Synthetic Collateral Debt Obligation Pricing

Transcription

1 Sngapore Management Unversty Insttutonal Knowledge at Sngapore Management Unversty Dssertatons and Theses Collecton (Open Access) Dssertatons and Theses 007 Synthetc Collateral Debt Oblgaton Prcng Zhanyong LIU Sngapore Management Unversty, Follow ths and addtonal works at: Part of the Fnance and Fnancal Management Commons, and the Portfolo and Securty Analyss Commons Ctaton LIU, Zhanyong. Synthetc Collateral Debt Oblgaton Prcng. (007). Dssertatons and Theses Collecton (Open Access). Avalable at: Ths Master Thess s brought to you for free and open access by the Dssertatons and Theses at Insttutonal Knowledge at Sngapore Management Unversty. It has been accepted for ncluson n Dssertatons and Theses Collecton (Open Access) by an authorzed admnstrator of Insttutonal Knowledge at Sngapore Management Unversty. For more nformaton, please emal lbir@smu.edu.sg.

2 SYTHETIC COLLATERAL DEBT OBLIGATION PRICING LIU ZHANYONG SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE REQUIREMENTS FOR THE DEGREE OF MASTER DEGREE OF MASTER OF SCIENCE IN FINANCE SINGAPORE MANAGEMENT UNIVERSITY 007

3 To my parents and my sster for ther Love and Support Copyrght 006 Zhanyong Lu All Rghts Reserved

4 Notaton B () t : Rsk free dscount factor at tme t B () t : Defaultable dscount factor at tme t assocate wth corporate bonds λ : y : Default ntensty / hazard rate of defaultable asset Tranche Spread τ Default tme { τ t} : Indcate functon of default R Recovery rate P( τ < T M = m) : Condtonal on M=m, the probablty of no default before tme T Accumulated Loss Functon n percentage, equally weghted credt portfolo: () l t N = ( R ) ( τ t ) N, N s the asset number n portfolo pdf: cdf: LHP: Probablty dstrbuton functon Cumulatve dstrbuton functon Large Homogeneous Portfolo N( 0, t ) The normal dstrbuton wth mean 0 and varance t

5 Abstract Portfolo credt products, such as CDO and Sngle Tranche CDO (STCDO) have ganed ther popularty n fnancal ndustry. The key problem facng by the fnancal engneers s how to prce these portfolo credt dervatves, especally how to model the dependent default structure. Copula model proposed by L (000) s wdely used n practce. Comparng wth smulaton, factor copula model and condtonal ndependent framework provde good balance between accuracy and computatonal effcency, but t s hard to acheve good performance f stckng to normal dstrbuton. There are a few ways to mprove t: ntroducng Levy dstrbutons, usng generc copula functons, and the sem parametrc estmaton. In ths paper the Levy dstrbuton and condtonal ndependent factor copula model are examned. The flexblty and accuracy mprovement comes from calbratng the skewness and heavy tal of Levy dstrbuton for the underlyng margnal dstrbutons. The smulaton result and short perod predcton result are dscussed too. One of the other benefts of ths model s that once calbratng to the standard market tranches spreads, the model can handle the customzed CDO, e.g. Sngle Tranche CDO,. JEL classfcaton: G3 Key words: factor copula model, portfolo credt dervatves, Levy process 3

6 ABSTRACT... 3 ACKNOWLEDGEMENT INTRODUCTION... 6 DOW JONES ITRAXX INDICES... 6 THE ECONOMIC OF CDO LITERAUTRE REVIEW CREDIT MODELS... 0 INTENSITY MODEL... 0 STRUCTURAL MODEL... 0 Drawbacks of Structural model... 3 CREDIT DEFAULT SWAP FACTOR COPULAE FUNCTIONS... 6 COMMON COPULA FUNCTIONS... 6 Gaussan Copula... 6 Student t Dstrbuton... 6 Double t Copula: SYNTHETIC CDO PRICING... 9 MODEL SPECIFICATION... 0 A Generc t Copula Model... 0 Condtonal Independent Model for synthetcs CDO Prcng... ASYMPTOTIC LARGE HOMOGENOUS PORTFOLIO APPROXIMATION... 3 FIRST RESULT FROM CONDITIONAL INDEPENDENT FACTOR MODEL SPREAD SENSITIVITY ANALYSIS SIMULATION METHOD FOR PRCING SYNTHETIC CDO... 4 SIMULATION PROCEDURE... 4 SIMULATION RESULT NUMERIAL RESULT FOR MARKET DATA SHORT TERM PREDICTION CONCLUSION FURTHER DEVELOPMENT HEDGING RISK MANAGEMENT APPENDIX THE THEORY OF COPULA FUNCTION Tal Dependence... 6 Archmedean Copula Exponental copula Sem Parametrc Method THE CONDUCTION OF ASYMPTOTIC LARGE HOMOGENOUS APPROXIMATION NIG DISTRIBUTION AND ITS PROPERTIES BASEL II ACCORD AND COPULA FUNCTION REFERENCE

7 Acknowledgement I am deeply ndebted to my supervsor, Professor Lm Kan Guan for hs tremendous support, encouragement and gudance. He ntroduced me to the nterestng area, fnance and nspred me for the academc research. I would also thank Professor Wu ChunCh, assstant Professor Tng Han Ann Chrstopher, vstng Professor Xue Hong and assstant Professor Dng Qng. They nspred me n varous ways from ther lectures and dscussons. Many fellow professors and frends n dfferent areas at SMU also helped me to fnsh ths dssertaton, such as Dem. I acknowledge the fnancal support from SMU. Wthout ths, ths thess would not be possble. 5

8 . INTRODUCTION Synthetc Collateralzed Debt Oblgatons (CDO) has ganed great attenton from both ndustry and academy due to the ncreasng traded CDO and the dffculty to prce them. CDO s basket credt dervatve based on Credt Default Swap (CDS). By poolng and tranchng some CDSs, CDO transfers the credt rsk of the reference credt portfolo to the nvestors wth dfferent senorty n tranches. The rsk of credt default losses on the reference credt portfolo s dvded nto tranches of ncreasng senorty. Losses wll frst affect the equty or frst tranche, next the mezzanne tranches, and fnally the senor tranches. The nvestors receve the premum payment every 3 month as the compensaton to bear the losses for outstandng tranche value ncurred by credt defaults. The spread of premum for dfferent tranches s determned by CDO prcng models. Dow Jones Traxx Indces Ths ndex compostes of 5 equally weghted enttes of CDS based on nvestment grade European bonds accordng to some lqud and dversfcaton crteron. The pool s outstandng amount wll reduce upon default events. And the pool s tranched and sold to the nvestors quoted by spread. The payment s made every 3 month and the payment s based on the outstandng amount for each tranche at payment tenor. There s a total ndex spread whch s the average of CDS spread n the pool, snce the CDS are equally chosen n the CDO. The most lqud synthetc CDOs are based on Traxx ndex. The attachment and detachment ponts for the DJ Traxx European CDO are 3%, 6%, 9%, %, %. They are called equty tranche, junor mezzanne tranche, senor mezzanne The equty tranched s quoted as upfront payment plus 5% annual payment. 6

9 tranche, senor tranche and most senor tranche. The tranche -00% s not quoted n market, but ts spread s mpled by the ndex and the other fve tranches. The CDO prcng models are calbrated by these fve tranches spreads and CDS spreads. DJ Traxx European CDO 5 year s the chosen nstrument snce t s most lqud wth the least bd ask dfference of quoted spreads. The Economc of CDO The CDO can help nvestors to hedge or speculate accordng to ther own rsk atttude and perspectve on default rsk and default correlaton. Here are some benefts ganed from CDO tradng: Some sngle name credt dervatves are not lqud, the bd ask spread for some snge name CDS could be prohbtve hgh for the potental nvestors. CDO provdes the lqudty and dversty for the portfolo credt dervatve market. For some nsttutes, such as commercal banks and nsurance companes, the loan and nsurance asset are not tradable. And they can dversfy ther portfolo to reduce the systematcal rsk by CDO products. The CDO also meets some nvestors ndvdual rsk averson atttude for credt rsk, whch can hardly meet wthout CDO products. The market for credt rsks s not complete, and CDO make some credt nvestment chance possble. Facng the new captal requrement n Basel II, the nsttutes such as banks or nsurance companes may fnd that t s more proftable to use CDO to transfer the asset wth credt rsk n ther balance sheets. The CDO ndces provde the transparence and lqudty of credt rsk market. 7

10 . LITERAUTRE REVIEW The benchmark of synthetc CDO model s the factor model (factor copula model) and condtonal ndependence concept, whch means that condtonal on the factors, ndvdual defaults are ndependent. Snce there are 5 CDSs n the Dow Jones Traxx tranched CDO, ths factor copula model can substantally reduce the prcng dmenson and make the CDO prcng model tractable. Homogenous portfolo assumpton further smplfes the factor copula model. The buldng block s copula method ntroduced by L (000). However the Gaussan copula factor model does not yeld a unque correlaton through all the tranches. Default events are rare and happen n the tal range of the dstrbuton; however, the Gaussan dstrbuton has a very thn tal to capture the dependent defaults. Other copula functons and other dstrbutons are proposed to mprove the CDO model s prcng capablty. Among factor copula models, Gaussan copula and student t copula model are popular, see [Hull (004, 005)]. And [Laurent (005)] provdes a good survey on comparson among the dfferent models. The man advantage n these Gaussan copula and student t copula factor models s that the correlaton coeffcents have economc meanng, hence can be communcated and nterpreted as dependence on common market factor or ndustry secton factor. The student t dstrbuton s condtonal normal wth heaver tal than Gaussan dstrbuton. However, the Gaussan copula factor model fals to catch the dependent structure of the rare default events; t results n correlaton smle for the quoted CDO spreads among dfferent tranches. In ths thess I try to apply other heavy tal asymmetrc dstrbuton to prce synthetc CDO whle stll keep the factor model and condtonal ndependence framework for tractablty. Condtonal on Ch-Square varable, student t dstrbuton s normal. 8

11 There are some further extended models beyond condtonal ndependent factor models whch are not so tractable, such as double student t [Hull & Whte (004)], Clayton copula model [Schönbucher (00)], random factor loadngs [Andersen & Sdenus (004)] and CDO prcng wth term structure of default ntensty [Schönbucher (005)]. In addton, Fast Fourer Transformaton (FFT) and Inverse FFT are proposed to facltate computatonal mplementaton n the CDO modelng. 9

12 3. CREDIT MODELS There are manly two knds of credt models: ntensty model and structural model. And sometmes they are both used as a hybrd model. Intensty Model Intensty model s also called reduced form model. It models the default hazard rate. Let λ () t be the hazard rate (default ntensty) functon gven no default up to current tme. Let τ be the default tme, S( t) be the survval probablty from tme 0, and p () t be the default probablty 3. Then: Pr ( ) S() t d S t () t = λ ( 0 ) t ( λ 0 ) t ( τ ) exp λ > t = S t = s ds () Pr ( τ ) () exp p t = < t = S t = s ds Structural Model Structural model orgnates from Merton s framework of valung corporate equty as an opton. The structural model assumes complete market and rsk neutral measure. Let the frm s value V () t follows the process of: () = + σ dv t V t r t dt t dw t r() t s nterest rate ; σ () t s volatlty of the frm; W ( t) N( 0, t) Brownan moton. N( 0, t) s the normal dstrbuton wth mean 0 and varance t. 3 The probablty measure used n ths thess s a rsk neutral measure. s the standard 0

13 By solvng ths SDE, If r() t and σ () t are constant: V t V r s s ds s dw s 0 t () = ( 0exp ) σ + σ In the factor copula model, assume W V () t = V ( 0exp ) rt σ t+ σw() t t be the random varable for the th company. Assume all the varables W () t are correlated by dependng on a common factor M ( t ) : () = ρ () + ρ () W t M t B t where ρ s correlaton coeffcent to the common factor M ( t ), B factor 4. M () t N( 0, t), B() t N( 0, t). Assume that M ( t) and W () t s also normal dstrbuton. t s dosyncratc B t are ndependent, It s worth to note that condtonal on the common factor M j condtonal varables W () t M = m and = m, gven jthe W t M = m are ndependent. For smple, I drop off the ndex. If M () t s jont normal dstrbuted vector, B ( t) s another ndependent normal varable, () = ρ () + ρ () W t M t B t Snce M () t and B () t are the normal unvarables N( 0, ) W() t = t W ; M ( t) = t M ; and B( t) t, and scale them by = t ε t, let: SoW, M and ε are ndependent normal varables wth standard normal dstrbuton: The above formula becomes:,, ε ( 0,) W t M t t N 4 If B () t =,, N have the same dstrbuton, ρ s same for all W ( t ).

14 W = ρm + ρε M s common factor and ε s ndvdual varable. If they follow other dstrbutons, the above expresson s smlar. For nstance, W, ε and M have the cumulatve dstrbuton functon of F, F and F 3 respectvely. The structural model assumes that the default happens when the frm value frst tme drops below an exogenous threshold determned by the frm s debt value. For long term bond the threshold s exponental ncrease functon of tme t. For short term, the smplest structural model assumes the default happens f the frm s value s less than ts debt value D when debt s mature, namely: V ( 0exp ) r σ t+ σwt < D The uncondtonal default for name occurs f: ( 0) V log r σ t D + W < K σ t K s called the default threshold. The condtonal default probablty becomes: ( = ) = Pr ( τ < = ) Pr{ W K} p t M m t M m = < { ρm ρ ε K} = Pr + < ( K ρm) =Φ ρ Φ s normal cumulatve dstrbuton functon. In general case, assume the uncondtonal cumulatve dstrbuton functon of a CDS default s F ( x ) and the default probablty p Recall: t at tme t, the default threshold s: K = F p t ()

15 = exp( λt) p t Condtonal on M=m, the default happens f: The condtonal default probablty s: ( ρ ρ ε) W = m+ < K ε < ( K ρm) ρ P( τ < t M = m) = F ( K ρm) ρ Where τ s the default tme for the frm. F s the cumulatve dstrbuton functon of ndvdual varable. In smple case, t follows standard normal dstrbuton Φ, N ( 0,). The condtonal default ndependent model s bult upon structured model, whle the default probablty s from ntensty model. The above conclusons play mportant role n later synthetc CDO prcng model. Drawbacks of Structural model The major drawback of the structural model s that the default probablty reduces to zero as tme horzon approachng zero. The structural model calculates very small spread for correspondng short tme horzon. However ths volates the observed market data. Snce there s some concern of credt event happens when the frm value drops suddenly n a short perod of tme. On the other hand, the ntensty model allows default jump n a very short tme horzon, and t s popular n the ndustry. In CDO prcng model, the condtonal default probablty s based on structural model, and the uncondtonal default probablty p ( t) at tme t s derved from ntensty model. 3

16 Credt Default Swap Frst the defaultable bond prce s nvestgated n the ntensty model. Let B ( t) be rsk free dscount factor at tme t. If nterest rate r s stochastc, then: ( 0 ) t B() t = E exp r( s) ds However, stochastc nterest rate r has lttle effect on the result of synthetc CDO model n ths thess, determnstc nterest rate r s used. Then B ( t ) becomes: ( r s ds 0 ) t () exp B t = If the bond recovery rate s 0, the correspondng defaultable dscount factor B () t s: ( 0 ) t () = () () = exp () +λ () B t B t S t r s s ds λ () t s the default ntensty and S( t) s the cumulatve survval probablty at tme t. Assume that the bond recovery rate R s based on par value, then B ( t) becomes: { } t exp 0 t s 0 0 λ ( ( τ λ τ ) τ) λ () exp = exp{ ( ) } t + + t exp s B t = r s + λ s ds + R λ s S s r τ dτ ds r s s ds R r d s ds In above expresson, the frst term s the payment f there s no default durng the CDS contract lfe tme. The second term s the expected recovery amount of the default bond f the default happens durng the CDS contract lfe tme. The above formula shows that B ( t) s not equal to B ( t) even when R =. Ths s due to the recovery payment s par once bond defaults whle the correspondng zero coupon bond value s less than par before maturty. So when B t s slghtly larger than B ( t ) ; assumng R=, B ( t) equals R =, B t only f rsk free nterest rate s zero. 4

17 For CDS wth 5 years to maturty or less, we can assume the nterest rate and default ntensty are constant, the above formula becomes: t ( r+ ) { } () exp B λτ t = r+ λ t + λ R e dτ Under no arbtrage opportunty assumpton, at the startng tme of CDS, the premum payment spread s determned to ensure the expected premum leg value equals to the expected default leg value. The CDS premum pays every three months. The followng s how to calculate the expected premum leg value Assumng y s the CDS premum spread, then: 0 m y ( t j t j ) exp ( λt j) B ( t j) + accrumed j= If default occurs, the CDS accrued nterest s pad wth the same spread rate for the nterval between default and last premum payment date. The expected accrued payment value approxmates to: m tj + tj t j tj accrumed = B { exp( λt ) } j exp λt j j= Addng the above two terms together, the followng ntegraton s a good approxmaton for the expected premum leg value: y t exp( λ ) 0 s B s ds On the other sde, the expected default leg value s: t s ( ) λ λ 0 R e B s ds Further f the CDS term s wthn 5 year, the credtworthness s relatvely stable. So we can assume that the hazard rate λ and recovery rate, R, are constant. At the begnnng of the CDS contact, the values for two legs are equal, so: t T λs λs ( ) λ = R e B s ds y e B s ds 0 0 5

18 In ths smplest case, hazard rate λ s determned by credt default y spread and recovery rate R: λ = y ( R) From the CDS spread, the hazard rate can be derved as above; hence the default probablty at any gven tme s obtaned. 4. FACTOR COPULAE FUNCTIONS The most commonly used copula functons are Gaussan and student t. Other copula functons nclude double t copula, exponental copula, Archmedean copula and Clayton copula functon, for detals see appendx. Common Copula functons Gaussan Copula As shown n prevous chapter, condtonal on common factor M, the default probablty s: ( K ρ m) P( τ < t M = m) =Φ ρ K =Φ ( p( t) ) Where Φ s normal cdf; M and ρ are ether scalar ρ or vector, where ρ becomes ρ. Student t Dstrbuton The student dstrbuton s the quotaton of a normal varable and square root of a Ch- Square dstrbuton scaled by ts degree of freedom, namely: X Where Z N χ X t Z / n n n 0, normal dstrbuton 6

19 Ths s a symmetrc t dstrbuton, t has smlar bell shape curve as normal dstrbuton, but wth heaver tal. Condtonal on each mplementaton of the random varable Z, the condtonal varable X Z = z follows normal dstrbuton. So the above condtonal ndependent Z / n Gaussan copula expresson s applcable. Double t Copula: In double t copula, the dstrbutons change from normal dstrbuton to student t dstrbuton. As shown n prevous chapter, assume the ndvdual frm random varable s correlated wth the common random varable: W n n n ρ n ξ ρ = + ξ Where t and t are ndependent t dstrbuton wth degree of freedom n and n. ξ s the market factor. The random varable s normalzed wth unt varance to get a unque expresson. n Km ρm n P( τ < t ξ = m) = tn ρ n n * KM = F pt F * s the cumulatve dstrbuton functon for W. 7

20 The double student t copula overcomes the thn tal problem n the Gaussan copula. By adjustng the degree freedom parameters, the double student t copula s capable to catch the correlated default structure more accurate than the Gaussan copula. However the determnant of the degree of freedom s a new problem. And the student t dstrbuton s not stable under convoluton 5 hence the double student t copula s computatonally costly. Generalzaton for Levy copula: Frst I wll generalze the expresson: W n n n ρ n ξ ρ = + ξ n n If we defne ξ and ξ as new random varables: common n n factor M and ndvdual varableε. If the cumulatve dstrbuton functon for ndvdual varableε s F, then the condtonal ndvdual default probablty s: P( τ < t X = m) = F ( F* ( p() t ) ρm) ρ Ths generc formula s also held n Levy one factor copula. For example, common factor M and ndvdual varableε follow Varance Gamma dstrbuton [Lucano (005)] or Normal Inverse Gaussan dstrbuton [Kalemanova (005)]. The dstrbuton parameters n these Levy factor models are calbrated to the quoted market tranches spreads. Ths calbraton s assocated wth the model n the followng chapters on synthetc CDO prcng. 5 The dstrbuton of sum of random varables s mplemented by convoluton. 8

21 5. SYNTHETIC CDO PRICING The general problem of prcng synthetc CDO s how to calculate the dynamc loss dstrbuton of the reference portfolo over dfferent tme horzon under some specfed default correlaton structure. The factor copula used n the synthetc CDO prcng model n ths thess s based on condtonal ndependence; namely condtonal on some common factor, the condtonal defaults are ndependent. The justfcaton for the factor copula model s that the correlaton coeffcents n the model have economy nterpretaton and are easy to commute. Smlar methods have been adopted: see [JPMorgan (004)] on base correlaton and [Elzalde (005)] for general revew on dfferent CDO prcng models. The frame work of prcng synthetc CDO n ths thess s:. Fnd out the margnal default dstrbuton under the rsk neutral measure. Identty default dependent structure 3. Dscount the default loss dstrbuton to calculate the expected default loss 4. Calculate the expected premum based on margnal default dstrbuton and default dependent structure 5. Fnd out premum payment spread for each tranche by dvdng expected default loss wth expected premum value The essental components n CDO prcng model are: ndvdual margnal default dstrbuton and default correlaton structure. If the default correlaton ncreases, the CDO s equty tranche spread decreases, whle the senor tranches spreads ncrease. The relatonshp between the spreads and correlaton for the mezzanne tranches s more complcated. 9

22 Snce the 5-year synthetc CDO tranches spreads are not senstve to nterest rate, the determnstc Euro nterest rate swap s used. Most synthetc CDOs are fnanced by nterest rate swap, so the swap rate s the proper choce for dscountng. And the recovery rate s assumed constant. Model Specfcaton I choose the condtonal ndependent copula model by adjustng margnal default dstrbuton to Levy dstrbuton. Frst a very generc t copula model s dscussed brefly and then condtonal ndependent factor copula model s elaborated. The condtonal ndependence approach s more parsmonous than the generc copula model. So t speeds up the computatonal tme and has clear economc nterpretaton. A Generc t Copula Model Consder a portfolo wth N dfferent names of CDS; ths generc CDO model does not requred equally weghted CDS pool. Let the random varableτ represents the random default tme for each name. Accordng to ntensty model, the rsk neutral default probablty for each name s: () = Pr ( τ ) = exp ( 0 ) t λ p t t u du The jont default tme dstrbuton can be expressed n the followng generc formula: Pr( τ T) = Pr( τ T τ T τ T ) =Φ,, N N ( φ ( p( T) ), φ ( p( T) ), φ ( p ( T ))) N N N N T s the tme vector; Φ N s a mult dmenson copula functon wth covarance matrx Σ and φ s the nverse of margnal cumulatve dstrbuton functon. 0

23 The above copula functon s an n-dmenson cumulatve dstrbuton functon. Under some loose condtons, dfferentatng ths n-dmenson jont cumulatve dstrbuton functon unquely determnes an n-dmenson jont probablty dstrbuton functon f ( T ) τ. 6 Here τ T meansτ T, τ T, τ T N N For mult dmenson Gaussan or t copula functon, there s explct expresson on ths jont default dstrbuton functon. For example, t copula wll gve the followng result on the mult dmenson probablty dstrbuton functon: 7 N+ v + v N T T z Σ z z Σ z f( τ T) = CNv, + λk( T) ( pk( Tk) ) + v k = C, v v Here C Nv, v+ N Γ = v Γ Σ ( vπ ) Γ( x) s Gamma functon. N, Here z t p ( T ), t,v s the cumulatve dstrbuton functon of one dmenson t k, v k k dstrbuton wth v degree of freedom; t,v s the correspondng nverse functon. After specfyng the expressons of premum and default value for the tranche n synthetc CDO wth attachment pont α and the detachment pont β, we can use ths jont default dstrbuton to compute the expected premum value and default value for that tranche. Ths approach s also applcable to cash CDO gven the cash flow structure. 6 See appendx on Sklar theorem 7 For detals, see [Andersen 003]

24 Condtonal Independent Model for synthetcs CDO Prcng In ths model, the synthetc CDO pool conssts of N equally weghted names of CDS. Snce the synthetc CDO s composte of equally weghted CDS, the condtonal default loss for each CDS s assumed homogeneous, hence exchangeable. The condtonal ndependent model can reduce the dmenson of covarance matrx Σ n the prevous generc t copula model and speed up the computaton. The condtonal ndependent model s also referred as sem analytcal parametrc model on dependent defaults, because there s an explct expresson for the condtonal cumulatve default loss functon of the CDS pool. The loss functon for each ndvdual CDS s: ( R) ( τ t ) N. Here R s recovery rate; random varableτ represents the random default tme for each name; and t s scaled by the equal weght N for each CDS. Defne: at tme t, the cumulatve portfolo percentage loss functon as () l t N = ( R) ( τ t ) The functon l() t s standardzed to range from 0 to. It takes the dscrete values of It s worthy to notce that: And () ( R) N k, k = 0,,, N. N N ( R N ) ( R) ( τ ) = = ( τ ) E l t = E t = Pr t N N ( τ t) = ( λ t) Pr exp

25 Ths shows the lnkage between the ndcaton functon and the ndvdual default probablty, whch s used to calculate the expected cumulatve portfolo percentage loss. Recalled that the condtonal default probablty for th CDS s: Where F, ( ) Pr( τ ) p t m < t M = m = F, ( K ρ m) ρ s the cumulatve dstrbuton functon of the th ndvdual varableε, t ( λ 0 ) K = F, exp s ds F, s the uncondtonal cumulatve dstrbuton functon of the th CDS default probablty. Snce the CDSs are equally weghted, the homogenety n the CDS pool s assumed, then the subscrpt s omtted. N R The cumulatve portfolo percentage loss functon l() t = τ t N dscrete values ( R) k, k = 0,,, N. N = takes the Notce that condtonal on the common factor, the ndvdual defaults are ndependent. Then the probablty of cumulatve portfolo percentage loss beng k ( R) k N Pr l() t = M = m = p t m p t m N k ( ) ( R) N k N k The condtonal default loss for each CDS s a bnary random varable; only two possble states are possble: ( R), wth probablty p ( t m) N 0, wth probablty p tm s: 3

26 And the CDS s assumed homogenous, so the condtonal default loss for each CDS s exchangeable. Thus the cumulatve portfolo percentage loss follows the above bnomal expresson. ( R) The uncondtonal probablty of cumulatve portfolo percentage loss beng k s N obtaned by ntegratng the product of the above condtonal expresson and probablty densty functon of common M over the value range of common factor M: ( R) k N k N k Pr l() t = = ( p( t m) ) p( t m) df m N k 3 Here, F3 ( m) s the cumulatve dstrbuton functon for common factor M at tme t. Now consder the probablty that the cumulatve portfolo percentage loss () l t N = ( R) ( τ t ) does not exceed x [0,]. x s percentage loss of the portfolo. N Defne F( x ) as the probablty that cumulatve portfolo percentage loss doesn t exceed x: F x = Pr l t x, x [0,] From here F( x) s called cumulatve portfolo percentage loss probablty functon or just cumulatve loss probablty for smple. F(x) s the cumulatve dstrbuton functon of portfolo percentage loss, both x and functon value range between 0 and. Notce that F ( x ) s a functon of tme t, and F ( x ) should be wrtten as (, ) however f there s no confuson n the context, I omt t n the expresson. F xt, Specfcally, n the above dscrete settng for portfolo default loss percentage s: 4

27 xn k = = k = 0 Pr l( t) F x Here xn s the maxmum ntegral less or equal to xn ( R) N Smlarly, condtonal cumulatve probablty of portfolo default loss percentage not exceedng x [0,] s defned as: Specfcally, F x m = Pr l t x M = m, x [0,] ( R) xn k F( x m) = Pr l( t) = M = m k = 0 N Generally, once we specfy ths cumulatve loss probablty, the portfolo s jont default dstrbuton s determned. So the tranches spreads n synthetc CDO are determned. The followng s the detals of the procedure to calculate the tranche spreads. Accordng to the CDO tranche structure, at tme t, the premum payment s the product of the spread and outstandng value for each tranche. At tme t, the outstandng value of tranche ( α, β ) takes the form of followng functon: [] ( β ()) ( α ()) + + Ht : l t l t Smlarly at tme t, the tranche loss value of tranche ( α, β ) s defned n ths functon: () () () ( () ) + + Q t : β α H t = l t α l t β Notce the fact that at any tme t the sum of tranche outstandng value and tranche loss value s the ntal tranche value: Q( t) + H( t) = ( β α ) Snce the portfolo percentage loss l( t) takes dscrete values, so the functons of Q() t and H() t are dscrete ncreasng functons; the ncrement s trggered y default occurrence. 5

28 Defne dq () t s as the ncrement of Q( t ) at tme t. dq ( t) s postve when default happens. Ths s for the later ntegraton. Let me further nvestgate the relatonshp between the values of Q( t ) and l() t : Q ( () β) () () α l() t α l() t l() t l() t + + t = l t l t, [ α, β] = β α, [ β,] 0, [0, α] Apply ths result, the expected value for tranche loss value Q( t ) can be rewrtten as: () () ( α) ( () β) + + E Q t = E l t l t β ( x α) df( x) df( x) α β α β β β ( F( x)( x α) α F( x) dx) ( β α) F( x) α = + = + β ( β )( β α) ( β α) ( β) = F F x dx+ F F β ()( β α) = F β ( F( x) ) α = dx α α F x dx The thrd lne s derved by ntegratng by parts. β The cumulatve portfolo percentage loss probablty functon F( x) s defned above as the probablty of cumulatve portfolo percentage loss not exceedng x: The last lne s due to the fact that F F x = Pr l t x, x [0,] = for any gven tme t, whch means the probablty of percentage loss less than s defntely for sure, wth probablty. And the expectaton of the outstandng tranche value can be calculated accordng to ths: 6

29 H = ( β α) Q ( β α) E Q( t) E t E t = β = F x dx α These two expectatons expresson hold over dfferent tme through the CDO lfe tme. The premum leg value PL() t s the present value of all spread payments made based on outstandng tranche value over the payment perod of tme: K = Δ PL t y t H t B t k = k k k And the expectaton of premum leg value PL( t) s: K = Δ E PL t y t E H t B t k = k k k Here t k, k =,, K are the premum spread payment dates, and t K = T s the maturty date of the synthetc CDO. Let t 0 = 0; Δ t = tk tk denotes the tme nterval between each payment; y s CDO spread rate; B ( t ) s the dscount factor. The expectaton s calculated under rsk neutral measure. If the accrued premum payment s consdered for the defaults between payment tmes, the more accurate premum expectaton of premum leg value should be: () E PL t = K K y Δ tke H( tk) B( tk) + y E H( tk) E H( tk ) B k= k= ( t ) k tk tk tk The second term s the accrued premum payment, assumng the defaults happen at the mddle ponts of each tme nterval. It s a frst order approxmaton. If hgher order approxmaton s appled, then the expectaton of premum leg value asymmetrcally approaches to the followng expresson: 7

30 T () = () () E PL t y B t E H t dt In the programmng the frst order approxmaton s used. But for concseness, n the followng part of the thess, I use the frst expresson wthout accrued payments. 0 Rephrase the defnton of dq() t as the ncrement of tranche loss value Q() t at tme t. dq() t s postve when default happens. The default leg value s the summery of product of dq( t) and the correspondng dscount factor at each default tme, namely: DL = T 0 () () B t dq t Ths s Remann-Steltjes ntegraton snce Q( t) s a dscrete ncreasng functon. And the expectaton of default leg value s: Then smplfy the DL expresson. T E[ DL] = E B() t dq() t 0 Frst ntegrate the default leg value DL by parts: Then apply the fact that: DL = T 0 () () B t dq t T t= T () () db() t Q() t = B t Q t f () t s the nstantaneous forward rate. So: t= 0 t () exp () t ( 0 ) B t = f s ds ; db t f ( t) B ( t) =. Then the above default leg value DL expresson becomes: 8

31 DL = T 0 () () B t dq t T t= T () () () () () = B t Q t t= 0 0 T () () () = B T Q T 0 f t B t Q t dt f t B t Q t dt The thrd lne s because that no default occurs at t=0, hence Q( t = 0) = 0. And the expectaton of default leg value DL becomes: T E[ DL] = E B() t dq() t 0 = E B T Q T T 0 T () () () 0 f t B t Q t dt = B T E Q T + E Q t f t B t dt Here the expectaton and ntegraton s assumed exchangeable. When synthetc CDO starts, y for the tranche ( αβ, ) the expectatons of default leg value and premum leg value are set equal: K [ ] E[ PL] E DL =. Snce E PL() t = y ΔtkE H( tk) B( tk), then the spread y of the tranche (, ) k = α β s: y = T + B T E Q T E Q t f t B t dt K k = 0 ΔtE H t Bt k k k If the forward rate s a constant r, the expectaton of default leg value E[ DL ] becomes: T T E[ DL] = E B() t dq() t = B( T) E Q( T) + E Q() t f () t B() t dt 0 0 Then the spread y s: T exp = exp rt E Q T + r rt E Q t dt 0 9

32 exp y = More detals on calculatng ( ) T exp 0 K te k H( tk) Bt ( k) rt E Q T + r rt E Q t dt k = Δ E H t and E Q t wll be conducted n the next secton. Cumulatve Loss Dstrbuton Functon of Homogenous Portfolo Snce the man problem n synthetc CDO prcng model s to derve the cumulatve loss dstrbuton of the correlated defaults, n ths secton detals on condtonal ndependent factor copula model s presented. Snce the synthetc CDO s composte of equally weghted CDS, the condtonal default loss for each CDS s assumed homogeneous, hence exchangeable.. Remnd that the default leg value s: T [ ] = + E DL B T E Q T E Q t f t B t dt 0 Snce E Q( t) s the only unknown, the followng shows how to calculate E Q ( t ). Remnd that for tranche( αβ, ), the tranche loss value functon s: l( t) () () l() t 0, [0, α] + + Q() t = ( l() t α ) ( l() t β) = l t α, l t [ α, β] β α, [ β,] Remnd the probablty that the cumulatve loss functon l( t) equals to ( R) k N k N k Pr l() t = = ( p( t m) ) p( t m) df m N k Snce l() t takes the dscrete values of loss value E Qt () becomes: ( ) k ( R) N k R, k =,, N, the expectaton of tranche N s: 30

33 Pr ( ) E Q t = ( ) N β N k R k R β α l t = + ( l() t α) Pr l() t = k= βn + N k= αn + N In there are one summaton and one ntegraton n ths expresson of E Q( t) ( R), snce k N k N k Pr l() t = = ( p( t m) ) ( p( t m) ) df( m) N requres ntegraton over k the range of common factor. T So n the expresson [ ] = + E DL B T E Q T E Q t f t B t dt, there are two ntegratons and one summaton. The addtonal ntegraton s over the tme horzon. 0 The ntegratons and summaton are assumed exchangeable. In the numercal mplementaton, t s more convenent to make the ntegraton over tme before the summaton n computng E Qt (). And the expectaton of tranche outstandng value can be derved by: Once E Q() t and E H( t) = ( β α) E Q( t) E H t are calculated, the tranche spread s delvered by: T + B T E Q T E 0 Q t f t B t dt y = K ΔtE k H ( tk) Bt ( k) k = The man dffculty and varety of the synthetc CDO prcng model le n how to compute cumulatve percentage loss probablty functon F x = Pr l t x, x [0,]. Ths crucal functon can be asymmetrc Levy dstrbutons provdng more flexblty and accuracy. But the parameters n Levy dstrbutons need to be calbrated to the market quoted spreads y. 3

34 Asymptotc Large Homogenous Portfolo Approxmaton The dea of asymptotc large homogenous portfolo approxmaton s from [Vascek, (987), (99)]. Accordng to the law of large number, when the number of CDS n the portfolo s very large, the dstrbuton of portfolo percentage loss approaches to the ndvdual default loss. However snce the accurate expresson on cumulatve portfolo percentage loss s shown above, ths asymptotc approxmaton s no more necessary. But, ths model can be modfed based on Levy processes, see [Albrecher (006)], [Baxter (006)] and [Moosbrucker (006a, 006b)], especally useful for rsk management 8. The Levy process n the model compensates the naccuracy ncurred by the asymptotc approxmaton. The adjusted parameters n Levy dstrbuton provde the capablty and flexblty to ft the market quoted spreads. Remnd the expectaton of tranche loss value functon: β ( Q() ) E t = F x dx α The cumulatve portfolo percentage loss probablty functon F( x ) s defned above as the probablty of cumulatve portfolo percentage loss not exceedng x: F x = Pr l t x, x [0,] Here the value of F( x) s the only unknown. As dscussed before, F( x) s a functon of tme t and should be expressed as F ( xt, ). Some researchers propose that n the fully dversfed portfolo of many equally weghted CDSs, the homogenety s assumed. Then by the law of large number, the portfolo loss dstrbuton l( t ) converges to the ndvdual t default probablty: p () t = Pr ( τ t) = exp ( λ( u) du 0 ) 8 The Basel Benchmark Rsk Weght s based ths asymptotc approxmaton. For detal, see appendx 4 3

35 One of the reprehensve papers s [Albrecher et al. (006)] : 9 F t, x = F ( x) F Portfolo Loss Ths s a functon of t snce C s a functon of t. M C ρ FZ x ρ In that paper Varance Gamma dstrbuton s proposed for FM ( x) and F ( x) based Portfolo Loss on [Schoutens (003)] and they also checked Normal Inverse Gaussan (NIG) dstrbuton under asymptotc large homogeneous approxmaton. In ths model, f F(x, t) s symmetrc, then: F ( x) = F Portfolo Loss M ρ FZ x C ρ Some authors gve ths result wthout clamng symmetrc assumpton. These loss dstrbuton functons F Z, FM and FPortfolo Loss ( x ) can be dfferent asymmetrc Levy dstrbutons such as Varance Gamma or NIG. However, snce exact expresson on prcng synthetc CDO s derved, I use the condtonal dependent copula factor model for the followng computatons. 9 See appendx for the conducton of asymptotc large homogenous portfolo approxmaton. 33

36 Frst Result from Condtonal Independent Factor Model Settngs: There are N = 5 CDS wth fve years to maturty n the synthetc CDO. The hypothetc tranches are 0-3%, 3-4% and 4-00%. The correlaton coeffcent s defned as ρ. Parameters: r = 0.05, R = 0.4, correlaton coeffcent ρ = 0.3, default ntensty λ = Normal dstrbuton N ( 0,) s assumed as underlyng dstrbuton n ths frst attempt. The followng table shows the tranches spreads obtaned from condtonal ndependent factor copula model shown n prevous context. Ths s exact number and later I wll compare the spreads values n ths table wth the smulated spreads values. Table Tranche Spreads from condtonal ndependent factor model Tranche 0%-3% 3%-4% 4%-00% Spread 4.48% 9.685% % 0 For the equty tranche 0-3%, the upfront payment based on a 5% annual spread s quoted; ths upfront payment effort mnmzes the counterparty rsk for the equty tranche. And the correspondng spread for the equty tranche s 67.3%. The upfront payment s used n all the followng examples for equty tranche 0-3%. The followng three fgures depct condtonal expectaton of premum leg value, ndvdual default probablty and condtonal cumulatve portfolo default probablty. 0 Namely bp, bp=

37 There two curves n Fgure. The upper curve shows the condtonal premum leg value for the 0-3% equty tranche condtonal on common factor M = m. Namely, the value of: K E PL M = m = Δ t E H t M = m B t [ ] k = k k k [ = m] E PL M Condtonal Expectaton of Premum Leg Value Fgure Condtonal Expectaton of Premum Leg Value wth respect to common factor M E PL M = m [ ] Upper curve (curve ): E[ PL M = m] Lower curve (curve ): E [ PL M = m] pdf ( m) Common Factor m Area below curve : uncondtonal expectaton of premum leg value The x-coordnaton s common factor value M=m. The lower bell shape curve (curve ) s the product of upper curve and the probablty densty functon of common factor M. Normal dstrbuton N ( 0,) s assumed as underlyng dstrbuton n ths frst smple attempt. Although N ( 0,) s symmetrc, curve s asymmetrc, skews to the rght. 35

38 The expectaton of premum leg value s the area below the bell shape curve. Ths expectaton s computed by ntegratng E [ PL M m] pdf ( m) possble range of common factor M: = numercal over the [ = ] E PL M m pdf m dm If the common factor M=m follows other asymmetrc Levy dstrbuton, n the above formula on the expectaton of premum leg value, the dstrbuton densty functon pdf ( m ) wll change accordngly. But these two curves stll have smlar shape. Fgure Indvdual Default Probablty wth respect to tme t Indvdual Default Probablty t n year These two curves depct ndvdual default probablty wth dfferent default ntensty. The above curve has hgher default ntenstyλ = 0.03, and 0.05 for lower curve. The tme ranges from 0 to 5 years. Fgure 3 shows the condtonal cumulatve loss probablty for the CDO portfolo percentage loss; t s condtonal on common factor M= 0. The common factor M s set as M=0. The x-coordnaton s the portfolo percentage loss. 36

39 ( m) F x M = s defned above as the condtonal probablty of cumulatve portfolo percentage loss not exceedng x: F x = Pr l t x, x [0,] It s called condtonal cumulatve percentage loss probablty functon. Fgure 3 Condtonal Cumulatve Percentage Loss Probablty wth respect to portfolo percentage loss x ( 0) F x M = F( x M = 0) x, (percentage loss) Snce F(x) s the cumulatve dstrbuton functon of portfolo percentage loss, both x and functon value range from 0 to. Spread Senstvty Analyss The followng tables and fgures show the senstvty of CDO tranches spreads wth respectve to the dfferent parameters n the condtonal ndependent factor copula model; the parameters nclude correlaton coeffcent, default ntensty, maturty, nterest rate and recovery rate. 37

40 . Spread Senstvty of Correlaton Coeffcent ρ for 0-3% tranche Table Spread Senstvty of Correlaton Coeffcent ρ for 0-3% tranche Correlaton Coeffcent Upfront Payment % 54.77% 4.48% 3.30% 5.45% 0.06% 5.57%.6% 7.69% For tranche 0-3%, the hgher correlaton coeffcent ρ, the lower the upfront payment. Fgure 4 Spread Senstvty of Correlaton Coeffcent ρ for 0-3% tranche Spread 0-3% tranche Correlaton Coeffcent The hgher coeffcent assgns hgher spread for senor tranche and lower spread or upfront payment to the equty tranche, vce versa. The relatonshp between the mezzanne tranche spread and correlaton coeffcent s more complcate. It s not monotonc functon, ncreasng wth correlaton coeffcent when coeffcent s at low level, but decreasng when coeffcent s hgh.. Spread Senstvty of Default Intensty λ for 0-3% tranche Table 3 Spread Senstvty of Default Intensty λ for 0-3% tranche Default Intensty Spread

41 The hgher default ntensty results n hgher premum spread; but the margnal ncrement decreases. Fgure 5 Spread Senstvty of Default Intensty λ for 0-3% tranche Spread 0-3% tranche Default Intensty λ 3. Spread Senstvty of Interest Rate r for 0-3% tranche Table 4 Spread Senstvty of Interest Rate r for 0-3% tranche Interest Rate Tranche Spread Fgure 6 Spread Senstvty of Interest Rate r for 0-3% tranche Spread 0-3% tranche Interest Rate 39

42 There s another llustraton on spread senstvty of nterest rate for the 9-% tranche. Table 5 Spread Senstvty of Interest Rate r for 9-% tranche Interest rate 0% % % 3% 4% Tranche Spread 45. bp bp bp bp 43.7 bp Here bp s base pont, bp = 0.0%. These two tables demonstrate that the tranches spreads are not senstve to the nterest rate. Nether of then are senstve to nterest rate. The results for other tranches on nterest rate senstvty are smlar. Ths justfes the usage of constant nterest rate n synthetc CDO prcng model. 4. Spread Senstvty of Maturty for 0-3% tranche Table 6 Spread Senstvty of Maturty for 0-3% tranche Maturty (n year) Spread Fgure 7 Spread Senstvty of Maturty for 0-3% tranche Spread 0-3% tranche {Upfront} {Payment} Maturty (year) From the model, based on 5% annual spread rate, as CDO contract tme ncreases from to 5 years, the upfront payment for equty tranche 0-3% decreases from to

43 5. Spread Senstvty of Recovery Rate for 0-3% tranche Table 7 Spread Senstvty of Recovery Rate for 0-3%Tranche Recovery Rate Tranche Spread Fgure 8 Spread Senstvty of Recovery Rate for tranche 0-3% Spread 0-3% tranche Maturty (year) From the above fgures, the spread of the equty tranche 0-3% s not senstve to nterest rate, but t s very senstve to the correlaton coeffcent, default ntensty and recovery rate. Hgh correlaton ncurs a low the spread for equty tranche. The spread for senor tranche s also senstve to correlaton coeffcent, default ntensty and recovery rate. But hgh correlaton ncurs a hgh spread for senor tranche, such as senor tranche 9-%. 4

44 6. SIMULATION METHOD FOR PRCING SYNTHETIC CDO Smulaton Procedure Monte Carlo smulaton s convenent to mplement. Snce there s no mperatve requrement for explct expresson, ths method s very flexble, but at the cost of long computatonal tme. Because convergence rate s qute slow for hgh dmenson smulaton on the jont default dstrbuton, smulaton tme s long. There are some new smulaton methods to reduce smulaton tme, such as Important Samplng and Sequental Monte Carlo. But they are not covered here, snce the condtonal ndependent factor model s mplemented n the next chapter. Model Settngs: Agan consder a portfolo wth N equally weghted CDS, each one wth notonal N. At tme t, gven default tmeτ, =... n, the cumulatve default loss functon s: And () l t ( R) N = { τ t} = N ( R N ) () = { τ t} E l t = Pr N { τ t} = ( λ t) Pr exp So the tranche loss functon and remaned outstandng are gven as: H Q ( s) = β l( s) α l s H Let t < t... < tn denote the payment dates. + + ( ) β α ( α) ( β) s = s = l s l s + + For each payment date, calculate the expected value of premum payment; then sum the product of these payment and rsk free dscounted factor over all dates to get premum Under rsk neutral measure. 4

45 leg PL. Smlar method s appled to for expected value of default leg DL. From default leg DL and premum leg PL, the spread y s derved. The smulaton s based on the followng one factor and mult factor copula models. One Factor Copula Model: The one factor model assumes that the th ndvdual varable W correlates wth each other by dependng on a common market factor M : W = ρ M + ρε Where ε s dosyncratc factor of frm ; t s ndependent of M. Snce,, T W W W N s ε and M are ndependent, covarance matrx of Mult Factor Copula Model:, = j Σ (, j) = ρρ j, j The mult factor model assumes that the th ndvdual varable W s correlated wth a few common market factors M k : W K K = ρ, kmk + ε ρ, k k= k= Where K s the number of common factors. ( ) and,, T ε ε N,, T M M M K are ndependent. So condtonal on the common factors M k, W s ndependent wth each other; the condtonal default s ndependent. In homogenous portfolo, the subscrpt s dropped, hence: W K K ρkmk ε ρk k= k= = + For example, f there are two common factors: W = ρ M + ρ M + Z ρ ρ 43

46 The common factor M s shared by all ndvdual varablesε ; M s dvded by sectors. In each sector, ε has same values of M and M ; cross dfferent sectors, only M keeps unchanged; M vares ndependently cross sectors. In ths two factor model, the covarance matrx of X becomes:, = j Σ (, j) = ρ + ρ, j name, j n same secton ρ, otherwse Accordng to the sngle or two factor model, I smulate the jont random varables τ based on the dstrbutons such as Normal or NIG. Then convert these random varables τ to the default tmes to compute synthetc CDO spreads, as shown followng. Frst generate the random varables accordng above descrpton. The n each mplementaton, calculate the premum and default leg value, the spread s ther quotent. Here are the steps for Monte Carlo smulaton: Generate a n jont random varablesw as above formulae Calculate the default tme by τ dstrbuton functon of ln ( u ) =, u F( W ) λ W. See followng recap for detals. =, F s the cumulatve Calculate the cumulatve default loss functon for each payment date based on default tmes Calculate the present values of premum and default payment Sum the present values to get the total value of premum leg payment PL and default leg payment DL Repeat the above steps, calculate the average of both premum leg PL and default leg DL. Calculate tranche spread from these two values. The second step s derved n the followng recap: 44

47 Recap: { τ } Pr( τ ) E t = t = u Let the cumulatve value u F( W ) = equal to default probablty: Pr ( τ ) exp( λ ) u = p t = t = t The default tme s an exponental dstrbuton. And t can be generated by: ln τ = λ ( u ) { τ } ( τ ) E t = Pr t = u Smulaton Result Sngle Factor Copula Model Smulaton Result Factor Copula model settng: N = 5, r = 0.05 and R = 0.4 correlaton ρ = 0.3and default ntensty λ = The tranches are 0-3%, 3-4% and 4-00%. The smulaton s based on Matlab software at a PC wth.0g CPU. Table 8 Smulaton Result n One Factor Gaussan Copula Model Iteraton 500 5,000 5,000 50,000 Smulaton Tme 5.4 second 5.75 second 66.7 second 530. second 0-3% 4.03% 4.8%, 4.69% 4.98% 3-4% 9.406% 9.798% 9.769% 9.78% 4-00% 0.355% 0.348% 0.359% 0.355% The equty tranche spread s quoted as the upfront payment based 5% annual premum rate. The absolute spreads wll be 67.5 %, 67.99%, 67.84% and 67.63% correspondngly. The followng s smulaton result comparson among one factor and two factor s model based on Gaussan and Normal Inverse Gaussan (NIG) dstrbutons. For more detals on NIG dstrbuton property, see appendx 3. 45

48 Mult Factor Copula Model Smulaton Result In ths mult factor model, two factor copula model s appled, the market factor and the secton factor. Settngs: N = 5, r = 0.05, R = 0.4, default ntensty λ = 0.03, correlaton ρ = 0.3, ρ = 0., or ρ = 0.3, ρ = 0.. Table 9 Two Factor Gaussan Copula Model Iteraton 500 5,000 5,000 50,000 Computng Tme (seconds) % 40.46% 39.48% 39.06% 39.06% 3-4% 9.775% 9.760% 9.55% 9.55% 4-00% 0.384% % % % By comparng the results wth one or two correlaton factors, t s shown that wth an addtonal factor the spread for equty tranche decrease and the spread for senor tranche ncrease. It s smlar to the effect of ncreasng correlaton coeffcent. Table 0 Smulaton Result Comparson among Factor Copula Model (Gaussan and NIG Factor Models) In one fact model, ρ = 0.3 (or ρ = 0.3 ) In two fact model, ρ = 0.3, ρ = 0.3 (or ρ = 0.3, ρ = 0. ) NIG dstrbuton settng: u = 0 andδ = Copula Model Factor NIG Factor NIG Factor Factor Gaussan Gaussan Iteratons 5,000 5,000 50,000 50,000 Computng Tme 00 mnutes 00 mnutes 6 mnutes 6 mnutes 0-3% (Spread) 76.36% 74.4% 67.63% 65.57% 3-4% 9.4% 0.00% 9.7% 9.58% 4-00% 0.43% 0.47% 0.35% 0.37% 46