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

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1 JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSIS VOL. 40, NO. 3, SEPTEMBER 2005 COPYRIGHT 2005, SCHOOL OF BUSINESS ADMINISTRATION, UNIVERSITY OF WASHINGTON, SEATTLE, WA 9895 The Performance of Alernaive Ineres Rae Risk Measures and Immunizaion Sraegies under a Heah-Jarrow-Moron Framework Senay Agca Λ Absrac Using a Mone Carlo simulaion, his sudy addresses he quesion of how radiional risk measures and immunizaion sraegies perform when he erm srucure evolves in a Heah- Jarrow-Moron (992) manner. The resuls sugges ha, for immunizaion purposes, immunizaion sraegies and porfolio formaion sraegies are more imporan han ineres rae risk measures. The performance of immunizaion sraegies depends more on he ransacion coss and he holding period han on he risk measures. Moreover, he immunizaion performance of bulle and barbell porfolios is no very sensiive o ineres rae risk measures. I. Inroducion Over he pas wo decades, developmen of a complicaed se of models for describing he evoluion of he erm srucure of ineres raes has occurred. Research in his area has been driven primarily by he fac ha ineres rae risk managemen is increasingly an imporan componen of he risk managemen pracices of financial insiuions. A he same ime, even he mos sophisicaed financial insiuions coninue o use radiional risk measures, such as he duraion measures developed by Macaulay (938) and Fisher and Weil (97), as well as convexiy measures. While one can model he erm srucure and hen use he corresponding ineres rae risk measures as an alernaive o radiional Λ Agca, sagca@gwu.edu, George Washingon Universiy, 2023 G S NW, Lisner Hall 540G, Washingon, DC This paper is par of my disseraion work a Virginia Tech, and has benefied from commens by Isabelle Bajeux-Besnainou, Randall Billingsley, Sephen Brown (he edior), Marin Day, Narasimhan Jegadeesh, Raman Kumar, Abon Mozumdar, and seminar paricipans a he 2003 Euro/Informs Conference, 200 FMA docoral suden seminar, Freddie Mac, George Washingon Universiy, Rugers Universiy, SUNY-Binghamon, and Virginia Tech. I am especially graeful o Don M. Chance, Saiyid Islam, and Pierluigi Balduzzi (he referee) for deailed and insighful commens. I hank Okan Kolak for echnical assisance and he Naional Crash Analysis Cener a George Washingon Universiy for providing addiional compuing resources. I acknowledge a summer research gran from he School of Business, George Washingon Universiy. Any remaining errors are my responsibiliy. The effec of ineres raes on he banking indusry is rigorously analyzed in he lieraure. Hasan and Sarkar (2002) review he lieraure in his area, and provide an analysis of he effec of ineres raes on bank porfolios. 645

2 646 Journal of Financial and Quaniaive Analysis risk measures, doing so is more cosly in erms of boh human capial and compuing echnology. Thus, he effor is useful only if here are subsanial benefis o be gained in using hese more complex risk measures insead of heir simpler radiional counerpars. This paper compares he immunizaion performance of radiional risk measures wih ha derived from specific erm srucure models. Previous sudies in a similar direcion provide inconclusive evidence. For insance, Cox, Ingersoll, and Ross (979), Brennan and Schwarz (983), and Nelson and Schaefer (983) develop erm srucure models and heir corresponding ineres rae risk measures. They hen compare he immunizaion performance of Macaulay and Fisher-Weil duraion measures wih he immunizaion performance of heir model-specific risk measures. The evidence suggess ha he risk measures of erm srucure models do no offer beer immunizaion performance han heir radiional counerpars. Gulekin and Rogalski (984) es he implicaions of alernaive duraion measures in erms of explaining he reurn variance of U.S. governmen bonds. They find ha none of he duraion measures analyzed are superior o he ohers. Babbel (983) and Carcano and Foresi (997) compare he immunizaion sraegies ha conrol for differing volailiy wih hose ha depend on simple duraion maching sraegies. While Babbel finds ha simple duraion maching sraegies perform as well as he proposed alernaives, Carcano and Foresi provide counerevidence by considering he correlaions beween ineres raes. Ilmanen (992) finds ha simple duraion measures explain 80% o 90% of he reurn variance of U.S. governmen bonds. Finally, Ho, Cadle, and Theobald (200), using Serling fuures daa, show ha he hedging performance of a specific Heah, Jarrow, and Moron (HJM hereafer) model-based risk measure is no beer han ha of modified duraion. These resuls are no comparable since hey are based on differen daa ses, porfolio formaion sraegies, and analysis periods. Addiionally, if he model or sochasic process conjecured o develop he risk measures does no represen he rue sochasic process underlying he daa a cerain imes, i would be hard o inerpre he immunizaion performance of he risk measures. A comparison of he immunizaion performance of risk measures based solely on erm srucure models wih empirical daa amouns o an analysis of which erm srucure bes fis he daa. Moreover, when he performance of cerain risk measures is poor, i is hard o infer he reasons for his poor performance only using empirical daa. To address hese problems, I use a seleced se of one-facor HJM (992) models as a benchmark. 2 Then I carry ou a Mone Carlo simulaion-based sudy o compare he immunizaion performance of radiional risk measures wih ha of benchmark risk measures. I esimae he parameers of he benchmark models using U.S. governmen bond daa, and I use hese parameers hroughou he simulaion. I examine he immunizaion performance boh wih and wihou rans- 2 I focus on one-facor HJM models for several reasons. Firs, radiional risk measures assume one random underlying facor, and herefore i is consisen o compare risk measures ha depend on only one random facor. Also Lierman and Scheinkman (99) and Chapman and Pearson (200) documen ha more han 85% of he variaion in he U.S. Treasury yield curve can be explained by one facor. These auhors refer o his facor as he level of ineres raes. Addiionally, all shor rae models can be represened as specific cases of one-facor HJM models, and shor rae models sill receive considerable aenion in he lieraure.

3 Agca 647 acion coss. Addiionally, o consider he impac of sampling variabiliy on he immunizaion performance of HJM risk measures, I calculae hese measures using parameers esimaed in he simulaion raher han hose used o simulae he daa. 3 I use bulle, barbell, and random porfolios for each scenario so ha he immunizaion performance of he risk measures is no overly dependen on he way he immunizaion porfolios are formed. 4 I consider wo widely used immunizaion sraegies in evaluaing he performance of each risk measure one ha maches only he duraion of he asses and liabiliies, and one ha maches boh heir duraion and convexiy. The evidence of his paper suggess ha major performance differences among immunizaion sraegies depend more on he immunizaion sraegy han on he risk measures. When a duraion maching sraegy is used, immunizaion performance wih radiional ineres rae risk measures is very similar o ha wih HJM risk measures. When here are no ransacion coss, a duraion and convexiy maching sraegy subsanially improves immunizaion performance. When he parameers of he HJM risk measures are esimaed in he simulaion, sampling error affecs he immunizaion performance of he HJM risk measures. Since radiional risk measures do no require any esimaion and hence are free of his sampling error under mos scenarios, hey provide similar or beer performance han HJM risk measures ha are based on esimaed parameers. When here are ransacion coss, a duraion and convexiy maching sraegy performs beer for shor holding periods, whereas a duraion maching sraegy performs beer for medium and long holding periods. The ransacion coss incurred increase wih he lengh of he holding period. Since a duraion maching sraegy incurs relaively fewer ransacion coss han a duraion and convexiy maching sraegy, he former sraegy performs beer over medium o long holding periods. Therefore, when ransacion coss are presen, immunizaion sraegies should be seleced wih respec o holding periods. Among porfolio formaion sraegies, boh bulle and barbell porfolios offer very good immunizaion performance over shor holding periods. Since bulle porfolios generally have a yield advanage over barbell porfolios, heir reurns over shor holding periods are closer o he respecive arge reurns han he corresponding reurns of barbell porfolios. However, as he holding period increases, he convexiy advanage of barbell porfolios overcomes he yield advanage of bulle porfolios. Therefore, over medium and long holding periods, barbell porfolios provide beer immunizaion han bulle porfolios. When here are no ransacion coss, a duraion and convexiy maching sraegy improves he immunizaion performance of boh bulle and barbell porfolios. When ransacion coss are presen, however, duraion and convexiy mached bulle porfolios offer beer performance over shor holding periods, and duraion mached barbell 3 I assume no ransacion coss for his se of simulaions. Since he aim is o examine he impac of sampling variabiliy on he performance of HJM risk measures, inroducion of ransacion coss would make he inference less clear. I would no be possible o associae he differences in performance solely wih he sampling variabiliy. 4 A barbell porfolio consiss of securiies ha have he lowes and highes duraions. A bulle porfolio, on he oher hand, consiss of securiies wih duraions closes o he arge holding period.

4 648 Journal of Financial and Quaniaive Analysis porfolios offer beer performance over long holding periods. In boh cases, he porfolio performance is no very sensiive o ineres rae risk measures. The overall evidence suggess ha immunizaion and porfolio formaion sraegies affec immunizaion performance more han risk measures. Addiionally, he lengh of holding period and ransacion coss has a considerable impac on he immunizaion performance of hese sraegies. HJM risk measures provide beer immunizaion only when used wih he appropriae immunizaion sraegy. When he parameers of he HJM risk measures are no known bu are esimaed, under mos scenarios, radiional risk measures ha do no require esimaion provide a similar or beer immunizaion performance han HJM risk measures. The paper proceeds as follows. Secion II gives a brief descripion of radiional duraion and convexiy measures, oulines he HJM framework, and presens he risk measures developed for specific one-facor HJM models. Secion III describes he simulaion framework. Secion IV compares he immunizaion performance of ineres rae risk measures and porfolio formaion sraegies. Secion V presens he conclusions. II. Ineres Rae Risk Measures A. Tradiional Ineres Rae Risk Measures The developmen of ineres rae risk measures daes back o Macaulay (938), Redingon (952), Hicks (939), and Samuelson (945). These auhors independenly developed an ineres rae risk measure ha assumes a consan yield and, herefore, heoreically provides accurae immunizaion under a fla yield curve and parallel shifs in he yield curve. This ineres rae risk measure is referred o as Macaulay s duraion. Fisher and Weil (97) relaxed he assumpion of a consan yield in Macaulay s duraion and developed a new duraion measure, henceforh referred o as he Fisher-Weil duraion. The discoun facors of he cash flows in he Fisher-Weil duraion are derived from he curren erm srucure. Theoreically, Fisher-Weil duraion provides accurae immunizaion under parallel shifs in he yield curve. These duraion measures a ime are summarized for a coupon bond mauring a T = fi n, wih coupon paymens a ime periods fi i and principal paymen a fi n, and wih price B(; T), 5;6 () D FW () = nx B(; T) C i (fi i )e y(;fii)(fii ) : Here, C i is he coupon paymen a dae fi i, where» fi < fi 2 < ::: < fi n» T, for a rading inerval of [; T]; P(;fi i ) is he price of a zero-coupon bond mauring a ime fi i, and y(;fi i ) is he yield o mauriy of P(;fi i ). In Macaulay s duraion, he yield o mauriy is consan and, herefore, y(;fi i )=y(). Duraion models are based on he assumpion of a linear relaion beween bond prices and ineres raes. Such an assumpion is valid for infiniesimal 5 None of he bonds considered in his sudy conain opions. 6 Since HJM models assume coninuously compounded forward raes, he yield o mauriy is considered coninuously compounded.

5 Agca 649 changes in he ineres rae. For non-infiniesimal changes, he nonlinear relaion beween bond prices and ineres raes mus be aken ino accoun in order o obain an accurae immunizaion. The second derivaive of he bond price wih respec o he ineres rae provides he expression, ofen called convexiy, o be considered in addiion o duraion, (2) Conv FW () = nx B(; T) C i (fi i ) 2 e y(;fii)(fii ) : The convexiy measure corresponding o Macaulay s duraion has a consan yield, y(;fi i )=y(), while ha corresponding o he Fisher-Weil duraion has a varying yield. B. HJM Ineres Rae Risk Measures HJM (992) develop an arbirage-free pricing framework for capuring he evoluion of forward raes and herefore modeling he erm srucure of ineres raes. The specificaion of differen volailiy funcions ogeher wih he inpu of he iniial forward curve resuls in differen erm srucure models. The general HJM model is a muli-facor model. An n-facor HJM model specifies he evoluion of he insananeous forward rae according o he sochasic process, (3) df (; T) = ff(; T; w)d + nx ff i (; T; w)dw i (): Here, f (; T) is he insananeous forward rae a ime for dae T, T > ; dw is Brownian moion a ime ; 7 ff(; T; w) is a random drif erm of he forward rae curve, and ff i (; T; w) is he volailiy funcion i for i = ;:::;n, where T > and w is a sample poin defined on Ω. The no-arbirage condiion pus a resricion on he relaion of drif o volailiies. Under he equivalen maringale measure, he drif resricion and corresponding forward rae process are, respecively, (4) ff(; T; w) = nx ff i (; T; w) Z T ff i (; v; w)dv and (5) df (; T) = 0 n ff i (; T; w) Z T ff i (; v; w)dv A d + nx ff i (; T; w)d W i (): In equaion (5), W() corresponds o he Wiener process under he equivalen maringale measure. In HJM models, differen volailiy funcions lead o dif- 7 A sochasic process W defined for 0on(Ω; F; P) is a Brownian moion if W 0 = 0, W is coninuous wih respec o ; he incremen W s+ W s is disribued as normal N(0; ), under P and is independen of F s.

6 650 Journal of Financial and Quaniaive Analysis feren erm srucure models. 8 The price a ime for a zero-coupon bond mauring a ime T, P(; T), is deermined by he forward rae dynamics, P(; T) = exp( R T f (; u)du). This sudy focuses on one-facor HJM models ha have only one random facor driving he forward rae dynamics. Afer specificaion of he volailiy funcions, he resuling erm srucure model provides he bond pricing framework. This bond pricing framework is used o derive he duraion and convexiy measures corresponding o he seleced HJM models. Au and Thurson (995) derive he duraion measures for cerain coninuousime one-facor HJM erm srucure models. Munk (999) and Jeffrey (2000) use an alernaive mehod o deermine duraion measures for HJM models. 9 The duraion measure for one-facor HJM models, D HJM (), wih deerminisic volailiy is (6) D HJM () = R fii C i ff(; v)dv ff(; ) C i P(;fi i ) P(;fi i ) For consan volailiy, ff(; T) =ff, D HJM () reduces o he Fisher-Weil duraion measure. Following Amin and Moron (994), I only consider one-facor HJM models ha have he ime-invariance propery. The volailiy funcions of hese HJM models depend only on he ime o mauriy, no he calendar dae. 0 In his respec, I consider he consan and exponenially decaying volailiy funcions analyzed by Au and Thurson (995). Addiionally, I analyze he volailiy funcion Mercurio and Moraleda (2000) develop. Mercurio and Moraleda develop a volailiy funcion o represen he humped erm srucure of volailiy someimes observed in he marke. The duraion measures ha correspond o hese one-facor HJM models are as follows. Consan Volailiy (ff(; T) =ff): (7) (8) D HJM () = C i (fi i )P(;fi i ) C i P(;fi i ) Exponenially Decaying Volailiy (ff(; T) =ffe (T ) ): D HJM () = C i P(;fi i )( e (fii ) ) C i P(;fi i ) 8 For example, in one-facor HJM models, if volailiy is consan, i.e., if ff(; T) =ff, he model reduces o he Ho and Lee (986) model. If volailiy is exponenially decaying, i.e., if ff(; T) = ff exp( (T )), he model reduces o he Hull and Whie (993) exended Vasicek model. Oher shor rae models can also be represened as special cases of one-facor HJM models. 9 Bierwag (2000) also derives he duraion measure of he Ho and Lee (986) model, i.e., he consan volailiy HJM model. Bierwag s duraion measure is differen han he one saed in his paper since Bierwag derives he measure for a binomial ineres rae sochasic process. 0 Since I only consider a resriced number of one-facor HJM models, i seems appropriae o selec he models ha have common properies. : ffi : :

7 Agca 65 (9) Humped Volailiy (ff(; T) =ff(+fl(t ))e (T ) ): D HJM () = nx C i P(;fi i ) fl» fl + C i P(;fi i ) e (fii ) (fii )e (fii ) : Following he framework used by Au and Thurson (995), I derive he convexiy measures of one-facor HJM models. 2 Under he HJM framework wih deerminisic volailiies, he convexiy of a coupon bond, Conv HJM,is 3 (0) Conv HJM () = = nx i ) fi i C i ff(; (; ) ff(; ) C i P(;fi i ) 0 NX Z P(;fi i ) C fii 2 B(; T) ff(; ) 2 ff(; v)dva : For consan volailiy, ff(; T) =ff, Conv HJM () reduces o he varying yield convexiy given in equaion (2). The convexiy measures for he seleced onefacor HJM models are summarized below. Consan Volailiy (ff(; T) =ff): () (2) (3) Conv HJM () = nx C i P(;fi i ) C i P(;fi i )(fi i ) 2 : Exponenially Decaying Volailiy (ff(; T) =ffe (T ) ): Conv HJM () = nx P 2 n C i P(;fi i ) C i P(;fi i )( e (fii ) ) 2 : Humped Volailiy (ff(; T) =ff(+fl(t ))e (T ) ): 4 Conv HJM () = fl2 2» fl + nx C i P(;fi i ) C i P(;fi i ) e (fii ) (fii )e (fii ) 2 : The derivaion of he humped volailiy HJM duraion measure is given in he Appendix. 2 Full derivaions are given in he Appendix. 3 Fruhwirh (2002) follows a similar approach in deriving he convexiy measures of one-facor HJM models. My resuls agree wih his. 4 The Appendix gives he derivaion of he humped volailiy HJM convexiy measure.

8 652 Journal of Financial and Quaniaive Analysis III. The Simulaion Framework Forward raes are based on he seleced HJM models; herefore, he risk measures of he HJM models provide he benchmarks. The inpus of he simulaions are esimaed using U.S. governmen bond daa. Forward rae daa up o 30 years of mauriy and he corresponding zero-coupon yield curve daa for he period January 947 o February 99 are obained from McCulloch and Kwon (993). 5 For he period January 947 o February 99, I form yield curve chars for each monh ha I caegorize by shape: fla, declining, increasing, and humped. Then, I selec one represenaive yield curve from each shape. The seleced yield curves are: for fla, Augus 989; for increasing, January 985; for decreasing, May 98; and for humped, Ocober 986. Figure shows hese yield curves and he corresponding forward rae curves. I use he cubic spline mehod o smooh he forward raes and o inerpolae missing monhly forward raes. 6;7 Forward raes corresponding o hese seleced daes are hen used as he iniial forward rae curves in he simulaions. Anoher necessary inpu for he simulaions is he forward rae volailiies. To deermine he volailiies, I use four years of hisorical forward rae daa corresponding o each of he seleced four yield curves. 8 Corresponding forward raes are smoohed and he missing monhly forward raes are inerpolaed using he cubic spline mehod. Then, I calculae changes in he forward raes. The sandard deviaion vecor of differen mauriy forward raes gives he forward rae volailiies. 9 Using regression mehods, I fi forward rae volailiies ino he volailiy funcions of cerain one-facor HJM models o esimae he parameers of each volailiy funcion. I hen use hese esimaed parameers in he simulaion of he forward raes. Table presens he resuls. The exponenial decay facor in boh exponenially decaying volailiy and humped volailiy models is negaive due o he increase in he respecive forward rae volailiies afer 20 years. The linear increase facor for he humped volailiy is posiive bu very small. As a resul, he humped and exponenially decaying volailiy funcions are deermined mainly by he negaive decay facor. In his respec, he forward rae volailiies increase 5 For his daa se, boh he zero-coupon bond yields and he forward raes are derived by McCulloch (975) using a ax-adjused cubic spline mehod. All forward raes are given as percenages per annum on a coninuously compounded basis. All daa are from he afernoon of he las business day of he monh indicaed. For each monh, 56 mauriies are available. The daa are monhly from zero o 8 monhs, hen quarerly o wo years, hen semi-annually o hree years, hen annually o 35 years; he daa hen make a five-year jump o 40 years. 6 Bliss (997) compares differen curve fiing procedures and finds ha he McCulloch (975) cubic spline procedure and he Fama and Bliss (987) boosrapping procedure ouperform oher exising mehods. 7 Since he forward rae daa of McCulloch and Kwon (994) are no available a he monhly frequency afer 8 monhs, I use he cubic spline mehod o inerpolae he missing monhly forward raes. 8 Rebonao ((998), pp. 5 74) and Amin and Ng (997) use four years of hisorical daa in heir volailiy esimaions. 9 Using a principal componen analysis, Lierman and Scheinkman (99) and Chapman and Pearson (200) show ha all variaion in he U.S. Treasury yield curve can be explained by hree facors, and more han 85% of i by one facor alone. Since he focus of his paper is one-facor HJM models, I assume ha all variaion in he yield curve is explained by one facor. In an earlier version, I employ a hree-facor principal componen analysis o esimae volailiies. I hen use he firs facor in he simulaions of he seleced one-facor HJM models. The resuls are similar o he ones repored here.

9 Agca 653 FIGURE Iniial Yield Curves and Forward Rae Curves Zero-Coupon Yield, Augus 989 Forward Raes, Augus 989 Yield Time o mauriy (years) Forward Rae Time o mauriy (years) Zero-Coupon Yield, January 985 Forward Raes, January Time o mauriy (years) Yield Forward Rae Time o mauriy (years) Zero-Coupon Yield, May 98 Forward Raes, May Time o mauriy (years) Yield Forward Rae Time o mauriy (years) Zero-Coupon Yield, Ocober 986 Forward Raes, Ocober 986 Yield Time o mauriy (years) Forward Rae Time o mauriy (years) wih he erm o mauriy in boh models. I plo hese volailiy funcions, along wih heir esimaed parameers, agains he observed forward rae volailiies in Figure 2. While he consan volailiy model fi is poor, he bes fi is for humped and exponenially decaying volailiies. The fis of hese wo volailiy models are indisinguishable from each oher. Hence, in he res of he sudy I focus mainly on he exponenially decaying volailiy HJM model. For he iniial ime period, I derive he zero-coupon bond prices from he forward raes, and I consruc he coupon bonds from he zero-coupon bond prices. To deermine he coupon raes, I use CRSP monhly Treasury daa. For each of he four daes of iniial yield curves, I compue he average coupon rae from he

10 654 Journal of Financial and Quaniaive Analysis TABLE Forward Rae Volailiy Funcions Fla Increasing Decreasing Humped (Aug. 989) (Jan. 985) (May 98) (Oc. 986) Panel A. Consan Volailiy (ff(; T) ff, a consan, and ff(; T)= ˆff + ") ˆff (4.57E 04) (3.0E 04) (8.53E 04) (6.76E 04) Panel B. Exponenial Decay Volailiy (ff(; T) ffe (T ) ;ff,, consans, and ln(ff(; T)) = ln( ˆff) ˆ (T ) + ") ln( ˆff).0769 (2.49E 02) (2.9E 02) (4.6E 02) (2.3E 02) ˆ (.54E 03) (.80E 03) (2.75E 03) (.43E 03) ˆff Panel C. Humped Volailiy (ff(; T) ff( +fl(t ))e (T ) ;ff,, and fl, consans, and ln(ff(; T)) = ln( ˆff) ˆ (T ) +ln( + ˆfl(T )) + ") ln( ˆff).0528 (3.73E 02) (4.37E 02) (6.90E 02) (3.49E 02) ˆ (.54E 03) (2.90E 03) (2.75E 03) (6.56E 03) ˆfl ( ) ( ) ( ) ( ) ˆff The one-facor Heah-Jarrow-Moron (HJM) model specifies he insananeous forward rae as 0 Z T) T C ff(; s)dsa d + ff(; T)d W ; df(; T) = where f(; T) is he insananeous forward rae a ime for dae T, T >, W is a Brownian moion under an equivalen Maringale measure a ime, and ff(; T) is he sochasic volailiy funcion. Three differen volailiy funcions are considered in Panels A, B, and C: consan, exponenially decaying, and humped volailiies, respecively. For each volailiy funcion, he parameers are esimaed using four years of hisorical monhly forward rae daa from McCulloch and Kwon (993). These esimaed volailiies are fied ino he above-menioned hree volailiy funcions in order o esimae he parameers. To esimae he parameers of humped volailiy, he Marquard nonlinear leas squares mehod is used. Oher volailiy parameers are esimaed using he linear regression mehod. Volailiy parameer esimaes are for Augus 989 (fla yield curve), January 985 (increasing yield curve), May 98 (decreasing yield curve), and Ocober 986 (humped yield curve). The volailiy esimaes presened are he annualized volailiies. In each panel, he esimaes of volailiy parameers are presened. Sandard errors are given in parenheses. coupon bonds rading on ha specific dae. I hen use hese average coupon raes o generae he coupon bonds. 20 The immunizaion performance of he ineres rae risk measures are analyzed over one-, five-, and 0-year holding periods. 2 The arge yield is he yield on a zero-coupon bond mauring a he end of he holding period. Table 2 gives he arge yields for he seleced iniial yield curves and holding periods. A porfolio is considered o be successfully immunized if he holding period reurn of he porfolio is close o he yield of he arge zero-coupon bond. 22 Using hese crieria, I analyze porfolios ha have holding period reurns wihin one, five, and 0 basis poins of he arge yield. 20 In Fisher and Weil (97), he coupon rae is fixed a 4%, whereas in Bierwag, Kaufman, Schweizer, and Toevs (98), he coupon rae is fixed a 5%. 2 Fisher and Weil (97) use five-, 0-, and 20-year holding periods. The holding periods in Cox, Ingersoll, and Ross (979) are five, 0, 5, 20, 35, and 50 years. Brennan and Schwarz (983) and Gulekin and Rogalski (984) use five- and 0-year holding periods. Nelson and Schaefer (983) use five-, 0-, and 5-year holding periods. 22 Brennan and Schwarz (983) and Nelson and Schaefer (983) use his sandard.

11 Agca 655 FIGURE 2 Forward Rae Volailiy Funcions Graph A. Forward Rae Volailiy Curve, Augus Volailiy Mauriy (Monhs) acual volailiy consan volailiy exponenial decay volailiy humped volailiy Graph B. Forward Rae Volailiy Curve, January Volailiy Mauriy (Monhs) acual volailiy consan volailiy exponenial decay volailiy humped volailiy

12 656 Journal of Financial and Quaniaive Analysis FIGURE 2 (coninued) Forward Rae Volailiy Funcions Graph C. Forward Rae Volailiy Curve, May Volailiy Mauriy (Monhs) acual volailiy consan volailiy exponenial decay volailiy humped volailiy Graph D. Forward Rae Volailiy Curve, Augus Volailiy Mauriy (Monhs) acual volailiy consan volailiy exponenial decay volailiy humped volailiy

13 Agca 657 TABLE 2 Yields of Targe Zero-Coupon Bond Prices Holding Period Yield One-Year Five-Year Ten-Year Fla (Aug. 989) 8.523% 8.546% 8.465% Increasing (Jan. 985) % %.0990% Decreasing (May 98) % % % Humped (Oc. 986) % % 7.627% Table 2 gives he yields of he arge zero-coupon bonds for each of he iniial yield curves and holding periods analyzed. Four iniial yield curve shapes are considered: fla (Augus 989), increasing (January 985), decreasing (May 98), and humped (Ocober 986). Zero-coupon bond prices are from McCulloch and Kwon (993). Three differen holding periods are considered: shor-erm (one-year), medium-erm (five-year), and long-erm (0-year). The arge bond is he zero-coupon bond ha has mauriy equal o he holding period. The yields of hese zero-coupon bonds are he arge yields used o es he immunizaion performance of alernaive ineres rae risk measures. I focus on he wo mos widely used immunizaion sraegies, namely he duraion maching and he duraion and convexiy maching sraegies. The duraion maching sraegy is expeced o perform well under small yield curve shifs. The duraion and convexiy maching sraegy is expeced o be successful for noninfiniesimal changes in he yield curve as well. The arge zero-coupon bond can be considered as he liabiliy porfolio and he coupon bond porfolio as he asse porfolio. The duraion of a coupon bond porfolio is mached o he duraion of he respecive arge zero-coupon bond o form a duraion mached porfolio. To form a duraion and convexiy mached porfolio, boh he duraion and he convexiy of a coupon bond porfolio are mached o hose of he respecive arge zero-coupon bond. Afer a porfolio is formed, he weighs of he securiies are changed a every rebalancing period o mach he duraion, or he duraion and convexiy, of he arge bond. The rebalancing period is one monh, since monhly forward rae daa are used. The immunizaion porfolios are self-financing. Porfolio formaion sraegies also affec immunizaion performance. To consider he impac of porfolio formaion sraegies on immunizaion performance, a bulle, a barbell, and 00 random porfolios are formed for each scenario. The random porfolios are formed o imiae porfolio formaion sraegies ha canno be observed or sraegies where he porfolio manager has o ac in response o exogenous facors. Uniformly disribued random numbers are used o deermine he mauriy of he coupon bonds in he random porfolios. The mauriies of he bonds in random porfolios vary beween he holding period and 30 years. 23 Two ses of 00 random porfolios are formed; one is for a duraion maching sraegy ha has wo coupon bonds in each porfolio; and he oher one is for a duraion and convexiy maching sraegy ha has hree coupon bonds in each porfolio. I use he same random porfolio se for each HJM model considered in he simulaions. Bulle and barbell porfolios are widely used sraegies. A barbell porfolio consiss of bonds wih he highes and lowes duraions available. Wih a duraion maching sraegy, a barbell porfolio is formed using wo coupon bonds; wih one mauring a he end of he holding period, and he oher mauring in 20 years. Rolling porfolios are no allowed. To mach boh duraion and convex- 23 The minimum mauriy of a coupon bond in he porfolio is he holding period, since rolling over is no allowed.

14 658 Journal of Financial and Quaniaive Analysis iy, an addiional coupon bond is necessary. This addiional coupon bond is a medium-erm bond. For a one-year holding period, I selec a 0-year bond. For five- and 0-year holding periods, he seleced coupon bond has a mauriy of 2 years and 5 years, respecively. A bulle porfolio consiss of bonds ha have duraions close o he arge duraion. As is he case wih oher porfolios, bulle porfolios are formed using wo securiies wih a duraion maching sraegy, and hree securiies wih a duraion and convexiy maching sraegy. 24 To deermine hese bonds, an opimizaion program is run o minimize he absolue difference beween he duraion of a bond porfolio and he arge duraion. The opimizaion program considers bond porfolios consising of all possible combinaions of bonds wih mauriies ranging beween he holding period and 360 monhs. The opimal porfolio has he closes duraion o ha of he arge bond, and hence is considered he bulle porfolio. I form separae bulle porfolios for each one-, five-, and 0-year holding periods, and for each risk measure considered in he sudy Macaulay, Fisher-Weil, and he benchmark HJM risk measures. I examine he immunizaion performance of duraion mached, and duraion and convexiy mached, porfolios boh wih and wihou ransacion coss. Theoreically, duraion and convexiy mached porfolios should provide beer immunizaion han duraion mached porfolios, since he duraion and convexiy maching sraegy akes he nonlineariy beween bond price and yield ino accoun. However, duraion and convexiy mached porfolios incur higher ransacion coss han duraion mached porfolios due o he greaer amoun of bond rebalancing in each period. 25 To analyze he impac of ransacion coss on he respecive immunizaion sraegies, I use he median effecive bid-ask spreads of differen mauriy on-he-run U.S. Treasury bonds repored by Fleming and Sarkar (999) for he year These median spread daa are duplicaed in Table 3. Missing mauriy spreads are linearly inerpolaed. Simulaed forward raes are assumed o generae he mid-price of he coupon bonds. Using he spread daa, bid and ask prices of he bonds are hen deermined. Throughou he analysis of he immunizaion sraegies wih ransacion coss, bonds are bough a ask and sold a bid a each rebalancing period. HJM risk measures depend on volailiy parameers, whereas radiional risk measures do no. While comparing he immunizaion performance of risk measures, he volailiy parameers I use in he simulaion of he forward raes are also used wih he HJM risk measures. Using HJM risk measures ha depend on he volailiy parameers esimaed in he simulaion raher han hose used in simulaing he forward raes can affec he immunizaion performance of he respecive HJM risk measures. 27 To examine he impac of sampling variabiliy on he performance of HJM risk measures, I run anoher se of simulaions under he 24 In Gulekin and Rogalski (984), random immunizaion porfolios are formed from wo o 0 securiies. Brennan and Schwarz (983) use 20 securiies in forming heir immunizaion porfolio. In mos sudies, such as Fisher and Weil (97), Bierwag, Kaufman, and Toevs (983), and Nelson and Schaefer (983), only wo securiies are used in forming he immunizaion porfolio. 25 I hank he referee for his consrucive commen. 26 Fleming and Sarkar (999) use GovPX, Inc. daa o deermine he effecive spreads of on-he-run Treasuries. Fleming and Sarkar give a deailed descripion of he daa. 27 I hank he referee for his consrucive commen.

15 Agca 659 TABLE 3 Median Bid-Ask Spreads in he U.S. Treasury Spo Marke Secor Effecive Spread 3-week bill week bill week bill year noe year noe year noe.00 7-year noe year noe year noe 2.73 Table 3 repors he median effecive spread of on-he-run U.S. Treasuries ha Fleming and Sarkar (999) documen for he year 993. Bid-ask spreads are measured in proporion o he bid-ask midpoin. All spreads are in basis poins. exponenially decaying volailiy HJM model and wih no ransacion coss. 28;29 In his se of simulaions, I esimae he volailiy parameers of HJM risk measures using four years of forward rae daa a each rebalancing period. 30 For simulaion purposes, he coninuous-ime HJM models are discreized. An Euler scheme provides a saisfacory discreizaion mehod for one-facor HJM models. 3;32 For a given discreizaion of rading inerval [ 0 ; T], 0 < < ::: < N = T, he forward rae equaion for a one-facor HJM model can be discreized using he Euler scheme as (4) f ( i+ ; T) = f ( i ; T) + 0 ff(; T) ff(; v)dv A (i+ i ) + ff( i ; T) W i+ W i ; i = 0; ;:::;N: Here, f ( i ; T) is he value of he approximaion of f ( i ; T) a discreizaion ime i The performance of HJM risk measures ha depend on esimaed volailiy parameers are analyzed when here are no ransacion coss. The inroducion of ransacion coss could aler he resuls, and i would no be possible o aribue he difference in he resuls solely o he sampling variabiliy corresponding o he parameer esimaion. 29 The analysis is carried ou only wih he exponenially decaying volailiy HJM model. Since he respecive fis o he daa of he humped volailiy and exponenially decaying volailiy HJM models are indisinguishable from one anoher, he paper focuses mainly on he exponenially decaying volailiy HJM model. Furhermore, since he humped volailiy HJM model has more parameers, he deerioraion in he immunizaion performance of he corresponding HJM risk measures due o sampling variabiliy should no be less han ha of he exponenially decaying volailiy HJM risk measures. 30 The forward rae daa used for parameer esimaion is composed of real and simulaed forward raes for he firs 48 monhs. Afer ha, he daa consiss of forward raes simulaed in he previous ime seps. 3 For muli-facor HJM models, more complex alernaives should be considered for higher order weak convergence. One of hese alernaives is he Milsein scheme, discussed in Kloeden and Plaen (992), pp , and James and Weber (2000), pp For one-facor models, he Euler scheme has saisfacory convergence properies (see Duanmu (994), pp , and James and Weber (2000), pp ). 33 Since I use monhly daa in his sudy, he ime incremen i+ i is aken as one monh.

16 660 Journal of Financial and Quaniaive Analysis I simulae forward rae evoluions over he holding period wih each seleced one-facor HJM model. 34;35 Following Duffie and Glynn (995), I conduc 20,000 simulaion rials. 36 I use hree one-facor HJM models (consan, exponenial decay volailiy, and humped volailiy), hree holding periods (one-, five-, and 0- year), four iniial yield curves (fla, increasing, decreasing, and humped), and wo immunizaion sraegies (duraion and duraion convexiy maching) o form he porfolios. I analyze he performance of immunizaion sraegies boh wih and wihou ransacion coss. When no ransacion coss are presen, exponenially decaying volailiy HJM risk measures are also formed using he volailiy parameers esimaed in he simulaion from he forward rae daa raher han hose used in simulaing he forward raes. 37 These simulaions reveal he condiions under which he immunizaion performance of he HJM and radiional risk measures differ. IV. The Immunizaion Performance of Ineres Rae Risk Measures and Porfolio Formaion Sraegies The immunizaion performance of radiional and HJM risk measures is analyzed under he seleced HJM models. 38 Firs, I discuss he performance of random porfolios. Then, I examine he bulle and barbell porfolios. A. The Immunizaion Performance of Random Porfolios Under a consan volailiy HJM model, I use he Fisher-Weil risk measures as benchmarks because he relaed HJM risk measures reduce o hem. The resuls under a consan volailiy HJM model show no subsanial difference beween he immunizaion performance of Macaulay and Fisher-Weil risk measures. 39 Therefore, he res of his secion focuses on he performance of risk measures and immunizaion sraegies under an exponenially decaying volailiy HJM model. 34 The random number generaor of L Ecuyer (999) is implemened for he generaion of uniform random numbers since i is more efficien han is similar counerpars. The polar mehod of Marsaglia and Bray (964) is used o generae normal random variaes from uniform random numbers. To increase he speed of convergence, I use aniheic variaes. Variance reducion echniques are discussed in deail in Nelson and Schmeiser (986), Duffie and Glynn (995), and Boyle, Broadie, and Glasserman (997). 35 If a simulaed forward rae is negaive, i is dropped and I simulae anoher forward rae. 36 Duffie and Glynn (995) provide an algorihm based on he rade-off beween he discreizaion and he Mone Carlo averaging errors. According o he proposed algorihm, for a given number of ime seps N, he opimal number of simulaions is N 2 wih an Euler scheme. In his paper, an Euler scheme is used for discreizaion, and monhly forward raes are simulaed for up o a 0-year holding period. This resuls in a maximum ime sep of 20 for any porfolio simulaion. Therefore, he opimal number of simulaions is 4,450. I use 20,000 simulaions for each porfolio irrespecive of he holding period. This crierion saisfies he number of simulaions suggesed by Duffie and Gylnn (995). 37 These combinaions lead o a oal of 24,848 porfolios. 38 I repor only he resuls corresponding o he exponenially decaying volailiy HJM model. The humped volailiy HJM model produces similar resuls. 39 These resuls are available from he auhor. As expeced, when here are no ransacion coss, a duraion and convexiy maching sraegy improves he immunizaion performance of boh risk measures. When here are ransacion coss, he immunizaion performance deerioraes regardless of he risk measure and immunizaion sraegy.

17 Agca 66 Table 4 shows he performance of he Fisher-Weil and HJM risk measures under he exponenially decaying volailiy HJM model. 40 Panel A in Table 4 shows he immunizaion performance of duraion mached, and duraion and convexiy mached, porfolios wih ransacion coss. Panel B in Table 4 shows he immunizaion performance of random porfolios when here are no ransacion coss and when he HJM risk measures depend on he esimaed volailiy parameers raher han he volailiy parameers used o simulae he forward rae daa. 4 TABLE 4 Performance of Random Porfolios Fisher-Weil HJM One-Year Five-Year Ten-Year One-Year Five-Year Ten-Year Panel A. (wih ransacion coss) Duraion Mached Porfolios Wihin basis poin 9.00% 7.00% 8.25% 0.50% 9.25% 7.25% Wihin 5 basis poins 33.50% 4.00% 54.75% 37.25% 46.75% 69.75% Wihin 0 basis poins 44.50% 55.50% 76.50% 52.25% 62.00% 79.25% MaxAD 6.57E E E E E E 03 MaxRD 7.4E E E E E E 02 Duraion and Convexiy Mached Porfolios Wihin basis poin 29.00% 2.50%.50% 44.00% 27.25% 22.25% Wihin 5 basis poins 4.50% 24.50% 7.75% 58.00% 40.25% 35.75% Wihin 0 basis poins 46.75% 37.50% 29.00% 63.00% 54.75% 47.00% MaxAD 6.3E E E E E E 03 MaxRD 5.72E E E E E E 02 Panel B. (wih parameer esimaion and no ransacion cos) Duraion Mached Porfolios Wihin basis poin 9.50% 22.75% 9.00% 8.25% 8.75% 6.50% Wihin 5 basis poins 35.75% 45.25% 62.25% 29.70% 40.00% 63.00% Wihin 0 basis poins 49.00% 62.25% 84.25% 44.25% 58.25% 85.00% MaxAD 5.27E E E E E E 03 MaxRD 5.66E E E E E E 02 Duraion and Convexiy Mached Porfolios Wihin basis poin 56.50% 38.50% 40.25% 6.75% 5.25% 49.25% Wihin 5 basis poins 70.75% 65.50% 77.00% 75.75% 65.00% 72.75% Wihin 0 basis poins 76.00% 78.50% 92.50% 84.75% 80.50% 90.25% MaxAD 5.78E E E E E E 03 MaxRD 6.20E E E E E E 02 Table 4 summarizes he immunizaion performance of random porfolios under he exponenially decaying volailiy HJM model. The benchmark risk measures are he exponenially decaying volailiy HJM risk measures. Panel A presens he immunizaion performance wih ransacion coss. In Panel B, he parameers of he HJM risk measures depend on he volailiy parameers esimaed in he simulaion a each rebalancing period raher han he parameers used in simulaing he forward rae daa. Two immunizaion sraegies are considered: a duraion maching sraegy, and a duraion and convexiy maching sraegy. Wih he duraion maching sraegy, porfolio duraions are mached wih he duraion of he arge zero-coupon bond using he HJM and radiional duraion measures. Wih he duraion and convexiy maching sraegy, boh duraion and convexiy of porfolios are mached wih hose of he arge zero-coupon bond by HJM and radiional risk measures. Three holding periods are considered: one-, five-, and 0-year. The arge zero-coupon bonds maure a he end of he respecive holding period. Immunizaion performance is assessed as he absolue difference beween he holding period reurns of porfolios and he yield on he respecive arge zero-coupon bond. The percenages of porfolios ha have holding period reurns wihin one, five, and 0 basis poins of he arge yield are repored. Also, he maximum absolue deviaion (MaxAD) and maximum relaive deviaion (MaxRD) of porfolio holding period reurns from he arge yield are repored. In he exponenially decaying volailiy HJM model, he esimaed decay facors are negaive for he analyzed periods. Thus, forward rae volailiies are 40 Since Macaulay and Fisher-Weil risk measures yield similar immunizaion performance, only he resuls of he Fisher-Weil risk measures are repored here. 4 The performance of random porfolios wihou ransacion coss and wih no parameer esimaion are comparable o he resuls repored in Table 4, Panels A and B. Specifically, he performance of risk measures is comparable o ha of wih ransacion coss are wih no parameer esimaion ha are repored in Table 4, Panel A. The performance of immunizaion sraegies is comparable o ha wihou ransacion coss and wih parameer esimaion repored in Table 4, Panel B.

18 662 Journal of Financial and Quaniaive Analysis higher for longer holding periods. Panel A in Table 4 shows ha, as expeced, here is some deerioraion in he immunizaion performance of all risk measures for five- o 0-year holding periods due o he higher volailiies over longer holding periods. The impac of ransacion coss on immunizaion performance is more prominen over longer holding periods. For shor holding periods, a duraion and convexiy maching sraegy provides beer immunizaion han a duraion maching sraegy, hough he difference is no large. For medium and long holding periods, a duraion maching sraegy produces beer immunizaion performance. Even hough HJM risk measures perform beer han Fisher-Weil risk measures when used wih he same immunizaion sraegy, his resul does no hold when hey are used wih differen immunizaion sraegies. 42 Therefore, a risk manager s main concern should be o choose he appropriae immunizaion sraegy. Noably, immunizaion sraegies should also be chosen wih respec o he holding period. The aforemenioned resuls compare radiional risk measures wih benchmark HJM risk measures where he laer are based on he exponenial decay parameer,, used in he simulaion of he forward raes. If he HJM risk measures depend on he esimaed exponenial decay parameer raher han he one used in he simulaion, he resulan sampling variabiliy could affec he performance of his risk measure. To examine he impac of he sampling variabiliy on he performance of HJM risk measures, he exponenial decay parameer,, is esimaed using four years of hisorical forward rae daa a each rebalancing period. The HJM risk measures are deermined according o he esimaed exponenial decay parameer. No ransacion cos is assumed. Panel B, Table 4 gives he resuls and shows ha wih a duraion maching sraegy, he Fisher-Weil risk measure provides slighly beer immunizaion han he HJM risk measure for shor and medium holding periods. For long holding periods, he immunizaion performance of HJM and Fisher-Weil risk measures is similar. The duraion and convexiy maching sraegy improves immunizaion performance of all risk measures when here are no ransacion coss. Wih a duraion and convexiy maching sraegy, HJM risk measures provide slighly beer immunizaion han Fisher-Weil risk measures, hough only for shor holding periods. For medium and long holding periods, Fisher-Weil and HJM risk measures exhibi similar immunizaion performance. These resuls sugges ha when one accouns for sampling variabiliy, HJM risk measures are no superior o radiional risk measures. In fac, in mos cases he immunizaion performance of he radiional risk measures is similar o or beer han ha of he HJM risk measures. Overall, he evidence suggess ha he performance of immunizaion sraegies depends more on he holding period and he ransacion coss han he risk measures. When he volailiy parameer of he HJM risk measure is esimaed, in mos cases he radiional risk measures are preferable o he HJM risk measures due o he sampling variabiliy in he HJM risk measures. 42 For example, according o Panel A in Table 4, duraion maching sraegy is he appropriae sraegy for long holding periods. If one uses a duraion maching sraegy wih he Fisher-Weil risk measure for a 0-year holding period, 76% of he porfolios are wihin 0 basis poins of he arge yield. On he oher hand, if one uses he benchmark HJM risk measure wih a duraion and convexiy maching sraegy for a 0-year holding period, only 47% of he porfolios are wihin 0 basis poins of he arge.

19 B. The Immunizaion Performance of Bulle and Barbell Porfolios Agca 663 The performance of bulle and barbell porfolios under he consan volailiy HJM model is similar o ha under he exponenially decaying volailiy HJM models wih he Fisher-Weil risk measures. 43 In his respec, he res of his secion examines he performance of bulle and barbell porfolios under he exponenially decaying volailiy HJM model. Table 5 gives evidence concerning he immunizaion performance of bulle and barbell porfolios under an exponenially decaying volailiy HJM model. Panel A of Table 5 shows he resuls wih ransacion coss. Panel B of Table 5 shows he resuls wihou ransacion coss when he HJM risk measures depend on he esimaed volailiy parameers raher han he volailiy parameers used o simulae he daa. 44 Panel A in Table 5 shows ha when ransacion coss are presen for boh bulle and barbell porfolios, a duraion and convexiy maching sraegy performs worse han a duraion maching sraegy over medium and long holding periods. Duraion and convexiy mached bulle porfolios are preferable for shor holding periods and duraion mached barbell porfolios are preferable for long holding periods. The immunizaion performance of Fisher-Weil and HJM risk measures is similar. As Panel B in Table 5 shows, when here are no ransacion coss and when he exponenial decay parameer of HJM risk measures is esimaed raher han aken as he one used in he simulaion of he forward rae daa, he immunizaion performance of he Fisher-Weil risk measure is comparable o ha of he HJM risk measure. In fac, he maximum errors corresponding o he Fisher-Weil risk measure are slighly less han hose corresponding o he HJM risk measure. When here are no ransacion coss, bulle and barbell porfolios exhibi very good immunizaion performance for shor holding periods, hough he holding period reurns of bulle porfolios are closer o he arge yield. For long holding periods, however, barbell porfolios exhibi a beer immunizaion performance. Furhermore, he duraion and convexiy maching sraegy improves immunizaion performance of boh bulle and barbell sraegies. These resuls suppor evidence concerning he immunizaion performance of bulle and barbell porfolios. 45 Bulle porfolios generally exhibi higher iniial yields han corresponding barbell porfolios. Barbell porfolios offer higher convexiies han bulle porfolios. The higher convexiy of barbell porfolios is desirable when here are large shifs in he yield curve. If he yield curve shifs are small, due o he yield advanage, bulle porfolios provide beer immunizaion han barbell porfolios. In his sudy, I simulae forward raes according o he seleced one-facor HJM models ha depend on a normal random shock. For 43 These resuls are available from he auhor. 44 The performance of bulle and barbell porfolios wihou ransacion coss and wih no parameer esimaion are comparable o he resuls repored in Table 5, Panels A and B. Specifically, he performance of risk measures is comparable o ha wih ransacion coss and wih no parameer esimaion ha Table 5, Panel A repors. The performance of immunizaion sraegies is comparable o ha wihou ransacion coss and wih parameer esimaion ha Table 5, Panel B repors. 45 Ilmanen (2000) provides a deailed analysis of he respecive immunizaion performance of bulle and barbell porfolios in erms of he yield advanage of he former and he convexiy advanage of he laer.