Assessment Methods for Measuring Value at Risk Based on the Hit Function
The aim of this paper is to provide tools to aid the management process of Financial Risk. Backtesting is the necessary procedure to choose and to evaluate the stability of a value at risk (VaR) models. This paper presents some typical, statistical methods based on the hit function. Advantages and d...
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Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України
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| Цитувати: | Assessment Methods for Measuring Value at Risk Based on the Hit Function / W. Skrodzka // Электронное моделирование. — 2009. — Т. 31, № 6. — С. 55-64. — Бібліогр.: 9 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860199069229514752 |
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| author | Skrodzka, W. |
| author_facet | Skrodzka, W. |
| citation_txt | Assessment Methods for Measuring Value at Risk Based on the Hit Function / W. Skrodzka // Электронное моделирование. — 2009. — Т. 31, № 6. — С. 55-64. — Бібліогр.: 9 назв. — англ. |
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| description | The aim of this paper is to provide tools to aid the management process of Financial Risk. Backtesting is the necessary procedure to choose and to evaluate the stability of a value at risk (VaR) models. This paper presents some typical, statistical methods based on the hit function. Advantages and disadvantages of those methods are discussed in this paper. In the proposed approach to risk measurement, special attention is paid to the use of the Monte Carlo simulation along with the copula relation function in VaR methodology.
Предложены средства управления финансовыми рисками. Обратное тестирование — необходимая процедура при выборе и оценивании пригодности моделей оценки риска. Представлены некоторые типичные статистические методы на основе функции совпадения. Описаны преимущества и недостатки этих методов. В предложенном подходе к управлению рисками особое внимание уделено моделированию методом Монте-Карло, а также функции связки в методологии оценки величины рисков.
Запропоновано способи управління фінансовими ризиками. Зворотнє тестування — необхідна процедура для вибору та оцінки придатності моделей оцінювання ризиків. Наведено деякі типові статистичні методи базовані на функції збігу. Показано переваги та недоліки цих методів. Окрему увагу приділено моделюванню методом Монте-Карло, а також функції зв’язки у методології оцінки величини ризиків.
|
| first_indexed | 2025-12-07T18:09:39Z |
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W. Skrodzka, PhD
Czestochowa University of Technology
(Al. Armii Krajowej 19B,42-200 Czestochowa, Poland)
Assessment Methods for Measuring
Value at Risk Based on the Hit Function
The aim of this paper is to provide tools to aid the management process of Financial Risk. Back-
testing is the necessary procedure to choose and to evaluate the stability of a value at risk (VaR)
models. This paper presents some typical, statistical methods based on the hit function. Advan-
tages and disadvantages of those methods are discussed in this paper. In the proposed approach to
risk measurement, special attention is paid to the use of the Monte Carlo simulation along with
the copula relation function in VaR methodology.
Ïðåäëîæåíû ñðåäñòâà óïðàâëåíèÿ ôèíàíñîâûìè ðèñêàìè. Îáðàòíîå òåñòèðîâàíèå —
íåîáõîäèìàÿ ïðîöåäóðà ïðè âûáîðå è îöåíèâàíèè ïðèãîäíîñòè ìîäåëåé îöåíêè ðèñêà.
Ïðåäñòàâëåíû íåêîòîðûå òèïè÷íûå ñòàòèñòè÷åñêèå ìåòîäû íà îñíîâå ôóíêöèè ñîâïà-
äåíèÿ. Îïèñàíû ïðåèìóùåñòâà è íåäîñòàòêè ýòèõ ìåòîäîâ. Â ïðåäëîæåííîì ïîäõîäå ê
óïðàâëåíèþ ðèñêàìè îñîáîå âíèìàíèå óäåëåíî ìîäåëèðîâàíèþ ìåòîäîì Ìîíòå-Êàðëî, à
òàêæå ôóíêöèè ñâÿçêè â ìåòîäîëîãèè îöåíêè âåëè÷èíû ðèñêîâ.
K e y w o r d s: back- testing, value at risk, hit function, copula function.
Introduction. In times of economic crisis, risk management gains on its impor-
tance. The outcome of this crisis can be observed in every branch of business ac-
tivity. No matter which economic sector we are dealing with, risk identification
and management, including its assessment, should be a stable element of com-
pany management in the strategic perspective. Last years have confirmed the ne-
cessity of applying an approach to risk management which relies on the analysis
of possible negative divergences from prices or from rates of return. The meth-
odology based on such an approach is commonly known as downside risk mea-
sures. The most popular of the downside risk measures described in literature is
referred to as value at risk (VaR). Computing the VaR boils down to forecasting
the p-quantile of a certain cumulative distribution function [1]. If the function
were known, computing the exact VaR with the use of mathematical analysis or
numerical methods would be an easy task.
Owing to the peculiar properties of financial time series, however, it is not
easy to estimate the extreme quantiles of the cumulative distribution function in
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 6 55
question. One also has to make an estimate of the VaR itself, using the analyses
of rates of return from the risk factors and at the same time making appropriate
assumptions concerning their distribution. This, in turn, means building up a
whole econometric model. During the initial phase of the measurement process,
one does not have the knowledge that follows from the methodology of VaR
measurement — then again, the model that has been applied is considered ac-
ceptable. What is necessary then is a procedure called back testing, which allows
for the verification of the risk level that has been estimated. It also answers the
question whether the approach that has been applied is optimal as regards the cri-
terion that has been assumed. That kind of action becomes particularly important
in the situation of increased instability of financial markets.
The paper presents selected methods of assessing the VaR results that make
use of hit function analysis. Back testing was applied to VaR assessment com-
puted with the use of classical methods: the method of historical simulation, the
method of variance-covariance, and the Monte Carlo method of simulation using
copula relationships. The risk factors assumed are WIBOR 6M and WIBOR T/N
interest rates (Warsaw Interbank Offered Rate), listed in the inter-bank market.
This paper uses �-stable distributions and different kinds of functions of copula
relationships for modeling the real distributions of rates of return for WIBOR in-
terest rates. The purpose of this paper is presenting the tests for assessing the
quality of VaR measurements, assessing the independence of VaR failures and
answering the question which of the measurement methods concerned allows for
the more successful risk assessment.
Classical tests of VaR model assessment. Value at risk has been defined in the
publications by the authors such as [1—3]. Value at risk is a maximal amount of
money that can be lost as a result of portfolio investments having a fixed time span
and an assumed level of relevance. It can be expressed in the following way:
P W W{ }� � �
0
VaR � ,
where W0 — the recent value of e.g. instrument, portfolio or institution; W —
the value of e. g. instrument, portfolio or institution at the end of the period; � —
tolerance level. In this paper, VaR is considered as the appropriate quantile of
the distribution of rates of return.
VaR r t r tq F q
, ,
( ) ( )� �
�1
.
The definitions quoted above give the possibility of choosing several different
approaches towards the measurement of the VaR. Among the classical methods
we can distinguish: the historical method, Monte Carlo simulation, variance-
covariance method and the methods based upon the theory of extreme values [3].
The existing methods of estimating the VaR can be divided in the following
W. Skrodzka
56 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 6
manner : parametrical methods, non-parametrical methods, simulations, analyti-
cal methods.
Widely-understood methodology can lead to different estimates of VaR. A
necessary procedure that allows for the verification of the correctness of VaR
models is the process of back-testing. The process makes it possible for one to
test the correctness of generated results and pin point every potential mistake,
which in turn allows for the verification of the models adopted. It contributes to
the improvement of the quality of risk measurement. Historical analysis also al-
lows one to assess the effectiveness of the decisions that have been made with re-
spect to past periods of business activity, and to introduce changes in already-ex-
isting models. That type of analysis is very useful in improving the deci-
sion-making process with respect to market risk management. Absolute agree-
ment of actual and forecast positions is never possible, although the differences
between the loss forecast with certain probability and actual results should be
confined within the limits of error. The degree of error is the measurement of ef-
fectiveness of the methods employed and of the potential verifications of the as-
sumption that have been made.
Value at Risk models can be analyzed via the comparison of calculation re-
sults with the actual losses that are discussed in this paper, or via the assessment
of the quality of the econometric models forming the base of the VaR model.
In order to assess the effectiveness of VaR estimates, several hit functions
are used — most commonly [4—6]. Hit functions have been defined in the fol-
lowing manner:
I
r F q
r F q
t
p t rp t
p t rp t
�
�
�
�
�
�
�
1
0
1
1
, ( ),
, ( ).
, ,
, ,
It assumes the value 1 if at the moment t the rate of return from the portfolio is
smaller or equals the appropriate quantile — that is to say, when the VaR has
been exceeded, and 0 if the VaR has not been exceeded.
The test that is employed most frequently is called Proportion of Failures
Test — POF, proposed in 1995 by P. Kupiec. For the purposes of the test one has
to assume that for a given size of the theoretical sample, the number of failures
has a binominal distribution. Appropriate test statistics would be Kupca’s test —
Kupca [5]:
LR
q q
q q
uc
T T
T T
� �
�
�
�
�
�
�
�
�
2
1
1
0 1
0 1
ln
( )
( �) �
,
where
�q
T
T T
�
�
1
0 1
,
Assessment Methods for Measuring Value at Risk Based on the Hit Function
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 6 57
in which Ti — the number of periods in which It = i; q — assumed level of VaR
tolerance.
The LRuc statistics has the distribution of �
2
with one degree of freedom.
The critical value of Kupiec’s test for the most commonly considered level of
relevance 0.05 equals CV = 3.8415. The zero hypothesis concerning the VaR
model is rejected if LRuc > CV.
Using the idea of Markov’s chains, Christoffersen proposed test statistics of
the independence of VaR overdrafts [4]:
LR
q q
q q q
ind
T T T T
T T
� �
�
� �
� �
2
1
1 1
00 10 01 11
00 01
01 01
ln
( )
( ) (
11 11
10 11)
T T
q
�
�
�
�
�
�
,
where
q
T
T T
ij
ij
i i
�
�
0 1
, q
T T
T T T T
�
�
� � �
01 11
00 01 10 11
.
Here Tij is the number of periods in which It = j if It –1 = i. LRind has also�
2
distri-
bution with one degree of freedom. Owing to the fact that LRuc and LRind statis-
tics are independent, some authors [6] propose a mixed test LRmix, which takes
into account both the number of VaR failures and the time span between those
failures:
LR LR LRmix uc ind� � .
Such statistics have the distribution�
2
with two degrees of freedom.
The tests presented here will constitute the basis of the assessment of VaR
models further in the paper. In his paper, Jorion also presents the Time Between
Failures Test [3]:
LR
q q
q q
q
v
vTBF
� �
�
�
�
�
�
�
�
�
� �
�
�
2
1
1
2
11
1
1
1 1
1
ln
( )
� ( � )
ln
( �
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
q
q q
v
i i
v
i
T i
i
)
� ( � )
1
1
1 1
1
, � /q vi i�1 ,
where vi — time between (i – 1) and ith failure.This statistics also has �
2
distri-
bution with Ti degree of freedom.
The test in which the failures occurring at the time t are independent of ear-
lier failures and of the information received at the time t – 1, and of the VaR, is
the test proposed by Engel and Manganelli in 2002 (Dynamic Quantile Test
(DQ)) [7].With the help of that test, one can identify the «non-acceptable» result
of value at risk measurement, defined below, which cannot be eliminated by
classical tests of failure number and independence. When VaR value is analyzed
VaR
with the probability
with the proba
t q
X q
X
( )
,
�
�
�
1
bility q
�
�
W. Skrodzka
58 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 6
the following regression equation is analyzed:
I q q I q q ft i t i p t
i
p
p j t j( ) ( ) ( ) (
,
� � � �
� �
�
� � ��0 1
1
1 1
� � � �VaR ) �
�
� � t
j
n
1
,
where�t j�1,
— information given to the investor at the time t – 1. Rejecting the
zero hypothesis: H q q
0 0
: � ,�i �0, i p n� � �1 2 1, , ..., means lack of the correct-
ness of the VaR model.
The DQ test makes it possible to detect any deviations from the independ-
ence of the failures described by Markov’s chains of the rank higher than one.
Estimating the VaR level with the use of Monte Carlo simulation in con-
nection with copula function. In this paper, the level of VaR has been estimated
with the use of Monte Carlo simulation in connection with copula function.
The definition of copula function as well as the description of its character-
istics is presented in detail by Nelsen in his work [8]. In a two-dimensional case
which can be generalized onto a multi-dimensional case, the copula function is
defined in the following manner.
Definition. A two-dimensional function C :[ . ] [ . ]01 01
2
� is called a copula
function on condition that it meets the following conditions:
C (u, v) is ascending as regards both the variable u and the variable v;
C (u, 0) = C (0, v) = 0;
C (u, 1) = u;
C (1, v) = v;
� �u u v v
1 2 1 2
01, , , [ . ] such, that u u
1 2
� and v v
1 2
� , then
C u v C u v C u v( , ) ( , ) ( , )
2 2 2 1 1 2
� � � �C u v( , )
1 1
0.
The calculations have been done for a two-summand portfolio with the risk
factors being WIBOR 6M i WIBOR T/N rates of interest. The calculations have
been based on the data taken from the inter-bank financial market. The VaR that
has been estimated for a two-summand portfolio at the time t has been compared
with the actual loss of the portfolio’s value during the period of [t, t + 1]. If the ac-
tual loss exceeds the VaR that has been calculated, the so-called failure is ob-
tained.
1
The sum of failures for the whole period of time divided by the length of
that period of time gives the relative number of failures. In a correctly-working
model, the number should more or less equal the level of VaR tolerance. Because
of the fact that estimating the parameters of �-stable distributions and of the cop-
ula functions was made once every 10 session days (once every two calendar
Assessment Methods for Measuring Value at Risk Based on the Hit Function
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 6 59
1
Committee on Banking Supervision in Poland (KNB), basing on the number of failures of a
1-day VaR (0.99), observed in the recent 250 days, lays down the spheres of VaR model quality.
weeks), the VaR was determined for all methods at the same time. VaR was de-
termined at the trust level of 0.95.
All calculations have been done with the use of the programme R: Version
2.2.0
1
. In the empirical research, the Bond functions of the Archimedean family
have been used. The functions employed include Clayton copula, Frank copula
and Gumbel copula. The empirical research has been conducted on the basis of
the data simulated by Monte Carlo method. Marginal distributions have been
generated with the use of �-stable distributions having the coefficients estimated
from historical data. The parameters of stable distributions have been estimated
with the use of a programme called STABLE written by J. Nolan ver. 3.04. Also,
the method of quantile estimation and the method of greatest plausibility have
been used. In the process of estimation as well as in the measurement itself, a
movable window was used.
That type of action allows for the comparison of the effectiveness of VaR
for different methods, but for the same amount of historical data used; 250 mar-
ket quotations have been assumed as the length of a movable window.
2
It is the
length of the period taken into account by KNB in defining the quality areas of
the VaR model for banking institutions. On the basis of 250 market quotations,
estimations of two �-stable distributions have been made for a series of rates of
return of two selected instruments of the financial market, constituting the sum-
mands of the investigated portfolio. In the case when the method of the greatest
plausibility has been used, the following assumptions have been made: � > 0.5;
when � < 0.8, the method is changed into a quantile-based one; when � �
�[0,993; 1] � = 1 is assumed; when ��� [1; 1,007] � = 1.007 is assumed.
The assumptions stem from the difficulties of numerical nature that arise during
the calculation of the density and the cumulative distribution function of an �-stable
distribution both in the case whaen � > 0.5 and when it approaches unity.
In the majority of cases, the parameters estimated with the use of particular
methods move within the same range of values. The location parameter�, whose
values calculated with the use of the quantile method differ as for the sign when
compared with the method of greatest plausibility, constitutes an exception. Of
some curiosity are also great estimation differences of almost every parameter
between the chosen subsamples, which indirectly indicates a change in market
mechanisms in particular subperiods. Also, the researchers have observed cer-
tain anomalies in the process of estimating � and � parameters with the use of
ML method. Such a situation occurs if the value of the estimated parameter � is
W. Skrodzka
60 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 6
1
The R Foundation for Statistical Computing. — Copyright 2005. ISBN 3-900051-07-0.
2
An average number of quotations per annum.
less than 0.8. Then, the process of estimation is broken. In such case, the empiri-
cal research has been made with the use of quantile estimation. The estimated
value of shape coefficient � turned out to be less than 2, which means the exis-
tence of the so-called «thick tails». Those results indicate that using the normal
distribution for the empirical approximation of the distribution of rates of return
of WIBOR interest rates can lead to the bias of the result.
Then, with the use of greatest plausibility method, the parameter � of a cho-
sen copula function has been estimated. The parameter has been determined
on the basis of Kendal calculated for a pair of rates of return of WIBOR
6M and WIBOR T/N interest rates. When comparing the results for different
bond functions, one can note higher values of the � coefficient for the Frank
copula bond function.
In the next stage, pairs of pseudo-random numbers have been sampled hav-
ing the overall distribution determined by the copula function with a pre-esti-
mated � parameter and with marginal distributions determined by the estimated
parameters of �-stable distributions. The size of the random sample generated
with the help of Monte Carlo Method amounted to 10000 two-dimensional quo-
Assessment Methods for Measuring Value at Risk Based on the Hit Function
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 6 61
2002 2003 2004 2005 2006
1.2
1.1
1.0
0.9
Value at Risk for the portfolio of WIBOR T/N and WIBOR 6M with the risk factors, simulated
with the Monte Carlo method with Clayton copula bond function and with stable marginal distri-
butions
Source: the calculations made by the author with the programme R: Version 2.2.0.
VaR
Years
tations for the portfolio and for each of the bond functions. By applying the for-
mula for the rate of return from the portfolio, 10000 simulations of the rate of
return from a two-dimensional portfolio have been obtained in each of the
subperiods.
The estimated VaR has been obtained on the basis of the simulated quantile
of the distribution of the rates of return, with the level of trust determined as 0.95.
This process has been repeated 95 times for each of the subperiods, created ana-
logically to the estimation process. The change of sampling parameters takes
place every 10 days. During the process of generating random numbers from a
multi-dimensional distribution with the use of copula bond functions, there ap-
peared a problem connected with the effectiveness of the calculations. The aver-
age time of the calculations connected with a single simulation amounted to 0.2
sec. For the purpose of increasing the effectiveness of the calculation procedure,
cubic splins have been applied.
Figure presents a sample simulation using Monte Carlo method with Clay-
ton copula bond function and with stable marginal distributions of the VaR in
particular subperiods for WIBOR T/N and for WIBOR 6M. The circles present
actual values of the portfolio calculated on the basis of empirical data. The black
line stands for the VaR level estimated with the help of a simulation. The circles
below the line stand for the failures of the simulated VaR.
Monte Carlo simulation method coupled with Clayton copula bond func-
tion, Frank copula and Gumbel copula for pairs of rates of return of WIBOR T/N
and WIBOR 6M interest rates gave similar result in the case of stable marginal
distributions.
Verifying the quality of the adopted models for the measurement of the
VaR — back-testing. On the basis of Monte Carlo simulation methods pre-
sented above coupled with the copula bond function and on the basis of classical
methods of measurement, we were able to calculate the VaR in particular
subperiods, identify the instances of failures and determine the value of the test
of the number of failures and their independence. Table presents the results of
empirical research as the values of the appropriate test statistics, for the pairs of
rates of return of WIBOR T/N and WIBOR 6M interest rates, for particular
methods of VaR calculation and with the tolerance level of 0.05.
In Table below, the first column shows the methods of VaR measurement
that have been taken into account in the research. The remaining columns pres-
ent the values of the statistics for Kupiec test, Christoffersen test and for the
mixed test. The values of the empirical statistics in bold stand for the cases in
which we have to reject the zero hypothesis of the correctness of the VaR model.
Furthermore, on the basis of Kupiec test we can claim that for the size of the sam-
ple amounting to T = 950, for the VaR tolerance level amounting to 0.05 and for
the relevance level of the test amounting to 0.05, the range of the number of fail-
W. Skrodzka
62 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 6
ures determining the non-critical area
1
equals [35; 61] with the expected value of
the number of failures amounting to 48. The underlined italics indicate the best
results that have been obtained.
For the investigated portfolio with the risk factors being the WIBOR T/N
and WIBOR 6M rates of interest, Monte Carlo simulation methods using copula
functions based on stable distributions gave similar results, approaching the
value of the expected number of failures. The best result was obtained for
Monte Carlo simulation with Clayton copula bond function and with stable mar-
ginal distributions. It means that within a given method the lowest result has
been obtained from the failures statistics tests and their independence, and from
the mixed test. In the table presented above, those results have been italicized
and underlined.
Correct results have also been obtained with the use of historical simulation.
In the case of variance-covariance method, the results obtained diverge signifi-
cantly from the expected number of failures. Those methods have noticeably in-
creased the value of the VaR parameter, and because of that, the number of fail-
ures was significantly lower than the expected one. This might be explained by
the fact that the assumption of normal distributions of rates of return has been
made in the situation when empirical distributions have much «thicker tails».
That is a very important observation from the point of interest rate risk measure-
ment. Using stable distributions as marginal distributions allows one to get
better results.
Assessment Methods for Measuring Value at Risk Based on the Hit Function
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 6 63
Method
Number
of failures
Relative
number
of failures
LRuc LRind LRmix
Historical simulation 45 0.0474 0.140869 0.00943 0.150299
Variance-covariance method 29 0.0305 8.757494 0.014963 8.772458
Monte Carlo simulation:
Gumbel copula; marginal
distributions stable 43 0.0453 0.462855 0.607275 1.07013
Monte Carlo simulation:
Frank copula; marginal
distributions stable 49 0.0516 0.049372 0.092143 0.141515
Monte Carlo simulation:
Clayton copula; marginal
distributions stable 47 0.0495 0.005559 0.053392 0.05895
Source: the calculations made by the author with the programme R: Version 2.2.0.
The values of test statistics for the rates of return of WIBOR T/N
and WIBOR 6M interest rates
1
The area of the assumption of the model correctness hypothesis.
Conclusions. In the paper we have compared the classical approach to-
wards the calculation of the VaR: with the use of historical simulation and with
the use of variance-covariance method with a much more complicated method of
determining the unknown quantile of a certain combined distribution with the
use of Monte Carlo simulation with copula bond function. Such a method gives
many possibilities of choosing the marginal distributions of the same or of dif-
ferent types. In this paper the author has used �-stable distributions, which, due
to their properties better fit the empirical data than the normal distributions.
In order to compare particular methods of VaR measurement, the back-test-
ing procedure has been applied. The results of the failure test that have been ob-
tained confirmed the effectiveness of Monte Carlo methods with copula bond
function and with �-stable distribution when it comes to measuring the risk of
the rates of interest.
The methodology of assessing the effectiveness of VaR measurement is
quite wide. The methods presented in the paper are the most popular ones. They
are not, however, free from drawbacks. The classical tests are characterized by
low power. Their undeniable merit is, however, their simplicity and intuitive in-
terpretation corresponding to the very definition of the VaR and to VaR failures.
Çàïðîïîíîâàíî ñïîñîáè óïðàâë³ííÿ ô³íàíñîâèìè ðèçèêàìè. Çâîðîòíº òåñòóâàííÿ — íåîá-
õ³äíà ïðîöåäóðà äëÿ âèáîðó òà îö³íêè ïðèäàòíîñò³ ìîäåëåé îö³íþâàííÿ ðèçèê³â. Íàâåäåíî
äåÿê³ òèïîâ³ ñòàòèñòè÷í³ ìåòîäè áàçîâàí³ íà ôóíêö³¿ çá³ãó. Ïîêàçàíî ïåðåâàãè òà íåäîë³êè
öèõ ìåòîä³â. Îêðåìó óâàãó ïðèä³ëåíî ìîäåëþâàííþ ìåòîäîì Ìîíòå-Êàðëî, à òàêîæ
ôóíêö³¿ çâ’ÿçêè ó ìåòîäîëî㳿 îö³íêè âåëè÷èíè ðèçèê³â.
1. Jajuga K. Metody Ekonometryczne i Statystyczne w Analizie Rynku Kapita owego. —
Wroclaw: AE, 2000.
2. Best P. Warto nara�zona na ryzyko. Obliczanie i wdra anie modelu VaR.— Krakow: Dom
Wydawniczy ABC, 2000.
3. Jorion P. Value at Risk. 2
nd
edition. — New Jersey: McGraw-Hill, 2001.
4. Christoffersen P. Evaluating Interval Forecasts // International Economic Review.—
1998. — 39.
5. Kupiec P. Techniques for Verifying the Accuracy of Risk Management Models//J. of
Derivatives. — 1995. — 2.
6. Piontek K. A Survey and a Comparison of Back Testing Procedures. / In P. Chrzan,
Matematyczne i Ekonometryczne Metody Oceny Ryzyka Finansowego/ Prace Naukowe AE
w Katowicach. — Katowice, 2007.
7. Sarma M., Thomas S., Shah A. Selection of Value-at-Risk Models, 2003. — ideas.repec.org/
s/jof/jforec.html
8. Nelsen R. B. An Introduction to Copulas. — N Y: Springer Verlag, 1999.
9. Piontek K. Wykorzystanie wielorownaniowych modeli AR-GARCH w pomiarze ryzyka
metod VaR, Modelowanie preferencji a ryzyko/Prace Naukowe AE w Katowicach. —
Katowice, 2005.
Submitted on 25.08.09
W. Skrodzka
64 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 6
|
| id | nasplib_isofts_kiev_ua-123456789-101528 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0204-3572 |
| language | English |
| last_indexed | 2025-12-07T18:09:39Z |
| publishDate | 2009 |
| publisher | Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України |
| record_format | dspace |
| spelling | Skrodzka, W. 2016-06-04T13:15:30Z 2016-06-04T13:15:30Z 2009 Assessment Methods for Measuring Value at Risk Based on the Hit Function / W. Skrodzka // Электронное моделирование. — 2009. — Т. 31, № 6. — С. 55-64. — Бібліогр.: 9 назв. — англ. 0204-3572 https://nasplib.isofts.kiev.ua/handle/123456789/101528 The aim of this paper is to provide tools to aid the management process of Financial Risk. Backtesting is the necessary procedure to choose and to evaluate the stability of a value at risk (VaR) models. This paper presents some typical, statistical methods based on the hit function. Advantages and disadvantages of those methods are discussed in this paper. In the proposed approach to risk measurement, special attention is paid to the use of the Monte Carlo simulation along with the copula relation function in VaR methodology. Предложены средства управления финансовыми рисками. Обратное тестирование — необходимая процедура при выборе и оценивании пригодности моделей оценки риска. Представлены некоторые типичные статистические методы на основе функции совпадения. Описаны преимущества и недостатки этих методов. В предложенном подходе к управлению рисками особое внимание уделено моделированию методом Монте-Карло, а также функции связки в методологии оценки величины рисков. Запропоновано способи управління фінансовими ризиками. Зворотнє тестування — необхідна процедура для вибору та оцінки придатності моделей оцінювання ризиків. Наведено деякі типові статистичні методи базовані на функції збігу. Показано переваги та недоліки цих методів. Окрему увагу приділено моделюванню методом Монте-Карло, а також функції зв’язки у методології оцінки величини ризиків. en Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України Электронное моделирование Assessment Methods for Measuring Value at Risk Based on the Hit Function Методы оценки для измерения цены риска на основе функции попадания Article published earlier |
| spellingShingle | Assessment Methods for Measuring Value at Risk Based on the Hit Function Skrodzka, W. |
| title | Assessment Methods for Measuring Value at Risk Based on the Hit Function |
| title_alt | Методы оценки для измерения цены риска на основе функции попадания |
| title_full | Assessment Methods for Measuring Value at Risk Based on the Hit Function |
| title_fullStr | Assessment Methods for Measuring Value at Risk Based on the Hit Function |
| title_full_unstemmed | Assessment Methods for Measuring Value at Risk Based on the Hit Function |
| title_short | Assessment Methods for Measuring Value at Risk Based on the Hit Function |
| title_sort | assessment methods for measuring value at risk based on the hit function |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/101528 |
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