XXXV ENCONTRO NACIONAL DE ENGENHARIA DE PRODUCAO
Perspectivas Globais para a Engenharia de Produção
Fortaleza, CE, Brasil, 13 a 16 de outubro de 2015.
PARAMETRIC EXTREME VAR WITH
LONG-RUN VOLATILITY: COMPARING
OIL AND GAS COMPANIES OF BRAZIL
AND USA.
RICARDO RYOITI KOJIMA (UFSCar)
[email protected]
Herick Fernando Moralles (UFSCar)
[email protected]
Andrei Aparecido de Albuquerque (UFSCar)
[email protected]
Parametric VaR seems to be extremely interesting because of its
computational simplicity and effectiveness when the distribution of
returns is properly specified. Thus, the present study proposes
goodness-of-fit test based on EDF statistics, such as KolmogorovSmirnov test, to find the best probability distribution of extreme returns
(worst losses), using GARCH approach to insert the average long-run
variance to the chosen distribution. The method was applied to Oil and
Gas companies assets for Brazil (developing market), and USA (a
developed market). The results demonstrate that in all assets VaR
violation were under the confidence interval given by the backtesting,
and the method proved to be more effective than the traditional
variance-covariance method.
Palavras-chave: VaR Parametrizado, testes goodness-of-fit, GARCH,
perdas extremas.
XXXV ENCONTRO NACIONAL DE ENGENHARIA DE PRODUCAO
Perspectivas Globais para a Engenharia de Produção
Fortaleza, CE, Brasil, 13 a 16 de outubro de 2015.
1. Introduction
Parametric methods for VaR calculation are extremely easy to implement even in a spreadsheet, thus been very
suitable for practitioners (ANGELIDIS, BENOS et al. 2004). However, such methods simply capture the
volatility by calculating the standard deviation of the return series, therefore not capturing effectively the
volatility clusters that are common to financial series.
Additionally, given that such methods are based on normality assumptions, this kind of simplification cannot
capture the leptocurtic nature (heavy tails and excess kurtosis) of financial data (KAWATA and KIJIMA 2007).
This fact is reflected by several studies that acknowledge different probability distributions appropriate for
computing VaR.
The GARCH (p, q) model successfully captures several characteristics of financial time series, such as thick
tailed returns and volatility clustering, therefore been widely used to model financial market volatilities in risk
management applications (ANGELIDIS, BENOS et al. 2004; HANG CHAN, DENG et al. 2007).
In fact, despite its advantages, flexible models for the innovation process of GARCH models have been limited,
because empirical evidence has demonstrated that the conditional normal time series models (e.g., the classical
GARCH models) are inadequate in estimating the tail quantiles of conditional return, and also, because of the
lack of freely available software for fitting of GARCH models (ORHAN 2012; NADARAJAH, AFUECHETA
et al. 2014). Despite all these arguments, parametric VaR seems to be extremely interesting because of its
computational simplicity and effectiveness when the distribution of returns is properly specified, and therefore it
should not be disregarded as a method for VaR calculation.
Consequently, the present study proposes a goodness-of-fit tests based on EDF statistics, such as KolmogorovSmirnov test, to find the best probability distribution of extreme returns (worst losses), using GARCH approach
to insert the average long-run variance to the chosen distribution. Hence, the proposed method configures a static
parametric approach with extreme value long-run average variance, but dynamic in terms of appropriate
probability distribution to model the return tails series.
The proposed method was applied to a sample of Oil and Gas companies for Brazil (developing market), and
USA (a developed market). This application will verify the effectiveness of the VaR model proposed for two
different realities and will identify the best goodness-of-fit to determine the most appropriate distributions for the
computation of parametric VaR. Finally, backtesting is used to verify that the VaR estimate is under control, and
compare it to the traditional covariance-variance parametric method.
2. Literature Review : GARCH methods and long-run average variance
A GARCH model is a popular time series model for weakly stationary financial data, and is composed by two
components: the volatility component and the innovation component. Its the simplest and the most commonly
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XXXV ENCONTRO NACIONAL DE ENGENHARIA DE PRODUCAO
Perspectivas Globais para a Engenharia de Produção
Fortaleza, CE, Brasil, 13 a 16 de outubro de 2015.
used model for volatility is of the order of (1, 1). The afore mentioned innovation is commonly assumed to come
from the Gaussian distribution, the Student’s t distribution or some skewed extension of these distributions
(NADARAJAH, AFUECHETA et al. 2014).
Since,
yt  ht  t
1/ 2
Where,
(1)
y t is an observed financial data,  t is the innovation process, and ht 1/ 2 is the residual volatility at time
“t”. Therefore, the GARCH (1,1) is presented at (2).
ht    yt21  ht 1
The restrictions of
Hence, if
(2)
  0 ,   0 , and   0
ensures that
ht 1  0 .
ht 1  0 α is the ARCH coefficient part, and β the GARCH term, and by adding yt 2 on both sides of
the equation,
yt2    (   ) yt21  ut 1  ut
(3)
Where, ut  yt  ht and ut 1  yt 1  ht 1
2
2
Here, the equation (3) is an second order stationary ARMA (1,1) model if
   1
(Aiube 2013). Also, the
GARCH model can be generalized to higher orders. The GARCH (p, q) is given by (4).
q
p
t 1
j 1
ht      i yt2i    j ht  j
(4)
Following this formulation, Engle (2001)states that the most widely used GARCH specification asserts that the
best predictor of the variance in the next period is a weighted average of the long-run average variance, the
predicted for this period, and the new information in this period that is captured by the most recent squared
residual. Thus, the long-run average variance is (5).
VL 

(1     )
, if
  0;   0;   0
(5)
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XXXV ENCONTRO NACIONAL DE ENGENHARIA DE PRODUCAO
Perspectivas Globais para a Engenharia de Produção
Fortaleza, CE, Brasil, 13 a 16 de outubro de 2015.
It should be noted that this only works if
    1,
because the final term in the equation becomes
progressively smaller as time increases. This means that the variance rate exhibits mean reversion with a
reversion level of VL and a reversion rate of 1 − α − β. Thus, the variance rate forecast tends toward VL, as the
risk manager tend to look further ahead. So, if α + β > 1, the weight given to the long-term average variance is
negative, and the process will not converge to the mean variance (HULL 2006).
The equation (6) forecasts the volatility on day n + k using the information available at the end of day n − 1. In
the EWMA model, α + β = 1 shows that the expected future variance rate equals the current variance rate. Thus,
the EWMA model configures a GARCH special case, used by the company RiskMetrics™, when they
introduced their analytic VaR methodology. The volatility forecast is the weighted average of the previous
period’s forecast and the current squared return, and the variance follows an exponentially declining process
(Angelidis, Benos et al. 2004).
So, this analysis emphasizes the point VaR calculation using the GARCH-based long run average volatility with
α +β < 1 in the context of extreme returns in a distributional flexible approach.
3. Data and Methods
The data of closing stock prices of the major d Oil & Gas companies traded on the São Paulo Stock Exchange
(BOVESPA) and the New York Stock Exchange (NYSE) were collected in the period from January of 2008
until August of 2014. This sample range choice aims to compare the proposed form of parametric VaR that
considers the stochastic volatility for emerging and developed countries, also being compared to a regular VaR
methodology that assumes the normality of the distribution of stock returns.
The choice of such a long period expresses the intention of specifying a rare or adverse event. This is, therefore,
the kind of information that interests managers that seek an analysis of cases of extreme loss or worst-case
scenario (MCNEIL and FREY 2000). The data collected were used to calculate the returns using natural
logarithms as in Equation (6). It can be observed that the variations found among the different days represent
daily returns, and other payments such as bonuses and dividends are not considered.
Rt ( x)  (ln rt / ln rt 1 )  ln rt  ln rt 1
(6)
As for the distributions that do not allow negative values in their domains, we used the exponential of returns in
(6).
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XXXV ENCONTRO NACIONAL DE ENGENHARIA DE PRODUCAO
Perspectivas Globais para a Engenharia de Produção
Fortaleza, CE, Brasil, 13 a 16 de outubro de 2015.
Figure 1. Residual and actual return representation of Ibovespa Index
.15
.10
.05
.00
.15
-.05
.10
-.10
.05
-.15
.00
-.05
-.10
-.15
250
500
750
Residual
1000
Actual
1250
1500
Fitted
From the series of returns, the tail of the empirical probability distribution to be parameterized is selected. A
series of techniques used to make the choice of the point where the tail starts is available in the literature.
However, the experiment performed in this study determines the choice of the tail length based on the
significance level to be used for VaR. Thus, for a VaR with 95% confidence, we recommend a value between 5
and 10% of the worst returns of the empirical distribution in order to obtain a sample value consistent with the
goodness-of-fit tests. However, this study uses the figure of 10% of the worst returns of the empirical
distribution (MORALLES, REBELATTO et al. 2013).
Once the returns and tails of each asset were calculated for the period, the Kolmogorov-Smirnov goodness-of-fit
test was performed for each asset considering the 17 probability distributions. Namely, Beta, Cauchy, Gumbel
max, Gumbel min, Logistics, Normal, Exponential, Frechét, Exponential, Gama, inverse Gaussian distribution,
Lognormal, Pareto, Rayleigh, Weibull, Three parameter Weibull, Generalized Extreme Values, and Generalized
Pareto. The description and format of the distributions mentioned above may be found in some specific studies
(KLEIBER and KOTZ 2003).
Therefore, the model for the calculation of VaR is (7).
VaR( x)  i1 ( p |  ,  ,  )  x :  ( x |  ,  ,  )
(7)
Where p is the significance level used to calculate VaR,
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XXXV ENCONTRO NACIONAL DE ENGENHARIA DE PRODUCAO
Perspectivas Globais para a Engenharia de Produção
Fortaleza, CE, Brasil, 13 a 16 de outubro de 2015.
,  , 
are respectively location, scale, and shape parameter, if it exists, which are estimated by maximizing
the likelihood function of the distribution
i
through iterative numerical methods. And
i1 ( x) is the inverse
cumulative density function of the probability distribution to be specified by the goodness-of-fit tests.
Thus, the distribution
i
to be used in the calculation of VaR is based on
KS  max x Fn ( x)  F ( x)
If,
(8)
KS  K1 , where K1 are critical values at a significance  .
In order to take into consideration the stochastic volatility, the assets and indices historical series collected were
tested for its stationarity and ARCH effect presence. For the stationarity test, the Dickey-Fuller test was
performed, and the ARCH test was performed for the presence of ARCH effect. The econometric Matlab®
toolbox was the software used to run the tests.
After the confirmation of stationarity and ARCH effect presence of the time series, the EViews® software was
used to forecast the stochastic volatility of each time series considering ARCH, GARCH, TARCH, EGARCH
and PARCH. For each asset, the Schwarz information criterion value was collected for each type of GARCH
derivation assumed. The GARCH models are estimated through maximizing the likelihood function based on
the t distribution, since its returns slightly better estimates than the Normal. But it is surprising to see the worst
performing specifications have t worse than the normal (ORHAN 2012).
Indeed, there are several possibilities to implement this maximum likelihood estimation like using Gaussian,
skewed Gaussian, the Student’s t, skewed Student’s t, generalized error, skewed generalized error, standardized
normal inverse Gaussian (NADARAJAH, AFUECHETA et al. 2014).
The BIC is a criterion for model selection based on the likelihood function. In this case, it is important to
compare the ARCH family models using this criterion as a parameter. Therefore, the best fitting model can be
selected. Regarding the methodology, there is a penalty term for the number of parameters in the model (over
fitting problem). For a large sample size, the BIC can be represented by (9).
BIC  2 ln Lˆ  k ln(n)
Where
(9)
L̂ = the maximized value of the likelihood function of the proposed model; k = number of free
parameters to be estimated; and n = number of data points in the sample.
According to the literature, the lower the value of the information criterion is the better is the adaptation to the
time series being analyzed. Finally, with the forecasted longrun variance obtained from EViews® and the best
fitted distribution of the tail, it is possible to change the distribution parameters and consider the estimated long
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XXXV ENCONTRO NACIONAL DE ENGENHARIA DE PRODUCAO
Perspectivas Globais para a Engenharia de Produção
Fortaleza, CE, Brasil, 13 a 16 de outubro de 2015.
.04
run volatility to calculate the VaR. An example of the long-run volatility for the Ibovespa index Forecast:
returns isIBOVF
presented at Figures 3.
Actual: IBOV
Forecast sample: 1 1652
Included observations: 1652
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.02
.00
-.02
-.04
250
500
750
1000
1250
1500
IBOVFof Ibovespa
± 2 S.E. Index Series of Returns
Figure 3. Long Run Variance Forecast
.00036
.00034
.00032
.00030
.00028
.00026
250
500
750
1000
1250
1500
Forecast of Variance
As a parameter of its accuracy, the proposed method for VaR calculation was compared to a regular VaR
methodology, which considers the returns of the assets as normally distributed. This comparison was performed
using the Backtesting, which additionally shows which distributions, are the most suitable for emerging and
developed countries.
4. Results
Results of VaR calculation are shown below. Only one Brazilian asset of the Petroleum Industry presented
extremely negative returns, and could not be analyzed by ARCH family models, since it does not present ARCH
effect. First, Tables 1 and 2 presents the proposed VaR method application for Brazilian and USA Gas & Oil
sectors.
Table 1 - Brazilian Oil&Gas assets calculation of VaR.
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XXXV ENCONTRO NACIONAL DE ENGENHARIA DE PRODUCAO
Perspectivas Globais para a Engenharia de Produção
Fortaleza, CE, Brasil, 13 a 16 de outubro de 2015.
OIL & GAS (BRAZIL)
Augmented Dickey-Fuller Test
ARCH Effect
Best Fitting Model
Schwarz - Best Fitting Model
Schwarz - GARCH (1,1)
Long Run Variance
Estimated Volatility
Tail Komogorov Smirnov Distribution
VaR - Proposed Method
VaR - Normal
ASSET 1
Stationary
Yes
GARCH (1,1) - GED
-3.484261
-3.299514
0.002543
0.050428167
GEV
-0.078
-0.082
ASSET 2
Stationary
No
ASSET 3
Stationary
Yes
PARCH(1,1)
-4.766323
-4.715696
0.000529
0.023
GEV
-0.0363
-0.0421
ASSET 4
ASSET 5
Stationary
Stationary
Yes
Yes
EGARCH(2,1) EGARCH(2,1)
-4.92233
-4.19321
-4.767995
-3.293229
0.000418
0.001801
0.020445048
0.04243819
GEV
GEV
-0.0324
-0.0698
-0.0397
-0.0923
Table 2 - USA Oil&Gas assets calculation of VaR.
OIL & GAS (USA)
Augmented Dickey-Fuller Test
ARCH Effect
Best Fitting Model
Schwarz - Best Fitting Model
Schwarz - GARCH (1,1)
Long Run Variance
Estimated Volatility
Tail Komogorov Smirnov Distribution
VaR - Proposed Method
VaR - Normal
ASSET 1
Stationary
Yes
PARCH (2,1)
-5.748532
-5.728664
0.000256
0.016
GEV
-0.0262
-0.0302
ASSET 2
Stationary
Yes
PARCH(1,1)
-5.402414
-5.260677
0.000427
0.020663978
GEV
-0.0297
-0.0349
ASSET 3
Stationary
Yes
PARCH(1,1)
-5.924748
-5.873361
0.000167
0.012922848
GEV
-0.0237
-0.0278
ASSET 4
Stationary
Yes
EGARCH(1,1)
-5.769653
-5.763524
0.000216
0.014696938
GEV
-0.0192
-0.0231
ASSET 5
Stationary
Yes
TARCH (1,1)
-4.917333
-4.908245
0.000462
0.021494185
GEV
-0.0384
-0.0421
The results shown at Tables 1 and 2 demonstrates that the GEV (Generalized Extreme Value) distribution is
pointed by the Kolmogorov-Smirnov test as the best one to model the tail returns, and the estimated long-run
volatility was based on GARCH, PARCH, and EGARCH for the Oil & Gas sector. As for the VaR computation,
the simple variance-covariance method tended to return risk measures above the proposed model. Also, it is
observable that the average VaR in Brazil (-0.03585) is higher than the average VaR of the US assets (-0.04767),
pointing to confirmation that developing economies tend to have more volatile markets.
In order to determine which methodology provides more precise estimates of VaR, the Kupiec (1995) test was
used, allows us to calculate the limit of the range in which the null hypothesis that the VaR actually "hits" 95%
of the cases is accepted, which is similar to solving the equation (12).
L  2 ln[(1  p)T  N p n ]  2 ln[(1  ( N / T ))T  N ( N / T ) N ]
(12)
The routine developed by MATLAB® establishes the confidence interval, i.e., it finds the number “n” in which
2ln (L)-2ln (L0) is closest to the tabulated value of chi-square with one degree of freedom, rounding the result to
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XXXV ENCONTRO NACIONAL DE ENGENHARIA DE PRODUCAO
Perspectivas Globais para a Engenharia de Produção
Fortaleza, CE, Brasil, 13 a 16 de outubro de 2015.
an integer. Thus, Table 3 show the acceptance ranges for the violations of the VaR calculation method used
herein.
Table 3 - Backtesting.
Backtesting
Asset
VaR - Proposed
Method
VaR
Normal
Sample
Size
Confidence
Region (0.95)
VaR Proposed
Method - Effective?
VaR Normal Effective?
52
88
54
85
94
87
80
52
88
45
61
32
50
67
69
60
33
73
976
1605
1603
865
1656
1677
1677
1677
892
1677
35 < T < 65
62 < T < 101
62 < T < 101
30 < T < 58
64 < T < 103
65 < T < 105
65 < T < 105
65 < T < 105
31 < T < 60
65 < T < 105
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
No
Yes
No
Yes
Yes
No
Yes
Yes
BR Oil&Gas Asset 1
BR Oil&Gas Asset 2
BR Oil&Gas Asset 3
BR Oil&Gas Asset 4
BR Oil&Gas Asset 5
US Oil&Gas Asset 1
US Oil&Gas Asset 2
US Oil&Gas Asset 3
US Oil&Gas Asset 4
US Oil&Gas Asset 5
Table 3 shows that the VaR violations for the proposed model, in all assets for US and Brazil, were under the
confidence interval given by the backtesting,
5. Conclusions
Regarding overall results, the proposed method for VaR calculation performed better results once it was not
effective in only one calculation (since data did not presented the ARCH effect). In comparison to the Normal
VaR calculation, the proposed method show a more precise market risk forecast, and the backtesting observed
that the VaR estimate was within the acceptable confidence interval, even been a static approach in long run time
series (2008 to 2014).
Also, the Kolmogorov-Smirnov test demonstrates that the GEV (Generalized Extreme Value) distribution was
the best distribution for tail modeling in both realities and sector. Further, the information criterion application
proved to be necessary given the multiplicity of GARCH family suitable models.
Observing the American scenario, the assets risk is measured more precisely with the proposed method, and the
majority of the VaR values calculated assuming the normality of returns is out of the confidence region. On the
other hand, the Oil & Gas industry assets presented most of the risk estimates inside the confidence region.
Depending on the economy sector, the assets VaR can be better predicted, especially if the returns have fat tails
of returns and non-Normal distributions.
The Brazilian scenario has a different situation compared to the American market. possibly due to its lack of
maturity, the stock market presents extreme returns and losses, and some equities, such as one of the Petroleum
Industry in the sample chosen to this study, cannot be described by ARCH family models because of extreme
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XXXV ENCONTRO NACIONAL DE ENGENHARIA DE PRODUCAO
Perspectivas Globais para a Engenharia de Produção
Fortaleza, CE, Brasil, 13 a 16 de outubro de 2015.
losses. Even though, the proposed method of VaR calculation presented better results compared to the Normal
VaR method (1 negative result compared to 4).
Regarding the main Indices of the stock markets, the proposed method performed satisfactory results, and for the
American Dow Jones Index the normality assumption of returns was not efficient in VaR calculation. Finally,
the parametric extreme value VaR based on EDF statistics and long run volatility presented satisfactory results
for both markets, even been a static approach.
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