ond Implied Spread and -ond asis Richard Zhou Augus 5, 8 Absrac We derive a simple formula for calculaing he spread implied by he bond mare price. Using no-arbirage argumen, he formula expresses he bond implied spread as he sum of bond price, bond coupon and ibor zero curve weighed by risy annuiies. We show ha he bond implied spread is consisen wih he sandard pricing model if he survival probabiliies and recovery are consisen wih he bond price.. Inroducion A conrac is an OC ransacion beween wo paries in which he proecion buyer pays a sream of coupon paymen o he proecion seller unil he earlier of mauriy or eniy defaul in exchange for a defaul coningen paymen. he common defaul selemen is he physical selemen where he proecion buyer delivers a bond from a pool of eligible bonds o he proecion seller in exchange for par. conracs can also be cash seled where he proecion buyer receives from he proecion seller he cash amoun of par less recovery. Wih he physical selemen, he proecion buyer holds a delivery opion where he can choose any bond from a pool of bonds o deliver ino he conrac. mpirical evidence shows ha bonds of he same eniy do no necessarily have he same mare value following defaul []. As a resul, he sandard pricing wih a fla recovery rae canno properly price in he value of he delivery opion embedded in he conracs. Given he issuer defaul probabiliy, he bond price is deermined by he recovery and oher fundamenal and mare echnical facors such as supply and demand, and liquidiy. From modeling perspecive, i is difficul o separae recovery, defaul probabiliy, and oher mare fundamenal and echnical facors since hey are inerwined. he recovery swap prices can be used as he expeced recovery rae bu he mare has no ye fully developed. ven if he recovery rae can be deermined independenly, he defaul probabiliies calibraed o he mare spreads or bond prices are sill conaminaed by oher facors such as supply and demand, funding cos, bond rading away from par (see [7] for a deailed exposiion on facors impacing and cash basis). mpirical sudies show ha he mares apears o price based on ibor curve raher han he reasury curve []. discouning should be based on ibor. Since he ibor Ris Managemen, he eposiory rus & Clearing Corporaion, mail: rzhou5@gmail.com. he opinions of his aricle are hose of he auhor and do no reflec in any way he views or business of his employer. All errors are auhor s own.
are he borrowing raes beween bans of AA raing, he ibor curve is implicily an AA raed yield curve. As a resul, we use ibor as he ris free ineres rae. An asse swap (ASW) is a pacage ransacion beween wo paries in which he ASW buyer purchases a bond from he oher pary and simulaneously eners ino an ineres rae swap ransacion, usually wih he same counerpary, o exchange he coupon on he bond for ibor plus a spread. he spread is called he asse swap spread. A common asse swap is he par asse swap where he buyer pays par a he incepion of he deal. Unlie, ASW coninues following bond defaul. -ond basis is he difference beween he spread and he ASW spread on he same bond. I is a general indicaor of relaive value of versus he cash bond. For example, when he spread is higher han he ASW spread, i.e. he basis is posiive, he is generally considered o be more aracive han he bond. he reverse is rue if he basis is negaive. ond implied spreads have been previously invesigaed. avies and ugachevsy proposed an approximaion mehod for calculaing he bond implied spread based on he Z spread adjused by he bond s mare price, duraion, convexiy and recovery rae [3]. In a series of paper, erd e al proposed o use he survival-based modeling as a consisen measure for credi-risy bond pricing and ris managemen [4]. In heir model, he survival probabiliy erm srucure of an issuer is esimaed by regressing model prices agains mare prices across all bonds of ha issuer under a consan recovery rae. he resuling survival probabiliy erm srucure is no necessarily he same as ha implied by he mare. he idiosyncraic fiing error is accouned for by he OASo-fi which is essenially a measure in spread of aggregae effec of mare facors on he bond. In his paper, we describe a simple model for esimaing he bond implied spread. he model concep is no new as i is based on survival probabiliy. u i is cas in such an explici way ha i is easy o analyze he effecs on he spread of bond coupon, recovery rae, bond price, and ineres rae and credi curves. hese effecs have been discussed in lieraure, bu an explici formula ha combines all hese facors seems needed and his paper provide a such formula. he paper is organized as follows. Secion describes he underlying heory and main formulas. Secion 3 gives some numerical examples, and secion 4 concludes he paper. he deailed model derivaion is given in Appendices A and.. and Asse Swap Spreads In his secion, we describe a model for calculaing he bond implied spread and he -ond basis. We define he bond implied spread as he spread ha equaes he bond mare price wih he bond fair heoreical price. We demonsrae he effecs of ibor curve, bond coupon and he price deviaion from par on he credi spread.
. Asse Swap Spread In a par asse swap ransacion, he invesor buys a pacage consising of a cash bond and a payer ineres swap where he invesor swaps he bond coupon for ibor plus a (ASW) spread. he price of he asse swap pacage is par (hence he erm par asse swap) which he invesor pays a he deal incepion. his means ha he bond diry price and he iniial swap value mus sum o par. For a discoun bond, he iniial swap value is posiive o he invesor. For a premium bond, i is negaive. he mare pracice is asse swap does no noc ou when he underlying bond defauls. As a resul, he invesor bears some ineres rae ris in ha he needs o see new funding o pay he fixed rae should he bond defaul. e be oday, and he bond coupon paymen daes be,,...,. Furhermore, we assume ha he ASW paymen daes coincide wih he bond coupon paymen daes. he spread of a par asse swap is given by M ( ) S ASW C A A () where M is he bond ris-free price, - is he bond s diry price, C is he bond coupon, is he mauriy forward ibor rae paid a, and A is he ris-free annuiy given by A where () is he discoun facor for mauriy,, and he ris-free average ibor weighed by he discoun facors is A ( ) (3) he firs form of ASW spread in formula () is well nown. I saes ha he ASW spread is he difference beween he ris-less price and he mare price amorized over he swap life. Since he bond ris-free price is always greaer han he mare price, ASW spread is always posiive. oe ha he spread would have o be negaive if he bond mare price is greaer han he bond ris-free price. he second form of formula () is less familiar, bu more inuiive. I shows ha he ASW spread is composed of hree componens: he bond coupon, he average ibor over he swap life and he difference of bond diry price and par scaled by he risless annuiy. he las componen can also be inerpreed as he bond discoun amoun amorized over he swap life. his discoun amoun is paid upfron by he invesor and recouped over he life of he swap as par of he swap spread. 3
Remars: ) If he bond is rading a par ( ), he ASW spread is he difference beween he bond coupon and he average ibor rae over he ASW erm. ) he ASW spread increases as he bond price decreases ( increases). Hence, he more deeply discouned is he bond, he higher is he ASW spread. 3) We will show in he nex secion ha he bond implied spreads given by formula (7) has characerisics similar o he ASW spread. he bond implied spread conains hree erms wih inerpreaions similar o hose of erms in formula (). And he bond implied spread increases wih decreasing bond price (increasing ) bu a a faser rae han ASW spread does, resuling in an increasing basis.. ond Implied Spread Suppose an invesor execues a so-called negaive basis rade in which he buys he cash bond on he ASW basis and buys proecion. he mare price of he bond is - where represens bond s discoun relaive o par. Assuming a recovery rae R, he invesor needs o buy -/(-R) noional proecion in order o mae he combined posiion defaul neural. Given a non-zero recovery, his amoun is less (more) han - for a discoun (premium) bond. e us assume ha he invesor borrows - o purchase a bond wih fixed coupon C. He hen buys -/(-R) noional proecion on he bond. enoing he recovery rae by R, he bond implied spread is given by (See Appendix A for derivaion and definiions of he erms in equaions (4), (5) and (6)). S W V C V Risy R oss V ( ) (4) R under he consrain R R where W /( ) ( ) C V R oss (5) quaion (4) shows ha he bond implied spread is he sum of conribuions from he bond coupon, he ibor curve, he difference beween he bond (diry) price and par, and he recovery rae augmened by he risy annuiies and W which is he raio of he noional amoun o he bond price. We will show in Appendix ha he bond implied spread calculaed using equaion (4) auomaically saisfies he sandard pricing equaion 4
R S oss V (6) his implies ha, if and only if he recovery and defaul probabiliy are consisen wih he bond price, is he model consisen wih he sandard pricing model (6). he reverse does no hold. Given a spread and a recovery rae R, he defaul probabiliy implied by equaion (6) is generally inconsisen wih equaion (5). Subsiuing equaion (6) ino equaion (4) and solve for S, we find a simplified form S V Risy C (7) V V I is clear from formula (7) ha for a par bond ( ) and a fla ibor curve, we have V S C which is slighly less han he C- implied by he risy par floaer V replicaion model. he difference is due o ha in our model, he bond accrued coupon is no paid bu he accrued premium and ibor ineres are paid upon defaul, while here is no such defaul paymen discrepancy in he par floaer replicaion. Given he bond s mare price -, he model framewor of (5-7) can be used in several ways: ) Single bond: In his siuaion, we assume a recovery rae R, and calculae he consan bond implied hazard rae using equaion (5). he bond implied spread is hen calculaed using equaion (7). he bond implied hazard rae is no necessarily consisen wih he hazard rae implied by he mare quoe of he same issuer. he bond implied hazard raes are deermined by he fundamenal and echnical facors in he cash mare while he hazard raes are deermined by he mare fundamenal and echnical facors. However, if we are concerned only wih he spread, equaion (7) is all ha maers. herefore, equaion (7) can be used o compare he bond implied spread o he mare quoe. ) Muliple bonds of differing mauriies: Assuming a consan recovery, we boosrap o obain a erm srucure of he bond implied hazard rae consisen wih he given bond prices. he bond implied spread for a mauriy can hen be calculaed from (7). 3) erm srucures of mare spread and bond price: We boosrap o calculae he hazard rae erm srucure and he bond implied recovery rae by simulaneously solving equaions (5) and (6). he resuling hazard rae and recovery rae erm srucures are consisen wih he bond mare prices and he mare spreads of he issuer. However, he recovery raes are influenced by he cash mare facors and are no necessarily he expeced percenage recovery amoun of par. 5
Remars: ) Ineresingly, he spread formula (7) can also be direcly obained if he noional is -. -/(-R) noional is defaul neural. - noional would resul in a small defaul payoff of R which is offse by he larger carry (-)*S. ) he form of equaion (7) is useful as i explicily expresses he spread in erms of bond coupon, ineres rae curve, and bond price augmened by he risy annuiies, allowing easy analysis of individual facors. 3) I is imporan o noe ha in equaions (4-7) we have no imposed any resricion on he shape of credi and ineres rae curves..3 ond asis he ond basis is defined as he difference beween he bond implied spread (7) and he par asse swap spread () Risy V V asis C (8) V V A he -ond basis consiss of hree erms: ) he firs erm is he difference beween he average ris-free ibor rae and he average risy ibor rae. his erm increases wih increasing bond discoun, or increasing defaul ris. I is posiive for an upward sloping ineres rae curve, and negaive for a downward sloping curve. I is zero when he forward curve is fla. herefore, he firs erm can be inerpreed as he impac of he yield curve slope o he basis. Since he normal yield curve shape is upward sloping, he ineres rae curve effec on he basis is generally posiive (see able ). ) he second erm is due o he paymen mismach beween he bond coupon and premium and borrowing cos upon defaul. While he invesor does no receive he bond s accrued ineres in he even of defaul, he sill needs o pay he accrued premium and ineres on he loan. his erm always conribues negaively o he -ond basis. 3) he hird erm represens he effec of bond s mare price on he -ond basis. I is posiive for discoun bond and negaive for premium bond. I explains why, rough speaing, -ond basis is posiive for discoun bond and negaive for premium bond. However, ables and show ha his is no sricly correc. hey show ha he -ond basis can be eiher posiive or negaive for par bond depending on he ineres rae curve shape. 3. umerical xamples We now show some pricing examples. he ineres rae curve is he swap zero curve for July 6, 8 aen from loomberg. he paymen frequency is semiannual. We linearly 6
inerpolae he swap zero curve o obain he zero raes for all paymen daes. For our purpose, linear inerpolaion is deemed adequae because we only need a forward ibor curve. However, differen inerpolaion scheme may and will resul in slighly differen bond implied spread. he impac of inerpolaion scheme on ASW spread seems o be smaller han on spread. e Z be he zero rae for mauriy ( ) Z, he forward ibor rae for period (, ) Z is, (9) able : ond implied spread and ASW spread as funcion of bond price - using mehod I. he bond has year o mauriy wih 7% semiannual coupon, 4% recovery. he day coun convenion is 3/36. oe ha all values are in percenage excep for W. - -5 5 5 W.6.3..96.93.88.83.96.6.3 3.9 3.97 4.97 6.3 ASW.5.67.9.9 3.54 4.6 4.79 asis -.9 -.7..7.43.8.34 λ.58.64 3.8 5.9 6.65 8.9. Risy.3.5.8..4.7. C( V / V ).3.5.7.9..4.8 / V / A -.9 -.7.5.4.78.3 able shows he bond implied spread (formula (7)), ASW spread (formula ()) and basis (formula (8) as a funcion of he bond price discoun. he bond pays 7% coupon semiannually and has years o mauriy. We assume 4% recovery rae and 3/36 day coun convenion. he spread, ASW spread and bond implied hazard rae all increase wih decreasing bond price (increasing ). As expeced, he - ond basis decreases wih increasing bond price. able also shows he hree erms in he -ond basis formula (7). he individual conribuion of he hree erms in formula (7) can be easily inferred from able. In his case, he firs erm slighly dominaes he second erm, and he ne effec is small due o offseing. u his is no always he case. For example, he firs erm in (7) is zero if he ibor curve is fla. For bond rading subsanially away from par, he 3 rd erm in formula (7) dominaes he basis. o demonsrae he curve effec, able shows he resuls for he same bond as in able bu wih a fla ibor curve of 4.7% which is he average ibor in able. We can see ha he shape of ineres rae curve has a small effec on he spreads and basis. 7
able : Fla ibor curve. he same bond as in able. All values are in percenage. - -5 5 5.9.55.4 3. 3.86 4.83 5.95 ASW.4.67.3.93 3.56 4. 4.83 asis -. -. -.6.7.9.63. λ.5.55 3.68 4.94 6.35 7.96 9.8 Risy....... C( V / V ).3.4.6.9..4.7 / V / A -.9 -.8..5.4.77.3 4. Conclusions We have presened a simple explici formula o calculae he spread implied by he bond mare price. he value of he model is ha i can be used eiher for issuers having a single bond ousanding or issuers having muliple bond issues. he formula explicily expresses he bond implied spread as he weighed sum of hree facors: bond coupon, bond discoun percenage and he ibor curve. A poenial use of he spread formula (7) is o explore he difference beween he mare spread quoe and he fair bond implied spread. 5. References [] R. ullirsch, R. Janowisch,. Veza, he elivery Opion in Credi efaul Swaps, Woring paper, Ocober 5, 7. [] J. Hull, M. redescu, A.Whie, he Relaionship eween Credi efaul Swap Spreads, ond Yields, and Credi Raing Announcemens, Journal aning and Finance, V8, pp 789-8, 4. [3] M. avies,. ugachevsy, ond spreads as a proxy for credi defaul swap spreads, Ris magazine, 5. [4] A. erd, R. Mashal,. Wang, efining, simaing and Using Credi erm Srucure, ar,, 3, ovember 4. [5]. ando, On Cox processes and credi risy securiies, woring paper, 998. [6] X. Guo, R. Jarrow, C. Menn, A oe on ando s Formula and Condiional Independence, woring paper May, 7. [7]. O ane, R. McAdie, xplaining he asis: Cash versus efaul Swaps, ehman rohers Repor, May. 8
Appendix A We derive he pricing formula (4) for he bond implied spread. As saed previously, given he bond discoun, he negaive rade invesor buys he bond and hedge wih buying -/(-R) noional proecion. his noional amoun maes he combined -bond posiion defaul neural. Suppose he invesor funds he purchase a ibor fla which is a reasonable assumpion because spread is based on ibor []. We assume ha he cash flow erminaes upon defaul. Furhermore, we assume he invesor will pay he accrued premium and loan ineres bu no receive bond accrued coupon when he bond defauls. ased on hese assumpions, he cash flow o he invesor is described in he following able. oan ond oal Iniial - -(-) aymen ae -(-/(-R))S -(-) C C-(-) - (-/(-R))S efaul (--R)-Accrued premium -(-)-accrued loan ineres R -Accrued and loan ineres Mauriy -(-) he no-arbirage condiion means ha he expeced presen value of all cash flow, iniial and fuure, o he invesor mus be zero. We arrive a C ( ) (, ) S ( ) ( ) (, ) ( ) ( ) In he above equaion, ( ) S R R ( < ) (A.), is he -mauriy forward ibor seen a ime, R is he expeced recovery rae, is he defaul ime, () is he money mare accoun and (A) is he indicaor funcion. he firs erm in equaion (A.) is he ne coupon paymen o he invesor on scheduled paymen daes. he second erm is he paymen upon defaul of accrued loan ineres and spread since he las scheduled coupon paymen dae. We assume he accrued bond coupon is no paid upon defaul. he hird erm is he payoff o he invesor a mauriy if he bond has no defauled. Adoping he usual assumpion of independence beween credi spread and ineres rae, using he ando formula (see [5]), and approximaing he inegral using rapezoidal rule, we ge 9
[ ] I d d < (A.) [ ] I d d < (A.3) Using he fac ha he, is a maringale under he forward measure, we have [ ] [ ] I d <,,, (A.4) where, is he forward rae. Subsiuing equaions (A.), (A.3) and (A.4) ino (A.), and rearranging erms resuls in Risy V S R V C. (A.5) where j j j Risy d xp oss V V V, /, λ (A.5)
Appendix ow we prove he equivalency beween equaions (4) and (6) under he consrain of equaion (5). roposiion: If equaion (5) is saisfied, he bond implied spread calculaed using equaion (4) saisfies equaion (6). roof: y virue of equaion (5), we can rewrie equaion (4) as R oss (.) Risy ( ) V W S V where, ( ). Furhermore, we rewrie he erm oss in equaion (A.) ino oss ( )( ) [ ] [ ] (.) oice ha and using (.), we have Risy Risy ( ) V oss V ( ) oss oss [( )( ) ( )] ( )( ) oss oss oss (.3) Subsiue (.3) ino (.) yields R oss S V (.4) W Since W /( ), we have /. R R W R his complees he proof.