
 Start

3504
 Prefix

In addition, as a primary source of energy in the
power industry, industrial production and transportation, volatile oil prices may lead to cost
uncertainties for other markets, thus extensively affecting the development of the economy.
A large number of studies have shown that oil price fluctuations could have considerable
impact on economic activities.
 Exact

Papapetrou (2001)
 Suffix

argues that the variability of oil prices
plays a critical role in affecting real economic activity and employment. Lardic and Mignon
(2008) explore the longterm relationship between oil prices and GDP, and find evidence that
aggregate economic activity seems to slow down particularly when oil prices increase.
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 Start

3646
 Prefix

A large number of studies have shown that oil price fluctuations could have considerable
impact on economic activities. Papapetrou (2001) argues that the variability of oil prices
plays a critical role in affecting real economic activity and employment.
 Exact

Lardic and Mignon (2008)
 Suffix

explore the longterm relationship between oil prices and GDP, and find evidence that
aggregate economic activity seems to slow down particularly when oil prices increase. This
asymmetry is found in both the U.
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 Start

4750
 Prefix

Although VaR is
recommended by Basel II and III and has been widely adopted by financial institutions, it
has been challenged by the Bank of International Settlements (BIS) Committee, who pointed
out that VaR cannot measure market risk as it fails to consider the extreme tail events of
a return distribution (see,
 Exact

Chen et al., 2012).
 Suffix

In addition, Artzner et al. (1999) argue that
VaR does not meet the requirements of subadditivity and thus is not a coherent measure
of risk. As an alternative, they proposed a conservative, but more coherent measure, called
Conditional VaR at risk (CVaR) or expected shortfall (ES), which considers the average loss
as that exceeding the VaR threshold.
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 Start

4784
 Prefix

Although VaR is
recommended by Basel II and III and has been widely adopted by financial institutions, it
has been challenged by the Bank of International Settlements (BIS) Committee, who pointed
out that VaR cannot measure market risk as it fails to consider the extreme tail events of
a return distribution (see, Chen et al., 2012). In addition,
 Exact

Artzner et al. (1999)
 Suffix

argue that
VaR does not meet the requirements of subadditivity and thus is not a coherent measure
of risk. As an alternative, they proposed a conservative, but more coherent measure, called
Conditional VaR at risk (CVaR) or expected shortfall (ES), which considers the average loss
as that exceeding the VaR threshold.
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 Start

6069
 Prefix

Therefore, in this paper we consider
risks affecting both oil supply and oil demand.
In recent years, the commodity price literature has shown that there is evidence of
leverage effects in various energy markets. More specifically,
 Exact

Chan and Grant (2016a),
 Suffix

considering lower frequency (weekly) commodity returns conclude that SV models (with
an MA component) are able to replicate the main features of the data more efficiently
than GARCH models.
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 Start

6421
 Prefix

More specifically, Chan and Grant (2016a), considering lower frequency (weekly) commodity returns conclude that SV models (with
an MA component) are able to replicate the main features of the data more efficiently
than GARCH models. At the same time, they find a significant negative leverage effect
in crude oil spot markets.
 Exact

Kristoufek (2014)
 Suffix

focuses on the leverage effect in commodity
futures markets and provides an extensive literature review in this area. Fan et al. (2008)
estimate VaR of crude oil prices using a GEDGARCH approach with daily WTI and Brent
prices spanning from 1987 to 2006.
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 Start

6949
 Prefix

They find that this type of model specification does
as well as the standard normal distribution at a 95% confidence level. They also test and
find evidence for asymmetric leverage effects without modelling them directly.
 Exact

Youssef et al. (2015)
 Suffix

evaluate VaR and CVar for crude oil and gasoline markets using a long memory
GARCHEVT approach. Their findings and backtesting exercise show that crude oil markets
are characterized by asymmetry, fat tails and long range memory.
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 Start

9210
 Prefix

Using the same logic, firms
who are on the supply side, would be better off not considering the leverage effect (SVN
model).
2. Stochastic volatility models
We use a general SV model to capture the volatility features for oil markets which has
been studied recently by
 Exact

Takahashi et al. (2009), Chai et al. (2011) and
 Suffix

Chan et al. (2016a):
yt=μ+σtzt(1)
lnσ2t=ht=δ+β(lnσ2t−1−δ) +ηtηt∼N(0,σ2η)(2)
whereytdenotes stock returns at timetwitht= 1,2,...,T,μdenotes the conditional mean,σt
is the stochastic volatility,lnσ2tfollows a stationary AR(1) process with persistence parameter
βhavingβ<1,ztandηtrepresent a series of independent identical (i.i.d.) random errors
in the return and volatility equation, respectively.2
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10214
 Prefix

of tdistribution.
(2) Standard Normal errors
zt∼N(0,1)
(3) Standard Asymmetric Laplace errors
zt∼ALD(0,κ,1)
whereσ= 1 andκis the coefficient driving the skewness of the distribution, is related toμ
andσas follows:
μ=
σ
√
2
(
1
κ
−κ
)
ht
2>0 (Symmetric Laplace Distribution).3
as a special caseκ= 1 forμ'0 andσ=e
2A number of original empirical works via extended SV models can be found from
 Exact

Breidt et al. (1998), So et al. (1998), Yu and Yang (2002),
 Suffix

Koopman and Uspensky (2002), Cappuccio et al. (2004), Chan (2013),
Chan and Hsiao (2013), Chan and Grant (2016c), Chan (2017).
3See appendix for the density of ALD.
(4) Standard Student t errors with leverage effect
yt=μ+σtzt
zt∼tν
lnσ2t=ht=δ+β
(
lnσ2t−1−δ
)
+ξt
ξt=ρzt+
√
1−ρ2ηt
ηt∼N
(
0,σ2η
)
where the coefficientρdrives the so calledleverage effect.
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10304
 Prefix

errors
zt∼ALD(0,κ,1)
whereσ= 1 andκis the coefficient driving the skewness of the distribution, is related toμ
andσas follows:
μ=
σ
√
2
(
1
κ
−κ
)
ht
2>0 (Symmetric Laplace Distribution).3
as a special caseκ= 1 forμ'0 andσ=e
2A number of original empirical works via extended SV models can be found from Breidt et al. (1998), So et al. (1998), Yu and Yang (2002), Koopman and Uspensky (2002),
 Exact

Cappuccio et al. (2004), Chan (2013),
 Suffix

Chan and Hsiao (2013), Chan and Grant (2016c), Chan (2017).
3See appendix for the density of ALD.
(4) Standard Student t errors with leverage effect
yt=μ+σtzt
zt∼tν
lnσ2t=ht=δ+β
(
lnσ2t−1−δ
)
+ξt
ξt=ρzt+
√
1−ρ2ηt
ηt∼N
(
0,σ2η
)
where the coefficientρdrives the so calledleverage effect.
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 Start

10365
 Prefix

the skewness of the distribution, is related toμ
andσas follows:
μ=
σ
√
2
(
1
κ
−κ
)
ht
2>0 (Symmetric Laplace Distribution).3
as a special caseκ= 1 forμ'0 andσ=e
2A number of original empirical works via extended SV models can be found from Breidt et al. (1998), So et al. (1998), Yu and Yang (2002), Koopman and Uspensky (2002), Cappuccio et al. (2004), Chan (2013), Chan and Hsiao (2013),
 Exact

Chan and Grant (2016c), Chan (2017).
 Suffix

3See appendix for the density of ALD.
(4) Standard Student t errors with leverage effect
yt=μ+σtzt
zt∼tν
lnσ2t=ht=δ+β
(
lnσ2t−1−δ
)
+ξt
ξt=ρzt+
√
1−ρ2ηt
ηt∼N
(
0,σ2η
)
where the coefficientρdrives the so calledleverage effect.
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13915
 Prefix

Estimation methodology of Bayesian MCMC
In order to improve the tractability of the ALD model, we introduce a modified Bayesian
MCMC method. That is, a new Scale Mixture of Uniform (SMU) representation for the AL
density (following
 Exact

Kotz et al., 2001)
 Suffix

is proposed to facilitate the estimation of the SVALD
model.
4See appendix B for the derivations ofV aRs,t,V aRd,t,CV aRs,tandCV aRd,t.
4.1. Scale mixture of uniform representation of ALD
Expressing the ALD via the representation can alleviate the computational burden when
using the Gibbs sampling algorithm in the MCMC approach and thus can simplify the
estimation method in Bayesian
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14500
 Prefix

of ALD
Expressing the ALD via the representation can alleviate the computational burden when
using the Gibbs sampling algorithm in the MCMC approach and thus can simplify the
estimation method in Bayesian analysis. To estimate the latent variables in the SV model,
we use the scaled ALD (SALD) which means that the ALD random variable is scaled by its
standard deviation (See
 Exact

Chen et al., 2009 and Wichitaksorn et al., 2015).
 Suffix

Proposition 1.Letztbe the ALD random variable withzt∼ALD(0,κ,1), then the random
variableεt=ztS.D.[z]has SALD withp.d.f.given by:
f(εtκ,σt) =
√
1 +κ4
1 +κ2
1
σt
exp(
−
√
1 +κ4
σt
εt)εt≥0
√
1 +κ4
1 +κ2
1
σt
exp(
√
1 +κ4
κ2σt
εt)εt<0
(7)
whereκis skewness parameter andσtis the standard deviation (or the timevarying volatility)
ofzt.5
Hence, the corresponding SMU of SALD can be obtained
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 Prefix

If the ratio is greater thanα, we conclude
that the model underestimate the risks, and vice versa. In addition, three likelihood ratio
backtesting criteria are implemented to test the statistical accuracy of the methods. These
criteria include unconditional coverage test (LRucby
 Exact

Kupiec, 1995),
 Suffix

independent test (LRind
by Christoffersen, 1998) and conditional coverage test (LRccby Christoffersen, 1998).
For backtesting CVaR in the SV model with Normal errors, we select the nominal risk
level at 5% and 1% as 1.96% and 0.38%, following the work by Chen et al. (2012).
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 Start

17210
 Prefix

In addition, three likelihood ratio
backtesting criteria are implemented to test the statistical accuracy of the methods. These
criteria include unconditional coverage test (LRucby Kupiec, 1995), independent test (LRind
by
 Exact

Christoffersen, 1998) and
 Suffix

conditional coverage test (LRccby Christoffersen, 1998).
For backtesting CVaR in the SV model with Normal errors, we select the nominal risk
level at 5% and 1% as 1.96% and 0.38%, following the work by Chen et al. (2012).
 (check this in PDF content)

 Start

17270
 Prefix

In addition, three likelihood ratio
backtesting criteria are implemented to test the statistical accuracy of the methods. These
criteria include unconditional coverage test (LRucby Kupiec, 1995), independent test (LRind
by Christoffersen, 1998) and conditional coverage test (LRccby
 Exact

Christoffersen, 1998).
 Suffix

For backtesting CVaR in the SV model with Normal errors, we select the nominal risk
level at 5% and 1% as 1.96% and 0.38%, following the work by Chen et al. (2012). To
identify the nominal risk level for ALD, we use a cumulative distribution function (c.d.f.) of
7Note that for the prior setting in the Bayesian inference, the WinBugs software uses a nonstandard
par
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 Start

17449
 Prefix

These
criteria include unconditional coverage test (LRucby Kupiec, 1995), independent test (LRind
by Christoffersen, 1998) and conditional coverage test (LRccby Christoffersen, 1998). For backtesting CVaR in the SV model with Normal errors, we select the nominal risk
level at 5% and 1% as 1.96% and 0.38%, following the work by
 Exact

Chen et al. (2012). To
 Suffix

identify the nominal risk level for ALD, we use a cumulative distribution function (c.d.f.) of
7Note that for the prior setting in the Bayesian inference, the WinBugs software uses a nonstandard
parametrization of Normal distribution in terms of the precision (1/variance) instead of the variance.
8See appendix E for the derivation of the parameters of full conditional posterior dist
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17909
 Prefix

ALD, we use a cumulative distribution function (c.d.f.) of
7Note that for the prior setting in the Bayesian inference, the WinBugs software uses a nonstandard
parametrization of Normal distribution in terms of the precision (1/variance) instead of the variance.
8See appendix E for the derivation of the parameters of full conditional posterior distributions.
ALD which, according to
 Exact

Kotz et al. (2001),
 Suffix

is given by:
F(zκ,θ,σ) =
1−
1
1 +κ2
exp
(
−
√
2κ
σ
z
)
z≥0
κ2
exp
(√
2
σκ
z
)
z <0
(13)
1 +κ2
Then, thisc.d.f.is evaluated at the point that equates to the CVaR level.As a
consequence, the probability ( ̃α) that CVaR occurs under ALD for oil supply and demand
can be mathematically expressed as:
Supply: ̃α=F(CV aRsα) =
α
e
(14)
Demand: ̃α=1−F(CV aRdα) =
α
e
(15)
whereeis
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18526
 Prefix

For both of oil supply and demand, the quantile level of
CVaR under ALD is simply a function ofαandeand does not depend on other parameters
in the AL density. This surprising finding is consistent with results of
 Exact

Chen et al. (2012)
 Suffix

although different ALD forms have been studied. Hence, according to formula (14) and (15),
the nominal risk level ̃αfor CVaR under ALD at 5% and 1% are obtained as 1.84% and
0.37%, respectively.
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19254
 Prefix

Efficiency measures
Adhering to the Basel Committee’s guidelines, supervisors are not only concerned with
the quantities of violations in a VaR model but also with the magnitude of those violations
(Basel Committee on Banking Supervision, 1996a, 1996b). Hence, we employ the regulatory
loss function (RLF) and firm’s loss function (FLF) of
 Exact

Sarma et al. (2003),
 Suffix

which considers
both the number of violations and their magnitude. This is a twostage model evaluation
procedure where the first stage aims to test the models in terms of statistical accuracy, while
in the second stage the models surviving the statistical accuracy tests are then evaluated for
efficiency (details see Sarma et al., 2003).
6.
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19592
 Prefix

This is a twostage model evaluation
procedure where the first stage aims to test the models in terms of statistical accuracy, while
in the second stage the models surviving the statistical accuracy tests are then evaluated for
efficiency (details see
 Exact

Sarma et al., 2003).
 Suffix

6. Simulation experiment
To examine the effectiveness of the proposed MCMC sampling procedure, we estimate
the SVALD model using simulated data. We generate 2874 observations from the SVALD model given by (1) and (2) with ALD errors by fixing parameter valuesδ=−7.587,
β= 0.9947,ση= 0.0889 andκ= 0.9956.
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21499
 Prefix

The posterior means and medians forβ,σηand
κare very close to the true values with small MC errors, which are all located inside the
95% credible intervals. The posterior means and medians forδvary more than others as it
would be expected since the variance parameter is quantiledependent (similar findings also
see
 Exact

Chen et al., 2009).
 Suffix

7. Empirical analysis
7.1. Data
We consider two major crude oil markets: West Texas Intermediate crude oil (WTI) and
Europe Brent crude oil (Brent). Daily closing spot prices, which are quoted in US dollars per
barrel, are obtained from the U.
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32387
 Prefix

that the estimated posterior mean ofσηfor the SVALD model is
lower comparing to the correspondingσηin the SVN and SVt model in both of the two oil
markets, andσηin the SVt model is lower than thatσηin the SVN model.
The estimate for theσηparameter from the SVALDL and SVtL model is lower than
the estimate coming from the SVNL model. These results are consistent with the findings
from
 Exact

Chib et al. (2002) and AbantoValle et al. (2010),
 Suffix

indicating that the introduction of
heavy tailed error distribution in the mean equation appears to explain excess returns, thus
decreasing the variance of the volatility process. More importantly, we find statistically
significant negative correlation (ρ <0) between shocks affecting oil returns and shocks
affecting volatility in the SVtL, SVNL and the SVALDL s
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32972
 Prefix

More importantly, we find statistically
significant negative correlation (ρ <0) between shocks affecting oil returns and shocks
affecting volatility in the SVtL, SVNL and the SVALDL specifications. Although
WinBugs can generate deviance information criterion (DIC) values straightforwardly, as
pointed out by
 Exact

Chan and Grant (2016),
 Suffix

conditional DIC typically favors overfitted models
in a series of Monte Carlo experiments. Therefore this cannot be used as a reliable criterion
to compare across models. For this reason, we use various comparison criteria.
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36197
 Prefix

The Shapiro Francia test (1972) for normality concurs with those
judgements for the standard errors coming from all the models. The Box Pierce portmanteau
(or Q) test for white noise rejects its null for both series of standard errors.
Diebold Mariano test.This test calculates a measure of predictive accuracy proposed
by
 Exact

Diebold and Mariano (1995).
 Suffix

We ran the test for each of 500 simulations per model
Table 10: WTI: Engle’s Lagrange multiplier test for autoregressive conditional heteroskedasticity for
standardised residuals and squared standardised residuals for SVt and SVtL models
1 lag pval 5 lags pval 10 lags pval 30 lags pval
SVt res2.040.158.980.1113.750.1847.150.02
SVt res squ0.190.660.820.981.
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44958
 Prefix

The model is said to be correctly specified if the calculated
ratio is equal to the prespecified VaR levelα(i.e. 5% and 1%). If the Failure Rate is
greater thanα, we can conclude that the model underestimates the risk, and vice versa (see
 Exact

Marimoutou et al., 2009; Aloui and Mabrouk, 2010; Louzis et al., 2014).
 Suffix

A criterion for evaluating our results comes from the consideration that a conservative
investor (see for example, Zhao et al., 2015 and Hung et al., 2008) might choose a greater
confidence level and estimate a relatively greater risk (corresponding toα= 5% in the
VaR definition), while a more speculative investor might estimate a smaller risk and face a
relatively smaller confidence l
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45142
 Prefix

If the Failure Rate is
greater thanα, we can conclude that the model underestimates the risk, and vice versa (see
Marimoutou et al., 2009; Aloui and Mabrouk, 2010; Louzis et al., 2014). A criterion for evaluating our results comes from the consideration that a conservative
investor (see for example,
 Exact

Zhao et al., 2015 and
 Suffix

Hung et al., 2008) might choose a greater
confidence level and estimate a relatively greater risk (corresponding toα= 5% in the
VaR definition), while a more speculative investor might estimate a smaller risk and face a
relatively smaller confidence level (corresponding toα= 1% in the VaR definition).
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46387
 Prefix

Modelling the data using SVN and SVNL models, the testLRucandLRccare passed, which indicates the capability of the models
of estimating tail risks. The SVALD and SVALDL models overestimate the tail risk and
the null hypothesis of testsLRucandLRccare rejected in WTI market and partly rejected in
12For details see
 Exact

Sarma et al. (2003).
 Suffix

Table 23: VaR backtesting results for WTI and Brent markets
αRisk
Failure timesFailure rateLRucLRindLRcc
WTI BrentWTIBrentWTIBrentWTIBrentWTIBrent
SVN
5%
VaRst1071224.248%4.839%0.07570.70990.8269 0.21140.19300.3869
VaRdt1111164.407%4.720%0.16340.37540.9598 0.49680.36150.9944
1%
VaRst22240.873%0.952%0.51380.80710.5334 0.53340.65980.7559
VaRdt16220.635%0.873%0.04860.51130.6510 0.659
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50666
 Prefix

% corresponds to 5% and 1% risk level of ALD.LRuccolumns show pvalues of Kupiec’s (1995)
unconditional coverage test,LRindcolumns are pvalues of Christoffersen’s (1998) independent test andLRcc
columns are pvalues of Christoffersen’s (1998) conditional coverage test, * denotes significance.
leads to overestimating risk for both type of investors.
Stage 2: Efficiency Measures.
 Exact

Lopez (1998, 1999)
 Suffix

was the first to propose the
comparison between VaR models on the basis of their ability to minimise some specific loss
function which reflected a specific objective of the risk manager.
Adhering to the Basel Committee’s guidelines, supervisors are not only concerned with
the number of violations in a VaR model but also with the magnitude of those violations
(Basel Committee on Banking Superv
 (check this in PDF content)

 Start

51148
 Prefix

Adhering to the Basel Committee’s guidelines, supervisors are not only concerned with
the number of violations in a VaR model but also with the magnitude of those violations
(Basel Committee on Banking Supervision, 1996a, 1996b). In order to address this aspect,
following
 Exact

Sarma et al. (2003),
 Suffix

we compare the relevant models in terms of theRegulatory Loss
Function(RLF) which focuses on the magnitude of the failure and in terms of theFirm’s
Loss Function(FLF) which, while giving relevance to the magnitude of failures, imposes an
Table 25: RLF and FLF Loss function approach applied to the models surviving the VaR backtesting stage
Volatility models
and VaR methods
RLFFLF
5%1%5%1%
WTIBrent
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54183
 Prefix

Considering the RLF criterion, this test shows that
the competing models (leverage vs noleverage models) are not significantly different from
13This criterion penalizes large failures more than small failures (See
 Exact

Sarma et al., 2003).
 Suffix

14For the sign test see Lehmann (1974), Diebold and Mariano (1995), Hollander and Wolfe (1999) and
Sarma et al. (2003).
Table 26: RLF and FLF Loss function approach applied to the models surviving the CVaR backtesting
stage
Volatility models
and CVaR methods
RLFFLF
1.84%/1.96%0.37%/0.38%1.84%/1.96%0.37%/0.38%
WTIBrentWTIBrentWTIBrentWTIBrent
Panel A: Average loss values
SVN
Suppl
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54232
 Prefix

Considering the RLF criterion, this test shows that
the competing models (leverage vs noleverage models) are not significantly different from
13This criterion penalizes large failures more than small failures (See Sarma et al., 2003). 14For the sign test see
 Exact

Lehmann (1974), Diebold and Mariano (1995), Hollander and Wolfe (1999) and Sarma et al. (2003).
 Suffix

Table 26: RLF and FLF Loss function approach applied to the models surviving the CVaR backtesting
stage
Volatility models
and CVaR methods
RLFFLF
1.84%/1.96%0.37%/0.38%1.84%/1.96%0.37%/0.38%
WTIBrentWTIBrentWTIBrentWTIBrent
Panel A: Average loss values
SVN
Supply0.0001670.000201 0.0001430.000136 0.0008770.0007960.0002220.000204
Demand0.0002840.0001670.0012210.000203
SVNL
Supply0.00
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