portfolio value-at-risk estimation with a time-varying copula approach
TRANSCRIPT
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1. IntroductionValue-at-risk (VaR) has become one of the most popular tools of risk
measurement as the topic of risk management dramatically exposed to both academia
and practical investment. The potential estimation biases and model risk in the VaR
model have been generally discussed in previous studies (Jorion 1996; Kupiec, 1999;
Rich, 2003; Miller and Liu, 2006; Brooks and Persand, 2002). Firms may maintain
either insufficient risk capital reserves to absorb large financial shocks or excessive
risk capital to reduce capital management efficiency. Citi group, Merrill Lynch,
Lehman Brothers, and Morgan Stanley are good examples of major financial
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Jorion(1996) first indicated that VaR estimates are themselves affected by their
sampling variation or estimation risk. Brooks and Persand (BP) (2002) investigated
a number of statistical modeling issues in the context of determining market-based
capital risk requirements. They highlighted several potential pitfalls in commonly
applied methodologies and concluded that model risk can be serious in VaR
Calculation. They found that when the actual data is fat-tailed, using critical values
from a normal distribution in conjunction with a parametric approach can lead to a
substantially less accurate VaR estimate than using a nonparametric approach.
However, the above analysis considers univariate distribution only.
The second p rpose of this paper is to ree amine hether a more acc rate VaR
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magnitude of the estimation error (model risk) by constructing confidence intervals
around the point PVaR estimation1.
VaR applications are somewhat limited in hedged portfolios because the PVaR
model cannot be stated in a closed form and can only be approximated with complex
computational algorithms. Miller and Liu (2006) criticized current PVaR approaches.
First, the assumption of normal joint distribution is unrealistic, though convenient to
use. Second, though nonparametric approaches are not subject to the criticism of
distributional assumptions, tail probability estimations based on the empirical
distribution function may be poor because the observed outcomes in the tails of
l t k ti di t ib ti t i ll Thi d PV R ti ti b d l l
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univariate return series. Second and more importantly, as indicated by Smith (2000),
multivariate EVT models may not rich enough to encompass all forms of tail
behaviors.
Copulas enable the modeler to construct flexible multivariate distributions
exhibiting rich patterns of tail behavior, ranging from tail independence to tail
dependence, and different kinds of asymmetry. They are an alternative to correlation
in the modeling of financial risks (Embrechts et al. (1999)). This paper employs
single-parameter conditional copula to represent the dependence between index
futures and spot returns, conditional upon historical information provided by a
i i f i d f t d t t Th t f th diti l
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Recently, Miller and Liu (2006) used a copula-mixed distribution (CMX)
approach to estimate the joint distribution of log returns for the Taiwan Stock
Exchange (TSE) index and its corresponding futures contract on SGX and TAIFEX.
This approach converges in probability to the true marginal return distribution, but is
based on weaker assumptions. PVaR estimates for various hedged portfolios are
computed from the fitted CMX model, and backtesting diagnostics show that CMX
outperforms the alternative PVaR estimators. Unfortunately, Miller and Liu (2006)
employed a static Gaussian copula to estimate the PVaR of a hedge portfolio. Using
Gaussian copula alone may be subject to the potential estimation bias because the
j i t di t ib ti t b l d th t d d t t b t
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copula model can be applied to PVaR estimation. In effect, a copula-based measure can
specify both the structure and the degree of dependence, which takes the non-linear
property into account and allows a more comprehensive understanding as well.
The third contribution of this paper is related to Rosenberg and Schuermann
(2006). They claim that during a period of financial collapse, portfolios are subject to
at least two of three types of risk: market, credit, and operational risk. The
distributional shape of each risk type varies considerably. They use the copula mixture
method to construct a joint risk distribution and compare it with several conventional
approaches to computing risk. Their hybrid copula approach is surprisingly accurate.
W b th i id d t d it t PV R ti ti t t th t f
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conventional models. In other words, the more accurately the PVaR can be estimated
by choosing a smaller coverage rate. Thus, it implies that, for the conventional risk
management models, using a smaller nominal coverage probability (say, 95% instead
of 99%) helps to ensure that virtually all probability losses are covered. Choosing an
extremely large coverage probability, say 99%, deteriorates model risk caused by
models inability to take tail dependences and the nature of time variation into
account.
The paper is organized as follows. In section 2, the time-varying copula
methodology and the portfolio VaR model are presented. Section 3 describes data and
th i i l lt S ti 4 f th b kt t f i PV R i d
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2. MethodologyPVaR estimation methods such as variance-covariance method, Monte Carlo
simulation, historical simulation, RiskMetrics method, Jorions method, and extreme
value theory (EVT) have been well established. The basic properties and limitations
of these methods are discussed by the earlier literature. Recently, copula method has
been emphasized because of its capability in modeling the contemporaneous
dependencies between portfolio asset returns. This method is increasing in popularity
because it can analyze dependence structures in portfolio components beyond linear
correlations. It also provides a higher degree of flexibility in estimation by separating
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static copula model and may be subjected to the potential estimation bias because the
dependence structures between portfolio components are time-varying. Therefore, a
time-varying specification must be established to capture the dynamics of dependence
structures. In a time-varying copula setting, the dependence parameters in the copula
function are modeled as a dynamic process conditional on currently available
information. This allows a non-linear, time dependant relationship. The PVaR is
therefore estimated conditional on previously estimated time-varying dependencies.
2.1.The conditional copula modelWe assume that the marginal distribution for each portfolio asset return (index
d it di f t ) i h t i d b GJR GARCH(1 1) AR(1) t
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{,}with ,1 = 1 when ,1 is negative and otherwise ,1 = 0. is the degreeof freedom.
Assume that the conditional cumulative distribution functions of and are
,,1 and , ,1, respectively. The conditional copula function,denoted as ( , |1) , is defined by the two time-varying cumulativedistribution functions of random variables = ,,1 and = , ,1. Let be the bivariate conditional cumulative distributionfunctions of , and ,. Using the Sklar theorem, we have
, , ,1 = ( ,|1)(2)
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Gumbel and the Clayton copula for specification and calibration. The Gaussian copula
is generally viewed as a benchmark for comparison, while the Gumbel and the
Clayton copula are used to capture the upper and lower tail dependence, respectively.
The Clayton copula is especially prevalent because the evidence indicates that equity
returns exhibit more joint negative extremes than joint positive extremes, leading to
the observation that stocks tend to crash together but not boom together (Poon et al.,
2004; Longin and Solnik, 2001; Bae et al., 2003).
2.2.Bivariate copula densityThe conditional Gaussian copula function is the density of joint standard uniform
i bl ( ) b th d i bl bi i t l ith th
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(
)1 + (
)1
1
2
+ (
1)
(
)1 + (
)1
12
(5)
where [1,) measures the degree of dependence between and . = 1implies an independent relationship and represents perfect dependence. Thedensity of the time-varying Clayton copula is
(, |) = ( + 1) + 12+1 11 (6)
where [0,) measures the degree of dependence between and . = 0implies an independent relationship and represents perfect dependence.2.3.A Hybrid method: The mixture of copulas
To find the copula that best estimates the PVaR, we consider some possible
i t f diff t l A i di t d b R b d S h (2006)
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futures are subject to at least two types of risk: market, credit and operational risk6
Hu (2006) pointed out that empirical applications so far have been limited to
using individual copulas, however, there is no single copula that applies to all
situations. Thus, a mixture model is better able to generate dependence structures that
do not belong to one particular existing copula family. By carefully choosing the
t l i th i t d l th t i i l t fl ibl h t
.
The distributional shape of each risk type varies considerably. Market risk typically
generates portfolio value distributions that are nearly symmetric and often
approximated as normal. Credit (and especially operational) risk generate more
skewed distributions because of occasional extreme losses.
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conditional Gumbel copula, and a conditional Clayton copula to capture all possibility
of dependence structures. The mixture copula can be defined as
( , | , ,) =
(
,
|
) +
(
,
|
) + (1
)
(
,
|
) (7)
Where wtClay
is the time-varying weight of the conditional Clayton copula, and
wtGum is the time-varying weight of the conditional Gumbel copula. Let
wt
Clay, w
t
Gum
[0,1] and w
t
Clay+ w
t
Gum
1 . These weights, named as shape
parameters8, reflect the structures of dependence and capture changes in tail
dependence. For instance, after an increase in wtClay
, the copula assigns more
b bili h l f il C d h d l f H (2006) Li (2000) d
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2.4.Parameterizing time-series variation in the conditional copulaA portfolio with time-invariant dependences among its components seems
unreasonable in reality. Recently, a conditional copula with a time-varying
dependence parameter has become prevalent in the literature (Patton, 2006a, b;
Bartram et al., 2007; Jondeau and Rochinger, 2006; Rodriguez, 2007). Following the
studies of Patton (2006a) and Bartram et al.(2007), we assume that the dependence
parameter is determined by previous information such as its previous dependence and
the historical absolute difference between cumulative probabilities of portfolio asset
returns10
A conditional dependence parameter can be modeled as an AR(1)-like process
.
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)(1 +
) =
2
, which is the modified logistic transformation to keep
in (-1,1) at all times (Patton, 2006a). Time-varying dependence processes for the
Gumbel copula and the Clayton copula are described as Eq. (9) and (10), respectively.
=
1 +
+
|
1
1| (9)
= 1 + + |1 1| (10)where [1,) measures the degree of dependence in the Gumbel copula and hasa lower bound equal to 1, indicating an independent relationship, whereas [1,) measures the degree of dependence in the Clayton copula. Boundaries ofparameters are set up in the estimation software.
2.5.Data and PVaR estimation with the time-varying copula
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component returns by constructing empirical distributions for each sample day. We
use historical data from the previous 60 and 90 trading days and roll it over. Thus,
given the estimated conditional joint distribution of asset returns, replicated samples
can be drawn for the portfolio components.
To demonstrate the application of this time-varying copula in PVaR estimation,
we constructed a hedged portfolio comprised of the S&P 500 index and its index
futures. The sample period covers 1 January 2004 to 29 October 2007, including the
period of the outbreak of the U.S. subprime market collapse from August to October
2007. 998 daily observations for the index and index futures are obtained.
Th i i i h d i (MVHR) hi h i h i f f
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conditional variance-covariance matrix of residual series from (
, ,
,) in (1a), is
denoted by
, , ,1 = ,2 , , ,2 ,Assume that the optimal weight of this hedge portfolio is its conditional
minimum-variance hedge ratio to keep all variables under a time-varying version. The
optimal conditional minimum-variance hedge ratio, {|}, can then be defined as
= ,
,2 (11)= , ,
and
d d
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conditional on a time-varying dependence between portfolio components is estimated
for each sample day.
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3. Empirical results of PVaR estimation3.1.Estimation results of the time-varying copula models
Table 1 reports the summary statistics for the S&P 500 index returns and its
futures returns. Table 2 shows the estimated parameters of the marginal distributions
characterized by a GJR-GARCH(1,1)-AR(1)-t model given by Eq. (1). As Table 2
indicates, all parameters are at least significant at the 5 percent level. Furthermore,
their residual series pass the goodness-of-fit test14
.
[Insert Table 1 here]
[Insert Table 2 here]
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asymmetric dependences specified by the Gumbel and the Clayton copula,
respectively. Eq. (9) and (10) are their time-varying dependence processes. For each
sample day, there are three types of time-varying dependences specified by the
Gaussian, the Gumbel and the Clayton copula.
[Insert Table 3 here]
Table 4 reports the summary statistics of the weight estimates of conditional
mixture copulas. These weights are estimated by MLE according to Eq. (7). Panel A
reports the weight estimates across the entire sample period, while Panel B focuses on
the period of the U.S. subprime market crash from August to October 2007. The
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Implementing the Monte Carlo simulation generates replicated samples for the
index and futures returns given the estimated parameters of the time-varying
dependence models in each sample day. As the optimal weight of a hedge portfolio is
assumed to be its conditional minimum-variance hedge ratio in Eq. (11), conditional
distributions for portfolio returns are generated for each sample day. Furthermore, 1%
and 5 % quantiles of the conditional portfolio distributions are used to estimate the
conditional PVaR. Table 5 summarizes the statistics of the conditional PVaR across
the sample period. D60 and D90 are the rolling horizons, and indicate that the
empirical distributions are constructed using historical data from the previous 60 and
90 trading days, respectively16
. Accordingly, 998 empirical distributions across
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assets exhibit asymmetric dependence, especially for lower tail dependence, the
stricter PVaR estimate of the Clayton copula should be used. Table 5 shows that
conditional PVaR estimates from the Clayton copula exhibit a lower frequency of
violation and magnitude of exceedances than other copulas. In particular, the number
of violations in the mixture copula is lowest, and its statistics are more modest than
others.
[Insert Table 5 here]
For comparison, Figure 2 shows the time series plots of the conditional PVaR
estimates with different significance levels (99% and 95%) and different rolling
horizons (60 and 90 trading days). In general, the conditional PVaR estimates from
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4. Do Copulas Improve the Estimation of PVaR: The Backtests4.1.Backtests
This section proposes three backtest methods to examine the accuracy of the
PVaR estimates and their sensitivity to various alternative copula models:(1)
unconditional and conditional coverage tests, (2) dynamic quantile test (Engle and
Manganelli(2004)), and (3) distribution and tail forecasts test (Berkowitz(2001)).
Unconditional and conditional coverage tests are statistical backtests based on the
frequency of exceedances. Dynamic quantile test is a statistical backtests based on the
whole of the distribution of exceedances, while distribution and tail forecast test is
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alternative hypotheses (Kupiec (1995), Christoffersen (1998) and Berkowitz (2001)).
To backtest the VaR results, we use the powerful tests developed by Christoffersen
(1998), who enables us to test both coverage and independence hypotheses at the
same time. More precisely, this study uses the conditional coverage test to determine
if the PVaR violations are not only violated at percent of the time, but they shouldalso be independent and identically distributed (i.i.d.) over time.
Using the same notation as Christoffersen (1998), we define the indicator
sequence of VaR violations as {}, where is a dummy variable equal to 1 if a VaRviolation occurs at time tand 0 if there is no VaR violation at time t. If , is thetransition probability for two successive dummy variables, i.e.,
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2 = (1 )0,0+1,0 0,1+1,1 which is then estimated as
2 = 1 0,1 + 1,10,0 + 1,0 + 0,1 + 1,10,0+1,0 0,1 + 1,10,0 + 1,0 + 0,1 + 1,1
0,1+1,1
The LR test statistic for (first-order) independence in the VaR violations is:
= 2 (21) ~2(1)If we condition on the first observant in the test for unconditional coverage, the
LR statistic for conditional coverage (i.e., the joint hypothesis of unconditional
coverage and independence) is:
= + ~2(2)where is the statistic for unconditional coverage (computed above for the
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where, under the null hypothesis,
0 =
, the target violation rate, and
= 0 ,
= 1, , + 1. In vector notation, we have = +
where 1 is a vector of ones. A good model should produce a sequence of unbiased and
uncorrelated variables, and the regressors should have no explanatory power, thatis H0: = . This regression model is estimated by ordinary least squares (OLS) andyields the test statistic
DQ = (1 ) +22 where is the OLS estimates of . The DQ test statistic has an asymptoticChi-square distribution with + 2 degrees of freedom. For the purpose of empirical
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of independent and identically distributed random variables. This idea is implemented
by applying probability integral transformation,
= () = ()where
is the ex post portfolio return realization and
(
) is the ex ante forecasted
portfolio density. If the underlying PVaR model is valid, then its {} series shouldbe iid and distributed uniformly on (0,1).
Berkowitz (2001) proposed the likelihood ratio test to evaluate density
forecast18
. For this test, the {} series is transformed using the inverse of thestandard normal cumulative distribution function, that is, = 1(). If the PVaRmodel is correctly specified, the { } series should be an iid N(0,1). This hypothesis
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Berkowitz (2001) specially allowed the user to evaluate tail forecast due to the
fact that risk managers are exclusively interested in an accurate description of large
losses or tail behavior. By considering a tail that is defined by the user, any
observations that do not fall in the tail will be intentionally truncated. Letting the
desired cutoff point PVaR = 1() , we choose PVaR = 1.64 and PVaR =2.33 to focus on the 5% and 1% lower tail, respectively. Then the log-likelihoodfunction for joint estimation is
L(,|) = ln 1
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we choose an extreme violation rate. Generally speaking, the copula-based PVaR
model, compared with the conventional GARCH model, exhibits superior
performance under extreme significance levels (say, 99%).
[Insert Table 6 here]
Figure 3 displays the time series of actual portfolio returns and corresponding
conditional PVaR estimates. The conditional PVaR estimate appears to track actual
portfolio return well. In addition, violations rarely occur during the U.S. subprime
market crash from August to October 2007. In Figure 4, we demonstrate the degree of
underestimation or overestimation from the conventional GARCH model by
comparing its estimates with those from the various conditional copula models.
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conditional coverage test. Under the 5% quantile level, however, the time-varying
copula models exhibit the superior performance relative to the benchmark model,
except for the Gaussian copula. Most test statistics generated from the time-varying
copula models do not reject null hypothesis. In particular, the time-varying Clayton
copula presents the best performance at 1% quantile due to its smallest test statistics.
[Insert Table 7 here]
4.2.3 Distribution and tail forecast test resultsTable 8 reports the results of distribution and tail forecast test. If we focus on the
interior of the distribution, most models predict the portfolio distribution well, except
for the Clayton copula model. If we turn to the 5% and 1% lower tail, all copula
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A popular issue in risk management practice concerns the coverage rate that
should be required by the minimum capital risk requirements (MCRRs). To ensure
adequate coverage, the Basle Committee chose to focus on the first percentile (1%) of
return distributions. In other words, risk managers are required to hold sufficient
capital to absorb all but 1 percent of expected losses, rather than the 5 percent level
used by Risk Metrics in the J.P. Morgan approach (1996)20
.
Brooks and Persand (BP) (2002) sought to determine whether a more accurate
VaR estimate could be derived by using the fifth percentile instead of the first
percentile of the return distribution (together with an appropriately enlarged scaling
factor)21
. They found that when the actual data is fat tailed, using critical values from
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accurately they can be estimated. Therefore, if a regulator has the desirable objective
of ensuring that virtually all probable losses can be covered, the use of a smaller
nominal coverage probability (say, 95% instead of 99%) combined with a larger
multiplier is preferable.
According to the conditional coverage test results presented in Table 6, although
the copula-based PVaR model performs better than the traditional model only under
backtesting criteria at the 1% (not for 5%) desired violation rate, from the results of
dynamic quantile test (Table 7) and tail forecast test (Table 8), our proposed copula
models outperforms the benchmark model at all coverage rates. In addition, both
dynamic quantile test and distribution forecast test are more powerful and robust than
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suggested for the conventional risk management models to avoid the model risk
arising from an inappropriate use of correlation measures.
4.4.Bootstrap:PVaR interval estimationAn interval estimate is often more useful and informative than just a point
estimate (Efron and Tibshirani, 1993). Jorion (1996) suggested that VaR should be
reported with confidence intervals. Christoffersen and Goncalves (2005) claim that
accurate confidence intervals reported along with the VaR point estimate will facilitate
the use of VaR in active portfolio management. Most importantly, the intervals widen
substantially as one moves to more extreme quantiles because fewer observations are
involved. Christoffersen and Goncalves (2005) illustrate that VAR confidence
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Table 9 shows the results of bootstrap. Lower limit means that the average of
lower bound of interval across sample day, while upper limit indicates that the
average of upper bound of interval across sample day. The upper limits of PVaR
estimates from the GARCH model are generally higher than those from time-varying
copula models, implying that PVaR estimates from the GARCH model tend to be
underestimated even considering an estimation error problem.
The uncertainty of the PVaR point estimate arising from estimation risk needs to
be identified. As Christoffersen and Goncalves (2005), we apply a bootstrap method
to calculate confidence intervals around the PVaR point estimate. Besides
demonstrating estimation error from the GARCH model, we also quantify the
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We also attempt to highlight that the upper limit of PVaR interval estimates may
fail in the backtest, even its point estimate passes the backtest. By taking estimation
error into account, we conduct backtest again for upper limit of PVaR interval
estimates and report results in Table 10. Apparently, the conventional GARCH has the
greatest violation rate. Under backtesting criteria at the 5% desired violation rate, the
conventional GARCH model is absolutely invalid based on unconditional,
independent and conditional coverage tests. Our proposed copula models, however,
are acceptable even with the estimation error problem. If we turn to backtesting
criteria at the 1% desired violation rate, the Gaussian, the Gumbel copula and the
GARCH models are failed to backtest, whereas the Clayton and the mixture copula
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This study uses PVaR estimation to illustrate that the model risk is attributable to the
inappropriate use of correlation coefficient and normal joint distribution. We also
attempt to improve the PVaR estimation and thus reduce model risk by relaxing the
conventional assumption of normal joint distribution and developing an empirical
model of time-varying PVaR conditional on time-varying dependencies between
portfolio components. To demonstrate the dynamic hedging version of the
time-varying PVaR comparisons, both single-parameter conditional copulas and
copula mixture models are applied to form a flexible joint distribution.
PVaR estimates for the optimal hedged portfolios are computed from various
copula models, and backtesting diagnostics indicate that the copula-based PVaR
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considering model risk. Second, to reduce the model risk, PVaR estimation using a
smaller nominal coverage rate (say, 95% instead of 99%) is preferable.
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Appendix A: Parameter Estimation of Conditional Copula
At time t, the log-likelihood function can be derived by taking the logarithm of
(3):
= + , + , (A1)Let the parameters in, and , be denoted as and , respectively, and
the other parameters in be denoted as . These parameters can be estimated bymaximizing the following log-likelihood function:
,() = () + () + () (A2)where = ( , ,) and represent the sum of the log-likelihood functionvalues across observations of the variable k.
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Patton (2006a) shows that the two-stage ML estimates = ; ; areasymptotically as efficient as one-stage ML estimates. The variance-covariance matrix
of must be obtained from numerical derivatives. We have only been able to obtainsatisfactory first derivatives, from which the fully efficient two-stage estimator
()1 of the variance-covariance matrix is given by = 1 =1 ,
where the score vector = log / is evaluated at = .
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Appendix B: Bootstrap Algorithm for Copula-based Risk Measures
Step 1 Generate a sample of T 200 bootstrapped portfolio returns {: t = 1, , T}by resampling with replacement from the conditional portfolio returns generated from
time-varying copula models.
Step 2 Compute the estimates of PVaR on the bootstrap sample
t = Q({}t200+1t200+200) t = 0, , (T 1)Step 3 Repeat Step 1 and 2 a large number of times, say 1000 times, and obtain a
sequence of bootstrapped copula-based risk measures, t(i): i = 1, , 1000Step 4 The 100(1 ) bootstrap prediction interval for PVaRt is given by
Q/2
(i) 1000, Q1 /2
(i) 1000
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Table 1 Summary statistics
This table shows summary statistics for the percentage log returns of the S&P 500 index and S&P 500
index futures. The sample period covers 1 January 2004 to 29 October 2007. 998 daily observations for
the index and index futures are collected.
Mean Standard
Deviation
Skewness Kurtosis
Index 0.03317 0.70894 -0.29684 1.78907
Index futures 0.03365 0.71083 -0.44807 2.07703
Table 2 Estimated parameters for GJR-GARCH(1,1)-AR-t marginal distributions
This table reports the estimated parameters of the marginal distributions for index and futures returns.
They are assumed to be characterized by an GJR-GARCH(1,1)-AR-t model given by Eq. (1). The
symbol * denotes significance at the 5% levels.
AR(1) GARCH
constant
Lagged
variance
Lagged
residual
Asymmetric
residual
Degree of
freedoma
Index -0.0664* 0.0090* 0.9535* -0.0397 0.1054* 7.5638
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Table 3 Estimated parameters of time-varying copula functions
This table shows the estimated parameters of time-varying dependencies in the chosen copulas. The
time-varying dependence models in Eq. (8), (9), (10) are estimated and calibrated. The parameter captures the degree of persistence in the dependence and captures the adjustment in the dependenceprocess. LLF(c) is the maximum copula component of the log-likelihood function. The symbol *
denotes significance at the 5% levels.
LLF(c)Panel A: Gaussian copula
0.94990* 0.71938* -0.96763* 907.88
Panel B: Gumbel copula
0.94743* 0.25587 -0.95841* 826.09
Panel C: Clayton copula
0.93638* 0.30843* -0.94724* 729.68
Table 4 Summary statistics of weight estimates of conditional mixture copulas
This table summarizes minimum, maximum, mean, 25% quantile, median, 75% quantile, and standard
deviation of weight estimates for conditional mixtures copulas. These weights are estimated by MLE
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Table 6 Unconditional and conditional coverage tests
Our backtest comprises unconditional coverage, independence and conditional coverage test, which
takes into account not only if violations occur, but they should also be independent and identically
distributed over time. Likelihood ratio statistics are reported in each test, and thep-values are displayed
in parentheses. The symbol * denotes significance at the 5% levels.
ViolationRate
UnconditionalCoverage (LRuc)
Independence(LRind)
ConditionalCoverage (LRcc)
Panel A: D90_SL0.05
Gaussian 0.02405 7.54228*(0.00603) 0.11650(0.73286) 7.67994*(0.02149)Gumbel 0.02104 9.69396*
(0.00185)0.39248
(0.53100)10.10492*(0.00639)
Clayton 0.01904 11.33691*(0.00076)
0.34766(0.55544)
11.70127*(0.00288)
Mixture 0.01804 12.22717*(0.00047)
0.22654(0.63410)
9.12764*(0.01042)
ConventionalGARCH
0.05711 0.57255(0.47924)
0.02101(0.88475)
0.64560(0.72411)
Panel B: D90_SL0.01Gaussian 0.00802 0.18488
(0.66721)0.05620
(0.81260)0.24808
(0.88335)Gumbel 0.00401 2.03328
(0.15389)0.01399
(0.90584)2.05077
(0.35866)Clayton 0.00301 2.95206
(0.08588)0.00786
(0.92935)2.96254
(0.22735)Mixture 0.00301 2.95206
(0.08588)0.00786
(0.92935)2.96254
(0.22735)
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Table 7 Dynamic quantile test
This table presents the results of the PVaR estimates evaluation using the dynamic quantile (DQ) test
proposed by Engle and Maganelli (2004). The DQ statistics and its asymptotic p-values for the
alternative PVaR models under 1%, 5% significance level are reported. The DQ statistics are
asymptotically distributed2(6). The cells in bold font indicate rejection of null hypothesis of correctPVaR estimates at the 5% significance level.
DQstatistics p-value
Panel A: D90_SL0.05Gaussian 13.22377 0.03961Gumbel 10.34363 0.11104Clayton 11.57783 0.07207Mixture 11.26154 0.08062Conventional GARCH 77.50367 0.00000
Panel B: D90_SL0.01Gaussian 7.02197 0.31881Gumbel 1.65849 0.94828Clayton 1.49195 0.96002Mixture 1.70745 0.94454Conventional GARCH 109.1632 0.00000
Panel C: D60_SL0.05Gaussian 16.51582 0.01123Gumbel 11.84296 0.06556Clayton 5.55835 0.47443Mixture 11.71187 0.06871
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Table 8 Distribution and tail forecast test
This table reports the results of distribution and tail forecast tests developed by Berkowitz (2001). The
LRdist test statistics evaluate the interior of the distribution, while LR5%tailand LR1%tailevaluate 5% and
1% lower tail of distribution, respectively. Their asymptotic p-values are shown in parentheses. The
cells in bold font indicate rejection of null hypothesis of correct PVaR estimates at the 5% significance
level.
LRdist LR5%tail LR1%tail
Gaussian 3.09180
(0.37768)
5.81881
(0.0545)
1.70678
(0.42596)
Gumbel 2.81602
(0.42086)
3.85452
(0.14554)
0.56714
(0.75308)
Clayton 13.34834
(0.00394)
4.16241
(0.12477)
3.62677
(0.16310)
Mixture 4.66271
(0.19822)
3.97142
(0.13728)
2.41872
(0.29838)
Convential GARCH 1.11626
(0.77314)
32.46121
(0.00000)
26.82990
(0.00000)
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Table 9 Bootstrapped interval properties for the conditional PVaR models
This table shows the results of bootstrap. Lower limit means that the average of lower bound of intervalacross sample day, while upper limit indicates that the average of upper bound of interval across
sample day. Length is defined as the difference between upper limit and lower limit.
Lower
Limit
Upper Limit Length
Panel A: D90_SL0.05
Gaussian -0.00404 -0.00264 0.00140
Gumbel -0.00414 -0.00245 0.00169
Clayton -0.00559 -0.00290 0.00269
Mixture -0.00478 -0.00267 0.00211
Conventional GARCH -0.00316 -0.00189 0.00127
Panel B: D90_SL0.01
Gaussian -0.00668 -0.00379 0.00289
Gumbel -0.00798 -0.00369 0.00429
Clayton -0.01057 -0.00548 0.00509
Mixture -0.00901 -0.00450 0.00451
Conventional GARCH -0.00452 -0.00257 0.00195
Panel C: D60_SL0.05
Gaussian -0.00413 -0.00270 0.00143
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Table 10 Backtests of the upper limit of conditional PVaR model
The backtest which comprises unconditional coverage, independence and conditional coverage test isconducted for upper limit of PVaR interval estimates. Likelihood ratio statistics are reported in each
test, and thep-values are displayed in parentheses. The symbol * denotes significance at the 5% levels.
ViolationRate
UnconditionalCoverage (LRuc)
Independence(LRind)
ConditionalCoverage (LRcc)
Panel A: D90_SL0.05Gaussian 0.06012 0.88023
(0.34814)0.46737
(0.49420)0.97803
(0.61323)
Gumbel 0.06513 1.91498(0.16642)
1.71047(0.19093)
3.68397(0.15850)
Clayton 0.05311 0.08637(0.76893)
0.73701(0.39062)
0.87079(0.64701)
Mixture 0.05311 0.08637(0.76893)
0.73701(0.39062)
0.87079(0.64701)
ConventionalGARCH
0.10621 22.11063*(0.00001)
4.71424*(0.02991)
23.24240*(0.00001)
Panel B: D90_SL0.01Gaussian 0.03607 17.81594*
(0.00001)0.11861
(0.73056)15.12604*(0.00051)
Gumbel 0.04409 27.66562*(0.00001)
0.64352(0.42245)
23.04649*(0.00001)
Clayton 0.01303 0.36601(0.54519)
0.86830(0.35143)
1.24570(0.53641)
Mixture 0.02004 3.41700(0.06453)
0.28938(0.59062)
3.72397(0.15537)
ConventionalGARCH
0.05511 43.33641*(0.00001)
0.01028(0.91924)
38.49426*(0.00001)
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Figure 1 Time-varying weight estimates of conditional mixture copulas
Figure 1 depicts the time series of the weight estimates of conditional mixture copulas across the
sample period. The Gaussian weight estimates are displayed as a red line. The weight estimates of the
Gumbel and Clayton copulas are represented by green and blue lines, respectively.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
4/01/01
4/04/01
4/07/01
4/10/01
5/01/01
5/04/01
5/07/01
5/10/01
6/01/01
6/04/01
6/07/01
6/10/01
7/01/01
7/04/01
7/07/01
7/10/01
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57
Figure 2 Comparisons between conditional Gaussian, Gumbel, Clayton and Mixture PVaR estimates
Conditional Gaussian (red line), Gumbel (green line), Clayton (blue line) and Mixture (black line) PVaR estimates are compared under a 90 or 60 rolling horizon and 95% or 99%
significance level.
-0.014
-0.012
-0.010
-0.008
-0.006
-0.004
-0.002
0.000
04/1
04/5
04/9
05/1
05/5
05/9
06/1
06/5
06/9
07/1
07/5
07/9
D90_SL0.05
-0.030
-0.025
-0.020
-0.015
-0.010
-0.005
0.000
04/1
04/5
04/9
05/1
05/5
05/9
06/1
06/5
06/9
07/1
07/5
07/9
D90_SL0.01
-0.016
-0.014
-0.012
-0.010
-0.008
-0.006
-0.004
-0.002
0.000
04/1
04/5
04/9
05/1
05/5
05/9
06/1
06/5
06/9
07/1
07/5
07/9
D60_SL0.05
-0.030
-0.025
-0.020
-0.015
-0.010
-0.005
0.000
04/1
04/5
04/9
05/1
05/5
05/9
06/1
06/5
06/9
07/1
07/5
07/9
D60_SL0.01
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58
Figure 3 Comparisons between realized portfolio return and conditional PVaR estimates
Time series of daily portfolio return from January 2004 to October 2007(solid lines) are plotted with conditional Gaussian, Gumbel, Clayton and Mixture PVAR estimates (dotted
lines).
Panel A: D90_SL0.05
Panel B: D90_SL0.01
-0.010
-0.008
-0.006
-0.004
-0.002
0.000
0.002
0.004
0.006
0.008
04/1
04/5
04/9
05/1
05/5
05/9
06/1
06/5
06/9
07/1
07/5
07/9
Conditional Gaussian PVaR
-0.010
-0.008
-0.006
-0.004
-0.002
0.000
0.002
0.004
0.006
0.008
04/1
04/5
04/9
05/1
05/5
05/9
06/1
06/5
06/9
07/1
07/5
07/9
Conditional Gumbel PVaR
-0.015
-0.010
-0.005
0.000
0.005
0.010
04/1
04/5
04/9
05/1
05/5
05/9
06/1
06/5
06/9
07/1
07/5
07/9
Conditional ClaytonPVaR
-0.010
-0.008
-0.006
-0.004
-0.002
0.000
0.002
0.004
0.006
0.008
04/1
04/5
04/9
05/1
05/5
05/9
06/1
06/5
06/9
07/1
07/5
07/9
Conditional Mixture PVaR
-0.020
-0.015
-0.010
-0.005
0.000
0.005
0.010
04/1
04/5
04/9
05/1
05/5
05/9
06/1
06/5
06/9
07/1
07/5
07/9
Conditional Gaussian PVaR
-0.030
-0.025
-0.020
-0.015
-0.010
-0.005
0.000
0.005
0.010
04/1
04/5
04/9
05/1
05/5
05/9
06/1
06/5
06/9
07/1
07/5
07/9
Conditional Gumbel PVaR
-0.025
-0.020
-0.015
-0.010
-0.005
0.000
0.005
0.010
04/1
04/5
04/9
05/1
05/5
05/9
06/1
06/5
06/9
07/1
07/5
07/9
Conditional Clayton PVaR
-0.020
-0.015
-0.010
-0.005
0.000
0.005
0.010
04/1
04/5
04/9
05/1
05/5
05/9
06/1
06/5
06/9
07/1
07/5
07/9
Conditional Mixture PVaR
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59
Panel C: D60_SL0.05
Panel D: D60_SL0.01
-0.010
-0.008
-0.006
-0.004
-0.002
0.000
0.002
0.004
0.006
0.008
04/1
04/5
04/9
05/1
05/5
05/9
06/1
06/5
06/9
07/1
07/5
07/9
Conditional Gaussian PVaR
-0.012
-0.010
-0.008
-0.006-0.004
-0.002
0.000
0.002
0.004
0.006
0.008
04/1
04/5
04/9
05/1
05/5
05/9
06/1
06/5
06/9
07/1
07/5
07/9
Conditional Gumbel PVaR
-0.015
-0.010
-0.005
0.000
0.005
0.010
04/1
04/5
04/9
05/1
05/5
05/9
06/1
06/5
06/9
07/1
07/5
07/9
Conditional Clayton PVaR
-0.010
-0.008
-0.006
-0.004
-0.002
0.000
0.002
0.004
0.006
0.008
04/1
04/5
04/9
05/1
05/5
05/9
06/1
06/5
06/9
07/1
07/5
07/9
Conditional Mixture PVaR
-0.025
-0.020
-0.015
-0.010
-0.005
0.000
0.005
0.010
04/1
04/5
04/9
05/1
05/5
05/9
06/1
06/5
06/9
07/1
07/5
07/9
Conditional Gaussian PVaR
-0.030
-0.025
-0.020
-0.015
-0.010
-0.005
0.000
0.005
0.010
04/1
04/5
04/9
05/1
05/5
05/9
06/1
06/5
06/9
07/1
07/5
07/9
Conditional Gumbel PVaR
-0.025
-0.020
-0.015
-0.010
-0.005
0.000
0.005
0.010
04/1
04/5
04/9
05/1
05/5
05/9
06/1
06/5
06/9
07/1
07/5
07/9
Conditional Clayton PVaR
-0.025
-0.020
-0.015
-0.010
-0.005
0.000
0.005
0.010
04/1
04/5
04/9
05/1
05/5
05/9
06/1
06/5
06/9
07/1
07/5
07/9
Conditional Mixture PVaR
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60
Figure 4 Differences between conditional PVaR estimates and the conventional GARCH estimates.
Conditional Gaussian (red line), Gumbel (green line), Clayton (blue line) and Mixture (black line) PVaR estimates are compared with that of the conventional GARCH estimates
under a 90 or 60 rolling horizon and 95% or 99% significance level. Positive difference means that the conventional GARCH estimates are underestimated while negative difference
indicates that they are overestimated.
-0.005
0.000
0.005
0.010
0.015
0.020
0.025
04
/1
04
/5
04
/9
05
/1
05
/5
05
/9
06
/1
06
/5
06
/9
07
/1
07
/5
07
/9
D90_SL0.05
-0.005
0.000
0.005
0.010
0.015
0.020
0.025
04
/1
04
/5
04
/9
05
/1
05
/5
05
/9
06
/1
06
/5
06
/9
07
/1
07
/5
07
/9
D90_SL0.01
-0.005
0.000
0.005
0.010
0.015
0.020
0.025
04/1
04/5
04/9
05/1
05/5
05/9
06/1
06/5
06/9
07/1
07/5
07/9
D60_SL0.05
-0.005
0.000
0.005
0.010
0.015
0.020
0.025
04/1
04/5
04/9
05/1
05/5
05/9
06/1
06/5
06/9
07/1
07/5
07/9
D60_SL0.01
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Figure 5 Bootstrapped confidence intervals for various PVaR models.
The bootstrapped 90% confidence intervals around the point estimates of time-varying PVaR are depicted in this figure. The estimation risk, measured by the length of confidence
interval, of first-percentile coverage rate is higher (around double) than that of fifth-percentile coverage ratio under various copula-based models and the conventional GARCH
model.
-0.012
-0.010
-0.008
-0.006
-0.004
-0.002
0.000
Gaussian Gumbel Clayton Mixture GARCH
D90_SL0.05
-0.012
-0.010
-0.008
-0.006
-0.004
-0.002
0.000
Gaussian Gumbel Clayton Mixture GARCH
D90_SL0.01
-0.012
-0.010
-0.008
-0.006
-0.004
-0.002
0.000
Gaussian Gumbel Clayton Mixture GARCH
D60_SL0.05
-0.012
-0.010
-0.008
-0.006
-0.004
-0.002
0.000
Gaussian Gumbel Clayton Mixture GARCH
D60_SL0.01