Прогнозування динамічних VaR і CVaR на основі квінтильної регресії з використанням металог розподілу
The paper proposes a new method of dynamic VaR and CVaR (ES) risk measures forecasting. Quantile linear GARCH model is chosen as the main forecasting model for time series quantiles. To build a forecast, the values of quantiles are approximated by the metalog distribution, which makes it possible to...
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| author | Zrazhevsky, Grigoriy Zrazhevska, Vira |
| author_facet | Zrazhevsky, Grigoriy Zrazhevska, Vira |
| author_institution_txt_mv | [
{
"author": "Grigoriy Zrazhevsky",
"institution": "Taras Shevchenko National University of Kyiv, Kyiv"
},
{
"author": "Vira Zrazhevska",
"institution": "National Technical University of Ukraine \"Igor Sikorsky Kyiv Polytechnic Institute\", Kyiv"
}
] |
| author_sort | Zrazhevsky, Grigoriy |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
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| datestamp_date | 2021-07-13T11:01:37Z |
| description | The paper proposes a new method of dynamic VaR and CVaR (ES) risk measures forecasting. Quantile linear GARCH model is chosen as the main forecasting model for time series quantiles. To build a forecast, the values of quantiles are approximated by the metalog distribution, which makes it possible to use analytical formulas to evaluate risk measures. The method of VaR and CVaR forecasting is formulated as a step-by-step algorithm. At the first stage, an initial model is built to obtain variance estimates. The predicted variance values obtained from the constructed model are used at the second stage to find the QLGARCH model coefficients by solving the minimization problem. At the third stage, the QLGARCH models are estimated on a non uniform quantile grid. The obtained predicted values of quantiles are used to estimate the approximating metalog distribution. The investigated theory is applied to VaR and CVaR forecasting for time series of daily log return of the DJI index. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2021.1.12 |
| first_indexed | 2025-07-17T10:27:19Z |
| format | Article |
| fulltext |
G. Zrazhevsky, V. Zrazhevska, 2021
Системні дослідження та інформаційні технології, 2021, № 1 139
UDC 519.6 : 519.81
DOI: 10.20535/SRIT.2308-8893.2021.1.12
QUINTILE REGRESSION BASED APPROACH FOR
DYNAMICAL VAR AND CVAR FORECASTING USING
METALOG DISTRIBUTION
G. ZRAZHEVSKY, V. ZRAZHEVSKA
Abstract. The paper proposes a new method of dynamic VaR and CVaR (ES) risk
measures forecasting. Quantile linear GARCH model is chosen as the main forecast-
ing model for time series quantiles. To build a forecast, the values of quantiles are
approximated by the metalog distribution, which makes it possible to use analytical
formulas to evaluate risk measures. The method of VaR and CVaR forecasting is
formulated as a step-by-step algorithm. At the first stage, an initial model is built to
obtain variance estimates. The predicted variance values obtained from the con-
structed model are used at the second stage to find the QLGARCH model coeffi-
cients by solving the minimization problem. At the third stage, the QLGARCH
models are estimated on a non uniform quantile grid. The obtained predicted values
of quantiles are used to estimate the approximating metalog distribution. The inves-
tigated theory is applied to VaR and CVaR forecasting for time series of daily log
return of the DJI index.
Keywords: VaR, CVaR, Expected Shortfall, dynamic risk measures, forecast,
Quantile LGARCH model, metalog distribution.
INTRODUCTION
The purpose of this study is to develop the new method of dynamic VaR and
CVaR risk measures estimation and forecasting. VaR and CVaR are classic
measures that are used in financial risk assessment [1]. In the practice of VaR and
CVaR estimating for a random variable that describes the profitability of a finan-
cial instrument, two main approaches can be distinguished. The first approach is a
nonparametric estimation method that is based on an empirical distribution func-
tion. The disadvantage of this estimation method is the critical dependence of the
effectiveness of the method on the presence in the initial data of values that arise
with low probabilities [1]. The second approach is parametric, based on a priori
estimation of the distribution function, which is the main disadvantage of this ap-
proach [1].
In time series analysis, in particular, in time series forecasting, in addition to
static risks measures, in practice it becomes necessary to build more complex risk
models, which take into account the changes of the series over time. In this case,
to estimate risk measures, various time series models can be used, such as, for
example, ARMA, GARCH. With this approach, the problem of risk measures
modeling is reduced to the estimation of a model for variance and finding static
risk measures for its residuals using parametric or nonparametric methods.
Examples of this approach are described in [2–4].
The described above approaches of evaluation of risk measures are based on
the construction of a cumulative distribution function on the full space of events.
G. Zrazhevsky, V. Zrazhevska
ISSN 1681–6048 System Research & Information Technologies, 2021, № 1 140
At the same time, to estimate risk measures, it is sufficient to evaluate only the
quantile of a given level (for VaR) or the distribution of values exceeding a given
level (for CVaR). At the same time, from a practical point of view, the most sig-
nificant is the distribution for a relatively small subset of the event space leading
to extreme consequences. Accordingly, it is possible to simplify the forecasting
task by using the quantile regression model proposed in [5, 6] contains a detailed
description of the theory of quantile regression estimation applicable to standard
time series models. Since financial time series, as a rule, are characterized by ra-
ther strong volatility, quantile GARCH models are popular for risk analysis. The
problem of building quantile models of the GARCH class and their application in
VaR forecasting for the series of log returns of stock market indices is considered,
for example, in [7, 8].
One of the possible approaches for fitting of the GARCH model residuals
distribution is to use the metalog distribution proposed in [9]. This choice is based
on the simplicity of quantile formulas and the availability of a sufficient set of
parameters of this distribution for an adequate fitting of empirical data of various
nature. Thus, in [10], the metalog distribution is used in the development of the
extended FAIR-BN combined approach for cyber security risk assessment. In [11],
the five-term metalog distribution is used to forecast fertility rates in Canada. SPT
(symmetric-percentile triplet) metalog distribution is used in [12] to statistically
compare the forecasts of annual production in the oil and gas industry in Norway.
In [13], the metalog distribution is used for dynamic risk measures VaR and
CVaR estimating based on a heteroscedastic time series model, taking into ac-
count the strong dependence of the data series. The method of smoothing of the
autocorrelation function is used for variance modeling. A metalog distribution is
proposed to use for risk measures model residuals estimating. The paper proposes
two methods of metalog distribution estimating and explicit analytical formulas
for VaR and CVaR modeling and forecasting with different numbers of members
in the metalog distribution.
A large number of publications devoted to the risk measures estimation and
forecasting testifies to the applied significance of this problem. At the same time,
the task of developing the new methods and approaches for risk modeling, which
more fully reflect the nature of the modeled series, remains relevant. Most of the
forecasting methods are based on the estimation of the entire distribution func-
tion. On the one hand, this is an overstated requirement for the model, and on the
other hand, it often leads to an incorrect description of tails of distribution. There-
fore, in this paper, it is proposed to build volatile models only for the tail parts of
the distribution. In this case, the obtained point values of the quantiles can be
smoothed, for example, by metalog distribution.
MATERIALS AND METHODS
On the probability space ),,( Pt a time series }, { Ttut with a finite mean is
considered ( t is the information set containing all available at the time t infor-
mation about the time series).
For a fixed confidence level α risk measure tVaR is defined as the condi-
tional — quantile of the CDF of tu : )(1
t
t FVaR . The risk measure
Quintile regression based approach for dynamical VaR and CVaR forecasting using metalog
Системні дослідження та інформаційні технології, 2021, № 1 141
tCVaR is defined as the integral:
dVaRVaRuuECVaR tt
ttt
t
1
][
0
~~ ,
)5.0( , where ][tE denotes expectation with respect to t . In this paper, the
continuity of the CDF is assumed.
As indicated in the introduction, most methods for dynamic VaR and CVaR
risk measures forecasting are based on time series modeling. The GARCH models
are among the models that describe volatility of financial time series. In this pa-
per, we consider the Linear GARCH model LGARCH ),( qp . This model is fre-
quently used for fitting log return volatility time series and appropriate for quan-
tile regression because of its linear structure [7].
The time series ...},2,1,0,{ tut follows LGARCH (p, q) process if:
,tttu jtj
p
j
iti
q
i
t u
11
0 , (1)
where }{ t are independent, identically distributed random variables with zero
mean and a conditional distribution function )(F , 00 ; q
q R T
21 ),...,,( .
Using the heteroscedastic time series model, the dynamic risk measures can
be found under the following formulas [1]:
t
t VaRVaR )( , t
t CVaRCVaR )( , (2)
where the model for t is defined in (1), )(VaR and )(CVaR are static risk
measures at time t. Then the P step forecast for dynamic risk measures can be
found by model (2) extrapolation:
Pt
Pt VaRVaR
)( , Pt
Pt CVaRCVaR
)( . (3)
In this work, the following methods are used to evaluate static risk measures
)(VaR , )(СVaR .
Historical simulation method. Let X be a random variable and its sample
values are NXXX , ... ,, 21 . In accordance with the historical simulation method,
an empirical distribution function is constructed on the sample values. Then ac-
cording to [14]:
])([ NXVaR , ])([/
][
1
NXCVaR i
N
i
, (4)
where )()2()1( ... NXXX .
Using Student’s t-distribution. If the random variable has the local scale
Student’s t-distribution with the parameters , and the degrees of freedom
2 , then the risk measures can be calculated as (see [14]):
)(1
tVaR ,
1
))((
))(( 211
ttg
CVaR , (5)
where )(g is the standard PDF and )(1
t is the inverse standard CDF value
at of t -distribution.
G. Zrazhevsky, V. Zrazhevska
ISSN 1681–6048 System Research & Information Technologies, 2021, № 1 142
Using metalog distribution. Suppose that X has a metalog distribution
)(xFX , that is defined by a quantile function )),(,( XaM n [9]:
.5evenfor,
1
ln)5,0(
;5oddfor,)5,0(
;4,)5,0(
1
ln)5,0(
1
ln
;3,
1
ln)5,0(
1
ln
;2,
1
ln
),(
1
2
1
2
1
1
4321
321
21
naM
naM
naaaa
naaa
naa
M
n
nn
n
nn
n a
The coefficients T
21 ),...,,( naaaa can be found as a solution of the system
of equations:
XYYYa nn
T1T ][ n , (6)
where T
21 ),...,,( NXXXX , the matrix nY is defined through a set of quantiles
with levels T
21
T
21 ))(,...),(),((),...,,( NXXXN XFXFXFα [9].
Following the definitions, risk measures can be found under the formulas:
))(,~()(~ XMXVaR n a ; (7)
.)) (,(~
1
)(
~
0
,~ dyXyMXCVaR nn a
Explicit formulas for )(,~ XCVaR n estimating with different number of
members of the metalog distribution were obtained in [13]:
nCVaR ,~
.6evenfor,~ln)1(~1ln
)1~2(arctg)1~2(2~
2
)(
;5oddfor,)5.0~()5,0()1(
1~
2
;4,
2
)1~(
)(
;3,~1
~
ln)1~(1
2
)(
;2,~1
~
ln~
)~1(ln
2
2
2
2
2
1,~
2
1
2
1
2
1
1,~
4
3,~
3
2,~
21
n
nG
n
a
XCVaR
n
n
a
XCVaR
n
a
XCVaR
n
a
XCVaR
naa
n
n
n
n
n
nnn
n
n
(8)
where
Quintile regression based approach for dynamical VaR and CVaR forecasting using metalog
Системні дослідження та інформаційні технології, 2021, № 1 143
]~22; 2,2; 21,1,1[ )1~(]2; 2,2; 21,1,1[ 2
32
3
nFnFG
]~2; 2,2; 21,1,1[ ~)1( 2
3
2 nF
n
,
]; ,; ,,[ 213212
3 zddcccF is generalized hypergeometric function.
For determining the parameters of the metalog distribution the Quantile
Metalog Method is proposed in [13]. This method uses the approximation of the
empirical distribution function by sample quantiles T),...,(
1 N X , where
;ger is inteif ,
;nteger is not i if ,
)()(
)1]([
i
i
i
in
nXX
nX
i
i
n
n i ][
,
n
n i ]1[
, )1,0(i .
Thus, the classical approach to risk measures forecasting (3) involves building
a model for variance estimating, obtaining model residuals, and using methods for
static risk measures )(VaR , )(CVaR estimating.
Another approach is based on estimating of quantile time series models,
which makes it possible to directly simulate the time series quantile of a given
level. In this paper, in accordance with (1), the QLGARCH model is considered
(see [6–8]):
tttut
Q zT
1 )()( , (9)
where )( 1 tut
Q is a conditional — quantile for }{u , tz
T
11 ),...,,,...,,1( qttptt uu , ),(()(),...,,,...,,()( 0
1
110
T Fqpt
))(),...,(),(),...,( 11 qp .
The paper proposes the following methodology for dynamic VaR and CVaR
forecasting. Model (9) is used to construct a set of quantiles. For a more detailed
description of the left tail of the CDF, the quantile levels can be found on a non
uniform grid. Assuming that the obtained set of quantiles can be fitted by the
metalog distribution, the quantile function )),(,( XM n a is estimated using (6).
To obtain forecasts of risk measures, formulas (7), (8) are used. The practical im-
plementation of the method is formulated as a step-by-step algorithm.
AN ALGORITHM FOR CONSTRUCTING THE DYNAMIC RISK MEASURES
VAR AND CVAR FORECAST BASED ON THE METHOD QLGARCH -
METALOG
1. Building a variance model to get estimates. The model LGARCH ),( qp
(1) is written as ARCH( ): jtj
j
t u
1
0 , where the coefficients j
G. Zrazhevsky, V. Zrazhevska
ISSN 1681–6048 System Research & Information Technologies, 2021, № 1 144
satisfy summability conditions implied by the regularity conditions [6]. Due to the
assumption of regularity, the coefficients decrease geometrically, therefore, the
model can be reduced to ARCH (m). The appropriate lag for reducted ARCH
model is chosen on the base of significant values of ACF and PACF for squared
values of returns (values that is more than confidence bounds). Estimates of j
can be obtained in various ways. QMLE is used in this work. The fitted model is
used to obtain estimates ptt ˆ,...,ˆ :
ijtj
m
j
it u
ˆˆˆ
1
0 , 0,i p . (10)
2. Building a set of quantiles predictive estimates. To obtain estimates of
the -quantile for tu , the QLGARCH model (9) is used. The orders of the model
qp, can be estimated using Akaike Inform Criteria (AIC) and Hannan-Quinn
Inform Criteria (HQIC). It is also possible to use Bayes Inform Criteria (BIC) and
Shibata Inform Criteria (SIC). To estimate the vector of parameters T)( , the
minimization problem is solved using the quantile regression estimation in the
form (see [8]):
)ˆ)((θ(min T
tt
t
u z
, (11)
where ))0(()( uIuu is a check function, ,...,ˆ,...,ˆ,1(ˆ 11 tpttt uz
T)..., qtu taking into account that the estimates ptt ˆ,...,ˆ 1 were obtained at
step 1 (10). The solution of the unconstrained minimization problem (11) makes it
possible to estimate the -quantile for tu in the form:
ttut
Q ẑ)(ˆ)(ˆ T
1 .
At this step, a grid of i -quantiles, Ni ,1 , is constructed and the problem
(11) is solved N times. For every quantile regression with Nii ,1, the quan-
tile estimates are t
T
itiut
Q ẑ)(ˆ)(ˆ , Ni ,1 . In this case, the predicted values
of the conditional quantiles are calculated by extrapolation
1
T ˆ)(ˆ)(ˆ
1
tiiut
Q z , where T
111 ),...,,ˆ,...,ˆ,1(ˆ qttpttt uuz , and it̂ ,
1,0 pi are obtained at the first step of algorithm (10).
3. Risk measures forecasting. The predictive quantiles ),...(ˆ
11
tuQ
)(ˆ...,
1 Nut
Q
(from previous step) are fitted using the metalog distribution. Esti-
mates of the metalog distribution parameters T
21 )ˆ,...,ˆ,ˆ(ˆ naaaa are found in ac-
cordance with (6), where T
1 ))(ˆ),...,(ˆ(
11 Nuu tt
QQ
X , T
21 ),...,,( N .
Specifying the quantile function )ˆ,~( anM for a given level ~ allows the use of
analytical expressions (7), (8) to find the predicted values 1~
tVaR and 1~
tСVaR .
Quintile regression based approach for dynamical VaR and CVaR forecasting using metalog
Системні дослідження та інформаційні технології, 2021, № 1 145
EXPERIMENT, RESULTS AND DISCUSSIONS
Proposed algorithm was applied for dynamic risk measures 1,0VaR , 1,0CVaR
forecasting for the time series of daily log returns of the Dow Jones Industrial
Average index (the DJI time series). The sample length was 3500 values from
2007/02/16 to 2021/01/11. The forecast model was based on 1500 historical val-
ues and was extrapolated one value forward. After that, the modeling window was
one step shifted, and the model was rebuilt. The procedure was repeated 2000
times (Rolling Forecast Method). The obtained one-step forecasts were compared
with real values for the corresponding period of time. To obtain variance esti-
mates (10), the model ARCH(30) was built. Based on the historical values of the
time series and the estimated variance values, quantile LGARCH ),( qp models
were built for different levels of quantiles. The orders of the models were found
using the AIC and HQIC criteria: 3p , 3q . For more detailed description of
the left tail of the distribution, the non uniform grid of quantiles was used: ihi ,
01,0h for 20,1i , and 05,0h for 19,5i . Estimates of the parameters were
obtained using QMLE. The estimated models were used to generate one-step
quantile predictions. The predicted quantile values were used to estimate the met-
alog distribution. Using (6), the estimates for the coefficients of the metalog dis-
tributions for 7,6,5,4n were obtained. Risk measures estimates were calculated
using formulas (7) and (8). The results were obtained using R packages rugarch
[15] and quantreg [16].
The results of dynamic VaR and CVaR risk measures forecasting using the
QLGARCH – Metalog (rq_met) method are shown in Fig. 1 along with the
historical values (the first 1500) and the real values (1501–3500) of the DJI time
series. As can be seen from the graph, the obtained forecast estimates describe the
dynamic behavior of the time series quite well.
1
32
D
at
a
Dynamic VaR and CVaR
Fig. 1. Historical data of the time series of daily log return of the DJI index (1 — TS re-
turns) from 2007/02/16 to 2021/01/11 and the forecast values for 1
1.0
tVaR (2 — )1,0(VaR )
and 1
1,0
tCVaR (3 — )1,0(CVaR ) obtained by the QLGARCH – Metalog method ( 5n )
for the period 2013/02/01 – 2021/01/11
time
G. Zrazhevsky, V. Zrazhevska
ISSN 1681–6048 System Research & Information Technologies, 2021, № 1 146
Historical data and the forecast values of dynamic risk measures over a short
period of time is shown in Fig. 2 for more convenient visual analysis.
For a comparative analysis of the effectiveness of the proposed method, a
forecast of dynamic risk measures for a given time series was built with standard
approach on the basis of a heteroscedastic model (3). To estimate the variance, the
LGARCH(3,3) model was considered. The AIC and HQIC criteria were used to
determine the orders of the model. The QMLE was used to estimate the model
coefficients. To determine the risk measures for the residuals of the
LGARCH(3,3) model, the following methods were used: the Historical simulation
method (4) (hist method), explicit formulas (5) under the assumption that the
model residuals have the local scale t-distribution (tLS method), explicit formulas
(7), (8) based on Quantile Metalog Method for 7,6,5,4n (metal method).
The analysis of the constructed forecast estimates was carried out using the
backtesting procedure. The following tests were used in the work:
for VaR estimates: the Kupiec test (LRuc), Christoffersen’s independence
test (LRind), PoE statistics [17];
for CVaR estimates: two tests proposed in [18]: one-sided simple condi-
tional calibration test (scc_1) and two-sided simple conditional calibration test
(scc_2); three regression based backtests proposed in [19]: the auxiliary ESR
backtest (Aux), the strict ESR backtest (Str), the intercept ESR backtest (Int).
The p-values of these tests are shown in Table 1. Table 2 shows the PoE sta-
tistic values for the forecast estimates of dynamic risk measure VaR.
1
2
3
D
at
a
Dynamic VaR and CVaR
time
Fig. 2. Historical data of the time series of daily log return of the DJI index (1 —
TS returns) and the forecast values for for 1
1.0
tVaR (2 — )1,0(VaR ) and 1
1,0
tCVaR
(3 — )1,0(CVaR ) obtained by the QLGARCH – Metalog method ( 5n ) for the
period 2019/10/31 – 2020/08/18
Quintile regression based approach for dynamical VaR and CVaR forecasting using metalog
Системні дослідження та інформаційні технології, 2021, № 1 147
T a b l e 1 . The results of the qualitative analysis of the forecast estimates of
dynamic risk measures VaR and CVaR for DJI time series
Risk meashure
VaR CVaR Method
LRuc LRind scc_1 scc_2 Aux Str Int
hist 0,0677 0,0091 0,0866 0,0076 0,6453 0,6191 0,5824
tLS 0,4582 0,3359 0,0657 0,1136 0,5364 0,5286 0,4710
metal, 4n 0,1497 0,0802 0,2301 0,0625 0,6980 0,7289 0,6267
metal, 5n 0,1726 0,1497 0,2411 0,0630 0,6883 0,7338 0,6461
metal, 6n 0,0945 0,0393 0,1251 0,0115 0,5707 0,6047 0,5355
metal, 7n 0,0945 0,0474 0,1245 0,0098 0,5261 0,5521 0,5180
rq_met, 4n 0,4983 0,7643 0,1572 0,0869 0,8454 0,8914 0,8072
rq_met, 5n 0,8810 0,5796 0,2741 0,1808 0,8842 0,9164 0,8475
rq_met, 6n 0,4272 0,4072 0,1863 0,1082 0,8566 0,8931 0,8121
rq_met, 7n 0,5476 0,3655 0,2688 0,1757 0,8671 0,8940 0,8450
Table 2. PoE statistic values for the forecast estimates of dynamic risk measure
VaR for DJI time series
metal rq_met
Method hist tLS
4n 5n 6n 7n 4n 5n 6n 7n
PoE 0,0879 0,1051 0,09045 0,09095 0,0889 0,0889 0,0954 0,0989 0,0944 0,0959
As follows from Table 1, the worst estimates were obtained using the
historical simulation method (hist). In particular, as a result of applying the
Christoffersen’s independence test (LRind) for VaR and the two-sided simple
conditional calibration test (scc_2) for CVaR, hypotheses with a significance level
of 0,05 were rejected. This indicates the inapplicability of the historical
simulation method for predicting the values of the DJI time series. At the same
time, all tests showed consistently good quality of forecasts obtained by the
QLGARCH — Metalog method (rq_met) and maximum p-value statistics
compared to other methods (tLS and metal).
In the article the metalog distribution with the different number of
parameters ( 7,6,5,4n ) was considered (see Table 1). An increase in the
number of parameters potentially increases the accuracy of the estimates, but can
lead to the problem of overfitting. As a result of the backtesting for forecasted
dynamic risk measures VaR and CVaR obtained with the Quantile Metalog
Method (metal), the choose of large n probably leads to overfitting. The estimates
of VaR and CVaR obtained using the metalog distribution for 4n and 5n are
consistently better than the estimates obtained using the same sample for 6n
and 7n . At the same time, the QLGARCH – Metalog (rq_met) method shows
less dependence on the number of parameters of the metalog distribution. The
results of the qualitative analysis for VaR and CVaR forecasts obtained by this
method are relatively uniform for all n . Although it should be noted that
according to the results of all tests, the highest quality forecasts for risk measures
were obtained at 5n . The best results of VaR forecasting (see Table 2)
according to PoE statistics is obtained also by rq_met method for 5n (has the
least deviation from the target value of 0,1).
G. Zrazhevsky, V. Zrazhevska
ISSN 1681–6048 System Research & Information Technologies, 2021, № 1 148
The paper proposes a method that is a natural continuation of existing re-
search and methods devoted to dynamic risk models developing. It combines pa-
rametric and nonparametric statistical approaches to time series modeling. Practi-
cal application of the method shows its effectiveness in the case of risk modeling
for highly volatile financial time series. The simplicity of the method and its
background make it possible to recommend it for using in various fields. How-
ever, the determination of the restrictions on applicability of this method, as well
as the automation of the procedure for estimation of its parameters, requires fur-
ther mathematical research, in particular, the construction of asymptotic estimates
of the convergence of the model.
CONCLUSIONS
The paper considers the problem of dynamic VaR and CVaR risk measures mod-
eling and forecasting for financial time series. Since the VaR measure is a condi-
tional quantile of the distribution function of a given level, and CVaR for continu-
ous distributions can be specified as the average of the quantile function it is
proposed to use QLGARCH as a model for risk measures forecasting. The advan-
tage of using of this model is the ability to estimate and predict not the full distri-
bution, but the values of the quantiles of the required levels. Since the risk meas-
ures are determined for the tail part of the distribution, an non uniform grid is
used in the work, which makes it possible to detail the quantiles with a low level.
To smooth point values, it is proposed to fit a set of quantiles with metalog distri-
bution. This approach is also convenient due to the presence of explicit analytical
expressions VaR and CVaR for the metalog distribution. The proposed method for
dynamic VaR and CVaR risk measures forecasting is formulated in the form of a
step-by-step algorithm.
The proposed methodology was tested on the time series of daily log return
of the Dow Jones Industrial Average (DJI) index for the period from 2007-02-16
to 2021-01-11. Using the formulated algorithm, a set of one-step forecasts of risk
measures was obtained. An analysis of the quality of the forecasts was carried out
using various standard backtesting techniques on real data. The results were com-
pared with the forecasts obtained by standard methods that are based on the
LGARCH model and various assumptions about distribution of the residuals. The
carried out qualitative analysis of the obtained predicted values showed the effec-
tiveness of using the method proposed in this work and its advantage in compari-
son with standard methods.
The results of the work can be directly applied in dynamic risk modeling for
highly volatile time series, in particular, financial time series, and also can serve
as the basis for the development of new methods and algorithms for random proc-
esses prediction.
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INFORMATION ON THE ARTICLE
Grigoriy M. Zrazhevsky, ORCID: 0000-0001-8475-2469, National Taras Shevchenko
University of Kyiv, Ukraine, e-mail: zgrig@univ.kiev.ua
Vira F. Zrazhevska, ORCID: 0000-0001-5117-8093, National Technical University
of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail:
vera.zrazhevska@gmail.com
ПРОГНОЗУВАННЯ ДИНАМІЧНИХ VAR І CVAR НА ОСНОВІ
КВІНТИЛЬНОЇ РЕГРЕСІЇ З ВИКОРИСТАННЯМ МЕТАЛОГ РОЗПОДІЛУ /
Г.М. Зражевський, В.Ф. Зражевська
Анотація. Запропоновано новий метод динамічного прогнозування мір ризику
VaR і CVaR (ES). Як основна модель прогнозування для квантилів часових ря-
дів обрано квантильну лінійну модель GARCH. Для побудови прогнозу зна-
чення квантилів апроксимуються розподілом металог, що дозволяє використо-
вувати аналітичні формули для оцінювання мір ризику. Методику
прогнозування VaR і CVaR сформульовано у вигляді покрокового алгоритму.
На першому етапі будується вихідна модель для отримання оцінок дисперсії.
Отримані за моделлю значення дисперсії використовуються на другому етапі
для знаходження коефіцієнтів моделі QLGARCH шляхом розв’язання задачі
мінімізації. На третьому етапі моделі QLGARCH оцінюються на неоднорідній
квантильній сітці. Отримані прогнозні значення квантилів використовуються
для оцінки параметрів розподілу металог. Розроблений метод застосовується
до прогнозування VaR і CVaR для часового ряду логарифмічної дохідності ін-
дексу DJI.
Ключові слова: VaR, CVaR, Expected Shortfall, динамічні міри ризику, про-
гноз, квантильна модель LGARCH, розподіл металог.
ПРОГНОЗИРОВАНИЕ ДИНАМИЧЕСКИХ VAR И CVAR НА ОСНОВЕ
КВИНТИЛЬНОЙ РЕГРЕССИИ С ИСПОЛЬЗОВАНИЕМ МЕТАЛОГ
РАСПРЕДЕЛЕНИЯ / Г.М. Зражевский, В.Ф. Зражевская
Аннотация. Предложен новый метод динамического прогнозирования мер
риска VaR и CVaR (ES). Квантильная линейная модель GARCH выбрана в ка-
честве основной модели прогнозирования для квантилей временных рядов.
Для построения прогноза значения квантилей аппроксимируются распределе-
нием металог, что позволяет использовать аналитические формулы для оценки
мер риска. Методика прогнозирования VaR и CVaR сформулирована в виде
пошагового алгоритма. На первом этапе строится исходная модель для полу-
чения оценок дисперсии. Полученные по модели значения дисперсии исполь-
зуются на втором этапе для нахождения коэффициентов модели QLGARCH
путем решения задачи минимизации. На третьем этапе модели QLGARCH
оцениваются на неоднородной квантильной сетке. Полученные прогнозные
значения квантилей используются для оценки параметров распределения ме-
талог. Разработанный метод применен к прогнозированию VaR и CVaR для
временного ряда логарифмической доходности индекса DJI.
Ключевые слова: VaR, CVaR, Expected Shortfall, динамические меры риска,
прогноз, квантильная модель LGARCH, распределение металог.
|
| id | journaliasakpiua-article-236942 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:27:19Z |
| publishDate | 2021 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/34/5165d7d920f3524d1d5640f5c3f6b234.pdf |
| spelling | journaliasakpiua-article-2369422021-07-13T11:01:37Z Quintile regression based approach for dynamical VaR and CVaR forecasting using metalog distribution Прогнозирование динамических VaR и CVaR на основе квинтильной регрессии с использованием металог распределения Прогнозування динамічних VaR і CVaR на основі квінтильної регресії з використанням металог розподілу Zrazhevsky, Grigoriy Zrazhevska, Vira VaR CVaR Expected Shortfall dynamic risk measures forecast Quantile LGARCH model metalog distribution VaR CVaR Expected Shortfall динамические меры риска прогноз квантильная модель LGARCH распределение металог VaR CVaR Expected Shortfall динамічні міри ризику прогноз квантильна модель LGARCH розподіл металог The paper proposes a new method of dynamic VaR and CVaR (ES) risk measures forecasting. Quantile linear GARCH model is chosen as the main forecasting model for time series quantiles. To build a forecast, the values of quantiles are approximated by the metalog distribution, which makes it possible to use analytical formulas to evaluate risk measures. The method of VaR and CVaR forecasting is formulated as a step-by-step algorithm. At the first stage, an initial model is built to obtain variance estimates. The predicted variance values obtained from the constructed model are used at the second stage to find the QLGARCH model coefficients by solving the minimization problem. At the third stage, the QLGARCH models are estimated on a non uniform quantile grid. The obtained predicted values of quantiles are used to estimate the approximating metalog distribution. The investigated theory is applied to VaR and CVaR forecasting for time series of daily log return of the DJI index. Предложен новый метод динамического прогнозирования мер риска VaR и CVaR (ES). Квантильная линейная модель GARCH выбрана в качестве основной модели прогнозирования для квантилей временных рядов. Для построения прогноза значения квантилей аппроксимируются распределением металог, что позволяет использовать аналитические формулы для оценки мер риска. Методика прогнозирования VaR и CVaR сформулирована в виде пошагового алгоритма. На первом этапе строится исходная модель для получения оценок дисперсии. Полученные по модели значения дисперсии используются на втором этапе для нахождения коэффициентов модели QLGARCH путем решения задачи минимизации. На третьем этапе модели QLGARCH оцениваются на неоднородной квантильной сетке. Полученные прогнозные значения квантилей используются для оценки параметров распределения металог. Разработанный метод применен к прогнозированию VaR и CVaR для временного ряда логарифмической доходности индекса DJI. Запропоновано новий метод динамічного прогнозування мір ризику VaR і CVaR (ES). Як основна модель прогнозування для квантилів часових рядів обрано квантильну лінійну модель GARCH. Для побудови прогнозу значення квантилів апроксимуються розподілом металог, що дозволяє використовувати аналітичні формули для оцінювання мір ризику. Методику прогнозування VaR і CVaR сформульовано у вигляді покрокового алгоритму. На першому етапі будується вихідна модель для отримання оцінок дисперсії. Отримані за моделлю значення дисперсії використовуються на другому етапі для знаходження коефіцієнтів моделі QLGARCH шляхом розв’язання задачі мінімізації. На третьому етапі моделі QLGARCH оцінюються на неоднорідній квантильній сітці. Отримані прогнозні значення квантилів використовуються для оцінки параметрів розподілу металог. Розроблений метод застосовується до прогнозування VaR і CVaR для часового ряду логарифмічної дохідності індексу DJI. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2021-07-13 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/236942 10.20535/SRIT.2308-8893.2021.1.12 System research and information technologies; No. 1 (2021); 139-150 Системные исследования и информационные технологии; № 1 (2021); 139-150 Системні дослідження та інформаційні технології; № 1 (2021); 139-150 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/236942/235501 |
| spellingShingle | VaR CVaR Expected Shortfall динамічні міри ризику прогноз квантильна модель LGARCH розподіл металог Zrazhevsky, Grigoriy Zrazhevska, Vira Прогнозування динамічних VaR і CVaR на основі квінтильної регресії з використанням металог розподілу |
| title | Прогнозування динамічних VaR і CVaR на основі квінтильної регресії з використанням металог розподілу |
| title_alt | Quintile regression based approach for dynamical VaR and CVaR forecasting using metalog distribution Прогнозирование динамических VaR и CVaR на основе квинтильной регрессии с использованием металог распределения |
| title_full | Прогнозування динамічних VaR і CVaR на основі квінтильної регресії з використанням металог розподілу |
| title_fullStr | Прогнозування динамічних VaR і CVaR на основі квінтильної регресії з використанням металог розподілу |
| title_full_unstemmed | Прогнозування динамічних VaR і CVaR на основі квінтильної регресії з використанням металог розподілу |
| title_short | Прогнозування динамічних VaR і CVaR на основі квінтильної регресії з використанням металог розподілу |
| title_sort | прогнозування динамічних var і cvar на основі квінтильної регресії з використанням металог розподілу |
| topic | VaR CVaR Expected Shortfall динамічні міри ризику прогноз квантильна модель LGARCH розподіл металог |
| topic_facet | VaR CVaR Expected Shortfall dynamic risk measures forecast Quantile LGARCH model metalog distribution VaR CVaR Expected Shortfall динамические меры риска прогноз квантильная модель LGARCH распределение металог VaR CVaR Expected Shortfall динамічні міри ризику прогноз квантильна модель LGARCH розподіл металог |
| url | https://journal.iasa.kpi.ua/article/view/236942 |
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