The article proposes an approach to assessing investment risks based on fuzzy modelling of the efficiency of innovative projects. The net present value indicator is considered as an efficiency model. In this indicator, the cash inflow parameter, considering its uncertainty, is specified by fuzzy linguistic estimates. A procedure for approximating linguistic estimates by fuzzy triangular and trapezoid numbers based on Gaussian membership functions is proposed. To simulate the efficiency indicator, the Neumann elimination method is used, in which Gaussian functions are considered as functions of the distribution density of cash inflow expectations. An example is provided to illustrate this approach.
The last decades have vividly demonstrated the scientific and practical significance of innovative
development as a key factor in the economic growth of national economies and all business entities
[
The uncertainty of the forecasted results leads to the risk that the goals set in the project may not be fully or partially achieved. Especially serious consequences can result from erroneous decisions regarding long-term investments. Therefore, when making decisions related to the implementation of innovative projects, risk assessment is one of the main components of investment analysis.
Many studies have been dedicated to the assessment of risks of innovative projects [
In order to increase the reliability of the results of forecast models under conditions of uncertainty, the values of their parameters can be specified by a standard min-max interval. However, such a representation is often unsatisfactory, since it is necessary to specify its boundaries. And these boundaries can be either overestimated or underestimated, which will raise doubts about the accuracy of the results.
Therefore, it is more appropriate to set these parameters fuzzy intervals. Firstly, such assessments
are psychologically easier to give under conditions of uncertainty, and secondly, the fuzzy interval
about the correctness of his assessment [
The use of fuzzy estimates leads to the need for fuzzy risk modeling and their assessment. Recently,
fuzzy modeling has become a promising area of research in the field of analysis and risk assessment
of innovation and investment projects [
This article further develops the ideas and approaches discussed in [
Assessing the efficiency of innovative projects is one of the main elements of investment analysis.
The more large-scale the innovative project and the more significant changes it brings to the business,
the more accurate the calculations of cash flows and the methods of evaluating the efficiency of such
projects must be [
In investment analysis, the net present value (NPV) of a project is most used as a predictive model
of project efficiency. It characterizes the overall economic effect of an innovative project and allows
one to assess the feasibility of investing funds. The net present value it is calculated by discounting
(reducing to the present value, i.e., at the time of investment) the expected cash flows (both income
and expenses) [
The interest rate p (Discount Rate) determines the investor's income from the implementation of an innovative project. Analyzing its possible options allows the investor to adopt an acceptable level of profitability for them as the discount rate.
Initial investments IC (Invest Capital) are associated with preparing production for the release of new or improved products. The goal of preparing new production is to transition the manufacturing process to a higher technical and socio-economic level, ensuring the effective operation of the enterprise. Therefore, to make an investment decision, a justified assessment of initial investments is necessary, which to some extent compensates for such risks.
The paper [
The main parameter on the basis of which the calculation is made is the annual cash flow . The
value of this parameter can be obtained by one of the known methods [
At the same time, it should be noted that the uncertainty of this parameter is due to both economic factors (fluctuations in market conditions, prices, exchange rates, inflation, etc.) that are independent of investors' efforts, and non-economic factors (climatic and natural conditions, political relations, etc.). And since these factors cannot always be precisely determined, it is more rational to set this parameter using fuzzy expert assessments.
As noted, in conditions of uncertainty, in order to increase the reliability of the results of forecast models, it is advisable to define their parameters by fuzzy numbers and intervals. Moreover, it is desirable to represent such numbers and intervals by fuzzy statements that are natural for a person. Therefore, we will specify the expected cash inflow by fuzzy linguistic estimates. A linguistic evaluation is a numerical evaluation that is specified by a statement with the quantifier Moreover, statements (2), as a rule, are used in the presence of minor uncertainty, and statements (3) are used in the presence of significant uncertainty.
Linguistic fuzzy assessments under conditions of uncertainty, of course, increase a person's confidence in their judgments, but at the same time they are subjective. Therefore, in such cases, a group examination is necessary, the reliability of the results of which depends on the consistency of the experts' assessments.
The issues of coordinating group expert assessments were considered in [
= √∑ =1( − )2
= ∑ =1
∑ =1
When estimates are expressed by statements (3), then the coefficient of variation (4) is calculated separately for the left and right boundaries of the intervals.
Let [ 1, 1]
[ , ] be the intervals of fuzzy linguistic estimates given by k experts. Then, for the left boundaries of these intervals, the coefficients (4) are calculated using the formula
= √∑ =1( − ) = ∑ =1 ; 2 and for the right boundaries by the formula Here where where
= √∑ =1( − ) = ∑ =1 .
2 In these formulas and are the variation coefficients, and are the standard deviations, and are the mean values of the interval boundaries [ , ].
= =
, = , (5) (6) (7) (8) good when V<0.2. These criteria are the basis for refining the assessments.
After the linguistic evaluations have been agreed upon, they are represented by fuzzy numbers of the (L-R) type. Statements (2) are represented by triangular numbers (a, ), and statements (3) are represented by trapezoid numbers (a, b, ). Since a and b are given by linguistic estimates, it is which determine the boundaries of the interval of possible values of cash inflow.
These coefficients determine the boundaries of the carriers of fuzzy sets (2) or (3). In practice, the standard and combined (double) Gaussian membership functions are most widely used to represent fuzzy sets.
The standard Gaussian function is used to define fuzzy sets ̃ ≜ "the number is approximately equal to a". According to [24], this function has the form:
̃ ( ) = (− ( − )2), (9) where = − 4 2(0.)5, and b(a) determines the distance between the transition points.
The carrier of the fuzzy set described by function (9) is unlimited, therefore in practice it is limited
to values at which the function is equal to 0.01. To find the fuzziness coefficients, one can either solve
the equation ̃ ( ) = 0.01 , or calculate them using a simpler
approximate method [
= − · (2 ), = + · (2 ), (10) where k is the scaling coefficient that determines the boundaries of the fuzzy set carrier for the
The combined function describes the fuzzy sets ̃ ≜ "the number is approximately in the interval from a to b". It has the form:
< , ̃ ( ) ̃( ) = { ≤ ≤ , 1 . (11)
> , ̃ ( )
Here ̃ ( )and ̃ ( )are the membership functions of the fuzzy set A ≜ "the number is near the number a" and the fuzzy set B ≜ "the number is near the number b", respectively. In this case, the μÃ(x) = 0.01 equation ̃ ( ) = 0.01 or are calculated using the formulas
= − · (2 ), = + · (2 ). (12)
Thus, the determination of fuzziness coefficients is reduced to calculating the distance between the transition points of function (9).
To calculate the distance between transition points, we will use the mechanism proposed in [24]. This mechanism is based on expert data, which for numbers approximately equal to N reflect obtained on the basis of this data (Table 1).
If >99, then the following algorithm is used.
Let the of least significant digit of a number have order q. Also let be the least significant digit of this number, and +1 be it digit whose order is one greater than the order of the digit . Let us define classes of numbers , ∈ {0,1,2}, where = q mod 3. Then: 1. if ∈ 0, then ( ) = ( )⋅ 10 −2, where = ⋅ 10, and ( ) is taken from Table 1. 2. if ∈ 1, then two cases are possible: 3. if ∈ 2, then two cases are also possible: a) if +1 = 0, then ( ) = ( )⋅ 10 −1, where = ; b) if +1 ≠ 0, then ( ) = ( )⋅ 10 −1, where = +1 ⋅ 10 + . a) if +1 = 0, then = ⋅ 10; ( ) = ( )⋅ 10 −2;
b) if +1 ≠ 0, then = +1 ⋅ 10 + ; ( ) = ( )⋅ 10 −1. (10) or (12). where 1, 2 are independent values =
( ).
≤ ≤ A random scenario is understood as a random value of the NPV indicator, which depends on the distribution considered in [25]. However, given that the basis for constructing intervals of possible values of cash inflow is the standard Gaussian membership function, it can therefore be considered as a density function of the normal distribution of cash inflow expectations. And since the normal distribution function is tabulated, therefore, in this case, for modeling random scenarios, it is more rational to use the Neumann elimination method [26].
Let X be a random variable whose density function f(x) on the interval [a,b] is bounded from above. = + ( − ) 1, = 2, account the above, the NPV indicator is modeled as follows.
Let be the number of periods (forecasting horizon). Also, let the range of possible values of the random variable X ≜ "cash inflow" in each year be represented by the following intervals [ 1, 1]
[ , ]. On each interval [ , ], the random variable X is modeled. Either the standard Gaussian function (9) or the combined function (11) is used as the density function ( ) on these intervals.
The values = + ( − ) 1 and = 2 are calculated, where M= 1. Then the condition < ( ) is checked and if it is true, then the value x is a realization of the random variable X and the ∑ =1 − scenarios { (1+ )
}. ( < 0) = , value = , where = . After all the values of are obtained, the indicator = is calculated. As a result of multiple run of the model, we obtain a set of random
the probability of the implementation of a scenario with a negative NPV value as an unprofitable result of the project implementation for the investor. This probability is defined as the ratio of the number of such scenarios to their total number: where is the number of negative values, is the number of experiments conducted.
If the risk P(NPV<0)>0 is accepted by the investor, then, as a rule, the question of expected profit arises. The profit is the average value of the positive scenarios of the set { }. Moreover, the discounted payback period (DPP) of the project occurs in the last year of its implementation. This is the period required to return the investments in the project due to the net cash flow, taking into account the discount rate.
If P(NPV<0)=0, then in this case the discounted payback period may occur before the end of the
project implementation. This period is calculated using the formula [
Let us illustrate the proposed approach with the following example. Let us estimate the risk of investing in some three-year innovation project. Let us assume that the initial investment volume and interest rate are IC=265 and p=5%, respectively. Also, one expert was involved to estimate the expected cash inflow in the first and second years, and three experts were involved to estimate the cash inflow in the third year, since the uncertainty of the cash inflow increases every year. Table 2 also presents estimates of the expected profit from the project.
First, we will check the consistency of the experts' estimates for the 3rd year, assuming their equal competence. For the boundaries of interval estimates, using formulas (5) (8), we will obtain the following correlation coefficients =0.03 and coordinated. Therefore, the average values of the boundaries of these estimates, i.e. 90 and 134, are taken as the boundaries of the cash inflow interval in the 3rd year.
Now we calculate the boundaries of the extended intervals of cash inflows for each year. These ). To calculate these coefficients, we find the values of b(95), b(115), b(90), and b(134).
The values of b(90) and b(95) are found directly from Table 1: b(90)= (0.357=(0.213-0.00067·95)·95=14.2. And the values of b(115) and b(134) are calculated according to the algorithm. Then, using formulas (10) and (12) the fuzziness coefficients of these numbers are calculated: ) - -2.55·142.2 7, =95+2.55·142.2 3.
) - -2.55·6.248 7, =125+2.55·6.248 3. for the number (90,134, , ) - = 90 − 2.55 · 19 2 = 66, = 134 + 2.55 · 4 = 139.
2
Therefore, the range of values of the random variable X ≜ by the following intervals [77, 113], [107, 123], [66, 139].
After this, a random variable X is simulated on these intervals. The standard Gaussian function (9) is used as the density function f(x) on the intervals of the 1st and 2nd years, and the combined function (11) is used on the interval of the 3rd year: on the 1st interval: ̃ ( ) = on the 2nd interval: ̃ ( ) = (− ( − 95)2), where = − 41 4.202.5 =0.014; (− ( − 115)2), where = − 46 .4802.5 = =0.067; < 90, ̃ ( ) on the 3rd interval: ̃( ) = {90 ≤ ≤ 134, 1 , where
> 134, ̃ ( ) ̃ ( ) = (− ( − 90)2), = − 4 1902.5 =0.008; ̃ ( ) = ( − ( − 134)2), = − 4 0.5 =0.17.
42 Also for these intervals, according to (13), we obtain the following modeling parameters: for the 1st: = 1, = 77 + 36 1, = 2, for the 2nd: = 1, = 107 + 16 1, = 2, for the 3rd: = 1, = 66 + 73 1, = 2.
Then, at each interval, a random variable X is being played out drawn, the values = , (1+ ) where = , are calculated and then the = ∑3=1 − indicator is calculated. 1000 runs of the model were made. The result of the statistical analysis of the obtained data is given in Table 3.
Indicator Investment size
Interest rate Minimum value of Maximum value of
Expected value of Cash inflow in first year 1 Cash inflow in second year 2 Cash inflow in third year 3
Num. cases when < 0
Investment risk
The following scale can be used for verbal expression the risk assessment of a project [
Thus, according to this table, the investment risk of the project is average. Let the received risk be accepted by the investor. Since the risk is greater than 0, the discounted payback period will occur in the third year of the project implementation, and the expected profit will be approximately 24 conventional units.
The approach to assessing investment risks based on fuzzy modeling of the effectiveness of innovative projects is considered. The net present value indicator serves as an efficiency model. In this indicator, the cash inflow parameter, taking into account its uncertainty, is specified by fuzzy linguistic estimates.
A procedure for approximating linguistic estimates by fuzzy triangular and trapezoid numbers based on Gaussian membership functions is proposed. An algorithm for calculating the distance between the transition points of these functions is considered, the use of which allows the approximation of linguistic estimates to be carried out automatically. Based on this algorithm, formulas for the approximate calculation of fuzziness coefficients of triangular and trapezoidal numbers with an error acceptable in practice are proposed.
To simulate the efficiency indicator, the Neumann elimination method is proposed, in which the Gaussian functions act as functions of the distribution density of cash inflow expectations. An example is given to illustrate this approach. This example demonstrated the practical feasibility of the approach, its simplicity and universality.
In general, the proposed approach, without claiming to be complete, can be used both as a basis for developing an appropriate methodological apparatus for assessing the risk of various innovative projects, and in a broader sense - for modeling random variables in various fields.
[4] N. Kaverina. Theoretical and methodical approaches to the analysis and evaluation of the risks -79. DOI: 10.15587/2313[24] A. Borisov, O. Krumberg, I. Fedorov, Decision making based on fuzzy models: examples of use.
Knowledge, 184 p., Riga (1994). [25] Y. Samokhvalov, (2020). Construction of the Job Duration Distribution in Network Models for a Set of Fuzzy Expert Estimates. In: Lytvynenko, V., Babichev, S., Wójcik, W., Vynokurova, O., Vyshemyrskaya, S., Radetskaya, S. (eds) Lecture Notes in Computational Intelligence and Decision Making. ISDMCI 2019. Advances in Intelligent Systems and Computing, vol 1020.
Springer, Cham. https://doi.org/10.1007/978-3-030-26474-1_8 [26] J. Von Neumann, Various Techniques Used in Connection with Random Digits, Applied Mathematics Series 12, National Bureau of Standards, Washington, DC, 1951, pp. 36-38.