Prices of Credit Default Swaps and the Term Structure of Credit Risk

Transcription

1 Prces of Cred Defaul Swaps and he Term Srucure of Cred Rsk by Mary Elzabeh Desrosers A Professonal Maser s Projec Submed o he Faculy of he WORCESTER POLYTECHNIC INSTITUTE n paral fulfllmen of he requremens for he Degree of Maser of Scence n Fnancal Mahemacs May 2007 APPROVED: Professor Domokos Vermes, Advsor

2 Execuve Summary Invesmens n any form of fnancal produc, oher han a bank depos a he rskfree neres rae, nvolve some sor of rsk due o he volaly of he economy. Ineres rae rsk s he mos crcal rsk facor affecng fxed ncome secures. However, he growng cred dervaves marke s based prmarly on cred or defaul rsk. Ths s he rsk caused by he possbly ha a company wll have fnancal roubles and wll have o defaul on paymens whch owes o s lenders. US reasury secures are consdered o be free of cred rsk because hey are backed by he governmen. In order o proec nvesors from hs rsk, he cred dervaves marke emerged wh varous producs whose sole purpose s o hedge cred rsk. A cred dervave s a conrac beween a proecon buyer and a proecon seller o ransfer he cred rsk of an asse whou he acual ransfer of he asse. The mos fundamenal cred dervave s he cred defaul swap. In a cred defaul swap, he proecon buyer makes perodc premum paymens o he proecon seller n exchange for he promse ha f defaul occurs, he proecon seller wll receve he defauled secury and repay he proecon buyer a percenage of wha was owed. The premums of he cred defaul swap conrac are deermned by he marke s vew of how lkely s ha defaul wll occur before he cred swap maures. Tme-o-defaul s a random varable whch characerzes he erm-srucure of cred rsk and affecs he prce of cred dervave producs. Ths projec quanfes he connecon beween he prces of he cred defaul swaps and he probably dsrbuon of he me-o-defaul n boh drecons. 2

3 1. We calculae he marke perceved probables and mng of possble defaul by a parcular borrower from he marke prces of a seres of raded cred defaul swaps referencng he same borrower s deb. 2. We calculae he far prces of he cred defaul swaps from he probably dsrbuon of he defaul me and of he recovery rae. The calculaons are mplemened n spreadshees of a Mcrosof Excel workbook. The resuls of he projec can also be used o deermne prces of more complex cred dervaes. The marke-mpled defaul probables deermne he cred rsk nheren n all secures dependng on he same borrower. They can hen be used as npu no more complcaed models for mul-name cred dervave producs, such as baske defaul swaps and collaeralzed deb oblgaons. 3

4 Absrac The objecve of hs projec s o nvesgae and model he quanave connecon beween marke prces of cred defaul swaps and he marke perceved probably and mng of defaul by he underlyng borrower. We quanfy he cred rsk of a borrower n a wo-way relaonshp: calculae he erm srucure of defaul probables from he marke prces of raded CDSs and calculae prces of CDSs from he probably dsrbuon of he me-o-defaul. 4

5 Table of Conens Execuve Summary... 2 Absrac... 4 Table of Conens Inroducon Tradable Asses and Rsk Facors Fxed Income Producs Bonds Swaps Asse-backed Secures Rsk Facors Ineres Rae Rsk Defaul/Cred Rsk Cred Dervaves Cred Dervave Producs Probably of Defaul Typcal defaul me dsrbuons Exponenal Dsrbuon Gamma Dsrbuon Webull Dsrbuon Approxmaon of Defaul Tme Dsrbuons Pecewse Consan CDF Pecewse Consan Densy Funcon Pecewse Consan Hazard Rae Funcon Cred Defaul Swaps (CDS CDS Prcng CDS Spreadshee Compung he Hazard Raes Compung he Prces Conclusons Appendx: Workbook User Manual Bblography

6 1. Inroducon Cred rsk s becomng an ncreasngly mporan opc for evaluaon n he fnancal ndusry. Up unl he recen growh of he cred dervaves marke, neres rae rsk was one of he only rsk facors aken no consderaon when evaluang fxedncome secures. Ineres rae rsk sll remans he mos mporan rsk facor o consder because affecs he enre marke, bu cred rsk s mporan when comes o deb nsrumens based srcly on cred. There are many dfferen ypes of cred dervave producs, all fallng no wo caegores: sngle-name cred dervaves and mul-name cred dervaves. Sngle-name cred dervaves are based on he defaul rsk of one parcular company, whle mul-name cred dervaves reference he correlaon beween he cred rsks of varous companes. The mos fundamenal sngle-name cred dervave and he bass for many more nrcae cred producs s he cred defaul swap. A cred defaul swap provdes nsurance o he buyer agans a cred even such as defaul. Probably of defaul plays an mporan role n prcng cred defaul swaps, bu hs probably s no always known. Ths paper nroduces mehods o derve he marke perceved probably of defaul whch can hen be used o prce cred defaul swaps or oher cred dervave producs. 2. Tradable Asses and Rsk Facors 2.1 Fxed Income Producs A common form of nvesmen s a fxed ncome secury. Fxed ncome secures come n many forms and dffer from oher varable-ncome secures, such as socks, n ha all paymens are known n advance. A fxed ncome nvesor lends s money n exchange for a promse of a pre-deermned sequence of paymens by he counerpary, 6

7 also known as he deb ssuer. Fxed-ncome secures are also known as deb or cred nsrumens; he nvesor creds s money o he ssuer, who assumes he deb. Fxed ncome producs are srucured based on he me value of money : a dollar receved oday s dfferen from a dollar o be receved one year from oday (Rsk Glossary, Bonds A bond s a form of securzed deb whch maures a a specfed dae n he fuure, pays neres perodcally n he form of coupon paymens, and repays s facevalue a maury. A zero-coupon s a specal knd of bond whch provdes only one paymen a he bond s maury dae conssng of he accrued neres and he prncpal poron of he bond. Bonds can be raded. A any pon n me, he far prce of a bond s he presen value of s fuure cash flows. The prce of a bond can flucuae due o many facors; he mos mporan beng neres-rae sensvy. As marke neres raes change, he presen value of fuure cash flows changes, affecng he marke prce of he bond. Anoher key facor n bond prce movemens s he perceved cred qualy of he bond ssuer. Fuure paymens are only ceran once receved, so f he marke senses an ncreased probably ha he ssuer wll defaul on some or all of he fuure paymens, he value of he bond deprecaes. Quanfyng hs cred sensvy of fxed-ncome secures s he man focus of hs projec Swaps A swap s an over-he-couner (OTC fnancal dervave n whch wo pares ener no an agreemen o exchange a seres of cash flows based on he value of an underlyng asse, bu ha underlyng asse s no drecly raded. The cash flows can be 7

8 deermned n any manner suable o boh pares objecves, as long as he presen values of boh cash flows are equal. Swaps have many uses such as hedgng, speculaon, and asse-lably managemen, and hey are classfed by he naure of he cash flow sreams beng exchanged. The mos mporan ypes are neres-rae swaps, foregn exchange swaps, and cred relaed swaps. An neres rae swap s useful for exchangng fxed rae fuure cash flows agans varable rae fuure cash flows. Foregn exchange swaps are agreemens o exchange fuure cash flows of dfferen currences. Cred relaed swaps are he man opc of hs projec and wll be explaned n furher deal below Asse-backed Secures An asse-backed secury s a fxed-ncome produc based on a specfed pool of underlyng asses. The asses, or collaeral, are pooled ogeher o form a sngle porfolo produc ha offers lower nvesmen rsk hrough dversfcaon. Typcal asse-backed secures are dfferen combnaons of hghly llqud asses such as bonds, loans, morgages, and cred nsrumens. A common asse-backed secury s a collaeralzed deb oblgaon (CDO. A CDO s a broad erm ha encompasses varous secures based on he specfc ype of deb by whch hey are backed. Some examples of specfc CDOs are: Collaeralzed Bond Oblgaons (CBOs, Collaeralzed Loan Oblgaons (CLOs, Collaeralzed Morgage Oblgaons (CMOs, ec. CDO nvesors assume he cred rsk of he pooled asses whou assumng he cred rsk of an ndvdual provder (Rsk Glossary,

9 2.2 Rsk Facors Rsk requres uncerany and exposure o ha uncerany. The level of uncerany and exposure deermnes he level of rsk (Rsk Glossary, Invesmens n any form of fnancal produc, oher han a bank depos a he rsk-free neres rae, nvolve some sor of rsk due o he volaly of he economy. Rsk comes n many forms and s a major facor nvolved n prcng fnancal producs and n nvesor decsonmakng. Normally he more rsk nvolved, he beer he reurn on he nvesmen and vce versa. Rsks on nvesmens can be grouped no wo caegores: sysemac and unsysemac. Sysemac rsks are rsks whch affec he enre marke, or a whole marke secor. Unsysemac rsk has an affec on a smaller, specfc group of nvesmens, or even one ndvdual secury. Whle unsysemac rsk can be reduced hrough mehods such as dversfcaon, hedgng, and leveragng, sysemac rsk can only be reduced by hedgng ( Rsk, Ineres Rae Rsk Ineres raes are a form of sysemac or marke rsk because any change affecs he enre marke. Ineres raes are consanly changng due o he economy and marke flucuaons. Fxed and floang neres raes pose rsks on nvesors. An nvesmen n a floang rae asse wll deprecae f neres raes drop over me. Rsks also arse wh fxed rae asses f maures on asses and lables n a porfolo are msmached. Once an asse or a lably maures, f neres raes have changed, hs has an affec on he overall porfolo value. Ineres rae rsk s he mos crcal rsk facor affecng fxed ncome secures. I s he prmary cause for marke prce flucuaons. The varyng level of exposure o neres rae rsk s he cause for he dfference beween he neres rae spreads on shor 9

10 and long erm bonds. The neres rae spread s he dfference beween he neres rae avalable on a US reasury secury of a gven maury and he rsk-free neres rae. The neres rae spread graphed as a funcon of maury me s known as he erm-srucure of neres raes. Ineres rae rsk nheren n a fxed ncome secury can be reduced, ncreased, or even elmnaed hrough hedgng; akng an offseng poson n a relaed secury. Commonly used hedgng nsrumens are neres rae swaps, neres rae opons, caps, floors, swapons, and oher neres rae dervaves Defaul/Cred Rsk Cred rsk s he second mos crcal rsk facor affecng deb nsrumens. Ths s he rsk caused by he possbly ha he ssuer of he bond may no be able o mee s oblgaons o pay neres or repay he prncpal of he loan. US reasury secures are consdered o be free of cred rsk. The dfference beween he neres rae offered on a bond of a parcular ssuer and he neres rae on he US reasury bond of he same maury s called he cred spread. The cred spread depends on he cred qualy of he ssuer and on he maury of he bond. The cred spread s he reward an nvesor receves for assumng he cred rsk nheren n he secury. Defaul rsk s an mporan facor o ake no consderaon when makng an nvesmen n a fxed ncome produc, such as a bond. Defaul occurs when he bond ssuer s unable o sele he remanng deb on a bond. Ths leads o he nvesor losng he remander of her fuure coupon paymens and he prncpal poron of he bond. Dependng on he erms se forh n he nal agreemen, he nvesor may be able o recover a percenage of her nvesmen based he specfc recovery rae nvolved. The 10

11 recovery rae of a bond s he fracon of he ousandng oblgaon expeced o be recovered hrough bankrupcy proceedngs or some oher form of selemen (Rsk Glossary Defaul rsk can be assessed pror o purchasng a bond by nvesgang he cred rangs of he bond ssuer. Sandard & Poor s and Moody s Invesors Servce are wo of he larges cred rang agences whch gve companes cred rangs based on hose companes ables o pay back her ousandng deb. These rangs reflec a company s rsk of defaul on her oblgaons, ulmaely reflecng he company s overall cred rsk. Fgure: Bond Rang Codes ( Bond Rangs, 2005 Rang S&P Moody's Hghes qualy AAA Aaa Hgh qualy AA Aa Upper medum qualy A A Medum grade BBB Baa Somewha speculave BB Ba Low grade, speculave B B Low grade, defaul possble CCC Caa Low grade, paral recovery possble CC Defaul, recovery unlkely C C Ca The cred rangs are based on he company s probably of defaul, her average recovery raes on prevous defauls, and he qualy and dversfcaon of her asses. The hgher he rsk of defaul or he lower he cred rang of a company, he hgher he yeld he nvesor should receve on he bond. Yeld s he annual rae of reurn of an nvesmen. The hghes qualy bonds, for example AAA, offer mnmal cred rsk and he lowes 11

12 yeld. As he qualy decreases, cred rsk ncreases, bu lower qualy bonds have much hgher yelds. Hgher rsk should gve hgher reurns. Cred rsk nheren n a deb nsrumen, and consequenly he cred spread, depends on he followng facors: 1. The probably of a defaul by he ssuer. 2. The mng of a possble fuure defaul 3. The probably dsrbuon of he recovery rae. Assumng a consan and known recovery rae, he erm-srucure of he cred spread (.e. he cred rsk has a one-o-one correspondence wh he probably dsrbuon of he me of defaul for he gven ssuer. One of he man goals of hs projec s o use marke prces of raded cred dervaves o recover he marke s percepon of he probably dsrbuon of he me of defaul for he ssuer. 3. Cred Dervaves Upon purchasng a fxed ncome produc, he nvesor faces he rsk of fnancal loss f he ssuer defauls on he oblgaon. In order o proec hemselves or o hedge hs rsk, nvesors have he opon of buyng a cred dervave. A cred dervave s a conrac beween a proecon buyer (for example, he owner of a bond and a proecon seller (a hrd pary fnancal nsuon o ransfer he cred rsk of an asse whou he acual ransfer of he asse. The dea s o avod drec ownershp of he asse n he ransacon n order o mnmze losses n he even of defaul. 12

13 3.1 Cred Dervave Producs There are many dfferen ypes of cred dervave producs, each based on he specfc rsk beng ransferred. The wo fundamenal caegores of cred dervaves are sngle-name cred dervaves and mul-name cred dervaves. Sngle-name cred dervaves offer proecon agans he defaul rsk of one parcular borrower. Examples are asse swaps, cred lnked noes, and cred defaul swaps. Mul-name cred dervaves are based on defauls of one or more borrowers from a group of borrowers. These nsrumens depend no only on he cred rsks posed by he ndvdual borrowers, bu also on he correlaon beween hem. Examples of mul-name cred dervaves are baske defaul swaps and CDOs. A oal reurn swap (TRS, also known as a oal rae of reurn swap (TRORS, s a cred dervave nended o proec agans deprecaon of an asse. The swap exchange s a combnaon of an underlyng asse and an neres rae swap. In he TRS agreemen, one pary receves he oal reurn, or he generaed ncome from he asse plus any capal gans, whle he oher pary receves paymens based on a se rae as par of he neres rae swap. The owner of he asse ges proecon agans any loss n value, whle he counerpary receves he benefs of he asse whou havng o pu he asse on s balance shee (Invesopeda, An asse swap s que smlar o a oal reurn swap n ha consss of a bond pared wh an neres rae swap. An nvesor purchases a bond and hen hedges ou he neres rae rsk wh an neres rae swap. The major dfference beween an asse swap and a oal reurn swap s ha n he even of defaul, 13

14 he oal reurn swap ermnaes whle he neres rae swap paymens of he asse swap connue unl maury. A cred-lnked noe (CLN, or cred defaul noe, s a produc ssued by a Specal Purpose Vehcle (SPV offerng nvesors perodc paymens plus he par value of he reference eny a maury, unless defaul occurs. The SPV also eners no a cred defaul swap wh a hrd pary whch pays he SPV an annual fee. Ths annual fee provdes hgher reurn o nvesors o compensae for he cred rsk nvolved. In he even of defaul, he nvesors receve a poron of he par value based on he recovery rae, and he SPV pays he hrd pary he par value mnus he recovery rae. A collaeralzed deb oblgaon (CDO s also a form of cred dervave. In a cash flow CDO, he nvesor faces cred rsk based on he pool of underlyng bonds or loans. A CDO s a pool of asses packaged no one porfolo, and hen ha porfolo s ranched. I s spl up no secons, each correspondng o a dfferen level of loss. The ranches provde he nvesor wh some flexbly n choosng he amoun of loss or cred rsk o whch hey are wllng o be exposed. In a synhec CDO, a CDO made up of cred defaul swaps, he nvesor faces cred rsk based on he cred worhness of he underlyng companes. There are many varaons of hese producs, bu he mos common and mporan cred dervave s he cred defaul swap (CDS whch wll be explored n deal n Chaper Probably of Defaul The premums of he cred defaul swap conrac are deermned by he marke s vew of how lkely s ha defaul wll occur before he reference eny maures. The 14

15 probably dsrbuon of he me-o-defaul s he erm-srucure of cred rsk and s one of he drvng facors behnd he cred dervaves marke. Defaul s an even whch s modeled usng probably heory and sascs. Tme-o-defaul s a random varable, τ wh non-negave values, whch can be characerzed by s cumulave probably dsrbuon funcon F, s probably densy funcon f, or hazard rae funcon h. Common probably dsrbuons ha are used o model he probably of defaul are he exponenal, gamma, and Webull dsrbuons. The cumulave dsrbuon funcon (cdf F( gves he probably ha he defaul occurs before me. F ( = P( τ < The probably densy funcon (pdf f( s he dervave of he cumulave dsrbuon funcon, whenever F s dfferenable. f ( = or F( = d d F( f ( s ds The probably ha defaul wll occur n a small me nerval of lengh Δ around me can be approxmaed as f( Δ. f ( Δ P( < τ < + Δ The hazard rae h( s he condonal densy funcon of he defaul me τ, condoned on he even ha no defaul has occurred before me. h( f ( lm P( < τ < + Δ τ > = Δ 0 1 F( = 15

16 The probably densy funcon can be recovered from he hazard rae funcon by he followng formula: = h( s ds 0 f ( h( e 4.1 Typcal defaul me dsrbuons Exponenal Dsrbuon λ CDF: F( = 1 e f > 0 λ PDF: f ( = λ e f > 0 16

17 Hazard: h ( = λ The exponenal dsrbuon s characerzed by a unque memoryless propery. In relaon o probably of defaul, memoryless ndcaes ha a any gven me, he probably of defaul s dsnc and does no depend on nformaon from he pas. Memorylessness s a form of condonal probably, ha for any posve real numbers s and, we have P ( T > + s T > = P( T > s Ths memorylessness propery mples ha he hazard rae s consan Gamma Dsrbuon The gamma dsrbuon s he sum of k>0 ndependen, exponenally dsrbued random varables. The gamma dsrbuon has wo parameers, k and β, where k s he shape parameer and β s he scale parameer. A specal case of he gamma dsrbuon s when k=1, we have he exponenal dsrbuon wh λ=1/β. CDF: F( = 0 0 x x k 1 k 1 e e x / β x / β dx dx 17

18 PDF: = 0 / 1 / 1 ( dx e x e f x k k k β β β Hazard: = x k k dx e x e h β / 1 1 ( The gamma dsrbuon s suable for defaul me modelng f s perceved ha a borrower has o go hrough a number of sages of crss before defauls. As ges 18

19 large, he gamma dsrbued defaul mes behave smlarly o exponenally dsrbued defaul mes,.e. lm h( = cons Webull Dsrbuon The Webull dsrbuon s a hree parameer dsrbuon wh α>0 as he shape parameer, λ>0 as he scale parameer, and γ as he locaon parameer wh - <γ<. CDF: F( = 1 e α γ λ PDF: α γ f ( = λ λ α 1 e γ λ α 19

20 Hazard: α γ h ( = λ λ α 1 The Webull dsrbuon s a form of exreme value dsrbuon. An exreme value dsrbuon s a lmng dsrbuon for he mnmum and maxmum of a large collecon of random observaons from he same dsrbuon. In erms of probably of defaul, he Webull dsrbuon governs he me unl defaul of he frs o defaul from a collecon of defaul mes. 4.2 Approxmaon of Defaul Tme Dsrbuons In real lfe, no enough nformaon s avalable abou he me-of-defaul o deermne s probably dsrbuon a every me. Usually, s possble o esmae he probably ha he defaul wll happen n varous me nervals of posve lengh (e.g. 6 monhs, 1 year, ec.. In such cases, he probably dsrbuon of he connuous random varable τ mus be deermned by some nerpolaon procedure. The man assumpon behnd hese echnques s ha defaul occurs a parcular dscree mes. Ths assumpon s suppored by he fac ha borrowers usually declare bankrupcy when hey are unable o mee an neres paymen, so defauls ofen occur on coupon paymen daes. Probablscally hs means ha he me-o-defaul s a sep funcon. 20

21 4.2.1 Pecewse Consan CDF A pecewse consan CDF jumps from one sep o he nex a he dscree mes when defaul s possble (e.g. he sem-annual coupon daes. The sze of each jump corresponds o he probably of defaul a ha parcular me. Pecewse consan CDFs are no dfferenable, hence densy and hazard funcons are no defned n such cases Pecewse Consan Densy Funcon Ths approxmaon assumes ha beween dscree jump pons of he probably densy funcon he defaul me s unformly dsrbued. In oher words, whn hose nervals of consancy, defaul s equally probable a any me. Pecewse consan densy funcons mply pecewse lnearly nerpolaed CDF. The correspondng hazard rae graph consss of adjonng hyperbolc curves Pecewse Consan Hazard Rae Funcon A pecewse consan hazard rae assumes ha beween dscree jump pons of he sep funcon, he defaul me follows he exponenal dsrbuon wh consan hazard rae. Pecewse consan hazard rae s he assumpon used n hs projec and he mehod used n he spreadshee for modelng me-o-defaul. 5. Cred Defaul Swaps (CDS A cred defaul swap s a conrac, ndexed o a sngle reference asse, whch provdes nsurance agans a defaul even on ha asse. There are hree pares nvolved n a cred defaul swap. The frs s he proecon buyer; hs s he nvesor and owner of he reference asse, for example a General Moors bond. The bond ssuer, General Moors n hs example, s he second pary ha plays a role, ndrecly, n he CDS. Based on he bond nvesmen, General Moors pays he nvesor perodc coupon paymens and 21

22 promses o pay he prncpal poron of he bond a a se maury dae. Afer purchasng he bond, he nvesor becomes nervous ha General Moors wll suffer a cred even and defaul on s promsed, fuure paymens. So, he bond owner purchases proecon agans he possbly of hs cred even n he form of a cred defaul swap. The CDS s a conrac beween he proecon buyer and a proecon seller. The laer s ypcally an nsurance company or a secures company, e.g. Morgan Sanley. In hs agreemen, he proecon buyer makes perodc premum paymens (perods are usually half year ncremens o he proecon seller, n hs case Morgan Sanley, and Morgan Sanley agrees o pay he enre face value of he bond o he proecon buyer f General Moors defauls on he bond. The CDS wll ermnae eher a he bond s maury or he dae a defaul even occurs, whchever comes frs. If defaul never occurs, General Moors connues o pay he perodc coupon paymens and a maury pays he prncpal poron of he bond o he nvesor, and he nvesor pays he CDS premum paymens o Morgan Sanley unl he bond maures. Morgan Sanley wll never have o make any paymens and profs for assumng he cred rsk of he bond. If a defaul even occurs before he se maury, Morgan Sanley nsanly compensaes he proecon buyer for s loss and has no furher oblgaons n he CDS conrac. In hs even, he nvesor would have mnmzed s losses by enerng no he cred defaul swap. The cred defaul swap s he bass for he cred dervaves marke. In 2001, cred defaul swaps accouned for 38% of he cred dervaves marke, whch was more han wo mes ha of he nex hghes conrbuor. Today, cred defaul swaps connue o domnae he marke, and are used as he foundaon of newer, more complcaed 22

23 producs. For example, a cred defaul swap ndex (CDSI s a sngle produc based on a baske of cred enes (Invesopeda, Fgure 1: Marke Share of Ousandng Noonal for Cred Dervave Producs (Cred Dervaves Explaned, 2001 Marke Share Cred Dervave Insrumen Type (% Noonal a End 1999 Cred Defaul Producs 38% Porfolo/CLOs 18% Asse Swaps 12% Toal Reurn swaps 11% Cred Lnked Noes 10% Baskes 6% Cred Spread producs 5% An nvesor n a CDS only assumes he cred rsk of defaul on he reference eny; all oher rsks such as neres rae movemens do no have an affec on he CDS agreemen. 5.1 CDS Prcng The man dea behnd prcng models for cred defaul swaps s ha hey are compleely ndependen from neres rae movemens. The only rsk assumed s ha of defaul, or cred rsk. The prce of he CDS s deermned by seng he presen value of he perodc premum paymens equal o he presen value of he reference eny a maury, or me of defaul. I s common o hnk of a CDS as havng wo opposng legs: he premum leg correspondng o he fxed premums paymens and he defaul leg correspondng o he conngen paymen upon defaul (Arvans, The premum leg s a sream of dscouned, fxed cashflows a fxed mes ( 0, 1, 2, n. These annualzed premum paymens, X, are pad unl maury, T = n, or 23

24 defaul, τ, whchever occurs frs. Ths sream of cashflows s dscouned back by he rsk-free dscoun facor, В(0,, and weghed by he nsananeous probably of defaul h(, or hazard rae, o acheve he presen value. Equaon 1: Presen Value of Premum Leg = > > = du u h P where P B X K PL PV 0 1 ( exp( ( ( (0, ( ( τ τ The defaul leg (DL s he paymen conngen upon defaul, (1-δ where δ s he assumed recovery rae, dscouned back usng he rsk-free dscoun facor and he condonal probably of defaul a me. Equaon 2: Presen Value of Defaul Leg ( (0, (1 ( 1 P B K DL PV < < = τ δ (K represens he noonal amoun and δ s he recovery rae. Equaon 3: Presen Value of Swap k k k k k Swap Swap h where X P B K PV so PL PV DL PV PV α α α δ τ = > = = = ( ] ( (1 [(1 ( (0, ( ( When a swap s naed, he premum paymens are deermned by seng he presen value of he premum leg and defaul leg equal o zero; n dong so, neher pary pays anyhng a he sar of he swap conrac. 0 ( (0, (1 0 ( (0, ( 1 1 = < < = > P B K P B X K τ δ τ 24

25 Equaon 4: Inal Presen Value of CDS PV wh Swap P( τ > = 0 = K k 1 = 1 k = 1 k = 1 α k B(0, k P( τ > k 1 [(1 δ (1 α ( k 1 X α ] k We now have an equaon ha we can use o solve for eher he premum paymens, X T, or he hazard raes, α, dependng on wha daa s known. 6. CDS Spreadshee The man goal of hs projec s o use cred defaul swaps o deermne he marke s percepon of he rsk-neural probably of defaul, usng a predeermned consan recovery rae. The workbook consruced for hs serves wo major purposes: 1. Derve he mpled marke hazard raes usng marke quoes for cred defaul swap premums. 2. Prce cred defaul swaps of dfferen maures usng derved marke hazard raes. The workbook draws from marke quoed premums of cred defaul swaps of dfferen maures on he same reference cred o deermne he mpled hazard rae whch models he defaul probably dsrbuon. The spreadshee user can hen use he marke s perceved defaul probably dsrbuon as a parameer o ge he rsk-neural prce of cred defaul swaps. Boh spreadshee applcaons requre he use of Equaon 4 and solvng a se of non-lnear equaons usng he Solver Add-In n Mcrosof Excel. (See Appendx: Workshee User Manual for nsrucons on how o operae he spreadshee. 25

26 6.1 Compung he Hazard Raes Boosrappng s a calbraon procedure used by he workbook o solve for he hazard raes. We began by gaherng marke premums for curren defaul swaps wh dfferen maures on he same reference eny. Then we assume he hazard raes, α for me nervals [ -1, ], are pecewse consan beween he maury daes of he ndvdual marke swaps. We exrac he hazard raes by solvng for he approprae α usng Equaon 4 and a consan recovery rae, δ. We solve for each α n order of ncreasng maury, usng he daa from he swap wh he frs maury T 1 o solve for α 1. Consequenly we know α 1 and have he daa from T 2 o solve for α 2 and so on. Ths probably srppng procedure gves us a sep-funcon for he hazard raes correspondng o he cred defaul swaps. These pecewse consan hazard raes form a sep funcon wh jumps a he dfferen maury daes. The workbook shows hs sep-funcon and hen akes hs sepfuncon and smoohes usng cubc splnes. Ths smoohed curve represens a connuous hazard of defaul. 6.2 Compung he Prces Assumng a consan recovery rae and usng gven or derved hazard raes, he workbook prces cred defaul swaps of varous maures. Smlarly o he above hazard rae procedure, he workbook uses Equaon 4 and he correspondng hazard raes o compue he premums. 26

27 7. Conclusons The prce of a cred defaul swap and he probably of defaul are drecly conneced. Quanfyng he defaul probably and erm srucure s useful for hedgng ou cred rsk nheren n fxed-ncome secures and s also helpful for calculang he rsk-neural prces of cred dervaves oher han CDSs. These marke perceved hazard raes whch hs prcng model compued can be negraed no more complcaed models for mul-name cred dervave producs, such as baske defaul swaps or CDOs. Oher exensons of hs projec could be o ncorporae sochasc recovery raes or sochasc hazard raes. 27

28 Appendx: Workbook User Manual The frs sep n usng he workbook s o gaher marke prces for cred defaul swaps on he reference eny for whch you wan o deermne he probably of defaul for as many dfferen maures ha are avalable. Also, deermne he followng parameers ha wll be used: sze of me nervals beween maury daes, he erm srucure of he rsk-free neres rae, your reference eny s noonal amoun, and he assumed consan recovery rae. The workbook conans macros, so before openng he workbook s necessary o se he macro secury o an approprae level whch allows for runnng hese macros. Upon openng he fle, he frs page you should see s he Inpu shee. If hs does no open up drecly, clck on he Inpu ab. Inpu ab 28

29 Inpu he followng parameers no he approprae blue cells n column H: sze of me nervals beween maury daes, rsk-free neres rae, your reference eny s noonal amoun, and he consan recovery rae. Inpu he marke premums you prevously gahered n he blue cells of Opon 1 for he approprae correspondng maury daes. To compue he hazard raes, go o he Hazard Raes ab. Open Solver by gong o he Tools drop down menu, and selecng Solver. If Solver s no prevously nsalled no your verson of Excel, you mus frs nsall by clckng on he Tools drop down menu, hen selec Add-Ins. Check he box nex o he Solver Add-In and clck OK. Clck on Yes when promped wh he opon o nsall solver now. Solver should now be lsed n he Tools drop down menu. Once you open solver, you wll be promped o ener he Solver Parameers. 29

30 You are gong o wan o solve for he approprae presen value equaon n row #8, begnnng wh C8 and gong n order o he rgh, unl you end on cell L8. In he Solver Parameers menu, you wan o Se arge cell: frs o cell C8. Equal o: he Value of: 0. And n he box where says By changng cells you wan o selec he cell of wha you are solvng for. For α 1, hs s cell C19. Then clck he solve buon. The correspondng hazard rae for he CDS wh maury 0.5 years should appear n he purple cell C19. You wan o repea hs process usng Solver 10 mes o solve for he hazard raes n row #19, sarng a C19 and movng o he rgh one cell a a me unl you solve for he fnal hazard rae, cell L19. 30

31 The compued hazard raes and resulng hazard rae sep funcon can be found by clckng he Hazard Rae Sep Funcon ab. A smoohed hazard rae funcon can be vewed on he Smooh Hazard Rae Funcon ab. And he correspondng probably dsrbuon can be vewed on he Probably Dsrbuon ab. To compue he Premums of he CDS, npu he hazard raes no he blue cells of Opon 2 nex o he approprae maury daes. Then go o he Premums ab, and you wll need o use Solver n he same manner as descrbed above o compue he hazard raes. The only dfference s ha you wll be solvng for he cells n row #13. So for he premum paymen for he frs CDS wh maury 0.5 years, you wll use Solver and Se arge cell C8 Equal o: he Value of: 0, By changng cells: C13. Repea hs 10 mes n ncreasng order unl you solve for cell L13. The resulng premums wll be shown n he purple cells n row #13. Any me you change any daa, you need o run he Solver over agan for every cell you wsh o solve for. 31

32 Bblography Arvans, Angelo and Jon Gregory. Cred: The Complee Gude o Prcng, Hedgng and Rsk Managemen. Rsk Waers Group Ld: London, "Bond Bascs." Invesopeda Invesopeda Inc. Feb.-Mar <hp:// "Bond Rangs." Fdely Invesmens FMR Corp. Jan.-Feb <hp://personal.fdely.com/producs/fxedncome/bondrangs.shml> Bowers, Gerber, Hckman, Jones, and Nesb, ed. Acuaral Mahemacs. Illnos: The Socey of Acuares, nd edon. Cred Dervaves Explaned: Marke, Producs, and Regulaons. Lehman Brohers Inernaonal (Europe. March 2001: Galan, Sefano S. Copula Funcons and her Applcaon n Prcng and Rsk Managng Mulname Cred Dervave Producs. Kng s College London. Sepember 2003: Gordy, Mchael B., ed. Cred Rsk Modellng: he Cung Edge Collecon. London: Rsk Books, Jackson, Mary and Mke Saunon. Advanced Modellng n Fnance usng Excel and VBA. New York: John Wley & Sons, Ld. L, Davd X. On Defaul Correlaon: A Copula Funcon Approach. The RskMercs Group. Aprl 2000: "Rsk Glossary." 1996-Curren. Conngency Analyss. 13 Feb.-Mar <hp:// Rsk. Invesopeda Invesopeda Inc. Feb.-Mar <hp:// Shmko, Davd, ed. Cred Rsk: Models and Managemen. London: Rsk Books,