Can we still benefit from international diversification? The case of the Czech and German stock markets
CCan we still benefit from international diversification? Thecase of the Czech and German stock markets (cid:73)
Krenar Avdulaj a,b , Jozef Barunik a,b, a Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic,Czech Republic b Institute of Economic Studies, Charles University, Prague, Czech Republic
Abstract
One of the findings of the recent literature is that the 2008 financial crisis causedreduction in international diversification benefits. To fully understand the possiblepotential from diversification, we build an empirical model which combines gener-alised autoregressive score copula functions with high frequency data, and allowsus to capture and forecast the conditional time-varying joint distribution of stockreturns. Using this novel methodology and fresh data covering five years after thecrisis, we compute the conditional diversification benefits to answer the question,whether it is still interesting for an international investor to diversify. As diversifi-cation tools, we consider the Czech PX and the German DAX broad stock indices,and we find that the diversification benefits strongly vary over the 2008–2013 crisisyears.
Keywords: portfolio diversification, dynamic correlations, high frequency data,time-varying copulas
JEL:
C14, C32, C51, F37, G11
1. Introduction
A proper quantification of the joint distribution allowing for the time-varyingdependence between assets is critical for asset pricing, portfolio allocation and risk (cid:73)
We are indebted to two anonymous referees for their helpful comments. Jozef Barunik gratefullyacknowledges the support of the Czech Science Foundation project No. P402/12/G097 DYME -“Dynamic Models in Economics”. Krenar Avdulaj gratefully acknowledges the support of the GrantAgency of the Charles University under the 852013 project.
Email address: [email protected] (Jozef Barunik)
Preprint submitted to Working Paper June 8, 2018 a r X i v : . [ q -f i n . P M ] S e p eduction. For a number of years, finance literature has been studying the riskreduction benefit from international diversification. After the recent 2008 financialcrisis, many researchers have documented possible reduction of these benefits dueto rising dependence between markets. The literature concentrating on the CentralEuropean Markets has been limited though, as it is widely believed that after theenlargement of the European Union, these markets became integrated with verylimited opportunities for diversification.In this paper, we revisit this line of research, and study the possible benefits fromdiversification between the Czech PX and the German DAX stock market indicesusing data covering the five years crisis period. While it is reasonable to believethat the Czech and German stock markets show large degree of dependence due tointegration of the Czech Republic into the euro area as well as large dependence ofthe Czech economy on the German one, we aim to study, whether the German andCzech stock indices can be considered for the reduction of risk of an internationalinvestor.A number of researchers have addressed the issue of Central and Eastern Euro-pean (CEE) markets integration with the euro area. Voronkova (2004) documentsincreasing stock integration between Central European (CE) markets and their ma-ture counterparts in Europe, and finds lower diversification opportunities at theaggregate stock index level. Syriopoulos (2004, 2006) finds long-run cointegratingrelationship and hence limited diversification opportunities between CEE marketsand Germany. Aslanidis and Savva (2011) confirm these findings using more recentdata. ´Egert and Koˇcenda (2007) analyze the intraday interdependence of WesternEuropean and Central and Eastern European markets using wide range of econo-metric techniques and find evidence of only short-term relationships among the CEEand Western European stock markets.In more recent study, ´Egert and Koˇcenda (2011) analyze the comovements ofthree developed and three emerging markets using DCC-GARCH model on high fre-quency data. They detect small correlation between the developed and emergingmarkets. This finding is important for investors, as it allows them to diversify theirportfolios by investing in the emerging markets. However, as the authors stress, thisdiversification opportunity may not be available more recently due to economic in-tegration with Western Europe. A study by Hanousek and Koˇcenda (2011) use highfrequency data to study the foreign macroeconomic announcements and spillover ef-fects on emerging CEE stock markets during 2004-2007. Among other findings itis of interest that Frankfurt stock market dominates the spillovers over the threeemerging markets, while the reaction to the New York market is smaller. Syllignakisand Kouretas (2011) study the time-varying conditional correlations among the US,2ermany, Russia, and CEE markets. The authors employ the DCC-GARCH modelfor correlations and use weekly data spanning from 1997-2009 and they find sig-nificant increase of correlation between the US and German markets and the CEEmarkets, especially during 2007-2009 financial crisis.In a recent work, Horv´ath and Petrovski (2013) analyze the comovements amongWestern Europe vis-`a-vis Central and South Eastern Europe (SEE). The analysisis carried on daily data for the period 2006-2011 using bivariate BEKK-GARCHmodel. Authors find the higher integration among CE and lower integration, withalmost zero correlations, among SEE countries. In another study, Gjika and Horv´ath(2013) employ asymmetric DCC-GARCH to study the comovements in CE markets.Using daily data spanning from 2001-2011 authors find increase in correlations afterthese countries joined the European Union, whereas asymmetric correlation effectswere found only for Hungary (BUX) and Poland (WIG) pair. In addition, positiverelation among conditional correlations and conditional variances suggesting lowerdiversifications in turbulent times is confirmed.A common feature of these studies is that they use cointegration, or multivariateGARCH to study the dependence, and with some exceptions, they use the databefore the 2008 financial crisis. We contribute to the literature by using a verydifferent approach proposed recently by Avdulaj and Barun´ık (2013), allowing us tomodel the time-varying joint distribution of stock market returns. Using recentlyproposed time-varying copula methodology and utilizing high frequency data, webuild an empirical model which allows us to study the time-varying benefits fromdiversification. In addition, we contribute to the understanding of the relationshipusing recent data covering crisis years. Using the fresh data, and state of the artmethodologies, we revisit the literature and uncover significant time-varying natureof the benefits from diversification between the PX and DAX markets. This findingis particularly interesting as previous literature generally reports decreasing potentialfor diversification.The work is organized as follows. The second section introduces our empiricalmodel composing from the realized GARCH and generalised autoregressive scoretime-varying copulas. The third section introduces the data we use, while fourthsection discusses the in-sample and out-of-sample fits of all model specifications,and chooses the one which best describes the data. Finally, the fifth section teststhe economic implications of our empirical model. We first evaluate the quantileforecasts, which are central to risk management, and then study the time-varyingdiversification benefits implied by our model. The last section concludes.3 . Dynamic copula realized GARCH modeling framework We introduce the empirical model used for describing the dependence betweenthe German DAX and Czech PX stock indices. Our modeling strategy utilizes highfrequency data to capture the dependence in the margins and recently proposeddynamic copulas to model the dynamic dependence. Final model is thus able todescribe the conditional time-varying joint distribution of returns.The methodology is based on the Sklar’s (1959) theorem extended to conditionaldistributions by Patton (2006b). The extended Sklar’s theorem allows to decomposea conditional joint distribution into marginal distributions and a time-varying copula.Consider the bivariate stochastic process { X t } Tt =1 with X t = ( X t , X t ) (cid:48) , which has aconditional joint distribution F t and conditional marginal distributions F t and F t .Then X t |F t − ∼ F t = C t ( F t , F t ) , (1)where C t is the time-varying conditional copula of X t containing all informationabout the dependence between X t and X t , and F t − available information set,usually F t = σ ( X t , X t − , . . . ). Due to Sklar’s theorem, we are able to construct adynamic joint distribution F t by linking together any two marginal distributions F t and F t with any copula function. Theoretically, there is limitless number of validjoint distribution functions that can be created by combining different copulas withdifferent margins, making this approach very flexible.
The first step in building an empirical model based on copulas is to model themargins. Since the largest part of the dependence in financial time series is in theirvariance, majority of researchers use the generalized autoregressive conditional het-eroscedasticity (GARCH) approach of Bollerslev (1986) in this step.We use the latest advances in the literature which improve volatility modellingby adding the realized volatility measure to the GARCH model. This approachutilizes high frequency data to help in explaining the latent volatility. Comparedwith standard GARCH(1,1) model where the conditional variance of i -th asset, h it = var ( X it |F t − ) is dependent on its past values h it − and past values of X it − , Hansenet al. (2012) propose to utilize realized volatility measure and make h it dependenton the realized variance as well. In this work, we restrict ourselves to the simple log-linear specification of the so-called realized GARCH(1,1). For the general framework Note that the information set for the margins and the copula conditional density is the same.
4f realized GARCH( p , q ) models we suggest to consult Hansen et al. (2012). Whileit is important to model conditional time-varying mean E ( X it |F t − ), we also includethe standard autoregressive (AR) term into the final modeling strategy. As we willfind later, autoregressive term of order no larger than two is appropriate for the DAXand PX return series, thus we restrict ourselves to specifying AR(2) with log-linearRealGARCH(1,1) model as in Hansen et al. (2012) X it = µ i + α X it − + α X it − + (cid:112) h it z it , for i = 1 , h it = ω i + β i log h it − + γ i log RV it − , (3)log RV it = ψ i + φ i log h it + τ i ( z it ) + u it , (4)where µ i is the constant mean, h it conditional variance, which is latent, RV it realized volatility measured from high frequency data, u it ∼ N (0 , σ iu ), and τ i ( z it ) = τ i z it + τ i ( z it −
1) leverage function. For the RV it , we use the high frequency data andcompute it as a sum of squared intraday returns (Andersen et al., 2003, Barndorff-Nielsen and Shephard, 2004). Innovations z it are modelled by the flexible skewed- t distribution of Hansen (1994). This distribution has two shape parameters, askewness parameter λ ∈ ( − ,
1) controlling the degree of asymmetry, and a degreeof freedom parameter ν ∈ (2 , ∞ ] controlling the thickness of tails. When λ = 0,the distribution reduces to the standard Student’s t distribution, and when ν → ∞ ,it becomes skewed Normal distribution, while for ν → ∞ and λ = 0, it becomes N (0 , z it = X it − ˆ µ i − ˆ α X it − + ˆ α X it − (cid:112) ˆ h it (5)ˆ z it |F t − ∼ F i (0 , , for i = 1 , . (6)which have a constant conditional distribution with zero mean and unit variance.Then the conditional copula of X t |F t − is equal to the conditional distribution of U t |F t − : U t |F t − ∼ C t ( γ ) , (7)with γ being copula parameters, and U t = [ U t , U t ] (cid:48) conditional probability integraltransform U it = F i (ˆ z it ; φ i, ) , for i = 1 , . (8) Since the probability integral transform is invertible, the copula function describes also thedependence of the returns X t |F t − . .2. Dynamic copulas: A “GAS” dynamics in parameters The notion of time-varying copula models was initially introduced by Patton(2006b). In further literature, Lee and Long (2009) develop a model where themultivariate GARCH is extended by copula functions to capture the remaining de-pendence. Recently, Hafner and Manner (2012), Manner and Segers (2011) proposea stochastic copula models, which allow parameters to evolve as a latent time se-ries. Another possibility is offered by ARCH-type models for volatility (Engle, 2002)and related models for copulas (Patton, 2006b, Creal et al., 2013), which allow theparameters to be some function of lagged observables. An advantage of the secondapproach is that it avoids the need to “integrate out” the innovation terms drivingthe latent time series processes.When working with time-varying copula models the driving dynamics of themodel is of crucial importance. For our empirical model, we therefore adopt thegeneralized autoregressive score (GAS) model of Creal et al. (2013), which specifiesthe time-varying copula parameter ( δ t ) as a function of the lagged copula param-eter and a forcing variable that is related to the standardized score of the copulalog-likelihood . This type of dynamics reduces the one-step-ahead prediction errorat current observation given the current parameter values of the copula function.Consider a copula with time-varying parameters: U t |F t − ∼ C t ( δ t ( γ )) . (9)Often, a copula parameter is required to fall within a specific range e.g. correla-tion for Normal or student’s t copula is required to fall in between values of -1 and1. To ensure this, Creal et al. (2013) suggest to transform copula parameter by anincreasing invertible function h ( · ) (e.g., logarithmic, logistic, etc.) to the parameter κ t = h ( δ t ) ⇐⇒ δ t = h − ( κ t ) (10)For a copula with transformed time-varying parameter κ t , a GAS(1,1) model isspecified as κ t = w + βκ t − + αI − / t s t − (11) s t − ≡ ∂ log c ( u t − ; δ t − ) ∂δ t − (12) I t ≡ E t − [ s t − s (cid:48) t − ] = I ( δ t ) . (13) Harvey (2013), Harvey and Sucarrat (2012) propose a similar method for modelling time-varying parameters, which they call a dynamic conditional score model. t . In addition we use constant copulas for comparison.To save the space, we do not provide functional forms of copula functions used inthis work. These can be found in Patton (2006b). The final dynamic copula realized GARCH model defines a dynamic parametricmodel for the joint distribution. The joint likelihood is defined as L ( θ ) ≡ T (cid:88) t =1 log f t ( X t ; θ ) = T (cid:88) t =1 log f t ( X t ; θ ) + T (cid:88) t =1 log f t ( X t ; θ ) (14)+ T (cid:88) t =1 log c t ( F t ( X t ; θ ) , F t ( X t ; θ ); θ c ) , (15)where θ = ( φ (cid:48) , γ (cid:48) ) (cid:48) is vector of all parameters to be estimated, including parametersof the marginal distributions φ and parameters of the copula, γ . The parametersare estimated using a two-step estimation procedure, generally known as multi-stagemaximum likelihood (MSML) estimation, first estimating the marginal distributionsand then estimating the copula model conditioning on the estimated marginal dis-tribution parameters. While this greatly simplifies the estimation, inference on theresulting copula parameter estimates is more difficult than usual as the estimationerror from the marginal distribution must be taken into account. In result, MSMLEis asymptotically less efficient than one-stage MLE, however as discussed by Patton(2006a), this loss is not great in many cases. Moreover, bootstrap methodology canbe used for statistical inference. One of the appealing alternatives to a fully parametric model is to estimate uni-variate distribution non parametrically, for example by using the empirical distribu-tion function. Combination of a nonparametric model for marginal distribution andparametric model for the copula results in a semiparametric copula model, whichwe use for comparison to its fully parametric counterpart. Forecasts based on a7emiparametric estimation where nonparametric marginal distribution is combinedwith parametric copula function are not common in economic literature thus it isinteresting to compare it in our modelling strategy. For the margins of the semi-parametric models, we use the non-parametric empirical distribution F i introducedby Genest et al. (1995) , which consists of modelling the marginal distributions bythe (rescaled) empirical distribution.ˆ F i ( z ) = 1 T + 1 T (cid:88) t =1 { ˆ z it ≤ z } (16)In this case, the parameter estimation is conducted via maximizing likelihood L ( γ ) ≡ T (cid:88) t =1 log c t ( ˆ U t , ˆ U t ; γ ) . (17)As it can be seen, the likelihood reduces in estimating the copula parameters only.However, we should note that the inference on parameters is more difficult thanusual, hence we rely on bootstrap inference as advocated in Patton (2006a). An important issue when working with copulas is the selection of the best copulafrom the pool. Several methods and tests have been proposed for selection proce-dure. The methods proposed by Durrleman et al. (2000) are based on distance fromempirical copula. Chen and Fan (2005) propose the use of pseudo-likelihood ratiotest for selecting semiparametric multivariate copula models. A test on conditionalpredictive ability (CPA) is proposed by Giacomini and White (2006). This is a robusttest which allows to accommodate both, unconditional and conditional objectives.Recently, Diks et al. (2010) have proposed a test for comparing predictive abilityof competing copulas. The test is based on Kullback-Leibler information criterion(KLIC) and its statistics is a special case of the unconditional version of Giacominiand White (2006).As our main aim is to use the model for forecasting, out-of-sample performanceof models will be tested by CPA, which considers the forecast performance of two The asymptotic properties of this estimator can be found in Chen and Fan (2006). Although some authors use AIC (or BIC) for choosing among two copula models, selectionbased on these indicators may hold only for the particular sample in consideration (due to theirrandomness) and not in general. Thus, proper statistical testing procedures are required [see Chenand Fan (2005)].
009 2010 2011 2012 2013020406080100 Prices of DAX and PX
DAXPX 2009 2010 2011 2012 2013020406080100 Realized volatility of DAX and PX
DAXPX
Figure 1: Normalized prices and annualized realized volatilities of the DAX and PX over the sampleperiod extending from January 3, 2008 until May 31, 2013. competing models conditional on their estimated parameters to be equal under thenull hypothesis H : E [ ˆ L ] = 0 (18) H A : E [ ˆ L ] > H A : E [ ˆ L ] < , (19)where ˆ L = log c ( ˆ U , ˆ γ t ) − log c ( ˆ U , ˆ γ t ). Other advantages of this test are the pos-sibilities to use it for both nested and non-nested models, and also for comparisonof parametric and semiparametric models. The asymptotic distribution of the teststatistic is N (0 ,
1) and we compute the asymptotic variance using HAC estimatesto correct for possible serial correlation and heteroskedasticity in the differences inlog-likelihoods.
3. Data description
The data set consists of the 5-minute prices of the Prague PX and German DAXcash indices over the period January 3, 2008 until May 31, 2013, covering the recentrecession. We synchronise the data using the time stamp matching, and eliminatetransactions executed on Saturdays, Sundays, holidays, December 24 to 26, andDecember 31 to January 2 due to low activity on these days, which could lead toestimation bias. Hence, in our analysis we work with data from 1349 days. For theempirical model, we need two time series, namely daily returns and realized varianceto be able to estimate the realized GARCH model in margins. For this, obtaindaily returns as a sum of logarithmic intraday returns, hence we work with open-close returns. Realized variance is computed as a sum of squared 5-minute intraday9eturns. Figure 1 plots the development of prices of the PX and DAX together withits realized volatility. Note that plot of prices is normalized so we can compare themovements, and for the plot of realized volatility, we use daily volatility annualizedaccording to the convention 100 × √ × RV t . Strong time-varying nature of thevolatility can be noticed immediately for both PX as well as DAX indices. Realizedvolatility of the DAX is larger in average when compared to the volatility of the PXindex. Otherwise the volatility has similar distributional properties for both indices.
4. Empirical Results
DAX PX DAX PXAR(2) AR(2) c c α - - 0.0842 (3.10) α - - 0.0842 (3.10) α - - -0.1069 (-3.94) α - - -0.1069 (-3.94)Realized GARCH(1,1) GARCH(1,1) ω κ β φ γ ψ ξ -0.5376 (-12.05) -0.5834 (-12.13) - - - - φ τ -0.1691 (-13.26) -0.1414 (-9.44) - - - - τ ν λ -0.1161 (-3.39) -0.0830 (-2.30) - - - - LL r,x -2461.00 -2545.41 - - LL r -1606.75 -1508.11 LL -1682.13 -1522.45 AIC r AIC
BIC r BIC log-linear
Realized GARCH(1,1) and benchmarkGARCH(1,1) with innovations distributed skew-t and normal respectively. t -statistics reportedin parentheses. Before modeling the dependence structure between the PX and DAX, we need tomodel their conditional marginal distributions first. Considering general AR modelsup to five lags, we find AR(2) to best capture the time-varying dependence in meanof PX, while DAX has a constant mean. These results are in line with previousresearch (Barun´ık, 2008). Table 1 summarizes the Realized-GARCH(1,1) fit for10oth PX and DAX. In addition, benchmark GARCH(1,1) model is fit to the data forcomparison. All the estimated parameters are significantly different from zero. Byobserving partial log-likelihood LL r as well as information criteria, we can see thatincluding realized measures into the GARCH model improves the fits significantly.This is crucial for copulas, as we need to specify the best possible model in themargins to make sure there is no univariate dependence left. If a misspecified modelis used for the marginal distributions, then the probability integral transforms willnot be uniformly distributed and this will result in copula misspecification. For theestimated standardized residuals from the AR(2) realized GARCH(1,1), we considerboth parametric and nonparametric distributions as motivated earlier in the text. Before specifying a functional form for time-varying copula function, we testfor the presence of time-varying dependence using the simple approach based onthe ARCH LM test. The test statistics is computed from the OLS estimate ofthe covariance matrix and critical values are obtained using i.i.d. bootstrap (fordetailed information, consult Patton (2012)). Computing the test for the time-varying dependence between the DAX and PX up to p = 10 lags, we find the jointsignificance of all coefficients. Thus we can conclude that there is evidence againstconstant correlation for the DAX and PX. Motivated by this finding, we estimatethree time-varying copula functions, namely Normal, rotated Gumbel and Student’s t using the GAS framework described in the methodology part. As a benchmark, wealso estimate the constant copulas to be able to compare the time-varying modelsagainst the constant ones. While semiparametric approach is empirically interestingand not often used in literature, we use it for all the estimated models as well.Table 2 shows the fit from all estimated models. Starting with constant copu-las, all the parameters are significantly different from zero and Normal copula seemsto describe the DAX and PX indices best according to the highest log-likelihood.Semiparametric specifications combining nonparametric distribution in margins withparametric copula function bring further improvement in the log-likelihoods. Impor-tantly, time-varying specifications bring large improvement in log-likelihoods andconfirm strong time-varying dependence between the DAX and PX indices. Due tothe large number of degrees of freedoms, t GAS copula in fact converges to the normalone N GAS , and thus time-varying normal copula again best describes the data.This is interesting finding, as it confirms that after proper models for the depen-dence in margins of the distribution, there is no asymmetry left and the PX-DAXbivariate distribution is standard normal. To study the goodness of fit for all the11 arametric Semiparametric
Constant copula
Est. Param log L Est. Param log L Normal ρ Clayton κ RGumb κ Student’s t ρ ν − Sym. Joe-Clayton τ L τ U Est. Param log L Est. Param log L RGumb
GAS ˆ ω -0.0466 (0.1245) -0.0037 (0.1103)ˆ α β N GAS ˆ ω α β t GAS ˆ ω α β ν − Table 2: Constant and time-varying copula model parameter estimates with AR(2)-RealizedGARCH(1,1) model for both fully parametric and semiparametric cases. Bootstrapped standarderrors are reported in parentheses. Kolmogorov-Smirnov (KS) and Cramer-von Mises (CvM)test statistics with p -values obtained from 1000 simulations. None of the fully para-metric models is rejected, while most of the semiparametric models are rejected withexception of constant student’s t , Sym. Joe-Clayton and time-varying student’s t .These results suggest that fully parametric models with realized GARCH and para-metric distribution in margins are all well-specified. Thus realized GARCH seemsto very well model all the dependence in margins, which is crucial for the goodspecification of the model in the copula-based approach. Semiparametric modelsare interestingly rejected and are not specified well, except for few mentioned cases.This is in line with results of Patton (2012), who finds rejections in semiparametricspecifications on the U.S. indices data. Still, both tests strongly support the realizedGARCH time-varying GAS copulas for modeling the joint distribution between DAXand PX. Forecasted correlation
Linear correlation from Normal−GAS copula for DAX−PX
Figure 2: Linear correlation from time-varying Normal GAS copula. The vertical dashed lineseparates the in-sample from the out-of-sample (forecasted) part.
While it is important to have a well-specified model which describes the data,most of the times we are interested in using the model for forecasting. Thus we The results of the in-sample goodness of fit tests are available upon request from authors. Wedo not include them in text to save the space. , we use the conditional predictive abil-ity (CPA) test of Giacomini and White (2006). The time-varying copula modelsoutperforms significantly the constant copula models in out-of-sample evaluation.This holds both for parametric and semiparametric cases. Thus time-varying copu-las have much stronger support for forecasting the dynamic distribution of the DAXand PX. When comparing the different time-varying copula functions, the test is notso conclusive. While student’s t and normal statistically outperform Rotated gumbel,the forecasts from student’s t can not be statistically distinguished from the normalcopula. Time-varying normal and student’s t copulas are thus best in the forecastingexercise. Finally, forecasts from parametric models statistically outperform thosefrom semiparametric ones.Thus, the out-of-sample results confirm the in-sample ones, which is a good sign ofproper model fit. The joint distribution of the PX and DAX indices is best modeledwith the AR(2)-realized GARCH(1,1) time-varying normal copula model.Having correctly specified the empirical model capturing the dynamic joint dis-tribution between the DAX and PX, we can proceed to studying the pair. Figure 2plots the time-varying correlations implied by our model with normal GAS copula.The dependence is generally strong, and also has strong time-varying nature duringthe studied period. During the last quarter of the year 2008, when stock marketswere declining due to the Lehman Brother’s crash, the correlation of the PX andDAX markets rose nearly to 0.7. In the following year, it dropped to 0.45 levels andfrom the year 2010 rose back to 0.7 again.This result has serious implications for investors as it suggests that diversifica-tion possibilities are rapidly changing over past few years during the financial crisis.We are going to utilize the results and study the possible economic benefits of themodeling strategy. The results of the out-of-sample forecast evaluation are available upon request from authors.We do not include them in text to save the space, and no to distract the reader from the mainresults.
009 2010 2011 2012 2013−8−6−4−202468
Out−of−sample
Parametric Value−at−Risk from Normal−GAS copula model, DAX−PX w=[0.5,0.5] qq=[0.05, 0.95]qq=[0.01, 0.99]Portfolio returns
May−12 Aug−12 Nov−12 Feb−13 May−13−3−2−10OOS Parametric Value−at−Risk from Normal−GAS copula model, DAX−PX w=[0.5,0.5] qq=0.05qq=0.01Portfolio returns
Figure 3: Value-at-Risk implied by Realized-GARCH time-varying Normal GAS copulas. Theportfolio consists of an equal-weight amount of DAX and PX and the estimation is made forquantiles 1% and 5%.
5. Economic implications: Time-varying diversification benefits and VaR
While it is important to have statistically correct fits, or even good out-of-sampleforecasts, the crucial question is whether it translates to economic benefits. Herewe test our proposed methodology for economic implications. First, we quantify therisk of an equally weighted portfolio composed from the DAX and PX, and second,we study the benefits from diversification to see how the strongly varying correlationaffect them. This is mainly interesting to the international investors considering theCzech PX stock market index and the German DAX index in the portfolio.
Quantile forecasts are central to risk management decisions due to a widespreadValue at risk (VaR) measurement. VaR is defined as the maximum expected losswhich may be incurred by a portfolio over some horizon with a given probability.Let q αt denote an α quantile of a distribution. VaR of a given portfolio at time t isthen simply q αt ≡ F − t ( α ) , for α ∈ (0 , . (20)Thus choice of the distribution is crucial to VaR calculation. For example assum-ing normal distribution may lead to underestimation of the VaR. Our objective isto estimate one-day-ahead VaR of an equally weighted portfolio composed from theDAX and PX returns as Y t = 0 . X t + 0 . X t , which has conditional time-varying15 able 3: Out-of-sample VaR evaluation. Empirical quantile ˆ C α , estimated Giacomini and Komunjer(2005) ˆ L , logit DQ statistics and its 1000 × simulated p -val are reported. ˆ L is moreover tested withDiebold-Mariano statistics with Newey-West estimator for variance. All models are compared to N GAS , while models with significantly less accurate forecasts at 95% level are reported in bold.
Parametric Semiparametric1% 5% 10% 90% 95% 99% 1% 5% 10% 90% 95% 99%
RGumb
GAS ˆ C α L p -val 0.972 0.541 0.430 0.038 0.343 0.999 0.972 0.541 0.068 0.053 0.343 0.999 t GAS ˆ C α L p -val 0.972 0.588 0.448 0.047 0.343 0.999 0.972 0.588 0.430 0.026 0.448 0.999 N GAS ˆ C α L p -val 0.999 0.588 0.068 0.053 0.343 0.999 0.999 0.541 0.430 0.070 0.448 0.999 joint distribution F t . In the previous analysis, we have found that the realizedGARCH model with time-varying normal GAS copula well fits and forecasts thedata, thus we use it in VaR forecasts to see whether it correctly forecasts also thejoint distribution. As there is no analytical formula, which can be used for this, werely on Monte Carlo approach, where we simply simulate the future conditional jointdistribution from the estimated models.While quantile forecasts can be readily evaluated by comparing their actual (es-timated) coverage ˆ C α = 1 /n (cid:80) Tn =1 y t,t +1 < ˆ q αt,t +1 ), against their nominal coveragesrate, C α = E [1( y t,t +1 < q αt,t +1 )], this approach is unconditional and does not cap-ture the possible dependence in the coverage rates. Number of approaches has beenproposed for testing the appropriateness of quantiles conditionally, for the best dis-cussion see Berkowitz et al. (2011). In our out-of-sample VaR testing, we use anapproach originally proposed by Engle and Manganelli (2004), who use the n -th or-der autoregression I t = ω + (cid:80) nk =1 β k I t − k + (cid:80) nk =1 β k q αt − k +1 + u t , where I t +1 is 1 if y t +1 < q αt and zero otherwise. While hit sequence I t is a binary sequence, u t is as-sumed to follow a logistic distribution and we can estimate it as a simple logit modeland test whether P r ( I t = 1) = q αt . To obtain the p -values, we rely on simulationsas suggested by Berkowitz et al. (2011) and we refer to this test as a DQ test in theresults. 16oreover, we evaluate the accuracy of VaR forecasts statistically by defining theexpected loss of VaR forecast made by a forecaster m as L α,m = E (cid:2) α − (cid:0) y t,t +1 < q α,mt,t +1 (cid:1)(cid:3) (cid:2) y t,t +1 − q α,mt,t +1 (cid:3) , (21)which has been proposed by Giacomini and Komunjer (2005). Then, differencesin the values of L α,m can be tested using Diebold and Mariano (2002) approach,where we test the null hypothesis that the loss function of a benchmark forecaster isthe same as the loss function of the tested forecaster m , under the alternative thatbenchmark forecaster is more accurate than the competing one.Table 3 reports out-of-sample VaR evaluation of all models, and Figure 3 il-lustrates the 1% and 5% estimated quantiles of the portfolio. We can see that allthe time-varying models are well specified and the conditional quantile forecasts fromthem are not rejected by the DQ test. For the statistical testing, we use time-varyingnormal copula as a benchmark forecaster and test all the other models against it.When looking at the loss functions ˆ L α,m , we can see that all the quantiles impliedby the different models can not be distinguished from each other statistically, exceptthe 1% quantile. This is mainly because student’s t copula has large number of de-grees of freedom basically converged to the normal one. Thus overall, AR(2)-realizedGARCH(1,1) with time-varying copula models are able to describe and forecast thequantiles of the PX-DAX distribution very well. In case the dependence of the assets is changing over time strongly, it needs totranslate to changing diversification benefits as well. Unlike VaR, expected shortfallsatisfies the sub-additivity property and is a coherent measure of risk. Motivatedby these properties, Christoffersen et al. (2012) propose a measure capturing thedynamics in diversification benefits based on expected shortfall. The conditionaldiversification benefit (CDB) for a given probability level α is defined by CDB αt = ES αt − ES αt ES αt − ES αt , (22)where ES αt is expected shortfall of the portfolio at hand, ES αt ≡ E [ Y t |F t − , Y t ≤ F − t ( α )] , for α ∈ (0 , , (23) ES αt is upper bound of the portfolio expected shortfall being the weighted averageof the asset’s individual expected shortfalls, and ES αt lower bound on the expectedshortfall being the inverse cumulative distribution function for the portfolio. In other17
009 2010 2011 2012 20130.20.250.30.350.40.450.5 Conditional diversification benefits for DAX−PX (alpha=0.05)
Normal−GASConstant CDB90% CI
Figure 4: Conditional diversification benefits,
CDB . t using time-varying normal copula togetherwith bootstraped confidence band for the constant conditional diversification benefits. words, this lower bound corresponds to the case where the portfolio never loses morethan its α distribution quantile. The measure is designed to stay within [0 ,
1] interval,and is increasing in the level of diversification benefits. When the CDB is equal tozero, there are literarily no benefits from diversification, when it equals one, thebenefits from diversification are highest possible.Figure 4 plots the conditional diversification benefits for the PX and DAX portfo-lio implied by our empirical model for α = 5%. Similarly to the VaR case, as there isno closed form solution to our empirical model available, we rely on the simulationsfor CDB computation. Encouraged by the previous results, we compute the CDB forthe AR(2) realized GARCH with time-varying normal copula model. Analysis couldbe taken step ahead by optimizing portfolio weights for the highest diversificationbenefits. This is done in Christoffersen et al. (2012), who basically find very smallincrease, implying that equally weighted portfolio is usually very close to optimalif CDB is used. Also please note that here we do not exploit the full potential ofdynamic asset allocation.Diversification benefits vary over time greatly. From the beginning of the sam-18le, the benefits from diversification between the DAX and the PX index are risinggradually until the end of the year 2009, when they start to decline. The lowestvalues are at the beginning of the year 2012, while from this point until the 2013,the benefits stay more or less lower.To support our results, we also report 90% bootstrapped confidence bands com-puted around a constant level of diversification benefits. Assuming the returns dataare independently distributed over time with the same unconditional correlation asthe PX and DAX pair, bootstrap confidence level can be conveniently computed viasimulations. We use 10.000 simulations, and report the mean value together withdistribution of constant conditional benefits in Figure 4. We can see, that the time-varying nature of the conditional diversification benefits is statistically significant,as it departs from the simulated constant distribution.Thus contrary to the general expectation of no diversification benefits for investorsconsidering the Czech PX index as a diversification tool for the German DAX dueto very high correlation between these two stocks, we find actual benefits which arevarying strongly in time.
6. Conclusions
This work revisits the Czech PX and German DAX stock markets dependencewith the aim to study the opportunities of these two assets in portfolio management.Using an empirical model utilizing high frequency data in the time varying copulas,we study the joint conditional distribution of the PX and DAX returns.The final AR(2) realized GARCH(1,1) with time-varying normal copula is able tocapture the dynamics accurately, yielding precise quantile forecasts. Using the crisisdata, we study the time-varying correlations between the PX and DAX returns. Moreimportant, we study how the time-varying dependence translates to the conditionaldiversification benefits. The main result is that the possible diversification benefitsare strongly varying over time, and hence even after the 2008 financial crisis, it may beeconomically interesting to use the DAX and PX returns for the risk diversification.This is important finding, as it is contrary to the belief that crisis caused reductionin international diversification benefits. Czech and German economies are stronglytightened as well, so one would expect that especially after the inclusion of the CzechRepublic into the euro area, diversification benefits will disappear.
References
Andersen, T., T. Bollerslev, F. Diebold, and P. Labys (2003). Modeling and forecasting realizedvolatility.
Econometrica (71), 579–625. slanidis, N. and C. S. Savva (2011). Are there still portfolio diversification benefits in easterneurope? aggregate versus sectoral stock market data. The Manchester School 79 (6), 1323–1352.Avdulaj, K. and J. Barun´ık (2013, July). Are benefits from oil - stocks diversification gone? a newevidence from a dynamic copulas and high frequency data. Papers 1307.5981, arXiv.org.Barndorff-Nielsen, O. and N. Shephard (2004). Econometric analysis of realized covariation:High frequency based covariance, regression, and correlation in financial economics.
Econo-metrica 72 (3), 885–925.Barun´ık, J. (2008). How do neural networks enhance the predictability of central european stockreturns?
Czech Journal of Economics and Finance (Finance a uver) 58 (07-08), 358–376.Berkowitz, J., P. Christoffersen, and D. Pelletier (2011). Evaluating value-at-risk models withdesk-level data.
Management Science 57 (12), 2213–2227.Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity.
Journal of Econo-metrics 31 (3), 307–327.Chen, X. and Y. Fan (2005, September). Pseudo-likelihood ratio tests for semiparametric mul-tivariate copula model selection.
The Canadian Journal of / La Revue Canadienne de Statis-tique 33 (3), 389–414.Chen, X. and Y. Fan (2006). Estimation of copula-based semiparametric time series models.
Journalof Econometrics 130 (2), 307 – 335.Christoffersen, P., V. Errunza, K. Jacobs, and H. Langlois (2012). Is the potential for internationaldiversification disappearing? a dynamic copula approach.
Review of Financial Studies 25 (12),3711–3751.Creal, D., S. J. Koopman, and A. Lucas (2013). Generalized autoregressive score models withapplications.
Journal of Applied Econometrics .Diebold, F. X. and R. S. Mariano (2002). Comparing predictive accuracy.
Journal of Business &economic statistics 20 (1).Diks, C., V. Panchenko, and D. van Dijk (2010, September). Out-of-sample comparison of cop-ula specifications in multivariate density forecasts.
Journal of Economic Dynamics and Con-trol 34 (9), 1596–1609.Durrleman, V., A. Nikeghbali, and T. Roncalli (2000). Which copula is the right one? Workingpaper, Groupe de Recherche Op´erationnelle Cr´edit Lyonnais, France.´Egert, B. and E. Koˇcenda (2007, June). Interdependence between eastern and western europeanstock markets: Evidence from intraday data.
Economic Systems 31 (2), 184–203.´Egert, B. and E. Koˇcenda (2011, April). Time-varying synchronization of european stock markets.
Empirical Economics 40 (2), 393–407.Engle, R. (2002, July). Dynamic conditional correlation: A simple class of multivariate gener-alized autoregressive conditional heteroskedasticity models.
Journal of Business & EconomicStatistics 20 (3), 339–50.Engle, R. F. and S. Manganelli (2004). Caviar: Conditional autoregressive value at risk by regressionquantiles.
Journal of Business & Economic Statistics 22 (4), 367–381.Genest, C., K. Ghoudi, and L.-P. Rivest (1995, September). A semiparametric estimation procedureof dependence parameters in multivariate families of distributions.
Biometrika 82 (3), 543–552. iacomini, R. and I. Komunjer (2005). Evaluation and combination of conditional quantile forecasts. Journal of Business & Economic Statistics 23 (4), 416–431.Giacomini, R. and H. White (2006). Tests of conditional predictive ability.
Econometrica , 1545–1578.Gjika, D. and R. Horv´ath (2013). Stock market comovements in central europe: Evidence from theasymmetric { DCC } model. Economic Modelling 33 (0), 55 – 64.Hafner, C. M. and H. Manner (2012). Dynamic stochastic copula models: Estimation, inferenceand applications.
Journal of Applied Econometrics 27 (2), 269–295.Hanousek, J. and E. Koˇcenda (2011, 02). Foreign news and spillovers in emerging european stockmarkets.
Review of International Economics 19 (1), 170–188.Hansen, B. E. (1994). Autoregressive conditional density estimation.
International EconomicReview 35 (3), 705–30.Hansen, P. R., Z. Huang, and H. H. Shek (2012). Realized garch: a joint model for returns andrealized measures of volatility.
Journal of Applied Econometrics 27 (6), 877–906.Harvey, A. (2013).
Dynamic Models for Volatility and Heavy Tails , Volume Econometric SocietyMonograph 52. Cambridge.: Cambridge University Press.Harvey, A. and G. Sucarrat (2012). Egarch models with fat tails, skewness and leverage. workingpaper CWPE 1236, Cambridge University .Horv´ath, R. and D. Petrovski (2013). International stock market integration: Central and southeastern europe compared.
Economic Systems 37 (1), 81 – 91.Lee, T.-H. and X. Long (2009, June). Copula-based multivariate garch model with uncorrelateddependent errors.
Journal of Econometrics 150 (2), 207–218.Manner, H. and J. Segers (2011). Tails of correlation mixtures of elliptical copulas.
Insurance:Mathematics and Economics 48 (1), 153–160.Patton, A. J. (2006a). Estimation of multivariate models for time series of possibly different lengths.
Journal of applied econometrics 21 (2), 147–173.Patton, A. J. (2006b, 05). Modelling asymmetric exchange rate dependence.
International EconomicReview 47 (2), 527–556.Patton, A. J. (2012). Copula methods for forecasting multivariate time series. Volume 2 of
Handbookof Economic Forecasting . Springer Verlag, forthcoming.Sklar, A. (1959). Fonctions de r´epartition `a n dimensions et leurs marges.
Publications de l’Institutde Statistique de l’Universit´e de Paris 8 , 229–231.Syllignakis, M. N. and G. P. Kouretas (2011). Dynamic correlation analysis of financial contagion:evidence from the central and eastern european markets.
International Review of Economics& Finance 20 (4), 717–732.Syriopoulos, T. (2004). International portfolio diversification to central european stock markets.
Applied Financial Economics 14 (17), 1253–1268.Syriopoulos, T. (2006). Risk and return implications from investing in emerging european stockmarkets.
Journal of International Financial Markets, Institutions and Money 16 (3), 283–299.Voronkova, S. (2004). Equity market integration in central european emerging markets: A cointegra-tion analysis with shifting regimes.
International Review of Financial Analysis 13 (5), 633–647.(5), 633–647.