E. Kalotychou - Quantitative and Qualitative Analysis in Social ...

Model**in**g Dependence **in** CDS **and** Equity Markets:Dynamic Copula with Markov-Switch**in**gFei Fei, Ana-Maria Fuertes ∗ , Elena **Kalotychou**Faculty of F**in**ance, Cass Bus**in**ess School, City University LondonMarch 2013AbstractWe propose a exible dynamic copula with Markov-switch**in**g to model the dependence betweenthe iTraxx Europe CDS market **and** the underly**in**g equity market. The model is able toreproduce extreme return cluster**in**g **and** asymmetry by allow**in**g for two time-vary**in**g dependenceregimes, low or normal **and** high or crash, both at the centre **and** tails of the bivariate distribution.In-sample statistical criteria support the Markov-switch**in**g dynamic copula. Empirically, weidentify high dependence regimes that co**in**cide with the recent credit crunch **and** the Greek **and**European sovereign debt crises. A portfolio Value-at-Risk simulation to generate one-day-aheadtrad**in**g limits highlights the economic signicance of the proposed copula through regulatory lossfunctions that consider the frequency **and** the magnitude of out-of-sample exceptions.JEL classications: C13; C41; G21; G28.Keywords: Credit spread; Copula; Regime switch**in**g; Tail dependence; Value-at-Risk.∗ Correspond**in**g author: Cass Bus**in**ess School, Faculty of F**in**ance, 106 Bunhill Row, London EC1Y 8TZ. Tel: +44 2070405259. E-mail address: a.fuertes@city.ac.uk1Electronic copy available at: http://ssrn.com/abstract=2161570

1 INTRODUCTIONAppropriately model**in**g the dependence structure of credit portfolios **and** systematic risk factors is importantfor risk managers **in** order to set trad**in**g limits, for traders **in** order to hedge the market risk of their creditpositions **and** for pric**in**g credit derivatives.In particular, the use of models that acknowledge shifts **in**the relationship between nancial **in**stitutions' credit exposures **and** the underly**in**g equity market can bebenecial towards the design of more adequate regulatory frameworks **and** reduce systemic risks dur**in**gstressed market conditions.Merton (1974)'s theory **in**directly suggests a l**in**k between credit derivativeprices **and** equity prices. Firm-value structural models orig**in**at**in**g from Merton's theoretical framework reston the fundamental asset value process, namely, a rm's default probability is driven by its leverage **and** bythe volatility of its assets. As asset value **and** volatility are latent, the implementation of structural credit riskmodels for publicly-traded rms relies on the observable equity return **and** a volatility proxy (e.g., historicalor implied), while the credit default swap (CDS) spread can be taken as a measure of rm default risk. 1CDS spreads can be argued to provide more reliable signals on the default risk**in**ess of corporate borrowersthan bond spreads as bond prices are often distorted by tax **and** liquidity issues. CDS contracts are highlyst**and**ardized **and** thus less likely to be **in**uenced by aspects of the contractual agreement such as seniority,coupon rates, embedded options **and** guarantees. Moreover, the CDS spread does not h**in**ge on the choiceof risk-free benchmark. Longsta et al. (2005) found that liquidity factors are a very important driver ofbond yield spreads. Blanco et al. (2005) showed that the CDS market leads the bond market **in** terms ofshort-run price discovery **and** attribute it to the higher liquidity **and** trad**in**g volume of the CDS marketwhich makes it **in**formationally more ecient. 2The perception of the CDS premium as a rather directmeasure of default risk together with the rapid development of the CDS market have spurred an enthusiasticdebate over the determ**in**ants of CDS spreads **and**, **in** particular, their sensitivity to structural factors suchas equity volatility, macrovariables, rm-specic balance sheet **in**formation **and** credit rat**in**gs.Norden **and** Weber (2009) **in**vestigate the l**in**k between changes **in** CDS spreads **and** stock returns, whileMadan **and** Unal (2000), Blanco et al. (2005), **and** Zhang et al. (2009) also consider stock return volatility.Ericsson et al. (2004) nd that volatility **and** leverage alone expla**in** a substantial proportion of the variation**in** CDS premia. Yu (2006) is the rst to document shifts between turbulent **and** calm regimes **in** thedynamics of CDS spreads. A common denom**in**ator to the above studies is that they focus on the determ**in**ants1 Similar to traditional **in**surance policies, the seller of a CDS contract must compensate the buyer if the underly**in**g lo**and**efaults. In return for this protection, the buyer is required to make xed periodic payments with predened premium (orspread) to the seller. In the event of default, the CDS buyer receives compensation **and** the seller takes possession of the loan.2 The global CDS market grew dramatically over a short period of time with a volume expansion from $300 billion **in** 1998to $25.9 trillion at the end of 2011 accord**in**g to the International Swaps **and** Derivatives Association (ISDA). This signicantgrowth can be primarily attributed to the development of CDS **in**dices. The market has slowed down **in** recent years; betweenJune 2011 **and** the end of the year volumes **in** the CDS market decl**in**ed by 12.5 percent, partly due to an **in**crease **in** centralclear**in**g, the eectiveness of nett**in**g **and** collateral, **and** portfolio compression.2Electronic copy available at: http://ssrn.com/abstract=2161570

of s**in**gle-name CDS spreads which are notably less liquid than CDS **in**dices. 3The launch of broad-basedCDS **in**dices **in** 2001 by JP Morgan **and** Morgan Stanley marks a new era **in** credit derivatives trad**in**gby oer**in**g more liquidity, tradeability **and** transparency. However, research **in**to the dependence structuredynamics between CDS **in**dex spreads **and** equity market **in**dicators is still sparse. Bystrom (2008) nds thatstock returns **and** stock market volatility are able to expla**in** most of the variation **in** iTraxx CDS spreads.Us**in**g Markov-switch**in**g regressions, Alex**and**er **and** Kaeck (2008) show that the determ**in**ants of CDS **in**dexspreads are regime-specic; implied volatility is strongly related to CDS spreads **in** the high volatility regimewhile stock returns play a bigger role **in** the tranquil regime.While all of the aforementioned empirical studies implicitly rely on the conventional l**in**ear Pearsoncorrelation as dependence measure, rm structural models **in**spired from Merton (1974) suggest that themarg**in**al eect of a fall **in** equity value is non-constant (as l**in**ear approaches would predict) but **in**steaddriven by rm fundamentals such as leverage. 4 Us**in**g an extension of Merton's model with realized volatility**and** jumps, Zhang et al. (2009) provide evidence that the strength of the relation between credit risk **and**equity value depends on the rm's credit rat**in**g. They document a nonl**in**ear convex relation between CDSspreads **and** equity volatility. Cao et al. (2010) nd that the l**in**k between the CDS market **and** impliedvolatility is stronger when CDS spreads are more volatile **and** credit rat**in**gs are lower. Empirical studieshave consistently suggested that credit spread predictions obta**in**ed from Merton-type structural credit riskmodels underestimate historical credit spreads; e.g., Jones et al. (1984), **and** Eom et al. (2004). This maypartly stem from the fact that the actual dependence structure of debt with equity has complex featuresthat l**in**ear correlation models fail to capture. Recent work supports this conjecture. Hull et al. (2004) showthat theoretical CDS spreads implied from Merton's model us**in**g equity value **and** volatility as **in**puts arenonl**in**early related to historical CDS spreads. Us**in**g adaptive nonparametric regressions, Giammar**in**o **and**Barrieu (2009) provide evidence that the relationship between iTraxx Europe CDS **in**dex returns **and** twosystematic factors, Euro Stoxx 50 returns **and** changes **in** the VStoxx 50 volatility **in**dex, suered severalstructural changes between November 2004 **and** January 2008.Our paper extends recent research on the nonl**in**ear relation between credit spreads **and** tradeable systematicrisk factors by adopt**in**g copulas which represent a very versatile framework to estimate multivariatedistributions. Although copulas have been employed **in** credit risk model**in**g before 5 , this is the rst applica-3 CDS **in**dices are pools of basic s**in**gle-name CDSs provid**in**g protection aga**in**st the basket of entities **in** the **in**dex. The rsttwo families of **in**dices (iBoxx **and** Trac-x) merged **in** 2004 to form the CDX **in** North America **and** iTraxx **in** Europe **and** Asiawhich comprise the most liquid s**in**gle-name CDSs. They were both acquired by Markit **in** 2007 which s**in**ce then adm**in**istratesboth CDS **in**dices. Unlike s**in**gle-name contracts, CDS **in**dex type contracts do not term**in**ate after a credit event but **in**steadthey cont**in**ue with the defaulted entity removed **and** the contract value reduced. For a comprehensive discussion of CDS **in**dexcomposition **and** performance, see Markit (2010).4 Zero correlation does not imply **in**dependence, it only rules out l**in**ear dependence.5 Crook (2011) use copula to analyze the dependence of default rates **in** consumer loans, Das **and** Geng (2006) focus oncorporate debt default dependence while Hull **and** White (2006) confront the task of credit-default option pric**in**g us**in**g copula.3

tion of copula to model nonl**in**earities **and** asymmetries **in** CDS-equity dependence. The ma**in** appeal of thecopula framework is that it facilitates separate model**in**g of the marg**in**al distributions **and** the dependence**and** thus, a variety of dependence structures can be captured with more exibility **and** parsimony than **in**compet**in**g frameworks (e.g., multivariate GARCH). Patton (2006) **in**troduces conditional or dynamic copulasto portray time-vary**in**g dependence structures which represent an important improvement upon the **in**itialstatic copula models. The orig**in**al dynamic copula framework is extended by Christoersen et al. (2012) **in**order to accommodate asymmetries **and** trends **in** time-vary**in**g cross-market dependence. Far less attentionhas been paid to the possibility of regime-switch**in**g (RS) behaviour **in** dependence structures. To the bestof our knowledge, the only few exceptions are Garcia **and** Tsafack (2011), Chollete et al. (2009), Okimoto(2008) **and** Rodriguez (2007). One can argue though that exist**in**g RS copula models have the limitation ofassum**in**g constant state-specic dependence, i.e. a dist**in**ct static copula governs each regime, despite thefact that a given regime or state could l**in**ger on for years.We provide both methodological **and** empirical contributions to the literature. On the former, we proposeexible Markov-switch**in**g dynamic (autoregressive) copulas which capture asymmetry **in** the form of highor crisis dependence **and** low or normal dependence.Our models generalize exist**in**g Markov-switch**in**gstatic copula by allow**in**g for dist**in**ct mean reversion **in** dependence with**in** each regime. Empirically, weseek to provide a better underst**and****in**g of the dynamic evolution of dependence **and** tail dependence for theEuropean credit market, proxied by the iTraxx Europe CDS **in**dex, **and** two underly**in**g systematic factorsproxied by the Stoxx equity **in**dex return **and** VStoxx implied volatility **in**dex, respectively. We carry out acomprehensive **in**-sample statistical comparison of various copula models **and** draw overall **in**ferences on crossmarket(i.e., CDS **and** equity) dependence at the centre **and** tails of the bivariate distributions. Given thatCDS **in**dices have become a very important **in**strument for risk hedg**in**g **and** arbitrage trad**in**g **and** therefore,a key component of **in**stitutional **in**vestors' portfolios, we assess the relevance of the proposed Markovswitch**in**gdynamic copulas **in** the context of CDS-equity portfolios from a risk management perspective. 6More specically, the economic signicance of our proposition is assessed through a Value at Risk (VaR)simulation to set 1-day-ahead trad**in**g limits for CDS-equity portfolios.We document various sudden changes **in** the dependence structure of CDS **and** equity markets over theperiod from September 2005 to March 2011. The identied transitions to the high dependence regime largelyreect the onset of the automotive **in**dustry **and** energy crises **in** 2005, the credit crunch **in** 2007 **and** the most6 CDS **in**dices facilitate the transfer of marketwide or sectoral credit risk by **in**stitutional **in**vestors like hedge funds **and****in**surance companies, **and** by capital structure arbitrageurs who can now use derivatives (CDS **in**dex options **and** futures) formanag**in**g the risk related to their CDS **in**dex positions. Yu (2006) provides evidence of capital structure arbitrage opportunities**in** the CDS market for **in**dustrials, i.e. it is possible to make prots out of a trad**in**g strategy that exploits the CDS mispric**in**gerror. The latter is dened as the dierence between observed CDS market spreads **and** predicted CDS spread predictions froma Merton-type structural model with **in**puts the observed equity prices **and** **in**formation about the obligor's capital structure.4

ecent Greek **and** European sovereign debt crises **in** late 2009. The Markov-switch**in**g dynamic copula modelreveals that, **in** crisis periods, shocks to dependence of CDS spreads with market volatility have longer-last**in**geects than shocks to dependence of CDS spreads with market returns. Both **in** crisis **and** normal periods,changes **in** CDS premia are more strongly l**in**ked with the evolution of equity returns than with marketvolatility. The two dist**in**ct regimes of dependence are more clearly identied at sectoral than marketwidelevel. The proposed Markov-switch**in**g dynamic copula models are supported over simpler nested copulas notonly by conventional **in**-sample statistical criteria but also by out-of-sample VaR forecast accuracy measures.Us**in**g regulatory loss functions that take **in**to account both the frequency **and** magnitude of exceptions,the VaR simulation highlights the economic relevance of our copula models by show**in**g that they lead tomore cautious 1-day-ahead trad**in**g limits. A mismatch is documented between **in**-sample statistical t **and**economic value of predictability regard**in**g the choice of specic copula function; log-likelihood values **and**Akaike Information Criteria support the Student's t copula but lower average regulatory losses are associatedto the VaR forecasts from the asymmetrically-tailed Gumbel copula.Our nd**in**gs have important implications. The proposed copula framework can be useful towards theBasel III macroprudential goal of mak**in**g the bank**in**g sector more resilient to stress conditions throughenhanced risk coverage. One of the reforms put forward by the Basel Committee on Bank**in**g Supervision(2011) is precisely about strengthen**in**g capital requirements for credit exposures aris**in**g from banks creditderivatives such as CDS positions, **and** **in**troduc**in**g stressed-VaR capital requirements for the trad**in**g book.Our study suggests that copula models that explicitly parameterize sudden shifts **in** the dependence structurebetween credit exposures **and** the equity market facilitate more conservative downside-risk measures. Hence,our results po**in**t **in**to a clear direction for improvement of stress test**in**g platforms **and** reduction of systemicrisk. The copula framework proposed can be useful too for capital structure arbitrageurs that seek to exploittemporary deviations between model-based CDS spread predictions **and** observed CDS market spreads.The rema**in**der of this paper is organized as follows. Section 2 outl**in**es the methodology **and** Section3 describes the data. Section 4 provides an **in**-sample statistical comparison of copulas **and** **in**ferences onCDS-equity dependence, followed by an evaluation of the economic signicance of the Markov-switch**in**gdynamic copula formulation proposed. Section 5 concludes. Technical details are conned to an Appendix.2 COPULA METHODOLOGYThis section presents the basel**in**e theory of copulas **in** a bivariate sett**in**g. Sklar's theorem **and** the marg**in**aldistribution models employed are outl**in**ed rst, then we discuss the specic copula models **and** estimationissues. Bold font denotes vectors **and** matrices, lowercase **and** uppercase are used for probability density5

function (pdf ) **and** cumulative distribution function (cdf ), respectively.2.1 Sklar's Theorem **and** Marg**in**al ProcessesLet x 1 **and** x 2 denote the whitened **and** st**and**ardized returns of two nancial assets which are realizationsof the r**and**om variables X 1 **and** X 2 , respectively. A copula is dened as follows.Denition (Copula). A function C : [0, 1] 2 → [0, 1] is a copula if it satises (i) C(u 1 , u 2 ) = 0 for u 1 = 0 oru 2 = 0; (ii) ∑ 2i=1∑ 2j=1 (−1)i+j C (u 1 (i) , u 2 (j)) ≥ 0 for all (u 1 (i) , u 2 (j)) **in** [0, 1] 2 with u 1 (1) < u 1 (2) **and**u 2 (1) < u 2 (2) ; **and** (iii) C(u 1 , 1) = u 1 , C(1, u 2 ) = u 2 for all u 1 , u 2 **in** [0, 1].Sklar (1959)'s theorem expressed formally below provides the theoretical foundation for copulas.Theorem (Sklar). A jo**in**t cdf H (x 1 , x 2 ) = P (X 1 ≤ x 1 , X 2 ≤ x 2 ) of the r**and**om variables X = (X 1 , X 2 )with respective marg**in**al cdf F 1 (x 1 ) = P (X 1 ≤ x 1 ) **and** F 2 (x 2 ) = P (X 2 ≤ x 2 ) can be written asH (x 1 , x 2 ) = C (F 1 (x 1 ) , F 2 (x 2 )) (1)where C is a copula. Given H, if F 1 **and** F 2 are cont**in**uous, then there exists a unique C satisfy**in**g (1).Conversely, given C **and** the marg**in**s F 1 (x 1 ) , F 2 (x 2 ) then the result**in**g C (F 1 (x 1 ) , F 2 (x 2 )) is a jo**in**t cdf.Copula is **in** essence a dependence function that maps two univariate pdf (or marg**in**s) **in**to a jo**in**t pdf. 7For cont**in**uous variables, the density c correspond**in**g to the copula C is given byc (F 1 (x 1 ), F 2 (x 2 )) = ∂2 C (F 1 (x 1 ) , F 2 (x 2 )). (2)∂F 1 (x 1 ) ∂F 2 (x 2 )The jo**in**t pdf denoted f X can be obta**in**ed as a function of the copula density as2∏f X (x 1 , x 2 ) = c (F 1 (x 1 ) , F 2 (x 2 )) f n (x n ) , (3)where f n (x n ) is the marg**in** or univariate pdf correspond**in**g to F n (x n ) = u n , n ∈ {1, 2}, which is distributedas Uniform(0, 1). The log-likelihood of the jo**in**t distribution can be conveniently expressed asn=1T∑2∑ T∑L (θ, φ) = log c (u 1t , u 2t ; θ) + log f n (x n ; φ n )t=1= L C (θ, φ) +n=1 t=12∑L n (φ n ) , (4)n=17 A thorough exposition of copula theory can be found **in** Nelsen (2006) **and** nancial applications **in** Cherub**in**i et al. (2004).6

where L (θ, φ) is the copula log-likelihood with φ = (φ 1 , φ 2 ) ′ , **and** L n (φ n ), n = 1, 2, are the log-likelihoodsof the marg**in**s. The vector θ gathers the copula parameters that govern the dependence structure.Let the r**and**om process r t denote the daily returns of a nancial asset which can be modeled asr t = a 0 +p∑a i r t−i +i=1σ 2 t = c 0 +q∑b j ε t−j + ε t (5)j=1r∑c i σt−i 2 +i=1s∑d i ε 2 t−i (6)where the ltered (whitened **and** st**and**ardized) returns x t = ε t /σ t , t = 1, ..., T, are skewed Student's t,denoted x ti.i.d.∼i=1skT (0, 1; ν, ζ), with ν > 2 **and** ζ denot**in**g the degrees of freedom (dof) **and** asymmetryparameters, respectively. Equations (5)-(6) represent an ARMA-GARCH-skT model. In our empirical work,we x r = s = 1, **and** select the best p **and** q among 1, 2, . . . , 10 by m**in**imiz**in**g the Akaike InformationCriterion (AIC). The model parameters φ n are estimated by quasi-maximum likelihood (QML). The skTdensity is quite general as it nests the Student's t density (ζ = 0) **and** the Gaussian density (ζ = 0,ν → ∞).Previous studies advocate this parameterization for the marg**in**s as able to capture the wellknownautocorrelation, volatility cluster**in**g, skewness **and** heavy tails of nancial returns; e.g., Jondeau**and** Rock**in**ger (2006) **and** Kuester et al. (2006). Uniform(0, 1) marg**in**s denoted u n = F n (x n ), n = 1, 2,can be obta**in**ed from each ltered return series via the probability **in**tegral transform.Once the vectoru = (u 1 , u 2 ) ′ is formed, the copula parameter vector can be estimated by maximiz**in**g the correspond**in**gcopula log-likelihood function L C (θ, φ). Further discussion on estimation can be found below.Copulas can be broadly grouped as elliptical (e.g., Gaussian **and** Student's t) **and** Archimedean (e.g.,Gumbel, Clayton **and** symmetrized Joy-Clayton denoted SJC). Unlike the Gaussian copula which is solelyparameterized by the l**in**ear Pearson's correlation ρ, the Student's t copula can capture extreme return comovementsvia the so-called tail dependence parameter which is determ**in**ed by the dof parameter ν alongsideρ; the smaller ν, the more prom**in**ent the tail dependence or cluster**in**g of extreme returns. The key advantageof elliptical copulas is tractability s**in**ce they can be easily extended from bivariate to high-dimensionalsett**in**gs, but their ma**in** shortcom**in**g is that they impose symmetry. Archimedean copulas can additionallycapture asymmetric tail dependence. Gumbel (Clayton) copula describes upper (lower) tail dependence but,by rotation, the opposite tail can be modeled. 8 The SJC copula can model asymmetrically the dependencestructure at both tails **and** hence, enables tests of symmetry. See Appendix B for further details on thecopula functions. In early nancial applications, the above copulas were ma**in**ly deployed **in** static sett**in**gs.8 Rotation is by 180 ◦ degrees, namely, the copulas are formulated on m**in**us the r**and**om variables; see Cherub**in**i et al. (2004).7

2.2 Dynamic CopulasIn a dynamic context the copula parameters are estimated conditionally (i.e., allowed to time-vary) **and** soare the rank correlation **and** tail dependence measures implied from them; see Appendix B. Patton (2006)sets the foundations for time-vary**in**g copulas by prov**in**g Sklar's theorem for conditional distributions, **and**suggests to allow the generic copula dependence parameter θ evolves **in** ARMA fashion as followsθ t = Λ (ω + ϕθ t−1 + ψΓ t ) (7)which permits mean-reversion **in** dependence. The forc**in**g variable Γ t is dened as⎧∑ ⎪⎨ 1 mm j=1Γ t =F 1 −1 (u 1,t−j ) F2 −1 (u 2,t−j ) elliptical∑ ⎪⎩ 1 mm j=1 | u 1,t−j − u 2,t−j | Archimedeanwhere Fn−1 (u n,t ), n = 1, 2 is the **in**verse cdf of the marg**in**s **and** Λ (·) is the (modied) logistic or exponentialfunction. 9 We use m = 10 as **in** Patton (2006).Engle (2002)'s dynamic conditional correlation (DCC) model **in**spired the copula formulation 10Q t = (1 − ϕ − ψ) ¯Q + ϕQ t−1 + ψɛ t−1 · ɛ ′ t−1, ϕ + ψ < 1; ϕ, ψ ∈ (0, 1) (8)R t =−1 ˜Q t Q ˜Q−1 t twhere all matrices are 2 × 2 **in** our bivariate sett**in**g; ¯Q is the unconditional covariance of ɛt = (ɛ 1,t , ɛ 2,t ) ′estimated as ¯Q = T −1 ∑ Tt=1 ɛ tɛ ′ t with ɛ 1,t ≡ F −11 (u 1,t ) **and** ɛ 2,t ≡ F −12 (u 2,t ); Q t is the conditional covariancematrix; ˜Q t is a diagonal matrix with elements the square root of diag(Q t ); **and** R t is a correlation matrixwith o-diagonal element ρ t which (for elliptical copula) relates to Kendall's τ t as shown **in** Appendix B.Both ARMA **and** DCC formulations have as common aspects: i) characteriz**in**g the dependence dynamicsas `autoregressive' type, **and** ii) nest**in**g static copulas under the restriction ϕ = ψ = 0. But they havedierent merits. The DCC copula formulation can be easily extended to multivariate contexts which is ratherchalleng**in**g with the ARMA formulation. On the other h**and**, the DCC formulation is not straightforwardto apply to non-elliptical copulas; see Manner **and** Reznikova (2012), for further comparative discussion.9 In the context of elliptical copulas, the dynamic parameter is the conventional correlation measure, θ t = ρ t, **and** Λ (y) =(1 − e −y ) ( 1 + e −y) −1 is the modied logistic transformation to ensure ρt ∈ (−1, 1). In the Gumbel copula θ t = η t **and**Λ (y) = e y to ensure η t ∈ (0, ∞) as dened **in** Appendix B. Once these dynamic parameters are estimated, they can be mapped**in**to time-vary**in**g rank-correlation **and** tail dependence measures, ˆτ t **and** ˆλ t, us**in**g the formulae tabulated **in** Appendix B. In theSJC copula, the parameter modeled **in** (7) is directly the upper tail dependence, θ t = λ U t (or lower tail dependence θ t = λ L t ),**and** Λ (y) = ( 1 + e −y) −1 is the logistic transformation to ensure λ Ut , λ L t ∈ (0, 1).10 The popular DCC model put forward by Engle (2002) can be cast as a Gaussian DCC copula. Jondeau **and** Rock**in**ger(2006) model the dependence between **in**ternational equity **in**dices us**in**g elliptical DCC copulas.8

2.3 Regime-Switch**in**g CopulasWe propose exible Markov-switch**in**g (RS) copula models which accommodate dynamic dependence with**in**each regime **and** hence, can capture regime-specic mean reversion. This feature represents a dist**in**ctionfrom conventional RS copulas where a static copula function is assumed to govern each regime regardlessof how long the given state prevails. Extant studies typically associate the low dependence regime withGaussian copula which assumes zero tail dependence **and** the high dependence regime with non-Gaussiancopula that permits tail dependence. For **in**stance, **in** the RS copula formulated by Rodriguez (2007) **and**Okimoto (2008) the means, variances **and** correlations switch together **and** each regime is dictated by adist**in**ct static copula. In a similar ve**in**, Chollete et al. (2009) **and** Garcia **and** Tsafack (2011) deploy RSdependence models which are parameterized by static Gaussian copula **in** the normal regime regime **and** amixture of static elliptical/Archimedean copulas **in** another regime.In order to outl**in**e our regime-switch**in**g (RS) copula framework, let S t be a state variable that dictatesthe prevail**in**g regime. The jo**in**t distribution of X 1t **and** X 2t conditional on be**in**g **in** regime s is dened as(X 1t , X 2t | X 1,t−1 , X 2,t−1 ; S t = s) ∼ C Stt(u 1t , u 2t | u 1,t−1 , u 2,t−1 ; θ Stt)with s ∈ {H, L} where H denotes the high dependence regime **and** L the low dependence regime.Ther**and**om variable S t follows a Markov cha**in** of order one characterized by the transition probability matrixπ =⎞⎛1 − π LL π LL⎜⎝ π HH 1 − π HH ⎟⎠ (9)where π HH **and** π LL are the so-called stay**in**g probabilities, namely, π HH (π LL ) is the probability of be**in**g**in** the high (low) dependence regime at time t conditional on be**in**g **in** the same regime at t − 1.First, we propose a regime-switch**in**g ARMA copula where the dependence structure evolves as followsθ Stt()= Λ ω St + ϕθ St−1t−1 + ψΓ t(10)**in** each regime, with Γ t **and** Λ(·) dened as **in** Section 2.2. We call this novel formulation RS-ARMA todist**in**guish it from conventional RS formulations where a static copula governs each regime.Second, we propose a regime-switch**in**g DCC (RS-DCC) dependence model where the time-vary**in**g copula9

function that governs each regime is of DCC type, formalized as 11Q Stt= ( 1 − ϕ St − ψ St) ¯Q + ϕS tQ St−1t−1 + ψSt ɛ t−1 · ɛ ′ t−1, ϕ St + ψ St < 1; ϕ St , ψ St ∈ (0, 1) (11)( ) −1 ( ) −1˜QStt Qt ˜QS ttR Stt =with Q Sttthe auxiliary matrix driv**in**g the rank correlation dynamics.In our empirical analysis below, the RS-ARMA **and** RS-DCC models employ the same copula function(e.g., Gumbel) for all regimes but allow for time-variation (mean reversion) **in** dependence **and** tail dependencewith**in** each regime. Put dierently, the RS-ARMA **and** RS-DCC copulas are exible enough to captureabrupt **in**creases (decreases) **in** dependence as nancial markets enter crisis (tranquil) regimes without impos**in**gthe restriction of static with**in**-regime dependence. If there is only one regime (i.e., π HH = π LL = 1)the RS-ARMA **and** RS-DCC copula collapse, respectively, to dynamic ARMA **and** DCC copulas formalizedas Eq. (7) **and** Eq. (8), respectively. Conventional RS copulas collapse **in**stead to static copulas.2.4 Estimation of Copula ParametersWe employ canonical maximum likelihood (CML) estimation to obta**in** the copula parameters.CML issimilar **in** spirit to the **in**ference functions for marg**in**s (IFM) method where the parameters of the marg**in**aldistributions are separated from each other **and** from those of the copula, **and** then multi-step ML estimationis applied. In the rst step, the parameters of the marg**in**s are estimated via univariate ML. In the second step,we estimate by ML the parameters of the copula conditional on the step-one marg**in**s. 12 One ma**in** advantageof CML (versus IFM) is that, by exploit**in**g the observed empirical distributions, it avoids hav**in**g to specifya priori the marg**in**s. More specically, CML relies on the concept of empirical marg**in**al transformationwhich approximates an unknown parametric marg**in** with the (uniform) empirical distribution function û 1t =ˆF 1 (x 1t ) = 1 T∑ Tt=1 1 {X 1t≤x 1t}, **and** likewise for û 2t = ˆF 2 (x 2t ), where (x 1t , x 2t ) , t = 1, . . . , T , are the lteredreturns. The CML estimator of the copula parameters is dened asˆθ ≡ arg maxθT∑t=1(L C(c û 1t = ˆF 1 (x 1t ) , û 2t = ˆF) )2 (x 2t ) , θ11 The RS-DCC copula can be seen as a generalization of Billio **and** Capor**in** (2005)'s regime-switch**in**g DCC model.12 Two-step estimation yields asymptotically ecient **and** multivariate normal parameters √ ( )T ˆθIF M − θ 0N ( 0, V −1 (θ 0 ) ) where V (θ 0 ) = H −1 M ( H −1) ′ is the Godambe Information Matrix dened as the covariance matrix of a parametervector's estimation error when the distribution is non-Gaussian. Given the score function g(θ) = , ∂L C( ∂LX1),∂θ Cthen the expectation of the log-likelihood Hessian, H −1= EM = E ( g (θ) g (θ) ′) , can be evaluated numerically at the optimum.( ∂g(θ)∂θ, ∂L X 2∂θ 1 ∂θ 2), **and** the covariance of the log likelihood scores,→10

which amounts to a ML estimator conditional on the empirical marg**in**s. 13Estimation of the RS copula parameters requires **in**ferences on the probabilistic evolution of the statevariable S t . Probability estimates based on **in**formation up to time t are called ltered probabilities **and**those based on full-sample **in**formation are smoothed probabilities. Our estimation approach builds onHamilton (1989)'s lter**in**g algorithm **and** Kim (1994)'s smooth**in**g algorithm; see Appendix C.3 DATA DESCRIPTIONOur analysis is based on daily midpo**in**t clos**in**g CDS spread quotes at 5-year maturity from Bloomberg onthree **in**dices: Markit iTraxx Europe, Markit iTraxx Europe Subord**in**ated F**in**ancials (SubF**in**) **and** MarkitiTraxx Europe Autos (Auto). As tradeable systematic equity factors we employ, respectively, the Dow JonesStoxx Europe 600 **in**dex, which comprises the 600 largest market capitalized companies **in** Europe, the StoxxEurope 600 F**in**ancials **in**dex **and** the Stoxx Europe 600 Automobiles & Parts **in**dex. We focus on the costof **in**sur**in**g aga**in**st default on automotive companies' debt as this sector has been severely hit by the recentnancial crisis; see 2000s crisis timel**in**e **in** Appendix D. F**in**ally, we employ the option-implied Dow JonesEuro VStoxx 50 **in**dex as proxy for the unobservable asset volatility. Implied volatility is based on marketprices of options on the Dow Jones Stoxx Europe 50 **and** is a forward-look**in**g volatility **in**dicator as it reectstraders' expectations about future movements of the underly**in**g equity **in**dex. The time period is September9, 2005 to March 11, 2011 mak**in**g a total of T = 1, 380 days. 14Figure 1 plots for each **in**dex the daily prices (Panel A) **and** daily logarithmic returns (Panel B).[Insert Figure 1 around here]CDS **and** VStoxx **in**dices move **in** t**and**em while Stoxx **in**dices move **in** the opposite direction. September2007 marks the start of a steady downward trend **in** equity prices, atta**in****in**g the lowest level **in** 2009, coupledwith **in**creased volatility **and** a steady rise **in** default risk premiums. Table 1 provides summary statistics.13 S**in**ce CML makes no a priori assumption on the parametric form of the marg**in**s, it provides superior t to IFM when themarg**in**s are mispecied. Another estimation approach is Exact Maximum Likelihood (EML) which provides the parameters ofthe copula **and** the marg**in**s simultaneously by maximiz**in**g the full likelihood. This is computationally rather burdensome. Fora detailed comparison of EML, IFM **and** CML, see Cherub**in**i et al. (2004).14 Markit iTraxx Europe comprises the 125 equally-weighted most liquid names **in** the European market. iTraxx SubF**in** iscomposed of the 25 most liquid CDS names on subord**in**ated debt issued by companies, e.g. ABN, Aegon, Allianz, AVIVA, AXA,Barclays, Deutsche Bank, Swiss Re, RBS, Zurich, **and** iTraxx Auto comprises the 10 most liquid CDS names **in** the automotive**in**dustry, BMW, Volvo, Michel**in**, Cont**in**ental, DaimlerChrysler, GKN, Peugeot, Renault, Valeo, Volkswagen. Markit iTraxx**in**dices trade at 3, 5, 7 **and** 10-year maturities **and** are reviewed every 6 months **in** March **and** September to form a new series(for each maturity) that reects changes **in** liquidity as determ**in**ed by polls of the lead**in**g CDS dealers while the old seriescont**in**ues trad**in**g; we use Markit Series 4 which was **in**troduced **in** September 2005. Index trad**in**g for 5-year maturity is themost liquid. Markit calculates the ocial mid-day (11am GMT) **and** clos**in**g (4pm GMT) levels for iTraxx **in**dices on a dailybasis. Stoxx Europe 600 F**in**ancials conta**in**s 23 out of the 25 entities that conform the iTraxx SubF**in**, while Stoxx Europe 600Automobiles & Parts **in**cludes 9 out of the 10 entities **in** iTraxx Autos. Stoxx Europe 600 **in**cludes all of the 125 rms **in** theiTraxx Europe **in**dex. Stoxx **and** VStoxx clos**in**g prices are downloaded from www.stoxx.com.11

[Insert Table 1 around here]CDS SubF**in** has the highest mean return of 0.17%. Both Figure 1 **and** Table 1 reveal that CDS **in**dices arenotably more volatile than equity **in**dices, **and** CDS Auto is by **and** large the most volatile. The Jarque-Bera test conrms that daily returns are non-Gaussian. The Ljung-Box Q test **and** Engle's ARCH LM testprovide strong evidence, respectively, of serial dependence **and** heteroskedasticity **in** daily returns. Both thecorrelation parameter ρ **and** Kendall's rank correlation parameter τ suggest that CDS returns are negatively(positively) associated with equity returns (volatility), **in** l**in**e with Merton (1974)'s model predictions: i)growth **in** rm value reduces the probability of default, **and** ii) higher equity volatility implies a largerprobability that the value of assets drops below the level of liabilities, trigger**in**g default.Table 2 reports parameter estimates **and** diagnostics of the (marg**in**s) ARMA-GARCH-skT models.[Insert Table 2 around here]The dof parameter ν of the skT density tted to the st**and**ardized residuals reveals substantial leptokurtosis;CDS **in**dex returns show fatter tails than the underly**in**g Stoxx **and** VStoxx **in**dex returns. The asymmetrycoecient ζ of the skT residual distribution is signicant for VStoxx, Stoxx, Stoxx F**in** **and** CDS SubF**in** **in**dexreturns. The KolmogorovSmirnov (KS) test, with p-values reported **in** Panel B of Table 2, cannot refutethe null hypothesis that the residuals conform to a skT distribution. The st**and**ardized ARMA-GARCH-skTresiduals, x t , t = 1, ..., T , are mapped **in**to Uniform(0,1) observations, via the probability **in**tegral transform;the result**in**g residual series û t = ˆF (x t ), t = 1, ..., T , are then **in**puts for copula estimation.In order toestablish further the goodness-of-t of the marg**in**s, follow**in**g Diebold et al. (1998) **and** Patton (2006) weapply the Ljung-Box Q test to various moments (û t − ū) m , m ∈ {1, 2, 3, 4}. The results reported **in** Panel Bsuggest that there is no rema**in****in**g serial dependence.4 EMPIRICAL RESULTS4.1 In-Sample Fit of Static, Dynamic **and** Regime-Switch**in**g CopulasWe beg**in** this section with a prelim**in**ary discussion of the ability of Student's t, Gumbel **and** SJC copulas topredict **in**-sample the dependence between CDS returns **and** tradeable systematic risk factors. We employthe dierent formulations discussed **in** Section 2: static, dynamic (ARMA **and** DCC), conventional regimeswitch**in**g(RS) which nests a static copula **in** each regime, **and** nally the RS-ARMA **and** RS-DCC that wepropose. Table 3 shows the AIC **and** log-likelihood (LL) values of the compet**in**g models.[Insert Table 3 around here]12

Student's t copulas, which account for tail dependence **in** a symmetric way, atta**in** higher LL **and** lower AICthan the compet**in**g Gumbel **and** SJC copulas, irrespective of the formulation employed (static, dynamicor regime-switch**in**g). 15Albeit **in** purely static formulations, Student's t copula models have been shownto succeed **in** previous competitions such as that conducted by Breymann et al. (2003) to characterizedependence among FX spot returns. This can be partially attributed to the ability of the Student's t copulato t well the central part of the jo**in**t distribution which more heavily contributes to the log-likelihood.Hence, for simplicity of exposition a large part of the subsequent discussion **in** this section focuses on**in**ferences from the Student's t copula function. The dynamic formulation (ARMA or DCC) clearly providesbetter **in**-sample t than the static formulation, irrespective of the underly**in**g copula function employed.The regime-switch**in**g formulation further enhances the copula's ability to describe the dependence structureof CDS-equity markets. Moreover, allow**in**g the dependence structure to display short-memory with**in** eachregime (RS-ARMA or RS-DCC) leads to the lowest AIC **and** largest LL among the compet**in**g formulations.Table 4 reports parameter estimates of static, dynamic **and** regime-switch**in**g Student's t copulas.[Insert Table 4 around here]The correlation parameter ρ of the static copula suggests signicantly negative dependence for all CDS **and**Stoxx pairs, **and** signicantly positive dependence for all CDS **and** VStoxx pairs **in** l**in**e with Merton (1974)'stheory; a rm's likelihood of default is a decreas**in**g function of asset value proxied by the market value ofits equity, **and** an **in**creas**in**g function of asset volatility proxied by the volatility of its equity returns. 16 Thecorrelation parameter ρ **in** the static copula formulation **and** (ρ U , ρ L ) ′ **in** the conventional RS copula revealthat, generally, changes **in** CDS spreads are more strongly associated with changes **in** equity returns thanwith changes **in** volatility. Furthermore, the CDS **and** equity return association is more prom**in**ent **in** theSubF**in** sector than the Auto sector, **in** l**in**e with extant evidence that structural credit risk models are moresuccessful **in** expla**in****in**g non-**in**vestment grade spreads; see Eom et al. (2004) **and** Zhang et al. (2009).The signicance of parameters ϕ **and** ψ **in** the dynamic ARMA **and** DCC formulations bears out thatrank correlations are time-vary**in**g. The persistence measure ϕ + ψ **in**ferred from the DCC copula suggeststhat the rank correlation of CDS **and** equity markets is more persistent at sector level than marketwide; thelargest persistence at 0.982 (Stoxx) **and** 0.987 (VStoxx) corresponds to the subord**in**ated nancial sector.15 Table 3 presents the AIC of the best-tted Gumbel copula among the various Gumbel copulas considered (0 ◦ , 90 ◦ , 180 ◦ **and**270 ◦ rotated). The CDS-Stoxx dependence is best captured by the 90 ◦ Gumbel copula which focuses on lower tail dependence.The CDS-Vstoxx dependence is best described by the 0 ◦ Gumbel (non-rotated) copula which models upper tail dependence.These results mirror the evidence of asymmetry reported **in** Long**in** **and** Solnik (2001) suggest**in**g that tails that describe adversemovements are strongly dependent whereas the opposite tails are essentially **in**dependent.16 We tested the hypothesis of zero tail dependence by means of a likelihood ratio (LR) test for H 0 : 1/ν = 0 which isessentially a test for the restriction that the dof parameter ν is large enough so that the Student's t copula eectively becomesthe Gaussian. The hypothesis is rejected for all pairs except CDS Auto **and** VStoxx. The AIC **and** LL criteria also favour theStudent's t copula over the Gaussian copula. Details are available from the authors upon request.13

Turn**in**g attention to the regime-switch**in**g models, the probabilities π HH **and** π LL consistently suggestslightly longer duration of the low dependence regime dur**in**g our sample period. The statistical signicanceof two dependence regimes can be tested by means of a LR test for the null hypothesis H 0 :ω H = ω L whichstates that the RS-ARMA copula has one regime, then becom**in**g the ARMA copula. An analogous LRtest is formulated as H 0 :ρ H = ρ L with the conventional RS copula, then becom**in**g the static copula. Thetraditional asymptotic theory for these LR test statistics does not apply because of the nuisance parameterproblem (i.e., unidentied parameters under the null such as the residual variance **in** each regime). However,the conventional p-values are very small, all below 0.008, provid**in**g prima facie evidence of regime-switch**in**geects. Parameters ρ H **and** ρ L of the RS copula conrm that both **in** crisis **and** normal regimes the levelof dependence is stronger for CDS returns with equity returns than for CDS returns with equity volatility.Regard**in**g the degree of dependence persistence, we learn from the RS-DCC copula (parameters ϕ + ψ) that**in** crisis regimes shocks to dependence between CDS returns **and** equity volatility die more slowly thanshocks to dependence between CDS returns **and** equity returns.Figure 2 plots the smoothed probabilities of the high dependence regime **in** the RS-ARMA copula. 17[Insert Figure 2 around here]In both sectors, Auto **and** SubF**in**, the dependence between CDS **and** equity markets enters a high or crashregime by late 2007 reect**in**g the onset of the credit crunch, **and** l**in**gers on for about a year. 18Althoughthe global credit crisis orig**in**ated from the huge losses of subprime CDS **in**vestment **in** the nancial sector,the automotive **in**dustry was badly by hit various illiquidity shocks, a sharp fall **in** consumer condence **and**soar**in**g oil prices; Appendix D provides a snapshot of the 2000s crisis timel**in**e. After a short pause, bothsectors enter aga**in** the high dependence regime by late 2009 possibly reect**in**g the breakout of the Europeansovereign debt crisis. Fung et al. (2008) also document a signicant **in**crease **in** dependence between theUS equity **and** CDS markets dur**in**g the 2007 credit crunch on the basis of l**in**ear vector autoregressions **and**common Pearson correlation coecients. Our nd**in**gs from exible copula models extend their evidenceto the European context. 19Altogether for the Auto **and** SubF**in** sectors, four transitions **in**to the highdependence regime are identied **in** Figure 2. One **in** 2005 roughly co**in**cid**in**g with the downgrade of two bigplayers **in** the auto **in**dustry (Ford **and** GM), another **in** 2006 reect**in**g the deterioration of the US hous**in**gmarket, a third entry **in** 2007 reect**in**g the credit crunch, **and** a fourth entry **in** 2009 concurrent with theEuropean debt crisis. The unsuccessful regime identication **in** the marketwide CDS-equity copulas possibly17 Inferences from the RS-DCC Student's t copula are qualitatively similar **and** therefore not reported to save space.18 Accord**in**g to the Euroco**in** **in**dicator, the eurozone was **in** recession from March 2007 to February 2009. Accord**in**g to theNBER bus**in**ess cycle **in**dicator, the US was **in** recession from December 2007 to June 2009.19 The relatively long high dependence regime identied for CDS Auto **and** VStoxx is not surpris**in**g given that the Europeanauto sector has suered various setbacks **in** recent years, e.g. energy crisis **and** sharp fall **in** consumer dem**and**.14

eects the fact that diversication, i.e. pool**in**g of entities from diverse sectors with dierent tim**in**gs oftransition between high **and** low dependence states, reduces the overall signal-to-noise ratio.The Kendall's rank correlation τ **in**ferred from various copula formulations is plotted **in** Figure 3.[Insert Figure 3 around here]Several observations can be made. First, the degree of dependence clearly varies over time. Second, the RS**and** RS-ARMA copula suggest upward shifts **in** dependence between CDS **and** equity markets at economicallyplausible time po**in**ts. For **in**stance, the sudden downgrade by S&P's of two important car manufacturers,GM **and** Ford, from BBB to BB **in** May 2005 **and** to B **in** December 2005 led to a dramatic **in**crease **in** CDS**and** equity dependence for the Auto **and** marketwide **in**dices which the ARMA copulas tend to smooth out.Crude oil prices reached historically high levels of over $77 per barrel **in** July 2006 which pushed the CDS-Stoxx (Auto) dependence to a high regime; aga**in** this pattern is better captured by the RS **and** RS-ARMAcopulas. For the SubF**in** CDS-equity pairs, the most dramatic **in**crease **in** dependence roughly occurs **in** late2009 when a credit rat**in**g downgrade from A- to BBB+ is announced by S&P's for Greece.The tail dependence parameter λ **in**ferred from the dierent copula formulations is plotted **in** Figure 4.[Insert Figure 4 around here]We can see evidence of high **and** low tail-dependence regimes which reect the presence of two dierentCDS-equity bivariate distributions correspond**in**g, respectively, to crisis **and** normal episodes. The **in**tuitionbeh**in**d this nd**in**g is that CDS spreads react more vigorously to `extreme' bad news **in** crises than **in** normalperiods, namely, the degree of tail dependence exacerbates dur**in**g periods of market stress. While the taildependence estimates may seem small, they are broadly aligned with those **in** Garcia **and** Tsafack (2011)for European equity-bond pairs **and** with those **in** Jondeau **and** Rock**in**ger (2006) for cross-country equitymarket pairs. The high tail dependence regime is most apparent for CDS SubF**in** **and** Stoxx F**in** returnswhich may **in**dicate that nancials are particularly sensitive to extreme market events. Regardless of thelevel of tail dependence, the graphs endorse the RS-ARMA copula as more adequate for captur**in**g suddenshifts **in** tail dependence **and** conrm the biases aris**in**g from the use of non-regime-switch**in**g copula, whichtend to smooth out the degree of dependence over time. The latter eectively implies overestimation of theextent of dependence **in** normal periods **and** underestimation dur**in**g crisis periods. The upshot is that us**in**gan implausible model of asset dependence that does not permit sudden changes of regime or that constra**in**sthe with**in**-regime dependence to be constant could be costly from a risk management perspective. Thisquestion is addressed **in** the next section.15

4.2 Out-of-Sample Copula Forecasts for Risk ManagementThe economic value of the proposed regime-switch**in**g dynamic copulas is demonstrated via a Monte Carlosimulation to set 1-day-ahead Value at Risk (VaR) trad**in**g limits for portfolios of equity **and** CDS **in**struments.20 S**in**ce the 1996 Market Risk Amendment (MRA) to the Basel Accord, the VaR measure has playeda central role **in** regulatory capital assessments **and** rema**in**s one of the most common portfolio risk controltools **in** banks **and** **in**surance rms. The MRA stipulates that banks should **in**ternally compute VaR on adaily basis for backtest**in**g purposes although regulators (usually, the central bank) requires 10-day-aheadVaR to be reported for establish**in**g the m**in**imum capital requirement, possibly to mitigate the costs of toofrequent monitor**in**g. The reason why the prescribed horizon for backtest**in**g purposes is 1 day is that it**in**creases the number of observations; **in** 1-day VaR the number of observations is 252 per year whereas with2-week VaR it reduces to 26. By now the commercial bank**in**g **in**dustry has settled on the 1-day horizon.The 1-day-ahead VaR is an α-quantile prediction of the future portfolio prot **and** loss (P/L) distribution.It provides a measure of the maximum future losses over a time span [t, t+1], which can be formalized asP [ R t+1 V aR α t+1|I t]= α (12)where R t+1 denotes the portfolio return on day t + 1, **and** I t is the **in**formation set available on day t. Thenom**in**al coverage 0 < α < 1 is typically set at 0.01 or 0.05 for long trad**in**g positions (i.e., left tail) mean**in**gthat the risk manager seeks a high degree of statistical condence, 99% **and** 95%, respectively, that theportfolio loss on trad**in**g day t+1 will not exceed the VaR extracted from **in**formation up to day t.VaR can be estimated us**in**g various methods, rang**in**g from non-parametric (simulation), semi-parametric(CAViaR) to fully parametric (location-scale) **and** optimal comb**in**ations thereof; e.g., Kuester et al. (2006)**and** Fuertes **and** Olmo (2012). Large banks **and** nancial **in**stitutions require multivariate VaR models forcaptur**in**g appropriately the asset dependence structure **in** their trad**in**g portfolios. We adopt a Monte Carlocopula-based approach via the Cholesky decomposition of the correlation matrix for simulat**in**g the portfoliovalue changes, or P/L distribution, **and** estimat**in**g the VaR at any condence level; see Appendix E.Various backtest**in**g methods can be used for assess**in**g the accuracy of VaR forecasts. Let H t+1 denote ahit or exception, namely, a day when the ex post portfolio return falls below the out-of-sample VaR forecast20 In this paper, we chose as evaluation method for the out-of-sample copula forecasts the economic loss function implicit **in**VaR backtest**in**g together with explicit regulatory-driven loss functions. However, other purely statistical methods are feasibleto compare the accuracy of out-of-sample copula density forecasts; e.g., see Diks **and** van Dijk (2010).16

(i.e., larger loss than the maximum loss anticipated). Formally, the hit sequence (0s or 1s) is given by⎧⎪⎨ 1 if R t+1 < V aRt+1αH t+1 =⎪⎩ 0 otherwise,Kupiec (1995)'s unconditional coverage (UC) test is designed to assess whether the expected hit rate is equalto the nom**in**al coverage rate, namely, the hypotheses are H 0 : E(H t+1 ) = α versus H A : E(H t+1 ) ≠ α. S**in**cethe r**and**om variable H t+1 is b**in**omial, the expected probability of observ**in**g N exceptions over an evaluationperiod of T 1 trad**in**g days is (1 − α) T1−N α N under H 0 . The correspond**in**g likelihood ratio statistic isLR UC = −2 ln()(1 − α) T1−N α N asy(1 − ˆα) T1−N ∼ χ 2ˆα N 1 (13)where ˆα = N T 1is the observed hit rate. A weakness of this test is its unconditional nature, i.e. it only countshits but disregards how clustered they are. A well-specied risk management model should eciently exploitall the available **in**formation I t so that VaR exceptions are unpredictable, i.e.E(H t+1 |I t ) = E(H t+1 ) = α.Christoersen (1998)'s conditional coverage (CC) test overcomes this drawback.Its aim is to assesswhether the correct out-of-sample VaR specication property,E(H t+1 |I t ) = α is met. An implication ofthis property is that H t+1 should be iid b**in**omial with mean α. Hence, this is essentially a test of the jo**in**thypothesis of correct unconditional coverage **and** **in**dependence of the hits via the LR statistic()(1 − α) T1−N α NLR CC = LR UC + LR Ind = −2 ln(1 − ˆπ 01 ) n00 ˆπ n0101 (1 − ˆπ 11) n10 ˆπ n1111asy∼ χ 2 2 (14)where n 10 denotes the number of transitions or **in**stances when an exception occurred on day t **and** not onday t − 1 **and** ˆπ 10 =n10n 10+n 11is the estimated probability of hav**in**g an exception on day t conditional onnot hav**in**g an exception on day t − 1. Thus the test can detect if the probability of observ**in**g an exception,under the assumption of **in**dependence, is equal to α which amounts to test**in**g that π 01 = π 11 = α.However, the condition of correct VaR specication E(H t+1 |I t ) = α is stronger than what Christoersen(1998)'s CC test can detect. The out-of-sample hits H t+1 should be uncorrelated with any variable **in** I t ,mean**in**g that H t+1 should be a completely unpredictable process. Christoersen (1998)'s test can only detectautocorrelation of order one because it is built upon a rst-order Markov cha**in** assumption for the hits.Engle **and** Manganelli (2004)'s dynamic quantile (DQ) test for conditional coverage was developed toaddress this shortcom**in**g. This is essentially a Wald test for the overall signicance of a l**in**ear probabilitymodel H −α1 = Xβ + ε where H −α1 with H = (H t+1 ) the demeaned hit variable, 1 a vector of ones, X =( )Ht , ..., H t−k , V aRt+1α ′the regressor vector, **and** β = (β1 , ..., β k+2 ) ′ the correspond**in**g slope coecients. .17

The null hypothesis is H 0 : β = 0 **and** it can be tested us**in**g the Wald type test statisticDQ = ˆβ ′ X ′ X ˆβα(1 − α)asy∼ χ 2 k+2. (15)Below we employ k = 4 as **in** Kuester et al. (2006).One drawback of these common backtest**in**g approaches is that they cannot provide a rank**in**g of VaRmodels. Accord**in**g to the requirements of the Basel Committee on Bank**in**g Supervision, the magnitude aswell as the number of exceptions are a matter of regulatory concern. The quadratic loss function suggestedby Lopez (1998) takes **in**to account both aspects by add**in**g a penalty based on the size of the exceptions⎧⎪⎨L Q t+1 = 1 + ( )R t+1 − V aR α 2t+1 if R t+1 < V aRt+1α⎪ ⎩ 0 otherwise, (16)**and** thus, larger tail losses get a disproportionately heavier penalty. However, the above loss function canbe subject to the criticism that squared monetary returns lack nancial **in**tuition. Blanco **and** Ihle (1999)suggest focus**in**g on the relative size of exceptions (percentage) via the loss function⎧⎪⎨R t+1−V aR α t+1L % V aR× 100 if Rt+1t+1 =α t+1 < V aRt+1α⎪⎩ 0 otherwise. (17)The average losses L Q = 1 ∑ T1T 1 t=1 LQ t+1 **and** L% = 1 ∑ T1T 1 t=1 L% t+1 conta**in** additional **in**formation on how goodthe VaR model is for predict**in**g tail behaviour of the portfolio P/L distribution. Therefore, they can rankthose VaR models that pass the **in**itial backtest**in**g stage accord**in**g to their potential cost to the risk manager.The out-of-sample or holdout period for the VaR forecast evaluation is March 11, 2010 to March 11,2011 (last sample year, T 1 = 256 days) which conforms with the Basel committee recommendation of us**in**gan evaluation period of at least 250 bus**in**ess days.The exercise is conducted for the six (CDS-equity)portfolios studied **in** the paper at two nom**in**al coverage levels α = {0.05, 0.01}. The simulation copulabasedVaR approach outl**in**ed **in** Appendix E is deployed sequentially over a roll**in**g w**in**dow of xed-length(T 0 = 1,124 days). Thus, the rst estimation w**in**dow runs from September 10, 2005 to March 10, 2010**and** the correspond**in**g one-step-ahead downside risk forecast V aRt+1 α is for March 11, 2010. This roll**in**gw**in**dow approach to generate VaR forecasts oers some "shield" for the simple static copula model aga**in**stchang**in**g market conditions. However, as the results below suggest, the dependence forecasts obta**in**ed frommodels that explicitly capture regime-switch**in**g behaviour are able to adapt faster **and** more eectively tochang**in**g market conditions. Table 5 summarizes the performance of VaR forecasts stemm**in**g from various18

formulations of the Student's t copula function. S**in**ce the two dynamic formulations, ARMA **and** DCC, didnot produce markedly dierent results to save space we only report the results for the former.[Insert Table 5 around here]The hit rate is above the desired nom**in**al coverage α **in** all four copula formulations but less so with RS-ARMA. For all portfolios, Kupiec's UC test **and** Chistoersen's CC test are comfortably passed by the fourcopula-based VaR models considered. In contrast, we observe various rejections of the relatively tough Engle**and** Manganelli's DQ test, however, none of them is associated with the RS-ARMA copula formulation. Forthe Stoxx-CDS portfolio, the null hypothesis of correct VaR model specication is rejected by the DQ test**in** the static, conventional RS **and** dynamic ARMA copula formulations.Tak**in**g **in**to account now both the frequency **and** magnitude of exceptions, both the quadratic loss function(16) **and**, more clearly, the percentage loss function (17) conrm that there is economic value **in** model**in**gCDS-equity portfolio risk with regime-switch**in**g dynamic copula. For all portfolios, the largest reduction**in** average out-of-sample losses relative to the static copula is atta**in**ed by the RS-ARMA copula whichalso improves upon the conventional RS copula. The economic benet of exibly model**in**g portfolio riskus**in**g regime-switch**in**g dynamic copula is most clearly seen for the nancial CDS-equity portfolios with apercentage loss reduction of over 40% **and** 70% for VaRs at the 0.05 **and** 0.01 nom**in**al coverages, respectively.The risk management exercise has thus far relied on the Student's t copula. We now consider the Gumbelcopula which also captures tail dependence but **in** an asymmetric manner. Our subsequent VaR analysis isbased on the Gumbel copula that describes dependence on the adverse tail; i.e., large CDS returns togetherwith low equity returns or with high equity volatility. In the dynamic ARMA formulation, the average levelof tail dependence λ t **in**ferred from Gumbel copula is about 0.25 for the six CDS-equity pairs **and** is stronglysignicant **in** each case whereas the tail dependence **in**ferred from Student's t copula is very low (order ofmagnitude 10 −3 ). This notable contrast is likely to have an impact on the VaR forecasts, that is, Gumbelcopula forecasts can be expected to yield more conservative VaRs than Student's t copula forecasts. Table6 summarizes the VaR forecast**in**g performance for the Gumbel copula.[Insert Table 6 around here]Like-for-like comparisons reveal that the Gumbel copula leads to a more reliable risk management modelthan the Student's t copula. For all portfolios, the DQ test is unable to reject the null hypothesis of correctVaR model specication us**in**g the Gumbel copula, irrespective of whether it is formulated **in** a purelystatic, dynamic or regime-switch**in**g framework. In l**in**e with our expectations based on the tail dependenceestimates, out-of-sample VaR forecasts from Gumbel copula are more conservative (higher) than those from19

Student's t copula. In fact, while the actual coverage levels of the Student's t VaRs always exceeded thenom**in**al levels (Table 5) those for Gumbel-based VaRs tend to be slighly below. Relatedly, the average outof-samplelosses **in** excess of VaR lessen **in** the Gumbel-based framework. Hence, relax**in**g the assumption ofsymmetric tail dependence can improve risk management practice. In this sense, our analysis is consistentwith Okimoto (2008) who documents for **in**ternational (U.S. **and** U.K.) equity **in**dex portfolios that ignor**in**gasymmetric tail-dependence eects dur**in**g bear market conditions tends to underestimate VaR.F**in**ally, we can see that when the underly**in**g copula function of choice is Gumbel the RS-ARMA formulationstill rema**in**s superior to the static, dynamic **and** conventional RS models, accord**in**g to the averageportfolio losses. Overall, it seems fair to conclude that exibly model**in**g the CDS-equity portfolio risk byallow**in**g not only for sudden regime-changes but also mean-reversion **in** dependence with**in** each regime canentail economic benets from a risk management perspective.5 CONCLUSIONAccurately describ**in**g the bivariate distribution of CDS **and** equity **in**struments is of relevance to risk managersfor sett**in**g VaR trad**in**g limits, to traders for hedg**in**g the market risk of their CDS positions, **and** toregulators **and** economic policymakers **in** order to set m**in**imum capital levels. Sudden changes from a low ornormal dependence regime to a high or crash dependence regime can occur as a reection of importantsystemic shocks. We propose exible copula models that explicitly capture regime-switch**in**g behaviour **and**allow for mean-reversion **in** dependence with**in** each regime. By means of the proposed Markov-switch**in**gdynamic copulas **and** simpler (nested) versions, we provide a comprehensive study of the dependence structure**in** CDS-equity markets. The evaluation **and** comparison of copulas is conducted both **in**-sample us**in**gcommon goodness-of-t measures **and** out-of-sample us**in**g Value at Risk (VaR) forecast accuracy measures.The proposed models conrm the presence of signicant negative comovement between CDS returns **and**stock returns, **and** signicantly positive comovement between CDS returns **and** stock return volatility overthe period from September 2005 to March 2011. They also **in**dicate that asset dependence is time-vary**in**g **and**nonl**in**ear. Signicant regime-switch**in**g dependence is revealed not only **in** the central part of the bivariatedistributions but also **in** the tails; namely, low **and** high dependence periods alternate over time. The latterbroadly co**in**cide with the automotive crisis, the subprime mortgage crisis **and** the Greek **and** Europeansovereign debt crises. The nd**in**gs suggest that dur**in**g periods of stress systematic factors play a strongerrole as drivers of default **and** volatility shocks have longer last**in**g eects than return shocks. Inadequatelymodel**in**g the bivariate distribution by ignor**in**g the time-variation **in** dependence with**in** each regime oraltogether neglect**in**g regime-switch**in**g eects implies too smooth rank correlation **and** tail dependence20

measures that under(over)-estimate the comovement of CDS **and** equity markets **in** crisis (normal) periods.Our assessment of the compet**in**g copula models **in** a risk-management exercise leads to two ma**in** conclusions.First, neglect**in**g regime-switch**in**g eects **in** dependence can be costly because it tends to underestimatethe maximum potential losses of CDS-equity porfolios. Second, relax**in**g the assumption that the dependencestructure rema**in**s constant with**in** each regime can be benecial for improv**in**g the accuracy of out-of-sampleVaR forecasts **and** produc**in**g smaller average regulatory losses. Lastly, the study provides yet another exampleof a disconnect between **in**-sample t **and** out-of-sample predictability; namely, the Student's t copulafunction is strongly supported by common **in**-sample statistical criteria but the Gumbel copula which focuseson the adverse tail leads to more conservative 1-day-ahead VaR trad**in**g limits.These nd**in**gs are relevant to **in**stitutional **in**vestors us**in**g CDS contracts to hedge their equity hold**in**gs**and** for capital structure arbitrageurs, particularly, those predom**in**antly exposed to nancial **and** auto sectors.The exible copula models proposed could prove useful for banks **in** order to address recommendations fromBasel III to carry out more rigorous stress test**in**g of the trad**in**g book. Our study oers important **in**sights**in**to the regime-switch**in**g dependence dynamics of CDS spreads **and** tradeable systematic risk factors. In thiscontext, extend**in**g the exible regime-switch**in**g copula models here proposed to ascribe a role to exogenousstructural variables as drivers of the regime-transition may be an **in**terest**in**g avenue of further research.AcknowledgementsWe thank Barbara Casu, Jerry Coakley, Carlos da Costa, Alex**and**ros Gabrielsen, Aurob**in**do Ghosh, MariosKalotychos, Aneel Keswani, Robert Sollis **and** participants at the 2011 Conference on Computational **and**F**in**ancial Econometrics at Birkbeck College, London, the 2012 F**in**ancial Management Association EuropeanConference, Istanbul, for helpful comments **and** suggestions.21

ReferencesAlex**and**er, C., Kaeck, A., 2008. Regime Dependent Determ**in**ants of Credit Default Swap Spreads. Journalof Bank**in**g & F**in**ance 31 (6), 10081021.Basel Committee on Bank**in**g Supervision, 2011. Basel III: A Global Regulatory Framework for More ResilientBanks **and** Bank**in**g Systems. Bank for International Settlements.Billio, M., Capor**in**, M., 2005. Multivariate Markov Switch**in**g Dynamic Conditional Correlation GARCHRepresentations for Contagion **Analysis**. Statistical Methods **and** Applications 14 (2), 145161.Blanco, C., Ihle, G., 1999. How Good is Your VaR? Us**in**g Backtest**in**g to Assess System Performance.F**in**ancial Eng**in**eer**in**g News, August, 12.Blanco, R., Brennan, S., Marsh, I. W., 2005. An Empirical **Analysis** of the Dynamic Relation betweenInvestment-grade Bonds **and** Credit Default Swaps. Journal of F**in**ance 60 (5), 22552281.Breymann, W., Dias, A., Embrechts, P., 2003. Dependence Structures for Multivariate High-Frequency Data**in** F**in**ance. **Quantitative** F**in**ance 3 (1), 114.Bystrom, H., 2008. Credit Default Swaps **and** Equity Prices: The iTraxx CDS Index Market. In: Wagner,N. (Ed.), Credit Risk - Models, Derivatives, **and** Management. Vol. 6. Chapman & Hall.Cao, C., Yu, F., Zhong, Z., 2010. The Information Content of Option-implied Volatility for Credit DefaultSwap Valuation. Journal of F**in**ancial Markets 13 (3), 321343.Cherub**in**i, U., Luciano, E., Vecchiato, W., 2004. Copula Methods **in** F**in**ance. John Wiley **and** Sons.Chollete, L., He**in**en, A., Valdesogo, A., 2009. Model**in**g International F**in**ancial Returns with a MultivariateRegime-switch**in**g Copula. Journal of F**in**ancial Econometrics 7 (4), 437480.Christoersen, P. F., 1998. Evaluat**in**g Interval Forecasts. International Economic Review 39 (4), 841862.Christoersen, P. F., Errunza, V., Jacobs, K., Langlois, H., 2012. Is the potential for **in**ternational diversicationdisappear**in**g? a dynamic copula approach. Review of F**in**ancial Studies 25, 37113751.Crook, J., F. M., 2011. Check**in**g for Asymmetric Default Dependence **in** a Credit Card Portfolio: A CopulaApproach. Journal of Empirical F**in**ance 18, 728742.Das, S., Geng, G., 2006. The Credit Market H**and**book. Advanced Model**in**g Issues. John Wiley & Sons, Inc.Diebold, F. X., Gunther, T. A., Tay, A. S., 1998. Evaluat**in**g Density Forecasts with Applications to F**in**ancialRisk Management. International Economic Review 39 (4), 863883.Diks, C., V. P., van Dijk, D., 2010. Out-of-sample Comparison of Copula Specications **in** MultivariateDensity Forecasts. Journal of Economic Dynamics **and** Control 34, 15961609.Engle, R., 2002. Dynamic Conditional Correlation: A Simple Class of Multivariate Generalized AutoregressiveConditional Heteroskedasticity Models. Journal of Bus**in**ess **and** Economic Statistics 20 (3), 339350.22

Engle, R. F., Manganelli, S., 2004. CAViaR: Conditional Autoregressive Value at Risk by Regression Quantiles.Journal of Bus**in**ess **and** Economic Statistics 22 (4), 367381.Eom, Y., Helwege, J., Huang, J., 2004. Structural Models of Corporate Bond Pric**in**g: An Empirical **Analysis**.Review of F**in**ancial Studies 17 (2), 499544.Ericsson, J., Jacobs, K., Oviedo-Helfenberger, R., 2004. The Determ**in**ants of CDS Premia. Journal of FixedIncome, 10598596.Fuertes, A., Olmo, J., 2012. Optimally Harness**in**g Inter-Day **and** Intra-Day Information for Daily Value-at-Risk Prediction. International Journal of Forecast**in**g 29 (1), 2842.Fung, H.-G., Sierra, G. E., Yau, J., Zhang, G., 2008. Are the US Stock Market **and** Credit Default SwapMarket Related? Evidence from the CDX Indices. Journal of Alternative Investments 11 (1), 4361.Garcia, R., Tsafack, G., 2011. Dependence Structure **and** Extreme Comovement **in** International Equity **and**Bond Markets. Journal of Bank**in**g & F**in**ance 35 (8), 19541970.Giammar**in**o, F., Barrieu, P., 2009. A Semiparametric Model for the Systematic Factors of Portfolio CreditRisk Premia. Journal of Empirical F**in**ance 16 (4), 655670.Hamilton, J. D., 1989. A New Approach to the Economic **Analysis** of Nonstationary Time Series **and** theBus**in**ess Cycle. Econometrica 57 (2), 357384.Hull, J., White, A., 2006. Valu**in**g credit derivatives us**in**g an implied copula approach. Journal of Derivatives14 (2), 828.Hull, J. C., Nelken, I., White, A. D., 2004. Merton's Model, Credit Risk **and** Volatility Skews. Journal ofCredit Risk 1 (1), 127.Jondeau, E., Rock**in**ger, M., 2006. The Copula-GARCH Model of Conditional Dependencies: An InternationalStock Market Application. Journal of International Money **and** F**in**ance 25 (5), 827853.Jones, E., Mason, S., Rosenfeld, E., 1984. Cont**in**gent Claims **Analysis** of Corporate Capital Structures.Journal of F**in**ance 39 (3), 611625.Kim, C.-J., 1994. Dynamic L**in**ear Models with Markov-switch**in**g. Journal of Econometrics 60 (1-2), 122.Kuester, K., Mittnik, S., Paolella, M., 2006. Value-at-Risk Prediction: A Comparison of Alternative Strategies.Journal of F**in**ancial Econometrics 4 (1), 5389.Kupiec, P. H., 1995. Techniques for Verify**in**g the Accuracy of Risk Management Models. Journal of Derivatives3 (2), 7384.Long**in**, F., Solnik, B., 2001. Extreme Correlation of International Equity Markets. Journal of F**in**ance 56 (2),649676.Longsta, F. A., Mithal, S., Neis, E., 2005. Corporate Yield Spreads: Default Risk or Liquidity?Evidence from the Credit Default Swap Market. Journal of F**in**ance 60 (5), 22132253.New23

Lopez, J. A., 1998. Methods for Evaluat**in**g Value-at-Risk Estimates. Economic Policy Review - FederalReserve Bank of New York 4 (3).Madan, D., Unal, H., 2000. A Two-factor Hazard Rate Model for Pric**in**g Risky Debt **and** the Term Structureof Credit Spreads. Journal of F**in**ancial **and** **Quantitative** **Analysis** 35 (1), 4365.Manner, H., Reznikova, O., 2012. A Survey on Time-vary**in**g Copulas: Specication, Simulations **and** Application.Econometric Reviews 31 (6), 654687.Markit, 2010. Markit Credit Indices - A Primer. Tech. rep., Markit Group Limited.McNeil, A. J., Frey, R., Embrechts, P., 2005. **Quantitative** Risk Management: Concepts, Techniques, Tools.Pr**in**ceton.Merton, R. C., 1974. On the Pric**in**g of Corporate Debt: The Risk Structure of Interest Rates. Journal ofF**in**ance 2 (2), 449470.Nelsen, R. B., 2006. An Introduction to Copulas. Spr**in**ger Verlag.Norden, L., Weber, M., 2009. The Co-movement of Credit Default Swap, Bond **and** Stock Markets: AnEmpirical **Analysis**. European F**in**ancial Management 15 (3), 529562.Okimoto, T., 2008. New evidence of asymmetric dependence structures **in** **in**ternational equity markets.Journal of F**in**ancial **and** **Quantitative** **Analysis** 43 (3), 787816.Patton, A. J., 2006. Modell**in**g Asymmetric Exchange Rate Dependence. International Economic Review47 (2), 527556.Rodriguez, J. C., 2007. Measur**in**g F**in**ancial Contagion: A Copula Approach. Journal of Empirical F**in**ance14 (3), 401423.Sklar, A., 1959. Fonctions de répartition á n dimensions et leurs marges. Publications de l'Institut de Statistiquede L'Université de Paris 8, 229231.Yu, F., 2006. How Protable is Capital Structure Arbitrage? F**in**ancial Analysts Journal 62 (5), 4762.Zhang, B. Y., Zhou, H., Zhu, H., 2009. Expla**in****in**g Credit Default Swap Spreads with Equity Volatility **and**Jump Risks of Individual Firms. Review of F**in**ancial Studies 22 (12), 50995131.24

APPENDIXADependence MeasuresPearson l**in**ear correlationLet X 1 **and** X 2 denote two cont**in**uous r**and**om variables represent**in**g thereturns of asset 1 **and** asset 2, respectively. Pearson correlation ρ ∈ [−1, 1] is dened byρ =Cov (X 1 , X 2 )√V ar (X1 ) √ V ar (X 2 )where Cov (X 1 , X 2 ) = E (X 1 , X 2 ) − E (X 1 ) E (X 2 ) is the covariance, **and** V ar (X 1 ) = Cov (X 1 , X 1 ) denotesthe variance. ρ is an appropriate measure of dependence only if the true jo**in**t distribution of the asset returnsis elliptical, **and** it is scale **in**variant only under strictly **in**creas**in**g l**in**ear transformations.Kendall's τ rank correlationKendall's τ ∈ [−1, 1] is a rank correlation measure based on the concordancenotion: two observation pairs (x 1t , x 2t ) **and** (x 1s , x 2s ) are concordant if (x 1t − x 1s ) (x 2t − x 2s ) > 0 **and**discordant if (x 1t − x 1s ) (x 2t − x 2s ) < 0, for t, s = 1, ..., T. The rank correlation of (X 1 , X 2 ) is a measure ofthe probability of concordance m**in**us the probability of discordance as given by#concordant pairs - #discordant pairsτ =12T (T − 1)where T is the number of observations for x 1 **and** x 2 , **and** the denom**in**ator **in**dicates the total number ofpairs (x 1t , x 2t ),(x 1s , x 2s ). Kendall's τ is scale **in**variant under strictly **in**creas**in**g (non)l**in**ear transformations.Tail dependenceTail dependence λ ∈ [0, 1] measures the concordance **in** the tails, or extreme values oftwo r**and**om variables, X 1 **and** X 2 , **and** is dened **in** terms of limit**in**g conditional probabilities of α−quantileexceedances as α → 1. Formally, for two positively correlated variables, upper tail dependence denoted λ Uis the probability that X 2 is above its α-quantile (X 2,α ) given that X 1 is above its α-quantile (X 1,α ), i.e.λ U ≡ P(X 2 > X 2,α |X 1 > X 1,α ), while lower tail dependence denoted λ L is the probability that X 2 is smallerthan its (1 − α)-quantile given that X 1 is smaller than its (1 − α)-quantile, i.e. λ L ≡ P(X 2 X 2,(1−α) |X 1 X 1,(1−α) ). For two negatively correlated variables, upper tail dependence is the probability that X 2 is aboveits α-quantile given that X 1 is below its (1 − α)-quantile, i.e. λ U ≡ P(X 2 > X 2,α | X 1 X 1,(1−α) ), whilelower tail dependence is the probability that X 2 is below its (1 − α)-quantile given that X 1 is above itsα-quantile, i.e. λ L ≡ P(X 2 X 2,(1−α) | X 1 > X 1,α ). Like the rank correlation, tail dependence depends only25

on the copula of X 1 **and** X 2 thus the roles of X 1 **and** X 2 are **in**terchangeable. 21BStatic CopulasGaussian copula (elliptical)The bivariate Gaussian copula pdf can be written as followsc (u 1 , u 2 ; θ = ρ) =1√(− exp 1 (1 − ρ2 2 Ψ′ R −1 − I ) )Ψwhere R is the correlation matrix with o-diagonal element the Pearson correlation ρ, **and** Ψ = ( Φ −1 (u 1 ) , Φ −1 (u 2 ) ) ′with Φ −1 the **in**verse of the univariate st**and**ardized Gaussian cdf. Tail dependence is not captured.Student's t copula (elliptical)The bivariate Student's t copula pdf can be written as followsc (u 1 , u 2 ; θ = (ν, ρ)) =1√1 − ρ2Γ ( ) (ν+22 Γ ν) (2 Π2i=1(t−1ν i(u i ) ) ) ν+12 21 + 1 ν( (Γν+1)) 2 (2 1 +1ν Ψ′ R −1 Ψ ) ν+22where R is the correlation matrix with ρ as o-diagonal element, Ψ = ( t −1ν 1(u 1 ) , t −1ν 2(u 2 ) ) ′**and** t−1ν (u) is the**in**verse cdf of the Student's t with ν > 2 degrees of freedom. Student's t copula captures tail dependencebut imposes the restriction of upper **and** lower tail symmetry. It converges to Gaussian copula as ν → ∞.Gumbel copula (Archimedean)The bivariate Gumbel copula pdf can be written as follows:⎧ [c (u 1 , u 2 ; θ = η) = (log u 1 log u 2 ) eη ⎨ 2∑] 1⎫e η +1 ⎬expu 1 u 2 ⎩ − (− log u n ) eη +1⎭n=1⎡ [ 2∑] 1 [ ]η+1 −2+ 1 ⎤η+12∑⎣e η + (− log u n ) eη +1(− log u n ) eη +1⎦n=1n=1where η ∈ [1, +∞). For positively correlated variables (e.g., CDS **and** VStoxx), the Gumbel copula cancapture upper tail dependence but 180 ◦ -anticlockwise-rotated (or survival) Gumbel can capture lower taildependence. For negatively correlated variables (e.g., CDS **and** Stoxx), the 90 ◦ (270 ◦ )-anticlockwise-rotatedGumbel copula captures upper (lower) tail dependence.SJC copula (Archimedean)The bivariate SJC copula pdf can be written as follows:c ( u 1 , u 2 ; θ = (λ U , λ L ) ) = 1 2[∂ 2 (C JC u1 , u 2 |λ U , λ L) +∂2 (C JC 1 − u1 , 1 − u 2 |λ U , λ L)]∂u 1 ∂u 2 ∂u 1 ∂u 221 For further discussion on dependence measures see Cherub**in**i et al. (2004), Ch.3.1 or McNeil et al. (2005) Ch. 5.2.26

where λ U (λ L ) is the upper (lower) tail dependence. C JC is the cdf of the Joe-Clayton copula.The follow**in**g table summarizes how the rank correlation **and** tail dependence are obta**in**ed:Copula Kendall's τ Lower tail dependence (λ L ) Upper tail dependence (λ U )2Gaussianπ arcs**in** ρ [0 0− √ ν + 1 √ ]√1−ρ1+ρStudent's t2π arcs**in** ρ 2t ν+1[2t ν+1 − √ ν + 1 √ ]√1−ρ1+ρGumbel 1 − 1 η- 2 − 2 1 ηSJC - λ L λ UElliptical copulas permit both positive dependence (i.e., τ > 0) **and** negative dependence (i.e., τ

The log-likelihood function of the RS-copula isL (u 1 , u 2 | θ) = L C(θ, ˆφ)+=2∑L Xi (φ i )i=1(T∑ 2∑)log c Stt (u 1t , u 2t | S t , I t−1 ) Pr [S t | I t−1 ]t=1 S t=1The parameters can be estimated either by the IFM or CML methods.+T∑t=1 i=12∑log (f Xi (x i,t ; φ i,t )) .Once the parameters are estimated **and** all ltered probabilities P [S t = s | I t ] for s ∈ {0, 1} **and** t =1, 2, . . . , T , are obta**in**ed, we follow Kim's algorithm to obta**in** the smoothed probabilitiesP [S t = s | I T ] =1∑k=0P [S t+1 = s | S t = k, I t ] P [S t = s | I t ] P [S t+1 = k | I T ]∑ 1j=0 P [S , t = T − 1, T − 2, ..., 1.t+1 = s | S t = k, I t ] P [S t = k | I t ]start**in**g from P [S T = s | I T ] **and** iterat**in**g backwards for t = T − 1, T − 2, . . . , 1.DTimel**in**e of late 2000s CrisisCredit Crunch• July 10, 2007: S&P announces it may cut rat**in**gs on $12bn of subprime debt.• August 9, 2007: ECB pumps 95bn euros **in**to the bank**in**g system to improve liquidity.• October 1, 2007: UBS announces $3.4bn losses from sub-prime related **in**vestments.• October 30, 2007: Merrill Lynch unveils $7.9bn exposure to bad debt.• January 19, 2008: World Bank predicts slowdown of global economic growth **in** 2008.• January 21, 2008: Global stock markets suer their largest fall s**in**ce September 2001.• February 17, 2008: UK government nationalizes Northern Rock.• March 17, 2008: Wall Street's 5th largest bank, Bear Stearns, is acquired by JP Morgan Chase.• April 8, 2008: IMF warns that potential losses from the credit crunch could reach $1tn.• September 7, 2008: Large US mortgage lenders Fannie Mae **and** Freddie Mac are nationalized.• September 15, 2008: Lehman Brothers les for Chapter 11 bankruptcy protection.• September 16, 2008: US Fed announces $85bn rescue package for AIG.• September 17, 2008: Lloyds TSB announces takeover of largest British mortgage lender HBOS.• October 13, 2008: UK government announces ¿37bn **in**jection to RBS, Lloyds TSB **and** HBOS.• November 6, 2008: Bank of Engl**and** cuts base **in**terest rate to lowest level s**in**ce 1955.28

Energy Crisis• March 5, 2005: Crude oil prices rose to new highs above $50 per barrel (bbl).• September 2005: US hurricane Katr**in**a pushes gasol**in**e prices to a record high.• August 11, 2005: Crude oil prices broke the psychological barrier of $60 bbl.• July 13, 2006: Israeli attacks on Lebanon pushed crude oil prices to historical highs above $78.40bbl.• October 19, 2007: US light crude rose to $90.02 bbl.• March 5, 2008: OPEC accused the US of economic "mismanagement" responsible for oil prices.• March 12, 2008: Oil prices surged above $110 bbl.Automotive Industry Crisis• May 5, 2005: S&P cut the debt rat**in**gs of GM **and** Ford to junk status.• February 12, 2008: GM announced its operat**in**g loss was $2bn.• October 7, 2008: SEAT cut production at its Martorell plant by 5%.• November 20, 2008: PSA Peugeot Citroen predicts sales volumes would fall by at least 10% **in** 2009,follow**in**g a 17% drop **in** the current quarter.• November 23, 2008: Jaguar L**and** Rover was seek**in**g a $1.5bn loan from the government.• December 11, 2008: The Swedish government **in**jected $3.5bn to rescue its troubled auto markers,Volvo **and** Saab.• December 19, 2008: US government said it would use up to $17.4bn to help the big three UScarmakers, General Motors, Ford **and** Chrysler.• December 20, 2008: GM **and** Chrysler receive CA$4bn government loans from Canada **and** theprov**in**ce of Ontario.• January 8, 2009: Nissan UK announced it was to shed 1200 jobs from its factories **in** North EastEngl**and**.• January 22, 2009: Fiat announces a 19% drop **in** revenues for 2008 Q3.• February 11, 2009: PSA Peugeot Citroen announced it would cut 11,000 jobs world wide.• February 12, 2009: Renault announces a 78% drop **in** prots for 2008.• April 22, 2009: GM admits it will default on a $1bn bond debt payment due **in** June.• April 30, 2009: Chrysler les for Chapter 11 bankruptcy protection.• June 1, 2009: GM les for Chapter 11 bankruptcy protection.29

European sovereign debt crisis• October 10, 2008: Fitch downgrades Icel**and** Sovereign debt from A+ to BBB-.• December 8, 2009: Fitch rat**in**gs agency downgraded Greece's credit rat**in**g from A- to BBB+.• April 23, 2010: Greek PM calls for eurozone-IMF rescue package. FTSE falls more than 600p.• May 18, 2010: Greece gets rst bailout of $18bn from EFSF, IMF **and** bilateral loans• November 29, 2010: Irel**and** receives $113bn bailout from EU, IMF **and** EFSF• January 5, 2010: S&P downgrades Icel**and**'s rat**in**g to junk grade.Sources: news.bbc.co.uk; www.reuters.com; www.bloomberg.com.ECopula VaR SimulationThe portfolio prot **and** loss (P/L) distribution is simulated us**in**g copula as follows:1. Obta**in** the 2 × 2 rank correlation matrix forecast ∑ us**in**g data up to day t. In static copula, the odiagonalentry is ˆτ t+1 = ˆτ t = ˆτ. In dynamic copula, ˆτ t+1 is the 1-day-ahead projection of the ARMAEq. (7) or DCC Eq. (8). In regime-switch**in**g copulas, the forecast h**in**ges on the ltered probabilities**and** estimated migration matrix π **in** Eq.(9) as ˆτ t+1 = (P [S t = H | I t ] , P [S t = L | I t ]) π (ˆτ H t+1, ˆτ L t+1) ′.2. Simulate two **in**dependent st**and**ard normal r**and**om variates z = (z 1 , z 2 ) ′ .3. Simulate a r**and**om variate s from a χ2ˆν distribution, **in**dependent of z, where ˆν is the degree-of-freedomestimated us**in**g data up to day t.4. Form the vectors b = Az **and** c = √ˆν √ sb where c=(c 1 , c 2 ) ′ **and** A is the Cholesky decomposition of ∑ .5. Determ**in**e the components (u 1 , u 2 ) = (tˆν (c 1 ) , tˆν (c 2 )) of copula where tˆν is the cdf of Student's tdistribution with degrees-of-freedom parameter ˆν.6. Obta**in** the st**and**ardized asset log-returns: Q = (q 1 , q 2 ) =( )ˆF−1 −11 (u 1 ) , ˆF 2 (u 2 ) , where**in**verse empirical cdf of st**and**ardized residuals,x n , n = 1, 2, of the **in**-sample data.7. Relocate **and** rescale the returns as (r 1,t+1 , r 2,t+1 ) =ˆF−1nis the()ˆµ 1,t+1 + q 1√ˆσ1,t+1, , ˆµ 2,t+1 + q j 2√ˆσ2,t+1 withˆµ t+1 **and** ˆσ t+1 denot**in**g ARMA-GARCH-skT forecasts of conditional mean **and** variance made at t.8. Obta**in** the 1-day-ahead P/L forecast of the equally-weighted portfolio as r t+1 = 0.5r 1,t+1 + 0.5r 2,t+1 .Repeat J = 100, 000 times the above steps to obta**in** the empirical or simulated 1-day-ahead P/L distribution{r t+1,j } J j=1 from which any α-quantiles (VaR) can be measured.30

Table 1: Descriptive Statistics of Daily ReturnsVstoxx Stoxx Stoxx Auto Stoxx F**in** CDS Europe CDS Auto CDS SubF**in**Mean 0.038 -0.005 0.024 -0.036 0.068 0.043 0.166Median -0.577 0.067 0.057 0.019 -0.193 -0.021 -0.057Maximum 32.767 9.410 40.817 14.666 41.745 199.212 47.500M**in**imum -24.919 -7.930 -35.427 -10.179 -40.297 -177.407 -43.987Std. Dev. 5.878 1.419 2.699 2.042 7.074 13.327 8.266Skewness 0.899 -0.053 3.192 0.303 0.394 2.610 0.343Kurtosis 6.432 9.741 97.591 10.081 11.520 106.320 10.418Number of obs. 1381 1381 1381 1381 1381 1381 1381Jarque-Bera test 0.001 0.001 0.001 0.001 0.001 0.001 0.001Ljung-Box(10) test 0.035 0.000 0.000 0.008 0.000 0.000 0.000ARCH(10) test 0.000 0.000 0.000 0.000 0.000 0.000 0.000Pearson l**in**ear correlation ρ :Vstoxx - - - - 0.362 0.148 0.302Stoxx - - - - -0.366 -0.157 -0.345Kendall's rank correlation τVstoxx - - - - 0.304 0.214 0.226Stoxx - - - - -0.352 -0.256 -0.274The table presents summary statistics of the logarithmic daily returns expressed **in** percentage for the volatility (VStoxx) **in**dex, equity(Stoxx) **in**dices, **and** CDS (iTraxx) **in**dices. p-values are reported for the tests. The reported correlation between CDS **and** Stoxx arefor matched Stoxx Europe 600 marketwide, Auto or F**in**ancial Services **in**dices.31

Table 2: Estimation Results for Marg**in**al ModelsVstoxx Stoxx Stoxx Auto Stoxx F**in** CDS Europe CDS Auto CDS SubF**in**Panel A: ARMA-GARCH-skT model estimatesConditional meanIntercept 0.015 0.062 ∗∗ 0.107 ∗∗ 0.052 ∗ −0.166 ∗∗ −0.143 ∗∗ −0.300 ∗∗(0.135) (0.025) (0.040) (0.028) (0.049) (0.049) (0.054)AR1 - - - - 0.079 ∗∗ - -- - - - (0.032) - -Conditional varianceIntercept 1.685 ∗∗ 0.017 ∗∗ 0.058 ∗∗ 0.019 ∗∗ 0.075 ∗∗ 0.080 ∗ 0.070 ∗∗(0.430) (0.007) (0.022) (0.009) (0.029) (0.047) (0.034)ARCH1 0.092 ∗∗ 0.112 ∗∗ 0.115 ∗∗ 0.113 ∗∗ 0.171 ∗∗ 0.216 ∗∗ 0.167 ∗∗(0.017) (0.021) (0.021) (0.021) (0.023) (0.028) (0.021)GARCH1 0.858 ∗∗ 0.881 ∗∗ 0.876 ∗∗ 0.887 ∗∗ 0.829 ∗∗ 0.784 ∗∗ 0.833 ∗∗(0.023) (0.019) (0.018) (0.019) (0.026) (0.036) (0.023)ν 6.885 8.959 8.024 7.685 5.005 3.524 4.806(1.206) (2.090) (1.949) (1.528) (0.485) (0.213) (0.436)ζ 0.291 ∗∗ −0.112 ∗∗ −0.005 −0.047 ∗ 0.030 −0.029 −0.051 ∗∗(0.039) (0.034) (0.008) (0.028) (0.026) (0.020) (0.023)Panel B: Goodness-of-t tests1 st Moment 0.242 0.873 0.611 0.551 0.606 0.934 0.1942 nd Moment 0.948 0.859 0.175 0.147 0.941 1.000 0.7383 rd Moment 1.000 0.998 0.613 0.980 0.999 1.000 0.6214 th Moment 1.000 1.000 0.999 0.860 1.000 1.000 0.996K-S test 0.950 0.071 0.306 0.137 0.275 0.109 0.101Panel A reports the parameter estimates for the conditional mean Eq. (5) **and** conditional variance Eq. (6); ν **and** ζ are the degrees-offreedom**and** asymmetry parameter of the skewed Student's t (skT ) distribution for the **in**novations. St**and**ard errors are **in** parentheses.** **and** * denote statistical signicance at the 5% **and** 10% levels, respectively. Panel B reports the p-value of the Ljung-Box Q-teston the rst four moments of the probability **in**tegral transformed series, (u t − ū) m , m = {1, 2, 3, 4}, obta**in**ed from the st**and**ardizedltered returns x t to assess the null that the the transformed uniform, u t = F (x t), has no autocorrelation up to 10 lags. The bottomrow reports the p-value of the Kolmogorov-Smirnov test to assess the null hypothesis that x t is skewed Student's t distributed.32

Student's t(Static)Gumbel(Static)SJC(Static)Student's t(RS)Gumbel(RS)SJC(RS)Student's t(DCC)Student's t(ARMA)Gumbel(ARMA)SJC(ARMA)Table 3: Goodness-of-Fit Measures for Compet**in**g Copula ModelsStoxx Vstoxx Stoxx Auto Vstoxx Stoxx F**in** VstoxxCDS Europe CDS Europe CDS Auto CDS Auto CDS SubF**in** CDS SubF**in**Static copulasAIC −379.413 −255.125 −167.542 −121.169 −214.191 −140.361LL 191.707 129.563 85.771 62.584 109.095 72.180AIC −362.281 −252.012 −166.509 −122.508 −218.837 −142.973LL 182.140 127.006 84.255 62.254 110.418 72.487AIC −360.039 −244.581 −159.234 −120.913 −216.928 −141.240LL 182.020 124.291 81.617 62.456 110.464 72.620Regime-switch**in**g Static CopulasAIC −404.514 −280.493 −193.126 −140.579 −223.487 −146.328LL 207.257 145.247 101.563 75.289 116.744 78.164AIC −384.918 −268.826 −187.505 −131.069 −223.458 −143.875LL 196.459 138.413 97.752 69.534 115.729 75.937AIC −381.162 −272.480 −184.326 −124.190 −222.882 −144.416LL 196.581 142.240 98.163 68.095 117.441 78.208Dynamic CopulasAIC −408.035 −287.043 −188.93 −145.561 −228.643 −149.081LL 207.018 146.521 97.467 75.780 117.321 77.540AIC −405.568 −287.440 −185.540 −141.770 −228.668 −147.691LL 206.784 147.720 96.770 74.885 118.334 77.845AIC −382.284 −274.081 −180.170 −128.327 −223.250 −140.876LL 194.142 140.040 93.083 67.163 114.625 73.438AIC −379.754 −276.367 −179.85 −121.389 −223.074 −145.683LL 195.877 144.183 95.925 66.695 117.537 78.842Student's t(RS-DCC)Student's t(RS-ARMA)Gumbel(RS-ARMA)SJC(RS-ARMA)Regime-switch**in**g Dynamic CopulasAIC −408.117 −287.094 −191.648 −141.984 −229.332 −148.110LL 211.058 150.547 102.824 77.992 121.656 81.055AIC −414.242 −289.265 −192.315 −142.914 −230.319 −149.380LL 214.121 151.623 103.158 78.457 122.160 81.690AIC −392.916 −278.836 −188.819 −135.819 −226.447 −148.821LL 202.458 145.418 100.410 73.909 119.224 80.411AIC −385.553 −281.532 −176.162 −124.429 −226.349 −146.612LL 202.776 150.766 98.081 72.214 123.175 83.306This table reports the goodness-of-t measures of Student's t, Gumbel **and** SJC copulas **in** a static, regime-switch**in**g, dynamic **and**regime-switch**in**g dynamic formulation. AIC denotes Akaike **in**formation criterion **and** LL denotes the optimized log-likelihood. Thebold font is used to signify the best case (largest LL or lowest AIC) with**in** each panel.33

StaticTable 4: Estimation Results for Static, Dynamic **and** RS Student's t CopulaStoxx Vstoxx Stoxx Auto Vstoxx Stoxx F**in** VstoxxCDS Europe CDS Europe CDS Auto CDS Auto CDS SubF**in** CDS SubF**in**ρ −0.495 ∗∗ 0.419 ∗∗ −0.347 ∗∗ 0.296 ∗∗ −0.381 ∗∗ 0.316 ∗∗(0.021) (0.022) (0.024) (0.025) (0.023) (0.025)ν 8.943 20.443 22.025 25.052 11.513 15.921(1.953) (8.970) (10.737) (17.066) (3.747) (6.430)RS-DCCARMADCCRSρ H −0.679 ∗∗ 0.642 ∗∗ −0.610 ∗∗ 0.383 ∗∗ −0.551 ∗∗ 0.475 ∗∗(0.033) (0.040) (0.050) (0.031) (0.048) (0.041)ρ L −0.273 ∗∗ 0.194 ∗∗ −0.214 ∗∗ 0.173 ∗∗ −0.267 ∗∗ 0.196 ∗∗(0.098) (0.057) (0.045) (0.141) (0.054) (0.043)ν 27.804 32.015 29.246 36.517 18.553 25.758(26.691) (18.568) (16.734) (10.658) (9.558) (32.067)π HH 0.969 ∗∗ 0.947 ∗∗ 0.981 ∗∗ 0.995 ∗∗ 0.991 ∗∗ 0.993 ∗∗(0.020) (0.024) (0.096) (0.034) (0.065) (0.042)π LL 0.966 ∗∗ 0.949 ∗∗ 0.992 ∗∗ 0.980 ∗∗ 0.995 ∗∗ 0.995 ∗∗(0.032) (0.020) (0.045) (0.018) (0.031) (0.026)ϕ 0.880 ∗∗ 0.829 ∗∗ 0.936 ∗∗ 0.940 ∗∗ 0.960 ∗∗ 0.972 ∗∗(0.038) (0.030) (0.027) (0.026) (0.012) (0.011)ψ 0.057 ∗∗ 0.078 ∗∗ 0.039 ∗∗ 0.034 ∗∗ 0.022 ∗∗ 0.015 ∗∗(0.016) (0.015) (0.015) (0.013) (0.008) (0.006)ν 10.299 29.609 22.854 39.324 13.465 18.122(2.535) (23.391) (11.297) (21.525) (5.067) (8.562)ω −0.087 0.297 ∗∗ −0.017 0.014 −0.002 0.010(0.063) (0.141) (0.015) (0.010) (0.022) (0.011)ϕ 1.809 ∗∗ 0.956 ∗ 1.941 ∗∗ 1.941 ∗∗ 2.030 ∗∗ 2.609 ∗∗(0.192) (0.529) (0.073) (0.059) (0.090) (0.056)ψ 0.211 ∗∗ 0.491 ∗∗ 0.105 ∗∗ 0.105 ∗∗ 0.082 ∗∗ 0.045 ∗∗(0.067) (0.189) (0.043) (0.034) (0.033) (0.021)ν 10.550 31.157 22.010 41.749 14.316 18.221(2.680) (27.071) (10.203) (72.621) (5.600) (8.520)ϕ H 0.346 0.646 ∗∗ 0.371 0.422 0.383 0.520(0.245) (0.204) (0.301) (1.804) (2.164) (0.527)ψ H 0.011 0.350 ∗∗ 0.015 0.019 0.011 0.109(0.031) (0.151) (0.096) (0.045) (0.014) (0.223)ϕ L 0.872 ∗∗ 0.841 ∗∗ 0.933 ∗∗ 0.950 ∗∗ 0.958 ∗∗ 0.971 ∗∗(0.038) (0.040) (0.039) (0.022) (0.013) (0.013)ψ L 0.065 ∗∗ 0.069 ∗∗ 0.054 ∗ 0.040 ∗∗ 0.027 ∗∗ 0.017 ∗∗(0.017) (0.022) (0.030) (0.013) (0.009) (0.007)ν 10.682 35.068 32.453 46.957 14.115 19.162(2.831) (17.996) (16.179) (7.457) (5.911) (9.233)π HH 0.927 ∗∗ 0.933 ∗∗ 0.958 ∗∗ 0.978 ∗∗ 0.926 ∗∗ 0.941 ∗∗(0.107) (0.066) (0.050) (0.014) (0.060) (0.065)π LL 0.990 ∗∗ 0.997 ∗∗ 0.978 ∗∗ 0.956 ∗∗ 0.983 ∗∗ 0.994 ∗∗(0.018) (0.006) (0.019) (0.034) (0.027) (0.007)RS-ARMAω H −1.839 ∗∗ 1.471 ∗∗ −0.194 ∗∗ 0.192 ∗ −1.548 ∗∗ 0.470(0.616) (0.563) (0.055) (0.106) (0.361) (0.408)ω L −0.076 ∗ 0.199 ∗ −0.002 0.050 ∗∗ −0.004 0.185(0.043) (0.110) (0.008) (0.021) (0.026) (0.165)ϕ 1.831 ∗∗ 1.224 ∗∗ 2.026 ∗∗ 1.488 ∗∗ 2.021 ∗∗ 1.056 ∗(0.194) (0.579) (0.053) (0.585) (0.103) (0.605)ψ 0.190 ∗∗ 0.399 ∗∗ 0.021 0.118 ∗ 0.078 ∗∗ 0.098(0.066) (0.198) (0.017) (0.068) (0.035) (0.136)ν 12.858 33.443 31.794 47.260 14.695 18.590(3.958) (19.219) (16.295) (18.302) (7.019) (10.030)π HH 0.900 ∗∗ 0.900 ∗∗ 0.979 ∗∗ 0.996 ∗∗ 0.901 ∗∗ 0.993 ∗∗(0.064) (0.076) (0.014) (0.004) (0.088) (0.004)π LL 0.996 ∗∗ 0.994 ∗∗ 0.993 ∗∗ 0.984 ∗∗ 0.998 ∗∗ 0.996 ∗∗(0.004) (0.009) (0.004) (0.017) (0.003) (0.002)This table gives the estimates of Student's t copula models **in** various formulations: static, regime-switch**in**g static, two dynamiccopulas (DCC **and** ARMA) **and** two regime-switch**in**g extensions (RS-DCC **and** RS-ARMA). Superscript H (L) **in**dicates the high(low) dependence regime. π HH (π LL ) is the probability of stay**in**g **in** the high (low) dependence regime. Numbers **in** parenthesesare st**and**ard errors. ** **and** * denote signicance at the 5% **and** 10% levels, respectively.34

Table 5: Value-at-Risk Portfolios of Stoxx **and** CDS Indices (Student's t copula)Static t copula RS t copula ARMA t copula RS-ARMA t copula5% VaR 1% VaR 5% VaR 1% VaR 5% VaR 1% VaR 5% VaR 1% VaRStoxx - CDS EuropeTotal exceptions 6.61% 2.33% 6.61% 2.33% 6.61% 2.33% 6.23% 2.33%LR UC test 0.257 0.067 0.257 0.067 0.257 0.067 0.384 0.067LR CC test 0.486 0.158 0.486 0.158 0.146 0.158 0.642 0.158DQ test 0.000 0.000 0.029 0.021 0.046 0.000 0.556 0.213Quadratic loss 2.011 0.985 1.942 0.878 1.542 0.356 1.412 0.480Percentage loss 6.493% 2.527% 5.76% 1.68% 4.531% 1.038% 4.469% 1.038%Stoxx Auto - CDS AutoTotal exceptions 7.00% 1.95% 6.61% 1.56% 6.61% 1.95% 6.23% 1.56%LR UC test 0.163 0.177 0.257 0.407 0.257 0.177 0.384 0.407LR CC test 0.090 0.358 0.486 0.655 0.355 0.358 0.642 0.655DQ test 0.090 0.145 0.267 0.892 0.149 0.323 0.836 0.558Quadratic loss 1.789 0.539 1.860 0.548 1.963 0.576 1.743 0.448Percentage loss 5.95% 1.68% 4.82% 1.44% 5.94% 1.52% 4.78% 1.40%Stoxx F**in** - CDS SubF**in**Total exceptions 6.23% 1.56% 5.84% 1.95% 5.45% 1.56% 5.06% 1.56%LR UC test 0.384 0.407 0.548 0.177 0.745 0.407 0.966 0.407LR CC test 0.407 0.655 0.424 0.358 0.864 0.655 0.473 0.655DQ test 0.246 0.833 0.248 0.646 0.883 0.800 0.868 0.439Quadratic loss 1.305 0.386 1.233 0.368 1.204 0.215 1.129 0.126Percentage loss 4.01% 1.00% 3.68% 0.78% 2.44% 0.38% 2.40% 0.28%Vstoxx - CDS EuropeTotal exceptions 5.45% 1.95% 5.06% 1.95% 5.06% 1.56% 5.06% 1.17%LR UC test 0.745 0.177 0.966 0.177 0.966 0.407 0.966 0.793LR CC test 0.399 0.358 0.473 0.358 0.871 0.655 0.473 0.921DQ test 0.378 0.282 0.341 0.074 0.832 0.415 0.457 0.721Quadratic loss 2.376 0.441 2.074 0.462 2.097 0.429 2.137 0.361Percentage loss 3.53% 0.73% 3.04% 0.73% 3.17% 0.75% 3.03% 0.61%Vstoxx - CDS AutoTotal exceptions 5.45% 1.95% 5.06% 1.17% 5.06% 1.17% 5.06% 1.17%LR UC test 0.745 0.177 0.966 0.793 0.966 0.793 0.966 0.793LR CC test 0.399 0.358 0.473 0.921 0.473 0.921 0.473 0.921DQ test 0.783 0.001 0.358 0.934 0.319 0.996 0.779 0.979Quadratic loss 2.578 0.355 2.544 0.459 3.761 1.352 3.246 0.981Percentage loss 2.77% 0.70% 2.78% 0.62% 2.64% 0.68% 2.55% 0.60%Vstoxx - CDS SubF**in**Total exceptions 5.45% 1.17% 5.45% 1.17% 5.45% 1.17% 5.06% 1.17%LR UC test 0.745 0.793 0.745 0.793 0.745 0.793 0.966 0.793LR CC test 0.399 0.921 0.864 0.921 0.399 0.921 0.338 0.921DQ test 0.468 0.997 0.769 0.990 0.417 0.992 0.399 0.977Quadratic loss 1.822 0.559 1.259 0.237 1.689 0.422 0.823 0.252Percentage loss 1.77% 0.42% 1.72% 0.36% 1.89% 0.39% 1.55% 0.30%This table reports the results of the comparison of out-of-sample VaR forecasts based on three Student's t copula formulations(static, dynamic **and** regime-switch**in**g) for portfolios of Stoxx/Vstoxx **and** CDS **in**dices. Total exceptions are percentage of days**in** out-of-sample period when the actual portfolio loss exceeds the VaR forecast. p-values are reported for Kupiec's unconditionalcoverage (UC), Christoersen's conditional coverage (CC) **and** Engle & Manganelli's dynamic quantile (DQ) tests. The quadratic**and** percentage losses reported are the out-of-sample averages of Eq. (16) **and** Eq. (17), respectively.35

Table 6: Value-at-Risk Porfolios of Stoxx **and** CDS Indices (Gumbel Copula)Static Gumbel RS Gumbel ARMA Gumbel RS-ARMA Gumbel5% VaR 1% VaR 5% VaR 1% VaR 5% VaR 1% VaR 5% VaR 1% VaRStoxx - CDS EuropeTotal exceptions 4.28% 1.56% 4.28% 1.56% 4.28% 1.56% 4.28% 1.56%LR UC test 0.588 0.407 0.588 0.407 0.588 0.407 0.588 0.407LR CC test 0.504 0.655 0.504 0.655 0.504 0.655 0.504 0.655DQ test 0.600 0.595 0.608 0.609 0.621 0.620 0.667 0.648Quadratic loss 1.254 0.452 1.250 0.447 1.245 0.436 1.227 0.432Percentage loss 3.06% 0.94% 3.05% 0.92% 3.01% 0.90% 2.93% 0.88%Stoxx Auto - CDS AutoTotal exceptions 4.67% 1.56% 4.67% 1.56% 4.28% 1.56% 3.89% 1.17%LR UC test 0.806 0.407 0.806 0.407 0.588 0.407 0.397 0.793LR CC test 0.512 0.655 0.512 0.655 0.504 0.655 0.447 0.921DQ test 0.767 0.455 0.770 0.491 0.808 0.479 0.825 0.879Quadratic loss 1.557 0.538 1.555 0.550 1.564 0.543 1.443 0.540Percentage loss 3.54% 0.95% 3.49% 0.91% 3.53% 0.90% 3.35% 0.92%Stoxx F**in** - CDS SubF**in**Total exceptions 5.06% 1.17% 4.28% 1.17% 4.67% 1.17% 4.28% 1.17%LR UC test 0.966 0.793 0.588 0.407 0.806 0.793 0.588 0.793LR CC test 0.871 0.921 0.644 0.921 0.792 0.921 0.644 0.921DQ test 0.920 0.995 0.901 0.987 0.861 0.996 0.790 0.997Quadratic loss 0.789 0.110 0.774 0.109 0.776 0.105 0.803 0.103Percentage loss 1.95% 0.27% 1.91% 0.27% 1.94% 0.27% 1.89% 0.26%Vstoxx - CDS EuropeTotal exceptions 5.45% 1.95% 5.45% 1.95% 5.45% 1.95% 5.06% 1.56%LR UC test 0.745 0.177 0.745 0.177 0.745 0.177 0.966 0.407LR CC test 0.399 0.358 0.399 0.358 0.399 0.358 0.473 0.655DQ test 0.414 0.123 0.422 0.130 0.422 0.128 0.821 0.260Quadratic loss 1.681 0.316 1.609 0.299 1.638 0.300 1.641 0.301Percentage loss 7.88% 2.94% 7.46% 2.53% 7.84% 2.92% 7.30% 2.38%Vstoxx - CDS AutoTotal exceptions 5.45% 1.17% 5.45% 1.17% 4.28% 1.17% 4.28% 0.78%LR UC test 0.745 0.793 0.745 0.793 0.588 0.793 0.588 0.710LR CC test 0.399 0.921 0.399 0.921 0.504 0.921 0.504 0.911DQ test 0.441 0.996 0.445 0.995 0.777 0.994 0.778 0.980Quadratic loss 2.766 0.360 2.639 0.331 2.661 0.272 2.217 0.211Percentage loss 2.44% 0.49% 2.39% 0.45% 2.18% 0.33% 1.80% 0.28%Vstoxx - CDS SubF**in**Total exceptions 5.45% 0.78% 5.06% 0.78% 5.06% 0.78% 5.06% 0.78%LR UC test 0.745 0.710 0.966 0.710 0.966 0.710 0.966 0.710LR CC test 0.864 0.911 0.871 0.911 0.871 0.911 0.871 0.911DQ test 0.660 0.999 0.735 0.998 0.784 0.999 0.779 0.999Quadratic loss 1.491 0.266 1.476 0.257 1.466 0.263 1.529 0.220Percentage loss 1.56% 0.21% 1.54% 0.19% 1.51% 0.21% 1.46% 0.18%See note to Table 5.36

Panel A: LevelsFigure 1: Daily Evolution of CDS **and** Equity Indices200Stoxx Stoxx Auto Stoxx F**in**15010050015002006 2007 2008 2009 2010 2011Vstoxx CDS Europe CDS Auto CDS SubF**in**100050002006 2007 2008 2009 2010 2011Panel B: Logarithmic Returns (%)50Vstoxx10Stoxx00−5009/2005 09/2006 09/2007 09/2008 09/2009 09/2010−1009/2005 09/2006 09/2007 09/2008 09/2009 09/201010Stoxx Auto10Stoxx F**in**00−1009/2005 09/2006 09/2007 09/2008 09/2009 09/2010−1009/2005 09/2006 09/2007 09/2008 09/2009 09/201050CDS Europe50CDS Auto00−5009/2005 09/2006 09/2007 09/2008 09/2009 09/2010−5009/2005 09/2006 09/2007 09/2008 09/2009 09/201050CDS SubF**in**0−5009/2005 09/2006 09/2007 09/2008 09/2009 09/2010The top graphs plot the daily levels of European equity market **in**dices (Stoxx, Stoxx Auto, Stoxx F**in** **and** Vstoxx) **and** CDS**in**dices with all series normalized to start at 100. The bottom graphs plot the daily logarithmic returns.37

Figure 2: Smoothed Probability of High Dependence RegimeStoxx−CDS EuropeVstoxx−CDS Europe110.80.80.60.60.40.40.20.2009/2005 09/2006 09/2007 09/2008 09/2009 09/20101Stoxx Auto−CDS Auto009/2005 09/2006 09/2007 09/2008 09/2009 09/20101Vstoxx−CDS Auto0.80.80.60.60.40.40.20.2009/2005 09/2006 09/2007 09/2008 09/2009 09/20101Stoxx F**in**−CDS SubF**in**009/2005 09/2006 09/2007 09/2008 09/2009 09/20101Vstoxx−CDS SubF**in**0.80.80.60.60.40.40.20.2009/2005 09/2006 09/2007 09/2008 09/2009 09/2010009/2005 09/2006 09/2007 09/2008 09/2009 09/2010The graphs show the smoothed probability of high dependence regimes for pairs of CDS **in**dices with equity market Stoxx **and** Vstoxx **in**dices **in**ferred from the RS-ARMA-Student's t copula.38

Figure 3: Rank Correlation EstimatesStoxx−CDS EuropeVstoxx−CDS Europe00.8−0.20.6−0.40.4−0.60.2−0.809/2005 09/2006 09/2007 09/2008 09/2009 09/20100.1Stoxx Auto−CDS Auto009/2005 09/2006 09/2007 09/2008 09/2009 09/20100.4Vstoxx−CDS Auto−0.10.2−0.30−0.509/2005 09/2006 09/2007 09/2008 09/2009 09/20100Stoxx F**in**−CDS SubF**in**09/2005 09/2006 09/2007 09/2008 09/2009 09/20100.4Vstoxx−CDS SubF**in**−0.10.3−0.30.20.1−0.509/2005 09/2006 09/2007 09/2008 09/2009 09/2010009/2005 09/2006 09/2007 09/2008 09/2009 09/2010Static RS DCC ARMA RS−ARMAThe graphs show the dynamics of rank correlation (Kendall's τ) estimated by Student's t copula for pairs of Stoxx-CDS **in**dices (left column) **and** Vstoxx-CDS **in**dices (rightcolumn). For RS **and** RS-ARMA copula models, the graphs show the weighted average τ t H pH t + τ t L pL t where the weights are given by the smoothed probability of each regime,i.e. p s t = P [St = s | I T ], s ∈ {H, L}, computed as detailed **in** Appendix C.39

Figure 4: Tail Dependence EstimatesStoxx−CDS EuropeVstoxx−CDS Europe0.30.030.150.015009/2005 09/2006 09/2007 09/2008 09/2009 09/20100.015Stoxx Auto−CDS Auto009/2005 09/2006 09/2007 09/2008 09/2009 09/20101 x 10−3 Vstoxx−CDS Auto0.010.80.60.0050.40.2009/2005 09/2006 09/2007 09/2008 09/2009 09/20100.1Stoxx F**in**−CDS SubF**in**009/2005 09/2006 09/2007 09/2008 09/2009 09/20100.02Vstoxx−CDS SubF**in**0.080.060.040.020.0150.010.005009/2005 09/2006 09/2007 09/2008 09/2009 09/2010009/2005 09/2006 09/2007 09/2008 09/2009 09/2010Static RS DCC ARMA RS−ARMAThe graphs show the dynamics of tail dependence (λ) estimated by Student's t copula for pairs of Stoxx-CDS **in**dices (left column) **and** Vstoxx-CDS **in**dices (right column).For RS **and** RS-ARMA copula models, the graphs show the weighted average λ H t pH t + λ L t pL t where the weights are given by the smoothed probability of each regime, i.e.p s t = P [St = s | I T ], s ∈ {H, L}, computed as detailed **in** Appendix C.40