Bond Relative Value Models and Term Structure of Credit Spreads: A Simplified Approach

Transcription

1 Bod Relve Vlue Models d erm Sruure of Cred Spreds A Smplfed Approh Kur Hess Seor Fellow Deprme of Fe Wko Mgeme Shool Uversy of Wko Hmlo New Zeld kurhess@wko..z VERSION 7-Ju-04 Co dels Kur Hess Uversy of Wko WMS Deprme of Fe Prve Bg 305 Hmlo New Zeld ph kurhess@wko..z - -

2 Bod Relve Vlue Models d erm Sruure of Cred Spreds A Smplfed Approh Absr Bod relve vlue models o dee mspred bods re wdely used he vesme ommuy. hese rge from smple yeld o mury omprsos o sophsed sohs models. he frs sep for my of hese models s he deermo of referee yeld urves. here re umerous publos o hese yeld urve fg pprohes wh reled emprl reserh ye few ully doume prl mplemeos for operol purposes. Aordgly he frs pr of hs rle desrbes d he llusres mplemeo of umber of hese behmrkg models. Wh suh fg frmework bods sube o red rsk ofe o be hdled se he umber of bods of equvle red quly s smply oo smll o derve relble referee urves. Here he rle proposes ovel pproh o prmeerze he erm sruure of red spred. Is m beef re uve model prmeers h rele o he oep of how mrke proers lke rders d sse mger ed o mesure red rsk of fxed ome seures. My of he models desrbed here hve bee mplemeed EXCEL/VBA some of whh re geerlzed versos of models h hve bee developed for prl bod relve vlue reserh. he fles og he models be dowloded from he followg webse hp// Keywords Bod Relve Vlue Models Ieres Re Models Cred Spreds Yeld Curve Modelg Exel VBA - -

3 Bod Relve Vlue Models d erm Sruure of Cred Spreds A Smplfed Approh Iroduo Relve vlue model o dee dos of poel exess wh uverse of fxed ome seures re wdely used he vesme ommuy. I he bod mrkes relve vlue usully refers o he proess of omprg reurs mog fxed ome seures bu wder sese hs defo be exeded o lude omprsos wh sy reled equy srumes. here re wo fudmel fors h prmrly ffe he prg of fxed ome seures. hese re frsly he prevlg mrke eres res d seodly he spef red rsk of he bod. hs pper wll del wh models h ddress hese wo prg spes. Wh regrd o he eres re for here re umber of heorel models my of hem versos of seml work by Vsek (977) d Cox Igersoll & Ross (985) h posule eres re proess s he drvg se vrble whh ur deermes he shpe of he yeld urve d hus he prg of bods. Uforuely her prl pplo for bod prg s lmed. he yeld urve shpe d dyms observed ofe o be expled wh hese pprohes s he rue sohs ure of he eres re proess rems elusve. he usul mehod s o lbre suh models wh observed yeld urves s for exmple he models of Ho & Lee (986) or Heh Jrrow & Moro (99). hs ur requres mehods o derve he erm Oher prg fors suh s xo d lqudy prem hve lso bee sube o reserh bu hese re yplly o osdered models used by mrke prps. See Blss (997 p.6) or Iodes (003 foooe p. 5) for referees o some of hese sudes. Reboo (998) desrbes mos of hese populr eres re models gre del

4 sruure from rded srumes or ors suh s bods or eres re swps. he frs pr of hs rle wll fous o hs lss of urve fg models whh hough her prmeers do o hve ul eoom meg hve muh greer sgfe for he mrke proer. Seo hrerzes hem log he les proposed by Blss (997) d he dels how some of hem hve bee mplemeed EXCEL/VBA. Exmples lude versos of MCulloh (97; 975) d Nelso & Segel (987) s well s umed smple pproh h s geerlzed verso of model esed rdg evrome. Whle be ssumed h smlr mplemeos re ppled dusry hey hve o bee formlly doumeed he dem lerure. Moreover Exel/VBA s wdely used for fl lyss.e. provdes very populr plform o prese hese models. I he oex of prg fxed ome seures red rsk s usully mesured by he solled red spred whh mesures he dfferee yeld offered for rsky bod ompred o equvle rskless goverme bod. Ever se he seml work of Mero (974) h poeered he sruurl prdgm red rsk modelg he ure d dyms of hs erm sruure of red spreds hs bee sube o subsl reserh. Oe puzzlg resul s h eve fer oug for possble xo effes oe fds h expeed deful d reovery res o bods expl oly pr of he yeld premum ully observed 3. here s o oly dem debe s o how o expl he ble bu Coll-Dufrese e l. (00) llusre h uses of spred hges re hrd o ppo oo. Alhough hey use lrge umber of proxes ffeg red rsk hey fl o expl mos of he observed dyms. hey olude h he dom 3 See for exmple Fos (994). Elo e l.(00) esme spred ompoes due o deful rsk xo wh he ble expled s rsk premum for sysem rsk. Duffe & Ldo (00) model he mperfe dsree ure of formo flowg o vesors o ou for hgher h expeed red spred observos

5 ompoe of red spred hges re lol supply/demd shoks o pked up by y of her proxes. I he bsee of y olusve models vesor hus g hve o rely o ppropre red spred fg models smlr o he oes dsussed erler hs seo 4. hese gve mrke proers lke rders whh wll geerlly be wre of suh lol supply/demd fors frs do of poel msprg. Uforuely he bske of omprble bods of he sme red quly s ofe lmed d hs preves fg relble erm sruures of red spreds o he d. hs rle hus preses heurs fg mehod h be ppled suh suos. Srg po s er rge red spred.e. he spred whh he vesor deems ppropre for he prulr bod or group of bods. hs s he omplemeed by umber of shpe prmeers. Whle he prulr rge spred s revewed regulrly e.g. by mes of ssl lyss he hrers of he shpe prmeers s more s d lso be ommoly se for lrger segme of he bod mrke for se for ll lower vesme grde bods 5. I s o he mbo of hs pproh o ompee wh y of he more dved fg mehods suh s he oes derved from equlbrum erm sruure models 6 bu o smply provde he proer ool o uover pprely mspred bods frs se. he deso proess s helped by he uve ure of rge red spred s he m model prmeer. he llusrve Exel/VBA 4 Ree developmes re ew o esmo ehques s preseed Houwelg Hoek & Kleberge (00). 5 A vesor mgh dede o defe lower vesme grde bods s bods wh rg BBB- up o BBB 6 e.g. see Aderso e l. (996 hper 4 p. 67) - 5 -

6 model mplemeo preseed seo 3 uses d for smple of Swss domes dusrl bods. Bod Relve Vlue wh Yeld Curve Fg Models As ded he roduo s yeld urve fg models s bss o dee mspred bods re very muh ppled he mrkes. Exmples re Merrll Lyh (004) dly Rh/Chep Repors for ouless bod mrkes d segmes. Krpper (003) p. lso lss JP Morg HSBC Bk d UBS Bk s suos produg bod relve vlue reserh bsed o yeld urve fg models. Evdee from vrous sudes suh s Seru & Wu (997) or more reely Iodes (003) deed suggess h here s usfo for pplyg suh models. For boh he Belg respevely UK goverme bod mrkes hese sudes foud sgf exess reurs for rdg sreges bsed o buyg (shorsellg) bods h re lssfed s udervlued (overvlued) relve o prulr esmed erm sruure model. he followg revews d lssfes hese models geerl whh s he followed by subseos doumeg he mplemeo of hree of hem.. Revew of S erm Sruure of Ieres Models he erm sruure of eres oep erl o eoom d fl heory plys key role o us for he prg of bods bu lso y eres re oge lm. hs seo wll fous o he work h hs bee doe he re of o-probbls yeld urve modelg. I follows frmework proposed by Blss (997 p. 4) who sees hree dmesos o suh models or rher here re hree desos requred o esme erm sruure of eres s he bss for bod relve vlue model he prg fuo he pproxmo fuo d flly he esmo mehod

7 .. Prg Fuo he mos srghforwrd prg fuo s erly he prese vlue (P) of he bod s promsed sh flows (C) P M m C m e r( m) ( m) () where M re he umber of remg sh flows umbered o m; r ( m) d ( m) re he spo res respevely mes whh hese sh flows wll our. hs s however o he oly prg relo used he mrkes. he muh smpler yeld o mury bsed bod vluo s sll very muh use by vesors. hs beuse he smple yeld mesure k o he well-kow erl re of reur s yplly he frs pee of bod lys lsed by fl d provders d he med. Some mrkes lke Ausrl d New Zeld eve refr from quog fxed ome seures by pre bu rher by yeld o mury whh ur s used o lule he ul seleme pre by mes of sdrdzed formul 7. A smplfed verso of suh formul usg ouously ompouded res d o osderg he omplexes of me mesureme oveos 8 would look lke he prg formul () bove wh os re r o loger depede o he me of he m h sh flow. Whever fuo s hose oe wll pre exly pre ll bods prulr referee bske. A ex relo for he pre P of prulr bod eeds o be formuled 7 See ppedx for he exmple of he New Zeld bod mrke formul s show RBNZ (997 p. ) 8 Chrse (003) provdes some deled desrpo of how me mesureme oveos ludg fors suh s ol holdy ledrs ffe bod yeld lulos

8 P [ Cm r( m) ] ε f () where he fuo f []. pures ll h we ssume wh deermes he pre of he bod d r ( m) s fed o mmze some fuo of he rdom resdul erm ε. I formulg f []. reserhers wll ofe dd erms (e.g. dummy vrbles) o he srgh prese vlue formul h emp o pure effes of fros he mrkes suh s x effes or lqudy prem 9. I s however he experee of hs uhor h hs s hrdly doe operol models beuse suh fors ed o hve less gble mp o he pre of he srume... Approxmo Fuo As ex sep oe mus dede o he fuol form o pproxme eher he dsou re fuo r ( m) or he dsou fuo d ( m). hs s eessry s here re lmed umbers of bods whh requres wy of erpolg he res respevely dsou fuo for rbrry me horzos. he usul pproh s o sele pproxmg fuo d he o esme he prmeers. We meo here us wo msrem mehods 0 prsmoous represeo defed by expoel dey erm poeered by Nelso & Segel (987) d Svesso (994) d ub sples rodued o fe by MCulloh (97; 975). he mos omprehesve omprve sudes of hese d oher fuol forms hve bee uderke by Blss (997) d Iodes (003). Whle here re dfferees bewee he very my mehods oe s dsqulfed by hese 9 See foooe roduo for referees o lerve prg fors. 0 Blss (997 p.6) d Iodes (003 p.3) ls referees o he mor lsses of suh models

9 reserhers for her goodess of f. Wekesses pper more oher spes e.g. dffuly o esme prmeers (see below) or usble respevely fluug forwrd res mpled. Wh regrd o resdul bsed bod relve models here s hus o use o dsou y of hem...3 Esmo Mehod Lsly he deso o he ppropre esmo ehque s more ehl bu everheless mpor ssue. he spe of esmo o oly ludes he hoe of umerl lgorhms for prmeer esmo. hese re ofe deermed by he ype of fuos hose before. Oe mus lso mke desos o error weghg fuos d how o hdle bd/sk spreds. Reled re d egry ssues. Oe mgh defe d flers o remove pprely erroeous d from he se. he rky spe rems h ors o emprl reserh wh hsorl d he d osello operol rdg pplo s o kow beforehd so oupu plusbly heks re essel.. Implemeos of Referee Curve Models hree referee yeld urve models re preseed hs pper. Frsly shows smple yeld o mury bsed behmrkg ool whh s llusred for bske of Swss goverme bods. Nex here s he JP Morg Dsou For Model (JPM) verso of MCulloh s (97; 975) ub sple mehod d flly bod relve vlue model usg exeded Nelso & Segel (987) pproh. he ler wo models re llusred wh d of he smll uverse of New Zeld A exepo s he Fsher Nyhk & Zervos (995) ub sple whh ws foud o be performg poorly by Blss (997 p. 6)

10 goverme bods. he fles og he models be dowloded from he webse hp// I followg ble he hree models re hrerzed le wh he frmework preseed he prevous seo. All he models re se up so hey esly be lked o rel me pre d soure. Noe h some ehl omplos my rse from he reme of rued eres whh depedg o mrke oveos hs o be pd upfro by he bod buyer. he models ssume h pres quoed re so-lled le pres exludg rued eres. hese d oher ssues reled o bod lys re dsussed spelzed fxed ome resoures suh s Fbozz (999)

11 ble Model Clssfo I II III Model Yeld o Mury Behmrkg JP Morg Model Dsou For Model Exeded Nelso & Segel Prg fuo Smplfed pre s fuo of yeld o mury.e. s model II/III bu P M C m m e r( m) ( m) ε os r ( m) r( ) see formuls for explo of prmeers Approxmo Model yeld o mury Model dsou fuo s Spo re modeled wh fuo erm sruure s polyoml. Verso of expoel form s desrbed polyoml (MCulloh 97; 975) Blss (997 p. ) Esmo mehod OLS of yeld o mury OLS of pre errors (equl Duro weghed les errors (equl weghg). weghg) squre mmzed wh Correspodg sysem of Correspodg sysem of Geerlzed Redued ler equos solved wh ler equos solved Grde (GRG) oler LU Deomposo Exel bul- LINES opmzo ode s Press e l. (99 p. 43) fuo. mplemeed Exel Solver - -

12 .. Smple yeld o mury bsed behmrkg model Yeld o mury lso lled redempo yeld bsed mesures o fd relve vlue uverse of bods s he rdol mehod sll used by my proers. As s ommoly kow yeld o mury ssumes h vesor holds he bod o mury d ll he bod s sh flows re revesed he ompued yeld o mury. I s foud by solvg for he eres re h wll eque he urre pre o ll sh flows from he bod o mury. I hs sese s he sme s he erl re of reur (IRR) defed my fe exbooks e.g. Relly & Brow (997 p. 59). Needless o sy h redempo yeld bsed bod lys hs mor shoromgs s ws doumeed my yers go by Shefer (977) d lso dsussed Aderso e l. (996 p. ). Revesg eh oupo he sme re s mou o ssumg fl erm sruure wh del spo res for eh mury. If spo res rese wh mury yeld o mury wll uderesme he spo re. Coversely overesmes dowwrd slopg spo-re urve. Hvg oed hs purely for bod relve vlue purposes rems useful mesure wh he eessry ves. Coupos should frsly be uform prulrly wh prulr mury rge. Smlrly errors re smller f mrke yeld levels ludg oupo res re low. here re my bod mrke segmes h hve omprbly low lqudy wh wde bd/sk spred. Applyg eser yeld o mury model hese ses s surely more hoes beuse msprg wll lso be deeed wh hs more rude pproh. he mplemeo of hs redempo yeld bsed relve model s sored he fle med CofBehmrk (Feb04).xls. I s geerlzo of model h hs bee used for prl relve vlue reserh he Swss bod mrke for umber of yers. I ws ppled o umber of homogeeous mrke segmes provdg formo regrdg he relve prg of hese - -

13 ssues. he followg brefly expls he mhems of he fg proedure d he elbores o seleed mplemeo ssues. Fgure Geer me / Yeld Chr wh Bod Yeld Curve Yeld 4.0% 3.5% 3.0%.5%.0%.5%.0% 0.5% 0.0% Yeld Behmrk Bods Behmrk Curve Yers o Mury Y Y Fgure llusres he prpl mehod of fg polyoml o he me / yeld o mury plo of behmrk bods. he yeld Y of bod ( bods referee bske) s pproxmed by Y m m m m... 0 where s me o mury of bod d 0... re he os oeffes of order m polyoml. m I ordry les squre regresso (OLS) mmzg he sum of squred yeld errors mes we hve o se prl dervve o zero M Y Y Y Y k 0 for k 0..m hs he yelds m equos for he ukow oeffes... 0 m

14 - 4 - Some lgebr shows h m... 0 mus be soluo of he followg sysem of ler equos usg mrx oo m m m m m m m m m Y Y Y Y I he llusro model hs sysem s solved umerlly by he well kow LU deomposo lgorhm s desrbed Press e l. (99 p. 43). As oeffes lke m beome exremely lrge umber for hgher dmesos of m he lgorhm wll lose ury. However hs s o ssue geerl beuse megful erpolos wll o exeed 3 rd o 4 h order polyomls. Oe he pproxmed yelds Y hve bee luled oe he deerme he orrespodg model pres P usg mrke oveo yeld formul (e.g. RBNZ 997 p. ). As he model wll derve behmrk polyomls for eh he bd d sk yeld here wll be boh bd d sk model pres. A buy (sell) sgl s geered f he bd (sk) pre he mrke exeeds (s below) he sk (bd) model pre foud by he model. here s feure o derese he sesvy of he model by rodug fler rule so reommedos re oly geered f hese pres re se bsolue mou pr. hese smple rules for geerg reommedos re llusred Fgure.

15 Fgure Buy /Sell Sgl Rules Smple Yeld o Mury Behmrkg Model Sesvy Mrke pre P Sesvy sk bd P sk bd P UNDERPRICED Fr Pred OVERPRICED Sgl BUY SELL sk bd P sk bd P sk bd P sk bd Aoher rely of operol models s h some d mgh suddely be mssg d hs hs o be hdled. I he soluo preseed here mssg d re repled by ls kow hsorl (e.g prevous dy) pres. I mkes sese o suppress orrespodg buy/sell sgls hese ses. he model s lso se up o hdle llble bods smplfed wy. For eh bod deermes he so-lled yeld o wors whh s he lower of eher bod o mury or he yeld o he ex ll. If he ll yeld s lower he bod s mury wll be se o he ex ll de for behmrkg purposes. I does hus o employ more dved ll feure lys suh s he ofe used opo dused spred lyss. hs mehodology s desrbed Wds (993) of Bloomberg bsed o he Blk Derm & oy (990) eres model. Uder hs pproh llble bod s vewed s log poso of opo-free bod plus shor ll o he bod (sold o he ssuer)

16 o meo fl feure he model os some uly mros for llusrve purposes. For operol use does o suffe o smply lk he model o rel me rdg pres. he bske wll be sube o ouous hge d so oe eeds ules o del wh ddos o d deleos from he referee bske. Suh mros would be d soure spef bu he srps show he mplemeo gve flvor of wh would hve o be uomed... JP Morg Dsou For Model (JPM) he JPM model llusres how o overome he wekess pure yeld o mury bsed lyss. he model s respevely ws kow he mrke s he JP Morg dsou for model (JPM) bu orgl doumeo ould o be uovered for he purposes of hs rle. A revew of he lerure reveled h hs model s ul f smple verso of he MCulloh (97; 975) sple pproh whou ode pos (s dsussed Aderso e l. 996 p.5). I le wh MCulloh he model works wh he dsou fuo whh mes he mmzg fuo be foud usg les squres s llusred below. A dvge of refrg from modelg he spo re urve s h he dsou urve muh beer behved o use olloqul erms. hs fuo hs ler boudry me zero d s moooly delg over me. he mplemeo of hs sple model s sored he fle med erm sruure JP Morg Model (Feb04).xls. I s more geerlzed h he erler yeld o mury model h us fs oe behmrk o he md pre whh s luled s he me of bd d sk pre. Aordgly here s o seleo of bods s show fgure bu smply lulo of he resduls. Smlrly o exmples of ules re luded. he followg expls he mhems of he dsou for fg hs se d he he m model prmeers o sree sho

17 - 7 - I le wh prese vlue prg fuo () he bods he referee bske wh mrke pres P [p p p ] should ll be equl o he prese vlue of fuure sh flows ( ) ( ) ( ) p p p d d d d d d d d d d d where s he fxed oupo re of bod ; d s he dsou for whh s he me of he h oupo of bod. he pproxmo of d s hose s he polyoml 0... m m m m. o smplfy he furher soluo s developed us for he hree dmesol se of ub sple. Noe however h he model mplemeo ope wh hgher order polyomls. Rewrg bove equos for m3 mrx oo oe fds p p p p We re hus lookg for he veor of oeffes [ ] A 3 0 h mmzes he sum of he squred dfferee bewee he mrke pre veor [ ] p p p P... d model pre veor p p p P....e. M P P

18 - 8 - Seg he prl dervves o zero k p p 0 for k 03 yelds four equos for he four ukow 3 0. Defg C C C C C C C C C C C C C oe mus hus solve he sysem of followg sysem of ler equos o fd he oeffe veor A ( ) ( ) P C C C A P C A C C he model pres re he foud by mulplyg mrx C wh he oeffe veor A P A C Bods pred below [bove] her orrespodg model pre re hep [ rh ]. he resdul veor of (uder prg)/over prg [ ] r r r R... 0 s he P A C P P R..3 Boudry odos yplly osrs re rodued o fore he dsou for me zero o oe whh s obvously mos resoble ssumpo. hs mes oeffe o s se o. Moreover oe ofe hooses he frs dervve of he dsou fuo me zero o f shor-erm re e.g. overgh bk re observed he mrke. For ssumed shor-erm re r o beomes mus he ouously ompouded equvle of he shor re respevely erms of he ul

19 equvle re - r - l( r ) where r s he shor-erm re expressed s ul equvle yeld d r s he shor-erm re expressed s ouously ompouded yeld. hs resul s derved foooe 3. Fgure 3 JPM model sreesho Seleme de 4-Feb-99 Coupo Mury Bd Ask Md JPM Fr Pre (hep) / rh JPM Coeffes 6.50% 5-Feb % 0.7% (0.60%) % 5-Feb % 04.48% (.66%) % 5-Mr %.34% (.87%) % 5-Apr % 97.7% (0.5%) % 5-Apr % 06.56% (.43%) 4 #N/A 8.00% 5-Nov % 06.06% 0.6% 5 #N/A 7.00% 5-Jul % 98.9%.8% 6 #N/A 6.00% 5-Nov % 9.83% (0.97%) 7 #N/A Model Prmeers deg resr frequey 3 Degree JP Morg polyoml o resros DF(0) oher vlues shor re Number of oupo pymes per yer Dsou Fuo Zero Res 8.0% 7.0% 6.0% 5.0% 4.0% 3.0%.0%.0% 0.0% he dsou for me ( ) expso of expoel fuo 0 ( ) ( ) ( )... e for r ouously ompouded yeld wh ylor r e 0 r 0r 0r 0r e r e

20 he prmeers s well pres re se yellow shded res. he mor prmeers re deg Allowg hoe of he degree of he dsou fuo polyoml. I s o megful o hoose muh lrger vlue for he smll bod uverse he exmple. resr 0 mes o resros o dsou fuo mes dsou fuo s fored o oe for 0 whh mes oeffe o s se o oe. Ay oher vlue s ssumed o be he shor-erm re (ul effeve yeld.e. o-ouous yeld). s luled wh he formul gve before ( o s fored o oe). frequey o se umber of oupo pymes per yer. for d omg seod d hgher order erms of r l... r r ( r )

21 ..4 Exeded Nelso & Segel Model Nelso d Segel (987) proposed prsmoous model of yeld urves h re ouous d smooh. Ulke he sple models lke JPM Nelso d Segel (N&S) model he forwrd or spo eres re drely whou modelg he dsou fuo frs. he model preseed here s sruurlly smlr o he JPM model us show h reles o he sme prg fuo d lso uses he sme smll d smple of New Zeld goverme bods. I does however o employ s ow esmo proedure usg Solver Exel s bul- muldmesol opmzo ool sed. he model s sored fle NelsoSegelYeldCurveModel.xls he followg frs expls he yeld urve fg ordg o he exeded N&S model s desrbed Blss (997) d he lks bou prulr mplemeo ssues. Uder he exeded N&S model he spo res r s fuo of me m re pproxmed by hs expoel fuo r ( m Θ) wh ε Θ ( 0 σ ) ( τ τ ) 0 m e 0 m τ ~ N τ m e m τ τ e m τ ε here re fve prmeers prmeer veor Θ whh be erpreed s follows. 0 represes he log-ru level of eres res s m d for very shor mes r wll overge o 0. ould be erpreed s he medum erm ompoe s wll ed o zero boh he shor d log ed of he me sle. Flly he wo dey prmeers τ d τ deerme how qukly he effe of he shor-erm respevely he medum-erm ompoe wll - -

22 ed o zero. τ d τ should boh be posve o esure overgee. I he bs verso of N&S (987) hey re se o equl vlues. Fgure 4 vsulzes he effe of hese hree ompoes. Fgure 4 Nelso & Segel (987) Spo Re Compoes 0.0% 8.0% 6.0% ol Comp 4.0%.0% Comp 3 0.0% Comp Corbuos of he hree N&S equo erms o he ol spo re r. Exmple prmeer vlues 0 5% % 8% τ τ. he prmeer veor Θ mus ow be hose o mmze he sum of squred pre errors N ˆ.e. ( ) P P m w ε where w D N D d ε P ˆ P Noe h hs se he squred errors re ully weghed wh he verse of he bod s Muly duro D. hs mes he pres of shor-erm bods re fed muh gher o ou for greer vrbly of shor-erm bod yelds. he esmo proedure eeds o be osred o esure re r rem posve d he mpled dsou fuo o-resg (o-egve forwrd res) r( ) d 0 r( m ) 0 m m where mm s smll vlue us slghly hgher h zero. (Noe h he N&S fuo s o defed for 0.) exp ( r( mk ) mk ) exp( r( mk ) mk ) mk < mmx - -

23 Flly smlrly o he JPM model s ofe megful o presrbe o us posve sor erm eres re.e. overgh ledg re o fx he urve he shor ed. hs s mou o presrbg osr for he sum of 0 d 0 ( ) r. m m Fgure 5 Sreesho of Nelso & Segel (987) Bod Relve Vlue Model Fg Exeded Nelso & Segel Spo Re wh Solver progrmmed by Kur Hess My 004 kurhess@wko..z me o mury m Log-ru levels of eres res 0 8.3% Shor-ru ompoe -3.8% Medum-erm ompoe 6.4% deermes mgude d he dreo of he hump Dey prmeer τ deermes dey of shor-erm ompoe mus be > 0 Dey prmeer τ deermes dey of medum-erm ompoe mus be > 0 Spo re me r 6.633% Obeve Fuos see formuls No-weghed obeve fuo x Iverse duro weghed fuo x Il Guess Vlues Deful Vlues Se Rdom Vlues Bod D Shor-erm re 4.50% Seleme de 4-Feb-99 Sep hrough opmzo Mmze Mmze Issuer Coupo Mury Bd Ask Md Cle Md Dry Before usg he mmzo mros you mus esblsh referee o he Solver dd-. Wh Vsul Bs module ve lk Referees o he ools meu d he sele he Solver.xl hek box uder Avlble Referees. If Solver.xl does' pper uder Avlble Referees lk Browse d ope Solver.xl he \Offe\Lbrry subfolder. 0.0% Model Pre Duro Weghs (w ) (hep) / rh NZ Goverme 6.50% 5-Feb % 03.80% 03.5% % NZ Goverme 8.00% 5-Feb % 06.05% 06.7% (0.07%) NZ Goverme 0.00% 5-Mr %.70% 3.0% (.40%) NZ Goverme 5.50% 5-Apr % 99.98% 97.84% % NZ Goverme 8.00% 5-Apr % 08.37% % (0.5%) NZ Goverme 8.00% 5-Nov % 09.90% 0.67% (0.78%) NZ Goverme 7.00% 5-Jul % 03.96% 03.83% % NZ Goverme 6.00% 5-Nov % 95.09% 95.37% (0.3%) 8.0% 6.0% 4.0%.0% 6.63% N&S Zero Re 0.0% Fgure 5 provdes sreesho of he N&S model mplemeo. here re some prerequses for he Exel seup order o use he Solver sofwre h mmzes he obeve fuo. Oe should o be ofused by he mrkedly dffere shpe of he zero yeld urve ompred o he urve foud hrough JPM for he sme smple uverse of New Zeld goverme bods. Fg model wh fve prmeers o purely llusrve bske of oly egh bods s boud o led o over fg problems. Aordgly he buy/sell sgls geered wll be very dffere

24 3 A Heurs Fg Mehod for he erm Sruure of Cred Spred Wh hs seo we kle he seod mor op hs pper. here s lso he eed for suble bod relve vluo whe red rsk he seod mor bod prg for beomes mpor deerm of mrke pre. hs seo wll frs expd o he dsusso he roduo o he ure d dyms of red spreds elborg frs o spes h ffe red spreds beyod fors geerlly ssumed sdrd dem models. hs s followed by revew of he shpe of he erm sruure of red spreds boh preded d observed he mrke. Purpose of hs dsusso s o provde he role d movo for heurs erm sruure of fg mehod of deeg mspred whh s preseed he fl subseo. 3. he ure of red spreds he red spred s he mos populr yrdsk for proers o ssess bods sube o deful rsk. Mesured bss pos s yplly us derved from redempo yeld dfferels o referee behmrk e.g. goverme bod urve. Oe hey hve doe so hey mus mke udgme wheher hs yeld premum ompeses hem dequely for he rsk hey ssume. hs ur s muh hrder d hey do o ge muh help from dem reserh where here s helhy debe o how o expl red spreds observed he mrke he frs ple. We meoed rles of Fos (994) d Elo e l. (00) he roduo who poed ou h pure deful rsk o possbly ou for bsolue yeld spreds loe. Whever he ompoes of he red spred proers re more oered bou he relve prg of he bods ompred o srumes of equvle red quly. Que urlly hey ur o red urves for he sme red rg egory bu hs s by o mes he oly - 4 -

25 deso rer. Reserh of Coll-Dufrese e l. (00) drew eo o he f h oly bou qurer of spred hges ully be rbued o fors oe would heorelly expe o fluee hem. Flg he defy he rue drver of spred hges hey o he expresso lol supply/demd shok h re depede of boh hges red rsk d ypl mesures of lqudy 4. Oe esly geerlze hese fdgs of Coll-Dufrese d se h o us he hges bu lso he bsolue level of red spreds re deermed by oher fors beyod he rsk s ssessed by offl red rgs. Adem reserh hs foused o xo d lqudy spes hs respe probbly beuse hese do led hemselves esly o sdrd emprl lyss 5. Whle would be beyod he sope of hs pper o kle hs ssue here s he professol work experee of hs uhor h red spreds prulr se re dfful o udersd my ses eve lk mmede rol explo. Here re some exmples. Household mes So-lled household mes rde o muh rrower spreds h ded by her red rgs. A exreme exmple ws Porshe s ured 0-yer bod ssued Aprl 997 whh 4 Whle Coll-Dufrese e l. (00) ofrmed h fors mpor for exmple Blk-Sholes (973) d Mero (974) oge lms frmework suh s frm s leverge equy reurs d volly deed hd sgf orrelo o spred hges o-frm spef rbues lke he reur of he whole shre mrke were muh sroger drvg fore. Overll her prpl ompoe lyss revels h here s lrge sysem ompoe h les ousde he model frmework. 5 e.g. V Ldshoo (003) for he Euro orpore bod mrke - 5 -

26 hs ofe rded below he Germ goverme urve. A reserh hypohess ould pos h suh omles wll our more frequely mrkes wh srog rel demd. Some spred levels my lso be rooed formol effees. hese re hghlghed Shulz (00 p. 678) for he US orpore bod mrke where he poel buyer o observe ll quoes erl loo. Smlrly f bod of he sme frm rdes dffere mrkes oe ofe observes dfferees whh o possbly be expled by urrey xo or oher fors. Spreds re ffeed by ew ssue supply A led bk my be pressured o move rso off her books hus ffeg seodry mrke spreds. Coversely srog demd for ew ssue wll ghe spreds of exsg dels. Rg ssessme of mrke dverges from offl gey rgs. Offl rgs re ofe o eped by he gross of mrke prps who olloqully spekg osder gees s behd he urve. Suh ses hve for se bee observed durg he 997 As rss bu gees lso ed o be slow o reogze mprovemes red quly. Due o he forml erl proesses volved gees frequely fd hrd o re promply. I hs lso bee oed h ofls of eres d he qus-moopoly of some gees my ree rg dsoros 6. All ll offl red rg osues us mosly qulve ssessme of well formed bu o 6 Suh oers re for se refleed SEC oep relese SEC (003) for he oversgh of red rg gees

27 fllble pry. All hese prg spes hghlgh h he prulr deso of ssessg relve vlue of bod volves qulve respevely mrke svvy ompoe o esly pured by y of he sdrd dem models. 3. he shpe of he erm sruure of red spreds We hve meoed he prevous seo h proers wll ur o red urves of omprble red quly s srg po for her relve vlue vluo. I hs seo we revew wh s geerlly preded d observed regrdg he shpe of hs erm sruure of red spreds. Fgure 6 erm Sruure of Ieres Rsky Dsou Bods he Logsff & Shwrz (995) Model erm Sruure of Ieres Rsky Dsou Bods 4% % 0% 8% 6% 4% % 0% me o mury Rsky Dsou Bod Modere Rsk Dsou Bod r 0 Yeld rskfree bod - 7 -

28 Fgure 6 llusres geer wys he erm sruure of spo res of rsky respevely less rsky dsou bod s preded by my of he msrem red rsk models. I hs se ws geered by he Logsff & Shwrz (995) model (L&S 95) whh for llusro s expled more del Appedx. L&S 95 s from he fmly of Mero s (974) oge lms models bu oe would fd smlr hump-shped dowwrd-slopg red yeld urves for exmple Jrrow Ldo & urbull s (997) redued form pproh. hs heorel predo ppers o be bked by emprl sudes suh s Fos (994 p. 30) who esmes ross-seol regresso of spreds o mury d fds sgfly egve oeffes for sgle B bods. hs resul s lso suppored by Moody s deful d where mrgl deful res of speulve grde bods (B rg) exhb delg red wh loger me horzos. he uo behd hs resul s h speulve frms beg very rsky ssue hve room o mprove.e. he loger he me o mury he more lkely he vlue of he frm wll rse subslly. Aoher erpreo would be h speulve bods ob he hrer of equy srume d re hus rded o brek-up vlue sed of yeld bss. Coversely hgh grde red oly beome worse hrough me d hus show upwrd slopg erm sruure of red spred. hese resuls re somewh gs he uo of proers who observe h where frm ssues wo sepre me rhes wll be sked offer hgher yeld premum for he loger mury rhe of he rso d hs pples eqully o weker d sroger reds. Helwege & urer (999) deed provde some emprl evdee for hs observo. hey rgue h dowwrd slopg red urves re resul of sfer speulve grde frms ssug loger-ded bods whh ur leds o smple seleo bs

29 Wh we olude s h here re wo shools of hough for modelg erm sruure of red spreds. A useful bod relve vlue ool for orpore bods mus hus be pble of ommodg boh of hem. 3.3 Heurs fg mehod for erm sruure of red spreds hs seo preses pproh for work ool so mrke proers frsly ssess he relve pre of bod vew of less gble prg fors (s show seo 3.) d seodly o presrbe erm sruure of red spreds hey feel s ppropre for he prulr srume d mrke odo (s expled seo 3.). he mplemeo of hs model s show fle Chep Rh Ls (Feb04).xls. he fle uses smple of Swss orpore dusrl bods o geere ls of hep respevely rh bods. I employs smple redempo yeld bsed referee urve of he ype h ws derved seo.. s bss o lule red spreds. hs yeld o mury bsed pproh ould esly be dped o spo re frmework usg urve derved seo.. or..3. Ye gve he heurs ure of he model hs s lkely o dd oly lmed vlue. he followg desrbes he model wh smplfed umerl exmple lso doumeed he Exel work book Defg he erm sruure of red spreds he model he model les he user hoose he desred erm sruure of red spreds for eh rg egory by mes of shpe prmeers. hs s llusred hrough umerl exmple Fgure 7 below. he erm sruure s bslly broke dow o wo sub-perods. A shor- o medum erm perod o (_def) whh s followed by he log-erm hrerss of he red spred

30 Fgure 7 Explo Shpe Prmeers erm Sruure of Cred Spreds bps/yr (Slope 0) erm Sruure for 4 3 erm Sruure for 4-3 Cred Spred S_def5 _def3 - bps/yr (Slope _def) Spred Lower Lm (70.0% of 5bps) 50 0 Shor-erm hrerss of red spreds Log-erm hrerss Yers o Mury 6.5 As o he frs perod ffh order polyoml s fed bewee zero d. 3 4 () Spred I mus mee he followg hree boudry odos mus be zero me 0 whh mes 0 0 mus reh S (S_def) me. slope po S / mus equl he slope of he log-erm urve beyod. (more dels o hs slope re below) he user ffes he shpe of he polyoml wh wo prmeers. he l slope ( bps per yer) me zero my be spefed he shpe of he hump s lso ffeed by prmeer 4 whh orrespods o he ffh order oeffe of he polyoml. Fgure 3 shows he erm sruure for 4 3 respevely mus

31 I he log-erm horzo he user spefy slope for he furher developme of red spreds. I mos ses wll be se o or slghly bove zero o ob os or moderely resg red spreds beyod me. here s lso he opo o spefy lower lm below whh he red spred my ever dele. hs prmeer s se s perege of S. If hs lower lm s spefed s umber greer h oe ully beomes upper lm spefo. Wh bove prmeers wde rge of shpe spefos beomes possble. Fgure 4 lss rge of poel shpes ludg some ommes s he rumses hese ould be ppropre. We g refer o seo 3. for dsusso of reserh o hs sube. Fgure 8 Seleo of erm Sruure Shpes Chr Smple os spred Chr No hump suble for hgh quly bods Cred Spred Slope Slope Lower S_def _def (0) ( ) 4 lm Yers o Mury Cred Spred Slope Slope Lower S_def _def (0) ( ) 4 lm _def Yers o Mury Chr 3 Smll hump suble for medum quly bods Chr 4 Prooued hump suble for o-vesme Cred Spred Slope Slope Lower S_def _def (0) ( ) 4 lm _def Yers o Mury Cred Spred _def Slope Slope Lower S_def _def (0) ( ) 4 lm Yers o Mury - 3 -

32 3.3. Deeg hep/rh bod wh heerogeeous red quly smplfed umerl exmple For llusro he followg bles (ble d 3) d fgures (Fgure 9) prese smplfed umerl exmple of hep/rh lyss. here re hree bods lyzed. Bod I hs AA rg whle bods II d IIb re red BBB. A buy [sell] sgl s geered f he model pre bsed o he rge spred s bove [below] he mrke pre. ble D Exmple Bods Bod I II IIb Rg AA BBB BBB Pre ($ per $fe vlue)* Coupo (pd oe ully) 4.00% 4.00% 4.00% me o Mury.79 yrs 5.0 yrs 7.79 yrs Yeld o Mury 3.4% 4.% 3.% Spred o Behmrk 36 bps 08 bps 58 bps rge Spred (derved from prmeers ble 3) 50 bps bps 0 bps Model Yeld.37% 3.36% 3.73% Model Pre (MP) * Reommedo MP > Pre MP > Pre MP < Pre Chep Bod Chep Bod Rh Bod Buy Buy Sell * Pre exludg rued eres - 3 -

33 Fgure 9 Geer Yeld o Mury Bsed Chep Rh Alyss Yeld 5.0% 4.0% 3.0%.0%.0% 3.4%.37% 4.% 3.36% 3.73% 3.% Yers o Mury Yeld exmple bods Model yeld ordg o rge spred Behmrk urve (polyoml degree 3) Bod Pre Buy Buy Sell Yers o Mury Pres exmple bods Model pres ordg o rge spred rge Spred ( bps) Yers o Mury Bod II&b Rg BBB Bod I Rg AA ble 3 erm Sruure of Cred Spred Shpe Prmeers Rg AA BBB S ( bps) Slope (0) Slope ( ) 0 - Hump prmeer 4-6 Lower lm (s % of S )

34 Compred o he m model he verso bove s smplfed hese spes. he llusrve exmple does o ke bd/sk spreds o osdero whe geerg buy/sell sgls. o lm he umber of reommedos he user my spefy sesvy prmeer o suppress reommedos where he pre s very lose o he model pre. he sesvy hose ould for se ke rso hrges o ou. hs he sme fler desrbed Fgure for he model seo... For shorer bods ypl pre hges less lqud mrkes my led o very err yeld moves h do rsle o megful buy/sell sgls. he user my hus exlude he lyss for very shor-erm bods. Flly he m model provdes sss o he red spreds observed he referee bskes. A exmple s show Fgure 0 below. Fgure 0 Geer Yeld o Mury Bsed Chep Rh Alyss Alyss red spreds Swss dusrl bods 6/0/ Ierl Rg AAA AA A BBB BB B Oher Grd ol Number of bods Averge spred o Swss Gov (bps) M of spred (bps) Mx of spred (bps) Yers o mury/ ex ll (verge) AAA AA A BBB BB B Oher Grd ol Averge Averge Cred Spred by Ierl Rg

35 3.3.3 Some remrks o how o pply he model he model desrpo hs o ddressed he ssue of lbro ye. I oher words how should oe deerme he shpe prmeers for prulr srume respevely group of srumes? Gve h hs s o fg mehod for emprl reserh purposes he followg prgm mehod s reommeded. I frs se oe would sele vlue for S. If we leve ll oher prmeers zero hs s ohg else h fl red spred we pply o referee urve.e. prllel upwrd shf of he referee urve. For my users hs s he d-ho mehod hey wll be used o d refleg o he emprl resuls of Fos (994) dsussed erler 7 hs s o uresoble pproh ll. o uome he seleo of S oe ould derve from me vlue for rg egory s show he lyss Fgure 0. Noe h he rg egory of prulr bod my be se by he user d would o eessrly ode wh offl gey rg. Wh regrd o he remg shpe prmeers s o megful o se hem for us oe rg lss. Oe would sele for exmple ommo vlues for ll BBB- o BBB red bods. As per seo 3. here wll be o hump for mos rg egores d we would smply se vlue for he rge of - yers some geer slope me 0 ( bps per yer) d 4 o zero. he prmeer of oer would be he slope me. hs ould be deermed by Fos (994 p. 30) ype regresso lyss lhough does mos probbly o mke sese o upde s regulrly s he m spred prmeer S. 7 Fos (994) fds rher fl slopes for mos rg egores d lhough -sss des vlues sgfly dffere from zero R vlues re que low

36 Flly respe o speulve grde rg egores here s obvously he opo o se presrbe hump-shped sruure uless he user prefers resg erm sruure le wh Helwege & urer (999). I mos bod mrke supply of hese segme s rher sprse d bllprk presrpo of prmeers whou ul lbro of vlues my well be ppropre

37 4 Coluso hs pper llusred he mplemeo of some models for yeld urve fg used rdg pplos somehg o ye formlly doumeed he dem lerure. For bods sube o red rsk preseed heurs model o ob formo o over- respevely uderpred bods. Boh ypes of model mplemeos doume he prgm ure of suh soluos. I he bsee of olusve resuls by dem reserhers hese models geere buy d sell sgls s resul of he m prg prmeer for fxed ome srumes whh re eres res respevely red spreds. he user he evlue hem vew of hs/her kowledge of lol supply d demd spes d oher qulve fors suh s he oes lsed seo 3.. here s oher more geerl lesso oe ould ler from hese models whh erms of omplexy re muh smpler h mos of he pprohes urrely dvoed quve lerure. Eve hough her pproh s s whou he mbo of usfyg hem dym or o-rbrge osse heorel frmework hey be hdled d udersood by he proers. More dved pprohes usully sr from dels premse bou how he world should look lke respevely how rol vesors should. Reserhers he fd h rely s dffere d emp o sve he pproh wh progressvely more sophsed medmes d exesos. hs ould lmos be ompred o mheml rms re where resgly omplex quve heores re ppled whou mrkedly mprovg he predve power of he models. Ieres re modelg s good exmple where oly models lbred ouously o urre mrke res do hve y megful pplos he mrke. I would perhps be me h dem reserh fe mde he requremes of mrke proers

38 srg po for mproved models d o fule hse for he ulme rue model h does o exs

39 5 Referees Aderso N. Breedo F. Deo M. Derry A. & Murphy G. (996). Esmg d erpreg he yeld urve. Chheser Joh Wley Seres Fl Eooms d Quve Alyss. Blk F. Derm E. & oy W. (990). A Oe-For Model of Ieres Res d Is Applo o resury Bod Opos. Fl Alyss Jourl 46() Blk F. & Sholes M. (973). he prg of opos d orpore lbles Jourl of Poll Eoomy (Vol. My - Jue 973 pp ). Blss R. R. (997). esg erm Sruure Esmo Mehods. Adves Fuures d Opos Reserh(9) Chrse D. (003). Arued Ieres & Yeld Clulos d Deermo of Holdy Cledrs Verso.. Swss Exhge. Rereved from he World Wde Web hp// Coll-Dufrese P. Goldse R. & Mr J. S. (00). he Deerms of Cred Spreds Chges. Jourl of Fe 56(6) Cox J. C. Igersoll J. E. Jr. & Ross S. A. (985). A Ieremporl Geerl Equlbrum Model of Asse Pres. Eoomer 53() p Duffe D. & Ldo D. (00). erm sruures of red spreds wh omplee oug formo. Eoomer 69(3) Elo E. J. Gruber M. J. Agrwl D. & M C. (00). Explg he Re Spred o Corpore Bods. Jourl of Fe 56() Fbozz F. J. (999). Bod Mrkes Alyss d Sreges (4h ed.). Eglewood Clffs N.J. USA Pree Hll. Fsher M. Nyhk D. & Zervos D. (995). Fg he erm sruure of eres res wh smoohg sples Fe d Eooms Dsusso Seres Federl Reserve Bord. Fos J. S. (994). Usg deful res o model he erm sruure of red rsk Fl Alyss Jourl (Vol. 50 pp. 5-3). Chrloesvlle. Heh D. Jrrow R. & Moro A. (99). Bod Prg d he erm Sruure of Ieres Res; A New Mehodology. Eoomer 60() Helwege J. & urer C. M. (999). he slope of he red yeld urve for speulve-grde ssuers. Jourl of Fe 54(5) Ho. S. & Lee S. B. (986). erm Sruure Movemes d Prg Ieres Re Coge Clms. Jourl of Fe

40 Houwelg P. Hoek J. & Kleberge F. (00). he o esmo of erm sruures d red spreds. Jourl of Emprl Fe Iodes M. (003). A omprso of yeld urve esmo ehques usg UK d. Jourl of Bkg & Fe 7() -6. Jrrow R. A. (997). A Mrkov Model for he erm Sruure of Cred Rsk Spreds. he Revew of Fl Sudes 0() Krpper L. (003). Opmsg fxed eres porfolos wh orhoormlsed Lguerre polyoml model of he yeld urve.upublshed musrp Leo.Krpper@mppl.om. Logsff F. A. & Shwrz E. F. (995). A Smple Approh o Vlug Rsky Fxed d Flog Re Deb. Jourl of Fe MCulloh J. H. (97). Mesurg he erm Sruure of Ieres Res. Jourl of Busess MCulloh J. H. (975). he x-adused Yeld Curve. Jourl of Fe Merrll Lyh. (004). Rh / Chep Repor Bod Ides Group Merrll Lyh Bod Ides & Alys. Mero R. C. (974). O he prg of orpore deb he rsk sruure of eres res Jourl of Fe (Vol. 9 pp ). Nelso C. R. & Segel A. F. (987). Prsmoous modelg of yeld urves Jourl of Busess (Vol. 60(4) pp ). Press W. H. Flery B. P. eukolsky S. A. & Veerlg W.. (99). Numerl repes C he r of sef ompug Cmbrdge Uversy Press. RBNZ. (997). Iformo Memordum New Zeld Goverme Bods. Reserve Bk of New Zeld / he resury New Zeld Deb Mgeme Offe. Rereved from he World Wde Web hp// Reboo R. (998). Ieres-re opo models udersdg lysg d usg models for exo eresre opos Wley Chheser ; New York. Relly F. K. & Brow K. C. (997). Ivesme Alyss d Porfolo Mgeme (5h ed.) Dryde Press. Shefer S. M. (977). he Problem wh Redempo Yelds. Fl Alyss Jourl 33(4) Shulz P. (00). Corpore Bod rdg Coss A Peek Behd he Cur. Jourl of Fe 56()

41 SEC. (003). Coep Relese Rg Agees d he Use of Cred Rgs uder he Federl Seures Lws Relese Nos U.S. Seures d Exhge Commsso. Rereved My from he World Wde Web hp// Seru P. & Wu X. (997). he formo oe bod model resduls A emprl sudy o he Belg bod mrke. Jourl of Bkg & Fe (5) Svesso L. (994). Esmg d erpreg forwrd eres res Swede Dsusso pper Cere for Eoom Poly Reserh(05). V Ldshoo A. (003). he erm Sruure of Cred Spreds o Euro Corpore Bods. Workg Pper Ceer for Eoom Reserh lburg Uversy. Rereved Februry 004 from he World Wde Web hp//greywww.kub.l080/greyfles/eer/003/do/46.pdf Vsek O. A. (977). A Equlbrum Chrerzo of he erm sruure Jourl of Fl Eooms (Vol. 5 pp ). Wds. (993). A Iroduo o Opo-Adused Spred Alyss. New York Bloomberg Mgze Publos

42 Appedx Seleme Pre Clulo New Zeld Domes Bod Mrke Aordg o he Reserve Bk of New Zeld Formul (RBNZ 997) C P b ( ) k 0 ( ) where k FV ( ) P Mrke Vlue of Bod FV Noml or fe vlue of bod Aul mrke yeld / ( %) Aul oupo re % C Coupo Pyme ( / * FV) sem-ul oupo Number of full oupo perods remg ul mury Number of dys from seleme o ex oupo de b Number of dys from ls o ex oupo de - 4 -

43 Appedx Logsff & Shwrz 995 Model I her 995 Jourl of Fe pper Logsff d Shwrz (L&S 95) show smple pproh o vlue rsky deb sube o boh deful d eres re rsk. I le wh he rdol Blk-Sholes (973) d Mero (974) oge lms-bsed frmework deful rsk s modeled usg opo prg heory. hs mes deful ours f he level of sse of frm (V) flls below bkrupy hreshold (K). V s ssumed he follow he followg sohs proess dv µ Vd σvdz where σ s he seous sdrd devo of he sse proess (os) d dzs sdrd Weer proess. follows dr where Smlrly eres res re ssumed o follow sdrd Vsek proess (Vsek 997) s ( ζ r) d ηdz ζ s he log-erm equlbrum of me reverg proess (os) s he "pull-bk" for - speed of dusme (os) η s he spo re volly (os) dz s sdrd Weer proess. he seous orrelo bewee dzd dz s ρ d L&S 95 he derve he vlue of rsky dsou bod s

44 ( ) ( ) r X Q wd r r D r X P ) ( ) ( hus he pre of hs bod s fuo of X whh orrespods o he ro of V/K he eres re r d he me o mury. he erms o be luled re expled below. ( ) ) ) ( ) ( r B A e r D s he vlue of rskfree (o red rsk) dsou bod ordg o Vsek (977) wh ( ) ( ) 4 ) ( 3 3 e e A η α η α η α e B ) ( Here α represes he sum of he prmeer ζ plus os represeg he mrke pre of rsk s defed bove he Q(Xr) erm be erpreed s probbly - uder rsk eurl mesure - h deful ours. I s he lm of ) ( r X Q s. ) ( r X Q s luled s follows q r X Q ) ( wh ( ) ( ) ( ) b N q N q N q... 3 where N(.) deoes he umulve sdrd orml dsrbuo d ( ) ( ) S M X l ( ) ( ) ( ) ( ) S S M M b

45 Expressos ) ( M d ) ( S re defed s follows ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) r M η η α η ρση σ η ρση α exp exp exp exp exp ) ( d ( ) ( ) ( ) ( ) S η η ρση σ η ρση exp exp ) ( 3 3 As remder ρ s he seous orrelo bewee he sse d eres re proesses. Flly he os prmeer w s he wre-dow se of deful pere of he fe vlue. I oher words s oe mus he reovery re se of deful. Oe he vlue of pure dsou bod s foud he vlue of oupo bod s smply vlued s seres of dsou bods ossg of oupos d prpl repyme. Noe h L&S 95 lso derve losed form soluo for perpeul flog re deb smlr fsho. he uhors olude her work wh emprl model vldo. hey odu regresso lyss of how hsorlly observed spreds (soured from Moody s) hve orreled wh he reur of shre des s proxy for he sse proess respevely hge eres res. hey deed fd sgf egve orrelos mos ses wh boh eres re hges d developme of sse pres. Jus for hgh grde bods (AAA d AA bod) hey deermed less sgfe for he sse orrelo oeffe. hs my be expeed hough beuse he well ushoed hgh grde reds wll be less ffeed by dowurs he shre mrke.

46 As llusro of he L&S 95 model oupus he hrs belwo show he vlue d yeld of dsou bod s fuo of me o mury for he prmeers ble A.. Fgure A. Rsky dsou bod pres s fuo of bod eor (me o mury) 00% 90% 80% 70% 60% 50% 40% 30% 0% 0% 0% Vlue rsky dsou bod (L&S 95) Vlue rsk free dsou bod (Vsek) me o mury Fgure A. erm Sruure of Ieres Rsky Dsou Bods erm Sruure of Ieres Rsky Dsou Bods 6% 4% % Rsky Dsou Bod 0% 8% 6% 4% % 0% me o mury r 0 Yeld rskfree bod

47 ble A. Prmeers L&S 95 Model Exmple Prmeer desrpo Symbol Vlue Re r 0 0 r 7.0% "Pullbk" for eres re.00 Iseous sdrd devo of shor re η 3.6% η α L&S ζ os α V 0 /K X (mesure of l red quly) X.30 Wredow - Reovery Re w 0.50 Volly of sse vlue proess σ 0.00% σ Iseous orrelo sse/eres re ρ Ieros for Q