=Paper= {{Paper |id=Vol-3101/Paper20 |storemode=property |title=Mathematical model and method of enterprise financial risk assessment based on threshold elements |pdfUrl=https://ceur-ws.org/Vol-3101/Paper20.pdf |volume=Vol-3101 |authors=Orken Mamyrbayev,Anzhelika Azarova,Liliia Nikiforova,Aliya Kalizhanova,Anatolii Shyian,Olga Ruzakova,Nataliia Savina |dblpUrl=https://dblp.org/rec/conf/citrisk/MamyrbayevANKSR21 }} ==Mathematical model and method of enterprise financial risk assessment based on threshold elements== https://ceur-ws.org/Vol-3101/Paper20.pdf

Mathematical Model and Method of Enterprise Financial
Risk Assessment Based on Threshold Elements
Orken Mamyrbayev1, Anzhelika Azarova2, Liliia Nikiforova2, Aliya Kalizhanova3, Anatolii
Shyian2, Olga Ruzakova4 and Natalia Savina5
1
Institute of Information and Computational Technologies CS MES RK, 28 Shevchenko Str., Almaty, 050010, Kazakhstan
2Vinnytsia National Technical University, 95 Khmelnitskoye shosse St., Vinnytsia, 21021, Ukraine
3
Institute of Information and Computational Technologies CS MES RK, University of Power Engineering and
Telecommunications, 126/1 Baytyrsynuly Str., Almaty, 050013, Kazakhstan
4
Vinnytsia National Agrarian University, 3 Sonyachna St., Vinnytsia, 21100, Ukraine
5
National University of Water and Environmental Engineering, Soborna Str. 11, Rivne, 33028, Ukraine

Abstract
To assess financial risk, it is necessary to restore a large set of many initial parameters, which aren’t only
determined by the criteria of authority, efficiency, and minimum capabilities, but also by using special
information and their display functions. The authors propose a model of financial risk assessment of the
enterprise, developed on the basis of mathematical apparatus of certain elements of decomposition
functions to coordinate parameters and functions of determining the level of risk of a potential investor,
which is a more accurate, unambiguous, and categorical during the assessment of financial risk by using
rigidly defined threshold elements of input parameters loaded into response classes. The solution to
the complex problem of accurate financial risk assessment becomes possible by obtaining several
quantitative estimates of all separate classes of input data. The accuracy of financial risk assessment for
the computer model developed in the article has been experimentally tested in small enterprises and is
the highest in comparison to the normative methods of financial risk assessment. The special proposal
of the proposed model consists of the restoration of many fundamental initial parameters aimed at
assessing the financial risk, which is determined by the relevant capabilities of the enterprise and expert
information. The model also uses the function of conversion of initial parameters for the assessment
and a set of functional decompositions for compiling parameters, to influence the identification of
financial risk which makes it universal for the use of enterprises operating in different sectors of the
economy. Also, the developed computer model can be used both offline and when using a cloud
environment.

Keywords1

CITRisk’2021: 2nd International Workshop on Computational & Information Technologies for Risk-Informed Systems, September
16–17, 2021, Kherson, Ukraine
EMAIL: morkenj@mail.ru (O.Mamyrbayev); azarova.angelika@gmail.com (A.Azarova); nikiforovalilia@gmail.com (L.Nikiforova);
kalizhanova_aliya@mail.ru (A.Kalizhanova); anatoliy.a.shiyan@gmail.com (A.Shyian); ruzakova@vsau.vin.ua (O.Ruzakova);
n.b.savina@nuwm.edu.ua (N.Savina)
ORCID: 0000-0001-8318-3794 (O.Mamyrbayev); 0000-0003-3340-5701 (A.Azarova); 0000-0002-7034-607X (L.Nikiforova); 0000-
0002-5979-9756 (A.Kalizhanova); 0000-0002-5418-1498 (A.Shyian); 0000-0002-4796-9703 (O.Ruzakova); 0000-0003-3678-9229
(N.Savina)
© 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
Risk, information technology, risk assessment, financial risk, computer model, financial condition, risk
function, threshold elements, 2-factor model of Gorvatov, multifactor regression analysis, experts,
algorithm, assessment, enterprise.

1. Introduction
Problems of various risks of the financial struggle are actual in any country, especially in the period
of its active digitalization and widespread introduction of information technology [1]. This is due,
in particular, to the fact that there is a general digitalization of all spheres of human activity and
society as a whole, which requires all the major integration of manufacturing enterprises with
modern information systems and technologies. Information technology and computer models
based on science-based methods are becoming increasingly important, which allow the assessment
of the risk level not only quickly and accurately, but also autonomously, especially when it comes
to the idea of assessing the financial risks of the enterprise. Financial risk not only affects the
development of the company itself, but also the economic development of society, so the
assessment of the financial risk of the company is partial.
The financial risks of the enterprise are determined by exogenous and endogenous factors.
Exogenous factors are usually well controlled with the help of modern information technology. In
contrast, endogenous factors are poorly taken into account in modern information systems to
support decision-making. Therefore, the development of computer models to identify the financial
risk of operating enterprises is an important direction in the development of modern information
technologies.
The presence of computer models that can be used to monitor the financial condition of both
the enterprise itself and its communicators in the future allows the creation of information systems
to improve the efficiency of the economy as a whole. These information systems can also be
localized in the cloud, which can significantly increase the number of their users.
An important circumstance that comes along this path is that computer models must be
accessible enough to cover a wide variety of business areas.
Problems of economic development and activity of enterprises in the crisis under the influence
of coronavirus in the world and Ukraine are exacerbated not only by uncertainty (demand, prices)
and fierce competition but also by the lack of adequate financial risk assessment model taking into
account current requirements. To survive in such conditions, business leaders need to make bold
and unconventional decisions about innovative investments to increase the competitiveness of the
company and improve its financial condition, which significantly increases not only the investment
but also the financial risk of the company. And, if the first type of risk and its assessment were
considered in previous works by individual authors [2, 3], the assessment of the financial risk of
the enterprise, which is to blame for the direct impact and investment risk, was not considered at
a sufficient level.
Traditionally, the decision-making problems involve creating a preference order to classify
alternatives to choose the best one. For these decision-making problems where alternatives are
compared to a single criterion, this can be easily done. However, in the most realistic cases, several
different criteria should be evaluated. That is why in this paper a model for calculating the financial
risk of the enterprise on the basis of the threshold elements of the tuple of selected financial criteria
is developed.
2. Related work
We will analyze related work and consider the main models used to assess the financial risk of
parameters in the works of both domestic and foreign scientists.
In paper [4] there is a proposed approach that integrates the decision-relevant information,
which is subject to uncertainty, to multi-criteria decision-making. An approach must enable
decision-makers to explore the uncertainty and risk involved in their decisions. It arose from the
theory of risk-based decision-making and the generalization of particular risk-based solutions in
different domains. The authors of this work consider the roar as a whole, without its financial
specification.
As a rule, the financial risk of the enterprise is considered as the risk of bankruptcy of the
enterprise. For example, the paper [5] presents the results of intelligent information system
development for enterprise bankruptcy risk estimation on the basis of fuzzy logic and neural
network technologies synthesis. The developed information system allows to make the current
estimation of the risk of bankruptcy of the enterprise and gives the chance to trace how it impacts
to separate indicators’ changes. The paper [6] develops a genetic bankrupt ratio analysis tool using
an agenetic bankrupt ratio analysis tool using a genetic algorithm to identify influencing ratios
from different bankruptcy models and their influences in a quantitative form.
The paper [7] proposes a novel financial risk assessment model for companies based on
heterogeneous multiple-criteria decision-making (MCDM) and historical data. Subjective and
objective indexes are comprehensively taken into consideration in the financial risk assessment
index system of the model, which combines fuzzy theory with quantitative data analysis.
Moreover, the assessment information obtained from historical financial information of the
company, credit rating agency, and decision-makers, including crisp numbers, triangular fuzzy
numbers, and neutrosophic numbers. However, the authors do not take into account the threshold
values of the selected criteria.
Thе article [8] discusses the bankruptcy prediction model using random forest based on the
most influential ratios needed to predict bankruptcy. These coefficients are selected based on a
genetic algorithm that filters out the most important of the various existing bankruptcy models.
The authors [9] developed the generalized computational models and methods for automated
assessments of the risks due to uncertainties of influencing factors in the complicated deterministic
stationary systems. The principal idea of these models and a method is based on computational
solving the finite set of boundary value problems modeling the considered systems to represent
the deterministic properties of researched possible risks in the general case.
The article [10] proves that the traditional practice of using a singular performance metric for
classifier evaluation is not sufficient for imbalanced classification credit and bankruptcy risk. This
paper proposes a multi-criteria decision-making (MCDM)-based approach to evaluate imbalanced
classifiers in credit and bankruptcy risk prediction by considering multiple performance metrics
simultaneously. Note that the estimation of crisis symptoms of the enterprise and diagnosing the
possibility of a financial crisis is carried out long before the detection of its obvious signs [11].
The article [12] considered the assessment of the risk of bankruptcy of an enterprise according
to indicators of financial and economic activity using Bayesian networks. The purpose of this study
is to develop a model for predicting the financial problems of enterprises. A Bayesian network has
been developed for analyzing and predicting the financial condition of industrial enterprises.
Financial statements were used to analyze 3000 industrial enterprises in Ukraine. Five integral
financial indicators were identified for building Bayesian networks (maneuvering coefficient,
debt-to-equity ratio, the coefficient of autonomy, current liquidity ratio, financial stability ratio).
The results obtained in the study show the forecast of the quality and the practical application
possibility of the developed Bayesian network in the decision support system for an intelligent
assessment of forecasting the bankruptcy probability of an enterprise.
The key point in assessing the financial risk of any organization is to determine the level of the
financial condition of the enterprise. The key point in assessing the financial risk of any
organization is to determine the level of the financial condition of the enterprise. For this, such
basic mathematical models as Horvathov's 2-factor model [13] and the Beaver model [14] are
used. But both of them have several disadvantages. For example, Horvathova's model was
developed for the US economy. In Ukraine there are high inflation rates, other cycles of
macroeconomics and microeconomics, levels of capital intensity, energy intensity, and labor
intensity of production, other taxation does not allow for a comprehensive estimation of the
financial condition of enterprises and therefore significant deviations in estimates in the model.
Beaver’s coefficient [2, 13, 14] also has a number of disadvantages. First, the normative values of
financial indicators do not take into account the industry specifics of enterprises. Secondly, the
efficiency of capital use in enterprises (turnover, profitability) is not taken into account. Third, the
calculation of the Beaver coefficient is carried out in statics, not taking into account the transient
external and internal environment of enterprise valuation.
In paper [15] an improved financial credit risk assessment approach is presented. Based on the
credit data from China Banking Regulatory Commission (CBRC), a multi-dimensional and multi-
level credit risk indicator system is constructed. In particular, we present an improved sequential
minimal optimization (SMO) learning algorithm, named four-variable SMO (FV-SMO), for the
credit risk classification model. At each iteration, it jointly selects four variables into the working
set and a theorem is proposed to guarantee the analytical solution of the sub-problem. The
assessment is made on the China credit dataset and two benchmark credit datasets from the UCI
database and CD-ROM database. Also, scientists from China are considering the assessment of
financial risks based on a factor analysis model [17]. Based on factor analysis, the presented paper
[17] establishes a financial risk assessment model at the company level, and determines the
influence degree of the solvency, operation ability, profitability, development ability, and the
ability to obtain cash flow, and collects a large amount of relevant information and data, calculates
the index weight. Finally, based on the analysis of the actual situation of each real estate company
in China, the SPSS software is used for empirical analysis and divides the risk levels of these 120
real estate companies. Unfortunately, these models cannot be used for Ukraine as they take into
account the specifics of China.
Modeling the interaction of risk factors using Copula functions was discussed in the article [17,
18]. The procedure is proposed for analyzing the risk factor interaction in financial systems. The
procedure is based upon the results of eigenvalues distribution analysis and distances between the
eigenvalues for empirical and theoretical dependency matrices. Some results of the theory of
random matrices are used to interpret the results achieved in the process of empirical studies for
the correlation matrices of a different kind. The results of computational experiments show that
for small eigenvalues the results of theoretical analysis for random matrices are similar to the
empirical matrices. This result provides a possibility for determining correctly the number of
principal factors to construct mathematical models necessary for practical applications.
Thus, an important part of the wider use of information technologies for the development of
the country's economy is the creation of information systems for managing the financial condition
of enterprises. Moreover, such systems should be flexible enough to, on the one hand, allow, from
a single point of view, to identify the financial risks of a large number of enterprises operating in
different sectors of the economy [19, 20]. However, on the other hand, these information tools
should allow taking into account the specifics of the enterprise.
These conditions are conflicting. This contradiction is resolved by creating a bank of computer
models. Users will take into account the conclusions of these models and will be able to select
those that will give the most adequate results.
3. Formal problem statement
The aim of the work is to develop a computer model as an element of information technology for
calculating financial risk, which is common enough for use by enterprises operating in various
sectors of the economy. To assess financial risk, it is necessary to take into account a large set of
many initial parameters, which are determined not only by the criteria of completeness, efficiency,
and minimality but also by using expert information, as well as their display function. The authors
propose a model for assessing the financial risk of an enterprise, developed on the basis of the
mathematical apparatus of threshold elements, taking into account the decomposition function for
folding the parameters and the function of determining the risk level of a potential investor, which
is more accurate, unambiguous and categorical in assessing the level of risk through the use of
rigidly defined threshold elements of the input parameters grouped into four classes. The solution
to the complex problem of accurately assessing financial risk becomes possible when obtaining a
quantitative assessment of all four classes of input data. The computer model proposed in the
article is quite general for use by enterprises operating in various sectors of the economy and can
be used both offline and in the cloud.

4. Building the structural and mathematical models of financial risk
assessment
The process of financial risk (FR) calculating belongs to the category of complex problems [22]
due to the need to take into account a powerful set of input parameters X and output parameters R,
and their transformation functions F: X → R. Promising is the way to decompose a complex
function into a sequence simpler so that the functions of lower levels unambiguously identify
certain parameters in the functions of higher levels [23].
The process of assessing FR is the consistent implementation of a number of functions. The
task of deciding on the evaluation of FR is to choose an adequate solution R from the set of
solutions Zj ( j = 1, J ). The choice is made with the help of FR estimates based on the set X of the
estimation parameters хі (і = 1, n , n∈ N).
To assess the FR, it is necessary to determine the criteria for assigning the company to a
particular class of financial condition. In addition, the specificity of the construction of such a
system is the need to take into account the set of initial input parameters, which are the basis for
calculating the evaluation parameters.
The peculiarity of the model is that it takes into account the set of initial input parameters K =
(kc) ( c = 1, C ), which is determined by the relevant reporting of the enterprise and expert
information; the set of evaluation parameters X = (xi) ( i = 1, n ) financial condition; function of
conversion of initial parameters into estimating F1: К → X; the set of decomposition functions D
= (Y, … S, Р) of the collapse of the parameters by which the identification of the financial condition
of the enterprise; the function of determining the level of risk F2: Zj → Rj of the potential investor,
which corresponds to the Zj level of FR; the set of output parameters R = (Rj) (j = 1, J ):
R = {K, F1, X, D, Z, F2}, (1)
To obtain the final result on the assessment of FR and the corresponding level of risk in
decision-making, based on the initial input evaluation parameters K, it is necessary to implement
the above functions in the following sequence:
F D F
K →
1
X → Z j →
2
R j. (2)

To determine the final assessment of the financial condition of the enterprise Zj and the
corresponding level of risk Rj of the potential investor, it is proposed to consider a combination of
complex functions - parameters Р1 ... Рq - financial condition, assessing groups of indicators of
the highest level of the hierarchy:
Z j = F ( P1 , Pq ) (3)

In turn, the input data for calculating the complex parameters P1 and Pq are a set of parameters
that evaluate certain groups of indicators ((S1 ... Sр), starting with financial stability (S1) and ending
with profitability (Sр), i.e.:
P1 = F ( S1 ...St ) , Pq = F ( S e ...S p ) , (4)

where t, e, p ∈ M, and M is the set of functionals of generalizing parameters of the P-th level.
Taking into account the influence of a constantly changing set of factors of the external and
internal environment means that the complex parameters of the penultimate level (Y1 ... Ym) are
functions of the corresponding evaluation parameters хі of the financial condition, in particular:
Y1 = f ( x1...xl ) ... Ym = f ( xk ...xn ) , (5)

where l, k, n ∈ N.
The estimation parameters xi are determined on the basis of the set of initial input
parameters K and the transformation function F1 : X = F1(K), K = (kc), c = 1, C ; X = (xi), i = 1, n .
Based on the composite functions (1) - (5), it is necessary to form a set X of corresponding
parameters for estimating FR. This set is formed using the set of initial input K parameters k1...ke),
where е ∈ N. The definition of this set K is carried out using the appropriate reporting of the
enterprise, including balance sheet, statement of financial performance, etc., as well as expert
estimates various issues.
The general structural model of the FR evaluation process (Fig. 1) consists from A levels, its
decomposition occurs as follows.
At the first level, the set K of the initial input parameters is formed. The second level involves
the formation of the set X of the evaluation parameters of the financial condition of the enterprise
on the basis of the set K of the initial input parameters.
At the third and subsequent intermediate levels, in particular S and P, there is a formation of
complex generalized indicators of evaluation of FR Y1...Ym ; S1...Sp; P 1...P q .At the highest level A,
the solution Zj, j = 1, J , is identified, which determines the FR from the set of possible states.
Figure 1: General structural model of the FR evaluation process

This model structure allows you to both add and remove indicators, taking into account the
industry, the state of economic development and the ever-changing impact of both external and
internal environments of the entity in a crisis. In addition, the hierarchy of the model proposed by
the authors simplifies the evaluation process using a modern mathematical apparatus.
To determine the functions (1) - (5) it is necessary to form sets of input and output parameters.
They must cover a wide range of influencing parameters, as well as meet the conditions of
completeness, effectiveness and minimality.
The set of evaluation parameters X provides the formation of such complex parameters as
financial stability (Y1), liquidity and solvency (Y2), business activity (Y3) and profitability (Y4).
Financial stability, which is a function Y1 = f ( x1...x5 ) ,, is determined by the parameters: x1 -
coefficient of independence; x2 - financial stability ratio; x3 - coefficient of financial stability; x4 –
coefficient of maneuverability of own funds; x5 - the ratio of own working capital.
Liquidity and solvency are a function Y2 = f ( x6 ...x10 ) . It is identified by the following
parameters: x6 - monetary solvency ratio; x7 - coefficient of estimated solvency; x8 - critical
liquidity ratio; x9 - the ratio of receivables and payables; x10 - asset mobility ratio.
Business activity, which is a function Y3 = f ( x11 ...x16 ) , is determined by a set of parameters: x11
- asset turnover ratio; x12 - receivables turnover ratio; x13 - turnover ratio of accounts payable; x14 -
turnover ratio of inventories; x15 - turnover ratio of fixed assets, x16 - turnover ratio of equity.
Profitability is a function Y4 = f ( x17 ...x 20 ) . It is determined on the basis of the following
parameters: x17 - cost-effectiveness; x18 - return on sales, x19 - return on all assets, x20 - return on
equity.
These evaluation parameters are calculated on the basis of the relevant reports of the enterprise
in accordance with the requirements of national legislation.
Based on these input parameters, a set of X quantitative parameters of the firm is formed
(table 1).

Table 1
The set of evaluation parameters of FR
The name of the indicator Formula for calculation
Financial stability
Coefficient of independence x1 k1/k2
Coefficient of financial stability x2 k1/k3
Coefficient of financial firmness x3 (k1+ k7)/k2
Coefficient of maneuverability of own means x4 (k1- k14)/k1
Ratio of own working capital x5 (k1- k14)/k3
Liquidity and solvency
Monetary solvency ratio x6 k4/k5
Estimated solvency ratio x7 k6/k5
Critical liquidity ratio x8 k8/k5
Ratio of receivables and payables x9 k10 /( k5+ k7)
Asset mobility ratio x10 k8/k14
Business activity
Asset turnover ratio x11 k9/k2
Receivables turnover ratio x12 k9/k10
Accounts payable turnover ratio x13 k9/k11
Inventory turnover ratio x14 k12/k13
Fixed assets turnover ratio x15 k9/k14
Equity turnover ratio x16 k9/k1
Profitability
Cost-effectiveness x17 k15/k12
Profitability of sales x18 k16/k9
Return on all assets x19 k16/k2
Return on equity x20 k16/k1
Thus, the set of estimating parameters xi і = 1,20 is determined, namely x1...x20), the values of
which are calculated on the basis of the input initial parameters k1... k16).
Define the set of initial parameters Z = (Z1,…,Zj), which determine the corresponding level of
financial risk of the enterprise Zj (j = 1, 3 ), as follows: Z1 - low level of FR; Z2 - average level of
FR; Z3 - high level of FR.
Given the above justification of the sets of parameters, the authors present the following refined
structural model of the FR evaluation process (Fig. 2).

Figure 2: Refined structural model of the process of evaluation of FR enterprise Rj
5. Method for assessing financial risk based on threshold elements
The specificity of the assessment of the level of risk for the financial condition of the enterprise
necessitates the need to take into account the evaluative financial parameters with varying degrees
of influence on the resulting decision. For this purpose, the mathematical apparatus of the threshold
elements will be optimal due to the presence of various influential parameters in the absence of
the possibility of their complete search in decision-making. This allows you to increase the speed
of information processing when building a risk assessment system for the financial condition of
the enterprise.
At the first stage, we determine the interval [xi min ; xi max ] j of change of estimation parameters xi,
i=1,20 for each level of financial condition Zj ( j = 1, J , J = 3).To assess the FR we will use the
following possible financial conditions: low level - j = 1, average level - j = 2, high level - j = 3.
Each financial condition is characterized by evaluative parameters xi, i= 1,20 .
With the help of expert assessments, the authors determined the ranges of change [xi min ... xi max ]
of these parameters in accordance with the possible FR Zj are given in table 2.

Table 2
Ranges of change of parameters x1...x10, x17...x20 for 3 levels of financial risk
Parameter Financial risk
low level Average level High level
[xi min ... xi max ] [xi min ... xi max ] [xi min ... xi max ]
x1 [0,4;+∞) [0,2–0,4) (-∞;0,2)
x2 [0,7;+∞) [0,25–0,7) (-∞;0,25)
x3 [0,4;+∞) [0,2–0,4) (-∞;0,2)
x4 [0,4;+∞) [0,2–0,4) (-∞;0,2)
x5 [0,1;+∞) [0,05–0,1) (-∞;0,05)
x6 [0,15;+∞) [0,05–0,15) (-∞;0.05)
x7 [1,5;+∞) [0,5–1,0) (-∞;0,5)
x8 [0,75;+∞) [0,25–0,75) (-∞;0,25)
x9 [0,6;+∞) [0,2–0,6) (-∞;0,2 )
x10 [0,4;+∞) [0,2–0,5) (-∞;0,2)
x17 [0,13,;+∞) [0,04–0,13) (-∞;0,04)
x18 [0,075;+∞) [0,025–0,075) (-∞;0,025)
x19 [0,1;+∞) [0,02–0,1) (-∞;0,02)
x20 [0,25;+∞) [0,8–0,25) (-∞;0,8)

For data processing, a mathematical apparatus based on threshold elements is used, which works
on Boolean algebra. To do this, the evaluation parameters must be presented in two-digit form, so
we convert the vector X to the vector G = (gi), where:
0, if x i ∉ [ x imin ; x imax ] j ;
gi =  (6)
1, if x i ∈ [ x imin ; x imax ] j .
In the second stage, to rank the evaluation parameters of хі using the knowledge of experts
whose competence was defined above.
I – g1, g2, g3, g5, g8 (high level),
II – g4, g7, g10, g18, g19, g20 (average level),
III – g6, g9, g11, g13, g16, g17 (low level),
IV – g12, g14, g15 (very low level),
Therefore, each of the parameters of I, II or III levels is equal to the set of parameters of the
lower level. For example, g6 is compensated by the parameters: {g12, g14, g15}; g4 – { g6, g9, g11,
g13, g16, g12, g14, g15 }; and g1 – { g4, g7, g10, g18, g19, g6, g9, g11, g13, g16, g12, g14, g15}.
In the third stage, determine the relationship between the weights wi, i = 1,20 by the following
rule: let gl,r have the weight wl,r, then for variables of the same rank g1,r, g2,r, ..., gp,r, the weights
will be the same:
w1,r = w2,r = ... = w p,r (7)

and the weights of the variables of the r-th rank are greater than the weights of the variables
(r+i)-th, i = 1, R − r ,

wl ,r > w j ,r +1 l = 1, p r , j = 1, p r +1 , ∀l , j (8)

Therefore, according to ranking (5) we define the ratio between the weights as follows:
w12 = w14 = w15 < w6 = w9 = w11 = w13 = w16 = w17 < w4 = w7 = w10 = w18 = = w19 = w20 < w1
= w2 = w3 = w5 = w8.
At the fourth stage we will make system of inequalities for weights of variables by rules:
1) Each rank forms an inequality. Let the set of evaluation parameters in the rank have the
form gk1 gk2…gkr. Each logical variable has its own weight gki → wki. Then the rank can be rewritten
as an inequality:
wk1 + wk 2 + ... + wkr ≥ Q (9)

where Q is the sum of the weights of the financial parameters at which the assessment of the
financial condition of the enterprise Zj becomes fair.
2) Each rank forms an inequality. In this case, for the implicants gl1gl2…gls the corresponding
inequality will look like:
wm1 + wm 2 + ... + wmt < Q (10)

where m1, m2,…, mt are indices of variables that are not included in this rank.
So, we get the following abbreviated system of inequalities:
w1+ w2+ w3+ w5+ w8 ≥ Q
w1+ w2+ w3+ w5+ w4+ w7+ w10+ w18+ w19+ w20 ≥ Q
w1+ w2+ w3+ w5+ w4+ w7+ w10+ w18+ w19+ w6+ w9+ w11+ w13+ w16+ w17 ≥ Q (11)
w1+ w2+ w3+ w5+ w4+ w7+ w10+ w18+ w19+ w6+ w9+ w11+ w13+ w16+ w12+ w14+ w15 ≥ Q
w12+ w14+ w6+ w9+ w11+ w13+ w16+ w4+ w7+ w10+ w18+ w19+ w1+ w2+ w3+ w5 q, where q = 0.4, the Enterprise 1 is characterized by a high
level of financial risk R3.
To verify the adequacy of the proposed methods of assessing FR, we compare the results of
existing regulatory approaches - the model of multivariate regression analysis, integrated indicator
and bankruptcy forecasting model and the proposed model.
Seven enterprises from different fields of activity (industry, transport, communications, etc.)
were considered. The results are summarized in table 7.
Table 7
Comparative characteristics of normative and proposed approaches to the assessment of FR
N for Normative methods The proposed The real
Enterp- method level
rise FR
Multifactorial Integral Bankruptcy Threshold
regression analysis indicator forecasting elements
1 average levels of FR low levels of average levels low levels of low levels
FR of FR FR of FR
2 average levels of FR average levels average levels average levels average
of FR of FR of FR levels of FR
3 low levels of FR average levels high levels of FR high levels of high levels
of FR FR of FR
4 average levels of FR low levels of average levels low levels of low levels
FR of FR FR of FR
5 high levels of FR high levels of high levels of FR high levels of high levels
FR FR of FR
6 average levels of FR average levels high levels of FR average levels average
of FR of FR levels of FR
7 high levels of FR high levels of high levels of FR high levels of high levels
FR FR of FR
Number of
erroneous 3 1 3 0 -
estimates

Table 7 shows the adequacy of the proposed models. At the same time, in contrast to the normative
ones, the developed approaches allow for a deeper analysis, accelerate the decision-making
process, reduce its risk and increase the efficiency of assessment for such poorly structured
decision-making tasks. This allows you to automatically display the set of input evaluation
parameters to the set of output results of the assessment of financial risk of the enterprise by
decomposing and formalizing the decision-making process based on the appropriate mathematical
apparatus.

7. Conclusion
The financial risks of the enterprise are caused by both exogenous and endogenous factors.
Exogenous factors are usually well controlled by modern information technology. On the contrary,
endogenous factors are poorly taken into account in modern information systems to support
decision-making. That is why the development of computer models to identify the financial risk
of existing enterprises is an important direction in the development of modern information
technology. An important factor in this is that computer models need to be general enough to cover
a wide variety of business areas.
A mathematical model for assessing the risk of the financial condition of the enterprise within
the principle of decomposition division of a complex function into a sequence of simpler ones is
constructed. The feature of this model is that it takes into account the set K of the initial input
parameters kc ( c = 1, C ), which is determined by the relevant reporting of the enterprise and expert
information; the set X of the evaluation parameters xi ( i = 1, n ) of the financial condition; function
of conversion of initial parameters into estimating F1: К → X; the set of decomposition functions
D = (Y, … S, Р) of the collapse of the parameters by which the identification of the financial
condition of the enterprise; the function of determining the level of risk of a potential investor F2:
Zj → Rj, which corresponds to Zj the level of the financial condition of the enterprise; the set of
initial parameters of financial risk Rj. This simplifies the process of assessing the financial risk of
the company, which will face a potential investor in a fast-paced external and internal environment.
The sets of estimating input and output parameters for the definition of risk of a financial
condition of the enterprise are made and substantiated. They take into account a wide range of
external and internal influencing factors. The choice of the set of evaluation parameters is justified
by the criteria of completeness, effectiveness, and minimality. The set of initial parameters allows
to identify the level of risk of the business subject and is determined by the criteria of completeness
and effectiveness, which allows to fully satisfy the consumers of the system.
A method of assessing the financial risk of the enterprise on the basis of the mathematical
apparatus of threshold elements, the use of which greatly simplifies and increases the accuracy of
assessing the financial condition of economic entities in various industries by taking into account
the importance of valuation parameters. The practical implementation of the proposed
mathematical models and methods of financial risk assessment on the example of real enterprises
is considered.
The advantages of these models and methods are that, unlike normative valuation methods,
they allow to take into account a wide range of valuation parameters, industry, and its specifics
and make decisions without considering all combinations of parameters, which simplifies the
process of assessing financial risk Rj and computer simulation. Also, the developed computer
model can be used both offline and when using a cloud environment.

References
[1] Y.Bulak, T.Lisovich, International market of it services and their impact on the digital
economy of Ukraine, The grail of science, 4, 2021, pp. 53-56. DOI.org/10.36074/grail-of-
science.07.05.2021.005
[2] A.Azarova, O.Royik, L.Kilymnyk, Mathematical model and method of risk level estimation
for capital structure by means of Hopfield neural network, Actual Problems of Economics,
1(103), 2010, pp. 245–253
[3] A.Azarova, L.Azarova, L.Nikiforova, V.Azarova, O.Teplova, N.Kryvinska, Neural network
technologies of investment risk estimation taking into account the legislative aspect, in: CEUR
Workshop Proceedings, CITRisk-2020,2805, 2020, pp. 308–323. http://ceur-ws.org/Vol-
2805/paper23.pdf
[4] V.Liubchenko, Risk-Oriented Approach in Multi-Criteria Decision-Making, in: CEUR
Workshop Proceedings, CITRisk-2020, 2805, 2020, pp. 308–323. http://ceur-ws.org/Vol-
2805/invited2.pdf
[5] E.Telipenko, S.Sopova, Software for Bankruptcy Risk Assessment of the Enterprise, In: Proc.
of 2018 IEEE 3rd Russian-Pacific Conference on Computer Technology and Applications
(RPC), Vladivostok, 2018, pp. 1-4. DOI: 10.1109/RPC.2018.8482206
[6] T.Lakshmi, A.Martin, V.Venkatesan, A Genetic Bankrupt Ratio Analysis Tool Using a
Genetic Algorithm to Identify Influencing Financial Ratios. In IEEE Transactions on
Evolutionary Computation, vol. 20, no. 1, 2016, pp. 38-51. DOI:10.1109/
TEVC.2015.2424313
[7] D.P.Li, S.J.Cheng, P.F.Cheng, J.Q.Wang, H.Y.Zhang, A novel financial risk assessment
model for companies based on heterogeneous information and aggregated historical data.
PLoS ONE 13(12), e0208166, 2018. DOI: doi.org/10.1371/journal.pone.0208166
[8] S.Joshi, R.Ramesh, S. Tahsildar, A Bankruptcy Prediction Model Using Random Forest, In:
Proc. of 2018 IEEE Second International Conference on Intelligent Computing and Control
Systems (ICICCS), Madurai, India, 2018, pp. 1-6. DOI: 10.1109/ICCONS.2018.8663128
[9] I.Izonin, I.Nevliudov, Yu.Romashov, Computational Models and Methods for Automated
Risks Assessments in Deterministic Stationary Systems, in: CEUR Workshop Proceedings,
CITRisk-2020, 2805, 2020, pp. 27-43
[10] Y.Song, Y.Peng. A MCDM-Based Evaluation Approach for Imbalanced Classification
Methods in Financial Risk Prediction, Іn IEEE Access, vol. 7, 2019, pp. 84897-84906. DOI:
10.1109/ACCESS.2019.2924923
[11] A Azarova, O Zhytkevych, Mathematical methods of identification of ukrainian enterprises
competitiveness level by fuzzy logic using. Economic Annals-XXI, 9-10(2), 2013, pp. 59-62
[12] P.Bidyuk, L.Petrenko, N.Savina, T.Ivchenko, M.Voronenko, Assessing Risk of Enterprise
Bankruptcy by Indicators of Financial and Economic Activity Using Bayesian Networks, in:
CEUR Workshop Proceedings, CITRisk-2020, 2805, 2020, pp. 59-73
[13] J.Horvаthovа, M.Mokrisova, Risk of Bankruptcy, Its Determinants and Models, Risks
6:117, 2018. DOI: 10.3390/risks6040117
[14] W.Beaver, Financial ratios as predictors of failures, Empirical Research in Accounting,
Supplement to Journal of Accounting Research, 1966, pp. 71-111
[15] Q.Zhang, J.Wang, A.Lu, Sh.Wang, J.Ma, An improved SMO algorithm for financial credit
risk assessment – Evidence from China’s banking. Neurocomputing, V. 272, 2017, pp. 314-
325. DOI:10.1016/j.neucom.2017.07.002
[16] H.Dan, R.Xinchang, Financial Risk Assessment Based on Factor Analysis Model, Journal
of Physics: Conference Series, V. 1616, 3rd International Symposium on Big Data and
Applied Statistics 10-12 July 2020, Kunming, China
[17] P.Bidyuk, О.Trofymchuk, І.Kalinina, А.Gozhyj, Modeling Risk Factor Interaction Using
Copula Functions, in: CEUR Workshop Proceedings, CITRisk-2020.2805, 2020, рр. 87-101
[18] R.B.Nelsen, An Introduction to Copulas. Springer, New York, 2006
[19] R.Conforti, M.de Leoni, M.La Rosa, IAalst, W.M.P. Supporting Risk-Informed Decisions
during Business Process Execution, In: Salinesi C., Norrie M.C., Pastor Ó. (eds), Advanced
Information Systems Engineering. CAiSE 2013, Lecture Notes in Computer Science,
Springer, Berlin, Heidelberg, vol 7908, 2013. DOI: doi.org/10.1007/978-3-642-38709-8_8
[20] M.Rausand, Risk Assessment: Theory, Methods and Applications, WILEY, USA, 2013
[21] A.A.Serov, E.N.Kuznetsova, Basic methodological approaches to assessing financial risks,
Bulletin of the Nizhny Novgorod State Agricultural Academy, v. 3, 2013, pp.282-487. URL:
https://www.elibrary.ru/item.asp?id=22988725
[22] K.Valaskova, T.Kliestik, L.Svabova, P.Adamko, Financial Risk Measurement and
Prediction Modelling for Sustainable Development of Business Entities Using Regression
Analysis, Sustainability 10, 2018, 2144. DOI: doi.org/10.3390/su10072144
[23] A.A.Azarova, O.M.Roik, A.V.Poplavsky, A.P.Tkachuk, Method of formalizing the
decision-making process based on the theory of threshold elements, Registration, storage and
data processing. 3, 2018, pp. 38–47