A Review of Term Structure Estimation Methods

Transcription

1 A Rvw of Trm Srucur Emaon Mhod Sanay Nawalkha, Ph.D Inbrg School of Managmn Glora M. Soo, Ph.D Unvry of Murca 67 Inroducon Th rm rucur of nr ra, or h TSIR, can b dfnd a h rlaonhp bwn h yld on an nvmn and h rm o maury of h nvmn. Many alrnav a uch a ral a, prva quy, and hdg fund nvmn ar llqud wh long-rm cah flow, whou a radly avalabl ourc for mark prc. Thu, a proprly mad rm rucur of nr ra nal for obanng h nrnc valu of h a. Du o h non-lnar convx rlaonhp bwn a prc and nr ra, any rror n h maon of nr ra n a low-yld nvronmn hav a largr mpac on h nrnc valuaon of h a. Thu, an accura maon of h rm rucur of nr ra aum vn grar mporanc n h currn low-yld nvronmn wh a yld around % on h hor nd, and a % yld on h 0-yar Traury bond. Morovr, h TSIR alo rlvan for macroconomc forca of hor-rm ra, and mplmnaon of monary polcy and db polcy by govrnmn ( Pazz [00]). A nod by Bl [997], h TSIR maon rqur makng hr mporan dcon. Fr, on mu condr h aumpon rlad o ax and lqudy prmum n h prcng funcon ha rla bond prc o nr ra or dcoun facor. Scond, on mu choo a pcfc funconal form o approxma h nr ra or h dcoun facor. Morovr, hrd, on mu choo an mprcal mhod for mang h paramr of h chon funconal form. Th papr focu on how o ma h dfaul-fr rm rucur of nr ra from bond daa ung hr mhod: h boorappng mhod, h McCulloch cubc-pln mhod, and h Nlon and Sgl mhod. Nlon and Sgl mhod hown o b mor robu han h ohr wo mhod. Th la wo mhod can b mplmnd ung h ur-frndly Excl pradh prpard by h auhor.

2 Th rucur of h papr a follow. Fr, w rvw h man concp abou h TSIR, uch a dcoun funcon, bond prc, yld o maury, vral dfnon of nr ra and a dcuon of h hap of h TSIR. Nx, w dcrb hr popular rm rucur maon mhod and pon ou h clu for a propr uag and hr lmaon.. Th Buldng Block: Bond Prc, Spo Ra, and Forward Ra Th TSIR can b xprd rgardng po ra, forward ra, or prc of dcoun bond. Th con how h rlaonhp bwn h concp... Th Dcoun Funcon Undr connuou compoundng, h prc (or prn valu) of a zro-coupon bond wh a fac valu of $00 and a rm o maury of yar can b wrn a: Quarr P d y () ( ) = = 00 = 00 ( ) y () whr y () h connuouly-compoundd ra corrpondng o h maury rm. Th funcon y () dfn h connuouly-compoundd rm rucur bad upon zrocoupon ra. Th xpron rfrrd o a h dcoun y ( ) funcon d. () Th ypcal hap of h dcoun funcon hown n Fgur. Th funcon ar a, nc h currn valu of a $ payabl oday $, and dcra wh ncrang maury du o h m valu of mony. If a r of dfaul-fr zro-coupon bond x for dffrng maur, hn pobl o xrac h rm rucur by mply nvrng quaon () o oban y. () Howvr, du o h lack of lqudy and unavalably of zro-coupon bond for all maur, h rm rucur canno b mply oband by ung zro-coupon bond uch a U.S. Traury STRIPS... Bond Prc and Accrud Inr A coupon bond can b vwd a a porfolo of zro-coupon bond. Ung dcoun funcon gvn abov, h prn valu of ach coupon pad prod from oday gvn by C d( ), whr C h coupon rcvd. Th approach can b ud o calcula h prn valu of all h paymn, coupon and fac valu. Th approach gv u P 0, whch calld h cah prc of a bond, and h prc ha purchar pay whn buyng h Fgur : Th dcoun funcon () bond. Howvr, bond prc ar no quod a cah prc. Th quod prc ar clan prc, whch xclud h accrud nr. Accrud nr h nr accumulad bwn h mo rcn nr paymn and h prn m. If 0 dno h currn m, p dno h da of h prvou coupon paymn, and q dno h da of h nx coupon paymn, hn h formula for accrud nr gvn a: AI C q 0 p = and h bond quod prc qual o h prn valu of h all h paymn mnu h accrud nr. Tha, p Quod Prc = P0 AI Compuaon of accrud nr rqur h day coun ba ud n h mark. Th day coun ba dfn how o maur h numbr of day n a yar and a wll a h numbr of day bwn coupon. No ha no h cah prc, bu h quod prc ha dpnd on h pcfc day coun convnon bng appld. Any ncra (dcra) n h accrud nr du o a pcfc day coun convnon ud xacly off by a corrpondng dcra (ncra) n h quod prc o ha h cah prc rman unchangd. Snc h TSIR compud ung cah prc, alo ndpndn of h day coun convnon ud. Of cour, ncary o know h day coun convnon o oban h cah prc ung h quod prc and h accrud nr... Yld o Maury Th yld o maury gvn a ha dcoun ra ha mak h um of h dcound valu of all fuur cah flow (hr of coupon or prncpal) from h bond qual o h cah prc of h bond, ha : N C F P = + y y N = No ha h yld o maury a complx wghd avrag of zro-coupon ra. Th z and mng of h coupon paymn nflunc h yld o maury, and h ffc calld h coupon ffc. In gnral, h coupon ffc wll mak wo bond wh dncal maur bu wh dffrn coupon ra or paymn frqunc hav dffrn yld o maury f h zro-coupon yld curv non-fla. Th coupon ffc mak h rm rucur of yld on coupon bond lowr (hghr) han h rm rucur of zro-coupon ra, whn h lar lopng upward (downward)..4. Spo Ra, Forward Ra and Fuur Ra Zro-coupon ra a dfnd abov ar po ra bcau hy ar nr ra for mmda nvmn a dffrn maur. Th forward ra bwn h fuur da and h annualzd nr ra ha can b conracually lockd n oday on an nvmn o b mad a m ha maur a m. Th forward ra dffrn from h fuur ra n ha h forward ra known wh crany oday, whl h fuur ra can b known only n fuur. () () (4) 68

3 Condr wo nvmn rag. Th fr ragy rqur makng a rkl nvmn of $ a a fuur da, whch rdmd a fuur da for an amoun qual o: 69 f (, )( ) Th varabl f(, ) whch known oday dfnd a h connuouly-compoundd annualzd forward ra, bwn da and. Now condr a cond nvmn ragy ha rqur horng oday (whch h am a borrowng and mmdaly llng) a $ fac valu rkl zro-coupon bond ha maur a m and nvng h procd from h hor al n a rkl nvmn maurng a m. Th procd of h hor al qual P ( ), h currn prc of $ fac valu rkl zrocoupon bond ha maur a m. Th nvmn co nohng oday, rqur covrng h hor poon a m by payng $, and rcvng h fuur valu of h procd from h hor al. Snc boh rkl nvmn rag rqur $ nvmn a m, and co nohng oday, h valu of h nvmn rag a m mu b dncal. Tha, hy mu offr h am compoundd ra of rurn. Th obrvaon can b ud o calcula h forward ra ha mpld by h rm rucurd obrvd oday. Thrfor, h compoundd forward ra of rurn bwn wo fuur da and gvn by: y ( ) y ( ) f(, ) = y ( ) + Th abov quaon mpl ha f h rm rucur of zrocoupon ra upward (downward) lopng, hn forward ra wll b hghr (lowr) han zro-coupon ra. For a fla rm rucur, zro-coupon ra and forward ra ar dncal and qual o a conan. In gnral, forward ra can b compud for any arbrary nrval lngh, and ach lngh mpl a dffrn rm rucur of forward ra. To avod h ndrmnacy, h rm rucur of forward ra uually dfnd ung nananou forward ra. Inananou forward ra ar oband whn h nrval lngh bcom nfnmally mall. Mahmacally, h nananou forward ra f(), h annualzd ra of rurn lockd-n oday, on mony o b nvd a a fuur m, for an nfnmally mall nrval. Th nananou forward ra can b nrprd a h margnal co of borrowng for an nfnmal prod bgnnng a m. By h am okn, h annualzd m zro coupon ra can b hown o b qual o h avrag of all forward ra bwn now and m : N y () f, + = 0 ( ) Th abov quaon gv a rlaonhp bwn zro-coupon ra and forward ra. I mpl ha h zro-coupon ra for rm an avrag of h nananou forward ra bgnnng (5) (6) (7) Fgur : Bac hap of h rm rucur from rm 0 o rm. Snc avragng rduc volaly, h rlaonhp ugg ha forward ra hould b n gnral mor volal han zro-coupon ra, pcally a h longr nd..5. Th Shap of h Trm Srucur of Inr Ra Emaon of h rm rucur nvolv obanng zro-coupon ra, or forward ra, or dcoun funcon from a of coupon bond prc. Gnrally, h rqur fng a funconal form ha flxbl n capurng ylzd fac rgardng h hap of h rm rucur. Th TSIR ypcally ak four dffrn hap gvn a h normal hap, h p hap, h humpd hap and h nvrd hap. Fgur how h four ypcal hap. Th normal hap ndcav of an conomy ha normally xpandng. Tha, h rm rucur nd o b lopng upward, rflcng h fac ha longr-rm nvmn ar rkr. A hghr rk mpl a hghr rk prmum and hnc, a hghr nr ra. Th p hap of h rm rucur ypcally occur a h rough of a bun cycl, whn afr many nr ra rducon by h cnral bank, h conomy m pod for a rcovry n h fuur. Th nvrd hap of h rm rucur ypcally occur a h pak of a bun cycl, whn afr many nr ra ncra by h cnral bank, h conomc boom or a bubbl may b followd by a rcon or a dpron. Fnally, h humpd hap ypcally occur whn h mark parcpan xpc a hor conomc rcovry followd by anohr rcon o ha hr ar dffrn xpcaon a dffrn rm. I could alo occur whn movng from a normal curv o an nvrd curvd or vc vra. 4 I alo worhy o hghlgh ha whavr h hap, h TSIR nd o b horzonal a long maur. Th raon for h wofold. Fr, alhough nvor can hold dffrn xpcaon abou h fuur of nr ra for h hor, mdum, and long rm, hr long rm hr xpcaon ar mor dffud, whch mak dffcul o ablh dffrnc bwn dffrn long ra. Scond, rk prmum nd o b mor abl for longr rm. Th ylzd fac hould b condrd whn mang h TSIR.. Thr Mhod for Trm Srucur Emaon Fr amp o ma h rm rucur rld on fng mooh funcon o h yld o maury of bond ung rgron analy. Howvr, h approach wa unafacory du o lmaon n dnfyng h zro-coupon yld, and

4 n dalng wh h coupon ffc. Th mnal work of J. Huon McCulloch n 97 uggd a nw mhod bad on quadrac pln, whch focud drcly on mang zro-coupon yld and dcoun facor. Much rarch ha xndd h work of McCulloch n h pa four dcad. Mhod for TSIR maon mu fnd a way o approxma h po ra, or h forward ra, or h dcoun funcon. Th rqur fng a parmonou funconal form ha flxbl n capurng ylzd fac rgardng h hap of h rm rucur. A good rm rucur maon mhod hould afy h followng rqurmn: Th mhod nur a uabl fng of h daa. Th mad zro-coupon ra and h forward ra rman pov ovr h nr maury pcrum. Th mad dcoun funcon, and h rm rucur of zro-coupon ra and forward ra ar connuou and mooh. Th mhod allow aympoc hap for h rm rucur of zro-coupon ra and forward ra a h long nd of h maury pcrum. Th commonly ud rm rucur maon mhod ar gvn a h boorappng mhod, h polynomal/xponnal pln mhod of McCulloch [97, 975] and Vack and Fong [98], and h xponnal funconal form mhod of Nlon and Sgl [987] and Svnon [994]. Exnon of h abov mhod ar gvn a h rror wghng modl uch a h B-pln mhod of Sly [99], h pnalzd pln mhod of Fhr, Nychka and Zrvo [995] and Jarrow, Ruppr, and Yu [004], and h conrand B-pln mhod of Pol and Moura [009], among ohr. 5 In h papr, w focu on h hr mo commonly ud rm rucur maon mhod: h boorappng mhod, h McCulloch polynomal cubc-pln mhod, and h Nlon and Sgl xponnal-form mhod... Boorappng Th boorappng mhod con of ravly xracng zrocoupon yld ung a qunc of ncrang maury coupon bond prc. 6 Th mhod rqur h xnc of a la on bond ha maur a ach boorappng da. To llura h mhod, condr a of K bond ha pay m-annual coupon. Th hor maury bond a x-monh bond, whch by dfnon do no hav any nrmda coupon paymn bwn now and x monh, nc coupon ar pad m-annually. Ung h 6-monh zro coupon ra, h prc of h bond gvn a: Quarr 07 P C + F (0.5) = y(0.5)0.5 whr F 0.5 h fac valu of h bond payabl a h maury of 0.5 yar, C 0.5 h m-annual coupon paymn a h maury, and y (0.5) h annualzd x-monh zro-coupon yld (undr connuouly-compoundng). Th x-monh zro-coupon yld can b calculad by akng logarhm of boh d of quaon (8), and mplfyng a follow: (8) F + C y(0.5) ln 0.5 P(0.5) = In ordr o compu h -yar zro-coupon yld, w can u h prc of a - yar coupon bond a follow: C F + C P() = + y y (0.5)0.5 () whr F h fac valu of h bond payabl a h bond -yar maury, C h m-annual coupon, whch pad a h nd of 0.5 yar and yar, and y () h annualzd -yar zro-coupon yld. By rarrangng h rm n quaon (0) and akng logarhm, w g h -yar zro-coupon yld a follow: y() ln F + C = C P() y(0.5)0.5 Snc w alrady know h x-monh yld, y (0.5) from quaon (9), h can b ubud n quaon () o olv for h -yar yld. Now, connung n h mannr, h x-monh yld, y (0.5), and h -yar yld, y (), can b boh ud o oban h.5-yar yld, y (.5), gvn h prc of a.5-yar maury coupon bond. Followng h am approach, h zro-coupon yld of all of h K maur (corrpondng o h maur of h bond n h ampl) ar compud ravly ung h zro-coupon yld of h prvou maur. Th zro-coupon yld corrpondng o h maur ha l bwn h K da can b compud by ung lnar or quadrac nrpolaon. Gnrally, abou 5 o 0 boorappng maur ar uffcn n producng h whol rm rucur of zro-coupon yld. Inad of olvng h zro-coupon yld qunally ung an rav approach a hown abov, on can ud h marx approach o olv for all K zro coupon ra mulanouly. Appndx dcu h approach. Th boorappng mhod ha wo man lmaon. Fr, nc h mhod do no prform opmzaon, compu zro-coupon yld ha xacly f h bond prc. Th lad o ovr-fng nc bond prc ofn conan doyncrac rror du o lack of lqudy, bd-ak prad, pcal ax ffc, c., and hnc, h rm rucur wll no b ncarly mooh a hown n Fgur. Scond, h boorappng mhod rqur ad-hoc adumn whn h numbr of bond no h am a h boorappng maur, and whn cah flow of dffrn bond do no fall on h am boorappng da. 7 Th nx wo mhod ovrcom h dffcul by mpong pcfc funconal form on h rm rucur... Cubc-pln mhod Condr h rlaonhp bwn h obrvd prc of a coupon bond maurng a m m, and h dcoun funcon. A dcud bfor, h prc of h bond can b xprd a h prn valu of ach coupon paymn ung zro coupon ra: (9) (0) () 70

5 m P( ) = CF d( ) + ε m = whr CF h oal cah flow from h bond (.., coupon, fac valu, or boh) on da ( =,,,m). Snc bond prc ar obrvd wh doyncrac rror, w nd o ma om funconal form for h dcoun funcon ha mnmz h rror. W fac wo problm n dong h. Fr, h dcoun funcon may b hghly non-lnar, uch ha w may nd a hgh-dmnonal funcon o mak h approxmaon work. Scond, h rror rm n quaon () may ncra wh h maury of h bond, nc longr maury bond hav hghr bd-ak prad, lowr lqudy, c. Du o h, maon of h dcoun funcon ung approach uch a la quar mnmzaon, gnrally f wll a long maur, bu provd a vry poor f a hor maur ( McCulloch [97] and Chambr Carlon and Waldman [984]). Th pln mhod addr h fr u by dvdng h rm rucur n many gmn ung a r of pon ha ar calld knopon. Dffrn funcon of h am cla (polynomal, xponnal, c.) ar hn ud o f h rm rucur ovr h gmn. Th famly of funcon conrand o b connuou and mooh around ach kno pon o nur h connuy and moohn of h fd curv, ung pln mhod. McCulloch ponrd h applcaon of pln o rm rucur maon by ung quadrac polynomal pln n 97 and cubc polynomal pln n 975. Th cubc pln mhod rman popular among praconr and xpland n Appndx. A rgard lmaon, a ponal crcm of h cubc-pln mhod h nvy of h dcoun funcon o h locaon of h knopon. Dffrn knopon rul n varaon n h dcoun funcon, whch can b omm gnfcan. Alo, oo many knopon may lad o ovrfng of h dcoun funcon. So, on mu b carful n h lcon of boh h numbr and h placng of h knopon. Anohr horcomng of cubc-pln ha hy gv unraonably curvd hap for h rm rucur a h long nd of h maury pcrum, a rgon whr h rm rucur mu hav vry ll curvaur. Addonally, h OLS rgron ud for h maon of h paramr n quaon (6), gv h am wgh o h prc rror of h bond wh hrognou characrc, uch a lqudy, bd-ak prad, maury, c. Ohr funcon can b ud for opmzaon o ovrcom h lmaon bu a h co of prcludng h u of OLS chnqu. 7 Fnally, h choc of polynomal a ba funcon alo conrovral. I argud ha h hap of h dcoun funcon mad ung cubc pln uually raonabl up o h maury of h long bond n h daa bu nd o b pov or ngav nfny whn xrapolad o longr rm. Th mpl ha pobl o gnra unboundd pov or ngav nr ra. Morovr, alhough h u of polynomal pln modra h wavy hap of mpl polynomal around h curv o b fd, h hap mgh no dappar complly and hnc, h fd dcoun funcon mgh wav around h ral dcoun funcon nroducng a gnfcan varably n boh po and forward ra. Dp h horcomng, h u 7 () of polynomal pln o ma h TSIR wdprad n h fnancal ndury... Nlon and Sgl Modl An alrnav approach ha ovrcom many of h horcomng of pln chnqu h mhodology of Nlon and Sgl. Th Nlon and Sgl [987] modl u a ngl xponnal funconal form ovr h nr maury rang. Nlon and Sgl ugg a parmonou paramrzaon of h nananou forward ra, whch hn ud o gv a mpl rprnaon of h zro coupon curv: β β y () = α + ( α + α ) ( ) α / / β Th Nlon and Sgl modl bad upon four paramr. Th paramr can b nrprd a follow: α + α h nananou hor ra,.., α + α = y (0) = f (0). α h conol ra. I gv h aympoc valu of h rm rucur of boh h zro-coupon ra and h nananou forward ra,.., α = y( ) = f ( ). Th prad bwn h conol ra and h nananou hor ra α, whch can b nrprd a h lop of h rm rucur of zrocoupon ra a wll a h rm rucur of forward ra. α affc h curvaur of h rm rucur ovr h nrmda rm. Whn α > 0, h rm rucur aan a maxmum valu ladng o a concav hap, and whn α < 0, h rm rucur aan mnmum valu ladng o a convx hap. () β > 0, h pd of convrgnc of h rm rucur oward h conol ra. A lowr β valu acclra h convrgnc of h rm rucur oward h conol ra, whl a hghr β valu mov h hump n h rm rucur clor o longr maur. Fgur llura how h paramr α, α, and α, affc h hap of h rm rucur of zro-coupon ra (gvn a Fgur : Influnc of h alpha paramr of Nlon and Sgl on h rm rucur of zro-coupon ra

6 α > 0 α + α > 0 β > 0 (5) Th fr conran n quaon (5) rqur ha h conol ra rman pov; h cond conran rqur ha h nananou hor ra rman pov; fnally, h hrd conran nur h convrgnc of h rm rucur o h conol ra. Snc h bond prcng quaon (4) a non-lnar funcon, h four paramr ar mad ung a non-lnar opmzaon chnqu. A non-lnar opmzaon chnqu ar uually nv o h arng valu of h paramr, h valu mu b carfully chon. Fgur 4: Influnc of h curvaur and hump poonng paramr of Nlon and Sgl conan β = ). A chang n α can b nrprd a h hgh or paralll chang, a chang n α can b nrprd a h lop chang (hough h paramr alo affc h curvaur chang lghly), and a chang n α can b nrprd a h curvaur chang n h rm rucur of zro-coupon ra. Fgur 4 dmonra ha Nlon and Sgl mhod conn wh a vary of rm rucur hap, ncludng monoonc and humpd, and allow aympoc bhavor of forward and po ra a h long nd. For llurav purpo, h conol and nananou ra hav bn a h am lvl. Th dcoun funcon aocad wh h rm rucur n () can b ud o oban a prcng formula for a coupon-barng bond, a follow: P( m) = CF Quarr 07 m = β ( )( ) / / β α βα+ α + α whr h bond maury and CF m h cah flow of h bond a m. (4) Th paramr n h quaon can b mad by mnmzng h um of quard rror bwn h lf hand and rgh hand d of quaon (4) ubc o h followng conran: Dp h compuaonal dffculy, h Nlon and Sgl modl, and xndd vron gvn by Svnon [994], hav a promnn poon among rm rucur maon mhod. Th moohn of h mad curv for boh po ra and forward ra, h aympoc bhavor of h rm rucur ovr h long nd, and hr robun o oulr and rror n mark daa ar h man advanag h mhod compard o pln mhod. In fac, a rpord n BIS [005], mo Cnral Bank u h mhod for rm rucur maon. Alo, n rcn yar, h modl ar aracng h nr of rarchr n h ara of nr modllng and porfolo rk managmn. Maznr-Løbr and Vlla [004] and Dbold and L [006], for xampl, rnrpr hm a modrn hr-facor modl of lvl, lop and curvaur facor n h mo pur radon of Lrman and Schnkman [99] and Bl [997] and oban mprcal vdnc n favor of hm. Morovr, Chrnn, Dbold and Rudbuh [0] provd horcal foundaon for h modl by obanng h affn arbrag-fr dynamc rm rucur vron of h modl, whch only dffr n h xnc of a yld-adumn rm, and Krppnr [0] how ha Nlon and Sgl modl can b nrprd from h prpcv of Gauan affn rm rucur modl. Fnally, Gürkaynak, Sack and Wrgh [007] provd h ma of h US TSIR a a daly frquncy from 96 o prn m ung h Nlon and Sgl pcfcaon for h prod bfor 980 (du o h lack of long rm bond) and h xnon of Svnon [994] afrward.. Concluon Inr ra play a cnral rol n valuaon of fnancal a and for makng macroconomc polcy. Howvr, hy ar no drcly obrvabl, and hould b mad from h mark prc of govrnmn cur wh dffrn maur. Many alrnav a uch a ral a, prva quy, and hdg fund nvmn ar llqud wh long-rm cah flow, whou a radly avalabl ourc for mark prc. Thu, a proprly mad rm rucur of nr ra nal for obanng h nrnc valu of h a. In h currn low-yld nvronmn, an accura maon of h rm rucur of nr ra aum vn grar mporanc du o h nonlnar convx rlaonhp bwn a prc and nr ra. Th papr focu on hr commonly ud rm rucur 7

7 mhod, gvn a h boorappng mhod, h McCulloch cubc pln mhod and h Nlon and Sgl mhod. W gv a mahmacally rgorou lluraon, xplanng h foundaon of h mhod, drvng h man quaon, and ponng ou h advanag and dadvanag of ach mhod. Appndx Th followng marx approach can b ud for obanng a drc oluon for h boorappng mhod. Condr K bond maurng a da,,,, and l CF K b h oal cah flow paymn of h h (for =,,,,K) bond on h da (for =,,, ). Thn h prc of h K bond ar gvn K by h followng ym of K mulanou quaon: P ( ) CF 0 0 d ( ) ( ) CF CF 0 P d ( ) = P ( K) CF d ( K) K CF K CF K K No ha h uppr rangl of h cah flow marx on h rghhand d of quaon (6) ha zro valu. By mulplyng boh d of quaon (6) by h nvr of h cah flow marx, h dcoun funcon corrpondng o maur,,..., can K b compud a follow: d ( ) CF 0 0 P ( ) d ( ) CF CF 0 P ( ) = d ( K) CF P ( K) K CF K CF K K Th abov oluon rqur ha h numbr of bond qual h numbr of cah flow maury da. 8 Th zro-coupon ra can b compud from h corrpondng dcoun funcon ung quaon (). Appndx Condr a of K bond wh maur of,,..., K. yar. Th rang of maur dvdd no - nrval dfnd by - kno pon T, T,..., T, whr T = 0 and T =. A cubc K polynomal pln of h dcoun funcon d () dfnd by h followng quaon: d () α g() = Th connuy and moohn of h dcoun funcon whn ach nrval nurd by h polynomal funconal form of ach g (). Th connuy and moohn a h knopon nurd by h rqurmn ha h polynomal funcon dfnd ovr adacn nrval ( T, T ) and ( T, T + ) hav a common 7 = + whr g (), g (),..., g() dfn a of ba pcw cubc funcon and α,..., α ar unknown paramr ha mu b mad. Snc h dcoun facor for m 0 by dfnon, w hav: (9) g (0) = 0 =,,, (6) (7) (8) valu and common fr and cond drvav a T. Th abov conran lad o h followng dfnon for h of ba funcon g (), g (),..., g() : Ca : < 0 < T ( T ) T < T 6( T T ) g () = ( T T ) ( T T )( T) ( T) ( T) + + T < T + 6 6( T+ T) T+ T T T+ ( T+ T ) + T+ 6 Ca : = g () = Subung quaon (8) no quaon (), w can rwr h prc of h bond maurng a da m a follow: m P( m) = CF + αg( ) + ε = = By rarrangng h rm, w oban: m m P( ) CF = α CF g ( ) + ε m = = = Th maon of h dcoun funcon rqur archng of h unknown paramr, α, α,, α, ha mnmz h um of quard rror acro all bond. Snc quaon () lnar wh rpc o h paramr α, α,, α, h can b achvd by an ordnary la quar (OLS) rgron. Th abov approach u numbr of maury gmn, numbr of knopon, and numbr of cubc polynomal funcon. An nuv choc for h maury gmn may b hor-rm, nrmda-rm, and long-rm, whch gv hr maury gmn of 0 o yar, o 5 yar, and 5 o 0 yar, four kno pon gvn a, 0,, 5, and 0 yar, and fv cubc polynomal funcon. McCulloch rcommnd choong knopon uch ha hr ar approxmaly qual numbr of daa pon (numbr of bond maur) whn ach maury gmn. Ung h approach, f h bond ar arrangd n acndng ordr of maury,..,... K, hn h kno pon ar gvn a follow: 0 = T = h + θ ( h+ h) K = whr h an ngr dfnd a: ( ) K h = INT (0) () () () (4)

8 and h paramr θ gvn a: Quarr 07 ( ) K θ = h McCulloch alo ugg ha h numbr of ba funcon may b o h ngr nar o h quar roo of h numbr of obrvaon, ha : = Round K Th choc of ha wo drd propr. Fr, a h numbr of obrvaon (bond) ncra, h numbr of ba funcon ncra. Scond, a h numbr of obrvaon ncra, h numbr of obrvaon whn ach nrval ncra, oo. Foono (5) (6). Th ofwar avalabl a ofwar.hml clckng on h lnk IRR. A Praccal Gud o Trm Srucur Emaon wh Excl n h Gud Sofwar con.. Whn compoundng dcr, ach xp(y) rplacd by k ( + y/ k). Snc cah prc ud n quaon (4), omm h dcoun ra alo calld h adud yld o maury.. An xclln vual xpoon of h dffrnc n h volal of h zro-coupon yld and ho of h nananou forward ra gvn n h xcl fl TSIRmov.xl avalabl a clckng on h lnk Trm Srucur Mov. 4. Th hap of h rm rucur alo xpland by ohr varabl no rlad o xpcaon uch a lqudy prmum, mark gmnaon, c. Alrnav rm rucur hypoh hav agnd dffrn rol o h varabl. For a brf dcuon abou h man hypoh, Nawalkha, Soo and Blava [005], pp Th mhod ud o ma h TSIR no only affc h ma, bu alo any daa drvd from hm. Daz, Jarño and Navarro [0] rpor h for ma of nr ra volaly. 6. Uually, no all h bond ha rad n h mark a a gvn m ar ud for h maon of h TSIR. Th bond lcd mu covr a wd pcrum of maur, hould hav an nough dgr of lqudy and hr prc houldn ncorpora hgh doron du o ax ffc or ohr mark frcon. Uually, h rqurmn ar fulflld by h ablhmn of flrng crra for drmnng h bond ha qualfy for ncluon n h ampl. 7. In fac, hr ar many alrnav rror-wghng chm whch mgh lad o mor robu ma of h rm rucur. For xampl, Bl [997] ugg wghng ach bond prc rror by h nvr of h bond duraon a a way o mprov h fng of long nr ra, whch mgh b poor. Th du o h fac ha n abnc of a wghng chm for prcng rror, h qualy of h f of h rm rucur dcra wh maury. To undrand h, condr h rlaonhp bwn prc, yld and maur. A am chang n prc mpl a much grar chang n yld n hor-rm bond compard o long-rm bond. Thrfor, followng a prc rror mnmzaon crron n h maon wll mak nr ra corrpondng o long-rm bond o b ovr-fd a h xpn of horrrm nr ra. Ohr approach nclud h u of pnaly funcon, a n Fhr, Nychka and Zrvo [995] or Jarrow, Ruppr, and Yu [004]. 8. For xampl, whn wo or mor bond maur on h am boorappng maury, h mad po ra rulng from ung ach of h bond ar uually avragd. In h oppo ca, whn no bond x a a rqurd boorappng maury, a common pracc o ma a par yld curv (ha, h yld o maur of bond prcd a par) ung mpl rgron modl ha mak h yld o maury on currn bond dpnd on a r of bond characrc ncludng h coupon ra and h m o maury. Thn, h yld on par bond ar mad by aumng ha h coupon ra of ach bond qual yld o maury. Rfrnc BIS, 005, Zro-coupon yld curv: Tchncal documnaon, Monary and Economc Dparmn, BIS Papr 5, Ocobr 005, Bank for Inrnaonal Slmn. Bl, R.R., 997, Movmn n h Trm Srucur of Inr Ra, Economc Rvw, FRB of Alana, fourh quarr, 6-. Chambr, D.R., W.T. Carlon and D.W. Waldman, 984, A Nw Approach o Emaon of h Trm Srucur of Inr Ra, Journal of Fnancal and Quanav Analy 9(), -5. Chrnn, J.H.E., F.X. Dbold and G. D. Rudbuch, 0, Th Affn Arbrag-fr Cla of Nlon Sgl Trm Srucur Modl, Journal of Economrc 64(), 4 0. Díaz, A., F. Jarño and E. Navarro, 0, Trm Srucur of Volal and Yld Curv Emaon Mhodology, Quanav Fnanc (4), Dbold, F.X., L. J. and C. L, 006, A Thr-Facor Yld Curv Modl: Non-Affn Srucur, Symac Rk Sourc, and Gnralzd Duraon, n L.R. Kln (d.), Long-Run Growh and Shor-Run Sablzaon: Eay n Mmory of Albr Ando. Chlnham, U.K., Edward Elgar, Fhr, M., D. Nychka and D. Zrvo, 995, Fng h Trm Srucur of Inr Ra wh Smoohng Spln, Workng Papr 95-, Fnanc and Economc Dcuon Sr, Fdral Rrv Board. Gürkaynak, R.S., B. Sack and J.H. Wrgh, 007, Th US Traury Yld Curv: 96 o h prn, Journal of Monary Economc 54, Jarrow, R., D. Ruppr and Y. Yu, 004, Emang h Trm Srucur of Corpora Db wh a Smparamrc Pnalzd Spln Modl, Journal of h Amrcan Sacal Aocaon 99, Krppnr, L., 0, A Thorcal Foundaon for h Nlon Sgl Cla of Yld Curv Modl, Journal of Appld Economrc, do: 0.00/a.60. Lrman, R. and J. Schnkman, 99, Common facor affcng bond rurn, Journal of Fxd Incom, Jun, Maznr-Løbr, E. C. and Vlla, 004, Funconal Prncpal Componn Analy of h Yld Curv, Inrnaonal Confrnc AFFI 004, Franc. McCulloch, J.H., 97, Maurng h Trm Srucur of Inr Ra, Journal of Bun 44, 9. 74

9 McCulloch, J.H., 975, Th Tax Adud Yld Curv, Journal of Fnanc 0, Nawalkha, S.K., G.M. Soo, and N.A. Blava, 005, Inr Ra Rk Modlng: Th Fxd Incom Valuaon Cour, Wly Fnanc, John Wly and Son, NJ. Nawalkha, S.K., N.A. Blava, and G.M. Soo, 007, Dynamc Trm Srucur Modlng: Th Fxd Incom Valuaon Cour, Wly Fnanc, John Wly and Son, NJ. Nlon, C.R. and A.F. Sgl, 987, Parmonou Modlng of Yld Curv, Journal of Bun 60(4), Pol, M. and M. Moura, 00, Conrand moohng B-pln for h rm rucur of nr ra, Inuranc: Mahmac and Economc 46(), Pazz, M., 00, Affn Trm Srucur Modl, n Y. Aï-Sahala and L. Hann (d.), Handbook of Fnancal Economrc. Norh Holland, Amrdam, Sly, J. M., 99, Emang h Gl-Edgd Trm Srucur: Ba Spln and Confdnc Inrval, Journal of Bankng, Fnanc and Accounng 8(4), Svnon, L.E.O., 994, Emang and Inrprng Forward Inr Ra: Swdn , Inu for Inrnaonal Economc Sud. Vack, O.A. and H.G. Fong, 98, Trm Srucur Modlng Ung Exponnal Spln, Journal of Fnanc 7(), Auhor 75 Sanay Nawalkha, Ph.D. Inbrg School of Managmn Dr. Sanay Nawalkha Profor and Charman of h Fnanc Dparmn, and h Rupndr Sdhu Faculy Fllow n Fnanc a h Inbrg School of Managmn a h Unvry of Maachu, Amhr. H ha auhord cholarly book n dffrn ara of fxd ncom, and ha publhd numrou arcl n manram fnanc ournal on opc rlad o a prcng, fxd ncom, and nr ra drvav. H ha prnd h work a varou naonal and nrnaonal confrnc, larg fnancal nuon, and forgn cnral bank. H rv a an aoca dor of h Journal of Invmn Managmn. H h co-foundr of h fnanc poral, fxdncomrk.com, whch provd fr downloadabl ofwar for valung fxd ncom drvav. Profor Nawalkha ach a vary of cour a h docoral lvl ncludng nr ra modlng, opon prcng, and crd rk and rurn modlng. Glora M. Soo Ph.D. Unvry of Murca Glora M. Soo, PhD, a Profor of Appld Economc a h Unvry of Murca, Span, whr h ach cour n fnancal mark and nuon and appld conomc. Dr. Soo ha publhd xnvly n boh Spanh and nrnaonal ournal n fnanc and conomc, pcally n h ara of nr ra rk managmn, bankng and monary polcy. Sh auhor of vral book on conomc and fnanc covrng dffrn lvl of ducaon, from condary ducaon o PhD cour ncludng profonal ranng and ducaon for oldr popl. Sh wa Vc dan of Economc Affar a h Faculy of Economc of h Unvrdad d Murca and currnly h dpuy drcor of h Cnro d Eudo Económco y Empraral (Cnr for Sud n Economc and Bun) n h Unvry. Sh alo workd for h Europan Common a an xpr advor from 008 o rcn day n h ara of bank crd.