Classification of methods for risk measures VaR and CVaR calculation and estimation
A systematic classification of the existing approaches for popular risk measures VaR and CVaR calculating and estimating is fulfilled. A review of the most used methods is done. For convenience, the considered methods are reduced to common econometric designations and concepts, guidance on the use o...
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Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України
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irk-123456789-1402492018-06-27T03:03:14Z Classification of methods for risk measures VaR and CVaR calculation and estimation Zrazhevska, N.G. Zrazhevsky, А.G. Математичні методи, моделі, проблеми і технології дослідження складних систем A systematic classification of the existing approaches for popular risk measures VaR and CVaR calculating and estimating is fulfilled. A review of the most used methods is done. For convenience, the considered methods are reduced to common econometric designations and concepts, guidance on the use of the methods is proposed. The correctness of the considered methods is numerically confirmed. Проведено системну класифікацію існуючих підходів знаходження і оцінювання популярних мір ризику VaR і CVaR. Проведено огляд найбільш поширених методів. Для зручності користування розглянуто методи, зведені до спільних економетричних позначень і понять, наведено рекомендації щодо використання методів. Коректність розглянутих методів підтверджено в результаті числової апробації. Проведена системная классификация существующих подходов нахождения и оценивания популярных мер риска VaR и CVaR. Проведен обзор наиболее используемых методов. Для удобства пользования рассмотренные методы сведены к общим эконометрическим обозначениям и понятиям, приведены рекомендации по использованию методов. Корректность предложенных методов подтверждена в результате численной апробации. 2016 Article Classification of methods for risk measures VaR and CVaR calculation and estimation / N.G. Zrazhevska, А.G. Zrazhevsky // Системні дослідження та інформаційні технології. — 2016. — № 3. — С. 126-141. — Бібліогр.: 24 назв. — англ. 1681–6048 DOI: 10.20535/SRIT.2308-8893.2016.3.11 http://dspace.nbuv.gov.ua/handle/123456789/140249 519.6:519.81 en Системні дослідження та інформаційні технології Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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Математичні методи, моделі, проблеми і технології дослідження складних систем Математичні методи, моделі, проблеми і технології дослідження складних систем |
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Математичні методи, моделі, проблеми і технології дослідження складних систем Математичні методи, моделі, проблеми і технології дослідження складних систем Zrazhevska, N.G. Zrazhevsky, А.G. Classification of methods for risk measures VaR and CVaR calculation and estimation Системні дослідження та інформаційні технології |
| description |
A systematic classification of the existing approaches for popular risk measures VaR and CVaR calculating and estimating is fulfilled. A review of the most used methods is done. For convenience, the considered methods are reduced to common econometric designations and concepts, guidance on the use of the methods is proposed. The correctness of the considered methods is numerically confirmed. |
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Article |
| author |
Zrazhevska, N.G. Zrazhevsky, А.G. |
| author_facet |
Zrazhevska, N.G. Zrazhevsky, А.G. |
| author_sort |
Zrazhevska, N.G. |
| title |
Classification of methods for risk measures VaR and CVaR calculation and estimation |
| title_short |
Classification of methods for risk measures VaR and CVaR calculation and estimation |
| title_full |
Classification of methods for risk measures VaR and CVaR calculation and estimation |
| title_fullStr |
Classification of methods for risk measures VaR and CVaR calculation and estimation |
| title_full_unstemmed |
Classification of methods for risk measures VaR and CVaR calculation and estimation |
| title_sort |
classification of methods for risk measures var and cvar calculation and estimation |
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Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України |
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2016 |
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Математичні методи, моделі, проблеми і технології дослідження складних систем |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/140249 |
| citation_txt |
Classification of methods for risk measures VaR and CVaR calculation and estimation / N.G. Zrazhevska, А.G. Zrazhevsky // Системні дослідження та інформаційні технології. — 2016. — № 3. — С. 126-141. — Бібліогр.: 24 назв. — англ. |
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Системні дослідження та інформаційні технології |
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AT zrazhevskang classificationofmethodsforriskmeasuresvarandcvarcalculationandestimation AT zrazhevskyag classificationofmethodsforriskmeasuresvarandcvarcalculationandestimation |
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2025-07-10T10:08:16Z |
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2025-07-10T10:08:16Z |
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| fulltext |
N.G. Zrazhevska, А.G. Zrazhevsky, 2016
126 ISSN 1681–6048 System Research & Information Technologies, 2016, № 3
126
UDC 519.6 : 519.81
DOI: 10.20535/SRIT.2308-8893.2016.3.11
CLASSIFICATION OF METHODS FOR RISK MEASURES VAR
AND CVAR CALCULATION AND ESTIMATION
N.G. ZRAZHEVSKA, А.G. ZRAZHEVSKY
Abstracts. A systematic classification of the existing approaches for popular risk
measures VaR and CVaR calculating and estimating is fulfilled. A review of the
most used methods is done. For convenience, the considered methods are reduced to
common econometric designations and concepts, guidance on the use of the methods
is proposed. The correctness of the considered methods is numerically confirmed.
Keywords: estimation, value-at-risk, conditional value-at-risk, structural-hierarchical
scheme, systematization, classification.
INTRODUCTION
Financial and economic crises of the end of the XX-beginning of the XXI century
shows the necessity of further development of the risk theory. Determination and
estimation of the possible risk arising from operational, financial and other activ-
ity of the company are among the main objectives of the risk management. Risk
measures VaR and CVaR are widely used to solve this problem.
There are many works dedicated to risk measure VaR. They analyze its
properties, advantages and disadvantages, the methods for its estimating [1–4 ].
VaR has become a standard, widely used risk measure because of its conceptual
simplicity, ease of calculation and the availability of a sufficiently large number
of standardized formulas and methods for calculation. At the same time, this risk
measure has two major drawbacks: VaR is not coherent in the sense of [5], it does
not have the sub-additivity property and VaR does not allow to determine the size
of the potential losses that exceed the given level [6]. CVaR (ConditionalVaR) or,
as it is called ES (Expected Shortfall), has been introduced to solve these prob-
lems [5, 6]. Unlike VaR, CVaR is a coherent measure. Being a convex function,
CVaR can be used in the optimization procedures [6]. The article [7] describes the
methods, based on analytical expressions for CVaR calculation. The articles [8, 9]
provide a detailed comparative analysis of risk measures VaR and CVaR. The
large number of approaches and methods complicates the selection of the optimal
way for solving the problem. Systematization, classification and comparison of
different methods for risk measures evaluation lead to the use of systematic meth-
odology [10].
This paper discusses the main approaches for risk measures VaR and CVaR
evaluation for a random variable, based on different statistical and econometric
methods. Prices and other characteristics of financial instruments are usually con-
sidered as random variables. It allows to use random models.
The methods of risk measures estimations may be classified on the basis of
statistical and stochastic approaches. This paper proposes the structural ― hierar-
chical scheme (Fig. 1) with classification of the most popular methods. The
scheme helps the user to choose a particular method.
Classification of methods for risk measures var and cvar CALCULATION ANd estimation
Системні дослідження та інформаційні технології, 2016, № 3 127
Analysis of methods for VaR and CVaR estimating allowed to formulate the
decision-making procedure for the choice of the method of static VaR and CVaR
evaluation, depending on the research objectives and the characteristics of the
analyzed data (Fig. 2).
Analytical
formulas
for VaR,
CVaR
The Monte
Carlo method
for VaR, CVaR
estimates
Analytical
formulas
for VaR,
CVaR
The Monte
Carlo method
for VaR, CVaR
estimates
Using of the
empirical
quantile
Using
the GPD
function
Using of the
empirical cdf
Using of the
analytical cdf
Kernel
estimating
«block maxima»
models and GEV
function
POT-models
Cdf
estimating
Quantile
estimating
Optimization
method
Decomposition
of the sample in
functional series
Full cdf
estimating
Estimating of
the distribution
tail
Wavelet
decomposition
Fourier
decomposition
Method for VaR and CVaR estimating
Fig.1. The structural ― hierarchical scheme for VaR and CVaR estimating
Analysis of the goal
of the study
Analysis of the data origin
Basic statistical data analysis
Data analysis for the presence of extreme values
Extreme values are statistically significant
and require separate analysis
Extreme values are ejections
and may be ignored
Wavelet –
decomposition
Optimization
method
Using of the
GEV, GPD
functions
Fourier
decomposition
Determination of
the type of cdf and the
parameters estimation
Explicit formulas
for VaR, CVaR
CVaR
Using the Monte
Carlo method
The Empirical
method
The type of cdf is
known, the parame-
ters are estimated
The type of cdf
is unknown
Explicit formulas
for VaR, CVaR
Using the Monte
Carlo method
The HS
method
Kernel
estimating
Fig. 2. The decision-making procedure for the choice of the method of static VaR and
CVaR evaluation
N.G. Zrazhevska, А.G. Zrazhevsky
ISSN 1681–6048 System Research & Information Technologies, 2016, № 3 128
Sections 2–4 contain the mathematical methods that we use to concretize the
methods of static VaR and CVaR estimating (Fig.1). Sections 5 and 6 show the
results of numerical testing of the methods and the comparative analysis of the
results.
KEY DEFINITIONS
Let Y be a random variable describing the portfolio returns. For a given confi-
dence level )1;0( let us consider the )1( th quantile of the returns distribu-
tion:
}.1)(|sup{)()1( yYPyYy
From the viewpoint of econometric )()1( Yy determines the minimum
value of the returns Y with probability )1( . If, for example, 95,0 then
)()1( Yy (may be negative) with probability 95%, determines the minimum re-
turns. For ease of understanding econometrics operate with the concept
)(VaR Y , defined as:
)()(VaR )1( YyY
.
So, )(VaR Y with probability )1( defines the limit value of the loss
(the sign «–»).
Some works operate with losses. Let X denotes a random variable describ-
ing the portfolio losses. Then YX , the quantile of the cumulative distribution
function (cdf) of the loss function is defines as })(|{inf)()( xXPxXx
and
)()(VaR )( XxX
. (1)
So )()(),(VaR)(VaR )1( YyXxXY
.
From the viewpoint of statistics from (1) we see that:
)()(VaR 1
XFX , (2)
where )(1
XF is the inverse function of the cdf of the random variable X .
For a given confidence level )1;0( the risk measure )(CVaR X can be
defined as an average expected value of the loss with probability :
)](VaR|[)(CVaR XXXEX . (3)
In the case if ));0((),(VaR X is integrability, )(CVaR X may be
defined as:
1
)(VaR
1
1
)(CVaR dXX .
The paper [11] gives the following definition of CVaR:
Classification of methods for risk measures var and cvar CALCULATION ANd estimation
Системні дослідження та інформаційні технології, 2016, № 3 129
)}](VaR{I[[
1
)(CVaR XXXEX
))](VaR()(VaR)(VaR XXPXX ,
where }I{ denotes the indicator function.
Consider an alternative definition of the risk measures [6]. Let X be a ran-
dom variable with the probability density function (pdf) )(xp , R is a scalar,
the function ),( xf for each fixed is a random variable with pdf )(xp . Let us
introduce the function ),( ― the probability that ),( xf will not exceed the
given level :
),(
)(),(
xf
dxxp .
Here ),( is a loss distribution function. Then )(VaR X and
)(CVaR X may be defined as:
}),(|min{)(VaR RX , (4)
)(VaR),(
)(),(
1
1
)(CVaR
Xxf
dxxpxfX . (5)
Comments. A great number of mathematical and econometric studies on
VaR and CVaR, leads to a problem of a confidence level choice. Note that the
events «loss is larger than given level» and «loss do not achieve the level» are a
complete group of events with probabilities and 1 respectively. In this pa-
per, the confidence level is the th quantile of the loss function (1), measures of
risk are designated as VaR , CVaR . In practice, the level is )1;9,0[ .
Nowadays there are many methods of VaR and CVaR estimating. In this pa-
per we have no goal to review all of them. We propose a classification system of
the best known and most commonly used approaches (Fig. 1) and briefly give the
mathematical methods (description) that is required for the hierarchical scheme
(Fig. 2) specification.
THE METHODS OF VAR AND CVAR ESTIMATING USING THE CDF
OF A RANDOM VARIABLE
Let X be a random variable. nXXX ,...,, 21 is a sample of its values,
)()2()1( ,..., nXXX is the order statistics in ascending order. We assume that
the distribution function of X can be defined in an analytical or empirical form.
Then definitions of VaR and CVaR (2) or (3) are used respectively.
VaR and CVaR estimating using the full CDF
Methods with the use of the empirical CDF
This group of methods includes, first of all, the Historical Simulation
methods - HS [7]. This method is based on the construction of empirical distribu-
tion function using historical data. Then:
N.G. Zrazhevska, А.G. Zrazhevsky
ISSN 1681–6048 System Research & Information Technologies, 2016, № 3 130
)/]([VaR nnX , (6)
])[(/CVaR
][
)(
nn
n
ni
iX . (7)
Estimate (7) can be modified. The paper [12] proposes the following esti-
mate:
;}I{
][1
1
CVaR
1
)1]([
n
i
nii XXX
n
the paper [13] (for returns) gives the estimate:
;
)1(
)]1([
1
)]1([
1
CVaR )1)]1(([
)]1([
1
)(
n
n
i
i Y
n
n
Y
n
and the paper [14] (for returns):
,)1(,)1(
,)1(,
CVaR
n:1)1(n:)1(
n:)1(
ZnYY
ZnY
nn
n
where nkYY
k
Y knk ,1),...(
1
)()1(: .
The advantage of the HS method is its simplicity in realization. Furthermore,
the method does not require a prior assumptions about the type of the cdf. The HS
method is very popular among economists and is implemented in many standard
packages for risk measures calculating, such as Matlab, PSG. However, using the
empirical cdf we suppose that the statistical characteristics of a random variable
will be stable. Empirical distribution function is smooth enough in the vicinity of
the mean value, and demonstrates jumps at the tails due to the relatively small
number of extreme values in the sample. To overcome this problem the large vol-
ume samples must be analyzed. The Bank of International Settlements recom-
mends to use samples consisting of not less than 250 data [15].
Let us consider the Rockafellar–Uryasev discrete method [16], based on the
empirical distribution. Let for a random variable we have a variation row
)()2()1( ... nXXX with corresponding empirical probabilities 0kp . Let k
is a single index such that
1
11
k
k
k
k
k
k pp . Then:
)(VaR
kX , .
1
1
CVaR
1
)()(
1
k
kk
kkk
k
k
k XpXp
Methods with the use of the analitycal cdf
If we can assume that a random variable has a certain type of cdf and we can
evaluate the parameters of the distribution, analytic formulas can be used to find
VaR and CVaR estimates.
Classification of methods for risk measures var and cvar CALCULATION ANd estimation
Системні дослідження та інформаційні технології, 2016, № 3 131
Let X denote a standard normal random variable with mean and vari-
ance 2 . Then [17]:
)(VaR 1
,
1
))((
CVaR
1
. (8)
Hereinafter, )( is the standard normal pdf and )( is the standard nor-
mal cdf.
The standard normal pdf is symmetrical. At the same time, many financial
instruments demonstrate the skewness of the distribution. The most popular and
the most widely used of skewed extensions of the normal distribution is the skew-
normal distribution due to Azzalini [18]. The cdf of this distribution is given by:
,,2)(
x
T
x
xF
where RRRx ,0,, ,
a
dx
x
xh
ahT
0
2
22
1
}2/)1(exp{
2
1
),( .
For a skew normal random variable, VaR is defined as the unique root of
the equation )(xF , and the CVaR can be expressed as:
)()(2)(
2
CVaR
yyz ,
where
yz
VaR
y 22 1,,1/ .
If X is a random variable with the pdf:
L
i i
i
i
x
xf
1
2
)(
i
i
i
x
, where the weights i are non-negative and sum to one
( 1,0
1
L
i
ii ), for CVaR takes place the formula:
L
i
iiiii
i
ii yyz
1
,,, )()(2)(
2
CVaR ,
where
i
ii
iiii
i
i
i
x
yyz
,
,,
2
,
2
,1,
1
, ix , is the root of
i
i
i
i
i x
T
x
,2 ,
i
i
i
i
ii
i T ,
VaR
2
VaR
.
As before VaR is the root of )(xF .
N.G. Zrazhevska, А.G. Zrazhevsky
ISSN 1681–6048 System Research & Information Technologies, 2016, № 3 132
Consider another version of representation of analytic formulas for VaR and
CVaR.
Consider a random variable
X
X , where a random variable X has
scailedlocalt distribution with parameters , and 2 degrees of free-
dom, X has the standard Student's t distribution with degrees of freedom and
2
)var(,0)(
XXE . Then [17]:
)(VaR 1
t ,
1
))((
1
))((
CVaR
211
ttg
, (9)
where )(),( tg denote, respectively, the pdf and the cdf of a Student's t random
variable.
To take into account skewed of the pdf the article [19] proposes the Az-
zalini's skewed t distribution. This distribution is given by the pdf
.0;,)
1
()(2)(
21
Rx
x
xgxtxf
Let (x)F be a corresponding cdf. Then VaR is the root of (x)F , and
VaR
dx
x
xgxtx
21
1
)(2CVaR .
A more complete list of analytical formulas for VaR and CVaR, for different
types of pdf can be found in [7].
The advantage of this approach is the availability of analytical formulas for
risk measures calculating. However, the definition of the type of the pdf and esti-
mation of its parameters may require considerable efforts.
Monte Carlo and Richardson’s method for risk measures estimating
If we know the type of a random variable distribution and can estimate its
parameters the Monte Carlo method can be used for VaR and CVaR estimating.
The Monte Carlo method is based on the obtaining of a large number of realiza-
tions of a random variable such that their probability characteristics coincide with
the estimates obtained by other methods. The principal difference between the
Monte Carlo method and the method of historical simulation is that the original
sample is generated on the basis of a model.
The Richardson's method for CVaR estimating on the basis of the Monte
Carlo method is described in [7]. The method is formulated as the following algo-
rithm.
1. Using the known cdf generate samples NXX ,...,1 by the Monte Carlo
method.
2. Estimate the
1
CVaR by the HS method.
Classification of methods for risk measures var and cvar CALCULATION ANd estimation
Системні дослідження та інформаційні технології, 2016, № 3 133
3. Repeat steps 1 and 2 M times and compute the estimates
j
CVaR ,
Mi ,1 ( M is the number of simulated samples).
4. Compute the mean: .CVaR
1
1
M
i
i
N M
m
5. Set 1,1, knmS
nNn for some k and 121 ..., kNNN .
6. Estimate CVaR as:
)1/(1...2/11
....
)1/(1...2/11
1...11
)1/(1...2/11
....
)1/(1...2/11
...
CVaR
121
k
k
k
k
SSS k
.
Kernel method
The kernel method and its modifications are nonparametric methods of pdf
estimation. They are based on the integral smoothing with a given kernel of em-
pirical histogram and do not require a priori information about the type of distri-
bution. The advantage of kernel methods is their independence from any informa-
tion about the data source, as well as the availability of simple semi-analytic
expressions for the estimates [12].
Let )(K denote a symmetric kernel, smoothing function
t
duuKtG )()( ,
)/()( htGtGh , where h is a suitable bandwidth. Then the smoothed estimate
of the loss distribution function is )(1)( xSxF hX , where )(xSh
n
t
th XxG
n 1
)(
1
. The kernel estimation for VaR is a root of equation:
1)(xSh , and CVaR estimation can be written as:
n
t
tht XGX
n 1
VaR
)1(
1
CVaR .
The Trimmed kernel method [7] gives the following estimate for CVaR .
Let }0{X
ttt IXX , a sequence }{ nk is such that, nk and ,0
n
kn
n . Then:
n
i
ii n
XXXIX
n 1
)(k })(VaR{
)1(
1
CVaR .
N.G. Zrazhevska, А.G. Zrazhevsky
ISSN 1681–6048 System Research & Information Technologies, 2016, № 3 134
Application of the extreme value theory to the risk measures var and cvar
assessment
As VaR is the quantile (and CVaR is the mean quantile), it is sufficient to study
only the right tail of the pdf to get their estimations. Very often loss distribu-
tions are skewed and have so-called fat tails. In this case, the use of methods
based on a priori assumptions about the normal distributions is untenable, and
it makes sense to use Extreme Value Theory (EVT). EVT allows to analyze
the extreme, and therefore relatively rare events in the historical data array.
Describe the mathematical formulation of EVT ([20]). Let we have a random
variable X with cdf )F(x , its sample ),...,( 1 nXX and },...,{max 1 nn XXM .
The EVT goal is to find a function )(xG , such that, ,)(xGx
a
bM
P
n
nn
n , where }0{ na , }{ nb are the sequences of constants. According to the
Fisher–Tippett theorem, function )(xG belongs to the generalized extreme value
(GEV) family of distributions. These distributions have the form:
.0,expexp
,0,1exp
)(
1
,,
x
x
xG
GEV involves three distributions: the Frechet distribution with «fat» tail if 0 ,
the Gambela distribution with «thin» tail if 0 and the Weibull distribution if
0 . The parameter /1 is called the tail index (for 0 ).
A popular method of parameter estimating is the Hill’s non-parametric
method [20]. Consider a sample of losses and define the order statistics as
)()2()1( ... nXXX .Then for integer 0l we have the estimation for as:
l
j
lnjn
Hill
XX
l
l
1
)()1( )ln(ln
1
)(
. It can be shown [3], that
)(ˆ lk Hill is asymptotically normally distributed );0( 2N . Parameter l can
be found from the plot ))(ˆ,( lk Hill : we select the value of l such that the evalua-
tion )(lHill
appears stable. A more accurate method for finding l using the boot-
strap procedure is described in [21]
The following formulas for risk measures estimating can be used [22]:
1
)1(
expVaR )(
ˆ
,
,
nl
n X
n
l
,
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Системні дослідження та інформаційні технології, 2016, № 3 135
,1exp
1
1
CVaR
1
)(
ˆ
,
,
dqX
nq
l
nl
n
where )]05,1([, nln .
Block maxima method
According to the block maxima method [20], the sample is divided into m
non-overlapping subsamples. For each subsample we find the maximum value
miM i
n ,1,)( . It is assumed that m is sufficiently large so that the Fisher–Tippett
theorem holds for }{ )(i
nM and the function )(,, xGEV is the distribution func-
tion of maximum values in the sample. To find the estimates of parameters ,,
maximum likelihood method can be used. Knowing the )(,, xGEV distribution
the Monte Carlo method or analytical formulas can be used. For example the arti-
cle [23] presents the next formulas:
]}ln{1[VaR
p , (10)
dx
xx
x
p
p
/1
VaR
1/1
1exp1
1
CVaR , (11)
where
n
m
p )1(1 .
The disadvantage of the block maxima method is a potential shortage of ex-
treme values because a single value is determined in each block. The POT (Peaks
Over Thresholds) method uses more complete information about the extreme
values.
POT–method
In the POT method [20] we choose some high enough threshold u and con-
sider only the sample values above it: uXi . It can be shown [20], that for large
enough u the excess distribution 0},|{P)( yuxyuXyFu is well ap-
proximated by the Generalized Pareto Distribution (GPD) given as:
;0,
)(
exp1
,0,
)(
11
)(
1
)(,
u
ux
u
ux
xG u
,0,
)(
,
,0),,[
u
uu
u
x 0)( u .
The advantage of the POT approach is that the method gives the explicit
formulas for VaR and CVaR [20]:
1
)1(
VaR
k
n
u ,
1
VaR
1
1
CVaR
u
.
N.G. Zrazhevska, А.G. Zrazhevsky
ISSN 1681–6048 System Research & Information Technologies, 2016, № 3 136
The article [24] proposes the POT method with the use of the empirical dis-
tribution function. The main idea f the method: the threshold value is displayed as
a horizontal line on the plot of sampled values, connected successively. The ex-
treme values are taken as the peak values between two distinct upcrossings. So we
get the extreme value sample .,1},{ NiX i To analyze the getting sample we
divide the whole interval into m intervals of the same length so that each interval
has at least one peak. Then
m
N
n is the average number of peaks for one inter-
val. So we can consider n independent, identically distributed random variables,
for which we have N realizations. Then the cdf of all peaks above u is:
n
n
i
i xFxXPxF )]([)()( pot
1
.
The function )(pot xF can be constructed as an empirical cdf and can be used
for VaR and CVaR estimating.
QUANTILE ESTIMATION
Empirical quantile estimation is one of the classic nonparametric methods for
VaR and CVaR estimating [3]. As before let X be a random variable.
nXXX ,...,, 21 is a sample of its values, )()2()1( ... nXXX is the order statis-
tics in ascending order, (x)f and (x)F are the pdf and the cdf respectively. The
method is based on the following theoretical result. If )(x is the th quantile of
)(xF and 0)( )( xf then
l
xfn
xNX l ,
)]([
)1(
,
2)(
)(
)( , (12)
where nl . Taking into account (1), the VaR estimate can be obtained
from (12) using )(lX .
If nl is notinteger, the interpolation can be used for quantile estimation.
Let 21, ll be the two neighbouring positive integers such 21 lnl and
2,1, i
n
l
p i
i . Then:
)(
12
1
)(
12
2
21
VaR ll X
pp
p
X
pp
p
,
n
i
ii XIX
N 1
)()( }VaR{
)1(
1
CVaR ,
where N is the number of order statistics larger than VaR .
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Системні дослідження та інформаційні технології, 2016, № 3 137
If the values of order statistics have different probabilities the risk measures
estimates can be obtained as follows. We consider the order statistics:
},{,...},,{ )()()1()1( nn PXPX , 1
1
)(
n
i
iP . Define
N
n
p )1(1 ,
k
i
iPp
1
)(
p , pPp
k
i
i
1
1
)( . Then:
)(VaR )()1(
)(
pp
pp
XX
X kk
k
,
1
1
)()(1
1
VaRCVaR
kn
i
ikik
PX
knpp
pp
.
Advantages of the empirical quantile estimation are its simplicity and using
no specific distributional assumption. However it has some drawbacks. It assumes
that the distribution under sample remains unchanged, that is often not so in prac-
tice. Furthermore, if is close enough to 1 the empirical quantile is not efficient
estimate of the theoretical quantile. In practice, VaR obtained by the empirical
quantile can serve as a lower bound for tis risk measure.
ROCKAFELLAR–URYASEV OPTIMIZATION METHOD
Using the definition of VaR and CVaR in the form of (4) and (5), respec-
tively, the paper [6] proposes the algorithm in the form of the optimization prob-
lem for risk measures estimating.
Define the function:
Rx
dxxpxfF )(]),([
1
1
),( , where
.0,0
,0,
][
t
tt
t (13)
It is shown that ),( F is a convex and continuously differentiable at
function. In this case:
),(argminVaR
F
R
,
),(minCVaR
F
R
.
( )(VaR X is the left point of the solution set of ),(argmin
F
R
).
The integral in (13) can be calculated approximately or exactly. For exam-
ple, in the case of a discrete sample niX i ,1},{ the rectangles method will lead
to the following formula:
N
k
kxf
N
xF
1
]),([
)1(
1
),( . To solve the
optimization problems standard optimization packages such as PSG, Gurobi can
be used.
N.G. Zrazhevska, А.G. Zrazhevsky
ISSN 1681–6048 System Research & Information Technologies, 2016, № 3 138
The optimization method can be used with practically no restrictions on the
type of the distribution, the sample size, noisy empirical data. However, the use of
optimization procedures requires high qualification of the user and access to spe-
cialized software (eg PSG). Unfortunately, the well-known packages Matlab and
Mathematica allow to solve effectively only linear problems.
APPLICATION OF METHODS OF VAR AND CVAR ESTIMATING`
In this section we apply the methods described above for obtaining numerical val-
ues of VaR and CVaR for artificial data. The first group of data (N) was modeled
on assumption of normal distribution with parameters: .5;5,0 For the
second group (T) we use the scailedlocalt distribution with the parameters
.4,5;5,0 10,000 values were generated in each group, the confidence
level was taken as .99,0;95,0 The results are shown in Tables 1, 2.
Table 1 shows the evaluations of the risk measures obtained with the use of
the full cdf. The column Exact in the table shows the exact values (formulas (8),
(9). As representatives of the methods based on the empirical distribution function
we use the Historical Simulated method (formulas (6), (7)) ― HS, and the
Rockafellar–Uryasev discrete method ((9)) ― R–U. Estimations PE we receive
using (8), where the parameters of the normal distribution are estimated with the
maximum likelihood method. Using the estimated distribution for the Monte
Carlo method we get the estimates PEMC, and The Richardson's method gives us
the estimates PEMCR.
T a b l e 1 . The results for VaRα and СVaRα estimating for different methods using the
full cdf for different confidence levels
Method/Risk Exact HS R-U PE PEMC PEMCR
VaR0,95 (N) 8,72 8,57 8,57 8,52 8,35 8,50
RE 0,00 0,02 0,02 0,02 0,04 0,03
CVaR0,95 (N) 10,81 10,57 10,57 10,58 10,49 10,56
RE 0,00 0,02 0,02 0,02 0,03 0,02
VaR0,99 (N) 12,13 12,26 12,25 12,13 12,11 12,17
RE 0,00 –0,01 –0,01 0,00 0,00 0,00
CVaR0,99 (N) 13,83 13,93 13,93 13,82 13,95 13,84
RE 0,00 –0,01 –0,01 0,00 –0,01 0,00
VaR0,95 (T) 11,16 10,99 10,98 11,02 10,95 11,05
RE 0,00 0,02 0,02 0,01 0,02 0,01
CVaR0,95 (T) 16,51 15,95 15,95 16,03 16,00 16,11
RE 0,00 0,03 0,03 0,03 0,03 0,02
VaR0,99 (T) 19,23 19,65 19,65 19,07 19,06 19,09
RE 0,00 –0,02 –0,02 0,01 0,01 0,01
C VaR0,99 (T) 26,60 26,67 26,67 26,17 26,92 26,12
RE 0,00 0,00 0,00 0,02 –0,01 0,02
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Системні дослідження та інформаційні технології, 2016, № 3 139
Table 2 shows the results obtained by the methods based on the analyzes of
the tail of the distribution. Using the method of maximum likelihood we estimate
the parameters for GEV (GPD) function. Estimates GEVQ (GPDQ) we get using
formulas (10), (11), and the Monte Carlo method gives the estimates GEVMC
(GPDMC). The empirical POT method gives POTE estimates.
RE means a relative error.
T a b l e 2 . The results for VaRα and СVaRα estimating for different methods using the
tail of cdf for different confidence levels
Method/Risk GEVQ GEVMC GPDQ GPDMC POTE
VaR0,95 (N) 7,89 7,88 8,83 8,90 8,71
RE 0,10 0,10 –0,01 –0,02 0,00
CVaR0,95 (N) 10,33 10,28 10,96 10,92 10,71
RE 0,05 0,05 –0,01 –0,01 0,01
VaR0,99 (N) 12,13 12,19 12,51 12,54 12,45
RE 0,00 –0,01 –0,03 –0,03 –0,03
CVaR0,99 (N) 13,82 13,79 13,88 13,78 14,10
RE 0,00 0,00 0,00 0,00 –0,02
VaR0,95 (T) 10,96 10,91 11,72 11,83 11,08
RE 0,02 0,02 –0,05 –0,06 0,01
CVaR0,95 (T) 15,92 15,95 16,60 16,49 16,06
RE 0,04 0,03 –0,01 0,00 0,03
VaR0,99 (T) 19,30 19,30 19,79 19,66 19,67
RE 0,00 0,00 –0,03 –0,02 –0,02
CVaR0,99 (T) 24,26 24,71 24,73 24,35 26,79
RE 0,09 0,07 0,07 0,08 –0,01
All calculations performed with Matlab package. The results indicate the
correctness of the formulas given in the article. The calculation error is caused by
the limited sample and data discretization. It should be noted that the existence of
analytical formulas allows to speed up the computation of risk measures, but does
not significantly reduce the accuracy of the calculations. For real data results may
vary, because of a priori hypotheses about the type of distributions (which can be
wrong) and limited sample data.
CONCLUSIONS
The article systematizes and classifies the most common used methods of the risk
measures VaR and CVaR calculation and estimation. In recent years, these risk
measures have been used to analyze a wide class of data, that caused to a large
number of different methodologies giving the ready formulas for their calculating
or indicating an algorithm for estimating. However, in most studies, these meth-
ods are applied to concrete data with own specifics. That's why despite their uni-
versality VaR and CVaR have got a tough bind to specific data. In this paper we
describe the methods without reference to the concrete data.
N.G. Zrazhevska, А.G. Zrazhevsky
ISSN 1681–6048 System Research & Information Technologies, 2016, № 3 140
Classification of methods presented in the form of a hierarchical table, that
helps to determine the sequence of operations required to obtain VaR and CVaR
values depending on the available information about analyzed data, the purpose of
analysis and the availability of information and computing resources needed to
get the result. Econometric concepts and designations are taken as a basis that
helps to use the results of the article in solving applied problems. The analysis
allowed to formulate the decision-making procedure for the choice of the method
of static VaR and CVaR evaluation. The procedure includes all necessary steps to
make a decision, from general statistical data analysis to the choice of a particular
method for risk measures estimating.
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Resieved 14.06.2016
From the Editorial Board: the article corresponds completely to submitted manuscript.
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