Fixed in Come

8/12/2019 Fixed in Come

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CS FIRSTBOSTON April 26, 1995

FIXED INCOME RESEARCH BULLETIN

Implied Default Pro a ilitie!" A Ne# Approa$% to Ri!&' De t

Clayton Perry 212/909-2682

Summary Implied default probabilities are a useful mechanism for e aluatin! ris"ydebt# $ather than discountin! promised cashflo%s at a hi!h rate %hich reflectsthe dan!er that they %ill not be recei ed& %e ha e more directly identified theimplied default probability in a set of 'r!entinian eurobonds by discountin!expected cashfio%s at the ris"-free rate# (urthermore& %e ha e remo ed someof the apparent inconsistencies inherent in assumin! different probabilities

for similar bonds in the same time period# )hus %e& ha e !enerated a set of probabilities a!ainst %hich in estors can measure their o%n opinions aboutthe ris"iness of the securities to decide upon relati e alue #

T%i! 'ear %a! #it(e!!ed turmoil i( emer)i() de t mar&et! o( a !$ale u(!ee( !i($et%e Brad' re!tru$turi()* Follo#i() t%e Me+i$a( de,aluatio( i( De$em er- !%o$& #a,e! !pread a! far afield a! Pola(d a(d T%aila(d a! i(,e!tor! fled for t%e !afet' of S#it.erla(d a(d /erma('* I( a( e(,iro(me(t $%ara$teri.ed ' t%i! !ort of 0ri!& redu$tio( at a(' pri$e0 it i! plau!i le t%at differe(t ri!&! ma' e pri$edi($o(!i!te(tl'- or i(deed t%at e1ui,ale(t ri!&! offer #idel' differi() retur(!*

Our approa$% to ide(tif'i() !u$% po!!i ilitie! i! to a(al'.e t%e pro a ilit' of

default #%i$% i(,e!tor! %a,e pri$ed i(to ea$% of a !et of !imilarl' ri!&' o(d!* T%ere!ult i! merel' a differe(t #a' of !tati() relati,e ,alue- ut %a! t%e ad,a(ta)e of e+pre!!io( i( term! of t%e !pe$ifi$ ri!& t%at i(,e!tor! are $o($er(ed #it% 2 t%e$%a($e! t%at t%e' #ill )et t%eir mo(e' a$&* T%e !tati!ti$ #e $al$ulate i! t%e rate of default o( t%e o(d #%i$% #ould )i,e a( e+pe$ted retur( 3(et of pro a ilit'#ei)%ted $redit lo!!e!4 e1ual to t%e ri!& free rate of retur(* 5e %a,e ta&e( a! our !ample t%e (o(2Brad' dollar o(d! i!!ued ' Ar)e(ti(a* T%e!e %a,e ee( fa,oredfor t%i! e+po!itio( pri($ipall' e$au!e t%ere are fi,e o(d! of t%e !ame $la!! #it% area!o(a l' re)ular maturit' !$%edule* Later ulleti(! #ill deal #it% Brad' o(d!a(d ot%er $la!!e! of ri!&' de t*

This memorandum is for informative purposes only. Under no circumstances is it to be used or considered as an offer to sell, or a solicitation of an offer to buyany security. Additional information is available upon request. esults based on hypothetical pro!ections or past performance have certain inherent limitations.

There is no certainty that the parameters and assumptions used can be duplicated "ith actual trades. Any results sho"n do not reflect the impact of commissionsand#or fees, unless stated. $inancial futures and options are not appropriate for all investors. Their relative merits should be carefully "ei%hed. &here informationhas been obtained from outside sources. 't is believed to be reliable but is not represented to be accurate or complete. This document may not be reproduced in"hole or in part or other"ise made available "ithout our "ritten consent This firm may from time to time perform investment ban(in% for, or solicit investmentban(in% or other business from any company mentioned. &e or our employees may from time to time have a lon% or short position in any contract or securitydiscussed.


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$i)ed 'ncome esearch

*ethodolo!y Suppo!e 'ou are due to re$ei,e a pa'me(t i( o(e 'ear6! time* If 'ou are 7889 !ure'ou #ill $olle$t- it i! !e(!i le to di!$ou(t t%at pa'me(t at t%e rele,a(t ri!& free rate-t%e 7 2'ear U*S* Trea!ur' ill 'ield* T%e le!! !ure 'ou are of re$ei,i() t%e pa'me(tt%e %i)%er #ill e t%e di!$ou(t rate applied a(d t%e le!! #ill e t%e pre!e(t ,alue of t%e pa'me(t* No# !uppo!e 'ou are 7889 $ertai( of )etti() either pa'me(t + or

pa'me(t '- #%ere ' i! a !maller amou(t* If 'ou di!$ou(t ot% at t%e ri!& free rate'ou )et t#o (et pre!e(t ,alue!- o(e of #%i$% #ill e $orre$t* To ,alue t%i! 0eit%er or0 pa'me(t t%e mo!t lo)i$al t%i() to do i! to a!!i)( pro a ilitie! to ea$% of

pa'me(t + a(d pa'me(t ' a(d ta&e a( a,era)e of t%e t#o NP:! #ei)%ted ' t%ea!!i)(ed pro a ilitie!*

I( our e+ample- pa'me(t + repre!e(t! t%e $a!%flo# !$%eduled o( a o(d- #%ile pa'me(t ' repre!e(t! t%e re$o,er' ,alue 2 t%e amou(t t%e o(d i! #ort% i( t%ee,e(t of a default* T%e mar&et pri$e of t%e o(d i( it! fi(al $oupo( period #illrepre!e(t t%e pro a ilit' #ei)%ted a,era)e 2 or e+pe$ted ,alue 2 of t%e!e t#oamou(t! di!$ou(ted at t%e ri!&2free rate* T%i! i! repre!e(ted al)e rai$all' a!follo#!"

c (1 - d) + Rd P= _____________

1 + r

#%ere P ; dirt' pri$e of o(d$ ; $a!%flo#d = the pro a ilit' of default i( t%i! period

R = recovery pro$eed! i( e,e(t of defaultr = risk free di!$ou(t rate

For e+ample- !uppo!e a ri!&' 72'ear .ero $oupo( o(d i! tradi() at a pri$e of


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default i( t%e fir!t period (1-d). 5e $a( &eep e+te(di() t%i! for ( !$%eduled $a!%flo#! i( t%e !amema((er* Note t%at ea$% period6! pro a ilitie! mu!t e multiplied ' t%e pro a ilit' of (o default i(all t%e pre,iou! period!*

If #e a!!ume t%at t%e pro a ilit' of default i( ea$% of t%e i(di,idual $oupo( period! i! t%e!ame #e $a( $ollap!e t%i! e1 itio( to t%e follo#i()@

T%i! i! re$o)(i.a le a! a( e+te(!io( of t%e !imple 'ield to maturit' $al$ulatio( #%i$% o(e#ould (ormall' !ol,e for t%e di!$ou(t rate- r- from t%e &(o#( pri$e a(d $a!%flo# !$%edule*T%i! di!$ou(t rate or 'ield #ould u!uall' refle$t opi(io(! of t%e ri!&i(e!! of t%e $a!%flo#!$%edule*+ur approach separates this discount factor from the ris" %ei!htin! by usin!the ris" free discount rate from the ,#S# )reasury spot cur e and confinin! e pectationsto the top line of the e.uation# If #e a!!ume a re$o,er' ,alue- #e $a( a$& out a ,alue for d

' iteratio( i( t%e !ame ma((er a! o(e #ould $o(,e(tio(all' $al$ulate 'ield to maturit' 'iterati() for r*

T%e met%odolo)' #e %a,e emplo'ed i! !et out dia)rammati$all' elo#* T%e pro a ilit' of )etti() to a poi(t o( t%i! i(omial tree i! t%e produ$t of t%e pro a ilitie! ide(tified o( ea$%

ra($% ta&e(*

hibit 1# Binomial Tree of Default Probabilities

Cumulati,e T%e $al$ulatio(! #e %a,e de!$ri ed a o,e $a( e !ol,ed for d to )i,e t%e defaultDefault pro a ilit' a!!umed for ea$% !i()le period* If a orro#er %a! de t #it% ma('Pro a ilit' maturitie! out!ta(di() or %a! de t #%i$% $apitali.e! i(tere!t or amorti.e!- t%e per

period default rate $a( e e!pe$iall' u!eful* For ma(' i!!uer! %o#e,er- of more=


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i(tere!t i! t%e $umulati,e default pro a ilit'- t%e $%a($e t%at a o(d #ill (ot ma&e it to a parti$ular poi(t o( our i(omial tree* For i(!ta($e- mo!t i(,e!tor! #ill e i(tere!ted i( t%e pro a ilit' t%at t%eir o(d #ill (ot default efore maturit'* T%i! i! t%e pro a ilit' t%at #e #ill)et to t%e ottom ri)%t o+ o( our i(omial tree- #%i$% i! u!t t%e produ$t of all t%e i(di,idual

per2period pro a ilitie! of !ur,i,al*

37 d 74372d >4 372d n4A)ai( if #e a!!ume $o(!ta(t per2period pro a ilitie!- t%e pro a ilit' of (o default up to a(di($ludi() period n is

372d 4 n

T%i! mea(! t%at t%e $umulati,e pro a ilit' of default #ill e

7 2 37 2d 4 (

A!!e!!i() a T%e a!!e!!me(t of t%e re$o,er' ,alue of t%e o(d i( t%e e,e(t of default i! ot%$eco ery alue diffi$ult a(d $ru$ial to t%e re!ult!* A $orporate o(d !%ould %a,e !ome ,alue i(

default from t%e $laim! t%at o(d2%older! %a,e o( t%e a!!et! of t%e $ompa('*Se$ured de t pre!e(t! fe# pro lem!- #%ile u(!e$ured de t #ill %a,e a $laim o(u(e($um ered a!!et! #%i$% #ill pre!uma l' e !old at !ome di!$ou(t to full ,aluei( t%e $a!e of li1uidatio(* A!!e!!i() t%e pri$e t%e!e a!!et! #ould fet$% i( a!ituatio( of di!tre!!- alo() #it% t%e $o!t! of li1uidati() t%e $ompa(' #ill pre!e(t!ome pro lem!- ut t%e de)ree of a$$ura$' !%ould e rea!o(a le*

So,erei)( de t- %o#e,er- pre!e(t! )reater diffi$ult'* Creditor! are (ot i( a po!itio(to $lo!e do#( t%e $ou(tr' a(d !ell off road!- rid)e! a(d dam!* O( t%e ot%er %a(d-a promi!e to pa' a future !erie! of $a!%flo#! a(d e,e( pa!t o li)atio(! i( default

pro a l' mea(! more from a !o,erei)( !tate t%a( from a $orporate e(tit'*T%erefore it i! diffi$ult to ima)i(e !u$% de t e,er tradi() at .ero* E,e( t%edefaulted paper of Li eria- o(e of t%e #orld6! poore!t $ou(trie!- trade! at arou(d>*?9 of fa$e ,alue*

T%i! i! $o(firmed ' %i!tori$al a(al'!i!* Prior to t%e mo!t re$e(t Brad're!tru$turi()- defaulted Lati( Ameri$a( de t traded at pri$e! et#ee( >


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(i ed Income $esearch

o(d- #%i$% appear! more t%eoreti$all' $o(!i!te(t t%a( a $o(!ta(t (um er* Of $our!e #e do (ot &(o# #%at di!$ou(t rate! #ill e pre,aili() at ea$% $oupo( date-a(d !i($e #e are ,alui() e,er't%i() toda' #e are i(tere!ted i( t%e di!$ou(ted,alue of ea$% of t%e R6! t

#%ere" a ; e+pe$ted per$e(ta)e pa'out i( e,e(t of default*

Our e+ample- Ar)e(ti(a- %a! a #ell2de,eloped i(fra!tru$ture a(d i! #ell e(do#ed #it% (aturalre!our$e!* Out!ta(di() de t i! o(l' =G9 of /DP* T%i! #ould lead u! to elie,e t%at a fi)ure of arou(d =8 $e(t! o( t%e dollar #ould e a rea!o(a le i(,e!tor e+pe$tatio( a! to t%e po!t2default

pa'out o( Ar)e(ti(a paper* Due to t%e ar itrar' (ature of !u$% a( e!timate a(d t%e !e(!iti,it' of our $al$ulatio(! to it- #e %a,e al!o deri,ed t%e implied pro a ilitie! a!!o$iated #it% pa'out rate! of 7?9 a(d G?9*

Some of t%e!e re!ult! de!er,e $omme(t* Implied per2period default pro a ilitie! i( )e(eral i($rea!e#it% maturit'- a fa$t #e deal #it% furt%er elo#* Note t%at t%e 7 - 7 a(d >888 o(d! all %a,e,er' !imilar rate! #it% a i) ump out to >88=* T%i! i! more appare(t from t%e plot elo# of t%e0implied default $ur,e*0


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Fixed Income Research

T%e mai( feature of t%e ta le i! t%at t%e (um er! appear fairl' !mall* Ho#e,er- #%e( #e loo& at t%ec # lative default rate implied- t%e ma)(itude ri!e! rapidl'* T%i! i! t%e pro a ilit' t%at a o(d #illdefault efore t%e )i,e( date- i( t%i! $a!e it! maturit'* T%e re!ult! are !et out elo#*

T%i! )i,e! a $learer ,ie# of #%ere t%e mar&et i! pla$i() it! et!* Naturall' t%e lo()er i! a o(d6!

maturit'- t%e )reater t%e $%a($e it #ill default efore maturit'* Note t%at e,e( i( t%e $a!e of t%e!malle!t re$o,er' ,alue i(,e!tor! elie,e t%ere i! (earl' a %alf $%a($e t%at t%e De$ >88= o(d#ill (ot pa' it! full !$%edule of $a!%flo#!*


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tendin! the T%e pri($ipal pro lem #it% t%i! approa$% i! o(e of i($o(!i!te($'* Be$au!e #e are*odel to on- a!!umi() $o(!ta(t per period default rate! t%rou)%out t%e life of a o(d- #e )et t%eConstant a(omalou! re!ult t%at t%e pro a ilit' of default ' t%e !ame i!!uer i( t%e !ame

efault $ates period #ill e differe(t #%e( mea!ured o( differe(t o(d!* Of $our!e t%i! ma' e are!ult of i($o(!i!te(t mar&et e%a,ior- ut e1uall' it ma' !%o# a maturit' effe$t*I( t%e !ame #a' a! 'ield2to2maturit' di!$ou(t! all $a!%flo#! at a u(iform rate i(!pite of a (o(2$o(!ta(t term !tru$ture- t%e a!!umptio( of $o(!ta(t per perioddefault rate! o,er time ma' e u(rea!o(a le* I(,e!tor! ma' e more $ertai( of a$ou(tr'6! e$o(omi$ pro!pe$t! i( t%e !%ort term t%a( a fe# 'ear! from (o#- or a!$%eduled ele$tio( $ould i($rea!e default pro a ilit' i( it! #a&e* A! for $orporatei!!uer!- o(e $a( e(,i!a)e a (e# produ$t lau($% or out$ome to liti)atio( #%i$% ma'dramati$all' alter t%eir $redit 1ualit'*

T%erefore #e %a,e e+te(ded t%e model to ta&e a$$ou(t of t%e po!!i ilit' of ,aria($e i( default rate! o,er time* T%e met%odolo)' i! merel' to a!!ume t%at t%e!%orte!t o(d determi(e! t%e implied default rate for it! lifetime- a(d t%at t%e$a!%flo#! o( all ot%er o(d! duri() t%at period !%ould e #ei)%ted ' t%at

pro a ilit'* T%e (e+t lo()e!t o(d t%e( determi(e! t%e pro a ilit' u(til it!maturit' a(d !o o(* T%u! if a o(d mature! after = $oupo( period! #e #ould #or& out t%e $o(!ta(t per2period default pro a ilit' for t%at !e$urit' a$$ordi() to our fir!t met%od* T%i! #ould )i,e u! d 1 % d &a(d d = for e,er' o(d o( t%e $ur,e* T%ee1uatio( for t%e (e+t o(d #ould u!e t%o!e t%ree rate! a! )i,e(- a(d #e #ould t%e(

e a le to a$& out a $o(!ta(t ,alue of d for t%e remai(i() period! i( t%at o(d*

T%i! pro$e!! $a( e repeated for e,er' o(d to )i,e t%e 0mar)i(al0 per2perioddefault rate for ea$%@ t%at i!- t%e rate ofd t%at applie! for t%e fi(al period! of t%at

o(d6! life* I( t%i! #a' #e $a( 0 oot!trap0 a $redit $ur,e i( mu$% t%e !ame #a't%at a !erie! of $oupo( o(d! $a( e u!ed to )e(erate a !pot $ur,e i( 'ield term!*

T%e re!tri$tio( i! t%at #e are o(l' deali() #it% a %a(dful of i!!ue!- a(d t%u! $a(

o(l' ,ar' t%e default pro a ilit' a fe# time!* Ne,ert%ele!! t%i! #ill )i,e area!o(a le i(di$atio( of t%e !%ape of t%e 0$redit $ur,e0 a(d ma' !%o# up ar itra)eopportu(itie!*

Mat%emati$all' t%i! i(trodu$e! (o(2$o(!ta(t default rate!* T%e e1uatio( (o# e$ome! more $ompli$ated ut !till ma(a)ea le"


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Be$au!e #e are 3t%eoreti$all'4 ,ar'i() t%e default rate! e,er' period- #e lo!e t%e a ilit' to u!ee+po(e(tial term! a(d mu!t i(trodu$e produ$t (otatio(* T%i! merel' %ar&! a$& to our i(omialtree a(d t%e (otio( of multipl'i() e,er' term ' all t%e pro a ilitie! t%at mu!t e !ati!fied torea$% t%at poi(t*

T%u! for t%e 7?9 re$o,er' rate- t%e implied pro a ilit' of default i( ea$% of t%e !emi2a((ual

period! up to Au)u!t 7 (d 1 %d & %d 3 ) i! =*7 9* T%ereafter t%e implied rate e$ome! G* 9 u(tilJul' 7 (d ' %d 5 ) a(d !o o( a! !%o#( i( t%e ta le elo#*

T%i! )i,e! a more a$$urate pi$ture of t%e time at #%i$% t%e mar&et i! etti() Ar)e(ti(a #ill failo( pa'me(t!* B' o($e more multipl'i() t%rou)% t%e!e per2period pro a ilitie!- #e $a()e(erate a more $o(!i!te(t pi$ture of $umulati,e pro a ilitie! a! follo#!*

T%ere are $a,eat! i( t%e i(terpretatio( of t%e!e (um er!- a! di!$u!!ed elo#- ut t%e' pro,idet%e mar&et6! e!t e!timate of #%e( Ar)e(ti(a i! mo!t li&el' to default o( it! o li)atio(!*T%e o,erall me!!a)e i! t%at if i(,e!tor! elie,e t%at t%ere i! le!! t%a( a ?89 $%a($e t%atAr)e(ti(a #ill default i( t%e (e+t ei)%t 'ear!- t%e' $a( ear( more premium t%a( t%e' re1uireo,er a ri!&le!! a!!et*


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Fi+ed I($ome Re!ear$%

5e !tre!! t%at t%ere are !implif'i() a!!umptio(! emplo'ed i( our model* Fir!tl' #e %a,e a !tra$ted fromall li1uidit' a(d ta+ effe$t! i( the mar&et- a!!umi() i(!tead t%at our u(i,er!e of o(d! i! %omo)e(eou! i(re!pe$t of t%e!e attri ute!* T%i! doe! (ot !eem too re!tri$ti,e a( a!!umptio( #%e( deali() #it%euro o(d! i!!ued i( rea!o(a le !i.e* More importa(tl' #e %a,e a!!umed (o $orrelatio( et#ee( U*S*i(tere!t rate! a(d rate! of default* I($orporate o(d! t%i! ma' (ot e ea!il' defe(da le- ut t%e !i.e ofAr)e(ti(a6! de t urde( relati,e to it! /DP 2 =G9 2 mea(! t%at rate! #ould %a,e to ri!e mu$% furt%ert%a( i! li&el' i( t%e (e+t $ouple of 'ear! for t%e i($rea!e i( de t !er,i$i() to %a,e a !i)(ifi$a(t impa$t o(Ar)e(ti(a6! fi(a($e!* B' ma&i() t%e!e a!!umptio(! #e %a,e ee( a le to reapprai!e t%e!e !e$uritie! i(term! t%at are more dire$tl' rele,a(t to t%e i(,e!tor! #%o u' t%em*

I


10/25

CS FIRST BOSTON *une 15,1995

FIXED INCOME RESEARCH BULLETIN

Implied Default Pro a ilitie! Part >"Appli$atio( to Brad' Bo(d!

Clayton Perry 212/322-4998

Summary (ollo%in! our earlier analysis of default probabilities priced into ris"y eurobonds& the purpose of this paper is to adapt the methodolo!y to collaterali5ed

rady bonds# y calculatin! the default probability& %e can more accuratelyalue the arious component parts of a rady bond and thus assess the true

return to country ris"# 7e illustrate this by sho%in! the considerable differ-ences bet%een con entionally calculated stripped yields and those %or"ed outusin! the implied default probability method#

T%i! paper refere($e! a(d e+te(d! t%e a(al'!i! de,eloped i( our earlier ulleti( #plied e!a lt Pro*a*ilities% ,e$ pproach to Risky e*t% CS Fir!t Bo!to(Fi+ed I($ome Re!ear$%- April 7 ?* T%e met%odolo)' emplo'ed %ere i! e+plai(edi( detail i( t%at pu li$atio( a(d re$apped o(l' riefl' %ere*

I( t%e pre,iou! paper #e di!$u!!ed t%e $o($ept of mar&et implied rate! of defaulta(d loo&ed at appl'i() t%i! to differe(t i!!ue! ' t%e !ame orro#er to $reate a

a!i$ term !tru$ture of default pro a ilit'* T%e aim of t%i! report i! to ame(d t%emet%odolo)' !li)%tl' to appl' to Brad' o(d! a(d to e+ami(e t%e ,aluatio(impli$atio(! a$ro!! differe(t i!!uer!* 5e e)i( ' re,i!iti() t%e a(al'!i! outli(ed

pre,iou!l'*

$ecappin! the Suppo!e 'ou are due to re$ei,e a pa'me(t i( o(e 'ear6! time* If 'ou are 7889 !ure*ethodolo!y 'ou #ill $olle$t- it i! !e(!i le to di!$ou(t t%at pa'me(t at t%e rele,a(t ri!& free rate-

t%e 72Kear U*S* Trea!ur' ill 'ield* T%e le!! !ure 'ou are of re$ei,i() t%e pa'me(t*t%e %i)%er #ill e t%e di!$ou(t rate applied a(d t%e le!! #ill e t%e pre!e(t ,alue of t%e pa'me(t* No# !uppo!e 'ou are 7889 $ertai( of )etti() either pa'me(t or

pa'me(t y- #%ere y i! a !maller amou(t* If 'ou di!$ou(t ot% at t%e ri!& free rate'ou )et t#o (et pre!e(t ,alue!- o(e of #%i$% #ill e $orre$t* To ,alue t%i! 0eit%er

This memorandum is for informative purposes only. Under no circumstances is it to be used or considered as an offer to sell, or a solicitation of an offer to buyany security. Additional information is available upon request. esults based on hypothetical pro!ections or past performance have certain inherent limitations.There is no certainty that the parameters and assumptions used can be duplicated "ith actual trades. Any results sho"n do not reflect the impact of commissionsand#or fees, unless stated. $inancial futures and options are not appropriate for all investors. Their relative merits should be carefully "ei%hed. &here informationhas been obtained from outside sources. 't is believed to be reliable but is not represented to be accurate or complete. This document may not be reproduced in"hole or in part or other"ise made available "ithout our "ritten consent This firm may from time to time perform investment ban(in% for, or solicit investmentban(in% or other business from any company mentioned. &e or our employees may from time to time have a lon% or short position in any contract or securitydiscussed.


11/25


or0 pa'me(t t%e mo!t lo)i$al t%i() to do i! to a!!i)( pro a ilitie! to ea$% of pa'me(t a(d pa'me(t y a(d ta&e a( a,era)e of t%e t#o NP:! #ei)%ted ' t%e a!!i)(ed pro a ilitie!*

I( our e+ample- pa'me(t repre!e(t! t%e $a!% flo# !$%eduled o( a o(d- #%ile pa'me(t y repre!e(t! t%e re$o,er' ,alue 2 t%e amou(t t%e o(d i! #ort% i( t%e e,e(t of a default* T%e

mar&et pri$e of t%e o(d i( it! fi(al $oupo( period #ill repre!e(t t%e pro a ilit' #ei)%teda,era)e 2 or e+pe$ted ,alue 2 of t%e!e t#o amou(t! di!$ou(ted at t%e ri!&2free rate* T%i! i!repre!e(ted al)e rai$all' a! follo#!"

P= c ( - d) +Rd + r

#%ere P ; pri$e i($ludi() a$$rued i(tere!t$ ; $a!% flo#d ; t%e pro a ilit' of default! i( t%i! period

R = re$o,er' pro$eed! i( e,e(t of defaultr = ri!& free di!$ou(t rate

For e+ample- !uppo!e a ri!&' 72'ear .ero $oupo( o(d i! tradi() at a pri$e of


12/25


If #e a!!ume t%at t%e pro a ilit' of default i( ea$% of t%e i(di,idual $oupo( period! i! t%e!ame #e $a( $ollap!e t%i! e1uatio( to t%e follo#i()@

T%e met%odolo)' #e %a,e emplo'ed i! !et out dia)rammati$all' elo#* T%e pro a ilit' of )etti() to a poi(t o( t%i!2 i(omial tree i! t%e produ$t of t%e pro a ilitie! ide(tified o(ea$% ra($% ta&e(*

hibit 1# Binomial Tree of Default Probabilities

)he rady Brad' o(d! are di!ti()ui!%ed from Euro o(d! mai(l' ' $ollaterali.atio( of ifference !ome pa'me(t!* U(der t%e term! of t%e !e$urit'- t%e i!!uer %a! defea!ed t%e

pri($ipal #it% U*S* Trea!ur' STRIPS of t%e !ame maturit'- a(d $ollaterali.ed!e,eral $oupo( pa'me(t! o( a rolli() a!i! #it% a %i)% 1ualit' !e$urit' 3u!uall'AA or etter4* T%e a$tual term! of t%e Brad' i!!ue! ,ar' ' i!!uer@ #e refer reader!#%o are u(familiar #it% t%e mar&et to n ntrod ction to #er"in" /o ntries

i ed nco#e nstr #ents% CS Fir!t Bo!to( Ma' 7 =*

T%e !imple #a' to i($orporate t%i! $ollaterali.atio( of !ome $a!% flo#! i(to our model i! to ad u!t t%e $al$ulatio( of a re$o,er' ,alue- R- to refle$t t%e i($lu!io( of t%e $ollaterali.ed $oupo(! a(d pri($ipal at t%eir full di!$ou(ted ,alue* U!i()Me+i$o a! t%e e+ample- t%i! mea(! t%at t%e re$o,er' ,alue for default i( t%e !i+t%$oupo( period #ould e t%e di!$ou(ted ,alue of t%e !i+t%- !e,e(t% a(d ei)%t%

$oupo(!- a(d t%e di!$ou(ted pri($ipal* I( )e(eral term! t%i! i! t%e follo#i()@

ote that %e ha e assumed no reco ery o er and abo e the collateral * T%i! i! a

,er' importa(t a(d !ome#%at ar itrar' a!!umptio(- a! t%e o(d i( default #ould


13/25


%a,e !ome ,alue attri uta le to t%e promi!ed $oupo( flo# out!ide of t%e $ollateral-a(d a(' ud)eme(t ma' a#ard $reditor! more t%a( u!t t%e $ollateral ,alue*Ho#e,er #e u!tif' our a!!umptio( o( t%e )rou(d! t%at it #ill repre!e(t t%e a!e$a!e- a(d t%at- %a,i() defea!ed !ome of t%e pa'me(t!- t%e i!!uer i! le!! li&el' tofeel $ompelled to pa' out more t%a( t%e' alread' %a,e*

Armed #it% t%i! i(formatio(- #e $a( (o# #or& t%rou)% t%e e1uatio( o( t%e pre,iou! pa)e to a$& out t%e implied default rate! for a !erie! of Brad' o(d!* Mo!t i!!uer!%a,e a Di!$ou(t a(d a Par o(d- a floati() rate a(d fi+ed rate =82'ear o(dre!pe$ti,el'- #%i$% %a,e rea!o(a l' u(iform !tru$ture! a$ro!! i!!uer! a(d t%u! le(dt%em!el,e! to $ro!!2mar&et $ompari!o(* For t%e floati() rate Di!$ou(t o(d!- #ea!!ume $oupo(! to e fi+ed a$$ordi() to LIBOR for#ard rate!*

T%e follo#i() ta le !ummari.e! our fi(di()!*

'pplications T%e $ollaterali.atio( of !ome $a!% flo#! allo#! t%e di,i!io( of t%e pri$e amo() t%ePricin! ,ariou! $ompo(e(t! a! follo#!*Compo(e(t!

P ; P ) P$ P p

#%ere P ; pri$e i($ludi() a$$rued i(tere!tP " ; pri$e of $oupo( )uara(teePc ; pri$e of ri!&' $oupo( !treamP p ; pri$e of pri($ipal redemptio(


14/25


Co(,e(tio(all' t%i! i! !plit a! !%o#( elo#* T%e fir!t part i! t%e $ollaterali.ed 3a(d t%u! $ertai(4fir!t t%ree $oupo(!- t%e !e$o(d t%e ri!&' !tream of $oupo(! follo#i() t%e fir!t t%ree- a(d t%e la!t parti! t%e pri($ipal* B' putti() a ,alue to t%e fir!t a(d la!t pie$e! #e $a( #or& out #%at pri$e i! $%ar)edfor t%e ri!&' $oupo( !tream 3P $4 a(d t%u! )e(erate t%e 'ield o( t%at pie$e of t%e o(d*

To !tart #it% t%e ea!ie!t- t%e pri($ipal of t%e o(d i! appro+imatel' e1ui,ale(t to off2t%e2ru(U*S* Trea!ur' STRIPS* 5e !a' appro+imatel' e$au!e alt%ou)% t%e STRIPS !e$ure t%eredemptio( pa'me(t- t%e!e !e$uritie! are (ot o tai(a le efore maturit' a(d $a((ot t%erefore etraded li&e a (ormal !e$urit'* Ho#e,er- for $urre(t purpo!e! t%e pri($ipal i! treated a! Trea!ur'STRIPS- a(d t%u! ,alued a! follo#!*

P p; 78837 r n4 n

T%e !plit et#ee( t%e fir!t part of t%e o(d- t%e $oupo( $ollateral- a(d t%e ri!&' $oupo( !tream i!t%e mo!t i(tere!ti()* Co(,e(tio(all' t%e approa$% %a! ee( to ta&e t%e $ollaterali.ed $oupo(! a!$ertai( $a!% flo#! a(d di!$ou(t t%em at t%e ri!& free rate* T%i! ,alue 3alo() #it% t%e pri($ipal,alue a o,e4 i! !u tra$ted from t%e pri$e a(d t%e remai(der i! t%e $o!t of t%e re!t of t%e $oupo(!tream*

T%e pro lem #it% t%i! approa$%- depi$ted i( e+%i it =- i! t%at it )i,e! (o ,alue to t%e rolli()$ollateral pa!t t%e fir!t t%ree $oupo(!* It implie! t%at i( ei)%tee( mo(t%!- if t%ere i! (o default-'ou %a,e a( u($ollaterali.ed $oupo( !tream #%e( i( fa$t 'ou #ill !till %a,e a partiall'$ollaterali.ed $oupo( !tream* T%u! t%e $o(,e(tio(al met%od $learl' o,er!implifie! t%e impa$t of t%e $ollateral u(le!! t%e o(d default! i( t%e fir!t period*

5e elie,e t%at t%e $orre$t #a' to ,alue t%e )uara(tee i! to pri$e it a! a !et of $o(ti()e(t $a!%flo#!* Loo& a)ai( at our i(omial tree o( t%e follo#i() pa)e* Ea$% of t%e o+e! repre!e(t! a$ertai( $a!% flo# at a $ertai( poi(t i( time- a(d t%u! %a! a &(o#( di!$ou(ted ,alue*Furt%ermore- from our earlier $al$ulatio( #e &(o# t%e ,alue! for d t%at t%e mar&et implie!- a(dt%u! t%e pro a ilit' of )etti() ea$% of t%e flo#!* T%e produ$t of t%e di!$ou(ted ,alue a(d t%e

pro a ilit' of re$ei,i() it )i,e! t%e e+pe$ted ,alue- #%i$% i! #%at t%e ri!& (eutral i(,e!tor #ould e #illi() to pa' for a $o(ti()e(t $a!% flo#*


15/25


)herefore the fair alue of the !uarantee P!: is the total e pected alue of t%etop half ofthe tree& lea in! the e pected alue of the bottom half of t%etree as the price of thescheduled coupon stream 3P-4*

Mat%emati$all'- t%e fair pri$e of t%e $oupo( )uara(tee $a( (o# e e+pre!!ed a! follo#!"

Remem er for t%e purpo!e! of t%i! $al$ulatio( #e %a,e alread' !u tra$ted t%e ,alue of t%e

pri($ipal a(d #e a!!ume .ero re$o,er' of (o(2$ollaterali.ed flo#!- !o t%e re$o,er' i! defi(ed"

T%i! )i,e! a ,alue for t%e $oupo( !trip a! follo#!"


16/25

Fixed income Research

No# #e $a( !u !titute t%e!e ,alue! for differe(t part! of t%e o(d 3P $- P) a(d P p4 a$& i(to t%e ori)i(al e1uatio( a(d $ollap!e term! to )i,e t%e follo#i()"

T%i! i!- of $our!e- our ori)i(al- ri!&' o(d pri$e e1uatio( #it% t%e pri($ipal e+pre!!ed !eparatel'*Ha,i() t%u! $lo!ed t%e loop- #e $a( loo& at t%e $urre(t pri$i() of Brad' o(d! a(d allo$ate ,alueto ea$% of t%e t%ree $ompo(e(t!- a! !%o#( i( t%e follo#i() ta le*

'pplications T%e reallo$atio( of ,alue i( t%e pri$e of a Brad' o(d %a! importa(t impli$atio(!)he )rue for mea!ure! of relati,e ,alue* O(e of t%e mo!t fre1ue(tl' u!ed i(di$ator! i( t%eStripped ;ield Brad' mar&et i! !tripped 'ield* T%i! i! t%e i(ter(al rate of retur( o( t%e ri!&' $a!%

flo#!- a(d i! t%u! $ompara le to a( u($ollaterali.ed Euro o(d i!!ued ' t%e !ame orro#er*

I( order to $al$ulate t%e !tripped 'ield- #e mu!t e a le to fairl' a!!i)( t%e proportio( of t%e pri$e of t%e o(d t%at i! attri uta le to t%e ri!&' $oupo( !tream a!o(l' t%e( $a( #e mea!ure t%e true retur( to $ou(tr'2!pe$ifi$ ri!&*

T%e follo#i() ta le !et! out t%e true !tripped 'ield! offered o( Brad'6! a(d$o(tra!t! t%em #it% t%e $o(,e(tio(all' $al$ulated mea!ure* T%e true !tripped 'ieldi! )e(erall' %i)%er ' arou(d 788 *p*- de!pite t%e fa$t t%at t%e $oupo( )uara(tee i!)e(erall' #ort% le!! 3a(d t%u! t%e $oupo( !tream more4 u(der our met%od* T%e

rea!o( for t%i! i! t%at t%e i($lu!io( i( t%e !tripped $oupo( !tream of a pro a ilit'#ei)%ted portio( of t%e fir!t t%ree $a!% flo#! out#ei)%! t%e %i)%er pri$e paid*


17/25


+ther T%rou)% our pre,iou!l' di!$u!!ed default pro a ilit' met%odolo)'- #e %a,e ee(Co(!ideratio(! a le to more a$$uratel' $apture t%e pri$e of t%e differe(t ri!&! i(%ere(t i( a Brad'

o(d* T%i!- i( tur(- %a! allo#ed u! to $al$ulate a true !tripped 'ield a(d t%u! amore $o(!i!te(t mea!ure of ,aluatio(*

T%i! a(al'!i! a!!ume! ri!& (eutralit' o( t%e part of i(,e!tor! !u$% t%at t%e' are#illi() to pa' 7 for ea$% dollar of e pected pre!e(t ,alue* Moreo,er #e %a,ea!!umed for t%i! paper t%at i( t%e e,e(t of default- t%ere #ould e (o pa'me(tot%er t%a( t%e $ollateral* T%i! a!!umptio( i! ope( to de ate- ut #e (ote t%atrela+i() it #ould $%a()e t%e (um er! )i,e( ut (ot t%e ,alidit' of t%e a(al'!i!*

Fi(all'- #e earlier me(tio(ed t%e li1uidit' pro lem a!!o$iated #it% t%e pri($ipal$ollateral* Alt%ou)% t%e redemptio( pa'me(t i! defea!ed #it% Trea!ur' STRIPS-t%ere i! (o #a' of rea&i() t%at out of t%e o(d efore t%e !$%eduled maturit'*

T%erefore- a(' ar itra)e #ould rel' o( !#appi() t%at amou(t a$& to it! pre!e(t,alue u!i() a =82'ear .ero2$oupo( !#ap* I( fa$t a(' reali.atio( of t%e e+pe$ted,alue of $oupo( pa'me(t! #ould al!o rel' o( !#appi() t%em- !o for ar itra)e

purpo!e! all flo#! !%ould e di!$ou(ted from t%e !#ap $ur,e*


18/25

0 ~

/onditional default ri!& refer! to t%e pro!pe$t of o(d default i( a $ertai( period- a!!umi() (o prior default*U(li&e $umulati,e default ri!&- $o(ditio(al default ri!& (eed (ot ri!e loo&i() for#ard* Co(,e(tio(all'- %o#e,er-emer)i() mar&et parti$ipa(t! pre!ume it doe!*

T%e pre!umptio( i! rea!o(a le for !tro() $redit!- e$au!e t%eir 1ualit' %a! !o mu$% more !$ope to de$a' t%a( toimpro,e* For #ea& $redit!- %o#e,er- t%e pre!umptio( !%ould e re,er!ed* O,er time- lo# )rade o(d! t%at do (otdefault are li&el' to impro,e* T%e!e ar)ume(t! $a( e refi(ed u!i() t%e t%eor' of Mar&o, $%ai(!*

Empiri$all'- default term !tru$ture! for U*S* $orporate o(d! are ri!i() for A rati()! ut !teepl' i(,erted for !i()le2B* A Mar&o, $%ai( approa$% e+plai(! t%e e,ide($e remar&a l' #ell- i($ludi() t%e %umped term !tru$ture!of Ba a(d Baa $redit!*

T%e di!$repa($' et#ee( mar&et a!!e!!me(t a(d ri!& !u))e!t! a( attra$ti,e ar itra)e opportu(it' for lo()er2termi(,e!tor!" u' $ale(dar !pread! i( lo#er2)rade o(d!*


19/25

)erm Structure of efault $is" ntrod ction For lo()er time period!- t%e re1uired premium $a(

e !%o#( to e1ual d(1+r 4 (1-d)%#%ere d i! t%eA! o(d! a)e- t%eir default ri!&! $%a()e* :ie#ed $o(ditio(al default rate a(d r i! t%e future ri!&2freefrom t%e date of pur$%a!e- ea$% !u$$e!!i,e $oupo( rate* I($orporati() po!iti,e po!t2default pa'out!i! le!! li&el' to e paid- !o t%e c # lative default a(d ri!& a,er!io( furt%er $ompli$ate! t%eri!& ri!e!* But #e mi)%t al!o a!&" %o# li&el' i! t%e relatio(!%ip- ut %i)%er future default ri!&! are !till

(e+t $oupo( to e defaulted- a!!umi() (o prior a!!o$iated #it% %i)%er future premium! o,er ri!&2default T%i! i! &(o#( a! t%e conditional default free i(tere!t rate!*ri!&* It! term !tru$ture i(di$ate! #%et%er t%at ri!& ri!e! or fall! o,er time* Conventional View of Forward Default Risk

I( emer)i() mar&et!- t%e term !tru$ture of default I( emer)i() mar&et!- default ri!& i! t'pi$all'ri!& i! t'pi$all' pri$ed a! if it #ere flat or ri!i()* pri$ed a! if it #ere 3a4 ri!i() for all t'pe! loo&i()T%i! paper mar!%al! e,ide($e !u))e!ti()- o( t%e for#ard- a(d 3 4 !teeper- t%e lo#er t%e i(itial $redit$o(trar'- t%at t%e term !tru$ture for #ea& $redit! rati()*(aturall' de$li(e!* 7

For e+ample- i( April 7 ?- o(e2'ear- fi,e2'ear-T%e term !tru$ture of default ri!& matter! to a(d ei)%t2'ear Ar)e(ti(e dollar Euro o(d! tradedi(,e!tor!- e$au!e it ou)%t to i(flue($e i(tere!t re!pe$ti,el' at 88 p- 88 p- a(d ? p !pread!rate!* A! a( illu!tratio(- !uppo!e t%e $o(ditio(al o,er U*S* Trea!urie!* A!!umi() a po!t2defaultdefault ri!&! are 78 a!i! poi(t! 3 p4 toda' a(d pa'out ratio of 7?9 of fa$e ,alue- t%e implied

p tomorro#- t%at t%ere i! (o po!t2default pa'out- $o(ditio(al default ri!&! e!$alate to 78*79 for a(d t%at ri!&2free i(tere!t rate! ru( > p per da'* 'ear! fi,e t%rou)% ei)%t from * 9 for 'ear o(e* > A ri!&2(eutral i(,e!tor 3i*e*- #illi() to a$$ept a(' For $ompari!o(- t%e 'ield differe(tial et#ee(

et e+pe$ted to rea& e,e(4 #ill $%ar)e 7> p triple2A U*S* $orporate o(d! a(d $orre!po(di()i(tere!t for a o(e2da' loa(- e$au!e %i! e+pe$ted U*S* Trea!urie! t'pi$all' ri!e! from 7827? p o(

pa' a$& per dollar i! * 89 time! 7*887> or t#o2'ear paper to G82?8 p o( =82'ear paper*rou)%l' t%e ri!&2free )ro!! retur( of 7*888>*Similarl'- %e #ill $%ar)e a( a,era)e 78 p per da' I(tuiti,el'- it ma&e! !e(!e for triple2A i(tere!tfor a loa( repa'a le i( t#o da'!- e$au!e %i! premium! to #ide( loo&i() for#ard* No $redit %a!e+pe$ted retur( per dollar i! * 89 time! * G9 e,er defaulted i( a 'ear it #a! rated triple2A*

time! 7*8878 !1uared- or rou)%l' t%e ri!&2free Ho#e,er- !ome triple2A $redit! %a,e defaulted)ro!! retur( of 7*888G* T%e dail' i(tere!t rate after %a,i() de$a'ed to lo#er $redit rati()!* I(fall! e,e( t%ou)% t%e $umulati,e default ri!& ri!e!* pro a ilit' term!- t%e' %a,e (o#%ere to )o ut

do#(* Note i( t%e pre$edi() e+ample t%at for t%e fir!tda'- t%e re1uired i(tere!t premium o,er t%e ri!&2 It al!o ma&e! !e(!e for default ri!&! to i(itiall' ri!efree rate i! 78 p- t%e !ame a! t%e default ri!&* For more 1ui$&l' for- !a'- !i()le2A $redit! t%a( for t%e !e$o(d da' of t%e t#o2da' loa(- t%e re1uired triple2A $redit!* Ha,i() lar)e !er,i$i() $u!%io(!-future i(tere!t rate i! < p- or a future premium of triple2A $redit! de$a' i( t%e fir!t i(!ta($e to ot%er

p- a)ai( t%e !ame a! t%e $o(ditio(al default ri!&* A2le,el!- #%ere immediate default ri!&! remai(lo#* Si()le2A $redit! te(d to de$a' !oo(er to !u

T%i! poi(t %a! ee( emp%a!i.ed '- amo() ot%er!- i(,e!tme(t )rade- #%ere prope(!itie! to default areJ*S* Fo(!* 0U!i() Default Rate! to Model t%e Term mu$% %i)%er*Stru$ture of Credit Ri!&-0 inancial nalysts o rnal.Septem er2O$to er 7 G* pp* >?2=>@a(d L*T* Niel!e(-J* Saa2Re1ue o- a(d P* Sa(ta2Clara- 0Default Ri!& a(d Cla'to( Perr'- #plied e!a lt Pro*a*ilities2 ,e$I(tere!t Rate Ri!&" T%e Term Stru$ture of Default pproach to Risky e*t. Fi+ed I($ome Re!ear$%Spread!0* INSEAD 5or&i() Paper* Ma' 7 =* Bulleti(- CS Fir!t Bo!to(* > April 7 ?*

Term Stru$ture of Default Ri!&


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T%e $%art a o,e o( t%e default %i!tor' of U*S* To !tre()t%e( t%e i(tuitio( for t%i! re!ult- ima)i(e ai(,e!tme(t2)rade $orporate o(d!- di!ti()ui!%ed ' li(dfolded ma( #a(deri() o( a( u(fe($ed roof*letter rati() out to a te(2'ear %ori.o(- le(d! T%e !tarti() po!itio( $orre!po(d! to a( i(itial!upport to t%i! ,ie#* Ea$% term !tru$ture i! rati()- #%ile falli() off t%e roof $orre!po(d! to)e(erall' ri!i()* Moreo,er- t%e lo#er t%e i(itial default* For a ma( !tarti() $lo!e to t%e ed)e-$redit 1ualit'- t%e !teeper t%e term !tru$ture* = di!a!ter ma' !eem immi(e(t* But if %e %a!

!ur,i,ed for a #%ile2#it%out faili()- t%e $%a($e! Revised ie$ o! or$ard e!a lt Risk are %e %a! #or&ed part #a' a$& to t%e $e(tre- a(d

%e($e i! le!! li&el' to fall i( t%e (e+t i(!ta(t*O( refle$tio(- t%e default term !tru$ture $a((ot ri!e Co(,er!el'- a ma( !tarti() at t%e $e(tre i! ou(d tofor t%e #ea&e!t $redit!* T%e latter are more li&el' $ome $lo!er to t%e ed)e- !o %i! $o(ditio(al ri!&! of to default i( t%e (ear2term- ut t%e' al!o %a,e more falli() mu!t i($rea!e*

up!ide* Moreo,er- t%e lo()er t%e' )o #it%outdefaulti()- t%e more li&el' t%e' are to %a,e Mat%emati$all'- #e are predi$ti() a for#ardimpro,ed i( 1ualit'* For e+ample- if a #ea& default ri!& $o(ditio(al o( a( i(itial rati() a(d (o!o,erei)( $redit li&e Bul)aria $a( !er,i$e it! Brad' i(terim default* O,er time- t%e i(terim %i!tor'

o(d! pu($tuall' for t%e (e+t 7? 'ear!- t%e $%a($e! $ou(t! for more- relati,e to t%e i(itial a!!e!!me(t*are $reditor! #ill (o lo()er dema(d t%e 7?88 p He($e- t%e differe(tial default ri!& et#ee( t%edefault !pread! t%e' $arr' toda'* Note t%at t%i! !tro()e!t a(d #ea&e!t $redit! !%ri(&! o,er time*re!ult doe! (ot %i()e o( our elief t%at Bul)aria i!ultimatel' 0emer)i()0 rat%er t%a( 0!u mer)i()0* To $o(!ider a (umeri$ e+ample- !uppo!e t%ere areIt i! !impl' t%e o !er,atio( t%at 7? 'ear! of t#o $redit 1ualitie!" Stro() 3S4 a(d 5ea& 354*!u$$e!!ful de t !er,i$i() i! more li&el' t%a( (ot to Curre(t 4 $redit! (e,er default- #%ile $urre(t 5i(di$ate impro,ed $redit 1ualit'* $redit! default 789 of t%e time* From o(e period

to t%e (e+t- a!!umi() (o prior or $urre(t default* 5$redit! %a,e a >89 $%a($e of !#it$%i() to S- #%ile

Co(ditio(al default ri!&! $al$ulated from data i( S $redit! %a,e a ?9 $%a($e of !#it$%i() to 5*Mood' ! I(,e!tor! Ser,i$e* /orporate ond e!a ltsa(d e!a lt Rates 1670- 166'% Ja(uar' 7 ?*


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)erm Structure of efault $is"

Loo&i() for#ard- t%e $o(ditio(al default ri!& for S G*>9 i( t%e fir!t 'ear to 7*=9 i( t%e te(t% a(di! 8*?9 3i*e*- ?9 $%a($e of !#it$%i() to 5- #%i$% 8*?9 i( t%e t#e(tiet%*%a! 789 default ri!&4 for t%e !e$o(d period a(d T%e default term !tru$ture i! !teepl' i(,erted8*< ?9 3?9 *?9 i( 'ear! t#o t%rou)% fi,e from a( i(itialStro() $redit!- a(d de$li(e loo&i() for#ard for 7* 9- t%e( de$li(e! )raduall' to 7*89*$urre(tl' 5ea& $redit!*

To furt%er te!t t%e re,i!ed ,ie#- t%e Appe(di+T%e Appe(di+ refi(e! t%e!e t%eme!- u!i() t%e e+plore! %o# #ell a Mar&o, $%ai( $a( repli$ate t%e

mat%emati$al t%eor' of Mar&o, $%ai(!* Ba!i$all'- o !er,ed default profile!* T%e fit i! e+$elle(t*t%e e,olutio( of $redit 1ualit' i! a Mar&o, $%ai( tot%e e+te(t t%at t%e $urre(t rati() !uper!ede! all T%i! i! (ot to $laim t%at forei)( o(d!- parti$ularl'

prior rati() i(formatio(* T%e a(al'!i! $o(firm! !o,erei)( o(d!- e+%i it t%e !ame default profile!t%at $o(ditio(al default ri!&! ultimatel' te(d to a! U*S* $orporate!* 5%ile t%e rati() a)e($ie!$o(,er)e* I( t%e !%ort ru(- %o#e,er- #it% more !tri,e to e(!ure t%at a!!i)(ed )rade! aret%a( t#o $redit 1ualitie!- for#ard default ri!&! $a( $ompara le a$ro!! i!!uer t'pe!- t%ere i! (ori!e a(d later fall or ,i$e2,er!a* )uara(tee t%at t%e' !u$$eed- or e,e( t%at it i!

po!!i le* E,e( for U*S* $orporate!- future defaultFor e,ide($e- #e tur( a)ai( to t%e default re$ord of rate! ma' depart from %i!tori$ ,alue!* Our $laimU*S* $orporate o(d! #T%e fa$i() $%art! i! rat%er t%at t%e 1ualitati,e patter(! predi$ted 'i($orporate !u 2i(,e!tme(t2)rade $redit! i(to t%e t%eor' a(d $o(firmed i( o(e importa(t data !et 2$ompari!o(! a(d e+te(d t%e %ori.o( to >8 'ear!* e!pe$iall'- t%e i(,erted default term !tru$ture for

Note t%at- $o(trar' to t%e $o(,e(tio(al ,ie#- ut #ea& $redit! 2 are li&el' to re$ur i( ot%er!*$ompati le #it% our re,i!ed ,ie#"

From a( i(,e!tme(t per!pe$ti,e- t%e mai( pra$ti$alTime ultimatel' (arro#! t%e ri!& )ap! implied impli$atio( i! to re$omme(d $ale(dar !pread! o(

' differe(t $redit rati()!* T%e a,era)e lo()2dated o(d! for emer)i() mar&et!* A follo#2differe($e i( $o(ditio(al default ri!&! et#ee( up paper ela orate! t%i! t%emQ*i(,e!tme(t a(d !pe$ulati,e )rade! !%ri(&! from


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)erm Structure of efault $is"

Appendix: Default as a arkov C!ain !impli$it'- #e $o(!trai( our !ear$% i( t%e follo#i()#a'!"

A Mar&o, $%ai( i! a !e1ue($e of tra(!itio(! fromo(e !tate to a(ot%er- #%ere t%e $o(ditio(al 2 a(' $redit $la!!e! lo#er t%a( B i)(ored*

pro a ilitie! of tra(!itio( depe(d o(l' o( t%e 2 $redit! allo#ed to mo,e at mo!t t#o $la!!e! uptra(!itio( it!elf- (ot o( %i!tor' or timi()* T%e or do#( per e,olutio( of $redit 1ualit' $a( e ,ie#ed a! a 2 1 a!!umed to e+a$tl' e1ual t%e o !er,edMar&o, $%ai( i( #%i$% t%e !tate! are t%e ,ariou! a,era)e! for t%e !ample 3#%i$%- amo() ot%er $redit rati()!- pro,ided t%at $urre(t rati()! ma&e t%i()!- re1uire! !i()le2A $redit! to %a,e 8*879

pa!t rati()! irrele,a(t* lo#er o(e2period default rate! t%a( Aa4

5e #ill u!e t%e follo#i() termi(olo)'" T%e!e $o(!trai(t! mea( t%at t%e P matri+ !ele$teddoe! (ot 'ield t%e e!t po!!i le fit* Moreo,er-

n (um er of fea!i le !tate!- la elled 7 to n e$au!e t%e e!timati() e1uatio(! are %i)%l' (o(2 p t i 0tra(!itio(0 pro a ilit'- )i,e( $urre(t !tate li(ear- #e do (ot tr' to $al$ulate !i)(ifi$a($e

i- of mo,i() to !tate i( e+a$tl' t period! le,el!* Ne,ert%ele!!- it i! $lear from t%e $%art! o( P t n+n matri+ #it% i eleme(t p t i * Al!o t%e t%e fa$i() pa)e t%at t%e e!timated term !tru$ture!

t 2time! produ$t of P #it% it!elf 3#%ere a tra$& t%e o !er,ed ,alue! remar&a l' #ell*matri+ produ$t P9 %a! i eleme(t p i1: 1 *** p in : n 4

dt $o(ditio(al default ri!& i( period t%)i,e(i(itial !tate i

t $olum( ,e$tor #it% n eleme(t! d i 3i4 lo()2ru( pro a ilit' of ei() i( !tate -

)i,e( i(itial !tate i

To tr' to repli$ate t%e $orporate default data 6m C%art ?u!i() a Mar&o, $%ai(- #e !ear$% for a P a(d 1 s ch t%at t%ee1uatio(! t = P t 1 appro+imatel' de!$ri e t%e o !er,atio(!* For


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