Financial risk analysis in multidimensional systems
A new approach is proposed for modeling the interdependence among factors of multivariate risks, represented as matrices of interdependence measures for numerical description and a family of copulas with parameter estimates for analytical description. The approach proposed to construct a multivariat...
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| Дата: | 2026 |
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| Автори: | , , , , |
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Kyiv National University of Construction and Architecture
2026
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Репозитарії
Environmental safety and natural resources| _version_ | 1868385038477819904 |
|---|---|
| author | Trofymchuk, Oleksandr Bidyuk, Petro Tymoshchuk, Oxana Huskova, Vira Kroptya, Arsen |
| author_facet | Trofymchuk, Oleksandr Bidyuk, Petro Tymoshchuk, Oxana Huskova, Vira Kroptya, Arsen |
| author_institution_txt_mv | [
{
"author": "Oleksandr Trofymchuk",
"institution": "Д.т.н., професор, член-кореспондент НАН України, директор Інституту телекомунікацій і глобального інформаційного простору НАН України, Київ"
},
{
"author": "Petro Bidyuk",
"institution": "Д.т.н., професор, професор кафедри математичних методів системного аналізу Інституту прикладного системного аналізу Національного технічного університету України «Київський політехнічний інститут імені Ігоря Сікорського», Київ"
},
{
"author": "Oxana Tymoshchuk",
"institution": "К.т.н., доцент, завідуюча кафедрою математичних методів системного аналізу Інституту прикладного системного аналізу Національного технічного університету України «Київський політехнічний інститут імені Ігоря Сікорського», Київ"
},
{
"author": "Vira Huskova",
"institution": "Доктор філософії, доцентка кафедри штучного інтелекту Інституту прикладного системного аналізу Національного технічного університету України «Київський політехнічний інститут імені Ігоря Сікорського», Київ"
},
{
"author": "Arsen Kroptya",
"institution": "К.т.н., доцент кафедри математичних методів системного аналізу Інституту прикладного системного аналізу Національного технічного університету України «Київський політехнічний інститут імені Ігоря Сікорського», Київ"
}
] |
| author_sort | Trofymchuk, Oleksandr |
| baseUrl_str | http://es-journal.in.ua/oai |
| collection | OJS |
| datestamp_date | 2026-06-18T11:17:53Z |
| description | A new approach is proposed for modeling the interdependence among factors of multivariate risks, represented as matrices of interdependence measures for numerical description and a family of copulas with parameter estimates for analytical description. The approach proposed to construct a multivariate risk model in which, marginal distributions are modeled separately using elliptical distributions for measurements at the center of the samples and extreme distributions in the tails, while the dependencies between risks are modeled by copulas. The joint distribution is modeled using marginal distributions and copulas and can be applied to the analysis of risk characteristics. An approach to determining risk dependencies using the concept of mutual information within the framework of Bayesian networks has been developed. A computational experiment involving two generated, theoretically well-known three-dimensional distributions and one empirical three-dimensional distribution for exchange rates demonstrated the applicability of the proposed approach to modeling multidimensional risk.The problem of identifying the optimal portfolio structure under active risk management and asset liquidity constraints, a multidimensional model for estimating tail risk measures is proposed. A computational experiment conducted to estimate risk measures by generating a sample yielded an estimation error of less than one percent for non-extreme quantiles. The quality of the estimation of risk deviation measures requires further refinement of the model. The quality of risk measure estimates for the tail regions of distributions indicates that the model based on a combination of marginal distributions using normal and Pareto distributions needs to be improved to describe central observations. |
| doi_str_mv | 10.32347/2411-4049.2026.2.117-134 |
| first_indexed | 2026-06-18T01:01:10Z |
| format | Article |
| fulltext |
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UDC 004.855:681.518
Oleksandr Trofymchuk1, Corresponding Member of the NASU, Dr. Sc., Professor,
Director of the Institute of Telecommunications and Global Information Space
ORCID ID: https://orcid.org/0000-0003-3358-6274 e-mail: Trofymchuk@nas.gov.ua
Petro Bidyuk2, Doctor of Technical Sciences, Professor, Professor of the Department of
Mathematical Methods of System Analysis at the NTUU “Igor Sikorsky Kyiv Polytechnic
Institute”
ORCID ID: https://orcid.org/0000-0002-7421-3565 e-mail: pbidyuke_00@ukr.net
Oxana Tymoshchuk2, Candidate of Technical Sciences, Associate Professor, the
Department of Mathematical Methods of System Analysis at the NTUU “Igor Sikorsky Kyiv
Polytechnic Institute”
ORCID ID: https://orcid.org/0000-0003-1863-3095 e-mail: oxana_tim@gmail.com
Vira Huskova2, PhD (Engineering), Associate Professor, the Department of Mathematical
Methods of System Analysis at the NTUU “Igor Sikorsky Kyiv Polytechnic Institute”
ORCID ID: https://orcid.org/0000-0001-7637-201X e-mail: guskovavera2009@gmail.com
Arsen Kroptya2, Candidate of Technical Sciences, Associate Professor, the Department of
Mathematical Methods of System Analysis at the NTUU “Igor Sikorsky Kyiv Polytechnic
Institute”
ORCID ID: https://orcid.org/0000-0003-1740-3837 e-mail: feodorit@ukr.net
1Institute of Telecommunications and Global Information Space of the National Academy of
Sciences of Ukraine, Kyiv, Ukraine
2NTUU “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine
FINANCIAL RISK ANALYSIS IN MULTIDIMENSIONAL SYSTEMS
Abstract. A new approach is proposed for modeling the interdependence among
factors of multivariate risks, represented as matrices of interdependence measures
for numerical description and a family of copulas with parameter estimates for
analytical description. The approach proposed to construct a multivariate risk
model in which, marginal distributions are modeled separately using elliptical
distributions for measurements at the center of the samples and extreme
distributions in the tails, while the dependencies between risks are modeled by
copulas. The joint distribution is modeled using marginal distributions and copulas
and can be applied to the analysis of risk characteristics. An approach to
determining risk dependencies using the concept of mutual information within the
framework of Bayesian networks has been developed. A computational experiment
involving two generated, theoretically well-known three-dimensional distributions
and one empirical three-dimensional distribution for exchange rates demonstrated
the applicability of the proposed approach to modeling multidimensional risk.
ІНФОРМАЦІЙНІ ТЕХНОЛОГІЇ
ТА МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ
INFORMATION TECHNOLOGY AND
MATHEMATICAL MODELING
© O. Trofymchuk, P. Bidyuk, O. Tymoshchuk, V. Huskova, A. Kroptya, 2026
https://www.scopus.com/redirect.uri?url=https://orcid.org/0000-0003-3358-6274&authorId=56110310300&origin=AuthorProfile&orcId=0000-0003-3358-6274&category=orcidLink%22
https://www.scopus.com/redirect.uri?url=https://orcid.org/0000-0002-7421-3565&authorId=6602445011&origin=AuthorProfile&orcId=0000-0002-7421-3565&category=orcidLink%22
https://orcid.org/0000-0003-1863-3095
mailto:oxana_tim@gmail.com
mailto:guskovavera2009@gmail.com
https://orcid.org/0000-0003-1740-3837
mailto:feodorit@ukr.net
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The problem of identifying the optimal portfolio structure under active risk
management and asset liquidity constraints, a multidimensional model for
estimating tail risk measures is proposed. A computational experiment conducted to
estimate risk measures by generating a sample yielded an estimation error of less
than one percent for non-extreme quantiles. The quality of the estimation of risk
deviation measures requires further refinement of the model. The quality of risk
measure estimates for the tail regions of distributions indicates that the model based
on a combination of marginal distributions using normal and Pareto distributions
needs to be improved to describe central observations.
Keywords: system analysis, financial risks, multidimensional distribution, copula
families, joint distribution, dependency measures, combined marginal distribution.
https://doi.org/10.32347/2411-4049.2026.2.117-134
Introduction
Financial risk management remains an important problem for modern economic
and financial systems, especially under conditions of non-stationary and nonlinear
processes [1–3]. Effective risk analysis requires not only estimation of separate risks
but also consideration of dependencies between multiple risk factors. Interaction
between factors in high-dimensional systems may significantly increase total losses
and complicate the process of risk estimation and decision-making [4, 5].
The behavior of financial positions depends on numerous exogenous and
endogenous factors related to market conditions, economic sectors, and internal
market dynamics [6]. As the number of factors increases, the probability of extreme
events and heavy-tailed distributions also grows. Traditional low-dimensional
models based on several principal factors remain popular due to their simplicity;
however, they are often insufficient for describing complex dependency structures
in large-scale systems [7].
To analyze dependencies between financial instruments, this study applies
approaches based on random matrix theory and dependency measures. Previous
studies demonstrated that empirical correlation matrices contain dominant
eigenvalues that differ from the theoretical spectra of random matrices, indicating
the existence of meaningful latent market factors [9–13]. Such analysis makes it
possible to separate informative dependencies from noise and improve multivariate
risk modeling.
Because the traditional linear correlation coefficient has limitations in describing
nonlinear and tail dependencies, this work considers alternative dependency
measures and copula-based approaches for modeling multivariate financial risks.
The proposed methodology combines dependency analysis, multivariate
distributions, and numerical simulation methods to support practical risk estimation
and portfolio analysis in multidimensional financial systems.
Problem statement
The study is aiming to solving the following problems:
⎯ to formulate systemic approach to analysis of risks for multivariate portfolio;
⎯ determine appropriate dependency measures and the limits of their application
to estimation the risk factors in numerical form;
⎯ application of random matrices theory to studying the properties of dependency
measures;
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⎯ to analyze the possibility of application the distributions of correlation matrices
eigenvalues for different dependency measures and distributions of distances
between the eigenvalues aiming to determine the number of principal factors in
a model;
⎯ to formulate the method for estimation risk characteristics using combined
multivariate model and illustrate practical application of the method.
Material
Dependency measures. First, consider the very notion of dependency for random
variables. Independent are random variables, 𝑋1, . . . , 𝑋𝑛, for which the following
equality is true:
𝑃(𝑋1 ≤ 𝑥1; . . . ; 𝑋𝑛 ≤ 𝑥𝑛) = 𝑃(𝑋1 ≤ 𝑥1) ⋅. . .⋅ 𝑃(𝑋𝑛 ≤ 𝑥𝑛) . (1)
It means that the knowledge regarding one of the variables does not provide new
knowledge about others. The dependency is inverse characteristic but the
possibilities of its formulation can differ dependently on the problem statement.
The dependency measure should be symmetric and universal, i.e. applicable to
any pair of continuous or discrete random variables, 𝑋 and 𝑌 that characterize risks
[15]. If dependence between risk factors does not exist the dependency measure
should be equal zero, it should be restricted by the values [-1; 1], and reach minimum
and maximum values when random variables are respectively, oppositely monotonic
and congruent monotonic. Linear correlation completely characterizes dependency
between normally distributed random values that model central part of combined
distribution of risk. That is why any dependency measure should be expressed
through Pearson coefficient of linear correlation in a case of two-dimensional normal
distribution.
The dependency measure used in the models of financial and economic risks
should be invariant with respect to continuous strictly increasing transforms and
indicate to availability of differences between random variables as well as to be a
measure of distance. If the distance measure has all mentioned properties it is called
metrics of dependency.
Linear correlation. Linear correlation is most widely used dependency measure
that is used in modeling multivariate economic and financial risks with the use of
elliptical distributions, for example, normal and Student t-distribution. The
coefficient of linear correlation between two random variables, 𝑋 and 𝑌, having
finite standard deviations is computed via the expression:
𝜌(𝑋, 𝑌) =
𝐸[𝑋𝑌]−𝐸[𝑋]𝐸[𝑌]
√𝜎2[𝑋𝜎2[𝑌]]
, (2)
where, 𝜎[𝑋] and 𝜎[𝑌] are standard deviation of 𝑋 and 𝑌, respectively.
The coefficient of linear correlation is not metric of dependency. Because of
usage of heavy tail distributions (and respectively infinite standard deviation) for
which the measure (1) has not been determined, the linear correlation coefficient is
not determined for all types of random values. This coefficient is invariant to strictly
increasing linear transforms but in general case it is not invariant to nonlinear strictly
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increasing transforms. The coefficient of linear correlation is commutative,
𝜌(𝑋, 𝑌) = 𝜌(𝑌, 𝑋) and restricted, −1 ≤ 𝜌(𝑋, 𝑌) ≤ 1 ; where equality is reached in
a case of complete linear dependency of random values. Completely dependent
random values can show correlation coefficient distinct from 1 or, -1. For
independent random values the equality is true: 𝜌(𝑋, 𝑌) = 0, however, from
𝜌(𝑋, 𝑌) = 0does not follows independence of linked to them risks (for example, in
the case of normally distributed 𝑋 and completely depending from it risk 𝑋2). The
correlation coefficient does not provide for description of dependency between risks,
especially in tails of a distribution (Fig. 1).
Fig. 1. Multivariate observations with similar normal marginal distributions and the same
correlation coefficients, 𝜌 = 0.14, but with different dependency structures
Concordance measures. Concordance of risks and corresponding variables
means a tendency to simultaneous increasing or decreasing their values. The
observations, (𝑥𝑖 , 𝑦𝑖) and (𝑥𝑗, 𝑦𝑗) of the random values vector, (𝑋, 𝑌), are
considered to be in concordance if, 𝑥𝑖 < 𝑥𝑗, 𝑦𝑖 < 𝑦𝑗 or 𝑥𝑖 > 𝑥𝑗, 𝑦𝑖 > 𝑦𝑗. The
conditions of concordance for observations can be also written as follows:
(𝑥𝑖 − 𝑥𝑗)(𝑦𝑖 − 𝑦𝑗) > 0; respectively, the condition of non-concordance is as
follows: (𝑥𝑖 − 𝑥𝑗)(𝑦𝑖 − 𝑦𝑗) < 0. The coefficient of Kendall rank correlation, 𝜏, and
coefficient of rank Spearman correlation, 𝜌𝑠, are numerical measures of
dependence that are linked to concordance measures. The concordance measure,
𝜏, for samples of two random variables, 𝑋 and 𝑌, is determined as a difference
between the number of pairs of two-dimensional observations that in concordance
and the number of pairs that are not in concordance, divided by the total number
of two-dimensional observation pairs.
If (𝑋′, 𝑌′) and (𝑋′′, 𝑌′′) are independent random vectors with similar distribution
functions, then concordance measure Kendall, 𝜏, is computed via the expression:
𝜏 = 𝑃[(𝑋′ − 𝑋′′)(𝑌′ − 𝑌′′) > 0] − 𝑃[(𝑋′ − 𝑋′′)(𝑌′ − 𝑌′′) < 0] . (3)
Under increasing transforms of 𝜓,𝜙 and with 𝑋′ ≥ 𝑋′′the following inequality
is true: 𝜓(𝑋′) ≥ 𝜓(𝑋′′); and in analogy, with 𝑌′ ≥ 𝑌′′ the following condition is
true: 𝜙(𝑌′) ≥ 𝜙(𝑌′′). Thus, the Kendall, 𝜏, is invariant to increasing transforms.
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For two-dimensional normal distribution and for any random variables having
dependency structure described by elliptical copula, the Kendall 𝜏 has an expression
linking it with coefficient of linear correlation, 𝜌, as follows:
𝜏 =
2
𝜋
arcsin (𝜌) . (4)
The Spearman rank correlation, 𝜌𝑠, is also supported by the notion of
concordance and non-concordance. However, this measure also takes into
consideration marginal distributions of random variables. Let, (𝑋′, 𝑌′), (𝑋′′, 𝑌′′) and
(𝑋′′′, 𝑌′′′), are independent random vectors with similar functions of joint
distributions; then the Spearman concordance measure, 𝜌𝑠, is defined as follows:
𝜌𝑆 = 𝑃[(𝑋′ − 𝑋′′)(𝑌′ − 𝑌′′) > 0] − 𝑃[(𝑋′ − 𝑋′′)(𝑌′ − 𝑌′′) < 0] . (5)
In the same way this measure of dependency can be determined using another
component of the third vector, 𝑋′′′. The coefficient of rank Spearman correlation is
linked to the coefficient of linear Pearson correlation, 𝜌 , as follows:
𝜌𝑆 =
𝐸(𝐹𝐺)−1/4
1/12
=
𝐸[𝐹𝐺]−𝐸[𝐹]𝐸[𝐺]
√𝜎2[𝐹]𝜎2[𝐺]
= 𝜌(𝐹, 𝐺), (6)
where, 𝐹 and 𝐺 are marginal distribution functions of random variables 𝑋 and 𝑌,
respectively.
The rank correlation coefficients, 𝜏 and, 𝜌𝑠, are commutative: 𝜏(𝑋, 𝑌) = 𝜏(𝑌, 𝑋),
𝜌𝑠(𝑋, 𝑌) = 𝜌𝑠(𝑌, 𝑋). For independent random variables, 𝜏(𝑋, 𝑌) = 𝜌𝑠(𝑋, 𝑌) = 0.
The values of the both coefficients of rank correlation belong to the range: [-1,1].
These two concordance measures can be expressed via copula. This approach
provides the possibility for using as numerical measure of dependence the
concordance measures, Kendall, , and Spearman, 𝜌𝑠, in cases where traditionally
is used linear correlation coefficient. In these cases it is also possible to remove the
restrictions regarding normal or elliptical joint distribution.
Matrices of correlation coefficients. Consider the problem of determining the
form of numerical description of dependency between more than two random
variables. The dependency measure characterizes dependency structure between two
random variables using one number. To model the risk of an organization the
dependency measure is generalized for the case of, 𝑁 > 2 (risks), i. e. the matrix,
𝑁 × 𝑁, of pairwise dependency measures is considered. Here empirical matrix of
linear correlations is a key part of a model for computing the Value-at-Risk (VaR)
measure for risks with normal distributions. According to Markowitz portfolio
theory optimal portfolio corresponds to small eigenvalues of correlation matrix [18].
VaR estimation for portfolio with normal distribution. To find risk measure
VaR for normal distribution of risk factors at given confidence level and known
portfolio cost it is necessary to compute standard deviation of return rate:
𝑉𝑎𝑅 = 𝛼𝜎𝑃0 . (7)
At the first step it is determined portfolio return rate, 𝑅𝑝, that is linear function of
return rates of its components:
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𝑅𝑝 = ∑ 𝜔𝑖
𝑁
𝑖=1 𝑅𝑖 , (8)
where, 𝑁 is number of portfolio components; 𝑅𝑖 is return rate for 𝑖-th component; 𝑤𝑖 =
𝑃𝑖/𝑃𝑝 is weighting coefficient for 𝑖-th component; where, 𝑃𝑝 is portfolio cost; 𝑃𝑖 is cost
of 𝑖th portfolio component. The matrix form of portfolio return rate is as follows:
𝑅𝑝 = [𝜔1, 𝜔2, … , 𝜔𝑁] [
𝑅1
𝑅2
…
𝑅𝑁
] . (9)
For the vector of weighting coefficients, 𝑤, and vector of return rates, 𝑅, we have:
𝑅𝑝 = 𝑤𝑇𝑅. The next step is determining standard deviation for return, 𝜎𝑝. Normal
distribution of linear sum of normal random values gives normal distribution for the
portfolio return rate, 𝑅𝑝. Thus, expected return rate is determined as follows:
𝜇𝑝 = ∑𝑁
𝑖=1 𝑤𝑖𝜇𝑖, and variance of the rate has the expression:
𝜎𝑝
2 = ∑ ∑ 𝜔𝑖𝜔𝑗
𝑁
𝑖=1
𝑁
𝑖=1 𝜎𝑖𝑗 = ∑ 𝜔𝑖
2𝜎𝑖
2 + 2𝑁
𝑖=1 ∑ ∑ 𝜔𝑖𝜔𝑗
𝑁
𝑗=1,𝑗<1
𝑁
𝑖=1 𝜎𝑖𝑗 , (10)
where, 𝜎𝑖𝑗 are elements of covariance matrix.
Properties of empirical correlation matrices. The correlation matrix is widely
used in theoretical studies but in systems of risk management its estimate is used in
the form of empirical correlation matrix. The model of averaged correlation in which
all element are equal, 𝜌, but for the “1s” on main diagonal, there exists one large
eigenvalue, 𝜆1 = 1 + (𝑁 − 1)𝜌, and all others eigenvalues are equal to, 𝜆𝑖≥1 = 1 − 𝜌.
Similar result was found in the case when non-diagonal elements of correlation
matrix are random values with mathematical expectation, 𝜌, and standard
deviation, 𝜎:
𝐸[𝜆1] = (𝑁 − 1)𝜌 +
𝜎2
𝜌
+ 1 + 𝜊(1) . (11)
Thus, when, 𝜌 > 0, maximum eigenvalue is increasing with increasing system
dimensionality, 𝑁. To the dominating eigenvalue corresponds uniformly distributed
on components eigenvector, 𝑣1(1/√𝑁). This vector has economic sense as a factor
of influence on all risk positions or generalized market index. The factor can be used
to explain large scale market crises. Such interpretation can be found in the studies
of empirical financial correlation matrices [12, 19].
The study [19] on empirical matrices of linear correlation for 406 stock rates in
the period of 1991-1996 showed correspondence between distribution of eigenvalues
to theoretical results from the theory of random matrices, but for 6% of maximum
eigenvalues. The work [10] points out to availability of concentration of particularly
large eigenvalues for random symmetric matrices. In practice of computing
experiments it was observed availability of several eigenvalues in the range that
exceeds for about 5-10 times the basic bulk of eigenvalues. This situation can be
explained by availability in the market besides basic generalized market factor other
factors, say sector factors that influence some positions.
The study [7] proposed the group model of fund markets. According to the model
assumptions the market includes several separate groups that use assets the prices of
which correlate with prices of other assets of this group. In this situation the
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correlation matrix becomes close to the block-diagonal form where each block
corresponds to the sector of economy with higher correlations in the frames of the
block and lower correlations outside of the block. Close to this case is correlation
matrix with, 𝑁1 × 𝑁1, diagonal blocks, in the frames of which the correlation
coefficients are equal, 𝜌1, with “1s” on diagonal, and 𝜌0, outside of the blocks. Then
maximum eigenvalue of the correlation matrix is estimated as follows:
𝜆1 = 1 + (𝑁1 − 1)𝜌1 + (𝑁 − 𝑁1)𝜌0 , (12)
the eigenvalues that correspond to the eigenvectors that characterize principal factors
influencing the branch of economy can be found as follows:
𝜆1 = 1 + (𝑁1 − 1)𝜌1 + (𝑁 − 𝑁1)𝜌0 , (13)
and other eigenvalues can be found as follows:
𝜆
𝑖=
𝑁
𝑁1
+1…𝑁
= 1 − 𝜌1 . (14)
The matrices of correlation coefficients that include (but for coefficient of linear
correlation because of its drawbacks regarding risk management) the concordance
measures, were studied with the methods of random matrix theory [13]. It was
pointed out in [12] to correspondence of results received for random symmetric
matrix to distribution of distances between eigenvalues of empirical matrix of linear
correlation for 1000 stocks of American companies for two-year period.
The matrices of Pearson, Kendall, and Spearman correlation coefficients are
symmetric and that is why the case of maximum statistical independence was
considered corresponding to the symmetry condition. The deviations from foresights
of the random matrix theory indicate to existence of dependences characteristic for
a specific system.
Methods
Numerical modeling using copulas. The complex structure of multivariate risk
distributions makes direct analytical estimation of risk measures difficult. Therefore,
numerical simulation methods, particularly Monte Carlo simulation, are used to
estimate risk measures and perform scenario analysis.
In the proposed approach, marginal distributions describe the behavior of
individual risk factors, while copulas define the dependency structure between them.
This allows the model to generate multivariate samples that preserve both individual
characteristics of risks and their joint behavior.
The generated samples are then used to estimate portfolio risk measures and
analyze possible scenarios of market behavior. Such an approach is especially useful
when dependencies between risk factors are nonlinear or become stronger in the tails
of distributions.
In the multivariate case, sample generation can be performed either sequentially
for each variable or directly for the whole multivariate distribution. The second
approach is more suitable for modeling complex dependency structures, since it
allows the joint behavior of all risk factors to be considered simultaneously.
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Fig. 2. Multivariate generalization of the generating method using cross-section
Application of models to actual data. Modeling of joint probabilistic
distribution of risk factors was performed for simulated three-dimensional
distributions of Cauchy, Student, normal, and exchange rate of currencies (with 15
min observation interval): EUR, CHF, GBP with respect to USD from 2009 to 2016.
For each dataset was performed estimation of Archimedean copulas from the
families: Gumball, Clayton, Frank and elliptical copulas from the Student family and
normal distribution.
Together with estimates of marginal distributions this experiment provided the
possibility for modeling the functions of joint distribution. Figs. 3–7 illustrate joint
distributions for the currency exchange rates modeled with the use of various
dependency structures.
Fig. 3. Joint distribution on the basis of normal copula
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Fig. 4. Joint distribution on the basis of t-Student copula
Fig. 5. Joint distribution on the basis of Frank copula
Fig. 6. Joint distribution on the basis of Gumball copula
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Fig. 7. Joint distribution on the basis of Clayton copula
Empirical joint distribution for the currencies mentioned is shown in Fig. 8.
Fig. 8. Empirical joint distribution the currencies exchange rate
Results
The quantitative measures of risk based on risk measures and qualitative based
upon scenario analysis provide researchers and risk managers with the integral
picture regarding level of risk of available positions and portfolios [26]. An active
risk management (control) requires constructing models that would allow to measure
risk and determine its acceptability for organization as well as establish the level of
risk by changing the structure of portfolio.
Estimation of risk through the market cost should also take into consideration the
liquidity risk. This component comes to being through impossibility to sell an asset
at definite moment using its market value. The liquidity risk is available on almost
all financial and commodity markets. On the markets with low volumes of trading
and during financial crisis the liquidity share reaches, 25%-30%. Volatility of
liquidity imposes its restrictions on possible changes of portfolio structure during
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active phase of risk control. An active risk control requires develop the approach that
would allow optimize with respect to estimates of risk measures the portfolio
structure under restrictions of liquidity with respect to separate portfolio positions.
To solve the problem of active control for portfolio risks it was proposed to use
the probabilistic-statistical model on the basis of combined distribution from normal
and generalized Pareto distribution, marginal distributions of separate positions and
multivariate distributions of portfolio loss. The multivariate distributions of loss are
constructed by linking marginal distributions using special link functions in the form
of copulas, and estimating risk measures related to tails and central parts of
distributions.
The problem of finding optimal portfolio structure with respect to the VaR
measure can be considered as the problem of optimizing estimate of VaR measure
using the model with restrictions that reflect volatility of market liquidity. To
estimate risk measures using the generated from the model of multivariate sample
the cost of separate positions were found corresponding values of portfolio cost. If
{𝑋𝑖:𝑗} is sample of costs for, 𝑛-dimensional portfolio, and reordered in the way that,
𝑋1:𝑛 ≤. . ≤ 𝑋𝑛:𝑛, then empirical estimate of VaR is the following:
𝑉𝑎𝑅𝛼(𝑋) = 𝑋max(𝑖 ∈ 𝑁|𝑖 ≤ 𝑛𝛼):𝑛∗ . (15)
In the computing experiment were used daily exchange rates of Swiss franc, GB
pound, Japanese yen and USD with respect to euro for nine years. Power of the
sample was 1643 observations after preliminary data processing. The parameters of
one-dimensional marginal distributions for each currency and copula parameters
were estimated using the method of maximum likelihood.
Table 1. Estimates of copula parameters
Copula Parameter Value MSE
Gumball 𝜃 1.6720 0.0158
Normal
𝜌1 0.5637 0.0118
𝜌2 0.3318 0.0136
𝜌3 0.5943 0.0120
𝜌4 0.8241 0.0054
𝜌5 0.8593 0.0051
𝜌6 0.8037 0.0061
Frank 𝛽 4.5874 0.0911
Empirical estimate of risk measure VaR with quintile, 0.03, i.e. for 50
observations that exceed the threshold is 3.497. For quintile, 0.01, i.e. for 16
observations that exceed the threshold the measure is 3.535. For quintile 0.03 there
is enough observations to have a possibility for use of empirical estimate; for
quintile, 0.01 the sample is too short and there is necessity to model risk distribution
and estimation of risk with the model.
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Table 2. Estimates of the risk measure VaR
Copula Quintile
Power of sample
100 1000 10000
Gumball
0,03 3,4140 3,4770 3,4896
0,01 3,4375 3,5665 3,6008
Normal
0,03 3,5006 3,5386 3,5346
0,01 3,6177 3,6880 3,6603
Frank
0,03 3,4986 3,4797 3,4959
0,01 3,5139 3,5632 3,5892
According to Table 2 the estimates of VaR measure using the models based upon
combined marginal distributions and linked to joint distribution with the Gumball
and Frank copulas have an error of about, 0.203%, and 0.022%, with respect to the
empirical value for quintile, 0.03. The model with normal copula has an error of
about, 1%. The results achieved provide the possibility for making conclusion that
all three models are adequate and have possibility for their practical application.
Thus, an estimate of the risk measure VaR can be considered 3.5892 for quintile
0.01. The same three models were used for estimating coherent risk measure ES
(Expected Shortfall).
Table 3. Estimates of the risk measure ES
Copula Quintile
Power of sample
100 1000 10000
Gumball
0.03 3.5121 3.5752 3.5835
0.01 3.5348 3.6479 3.6737
Normal
0.03 3.6104 3.6475 3.6438
0.01 3.7124 3.7733 3.7523
Frank
0.03 3.5857 3.5662 3.5836
0.01 3.6017 3.6548 3.6822
Empirical estimate of ES with quintile 0.03 is 3.6074. The estimates in Table 3
show that the most adequate model for estimating this measure of risk is the model
with Frank copula. Thus the model estimate for ES with quintile 0.01 is 3.682.
All three models showed worse results for the Markowitz deviation measure
comparing to the empirical result. The models proposed are also used for active risk
control through changing portfolio structure to optimize selected measure of risk.
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Table 4. Estimates of the Markowitz risk measure
Copula 𝜎+
Gumball 0.1330
Normal 0.1462
Frank 0.1370
Empirical estimate 0.1733
The constructed four-dimensional probabilistic distribution model was used to
develop active risk control methodology by finding optimal portfolio structure on
the basis of selected tail risk measure VaR. Fig. 9 illustrates the values of VaR
estimates for various relationships between portfolio positions and liquidity
restrictions that allowed for changings in positions of Swiss franc and GB pounds.
Fig. 9. An estimate of risk measure VaR depending on relation between positions in Swiss
franc and GB pounds (accepted as 1); with step 0.1
It was also established that the values of tail risk measures for multivariate models
of financial data exhibit nonlinear character with multiple local extremes depending
on the portfolio structure. This character of dependency requires development
effective optimization algorithm.
Table 5 demonstrates that empirical correlation matrices contain several
dominant eigenvalues that substantially exceed the remaining part of the spectrum.
These eigenvalues indicate the presence of key latent factors that determine the
dependency structure of the analyzed financial instruments. The similarity of results
for Kendall, Pearson and Spearman measures confirms the stability of the detected
dependency pattern. At the same time, the remaining eigenvalues are close to the
random-matrix range, which makes it possible to separate meaningful risk factors
from noise.
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Table 5. Maximum eigenvalues of empirical correlation matrices
Name 𝜆1 𝜆2 𝜆3 𝜆4
Kendall 236,2 60,4 24,4 14,8
Pearson 318,7 69,6 16,6 12,5
Spearmen 313,4 70,0 17,7 13,0
For example, for the matrix of linear correlations, 𝜆𝑚𝑎𝑥 = 8.847487, and four
maximum eigenvalues are: 318.7, 69.6, 16.6 and 12.5, other 480 eigenvalues are
positive and less than 1.02.
For empirical correlation matrices was estimated empirical distribution of
distances between eigenvalues expanded with Gaussian unfolding and theoretical
distribution of distances for corresponding symmetric random matrix from (5). The
empirical distributions for correlation matrices and theoretical distributions for
random matrices turned out to be similar for majority of eigenvalues except for
maximum eigenvalue of empirical correlation matrix which turned out to be
substantially larger than proposes theoretical distribution.
For linear correlation it corresponds to the level of about, 99.9992%. In right tail
of distribution theoretical threshold for 95% of observations exceed 5% of
eigenvalues; and the threshold of 97% exceed 2.7% of eigenvalues (Fig. 10).
Fig. 10. Empirical density distribution of distances between eigenvalues of empirical matrix
of linear Pearson correlation coefficients and theoretical density of distances distribution
The distributions of eigenvalues and distances between eigenvalues for empirical
correlation matrices and different dependency measures used in the experiment
demonstrate similar behavior. The eigenvalue distributions provide the possibilities
for determining the number of basic factors in the model.
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Discussion
The number of risk factors in large scale financial systems can be very high, and
that is why the problem of multivariate risk analysis attracts attention of scientists,
engineers and risk managers the world over. Interaction and simultaneous influence
of multiple risk factors in high-dimension systems can result in substantial increase
of total loss comparing to the cases when interaction between elements of the
systems is taken into consideration.
The main task of the studies is to identify principal factors making substantial
influence on the value of possible loss. The theory of random matrices provides the
possibilities for analysis of eigenvalues distribution regarding correlation matrices
of dependency measures. The results of various studies show that this is the
possibility for receiving practically useful information to be further used in
management of multivariate risks. It is possible to carry out the studies in the future
directed to improvement of the results using theoretical distributions of eigenvalues
and distances between eigenvalues for symmetric positively defined matrices. Also
has perspective analysis of an influence of non-linear strictly increasing transforms
on distribution of eigenvalues of dependency measures for empirical matrices.
Conclusions
The proposed system-analysis approach enables modeling dependencies between
risk factors using matrices of dependency measures and copula families with
estimated parameters. This makes it possible to construct a multivariate risk model
in which marginal distributions and dependency structures are modeled separately.
Methods for estimating copula parameters were considered, including a two-step
maximum likelihood procedure for joint distribution modeling. The results confirm
that this approach can be applied to practical risk management tasks and scenario
analysis. The study also used mutual information within a Bayesian network
framework to determine risk dependencies and account for expert knowledge and
new information during risk management. The analysis of Pearson, Kendall and
Spearman correlation matrices showed that dominant eigenvalues exceed theoretical
random-matrix limits. This indicates the presence of key latent factors, while smaller
eigenvalues mainly correspond to noise. Therefore, portfolio optimization based on
Markowitz theory should be performed after filtering noisy data. Computational
experiments with generated and empirical three-dimensional distributions confirmed
the applicability of the proposed approach to multivariate risk modeling. For tail risk
estimation, the proposed model supported portfolio structure optimization under
liquidity constraints. The estimation error for non-extreme quantiles was less than
one percent, while tail and deviation risk measures require further model
improvement.
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The article was received 23.01.2026, received after revision 19.03.2026, accepted
15.04.2026
О.М. Трофимчук, П.І. Бідюк, О.Л. Тимощук, В.Г. Гуськова, А.В. Кроптя
АНАЛІЗ ФІНАНСОВИХ РИЗИКІВ У БАГАТОВИМІРНИХ СИСТЕМАХ
Анотація. Запропоновано новий підхід для моделювання залежності між
факторами багатовимірних ризиків, представлених у вигляді матриць мір залежності
для чисельного опису і сімейства копул з оцінками їх параметрів для аналітичного
опису, що дало можливість побудувати багатовимірну модель ризиків, за якої, окремо
моделюються маргінальні розподіли з використанням еліптичних розподілів для
вимірів у центрі вибірок та екстремальні розподіли у хвостових частинах, залежності
між ризиками моделюються копулами. Спільний розподіл моделюється за допомогою
маргінальних розподілів і копул, може бути застосований для аналізу характеристик
ризиків. Розроблено підхід до визначення залежності ризиків з використанням поняття
взаємної інформації в межах побудови байєсівських мереж. Обчислювальний
експеримент з двома згенерованими, відомими з точки зору теорії, тривимірними
розподілами та одним емпіричним тривимірним розподілом для курсів обміну валют
продемонстрували можливість застосування запропонованого підходу до
моделювання багатовимірного ризику.
Для вирішення проблеми пошуку структури оптимального портфеля в умовах
активного керування ризиками та обмежень на ліквідність активів, запропоновано
багатовимірну модель для оцінювання хвостових мір ризику. Обчислювальний
експеримент, виконаний для оцінювання мір ризику шляхом генерування вибірки,
забезпечив похибку оцінювання, меншу одного процента для неекстремальних
квантилів. Якість оцінювання мір відхилення ризику вимагає подальшого
удосконалення моделі. Якість оцінювання мір ризику для хвостових частин розподілів
свідчить, що модель на основі комбінації маргінальних розподілів з використанням
нормального і Парето розподілів потрібно покращити для опису центральних
спостережень.
Ключові слова: системний аналіз, фінансові ризики, багатовимірний розподіл,
сімейства копул, спільний розподіл, міри залежності, комбінований маргінальний
розподіл.
Стаття надійшла до редакції 23.01.2026, надійшла після рецензування 19.03.2026,
прийнята 15.04.2026
Трофимчук Олександр Миколайович
д.т.н., професор, член-кореспондент НАН України, директор
Інститут телекомунікацій і глобального інформаційного простору НАН України
Адреса робоча: 0186, м. Київ, Чоколівський бульв., 13
ORCID ID: https://orcid.org/0000-0003-3358-6274 e-mail: Trofymchuk@nas.gov.ua
Бідюк Петро Іванович
д.т.н., професор, професор кафедри математичних методів системного аналізу
Інститут прикладного системного аналізу Національного технічного університету
України «Київський політехнічний інститут імені Ігоря Сікорського»
Адреса робоча: 03056, м. Київ, Берестейський просп., 37
ORCID ID: https://orcid.org/0000-0002-7421-3565 e-mail: pbidyuke_00@ukr.net
http://dx.doi.org/10.2139/ssrn.731784
https://www.scopus.com/redirect.uri?url=https://orcid.org/0000-0003-3358-6274&authorId=56110310300&origin=AuthorProfile&orcId=0000-0003-3358-6274&category=orcidLink%22
https://www.scopus.com/redirect.uri?url=https://orcid.org/0000-0002-7421-3565&authorId=6602445011&origin=AuthorProfile&orcId=0000-0002-7421-3565&category=orcidLink%22
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Тимощук Оксана Леонідівна
к.т.н., доцент, завідуюча кафедрою математичних методів системного аналізу
Інститут прикладного системного аналізу Національного технічного університету
України «Київський політехнічний інститут імені Ігоря Сікорського»
Адреса робоча: 03056, м. Київ, Берестейський просп., 37
ORCID ID: https://orcid.org/0000-0003-1863-3095 e-mail: oxana_tim@gmail.com
Гуськова Віра Геннадіївна
доктор філософії, доцентка кафедри штучного інтелекту
Інститут прикладного системного аналізу Національного технічного університету
України «Київський політехнічний інститут імені Ігоря Сікорського»
Адреса робоча: 03056, м. Київ, Берестейський просп., 37
ORCID ID: https://orcid.org/0000-0001-7637-201X e-mail: guskovavera2009@gmail.com
Кроптя Арсен Володимирович
к.т н., доцент кафедри математичних методів системного аналізу
Інститут прикладного системного аналізу Національного технічного університету
України «Київський політехнічний інститут імені Ігоря Сікорського»
Адреса робоча: 03056, м. Київ, Берестейський просп., 37
ORCID ID: https://orcid.org/0000-0003-1740-3837 e-mail: feodorit@ukr.net
https://orcid.org/0000-0003-1863-3095
mailto:oxana_tim@gmail.com
https://orcid.org/0000-0001-7637-201X
mailto:guskovavera2009@gmail.com
mailto:feodorit@ukr.net
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X 1 : n ≤ . . ≤ X n : n
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{ X i : j }
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𝜆 1 = 1 + ( N 1 − 1 ) 𝜌 1 + ( N − N 1 ) 𝜌 0
𝜆 1 = 1 + ( N 1 − 1 ) 𝜌 1 + ( N − N 1 ) 𝜌 0
𝜌 0
𝜌 1
N 1 × N 1
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𝜌 > 0
E [ 𝜆 1 ] = ( N − 1 ) 𝜌 + 𝜎 2 𝜌 + 1 + 𝜊 ( 1 )
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𝜌
𝜆 i ≥ 1 = 1 − 𝜌
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𝜌
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𝜎 p 2 = ∑ i = 1 N ∑ i = 1 N 𝜔 i 𝜔 j 𝜎 i j = ∑ i = 1 N 𝜔 i 2 𝜎 i 2 + 2 ∑ i = 1 N ∑ j = 1 , j < 1 N 𝜔 i 𝜔 j 𝜎 i j
𝜇 p = ∑ i = 1 N w i 𝜇 i
R p
𝜎 p
R p = w T R
R
w
R p = [ 𝜔 1 , 𝜔 2 , … , 𝜔 N ] [ R 1 R 2 … R N ]
i
P i
P p
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w i = P i / P p
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R p = ∑ i = 1 N 𝜔 i R i
R p
V a R = 𝛼 𝜎 P 0
N × N
N > 2
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𝜌 s
𝜏
Y
X
G
F
𝜌 S = E ( F G ) − 1 / 4 1 / 12 = E [ F G ] − E [ F ] E [ G ] 𝜎 2 [ F ] 𝜎 2 [ G ] = 𝜌 ( F , G ) ,
𝜌
X ′ ′ ′
𝜌 S = P [ ( X ′ − X ′ ′ ) ( Y ′ − Y ′ ′ ) > 0 ] − P [ ( X ′ − X ′ ′ ) ( Y ′ − Y ′ ′ ) < 0 ]
𝜌 s
( X ′ ′ ′ , Y ′ ′ ′ )
( X ′ ′ , Y ′ ′ )
( X ′ , Y ′ )
𝜌 s
𝜏 = 2 𝜋 arcsin ( 𝜌 )
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𝜏
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Y ′ ≥ Y ′ ′
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𝜓 , 𝜙
𝜏 = P [ ( X ′ − X ′ ′ ) ( Y ′ − Y ′ ′ ) > 0 ] − P [ ( X ′ − X ′ ′ ) ( Y ′ − Y ′ ′ ) < 0 ]
𝜏
( X ′ ′ , Y ′ ′ )
( X ′ , Y ′ )
Y
X
𝜏
𝜌 s
𝜏
( x i − x j ) ( y i − y j ) < 0
( x i − x j ) ( y i − y j ) > 0
y i > y j
x i > x j
y i < y j
x i < x j
( X , Y )
( x j , y j )
( x i , y i )
𝜌 = 0 . 14
X 2
X
𝜌 ( X , Y ) = 0
𝜌 ( X , Y ) = 0
− 1 ≤ 𝜌 ( X , Y ) ≤ 1
𝜌 ( X , Y ) = 𝜌 ( Y , X )
Y
X
𝜎 [ Y ]
𝜎 [ X ]
𝜌 ( X , Y ) = E [ X Y ] − E [ X ] E [ Y ] 𝜎 2 [ X 𝜎 2 [ Y ] ]
Y
X
Y
X
P ( X 1 ≤ x 1 ; . . . ; X n ≤ x n ) = P ( X 1 ≤ x 1 ) ⋅ . . . ⋅ P ( X n ≤ x n )
X 1 , . . . , X n
|
| id | es-journalinua-article-364961 |
| institution | Environmental safety and natural resources |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-06-19T01:00:24Z |
| publishDate | 2026 |
| publisher | Kyiv National University of Construction and Architecture |
| record_format | ojs |
| resource_txt_mv | es-journalinua/36/319e1320c98c3308e796a8394d7db336.pdf |
| spelling | es-journalinua-article-3649612026-06-18T11:17:53Z Financial risk analysis in multidimensional systems Financial risk analysis in multidimensional systems Trofymchuk, Oleksandr Bidyuk, Petro Tymoshchuk, Oxana Huskova, Vira Kroptya, Arsen system analysis financial risks multidimensional distribution copula families joint distribution dependency measures combined marginal distribution системний аналіз фінансові ризики багатовимірний розподіл сімейства копул спільний розподіл міри залежності комбінований маргінальний розподіл A new approach is proposed for modeling the interdependence among factors of multivariate risks, represented as matrices of interdependence measures for numerical description and a family of copulas with parameter estimates for analytical description. The approach proposed to construct a multivariate risk model in which, marginal distributions are modeled separately using elliptical distributions for measurements at the center of the samples and extreme distributions in the tails, while the dependencies between risks are modeled by copulas. The joint distribution is modeled using marginal distributions and copulas and can be applied to the analysis of risk characteristics. An approach to determining risk dependencies using the concept of mutual information within the framework of Bayesian networks has been developed. A computational experiment involving two generated, theoretically well-known three-dimensional distributions and one empirical three-dimensional distribution for exchange rates demonstrated the applicability of the proposed approach to modeling multidimensional risk.The problem of identifying the optimal portfolio structure under active risk management and asset liquidity constraints, a multidimensional model for estimating tail risk measures is proposed. A computational experiment conducted to estimate risk measures by generating a sample yielded an estimation error of less than one percent for non-extreme quantiles. The quality of the estimation of risk deviation measures requires further refinement of the model. The quality of risk measure estimates for the tail regions of distributions indicates that the model based on a combination of marginal distributions using normal and Pareto distributions needs to be improved to describe central observations. Запропоновано новий підхід для моделювання залежності між факторами багатовимірних ризиків, представлених у вигляді матриць мір залежності для чисельного опису і сімейства копул з оцінками їх параметрів для аналітичного опису, що дало можливість побудувати багатовимірну модель ризиків, за якої, окремо моделюються маргінальні розподіли з використанням еліптичних розподілів для вимірів у центрі вибірок та екстремальні розподіли у хвостових частинах, залежності між ризиками моделюються копулами. Спільний розподіл моделюється за допомогою маргінальних розподілів і копул, може бути застосований для аналізу характеристик ризиків. Розроблено підхід до визначення залежності ризиків з використанням поняття взаємної інформації в межах побудови байєсівських мереж. Обчислювальний експеримент з двома згенерованими, відомими з точки зору теорії, тривимірними розподілами та одним емпіричним тривимірним розподілом для курсів обміну валют продемонстрували можливість застосування запропонованого підходу до моделювання багатовимірного ризику.Для вирішення проблеми пошуку структури оптимального портфеля в умовах активного керування ризиками та обмежень на ліквідність активів, запропоновано багатовимірну модель для оцінювання хвостових мір ризику. Обчислювальний експеримент, виконаний для оцінювання мір ризику шляхом генерування вибірки, забезпечив похибку оцінювання, меншу одного процента для неекстремальних квантилів. Якість оцінювання мір відхилення ризику вимагає подальшого удосконалення моделі. Якість оцінювання мір ризику для хвостових частин розподілів свідчить, що модель на основі комбінації маргінальних розподілів з використанням нормального і Парето розподілів потрібно покращити для опису центральних спостережень. Kyiv National University of Construction and Architecture 2026-06-18 Article Article application/pdf https://es-journal.in.ua/article/view/364961 10.32347/2411-4049.2026.2.117-134 Environmental safety and natural resources; Vol. 58 No. 2 (2026): Environmental safety and natural resources; 117-134 Екологічна безпека та природокористування; Том 58 № 2 (2026): Екологічна безпека та природокористування; 117-134 2616-2121 2411-4049 10.32347/2411-4049.2026.2 en https://es-journal.in.ua/article/view/364961/350487 Copyright (c) 2026 О.М. Трофимчук, П.І. Бідюк, О.Л. Тимощук, В.Г. Гуськова, А.В. Кроптя http://creativecommons.org/licenses/by/4.0 |
| spellingShingle | system analysis financial risks multidimensional distribution copula families joint distribution dependency measures combined marginal distribution Trofymchuk, Oleksandr Bidyuk, Petro Tymoshchuk, Oxana Huskova, Vira Kroptya, Arsen Financial risk analysis in multidimensional systems |
| title | Financial risk analysis in multidimensional systems |
| title_alt | Financial risk analysis in multidimensional systems |
| title_full | Financial risk analysis in multidimensional systems |
| title_fullStr | Financial risk analysis in multidimensional systems |
| title_full_unstemmed | Financial risk analysis in multidimensional systems |
| title_short | Financial risk analysis in multidimensional systems |
| title_sort | financial risk analysis in multidimensional systems |
| topic | system analysis financial risks multidimensional distribution copula families joint distribution dependency measures combined marginal distribution |
| topic_facet | system analysis financial risks multidimensional distribution copula families joint distribution dependency measures combined marginal distribution системний аналіз фінансові ризики багатовимірний розподіл сімейства копул спільний розподіл міри залежності комбінований маргінальний розподіл |
| url | https://es-journal.in.ua/article/view/364961 |
| work_keys_str_mv | AT trofymchukoleksandr financialriskanalysisinmultidimensionalsystems AT bidyukpetro financialriskanalysisinmultidimensionalsystems AT tymoshchukoxana financialriskanalysisinmultidimensionalsystems AT huskovavira financialriskanalysisinmultidimensionalsystems AT kroptyaarsen financialriskanalysisinmultidimensionalsystems |