Pricing Convertible Bonds with Interest Rate, Equity, Credit and FX Risk

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1 The Uiesiy o Readig THE BUINE CHOO FOR FINANCIA MARKET Piig Coeibe Bods ih Iees Rae Eqiy Cedi ad F Risk IMA Cee Disssio Papes I Fiae 00-4 Fis Da: 30 h Noebe 00 Ai Boa Yigibasiog IMA Cee Uiesiy o Readig Copyigh 00 IMA Cee. A ighs eseed. The Uiesiy o Readig IMA Cee Whiekighs PO Box 4 Readig RG6 6BA UK Te: Fax: Eai: eseah@isaee.dg.a.k Web:.isaee.dg.a.k Dieo: Poesso Bia o-qi IMA Chai i Iese Bakig The IMA Cee is sppoed by he Ieaioa eiies Make Assoiaio

2 Absa Coeibe bods ae hybid seiies hose piig eies o a se o opex iedepedeies de o he sesiiiy o iees ae isk deyig eqiy isk F isk ad edi isk ad de o he oeibe bod s eay exeise Aeia eae. We pese a o ao ode o iees ae ad eqiy isk ha is ipeeed sig he Cak-Nihoso ehiqe o he diseized piig eqaio ih pojeie sessie oe-eaxaio. This pape exeds a ehodoogy poposed i he ieae TF[98] o dea ih edi isk i a seosise ay ad poposes a e ehodoogy o dea ih F sesiie oss-ey oeibes. A ehiqe o exaig he pie o aia opios sk o a syhei asse he oeig eqiy i doesi ey is epoyed o obai he ipied oaiiy o hese opios. These ipied oaiiies ae he sed o obai he oa oaiiy o se i he eia oie. The ode is desiged o dea ih os o he sa oaa eaes sh as opos diideds oios ad/o Beda a ad p ases. We sgges ha edi spead adjses i he boday odiios a be ade o ao o he egaie oeaio beee speads ad eqiy. Deaied desipio o he eia ehods ad he diseizaio shees ogehe ih hei aay ae poided. Aho Coas: Ai Boa Yigibasiog IMA Cee Uiesiy o Readig Box 4 Whiekighs Readig UK RG6 6BA E-Mai: a.yigibasiog@isaee.eadig.a.k JE Cassiiaio: C63 G3 G5 Keyods: oss-ey oeibes edi spead iees ae isk Aeia eae oa oaiiy Cak-Nihoso. I od ike o hak y speiso Po. Cao Aexade ad D. Jaqes Pezie o hei aabe oes ad sggesios. I od aso ike o hak ee Heig Deshe Bak AG o ay siaig disssios o his sbje ad o joi opaioa ok. A eos ae y esposibiiy. This disssio pape is a peiiay esio desiged o geeae ideas ad osie oe. Pease do o qoe iho peissio. The oes o he pape ae peseed o he eade i good aih ad eihe he aho he IMA Cee o he Uiesiy i be hed esposibe o ay osses iaia o oheise esig o aios ake o he basis o is oe. Ay pesos eadig he pape ae deeed o hae aeped his.

3 Disssio Papes i Fiae: Iodio Coeibe bods ae hybid seiies hih oe po he oe he igh o eeie a ixed ioe sea dig he ie o he oeibe ih he ebedded igh o oego he ixed ioe sea ad oe io a pesibed ao o deyig eqiy ay ie dig he ie o he ise. Bease oeibes i be exeised he he isse does e oeibe deb as o oe he isse s os o deb iaig hogh he ipii a opio o he isse s sok ha is ipii i he oeibe. I essee he oeibe bod is a ixe o a opoae bod ad a aa o he deyig eqiy. The deisio o exeise he aa eiaes he bod opoe o he ise. Is opoae deb ad eqiy ae edes i sesiie o eqiy iees ae ad edi isk. Fheoe he speia ase o F isk aises o hose oeibes ha pay opos ad ae ae i soe oeig ey b oe io soe doesi eqiy. The odide oeibes ake by isse size is i exess o U$ 400 biio ad highy deeoped akes o oeibes exis i he Uied aes Japa Egad Fae Asaia Caada ede ad izead. The ajoiy o isses i hese akes osis o doesi oeibes deoiaed i a oa ey ad oeibe io doesi eqiy. Deais o osadig isse size aiy iiia pei o he U ake ase ae poided i Appedix. I addiio o doesi oeibe akes hee is a sizeabe Eobod oeibe ake i hih ey sesiie oeibes ae aiey issed ad aded. 3 This ake as a hoe ge o U$ 8. biio i 990 o U$ 4 biio i 995 C[98]. The Eobod oeibe ake oais as a sbse oeibe isses ha oe io he isse s doesi eqiy b ae deoiaed i.e. pay opos ad ae ae i soe odoesi ey sh as UD o EUR. These ises ae aay sesiie o F isk. Oe o he ages oeibe bod akes by isse size is Japa hih as o 998 had a 40% shae o he od ake i oeibes C[98]. The Japaese ake aso poded oe o he ages ee isses o F-sesiie oeibes. I epebe 995 he Misbishi Bak issed a U$ biio eseabe adaoy exhageabe oeibe seiy aig I soe ases he oeibe ay oy be oeed dig a pesibed ie peiod dig is ie. Deied as he peeage dieee beee he oesio pie o eqiy ad he isse pie. 3 Eobod oeibe ees o he boad ake aegoy eopassig a oeibes issed oside he isse s doesi ake. The deoiaio ey a be isse s doesi ey o F isk o soe oeig ey eadig o F isk. 00 Ai Boa Yigibasiog

4 Disssio Papes i Fiae: 00-4 No. 00 adaoiy oeig io Japaese eqiy AD a aiy. The adaoy oesio as seeeed ih a pad eseig oesio aio oe ie shod he doa ae o he Japaese eqiy depeiae beo a eai ee o ixed ese daes. Ceay a depeiaig Japaese Ye od deease he doa ae o he Japaese eqiy ad heeby ipa he ae o he oeibe. The deais o his ise ae poided i Appedix 3. A apidy goig oiea Eobod oeibe ake is he iss ake hih is iqe i ha he ajoiy o opaies ha isse oeibe deb i iss as ae o doesi is 4. Exapes o osadig oeibe isses o he iss akes ide he UD deoiaed Hhiso Whapoa Ieaioa % opo oeibe io seig deoiaed shaes i Ja Ths he iss ake poides aohe ieie o he isio o F isk i a oeibe piig ode. Addiioa eaes opiae he piig pobe. These ide he Aeia eay exeise eae ie depede a by he isse ad Beda o oios p by he ieso eseig oesio aios. Fhe opexiy is ased by he hybid edi isk ae o oeibes. The oeibiiy eae o he oeibe bod poses o edi isk o he ieso. This is bease he isse a aays deie is o eqiy 5 hie he ixed ioe haaeisis sh as iees payes he ae ae epaye ad he p paye assoiaed ih he p ee is sbje o edi isk as he isse eeds ash o i is paye obigaios see TF[98]. I is possibe o ie he edi spead assoiaed ih he sae isse s ooeibe deb as a ppe bod o he edi isk assoiaed ih he oeibe bod a a ha is oe sed by paiioes o aig oeibes ih a e o hb appoah. Usig his ie a ade igh se say 60% o he edi spead o ooeibe deb as he appopiae diso ae o he oeibe bod i he piig eqaio. The pias o sig sh adho appoahes is disssed i TF[98] ho sgges a oe ao oped PDE appoah osise ih Bak-hoes o dea ih edi isk. This pape exeds hei ode ad ides iees ae isk i he oped piig PDE. 4 C[98] epos ha appoxiaey 90% o isses ae oeig is. 5 Uess he oeibe is exhageabe io he eqiy o a i ohe ha he isse i hih ase he issig i has o phase he eqiy ih ash. 00 Ai Boa Yigibasiog

5 Disssio Papes i Fiae: We popose a hage o eaie ehiqe o ao o he F isk i oss-ey oeibe bods iho ieasig he diesioaiy o he pobe. I is ieesig ha he piig o F-sesiie oeibes has aogehe o aaed a high degee o iees i he ieae o dae. The ode e deeop o ieaioa oeibes has he popey ha doesi oeibes ae a sbse ad a easiy be aoodaed i he sae aeok. This pape peses a o ao eqiy ad iees ae paaboi oped PDE appoah ih hage o eaie o pie ieaioa oeibes ih edi isk. The edi isk o oeibes is dea ih by iodig a axiiay asse aed he ash oy pa o he oeibe COCB hih pays sae oige ixed iees payes oy ad exeds he appoah ioded by T[98] hih aso saisies a siia piig eqaio ih is boday ad iiia odiios oped o he oeibe seiy. The o piig eqaios ae soed siaeosy de o his opig. As i he basi TF[98] ode e asse ha he apia se o he isse osiss o seio opoae deb ih assoiaed edi spead oe easy es ad he jio oeibe isse. This asspio i he ode a be eaxed iho diiy as disssed i TF[98]. Fheoe e popose o se he oa oaiiy appoah i ie o osa eqiy oaiiy ihi he hage o eaie aeok. To o koedge his appoah has o bee sed i he oeibe piig ieae o dae ad ahogh i is eioed i a oooe as a possibiiy i a ee pape HNB[0] he sggesio eaes o he doesi oeibe bod pobe. The hage o eaie appoah eies o he sohasi poess ooed by he oeig eqiy i doesi ey. As a es hee is o ake-gie oaiiy sie o opios sk o his syhei poess. We popose o bak o he pie o opios sk o he poess ooed by he oeig eqiy i doesi ey sig he ipied oaiiy o aia F ad oeig eqiy i oeig ey opios. This is aopished eiay by eaaig a expeaio oe he joi desiy o he F ae ad he oeig eqiy poesses oe a possibe obiaios o sikes o F ad oeig eqiy opios. The pies so obaied ae sed o bak o he ipied oaiiy o he oeig eqiy i doesi ey. These ipied oaiiies ae he sed o obai oa oaiiy o se i he eia oie. 00 Ai Boa Yigibasiog

6 Disssio Papes i Fiae: A isse pagig he TF[98] appoah hih is exeded i his pape is he oeaio beee edi speads ad eqiy hih is aoed o i he TF[98] aeok. Cedi speads ae sesiie o i ae ad i ed o iease he eqiy pies deease ad ie esa. TF[98] se a osa edi spead i hei pape hih is sed o diso he COCB. By hei osio his spead is iaia o hages i he edi qaiy o he isse haee he ae o he isse s eqiy i he e. O aeok a be sed o adjs he edi spead o o eqiy aes i he boday odiios o he piig PDE. To his ed e poide a desipio o he piig PDE i a he egios o he gid. O he ohe had he edi spead o high aes o eqiy i i ee dop o o he piig PDE de o he a ha he COCB i be aed a zeo i high eqiy egios i he gid. A Cak-Nihoso shee ih adapie gid o ah ie seps o jp ees sh as opo payes is epoyed. The Aeia eay exeise eae is posed as a iea opeeaiy pobe ad is aked sig he pojeie sessie oe-eaxaio POR ehod. This ehod has soe shooigs as deaied i he ieae see Z[99] ad WK[97] ad seio o his pape i ha i ao gie he exa oaio o he ee boday a hih i is opia o oe he oeibe bods. By epoyig sae eqiy seps i o gid he adese ee o he POR ehod is eded: he POR shee eqies a diagoay doia popagao aix o oegee hih is eoed sig sa ie seps ad he high be o eqiy seps edes he iaay de o pojeio. We aso poide deais o he piig PDE ad is diseizaio i aios egios o he gid i eaio o is boday/iiia odiios. This sees o pposes. Fis his poides a sope o iesigae he sabiiy o he diseizaio shee ad o shed igh o he ipeeaio deais. The seod ppose is o poide a opee desipio o he gid sed: his is hee edi spead adjses i he boday odiios a be ade de o he egaie oeaio beee speads ad eqiy. The pape is ogaised as oos. eio poides a bie iia eie o ieae o piig oeibes. eio 3 deais he basi ode o F eqiy ad iees ae; i poides deais o he hage o eaie ehiqe he oa oaiiy ehiqe sed ad he edi isk appoah i addiio o he piig eqaios ad boday ad iiia odiios o he gid. eio 4 poides he eia deais o he ode. eio 5 00 Ai Boa Yigibasiog

7 Disssio Papes i Fiae: odes. A deaied exposiio o he eia ehiqes he diseizaio ad aay o he shees ae poided i he appedix.. ieae Reie The ieae o eia ehods o soig he oeibe bod pobe i he B[73] aeok oigiaes i Bh[77] ad I[77]. They se i ae odes o doesi oeibes iho edi isk sig oy oe ao. Bh[77] se he expii iie dieee shee o soe he pobe eiay. Bh[80] he exed hei eaie ode o ide iees ae isk. A hee eaie papes ege o eio he Aeia eay exeise eae o oeibes o do hey ao o edi isk oaiiy aibaio o F isk. WDH[93] sho ha he Aeia eae o oeibe bods a be iepeed as a iea opeeaiy pobe hih ae diseizaio a be soed sig a so-aed pojeio esio o he sessie oe-eaxaio ehiqe o eia aaysis. This pojeed sessie oe-eaxaio POR ehiqe aoids soig o he eay exeise boday. WK[97] se a asoaio ehod o soe expiiy o he eay exeise boday o a Aeia p ih diideds sig a oe ao PDE. Coseqey Z[99] exed he WK[97] ehiqe o o aos ad sdy oeibe bods. Thei pape eges o iopoae he a ad p eaes o oeibes eah o hih osies a he oped o-iea PDE opiaig he eia ipeeaio eoosy. TF[98] ae he is o ode edi isk expiiy i a Bak-hoes od. They soe o he oeibe bod ae sig a oe ao oped PDE ode ih he expii iie dieee ehiqe. D[99] popose a o ad a ha ao aie ode ih sohasi iees ae eqiy ad hazad ae hih aibaes o he iiia e se ia a HW[90] iees ae aeok ad o he e se o edi speads. The D[99] aeok does o addess F isk ad ses a osa eqiy oaiiy asspio. Reey HN[0] pese a oe ao ode o pie Japaese eseig oeibes ad popose a ehod o aoig o he a oie peiod ad he ese eae iho ieasig he diesio o he pobe. 00 Ai Boa Yigibasiog

8 Disssio Papes i Fiae: The Mode 3. Mode Basis We a se p a ake osisig o a oeig asse payig oios diided yied γ a oa isk-ee bod a oeig isk-ee bod hee he oeaed asses oo he ooig poesses i he objeie P-ease: 6 d db db d d α B B d d d γd dw Hee dw is a doesi idiesioa Boia oio de he objeie ease P ad is soe oa oaiiy io depedig o he oaiiy sie o oeig eqiy aia opios. We hoose he ooig oe ao CIR[85] ode o haaeise doesi iees ae dyais ahogh he piig aeok is geea ad a aoodae ohe iees ae odes idig hose hih aibae o he iiia e se sh as HW[90]. dd ad d d dw I is poposed ha he exhage ae he be o oeig is o ey pe oe i o doesi ey oos he geoei Boia oio d α d dw hee is oa oaiiy io depedig o he oaiiy sie o F aia opios. Voaiiy is ioay depede o he deyig ad ie ad hee oige po he eaized ae o he deyig i he e is y deeied see R[99]. o e s 6 Ideia oaio o Bjok998 is sed o he basi aeok. 00 Ai Boa Yigibasiog

9 Disssio Papes i Fiae: ie ad as shohad o he oa oaiiies o hee o sppessig depedee o deyig ad sike o oaioa oeiee. We oe ha: W W ad W 3 hee ad ae x eos i he aboe deiiio 3. I paia oeaios ae expessed i es o a eo pod i i i ρ T As he ake ih oa oaiiy is opee Reboao 999 e a se abiage ages o deie he poesses o he F ae ad he oeig eqiy i doesi ey de he doesi isk ea ease Q. I is easy o sho ha he oeig exhage ae oos he geoei Boia oio ih oa ae o e i he Q ease: d d d d dw hee dw is a idiesioa Boia oio i he doesi Q ease. 3. Mease hage To deie he doesi pie o he oeig eqiy deoed Iô s ea is ioked: 00 Ai Boa Yigibasiog

10 Disssio Papes i Fiae: d d α d d dw d γd dw α α γ d dw T α α γ d dw α dd d Fo o hae he oe isk ea di o d γ iode he ease asoaio: T dw dw d α α d d γd hee dw is he doesi isk ea ease d 4 Hee dw is he doesi idiesioa Boia oio assoiaed ih he isk ea pobabiiy ease. Fheoe e asse ha he iees ae DE ad he DE o ae oeaed hogh d d ρ d. 3.3 Risk eaiy ad he basi piig eqaio Iô s ea appied o he oeibe bod hih depeds o he oeig eqiy i doesi ey he doesi iees ae ad ie yieds he DE ooed by he dieeia poess d: a γ dw d ρ ρ d dw 00 Ai Boa Yigibasiog

11 Disssio Papes i Fiae: Risk-eaiy ages he ead o he piig eqaio o he oeibe bod hih pays doesi iees b oes io oeig eqiy a γ ρ 0 ρ 5 This eqaio is aid oy p o he eay exeise boday a hih CR. poided he oesio pie is beo he a pie ps aed iees. Hee CR ees o he oesio aio. I is assed ha he oeibe bodhode he doesi ieso oes oy he CR. is high eogh opaed o: V axppai Bod_Vae hee PP is he peaiig p pie a ie a sep io o ie AI is he aed iees a ie sie he as opo paye ad Bod_Vae is he ie pese ae o he ixed ioe os o he oeibe bod. Ths V osies a oe ae o he oeibe. The a ae o he oeibe is assed o be eseig ad is as as a sep io possiby ih had a poeio 7 ad a be as as a ppe ae o he oeibe: V U V a Eqiaey his is a ppe ae o ise o he iie dieee gid. heeby osiig a ppe pie bod o he 00 Ai Boa Yigibasiog

12 Disssio Papes i Fiae: oa Voaiiy I is spposed ha pies o OTC a/p opios sk o ad ae aaiabe p o he aiy ie o he oeibe bod 8 ad hee hei ipied oaiiies ae aso aaiabe p o he aiy o he oeibe bod. The he oa oaiiy Φ o aia opios sk o a be od eiay o he ipied oaiiy o oaiiy o is sppied o he oeibe bod PDE i ie o osa oaiiy.. This oa The pobe is ha ypiay hee i be o iqid opios pies sk o ad e i IMP eed o os he ipied oaiiy o hese osees. Deoe by Φ he ipied oaiiy o as sk o hih is sed sbseqey o obai he oa oaiiy ioa i he disio eqaios. Thee ae ay ehiqes i he ieae o IMP soig he i-posed io appoxiaio Φ Φ oe IMP Φ has bee od. These ide a PDE appoah ABR[97] ad spie ioa ehods CV[99]. IMP To obai Φ ea ha C TΦ e IMP dt e K K K dt [ TT K0 I0 ] E TT K ψ T T;0 0d ψt 0d 6 hee ψ T T; 0 0 is he odiioa ogoa pobabiiy desiy io o gie a aiy gie by: 7 A oeibe ha is o aabe o a eai be o ohs o yeas is said o hae had a poeio. This is i oas o so a poeio he he oeibe beoes aabe ae he deyig eqiy hee ades a a pedeeied ee aboe iiia pie. 8 This is ypiay e o eqiy ad F p o 5 yeas. F ipied oaiiies o o oie syses sh as MURE ae aaiabe ypiay p o yeas ad oge aiies p o 5 yeas a be obaied o bokes. Beyod 5 yeas hoee F opios pies ae had o oe by ad he ehodoogy sggesed hee is ess se. 00 Ai Boa Yigibasiog

13 Disssio Papes i Fiae: 00-4 ψ ψ T T;0 0 TT 0 ψt 0 0 ψ T 0 ψt 0 ad ψ TT 0 0 is he joi pd o ad ad a be obaied o he ogoa disio epeseaio o : ψ TT 0 0 exp 0.5[ γ] d IMP IMP IMP IMP IMP IMP ρ 7 ih: 0.5[ d ] ψt 0 exp 8 IMP IMP As he dobe iega depeds o he sie o oeig eqiy i oeig ey ad o he exhage ae he eia soio o he iega gies he ipied-ipied oaiiy o he asse. 3.5 Cedi Risk Cedi isk is ided by iodig he ash oy pa o he oeibe bod COCB a syhei seiy disssed is i TF[98]. The payo o he COCB depeds o he deyig oeibe seiy ad he hode o he COCB is paid a he ixed ioe os poided he oeibe eais oeed ad ay p ash os i he hode o he oeibe deides o p he bod bak o he isse. 00 Ai Boa Yigibasiog

14 Disssio Papes i Fiae: C C C C C CFV 0. T- T COCB ih o oesio 0 C C C K. T- T COCB ih oesio 0 C C CF. di di di 0 K. T- T Coeibe ih oesio a ie K 0 C C T- T COCB he CB is oeed 0 C C PPK K. T- T COCB he CB is p bak o isse a ie K As o a oie peiods ad isaaeos oesio is assed oe he oeibe pie eahes he a pie ps aed iees i is iposed ha a a he COCB is kied as he oeibe is oeed io oeig eqiy. As he COCB ixed ioe ash os ae ied i ih he isse s abiiy o poe he eessay ds o hoo is opo/ae epaye ad p paye he appopiae diso ae o he COCB i he piig eqaio is he isk ee ae ps a edi spead ha he ake aibes o he isse s ooeibe opoae bods. 00 Ai Boa Yigibasiog

15 Disssio Papes i Fiae: Fooig he oaio o TF[98] e se o deoe he COCB ad o deoe he oeibe bod. The COCB o he ieaioa oeibe bod he oos he piig eqaio a γ ρ ih H 0 o ρ > 0 ee boday 9 hee is he soe spead aboe easy aes ha is eaed b o he sae as he spead o ooeibe opoae deb by he sae isse. TF[98] se he ooeibe deb spead o diso he COCB. This appoah is poe o eo bease opoae deb is eay aays seio o oeibe deb. Ths he add-o o he isk-ee ae ha appied o he COCB hih e a pe spead o a o a bee od a e be aboe he seio opoae bod s edi spead. The esida seiy - is he eqiy opoe o he oeibe bod i.e. ha pa o he oeibe hose payo is oige o he oigia oeibe beig oeed. Fooig TF[98] his shod be disoed a he isk-ee ae as he isse o he ieaioa oeibe bod a aays deie is o eqiy ee he i is ohess. As - aso depeds o ad he piig PDE ha - saisies is a γ ih H 0 o > ee boday ρ ρ 0 0 Reaagig he aboe eads o: 00 Ai Boa Yigibasiog

16 Disssio Papes i Fiae: a ρ γ ρ ρ a γ ρ The e had side o he eqaio aboe is eqa o. Theeoe he ieaioa oeibe bod piig PDE a be eaaged o yied a γ ρ 0 ρ 3.6 Boday ad Iiia odiios 3.6. COCB ase The opig i he eqaios is appae hogh he e hih s be aaiabe a he ie o soig o he e ie sep gie he ae a. Cak-Nihoso ih POR is sed o soe o aboe b sig he peios ae o i.e.. The agoih o is a sepaae iie dieee ih obaied o sig he Cak-Nihoso shee ih he boday/iiia odiios obaied o he soio o he oeibe bod. Tha is he boday/iiia odiios o ae gie by: Iiia odiios: T T T 0 i he bod as oeed pio o o a aiy axppt AIT Fae Vae AIT axppt opo FV opo i.e. TTT ax[ppfv] CopoI > axpp FV Boday odiios: ax 0 hee ees o he ie sep ees o he asse sep ad o he iees ae sep. The oped boday odiios ha ae sed o he gid ae: 00 Ai Boa Yigibasiog

17 Disssio Papes i Fiae: PP AI i PP AI 0 i CP AI 0 i AI s AI 3 These ae obied ih he oeibe bod iiia/boday odiios o beo CB ase Iiia odiio: TTT axfv CopoT PPT CopoT icr. TCPT Copo Boday odiio: ax CR. ax o o - aabe ax 0 o aabe o o - aabe s ax 0 o aabe 0 eeyhee o ax We aso ipose he oios ppe ad oe osais: ax[pp AI Mi[CP Ai CR. AI ]] 4 The soio poeeds as oos: is T- is soed o sig CN ih T ad T ko o iiia odiios. The soe o T- sig he boday odiios o T- hih depeds o T- o aboe. I his ay ieae beee ad ad soe i 0 is obaied. 4. Neia Mehods Fo paaboi PDE s he ie ad spae dyais do o eoe joiy. Tha is hee ae o ixed ie ad so-aed spae eqiy ad iees ae paia deiaies i he PDE. We a heeoe soe he oeibe bod pobe i o disi seps: is e os a gid oespodig o he spae diseizaio; he e poeed o he seod sep ia a 00 Ai Boa Yigibasiog

18 Disssio Papes i Fiae: bakad ie-seppig poede hee eay a p disee diideds ad ay ohe ie-depede jp odiios ae appied i he sessie oe-eaxaio oie. This sas ih he as ie-sep ae o he oeibe bod as a iiia ieae ad oies he ieaio oads a e soio i.e. he peios ie sep ae. Foay e hae he eaioship D DU ih DU D hee: D a γ ρ ρ The ie eqiy ad iees ae gid pois ae idexed espeiey as ad ih oespodig sep sizes T N ax / / / M o N M ad akig ax aes i he se o posiie ieges. The oaio he oespods o he ae o he oeibe a he gid pois.... I he ooig i is ephasised ha he ie idex ees o aeda ie o iese ie 9. The Cak-Nihoso o-sep poede is he as as a se o o odiay dieeia eqaios: d ˆ D ˆ U D U d d hee ˆ DU d i j i j i j i j Dˆ U i j i j 5 Hee ˆ ees o he spae diseizaio aix epeseig osy seod ode D U ; i j aae ad diseizaio o he paia dieeias. The i ad j idies ee o he ag ad ead es i he spae diseizaio ad ae ypiay i he age ± ad ± 9 By iese ie e ee o ie eoig bakads aay o he aiy o he oeibe. 00 Ai Boa Yigibasiog

19 Disssio Papes i Fiae: depedig o hehe eed i he ee o he gid o o-eed i he bodaies shees ae epoyed. The eea aix o o he pobe is: Dˆ U - Dˆ U 6 As he ia payo o he oeibe ad he COCB is ko he pobe osiss i soig he eqaio aboe bakad i ie o obai oday s aes sbje o boday odiios o he oped PDE s disssed beoe. Taio pois o he gid ae aso hose ih he ie o deii he gid o hose egios ha ae o iees. esibe hoies ae ax a 3 sadad deiaios o is expeed e ae ypiay i he agide o 00% o iiia ae o a ie yea oeibe bod i 0 5% ad i 0. ax 0- The soio o he aix eqaios ad he deyig PDE depeds ey h o he hoie o he spae diseizaio aix Dˆ aboe. High ode aay o he iie dieee appoxiaios depeds o he hoie o shee a he bodaies ad he ieio o he gid. I he appedix e oside he diseized PDE i aios egios o he gid ad epo o he aay o he shees. Noe ha i is saighoad o ipose ha he edi... depede o he oaio i he gid. spead is soe io The ee o egaie oeaio beee edi speads ad eqiy a he be iopoaed. 00 Ai Boa Yigibasiog

20 Disssio Papes i Fiae: Cosio This pape has peseed a opehesie heoeia aeok o piig oeibe bods i he pesee o iees ae eqiy edi ad F isk i a o ao ode iopoaig os o he oaa eaes. We hae sho ha F isk o ieaioa oeibe bods does o iode a exa ao i he PDE sed o pie i. Fheoe e popose a poede o bak o he pie o aia opios ad hee he oa oaiiy o he oeig eqiy i doesi ey. The ode e popose aos o oeibe edi isk by exedig he TF[98] ode o iees ae isk. ggesios ae ade o ake he eae o edi isk oe obs i paia o ape he egaie oeaio beee eqiy ad edi speads by iposig a ioa depedee o he spead i he aios egios i.e. bodaies o he gid. To his ed ad o poide diseizaio aay ess e pese he opee diseizaio shee ha haaeises he gid i a egios. 00 Ai Boa Yigibasiog

21 Disssio Papes i Fiae: Reeees ABR[97]: Adese.B.G ad R. Boheo-Raie. The eqiy opio oaiiy sie: a ipii iie-dieee appoah Joa o Copaioa Fiae Wie 997/ Bh[80]: Bea M. ad E. haz. Aayzig oeibe bods Joa o Fiaia ad Qaiaie Aaysis Bh[77]: Bea M. ad E. haz. Coeibe Bods: Vaaio ad Opia aegies o Ca ad Coesio Joa o Fiae C[98]: Caaos J.P. Coeibe eiies MGa-Hi 998. D[94]: Dpie B. Piig ih a sie Risk Jy D[99]: Dais M. ad F.R. ishka. Coeibe bods ih ake isk ad edi isk Tokyo-Misbishi Ieaioa P 999. HW[90]: H J. ad A. Whie Piig iees-ae deiaie seiies Reie o Fiaia dies CIR[85]: Cox J.C. Igeso J.E. ad.a Ross. A Theoy o he Te e o Iees Raes Eooeia HNB[0]: Hoogad J. Nea D. ad D. Boh. Coeig he es okig pape Jy 00. Ø[00]: Økseda B. ohasi Dieeia Eqaios pige 000 R[99] Reboao R. Voaiiy ad Coeaio Wiey 999. TF[98] Tsieiois K. ad C. Feades. Vaig Coeibe Bods ih Cedi Risk The Joa o Fixed Ioe epebe TR[00] Taea D. ad C. Rada. Piig Fiaia Ises: The Fiie Dieee Mehod Joh Wiey & os 000. WDH[93] Wio P. J. Deye ad. Hoiso Opio Piig. Oxod Uiesiy Pess 993. WK[97] W. ad Y-K Kok A Fo-ixig Fiie Dieee Mehod o he Vaaio o Aeia Opios Joa o Fiaia Egieeig Z[99] Zh Y-. ad Y. The sigaiy-sepaaig ehod o o-ao oeibe bods Joa o Copaioa Fiae 3 Fa Ai Boa Yigibasiog

22 Disssio Papes i Fiae: Ai Boa Yigibasiog 0 Appedix o he Diseizaio hees i Fo 0 0 hih oespods o 0 0 he paia dieeia eqaio beoes The spaia diseizaio is he gie by τ d d This shee possesses seod ode aay i boh ad see TR[00] ii Fo 0 - hih oespods o 0 ad 0 he paia dieeia eqaio edes o: a The spaia diseizaio is he gie by a a d d τ This is seod ode aae i boh ad. iii Fo 0 0 ax 0 he paia dieeia eqaio sipiies o: a The spaia diseizaio is he gie by a a d d τ hih is seod ode aae i ad.

23 Disssio Papes i Fiae: Ai Boa Yigibasiog i Fo M he paia dieeia eqaio a be ie as γ γ ρ hee ees o he oa oaiiy o oeig eqiy i doesi ey obaied eiay o he iegaio add e o seio. The spaia diseizaio is he gie by d d γ γ τ hih is seod ode aae i ad. Fo M he paia dieeia eqaio a be ie as: a γ ρ The spaia diseizaio is he gie by:

24 Disssio Papes i Fiae: Ai Boa Yigibasiog a a d d 4 4 γ γ ρ ρ τ hih is seod ode aae i ad. i Fo M- 0 ax 0 he paia dieeia eqaio a be ie as: a γ ρ The spaia diseizaio is he gie by:

25 Disssio Papes i Fiae: Ai Boa Yigibasiog 3 a a d d γ γ ρ ρ τ This is seod ode aae i ad is ode aae i. ii Fo M 0 oespodig o ax 0 he paia dieeia eqaio a be ie as: 0 e assed ha a as ax γ The spaia diseizaio is he gie by: M0 M M M0 M M M0 M -0 M -0 M0 M M -0 M -0 M0 M M0 M0 M γ 4 3 γ τ d d hih is seod ode aae i ad

26 Disssio Papes i Fiae: iii Fo M - ax 0 0 he paia dieeia eqaio sipiies o γ a as 0 a he ax boday The spaia diseizaio is he gie by: d M d γ a M M 3 M 4 M - M M M M - M a M - This is seod ode aae i ad M M γ M - M M 3 M 4 M - M M - ix Fo M ax 0 ax 0 he paia dieeia eqaio beoes: γ a ad he is ode diseizaio is: d M d γ a M M 3 M 4 M - 3 M 4 M - M M M - M M - M - M - a γ M M M - 3 M 4 M - 3 M 4 M - M - M - M - hih is seod ode aae i ad is ode aae i 00 Ai Boa Yigibasiog

27 Disssio Papes i Fiae: Appedix : Coeibe bods by ao osadig i he U: Coeibe bods by iiia pei i he U: 00 Ai Boa Yigibasiog

28 Disssio Papes i Fiae: Coeibe by aiy dae i he U: soe: Coebod.o diisio o Moga aey & Co. Iopoaed opyigh Ai Boa Yigibasiog

29 Disssio Papes i Fiae: Appedix 3- e shee o oeibe ih expii F sesiiiy Misbishi Bak eseabe adaoy exhageabe oeibe seiy 0 Fixed Issae Tes Isse Dae: ep-95 Isse ize: U$ biio ead Maage: Moga aey Copo: 3% Copo Feq./daes: ei No30/ May3 Maiy: No. 00 Coesio Peiod: Ap.96- No.0 Eay Madaoy Coesio Begiig Dae: No Aay Oads: No. 30 Peoae Cieia: AD> U$6.5 Rese Feaes: s Rese Dae: May Aa Rese Dae heeae: No. 30 Fia Rese Dae: No Rese Caaio peiod: 0-day peiod ex. hos. Japa NY odo -5 days Caaio ype: sipe aeage F Ye/U$ behak: p NY Fed ae Oigia Coesio Pie: U$ Iiia ese oo%: 65% Iiia ese oo U$: U$3.6 Fia ese oo %: 50% Fia ese oo U$: U$ This e shee a be od i C[98]. 00 Ai Boa Yigibasiog