Portfolio Value-At-Risk Estimation With a Time-Varying Copula Approach

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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

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    06

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    06

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    07

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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