An Analysis of the Determinants of the itraxx CDS Spreads. using the Skewed Student s t AR-GARCH Model

Transcription

1 An Analysis of he Deerminans of he itraxx CDS Spreads using he Skewed Suden s AR-GARCH Model Yuan-Sung Chu * Nick Consaninou John O Hara Absrac This paper examines he volailiy clusering behaviour beween he changes in he itraxx 5-year Europe Credi Defaul Swap (CDS) index and is heoreical deerminans using he AR-GARCH model wih he skewed suden s disribuion. Financial ime series are known o be condiionally heeroskedasic, fa-ailed, and ofen skewed. I is demonsraed ha he AR-GARCH model wih he skewed suden s marginal densiy funcion provides boh beer in-sample fi and more accurae ou-of-sample forecass han hose based on he symmeric disribued assumpions. Moreover, o es he explanaory power of he heoreical deerminans we perform regression analysis on daa from he pre-crisis ( ) and crisis periods (007-00). We show ha sock reurns and equiy volailiy have saisically significan explanaory power on he itraxx CDS index during each period, while he risk-free ineres rae has no saisically significan effec on he itraxx CDS index during he crisis period. JEL: G0, G7, C, C46 Keywords: itraxx; CDS; Credi defaul swap; AR-GARCH; Skewed Suden s. Inroducion The credi derivaives marke has grown rapidly over he las five years. A credi defaul swap (CDS), one of he mos imporan credi derivaives, is creaed o help banking and non-banking insiuions efficienly diversify credi exposure. Principally, CDSs can be classified ino wo ypes: single-name and muli-name conracs. Single-name CDSs involve only one underlying eniy whereas muli-name CDSs (e.g. CDS indices) comprise of a se of underlying eniies in a porfolio pool. Our work focuses on CDS indices ha are more efficien and liquid han holding a group of * Cenre for Compuaional Finance and Economic Agens (CCFEA), Universiy of Essex, Wivenhoe Park, Colcheser CO4 3SQ, UK. ychud@essex.ac.uk Essex Business School, Universiy of Essex, Wivenhoe Park, Colcheser CO4 3SQ, UK. ncons@essex.ac.uk. Cenre for Compuaional Finance and Economic Agens (CCFEA), Universiy of Essex, Wivenhoe Park, Colcheser CO4 3SQ, UK. johara@essex.ac.uk

2 single-name CDS conracs as hey allow invesors o shor or long he indices in order o adjus heir credi risk exposure. The Dow Jones itraxx family, one of he mos liquid families of CDS indices, includes global companies wih he excepion of hose based in Norh America. The mos widely raded CDS index of he Dow Jones itraxx family is he Dow Jones itraxx Europe index, a porfolio made up of he 5 mos liquid invesmen grade European companies in accordance wih CDS volume raded. Each company has equally weighed credi exposure (noional principal). Undersanding he relaionship beween CDS spreads and is heoreical deerminans has become an imporan issue. There are wo pricing models for CDSs, srucural form and reduced form models, of which our work focuses on he former o invesigae he deerminans of he mos liquid itraxx Europe 5-year CDS index spreads. Srucural form models (or firm s value models) originaed from he opion pricing heory of Black and Scholes (973), and were firs formulaed by Meron (974), and subsequenly exended by researchers such as Black and Cox (976), Geske (977), Leland and Tof (996), Zhou (00), Collin-Dufresne e al.(00), and amongs ohers. The core concep of a srucural form models is ha an obligor defauls when he asse value of a company his a specific hreshold level. Based on he srucural form models, several earlier empirical sudies have been applied o defaul swap markes. Collin-Dufresne e al.(00), Longsaff e al. (005), and Zhang e al. (006) use linear regression o invesigae he link beween he key economic variables and bond yields. Recenly, work has focused on exending earlier empirical sudies on he itraxx CDS marke since itraxx Europe CDS indices are more liquid han oher relevan credi defaul insrumens. Bysröm (005), he firs sudy exended o muli-name CDS spreads, ess he link beween itraxx spread changes and sock marke using linear regression. This analysis shows ha he itraxx CDS marke ends o be led by he sock marke. Furhermore, he also demonsraes a significanly posiive auocorrelaion in he itraxx marke. Alexander and Kaeck (008) and Ericsson e al. (009) find ha he changes in he CDS spreads are no only significanly affeced by he firm s equiy value, bu also influenced by he equiy volailiy and risk-free ineres raes. They observe ha he higher he equiy volailiy, he higher he firm s value volailiy, resuling in an upward rend in he CDS spread. Moreover, a low risk-free rae increases he defaul probabiliy in he itraxx CDS marke. However, all hese earlier empirical sudies assume he error erm of he linear regression model is normally disribued. Financial ime series are ypified by volailiy clusering behaviour; hence he auoregressive condiional heeroskedasiciy (ARCH) model proposed by Engle (98) and he generalized auoregressive condiional heeroskedasiciy (GARCH) inroduced by Bollerslev (986) are very popular for modelling financial volailiy.

3 These models are also being used o develop a more flexible and robus model o forecas financial ime series in order o provide an explanaion for volailiy clusering and, implicily, heavy-ailedness. However, numerous empirical sudies show ha financial ime series ofen show skewed disribuion condiionally as well as uncondiionally, which implies ha very large changes in reurns occur wih a higher frequency han under he symmerically disribued assumpion. A common assumpion in applying sandard GARCH models o financial ime series is ha he reurn series is condiionally symmerically disribued (i.e. normally or suden s disribued), which are rarely able o accommodae he excess of skewness. In his sudy he AR-GARCH model wih he skewed suden s disribuion proposed by Hansen (994) is implemened o invesigae he volailiy clusering feaure of he itraxx Europe CDS index variaion agains is deerminans. The skewed suden s disribuion involves higher-order feaures of he condiional disribuion, allowing he asymmeric behaviours of he error erm o be capured. The symmeric suden s disribuion can be seen as a special case in which he skewness parameer is se o zero. As more skewness is imposed, he skewed suden s disribuion can be used o describe various combinaions of asymmeric ail behaviours. The res of he paper is organized as follows. In Secion, he heoreical deerminans of CDS spreads are discussed. In Secion 3, he AR-GARCH model wih differen disribued assumpions and economic mehodology of volailiy forecasing performance will be inroduced. In Secion 4, we summarize he empirical analyses and resuls using he AR-GARCH wih differen marginal densiy funcions. Conclusions are presened in Secion 5.. The Theoreical Deerminans of CDS Spreads Theoreically, he main economic deerminans of srucural form pricing models are firm value, equiy volailiy, and he risk-free ineres rae. Numerous srucural form models have been exended from he basic srucural form model suggesed by Meron (974), and all he exended sudies are cenral o hese hree key deerminans. Hence, in his sudy, we use linear regression analysis o invesigae he relaionship beween he itraxx spread changes wih hese hree key variables, raher han including all sysemaic facors suggesed by oher exensive sudies. A decrease in he marke value of underlying eniy resuls in an increase in he defaul probabiliy, since he underlying eniy is approaching he defaul barrier. However, esimaing he value of firm asses and liabiliies is no easy because invesors can no ge all he informaion analogous o ha of he company s managers. Since he changes in he firm s sock price can reflec on he condiion of he firm s 3

4 operaion, we use he Dow Jones Euro STOXX 600 index o represen he performance of he Euro zone equiy marke. The Dow Jones Euro STOXX 600 reflecs he performance of he 600 larges marke capialized companies in he European economic region. The variaion of credi risk premia is an increasing funcion of equiy volailiy, since he likelihood of hiing he defaul barrier will increase while he flucuaion of firm value widens. The equiy volailiy can be esimaed using eiher hisorical daa or implied volailiy based on sock opions. Benker (004) demonsraes ha explaining he variaion in CDS spreads using implied volailiy has a sronger impac han using hisorical volailiy. However, considering he lack of raded opions on some reference eniies of he itraxx Europe CDS index and illiquidiy on some raded opions, he implied volailiy consruced by all available single-name underlying eniies may no reflec all he curren business risk in he European marke. In his sudy, we choose he opion-implied volailiy index, he Dow Jones Euro VSTOXX 50 index, as a proxy of equiy volailiy insead. The Dow Jones Euro VSTOXX 50 index is based on Dow Jones Euro STOXX 50 opions prices, which is designed o reflec near-erm volailiy marke expecaions by deermining he implied volailiy across all underlying equiy opions wih a given ime o mauriy. The heoreical argumens suppor ha he credi defaul spread is inversely relaed o he risk-free ineres rae (see Duffie, 999). The risk-free rae deermines he risk neural drif in firm value, ha is, an increase in he risk-free ineres rae ends o lower he risk-neural defaul probabiliy. Houweling and Vors (005) use he swap rae as a proxy of he risk-free ineres rae and find a sronger impac of he swap rae on he CDS marke han ha of he Treasury rae. Thus, in our approach we use he -year Euro Swap middle raes as a proxy for risk-free raes, which may provide near-erm expecaions on he European economic and credi environmen. 3. Volailiy Modelling and Evoluion of Volailiy Forecass 3. The AR-GARCH Model wih Differen Disribued Assumpions An auoregressive (AR) model is chosen for he condiional mean o allow for possible auocorrelaion in he lagged itraxx index spreads. The condiional mean equaion of a univariae ime series y can be expressed as: y s = Ai y i= i + ε ε = σ z, z ~ N(0,) () () where z is a sequence of independenly idenically disribued random variables, 4

5 σ is condiional variance, and ε denoes he error erm of he ime series. Therefore, he condiional variance equaion of Bollerslev (986) can be expressed as: where ω, α i and q p = + i i i= j= σ ω + α ε β σ, (3) j j q p β j are non-negaive inegers wih α i + i= j= β < o ensure he saisfacion wih he sufficien sable condiion of sricly saionary in he GARCH process. Moreover, p and q are also resriced o be non-negaive inegers. The inabiliy of he symmeric GARCH model o accommodae he excess of skewness in financial series is well-known. The skewed suden s densiy funcion proposed by Hansen (994) has he addiional benefi of including igher-order feaures of he condiional disribuion. Therefore, he skewed suden s densiy funcion, presen in Equaion (4), will be used o accoun for he excess of skewness in his sudy: j g ( ν, λ) b + a bc( + ( ) ) ν λ = b + a bc( + ( ) ) ν + λ ( ν + ) ( ν + ) a < b a b (4) wih a, b, and c defined as: ν a = 4λc( ), ν (5) b = + 3λ a, (6) ν + Γ( ) c =, (7) ν π ( ν ) Γ( ) where ν is he ail index and λ is he skewness parameer, which are used o conrol he differen shape of he densiy funcion and Γ ( ) is he gamma funcion. The wo shape parameers need o be resriced wih ν > and < λ < respecively. If < λ < 0 he densiy funcion skews oward o he righ, conversely if 0 > λ > he densiy funcion skews oward o he lef. Furhermore, he skewed suden s disribuion ransforms o he normal disribuion by seing ν = and λ = 0 and o symmeric suden s disribuions by seing λ = 0. 5

6 3. Evaluaion of Volailiy Forecasing Performance 3.. Loss Funcions A common way o compare differen forecas models is given by he minimizaion of a saisic loss funcion. AR-GARCH models describe he evoluion of he condiional mean and condiional variance. Once he parameers of he AR-GARCH models have been esimaed using in-sample period daa, and condiional variance forecass based on hese esimaes can be generaed over he ou-of-sample period. In order o evaluae he forecas accuracy of various compeing models, saisical loss funcions have o be used o measure forecas errors. Mean squared error (MSE) is he mos commonly used loss funcion, however, he MSE crierion is very sensiive o ouliers. Hence, a more robus loss funcion o possible presence of ouliers, mean absolue error (MAE) is aken ino accoun in his sudy. Moreover, he logarihmic loss (LL) funcion, proposed by Pagan and Schwer(990), penalizes volailiy forecass asymmerically in low volailiy and high volailiy periods, which can be used o deal wih he second shorcoming of he MSE crierion. These hree loss funcions are defined as follows: ( ), T n MSE = h σ (8) n + = T + T n MAE = h σ, (9) n + = T + [ ln( ) ln( )], T n LL = h σ (0) n + = T + where n is he number of forecas daa poins, T is he enire in-sample period, h is forecas condiional variance a ime, and σ is realized condiional variance a ime. 3.. Diebold-Mariano Tes (DM Tes) Diebold and Mariano (995) propose a es o evaluae he accuracy of L-seps ahead forecass of one model in comparison o anoher. The null hypohesis of he es is ha wo compeing models have equal forecas accuracy agains he alernaive hypohesis of significan difference in he accuracy of he wo compeing forecass. The DM es is based on a -es ha E( δ ) = 0 where E ( ) is he expecaion A B A B operaor, δ = l l. l, and l are forecas errors of wo compeing models A and B respecively. The DM es saisic can be expressed as: 6

7 δ DM = () V where δ and V (δ ) is mean and variance of forecas errors over he forecasing period respecively. However, δ can no be assumed o be uncorrelaed. The DM es assumes ha he auocorrelaions of order L or higher are zero of δ, so V (δ ) can be esimaed asympoically as: + L V δ γ 0 γ j () n j= ( δ ) ( ) where δ is serially correlaed for n > and γ j denoes he j-h auocovariance of δ. 4. Empirical Analysis 4. Daa Descripion Our daa se on he mos liquid itraxx 5-year Europe CDS index, he STOXX 600 index, he VSTOXX 50 index, and he -year Euro swap rae consiss of 375 daily observaions; i covers he period from Augus 6, 004 o January 8, 00. Alhough he itraxx 5-year Europe CDS index was launched in June 004, our daa se lacks some rading day observaions on he itraxx CDS index before mid-augus 004. As a resul, we use daa from Augus 6, 004 onward in his sudy. Since he new series of he itraxx Europe CDS index is launched every six monhs in order o adjus he underlying companies, we use he mos curren series daa of he itraxx index a any poin in ime o ensure our analysis is always based on he mos liquid underlying eniies. The daa se on -year Euro Swap rae is obained from Daasream, he daa for he STOXX 600 and he VSTOXX 50 indices are obained from Soxx Ld, and he itraxx CDS index daa is available from Marki Group. 4. Descripive Saisics Before invesigaing he relaionship beween he itraxx CDS index and he explanaory variables, we provide summary saisics for he itraxx CDS index and he key variables. We consider he naural logarihmic reurns of all he ime series. In order o deermine he presence of mean reversion in he logarihm of reurns, he uni roo ess (Augmened Dickey-Fuller es (ADF es)) can be used o verify he behaviour of he ime series. The summary saisics and he uni es resuls are 7

8 presened in Table. The resuls in Table show ha he disribuions of log reurns of he itraxx index and is heoreical deerminans are skewed. Moreover, he itraxx CDS and he VSTOXX 50 indices are more volaile han he STOXX 600 index and he Euro swap rae. The ADF es is run on he level of he log reurns of all indices. One can see from he resuls ha all he null hypoheses under he ADF es can be rejeced a he 5% level. The mean-revering coefficiens in he ADF es is saisically significanly negaive, confirming he saionariy of periodically mean-revering cycles. Table Summary saisics of daily logarihmic reurns itraxx STOXX600 VSTOXX 50 Euro Swap Mean (%) Median (%) Maximum (%) Minimum (%) Sd. Dev. (%) Skewness Kurosis Jarque-Bera Tes * * * * ADF Tes * * * * * denoe significanly a 5% significance level. Figure. The itraxx Europe 5-year index and is deerminans, normalized o sar a itraxx Index Soxx 600 Index Vsoxx Index Euro Swap Rae 0 Aug-04 Feb-05 Aug-05 Feb-06 Aug-06 Feb-07 Aug-07 Feb-08 Aug-08 Feb-09 Aug-09 Bysröm (005, 006), and Alexander and Kaeck (008) poin ou ha he itraxx Europe CDS index exhibis significan auocorrelaion in is previous day s 8

9 changes, resuling in he inefficiency in he European 5-year CDS marke. According o he resuls of he Ljung Box es, Q(0) and Q(30) are and respecively, supporing earlier empirical sudies. In order o deal wih he auocorrelaion problem, he effec of he lagged reurns on previous day s itraxx CDS spread will be included in he nex secion. As seen in Figure, before 6 July 007 he itraxx CDS index demonsraes sable downward rend, and displays posiive correlaion wih he VSTOXX 50 index and negaive correlaion wih he Euro swap rae and he STOXX 600 index. Afer 6 July 007, all key marke variables and he itraxx CDS index flucuaed dramaically. I's worh menioning ha he Euro swap rae seems no o be significanly negaively correlaed o he itraxx index afer 6 July Linear Regression Analysis In his secion, we use he following linear regression (Equaion (3)) o analyze he dependence beween he daily log reurns of he itraxx CDS spreads and daily log reurns of he explanaory variables. ΔCDS = θ + θ ΔS + θ ΔV + θ ΔCDS + θ Δr + ε, (3) The regression will be esed in hree sages: he whole period, he pre-crisis period from Augus o July 5 007, and he crisis period beween July and January 8, 00. Le Δ CDS denoe he logarihmic reurns of he itraxx CDS spreads, Δ V denoe logarihmic reurns of he VSTOXX 50 index, Δ r denoe he logarihmic reurns of he -year Euro swap rae, and Δ S denoe he logarihmic reurns of he STOXX 600 index. The error erm, ε, are i.i.d. random errors wih zero mean and uni variance, and will be assumed o be normally, symmeric suden s and skewed suden s disribued respecively. The regression resuls are given in Table. All he explanaory variables are saisically significan a he 5% level in he main and wo sub-period regressions excep for he Euro swap rae during he crisis period. Bysröm (005, 006) and Alexander and Kaeck (008) discover significan predicive abiliy for lagged daily itraxx spread changes by using previous day s itraxx spreads, he resuls in Table suppor his in ha he European CDS marke exhibis significan posiive firs-order auocorrelaion. However, he impac of he posiive firs-order auocorrelaion in he crisis period is no as sensiive as in he pre-crisis period. The flucuaion of he itraxx CDS index has he negaive correlaion wih he lagged reurns in he STOXX 600 index and he posiive correlaion wih he lagged 9

10 reurns in VSTOXX 50 index, which is consisen wih previous invesigaions. The flucuaion of he itraxx CDS index is more sensiive o lagged reurns in he STOXX 600 and he VSTOXX 50 indices during he crisis period. The Euro swap rae has a significan effec on he itraxx CDS spreads during he pre-crisis period only. Alexander and Kaeck (008) indicae ha he risk-free rae has no saisically significan impac on he itraxx CDS spreads during he urbulen period, consisen wih our finding in he crisis period. We also use he -year Euribor rae as he proxy of he risk-free ineres rae and he resuls appear similar o hose displayed in Table. However, he values of R for hree esing periods are no as good as using he Euro swap rae. For his reason, we do no provide he resuls of using he -year Euribor rae in he sudy. Table Regressions wih all explanaory variables Whole Period Pre-crisis Period Crisis Period θ (0.3834) (.0945) ( ) θ (-3.930) (-4.65) ( ) θ (4.038) (3.437) (4.0849) θ Adjused (7.60) (9.364) (3.9387) θ ( ) (-4.336) (-.653) R R Each column repors he esimaes of he regression by Eq. (3), wih -saisics below in brackes. Comparing he wo sub-periods, he marke facors have a sronger effec during he crisis period han in he pre-crisis period. The degree of explanaion, R, can be used o measure how well he regression esimaes he ime series. In he crisis period, he itraxx CDS spreads were more volaile and R is much higher han ha in he pre-crisis period, which may indicae ha hese heoreical deerminans have sronger explanaory power in he period of high volailiy. Using he proxy of firm value suggesed by Alexander and Kaeck (008) in Equaion (3), he values of R are around 3% for he whole period and 36% for he crisis period. The main difference is ha Alexander and Kaeck (008) generae an equally weighed sock porfolio comprised of mos of he underlying eniies from he itraxx 5-year CDS Europe 0

11 index as he proxy firm value. However, he itraxx CDS index only covers 5 invesmen grade European companies, which may no reflec in enirey he European economic siuaion. On he conrary, he STOXX 600 index can represen he performance of he Euro zone equiy marke, which may explain he daily flucuaion of he itraxx CDS index beer. 4.4 Residual Analysis The AR()-GARCH(,) model wih he skewed suden s marginal disribuion is used o esimae he behaviour of he regression residuals insead of he sandard AR-GARCH models. In order o assess he pracical relevance of his disribuion, he comparison among he skewed suden s disribuion, he normal and he symmery suden s disribuions are aken ino accoun. In addiion, he Ljung Box Q-Tes, he ARCH-LM es, and he Pearson goodness-of-fi (GOF) es are used o address esimaion risk. The Ljung Box Q-Tes for squared sandardized residuals deecs any remaining serial correlaion in he condiional variance equaion. If he condiional variance equaion is correcly specified, all Q-saisics of sandardized residuals should be insignifican wih no observable auocorrelaion. The ARCH-LM ess he null hypohesis ha here is no ARCH effec up o order q in he squared residuals. Moreover, we employ he Pearson goodness-of-fi es for sandardized residuals as a saisical diagnosic in order o es he null hypohesis of he disribuion of cerain ime series is consisen wih a paricular heoreical disribuion. Table 3 shows he AR()-GARCH(,) esimaions and he ess resuls for he whole period. The AR()-GARCH(,) model can be adequae o capure he dynamics of he residual series of Equaion (3). All he ARCH-LM ess are highly saisically significan a he 5% level, meaning he heeroskedasiciy is removed. The resuls of he Ljung Box Q-saisic wih 30 lags es accep he null hypohesis of no auocorrelaion, meaning he AR()-GARCH (,) model is adequae o capure he condiional mean and he condiional variance. The esimaed numbers of degrees of freedom ν are saisically significan lying on 5.9 and 5.68, which indicae residual series of he whole period o be fa-ailed. The skewness parameer λ is posiively significanly differen from 0 a he 5% level, exhibiing he disribuions of residual series seem o be skewed. According o he Pearson goodness-of-fi saisics, he skewed suden s disribuion is more suiable o capure he volailiy clusering due o he insignifican Pearson goodness-of-fi saisics of he normally and he symmeric suden s disribued assumpions. Moreover, he Akaike informaion crieria (AIC) and he

12 log-likelihood values highligh he fac ha AR()-GARCH(,) wih skewed suden s marginal densiy funcion beer esimae he residual ime series han symmeric AR()-GARCH(,) model. Table 4 presens he Jarque-Bera and he Kolmogorov-Smirnov (KS) ess of he residual series for he whole period. The resuls show ha he residual is neiher normally disribued nor symmeric suden s disribued bu skewed suden s disribued, supporing he heory ha he AR()-GARCH(,) wih skewed suden s marginal densiy funcion is more suiable o capure he volailiy clusering feaure of he regression residual. Table 3 Esimaion resuls of AR()-GARCH(,) model for he whole period Normal Suden s Skewed Suden s Cons ( ) (59.75) ( ) ARCH () ( ) (4.834) ( ) GARCH () (4.980) (80.79) ( ) ν (6.9997) (9.6586) λ (5.53) AIC Log-Lik GOF(0) [0.0000] [0.0000] [0.0] ARCH-LM Tes [0.83] [0.588] [0.9769] Ljung Box Q-Tes [0.97] [0.9905] [0.989] Each column repors he esimaes of he model defined in secion., wih -saisics underneah in parenheses. The saisics of all ess are repored wih p-value underneah in brackes. AIC and Log-Lik are he Akaike Informaion crierion and Log-Likelihood value Table 4 Disribuion es of he residual series in he whole period Jarque-Bera Tes KS (Symmeric ) KS(Skewed ) Saisic p-value

13 As is ypical of AR-GARCH model esimaes, he sum of he coefficiens on he lagged squared error and lagged condiional variance for he whole period wih he skewed suden s marginal densiy funcion is very close o uniy (approximae ). This implies ha shocks o he condiional variance will be highly persisen. This can be demonsraed by considering he equaions for forecasing he sandardized regression residuals of he condiional variance using he AR()-GARCH(,) model based on skewed suden s disribuion. A large sum of hese coefficiens implies ha a large posiive or a large negaive change will lead o high fuure variance forecas for a proraced period. 4.5 Ou-of-Sample Forecas Evaluaion An ou-of-sample forecas evaluaion provides a powerful framework o evaluae he performances of compeing models. Many empirical sudies (e.g. Pagan and Schwer(990), Davis and Kuan(003), Alberg, Shali and Rosef(008), and so on) demonsrae ha he asymmeric assumpion of condiional variance provides a beer in-sample fi and beer ou-of-sample predicive power. In his secion, he whole daa se is divided ino an in-sample period from Augus 6, 004 o Sepember 008 and an ou-of-sample period from Sepember o January 8 00 for he forecasing evaluaion. Table 5 repors he ou-of-sample forecas evaluaion resuls using he saisical loss funcion from secion 3.. of his sudy ogeher wih various disribuions. All of he loss funcions clearly indicae ha he AR()-GARCH(,) model wih skewed suden s marginal densiy funcion performs bes for all of he four forecas horizons, and he second bes is based on he suden s disribuion. In order o examine if he forecas accuracy among he hree assumpions on disribuion are significanly differen, he DM es is adoped o check he saisical significance when all models are compared in pairwise. Table 6 presens he DM es resuls using he skewed suden s disribuion as he benchmark, and he comparison is carried ou wih regards o all he saisical loss funcions discussed in secion 3... As can be seen in Table 6, he skewed suden s disribuion significanly ouperforms he symmeric models a he 5% level. The signs of he DM saisics of he comparison beween he skewed suden s disribuion and he symmeric disribuions are always significanly posiive; indicaing he loss of he skewed suden s is significanly lower han ha of symmeric disribuions. In oher words, he AR()-GARCH(,) wih skewed suden s marginal densiy funcion provides a more accurae volailiy forecas, demonsraing he skewed suden s disribued assumpion of regression residual 3

14 ouperforms he symmeric disribued assumpion for he forecas evaluaion. Table 5 Ou-of-sample forecas evaluaion resuls for saisical loss funcions: -,5-,0- and 0-day forecas horizions Normal Suden's Skewed day MSE (%) MAE (%) LL (%) days MSE (%) MAE (%) LL (%) days MSE (%) MAE (%) LL (%) days MSE (%) MAE (%) LL (%) Table 6 Diebold-Mariano Tes MSE MAE LL Saisiic p-value Saisiic p-value Saisiic p-value day Normal v.s. Skewed Suden's v.s. Skewed days Normal v.s. Skewed Suden's v.s. Skewed days Normal v.s. Skewed Suden's v.s. Skewed days Normal v.s. Skewed Suden's v.s. Skewed

15 5. Conclusion This sudy invesigaes he linear relaionship beween heoreical deerminans and he itraxx Europe CDS index. The heoreical deerminans of he itraxx CDS index have a significan impac on he itraxx CDS marke, excep for -year Euro Swap rae, which has no saisically significan effec on he itraxx CDS marke in he crisis period. This finding indicaes ha he risk-free rae yield curve has no saisically significan effec on he itraxx CDS marke in he period of high volailiy. Moreover, all he explanaory variables in his sudy have higher explanaory power han previous empirical sudies due o a beer proxy of firm s value suggesed in his sudy. Furhermore, financial ime series are known o be condiionally heeroskedasic, fa-ailed, and ofen skewed. To accommodae for his we furher relaxed he assumpion of ime series of regression residuals from symmeric disribuions o an asymmeric disribuion. Our resuls show ha volailiy clusering feaures in he itraxx CDS marke can no be fully capured by using he sandard AR()-GARCH(,) model. The AR()-GARCH(,) model wih he skewed suden s marginal densiy funcion is much more suiable o capure mean-revering condiionally heeroscedasic process for innovaions. In addiion, he sums of he coefficiens on he lagged squared error and lagged condiional variance are very close o uniy, indicaing a large posiive or a large negaive change will lead o high fuure variance forecas for a proraced period. Ou-of-sample forecasing comparison is carried ou by comprising -day, 5-day, 0-day and 0-day ahead volailiy forecass. The saisical loss funcions demonsrae ha AR()-GARCH(,) model wih skewed suden s marginal densiy funcion ouperforms he sandard AR()-GARCH(,) models in forecasing volailiy for all four forecas horizons. The Diebold-Mariano is applied o furher examine wheher he differences in he performances among hree disribued assumpions are significan, and he resuls show ha he AR()-GARCH (,) model based on he skewed suden s disribued assumpion significanly ouperforms oher wo compeing models. Overall, he inclusion of AR()-GARCH(,) effecs wih fa-ailedness and skewness provides no only a beer in-sample fi bu also more accurae ou-of-sample forecass. 5

16 References Alexander, C., Kaeck, A., (008), Regimes Dependen Deerminans of Credi Defaul Swap Spreads, Journal of Banking & Finance, Vol. 3, pp Alberg, D., Shali, H., Rosef, R., (008), Esimaing Sock Marke Volailiy Using Asymmeric GARCH Models, Applied Financial Economics, Vol. 8, pp Benker, C., (004), Explaining Credi Defaul Swap Premia, Journal of Fuures Markes, Vol. 4, pp Black, F., Cox J. C., (976), Valuing Corporae Securiies: Some Effecs of Bond Indenure Provisions, Journal of Finance, Vol. 3, pp Black, F., Scholes, M., (973), The Pricing of Opions and Corporae Liabiliies, Journal of Poliical Economy, Vol. 8, pp Bollerslev, T., (986), Generalized Auoregressive Condiional Heeroscedasiciy, Journal of Economerics, Vol. 3, pp Bysröm, H., (005), Credi Defaul Swaps and Equiy Prices: The itraxx CDS Index Marke, Working Papers, Lund Universiy, Deparmen of Economics. Bysröm, H., (006), CrediGrades and he itraxx CDS Index Marke. Financial Analyss Journal, Vol.6, pp Collin-Dufresne, P., Goldsein, R.S., Marin, J.S., (00), The Deerminans of Credi Spread Changes, Journal of Finance, Vol.56, pp Davis, N., Kuan, M., (003), Inflaion and Oupu as Predicors of Sock Reurns and Volailiy: Inernaional Evidence, Applied Financial Economics, Vol.3, pp Diebold, F., Mariano, R., (995), Comparing Predicive Accuracy. Journal of Business and Economic Saisics, Vol.3, pp Duffie, D., (999), Credi Swap Valuaion, Financial Analys Journal, Vol.55, pp Engle, R., (98), Auoregressive Condiional Heeroscedasiciy wih Esimaes of Variance of Unied Kingdom Inflaion, Economerica Vol. 50, pp Ericsson, J., Jacobs, K., Oviedo-Helfenberger, R., (009), The Deerminans of Credi Defaul Swap Premia, Journal of Financial and Quaniaive Analysis, Vo.44, pp Geske, R., (977), The Valuaion of Corporae Liabiliies as Compound Opions, Journal of Financial and Quaniaive Analysis, Vol.5, pp Hansen, B., (994), Auoregressive Condiional Densiy Esimaion, Inernaional Economic Review, Vol. 35, pp Houweling, P., Vors, T., (005), Pricing Defaul Swaps: Empirical Evidence, Journal of Inernaional Money and Finance, Vol. 4, pp

17 Leland, H., Tof, K., (996), Opimal Capial Srucure, Endogenous Bankrupcy, and he Term Srucure of Credi Spreads, Journal of Finance, Vol.5, pp Longsaff, F.A., Mihal, S., Neis, E., (005), Corporae Yield Spreads: Defaul Risk or Liquidiy? New Evidence from he Credi Defaul Swap Marke, Journal of Finance, Vol.60, pp Meron, R. C., (974), On he Pricing of Corporae Deb: The Risk Srucure of Ineres Raes, Journal of Finance, Vol. 9, pp Pagan, R., Schwer, W., (990), Alernaive Models for Condiional Sock Volailiy, Journal of Economerics, Vol.45, pp Zhou C., (00), The Term Srucure of Credi Spreads wih Jump Risk, Journal of Banking & Finance, Vol.5, pp Zhang, B., Zhou, H., Zhu, H., (006), Explaining Credi Defaul Swap Spreads wih he Equiy Volailiy and Jump Risks of Individual Firms, Working Paper, BIS. 7