IMES DISCUSSION PAPER SERIES

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1 IMES DISCUSSIO ER SERIES Deg d Emo of udr Term Sruure Model wh Mxure of orml Druo Kero Kuh Duo er o -E-8 ISTITUTE FOR MOETR D ECOOMIC STUDIES BK OF J -- IHOBSHI-HOGOKUCHO CHUO-KU TOKO J ou dowlod h d oher er he IMES We e: h://wwwmeojorj Do o rer or rerodue whou ermo

2 OTE: IMES Duo er Sere ruled order o mule duo d omme Vew exreed Duo er Sere re hoe of uhor d do o eerly refle hoe of he B of J or he Iue for Moery d Eoom Sude

3 IMES Duo er Sere -E-8 Jue Deg d Emo of udr Term Sruure Model wh Mxure of orml Druo Kero Kuh* r To ee yeld o-egve qudr Gu erm ruure model GTM he hor re rereeed y he qudr form of he Gu e vrle The GTM mog he mo rve dde ool for lyzg yeld urve for oure wh low ere re However he model ulely o ure he f-led feure of hge yeld oerved ul od mre Th udy exed he GTM y rodug e vrle whoe fuure druo follow mxure of orml druo Th exeo llow our model o ommode v hge o-egve yeld llurve emrl udy we led our model o Jee goverme od JGB yeld ug he ueed Klm fler We he ued he rmeer oed o vege mre vew o JGB ere re y mulg fuure ere re roly druo uder he hyl meure d y deomog ere re o exeed hor re d erm rem Keyword: ffe erm ruure model; qudr Gu erm ruure model; mxure of orml druo; ueed Klm fler; mxmum lelhood mehod JEL lfo: C3 E43 G *Deuy Dreor d Eoom Iue for Moery d Eoom Sude B of J E-ml: erouuh@ojorj The uhor would le o h Do H Km oe Uvery r of he Ooer JFEE Worho d he Deemer ohm Worho o Mheml Fe d he ff of he B of J for her ueful omme Vew exreed h er re hoe of he uhor d do o eerly refle he offl vew of he B of J

4 Iroduo I emrl ude of he erm ruure of ere re he ffe Gu erm ruure model GTM oe of he ffe erm ruure model orglly rodued y Duffe d K [997] he model mo ommoly led due o he ee of emo offer I he GTM he hor re rereeed he ffe fuo of e vrle followg Gu roee Togeher wh he ove eg he o-rrge odo led o yeld h re lo rereeed he ffe fuo of hee Gu e vrle Model rmeer d e vrle re redly emed e h model h Gu ffe feure The mxmum lelhood mehod wh he Klm fler ofe led o eme he GTM Th mehod lylly ree d doe o requre roxmo Th oe lely reo why my emrl ude of yeld urve re ed o he GTM The GTM h er drw The e-ow h o eure he o-egvy of heorel yeld Th ugge he GTM would e hrd o ue o urely lyze yeld urve for oure wh low ere re We reor o ffe erm ruure model wh qure roo dffuo roe o rumve h rolem However oed ou y D d Sgleo [] he qure roo ffe model moe o-egve orrelo ewee e vrle o geere well-defed od re Th or rerve d me he model le flexle for lo o ul d h Dmr d Gll [] d Leold d Wu [3] develoed he qudr Gu erm ruure model GTM o ddre he drw of he GTM I he GTM he hor re rereeed y he qudr form of he e vrle h follow Gu roee The uhor of he reedg wo er derved lyl rg formul for od uder he o-rrge odo ed o h eg Severl emrl ude ed o he GTM hve ered reely Km d Sgleo [] d yholm d Vdov-Kolev [] If we em o eme he GTM he Klm fler vld e heorel yeld re he oler fuo of e vrle he model Ied oler flerg mehod uh he exeded Klm fler d he ueed Klm fler develoed y Juler d Uhlm [997] re ued o eme he model rmeer d e vrle of

5 he model The GTM reul o-egve yeld dvge over he GTM However he GTM ulely o dequely ure he f-led feure of hge yeld Th feure ofe oerved od d ere re mre I f mohly hge Jee goverme od JGB yeld exh he f-led feure how orml - lo Fgure Fgure : orml - lo of mohly hge -yer JGB yeld 4 orml qule d orml heorel qule oe: I h lo mohly hge -yer JGB yeld re drdzed o me zero d u vre for he erod from Jury 996 o Deemer The rgh le wh 45 degree loe he ove fgure de he d follow orml druo Th udy exed he GTM o exlly orore h f-led feure o he erm ruure model I our model he hor re rereeed y he qudr form of he e vrle whoe fuure odol druo follow mxure of orml druo We ll h model he qudr mxure of Gu erm ruure model MGTM Uder he o-rrge odo we derve he od rg formul ed o he MGTM ug log-ler roxmo The model geere o-egve yeld d exh he f-led feure of hge yeld Our model rovde more ure yeld eme from ul d Moreover eue le u ure roly druo for fuure ere re uder he hyl roly meure our model le u lyze mre vew o ere re We led h model o vege mre vew o JGB yeld mulg fuure ere re roly druo uder he hyl meure d deomog

6 ere re o exeed fuure hor re d erm rem The remder of h er orgzed follow Seo dere our model he MGTM well he GTM d he GTM Seo 3 dere he emo mehodology Seo 4 ree erforme omro of he GTM GTM d MGTM d he reul of emrl udy for JGB yeld Seo 5 ree our oluo Term Sruure Model Th eo dere he ruure of he GTM GTM d our model he MGTM I lo dere deled eg for rmeer he dree me frmewor for eh model ffe Gu Term Sruure Model I he GTM he hor re rereeed he ffe fuo of e vrle followg Gu roee To exr mre vew from oerved yeld ovee o reree e vrle roee uder oh he r eurl meure d he hyl meure follow: I ~ d I ~ d where deoe veor of e vrle me d I he orml druo wh zero me d ovre mrx I he deoe ho me uder y ho of r d he ho : u mrx We d ho me uder y The hve he followg reloh meded y mre re 3 Throughou h er Duffee [] we ume h of veor of e vrle : ler fuo 4 3

7 Th he eelly ffe umo From equo 4 we fd h I he GTM he hor re r rereeed he ffe fuo of how elow: r 5 where lr d veor I ddo reree he roe of Herefer we ume h deoe he roe of veor or mrx Uder he o-rrge odo ddo o he ove eg he followg relo mu hold ewee zero ouo od re d zero ouo od re wh erod mury me wh erod mury me + : E [ e r ] 6 where E [] deoe he odol exeo oeror uder meure odoed o formo oed me The oudry odo of equo 6 Tg o odero h oudry odo we derve he followg lyl od rg formul 7 from equo 5 d 6 ex 7 Therefore he heorel yeld fuo of he GTM: y wh erod mury wre he ffe y 8 mled y he form of he yeld equo 8 yeld uder he GTM re lely o e egve Whe we em eme ed o ul oerved yeld we erre he 4

8 oerved yeld zero me Th y~ he um of he heorel yeld y d Gu oe wh ~ y y 9 We wll roeed o efy deled rmeer eg for he GTM requred for eme of model rmeer The umer of for for ll model lyzed h er umed o e hree Ug he vr rformo dued y D d Sgleo [] we e everl rmeer for he GTM whou lo of geerly follow: Furhermore he rmeer for he mre re of r d equo 4 re umed o e hree-dmeol full veor or full mrx: udr Gu Term Sruure Model oed ou ove yeld oed from he GTM e egve vlue To ddre h drw h Dmr d Gll [] d Leold d Wu [3] develoed he GTM We ume h he oh roe of he veor of e vrle of he me form equo d We lo ume h he mre re of r of he me equo 4 Th he eelly ffe mre re of r I h model he hor re r rereeed y he followg qudr form of : r 5

9 6 To eure he o-egvy of he hor re he mrx equo umed o e ove em-defe Uder he o-rrge odo we derve he od rg formul from equo 6 d follow: where 5 5log 5log ex 3 Theorel yeld y wh erod mury uder he GTM wre he qudr form of ug d defed equo 3 follow: y 4 mled y he form of he yeld equo 4 he yeld mu e o-egve We wll efy deled rmeer eg for he GTM requred y he eme Ug vr rformo for he GTM dued y Leold d Wu [3] we e everl rmeer whou lo of geerly follow: Bed o he gulr vlue deomoo of he ove em-defe mrx we reree equo he rodu of er mre: d d d SDS d d d T 6 where T S deoe he roe of mrx S For he rmeer of mre re of r he GTM we ume h he eg

10 7 he me equo 3 Our Model: udr Mxure of Gu Term Sruure Model I h ueo we rooe ew erm ruure model exeo of he GTM Whle he GTM eure o-egve yeld he model my o ure exremely he lrge hge yeld frequely oerved od mre Thu we model he oh roe of veor of e vrle o Gu roe u roe whoe mrgl dee follow mxure of orml druo More eflly he roe of veor of e vrle defed ug ew oh roe uder he hyl meure follow: re ~ d I I I d 7 We deoe he formo e oed me y I Equo 7 mle h oe-erod odol druo of fuure e vrle me uder follow mxure of orml druo Th model le u orore he f-led feure of yeld o erm ruure model Leme [6] ume he me eg for dym equo 7 However he hor re here umed o e ffe fuo; or we e r he qudr fuo of We ume h he mre re of r of our model lo follow he eelly ffe form of equo 4 wh he GTM d he GTM Equo 3 4 d 7 gve he dym of veor of e vrle uder he r eurl meure follow: ~ I d 8 I ddo we ume h he odol roly for uder me follow equo 9: I I I I 9

11 We lo ume h he hor re h he me form equo he GTM Th umo llow he hor re o rem o-egve Uder he er umo ded ove d he o-rrge odo we ee o derve he zero ouo od rg formul However we mu rely o roxmo due o he dffuly of dervg he loed-form oluo wh h frmewor Thu we ue he log-ler roxmo ued y Bl d Zhou [] who uded he ffe erm ruure model regme whg Fr we ume h zero ouo od re e dered ex equo 6 we o he followg equo Uder h umo d he o-rrge odo ex ex E [ex r ex E [ex ] ] ex whe we ry o derve he reurve reloh of oeffe from equo we uue equo 8 o he rgh hd de of equo Here we ue log-ler roxmo edx gve he reurve oluo of oeffe 8 d ex x x o olve equo d ed o h roxmo he mury of yeld loger h degrde he reo of he roxmo everhele e yeld emed from our model demore good f wh oervo yeld wh mure of x moh d wo fve d yer ded Seo 4 we ee h uffe for our model o exr formo luded he oerved yeld For h reo our ue of he roxmo formul vld for lye of hor re d -yer yeld over he ex x moh d erm rem of wo d -yer yeld Seo 4 Herefer we refer o our model he qudr mxure of Gu erm ruure model MGTM We wll ow roeed o efy he del of he MGTM o eme model rmeer d e vrle Fr he rmeer equo 7 re e ug he followg vr We omred yeld ed o our model roxmo formul o hoe gve y Moe Crlo mulo ug e vrle d rmeer emed from JGB zero ouo yeld ded edx Error ro verged durg he mle erod were 4% for x-moh yeld 5% for wo-yer yeld 74% for fve-yer yeld 3% for -yer yeld d 55% for -yer yeld

12 rformo rooed y Leold d Wu []: The vr rformo ele d equo 8 o ur o he form of equo oe h geerlly rereeed rgulr mrx ex we ume h he mxure roly of he e vrle ro equo 7 fxed vlue loe o o free rmeer Th eue delg wh free rmeer delze model rmeer eme Seflly we e 95 fxed vlue loe o Th llow u o ure he f-led feure of hge yeld wh low roly of ourree 3 Bed o h Seo 4 we omre MGTM erforme o GTM d GTM erforme Flly for he rmeer of mre re of r he MGTM we ume h he eg he me for equo Seo 4 le hee model o ul d d omre he erforme of our model o he wo oher model We wll lo oder eme of rmeer d e vrle of he model from he ul d 3 Emo Mehodology well-ow whe he erm ruure model dered e e model wh ffe Gu ruure he mxmum lelhood emo wh Klm fler ueful for muleou emo of le e vrle d model rmeer However oe he e e model o-gu or oler we I he MGTM eg he e whh d he oher e whh or oh reree he GTM Thee wo e re dguhle he emo If he GTM ure ul d urely we ofro he rolem of dguhg ewee model rmeer he MGTM 3 We lo emed he model rmeer of he MGTM whe 9 de from he e whh 95 The reul wh 9 d 95 were er-del 9

13 o ue he mxmum lelhood emo wh Klm fler We mu rely ed o oler flerg mehod oler flerg mehod requred for eme of he GTM d MGTM e hee wo model re o ffe I h udy we ue he ueed Klm fler develoed y Juler d Uhlm [997] for emo of he GTM d MGTM The hree model dered he revou eo re ll regrded e e model Se ro equo for he GTM d he GTM re wre he followg geerl form: ~ ~ ~ d f I where f deoe ome ffe fuo of The e ro equo for he MGTM erly del o equo However deed o he rdom e vrle whh e he vlue of wh roly or wh roly rereeed follow: ~ f I The e dym of he MGTM ~ I ~ ~ d I d The oervo equo for he GTM GTM d MGTM re wre he followg ommo form: where ~ 3 ~ G ~ d I 4 veor wh eleme of ome yeld oerved od mre or omued from od re; G veor wh eleme of heorel yeld orreodg o oervol yeld; d oervol error veor h follow mulvre orml druo Here we oe h he form of he ove fuo G vre deedg o he hoe of model For he GTM G he ffe fuo of For he GTM d he MGTM G he oler fuo of We wll du he ueed Klm fler lgorhm g he MGTM exmle The fr e of he lgorhm lzo:

14 ] [ : ] [ : T E E 5 I h lzo e we omue he uodol exeo ] [ E d vre of e vrle ex we lule gm o We deoe he e dmeo y For T where T he umer of mle erod we lule o lled gm o: ] [ 6 where mrx d reree he qure roo mrx of For exmle he qure roo mrx omued ug Choley deomoo We defe of equo 6 follow: : where lled he lg rmeer We defe mlrly For he emo reeed h er we ume h d The ex e he me ude of e vrle I he me ude we rform gm o equo 6 hrough equo 3 ] [ ] [ * * f f f f f f 7 Ug eleme of he mre equo 7 we lule he me d ovre of he e vrle elow * * * * * * T T T T m m W W W W 8 where wegh re ged follow:

15 W W W W W m m m 9 where ove oegve weghg rmeer rodued o dju he weghg of he fr gm o for he ovre Th rmeer e ued o orore owledge of hgher order druo mome I h er we e We defe gm o of he reded e vrle follow: ] [ 3 ex we rform hoe gm o o reded gm o of he yeld ] [ m W G G G 3 Flly we o he followg meureme ude equo: I W T 3 T W 33 K 34 K 35 T K K 36 The ueed Klm fler roxme he druo of e vrle relvely urely e ue more formo hrough he gm o ded ove Some revou ude hve vlded he erforme of he ueed Klm fler emrl ude eg Chroffere e l [8] Through he ueed Klm fler we omue he odol exeo d ovre mrx of he yeld veor me : G odoed o oervo Ug he odol exeo d ovre mrx of we

16 he ly he qu-mxmum lelhood mehod o rmeer emo The log lelhood fuo L omued follow: T ~ T ~ L log ~ where T deoe he umer of oervo d veor o of oerved yeld me We hooe he model rmeer o mxmze he ove log lelhood of he d ere 37 4 Emo Reul I h eo we ly he GTM GTM d he MGTM o Jee goverme od JGB yeld d d du he reul of he emo We fr omre how dffere model f he d he mule he roly druo of fuure ere re uder he hyl meure ug emed le vrle d model rmeer I rulr we exme he exeo well erele o of he fuure ere re for he JGB ere re whh ur re luled from he muled druo Flly we vege he develome of he JGB yeld y deomog hem o exeed fuure hor re d erm rem for he mle erod 4 D d Model F We ue mohly d from Jury 996 hrough Deemer for JGB zero ouo yeld Th d omued from he Broer Broer JGB re y he mehod reeed MCulloh [975] The mure luded re x moh d wo fve d yer rmeer d e vrle emo for he GTM GTM d he MGTM re mlemeed ed o he ueed Klm fler wh he qu-mxmum lelhood mehod efed he revou eo lhough he mxmum lelhood mehod eme wh he Klm fler wh for he GTM e heorelly ole we lo ly he ueed Klm fler o he GTM o omre he hree model Tle B d C edx dly he emed rmeer for eh model Fgure edx gve he me ere of he emed e vrle for 3

17 eh model We omre he f o he d for he hree model Tle ree omrve of f for he hree model We omre model o ju hrough lelhood u wo formo rer Thee formo rer re deermed ed o he wy whh he log-lelhood elzed y he umer of model rmeer Tle de h he MGTM y y rero gve he e erforme of he hree model: Tle : Comrve for f GTM GTM MGTM log L IC BIC oe: L lelhood IC e formo rero BIC Bye formo rero ex we ee he reul of model fg for eh mury Tle ree o redo error ewee oerved yeld d reded yeld for eh mury Comrg model f for he hree model we fd h he me qured error MSE of he MGTM for eh mury he lowe of he hree model Th rule o low drd devo of he redo error of he MGTM for eh mury omred o hoe of he wo oher model The mxmum vlue d he olue vlue of he mmum vlue of he MGTM redo error re lo gfly lower h for he wo oher model Tle : redo error Sx-moh Two-yer GTM GTM MGTM GTM GTM MGTM Me Me Sdev Sdev 7 3 Mx Mx M M MSE MSE

18 Fve-yer -yer GTM GTM MGTM GTM GTM MGTM Me Me Sdev Sdev Mx Mx M M MSE MSE yer GTM GTM MGTM Me Sdev Mx M MSE oe: Sdev he drd devo of he redo error MSE he me qured error U exludg MSE re o The u for MSE qure of ere 4 roly Druo for Fuure Iere Re uder he hyl Meure Th ueo due mulo of roly druo for fuure ere re uder he hyl meure ed o emed e vrle d model rmeer The GTM d MGTM re more lely o ure fuure ere re urely h he GTM eue hey eure he o-egvy of yeld Moreover he GTM lerly uderle for h ly e druo for he fuure ere re lwy follow he orml druo he GTM d drd devo lo rem o for he whole erod rereeed y he d mle Hee we exlude he GTM from our ly d ddre oly he GTM d he MGTM o mule roly druo of fuure ere re Smulo of roly druo for ere re over he ex x moh uder he hyl meure re mlemeed ed o eme of model rmeer d flered e vrle ] The mulo roe dered follow E [ 5

19 Fr we mule he fuure ho of 6 me Here he umer of mulo e o 3 The ex e for he GTM o omue 6 from he muled ho 6 ed o equo I or for he MGTM he ex e o omue 6 from he muled ho 6 d he muled vrle 6 ug equo 7 Oe we o he muled le vrle 6 we omue muled yeld over he ex x moh ed o he zero ouo od rg formul equo 3 for he GTM d equo for he MGTM Fgure : Druo for -yer yeld over he ueque x moh 5 of Deemer Frequey 5 MGTM GTM ere Fgure dly roly druo uder for -yer yeld over he ex x moh uder he GTM d he MGTM evlued o Deemer hogrm Comug he uro for eh druo we fd h he GTM druo uro equl o 3 whle h of MGTM equl o 59 Th mle h he roly druo uder he MGTM h fer l h h of he GTM Th oe wh our redo h he MGTM ure f-led feure of hge yeld Fgure 3 dly he me ere o ju of exeo u d 99h erele o of roly druo uder he hyl meure for oe-moh yeld over he ex x moh ed o wo model We fou o he re he exeed oe-moh yeld ummer 3 uder he GTM Durg h erod roe for hor-erm ere re he er fuure dd o lmo hge Th ree 6

20 exeo he GTM er rge lgh of oemorry mre odo I or he exeo oed ed o he MGTM h me er-zero Th er eer h he GTM reul Loog he MGTM 99h erele o from he ler hlf of o he fr hlf of 5 whh ou for mo of he erod durg he me of quve eg we fd h remed erly edy Th me he oel r umed y mre r for he hor erm ere re over he ex x moh dd o hge durg h erod Fgure 3: Exeo d erele o of druo for oe-moh yeld over he ueque x moh 5 GTM ere h erele o exeed hor re erele o 5 MGTM ere h erele o exeed hor re erele o 7

21 43 Exeed Shor Re d Term remum Th ueo due he modfed vero of he exeo hyohe whh orore he me-vryg erm remum Uder h hyohe he medum o log-erm ere re re deomoed o he verge of exeed fuure hor re uder he hyl meure d he erm remum We deomoe wo- d -yer yeld o he exeed hor re d he erm rem uder h hyohe for he MGTM d he GTM d vege he develome of JGB yeld d he deomoed eleme Fgure 4: Two-yer yeld deomoo GTM 5 ere Two er eld Exeed Shor Re Term remum MGTM 5 ere Two er eld Exeed Shor Re Term remum Fgure 4 how he deomoo of wo-yer yeld Oe feure of Fgure 4 8

22 h he develome of he exeed fuure hor re d he erm rem dffer gfly ewee he GTM d he MGTM from he ler hlf of 5 u o he ler hlf of 7 Boh he exeed fuure hor re d he erm remum of he MGTM roe from July 5 o My 6 I or he exeed fuure hor re of he GTM roe durg h erod whle he erm remum of he GTM remed er-zero From Jue 6 ul July 7 he exeed fuure hor re of he MGTM roe whle h of he GTM moved wh rrow rge I rulr he exeed fuure hor re for he GTM July 7 w erly he me level he eme Mrh 6 whe he quve eg oly eded Th er rge e he B of J red he oly re we durg h erod; my mre r eleved he B of J would re he oly re e rllelg eoom reovery over he ex oe or wo yer Deomoo ug he MGTM would herefore er more vld h he GTM Regrdg he erm remum we erre he develome of he erm remum from July 5 o July 7 follow: Fr he ree he MGTM erm remum from July 5 o My 6 ould e rule o he ree uere oerg he develome of he hor erm ere re fer he ed of quve eg The deree he erm remum from Jue 6 o July 7 my mly h uere ou he develome of he hor erm ere re f deled fer he ed of he quve eg d zero ere re ole Fgure 5 how deomoo of -yer yeld uder he modfed vero of exeo hyohe for he GTM d he MGTM I ummer 3 medum- o log-erm ere re roe hrly reul of oe-de ellg y he whh rehed he erl r lm erm of vlue r VR due o he ree he JGB volly Th eode lled he VR ho Fgure 5 how h -yer yeld hge efore d fer he VR ho w ued y he develome of erm remum uder oh wo model I hoe dy eoom odo howed lle hge; herefore he ove meoed develome of erm remum er oe wh he rely The develome he fr hlf of 9 Fgure 5 o o ey dfferee ewee he GTM d he MGTM I he GTM he exeed fuure hor re d 9

23 he erm remum yed erly o I or he MGTM he exeed fuure hor re roe whle he erm remum deled ordg o B of J [9] he hr deeroro eoom odo led dowwrd reure whle he uly d demd odo of JGB refleg he growg fl def led uwrd reure o JGB log erm ere re The deree he exeed fuure hor re he MGTM e exled y he hr deeroro eoom odo The ree he erm remum he MGTM my refle he mle ewee uly d demd odo of JGB Fgure 5: -yer yeld deomoo 4 GTM 3 ere er eld Exeed Shor Re Term remum 4 MGTM 3 ere er eld Exeed Shor Re Term remum

24 5 Coluo Our er rooe ew erm ruure model h llow ere re o rem o-egve d ure he f-led feure of hge ere re I our model he hor re formuled he qudr form of he e vrle whoe fuure roly druo follow mxure of orml druo uder oh r eurl d hyl meure dree me eg I h frmewor we derved he roxmed rg formul for zero ouo od re y log-ler roxmo We he emed e vrle d model rmeer ug he ueed Klm fler jo wh he mxmum lelhood mehod Our emo reul howed he MGTM h he gree lelhood I lo rovded he e IC d BIC of ll hree model The MGTM lo oued for he mlle me qured error for eh oerved mury The eme of he le e vrle d model rmeer llow u o omue deomoo of JGB ere re o exeed fuure hor re d erm rem d o mule roly druo of fuure ere re uder he hyl meure Our model er o exl ul red eer h oher model For exmle he deomoo of wo-yer yeld ed o our model howed oully reg exeed hor re from Jue 6 o July 7 The exeed hor re uder he GTM dd o hge durg h me Durg h erod he B of J red he oly re we Our model exeed hor re hroughou h erod er le o refle ul mre vew Severl oher exmle dued h er de h our model er o ure ul mre odo more urely h he oher model Wh regrd o he emrl e of JGB ere re emo reul ugge he followg oervo Fr oh he 99h erele o d he exeo of he hor re over x-moh erod remed edy durg mo of he quve eg oly erod from o 6 Th mle he oly duro w hghly effeve lzg mre exeo regrdg he fuure hor re Seod muh of he develome of he JGB log erm ere re ummer 3 exled y hge erm remum o y hge exeed hor re Thrd he erm remum wh wo yer yeld roe from Jue 5 o My 6 d

25 deled from Jue 6 hrough ummer 7 The re my refle uere regrdg fuure oly re develome fer he ed of he erod of quve eg The dele he erm remum from Jue 6 my e rule o reedg uere regrdg he fuure oly re g he dro of he ul ed of quve eg d zero ere re ole Flly he fr hlf of 9 he exeed hor re of -yer yeld deled whle erm remum reed Th my e eue he hr deeroro eoom odo led dowwrd reure o he exeed hor re whle he deerorg uly d demd odo of JGB led uwrd reure o he erm remum lhough our udy foued o he JGB yeld our model e led o yeld oher oure I rulr our model hould e hghly effeve for exmg yeld oure wh low ere re The Ued Se d Euroe oure hve med exory moery oly e he emergee of he fl r 7 d ere re hee oure remed low Oe ole dreo for fuure reerh o exlore he effe of uoveol moery ole ree yer ed o our model y mulg roly druo of he fuure ere re uder he hyl meure or y deomog he ere re o he exeed hor re d erm remum

26 Referee h Dog-Hyu Roer F Dmr d Rold Gll udr Term Sruure Model: Theory d Evdee Revew of Fl Sude B of J Fl Mre Reor Develome he fr hlf of 9 BOJ Reor d Reerh er 9 Bl Rv d Ho Zhou Term Sruure of Iere Re wh Regme Shf Jourl of Fe Chroffere eer F Kr Jo Lof Krou d Krm Mmou oler Flerg ffe Term Sruure Model: Evdee from he Term Sruure of Sw Re worg er 8 D g d Keeh J Sgleo Sefo ly of ffe Term Sruure Model Jourl of Fe Duffee Gregory Term rem d Iere Re Fore ffe Model Jourl of Fe Duffe Drell d Ru K eld For Model of Iere Re Mheml Fe Juler J Smo d Jeffrey K Uhlm ew Exeo of he Klm Fler o oler Syem The h Ierol Symoum o eroe/defee Seg Smulo d Corol 997 Km Do H d Keeh J Sgleo Term Sruure Model wh Zero Boud: Emrl Ivego of Jee eld worg er Leold Mru d Lure Wu Deg d Emo of udr Term Sruure Model Euroe Fe Revew Leme Wolfgg Term Sruure Modelg d Emo Se Se Frmewor Leure oe Eoom d Mheml Syem 565 Srger 6 MCulloh J Huo The Tx-djued eld Curve Jourl of Fe 33 3

27 yholm Ke d Ro Vdov-Kolev elo Segel ffe d udr eld Curve Sefo Whh Oe Beer Foreg? Euroe Cerl B Worg er 5 4

28 5 edx We derve he roxmed rg formul for he zero ouo od ed o he MGTM The model rereeed from equo 8 d 9 follow: r - ~ I d - I I -3 Uder he o-rrge odo he followg formul mu hold: ] [ r e E -4 From equo ---4 we w o derve he loed rg formul for ; however o ole o do h Thu we derve he roxmed rg formul for The re for umed o hve he form ex Suug h o equo -4 we o he followg equo: ex ex ex ex ] [ex ex E e E e e E E e E -5 The followg formul helful reolvg he rgh-hd de of equo -5:

29 6 5 5 ex ] [ex K d K d d K E -6 where ~ I Ug equo -6 equo -5 omued follow: ex ex ex e e e -7 Irodug he log-ler roxmo ued Bl d Zhou [] equo -7 we o: log log Hee we o he followg reurve equo for oeffe :

30 7 5 5log 5log 5 5log 5log ex -9

31 edx Here we eg y howg emo reul for model rmeer We he how he emed e vrle for eh model Tle B d C reevely gve he emed rmeer for he GTM GTM d he MGTM : Tle : Emed rmeer of he GTM : : : : oe: Sdrd error re gve rehee : : S : D : Tle B: Emed rmeer of he GTM

32 : : oe: Sdrd error re gve rehee S D Tle C: Emed rmeer of he MGTM oe: Sdrd error re gve rehee Fgure gve he emed e vrle for eh model 9

33 Fgure : Emed Se Vrle ] E [ 3 GTM he fr for he eod for he hrd for GTM he fr for he eod for he hrd for 3 MGTM he fr for he eod for he hrd for 3