The Performance of Alternative Interest Rate Risk Measures and. Immunization Strategies under a Heath-Jarrow-Morton Framework

Transcription

1 Th Prformanc of Alrnav Inrs Ra Rsk Masurs and Immunzaon Srags undr a Hah-Jarrow-Moron Framwork Şnay Ağca * Ths vrson: January * Asssan Profssor of Fnanc Dparmn of Fnanc Gorg Washngon Unvrsy 2023 G Sr NW Lsnr Hall 540G Washngon DC Tl: Fax: Emal: sagca@gwu.du. Ths papr s par of my dssraon work a Vrgna Tch. Ths papr has bnfd from commns by Don M. Chanc Raman Kumar Abon Mozumdar Randall Bllngsly Marn Day N. Jgadsh Sayd Islam Isabll Bajux-Bsnanou h smnar parcpans a h 200 FMA docoral sudn smnar Frdd Mac Gorg Washngon Unvrsy Rugrs Unvrsy SUNY-Bnghamon and Vrgna Tch. I hank Okan Kolak for chncal asssanc and Naonal Crash Analyss Cnr a h Gorg Washngon Unvrsy for provdng addonal compung rsourcs. I acknowldg a summr rsarch gran from h School of Busnss and Publc Admnsraon Gorg Washngon Unvrsy. Any rmanng rrors ar my rsponsbly.

2 ABSTRACT Usng a Mon-Carlo smulaon hs sudy addrsss h quson of how radonal rsk masurs and mmunzaon srags prform whn h rm srucur volvs n a Hah- Jarrow-Moron HJM mannr. Th rsuls suggs ha for mmunzaon purposs mmunzaon srags and porfolo formaon srags ar mor mporan han nrs ra rsk masurs. Th prformanc of mmunzaon srags dpnds mor on h ransacon coss and h holdng prod han on h rsk masurs. Morovr mmunzaon prformancs of bull and barbll porfolos ar no vry snsv o nrs ra rsk masurs. 2

3 I. Inroducon Th pas coupl of dcads hav wnssd h dvlopmn of a complcad s of modls for dscrbng h voluon of h rm srucur of nrs ras. Rsarch n hs ara has bn drvn prmarly by h fac ha nrs ra rsk managmn has bcom an mporan componn of h rsk managmn praccs of fnancal nsuons. A h sam m vn h mos sophscad fnancal nsuons connu o us radonal rsk masurs such as h duraon masurs dvlopd by Macaulay 938 and Fshr and Wl 97 as wll as convxy masurs. Whl on can modl h rm srucur and hn us h corrspondng nrs ra rsk masurs as an alrnav o radonal rsk masurs s mor cosly boh n rms of human capal and compung chnology. Thus h ffor would only b usful f hr ar subsanal bnfs o b gand n usng hs mor complx rsk masurs nsad of hr smplr radonal counrpars. Ths papr compars h mmunzaon prformancs of radonal rsk masurs wh hos drvd from spcfc rm srucur modls. Prvous suds n a somwha smlar drcon hav provdd nconclusv vdnc. For nsanc Cox Ingrsoll and Ross 979 Brnnan and Schwarz 983 and Nlson and Schafr 983 dvlop rm srucur modls and hr corrspondng nrs ra rsk masurs. Thy hn compar h mmunzaon prformancs of Macaulay s and Fshr-Wl duraon masurs wh h mmunzaon prformanc of hr modl-spcfc rsk masurs. Th vdnc suggss ha h rsk masurs of rm srucur modls do no hav br mmunzaon prformancs han hr Th ffc of nrs ras on h bankng ndusry s rgorously analyzd n h lraur. Hasan and Sarkar 2002 rvw h lraur n hs ara and provd an analyss of h ffc of nrs ras on bank porfolos. 3

4 radonal counrpars. Gulkn and Rogalsk 984 s h mplcaons of alrnav duraon masurs n rms of xplanng h rurn varanc of U.S. govrnmn bonds. Thy fnd non of h duraon masurs analyzd o b supror o h ohrs. Babbl 983 and Carcano and Fors 997 compar h mmunzaon srags ha conrol for dffrng volaly wh hos ha dpnd on smpl duraon-machng sragy. Whl Babbl fnds ha smpl duraon machng srags prform as wll as h proposd alrnavs Carcano and Fors provd counrvdnc by consdrng h corrlaons bwn nrs ras. Ilmann 992 fnds ha smpl duraon masurs xplan 80 o 90 prcn of h rurn varanc of US govrnmn bonds. Fnally Ho Cadl and Thobald 200 usng Srlng fuurs daa show ha h hdgng prformanc of a spcfc HJM modl-basd rsk masur s no br han ha of modfd duraon. Ths rsuls ar no comparabl snc hy ar basd on dffrn daa ss porfolo formaon srags and analyss prods. Addonally f h chosn modl or sochasc procss conjcurd o dvlop h rsk masurs dos no rprsn h ru sochasc procss of h daa a cran ms hn would b hard o nrpr h mmunzaon prformanc of h rsk masurs. In a way a comparson of h mmunzaon prformancs of rsk masurs basd on rm srucur modls solly wh mprcal daa amouns o an analyss of whch rm srucur fs h daa bs. Morovr whn h prformanc of cran rsk masurs s poor s hard o nfr h rasons for hs poor prformanc usng only mprcal daa. To addrss hs problms I us a slcd s of on-facor Hah Jarrow and Moron 992 modls 2 HJM modls hncforh as a bnchmark. Thn I carry ou a 2 I focus on on-facor HJM modls for svral rasons: Tradonal rsk masurs assum on random undrlyng facor and hrfor would b conssn o compar rsk masurs ha dpnd on only on random 4

5 Mon-Carlo smulaon-basd sudy o compar h mmunzaon prformancs of radonal rsk masurs wh hos of bnchmark rsk masurs. I sma h paramrs of h bnchmark modls and h corrspondng nrs ra rsk masurs usng U.S. Govrnmn Bond daa. I us hs paramrs hroughou h smulaon. To consdr h mpac of samplng varably on h mmunzaon prformanc of HJM rsk masurs I also calcula HJM rsk masurs usng paramrs smad n h smulaon rahr han hos usd o smula h daa. I us bull barbll and random porfolos 3 for ach scnaro so ha h mmunzaon prformancs of h rsk masurs ar no ovrly dpndn on h way h mmunzaon porfolos ar formd. I consdr wo wdly usd mmunzaon srags n valuang h prformanc of ach rsk masur; on machng only h duraon of h asss and labls and h ohr machng boh hr duraon and convxy. I xamn h mmunzaon prformancs boh wh and whou ransacon coss. Th vdnc of hs papr suggss ha major dffrncs among mmunzaon prformancs dpnd mor on h mmunzaon sragy han on h rsk masurs. Whn a duraon machng sragy s usd mmunzaon prformancs wh radonal nrs ra rsk masurs ar vry smlar o hos wh HJM rsk masurs. Whn hr ar no ransacon coss a duraon and convxy machng sragy mprovs mmunzaon facor. Also as documnd n Lrman and Schnkman 99 and Chapman and Parson 200 mor han 85 prcn of h varaon n h U.S. Trasury yld curv can b xpland by on facor. Thy rfr o hs facor as h lvl of nrs ras. Addonally all shor ra modls can b rprsnd as spcfc cass of onfacor HJM modls and shor ra modls sll rcv consdrabl anon n h lraur. 3 A barbll porfolo consss of scurs ha hav h lows and hghs duraons. A bull porfolo on h ohr hand consss of scurs wh h duraons closs o h arg holdng prod. 5

6 prformanc subsanally spcally wh HJM rsk masurs basd on h paramrs usd n h smulaon of h forward ra daa. Whn h paramrs of h HJM rsk masurs ar smad n h smulaon howvr samplng rror lads o a droraon n h mmunzaon prformanc of h HJM rsk masurs. Snc radonal rsk masurs do no rqur any smaon and hnc ar fr of hs samplng rror undr mos scnaros hy provd smlar or br prformanc han HJM rsk masurs ha ar basd on smad paramrs. Whn hr ar ransacon coss a duraon and convxy machng sragy prforms br for shor holdng prods whras a duraon machng sragy prforms br for mdum and long holdng prods. Th ransacon coss ncurrd ncras wh h lngh of holdng prod. Snc a duraon machng sragy ncurs rlavly lss ransacon coss han a duraon and convxy machng sragy h formr sragy prforms br ovr mdum o long holdng prods. Thrfor whn ransacon coss ar prsn mmunzaon srags should b slcd wh rspc o holdng prods. Among porfolo formaon srags boh bull and barbll porfolos hav vry good mmunzaon prformancs ovr shor holdng prods. Snc bull porfolos gnrally hav a yld advanag ovr barbll porfolos hr rurns ovr shor holdng prods ar closr o h rspcv arg rurns han h corrspondng rurns of barbll porfolos. Howvr as h holdng prod ncrass h convxy advanag of barbll porfolos ovrcoms h yld advanag of bull porfolos. Thrfor ovr mdum and long holdng prods barbll porfolos provd br mmunzaon han bull porfolos. Whn hr ar no ransacon coss a duraon and convxy machng sragy mprovs h mmunzaon prformanc of boh bull and barbll porfolos. Whn ransacon coss ar prsn 6

7 howvr duraon and convxy machd bull porfolos hav br prformancs ovr shor holdng prods and duraon machd barbll porfolos hav br prformancs ovr long holdng prods. In boh cass h porfolo prformanc s no vry snsv o nrs ra rsk masurs. Th ovrall vdnc suggss ha mmunzaon and porfolo formaon srags affc mmunzaon prformanc mor han rsk masurs. Addonally h lngh of holdng prod and ransacon coss hav a consdrabl mpac on h mmunzaon prformancs of hs srags. Whn HJM rsk masurs ar h corrc rsk masurs hy provd br mmunzaon only whn usd wh h appropra mmunzaon sragy. Morovr f h paramrs of h HJM rsk masurs ar no known bu ar smad hr prformanc droras. Thus undr mos scnaros radonal rsk masurs ha do no rqur smaon provd a smlar or br mmunzaon prformanc han HJM rsk masurs. Th papr procds as follows. Scon 2 gvs a brf dscrpon of radonal duraon and convxy masurs oulns h HJM framwork and prsns h rsk masurs dvlopd for spcfc on-facor HJM modls. Scon 3 dscrbs h smulaon framwork. Scon 4 compars h mmunzaon prformancs of nrs ra rsk masurs and porfolo formaon srags. Scon 5 prsns h conclusons. II. Inrs Ra Rsk Masurs A. Tradonal Inrs Ra Rsk Masurs Th dvlopmn of nrs ra rsk masurs das back o Macaulay 938 Rdngon 952 Hcks 939 and Samulson 945. Ths auhors ndpndnly dvlopd an nrs ra rsk masur ha assums a consan yld and hrfor horcally provds accura mmunzaon undr a fla yld curv and paralll shfs n h 7

8 yld curv. Ths nrs ra rsk masur s rfrrd o as Macaulay s duraon. Fshr and Wl 97 rlaxd h assumpon of a consan yld n Macaulay s duraon and dvlopd a nw duraon masur hncforh Fshr-Wl duraon. Th dscoun facors of h cash flows n h Fshr-Wl duraon ar drvd from h currn rm srucur. Thorcally Fshr-Wl duraon provds accura mmunzaon undr paralll shfs n h yld curv. Ths duraon masurs a m ar summarzd blow for a coupon bond maurng a T n wh coupon paymns a m prods and prncpal paymn a n and wh prc BT 45. Macaulay s duraon D M : D M B T n C y Fshr-Wl duraon D FW : 2 D M B T n C y Hr C s h coupon paymn a da whr < 2 < < n T for a radng nrval of [T]; P s h prc of a zro coupon bond maurng a m. and y s h yld o maury of P. In Macaulay s duraon h yld o maury s consan and hrfor y y. Duraon modls ar basd on h assumpon of a lnar rlaon bwn bond prcs and nrs ras. Such an assumpon s vald for nfnsmal changs n h nrs ra. For non-nfnsmal changs h non-lnar rlaon bwn bond prcs and nrs ras 4 Non of h bonds consdrd n hs sudy conan opons. 5 Snc HJM modls assum connuously compoundd forward ras h yld o maury s consdrd o b connuously compoundd. 8

9 9 mus b akn no accoun n ordr o hav an accura mmunzaon. Th scond drvav of h bond prc wh rspc o h nrs ra provds h xprsson ofn calld convxy o b consdrd n addon o duraon. Th convxy masur corrspondng o Macaulay s duraon has a consan yld whl ha corrspondng o Fshr-Wl duraon has a varyng yld. Consan yld convxy Conv M : 3 2 y n M C T B Conv Varyng yld convxy Conv FW : 4 2 y n FW C T B Conv B. HJM Inrs Ra Rsk Masurs Hah Jarrow and Moron 992 dvlopd an arbrag-fr prcng framwork for capurng h voluon of forward ras and hrfor modlng h rm srucur of nrs ras. Th drmnaon of dffrn volaly funcons oghr wh h npu of h nal forward curv rsuls n dffrn rm srucur modls. Th gnral HJM modl s a mulfacor modl. An n-facor HJM modl spcfs h voluon of h nsananous forward ra accordng o h followng sochasc procss: 5 + n dw w T d w T T df σ α

10 Hr f T s h nsananous forward ra a m for da T T > ; dw s Brownan moon a m 6 ; α T w s a random drf rm of h forward ra curv; and σ T w s volaly funcon for..n whr T > and w s a sampl pon dfnd on Ω. Th no-arbrag condon pus a rsrcon on h rlaon of drf o volals. Undr h quvaln marngal masur h drf rsrcon and h corrspondng forward ra procss ar as follows: 6 α T w σ T w σ v w dv n n T n 7 ~ df T σ T w σ v w dv d + σ T w dw ~ In h abov quaon W corrsponds o h Wnr procss undr h quvaln marngal masur. In HJM modls dffrn volaly funcons lad o dffrn rm srucur modls. For xampl n on-facor HJM modls f volaly s consan σ T σ h modl rducs o h Ho and L 986 modl. If volaly s xponnally dcayng T σ xp T σ h modl rducs o h Hull and Wh 993 xndd Vasck modl. Ohr shor ra modls can also b rprsnd as spcal cass of on-facor HJM modls. Th prc a m for a zro coupon bond maurng a m T PT T s drmnd by h forward ra dynamcs T P T xp f u du. 6 A sochasc procss W dfnd for 0 on Ω F Ρ s a Brownan moon f W 0 0 W s connuous wh rspc o ; h ncrmn W s+ W s s dsrbud as normal N0 undr Ρ; and s ndpndn of F. s 0

11 Ths sudy focuss on on-facor HJM modls whch hav only on random facor drvng h forward ra dynamcs. Afr spcfcaon of h volaly funcons h rsulng rm srucur modl provds h bond prcng framwork. Ths bond prcng framwork s usd o drv h duraon and convxy masurs corrspondng o h slcd HJM modls. Au and Thurson 995 drvd h duraon masurs for cran connuous m on-facor HJM rm srucur modls. Munk 999 and Jffry 2000 usd an alrnav mhod for drmnng duraon masurs for HJM modls 7. Th duraon masur for onfacor HJM modls D HJM wh drmnsc volaly s as follows: 8 D HJM n C σ v dv P n σ C P For consan volaly σt σ D HJM rducs o h Fshr-Wl duraon masur. Followng Amn and Moron 994 I consdr only on-facor HJM modls ha hav h m-nvaranc propry. Th volaly funcons of hs HJM modls dpnd only on h m o maury no on h calndar da. 8 In hs rspc h consan and xponnally dcayng volaly funcons analyzd by Au and Thurson ar consdrd. Addonally h volaly funcon dvlopd by Mrcuro and Moralda 2000 s analyzd. 7 Brwag 2000 also drvs h duraon masur of h Ho and L 986 modl.. h consan volaly HJM modl. Brwag s duraon masur s dffrn han h on sad n hs papr snc Brwag drvs h masur for a bnomal nrs ra sochasc procss. 8 Snc only a rsrcd numbr of on-facor HJM modls ar consdrd sms mor appropra o slc h modls ha hav common proprs.

12 2 Mrcuro and Moralda dvlopd a volaly funcon o rprsn h humpd rm srucur of volaly somms obsrvd n h mark. Th duraon masurs corrspondng o hs on-facor HJM modls ar as follows: Consan Volaly σt σ: 9 n n HJM C P P C D Exponnally Dcayng Volaly σt T σ : 0 D CP CP HJM n n / Humpd Volaly T T T + γ σ σ : 9 + n n HJM C P C P D γ γ Followng h framwork usd by Au and Thurson 995 I drv h convxy masurs of on-facor HJM modls 0. Undr h HJM framwork wh drmnsc volals h convxy of a coupon bond Conv HJM s as follows : 9 Th drvaon of h humpd volaly HJM duraon masur s gvn n Appndx 2. 0 Full drvaons ar gvn n Appndx. Fruhwrh 2002 follows a smlar approach n drvng h convxy masurs of on-facor HJM modls. Our rsuls agr wh hs.

13 3 2 n n HJM dv v f P C C P Conv σ σ 2 2 N dv v P C T B σ σ For consan volaly σt σ Conv HJM rducs o h varyng yld convxy gvn n quaon 4. Th convxy masurs for h slcd on-facor HJM modls ar summarzd blow. Consan Volaly σt σ: 3 Conv CP CP HJM n n 2 Exponnally Dcayng Volaly σt σ T : 4 Conv CP CP HJM n n 2 2 Humpd Volaly T T T + γ σ σ : n n HJM C P C P Conv γ γ III. Th Smulaon Framwork Forward ras ar basd on h slcd HJM modls and hrfor h rsk masurs of h HJM modls provd h bnchmarks. Th npus of h smulaons ar smad usng 2 Th drvaon of h humpd volaly HJM convxy masur s gvn n Appndx 2.

14 U.S. govrnmn bond daa. Forward ra daa up o hry yars of maury and h corrspondng zro-coupon yld curv daa for h prod January 947 o Fbruary 99 ar oband from McCulloch and Kwon For h prod January 947 o Fbruary 99 yld curv chars ar formd for ach monh and ar cagorzd by shap: fla dclnng ncrasng and humpd. On rprsnav yld curv from ach shap s hn slcd. Th slcd yld curvs ar: for fla Augus 989; for ncrasng January 985; for dcrasng May 98; and for humpd Ocobr 986. Ths yld curvs and h corrspondng forward ra curvs ar shown n Fgur. Th cubc spln mhod s usd o smooh h forward ras and o nrpola mssng monhly forward ras. 4 5 Forward ras corrspondng o hs slcd das ar hn usd as h nal forward ra curvs n h smulaons. Anohr ncssary npu for h smulaons s h forward ra volals. To drmn h volals four yars of hsorcal forward ra daa corrspondng o ach of h slcd four yld curvs ar usd. 6 Corrspondng forward ras ar smoohd and h 3 For hs daa s boh h zro-coupon ylds and h forward ras ar drvd by McCulloch 975 usng a ax-adjusd cubc spln mhod. All forward ras ar gvn as prcnags pr annum and ar on a connuously compoundd bass. All daa ar from h afrnoon of h las busnss day of h monh ndcad. For ach monh 56 maurs ar avalabl. Th daa ar monhly from 0 o 8 monhs hn quarrly o 2 yars hn sm-annually o 3 yars hn annually o 35 yars and hn mak a 5-yar jump o 40 yars. 4 Blss 997 compars dffrn curv fng procdurs and fnds ha McCulloch 975 cubc spln procdur and Fama and Blss 987 boosrappng procdur ouprform ohr xsng mhods. 5 Snc h forward ra daa of McCulloch and Kwon 994 ar no avalabl wh monhly frquncy afr 8 monhs h cubc spln mhod s usd o nrpola h mssng monhly forward ras.. 6 Rbonao 998 pp and Amn and Ng 997 us four yars of hsorcal daa n hr volaly smaons. 4

15 mssng monhly forward ras ar nrpolad usng h cubc spln mhod. Thn changs n h forward ras ar calculad. Th sandard dvaon vcor of dffrn maury forward ras gvs h forward ra volals. 7 Usng rgrsson mhods forward ra volals ar fd no h volaly funcons of cran on-facor HJM modls n ordr o sma h paramrs of ach volaly funcon. Ths smad paramrs ar hn usd n h smulaon of h forward ras and n h calculaon of HJM rsk masurs for h volaly funcons. Th rsuls ar prsnd n Tabl. Th xponnal dcay facor n boh xponnally dcayng volaly and humpd volaly modls s ngav du o h ncras n h rspcv forward ra volals afr wny-yars. Th lnar ncras facor for h humpd volaly s posv bu vry small. As a rsul h humpd and xponnally dcayng volaly funcons ar drmnd manly by h ngav dcay facor. In hs rspc h forward ra volals ncras wh h rm o maury n boh modls. Ths volaly funcons along wh hr smad paramrs ar plod agans h obsrvd forward ra volals n Fgur 2. Th bs f s for humpd and xponnally dcayng volals. Th fs of hs wo volaly modls ar ndsngushabl from ach ohr. Hnc h rs of h sudy focuss manly on consan and xponnally dcayng volaly HJM modls. 7 Usng a prncpal componn analyss Lrman and Schnkman 99 and Chapman and Parson 200 show ha all varaon n h U.S. Trasury yld curv can b xpland by hr facors and mor han 85 prcn of by on facor alon. Snc h focus of hs papr s on-facor HJM modls I assum ha all varaon n h yld curv s xpland by on facor. In an arlr vrson I mployd a hr-facor prncpal componn analyss o sma volals. I hn usd h frs facor n h smulaons of h slcd onfacor HJM modls. Th rsuls wr smlar o h ons rpord hr. 5

16 For h nal m prod h zro coupon bond prcs ar drvd from h forward ras. Coupon bonds ar consrucd from h zro coupon bond prcs. To drmn h coupon ras CRSP monhly Trasury daa s usd. For ach of h four das of nal yld curvs h avrag coupon ra s compud from h coupon bonds radng on ha spcfc da. Ths avrag coupon ras ar hn usd o gnra h coupon bonds. 8 Th mmunzaon prformancs of h nrs ra rsk masurs ar analyzd ovr on- fv- and n-yar holdng prods. 9 Th arg yld s h yld on a zro coupon bond maurng a h nd of h holdng prod. Th arg ylds for h slcd nal yld curvs and holdng prods ar gvn n Tabl 2. A porfolo s consdrd o b succssfully mmunzd f h holdng prod rurn of h porfolo s clos o h yld of h arg zrocoupon bond. 20 Usng hs crra I analyz hos porfolos ha hav holdng prod rurns whn on fv and n bass pons of h arg yld. I focus on h wo mos wdly usd mmunzaon srags. Ths ar h duraon machng and h duraon and convxy machng srags. Th duraon machng sragy s xpcd o prform wll undr small yld curv shfs. Th duraon and convxy machng sragy s xpcd o b succssful for non-nfnsmal changs n h yld curv as wll. Th arg zro-coupon bond can b consdrd as h lably porfolo and h 8 In Fshr and Wl 97 h coupon ra s fxd as 4 prcn whras n Brwag Kaufman Schwzr and Tovs 98 h coupon ra s fxd as 5 prcn. 9 Fshr and Wl 97 us 5-0- and 20-yar holdng prods. Th holdng prods n Cox Ingrsoll and Ross 979 ar and 50 yars. Brnnan and Schwarz 983 and Gulkn and Rogalsk 984 us 5- and 0-yar holdng prods. Nlson and Schafr 983 us 5-0- and 5-yar holdng prods. 20 Ths sandard s usd by Brnnan and Schwarz 983 as wll as by Nlson and Schafr

17 coupon bond porfolo as h ass porfolo. Th duraon of a coupon bond porfolo s machd o h duraon of h rspcv arg zro-coupon bond o form a duraon-machd porfolo. To form a duraon and convxy machd porfolo boh h duraon and h convxy of a coupon bond porfolo ar machd o hos of h rspcv arg zrocoupon bond. Afr a porfolo s formd h wghs of h scurs ar changd a vry rbalancng prod o mach h duraon or h duraon and convxy of h arg bond. Th rbalancng prod s on monh snc monhly forward ra daa ar usd. Th mmunzaon porfolos ar slf-fnancng. Porfolo formaon srags also affc mmunzaon prformanc. In ordr o consdr h mpac of porfolo formaon srags on mmunzaon prformanc a bull a barbll and 00 random porfolos ar formd for ach scnaro. Th random porfolos ar formd o ma porfolo formaon srags ha w canno obsrv or srags whr h porfolo managr has o ac n rspons o xognous facors. Unformly dsrbud random numbrs ar usd o drmn h maury of h coupon bonds n h random porfolos. Th maurs of h bonds n random porfolos vary bwn h holdng prod and hry yars 2. Two ss of 00 random porfolos ar formd; on for a duraon machng sragy has wo coupon bonds n ach porfolo; and on for a duraon and convxy machng sragy has hr coupon bonds n ach porfolo. Th sam random porfolo s s usd for ach HJM modl consdrd n h smulaons. Bull and barbll porfolos ar wdly usd srags. A barbll porfolo consss of bonds wh h hghs and lows duraons avalabl. Wh a duraon machng sragy a barbll porfolo s formd usng wo coupon 2 Th mnmum maury of a coupon bond n h porfolo s h holdng prod snc rollng ovr s no allowd. 7

18 bonds; on maurng a h nd of h holdng prod and h ohr maurng n wny yars. Rollng porfolos ar no allowd. To mach boh duraon and convxy an addonal coupon bond s ncssary. Ths addonal coupon bond s a mdum-rm bond. For a onyar holdng prod a n-yar bond s slcd. For fv- and n-yar holdng prods h slcd coupon bond has a maury of wlv yars and ffn yars rspcvly. A bull porfolo consss of bonds ha hav duraons clos o h arg duraon. As s h cas wh ohr porfolos bull porfolos ar formd usng wo scurs wh a duraon machng sragy and hr scurs wh a duraon and convxy machng sragy 22. To drmn hs bonds an opmzaon program s run o mnmz h absolu dffrnc bwn h duraon of a bond porfolo and h arg duraon. Th opmzaon program consdrs bond porfolos conssng of all possbl combnaons of bonds wh maurs rangng bwn h holdng prod and 360 monhs. Th opmal porfolo has h closs duraon o ha of h arg bond and hnc s consdrd as h bull porfolo. Spara bull porfolos ar formd for ach holdng prod on- fv- and n-yar and for ach rsk masur consdrd n h sudy Macaulay Fshr-Wl and h bnchmark HJM rsk masurs. Th mmunzaon prformancs of duraon machd and duraon and convxy machd porfolos ar xamnd boh wh and whou ransacon coss. Thorcally duraon and convxy machd porfolos should provd br mmunzaon han duraon machd porfolos snc duraon and convxy machng sragy aks h nonlnary 22 In Gulkn and Rogalsk 984 random mmunzaon porfolos ar formd from 2 o 0 scurs. Brnnan and Schwarz 983 us 20 scurs n formng hr mmunzaon porfolo. In mos suds such as Fshr and Wl 97 Brwag Kaufman and Tovs 98 and Nlson and Schafr 983 only wo scurs ar usd n formng h mmunzaon porfolo. 8

19 bwn bond prc and yld no accoun. Howvr duraon and convxy machd porfolos ncur hghr ransacon coss han duraon machd porfolos du o h grar amoun of bonds rbalancd n ach prod. 23 To analyz h mpac of ransacon coss on h rspcv mmunzaon srags I us h mdan ffcv bd-ask sprads of dffrn maury on-h-run U.S. Trasury bonds rpord by Flmng and Sarkar 999 for h yar Ths mdan sprad daa ar duplcad n Tabl 3. Mssng maury sprads ar lnarly nrpolad. Smulad forward ras ar assumd o gnra h md-prc of h coupon bonds. Usng h sprad daa bd and ask prcs of h bonds ar hn drmnd. Throughou h analyss of h mmunzaon srags wh ransacon coss bonds ar bough a ask and sold a bd a ach rbalancng prod. HJM rsk masurs dpnd on volaly paramrs whras radonal rsk masurs do no. Whl comparng h mmunzaon prformancs of rsk masurs h volaly paramrs usd n h smulaon of h forward ras ar also usd wh h HJM rsk masurs. Usng HJM rsk masurs ha dpnd on h volaly paramrs smad n h smulaon rahr han hos usd n smulang h forward ras can affc h mmunzaon prformanc of h rspcv HJM rsk masurs. 25 To xamn h mpac of samplng varably on h prformanc of HJM rsk masurs anohr s of smulaons s run undr 23 I hank h rfr for hs consrucv commn. 24 Flmng and Sarkar 999 us GovPX Inc. daa o drmn h ffcv sprads of on-h-run Trasurs. Flmng and Sarkar gv a dald dscrpon of h daa. 25 I hank h rfr for hs consrucv commn. 9

20 h xponnally dcayng volaly HJM modl and wh no ransacon coss In hs s of smulaons h volaly paramrs of HJM rsk masurs ar smad usng four yars of forward ra daa a ach rbalancng prod. 28 For smulaon purposs h connuous m HJM modls ar dscrzd. An Eulr schm provds a sasfacory dscrzaon mhod for on-facor HJM modls For a gvn dscrzaon of radng nrval [ 0 T] 0 < < < N T h forward ra quaon for a on-facor HJM modl can b dscrzd usng h Eulr schm as follows: 26 Th prformanc of HJM rsk masurs ha dpnd on smad volaly paramrs ar analyzd whn hr ar no ransacon coss. Th nroducon of ransacon coss could alr h rsuls and would no b possbl o arbu h dffrnc n h rsuls solly o h samplng varably corrspondng o h paramr smaon. 27 Th analyss s carrd ou only wh h xponnally dcayng volaly HJM modl. Snc h rspcv fs o h daa of h humpd volaly and xponnally dcayng volaly HJM modls ar ndsngushabl from on anohr h papr focuss manly on h xponnally dcayng volaly HJM modl. Furhrmor snc h humpd volaly HJM modl has mor paramrs h droraon n h mmunzaon prformanc of h corrspondng HJM rsk masurs du o samplng varably should no b lss han ha of h xponnally dcayng volaly HJM rsk masurs. 28 Th forward ra daa usd for paramr smaon s composd of ral and smulad forward ras for h frs 48 monhs. Afr ha consss of forward ras smulad n h prvous m sps. 29 For mul-facor HJM modls mor complx alrnavs should b consdrd for hghr ordr wak convrgnc. On of hs alrnavs s h Mlsn schm dscussd n Klodn and Plan 992 and Jams and Wbr S Klodn and Plan 992 pp and Jams and Wbr 2000 pp For on-facor modls h Eulr schm has sasfacory convrgnc proprs. S Duanmu 994 pp and Jams and Wbr 2000 pp

21 6 ~ f T ~ T ~ ~ f T T v dv + + σ σ + + σ T W W 0... N + ~ Hr f T s h valu of h approxmaon of f T a dscrzaon m. 3 Forward ra voluons ar smulad ovr h holdng prod wh ach slcd onfacor HJM modl Followng Duff and Glynn smulaon rals ar conducd. 34 Thr on-facor HJM modls consan xponnal dcay volaly and humpd volaly hr holdng prods on- fv- and n-yar four nal yld curvs fla ncrasng dcrasng and humpd and wo mmunzaon srags duraon and duraon-convxy machng ar usd o form h porfolos. Th prformancs of mmunzaon srags ar analyzd boh wh and whou ransacon coss. Whn no 3 Snc monhly daa ar usd n hs sudy h m ncrmn + - s akn as on monh. 32 Th random gnraor of L Ecuyr 999 s mplmnd for h gnraon of unform random numbrs snc s mor ffcn han s smlar counrpars. Th polar mhod of Marsagla and Bray 964 s usd o gnra normal random varas from unform random numbrs. To ncras h spd of convrgnc anhc varas ar usd. Varanc rducon chnqus ar dscussd n dal n Nlson and Schmsr 986 Duff and Glynn 995 and Boyl Broad and Glassrman If a smulad forward ra s ngav s droppd and anohr forward ra s smulad. 34 Duff and Glynn 995 provd an algorhm basd on h radoff bwn h dscrzaon and h Mon Carlo avragng rrors. Accordng o h proposd algorhm for a gvn numbr of m sps N h opmal numbr of smulaons s N 2 wh an Eulr schm. In hs papr an Eulr schm s usd for dscrzaon and monhly forward ras ar smulad for up o a n-yar holdng prod. Ths rsuls n a maxmum m sp of 20 for any porfolo smulaon. Thrfor h opmal numbr of smulaons s I us smulaons for ach porfolo rrspcv of h holdng prod. Ths crron sasfs h numbr of smulaons suggsd by Duff and Gylnn

22 ransacon coss ar prsn xponnally dcayng volaly HJM rsk masurs ar also formd usng h volaly paramrs smad n h smulaon from h forward ra daa rahr han hos usd n smulang h forward ras. 35 Ths smulaons rval h condons undr whch h mmunzaon prformancs of h HJM and radonal rsk masurs dffr. IV. Th Immunzaon Prformancs of Inrs Ra Rsk Masurs and Porfolo Formaon Srags Th mmunzaon prformancs of radonal and HJM rsk masurs ar analyzd undr h slcd HJM modls. 36 In ach scon frs h prformanc of random porfolos s dscussd. Thn bull and barbll porfolos ar xamnd. 35 Ths combnaons lad o 44 3*3*4*2*2 scnaros combnd undr all HJM modls usd. Undr h xponnally dcayng HJM modl hr ar an addonal 24 3*4*2 scnaros rlad o h HJM rsk masurs basd on smad paramrs. For ach of hs scnaros on bull on barbll and on hundrd random porfolos ar analyzd gvng a oal of *44+02* Boh radonal and HJM rsk masurs ar usd. Undr h consan volaly HJM modl only wo radonal rsk masurs ar usd snc h HJM rsk masurs rduc o h Fshr-Wl rsk masurs. Undr h rmanng wo HJM modls wo radonal rsk masurs Macaulay and Fshr-Wl and on HJM rsk masur ar usd for mmunzaon. Thus *2 porfolos ar analyzd undr h consan volaly modl and *2*3 porfolos undr h rmanng wo. Undr h xponnally dcayng volaly HJM modl an addonal *3 porfolos ar xamnd. Aloghr hn a oal of porfolos ar xamnd. 36 Only h rsuls corrspondng o h xponnally dcayng volaly HJM modl ar rpord. Th humpd volaly HJM modl producd smlar rsuls. 22

23 A. A Consan Volaly HJM Modl. Th Immunzaon Prformancs of Random Porfolos Undr h consan volaly HJM modl h Fshr-Wl rsk masurs ar bnchmarks bcaus h rlad HJM rsk masurs rduc o hm. Tabl 4 gvs h mmunzaon prformancs of h Macaulay and Fshr-Wl rsk masurs for h consan volaly HJM modl 37. Panls A and B n Tabl 4 gv h rsuls whou and wh ransacon coss rspcvly. Th rsuls n Tabl 4 Panl A suggs ha wh a duraon machng sragy and whn hr s no ransacon cos nvolvd hr s no subsanal dffrnc bwn h mmunzaon prformancs of Macaulay and Fshr-Wl rsk masurs. Lss han 25 prcn of h porfolos hav holdng prod rurns whn on bass pon of h yld on h arg zro-coupon bond. Assumng a lnar rlaon bwn h chang n h bond prc and h nrs ra and hrfor usng a duraon machng sragy lads o a poor mmunzaon prformanc vn f s usd wh h bnchmark nrs ra rsk masur. A duraon and convxy machng sragy mprovs h mmunzaon prformanc of boh rsk masurs. As xpcd h mprovmn s grar wh h bnchmark Fshr-Wl rsk masurs. Wh a duraon and convxy machng sragy 78 o 88 prcn of h porfolos machd wh h Fshr-Wl rsk masurs hav holdng prod rurns whn on bass pon of h arg yld. Wh Macaulay s duraon h prcnag rangs bwn 38 and 53 prcn. Th nonlnar rlaon bwn h changs n h bond prc and h nrs ras has consdrabl mpac on h mmunzaon prformanc. Usng h corrc duraon masur s 37 Th rsuls ovr all scnaros ar summarzd n h abls. Ths rsuls ar also analyzd sparaly for ach of h four nal yld curvs. Th man rsuls ar smlar o h ons rpord. 23

24 no suffcn for mmunzaon. For xampl whn a duraon and convxy machng sragy s usd wh h Macaulay rsk masur around 70 prcn of h porfolos nd up wh holdng prod rurns whn fv bass pons of h arg yld. On h ohr hand whn a duraon machng sragy s usd wh h corrc rsk masur.. h Fshr- Wl rsk masur- lss han 65 prcn of h porfolos nd up wh holdng prod rurns whn fv bass pons of h arg yld. 38 Panl B n Tabl 4 shows h mpac of ransacon coss on boh duraon machd and duraon and convxy machd porfolos. As xpcd whn hr ar ransacon coss h mmunzaon prformanc droras rgardlss of h rsk masurs usd and wh boh duraon machng and duraon and convxy machng srags. Wh a duraon machng sragy h nroducon of ransacon coss dcrass h prcnag of porfolos wh holdng prod rurns whn n bass pon of h arg yld by 8 o 22 prcn rlav o wha s h cas whou ransacon coss. Howvr 33 o 75 prcn of h porfolos sll hav holdng prod rurns whn n bass pon of h arg. Th droraon n h mmunzaon prformanc s subsanal wh a duraon and convxy machng sragy. Wh hs sragy and wh ransacon coss only 22 o 57 prcn of h porfolos hav rurns whn n bass pon of h arg yld. Thus hr s a 32 o 66 prcn dcras n h prcnag of porfolos wh holdng prod rurns whn n bass pons of h arg yld rlav o wha s h cas whou ransacon coss. Nvrhlss for a on-yar holdng prod a duraon and convxy machng sragy sll has a br mmunzaon prformanc han a duraon machng sragy. For a n-yar holdng prod on h ohr 38 Ths rsuls ar no drvn by changs n volals bcaus undr hs HJM modl volaly s consan ovr m. 24

25 hand h mmunzaon prformanc of a duraon machng sragy s br han ha of a duraon and convxy machng sragy. For a fv-yar holdng prod h rspcv prformancs of boh srags ar no vry dffrn from ach ohr. Fshr-Wl rsk masur provds br mmunzaon han Macaulay rsk masur wh ransacon coss as wll. Y h dffrnc n h mmunzaon prformancs of hs wo masurs s no subsanal. Th mpac of ransacon coss on mmunzaon prformanc ncrass wh h lngh of h holdng prod. Snc h amoun of rbalancng s grar wh a duraon and convxy machng sragy han wh a duraon machng sragy h mmunzaon prformanc of h lar s br ovr longr holdng prods. Ovr shor holdng prods howvr alhough ransacon coss ar hghr wh a duraon and convxy machng sragy sll prforms br han a duraon machng sragy. Ths fndng suggss ha h nonlnar rlaon bwn bond prc and yld s mporan; morovr ha hs mporanc s no ovrcom by h mpac of ransacon coss ovr shor holdng prods. In hs rspc a rsk managr should bas hs dcson on whch mmunzaon sragy o us on h holdng prod. Ovr shor holdng prods a duraon and convxy machng sragy s prfrabl o a duraon machng sragy. Ovr mdum and long holdng prods a duraon machng sragy s prfrabl. 2. Th Immunzaon Prformancs of Bull and Barbll Porfolos Tabl 5 shows h mmunzaon prformancs of h bull and barbll porfolos undr h consan volaly HJM modl. Panls A and B n Tabl 5 show h rsuls whou and wh ransacon coss rspcvly. 25

26 As obsrvd n Tabl 5 Panl A whn hr ar no ransacon coss boh bull and barbll porfolos hav holdng prod rurns whn on bass pon of h arg yld for on-yar holdng prod. For shor holdng prods h rurns of bull porfolos ar closr o h arg yld han hos of barbll porfolos. Convrsly for long holdng prods barbll porfolos provd br mmunzaon han bull porfolos. Ths rsuls suppor prvous vdnc concrnng h mmunzaon prformancs of bull and barbll porfolos. 39 Bull porfolos gnrally hav hghr nal ylds han corrspondng barbll porfolos. On h ohr hand barbll porfolos hav hghr convxs han bull porfolos. Th hghr convxy of barbll porfolos s dsrabl whn hr ar larg shfs n h yld curv. If h yld curv shfs ar small du o h yld advanag bull porfolos provd br mmunzaon han barbll porfolos. In hs sudy forward ras ar smulad accordng o h slcd on-facor HJM modls ha dpnd on a normal random shock. For a gvn probably h numbr of larg shocks ncrass wh h lngh of h holdng prod snc h numbr of smulad m sps ncrass. 40 Thrfor s mor lkly o obsrv larg yld curv shfs for a n-yar holdng prod. In hs rspc barbll porfolos ar xpcd o prform br han bull porfolos for a n-yar holdng 39 Ilmann 2000 provds a dald analyss of h rspcv mmunzaon prformancs of bull and barbll porfolos n rms of h yld advanag of h formr and h convxy advanag of h lar. 40 As an xampl consdr ha h uppr and lowr 2.5 prcn of h dsrbuon gvs h larg shfs. Thn h probably of obsrvng a larg shf s 5 prcn. For a on-yar holdng prod hr ar 2 smulad monhly forward ra pahs. Thrfor for hs holdng prod h xpcd numbr of larg shfs s 0.6;.. hr 0 or. For a n-yar holdng prod on h ohr hand hr ar 20 smulad monhly forward ra pahs. Thus h xpcd numbr of larg shfs s 6. 26

27 prod. For a on-yar holdng prod on h ohr hand s lss lkly o obsrv larg shfs n h yld curv. Hnc h yld advanag of bull porfolos ovrcoms h convxy advanag of barbll porfolos for on-yar holdng prod. Whn hr ar no ransacon coss h mmunzaon prformancs of boh bull and barbll porfolos mprov whn a duraon and convxy machng sragy s usd. Wh a duraon and convxy machng sragy h convxy advanag of barbll porfolos should dsappar snc boh h duraon and h convxy of h porfolos ar machd o hos of h arg. Ths rsul s obsrvd wh h bnchmark HJM rsk masurs. Wh radonal rsk masurs howvr barbll porfolos sll hav a slghly br prformanc han bull porfolos for long holdng prods. Snc h radonal convxy masurs ar no h ru convxy masurs corrspondng o h sochasc procss of h forward ras n h smulaons barbll porfolos may sll hav a margnal convxy no capurd by h radonal masurs. As obsrvd n Panl B n Tabl 5 whn hr ar ransacon coss h mmunzaon prformancs of boh bull and barbll porfolos drora as h lngh of h holdng prod ncrass spcally whn a duraon and convxy machng sragy s usd. Whn no ransacon coss ar prsn almos all duraon and convxy machd bull and barbll porfolos hav holdng prod rurns whn fv bass pon of h arg. Whn ransacon coss ar prsn howvr h numbr of duraon and convxy machd bull and barbll porfolos ha hav rurns whn fv bass pons of h arg yld s lss han four n all cass xcp ovr shor holdng prods. For long holdng prods a duraon machng sragy prforms br han a duraon and convxy machng sragy. For shor holdng prods on h ohr hand a duraon and convxy machng sragy sll producs holdng 27

28 prod rurns closr o h arg yld han a duraon machng sragy. Bull porfolos prform br han barbll porfolos ovr shor holdng prods. Ovr mdum and long holdng prods barbll porfolos prform br han bull porfolos wh a duraon machng sragy. Snc h mmunzaon prformanc of a duraon and convxy machng sragy droras subsanally ovr long holdng prods on should prfr duraon machd barbll porfolos ovr mdum and long holdng prods and duraon and convxy machd bull porfolos ovr shor holdng prods. Ovrall h vdnc suggss ha a rsk managr should dcd on mmunzaon and porfolo formaon srags oghr. B. An Exponnally Dcayng Volaly HJM Modl. Th Immunzaon Prformancs of Random Porfolos Tabl 6 shows h prformancs of h Fshr-Wl and HJM rsk masurs undr h xponnally dcayng volaly HJM modl. 4 Panls A and B n Tabl 6 show h mmunzaon prformancs of duraon machd and duraon and convxy machd porfolos whou and wh ransacon coss rspcvly. Panl C n Tabl 6 shows h mmunzaon prformancs of random porfolos whn h HJM rsk masurs dpnd on h smad volaly paramrs rahr han h volaly paramrs usd o smula h forward ra daa. In h xponnally dcayng volaly HJM modl h dcay facors ar smad as bng ngav for h analyzd prods. Thus forward ra volals ar hghr for longr holdng prods. Tabl 6 Panl A shows ha as xpcd hr s som droraon n h mmunzaon prformancs of all rsk masurs for fv- o n-yar holdng prods du o 4 Snc Macaulay and Fshr-Wl rsk masurs hav smlar mmunzaon prformancs only h rsuls of h Fshr-Wl rsk masurs ar rpord hr. 28

29 h hghr volals ovr longr holdng prods. Th bnchmark HJM rsk masurs hav a br prformanc han h Fshr-Wl rsk masurs. Whn hr ar no ransacon coss rrspcv of h nrs ra rsk masur usd wh a duraon machng sragy lss han 28 prcn of h porfolos hav holdng prod rurns whn on bass pon of h arg yld. Wh a duraon and convxy machng sragy h mmunzaon prformanc of h HJM rsk masurs mprovs subsanally. Thr s mprovmn n h prformanc h Fshr-Wl rsk masurs as wll. Wh h HJM rsk masurs mor han 85 prcn of h porfolos hav holdng prod rurns whn on bass pon of h arg yld. Wh radonal rsk masurs 38 o 58 prcn of h porfolos hav holdng prod rurns whn on bass pon of h arg. If a duraon machng sragy s usd and f hr ar no ransacon coss h corrc rsk masur s sll nsuffcn for mmunzaon. Wh hs mmunzaon sragy radonal rsk masurs ar as good as HJM ons. Usng h corrc rsk masur dos hav addonal bnfs whn usng a duraon and convxy machng sragy snc hs sragy ncorporas h nonlnary bwn h chang n h bond prc and h nrs ra. Panl B n Tabl 6 shows ha h nroducon of ransacon coss lads o a droraon n h mmunzaon prformancs of boh duraon machng and duraon and convxy machng srags. Droraon s subsanal whn usng a duraon and convxy machng sragy snc hs sragy rqurs a grar amoun of rbalancng. Ths rsuls suppor h ons corrspondng o h consan volaly HJM modl. Th mpac of ransacon coss on mmunzaon prformanc s mor promnn ovr longr holdng prods. For shor holdng prods a duraon and convxy machng sragy sll provds br mmunzaon han a duraon machng sragy hough h dffrnc s no 29

30 larg. For mdum and long holdng prods a duraon machng sragy has a br mmunzaon prformanc. Knowng h corrc rsk masur has addonal bnfs whn usd wh h appropra mmunzaon sragy. Evn hough HJM rsk masurs prform br han Fshr-Wl rsk masurs whn usd wh h sam mmunzaon sragy hs rsul dos no hold whn hy ar usd wh dffrn mmunzaon srags. If on uss a duraon machng sragy wh h Fshr-Wl rsk masur for a n-yar holdng prod 76 prcn of h porfolos ar whn n bass pons of h arg yld. On h ohr hand f on uss h bnchmark HJM rsk masur wh a duraon and convxy machng sragy for a n-yar holdng prod only 47 prcn of h porfolos ar whn n bass pons of h arg. Thrfor h man concrn of a rsk managr should b choosng h appropra mmunzaon sragy. Noably mmunzaon srags should also b chosn wh rspc o h holdng prod. Th aformnond rsuls compar radonal rsk masurs wh bnchmark HJM rsk masurs whr h lar ar basd on h xponnal dcay paramr usd n h smulaon of h forward ras. If h HJM rsk masurs dpnd on h smad xponnal dcay paramr rahr han h on usd n h smulaon h rsulan samplng varably could affc h prformanc of hs rsk masur. To xamn h mpac of h samplng varably on h prformanc of HJM rsk masurs h xponnal dcay paramr s smad usng four yars of hsorcal forward ra daa a ach rbalancng prod. Th HJM rsk masurs ar drmnd accordng o h smad xponnal dcay paramr. No ransacon cos s assumd. Th rsuls ar gvn n Panl C n Tabl 6. Panl C n Tabl 6 shows ha h prformanc of h Fshr-Wl rsk masur s no dffrn from ha prsnd n Panl A of h sam abl. Ths s xpcd snc hs 30

31 masur dos no dpnd on h paramr smaon. Th prformanc of h HJM rsk masurs droras compard o ha prsnd n Panl A. Wh a duraon machng sragy Fshr-Wl rsk masur provds slghly br mmunzaon han HJM rsk masur for shor and mdum holdng prods. For long holdng prods h mmunzaon prformancs of HJM and Fshr-Wl rsk masurs ar smlar. Wh a duraon and convxy machng sragy HJM rsk masurs provd slghly br mmunzaon han Fshr-Wl rsk masurs hough only for shor holdng prods. For mdum and long holdng prods Fshr-Wl and HJM rsk masurs xhb smlar mmunzaon prformancs. Ths rsuls suggs ha whn samplng varably s accound for HJM rsk masurs ar no supror o radonal rsk masurs. In fac n mos cass h mmunzaon prformanc of h radonal rsk masurs s smlar o or br han ha of h HJM rsk masurs. Ovrall h vdnc suggss ha h prformanc of mmunzaon srags dpnds on h holdng prod and h ransacon coss. Knowng h corrc rsk masur h bnchmark HJM rsk masur wh known volaly paramrs mprovs mmunzaon prformanc only whn s usd wh h appropra mmunzaon sragy. Whn h volaly paramr of h HJM rsk masur s smad rahr han known for cran h samplng varably droras h mmunzaon prformanc of h HJM rsk masur such ha n mos cass h radonal rsk masur ar prfrabl o h HJM rsk masurs. 2. Th Immunzaon Prformancs of Bull and Barbll Porfolos Th vdnc concrnng h mmunzaon prformanc of bull and barbll porfolos undr h xponnally dcayng volaly HJM modl s gvn n Tabl 7. Panls 3

32 A and B n Tabl 7 show h rsuls whou and wh ransacon coss rspcvly. Tabl 7 Panl C shows h rsuls whn h HJM rsk masurs dpnd on h smad volaly paramrs rahr han h volaly paramrs usd o smula h daa. Whn hr ar no ransacon coss h rsuls ar smlar o hos rpord for h consan volaly HJM modl. Tabl 7 Panl A shows ha boh bull and barbll porfolos xhb vry good mmunzaon prformancs for shor holdng prods hough h holdng prod rurns of bull porfolos ar closr o h arg yld. For long holdng prods howvr barbll porfolos hav a br mmunzaon prformanc. Whn hr ar no ransacon coss a duraon and convxy machng sragy mprovs h mmunzaon prformanc for all cass. Addonally h mmunzaon prformancs of boh bull and barbll porfolos ar no vry snsv o h nrs ra rsk masur usd n formng h porfolos. Thrfor radonal rsk masurs rahr han bnchmark HJM rsk masurs can b usd whn formng hm. Thy ar howvr snsv o h lngh of h holdng prod and h mmunzaon sragy adopd. Panl B n Tabl 7 shows ha whn hr ar ransacon coss h mmunzaon prformancs of boh bull and barbll porfolos drora as h lngh of h holdng prod ncrass. Morovr n suppor of prvous vdnc whn ransacon coss ar prsn for boh bull and barbll porfolos a duraon and convxy machng sragy prforms wors han a duraon machng sragy ovr mdum and long holdng prods. Duraon and convxy machd bull porfolos ar prfrabl for shor holdng prods and duraon machd barbll porfolos for long holdng prods. As obsrvd n Panl C n Tabl 7 whn h xponnal dcay paramr of HJM rsk masurs s smad rahr han akn as h on usd n h smulaon of h forward ra 32

33 daa h mmunzaon prformanc of h HJM rsk masur droras rlav o h prformanc prsnd n Panl A. Th prformanc of h Fshr-Wl rsk masur s smlar o ha obsrvd n Panl A snc hs masur dos no dpnd on paramr smaon. Th mmunzaon prformancs of Fshr-Wl and HJM rsk masurs ar smlar. In fac h maxmum rrors corrspondng o Fshr-Wl rsk masur ar slghly lss han hos corrspondng o h HJM rsk masur. Ths rsuls suggs ha on should dcd on h porfolo formaon and mmunzaon sragy oghr wh rspc o h holdng prod and h ransacon coss. Also h mmunzaon prformancs of bull and barbll porfolos ar no vry snsv o h nrs ra rsk masur usd. V. Concluson Th rsuls of hs papr suggs ha for mmunzaon purposs drmnng and usng h corrc rm srucur modl and h corrspondng nrs ra rsk masur s no as ffcv n mmunzaon as usng h rgh mmunzaon sragy. Whn hr ar no ransacon coss wh a duraon machng sragy h prformanc of all rsk masurs droras undr hgh volals. Furhrmor any advanag gand by dnfyng and usng h corrc rm srucur modl and s corrspondng rsk masurs nsad of h radonal ons s margnal whn usng hs mmunzaon sragy. As xpcd a duraon and convxy machng sragy s parcularly bnfcal undr hgh volals. Alhough usng bnchmark rsk masurs lads o br mmunzaon wh a duraon and convxy machng sragy h mmunzaon prformancs of radonal rsk masurs ar clos o ha of h bnchmark masurs for mos cass. Whn HJM rsk masurs dpnd on h smad volaly paramrs rahr han h paramrs usd n h smulaon of h 33

34 forward ra daa radonal rsk masurs provd n mos cass smlar or br mmunzaon prformancs han HJM rsk masurs. Whn ransacon coss ar prsn h mmunzaon prformancs of boh duraon machng and duraon and convxy machng srags drora. Snc a duraon and convxy machng sragy rqurs a grar amoun of rbalancng han a duraon machng sragy h droraon n s mmunzaon prformanc s subsanal spcally for long holdng prods. For shor holdng prods a duraon and convxy machng sragy sll provds br mmunzaon. For mdum and long holdng prods howvr a duraon machng sragy mmunzs br han a duraon and convxy machng sragy. Ths rsul holds rrspcv of h rsk masur usd. Also usng h bnchmark HJM rsk masurs wh an nappropra mmunzaon sragy lads o a wors mmunzaon prformanc han usng a radonal rsk masur wh an appropra mmunzaon sragy. Th mmunzaon prformancs of bull and barbll porfolos ar no vry snsv o h nrs ra rsk masurs. Thy ar howvr snsv o h mmunzaon srags and holdng prods. Whn hr ar no ransacon coss a duraon and convxy machng sragy mprovs h mmunzaon prformanc of hs porfolos. Bull porfolos prform br for shor holdng prods and barbll porfolos prform br for long holdng prods. Whn ransacon coss ar prsn duraon and convxy machd bull porfolos mmunz mor ffcvly for shor holdng prods whl duraon machd barbll porfolos ar mor ffcv for mdum and long holdng prods. Usng radonal rsk masurs n porfolo formaon rahr han HJM rsk masurs provds suffcn mmunzaon. 34

35 Th prsnd vdnc suggss ha mmunzaon srags affc mmunzaon prformanc mor han rsk masurs. Immunzaon and porfolo formaon srags should b chosn oghr accordng o h holdng prod and h ransacon coss. Addonally n mos cass radonal rsk masurs ar prfrabl o HJM rsk masurs whn h paramrs of h lar hav o b smad. Snc radonal rsk masurs do no dpnd on any paramr smaon hy ar fr of any samplng rror ha mgh caus h mmunzaon prformanc of h HJM rsk masurs o drora. 35

36 Rfrncs Amn K. L. and A. Moron. Impld Volaly Funcons n Arbrag Fr Trm Srucur Modls. Journal of Fnancal Economcs Amn K. L. and V. K. Ng. Infrrng Fuur Volaly from h Informaon n Impld Volaly n Eurodollar Opons: A Nw Approach. Rvw of Fnancal Suds 0 Summr Au K. T. and D. C. Thurson. A Nw Class of Duraon Masurs. Economcs Lrs Babbl D. F. Duraon and h Trm Srucur of Inrs Ras Volaly. In Innovaons n Bond Porfolo Managmn: Duraon Analyss and Immunzaon G. G. Kaufman G. O. Brwag and A. Tovs dors Grnwch CT: JAI Prss 983. Brwag G. O. Duraon and Inrs Ra Rsk for a Bnomal Inrs Ra Sochasc Procss. Canadan Journal of Admnsrav Scncs 7 Jun Brwag G. O. G. G. Kaufman and C. M. Laa. Bond Porfolo Immunzaon: Ts of Maury On- and Two-Facor Duraon Machng Srags. Fnancal Rvw 22 May Brwag G. O. G. G. Kaufman R. Schwzr and A. Tovs. Th Ar of Rsk Managmn n Bond Porfolos. Journal of Porfolo Managmn Sprng Brwag G. O. G. G. Kaufman and A. Tovs. Immunzaon Srags for Fundng Mulpl Labls. Journal of Fnancal and Quanav Analyss Blss R. Tsng Trm Srucur Esmaon Mhods. Advancs n Fuurs and Opons Rsarch Boyl P. M. Broad and P. Glassrman. Mon Carlo Mhods for Scury Prcng. Journal of Economc Dynamcs and Conrol Brnnan M. J. and E. S. Schwarz. Duraon Bond Prcng and Porfolo Managmn. In Innovaons n Bond Porfolo Managmn: Duraon Analyss and Immunzaon G. G. Kaufman G. O. Brwag and A. Tovs dors Grnwch CT: JAI Prss 983. Carcano N. and S. Fors. Hdgng Agans Inrs Ra Rsk: Rconsdrng Volaly- Adjusd Immunzaon. Journal of Bankng and Fnanc Chapman D. A. and N. D. Parson. Rcn Advancs n Esmang Trm-Srucur Modls. Fnancal Analyss Journal 57 July/Augus

37 Cox J. J. E. Ingrsoll and S. A. Ross. Duraon and Masurmn of Bass Rsk. Journal of Busnss January Cox J. J. E. Ingrsoll and S. A. Ross. A Thory of h Trm Srucur of Inrs Ras. Ecnomrca 53 March Duff D. and P. Glynn. Effcn Mon Carlo Smulaon of Scury Prcs. Annals of Appld Probably Duanmu Z. Frs Passag Tm Dnsy Approach o Prcng Barrr Opons and Mon Carlo Smulaon of HJM Inrs Ra Modl Ph.D. Dssraon Cornll Unvrsy 994. Fama E. and R. Blss. Th Informaon n Long-Maury Forward Ras. Amrcan Economc Rvw Fshr L. and R. L. Wl. Copng wh h Rsk of Mark Ra Flucuaons: Rurns o Bondholdrs from Nav and Opmal Srags. Journal of Busnss 44 Ocobr Flmng M. and A. Sarkar. Lqudy n U.S. Trasury Spo and Fuurs Marks. In Mark Lqudy: Rsarch Fndngs and Slcd Polcy Implcaons. Rpor No. by h Comm on h Global Fnancal Sysm Bank of Inrnaonal Slmns May 999. Fong H. G. and O. Vasck. Rurn Maxmzaon for Immunzd Porfolos. Innovaons n Bond Porfolo Managmn: Duraon Analyss and Immunzaon G. G. Kaufman G. O. Brwag and A. Tovs dors Grnwch CT: JAI Prss 983. Fruhwrh M. Th Hah-Jarrow-Moron Duraon and Convxy: A Gnralzd Approach. Inrnaonal Journal of Thorcal and Appld Fnanc 5 Novmbr Gulkn N. B. and R. J. Rogalsk. Alrnav Duraon Spcfcaons and h Masurmn of Bass Rsk: Emprcal Tss. Journal of Busnss Hasan I. and S. Sarkar. Banks Opon o Lnd Inrs Ra Snsvy and Crd Avalably. Rvw of Drvavs Rsarch Hah D. R. Jarrow and A. Moron. Bond Prcng and h Trm Srucur of Inrs Ras: A Dscr Tm Approxmaon. Journal of Fnancal and Quanav Analyss 25 Dcmbr Hah D. R. Jarrow and A. Moron. Bond Prcng and h Trm Srucur of Inrs Ras: A Nw Mhodology for Conngn Clams Valuaon. Economrca Hcks J. R. Valu and Capal Cladron Prss Oxford

38 Ho L. Cadl J. and M. Thobald. Esmaon and Hdgng wh a On-Facor Hah- Jarrow-Moron Modl. Journal of Drvavs Summr Ho T. S. Y. and S. L. Trm Srucur Movmns and Prcng Inrs Ra Conngn Clams. Journal of Fnanc 4 Dcmbr Hull J. C. and A. D. Wh. Prcng Inrs-Ra-Drvav Scurs. Rvw of Fnancal Suds Hull J. C. and A. D. Wh. Bond Opon Prcng Basd on a Modl for h Evoluon of Bond Prcs. Advancs n Fuurs and Opons Rsarch Ilmann A. How Wll Dos Duraon Masur Inrs Ra Rsk? Journal of Fxd Incom March Ilmann A. Convxy Bas and h Yld Curv. In Advancd Fxd-Incom Valuaon Tools N. Jgadsh and B. Tuckman dors Nw York NY: John Wly & Sons Inc Jams J. and N. Wbbr. Inrs Ra Modlng John Wly & Sons Ld Jarrow R. A. Modlng Inrs Ra Scurs and Inrs Ra Opons McGraw-Hll Compans Inc Jffry A. Duraon Convxy and Hghr Ordr Hdgng Rvsd. Workng Papr Yal School of Managmn Novmbr Klodn P. E. and E. Plan. Numrcal Soluon of Sochasc Dffrn Equaons Sprngr- Vrlang 992. L Ecuyr P. Good Paramrs and Implmnaons for Combnd Mulpl Rcursv Random Numbr Gnraors. Opraons Rsarch Lrman R. and J. Schnkman. Common Facors Affcng Bond Rurns. Journal of Fxd Incom 3 Dcmbr Macaulay F. R. Som Thorcal Problms Suggsd by h Movmns of Inrs Ras Bond Ylds and Sock Prcs n h Und Sas Snc 856. Columba Unvrsy Prss Nw York 939. Marsagla G. and T. A. Bray. A Convnn Mhod for Gnrang Random Varabls. SIAM Rvw