Forecasting value-at-risk using time varying copulas and EVT return distributions

International Economics 133 (2013) 93–106

Contents lists available at SciVerse ScienceDirect

International Economics journal homepage: www.elsevier.com/locate/inteco

Forecasting value-at-risk using time varying copulas and EVT return distributions Theo Berger University of Bremen, Empirical Economics and Applied Statistics, Germany

a r t i c l e in f o

a b s t r a c t

Available online 23 April 2013

Forecasting portfolio risk requires both, estimation of marginal return distributions for individual assets and the dependence structure of returns as well. In this paper, we concentrate on Value at Risk as a popular risk measure and combine elliptical copulas with time varying Dynamic Conditional Correlation (DCC) matrices and Extreme Value Theory (EVT) based models for the marginal return distributions. The approach leads to reliable Value-at-Risk figures with respect to several backtesting criteria. Feasibility and accuracy of the approach is corroborated by an extensive empirical application to different financial portfolios consisting of stocks, market indices and FX-rates. & 2013 CEPII (Centre d’Etudes Prospectives et d’Informations Internationales), a center for research and expertise on the world economy. Published by Elsevier Ltd. All rights reserved.

JEL classifications: C58 G01 G11 Keywords: Portfolio value-at-risk Elliptical copulas Dynamic conditional correlations Extreme value theory

1. Introduction The recent financial crisis dramatically emphasized the importance of adequate risk measurement for financial as well as non-financial institutions. Concerning risk measurement, Value-at-Risk (VaR) as a quantile of the return distribution became a widely accepted risk figure for managing and communicating risk, not least due to its conceptual simplicity. VaR is used as a key measure by both financial regulators and financial companies as well: For instance, the Basel Committee on Banking Supervision (1996) bases the standards on capital required to meet market risk on VaR, and many of financial institutions and listed companies publish VaR figures as an additional information to their shareholders. Consequently, the appropriate modeling of VaR, especially in turbulent market times,

E-mail address: [email protected] 2110-7017/$ - see front matter & 2013 CEPII (Centre d’Etudes Prospectives et d’Informations Internationales), a center for research and expertise on the world economy. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.inteco.2013.04.002

94

T. Berger / International Economics 133 (2013) 93–106

represents a staggering task for financial risk measurement and management (see e.g. Jorion, 2006; Das, 2007). However, current research regarding VaR predictors is twofold, on the one hand finding the optimal VaR model is addressed (see Roy, 2011; Aloui et al., 2011; Lee and Lin, 2011) and on the other hand appropriate quality criteria need to be considered (see Hurlin and Tokpavi, 2007; Virdi, 2011). In this paper, we investigate a dynamic model for forecasting daily VaR. It combines two model classes extensively used to analyze the distribution of portfolio returns and portfolio risk measures in the recent past: Dynamic conditional correlations models (DCC, Engle, 2002, 2009) and copulas (Sklar, 1959). Both model classes allow for a two stage modeling of portfolio returns: In the first step marginal return distributions are specified, and in the second step the marginals are linked to a joint distribution either via time variant correlations (DCC) or a time invariant link function (copula). In case of elliptical copulas, the two stage decomposition offers a feasible way to combine both models, leading to a “time varying” DCC-copula. Even though time varying copulas have been suggested earlier (e.g. Patton (2004, 2006) for an autoregressive and Jondeau and Rockinger (2006) for a GARCHtype specification), applications to VaR have not been addressed until recently (Manner and Reznikova, 2012). We use the hybrid DCC-Copula approach to empirically analyze different portfolios consisting of stocks, indexes and foreign exchange rates in a period covering the outbreak of the actual financial crisis. Moreover, we use Extreme Value Theory (EVT) to model marginal distributions of assets. The resulting model leads to reliable Value-at-Risk forecasts with respect to several backtesting criteria. Hence, the contribution of this paper is twofold: on the one hand it presents an evaluation of several dependency models drawing on a rather comprehensive data set. On the other hand, it proposes the hybrid DCC-Copula approach with EVT margins as an highly efficient and flexible tool for VaR calculations even for higher order portfolios. The paper is organized as follows: Section 2 reviews the literature. Section 3 describes the methodology. The empirical application is presented in Sections 4 and 5 concludes.

2. Literature review In our analysis, we use different (marginal) distributions for individual asset returns frequently studied in risk analyses: normally and t-distributed returns (see Angelidis et al., 2004) and Generalized Pareto (GPD) distributed returns (see Longin and Solnik, 2001; Aloui et al., 2011) in the EVT framework. For both of the dependency approaches, various specifications have been proposed and applied empirically. For a survey on DCC-Models see Bauer (2011) and on copulas Embrechts et al. (2002), Yener (2011). Time varying copulas have been addressed by Patton (2004, 2006) using an ARMA specification for dynamic correlations and a particular transformation of variables to insure that conditional correlations fall in the interval ½−1; 1. Jondeau and Rockinger (2006) integrate the time varying correlations approach of Tse and Tsui (2002), which is similar to the DCC approach of Engle (2002), in their copula model. In a hierarchical copula model representing a cascade of bivariate conditional copulas, Heinen and Valdesogo Robles (2009) make use of dynamic correlations according to the DCC. For a recent survey on time varying copulas see Manner and Reznikova (2012). Moreover, due to different asset classes and different sample periods, empirical results are hardly comparable and it still is a moot point which model to choose for addressing risk related problems. This problem carries over to the adequate choice of backtesting criteria: as typically every criterion emphasizes one specific feature of VaR forecasts, usually several criteria are applied in empirical studies. However, finding criteria considering multiple aspects of VaR forecast adequacy is currently a busy area of research (Candelon et al., 2012; Sener et al., 2012) However, the adequate choice of backtesting methods is not that clear so far. For a broader discussion of the shortcomings of the applied backtesting criteria we refer to Campbell (2006) and Hurlin and Tokpavi (2007). In this paper, we restrict our attention to elliptical copulas, as only the Gauss and the t-copula turn out to be tractable copula models for multidimensional portfolios consisting of more than three assets (see also Nelsen, 2006; Patton, 2009). Even for this subclass of copula models, mixed empirical

T. Berger / International Economics 133 (2013) 93–106

95

evidence is reported: Malevergne and Sornette (2003) conclude for bivariate portfolios that Gaussian copulas cannot be rejected against t copulas for currencies and stocks. In contradiction, Kole et al. (2007) underline the quality of t copula performance compared to Gaussian copula for stocks and bonds and Junker and May (2005) show that the classical gauss copula in combination with time varying margin models portfolio risk appropriately. Up to now only few applications, addressing both, DCC and copulas in the context of VaR estimation and VaR prediction are reported. In their survey on time-varying copulas, Manner and Reznikova (2012) illustrate application of the copula models presented on VaR forecasts in the bivariate case. Hakim et al. (2007) and Hakim and McAleer (2009) address parametric VaR forecasts for stocks, bonds and FX-rates. Palaro and Hotta (2006) use a copula approach to model VaR for a portfolio consisting of Nasdaq and S&P 500 indices. Ozun and Cifter (2007) investigate Latin American markets, Hsu et al. (2011) investigate the application of EVT for bivariate portfolios covering Asian emerging markets as well as Aloui et al. (2011), who address several bivariate portfolios consisting of emerging and the US markets. An interesting application of EVT and elliptical copulas in the context of hedge fund indexes is reported in Viebig and Poddig (2010).

3. Methodology 3.1. Dependency modeling 3.1.1. Dynamic conditional correlations The DCC model of Engle (2002) belongs to the class of multivariate GARCH models. The approach separates variance modeling from correlation modeling. Let the Nx1 vector rt be a set of N asset log returns at time t. Volatilities are calculated in order to construct volatility adjusted residuals ϵt . For our research, we assume that each return follows a univariate GARCH(1,1) process. The correlations are estimated based on the standardized residuals. Let Rt denotes the correlation matrix and Dt the diagonal matrix with conditional standard deviations at time t. The full DCC setup is given by r t jIt−1 ∼Nð0; Dt Rt Dt Þ;

ð1Þ

D2t ¼ diagfHt g;

ð2Þ

H i;i;t ¼ wi þ αi r 2i;t−1 þ βi H i;i;t−1 ;

ð3Þ

ϵt ¼ D−1 t rt ;

ð4Þ 1=2

Rt ¼ diagfQ t

1=2

gQ t diagfQ t

g;

Q t ¼ Ω þ αϵt−1 ϵ′t−1 þ βQ t−1 ;

ð5Þ ð6Þ

whereas Ω ¼ ð1−α−βÞR and α and β are always positive and their sum is less than one. The log-likelihood is given by lt ¼ −

1 T ∑ ðn logð2πÞ þ 2 logjDt j þ r′t D2t r t −ϵ′t ϵt þ jRt j þ ϵ′t R−1 t ϵt Þ: 2t ¼1

ð7Þ

According to Engle (2002), the full ML-estimates may be replaced by a consistent two step estimator: In the first step the variance parameters are estimated and in the second step the correlation parameters based on volatility adjusted data. So that the second step log likelihood is given by l2;t ¼ −



1 T ∑ ½log
Rt
þϵ′t Rt ϵt : 2t ¼1

ð8Þ

96

T. Berger / International Economics 133 (2013) 93–106

3.1.2. Copulas The copula approach is based on Sklar's Theorem (1959): Let X 1 ; …:X n be random variables, F 1 ; …; F n the corresponding marginal distributions and H the joint distribution, then there exists a copula C: ½0; 1n -½0; 1 such that Hðr 1 ; …; r n Þ ¼ CðF 1 ðr 1 Þ; …F n ðr n ÞÞ: Conversely if C is a copula and F 1 ; …; F n are distribution functions, then H (as defined above) is a joint distribution with margins F 1 ; …F n . The Gaussian and t copula belong to the family of elliptical copulas and are derived from the multivariate normal and t distribution respectively. The setup of the Gaussian copula is given by: C Ga ðr 1 ; …; r n Þ ¼ Φρ ðΦ−1 ðr 1 Þ; …; Φ−1 ðr n ÞÞ   Z Φ−1 ðr1 Þ Z Φ−1 ðrn Þ 1 1 T −1 z ¼ … exp − ρ z dz1 …dzn 2 2ðπÞn=2 jρj1=2 −∞ −∞

ð9Þ

whereas Φρ stands for the multivariate normal distribution with correlation matrix ρ and Φ−1 symbolizes the inverse of univariate normal distribution. The t copula is given by −1 C t ðr 1 ; …; r n Þ ¼ t ρ;v ðt −1 v ðr 1 Þ; …; ; t v ðr n ÞÞ     Z t −1 ðr1 Þ Z t−1 ðrn Þ Γ vþn 1 T −1 −ðvþnÞ=2 2 z … 1 þ ρ z dz1 …dzn ; ¼  v v Γ 2 ðvπÞn=2 jρj1=2 −∞ −∞

ð10Þ

In this setup t ρ;v stands for the multivariate t distribution with correlation matrix ρ and v degrees of freedom (d.o.f.). t −1 stands for the inverse of the univariate t distribution and v influences tail v dependency. For v -∞ the t distribution approximates a Gaussian. Due to the fact that estimating parameters for higher order copulas might be computationally cumbersome, all parameters are estimated in a two step maximum likelihood method given by Joe (1996). In the first step the parameters θ^1 related to the univariate margins are estimated by 1. T

n

θ^1 ¼ ArgMaxθ1 ∑ ∑ ln f j ðr jt;θ1 Þ:

ð11Þ

t ¼1j¼1

Based on θ^1 the copula parameters θ^2 are estimated in the second step using 2. T

θ^2 ¼ ArgMaxθ2 ∑ ln cðF 1 ðr 1t Þ; F 2 ðr 2t Þ; …F n ðr nt Þ; θ2 ; θ^1 Þ:

ð12Þ

t¼1

This method is also known as Inference for the margins (IFM), θ^ IFM ¼ ðθ^ 1 ; θ^ 2 Þ′.

3.1.3. Time-varying DCC-copula Due to the fact that the classical copula approach does not account for time varying dependency, we combine the conditional correlation parameters modeled by DCC (Eq. (16)) with the copula setup (Eqs. (9) and (10)). For elliptical copulas, this operation is straightforward and leads to the conditional DCC-Copula setup (firstly addressed by Patton (2006)). The DCC-Gauss copula is given by −1 −1 C Ga DCC ðr 1 ; …; r n Þ ¼ Φρt ðΦ ðr 1 Þ; …; Φ ðr n ÞÞ;

ð13Þ

and the DCC-t copula by −1 C tDCC ðr 1 ; …; r n Þ ¼ t ρt ;v ðt −1 v ðr 1 Þ; …; t v ðr n ÞÞ:

ð14Þ

T. Berger / International Economics 133 (2013) 93–106

97

The estimation of the DCC-Copula is tractable by treating θ2 as an observable function of α, β (see (14)) and the information at t−1: 1=2

Rt ¼ diagfQ t

1=2

gQ t diagfQ t

g;

Q t ¼ Ω þ αϵt−1 ϵ′t−1 þ βQ t−1 ;

ð15Þ ð16Þ

The copula likelihood function (Eq. (12)) can than be maximized over α and β that drive the dependence parameter (see Manner and Reznikova, 2012), i.e. T

θ^2 ¼ ArgMaxθ2 ðα;βÞ ∑ ln cðF 1 ðr 1t Þ; F 2 ðr 2t Þ; …F n ðr nt Þ; θ2 ðα; βÞ; θ^1 Þ:

ð17Þ

t¼1

3.2. Marginal distributions The DCC approach generally relies on elliptical marginal return distributions. Hence, we assume both normally and t distributed returns in the subsequent empirical analysis. However, the copula approach allows for more flexible return models. With respect to VaR forecasts, the tail behavior of returns matters. Therefore, using Extreme Value Theory (EVT) to model exceptional returns (and using the empirical distribution for the “interior” part of the marginals) seems to be a sensible alternative. Based on the GARCH filtered i.i.d. residuals, additionally to parametric distributions, the peaks over threshold approach, will be applied in connection with copula models. The data that exceeds a predefined threshold at level α, is modeled via the Generalize Pareto Distribution (GPD). In the succeeding empirical analysis, α ¼ 10% will be the threshold value. So to say, the extreme observations in the tails of the empirical distribution are modeled via EVT (see Longin and Solnik, 2001) and the cumulative distribution function F ξ is given by !−1=ξ x FðxÞξ;β ¼ 1− 1 þ ξ ð18Þ β with x≥0, β 4 0 and ξ 4 −0; 5. In this context, x represents the exceedances, ξ the tail index (shape) parameter and β the scale parameter respectively. Let g ξ;β denote the density function, Nu the number over the threshold u, and Xj the observed values over the threshold u and Y j ¼ X j −u. Then the log likelihood is given by Nu

lt ¼ ∑ ln g ξ;β ðY i Þ

ð19Þ

j¼1

1 Nu lt ¼ −Nu ln β− 1 þ ∑ ln ξj¼1

 ! Yi ; 1þξ β

ð20Þ

3.3. Value-at-risk and backtesting 3.3.1. Value-at-risk VaR is defined as the quantile at level α of the distribution of portfolio returns (denoted by r): Z VaRα f ðrÞ dr ¼ Pðr≤VaRα Þ: ð21Þ VaRα ¼ F −1 ðαÞ ¼ −∞

For elliptical distributions, quantiles are direct functions of the variances and we can directly translate the quantiles of the estimated portfolio variances into VaR. Let α be the quantile, VaR at time t for both normal and t distributions is given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi VaRα;t ¼ −zα w′H t w; ð22Þ where Ht represents the covariance matrix, w is the vector of portfolio weights and zα the quantile of order α.

98

T. Berger / International Economics 133 (2013) 93–106

Using simulation methods as in the context of copulas (Section 3.1.2), N return scenarios are simulated based on information available at time t, so that the estimated one period VaR is represented by the empirical α-quantile of the N simulated returns.1

3.3.2. Backtesting criteria Conditional coverage criteria: Additionally to the absolute amount of misspecifications, unconditional coverage (UC), independence (IND) and conditional coverage (CC) are applied. The unconditional coverage test, proposed by Kupiec (1995), adds to the Basel II backtesting framework, and checks if the expected failure rate α of a VaR model is statistically different from its realized failure rate α: ^ LRUC ¼ −2 ln½ð1−αÞT−N αN  þ 2 ln½ð1−ðN=TÞÞT−NðN=TÞN ;

ð23Þ

where α stands for the percent left tail level, T for the total days and N for the number of ^ ¼ α is LRUC ∼χ 2 ð1Þ. Due to the fact misspecifications. The likelihood ratio statistic for testing H0 ¼ E½α that the UC method exclusively tests the equality between empirical VaR violations and the chosen confidence level, also Christoffersen (1998) developed a likelihood ratio statistic to test whether the VaR misspecifications are correlated in time. The LR-Statistic ratio for testing time dependence versus time independence (H0) is defined as LRIND ¼ −2 ln½ð1−πÞðT 00 þT 01 Þ π ðT 01 þT 11 Þ  þ 2 ln½ð1−π 0 ÞT 00 π T001 ð1−π 1 ÞT 10 π T111 :

ð24Þ

Tij stands for the number of observed values i followed by j. Whereas 1 represents a misspecification and 0 a correct estimation. π represents the probability of observing an exception and π i the probability of observing an exception conditional on state i. Thus, this approach rejects a model that either creates too many or too few clustered VaR violations with LRIND ∼χ 2 ð1Þ. The CC test combines both test statistics with H0 covering the empirical failure rate and time independence. The likelihood ratio statistic is given by LRCC ¼ LRUC þ LRIND :

ð25Þ

Given the stochastic independence of LRUC and LRIND, LRCC ∼χ 2 ð2Þ. Dynamic quantiles: The dynamic quantile backtesting approach by Engle and Manganelli (2004) represents an out of sample approach (see Dumitrescu et al., 2012) and overcomes a shortcoming of the LRIND test, since it allows testing for autocorrelation with more than just one lag. Let the hit function be defined by Hit t ¼ Iðr t o −VaRα;t Þ−α;

ð26Þ

with indicator function I. Then the hit sequence should neither have a nonzero mean nor should it be autocorrelated. Engle and Manganelli (2004) test the Nullhypothesis that Hitt is orthogonal to a predefined vector of instruments which includes for instance past lags of Hitt and VaRt. Under the null, the test statistic for T out of sample forecasts DQ OOS ¼

Hit′t X t ½X′t X t −1 X′t Hit′t Tαð1−αÞ;

ð27Þ

is asymptotically χ 2 ðnÞ distributed with n degrees of freedom.2 1 More precisely, if the generated N return scenarios are ordered in an ascendant order, the VaR is given by the ½ð1−αÞN−1th observation. So to say, it is simply the empirical quantile of the vector of simulated portfolio returns based on the information available at time t. In this study, a rolling window approach is applied to forecast the one-day ahead VaR thresholds based on the given dependence. The rolling window size is at 1000 observations for all data sets and 10,000 scenarios are simulated for each day. 2 According to Gaglianone et al. (2009) this test can be applied to any VaR model, its application is not restricted to the CAViaR model of Engle and Manganelli (2004) where it has been initially proposed. In our work, the vector of instruments Xt includes a constant, Hit t−j with j ¼ 0; 1; …; 3 and VaRtjt−1 .

T. Berger / International Economics 133 (2013) 93–106

99

3.3.3. Conditional predictive ability As we aim at comparing different models for VaR forecasts, tests of predictive ability seem to be natural candidates for model evaluation. Typically, those tests compare ex ante model predictions with their respective realizations. This approach proves to be convenient whenever realized values are observable ex post, for example when it comes to forecast return figures. In case of VaR, realizations are not observable and finding a good proxy for the true unknown VaR is a rather demanding challenge.3 Therefore, in applied VaR forecast evaluation, measures of predictive ability are usually based on the difference between predicted VaRtjt−1 and realized portfolio returns rt, i. e. a loss function lt ¼ f ðr t −VaRt Þ is used for model evaluation. This choice shifts the focus from a statistical to an institutional perspective: Typically financial companies prefer models that – if no exceptional loss occurs – predict VaR values not too far from realized returns, as this lowers capital requirements. Regulators instead prefer models that – if exceptional losses occur – predict VaR values not too far from realized return, as losses are backed by required capital to a large extent and financial stability is largely guaranteed in this case. Hence, the sign of ðrt −VaRt Þ matters, if model performance is evaluated from a certain perspective.4 Loss functions can be categorized according to whether they treat deviations ðr t −VaRt Þ symmetrically or asymmetrically. Sener et al. (2012) discuss candidate loss functions for both cases. As we do not aim at taking any particular point of view from an institutional perspective, we use the symmetric absolute error loss function lt ¼ jr t −VaRt j in this study. Additionally, from a statistical point of view and due to the focus on conditional α-quantiles of the return distributions, we also apply the asymmetric linear “tick” loss function T α ðlt Þ ¼ ðα−1ðlt o 0Þ Þlt with lt ¼ r t −VaRt (see Giacomini and Komunjer, 2005). Two competing VaR models can be compared by testing the null hypothesis of equal predictive ability H 0 : E½f ðr t −VaRi;t Þ−f ðr t −VaRj;t ÞjΓ t−1  ¼ 0

ð28Þ

with VaRi;t and VaRj;t denoting VaR forecasts from model i and model j, respectively. Giacomini and White (2006) propose a test procedure that does only depend on model estimates and is robust with respect to parameter uncertainty.5 The test statistic, as given in Giacomini and White (2006) is asymptotically χ 2 ðnÞ distributed with n degrees of freedom.

4. Empirical application 4.1. Data Portfolios covering different asset classes are investigated, to avoid drawbacks flawed by assetspecific properties. Namely, we analyze stocks, indexes and currency exchange rates (FX-rates). Moreover, due to the fact that the focus of our research lies exclusively on dependency modeling, we avoid issues stemming from non-synchronized timezones (see Martens and Poon, 2001), thus we investigate assets from the European timezone to ensure that news-impacts on markets are captured by the same day. Additionally, in case of stock and index returns, we avoid return distortions stemming from currency translations by solely focusing on return series which are all denominated against the EURO.6 3 Actually, this is the reason why usually coverage criteria are applied when it comes to VaR model evaluation, see Hurlin and Tokpavi (2007). 4 Note that loss functions with argument ðr t −VaRt Þ usually do not account for correct coverage. Hence, their application has to be restricted to VaR models conforming to the specified VaR level. 5 We thank an anonymous referee for proposing this test and emphasizing its robustness property, as in an application to copula models, the test is robust in particular with respect to potential misspecification of marginal return distributions. 6 However, in order to include one asset which incorporates all those properties, we add the NIKKEI index to a second portfolio consisting of indexes.

100

T. Berger / International Economics 133 (2013) 93–106

Four equally weighted portfolios are investigated: Two portfolios comprising national stock indexes, one portfolio of individual German stocks and one currency portfolio. The examined data of Portfolio I are the daily closing prices of five national stock markets namely CAC (France), DAX (Germany), IBEX (Spain), MIB (Italy) and PSI (Portugal). Whereas the first four represent larger EURO-denominated economies and Portugal represents a small economy. Portfolio II incorporates smaller indexes plus one exotic index: DAX, AEX (Netherlands), ISEQ (Ireland), MIB (Italy) and Nikkei (Japan). Portfolio III covers German DAX-listed companies, whereas every company represents a different business segment to address the idea of diversification: Deutsche Bank, Daimler, EON, BASF and Lufthansa. Portfolio IV covers currencies denominated against the EURO, the USD is combined with European currencies: GBP/EUR, CHF/EUR, CZK/EUR and NOK/EUR.7 Data on all portfolios cover the 2000 day period from the beginning of 2003 until the end 2009. Using a rolling window of 1000 observations leads to a set of 1000 successive daily (one step) VaR forecasts for every portfolio. 4.2. Results The data sample covers the outburst of the financial crisis. Hence, return volatilities show pronounced clustering effects. In Fig. 1, estimated GARCH volatilities are plotted. Addressing the accuracy of VaR forecasts for Portfolio I consisting of European stock indexes, results are given in Table 1. As we calculate 1000 VaR forecasts, we expect 10 portfolio returns to exceed the respective VaR figure. Additionally, test statistics for the backtesting criteria discussed in Section 3.3.2 are reported. Corroborating the results of the simulation study, figures for the DCC-Copula are very similar to those of the benchmark DCC model, which comes as no surprise as we restrict our attention to one day ahead VaR forecasts. Obviously, it is the number of VaR exceedances that leads to rejection of the models by the LR-test statistics, as can be seen from the LRUC and the LRIND statistics, respectively. However, as soon as we take advantage of the flexibility of the copula approach and use EVT margins instead of elliptical return distributions, the hybrid DCC-copula approach clearly outperforms the DCC model in terms of VaR accuracy, represented by the relative hit rate of VaR misspecifications (% VaR). Both the DCC-Gauss and the DCC-t-Copula models are accepted with respect to VaR estimates and the EVT based VaR forecasts are in line with the LRUC, LRIND and LRCC criteria. As well, the DQ teststatistic accepts, however, the lower p-values are caused by the fact, that clustered misspecifications do occur with lag ¼3. Note that the DQ-test significantly rejects the null for t distributed returns, albeit the relative hit rate gets rejected by LRUC. Results regarding the empirical failure rate and LR-statistics turn out to be pretty much the same for Portfolio II that comprises smaller European stock indexes and the NIKKEI, see Table 2. Again, copulas based on EVT margins show the best “relative hit rate” concerning the VaR forecasts and are clearly accepted by the LR test criteria as well as the DQ test statistics. Both normal and t marginal distribution functions result in an inadequate number of misspecifications, and consequently get rejected by LRUC and DQ. Turning to the single stocks portfolio (Portfolio III), one can see that the application of the DCC copula approach is not recommendable without fail if one sticks to elliptical marginal distributions. As can be seen from Table 3, assuming normally and t distributed returns leads to a inappropriate VaR failure rate in the DCC-copula framework (both DDC-Gauss-Copula and t-Copula are rejected by the LRCC criteria with 90% confidence). In addition to the LRCC criteria regarding the elliptical distributed returns, the DQ test statistics only accepts the EVT-DCC-Copula approach. So that, the hybrid approach leads to sensible VaR estimates if marginal returns are modeled by a generalized Pareto distribution, and again it is the DCC-Copula that performs best regarding the accuracy of the relative VaR failure rate (accepted by LRCC and DQ). Findings for Portfolio IV, the FX-rates portfolio, contradicts the results of the third portfolio: The advantage of assuming normally distributed exchange rates is overwhelming, all dependency 7

CHF¼ Swiss franc, CZK ¼Czech Koruna, GBP ¼ Pound sterling, NOK ¼Norwegian krone and USD¼ US dollar.

T. Berger / International Economics 133 (2013) 93–106

101

Fig. 1. Univariate GARCH (1,1) volatilities, Portfolio I.

Table 1 Portfolio I: Backtesting 99% VaR. Portfolio I

P-Values

Marg. Dist.

Dep. Model

% VaR

LRUC

LRIND

LRCC

DQ

Normal Normal Normal t dist t dist t dist EVT EVT

DCC DCC-G-Cop. DCC-t-Cop. DCC DCC-G-Cop. DCC-t-Cop. DCC-G-Cop. DCC-t-Cop.

2.5 2.4 2.4 0.3 0.4 0.3 1.3 1.2

0.000 0.000 0.000 0.009 0.029 0.009 0.362 0.362

0.597 0.603 0.603 1.000 1.000 0.841 1.000 1.000

0.000 0.000 0.000 0.032 0.094 0.032 0.660 0.827

0.000 0.040 0.039 0.420 0.422 0.602 0.221 0.223

Marg. Dist.¼ Margin Distribution, % VaR¼ Relative Amount of VaR exceedances, Dep. Model¼ Dependency Model.

Table 2 Portfolio II: backtesting 99% VaR. Portfolio II

P-Values

Marg. Dist.

Dep. Model

% VaR

LRUC

LRIND

LRCC

DQ

Normal Normal Normal t dist t dist t dist EVT EVT

DCC DCC-G-Cop. DCC-t-Cop. DCC DCC-G-Cop. DCC-t-Cop. DCC-G-Cop. DCC-t-Cop.

2.5 2.8 2.0 0.4 0.4 0.4 1.4 1.2

0.000 0.000 0.005 0.03 0.03 0.03 0.230 0.538

0.654 0.806 0.413 1.000 0.009 0.009 1.000 1.000

0.000 0.000 0.014 0.095 0.003 0.003 0.487 0.827

0.012 0.043 0.056 0.000 0.000 0.000 0.847 0.865

Marg. Dist.¼ Margin Distribution, % VaR¼ Relative Amount of VaR exceedances, Dep. Model¼ Dependency Model.

approaches lead to very accurate failure rates, whereas the model with t distributed daily FX rates seems to be clearly misspecified, irrespective of the dependency model. However, the hybrid models with EVT margins again perform excellent with respect to the hit rates, even though the LRInd statistics – albeit

102

T. Berger / International Economics 133 (2013) 93–106

Table 3 Portfolio III: Backtesting 99% VaR. Portfolio III

P-Values

Marg. Dist.

Dep. Model

% VaR

LRUC

LRIND

LRCC

DQ

Normal Normal Normal t dist t dist t dist EVT EVT

DCC DCC-G-Cop. DCC-t-Cop. DCC DCC-G-Cop. DCC-t-Cop. DCC-G-Cop. DCC-t-Cop.

2.0 1.7 1.6 0.4 0.3 0.3 1.4 1.3

0.005 0.043 0.079 0.030 0.009 0.009 0.079 0.362

0.368 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.013 0.075 0.037 0.095 0.033 0.033 0.214 0.660

0.004 0.009 0.011 0.000 0.002 0.003 0.935 0.959

Marg. Dist.¼ Margin Distribution, % VaR¼ Relative Amount of VaR exceedances, Dep. Model¼ Dependency Model.

Table 4 Portfolio IV: Backtesting 99% VaR. Portfolio IV

P-Values

Marg. Dist.

Dep. Model

% VaR

LRUC

LRIND

LRCC

DQ

Normal Normal Normal t dist t dist t dist EVT EVT

DCC DCC-G-Cop. DCC-t-Cop. DCC DCC-G-Cop. DCC-t-Cop. DCC-G-Cop. DCC-t-Cop.

1.1 1.2 1.1 0.1 0.6 0.1 1.2 1.0

0.752 0.538 0.764 0.000 0.169 0.000 0.538 1.000

1.000 1.000 1.000 1.000 0.025 1.000 0.130 0.085

0.951 0.827 0.956 0.001 0.031 0.001 0.263 0.227

0.182 0.274 0.173 0.000 0.000 0.000 0.264 0.270

Marg. Dist.¼ Margin Distribution, % VaR¼ Relative Amount of VaR exceedances, Dep. Model¼ Dependency Model.

remaining insignificant – do raise in the DCC-Copula approach. The DQ criteria accept the models based on normal and EVT margins (Table 4). So far, the results have shown that the specification of marginal distributions turns out to be crucial for VaR forecast quality. Hence, the copula approach is favorable as it does not restrict the choice of marginals and allows i.e. to use EVT margins, which lead to reasonable results for the 99% VaR. The question whether the DCC copula outperforms the static copula model remains to be answered. Two approaches are possible: By means of a simulation study, influence of the dynamic copula behavior might be decoupled from the influence of marginal return distributions on VaR forecasts. This approach has been analyzed in Berger (2013). By controlling the margins, both competing dependency approaches were compared in an extensive Monte Carlo simulation, and the dynamic copula approach clearly dominates the static copula model. Likewise, both approaches may be compared using the test of conditional predictive ability (CPA) by Giacomini and White (2006) as discussed in Section 3.3.2. For the pairwise comparison of forecast models we use both the “tick” loss function T α ðlt Þ ¼ ðα−1ðlt o 0Þ Þlt and the absolute error loss function lt ¼ jr t −VaRt j. As the latter loss function does not account for the gap between VaR confidence level and the observed share of VaR exceedances, application of the test should be restricted to models with equal performance with respect to unconditional coverage. Indeed, both copula specifications the static one and the DCC approach, show nearly exactly the same performance with respect to this criterion. This can be seen from column 3 in Table 5 showing the pairwise difference in the number of VaR exceedances. The results based on the “tick” loss function are in line with the LRUC backtesting, since both models show the same rate of VaR misspecifications.

T. Berger / International Economics 133 (2013) 93–106

103

Table 5 Conditional predictive ability tests, “tick” loss function for portfolios I–IV (based on EVT). Portfolio

VaR

Dependency models

VaR Diff.

CPA (P-Values)

PF I

95%VaR

DCC-G Cop vs. G Cop

+1

0:23ðþÞ (0.628)

95%VaR

DCC-t Cop vs. t Cop

99%VaR 99%VaR PF II

PF III

PF IV

0

2:70ðþÞ (0.150)

DCC-G Cop vs. G Cop

−1

1:02ðþÞ (0.312)

DCC-t Cop vs. t Cop

−2

1:54ðþÞ (0.214)

+3

2:43ð−Þ (0.119)

95%VaR

DCC-G Cop vs. G Cop

95%VaR

DCC-t Cop vs. t Cop

99%VaR 99%VaR

0

0:58ð−Þ (0.448)

DCC-G Cop vs. G Cop

+1

0:51ð−Þ (0.473)

DCC-t Cop vs. t Cop

+1

1:01ð−Þ (0.315)

95%VaR

DCC-G Cop vs. G Cop

+2

0:11ð−Þ (0.742)

95%VaR

DCC-t Cop vs. t Cop

0

1:04ð−Þ (0.307)

99%VaR

DCC-G Cop vs. G Cop

0

3:57ð−Þ (0.059)

99%VaR

DCC-t Cop vs. t Cop

+1

0:09ðþÞ (0.770)

95%VaR

DCC-G Cop vs. G Cop

+4

0:01ðþÞ (0.914)

95%VaR

DCC-t Cop vs. t Cop

99%VaR

DCC-G Cop vs. G Cop

99%VaR

DCC-t Cop vs. t Cop

0

4; 31ð−Þ (0.038)

+1

0:81ð−Þ (0.367)

0

61:17ð−Þ (0.000)

VaR Diff.¼ Absolute Difference of VaR Misspecifications (#DCC-Copula - #Copula) The entries are the CPA test statistics and (the respective p-values). A (+)/(-) indicates that the model in the first column gets outperformed/outperforms the model in the second column at a 5% significance level.

The CPA tests based on absolute error loss functions are given in Table 6 and clearly prefer the DCC copulas over their respective static counterparts for 95% and 99% VaR forecasts.8 This finding corroborates the simulation results in Berger (2013). All in all, the results of the empirical study can be summarized as follows: For one step ahead forecasts, choice of the dependence model determining the intercorrelation of assets seems to be of minor importance once the marginals are fixed. Based on several competing backtesting criteria, no dependency approach categorically outperformed the others. We showed, that with respect to LRCC and DQ criteria the DCC-EVT-Copula approach outperformed all other approaches in terms of compatibility. However, for a discussion about the shortcomings of the applied backtesting criteria we refer to Hurlin and Tokpavi (2007). Nevertheless, the DQ test statistics also confirms the idea that dependency measure itself does not seem to have notable impact on the empirical hit rate of VaR forecasts, since based on a unified setup regarding the margins, the rejection rate of the dependency approaches is similar. However, even though the popular DCC model does only allow for elliptical marginals, it can lead to reasonable VaR forecasts regarding the VaR failure rate. Alas, applying the DCC to a particular portfolio requires a rather careful and individual choice of return distributions, as results can be rather misleading otherwise. With respect to VaR forecasts, modeling the marginals drawing on EVT turns out to be a promising alternative. When combined with elliptical copulas, the resulting model proves to be a highly efficient and universal tool for inferences concerning VaR estimates, irrespective of the underlying portfolio's composition. So that, although the dependency modeling seems to be of minor empirical importance for VaR modeling, the flexibility provided by the copula setup proved to be rather fruitful, especially when it comes to evaluate the overall performance throughout different asset classes. By merging the DCC and the classical copula approach to a hybrid DCC-Copula approach, we showed that the classical static copula setup should be expanded to capture the dynamics of conditional correlations. Although the empirical hit rate of VaR is not affected by this expansion, this approach leads to significantly smaller deviations from the realized returns.

8

The empirical backtesting performances of 95% VaR forecasts for all investigated portfolios are available upon request.

104

T. Berger / International Economics 133 (2013) 93–106

Table 6 Conditional predictive ability tests, absolute error loss function for portfolios I–IV (based on EVT). Portfolio

VaR

Dependency models

VaR Diff.

CPA (P-Values)

PF I

95%VaR

DCC-G Cop vs. G Cop

+1

0:55ðþÞ (0.457)

95%VaR

DCC-t Cop vs. t Cop

0

0:44ðþÞ (0.506)

99%VaR

DCC-G Cop vs. G Cop

−1

0:54ðþÞ (0.463)

99%VaR

DCC-t Cop vs. t Cop

−2

0:40ðþÞ (0.527)

+3

PF II

PF III

PF IV

95%VaR

DCC-G Cop vs. G Cop

95%VaR

DCC-t Cop vs. t Cop

99%VaR 99%VaR

117:74ð−Þ (0.000)

0

11:27ð−Þ (0.001)

DCC-G Cop vs. G Cop

+1

37:66ð−Þ (0.000)

DCC-t Cop vs. t Cop

+1

93:59ð−Þ (0.000)

95%VaR

DCC-G Cop vs. G Cop

+2

0:67ðþÞ (0.412)

95%VaR

DCC-t Cop vs. t Cop

0

4:19ð−Þ (0.041)

99%VaR

DCC-G Cop vs. G Cop

0

2:40ðþÞ (0.121)

99%VaR

DCC-t Cop vs. t Cop

+1

166:18ð−Þ (0.000)

95%VaR

DCC-G Cop vs. G Cop

+4

95:73ð−Þ (0.000)

95%VaR

DCC-t Cop vs. t Cop

99%VaR

DCC-G Cop vs. G Cop

99%VaR

DCC-t Cop vs. t Cop

0

86:92ð−Þ (0.000)

+1

93:61ð−Þ (0.000)

0

60:77ð−Þ (0.000)

VaR Diff. ¼Absolute Difference of VaR Misspecifications (#DCC-Copula - #Copula). The entries are the CPA test statistics and (the respective p-values). A (+)/(−) indicates that the model in the first column gets outperformed/outperforms the model in the second column at a 5% significance level.

5. Conclusion When it comes to forecasting portfolio VaR, finding the “right” portfolio return distribution is crucial. Both the distribution of individual assets' returns and their dependency structure determine portfolio returns and, therefore, VaR. The DCC and the Copula model are popular approaches to deal with both of these issues empirically in a two step procedure. Both approaches, DCC and copula, have particular drawbacks: The DCC sticks to linear correlations and severely restricts the class of admissible marginals, whereas the copula is a basically static concept. However, both models can be easily combined whenever one restricts attention to elliptical copulas. The resulting DCC-copula model is feasible even for higher order portfolios. The empirical analysis of daily data in the years 2005–2009 covering portfolios of rather different composition unrevealed that mainly the assumptions concerning marginal return distributions determine model performance, i.e. the accuracy of VaR forecasts. Hence, an adequate dependence model first and foremost must not pose any restriction on (marginal) return distributions. This claim favors the copula approach over the DCC model, as the latter only allows for elliptical return distributions. But, the static copula approach suffers from its lacking responsiveness to new information, a feature that may be crucial in turmoil market times. The hybrid DCC copula approach proved to resolve this deficiency in our empirical application and turned out to be a very powerful tool for VaR estimation when combined with EVT-based margins. As the resulting model produced favorable results concerning several backtesting criteria for the portfolios analyzed, we can generally recommend it to forecast portfolio VaR, especially in periods of turbulent financial markets. Concerning further studies on VaR forecasts, the empirical analysis has underlined that particular attention and care should be payed to modeling marginal return distributions as the key drivers of VaR estimates. Moreover, it became clear that it is worth the effort to use a portfolio of backtesting and model selection criteria. Even if different criteria may point in different directions, a single criterion usually focuses on particular features of the model and hence assesses model performance

T. Berger / International Economics 133 (2013) 93–106

105

only partially. Therefore, derivation and application of more general, “all-embracing”, goodness of fit measures concerning VaR predictions (e.g. along the lines of Hurlin and Tokpavi (2007) or Sener et al. (2012)) seems to be both relevant and promising and is left for future work. References Angelidis T, Benos A, Degiannakis S. The use of GARCH models in VaR estimation. Statistical Methodology 2004;1:105–28. Aloui R, Aissa M, Nguyen D. Global financial crisis, extreme interdependences, and contagion effects: the role of economic structure?. Journal of Banking and Finance 2011;35:130–41. Bauer F. Dynamic conditional correlation models and portfolio management. Shaker Verlag; 2011. Berger T., 2013. On the impact of dependency modeling on multivariate value-at-risk forecasts. Verlag Dr. Kovac. Candelon B, Colletaz G, Hurlin C, Tokpavi S. Backtesting value-at-risk: a GMM duration-based test. Journal of Financial Econometrics 2011;9:314–43. Campbell SD. A review of backtesting and backtesting procedures. Journal of Risk 2007;9(2):1–18. Christoffersen PF. Evaluating interval forecasts. International Economic Review 1998;39(4):841–62. Das SR (2007), Basel II Correlation Related Issues, Journal of Financial Services Research, 32, 17–38. Dumitrescu E-I, Hurlin C, Pham V. 2012. Working paper, halshs-00671658, version 1. Embrechts P, McNeil A, Straumann D. 2002. Correlation and dependence in risk management: properties and pitfalls. Risk management: value at risk and beyond. Cambridge University Press; 176–223. Engle R, Manganelli S. CAViaR: conditional autoregressive value at risk by regression quantiles. Journal of Business and Economics Statistics 2004;22:367–81. Engle R. Dynamic conditional correlation: a simple class of multivariate generalized autoregressive conditional heteroscedasticity models. Journal of Business and Statistics 2002;20:339–50. Engle R. Anticipating correlations: a new paradigm for risk management. Prindeton University Press; 2009. Gaglianone Q, Linton O, de Oliveira Lima LR, Smith D. 2009. Evaluating value-at-risk models via quantlie regressions. Working paper 09-46, Economic Series (25), 〈http://www.eco.uc3m.es/temp/09-46-25.pdf〉. Giacomini R, Komunjer I. Evaluation and combination of conditional quantile forecasts. Journal of Business and Economic Statistics 2005;23:416–31. Giacomini R, White H. Tests of conditional predictive ability. Econometrica 2006;74(6):1545–78. Hakim A, McAleer M, Chan F. 2007. Forecasting portfolio value-at-risk for international stocks, bonds and foreign exchange. Working paper, School of Economics and Commerce, University of Western Australia. Hakim A, McAleer M. 2009. VaR forecasts and dynamic conditional correlations for spot and future returns on stocks and bonds. Econometrics Institute report EI32, Erasmus University Rotterdam. Heinen A, Valdesogo Robles A. 2009. Asymmetric CAPM dependence for large dimensions: the Canonical Vine Autoregressive Model. Working paper (2009069), Universite Catholique de Louvain. Hsu C, Huang C, Chiou W. Effectiveness of copula-extreme value theory in estimating value-at-risk: empirical evidence from Asian emerging markets. Review of Quantitative Finance Accounting 2011;10:1007. Hurlin C, Tokpavi S. Backtesting value-at-risk accuracy: a simple new test. The Journal of Risk 2007;9(2):19–37. Joe H. 1996. Families of m-variate distributions with given margins and m (m-1)/2 bivariate dependence parameters. In: Ruschendorf L, Schweizer B, Taylor MD, editors. Distributions with fixed margins and related topics. In: IMS lecture notes monograph seriex, 28, p. 120–41. Jondeau, E and M. Rockinger (2006), The Copula-GARCH Model of Conditional Dependencies: An International Stock Market Application, Journal of International Money and Finance. Jorion, P. (2006), "Value at Risk. The new Benchmark for Controlling Derivatives Risk", McGraw, New York. Junker, M. and A. May (2005), "Measurement of aggregate risk with copulas", Econometrics Journal, 428-454. Kole E, Koedijk K, Verbeek M. Selecting copulas for risk management. Journal of Banking and Finance 2007:2405–23. Kupiec P. Techniques for verifying the accuracy of risk measurement models. Journal of Derivatives 1995;2:73–84. Lee W, Lin H. Portfolios value at risk with copula-Armax-GJR-GARCH model: evidence from the gold and silver futures. African Journal of Business Management 2011;5(5):1650–62. Longin, F. and B. Solnik (2001), "Extreme Correlation of International Stock Equity Markets", Journal of Finance, 56, 649-676. Malevergne Y, Sornette D. Testing the Gaussian copula hypothesis for financial asset returns. Quantitative Finance 3 2003;3(4): 231–50. Manner H, Reznikova O. A survey on time-varying copulas: specification, simulation and application. Economic Reviews 2012;31(6):100–20. Martens, M. and S. Poon (2001), "Returns Synchronization and Daily Correlation Dynamics between International Stock Markets", Journal of Banking and Finance, 25, 1805-1827. Nelsen B. PAn introduction to Copulas. Springer; 2006. Ozun A, Cifter A. 2007. Portolio valueatrisk with timevarying copula: evidence from the americas. Marmara University, MPRA Paper No. 2711. Palaro H, Hotta L. Using conditional copula to estimate value at risk. Journal of Data Science 2006;4:93–15. Patton A. On the out-of-sample importance of skewness and asymmetric dependence for asset allocation. Journal of Financial Econometrics 2004;2:130–68. Patton A. Modelling asymmetric exchange rate dependence. International Economic Review 2006;47:527–56. Patton A. Copula based models for financial time series. Handbook of financial time series. Springer Verlag; 2009. Roy I. 2011. Estimation of portfolio value at risk using copulas. RBI working paper series, 4/2011. Sklar C. Fonctions de repartition a n dimensions et leurs marges. Publicationss de Institut Statistique de Universite de Paris 1959;8:229–31.

106

T. Berger / International Economics 133 (2013) 93–106

Sener E, Baronyan S, Menguturk LA. 2012. Ranking the predictive performances of value-at-risk estimation methods. International Journal of Forecasting. Tse YK, Tsui AKC. A multivariate generalized autoregressive conditional heteroscedasticity model with time-varying correlations. Journal of Business and Economic Statistics 2002;20:351–62. Viebig J, Poddig T. Modeling extreme returns and asymmetric dependence structures of hedge fund strategies using extreme value theory and copula theory. Journal of Risk 2010;13:23–55. Virdi NK. A review of backtesting methods for evaluating value at risk. International Review of Business Research 2011;7:14–24. Yener T. Risk management beyond correlation. Pro Business Verlag; 2011.