, IRR} or of the triplet {NPV, PI, IRR}, differ for seven groups of alternatives of initial data is determined. Based on done calculations, some properties of indices were identified, including: the quantitative features and the character of dependences on initial data; the average percentage of cases with different solutions, which is of approx. 9 % for the pair of indices PI and IRR, and of 34-35 % for the other two pairs of indices specified above. On average, the solutions of comparing the efficiency of projects with equal lives, obtained using the NPV, PI and IRR indices, does not coincide in more than 1/3


Introduction
As is well known, offered advantages impose the computerization of diverse activities implying respective investments. A decision of investment in an IT project is usually made on the basis of efficiency criteria/indices.
Depending on project product and its field of use, the set of applied indices may differ. In a specific project, a small set of indices is usually applied. It is recommended to analyze 7 ± 2 indices [7]. Typically, 1-3 core indices and a few auxiliary indices are used. According to [4], the NPV, IRR, and discounted payback period (DPP) indices are most often recommended to be used. Along with the NPV, PI, IRR, and DPP ones, in [10] the Finite Value of the project and Modified Internal Rate of Return indices are explored; for a concrete project, using all these five indices leads to the same decision -it is appropriate to invest. But, of course, there may be many cases where the results differ. How often such situations occur? Known theoretical results do not give an unambiguous answer to this question. At the same time, to identify them computer simulation can be used.
Monte-Carlo method is largely used to assess financial risks in investment projects. For example, risk assessment for environmental projects using this method is provided in [11]. To select a project for the research, characteristics of 63 projects in the field were analyzed. By computer simulation it was determined the cumulated probability that the project value and execution period will be higher than the initially estimated values. A Monte-Carlo approach to assess financial risk in investment projects is used also in [12]. As a result, a new contribution to the field is made: the proposed risk scale offers five classifications regarding the degree of loss. In [13], a multiple criteria procedure based on stochastic dominance and PROMETEE II methodology is proposed. The first step of this procedure is computer simulation and the uncertainty of processes is taken into account by special stochastic dominance rule. There are many other aspects regarding the selection of investment projects which are explored by computer simulation.
In order to extend the theoretical results regarding the estimation of efficiency of investment in i-projects with equal lives, in this paper the net present value, profitability and internal rate of return indices are researched comparatively by computer simulation mainly to identify the frequency of non-coincidence of the obtained solutions.
of the characterized aspects, show that as basic indices, for projects the revenues from the implementation of which can be estimated with reasonable efforts, it is opportune to use three: NPV, IRR and PI, eventually in conjunction with the equivalent annual value method. The last method allows the appropriate comparison of projects with different lifetimes that is not the case of this paper.
Below, the approach defined in [16] is followed, but with adaptations for projects of equal lives. Let I are investments and CF t are cash flows in year t related to the project. Then NPV, IRR and PI indices are determined as [1,7]: where d is the discount rate. These three indices form a Pareto set: no one of the three can always replace the use of one or two of the other indices, in sense of obtaining the same solutions when comparing projects. At the same time, there are particular cases when the use of all or two of the three indices for comparing two projects leads to the same solution. It is of interest how frequently such cases take place. To this and some other aspects, the answer can be obtained by computer simulation.
Let's compare two i-projects, 1 and 2, with equal lifetimes D 1 = D2 = D the revenues from the implementation of which can be estimated with reasonable efforts. When updating the values of indices, as time reference point will be used the time of projects launch in operation; this time is the same for both projects. It is required to identify, by computer simulation, the percentages of cases when the solutions, obtained using indices of each of the pairs {NPV, PI} (NP) -q NP, {NPV, IRR} (NR) -qNR, {PI, IRR} (PR) -qPR and also of at least one of these three pairs -qNPR, leads to different solutions. Obviously, the percentage of coincidence of all solutions when applying the three indices (NPV, PI and IRR) is equal to 100 -q NPR.
The discount rate d will be considered constant and equal for the two projects, but the values of CF t and also those of I can be different for the two projects. They are also introduced two parameters, g and v. Parameter g value is determined for reasons of ensuring a given value r for the IRR index. So, from Eq.(2) at CF t = CF, t = 1, 2, …, D, one has Thus, g depends on r and D and, at the same time, it establishes the relation between the value I of investment and the average value CF of cash flows CF t, t = 1, 2, …, D. Of course, at CFt ≠ CF, t = 1, 2, …, D the IRR value isn't equal to r, but it is relatively close to it.
In its turn, parameter v characterizes the range of relative variation of CFt with respect to CF. Therefore, the value of v is assigned according to the value CF = gI, namely v = (CF -CFmin)/CF = (CFmax -CF)/CF. So, and CFt ∈ [CFmin; CFmax], t = 1, 2, …, D.
In calculations, for parameters d, r, v, D and I will be used values from the ranges argued and used in [16], namely: d ∈ [0.05; 0.14], r ∈ [0.1; 0.9], v ∈ [0.1; 0.5], D ∈ [1; 10] and I ∈ [100; 1000]. Using these ranges of values, a very large number of alternatives of initial data can be formed. From these, as in [16], seven groups of alternatives, a1-a7, are selected. In all of them, the CF t values are generated randomly at uniform repartition in the respective range as follows (taking into account Eq.(6)-Eq.(8)): In alternative a6, the values of I and D are also generated randomly at uniform repartition in the respective range: For each of the seven alternatives, the respective percentages qNP, qNR, qPR, qNPR and f have to be determined. Here f is the dependence on respective parameter (parameters) of the percentage of generated sets of initial data for which at least one of the following requirements take place: NPV 1 < 0, NPV2 < 0 or |IRR1 -IRR2| > ε (percentage of failure cases).  D max]; I1∈ [Imin; Imax], I2 ∈ [Imin; Imax] and g: Similar, with respective adaptations, are the algorithms for the groups of alternatives a1-a6. To implement the seven algorithms, the computer application SIMINV in C ++ was made up.

Results and Discussion
To achieve the goal defined in Section 1, respective calculations were performed using the computer application SIMINV. Some of the obtained results are systemized in this section. Each set of initial data characterizes two concrete projects, 1 and 2. According to the algorithm and the seven groups of alternatives described in Section 2, a sample of 100000 was generated. So, were generated for the group of alternatives: a1, a6 and a7 by 10 × 10 5 = 1 mil sets of initial data; a2 and a3 by 10 × 10 × 10 5 = 10 mil sets of initial data; a4 and a5 by 10 × 9 × 10 5 = 9 mil sets of initial data.

The number of initial data generation failures
The approach, used to establish and generate initial data sets, doesn't ensure the requirements of NPV 1 > 0 and NPV2 > 0. Also there exists an error when calculating the IRR1 and IRR2 values using the dichotomy method whithin the algorithm described in Section 2. That is why the algorithm counters the total number of cases of failure m f (if takes place at least one of the inequalities: NPV1 < 0, NPV2 < 0 or |IRR1 -IRR2| > ε). This number is used when calculating the values of percentages qNP(⋅), qNR(⋅), qPR(⋅), qNPR(⋅) and f(⋅). If this number is too large, then the calculation errors of obtained percentages are also significant. Therefore it is important to know its value.
In Figure 1, the dependences of f on d for the groups of alternatives of initial data a1, a6 and a7 are shown. The character of these dependencies is largely similar to those for the case of unequal lives described in [16], however the absolute value is higher, but not exceeding 48.2 %. The results of performed calculations show also that for the group of alternatives of initial data: a2  3] %, except the case of group a4 at r = 0.1 when the high limit is of 97.7%. Thus, in case of group a4 at r = 0.1, the sample of initial data is of 100000(100 -97.7)/100 = 2300 alternatives and usually is sufficient. In all other cases, the sample of initial data exceeds 100000(100 -74.3)/100 = 25700 alternatives and is good.

Frequency of cases for which the obtained solutions differ
Computer simulation using SIMINV was performed for all seven groups of alternatives defined in Section 2. Some results are described below.  Figure 2 shows that all mentioned dependences are decreasing on d. At the same time, dependences qNP(d) and qNR(d) practically coincide, and dependence qNPR(d) is close to the first two. Also, one has: The obtained ranges of values for the four dependences are specified in Table 1.   The obtained ranges of values for the four dependences are specified in Table 2.   Table 3.
One can see that all four examined dependencies are increasing on r and those of qNP(r) and q NR(r) practically coinciding with each other (qNP(r) ≈ qNR(r)). It is also increasing on r the discrepancy between qNP(r) ≈ qNR(r) and qNPR(r). Compared to the previous three groups of alternatives, the increase on r of qPR(r) is stronger. At the same time, take place qNR(r) < qNP(r) ≈ q NPE(r) < qNPR(r) and qPR(r) = 0 at {r = 0.1, d = 0.14}. The obtained ranges of values, for the four dependences at d ∈ [0.05; 0.14], are systemized in Table 4.
As in previous three groups of alternatives, there can be a considerable number of cases when the use of any two of the three examined indices leads to different solutions. The largest range of values is that of q NPR(d) equal to 58. 7      According to Figure 6, three of the four dependences, namely qNP(v), qNR(v) and qNPR(v), are decreasing, and the qPR(v) one is slightly increasing on v. At the same time, at v ∈ [0.1;

Journal of Social Sciences
September, 2022, Vol. 5 0.2] take place qNP(v) ≈ qNR(v) ≈ qNPR(v), and at v > 0.1 the discrepancy between qNP(v) ≈ qNR(v) and q NPR(v) is slightly increasing, but is relatively small. The obtained ranges of values for the four dependences on v at d ∈ [0.05; 0.14] are specified in Table 5. Based on data of Table 5 Figure 7.
Similar to the group of alternatives a1 (dependence on d), for group a6 all four dependences are decreasing on d, and the ones for the pairs q NP(d+) and qNR(d+) practically coinciding. At the same time, the discrepancy between percentages qNP(d+) ≈ qNR(d+) and qNPR(d+) is slightly decreasing on d. Also, take place the relations qPR(d+) < qNP(d+) ≈ qNR(d+) < q NPR(d+). The obtained ranges of values for the four dependences at d ∈ [0.05; 0.14] are specified in Table 6.  On average, there are a significant number of cases when the use of investigated pairs of indices leads to different solutions; for example q NPR(d+) ∈ [20.9; 30.4] %. The largest range of values is that of qNPR(d+) equal to 30.4 -20.9 = 9.5%, and the narrowest range is that of qPR(d+) equal to 4.  Table 7.
On average, for the group of alternatives of initial data a7 the number of cases when the use of indices of researched pairs leads to different solutions is less than 35.7 %, and overall, that is when at least two of the three examined indices leads to different solutions is less than 40.7 %. The largest range of values is that of qNPR(d⋅) equal to 40.7 -37.9 = 2.8 % (qPR(d⋅) ∈ [37.9; 40.7]%), and the narrowest range is that of qNP(d⋅) equal to 34.7 -33.4 = 1.3 %. As in previous six groups of alternatives, because of the smallest values of percentage qPR(d) ∈ [8.3; 11.0]%, from the three compared indices, the PI and IRR are the closest to each other. The obtained dependences qNP(d⋅), qNR(d⋅), qPR(d⋅) and qNPE(d⋅)) are shown in Figure 8.  Like the groups of alternatives a1 (dependence on d) and a6 (dependence on d+), for the group a7 all four dependences are decreasing on d, but slightly than the qPR(v) for nominated two. At the same time, this is the only group of the seven examined for which clearly occurs qNP(d⋅) < qNR(d⋅), and the discrepancy between qNP(d⋅) and qNPR(d⋅) as well as the one between qNR(d⋅) and qNPR(d⋅) are relatively large at d ∈ [0.05; 0.14].  (Figure 8) are decreasing or slightly decreasing, except that: a) q PR(I2) is, practically, invariable ( Figure 4); b) q PR(v) is slightly increasing ( Figure 6). Are increasing also dependences: qPR(D) ( Figure 3); qNP(r), qNR(r), qPR(r) and qPR(r) ( Figure  5). At the same time, dependences qNP(D), qNR(D) and qNPR(D) are initially decreasing and after increasing ( Figure 3).

Generalization of the results of computer simulation
By pairs, in groups a1-a6 of alternatives of initial data, the dependences q NP(⋅) and qNR(⋅) practically coincide, and in group a7 they are very close to each other. Relatively close to them is also the dependence q NPR(⋅). With refer to percentages qPR(⋅), usually these are considerable smaller than the qNP(⋅), qNR(⋅) and qNPR(⋅) ones. Thus, from the NPV, PI and IRR indices, the last two are the closest to each other regarding the solutions of comparing the efficiency of projects obtained. A comparative analysis of the range of values for the four percentages can be done based on data of Table 8.
At the same time, there are categories of sets of initial data when examined indices in pairs always lead to the same solution, including the pairs:  {NPV, PI} for group a3 (dependence on I2) at I1 = I2 = 1000, that is obvious;  {PI, IRR} for group a2 (dependence on D) at D = 1, for group a4 (dependence on r) at {r = 0.1, d = 0.14} and for group a5 (dependence on v) at {v = 0.1, d ∈ [0.12, 0.14]}. But there were not identified such categories of sets of initial data when using the NPV and IRR indices or, as a result, all three examined indices (NPV, PI and IRR) together.
It is useful also to mention that, based on group a7 of alternatives of initial data (general group -dependence on d when D 1 = D2, I1, I2, r and v values are generated randomly), the average percentage of cases with different solutions is approx. (in the increasing order): 9.1 % for q PR(⋅), 34.1% for qNP(⋅), 34.9 % for qNR(⋅) and 39.3 % for qNPR(⋅) (see Table 7). Thus, on average, the solutions of comparing the efficiency of projects obtained, when using the NPV, PI and IRR indices, does not coincide in more than 1/3 of cases.

Conclusions
To research comparatively by computer simulation the NPV, PI and IRR indices, used when selecting investment i-projects with equal lives, a model of comparative analysis of projects is defined and the SIMINV application is made up.
Each of the two compared projects is characterized by: discount rate d, duration D, volume of investment I and cash flows CF t, t = 1, 2, …, D. From these characteristics, only the values of d and D are common for both projects. The other characteristics in some cases have fixed value and in other cases are generated randomly, in such a way forming seven groups of alternatives of initial data.
By computer simulation, the percentages of cases when the solutions, obtained using indices of each of the pairs {NPV, PI} -q NP, {NPV, IRR} -qNR, {PI, IRR} -qPR or at least two of the NPV, PI and IRR indices -qNPR, does not coincide is determined. These results complement, to some extent, the known theoretical ones in the domain.
So, for all seven groups of alternatives of initial data are determined the quantitative values and the character of dependencies q NP(⋅), qNR(⋅), qPR(⋅) and qNPR(⋅). There are categories of sets of initial data when examined indices in pairs always lead to the same solution. But there were not identified such categories of sets of initial data when using the NPV and IRR indices or, as a result, all three examined indices (NPV, PI and IRR) together.
The average percentage of cases, for which the obtained solutions does not coincide, is of approx. (in the increasing order): 9.1 % for q PR(⋅), 34.1% for qNP(⋅), 34.9 % for qNR(⋅) and 39.3 % for qNPR(⋅), being considerable. Thus, from the NPV, PI and IRR indices, the last two are the closest to each other regarding the solutions of comparing the efficiency of projects obtained. Also, on average, the solutions of comparing the efficiency of projects, obtained when using the NPV, PI and IRR indices, does not coincide in more than 1/3 of cases.

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