CDS index tranches and the pricing of credit risk correlations 1

Transcription

1 Jeffery D Amao [email protected] Jacob Gynelberg [email protected] CDS index ranches and he pricing of credi risk correlaions 1 Sandardised loss ranches based on credi defaul swap (CDS) indices have increased liquidiy in he marke for credi risk correlaions. Alhough progress is being made, quaniaive modelling of hese correlaions is complex and no ye fully developed. JEL classificaion: G12, G13, G14. One of he mos significan developmens in financial markes in recen years has been he creaion of liquid insrumens ha allow for he rading of credi risk correlaions. Prime among hese insrumens are CDS index ranches. Broadly pu, index ranches give invesors, ie sellers of credi proecion, he opporuniy o ake on exposures o specific segmens of he CDS index defaul loss disribuion. Each ranche has a differen sensiiviy o credi risk correlaions among eniies in he index. One of he main benefis of index ranches is higher liquidiy. This has been achieved mainly hrough sandardisaion, ye i is also due o he liquidiy in he single-name CDS and CDS index markes. In conras, possibly owing o he limied liquidiy in he corporae bond marke, securiies referencing corporae bond indices have no been acively raded. The sandardisaion of index ranches may prove o be a significan furher sep owards more complee markes. Credi risk correlaions have always been key risk componens in porfolios of credi-risky securiies. However, up unil now, sandardised producs for he rading of credi risk correlaions have no been available. The emergence of index ranches herefore fills a gap in he abiliy of he markes o ransfer cerain ypes of credi risks across individuals and insiuions. We examine CDS index ranches in his aricle. In he firs secion we inroduce hese securiies, focusing on he mechanics of CDS-based conracs and marke liquidiy. In he second secion we discuss he pricing of CDS index 1 We hank JPMorgan Chase for providing us wih daa; Rishad Ahluwalia, Jakob Due and Mike Harris of JPMorgan Chase for useful discussions; Henrik Baun, Claudio Borio, Ingo Fender, Frank Packer and Eli Remolona for helpful commens; and Marian Micu for research assisance. The views expressed in his aricle are hose of he auhors and do no necessarily reflec hose of he BIS. BIS Quarerly Review, March

2 ranches, wih an emphasis on how hese insrumens allow for he rading of credi risk correlaions. CDS-based conracs: characerisics and liquidiy To undersand he advanages offered by CDS index ranches for he rading of credi risk correlaions, i is firs necessary o undersand heir composiion, namely, he srucure of CDS indices and he underlying single-name CDS conracs. CDS conracs A single-name CDS conrac is an insurance conrac covering he risk ha a specified credi defauls. Following a defined credi even, he proecion buyer receives a paymen from he proecion seller o compensae for credi losses. In reurn, he proecion buyer pays a premium o he proecion seller over he life of he conrac. 2 There are wo main reasons why CDS conracs are more liquid han mos corporae bonds. Firs, hey are more sandardised. For insance, he credi evens ha rigger paymen o he proecion buyer are now clearly defined in he ISDA credi derivaives definiions (ISDA (2003)). 3 This is also he case for he selemen mehod. 4 Second, CDS conracs allow marke paricipans o go long credi risk wihou a cash paymen, as well as go shor credi risk wih less difficuly and a lower cos han wih corporae bonds. Single-name CDSs are he building blocks CDS indices A CDS index conrac is an insurance conrac covering defaul risk on he pool of names in he index. Index conracs differ slighly from single-name securiies. The main difference is ha a buyer of proecion on he index is implicily obligaed o pay he same premium, called he fixed rae, on all he names in he index. In addiion, index conracs resric he eligible ypes of credi evens o bankrupcy or failure o pay. 5 In he case of a credi even, he eniy is removed from he index and he conrac coninues (wih a reduced noional amoun) unil mauriy. The marke liquidiy of CDS index conracs is enhanced by: (1) he emergence of widely acceped benchmark indices, which comprise he mos Liquidiy of CDS indices is enhanced by 2 Several sources conain descripions of CDS conracs and heir feaures (eg Anson e al (2003) and O Kane, Naldi e al (2003)). 3 Credi evens include bankrupcy, failure o pay, repudiaion and maerial resrucuring of deb (including acceleraion). 4 Payoffs can be seled eiher by cash (wih he proecion buyer receiving par minus he defaul price of he reference asse) or in physical form (where he proecion buyer delivers he defauled securiy o he proecion seller in reurn for a cash paymen of par). 5 This corresponds o he no-resrucuring (XR) documenaion clause in single-name CDS conracs, ie excluding deb resrucuring as a riggering even (see ISDA (2003) for a descripion of documenaion clauses). See O Kane, Pedersen and Turnbull (2003) for a discussion of common marke pracices, as well as Packer and Zhu (his issue of he Quarerly Review). 74 BIS Quarerly Review, March 2005

3 he creaion of benchmark indices across across regions regions and and secors secors liquid single-name CDS conracs in he marke and have a group of global dealers commied o marke-making; (2) a clear geographical focus, relaively sable secor-raing composiion and sandardised mauriies for each index; and (3) he availabiliy of wo differen conrac formas. We consider each elemen in urn. Firs, he main raded CDS indices have now been consolidaed ino a single family under he names DJ CDX (for Norh America and emerging markes) and DJ itraxx (for Europe and Asia); see Table 1. 6 The composiion of he new indices is chosen by paricipaing dealers based on he liquidiy of individual conracs, ie he mos acively raded names are included. Once formed, an index remains saic over is lifeime, excep for eniies ha defaul, which are eliminaed from he index. However, every six monhs a new rebalanced index is launched and associaed on-he-run securiies are issued. Second, indices have been creaed for he main currencies, invesmen grade and non-invesmen grade credis and he main secors. A he CDS indices 1 By region Norh America Europe Japan Asia excl Japan Ausralia Emerging markes Maser CDX.NA.IG (125) CDX.NA.HY (100) itraxx Europe (125) itraxx Corporae (52) 4 itraxx CJ (50) 2 itraxx Asia (30) itraxx Ausralia (25) CDX.EM (14) 3 itraxx Crossover (30) 5 Sub-indices Financials (24) Consumer (34) Energy (15) Indusrials (30) TMT (22) HiVol (30) B (44) BB (43) HB (30) Financials (15) Auos (10) Consumer cyclicals (15) Consumer noncyclicals (15) Energy (20) Indusrials (20) TMT (20) HiVol (30) Financials (10) Capial goods (10) Tech (10) HiVol (10) Korea (8) Greaer China (9) 6 Res of Asia (13) 7 None None 1 Earlier generaions of DJ Trac-x and iboxx indices are sill raded. This able summarises he composiion of he mos recenly issued series, DJ CDX and DJ itraxx, which are a by-produc of he merger beween he DJ Trac-x and iboxx families. The number of reference eniies in each index is given in parenheses. 2 Maximum of 10 names in a given secor. 3 Includes only sovereigns: Brazil, Bulgaria, Colombia, Korea, Malaysia, Mexico, Panama, Peru, he Philippines, Romania, Russia, Souh Africa, Turkey and Venezuela. 4 Includes he larges, mos liquid non-financial names from he iboxx EUR Corporae bond index. 5 Mos liquid non-financial names raed BBB/Baa3 or lower and on negaive oulook. 6 Includes China, Hong Kong SAR and Taiwan (China), wih a leas wo names from each. 7 Includes India, Malaysia, he Philippines, Singapore and Thailand. Table 1 6 Two compeing families of indices (Trac-x and iboxx), suppored by differen dealers, were iniially launched in Las year hese indices were merged o form he new indices, which are adminisered by Dow Jones. BIS Quarerly Review, March

4 Secors and raings disribuions Secors Raings 3 Norh America¹ Europe² CNS EN FIN IND TMT 0 AAA AA A BBB Noe: CNS = consumer, EN = energy, FIN = financials, IND = indusrials, TMT = echnology, media and elecommunicaions. For Europe, CNS includes consumer cyclicals and consumer noncyclicals, and IND includes indusrials and auos. 0 1 DJ CDX.NA.IG.3. 2 DJ itraxx Europe Series 2. Poor s and Fich (where available). 3 Average raing from Moody s, Sandard & Sources: Bloomberg; BIS calculaions. Graph 1 invesmen grade level, he broad indices in Norh America (CDX.NA.IG) and Europe (itraxx Europe), which are he mos acively raded, are each composed of 125 reference eniies, wih an equal weighing given o each. There are also indices for seleced secors; an index based on names wih high sysemaic exposures (ie high marke beas); indices composed of speculaive grade firms; and indices for regions oher han Norh America and Europe, such as Japan, Asia (excluding Japan), Ausralia and a selecion of emerging marke counries. Graph 1 shows he disribuion across secors and raings in he mos recenly issued versions of CDX.NA.IG and itraxx Europe. Securiies on he main indices are available a five- and 10-year mauriies. Third, wo ypes of index conracs, unfunded and funded, are raded o beer ailor he securiies o invesors preferences wih respec o funding forma and counerpary risk exposure. An unfunded conrac is simply a muliname CDS; he funded version is a bond, where, a originaion, he buyer of proecion receives a pool of collaeral securiies from he proecion seller and pays an upfron noional, in addiion o paying a quarerly premium. In an unfunded conrac, he proecion buyer is exposed o counerpary risk, whereas in a funded ransacion he proecion buyer is exposed o he risk of credi deerioraion in he collaeral pool (bu no o counerpary risk). 7 The relaively liquid naure of hese insrumens, compared o oher credi producs, has been refleced in fairly igh bid-offer spreads, a leas on he mos acively raded conracs. For insance, bid-offer spreads on five-year unfunded conracs on he CDX.NA.IG index have ypically been in he range and availabiliy of differen conrac ypes Bid-offer spreads have been igh 7 In he even of defauls in he index, he proecion buyer sells he collaeral o recover losses on he CDS index. 76 BIS Quarerly Review, March 2005

5 0.5 4 basis poins. To pu he size of his bid-offer differenial in conex, spreads on he broad invesmen grade indices have averaged abou 62 basis poins in Norh America and 45 basis poins in Europe since January 2004 (Graph 2, lef-hand panel). 8 CDS index ranches Compared o oher CDOs, index ranches are sandardised and more liquid CDS index ranches are synheic collaeralised deb obligaions (CDOs) based on a CDS index, where each ranche references a differen segmen of he loss disribuion of he underlying CDS index. 9 The main advanage of index ranches relaive o oher CDOs is ha hey are sandardised. Sandardisaion applies o boh he composiion of he reference pool and he srucure ( widh ) of he ranches. Sandardisaion helps o foser greaer liquidiy in he secondary marke. The developmen of a liquid secondary marke for he rading of oher CDO ranches has hus far been elusive largely because he srucure of mos CDOs has been highly cusomised. 10 CDS index spreads 1 Maser invesmen grade indices Tranches 2 3-7% 7-10% 10-15% 15-30% Norh America Europe Jan 04 Mar 04 May 04 Jul 04 Sep 04 1 On-he-run five-year swap spreads, in basis poins Nov 03 Feb 04 May 04 Aug 04 2 Norh America maser invesmen grade Source: JPMorgan Chase. Graph 2 8 A originaion, he fixed spread for he index swap is se o be roughly equal o he average CDS spread for he names in he index. As ime progresses, he index swap will have a posiive value o he proecion buyer when average spreads on individual names are high compared wih he fixed rae. In his case, new buyers of proecion would make a paymen o he proecion seller equal o his difference (and vice versa when average spreads are lower han he fixed rae). 9 In general, a CDO is a srucured finance produc in which he credi risk on a pool of asses is sold o invesors. The claims issued agains he asses in a CDO are prioriised (srucured) in order of senioriy, ie here are differen levels or ranches of deb securiies. This ypically includes one or more invesmen grade classes and an equiy (firs loss) ranche. See CGFS (2005) for more deail on CDOs and heir economics, and Gibson (2004) for a discussion of he risks in synheic CDOs. 10 One of he main growh areas in he CDO marke over he pas couple of years has been socalled bespoke single-ranche CDOs. These are designed in accordance wih a specific invesor s wishes. I could be argued ha marke forces are pushing owards wo exremes: BIS Quarerly Review, March

6 Tranches have been issued on several indices, hough mos rading o dae has been concenraed in he CDX.NA.IG index. 11 There are five ranches based on his index. The lowes ranche, known as he equiy ranche, absorbs he firs 3% of losses on he index due o defauls. If defauls occur over he lifeime of he ranche conrac, he invesor in an equiy ranche is obliged o pay is counerpary an amoun equal o he losses from defaul (he difference beween par and he recovery price of he defauled asse) up o a maximum of 3% of he oal index. The nex ranche (mezzanine) absorbs losses of 3 7% and is herefore fully insulaed, by he equiy ranche, from losses up o 3%. Furher losses are absorbed by higher-ranking ranches. The 7 10% and 10 15% ranches are known as he senior ranches, while he super-senior ranche covers losses of 15 30%. 12 In reurn for bearing he risk of losses, invesors receive a quarerly paymen from buyers of proecion equal o a premium imes he effecive ousanding noional amoun of a given ranche. 13 The premiums on he mezzanine and senior ranches are a running spread wih no upfron paymen. By conras, buyers of proecion on an equiy ranche make an upfron paymen ha is a percenage of he original noional of he conrac, in addiion o paying a running spread premium of 500 basis poins. 14 The presence of a (relaively large) upfron paymen changes he prospecive iming of cash flows o he invesor in an equiy ranche compared o he case of receiving a running spread only, and herefore he equiy invesor s exposure o he iming of defauls is differen. Marke quoes of he premiums on he mezzanine and higher ranches are shown in Graph 2 (righ-hand panel). 15 Index ranches arge segmens of he defaul loss disribuion Trading credi risk correlaions: pricing he ranches Credi risk correlaions among he names in he index have a large impac on he riskiness of CDS index ranches. The high degree of sensiiviy o credi risk correlaions is clearly refleced in he pricing of he ranches. This implies Credi risk correlaions affec he riskiness of index ranches sandardised index ranches (which can be used in acive rading) and bespoke ranches (which are designed for buy-and-hold purposes). 11 Crediflux repors ransacions volume of $10.2 billion in he second quarer of 2004, wih 82% of his oal referencing iboxx CDX.NA.IG Series 2 and Trac-x NA combined. 12 Conracs for insuring agains losses greaer han 30% of he index currenly do no exis. 13 The effecive noional is he original noional less any losses incurred due o defauls ha have impaced on he ranche (wih a floor a zero). 14 A conrac wih an upfron paymen can be convered ino a conrac wih a running spread and no upfron paymen. This is done by dividing he upfron paymen by he (risky) duraion of he ranche and adding any running spread. Thus, an equiy ranche wih an upfron paymen of 37.5%, a running spread of 500 basis poins and risky duraion of 3.75 is equivalen o a conrac wih a running spread of (37.5*100/3.75) basis poins = 1,500 basis poins. See O Kane and Sen (2003) for an analysis of upfron versus running spread quoing convenions. 15 Bid-ask spreads have been 1 2 basis poins for he mos senior ranche and 5 10 basis poins for he mezzanine ranches, while hey have been basis poins for he equiy ranche. 78 BIS Quarerly Review, March 2005

7 The pricing of index ranches has focused on defaul ime correlaions ha, in conjuncion wih he greaer liquidiy of hese insrumens relaive o oher muli-name credi producs, hese securiies offer a relaively efficien way of rading his form of risk. To illusrae he imporance of credi risk correlaions on he value of he ranches, consider ranches wih a five-year mauriy on a CDS index consising of 125 names whose characerisics are similar o he average credi in CDX.NA.IG Series The lef-hand panel of Graph 3 shows he five-year loss rae disribuion, as a percenage of ranche size, from he equiy o junior mezzanine ranche. The righ-hand panel repors he expeced loss as a percenage of he oal index, on each ranche. This clearly illusraes ha loss, boh relaive and absolue, is declining in ranche senioriy. Indeed, he expeced loss on he equiy ranche is abou 40 50% of noional in he cases shown in he graph. This example indicaes ha he marke value of a given CDS index ranche will depend upon he join defaul loss probabiliy disribuion for he reference eniies in he index. In general, he join defaul loss disribuion incorporaes boh he correlaions beween individual defaul probabiliy levels and he correlaions beween individual defaul imes. In addiion, he rue loss disribuion also incorporaes correlaions beween losses-given-defaul and defaul probabiliy levels (eg losses end o be larger when he overall risk of defaul is higher, such as in recessions) and correlaions beween lossesgiven-defaul and defaul imes (eg losses may be larger when defauls are clusered, such as when here are muliple defauls in an indusry over a shor period of ime). Index loss rae disribuions and expeced losses on index ranches 1 Loss disribuions Expeced losses Correlaion = 0.05 Correlaion = Correlaion = 0.05 Correlaion = % 5% 10% 15% 20% 25% 30% 0-3% 3-7% 7-10% 10-15% 15-30% Noe: he x-axis is percenage loss (lef-hand panel) and ranche (righ-hand panel). 0 1 In per cen. See ex for deails on compuaions. Source: BIS calculaions. Graph 3 16 To calculae he loss disribuion, we use a one-facor Gaussian copula model (see below) and assume idenical five-year defaul probabiliies (2.97%), consan recovery raes (40%) and consan idenical pairwise defaul ime correlaions (0.05 or 0.3). The defaul rae is esimaed using Moody s daa for US Baa-raed corporae issuers over he period The recovery rae is he average for defauled senior unsecured US corporae bonds. The chosen values of defaul ime correlaions are roughly in he range used by he raing agencies. BIS Quarerly Review, March

8 Pricing index ranches The premium on an index ranche is he spread paid by he proecion buyer ha equaes he expeced presen value of defaul coss o be borne by he proecion seller ( proecion leg ) o he expeced presen value of invesing in he ranche ( premium leg ). The value of he premium leg is he presen value of he spread paymens he proecion seller receives from he proecion buyer. Index conracs specify M quarerly paymen daes, = 1, 2,, M, on which he buyer of proecion makes paymens o he seller. Noe ha paymens are only made as long as he (uncerain) effecive noional of he ranche a ime i, denoed by N( i ), is posiive. Assume also ha invesors discoun expeced fuure income sreams using he (uncerain) discoun facors D(0, i ). Given he ranche premium S, he expeced presen value of he premium leg is: V M [ D(0, i ) N( i ] prem = S E i = ) 1 The expeced ranche sizes depend on he number and iming of any fuure defauls and he expeced coss of hese fuure defauls (ie recovery raes). The presen value of he premium leg is lower if: he premium is low; he recovery rae is low; and defaul losses are incurred early. The expeced presen value of he proecion leg is: V M [ D(0, i ) ( N( i ) N( i )] pro = E i = 1 1) The presen value of he proecion leg is lower if: he ranche size does no change; he recovery rae is high; and defauls occur lae during he conrac period. The ranche premium is found by solving V prem = V pro for S: Implemenaion S E M [ D(0, ] i = i ) ( N( i ) N( i 1)) 1 E[ D(0, i ) N( i )] = M i = 1 As can be seen from he equaions above, wo key facors are required o deermine S: fuure effecive ranche sizes and discoun facors. Discoun facors can be found via mehods also used for oher financial insrumens (see Rebonao (2002)). To evaluae fuure ranche sizes, however, several inpus are needed: (1) he losses-given-defaul; (2) he number of defauls; and (3) he iming of defauls. All of hese quaniies are uncerain, and herefore expecaions of hem mus be formed. For he loss-given-defaul (or one minus he recovery rae), a simple approach is o assume ha recovery raes are consan and equal o he average hisorical recovery rae on senior unsecured bonds for US corporaions (ypically around 40%). Recovery raes can also be esimaed from CDS spreads. Individual defaul probabiliies for he names in he index can be esimaed direcly from singlename CDS spreads. Alernaively, hey can be inferred indirecly from equiy prices (eg Moody s KMV s expeced defaul frequencies). Noe ha a recovery rae assumpion is needed o exrac defaul probabiliies from CDS spreads. The iming of defauls for he N eniies over he lifeime of he conrac can be calculaed from a join defaul ime probabiliy disribuion. As his is unknown, a common approach is o assume ha defaul imes follow an N-dimensional mulivariae normal disribuion, ie he so-called Gaussian copula (see Nelsen (1999), Li (2000) and Cherubini e al (2004)). In pracice, when defauls occur beween paymen daes, sellers of proecion receive an accrual paymen a he nex paymen dae based on he previous effecive ranche size. Noe ha any upfron paymen on he equiy ranche can be included in he presen value of he premium leg by adding a consan. Expecaions are aken under a risk neural measure, ie risk-adjused expecaions. Assuming proecion buyers receive compensaion a he nex scheduled paymen dae afer a defaul has occurred. 80 BIS Quarerly Review, March 2005

9 In a one-facor Gaussian copula model, he correlaions in defaul imes are assumed o be equal and consan across eniies. This is equivalen o assuming ha here is a direc mapping from a laen random variable X i o defaul imes, where he evoluion of X i is given by: X = ρ M + 1 ρ i Z i where M is a normally disribued random variable, he Z i s are muually uncorrelaed and normally disribued random variables and 1 < ρ < 1 is he consan pairwise correlaion beween defaul imes (see Hull and Whie (2004) for furher deails). One inerpreaion of he one-facor Gaussian copula approach is ha X i is he value of asses held by eniy i, and eniy i defauls if is asses fall below some hreshold. This is similar in spiri o a Meron-ype model, where he opion o no repay deb is exercised when asse value reaches a given hreshold. Wih his inerpreaion, M can be seen as he single common risk facor, while he Z i s are N idiosyncraic risk facors, driving he values of firms asses, and hus defaul imes. The correlaion parameer ρ can be esimaed from correlaions of equiy reurns, which are ypically in he range 0 30%. Up unil now, he pricing of index ranches has focused on capuring he implicaions of defaul ime correlaions (see box). For his purpose, he socalled one-facor Gaussian copula model has become he marke sandard for gauging he prices on index ranches, similar o he Black-Scholes model for rading opions. The erm copula is mean o emphasise ha his ype of model couples individual-name defaul probabiliy disribuions ogeher o form a join defaul probabiliy disribuion (see Nelsen (1999)). The one-facor Gaussian copula assumes idenical consan pairwise defaul ime correlaions across all firms, normally disribued defaul imes and a normal join defaul probabiliy disribuion. These simplifying assumpions make he one-facor Gaussian copula relaively easy o use o calculae valuaions, which is one of he main reasons for is populariy. Defaul ime correlaions and ranche pricing Higher correlaions imply more defaul clusering The imporance of defaul ime correlaion for he riskiness of he differen index ranches is apparen in Graph 3. I is shown in he lef-hand panel ha, depending on he ranche, he probabiliies of having eiher very small or very large loss raes are higher when defaul ime correlaion is higher. This can easily be seen by comparing wo exreme, albei unrealisic, cases. Firs, if correlaion is zero, he probabiliy of zero names (ou of 125) defauling wihin a five-year period is ( ) 125 = 2.31%, where 2.97% is he average hisorical five-year-ahead defaul rae of Baa-raed firms. By conras, if correlaion is equal o one (ie if he porfolio can be viewed as a single credi), he probabiliy of zero names defauling is 97.03%. Ye he index could lose one minus he recovery rae (= 1 0.4) wih probabiliy 2.97%, making he expeced loss equal o 1.78%. 17 The righ-hand panel of Graph 3 shows ha he expeced loss on he equiy ranche is higher wih low 17 Increasing defaul ime correlaion is equivalen o making he defaul probabiliy random bu wih he same mean defaul probabiliy. Noe ha a mean-preserving disribuion of his ype implies a higher average join survival rae due o convexiy of he join survival probabiliy disribuion. See Lando (2004) for furher discussion. BIS Quarerly Review, March

10 Price sensiiviy of CDX ranches o defaul ime correlaion 1 0 3% ranche 2 3 7% ranche % ranche Based on correlaion sensiiviies repored in Hull and Whie (2004). 2 Upfron paymen, in per cen. 3 Spread, in basis poins. Sources: Hull and Whie (2004); BIS calculaions. Graph 4 correlaion. This is no he case for he mezzanine and senior ranches. Indeed, expeced losses are higher on he senior ranches when correlaion is higher. As he risk of differen ranches varies wih defaul ime correlaion, so does he pricing of he ranches. This is illusraed in Graph 4, which plos he model-implied upfron paymen on he equiy ranche and spreads on he mezzanine and super-senior ranches as a funcion of defaul ime correlaion. 18 Consider he equiy ranche. More defaul clusering has lile negaive impac on he value of his ranche, as only few defauls are needed for his ranche o incur subsanial losses. A he same ime, a higher defaul ime correlaion increases he chance ha no defauls will occur. Therefore, he upfron paymen on he equiy ranche declines as defaul ime correlaion increases. By conras, he pricing of he senior ranche reflecs is greaer exposure o he risk of losses when defauls are more clusered. Unlike he equiy and senior ranches, he price of he mezzanine ranche is generally no a monoonic funcion of defaul ime correlaion. Wih boh high and low correlaions, here is a high probabiliy ha his ranche will survive inac. However, for medium levels of defaul ime correlaion, here is a high risk ha he mezzanine ranche will suffer subsanial losses. As correlaion increases, he spread on he equiy ranche declines whereas i increases on he senior ranche Marke prices and implied defaul ime correlaions Evidence of he marke s view on defaul ime correlaions can be inferred from marke prices on CDS index ranches. This can be done by specifying a pricing model and all he necessary inpus for he model excep he defaul ime correlaion. For insance, by specifying values for all of he inpus in he onefacor Gaussian copula model excep for he consan pairwise defaul ime correlaion, i is possible o back ou an implied correlaion using marke Defaul ime correlaions can be inferred from marke prices 18 Tranche prices are based on Hull and Whie (2004). 82 BIS Quarerly Review, March 2005

11 Marke-implied defaul ime correlaions have a smile... and a skew quoes. 19 This is illusraed in he lef-hand panel of Graph 5, which plos implied defaul ime correlaions for he index ranches over ime. The lef-hand panel in Graph 5 illusraes one of he puzzles observed in marke quoes: he so-called correlaion smile. 20 The correlaion smile illusraes ha, when using a one-facor Gaussian copula, marke spreads on he mezzanine ranche (ypically) imply a lower defaul ime correlaion han is implied by he spreads on equiy and senior ranches. Thus, he degree of defaul clusering assumed by he marke appears o be higher for he equiy and senior ranches. If he one-facor Gaussian model is indeed he correc descripion of join defaul dependence, hen he same implied correlaion value should be inferred for all ranches. The righ-hand panel in Graph 5 illusraes anoher implicaion of marke quoes: he so-called correlaion skew. I plos he marke-implied base correlaion agains he upper bound for each ranche. For example, in he case of he CDX.NA.IG index, he base correlaion for he 0 10% inerval would be defined as he correlaion which equaes he price of his synheic firs loss ranche o he combined observed marke values of he 0 3%, 3 7% and 7 10% ranches. The base correlaion can be inerpreed, from he perspecive of he proecion buyer, as he correlaion in an insurance conrac which pays ou up unil a given level of losses is reached. The fac ha he base correlaion curve is upward-sloping, or skewed, shows ha marke prices for index ranches imply ha defaul ime correlaion is increasing wih ranche senioriy. Implied and base defaul ime correlaions 1 Implied correlaions 2 Base correlaions % 3-7% 7-10% 10-15% 15-30% % 7% 10% 15% 30% Noe: he x-axis is ranche (lef-hand panel) and ranche size (righ-hand panel). 1 Based on on-he-run five-year conracs. 2 Averages based on correlaion sensiiviies repored in Hull and Whie (2004) and marke quoes from JPMorgan Chase spanning he period 13 November 2003 o 28 Sepember Sources: Hull and Whie (2004); JPMorgan Chase; BIS calculaions. Graph 5 19 Index ranches are someimes quoed in erms of implied correlaion insead of spread. 20 The correlaion smile is reminiscen of he volailiy smile wih respec o srike prices exraced from equiy opions using he Black-Scholes model. BIS Quarerly Review, March

12 This reflecs he fac ha spreads are high on he senior ranches, a leas relaive o he low level of expeced losses on hese ranches implied by he model. This is reminiscen of he posiive relaionship beween risk premia and credi qualiy observed for corporae bonds. 21 There are several possible explanaions for he correlaion smile (and skew). 22 One is ha here is segmenaion among invesors across ranches and ha hese differen invesor groups hold differen views abou correlaions. For insance, he views of sellers of proecion on equiy ranches (eg hedge funds) may differ from sellers of proecion on mezzanine ranches (eg banks and securiies firms). However, here is no compelling reason why differen invesor groups would sysemaically hold differen views abou correlaions. A second possible explanaion is ha he smile reflecs marke paricipans uncerainy abou how bes o model credi risk correlaions. The implicaion is ha he equiy and senior ranches, which are more sensiive o correlaions, conain a model risk premium embedded in heir prices. While his explanaion can accoun for he relaively large premium on he senior ranche, i is no consisen wih he relaively low equiy ranche premium. A hird explanaion is ha, even hough he index ranche marke has grown significanly over he pas year, prices migh sill be subjec o local demand condiions. For example, he implied correlaion on he mezzanine ranche may reflec srong ineres by banks in selling proecion on his segmen of he index loss disribuion. This could be due o he hedging demands of banks, which may be shor credi risk of his ype as a resul of heir role as originaors of oher, noably single-ranche, CDOs. A fourh explanaion is ha marke paricipans may, in fac, use oher models for pricing han he one-facor Gaussian copula. Possibiliies include: (1) using faer-ailed disribuions (eg Suden s-); (2) relaxing he resricion of consan pairwise correlaions; (3) allowing individual defaul probabiliies o depend on macroeconomic risk facors; and (4) leing recovery raes vary over ime and be correlaed wih defaul imes and defaul probabiliies. 23 For insance, he impac on pricing from using a faer-ailed disribuion, which implies more clusering of defauls, increases break-even spreads for senior ranches and lowers hem for junior ranches. Alernaively, a posiive correlaion beween losses-given-defaul and clusering of defaul imes would lower he price on he mos senior ranches for a given level of defaul ime correlaion. In his case, he implied correlaion inferred from senior ranches (under a consan recovery rae assumpion) would be upward biased. This could also explain he pricing of equiy ranches, as higher recovery raes during imes of lile defaul clusering would imply ha his ranche is more valuable. The correlaion smile migh reflec marke segmenaion uncerainy abou credi risk correlaions local demand condiions or he use of differen pricing models 21 For furher discussion of his, see Amao and Remolona (2004). 22 See also Bernand e al (2004). 23 The imporance of hese elemens for he modelling of credi risk have been discussed, respecively, by Hull and Whie (2004), Gregory and Lauren (2004), Duffie and Singleon (2003) and Alman e al (2004). 84 BIS Quarerly Review, March 2005

13 Looking forward For CDS indexbased markes o maure diversificaion mus increase and credi risk modelling should improve o capure more ypes of credi risk codependencies Despie rapid growh, he marke for CDS index ranches is sill relaively small. Furhermore, even hough hey have improved diversificaion opporuniies a a lower cos o invesors, hese insrumens sill conain significan idiosyncraic risk because hey only reference 125 names in five differen secors. 24 However, as hese markes coninue o maure, he number of underlying names is likely o increase and improve diversificaion. Thus, in fuure, index ranches should provide furher scope for more efficien rading of credi risk correlaions. To improve marke efficiency and limi he risk ha exposures are accumulaed in ways ha are no fully appreciaed, i is imporan for credi risk modelling o develop furher. The main challenge appears o be developing frameworks ha realisically capure credi risk correlaions (see Duffie (2004)). As noed above, he valuaion of CDS index ranches has so far mainly focused on modelling he correlaion of defaul imes. By conras, correlaions among defaul probabiliies and losses-given-defaul (ie credi spread correlaions), have received less aenion. No doub, progress is being made in developing more general models o capure credi risk codependencies. 25 For insance, some models incorporae conagion effecs, which allow hem o capure he impac on credi risk from declines in overall marke liquidiy, he failure of large firms or adverse indusry-level developmens. 26 Examples of large defauls ha have had a marke-wide impac include Enron and WorldCom; a recen example of an adverse indusry developmen is he invesigaion by he New York Aorney General s office ino insurance indusry pracices in he Unied Saes. Looking ahead, praciioners, as well as policymakers monioring hese markes, will face he challenge of designing robus models ha capure hese ypes of sysemaic and sysemic evens. References Alman, E I, B Brady, A Resi and A Sironi (2004): The link beween defaul and recovery raes: heory, empirical evidence and implicaions, Journal of Business, forhcoming. Amao, J and E Remolona (2004): The pricing of unexpeced credi losses, Bank for Inernaional Selemens, mimeo. Anson, M, F Fabozzi, M Choudhry and R-R Chen (2003): Credi derivaives: insrumens, applicaions and pricing, Wiley Finance. 24 For a discussion of he imporance of idiosyncraic risk in credi porfolios, see Amao and Remolona (2004). 25 These dependencies could also include correlaions beween discoun facors and credi risk. 26 See Davis and Lo (2001) and Collin-Dufresne e al (2003) for heoreical models of credi risk conagion. Schönbucher and Schuber (2001) show how cerain ypes of more general copulas are able o capure hese general credi risk codependencies. BIS Quarerly Review, March

14 Bernand, A, F Pourmokhar, B Jacquard, D Baum, L Gibson, L Andersen and J Sidenius (2004): The Bank of America guide o advanced correlaion producs, supplemen, Risk magazine, May. Cherubini, U, E Luciano and W Vecchiao (2004): Copula mehods in finance, Wiley, New York. Collin-Dufresne, P, R Goldsein and J Helwege (2003): Is credi even risk priced? Modeling conagion via he updaing of beliefs, Carnegie Mellon Universiy, mimeo. Commiee on he Global Financial Sysem (2005): The role of raings in srucured finance: issues and implicaions, Bank for Inernaional Selemens, Basel. Davis, M and V Lo (2001): Infecious defauls, Quaniaive Finance, 1, pp Duffie, D (2004): Time o adap copula mehods for modelling credi risk correlaion, Risk magazine, April, p 77. Duffie, D and K J Singleon (2003): Credi risk: pricing, measuremen and managemen, Princeon Universiy Press. Gibson, M (2004): Undersanding he risk of synheic CDOs, FEDS Discussion Papers, no , Board of Governors of he Federal Reserve Sysem. Gregory, J and J-P Lauren (2004): In he core of correlaion, Risk magazine, Ocober, pp Hull, J and A Whie (2004): Valuaion of a CDO and an n-h-o-defaul CDS wihou Mone Carlo simulaion, Journal of Derivaives, forhcoming. Inernaional Swaps and Derivaives Associaion (2003): ISDA Credi Derivaives Definiions, Supplemens and Commenaries. Lando, D (2004): Credi risk modeling: heory and applicaions, Princeon Universiy Press. Li, D (2000): On defaul correlaion: a copula funcion approach, Journal of Fixed Income, March, pp Nelsen, R (1999): An inroducion o copulas, Lecure Noes in Saisics, Springer, Berlin. O Kane, D, M Naldi, S Ganapai, A Berd, C Pedersen, L Schloegl and R Mashal (2003): The Lehman Brohers guide o exoic credi derivaives, supplemen, Risk magazine, November. O Kane, D, C Pedersen and S Turnbull (2003): The resrucuring clause in credi defaul swap conracs, Fixed income quaniaive credi research, Lehman Brohers, April. O Kane, D and S Sen (2003): Up-fron credi defaul swaps, Quaniaive Credi Research Quarerly, Lehman Brohers, Third Quarer. 86 BIS Quarerly Review, March 2005

15 Packer, F and H Zhu (2005): Conracual erms and CDS pricing, BIS Quarerly Review, March. Rebonao, R (2002): Modern pricing of ineres-rae derivaives, Princeon Universiy Press. Schönbucher, P and D Schuber (2001): Copula-dependen defaul risk in inensiy models, Deparmen of Saisics, Bonn Universiy, mimeo. BIS Quarerly Review, March

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