THE PERFORMANCE OF ALTERNATIVE INTEREST RATE RISK MEASURES AND IMMUNIZATION STRATEGIES UNDER A HEATH-JARROW-MORTON FRAMEWORK

Transcription

1 THE PERFORMANCE OF ALTERNATIVE INTEREST RATE RISK MEASURES AND IMMUNIZATION STRATEGIES UNDER A HEATH-JARROW-MORTON FRAMEWORK By Şenay Ağca Dsseraon submed o he faculy of Vrgna Polyechnc Insue and Sae Unversy n paral fulfllmen of he degree of Docor of Phlosophy n Fnance Approved By: Don M. Chance, Char Randall Bllngsley Marn Day Raman Kumar Abon Mozumdar March 25, 2002 Blacksburg, VA Keywords: Term Srucure Models, Arbrage-free Prcng, Ineres Rae Rsk Measures, Immunzaon Sraeges, Porfolo Formaon Sraeges

2 THE PERFORMANCE OF ALTERNATIVE INTEREST RATE RISK MEASURES AND IMMUNIZATION STRATEGIES UNDER A HEATH-JARROW-MORTON FRAMEWORK By Şenay Ağca Commee: Don M. Chance, Charman Randall Bllngsley, Marn Day Raman Kumar, Abon Mozumdar ABSTRACT The Heah-Jarrow-Moron HJM model represens he laes n powerful arbragefree echnology for modelng he erm srucure and managng neres rae rsk. Ye rsk managemen sraeges n he form of mmunzaon porfolos usng duraon, convexy, and M-square are sll wdely used n bond porfolo managemen oday. Ths sudy addresses he queson of how radonal rsk measures and mmunzaon sraeges perform when he erm srucure evolves n he HJM manner. Usng Mone Carlo smulaon, I analyze four HJM volaly srucures, four nal erm srucure shapes, hree holdng perods, and wo radonal mmunzaon approaches duraon-machng and duraon-and-convexymachng. I also examne duraon and convexy measures derved specfcally for he HJM framework. In addon I look a wheher porfolos should be consruced randomly, by mnmzng her M-squares or usng barbell or bulle srucures. I assess mmunzaon performance accordng o hree crera. One of hese crera corresponds o acve porfolo managemen, and he oher wo correspond o passve porfolo managemen. Under acve porfolo managemen, an asse porfolo s successfully mmunzed f s holdng perod reurn s greaer han or equal o he holdng perod reurn of he lably porfolo. Under passve

3 porfolo managemen, he closer he reurns of he asse porfolo o he reurns of he lably porfolo, he beer he mmunzaon performance. The resuls of he sudy sugges ha, under he acve porfolo managemen creron, and wh he duraon machng sraegy, HJM and radonal duraon measures have smlar mmunzaon performance when forward rae volales are low. There s a subsanal deeroraon n he mmunzaon performance of radonal rsk measures when here s hgh volaly. Ths deeroraon s no observed wh HJM duraon measures. These resuls could be due o wo facors. Tradonal rsk measures could be poor rsk measures, or he duraon machng sraegy s no he mos approprae mmunzaon approach when here s hgh volaly because yeld curve shfs would ofen be large. Under he acve porfolo managemen creron and wh he duraon and convexy machng sraegy, he mmunzaon performance of radonal rsk measures mproves consderably a he hgh volaly segmens of he yeld curve. The mprovemen n he performance of he HJM rsk measures s no as dramac. The mmunzaon performance of radonal duraon and convexy measures, however, deeroraes a he low volaly segmens of he yeld curve. Ths deeroraon s no observed when HJM rsk measures are used. Overall, wh he duraon and convexy machng sraegy, he mmunzaon performance of porfolos mached wh radonal rsk measures s very close o ha of porfolos mached wh he HJM rsk measures. Ths resul suggess ha he duraon and convexy machng approach should be preferred o duraon machng alone. Also he resul shows ha he underperformance of radonal rsk measures under hgh volaly s no due o her beng poor rsk measures, bu raher due o he reason ha he duraon machng sraegy s no an approprae mmunzaon approach when here s hgh volaly n he marke. Under he passve porfolo managemen crera, he performances of radonal and HJM measures are smlar wh he duraon machng sraegy. Less han 29% of he duraon mached porfolos have reurns whn one bass pon of he arge yeld, whereas almos all are whn 100 bass pons of he arge yeld. These resuls sugges ha he duraon machng sraegy mgh no be suffcen o generae cash flows close o hose of he arge bond. The

4 duraon measure assumes a lnear relaon beween he bond prce and he yeld change, and he nonlneares ha are no capured by he duraon measure mgh be mporan. When he duraon and convexy machng sraegy s used, more han 36% of he porfolos are whn one bass pon of he arge wh HJM rsk measures. Ths dramac mprovemen n he mmunzaon performance of HJM measures s no guaraneed for radonal rsk measures. In fac, here are ceran cases n whch he performance of radonal rsk measures deeroraes wh he duraon and convexy machng sraegy. In hs respec, choosng he correc rsk measure s more mporan han he mmunzaon sraegy when passve porfolo managemen s pursued. Under acve porfolo managemen creron, here s no sgnfcan dfference among bulle, barbell, mnmum M-square, and random porfolos wh boh duraon machng and duraon and convexy machng sraeges. Under he passve porfolo managemen creron, bulle porfolos produce closer reurns o he arge for shor holdng perods when he duraon machng sraegy s used. Wh he duraon and convexy machng sraegy, bulle, barbell and mnmum M-square porfolos produce closer reurns o he arge for shor holdng perods. Random porfolos perform as well as bulle, barbell and mnmum M-square porfolos for medum o long holdng perods. These resuls sugges ha when he duraon machng sraegy s used, bulle porfolos are preferable o oher porfolo formaon sraeges for shor holdng perods. When he duraon and convexy machng sraegy s used, no porfolo formaon sraegy s beer han he oher. Under he acve porfolo managemen creron, mnmum M-square porfolos are successfully mmunzed under each yeld curve shape and volaly srucure consdered. Under he passve porfolo managemen creron, mnmum M-square porfolos perform beer for shor holdng perods, and her performance deeroraes as he holdng perod ncreases, rrespecve of he volaly level. Ths suggess ha he performance of mnmum M-square porfolos s more sensve o he holdng perod raher han he volaly. Therefore, mnmum M-square porfolos would be preferred n he markes when here are large changes n volaly. v

5 Overall, he resuls of he sudy sugges ha, under he acve porfolo managemen creron and wh he duraon machng sraegy, radonal duraon measures underperform her HJM counerpars when forward rae volales are hgh. Wh he duraon and convexy machng sraegy, hs underperformance s no as dramac. Also no parcular porfolo formaon sraegy s beer han he oher under he acve porfolo managemen creron. Under he passve porfolo managemen creron, he duraon machng sraegy s no suffcen o generae cash flows closer o hose of he arge bond. The duraon and convexy machng sraegy, however, leads o subsanal mprovemen n he mmunzaon performance of he HJM rsk measures. Ths mprovemen s no guaraneed for he radonal rsk measures. Under he passve porfolo managemen creron, bulle porfolos are preferred o oher porfolo formaon sraeges for shor holdng perods. For medum o long holdng perods, however, he porfolo formaon sraegy does no sgnfcanly affec mmunzaon performance. Also, he mmunzaon performance of mnmum M-square porfolos s more sensve o he holdng perod raher han he volaly. v

6 DEDICATION I DEDICATE THIS DISSERTATION TO MY GRANDMOTHER ZELİHA INAK WHO PASSED AWAY IN v

7 ACKNOWLEDGEMENTS I would lke o hank my commee char Don M Chance for all hs gudance hroughou he Ph.D. program and helpng me prepare for he job marke. He, as a commee char, a coauhor, and a professor for whom I have worked as a graduae asssan for fve years, was a bg par of my lfe a Vrgna Tech. I have learned a lo n workng wh hm, and I wll always admre hs work ehcs. I would also lke o hank Raman Kumar for hs gudance durng he Ph.D. program as Ph.D. drecor and also as my commee member. Hs commens on my dsseraon as well as npus from hm regardng my job marke preparaon were exremely helpful. I also apprecae hs suggesons and flexbly as Ph.D. drecor n helpng me selec he approprae courses durng he Ph.D. program. I am graeful o Randy Bllngsley, Abon Mozumdar, and Marn Day, for her nsghful commens as my commee members. I really enjoyed workng wh hem. The umos hanks and apprecaons are for my moher Ülkü, my faher Casm, my sser Tülay, and my broher Barõş. They were here whenever I needed hem. Whou her suppor, none of hs would be possble. I am lucky o have such grea famly members. My moher deserves a lo more han my hanks. She s he mos wonderful person I have ever known. Whoever I am, and whaever I have accomplshed, I owe enrely o her. I am blessed o have such a marvelous moher and a role model o follow when and f I have my own chldren. I also wan o hank Sayd Islam. Hopefully, we would be fnshng he Ph.D. program ogeher. We wll always be n ouch alhough he s ryng o run away from me by gong o he Wes Coas. Sayd was a wonderful frend and colleague. Whenever I was no able o reach my famly n Turkey, he was here for me. I canno hank hm enough for hs frendshp and suppor hroughou he years. v

8 Also, my wonderful housemae Özlem Armuçuoğlu deserves my hanks. She helped me a lo, especally durng he job marke process, from selecng wha o wear o soohng my nerves whenever I fel anxous. In her company, I always fel a home and yes, she, for me, s lke a famly member. A specal hanks goes ou o Mahesh Praman. In he early sages of my Ph.D., hs suppor helped me ge ou of several predcamens. Alhough he s very humble no o recognze as anyhng mporan, hs help and suppor was exremely valuable o me. Among all my frends ha I me hroughou he years, I especally wan o hank a few. I am hankful o Tunde for pckng my nervew calls n he offce. Her help was very sgnfcan hroughou my job search. Promod, Al, Fane, and Özge were wonderful asssans who reduced my eachng load wh her valuable help. Andrew s encouragemens durng he job marke were precous. Fnally, I would lke o hank Erk for he hours-long phone conversaons ha helped me hnk and alk abou opcs oher han fnance. v

9 TABLE OF CONTENTS CHAPTER1 INTRODUCTION 1 CHAPTER 2 LITERATURE REVIEW Tradonal Rsk Measures Rsk Measures Derved from Specfc Term Srucure Models Ineres Rae Rsk Measures Derved from Equlbrum Term Srucure Models Heah, Jarrow and Moron Framework Ineres Rae Measures Derved from Specfc HJM Models Emprcal Evdence on he Performance of Alernave Ineres Rae Rsk Measures Porfolo Formaon Sraeges Random Porfolos Bulle and Barbell Porfolos Mnmum M-Square Porfolos 26 CHAPTER 3 TRADITIONAL RISK MEASURES Duraon Summary of Duraon Measures Convexy Summary of Convexy Measures M-Square 32 CHAPTER 4 THE HEATH-JARROW-MORTON FRAMEWORK 35 CHAPTER 5 INTEREST RATE RISK MEASURES OF ONE-FACTOR HEATH-JARROW-MORTON MODELS Duraon Measures of One-Facor HJM Models Summary of Duraon Measures for Ceran One-Facor HJM Models D HJM Convexy Measures of One-Facor HJM Models Summary of Convexy Measures for Ceran One-Facor HJM Models Conv HJM 43 CHAPTER 6 DATA AND METHODOLOGY Forward Rae Daa Inal Yeld Curves 45 x

10 6.3. Forward Rae Volales Coupon Bonds Holdng Perods Immunzaon Crera Porfolo Formaon Sraeges Barbell Porfolos Bulle Porfolos Random Porfolos Mnmum M-Square Porfolos Dscrezaon of One-Facor HJM Models Random Varables Smulaon 79 CHAPTER 7 PERFORMANCE OF DURATION-MATCHED PORTFOLIOS Consan Volaly One-Facor HJM Framework Exponenal Decay Volaly One-Facor HJM Framework Consan Decay Volaly One-Facor HJM Framework Humped Volaly One-Facor HJM Framework Dscusson on he Performances of Duraon Machng Porfolos 117 CHAPTER 8 PERFORMANCE OF DURATION AND CONVEXITY MATCHED PORTFOLIOS Consan Volaly One-Facor HJM Framework Exponenal Volaly One-Facor HJM Framework Consan Decay Volaly One-Facor HJM Framework Humped Volaly One-Facor HJM Framework Dscusson on he Performances of Duraon and Convexy Mached Porfolos 152 CHAPTER 9 PERFORMANCE OF MINIMUM M-SQUARE PORTFOLIOS Consan Volaly One-Facor HJM Framework Exponenal Decay Volaly One-Facor HJM Framework Consan Decay Volaly One-Facor HJM Framework Humped Volaly One-Facor HJM Framework Dscusson of he Performance of Mnmum M-Square Porfolos 175 CHAPTER 10 CONCLUSION 177 REFERENCES 185 APPENDIX x

11 TECHNICAL APPENDIX TECHNICAL APPENDIX VITA 209 x

12 LIST OF TABLES Table 1 Forward Rae Volaly Funcons 55 Table 2 Coupon Raes, Zero Coupon Bond Prces and Yelds.. 67 Table 3 Bulle Porfolos for Duraon Machng Sraegy 72 Table 4 Bulle Porfolos for Duraon and Convexy Machng Sraegy 73 Table 5 Mnmum M-Square Porfolos for Duraon Machng Sraegy. 75 Table 6 Mnmum M-Square Porfolos for Duraon and Convexy Machng Sraegy. 76 Table 7 Performance of Duraon Mached Porfolos under Consan Volaly HJM Framework Table 8 Performance of Duraon Mached Bulle and Barbell Porfolos under Consan 91 Volaly HJM Framework.. Table 9 Performance of Duraon Mached Porfolos under Exponenal Decay Volaly HJM Framework Table 10 Performance of Duraon Mached Bulle and Barbell Porfolos under Exponenal Decay Volaly HJM Framework 100 Table 11 Performance of Duraon Mached Porfolos under Consan Decay Volaly HJM Framework 102 Table 12 Performance of Duraon Mached Bulle and Barbell Porfolos under Consan Decay Volaly HJM Framework. 108 Table 13 Performance of Duraon Mached Porfolos under Humped Volaly HJM 111 Framework.... Table 14 Performance of Duraon Mached Bulle and Barbell Porfolos under Humped Volaly HJM Framework x

13 Table 15 Performance of Duraon and Convexy Mached Porfolos under Consan Volaly HJM Framework Table 16 Performance of Duraon and Convexy Mached Bulle and Barbell Porfolos under Consan Volaly HJM Framework Table 17 Performance of Duraon and Convexy Machng Porfolos under Exponenal Decay Volaly HJM Framework Table 18 Performance of Duraon and Convexy Mached Bulle and Barbell Porfolos under Exponenal Decay Volaly HJM Framework. 136 Table 19 Performance of Duraon and Convexy Mached Porfolos under Consan Decay Volaly HJM Framework 139 Table 20 Performance of Duraon and Convexy Mached Bulle and Barbell Porfolos under Consan Decay Volaly HJM Framework Table 21 Performance of Duraon and Convexy Mached Porfolos under Humped Volaly HJM Framework Table 22 Performance of Duraon and Convexy Mached Bulle and Barbell Porfolos under Humped Volaly HJM Framework Table 23 Performance of Mnmum M-Square Porfolos under Consan Volaly HJM 159 Framework.... Table 24 Performance of Mnmum M-Square Porfolos under Exponenal Decay Volaly HJM Framework Table 25 Performance of Mnmum M-Square Porfolos under Consan Decay Volaly HJM Framework Table 26 Performance of Mnmum M-Square Porfolos under Humped Volaly HJM Framework 172 x

14 LIST OF FIGURES Fgure 1. Inal Yeld Curves and Forward Rae Curves Fgure 2. Forward Rae Volaly Curves 51 Fgure 3. Forward Rae Volaly Funcons.. 60 Fgure 4. Hsogram of Forward Rae Volales. 65 Fgure 5. Yeld Curve Shfs. 81 xv

15 CHAPTER 1 INTRODUCTION Fnancal nsuons nermedae beween pares by exposng hemselves o ceran rsks. One of he major rsks s caused by he msmach of he maures of asses and lables. Banks usually generae funds wh shor-erm lables, whereas her asses are by and large long-erm. The lables of penson funds, on he oher hand, are long erm, and her asses may no mach hese long-erm lables. Lfe nsurance companes sell guaraneed nvesmen conracs by whch hey are oblged o make ceran neres paymens a ceran mes. Therefore nsurance frms wan o make sure ha hey wll have he asses o cover hese lables. When asses and lables do no maure a he same me, hese fnancal nsuons expose hemselves o neres rae rsk. When neres raes change, he response of dfferen maury asses and lables o hs change wll be dfferen. These nsuons wan o guaranee ha hey wll be able o cover her lables wh her asses when neres raes change. The process of asse-lably managemen provdes hs proecon by measurng and managng neres rae rsk. The earles neres rae rsk measure s developed by Macaulay He called hs measure duraon snce s a weghed average of he me o each cash paymen. Hcks 1952 derved he same measure and showed ha duraon measures he change n he prce of a bond when he yeld changes. The duraon measure assumes a lnear relaon beween he bond prce and he yeld change. Therefore, s an approprae rsk measure only when yeld curve shfs are nfnesmal. Addonally, he duraon measure of Macaulay Macaulay s duraon, henceforh s resrcve, snce s heorecally vald only for fla yeld curves and parallel shfs n he yeld curve. Fsher and Wel 1971 relaxed he assumpon of a fla yeld curve and develop anoher duraon measure Fsher-Wel duraon, henceforh. Smlar o Macaulay s duraon, Fsher-Wel duraon assumes parallel yeld curve shfs. These wo rsk 1

16 measures are generally referred o as radonal rsk measures and are sll wdely used by asse-lably managers. Snce duraon assumes a lnear relaon beween he bond prce and he yeld change, does no always perform well when yeld curve shfs are non-nfnesmal. To capure he nonlnear relaon beween he bond prce and he yeld change, anoher measure s used n addon o he duraon. Ths hgher order measure s called convexy. Convexy measures he change n he bass rsk when he yeld changes. I s parcularly mporan when yeld curve shfs are large and, herefore, duraon alone s a poor rsk measure. Alhough hese radonal rsk measures are sll wdely used, over he las decade, we have observed he nroducon of more advanced rsk measures. The rapd developmen n he fxed ncome markes leraure, and avalably and sophscaon of echnology, creaed an opporuny o develop hese new measures. Snce hese rsk measures depend on ceran erm srucure models, erm srucure modelng consues one of he cornersones of neres rae rsk managemen oday. Term srucure models can be caegorzed as equlbrum models and arbrage-free models 1. One of he mos well known arbrage-free models s Heah, Jarrow, and Moron 1992 HJM, henceforh. In HJM models, he nal forward rae curve s exogenously gven and he evoluon of forward raes over me s deermned by a specfc connuous me sochasc process. HJM gves he condons ha have o be sasfed o ensure he exsence of a unque equvalen marngale measure under whch dscouned bond prces are marngales. HJM s a popular model snce s very flexble n erms of he number of random facors ha can be used n he model and dfferen volaly srucures ha can be assumed for dfferen maury forward raes. In one-facor HJM models, here s only one source of randomness. Hence, bond prces are perfecly correlaed wh each oher. Neverheless, one-facor models are sll wdely used, and mos of he avalable models ha are based on a sngle, shor-erm spo rae, called shor rae models, can be represened as specfc cases of one-facor HJM models. An addonal aracon of one-facor models s ha alhough hey are relavely smple wh respec o her 1 Ineresed readers may refer o Ho 1995 and Yan 2001 for a dscusson of equlbrum versus arbrage-free neres rae models 2

17 mul-facor counerpars, Lerman and Schenkman 1991 and Chapman and Pearson 2001 show ha 88% of he varaon n he yeld curve can be explaned by only one facor. Gven ha shor rae models are sll popular, we focus on he rsk measures developed from specfc one-facor HJM models. Snce volaly funcons are mporan deermnans of HJM models, we selec four wdely used forward rae volaly funcons. These are consan, exponenal decay, consan decay, and humped volaly srucures 2. We deermne he duraon and convexy measures correspondng o hese four specfc HJM models 3. Our am s o examne wheher hese more complex bu more advanced rsk measures provde any addonal benef over he smple radonal counerpars. If s so, we would lke o analyze under wha condons hese benefs are observed. Alhough compuaonal effcency has ncreased hrough me, s sll cosly for a frm o mplemen a more advanced rsk measure ha depends on a ceran erm srucure model. The frm should have he necessary compuer nework ha wll make he updaes n he erm srucure parameers avalable mmedaely a dfferen branches a he same me, so ha he same measure can be used for dfferen acves. For example, a bank would lke o use he same neres rae rsk measure n he radng as well as bankng book, and herefore he parameers ha wll be used n he rsk measure should be avalable a he same me for boh he bankng and radng relaed saff. Anoher cos assocaed wh he complex rsk measures s ha specalzed personnel are needed o mplemen he erm srucure models. In hese respecs, he frm has o nves boh n compuer and human resources o use hese complex rsk measures. Therefore, s mporan for a frm o know f s benefcal o ncur hese coss and use complex rsk measures nsead of he smple radonal counerpars. Emprcal sudes of Brennan and Schwarz 1983, Nelson and Schaefer 1983, Gulekn and Rogalsk 1984, and Ho, Cadle and Theobald 2001 compare he performances of radonal rsk measures wh hose of he complex rsk measures derved from ceran erm 2 By choosng specfc volaly funcons, we can exacly specfy he duraon and convexy measures correspondng o he HJM models wh he seleced volaly funcons. We would no be able o specfy hese measures exacly f we were usng he acual erm srucure of volales nsead of specfc volaly srucures. 3 These duraon and convexy measures are eher aken from he exsng leraure or derved f was no avalable n he leraure. 3

18 srucure models. All bu Gulekn and Rogalsk fnd ha radonal rsk measures perform as well as complex counerpars. Gulekn and Rogalsk presen some counerevdence. Some of he reasons for hs mxed evdence are due o dfferen daa ses, dfferen porfolo formaon sraeges, and dfferen analyss perods used by hese sudes as well as possble esmaon and measuremen errors. Ye anoher mporan reason s model rsk, whch s nroduced as sochasc process rsk n Berwag Model rsk s he rsk of selecng a erm srucure model ha does no capure he evoluon of neres raes. Ths can happen f he sochasc process conjecured o develop he duraon measure s no he sochasc process of he real daa. Any devaons from hs conjecure wll affec he resuls of he performance of alernave rsk measures. Under dfferen daa ses and dfferen analyss perods, sochasc process rsk can lead o conflcng emprcal evdence. The porfolo formaon sraeges used n comparave and emprcal sudes are also dfferen and hese dfferences may also affec he mmunzaon performances. The mos common porfolo formaon sraeges consdered are random, bulle and barbell porfolos. To form a random porfolo, secures are seleced wh a random drawng. A barbell porfolo consss of secures ha have he lowes and hghes duraons. A bulle porfolo, on he oher hand, has secures wh he closes duraons o he arge holdng perod 4. Fong and Vascek 1983 propose anoher sraegy. They develop a measure called M-square and show ha f a duraon-mached porfolo has he mnmum M-square, would have he bes mmunzaon performance. When we examne he emprcal sudes ha compare alernave rsk measures, we observe ha some sudes form random porfolos and some use bulle or barbell porfolos. Anoher group of sudes uses mnmum M-square porfolos o conrol he sochasc process rsk. In hs respec, even f he daa se and he analyss perod are he same, emprcal sudes 4 Holdng perod corresponds o he me perod n whch a arge zero-coupon bond s held unl he maury. Therefore, holdng perod s he same as he duraon of a arge zero-coupon bond. In some sudes such as Gulekn and Rogalsk 1984, and Berwag, Kaufman, Laa, and Robers 1987, he erm holdng perod s used as he rebalancng perod. In hs sudy, he duraon of a arge zero-coupon bond, whch s s me o maury, s equal o he holdng perod. 4

19 can show dfferen performances of radonal rsk measures f dfferen porfolo formaon sraeges are used. To parally address hese problems, we carry ou smulaons usng real daa. Inal forward raes and her volales as well as coupon raes of Treasury secures are npus from real daa. Usng hese npus, forward raes are smulaed under he seleced HJM models and he rsk measures correspondng o hese models are used as benchmarks. For each scenaro consdered, one bulle, one barbell, one mnmum M-square, and 100 random porfolos are formed. Ths approach allows us o analyze he effecs of porfolo formaon sraeges on mmunzaon performance. The mmunzaon performance s examned accordng o hree crera. One of hese crera corresponds o acve porfolo managemen and he oher wo correspond o passve porfolo managemen. Under acve porfolo managemen, he am of he asse-lably manager s o ensure ha he reurn of he asse porfolo wll be equal o or greaer han he reurn on he lably porfolo. In hs respec, we consder a porfolo successfully mmunzed under acve porfolo managemen f s reurn s greaer han or equal o ha of he arge bond. Under he passve porfolo managemen creron, he am of an asse-lably manager s o ensure ha he reurn of he asse porfolo s as close as possble o he reurn on he lables. We consder wo crera ha correspond o passve porfolo managemen. We analyze he number of porfolos ha have reurns whn 100 bass pons, and whn one bass pon of he arge bond reurn. Snce we are generang neres rae changes usng he seleced HJM models, he sudy s srongly based n favor of he HJM rsk measures. In hs respec, accepable performance wh radonal rsk measures would sugges ha frms are beer off wh radonal rsk measures raher han he HJM counerpars. On he oher hand, f n mos of he cases, radonal rsk measures underperform HJM rsk measures, hen he frms should prefer HJM rsk measures o radonal ones. Our resuls show ha, accordng o he acve porfolo managemen creron, radonal duraon measures perform as well as her HJM counerpars n 67% of he scenaros consdered. These scenaros correspond o he cases where forward rae volales 5

20 are relavely low 5. Ths resul suggess ha when he yeld curve shfs are small, duraon s an accepable rsk measure. We also confrm he resuls of Lerman and Schenkman 1991, and Chapman and Pearson 2001 n he sense ha capurng he change n he level of he yeld curve by radonal duraon measures s suffcen n 67% of he cases. When he forward rae volales are hgh, whch s observed n he remanng 33% of he cases, he number of successfully mmunzed porfolos formed by HJM rsk measures s a leas 50% more han hose formed by radonal rsk measures. Ths suggess a sgnfcan underperformance of radonal rsk measures. Ths underperformance can be due o wo reasons. Tradonal duraon measures mgh be poor neres rae rsk measures under ceran yeld curve shfs, or machng he porfolos only wh he duraon measure alone mgh no be suffcen when yeld curve shfs are large. When forward rae volales are hgh, we are more lkely o observe more frequen large yeld curve shfs. In hese cases, we mgh consder hgher order rsk measures lke convexy n addon o duraon o capure he nonlnear relaon beween he bond prce and he yeld change. Our resuls show ha when we form porfolos by machng wh boh duraon and convexy, he performances of radonal rsk measures mprove subsanally. We also observe mprovemen n he mmunzaon performance of he HJM rsk measures, bu hs mprovemen s no as grea. On average, he dfference n he number of successfully mmunzed porfolos formed by he HJM and radonal rsk measures vares n a range of 10% o 20%. Our resuls sugges ha here s a rade-off for he frm. If an asse-lably manager who follows an acve porfolo managemen creron forms he porfolos by machng wh boh duraon and convexy, hen here s a possble unceran gan of 10% o 20% f he HJM rsk measures are used. Ths gan s unceran n he sense ha, when HJM rsk measures are used, here are possble esmaon and calbraon problems ha mgh occur durng he mplemenaon of he HJM model. These problems mgh reduce he performance of hese HJM rsk measures. One should keep n mnd ha our resuls are srongly based for he HJM 5 We consder forward rae volaly as low f s less han 2.25%. To deermne hs volaly level, we analyze he forward rae volaly dsrbuon and selec he level ha wll make our resuls he mos conservave. Any volaly level hgher han 2.25% makes he resuls sronger n favor of he radonal neres rae rsk measures. 6

21 rsk measures. In our sudy, we smulae he forward raes accordng o he seleced HJM models, and herefore HJM rsk measures are expeced o perform well. Under he passve porfolo managemen crera, almos all duraon-mached porfolos are whn 100 bass pons of he arge yeld. On he oher hand, less han 29% of hese porfolos are whn one bass pon of he arge yeld wh boh HJM and radonal rsk measures. Ths suggess ha he duraon machng sraegy alone s no suffcen o generae reurns close o ha of he arge yeld. Ths resul may be due o assumng a lnear relaon beween he bond prce and he yeld change by usng only he duraon measure. The nonlneares ha could no be capured by duraon mgh be mporan for a porfolo o have cash flows closer o hose of he arge porfolo. When he porfolos are mached wh boh he HJM duraon and convexy measures, more han 36% of he porfolos have reurns whn one bass pon of he arge. Ths drasc mprovemen of HJM rsk measures s no guaraneed for he radonal rsk measures. There are cases where he performance of radonal rsk measures deeroraes wh he duraon and convexy machng sraegy. In hs respec, under passve porfolo managemen, deermnng he correc rsk measure s more mporan han he selecon of an mmunzaon sraegy. An asse-lably manager wll benef sgnfcanly wh he duraon and convexy machng sraegy f he correc neres rae rsk measure s known. Oherwse, usng he duraon machng sraegy mgh be a beer alernave. As prevously noed, we also analyze he performance of alernave porfolo formaon sraeges. When we examne duraon-mached porfolos, we fnd ha bulle porfolos provde he closes reurns o he arge. For medum o long holdng perods, he mmunzaon performances of bulle, barbell and mnmum M-square porfolos are smlar o hose of he random porfolos. These resuls sugges ha for shor holdng perods, he mos mporan deermnan of he duraon measure s me o maury of he coupon bonds. Snce bulle porfolos generally conss of bonds ha have he closes mes o maury o ha of he arge, hese porfolos have he closes reurns o he arge. When we analyze porfolos ha are mached wh boh duraon and convexy, we do no observe sgnfcan dfferences among he performances of bulle, barbell, and mnmum 7

22 M-square porfolos. All produce very close reurns o he arge for shor holdng perods rrespecve of he rsk measure and he HJM model. For medum o long holdng perods, he performances of hese specfc porfolo formaon sraeges are no sgnfcanly dfferen han he performances of random porfolos. These resuls sugges ha asse-lably managers mgh prefer bulle, barbell, or mnmum M-square wh he duraon and convexy machng sraegy for shor holdng perods. Snce M-square measures he me-o-paymen varance of he bond prce around he duraon, mnmzng he M-square of duraon mached, or duraon and convexy mached porfolos should reduce he varance around he duraon. We expec ha mnmzng M- square should mprove he mmunzaon performance f yeld curve shfs are no capured by he duraon and convexy measures. Our resuls show ha he performance of mnmum M- square porfolos s very sensve o he holdng perod bu no sensve o he volaly level. Even n he cases where volaly s nversely relaed o holdng perod, we observe ha mnmum M-square porfolos produce reurns closer o he arge yeld for shor holdng perods. In hese respecs, asse-lably managers mgh prefer mnmum M-square porfolos o oher porfolo formaon sraeges when he dfferences n volales across he erm srucure are large. Also, all mnmum M-square porfolos are successful rrespecve of he rsk measure used for duraon and convexy. Neverheless, mnmum M-square porfolos formed by HJM rsk measures have closer reurns o he arge han do he ones formed by radonal rsk measures. Therefore, usng he correc neres rae rsk measure sll has addonal benefs, alhough hese benefs are no very large. The comparson of HJM rsk measures wh her smpler counerpars ndcaes he srenghs and weaknesses of radonal rsk measures. The sudy also provdes a beer undersandng of he conrbuon of HJM models o mmunzaon, and wheher he addonal complexy s worhwhle. The dsseraon s organzed as follows. Chaper 2 surveys he leraure on he radonal rsk measures, he HJM framework, he HJM rsk measures and porfolo formaon sraeges. Chaper 3 dscusses radonal rsk measures. Chaper 4 analyzes he HJM framework. Chaper 5 s on he HJM duraon and convexy measures. Chaper 6 descrbes 8

23 he daa and he mehodology. Chaper 7 and Chaper 8 examne he performances of duraon-mached, and duraon- and convexy-mached porfolos, respecvely. Chaper 9 dscusses he resuls for mnmum M-square porfolos. Chaper 10 concludes he dsseraon. 9

24 CHAPTER 2 LITERATURE REVIEW 2.1. Tradonal Rsk Measures The developmen of neres rae rsk measures daes back o Macaulay 1938, Redngon 1952, Hcks 1939, and Samuelson Macaulay nroduced he concep of duraon as a summary measure of he lfe of a bond. Duraon s a weghed average of he me o each cash paymen. The wegh of each paymen s s presen value dvded by he oal presen value of he cash paymens. Independen of Macaulay, Hcks developed he same measure as he elascy of capal value wh respec o he dscoun facor and called he average perod. Whle analyzng he mpac of neres rae changes on he value of fnancal nsuons, Samuelson derved an average me perod, whch corresponds o Macaulay s duraon. In he analyss of lably and asse allocaon of lfe nsurance frms, Redngon developed he same measure and named he mean erm. In he same arcle, Redngon nroduced he erm mmunzaon as machng he mean erm Macaulay s duraon of asses and lables such ha he porfolo wll be mmune o neres rae changes. In all of hese early works, he duraon measure assumes a consan yeld as he dscoun facor. Under a fla yeld curve and parallel shf of neres raes for all erms, Macaulay s duraon provdes an accurae mmunzaon sraegy. Fsher and Wel 1971 relaxed he assumpon of consan yeld n Macaulay s duraon and developed a new duraon measure, henceforh Fsher-Wel duraon. The dscoun facors of he cash flows n Fsher-Wel duraon are derved from he curren erm srucure. Accordng o Fsher and Wel, a bond porfolo s mmunzed agans neres rae changes f he holdng perod reurn of he porfolo s a leas as large as he holdng perod reurn of he arge bond 6. Fsher-Wel duraon does no assume a fla yeld curve and provdes an accurae hedgng sraegy for parallel shfs. 6 Targe bond s he zero coupon bond ha maches he nvesmen preferences of a ceran nvesor. An example gven n he Fsher and Wel sudy s a desre o have $1,000 en years hence by nvesng $ oday. Ths 10

25 Duraon models are based on a lnear relaon beween bond prces and neres raes. Ths assumpon s vald for nfnesmal changes n neres rae. For non-nfnesmal changes he rue non-lnear relaon beween bond prces and neres raes should be aken no accoun o acheve an accurae mmunzaon. In hs respec, he second dervave of he bond prce wh respec o he neres rae provdes he erm, ofen called convexy, o be consdered n addon o duraon. Duraon and convexy ogeher provde a more accurae rsk measure for non-nfnesmal changes n neres raes. In addon o convexy, hgher order duraon measures are proposed o mmunze a bond porfolo for non-parallel shfs as well as parallel shfs. One of hese measures s M- square, whch s developed by Fong and Vascek 1983 as a measure of mmunzaon rsk agans arbrary neres rae changes. M-square measures he weghed varance of he me o paymens around he holdng perod, where he weghs are he presen values of he paymens. Porfolos wh low M-square wll have low exposure o neres rae changes, and vce versa. Thus, has been proposed o choose he porfolo wh mnmum M-square among he duraon-mached porfolos. Berwag, Foolad, and Robers 1993 analyze M-square from he perspecve of he wo-facor model of Berwag 1987 ha uses wo duraon measures. The second order duraon measure of Berwag can be mapped ono M-square. Berwag e al show ha mnmzng M-square corresponds o machng Berwag s second order duraon measure o he square of he holdng horzon Rsk Measures Derved from Specfc Term Srucure Models Afer he Fsher and Wel sudy, he body of knowledge on neres rae rsk measures acceleraed manly n wo drecons. One group of sudes s devoed o he developmen of rsk measures for dfferen and more complex yeld curve shf behavor such as mulplcave, or combned addve and mulplcave shfs. The oher group focuses on specfc erm srucure models and derves he rsk measures mpled by he assumed erm srucure model. Boh of hese groups consder sngle facor and mul facor duraon example corresponds o a arge zero coupon bond ha has maury of en years and promsed yeld o maury of 6% per annum. The arge bond s free of defaul rsk and s no callable. 11

26 measures. In hs sudy, we focus on he rsk measures derved from specfc neres rae models 7. As surveyed by Ho 1995 and Yan 2001, erm srucure models can be caegorzed as equlbrum models and arbrage-free prcng models. Equlbrum models specfy he marke prce of rsk and esmae he parameers of he model from hsorcal daa on neres raes usng curren marke fundamenal or a combnaon of hsorcal and fundamenal analyss. These models use he esmaed marke prce of rsk o value bonds. Therefore, bond prces deermned by equlbrum models may no mach curren bond prces. These models are useful for forecasng purposes as well as percevng possble msprcng n bonds. They are, however, no praccal for prcng purposes. To prce bonds and oher neres rae producs, we prefer arbrage-free prcng models. Arbrage-free prcng models assume ha he underlyng secures are correcly prced n he marke and ake he prces of he underlyng secures as gven. These models are calbraed o he observed prces. Snce bond prces deermned by arbrage-free prcng models do no dffer from he observed prces, hese models are preferred for prcng purposes. One of he mos wdely referred arbrage-free prcng model s Heah, Jarrow, and Moron Heah, Jarrow, and Moron develop a general arbrage-free prcng framework ha depends on he evoluon of he nsananeous forward rae curve. HJM framework s appealng because of s flexbly o ncorporae dfferen volaly srucures and dfferen random facors. In hs sudy, we focus on a class of one-facor HJM models. For compleeness, however, we revew neres rae rsk measures derved from boh equlbrum and arbragefree models. In hs respec, he followng subsecons are organzed as follows. Frs, we revew he leraure on he rsk measures derved from equlbrum models. Then we survey exsng leraure on HJM models. The las subsecon s devoed o he rsk measures derved from HJM models. 7 The reader neresed n he leraure relaed o he rsk measures developed for ceran yeld curve behavor can refer o Berwag, Kaufman, and Toevs

27 Ineres Rae Rsk Measures Derved from Equlbrum Term Srucure Models Ingersoll, Skelon, and Wel 1978 nroduce bass rsk as he relave change n he bond prce for an unexpeced change n he neres raes, ceers parbus. The spo neres rae s aken as a sngle sochasc facor dervng he erm srucure. The neres rae s assumed o follow he sochasc process: dr = [ γ + k α r ] d + σdw, 1 where dγ = βγ d + ησdw Here, α s he drf, k s he pullng facor, γ s an exrapolave velocy, and W s a sandard Brownan moon or Wener process. Under hs sochasc, process Ingersoll e al derve he measure for he bass rsk. The well-known Vascek 1977 model s a specal case of he above where γ, β, and η equal zero. Accordng o he developed measure, f bass rsk ncreases lnearly wh maury, Fsher-Wel duraon s an accurae bass rsk measure. If bass rsk ncreases decreases wh maury a an ncreasng rae, Fsher-Wel duraon s hgher han he bass rsk of shor long maury bonds. Cox, Ingersoll, and Ross CIR, henceforh 1979 nroduce he erm sochasc duraon o measure he relave bass rsk of bonds. Sochasc duraon s defned as he me o maury of a zero coupon bond wh he same bass rsk as he coupon bond. Sochasc duraon accommodaes mulple neres rae shocks for boh shape and locaon changes n he yeld curve. The posulaed sochasc process s he square roo process of CIR 1985 where he spo rae s he only underlyng sochasc facor s dr = k α r d + σ r dw 2 13

28 CIR 1979 deermne he equaon for prcng zero coupon bonds conssen wh he above sochasc process. Ther sochasc duraon measure s aaned by akng he paral dervave of he bond prce wh respec o he spo rae and dvdng by he bond prce. The sochasc duraon measure depends on he parameers of he above saed neres rae process as well as he lqudy preferences of ndvduals. Brennan and Schwarz 1983 develop a wo-facor equlbrum model o deermne he bond prce. The wo facors are he spo rae, r, and he yeld on a consol bond, l. The sochasc processes of hese wo facors are jonly deermned: dr = α r, l, d + σ r, l, dw dl = α r, l, d + σ r, l, dw, where dw 1 2 dw = ρd In hs sochasc process, α 1 and α 2 are he drfs, andσ 1 and σ 2 are he volales of he spo rae and consol yeld, respecvely. Boh drf and volaly are funcons of me, spo rae, and consol yeld. The dervaon of he bond prce accordng o above sochasc processes yelds a paral dfferenal equaon wh he boundary condon ha he bond prce a maury s s face value. The approprae hedgng sraegy s o consruc he bond porfolo such ha s esmaed response he changes n he spo rae and consol yeld wll be he same as ha of a dscoun bond wh he approprae maury. Nelson and Schaefer 1983 develop a K-facor model. The zero coupon bond prce s a funcon of hese K facors, as well as me, and me o maury. The facors are assumed o follow Gauss-Wener processes. Each facor k has drf of α k ha s a funcon of me and maury, volaly of σ k, and correlaon ρ jk wh facor j. The sensvy of a bond s reurn o facor k s consdered o be a generalzaon of duraon. To mmunze a porfolo wh he K facor model, K+1 bonds are requred. Immunzaon s acheved when he weghed sum of he sensves of each bond reurn wh respec o facor k s equal o he sensvy of he arge asse s reurn o facor k. Ths condon should be sasfed for all K facors, and he 14

29 weghs n he summaon should no change from one facor o he oher. In hs case, he more facors here are, he more wll be he resrcons on he mmunzaon sraegy. Ths may lm he usefulness of he proposed sraegy. Nelson and Schaefer analyze a wo-facor sraegy where he facors are he spo rae and he spread beween he spo rae and he long rae. Ths wo-facor sraegy s based on he fve-year rae, and he spread beween he hreen-year rae and he fve-year rae. The sochasc process of he wo-facor model of Nelson and Schaefer s as follows: dl = α d + σ dw ds = k α S d + σ dw where dw 1 dw2 = 0 In hs wo-facor model, α 1 andα 2 are he drfs of a long rae and he spread beween a long and a shor rae, respecvely. The volales of hese wo facors are gven by σ1 and σ 2. The spread s mean reverng, and he pull facor s k. Boh nnovaons underlyng he sochasc processes of he long rae and he spread are drven by Wener processes. Noe ha he Wener processes are uncorrelaed Heah, Jarrow and Moron Framework Heah, Jarrow, and Moron 1992 esablsh an arbrage-free prcng framework based on he evoluon of nsananeous forward raes over me. In HJM models, he nal forward rae curve s exogenously gven, and here s a drf resrcon ha mus be sasfed o guaranee he no arbrage condon. HJM s a popular model snce s very flexble n erms of he number of random facors ha can be used n he model and dfferen volaly srucures ha can be assumed for dfferen maury forward raes 8. 8 The HJM framework and one-facor HJM models are analyzed n deal n Chaper 4. Therefore, we do no presen any mahemacal deals relaed o he HJM models n hs secon 15

30 In one-facor HJM models, here s only one source of randomness, a smple Brownan moon ha drves all forward raes. Therefore, yelds, bond prces, and changes n forward raes are perfecly correlaed n one-facor models. Neverheless, one-facor HJM models are flexble n he sense ha all shor rae models can be represened as specal cases of one-facor HJM models. Baxer and Renne show Ho and Lee 1986, Vascek 1977, Hull and Whe 1993, Cox, Ingersoll, and Ross 1985, and Black and Karasnsk 1991 models as specal cases of one-facor HJM models. Mul-facor HJM models nvolve mulple Brownan moons, whch allows dfferen bonds o be affeced by shocks n dfferen ways. Recen developmens relaed o HJM models are random feld models of Kennedy 1994, Kennedy 1997, and Goldsen 2000, and he sochasc srng model of Sana-Clara and Sornee In hese models, nsananeous forward raes are mperfecly correlaed wh he forward raes of dfferen maures and each forward rae has s own shock. Anoher exenson s he jump dffuson HJM models. Björk e al 1997 dscuss he HJM mehodology for jump dffusons and derve he condons under whch arbrage s no permed. Colwell 2001 exends he consan volaly one-facor HJM model 10 o a jump dffuson process and dscusses he assumpons under whch opon prces have explc formulas. Alhough HJM models are flexble enough o conan varous erm srucure models as specal cases, hey are no whou lmaons. The man resrcon comes from he fac ha he larger he number of facors n he erm srucure model, he greaer are he esmaon and mplemenaon problems. Only ceran cases of one-facor HJM models are Markov 11. Therefore, n mul-facor HJM models and n mos cases of one-facor HJM models, he mulnomal and bnomal rees, whch are ofen used for fng he model, may no recombne and compuaonal effcency can be low. Shor rae models can be represened as one-facor HJM models. Specfc volaly funcons n he one-facor HJM models lead o ceran shor rae models ha are popular n 9 An excellen overvew of he popular shor rae models n he HJM erms can be found on pages of Baxer and Renne Ths s he same as Ho and Lee 1986 model. 11 Carverhll 1994, Jeffrey 1995, Rchken and Sankarasubramanan 1995, Blss and Rchken 1996, and Inu and Kjma 1998 mpose ceran resrcons on he volaly srucure of he HJM models o assure ha HJM models are Markovan. 16

31 he marke. As menoned n Yan and Chapman and Pearson , mos of he recen academc leraure focuses on one-facor models. As documened n Lerman and Schenkman % of he varaon n U.S. Treasury raes can be explaned by he frs facor, generally beleved o be he level of neres raes. Smlar o Lerman and Schenkman, Chapman and Pearson 2001 carry ou prncpal componen analyss on weekly changes n Treasury yelds, and suppor he fndngs of Lerman and Schenkman. Chapman and Pearson fnd ha hree facors explan 99% of yeld curve varaon bu frs facor alone explans 88% of varaon. Flesaker 1993 ess he performance of a one-facor consan volaly HJM model o explan he cross-seconal varaon n daly marke prces of Eurodollar opons. Flesaker fnds ha shor-erm opons are overvalued relave o long erm opons, exacly oppose of he mean reverson feaures of Vascek 1977 and Cox, Ingersoll, and Ross 1985 models. Flesaker suggess ha one should allow a humped volaly forward rae erm srucure. Mercuro and Moraleda 2000 develop a volaly funcon for nsananeous forward raes ha provdes a humped volaly srucure n he HJM framework. They compare hs model wh an exponenal decay volaly HJM model by esmang mpled volales usng caps and floors. The prcng errors of he humped volaly HJM model are found o be less han hose of he exponenal decay HJM model. Bühler e al 1999 analyze he prcng errors of wo one-facor and wo wo-facor HJM models, a spo rae model and wo wo-facor spo rae models usng German daa on neres rae warrans. They fnd ha he model wh lowes absolue devaon s he lnear-proporonal one-facor HJM model. One-facor HJM models are also used o esmae mpled volales. Amn and Moron 1994 fnd mpled neres rae volales of sx one-facor HJM models usng Eurodollar opons. They consder he Ho and Lee 1986, square roo, proporonal, lnear absolue, exponenal decay and lnear proporonal models. The frs hree are one-parameer and he las hree are wo-parameer models. They fnd ha wo-parameer models f prces beer bu one-parameer esmaes are more sable. Among one-parameer models, he Ho and 12 Yan 2001 revews dynamc defaul erm srucure models. 13 Chapman and Pearson 2001 dscuss he esmaon of erm srucure models and provde an excellen revew of recen advances n research relaed o shor rae models. 17

32 Lee model performs beer. Among wo-parameer models, he lnear proporonal model s beer. Amn and Ng 1997 esmae mpled volales from Eurodollar opons prces by ncorporang dfferen spo rae models n he HJM framework 14. They fnd ha hese mpled volaly esmaes explan mos of he realzed neres rae volaly. They also fnd ha, he Vascek 1977 and lnear proporonal models perform beer han ohers. Moraleda and Pelsser 2000 compare hree spo rae models and hree Markov forward rae models usng hree-monh caps and floors 15. They fnd ha spo rae models provde a beer f o he cap and floor prces han he forward rae models. Gupa and Subrahmanyam 2000 use one-facor models o analyze convexy bas n prcng neres rae swaps. The convexy bas s he upward bas n he mpled swap raes ha s a resul of usng Eurodollar fuures raes ha are hgher han he forward raes ha should be used o deermne he swap raes. They consder he Vascek, Cox-Ingersoll-Ross, Hull-Whe, and Black-Karasnsk models. They also consder a wo-facor HJM model. Convexy curves are smlar for he Hull-Whe, and Black-Karasnsk models. The Vascek model produced a convexy curve close o ha of he Hull-Whe model for up o hree years of maury. Snce one-facor models are sll wdely used and shor rae models can be represened as one-facor HJM models, we focus on one-facor HJM models. We consder consan volaly Ho and Lee 1986 and exponenal decay volaly models Vascek 1977, a consan decay volaly model used n Au and Thurson 1995, and he humped volaly model proposed by Mercuro and Moraleda Ineres Rae Measures Derved from Specfc HJM Models Au and Thurson 1995 develop duraon measures for ceran one-facor HJM models. Assumng ha all bond prces are funcons of he spo rae, Au and Thurson derve 14 The spo rae models ncorporaed n he HJM framework are Couradon 1982, Cox, Ingersoll, and Ross 1985, Ho and Lee 1986, Vascek 1977 and he lnear proporonal model of Amn and Moron The spo rae models consdered n he sudy are Hull and Whe 1994, Black and Karasnsk 1991 and Pelsser They use L e. al characerzaon for Markovan forward rae models and hey analyze forward rae models correspondng o Hull and Whe 1994, square roo model and a proporonal volaly model. 18

33 he bond prce process of he forward rae dynamcs of one-facor HJM models. The resulng expresson shows ha he bass rsk of a coupon bond derved from an HJM model s a funcon of he forward rae volaly. Usng hs bass rsk expresson, Au and Thurson derve he correspondng duraon measure for a coupon bond. The HJM duraon measure developed by Au and Thurson s a funcon of forward rae volaly, coupon paymens and he zero coupon bond prce. Snce he HJM duraon s a funcon of forward rae volaly, HJM duraon measures dffer for dfferen volaly funcons. Au and Thurson specfcally derve he duraon measures for consan, consan decay, and exponenal decay volaly srucures. Jeffrey 2000 also shows he relaon of duraon measures o he forward rae volaly srucure of HJM models. Jeffrey provdes schemes o deermne duraon measures for sngle-facor HJM models. Alhough he approach of Jeffrey s dfferen han ha of Au and Thurson, boh arrve a he same HJM duraon measure. Jeffrey derves he sochasc process of a ceran maury asse and lably by assurng ha he asse-lably porfolo s self-fnancng. Then Jeffrey maches he bass rsk of he asse o he bass rsk of he lably. Replacng he asse porfolo wh a zero coupon bond maurng a a specfc dae provdes he duraon measure for he one-facor HJM models. Alhough duraon measures of ceran one-facor HJM models are avalable n he leraure, he convexy measures of one-facor HJM models are no avalable n he leraure. Therefore, we exend he approach of Au and Thurson and derve he convexy measures of one-facor HJM models. Fruhwrh 2001 also follows a smlar procedure and arrves a he same convexy measures wh ours. 2.3 Emprcal Evdence on he Performance of Alernave Ineres Rae Rsk Measures Theorecally, Macaulay s duraon s accurae under a fla erm srucure and parallel shfs, and Fsher-Wel duraon s a vald rsk measure for parallel shfs. Emprcal evdence, however, s mxed on he performance of hese measures. 19