CoVaR with volatility clustering, heavy tails and non-linear dependence
Michele Leonardo Bianchi, Giovanni De Luca, Giorgia Rivieccio
CCoVaR with volatility clustering, heavy tailsand non-linear dependence
Michele Leonardo Bianchi, a,1
Giovanni De Luca b , and Giorgia Rivieccio ba Regulation and Macroprudential Analysis Directorate, Bank of Italy, Rome, Italy b Department of Management and Quantitative Studies, Univerisity of Naples Parthenope,Naples, Italy
This version: September 24, 2020
Abstract.
In this paper we estimate the conditional value-at-risk by fitting differentmultivariate parametric models capturing some stylized facts about multivariate finan-cial time series of equity returns: heavy tails, negative skew, asymmetric dependence, andvolatility clustering. While the volatility clustering effect is got by AR-GARCH dynamicsof the GJR type, the other stylized facts are captured through non-Gaussian multivariatemodels and copula functions. The CoVaR ≤ is computed on the basis on the multivariatenormal model, the multivariate normal tempered stable (MNTS) model, the multivariategeneralized hyperbolic model (MGH) and four possible copula functions. These risk mea-sure estimates are compared to the CoVaR = based on the multivariate normal GARCHmodel. The comparison is conducted by backtesting the competitor models over the timespan from January 2007 to March 2020. In the empirical study we consider a sample oflisted banks of the euro area belonging to the main or to the additional global systemicallyimportant banks (GSIBs) assessment sample. Key words: systemic risk, value-at-risk, conditional value-at-risk, heavy tails, non-lineardependence, copula functions, backtesting.
The contribution of financial institutions to systemic risks in financial markets is a largelydebated topic in literature. Relevant articles include Billio et al. (2012), Adrian andBrunnermeier (2016), Girardi and Erg¨un (2013), Lin et al. (2018).In Adrian and Brunnermeier (2016) the distress of a financial institution is defined asthe event ( y j = VaR jα ), where y j is the random variable representing the log-returns ofthe financial institution j and VaR jα the corresponding value-at-risk (VaR) at tail level α .Here we consider the conditional value-at-risk (CoVaR ≤ ) measure, that is the conditionalvalue-at-risk where the conditioning event is the distress of a financial institution repre-sented through the inequality y j ≤ VaR jα . This allows us to have a robust systemic riskmeasure which can be backtested without a great effort (see Girardi and Erg¨un (2013)and Banulescu et al. (2020)). Conversely the original CoVaR of Adrian and Brunnermeier This publication should not be reported as representing the views of the Bank of Italy. The viewsexpressed are those of the authors and do not necessarily reflect those of the Bank of Italy. a r X i v : . [ q -f i n . R M ] S e p ≤ is a proper systemic risk measure, we explore to whichextent the model assumptions on the univariate financial institution log-returns and onthe dependence structure affect the estimates of this risk measure. Additional, we conducta backtesting analysis to obtain a robust model comparison.It is widely known that the amplitude of daily returns varies over time and that ifthe volatility is high, it tends to remain high, and if it is low, it tends to remain low.This means that volatility moves in clusters and for this reason it is necessary to capturesuch observed behaviour (Rachev et al. (2011)). Additionally the CoVaR estimation isby definition a multi-dimensional problem. The multivariate normal model is usuallyapplied in practical applications to finance, mainly because both the theoretical andpractical complexity of a model increases if one moves from a normal to a non-normalframework. However, the multivariate normal distribution has two main drawbacks: (1)its margins are normally distributed, therefore it does not capture empirically observedskewness and kurtosis; (2) its dependence structure is symmetric, it does not captureasymmetry of dependence during extreme market movements and the dependence of tailevents. For the reasons above, in this work we implement multivariate non-normal modelswith volatility clustering for the CoVaR estimation.More in details, we assume that the univariate time series have AR-GARCH dynamicswith Glosten-Jagannathan-Runkle (GJR) volatility (see Glosten et al. (1993)) and thenwe analyze different dependence structures. Among possible multivariate parametricmodels applied to finance (see Bianchi et al. (2020)), we select the multivariate normaltempered stable (MNTS) and the multivariate generalized hyperbolic (MGH) model,and four copula functions, normal, t and BB1 and BB7, as described in Jaworski (2017),De Luca and Rivieccio (2018) and De Luca et al. (2019). Both non-normal multivariatedistributions and copula functions are widely known in the financial literature. Bianchiet al. (2016) and Bianchi and Tassinari (2020) analyzed both MNTS e MGH modelsapplied to risk assessment and portfolio optimizations (see also Fallahgoul and Loeper(2019) and Bianchi et al. (2019)). Kurosaki and Kim (2013b) developed a model basedon the MNTS distribution to estimate the CoVaR, and subsequently Kurosaki and Kim(2013a) and Biglova et al. (2014) studied a mean-CoAVaR strategy applied to portfoliooptimization to mitigate the potential loss arising from systemic risk.The remainder of the paper is organized as follows. In Section 2 we describe themethodology implemented in this work: we define both the univariate model and thedependence structure, we show the necessary formulas to compute the CoVaR ≤ . Afterhaving described in Section 3 the market data considered in this study, the main empiricalresults are discussed in Section 4. In Section 5 we compare the different distributionalassumptions through a backtesting exercise. In Section 6 we compare the ∆CoVaR withthe score defined by the Financial Stability Board (FSB) for the global systemicallyimportant banks (GSIBs) bucket allocation and we introduce a score adjusted for theinformation coming from the stock markets.. Section 7 concludes.2 Methodology
For each institution j , the random variable y jt represents the log-returns of the marketvalue of equity. Superscript sys denotes the entire financial system, i.e. the capitalization-weighted portfolio of all financial institutions in the selected sample or an index repre-sentative of the stock market and frequently used by financial professionals (i.e. the S&P500 index or the Euro Stoxx 50 index).At time t , given the VaR of the financial institution j , with tail level α (VaR jα,t ),for a given tail level β , the CoVaR ≤ jβ,α,t of the financial system conditional on financialinstitution j being in distress (i.e. market returns of bank j are less or equal to its VaR α )is equal to P (cid:16) y syst ≤ CoVaR ≤ jβ,α,t | y jt ≤ VaR jα,t (cid:17) = β. (2.1)A tail level α equal to 1% (2.5% or 5%) denotes a distress state of the world, while alevel α equal to 50% denotes a normal, or median, state. The financial institution j contribution to systemic risk is defined by∆CoVaR ≤ jβ,α,t = CoVaR ≤ jβ,α,t − CoVaR ≤ jβ, . ,t . (2.2)∆CoVaR ≤ jβ,α,t captures the negative externality that financial institution j imposes onthe financial system. As described in the following sections, we estimate the CoVaR andthe ∆CoVaR of main listed European banks over the time span from January 2007 toMarch 2020 by using different dependence structure.The systemic risk measure estimation is divided in three steps. In the first step weestimate a univariate AR-GARCH model to the time series of log-returns and computethe VaR at the given tail level α . In the second step we calibrate the dependence structureby applying different multivariate approaches. In the third step we estimate the systemicrisk measure by means of a numerical integration and (or) a numerical inversion.Let S t be the stock price process of a given financial institution and y t = log S t S t − be its log-return process. We assume for the log-return process an AR-GARCH modelwith GJR dynamics for the volatility, that is y t = ay t − + σ t ε t + cσ t = α + α ( | σ t − ε t − | − γ ( σ t − ε t − )) + β σ t − (2.3)where the innovation ε t are independent and identically distributed random variableswith zero mean and unit variance. As observed in Kim et al. (2011), the follow equalityholds VaR α ( y t +1 ) = ay t + σ t (VaR α ( ε t +1 )) + c. (2.4)that in practice means that it is possible to compute the VaR on the basis of the quantileof a standardized random variable. It should be noted that a numerical inversion isusually needed to compute these quantiles. 3fter having estimated for each bank and for the system the univariate discrete-timedynamic volatility model defined in equation (2.3), we extract the innovations and esti-mate different dependence structures. We consider the multivariate normal tempered sta-ble (MNTS) model, the multivariate generalized hyperbolic (MGH) model, as describedin Bianchi et al. (2019), and the best copula function in terms of AIC among normal, t , BB1 and BB7 copulas, as described in De Luca and Rivieccio (2018) and De Lucaet al. (2019). For the MNTS and MGH models we estimate a 13-dimensional model byusing an ad-hoc procedure implemented in R considering an expectation-maximizationmaximum-likelihood approach. Differently, the copulas are calibrated on bivariate timeseries (i.e. for each couple j , the couple ε sys and ε j is considered) and they are esti-mated through the VineCopula package of R. While the estimation of both the MNTSadn MGH is time-consuming from a computational point of view and for this reason wedecide to run few estimation procedures as possible, thus instead of estimating a bivari-ate model we perform a 13-dimensional estimation, the estimation of copula functions inlarge dimension can be problematic from a numerical error perspective, thus instead ofestimating a 13-dimensional model we perform a bivariate estimation.After these two estimation steps, we forecast the one-day ahead volatility obtainedfrom the estimated AR-GARCH parameters. While in the MNTS (MGH) model ε sys and ε j are assumed NTS (GH) distributed, in the copula model they are assumed skew- t distributed. This means that to evaluate the univariate VaR for the each single bank j , while in the NTS (GH) case one needs to invert the cumulative distribution functionobtained by means of the fast Fourier transform algorithm (Bianchi et al. (2019)), in theskew- t case the VaR can be directly obtained through the qsstd function of the fGarch package of R.On the basis of the univariate and multivariate estimates, it is possible to evaluate theCoVaR for each financial institution j and time t . While closed formula for the CoVaRare available under normal distributional assumptions (see Bernard et al. (2012)), fornon-normal models a numerical integration procedure is needed (see Girardi and Erg¨un(2013) and Bernard and Czado (2015)). More in details, equation (2.1) can be writtenas P (cid:16) y syst ≤ CoVaR ≤ jβ,α,t , y jt ≤ VaR jα,t (cid:17) P (cid:0) y it ≤ VaR jα,t (cid:1) = β (2.5)and by the definition of VaR, it follows that P (cid:16) y syst ≤ CoVaR ≤ jβ,α,t , y it ≤ VaR jα,t (cid:17) = αβ. In the MNTS and MGH cases, given the density f jt of the bivariate random variabledefined by y syst and y jt , then the following equality can be considered (cid:90) CoVaR ≤ jβ,α,t −∞ (cid:90) V aR jα,t −∞ f jt ( x, y ) dxdy = αβ. (2.6)to obtain an estimate of CoVaR ≤ . The integral in equation (2.6) is evaluated by thenumerical integration algorithm implemented in the quadrature function of the mvQuad package of R. Thus, the CoVaR estimate is obtained through the one dimensional root4nding algorithm implemented in the uniroot function of the stats package of R. Asobserved by Banulescu et al. (2020), it is interesting to highlight that by slightly modifyingthe integral in equation (2.6) it is possible to obtain an estimate of the marginal expectedshortfall.It should be noted that, as in equation (2.4), the CoVaR can be computed by con-sidering the density of the bivariate random variable defined by ε syst and ε jt and thenby performing a location and scale transformation based on the estimated AR-GARCHparameters.In the copula function cases, as shown in Bernard et al. (2012), if one considers thedependence structure between y syst and y jt , then the following equality can be considered P ( y syst ≤ x, y jt ≤ y ) = C (cid:16) F y syst ( x ) , F y jt ( y ) (cid:17) and equation (2.5) can be rewritten as C (cid:16) F y syst ( CoV aR ≤ jβ,α,t ) , F y jt ( V aR jα,t ) (cid:17) = αβ where C is a given copula function. By following the approach described in Bernard et al.(2012), it is possible to obtain an estimate of the CoVaR by means of the inversion ofthe copula function. After having estimated the bivariate copula function the CoVaRestimate is obtained through the one dimensional root finding algorithm implemented inthe uniroot function of the stats package of R.As we will show in Section 4 the CoVaR ≤ based on these multivariate models iscompared to the CoVaR = based on the multivariate normal GARCH model as definedby Adrian and Brunnermeier (2016) and empirically studied by Bianchi and Sorrentino(2020). Before starting with the empirical analysis, in this section we provide a description of thedata used in this study. We obtained from Thomson Reuters Datastream daily dividend-adjusted closing prices from January 2002 to March 2020 for a sample of listed banksof the euro area belonging to the main or to the additional GSIB assessment samplefor a total of twelve banks. These banks are: Deutsche Bank (D:DBK), Commerzbank(D:CBK), Unicredit (I:UCG), Intesa (I:ISP), BNP Paribas (F:BNP), Soci´et´e G´en´erale(F:SGE), Cr´edit Agricole (F:CRDA), BBVA (E:BBVA), Santander (E:SAN), Banco deSabadell (E:BSAB), KBC Group (B:KC) and ING Bank (H:INGA). We refer to theseEuropean banks as GSIBs, even if for some of these banks there is no additional capitalrequirements. In this study the system is the Euro Stoxx 50 index. The time periodin this analysis includes the high volatility period after the Lehman Brothers filing forChapter 11 bankruptcy protection (September 15, 2008), the eurozone sovereign debtcrisis, during which, in November 2011, the spread between the 10-year Italian BTP andthe German Bund with the same maturity exceeded 500 basis points, the turmoil afterthe Italian political elections in 2018 and the recent financial market crash at the end ofthe first quarter of 2020. 5he CoVaR (∆CoVaR) is estimated on the basis of the time series from the previousfive years. For example, the CoVaR (∆CoVaR) for July 16, 2007 is estimated from thedata for the period from July 16, 2002 to July 16, 2007. For each bank and each modelwe consider 3,389 estimations from January 2, 2007 to March 30, 2020.
In this section we compare the different non-normal models with the multivariate normalone to which we refer to as MNormal. Additionally, we estimate both the CoVaR ≤ and CoVaR = under this multivariate assumption. Recall that the CoVaR = under theMNormal assumption is the systemic risk measure originally proposed by Adrian andBrunnermeier (2016).As observed in Section 2, the first step is the estimation of the univariate autoregres-sive GARCH model with GJR dynamics for both system and banks log-returns. Thisstep is performed through the garchFit function of the package fGarch of R. While for themultivariate models we consider a normal distributional assumption to extract the inno-vations, for the copula model we directly consider the skew-t distributional assumption.The former approach can be viewed as a quasi maximum-likelihood-estimation (QMLE)approach (see Goode et al. (2015)). Additionally, at each estimation step we verify if theautoregressive component is statistically significant: if it is not, we estimate the modelwithout the autoregressive component.After estimating the univariate dynamic volatility models, we extract the innovationsand estimate the multivariate models (MNTS and MGH) and the copula ones. Thereare remarkable differences in terms of computational time between the two approaches.While for a one-step estimation of the four competitor bivariate copulas 0.9 seconds areneeded, for a one-step estimation of the multivariate MNTS (MGH) model around 150(10) seconds are needed. For each estimation day, in the copula case we have 12 inde-pendent bivariate estimations and in the multivariate case we have a single multivariateestimation. This means that the computing time of the copula and MGH model is similarto the copula one, but the computing time of the MNTS model is 15 times larger. Byconsidering that the overall computing time is around 24 hours for the faster models, todeal with the MNTS estimation problem we relied on an efficient R code making use ofthe packages foreach and doParallel and run it on a multi-core platform.As far as the copula model is concerned, on the basis of the AIC criterion the t -copulais selected in most of the cases (around 88% over the 40,668 bivariate estimation), invery few cases the normal copula (0.5%) and in all other cases the BB1 (around 12%).As observed above, it should be noted that only 0.9 seconds are needed to perform aone-step estimation of four bivariate copulas.For each model and each margin (i.e. each bank) Figure 1 shows the average valueover the entire estimation window from January 2, 2007 to March 30, 2020 of the p -valueof the KS test. Even if the normal model has a satisfactory performance, in all cases thenon-normal models are better than the normal one. The overall performance of the threenon-normal models is comparable.While for the copula model the computing time for VaR and CoVaR is instantaneousand the overall computing time is a few minutes, this is not the case for multivariate6odels. The time needed to estimate the CoVaR for the MNTS, the MGH and theMNormal model for a given day is around 8 minutes. For this reason also in this case werelied on a multi-core implementation. However, the VaR estimations are 60 times fasterthan the CoVaR ones. The main bottleneck is the numerical evaluation of the doubleintegral in the MNTS case. It should be noted that the evaluation of the density functionof a bivariate MNTS random variable is based on both a numerical integration and theFFT algorithm and, in the evaluation of the double integral, a large number of points isneeded for the bidimensional grid in order to avoid numerical errors .In order to show the differences between models, we show in Figure 2 the average timeseries computed across all banks from January 2, 2007 to March 30, 2020 of the CoVaR = based on the multivariate normal model and the CoVaR ≤ based on the MNormal, theMGH, the MNTS and the copula model for various level of α and β . In the MNormaland the three non-normal CoVaR ≤ cases we report the differences with respect to theCoVaR = .In Figure 3 we report the time series of the average ∆CoVaR = based on the multivari-ate normal model. We consider α and β equal to 0.05 in equation (2.2). The average iscomputed across all banks. Then, we compare the ∆ CoV aR = to the ∆CoVaR ≤ evaluatedunder different distributional assumptions. In the MNormal case in panel (b) it appearsa large difference between CoVaR ≤ and CoVaR = . This difference increases and changesin sign in non-normal cases. In all cases this difference varies over time. As shown inthe violin plot on the right side of panel (b), it ranges from an average value of -0.55%in MNormal case to 0.035% in MGH case. The maximum distance is reached in March2020 and it ranges from -2.8% in the MNormal case to 5.3% in the skew-t case. Recallthat the violin plot is a method of drawing numeric data and combining a box plot witha kernel density plot. The definition (2.1) allows one to perform a two-steps backtesting. In both steps it ispossible to follow the approach proposed in Christoffersen (2010). First we conduct apreliminary VaR back-test by considering the entire observation window and defining afirst hit sequence (1 if the loss of the financial institution on that day was larger than itspredicted VaR level, and zero otherwise). Then we define a subset of observations on thebasis on the distress of the financial institution j (i.e. y it ≤ VaR jα,t ). Thus, by looking atthis subset, we can backtest the CoVaR. More in details, we compare the CoVaR forecastwith the ex-post loss of the financial system and define a second hit sequence which is 1if the loss of the financial system on that day is larger than its predicted CoVaR level,and zero otherwise.For evaluating the accuracy of forecasted VaR and CoVaR for the models analyzed inthis paper, we perform the likelihood ratio (LR) tests proposed by Christoffersen (1998)and Christoffersen (2010). The LR tests use the number of violations (i.e. the hit se-quences defined above), where violations occur when the actual loss exceeds the estimatedVaR (CoVaR). The LR test consists of three parts: (1) the LR test of unconditional cov-erage (LR uc ), which is the same as the proportion of failures test by Kupiec (1995), (2)the LR test of independence (LR ind ), and (3) the joint test of coverage and independence7LR cc ). In Figure 4 we report the p -values of the LR test of unconditional coverage (uc)and coverage and independence (cc) for both VaR and CoVaR for all analyzed modelsand various values of α and β . We do not report the p -values for the LR test of indepen-dence. The backtest for α = 0 . β = 0 .
05 is not reported, even if it shows satisfactoryresults. By equation (2.2), the tail level α = 0 . α , but it is not possible to backtest all meaningful tail levels β . Forexample, even if we are considering more than 18 years of daily data, it is not possible tostudy the CoVaR for α and β equal to 0.01, because in this case the theoretical numberof exceedances is less than one.We further consider the dynamic quantile (dq) test proposed by Engle and Manganelli(2004), that can be viewed as a more general formulation of the tests proposed above andtwo loss functions, that is the magnitude loss (LM) and the asymmetric magnitude loss(LA) function as defined by Amendola and Candila (2016). The farther the actual numberof violations is from the expected one, the larger is the value of these loss functions.The LA function is built to penalize more heavily the models with a higher number ofviolations with respect to the number of expected violations. The two functions are LM = (cid:26) y t − VaR t ) if y t < VaR t y t ≥ VaR t and LA = (cid:26) P (1 + | y t − VaR t | ) if y t < VaR t | y t − VaR t | if y t ≥ VaR t with P = exp(( ˆ α − α ) /α ) if ˆ α > α and P = 1 in the opposite case, where ˆ α is theempirical coverage. In Tables from 1 to 3 we report for each bank and for all α and β considered in this study the p -values of the three tests (uc, cc and dq) and the values ofthe two loss functions (LM and LA).As far as the VaR backtesting is concerned, the non-normal models (MNTS, MGHand copula) largely outperform the MNormal model, at least for the tail probability levelsthat are usually of interest (i.e. for α equal to 0.01 and 0.025). However, for α equalto 0.05, the MNormal model shows a better performance than the copula model, even ifin the copula model case the null hypothesis is never rejected. This in practice meansthat capturing the volatility clustering effect is of paramount importance in evaluatingrisk measures, at least for the dataset analyzed in this study. Differently, in the CoVaRbacktesting the MNormal model is always rejected and the non-normal model show alwaysa satisfactory performance. By looking at the p -values of the three tests the MNTS andthe MGH model seems slightly better in comparison with their competitor models.As far as the VaR estimates are concerned, the results in Tables from 1 to 3 showthat the LM function is smaller for the MGH and the MNTS models with respect to thecopula model in 12 cases out of 12 when α = 0 .
05, in 9 cases out of 12 when α = 0 . α = 0 .
01. Similar findings hold for the LA function, withthe exception of the α = 0 .
01 case. However, it should be noted that, on average, alsothe MNormal model show satisfactory results in terms of loss functions.As far as the CoVaR estimates are concerned, the results in Tables from 1 to 3 showthat the LM function is smaller for the MGH and the MNTS models with respect to the8opula model in 23 cases out of 36 when α = 0 .
05, in 16 cases out of 24 when α = 0 . α = 0 .
01. Similar findings hold for the LA function.However, it should be noted that while, on average, there are not remarkable differencesbetween the three non-normal models in terms of loss functions, the performance of theMNormal model is not satisfactory.At least for the time-series and the values of α and β analyzed in this study, the non-normal multivariate models (i.e. the MGH and the MNTS model) slightly outperform thecopula from both a VaR and a CoVaR perspective. The MNormal approach is satisfactoryonly in the VaR case, but its performance is not good enough in the CoVaR backtesting. In this section, we compare the ranking provided by the ∆CoVaR with the ranking iden-tified by the FSB for the GSIBs as defined in the Financial Stability Board (2011) policydocument and on its yearly updates. The FSB approach relies on firm-specific infor-mation on size, interconnectedness, substitutability, complexity, and cross-jurisdictionalactivity and it considers annual accounting and other data provided to regulators by fi-nancial institutions (see also Basel Committee on Banking Supervision (2011) and BaselCommittee on Banking Supervision (2013)).More precisely, the GSIB framework analyze bank activities over 12 indicators. Eachbank indicator is compared with the aggregate indicators of all banks in the specifiedsample (Basel Committee on Banking Supervision (2014)). These indicators are groupedinto five categories (i.e. size, interconnectedness, substitutability, complexity and cross-jurisdictional activity). To calculate the score needed to determine the additional require-ment, for each given indicator, the bank reported value for that indicator is divided bythe corresponding sample total, and the resulting value is then expressed in basis points(bps). The final score is obtained as weighted average of the 12 indicators or as simpleaverage of the five category scores (the average is rounded to the nearest whole basispoint). Both specific bank and aggregated indicators are available on the FSB website. The score represents a bank activities as a percentage of the sample total and is used todetermine the bank additional requirement. A higher score results in a higher require-ment. On the basis on the indicators obtained from the FSB website we compute thescore for a subsample of listed European banks for which these indicators are available(for Banco de Sabadell and KBC Group the indicators are not available). These scoresare not equal to the official ones, because here we are considering only a subsample ofbanks.By following the same approach implemented for the evaluation of the GSIBs indi-cators, for each model, we compute the annual average of the ∆CoVaR and rescale it inorder to obtain the corresponding score expressed in basis points. We consider α and β equal to 0.05. Both indicators and scores are reported in Table 4, where the sum of eachcolumn is by construction equal to 10,000 basis points (i.e. 100%). The banks are orderedfrom the most to the less systemic according the GSIBs score. By looking at the valuesand at the colours of these scores, it appears evident that the variability across banksof the score defined on the basis of the ∆CoVaR is not high. Additionally, as shown in See
ARP E = 1 nobs (cid:88) j,t | score GSIBs j,t − score ∆ CoVaR j,t | score GSIBs j,t (6.1)where nobs is the number of observations and score
GSIBs j,t and score ∆ CoVaR j,t representthe GSIBs score and the score computed on the basis of the ∆CoVaR estimates. TheARPE from 2013 to 2018 across the ten banks in the sample is around 50%. As observedabove, the GSIBs score is computed as score
GSIBs j,t = (cid:88) i ω i Category i where i goes from 1 to 5 (size, interconnectedness, substitutability, complexity and cross-jurisdictional activity) and the weights ω i are all equal to 0.2.Since the ARPE value is high, we try to modify the weights ω i of the five categoriesin order to minimize the percentage difference in equation (6.1), and we refer to them as minimum distance weights . By considering these weights, the ARPE with respect to theGSIBs score decreases to around 30%.However, for all models only size and interconnectedness have minimum distanceweights different from zero. The size weight range from 18% for the MNTS model to 30%for the normal model, and the interconnectedness ranges from 70% for the normal modelto 82% for the MNTS model. It seems that the ∆CoVaR is correlated only with the sizeand interconnectedness. Conversely, it is not correlated with the other three categories.Once we have these new weights, we can compute the corresponding score which is asort of adjusted GSIBs score, as reported in Table 4. This adjusted score can be view asa GSIBs score adjusted for the information coming from the stock market. In this paper we compute the CoVaR and the ∆CoVaR under different distributionalassumptions and different CoVaR definitions (i.e. CoVaR = and CoVaR ≤ ). In order tobacktest the CoVaR ≤ , we conduct an empirical test over more than 18 years of daily stock10og-returns. We calibrate three different models allowing for volatility clustering, heavytails and non-linear dependence. These multivariate non-normal models outperform thenormal one in term of log-returns fitting, VaR and CoVaR backtesting for all tail levelsanalyzed in this study. The backtesting exercise shows that the performance of the MGH(MNTS) model is slightly better compared to the copula model, however this latterapproach is very promising from a computational point of view. Finally, we compare thescore obtained through the ∆CoVaR measure with the GSIBs score defined by the FSBand introduce a score adjusted for the information coming from the stock markets. References
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Financial models with L´evyprocesses and volatility clustering . Wiley, Hoboken, NJ, 2011.13 N o r m a l M G H M N T S C o pu l a u ccc d q L M L A u ccc d q L M L A u ccc d q L M L A u ccc d q L M L A V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . D : D B K C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . D : C B K V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . I : U C G C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . I : I S P V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . F : B N P C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . F : S G E V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . F : C R D A V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . F : C R D A C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . E : BB VA V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . E : S AN C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . E : B S A B V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . B : K B C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . H : I N G A V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . T a b l e : P - v a l u e s o f t h e L R t e s t o f un c o nd i t i o n a l c o v e r ag e ( u c ) , c o v e r ag e a nd i nd e p e nd e n ce ( cc ) a ndd y n a m i c q u a n t il e t e s t s ( d q ) f o r b o t h V a R a nd C o V a R w i t h α = . f o r a ll a n a l y ze d m o d e l s a ndb a n k s . T h e v a l u e s o f t h e m ag n i t ud e ( L M ) a nd o f t h e a s y mm e t r i c l o ss ( L A ) f un c t i o n s a r e a l s o r e p o r t e d . N o r m a l M G H M N T S C o pu l a u ccc d q L M L A u ccc d q L M L A u ccc d q L M L A u ccc d q L M L A V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . D : D B K C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . D : C B K V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . I : U C G C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . I : I S P V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . F : B N P C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . F : S G E V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . F : C R D A C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . E : BB VA V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . E : S AN C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . E : B S A B V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . B : K B C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . H : I N G A V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . T a b l e : P - v a l u e s o f t h e L R t e s t o f un c o nd i t i o n a l c o v e r ag e ( u c ) , c o v e r ag e a nd i nd e p e nd e n ce ( cc ) a ndd y n a m i c q u a n t il e t e s t s ( d q ) f o r b o t h V a R a nd C o V a R w i t h α = . f o r a ll a n a l y ze d m o d e l s a ndb a n k s . T h e v a l u e s o f t h e m ag n i t ud e ( L M ) a nd o f t h e a s y mm e t r i c l o ss ( L A ) f un c t i o n s a r e a l s o r e p o r t e d . N o r m a l M G H M N T S C o pu l a u ccc d q L M L A u ccc d q L M L A u ccc d q L M L A u ccc d q L M L A V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . D : D B K C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . D : C B K V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . I : U C G C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . I : I S P V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . F : B N P C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . F : S G E V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . F : C R D A C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . E : BB VA V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . E : S AN C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . E : B S A B V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . B : K B C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . H : I N G A V a R ( α = . ) . . . . . . . . . . . . . . . . . . . . C o V a R ( β = . ) . . . . . . . . . . . . . . . . . . . . T a b l e : P - v a l u e s o f t h e L R t e s t o f un c o nd i t i o n a l c o v e r ag e ( u c ) , c o v e r ag e a nd i nd e p e nd e n ce ( cc ) a ndd y n a m i c q u a n t il e t e s t s ( d q ) f o r b o t h V a R a nd C o V a R w i t h α = . f o r a ll a n a l y ze d m o d e l s a ndb a n k s . T h e v a l u e s o f t h e m ag n i t ud e ( L M ) a nd o f t h e a s y mm e t r i c l o ss ( L A ) f un c t i o n s a r e a l s o r e p o r t e d . i c k e r ∆ C o V a R i nd i c a t o r s G S I B s i nd i c a t o r s a nd s c o r e s N o r m a l M N o r - m a l M G H M N T S C o pu l a S i ze I n t e Sub s C o m p C j a S c o r e A d . s c o r e D : D B K F : B N P F : S G E F : C R D A E : S AN I : U C G H : I N G A D : C B K E : BB VA I : I S P T a b l e : A v e r ag e i nd i c a t o r s a nd s c o r e s o v e r t h e y e a r s f r o m t o2018 . T h e a d j u s t e d s c o r e i s c o m pu t e d o n t h e b a s i s o f t h e w e i g h t s m i n i m i z i n g t h e a v e r ag e r e l a t i v e p e r ce n t ag ee rr o r b e t w ee n t h e G S I B s a nd t h e ∆ C o V a R s c o r e i n t h e M N T S c a s e . F o r t h e ∆ C o V a R , w ec o n s i d e r α a nd β e q u a l t o0 . . For each margin and each model we report the average p -value of the KS test from January2, 2007 to March 30, 2020. We report the average time series computed across all banks from January 2, 2007 to March30, 2020 of the CoVaR = based on the multivariate normal model ( normal ) and the CoVaR ≤ based on theMNormal, the MGH, the MNTS and the copula model for various level of α and β , that is (a) α = 0 . β = 0 . α = β = 0 .
05, (c) α = 0 .
05 and β = 0 . α = 0 .
05 and β = 0 .
01, (e) α = 0 . β = 0 .
05, (f) α = β = 0 . α = 0 .
01 and β = 0 .
05. We show the differences with respectto the (a) case. All values are changed in sign.
In the panel (a) we report the time series from January 2, 2007 to March 30, 2020 of the∆CoVaR = based on the multivariate normal GARCH model with GJR dynamics. In the panel (b) foreach model we report for the same time window the time series and the violin plot of the differencebetween the ∆CoVaR ≤ and the above ∆CoVaR = . In all cases we consider α and β equal to 0.05. Allvalues are changed in sign. Boxplots across all banks of the p -values of the LR test of unconditional coverage (uc) andcoverage and independence (cc) for both VaR and CoVaR for all analyzed models.-values of the LR test of unconditional coverage (uc) andcoverage and independence (cc) for both VaR and CoVaR for all analyzed models.