Оцінювання операційного ризику з використанням методології системного аналізу
Financial risks are considered today as popular research topics due to the existing practical necessity for the use of their mathematical models, estimates of possible loss in many areas of human activities, forecasting, and respective managerial decisions in financial and other spheres where capita...
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| author | Bidyuk, Petro Tymoshchuk, Oxana Levenchuk, Liudmyla |
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| description | Financial risks are considered today as popular research topics due to the existing practical necessity for the use of their mathematical models, estimates of possible loss in many areas of human activities, forecasting, and respective managerial decisions in financial and other spheres where capital, obligations, stocks, bonds, and other activities are circulating successfully. Financial processes today exhibit sophisticated forms of evolution in time that require the application of sophisticated modeling, risk estimating, forecasting, and decision-making/support methods, techniques, and procedures. The system analysis approach is applied to solving such problems as a unique and universal research methodology. The financial risks, specifically the operational ones in the study considered, are classified as nonlinear and nonstationary processes that require appropriate methods for analysis and a rather sophisticated analytical description to estimate and forecast possible loss. The results of operational risk analysis are achieved in the form of systemic methodology, models constructed with statistical data, regression analysis, and Bayesian techniques, and estimated loss with the models. The models and system analysis approach proposed for analyzing financial processes are suitable for practical applications, provided the users have appropriate statistical data and expert estimates. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2024.1.04 |
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P.I. Bidyuk, O.L. Tymoshchuk, L.B. Levenchuk, 2024
42 ISSN 1681–6048 System Research & Information Technologies, 2024, № 1
TIДC
ПРОБЛЕМИ ПРИЙНЯТТЯ РІШЕНЬ ТА
УПРАВЛІННЯ В ЕКОНОМІЧНИХ, ТЕХНІЧНИХ,
ЕКОЛОГІЧНИХ І СОЦІАЛЬНИХ СИСТЕМАХ
UDC 519.766.4:004.942
DOI: 10.20535/SRIT.2308-8893.2024.1.04
OPERATIONAL RISK ESTIMATION USING
SYSTEM ANALYSIS METHODOLOGY
P.I. BIDYUK, O.L. TYMOSHCHUK, L.B. LEVENCHUK
Abstract. Financial risks are considered today as popular research topics due to the
existing practical necessity for the use of their mathematical models, estimates of
possible loss in many areas of human activities, forecasting, and respective manage-
rial decisions in financial and other spheres where capital, obligations, stocks,
bonds, and other activities are circulating successfully. Financial processes today
exhibit sophisticated forms of evolution in time that require the application of so-
phisticated modeling, risk estimating, forecasting, and decision-making/support
methods, techniques, and procedures. The system analysis approach is applied to
solving such problems as a unique and universal research methodology. The finan-
cial risks, specifically the operational ones in the study considered, are classified as
nonlinear and nonstationary processes that require appropriate methods for analysis
and a rather sophisticated analytical description to estimate and forecast possible
loss. The results of operational risk analysis are achieved in the form of systemic
methodology, models constructed with statistical data, regression analysis, and
Bayesian techniques, and estimated loss with the models. The models and system
analysis approach proposed for analyzing financial processes are suitable for practical
applications, provided the users have appropriate statistical data and expert estimates.
Keywords: financial operational risk, mathematical model, statistical data, system
analysis methodology, loss estimation, decision support system.
INTRODUCTION
Financial risks are related to very popular research topics due to existing practical
necessity for the use of their mathematical models, estimates of possible loss in
many areas of human activities, forecasting and respective managerial decisions
in financial and many other spheres where capital, obligations, stocks, bonds and
other activities are circulating successfully [1]. Generally financial processes of-
ten exhibit rather sophisticated forms of evolution in time what requires applica-
tion of modern modeling, forecasting and decision making/support methods,
technics and procedures. The effective system analysis approach should be ap-
plied to solving such problems successfully [2]. The most known financial risks
can be classified as nonlinear and nonstationary processes (NNP) that require ap-
propriate attention for analysis and rather sophisticated analytical description to
estimate and forecast possible loss.
Operational risk estimation using system analysis methodology
Системні дослідження та інформаційні технології, 2024, № 1 43
The possible process nonstationarities appear due to availability of determi-
nistic and stochastic trends, changing in time variance, and possible nonlinearities
can be divided into two wide classes: nonlinearities with respect to variables and
nonlinearities with respect to parameters. The nonlinearities with respect to vari-
ables are easier to cope with because the models (parameters) containing them
can be estimated easier than models nonlinear with respect to parameters. For ex-
ample, the model describing process with nonlinear trend (say, polynomial trend
of second or higher order) can be correctly estimated using ordinary or nonlinear
LS (NLS) or maximum likelihood (ML) methods. On the other side, the models
nonlinear with respect to parameters may require for their estimation application
of sophisticated optimization procedures like NLS, including Bayesian techniques
such as Monte Carlo for Markov Chain (MCMC) or Bayesian optimization ap-
proach.
Thus, most of the financial processes that researchers have to cope with to-
day are nonlinear and nonstationary what requires paying special attention to
modeling their structure and parameter estimation. Generally, most of the proc-
esses around us (in economy, finances, industrial production, weather forecasting,
description of natural phenomena) are related to NNPs, and they should be mod-
eled, forecasted and governed correctly using respective theoretical basis pro-
vided by the estimation theory [3]. Usually specialized intellectual decision sup-
port systems (IDSS), based upon system analysis principles, are constructed to
analyze the processes that help substantially to improve constructed model ade-
quacy, quality of forecasts, respective decisions, and increase general understand-
ing of practical situations related to the modeling and forecast estimation prob-
lems. This is especially true regarding financial risk analysis where different
model types are to be applied to reach high quality results of possible loss estima-
tion/forecasting.
Among often met financial risks that researchers today have to cope with the
following ones should be mentioned: market risk, operational risk, risk of a coun-
try, credit and liquidity risks, risk of structured products sale, Exchange-traded
products (ETP) risk, Over-the-counter (OTC) derivatives risk and some others. In
this study we provide for a short review of modern methods for modeling selected
types of financial risks, and estimation of possible loss with special attention to
active usage of IDSS. Comparison of some approaches to risk estimation will also
be given. The studies [4, 5, 6] provide for grounded notion of “risk”, consider
possible instruments for operational risk analysis, and describe basic problems
met on the way of development and practical implementation of risk estimation
and prediction systems in financial institutions. The study [7] touches upon analy-
sis of key risks inherent to Ukrainian banking system. Many studies are devoted
to analysis and use of loss distributions approach (LDA) because it provides the
possibility for estimating the volume of capital necessary for covering operational
loss. The paper [8] is completely dedicated to analysis and practical use of the
popular LDA approach.
PROBLEM STATEMENT
The purposes of the study are as follows: to provide a short review of operational
financial risk modeling methods and respective loss estimation; to consider possi-
bility of financial risk estimation by making use of regression techniques (gener-
alized linear models) and Bayesian approach to data analysis; to compare quality
P.I. Bidyuk, O.L. Tymoshchuk, L.B. Levenchuk
ISSN 1681–6048 System Research & Information Technologies, 2024, № 1 44
of risk estimation by the methods selected; to give necessary illustrative examples
of financial operational risk estimation; to stress necessity of constructing and
practical application of IDSS to estimation/prediction of possible financial loss.
SOME GENERAL APPROACHES TO FINANCIAL RISK ESTIMATION
Here we start analysis with operational risk that is available practically in every
activity when we consider capital circulation problems on different levels of
economy and finances. The term “operational risk” does not have clearly estab-
lished definition. Some financial institutions define it as non-measurable risk
(what is also correct). In 2001 the Basel Committee on Banking Supervision for-
mulated general supervision standards and governing principles for banking sys-
tem, and gave definition for operational risk: “operational is the risk of loss due to
inadequate or erroneous internal processes, actions of co-workers or systems, or
influence of external events” [1].
The Basel Committee also proposed some methods for operational risk
estimation, namely: the basic indicator approach (BIA); standardized indicator
approach (SIA); internal estimation approach (IMA); an approach based upon loss
distribution curve (LDA); scoring cards, and scenario analysis techniques. The
first three methods are relatively simple, and can be used in various financial
organizations though they do not take into consideration special features of an
enterprise, its size and principles if risk management implemented within its
structure. Using the methods mentioned “good” and “bad” financial company will
be keeping the same amount of capital to cover operational risks. The last three
methods are considered as advanced approaches to operational risk analysis and
estimation. They are based upon mathematical models and advanced information
technologies, and provide the possibility for decreasing the capital necessary for
covering operational risk loss. Now consider some types of models used in
practice of operational and some other risks analysis.
GENERALIZED LINEAR MODELS
Generalized linear models (GLM) create to some extent universal approach to
modeling various processes including linear and nonlinear, stationary and nonsta-
tionary ones. This is possible thanks to the fact that this group of models includes
the models of different structures: pure linear Bayesian multiple regression; Pois-
son regression; variance and covariance analysis; nonlinear structures including
nonlinear polynomial elements; nonlinear structures like logit and probit models
that are capable to capture nonlinear dependences in the processes under study.
Combination of linear and nonlinear structural elements in one model is directed
to describing complex nonlinear nonstationary dependences inherent to many fi-
nancial processes. For example, nonlinear logit and probit models can be com-
bined with paired or multiple linear regression, Bayesian networks (static and/or
dynamic), Bayesian data filtering procedures that can be linear or nonlinear.
ESTIMATION OF REGRESSION MODEL STRUCTURE
The basics of regression model structure estimation were proposed by Box and
Junkins in the 1960s. Today the modeling methodology proposed, based on sys-
tem analysis approach, includes the following steps:
Operational risk estimation using system analysis methodology
Системні дослідження та інформаційні технології, 2024, № 1 45
systemic analysis of a process under study on the basis of available basic
theory and expert knowledge, analysis of inputs and outputs behavior, represented
by the time series information, study of earlier existing model structures, and oth-
er accessible information;
preliminary processing of available data that is based upon application of
appropriate normalizing and filtering procedures, appropriate analysis of possible
extreme values, filling in missing values, structuring the data etc;
analysis of statistical data for stationarity and nonlinearity using available
statistical tests;
generation and estimation of candidate model structures with taking into
consideration possible structure of GLM: identification of stochastic and systemic
components and link function; computing descriptive characteristics; determining
statistical significance of independent variables using Wald statistic; estimate sta-
tistical characteristics of other structural elements of constructed mathematical
model (for example, residuals);
selection/development and application of model parameter estimation
method (or methods); it can be linear or nonlinear least squares (NLS) method,
maximum likelihood (ML) or Monte Carlo based Bayesian approach;
selection of the best fitting model in the set of estimated candidates using
statistical quality criteria.
Now consider some details of the stages mentioned.
PRELIMINARY DATA PROCESSING
The time series theory considers observations as random variables containing de-
terministic component. Generally we have practically always process the data
measurements influenced by some random external disturbances and measure-
ment errors. The purpose of preliminary data processing is in eliminating possible
errors from available measurements and improving conditions for determining
their distribution and further modeling. According to current requirements when
GLMs are hired for formal describing the processes under study the data should
correspond to the family of exponential distributions [9; 10].
Preliminary data processing usually includes the following operations:
normalization and possible correcting of measurements; normalization
means application of logarithmic operator or transforming the data to acceptable
and convenient amplitude interval;
possible data correction may include imputation of missing values, appli-
cation of filtering techniques, and processing extreme values;
computing, when necessary, first and higher order differences that are
necessary for analysis of corresponding effects of a time series under study.
Sample mean is subtracted sometimes from the observations to get a possi-
bility for constructing a model for deviations. It is clear that application of spe-
cific data preprocessing technique depends on specific features of the data avail-
able and established purpose of modeling (forecasting, deeper analysis of a
process or control).
Nonlinear deterministic trend available in data can be identified by estimat-
ing the following time polynomial:
,...)( 2
210
m
mkckckcaky
P.I. Bidyuk, O.L. Tymoshchuk, L.B. Levenchuk
ISSN 1681–6048 System Research & Information Technologies, 2024, № 1 46
where ...,2,1,0k is discrete time )( sTkt , where t is continuous time, and
sT is sampling interval for measurements. If one of the coefficients mici ,...,2,
is statistically significant, then hypothesis on trend linearity is rejected. In the case
when trend is quickly changing it time and its functional description is not ade-
quate enough, then model of stochastic trend should be constructed that is based
upon combinations of random processes.
ESTIMATION OF OTHER ELEMENTS OF GLM MODEL STRUCTURE
Taking into consideration possible GLM distributions there are several GLM
types presented in Table 1.
T a b l e 1 . GLM types considered in the study
Type of model Link function Distribution of dependent variable
GLM )(g Normal
Log-linear model )ln()( g Poisson
Logistic model
1
ln)(g Binomial
Probit-analysis 1)(g Binomial
“Survival” analysis 1)( g Gamma-distribution, exponential
Considering GLM, model structure is estimated from the point of view of its
components and includes the following elements:
stochastic component: this is independent variable that is characterized by
distribution related to the exponential family;
systematic component that includes p independent variables creating so
called “linear predictor”:
X ;
estimating features of the link function with taking into consideration
their classification.
Besides the definition mentioned the model structure includes the following
elements: model order (maximum order of equations creating the model); model
dimension or number of equations creating the model; selection of a link function,
variance function and elements of the linear predictor.
MODEL PARAMETER ESTIMATION
After estimating model structure the next problem is to compute model parame-
ters using available statistical/experimental data. Usually, the model structure es-
timated provides information on a number of parameters to be estimated and the
estimation method to be applied in concrete case. In some cases, it is reasonable
to use the saving principle that supposes the following: number of parameters to
be estimated should not exceed their necessary quantity. Here “necessity” can be
viewed as necessity to preserve in the model constructed basic statistical charac-
teristics of the process being studied: mean value, variance and covariance.
Operational risk estimation using system analysis methodology
Системні дослідження та інформаційні технології, 2024, № 1 47
Very often parameters of a generalized linear model can be estimated using
ordinary LS (OLS) method. In a case of normal data distribution this is equivalent
to application of maximum likelihood (ML) method. Generally, widely used
methods for estimation of GLM parameters are the following: OLS, weighted LS
(WLS), ML, method of moments and Monte Carlo techniques (iterative and non-
iterative). As far as the OLS in some cases results in biased parameter estimates
due to its sensitivity to data outliers an alternative procedure for estimation is ML.
In the case of normal residuals logarithmic likelihood function, l, can be rep-
resented for n measurements as follows:
.
)(
)2(log2
1
2
2
2
n
i
iiy
nl
For a fixed, σ2, maximization of l is equivalent to minimization of quadratic
deviations from mean: 2])([ ky ; thus, for linear model we have:
.
1
j
p
j
ijii x
Very popular today method for GLM parameter estimation is Markov Chain
Monte Carlo (MCMC) procedure that can be applied to linear and nonlinear mod-
els. The estimation algorithm based on this method is functioning as follows [11]:
sampling 1i
uX from the distribution ),|( 0
i
u XXP , where 0X is ex-
perimental data;
sampling parameter estimate, 1i , from distribution ),|( 0
i
uXXP .
The statistical series generated this way under weak regularity conditions has
marginal stationary distribution, )|,( 0XXP u , that can be trivially transformed
into )|( 0XP . The MCMC techniques require rather high computational re-
sources but they are very popular due to their universality, good scaling character-
istics, capability to take into account non-observable variables, low estimation
errors, and possibilities for parallel computing. Correctness of the conditions nec-
essary for parameter estimation can be analyzed after computing the parameters.
After computing the parameters, we get estimates of the random process influenc-
ing the process under study as follows:
)(ˆ)()()(ˆ kykykek ,
and analyze its statistical characteristics indicating correctness of the parameter values.
MODEL DIAGNOSTICS: SELECTION OF THE BEST MODEL OF THE
CANDIDATES CONSTRUCTED
At this stage model adequacy is analyzed that includes the steps given below.
a) Visual study of residual graph, )(ˆ)()( kykyke , where )(ˆ ky is de-
pendent variable estimate computed with the model constructed. The graph
should not exhibit outliers and long periods of time with large values of residuals
(i.e., long periods of model inadequacy). In the case of constructing GLM such
model characteristics can result from random selection of distribution law for de-
pendent variable without substantial argumentation. It can result in substantial
P.I. Bidyuk, O.L. Tymoshchuk, L.B. Levenchuk
ISSN 1681–6048 System Research & Information Technologies, 2024, № 1 48
deviations of forecast estimates computed with the model from actual observa-
tions.
b) The estimated model residuals should not be auto-correlated. To analyze
the possible autocorrelation level it is necessary to compute autocorrelation func-
tion (ACF) and partial ACF (PACF) for the series, )}({ ke , and analyze statistical
significance of the functions with Q -statistics. Besides, the level of residuals au-
tocorrelation can be additionally tested with Durbin–Watson statistic,
22DW , where, 2/])1()([ ekekeE , is correlation between neighboring
values of residuals; 2
e is variance of the residual process; if 0 , auto-
correlation between residuals is absent, and ideal value for the DW statistic is 2.
c) Statistical significance of the model parameter estimates can be tested
with Student t statistic, aSEaat ˆ
0 /]ˆ[ , where â is parameter estimate; 0a is
zero-hypothesis regarding the estimate; aSE ˆ is standard error for the estimate.
Usually all the computer systems that include functionality for time series analy-
sis provide all necessary information for a user regarding analysis of statistical
significance of parameter estimates.
d) Determination coefficient, ][var/]ˆ[var2 yyR , where, )ˆ(var y is vari-
ance of dependent variable estimated via the model constructed; )(var y is actual
sample variance of dependent variable. The ideal value of the coefficient is,
12 R , when the values of variance in nominator and denominator are the same.
This statistical parameter can be interpreted as a measure of information con-
tained in a sample and in a model. From this point of view 2R compares volume
of information represented by a model to the volume of information represented
by data sample used for model constructing.
e) The sum of squared errors should take minimum value for the best candi-
date model constructed:
N
k
N
k
kykyke
1 1
ˆ
22 min)]()(ˆ[)( .
f) For estimating constructed model adequacy Akaike information criterion
(AIC) is used:
nkeNAIC
N
k
2)(ln
1
2
,
and Bayes–Schwarz criterion:
),ln()(ln
1
2 NnkeNBSC
N
k
where 1 qpn , is a number of estimated model parameters (p is a number of
parameters for auto-regression part; q is a number of parameters for the moving
average part; 1 is added in a case when the bias, 0a , is added). Both criteria ex-
hibit minimum values for the best model among the candidates estimated.
g) Besides the statistics mentioned, Fisher statistic )1/(~ 22 RRF is ap-
plied that may show adequacy of a model as a whole after testing it for statistical
significance.
Operational risk estimation using system analysis methodology
Системні дослідження та інформаційні технології, 2024, № 1 49
To formulate statistical inference regarding the model constructed Bayesian
factor, ),( jiBF , is also used that is represented by the ratio of posterior probabili-
ties to prior [12; 13; 14]. The criteria of maximum marginal density of the distribu-
tion, )|( iMxp , is used that corresponds to the following condition: 1),( jiBF .
Correct application of the methodology presented provides a possibility for
constructing adequate mathematical model in the class of generalized linear mod-
els if the statistical/experimental data hired corresponds to the requirement of sys-
tem representation and contain necessary information.
EXAMPLE 1. GLM MODEL CONSTRUCTING
Structure of statistical data hired for model constructing is shown in Table 2. The
data shows loss due to car insurance for three selected regions of Ukraine.
T a b l e 2 . The structure of statistical data
N Characteristic of data Value of the characteristic
1 Power of sample 9546
2 Dependent variable Loss due to insurance
3 Region where police was sold Kyiv, Crimea, Odesa
4 Year of a car production Starting from 2006
5 Car brend Mitsubishi, Toyota, VAZ
For the data described in Table 2, suppose that dependent variable is nor-
mally distributed, and accept as a link log-function. The histogram for dependent
variable “loss” is presented in Fig. 1.
Fig. 1. The histogram for normally distributed dependent variable
P.I. Bidyuk, O.L. Tymoshchuk, L.B. Levenchuk
ISSN 1681–6048 System Research & Information Technologies, 2024, № 1 50
The mean value of dependent variable “Loss” is 1877.531. To select the best
model from the set of possible models constructed Akaike information criterion
was used. Its minimum value corresponds to the best model. For log-normal mod-
el value of the criterion is 20.666. The drawback of the criterion is that its esti-
mate asymptotically overestimates true value with non-zero probability. An alter-
native is Hannan–Quinn criterion that is based upon minimizing of corresponding
sum instead of value itself.
Information Schwarz criterion usually selects the best model with a number
of parameters that does not exceed number of parameters in the model selected by
the Akaike criterion. The Schwarz criterion is asymptotically more reliable, and
the Akaike criterion has a tendency to bias closer to selection of parametric mod-
els. However, for the example under consideration the values of Akaike, Hannan–
Quinn, and Schwarz criteria are practically equal; that is why any of them is suit-
able for use. The standard deviation for dependent variable, “Loss”, is 7492.245.
The Likelihood Ratio Test is used for testing restrictions regarding parame-
ters of statistical model, estimated with sample statistical data. If the value of LR-
statistic exceeds critical value from χ-squared distribution at given significance
level then the restrictions are neglected, and the model without restrictions is
hired; otherwise, the model with restrictions is used.
The variance calculated shows how much the mean random value deviates
from mathematical expectation; here it is important that this is quadratic value. At
the same time the variance itself is not very convenient for practical analysis be-
cause it has quadratic dimensionality of a random value. That is why standard
deviation is used further on as a measure of risk. From the point of view of finan-
cial analysis, the standard deviation is more important because the mean deviation
of insurance results from expected return shows actual financial results and insur-
ance risk. As a risk measure can also be used mean absolute deviation (mad). In
practice the value of standard deviation is greater than mean absolute deviation
but these values have the same order, and here the following relation holds:
Smad 7979.0 . The result of forecasting loss and risk is presented in Table 3.
The relative error of forecasting with log-normal model amounted to 1.06%, what
is high quality result for the model hired. The total forecasted loss amounts to
18111231.380, and actual loss was 17921032.581, what supports the fact that the
variable “Loss” is normally distributed. The proposal to use logarithmic link func-
tion is acceptable for subsequent analysis of alternative models.
T a b l e 3 . Forecasting results with log-normal model
Value Total loss Mean Std. deviance Max Min Variance
Actual 17921032.58 18577.53 7492.24 151771.4 0.00
Forecasted 18111231.38 18976.45 9379.910 14010.97 634.0 49.535
Thus, the log-normal model is acceptable but it is not the best for the dataset
used. That is why the search for the better model was continued. Other modeling
results are given in Table 4.
Operational risk estimation using system analysis methodology
Системні дослідження та інформаційні технології, 2024, № 1 51
T a b l e 4 . Other modeling results with application of alternative data distribu-
tions and link functions
Model characteristics
N Dependent
variable
distribution
Link
function
Total
forecasted loss
Actual
total loss
Deviation of
actual data
from forecasts
Risk of loss
1 Gamma LOG 102008320.905 84087288.32 1.003
2 Normal LOG 18111231.380 190198.799 0.495
3 Poisson LOG 17921032.574 0.007 0.547
4 Normal identity 17921032.589
17921032.581
0.009 0.532
Results of model parameters estimation with the use of classic and Bayesian
approach are presented in Table 5.
T a b l e 5 . Results of model parameters estimation with the use of classic and
Bayesian approach
Classic approach Bayesian approach
N
Mean Std.
deviance
Variance,
% Mean Std.
deviance
Variance,
% R-squared
1 11805.7 15358.1 130.091 11804.34 15247.23 128.669 0.897
2 1897.45 939.91 49.535 1897.294 939.94 49.4 0.998
3 1877.53 1027.56 54.73 1877.301 1027.552 55.679 0.998
4 1877.53 999.302 53.224 1876.909 999.751 53.809 1.0
Thus, after constructing models using various proposals regarding initial dis-
tributions of dependent variable and link function the following results of comput-
ing were achieved:
1) the best data approximating model turned out to be the model with initial
Poisson distribution of dependent variable and logarithmic link function what is
proved by the forecasted total loss of 17921032.574 with practically zero errors of
forecasting;
2) the value at risk for the models constructed was in the range 40–60%,
what requires extra measures for minimizing this value;
3) the model based on normal distribution of data showed the risk about 49–50 %,
but there were observed substantial deviations of forecasts from actual data;
4) comparing the models based upon normal data distribution with logarith-
mic and identity link functions it was established that Akaike criterion accepts the
same value of about 20.66; that is why the model selection should be based upon
forecasts of total loss;
5) the model based upon gamma-distribution with logarithmic link function
showed maximum deviation from actual data (84087288.324) and maximum er-
ror: about 100%;
6) it is clear from results in Table 4 that the quality of model estimation with
Bayesian approach are close to classic estimation with the maximum likelihood
method but with better estimates of variance and standard deviation.
Thus, it can be concluded that the model based upon Poisson distribution and
exponential link function is better for practical use from the point of view of fore-
casting, risk estimation, and parameter estimation.
P.I. Bidyuk, O.L. Tymoshchuk, L.B. Levenchuk
ISSN 1681–6048 System Research & Information Technologies, 2024, № 1 52
BAYESIAN NETWORK APPLICATION TO OPERATIONAL RISK
ANALYSIS
The next approach to operational risk estimation that we consider in the study is
based upon application of Bayesian networks (BN) that is constructed for a com-
mercial enterprise. The bases for the model create expert estimates though statis-
tical data can also be hired successfully. Comparing to regression approach, BN
provide a possibility for taking into consideration not only risk dependence on
risk factors, but also the dependences of between risk factors [15]. The probabilis-
tic inference also can be computed using incomplete data.
Mathematically, BN is directed acyclic graph nodes of which correspond to
risk factors and environment variables, and edges correspond to possible relations
between nodes. Each BN is also described by the set of conditional distributions
characterizing risk factors and environment variables.
Formally, BN is a triple JGVN ,, , where V stands for a set of net-
work variables; G is directed acyclic graph nodes of which represent variables of
the process being modeled; J represents joint probability distribution for the
network variables, }...,,,{ 21 nXXXV [15; 16]. Here also Markov condition
should be fulfilled: each network variable does not depend on all other variables
except for the parent nodes of the variable. To construct the BN model first mu-
tual information between all nodes is computed to discover possible dependences,
then optimal network structure is estimated using appropriate criterion, say mini-
mum description length (MDL) that is analyzed and re-computed at each iteration
of the structure learning algorithm.
For two independent discrete events D and S Bayes theorem can be formu-
lated as follows:
)(
)|()(
)|(
Sp
DSpDp
SDp .
Suppose that state variable D can accept two values: tD is true probability
that system under study accepted one of possible state; fD its opposite state.
These two probabilities sum to 1 independently on the value that accepts variable S :
1)|()|( SDpSDp ft .
Apply to this equality Bayes theorem:
1
)(
)|()(
)(
)|()(
Sp
DSpDp
Sp
DSpDp fftt ,
or
)|()()|()()( fftt DSpDpDSpDpSp .
Thus, known estimate )(Sp can be eliminated from consideration. In this
example variable D has two states only, however, it is clear that )(Sp can be
eliminated with arbitrary number of states for D .
Procedure of constructing Bayesian network. To estimate the degree of
dependency between two random variables ix and jx the following expression
was proposed in [17]:
Operational risk estimation using system analysis methodology
Системні дослідження та інформаційні технології, 2024, № 1 53
ji xx
ji
ji
jiji
xPxp
xxp
xxpxxMI
, )()(
),(
log),(),( .
This is mutual information that shows volume of information contained
in the variable ix about variable jx . It accepts nonnegative values only,
0),( ji xxMI , and in the case of complete independence between the variables it
accepts zero value 0),( ji xxMI , because )()(),( jiji xPxpxxp and
0)1log(
)()(
)()(
log
)()(
),(
log
ji
ji
ji
ji
xPxp
xPxp
xPxp
xxp
.
In the case BN contains N nodes, estimation of ),( ji xxMI for all possible
pairs of ix and jx it is necessary to perform
2
)1( NN
computations under
condition ),(),( ijji xxMIxxMI .
The use of minimum description length (MDL) principle to estimate BN
structure. According to the Shannon coding theory, for known distribution )(XP
for variable X the optimal length of code for transmitting specific value of ran-
dom variable X tends to the value of )(log)( xPxL [18]. The source entropy
x
xPxPPS )(log)()( is minimum expected length of the coded message.
Any other code that is based upon incorrect representation of information source
will result in longer expected length of message. In other words, better model of
message source results in more compact code of data.
In the problem of BN learning as a data source is functioning some unknown
distribution function )( 0hDP , where },...,{ 1 NddD is dataset; h is hypothesis
regarding probabilistic origin of the data; )(log)( hDPhDL is empirical risk
that is additive regarding the number of observations and proportional to empiri-
cal errors [19]. The difference between )( 0hDP and modeled distribution,
)( hDP , according to the Kullbak–Leibler measure is determined as follows:
D hDP
hDP
hDPhDPhDP
)(
)(
log)()()( 0
00
D
hDLhDLhDP 0)()()( 00 .
In fact, this is the difference between hypothetically expected data code
length and minimum possible length. This difference is always nonnegative and it
equals to zero in the case of complete equality between distributions only. The
MDL principle in its general formulation declares: from the set of possible candi-
date models it is necessary to select the one that describes data in short and com-
pletely (without loss of information) [19].
Thus, in general form the problem of forming MDL is formulated as follows:
for the set of learning data },...,{ 1 nddD , }...{ )()2()1( N
iiii xxxd (here upper in-
P.I. Bidyuk, O.L. Tymoshchuk, L.B. Levenchuk
ISSN 1681–6048 System Research & Information Technologies, 2024, № 1 54
dex is variable number); n is a number of observations; each observation includes
N ( 2N ) variables )()2()1( ,...,, NXXX . Each j -th variable ( nj ,...,1 ) has
}1,...,1,0{ )()( jjA ( 2)( j ) states, and each structure, Gg , of BN is rep-
resented by N sets of parents, ( )()1( ,..., N ), i.e., for each node, nj ,...,1 ,
)( j , this is a set of parent variables such, that }{\},...,{ )()()1()( jNj XXX
(any node cannot be a parent for itself, what means that graph does contain cy-
cles). Thus, MDL of the structure Gg for n observations, n
n dddx ...21 , is
computed as follows:
)log(
2
)(
),(),( n
gk
xgHxgL nn ,
where )(gk is a number of independent conditional probabilities in the net struc-
ture, g , and ),( nxgH is empirical entropy:
Jj
nn xgjHxgH ),,(),( ,
Jj
gjkgk ),()( ,
where MDL for j -th node is computed as follows:
)log(
2
),(
),,(),,( n
gjk
xgjHxgjL nn ;
),( gjk is a number of conditional probabilities for j -th node:
)(
)( )1(),(
jk
kjgjk ,
where },...,1,1,..,1{)( Njjj is such a set that }:{ )()()( jkj kX .
The empirical entropy of j -th node is computed as follows:
),( )( ],,[
],,,[
log],,,[),,(
gjSs Aq
n
j gjsn
gjsqn
gjsqnxgjH ,
n
i
j
i sIgjsn
1
)( )(),,( ;
n
i
j
ii sqxIgjsqn
1
)( ),(],,,[ ,
where )()( jj means that )()()( , jkk kxX ; the function 1)( EI , if the
predicate, trueE , otherwise 0)( EI . The simplified BN structure learning
algorithm is shown in Fig. 2; it is performing in a cycle analysis of all possible
acyclic network structures. The structure *g contains optimal network structure.
The optimal structure accepts minimum value for the function ),( nxgL .
BN learning algorithm using MDL principle
1. )(0
* Ggg ;
2. for }{ 0gGg : if ),(),( * nn xgLxgL then gg * ;
3. solution is contained in *g .
Operational risk estimation using system analysis methodology
Системні дослідження та інформаційні технології, 2024, № 1 55
EXAMPLE 2. BN MODEL CONSTRUCTION FOR OPERATIONAL RISK
ESTIMATION AT COMMERCIAL ENTERPRISE
Consider commercial enterprise that provides for the services in entertainment
sphere, and has obligations to other commercial enterprises for established possi-
ble services. These enterprises are banks, owners of cloud technologies and serv-
ers, and outsourcing companies that perform some separate functions for the en-
tertainment enterprise. The obligations suppose transfer of money to other
enterprises. That is why the main enterprise income should cover all obligations.
Suppose that the main enterprise income is equal to one million euros.
Certainly, during the period of the main enterprise functioning various risk
may arise that influence negatively its income. A substantial part of the risks cre-
ate operational risks. To have a possibility for covering the risks the enterprise
should direct a part of its capital (income) on covering operational risks, this is so
called Capital at Risk (CaR). The use of statistical data in such case is practically
impossible due to impossibility of separating operational loss from other risks.
That is why the basic source of information for modeling operational risks pro-
vide the expert estimates.
According to recommendations provided for researchers by Basel there exist
four basic factors influencing generation of operational risks:
Risk related to working personnel: this type of risk is adhered to errors
and incorrect actions of personnel, their insufficient qualification, sometimes
overload at work place, possible irrational organization of working activities etc.
Risk related to systems and work technologies. This type of risk is related
to insufficiently high information technologies used at enterprise, the hardware sys-
tem characteristics can be inadequate to performing operations, data processing
techniques can be inadequate or data itself may exhibit unaccepted characteristics.
Processes risk: this is risk of loss due to errors in the processes of per-
forming various financial operations and corresponding calculations, forming re-
ports, price forming operations etc.
Risk induced by external influences: this type of risk is generated by
events from external environment; for example, by legislative changes, general
policy of state, economy, as well as the risks of external physical influences into
activities of financial organization.
Each of the four named factors is also influenced by some other reasons of
coming to being operational risk that can be analyzed by the experts of specific
commercial enterprise. Table 6 illustrates the nodes (variables) selected for esti-
mating the structure of Bayesian network. The nodes and respective values were
selected using expert estimates.
The central point regarding the Bayesian network is touching upon variable
Loss that is described by the quantitative values in euros as follows: {0, 1000,
10000, 50000, 100000}. The specific numbers are used to define the variable be-
cause of necessity to perform linear interpolation. The numbers were selected by
an expert taking into consideration the loss at enterprise due to operational risks
that took place earlier. Using the analysis being performed the enterprise manag-
ers can estimate the loss say within a week/month and accumulate necessary capi-
tal to cover the loss.
P.I. Bidyuk, O.L. Tymoshchuk, L.B. Levenchuk
ISSN 1681–6048 System Research & Information Technologies, 2024, № 1 56
T a b l e 6 . Nodes of Bayesian network and their possible values
N Variable name Name of node in network Possible values
1 Possible loss Loss 0, 1000, 10000,
50000, 10000 ($)
2 Risk of external environment Risk of External
Environment Low, Medium, High
3 Risk of working personnel Risk of Staff Low, Medium, High
4 Risk of hardware system used Risk of System Low, Medium, High
5 Risk of information processes Risk of Process Low, Medium, High
6 Staff qualification level Qualified staff Low, Medium, High
7 Compliance of Compliance to staff Low, Medium, High
8 Possible changes in legislation Changes in legislation Low, Medium, High
9 Reliability of payment systems
used and partner banks
Reliability of payment
systems and partner banks Low, Medium, High
10 Possible hacker attacks and viruses Hacker attacks and viruses Low, Medium, High
11 Loss of information Loss of information 0, 50, 100 (%)
12 Using workflow automation Using workflow automation Low, Medium, High
13 Possible crashes of servers Server crashes Low, Medium, High
14 Network failures Network failures Low, Medium, High
15 Using of firewall in the system Using of Firewall Yes, No
16 Using of UPS to support system Using of UPS Yes, No
The Loss variable is influenced by the four basic factors mentioned above:
Risk of Staff, Risk of System, Risk of Processes and Risk of External Environment.
These factors have the following levels: Low, Medium, and High, that can be in-
terpreted as probability of realizing specific type of risk.
All other nodes and connections between them were identified on the basis
of expert data, and reflect real reasons that influence specific type of risk:
Qualified staff — describes the level of competence and qualification of
personnel working at the enterprise. This node influences Risk of Staff, and Loss
of information.
Compliance staff — is related to correspondence of actually working staff
to the necessary staff for functioning of an enterprise. This node has substantial
influence on the variable Risk of Staff.
Changes in legislation — describes possible level of legislative changes
in a country. This is substantial reason for possible material loss because the pos-
sibilities for entertainment at any moment can be restricted or forbidden at all.
This node influences the Risk of External Environment.
Reliability of payment systems and partner banks — this factor can be a
substantial reason for possible loss due to low quality of financial transactions
(prolonged transaction time and possible errors). The factor also influences the
Risk of External Environment.
Hacker attacks and viruses — describes the level of hacker and viruses
attacks to the enterprise servers. This influence can result in server reloading and
system rejects. The node influences the Risk of External Environment.
Loss of information — the loss is usually described in percentage form
and reflects the portion of information that could be lost in the process of func-
tioning of an enterprise. The node influences the Risk of Processes.
Operational risk estimation using system analysis methodology
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Using workflow automation — this factor describes the level of usage
specialized software that provides the possibility for performing workflow auto-
mation. This approach influences the Risk of Processes.
Network failures — describes the level of computer network reliability
that usually does not depend on enterprise (under study) itself. The level can be
changed by changing network provider. This factor influences the Risk of Exter-
nal Environment.
Using of Firewall — this factor has two states: Yes and No, and is de-
scribed by probabilities that are related to the time intervals when firewall is used
or not. It influences on Hacker attacks and viruses and Server crashes.
Using of UPS — this factor is also characterized by two states and it is
also described by the probabilities related to time intervals when uninterrupted
power source was used or not. In its turn the factor influences on possible Server
crashes.
Server Crashes — the factor describes possible level of server crashes of
an enterprise and depends on the Use of firewall and Uninterrupted power sup-
plies. The reason is that loss of power or virus/hacker/ddos attack may result in
server turnoff. The factor influences on the Risk of External Environment.
The next step of estimating Bayesian network structure is filling in probabil-
ity tables for nodes. All probabilities in this example were estimated using expert
estimates. On the other side, availability of necessary statistical information at the
enterprise (i.e., storing the information during its functioning) will give the possi-
bility for the fast replacing expert estimates by actual data. This way work adap-
tive risk analyzing systems.
For the nodes that do not have parents it will be unconditional probabilities
for each possible state. The nodes that have parents require estimating conditional
probabilities. The problem is that number of required conditional probabilities is
growing fast with growing number of parents and their possible states. In case of
discrete variables the number of values in conditional probability tables (CPT)
can be estimated via the expression:
Parents
i
iesParentStatNodeStatesCount ,
where Count is the number of values necessary for CPTs; NodeStates is number
of states for a specific node under consideration; Parents is the number of parents
for a specific node being analyzed; ParentStatesi represents the number of states
for –і-th parent. Using this formula it was estimated that for the node Loss it were
required about 400 values. Example of CPT for the node of “Loss of information”
is given below in Table 7.
T a b l e 7 . Conditional probabilities for the node “Loss of Information”
Using workflow
automation
Low Low Low Medium Medium Medium High High High
Loss of information 0% 50% 100% 0% 50% 100% 0% 50% 100%
Low 0.25 0.1 0.03 0.55 0.42 0.31 0.95 0.8 0.6
Medium 0.3 0.22 0.1 0.25 0.3 0.38 0.04 0.12 0.23
High 0.45 0.68 0.87 0.2 0.28 0.31 0.01 0.08 0.17
P.I. Bidyuk, O.L. Tymoshchuk, L.B. Levenchuk
ISSN 1681–6048 System Research & Information Technologies, 2024, № 1 58
As a result of the model parameter estimation the BN was constructed for
operational risk analysis for commercial enterprise given in Fig. 2.
ANALYSIS OF THE RESULTS
Table 8 contains the capital necessary for covering the operational risks: the 90,
95 і 99 quintiles reflect the probabilities 0.9, 0.95 and 0.99, correspondingly.
T a b l e 8 . Capital required for covering the risks at selected level of confidence
Quintile, % 90 95 99
CaR, $ 38476 56833 97554
It can be easily tracked that Capital-at-Risk (CaR) is substantially growing
with growing confidence level from 90 to 99. According to the proposal by the
Basel Basic Indicator Approach (BIA) acceptable level of risk for banking sphere
Fig. 2. Possible probabilistic inference estimation using the BN constructed
Operational risk estimation using system analysis methodology
Системні дослідження та інформаційні технології, 2024, № 1 59
is 15%. Thus, the enterprise with monthly income of about one million euros
should keep CaR of 150000 euros. This is about three times as much than the re-
sult achieved in the example above. This result stresses that enterprises in banking
sphere should use the Advanced Measurement Approaches (AMA) for analysis of
operational risks. The volume of CaR can be reduced substantially this way and
available extra capital could be used more rationally to achieve higher income.
CONCLUSIONS
The reasons for emerging financial operational risks in financial organizations
were presented. It was shown that an urgent task for the organizations is devel-
opment and practical use of the risk analysis and management systems on the ba-
sis of modern system analysis approach, providing the possibility for constructing
mathematical models, more specifically probabilistic models of Bayesian type
that provide the possibility for identification and taking into consideration uncer-
tainties related to the random nature of multiple events in financial analysis.
Improved systemic multistep methodology of constructing models for finan-
cial processes and financial risks of arbitrary origin was proposed. An example of
the methodology application is given that shows effectiveness of the methodology
in application to operational risk analysis. Application of the methodology pro-
posed for modeling financial processes with the use of generalized linear models
and Bayesian parameter estimation guaranties high quality of risk estimation with
minimum errors.
Also simplified methodology was proposed for constructing models of risk
in the form of Bayesian networks using the notion of mutual information between
variables of network, and criteria of estimation the quality of model structure on
the basis of minimum description length. An example of BN model constructing
was given for a commercial enterprise on the basis of expert estimates. The model
was used for estimating distribution of loss in the case of operational risk avail-
ability. Analysis of estimation results provided by Bayesian network proves that
models of the type selected can be successfully applied for estimating possible
loss in financial organizations.
In future studies it is reasonable to create specialized commercial intellectual
decision support system for performing mathematical modeling and financial
risks estimation. The system should provide a possibility for the use of statistical
data, expert estimates and generated (simulated) variables of continuous type
convenient for the use in Bayesian framework. Also possible data uncertainties
should be identified and taken into consideration to improve the results of all the
stages of expert estimation and statistical data processing. All the stages of data
and expert estimates processing should be controlled by appropriate sets of statis-
tical quality analysis criteria to reach high quality of intermediate and final results
of computing.
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Received 15.10.2023
INFORMATION ON THE ARTICLE
Petro I. Bidyuk, ORCID: 0000-0002-7421-3565, Educational and Research Institute for
Applied System Analysis of the National Technical University of Ukraine “Igor Sikorsky
Kyiv Polytechnic Institute”, Ukraine, e-mail: pbidyuke_00@ukr.net
Oxana L. Tymoshchuk, ORCID: 0000-0003-1863-3095, Educational and Research Insti-
tute for Applied System Analysis of the National Technical University of Ukraine “Igor
Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: oxana.tim@gmail.com
Operational risk estimation using system analysis methodology
Системні дослідження та інформаційні технології, 2024, № 1 61
Liudmyla B. Levenchuk, ORCID: 0000-0002-8600-0890, Educational and Research
Institute for Applied System Analysis of the National Technical University of Ukraine
“Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: lusi.levenchuk@gmail.com
ОЦІНЮВАННЯ ОПЕРАЦІЙНОГО РИЗИКУ З ВИКОРИСТАННЯМ
МЕТОДОЛОГІЇ СИСТЕМНОГО АНАЛІЗУ / П.І. Бідюк, О.Л. Тимощук,
Л.Б. Левенчук
Анотація. Фінансові ризики — популярна тема досліджень практичної необ-
хідності використання їх математичних моделей, оцінювання можливих втрат
у багатьох напрямах людської діяльності, прогнозуванні і відповідних управ-
лінських рішеннях у фінансовій та інших сферах, де капітал, облігації, біржові
акції та інші активи успішно використовуються. Фінансові процеси характери-
зуються складною формою розвитку у часі, що потребує застосування відпові-
дних методів, прийомів та процедур моделювання, оцінювання ризиків, про-
гнозування і підтримання/прийняття рішень. Для вирішення цих проблем
застосовано підхід на основі системного аналізу як унікальну та універсальну
методологію дослідження. Фінансові ризики, зокрема операційні, які розгля-
даються у роботі, класифікуються як нелінійні нестаціонарні процеси, що по-
требують застосування належних методів для аналізу і досить складного ана-
літичного опису для оцінювання і прогнозування можливих втрат. Результати
аналізу операційних ризиків отримано у формі системної методології, нових
моделей, побудованих з використанням статистичних даних, регресійного ана-
лізу та байєсівських методів, а також оцінок втрат, отриманих за допомогою
створених моделей. Побудовано моделі і запропоновано системний підхід до
аналізу фінансових процесів, придатний для практичного використання за
умови, що користувачі матимуть належні статистичні дані та експертні оцінки.
Ключові слова: фінансовий операційний ризик, математична модель, статис-
тичні дані, методологія системного аналізу, оцінювання втрат, система під-
тримання прийняття рішень.
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| id | journaliasakpiua-article-304385 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:28:30Z |
| publishDate | 2024 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
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| spelling | journaliasakpiua-article-3043852024-05-23T07:09:36Z Operational risk estimation using system analysis methodology Оцінювання операційного ризику з використанням методології системного аналізу Bidyuk, Petro Tymoshchuk, Oxana Levenchuk, Liudmyla фінансовий операційний ризик математична модель статистичні дані методологія системного аналізу оцінювання втрат система підтримання прийняття рішень financial operational risk mathematical model statistical data system analysis methodology loss estimation decision support system Financial risks are considered today as popular research topics due to the existing practical necessity for the use of their mathematical models, estimates of possible loss in many areas of human activities, forecasting, and respective managerial decisions in financial and other spheres where capital, obligations, stocks, bonds, and other activities are circulating successfully. Financial processes today exhibit sophisticated forms of evolution in time that require the application of sophisticated modeling, risk estimating, forecasting, and decision-making/support methods, techniques, and procedures. The system analysis approach is applied to solving such problems as a unique and universal research methodology. The financial risks, specifically the operational ones in the study considered, are classified as nonlinear and nonstationary processes that require appropriate methods for analysis and a rather sophisticated analytical description to estimate and forecast possible loss. The results of operational risk analysis are achieved in the form of systemic methodology, models constructed with statistical data, regression analysis, and Bayesian techniques, and estimated loss with the models. The models and system analysis approach proposed for analyzing financial processes are suitable for practical applications, provided the users have appropriate statistical data and expert estimates. Фінансові ризики — популярна тема досліджень практичної необхідності використання їх математичних моделей, оцінювання можливих втрат у багатьох напрямах людської діяльності, прогнозуванні і відповідних управлінських рішеннях у фінансовій та інших сферах, де капітал, облігації, біржові акції та інші активи успішно використовуються. Фінансові процеси характеризуються складною формою розвитку у часі, що потребує застосування відповідних методів, прийомів та процедур моделювання, оцінювання ризиків, прогнозування і підтримання/прийняття рішень. Для вирішення цих проблем застосовано підхід на основі системного аналізу як унікальну та універсальну методологію дослідження. Фінансові ризики, зокрема операційні, які розглядаються у роботі, класифікуються як нелінійні нестаціонарні процеси, що потребують застосування належних методів для аналізу і досить складного аналітичного опису для оцінювання і прогнозування можливих втрат. Результати аналізу операційних ризиків отримано у формі системної методології, нових моделей, побудованих з використанням статистичних даних, регресійного аналізу та байєсівських методів, а також оцінок втрат, отриманих за допомогою створених моделей. Побудовано моделі і запропоновано системний підхід до аналізу фінансових процесів, придатний для практичного використання за умови, що користувачі матимуть належні статистичні дані та експертні оцінки. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2024-03-29 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/304385 10.20535/SRIT.2308-8893.2024.1.04 System research and information technologies; No. 1 (2024); 42-61 Системные исследования и информационные технологии; № 1 (2024); 42-61 Системні дослідження та інформаційні технології; № 1 (2024); 42-61 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/304385/296305 |
| spellingShingle | фінансовий операційний ризик математична модель статистичні дані методологія системного аналізу оцінювання втрат система підтримання прийняття рішень Bidyuk, Petro Tymoshchuk, Oxana Levenchuk, Liudmyla Оцінювання операційного ризику з використанням методології системного аналізу |
| title | Оцінювання операційного ризику з використанням методології системного аналізу |
| title_alt | Operational risk estimation using system analysis methodology |
| title_full | Оцінювання операційного ризику з використанням методології системного аналізу |
| title_fullStr | Оцінювання операційного ризику з використанням методології системного аналізу |
| title_full_unstemmed | Оцінювання операційного ризику з використанням методології системного аналізу |
| title_short | Оцінювання операційного ризику з використанням методології системного аналізу |
| title_sort | оцінювання операційного ризику з використанням методології системного аналізу |
| topic | фінансовий операційний ризик математична модель статистичні дані методологія системного аналізу оцінювання втрат система підтримання прийняття рішень |
| topic_facet | фінансовий операційний ризик математична модель статистичні дані методологія системного аналізу оцінювання втрат система підтримання прийняття рішень financial operational risk mathematical model statistical data system analysis methodology loss estimation decision support system |
| url | https://journal.iasa.kpi.ua/article/view/304385 |
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