WP 2000-9 Opimal Consumpion and Invesmen Sraegies wih Sochasic Ineres Raes af Claus Munk & Carsen Sørensen INSTITUT FOR FINANSIERING, Handelshøjskolen i København Solbjerg Plads 3, 2000 Frederiksberg C lf.: 38 15 36 15 fax: 38 15 36 00 DEPARTMENT OF FINANCE, Copenhagen Business School Solbjerg Plads 3, DK - 2000 Frederiksberg C, Denmark Phone (+45)38153615, Fax (+45)38153600 www.cbs.dk/deparmens/finance ISBN 87-90705-43-2 ISSN 0903-0352
Opimal Consumpion and Invesmen Sraegies wih Sochasic Ineres Raes Λ Claus Munk Dep. of Accouning, Finance & Law SDU Odense Universiy Campusvej 55 DK-5230 Odense M Denmark Carsen Sfirensen Deparmen offinance Copenhagen Business School Solbjerg Plads 3 DK-2000 Frederiksberg Denmark Λ We hank for commens and suggesions from seminar paricipans a Copenhagen Business School. Boh auhors graefully acknowledge financial suppor from he Danish Research Council for Social Sciences.
Opimal Consumpion and Invesmen Sraegies wih Sochasic Ineres Raes Absrac We sudy he consumpion and invesmen choice of a ime-addiive power uiliy invesor and demonsrae how heinvesor should opimally hedge changes in he opporuniy se. The invesor is allowed o inves in socks and ineres rae dependen asses in a coninuous-ime dynamically complee marke. In paricular, we demonsrae ha under sochasic ineres raes he invesor opimally hedges changes in he erm srucure of ineres raes by invesing inacoupon bond, or porfolio of bonds, wih a paymen schedule ha maches he forward-expeced (i.e cerainy equivalen) consumpion paern. This is of concepual imporance since he hedge porfolio does no depend on he specific erm srucure dynamics (only hrough he consequences for he opimal consumpion paern). We consider wo explici examples where he dynamics of he erm srucure of ineres raes are given by hevasicek-model and a hree-facor non-markovian Heah-Jarrow-Moron model.
1 Inroducion In his paper, we esablish specific resuls on how invesors in a coninuous-ime model should opimally hedge changes in he invesmen opporuniy se, in he sense of Meron (1971,1973). In paricular, under specialized (Gaussian) erm srucure dynamics and dynamically complee markes, we demonsrae ha he opimal way o hedge changes in he erm srucure of ineres raes is by invesing in a coupon bond, or porfolio of bonds, wih a paymen schedule ha maches he forward-expeced (i.e cerainy equivalen) consumpion paern. This provides a concepual way o hink abou how ohedgechanges in he opporuniy se where he focus is on he opimal consumpion paern and no he assumed dynamics of he opporuniy se. We consider he opimal consumpion and invesmen problem of an invesor wih imeaddiive power uiliy, i.e. wih a Consan Relaive Risk Aversion (CCRA) uiliy funcion. Throughou he paper we allow for non-markovian dynamics of asse prices and he erm srucure of ineres raes. In paricular, under specialized Gaussian erm srucure dynamics we use he framework of Heah, Jarrow and Moron (1992) (HJM) which as inpu uses he curren form of he erm srucure of ineres raes. This allows us o address quesions like: Is he curren form of he erm srucure imporan forhow o hedge changes in he opporuniy se or does only he dynamics of he erm srucure maer? We address his quesion by considering wo explici examples based on he Vasicek (1977) model as well as a HJM hree-facor model where he erm srucure can exhibi hree kinds of changes: a parallel shif, a slope change, and a curvaure change. Our resuls sugges ha he form of he iniial erm srucure of ineres raes is of crucial imporance for he (forward-expeced) consumpion paern and, hence, imporan for he relevan bond o hedge changes in he opporuniy se while he specific dynamics of he erm srucure is of minor imporance. Our specific resuls and examples also provide a conribuion on is own since hese are, o our knowledge, he firs explici resuls o an ineremporal consumpion and invesmen problem where he dynamics of he opporuniy se is non-markovian and he invesor has non-logarihmic uiliy. The ineremporal consumpion and invesmen decision of a uiliy-maximizing invesor is a classical problem of financial economics. In wo pahbreaking papers, Meron (1969, 1971) sudies his problem in a coninuous-ime framework using dynamic programming. He shows ha he opimal invesmen sraegy combines a myopic, or speculaive, porfolio and some porfolio which hedges changes in he invesmen opporuniy se (i.e. ime-varying reurns and volailiies). Also, Meron (1973) addresses he opimizaion problem under a sochasic invesmen opporuniy se where he drif and diffusion coefficiens of he asse prices depend 1
on he curren shor-erm ineres rae, bu he does no derive explici resuls for he hedge porfolio in his case. Our paper deviaes from he classical Meron (1969,1971,1973) papers in a leas hree mehodological aspecs: (i) We do no assume ha asse prices are Markovian, (ii) we assume complee markes, and (iii) insead of using a dynamic programming approach we use he maringale soluion approach suggesed and formalized by Cox and Huang (1989,1991) and Karazas, Lehoczky, and Shreve (1987). In he lieraure here has recenly been a number of sudies of opimal invesmen sraegies in dynamic markes wih specific assumpions on he changes in he invesmen opporuniy se. Brennan, Schwarz and Lagnado (1997) solve numerically for he opimal invesmen sraegy of an invesor wih CRRA uiliy from erminal wealh in a marke wih socks and bonds driven by hree variables: he shor-erm and he long-erm ineres raes and a sochasic dividend yield on he sock. They invesigae numerically how heopimal invesmen sraegy deviaes from he myopic sraegy for differen invesmen horizons. Brennan and Xia (1998) and Sfirensen (1999) consider he invesmen problem of a CRRA uiliy invesor wih uiliy from erminal wealh only. They assume complee markes and show ha in he case where he erm srucure of ineres raes is described by avasicek-model and marke prices on risk (and excess reurns) are consan, he opimal hedge porfolio is he zerocoupon bond ha expires a he invesmen horizon. Also, Liu (1999) provide a similar resul in an example and his paricular resul is also a very special case of he resuls obained in his paper. Campbell and Viceira (1998) consider a dynamic invesmen problem ha allows for ineremporal consumpion. In fac, heir (recursive) preferences are more general han he CRRA uiliy assumed in his paper. On he oher hand we allow for much more general dynamics of he opporuniy se. While Campbell and Viceira (1998) provide explici (bu approximae) resulsonhow o hedge changes in ineres raes, hey do no explicily link he opimal hedge porfolio o he opimal consumpion paern. Deemple, Garcia and Rindisbacher (1999) consider a complee marke where changes in he invesmen opporuniy se are driven by amuli-dimensional sae variable following a Markov diffusion. For an invesor maximizing uiliy of erminal wealh, hey are able o express he opimal invesmen sraegy as a combinaion of he myopic, growh-opimal sraegy and wo erms represening he hedge agains changes in he shor-erm ineres rae and he marke prices of risk, respecively. The wo hedge erms involves Malliavin derivaives and are based on he Clark-Ocone formula, cf. Ocone and Karazas (1991). In he special seing of a CRRA 2
invesor in a complee marke wih a single risky asse where he shor-erm ineres rae and he marke price of risk follow wo paricular diffusion processes ha are boh perfecly correlaed wih he risky asse price, hey are able o derive he hedge sraegies in closed-form. For oher seings hey demonsrae how o compue he hedge componens numerically by simulaions using heir Malliavin formulaion. Again, hey do no explicily link he opimal hedge porfolio o he opimal consumpion paern, as in his paper. Kim and Omberg (1996), Campbell and Viceira (1999), Chacko and Viceira (1999), and oher recen papers consider dynamic opimizaion problems where ineres raes are consan bu where risk premia are sochasic and/or markes are incomplee; see also Campbell (2000) for a survey of he lieraure. In conras o all he papers referred above, we allow for non-markovian dynamics of he opporuniy se and link he opimal hedge porfolio o he opimal consumpion paern of he invesor. The res of he paper is organized as follows. In Secion 2 we se up he general coninuousime consumpion and invesmen problem in a dynamically complee marke and provide a general characerizaion of he opimal consumpion and invesmen policy for a CRRA invesor in a possibly non-markovian seing. In Secion 3 we derive explici resuls on how o hedge changes in he erm srucure of ineres raes using coupon bonds in a specialized HJM mulifacor Gaussian erm srucure seing. In Secion 4 we consider wo specificnumerical examples based on he Vasicek-model and an HJM hree-facor model. Secion 5 concludes and proofs are given in an Appendix. 2 The invesmen problem We consider a fricionless economy where he dynamics are generaed by ad-dimensional Wiener process, w = (w 1 ;:::;w d ), defined on a probabiliy space(ω; F; IP). F = ff : 0g denoes he sandard filraion of w and, formally, (Ω; F; F; IP) is he basic model for uncerainy and informaion arrival in he following. 2.1 Preferences We will consider he invesmen problem of an expeced uiliy maximizing invesor wih a ime-separable consan relaive riskaversion uiliy funcion on he form, " Z T K E 0 U 1 (C ;)d +(1 K) E 0 [U 2 (W T )] (1) 0 # 3
where U 1 (C; ) =e fi C 1 1 1 and U 2 (W )=e fit W 1 1 1 and where fi is a consan subjecive ime discoun rae and is a consan relaive riskaversion parameer. The preference parameer K conrols he relaive weigh of inermediae consumpion, C, and erminal wealh, W T, in he agen's uiliy funcion. The special case where = 1 is he logarihmic uiliy case: U 1 (C; ) =e fi log C and U 2 (W )=e fit log W. 2.2 Invesmen asses The agen can inves in a se of financial securiies. One of hese financial asses is assumed o be an insananeously" risk-free bank accoun which has a reurn equal o he shor-erm ineres rae r. In addiion, he agen can inves in d risky asses wih prices described by he vecor V =(V 1;:::;V d ) 0. The price dynamics of he risky asses (cum dividend) are given by dv =diag(v )[(r 1 d + ff ) d + ff dw ] (2) where is an IR d -valued sochasic process of marke prices of risk, ff is an IR d d -valued sochasic process of volailiies, 1 d is a d-dimensional vecor of ones, and diag(v )isa(d d)- dimensional marix wih V in he diagonal (and zeros off he diagonal). I is assumed ha ff has full rank d implying ha markes are dynamically complee, c.f. Duffie and Huang (1985). As a consequence of markes being dynamically complee, he pricing kernel (or sae-price deaor) is uniquely deermined and given by (see, e.g., Duffie (1996), chaper 6) ρ Z =exp or, equivalenly, in differenial form, 0 Z r s ds 0 0 s dw s 1 2 Z 0 ff k s k 2 ds ; 0 (3) d = [ r d dw ] ; 0 =1: (4) The presen value of any sochasic payoff, X, paid a some fuure ime poin s can be deermined by evaluaing he pricing kernel weighed payoff. In paricular, we have» s PV [X] =E X = P (s)^e s [X] (5) where P (s) is he ime price on a zero-coupon bond ha expires a ime s. The las equaliy defines he so-called cerainy-equivalen or forward-expeced payoff, ^E s [X]; see e.g. Jamshidian (1987,1989) and Geman (1989) who inroduce he noion of he forward-expeced maringale measure, as being disinc from he usual risk-neural maringale measure in he conex of ineres rae models. 4
2.3 The problem and he general soluion Le ß be an IR d -valued process ha describes he fracions of wealh ha he agen allocaes ino he d differen risky asses. The wealh of he agen hen evolves according o dw = (r + ß 0 ff )W C Λ d + W ß 0 ff dw : (6) The agen's problem is o choose a dynamic consumpion sraegy, C, and porfolio policy, ß, in order o maximize he expeced uiliy in (1). The main idea of he maringale soluion approach suggesed and formalized by Cox and Huang (1989,1991) and Karazas, Lehoczky, and Shreve (1987) is o alernaively consider he saic problem " Z # T sup K E 0 U 1 (C ;)d +(1 K) E 0 [U 2 (W T )] (7) fc ;W T g 0 subjec o " Z T E 0 0 0 # T C d + W T» W 0 : (8) 0 The problem in (7) and (8) is a sandard Lagrangian opimizaion problem which can be solved using he Saddle Poin Theorem (see e.g., Duffie (1996), pp. 205 208) o deermine he opimal consumpion process, C, and erminal wealh, W T. In principle, he problem is o maximize expeced uiliy subjec o he budge consrain (8) which saes ha he presen value of he consumpion sream and erminal wealh canno exceed he agen's curren wealh. The value funcion, or indirec uiliy, J from he opimizaion problem is he maximum expeced remaining life-ime uiliy which can be achieved by he opimal consumpion and erminal wealh plan following any ime poin, 0»» T. As shown by Cox and Huang (1989,1991) and Karazas, Lehoczky, and Shreve (1987), he soluion o his problem also provides he soluion o he dynamic problem of choosing he opimal consumpion sraegy and porfolio policy. While he consumpion policy is given explicily when solving (7) and (8), he opimal porfolio policy is only given implicily as he policy which replicaes he opimal erminal wealh from he above problem and in accordance wih (6). As described by Meron (1971,1973), he opimal invesmen policy can in a general Markovian seing be decomposed ino a speculaive porfolio (chosen by amyopic or logarihmic uiliy invesor) and a erm which hedges changes in he opporuniy se. For general uiliy funcions, he opimal invesmen sraegy can be represened raher absracly in complee markes in erms of sochasic inegrals of Malliavin derivaives by he Clark-Ocone formula, cf. Ocone and Karazas (1991). However, in order o derive an explici expression for he opimal porfolio 5
for non-logarihmic uiliy funcions i is generally recognized ha he price dynamics mus be specialized. Cox and Huang (1989) show ha when he sae-price deaor and he risky asse prices consiue a Markovian sysem, he opimal invesmen sraegy can be represened in erms of he soluion of a linear second order parial differenial equaion. On he oher hand, in he following proposiion we provide a closed-form expression for he opimal invesmen sraegy for a power uiliy invesor in a general non-markovian complee marke seing for a CRRA invesor. The (invesor specific) sochasic process Q = K 1 2 3 2 Z T 1 e fi s (s ) E 4 5 ds +(1 K) 1 e fi (T T ) E 4 3 1 5 (9) is crucial for how o hedge changes in he opporuniy se, as will be formalized in he proposiion below. Since Q is a posiive sochasic process adaped o he filraion generaed by w, i follows from he Maringale Represenaion Theorem (see e.g., Duffie (1996)), ha he dynamics of Q can be described on he form for some drif process μ Q and some volailiy process ff Q. dq = Q [μ Q d + ff Q dw ] (10) Proposiion 1 The value funcion of he general problem in (7) and (8) has he form J = Q W 1 A() 1 (11) where A() = K fi 1 e fi(t ) fi(t ) +(1 K)e and Q is defined in equaion (9). The opimal consumpion choice and he opimal porfolio policy a ime are given by C = K 1 W Q (12) and ß = 1 (ff 0 ) 1 +(ff 0 ) 1 ff Q : (13) Proof: See he appendix. Proposiion 1 saes he opimal porfolio policy has he same form as in Meron (1971). The porfolio policy can be decomposed ino a speculaive porfolio (he firs erm in (13)) and a hedge porfolio ha describes how he invesor should opimally hedge changes in he 6
invesmen opporuniy se (he las erm in (13)). As usual he hedge porfolio vanishes in he special case where he invesor has logarihmic uiliy; he precise resul for he logarihmic invesor benchmark case is saed explicily in a corollary below. Also, noe ha he opimal consumpion choice in (12) is on he same feed-back form" as in, e.g., Meron (1971) even hough Proposiion 1 holds for a general non-markovian seing where dynamic programming is no direcly applicable. I is seen from Proposiion 1 ha in order o hedge changes in he opporuniy se, he invesor mus form a hedge porfolio ha basically mimics he dynamics of Q. Hence, Q reecs everyhing of imporance for how o hedge changes in he invesmen opporuniy se. For a given invesor i can hus be inferred from (9) ha only processes included in he descripion of (momens of) he pricing kernel saed in (3) are relevan for ineremporal hedging purposes. In general, he invesor should alone consider o hedge changes in ineres raes and changes in prices on risk in he economy while changes in, say, volailiies on markeed securiies should be of no concern in our complee marke seing. In order o discuss he implicaions of he resuls in Proposiion 1 we will focus on wo benchmark cases: he log-uiliy case( = 1) and he case of an infiniely risk averse invesor ( = 1). 1 These wo invesor ypes represen imporan polar cases since he logarihmic invesor does no hedge changes in he opporuniy se a all while he infiniely risk averse invesor has no speculaive demand for securiies a all. Moreover, in he nex secion we will demonsrae ha he forward expeced consumpion paern of he agen is imporan for how o hedge changes in he erm srucure of ineres raes in a specialized marke seing. As formalized and explicily saed in he following corollaries, he forward-expeced consumpion paerns of he benchmark cases of log-uiliy invesors and infiniely risk averse invesors do no depend on he dynamics in he invesmen opporuniy se bu only on he curren form of he erm srucure of ineres raes. Corollary 1 For he special case of an invesor wih logarihmic uiliy, i.e. = 1, he value funcion general problem in (7) and (8) has he form J = Q log W (14) wih Q = A() and where A() is defined in (11). The opimal consumpion choice and he opimal porfolio policy a ime are given by C = K W Q (15) 1 Formally, he resuls for an infiniely risk averse invesor are defined as he limiing resuls as!1. 7
and ß =(ff 0 ) 1 : (16) Furhermore, a ime he forward-expeced consumpion raes a fuure ime poins s 2 [; T ) are given by ^E s [C s ]=C (P (s)) 1 e fi(s ) ; 0»» s<t; (17) and he forward-expeced erminal wealh a ime T is given by Proof: See he appendix. ^E T [W T ]=C 1 K K (P (T )) 1 e fi(t ) : (18) Corollary 2 For he special case of an infiniely risk averse invesor, i.e. = 1, he opimal consumpion choice is consan and given by and he opimal porfolio policy a ime is given by wih C = W Q = W 0 Q 0 (19) ß =(ff 0 ) 1 ff Q (20) Z T Q = P (s) ds + P (T ): (21) In paricular, he forward-expeced consumpion raes a fuure ime poins s 2 [; T ) are given by and he forward-expeced erminal wealh a ime T is given by Proof: See he appendix. ^E s [C s ]=C ; 0»» s<t; (22) ^E T [W T ]=C ; 0»» s<t: (23) Corollary 1 saes ha he opimal porfolio choice of a logarihmic uiliy invesor is described enirely by he speculaive porfolio while Corollary 2 saes ha he opimal porfolio choice of an infiniely risk averse invesor is described enirely by he hedge porfolio. The hedge porfolio, as reeced in Q, will in his case be an annuiy bond 2 and he opimal consumpion will be cerain and a" and basically be generaed by he cerain paymens on he annuiy bond. 3 2 An annuiy bond is a coupon bond where he cerain cash ows (coupon + principal repaymen) from he bond are he same hroughou he finie life of he bond (and including he las paymen on he bond). 3 Wacher (1999) provides similar resuls for an infiniely risk averse invesors in a Markovian marke seing. 8
3 Hedging changes in ineres raes In he following, we are especially ineresed in how he agen should allocae invesmen funds ino wo general classes of securiies: socks and bonds. Therefore, we will inroduce addiional noaion and separae he invesmen asses ino socks and bonds. Formally, we will spli he d-dimensional Wiener process generaing he financial asse reurns as w =(w B ;w S ), where w B is of dimension k and w S of dimension l = d k. We assume ha he dynamics of he erm srucure of ineres raes, and, hence, he dynamics of prices on bonds and oher erm srucure derivaives raded a he bond marke, are affeced only by w B. The dynamics of he prices of he socks may depend on boh w B and w S which allows for correlaion beween socks and erm srucure derivaives. Specifically, he invesor can inves in he insananeously" risk-free bank accoun, k erm-srucure derivaives, and l socks. The asse price dynamics are given by db = diag(b )[(r 1 k + ff B B ) d + ff B dw B ] (24) and ds = diag(s )[(r 1 l + ' S ) d + ff S1 dw B + ff S2 dw S ] (25) where ff B, ff S1,andff S2 are marix valued processes of dimension k k, l k,andl l, respecively. Again, ff B and ff S2 are assumed non-singular so ha markes are complee. Changes in he reurns of he erm srucure derivaives and he socks are correlaed wih k l covariance marix ff B ffs1 0. The marke price of risk process (which is no dependen on he paricular se of asses chosen) has he form where =( B ; S ) 0 S = ff 1 S2 ' S ff 1 S2 ff S1 B : Noe ha we have inroduced he IR l -valued sochasic process ' S (= ff S1 B + ff S2 S ) which can be inerpreed as he expeced excess reurn on he socks. 3.1 Term srucure dynamics and porfolio choice In he following we will inroduce addiional noaion in order o be specific abou he erm srucure dynamics in he economy and in order o provide a framework for casing he specific examples considered in a laer secion. Specifically, we assume ha he dynamics of he erm srucure of ineres raes can be described by ak-facor model of he HJM-class inroduced by Heah, Jarrow, and Moron (1992). Moreover, we will assume ha forward rae volailiies are 9
deerminisic which implies ha we only consider erm srucure dynamics wihin he so-called Gaussian HJM-class. For any mauriy dae fi he dynamics of he fi-mauriy insananeous forward rae are Z f (fi) =f 0 (fi)+ 0 Z ff(s; fi) ds + 0 ff f (s; fi) 0 dw Bs ; (26) where ff f ( ;fi) is an IR k -valued deerminisic funcion and f 0 (fi) is he fi-mauriy forward rae observed iniially a ime 0. The mos imporan feaure of he HJM-modeling is ha in he absence of arbirage, one only has o specify he iniial erm srucure of forward raes and he volailiy srucure ff f (; fi) for all and fi in order o have awell-specified erm srucure model; in paricular, as a no-arbirage drif resricion, we have ha Z fi ff(; fi) =ff f (; fi) 0 B ()+ ff f (; u) du : While equaion (26) (and he no-arbirage drif resricion) describes he evolvemen over ime of he enire forward rae curve, he dynamics of he shor-erm ineres rae is given as he special case where fi =, i.e. r = f (). The dynamics of he shor-erm ineres rae are hus described by Z r = f 0 ()+ 0 Z ff(s; ) ds + 0 ff f (s; ) 0 dw Bs : (27) Among he many erm-srucure derivaives, we focus on defaul-free bonds. The dynamics of he price P (fi) =exp( R fi f (s) ds) of he zero-coupon bond mauring a ime fi is given by dp (fi) =P (fi) r + ff P (; fi) 0 B () d + ff P (; fi) 0 dw B Λ (28) where ff P (; fi) = R fi ff f(; u) du. For laer use we will also consider a bond paying a coninuous coupon of k() up o ime T and a lump sum paymen of k(t ) a ime T. The ime price of such a bond is Z T B cpn = k(s)p (s) ds + k(t )P (T ): Applying he Leibniz-ype rule for sochasic processes saed in Lemma 1 in he Appendix, we see ha he coupon bond price evolves according o where db cpn h = k() d + B cpn ff B cpn = r + ff 0 B cpn B () d + ff 0 i B dw cpn B ; R T k(s)p (s)ff P (; s) ds + k(t )P (T )ff P (; T ) R T k(s)p (s) ds + k(t )P (T ) : (29) Our specific resuls on how o hedge changes in ineres raes, as saed in he following Proposiion 2, are based on he following assumpion. 10
Assumpion 1 The relaive risk process () and he forward rae volailiies ff f (; fi) are deerminisic funcions of ime. The implicaions of he assumpion ha prices of risk and forward rae volailiies are deerminisic are imporan since we only allow ineres raes o change and, hence, here are no reasons o hedge changes in prices of risk nor forward rae volailiies. Also, as a consequence of Assumpion 1 he following analysis is limied o Gaussian models of he erm srucure of ineres raes. However, noe ha we do no assume ha he diffusion coefficiens ff B, ff S1, and ff S2 of he invesmen asses are deerminisic and, in fac, hey may be described by non-markovian processes. Despie shorcomings of Gaussian erm srucure models, such as no ruling ou negaive ineres raes, muli-facor Gaussian models are ofen used for derivaive pricing since hey allow closed-form soluion for mos European-ype erm srucure coningen claims, cf., e.g., Amin and Jarrow (1992) and Brace and Musiela (1994). As we shall see in he following, he Gaussian assumpion also allows closed-form expressions for opimal invesmen sraegies. Furhermore, i is imporan o poin ou ha no even in Gaussian HJM-models is he shor rae process necessarily Markovian. Only if ff f (; fi) can be separaed as ff f (; fi) =G()H(fi), where H is a real-valued coninuously differeniable funcion ha never changes sign and G is an IR k -valued coninuously differeniable funcion, is he shor rae Markovian, cf. Carverhill (1994). This will e.g. no be he case in he HJM hree-facor example considered in a subsequen secion and, hence, he shor-erm ineres rae will no be a Markovian process in he specific example. From he assumpion ha prices of risk and forward rae volailiies are deerminisic, i follows ha he shor-erm ineres rae in (27) is normally disribued (Gaussian) and ha he pricing kernel, as saed in (3), is lognormally disribued. I is hus possible o compue in closed-form he expecaions in he definiion of Q in (9) and, hence, obain an analyical expression for Q. The proof of he following proposiion is based on his feaure. Proposiion 2 Assuming ha he relaive risk process () and he forward rae volailiies ff f (; fi) are deerminisic funcions of ime, he value funcion and he opimal consumpion sraegy are given by (11) and (12) in Proposiion 1 where in his case Z T Q = Z (s) ds + Z (T ) (30) wih Z (s) = K 1 (P (s)) 1 ρ exp fi Z (T ) = (1 K) 1 (P (T )) 1 ff (s )+1 2 g(; s) ; 0»» s<t 2 (31) ρ exp fi ff (T )+1 2 g(; T ) 2 (32) 11
and g(; s) = Z s Z s Z s k (u)k 2 du + kff P (u; s)k 2 du 2 B (u) 0 ff P (u; s) du: (33) The opimal porfolio policy a ime is described by where ff B cpn coupon according o 1 1 ß = ff 1 ()+ ff 1 0 @ ff B cpn 0 1 A (34) is he volailiy vecor of a bond, as defined in equaion (29), which pays coninuous k(s) =^E s [C s ]=K 1 W Q (P (s)) 1 exp ρ fi and has a erminal lump sum paymen a ime T of ff (s )+1 2 g(; s) ; 0»» s<t; (35) 2 k(t )=^E T [W T ]=(1 K) 1 W Q (P (T )) 1 exp ρ fi ff (T )+1 2 g(; T ) 2 (36) Proof: See he appendix. Proposiion 2 saes ha in order o hedge changes in he opporuniy se, i.e. changes in he erm srucure of ineres raes in he presen conex, he relevan hedge porfolio is a coupon bond wih coupons chosen o mach he (forward-expeced) consumpion paern. Specifically, Proposiion 2 saes ha he opimal invesmen sraegy allocaes a fracion of wealh (1=) ino he speculaive porfolio and a fracion of wealh (1 1=) ino he specific coupon bond. Since he porfolio weighs in ß only describe he porfolio weighs on he individual risky asses in he economy, he individual componens in ß are no resriced o sum o one, and any excess wealh will be invesed in he bank accoun. Furhermore, i follows direcly ha in he special case of invesors wih logarihmic uiliy ( = 1) here are no desire for hedging changes in he opporuniy se while on he oher hand for very risk averse invesors (i.e. as!1) here are no speculaive demand for risky asses, in accordance wih Corollary 1 and Corollary 2. The special case K = 0 corresponds o uiliy from erminal wealh only and in his case i follows from Proposiion 2 ha he relevan bond for hedging changes in he erm srucure of ineres raes is a bond ha only has a lump sum paymen a ime T. In his special case Proposiion 2 generalizes he insighs of Brennan and Xia (1998) and Sfirensen (1999) who in Vasicek seings demonsrae ha, in he case of uiliy from erminal wealh only, he relevan bond for hedging changes in he opporuniy se is he zero-coupon bond ha expires a he invesmen horizon. In our view he hedging sraegy in Proposiion 2 is of concepual imporance since he opimal way o hedge changes in he opporuniy se is model-independen in he sense ha 12
he invesor should alone aim a buying a coupon bond, or porfolio of bonds, such ha he (forward-expeced) consumpion paern is mached. In general, Proposiion 2 saes ha he specific dynamics of he erm srucure of ineres raes is of imporance for how o hedge changes in he opporuniy se only hrough is effec on he opimal (forward-expeced) consumpion paern. In he following examples, we will focus on he deerminans of he opimal (forwardexpeced) consumpion paerns and, in paricular, we will focus on wheher he curren form of he erm srucure or he dynamics of he erm srucure are of crucial imporance for he opimal (forward-expeced) consumpion paern. A his poin i can be noed ha in general only he form of he erm srucure of ineres raes maers for he opimal (forward-expeced) consumpion paerns for he benchmark cases of logarihmic uiliy invesors and infiniely risk averse invesors, as described in Corollary 1 and Corollary 2. 4 Specific examples In his secion we describe wo specific examples and presen numerical resuls on how o hedge muli-facor ineres rae risk in a dynamic seing. The firs example is based on he erm srucure dynamics from he Vasicek (1977) model while he second example considers a non- Markovian hree-facor HJM-erm srucure model where he erm srucure can exhibi hree differen kinds of changes: a parallel level change, a slope change, or a curvaure change. 4.1 Vasicek example In he following example we allow for uiliy from inermediae consumpion by seing he preference parameer K equal o 1 2 in he specificaion of he uiliy funcion in (1) so ha uiliy from inermediae consumpion and uiliy from erminal wealh are equally weighed. On he oher hand, he se-up for invesmen asses in he following example is basically as in Brennan and Xia (1998) and Sfirensen (1999) who only consider uiliy from erminal wealh. The agen can hus inves in a single sock and a single bond as well as he insananeously" riskfree bank accoun. The erm srucure dynamics are described by he one-facor erm srucure model originally suggesed by Vasicek (1977). In paricular, he dynamics of he shor-erm risk-free ineres rae is described by an Ornsein-Uhlenbeck process of he form, dr =»( r )d ff r dw B (37) where he parameer describes he long-run level for he shor-erm ineres rae,» is a mean-reversion parameer ha deermines he srengh of endency o he long-run level, and he parameer ff r describes he ineres rae volailiy. Besides he parameers describing he 13
ineres rae dynamics, he parameer denoed B in he conex of secion 3 deermines he price of ineres rae risk. Using sandard no-arbirage argumens, one can solve for prices on ineres rae coningen claims in he Vasicek-model. The possible forms of he erm srucure of forward ineres raes can hus be deermined by solving for prices on zero-coupon bonds. The fi-mauriy forward rae a ime in he Vasicek-model is given by f (fi) =e»(fi ) r + r 1 1 e»(fi ) + ff2 r 2» e»(fi ) b(fi ) (38) where r 1 = + Bff r» ff2 r 2» 2 b(s) = 1» (1 e»s ) : The dynamics of he fi-mauriy forward rae can be deermined from (38) and (37) and an applicaion of Io's lemma. In paricular, i is seen ha he forward rae volailiy srucure in his example has he form ff f (; fi) = ff r e»(fi ). Wihin he HJM-framework of secion 3, his volailiy srucure and an iniial erm srucure of forward raes of he form in (38) provide a complee specificaion of he Vasicek (1977) erm srucure model. The agen can inves in a single sock aswell as bonds and he bank accoun. In he specific case of a one-facor ineres rae model i is sufficien ha he agen caninves in a single bond besides he sock and he bank accoun in order o implemen he complee markes opimal soluion. The price process of he single sock is described in equaion (25) where in his case ff S1 and ff S2 are scalars (i.e., of dimension 1 1). The specific parameer values used in he following numerical example are chosen as follows: =0:04;» =0:15; ff r =0:015; ff S1 =0:0625; ff S2 =0:2421; ' S =0:05; S =0:19365; B =0:05: (39) In paricular, he parameers»,, and ff r ha describe he ineres rae dynamics are chosen so ha hey are close o hose obained by Chan, Karolyi, Longsaff, and Sanders (1992) for he Vasicek ineres rae process. The parameers for he sock process are chosen such ha he expeced excess reurn on he sock is' S =5%,hevolailiy of he sock is consan 25% = (ff 2 S1 + ff2 S2 )1=2, and he insananeous" correlaion coefficien beween he sock and he shor-erm ineres rae is consan 25% (and, hence, he correlaion beween he sock and any bond in he one-facor Vasicek-model is 25%). The 5% expeced rae of excess reurn on he sock is below he 8.4% poin esimae suggesed by he Ibboson Associaes 1926-1994 hisorical reurns daa on socks (see, e.g., Brealey and Myers (1996), chaper 7, Table 7-1). Though, as 14
poined ou by Brown, Goezmann and Ross (1995), he use of realized mean reurns in his conex is likely o involve a survival bias which could be as high as 400 basis poins per year. The 25% volailiy of he sock is slighly higher han 20.2% hisorical volailiy esimae on he S%P 500 index based on he Ibboson and Associaes reurns daa (see, e.g., Brealey and Myers (1996), chaper 7) bu well in accordance wih, say, volailiies on individual socks and less diversified porfolios of socks. Furhermore, he 25% posiive correlaion beween he sock and bonds is consisen wih he empirical resuls in, e.g., Campbell (1987), Fama and French (1989), and Shiller and Belrai (1992). Finally, he risk premia on bonds, B = 0.05, is se so ha, e.g., he expeced excess reurn on a 10-year zero-coupon bond in he Vasicek-model is 0.39%. 4 The above parameer values imply ha an agen wih logarihmic uiliy invess an 80% fracion of wealh in he sock, a fracion of 0% in bonds, and he residual 20% of wealh in he bank accoun. On he oher hand, agens wih non-logarihmic uiliy wan o inves in bonds ha have payoffs ha mach heir (forward-expeced) consumpion paern in order o hedge changes in he opporuniy se, as described in Proposiion 2. The infiniely risk averse invesors inves 100% in he hedge bond while, e.g., an invesor wih consan relaive risk aversion,, equal o 2 will inves 50% of wealh in he speculaive porfolio and 50% of wealh in he hedge bond; i.e. he porfolio composiion in his case is: 40% in he sock, 10% in he bank accoun, and 50% in he hedge bond. We will consider hree cases wih differen iniial erm srucures of forward raes. These hree forms are given by seing he shor-erm ineres rae equal o 0.01, 0.04, and 0.07, respecively. The hree forms of he iniial erm srucure of forward ineres raes are displayed in Figure 1. [ INSERT FIGURE 1 ABOUT HERE ] As formalized in Proposiion 2, he (forward-expeced) consumpion paern of he agen is crucial for how o hedge changes in ineres raes. The (forward-expeced) consumpion paern and he (forward-expeced) erminal wealh of he agen can be deermined by insering in he expressions in (35) and (36). In paricular, he consumpion paern over ime depends on he erm srucure of forward raes hrough he occurrence of he zero-coupon price P (; fi) = exp ( R fi f(; s) ds) in he expressions. Also, he consumpion paern over ime depend on he prices on risk in he economy hrough he expression for he variance of he log-pricing kernel, g(; s) in (33). Using ha he zero-coupon bond volailiy isff P (; fi) = R fi ff f (; u) du 4 Again, Brealey and Myers (1996, chaper 7) abulae he average hisorical excess reurn on governmen bonds o be slighly higher, 1.4%, based on he Ibboson and Associaes reurns daa. 15
= ff r b(fi ) and by evaluaing he inegrals in (33), one obains g(; s) =( 2 B + 2 S)(s )+2(r 1 )(b(s ) (s )) ff2 r 2» (b(s ))2 (40) The (forward-expeced) consumpion paerns are displayed in Figure 2 o Figure 6 for differen degrees of relaive risk aversion and for a subjecive ime discoun rae of fi = 0.03 and ime horizon of T = 25(years). [INSERT FIGURE 2 TO FIGURE 6 ABOUT HERE ] The consumpion paerns in he figures describe he specific paymen schedules for he relevan coupon-bonds ha he differen invesors should use in order o hedge changes in he erm srucure of ineres raes. The log-uiliy invesors and he infiniely risk averse invesors are polar benchmark cases where eiher he demand for he hedge bond is exacly 0% or exacly 100%. Invesors in beween hese wo polar cases will inves a fracion of wealh beween 0% and 100% in he specific bonds in order o hedge changes in he opporuniy se. For example, invesors wih relaive risk aversion of 4/3, 2, and 4 should opimally inves 25%, 50%, and 75%, respecively, in heir specific hedge bonds. For a log-uiliy invesor and for an invesor wih = 1, he forward-expeced consumpion and erminal wealh paerns, as saed in he general expressions in Corollary 1 and Corollary 2, only depend on he iniial erm srucure of ineres raes. In paricular, for = 1 he forwardexpeced consumpion paern is always a, as displayed in Figure 6, while he forward-expeced consumpion paern for a log-uiliy invesor in Figure 2 depends on he subjecive discoun rae fi and he specific form of he curren erm srucure. From (17) i follows ha in he logarihmic uiliy case, = 1, he forward-expeced consumpion rae k(s) mus saisfy k 0 (s) =(f (s) fi) k(s) and, hence, ha he forward-expeced consumpion rae as a funcion of he ime o consumpion is increasing whenever he forward rae is higher han he subjecive discoun rae fi = 0:03, and vice versa. The consumpion paerns for he invesors in Figure 3 o Figure 5 are basically in beween he wo polar benchmark cases of invesors wih logarihmic uiliy and infiniely risk averse invesors. 4.2 A non-markovian hree-facor HJM-model This example feaures non-markovian dynamics of he opporuniy se. We consider hree differen iniial erm srucures of forward raes; hese are he iniial erm srucures from he above Vasicek example, as displayed in Figure 1. 16
The erm srucure can exhibi basically hree kinds of changes: a parallel level change, a slope change, and a curvaure change. assumed o have he form Specifically, he forward rae volailiy srucure is ff f (; fi) 0 = ff 1 ;ff 2 e» 2(fi ) ;ff 3 (fi )e» 3(fi ) ; 0»» fi» T: (41) The dynamics of he forward rae curve is described by insering he volailiy srucure (41) in (26). In paricular, a change in he Wiener-process ha governs movemens in he firs facor will resul in an equal change in all forward raes for differen mauriies; hence, his causes a parallel level change of he forward curve. Likewise, a change in he Wiener-process ha governs movemens in he second facor will significanly affec forward raes wih shor mauriies bu no forward raes wih long mauriies; hence, his causes a slope change of he forward curve. Finally, a change in he Wiener-process ha governs movemens in he hird facor will affec forward raes wih medium mauriies bu neiher forward raes wih shor and long mauriies; hence, his causes a change in he curvaure of he forward curve. The hree facors are similar o he fundamenal hree componens in he Nelson and Siegel (1986) srucural forms widely used in pracice for calibraion of erm srucures of ineres raes and also consisen wih he erm srucure facors deermined empirically by, e.g., Lierman and Scheinkman (1991). The volailiy ofany zero-coupon bond is described by ff P (; fi) = R fi ff f(; u) du and under he above specificaion of forward curve volailiy, we have ff P (; fi) 0 = where b j (fi) = 1» j (1 e» jfi ) for j =2; 3. ff 1 (fi );ff 2 b 2 (fi ); ff 3» 3 (b 3 (fi ) (fi )e» 3 (fi ) ) As in he Vasicek-example above, i is possible o deermine he opimal (forward-expeced) consumpion paern and, hence, he relevan coupon bond o hedge changes in he opporuniy se using he general resuls in Proposiion 2. Besides he form of he iniial erm srucure of ineres raes he variance of he (log-) pricing kernel, g(; s), is deermining he relevan consumpion paerns in (35) and (36). Sraighforward calculaions using (33) show ha g(; s) = ( 2 B1 + 2 B2 + 2 B3 + 2 S )(s ) B1ff 1 (s ) 2 + 1 3 ff2 1(s ) 3 2 + B2 ff 2 4 B3 ff 3» 2 ff2 2» 2 3 4 + B3 ff 3 ff2 3» 2 3» 2 3 3 2 3 2» 2 2 (b 2 (s ) (s )) ff2 2 2» 2 (b 2 (s )) 2 ff 2 3» 4 3 ff 2 3 +» 4 3 (s ) 1 2 ff 2 3» 3 3 2 B3 ff 3» 3 + ff2 3» 3 3 (s ) 2 5 4» 3 + 3(s )+ 1» 2 2 3(s ) 2 (s )+ ff2 3 (s ) 2 b» 2 3 (s ) 3 (b 3 (s )) 2 In he following, we will abulae numerical resuls for hree differen ses of parameers for he hree-facor HJM-model. Our base case se of parameers are chosen such ha he volailiies 17 (42) (43)
of shor erm and long erm bonds as well as he expeced excess reurns on socks and bonds are of he same magniude as in he Vasicek-example above. Below, we will commen furher on how hisisachieved bu, specifically, he parameers values in he base case are:» 2 =1:00;» 3 =0:50; ff 1 =0:00325; ff 2 =0:01184; ff 3 =0:00869; ff S1 =(0:03187; 0:02305; 0:04857) 0 ; ff S2 =0:24206; (44) ' S =0:05; S =0:19365; B =(0:02549; 0:01844; 0:03886) 0 : In choosing he parameers in (44) we firsfixed» 2 and» 3 so ha i makes sense o alk abou a slope effec and a curvaure effec in he dynamics of he forward rae curve in (26). In he presen conex, he innovaions in he forward curve are generaed by a hree-dimensional Wiener process, w B = (w B1 ;w B2 ;w B3 ) 0. As described above, an innovaion in w B1 affecs all forward raes equally while, e.g., an innovaion in w B2 affecs shor raes bu no very long raes. For example,» 2 = 1.00 implies ha if an innovaion in w B2 increases he spo rae wih 100 basis poin, he 1-year forward rae is increased only by (100 e» 2 1 =) 36.79 basis poins, and he 5-year forward rae is only increased by 0.67 basis poins; hence, an innovaion in w B2 will significanly change he slope of he forward rae curve. Likewise, an innovaion in w B3 will no affec he very near forward raes nor he very disan forward raes bu will change he curvaure of he forward rae curve. The maximum ampliude in he forward rae curve caused by an innovaion in w B3 occurs for a medium disan forward rae; specifically, for» 3 = 0.50 he maximum ampliude occurs for he (1=» 3 =) 2-year forward rae. While he parameers» 2 and» 3 are specified exogenously, he forward rae volailiy parameers ff 1, ff 2, and ff 3 are calibraed in order o ensure ha he volailiies of zero-coupon bonds wih imes o mauriy equal o 0.25 years, 2 years, and 10 years, respecively, are idenical o hose in he Vasicek example. 5 Nex, ff S1 and ff S2 are chosen such ha he volailiy on he sock is 25% and such ha he correlaion coefficiens beween he sock and any of he hree erm srucure facors are 25% which corresponds o he 25% correlaion beween he sock and he shor-erm risk-free ineres rae in he Vasicek example. Finally, risk premia are also calibraed o be similar o hose in he Vasicek example. In paricular, he expeced excess reurn on he sock is 5% while he risk premia on bonds, as reeced in B, are calibraed so ha here are no speculaive demand for bonds (also, jj B jj = 0.05, as in he Vasicek example). The porfolio choice of a logarihmic invesor is, hence, o inves 80% of wealh in he sock, 0% in bonds, and 20% in he bank accoun, as in he Vasicek example. Likewise, oher invesors allocae he same fracion of wealh ino he sock, he bank accoun, and a hedge bond as in 5 This is done by equaing he relevan zero-coupon bond volailiies from (42)) o hose in he Vasicek example. 18
he Vasicek example. The speculaive demand for securiies in his example is exacly similar o he speculaive demand in he above Vasicek example. The way he invesors wan o hedge changes in he opporuniy se, however, may be quie differen due o he more complex dynamics of he erm srucure of ineres raes in his HJM hree-facor seing. In our view, a comparison beween he hedge choice in he Vasicek example and in his HJM hree-facor seing using he base case parameers in (44) is relevan for addressing quesions such as: (i) is he presen form of he erm srucure of ineres raes imporan forhow o hedge changes in he opporuniy se, and (ii) is he exibiliy and dynamics of he erm srucure of ineres raes imporan for how o hedge changes in he opporuniy se? As formalized in Proposiion 2 he forward-expeced consumpion paern is crucial for he hedging behavior since he appropriae bond (or bond porfolio) for hedging changes in he opporuniy se is one ha has a paymen schedule similar o he opimal forward-expeced consumpion paern. Hence, he quesions above can be answered by comparing he opimal consumpion paerns across he wo differen examples. The opimal consumpion paerns are abulaed in Table 1 for invesors wih differen degrees of relaive risk aversion. As in he Vasicek example, he invesors have invesmen horizon of 25 years, a subjecive ime discoun rae of fi = 0:03, and hey equally weigh uiliy from inermediae consumpion and final wealh, i.e., K = 1 in he general uiliy funcion specificaion in (1). 2 [ INSERT TABLE 1 ABOUT HERE ] The (forward-expeced) consumpion paerns for he Vasicek dynamics are exacly idenical o hose displayed in Figure 2 o Figure 6 in he Vasicek example above. The (forward-expeced) consumpion paerns for he HJM hree-facor model are for he benchmark parameers in (44). Of course, he resuls for log-uiliy invesors and infiniely risk averse invesors in Table 1 are exacly idenical since he (forward-expeced) consumpion paerns of hese invesors depend only on he curren form of he erm srucure of ineres raes, as shown in Corollary 1 and Corollary 2. However, also for invesors wih relaive riskaversion in beween hese benchmark invesors he differences beween he consumpion paerns in he Vasicek example and in he base case HJM hree-facor model seems basically ignorable. The conclusion from observing similar (forward-expeced) consumpion paerns from he Vasicek example and he base case HJM hree-facor model is ha invesors need no care abou he dynamics of he erm srucure of ineres raes since in boh cases he invesors should hedge changes in he invesmen opporuniy se by basically buying he same coupon bond. On he oher hand, he curren form of he erm srucure is imporan for he opimal consumpion paerns of he invesors and, 19
hence, imporan for he precise paymen schedule of he relevan bond for hedging changes in he opporuniy se. In Table 2 we have abulaed resuls for wo oher ses of parameer values for he HJM hree-facor model. [ INSERT TABLE 2 ABOUT HERE ] In he discussion of Proposiion 1, i was noed ha opimal consumpion choices are only alered if one change parameers ha ener he dynamics (or paricular momens) of he pricing kernel process. Hence, e.g., changing he volailiies ff S1 and ff S2 of he invesmen asses will have no consequences for he opimal forward-expeced consumpion paern and, hence, no consequences for he relevan coupon bond o hedge changes in he opporuniy se. On he oher hand, if one changes risk premia or parameers in he descripion of he erm srucure dynamics he opimal consumpion paern will in general be affeced. Therefore, we only consider wo oher ses of parameers: one in which forward-rae volailiy parameers are changed and one in which risk premia parameers are changed. The wo ses of alernaive parameers considered in Table 2 are:» 2 =1:00;» 3 =0:50; ff 1 =0:00650; ff 2 =0:02367; ff 3 =0:01738; ff S1 =(0:03187; 0:02305; 0:04857) 0 ; ff S2 =0:24206; (45) ' S =0:05; S =0:19365; B =(0:02549; 0:01844; 0:03886) 0 and» 2 =1:00;» 3 =0:50; ff 1 =0:00325; ff 2 =0:01184; ff 3 =0:00869; ff S1 =(0:03187; 0:02305; 0:04857) 0 ; ff S2 =0:24206; (46) ' S =0; S =0; B =(0; 0; 0) 0 ; respecively. The parameer se in (45) differs from he base se of parameers in (44) alone by higher volailiies on he forward rae curve; specifically, he parameers in (45) are chosen such ha he volailiies on zero-coupon bonds wih ime o mauriy equal o 0.25 years, 2 years, and 10 years, respecively, are exacly wice as large as in he HJM base case parameers se and, hence, wice as large as in he Vasicek example. The speculaive demands for socks and bonds are similar o hose in he base case, i.e. a logarihmic uiliy invesor inves an 80% fracion of wealh in he sock, 0% in bonds, and 20% in he bank accoun. The parameer se in (46) differs from he base se of parameers in (44) alone by having zero prices on risk so ha he speculaive demands for socks and bonds are zero, i.e. a logarihmic 20
uiliy invesor in his case invess a 0% fracion of wealh in he sock, 0% in bonds, and 100% in he bank accoun. The opimal (forward-expeced) consumpion paerns in he HJM hree-facor example wih he above parameer choices are abulaed in Table 2 under he labels HJM-2" and HJM-3", respecively. The opimal (forward-expeced) consumpion paerns for he benchmark parameer se in (44) are idenical o hose in Table 1 and abulaed under he label HJM-1" in Table 2. For he polar cases of log-uiliy invesors and infiniely risk averse invesors he opimal consumpion paerns are unalered across he differen parameer ses since hese only depend on he iniial form of he erm srucure; hese cases are, herefore, no abulaed in Table 2. For invesors wih preferences in beween he polar cases of logarihmic uiliy and infinie risk aversion, he (forward-expeced) consumpion paerns do depend on he specific se of parameers applied, as can be seen from Table 2. However, i seems ha he opimal consumpion paerns do no change dramaically across he differen parameer ses. In paricular, he consumpion paerns in he case of higher forward rae volailiies are basically similar o hose in he benchmark parameer case (44) and in he Vasicek-example. In order o have an objecive measure of he disance beween he differen consumpion plans in Table 2 and, hence, of he relevan bonds o hedge changes in he opporuniy se, we have also abulaed Fisher-Weil duraions in Table 2. The Fisher-Weil duraion measure is in his conex defined by R T (s )k(s)p (s)ds +(T )k(t )P (T ) R T k(s)p (s)ds + k(t )P (T ) and is a measure of he average ime o he paymens of any paricular bond. Even for he case of zero risk premia, he duraions of he relevan coupon bond for hedging changes in he opporuniy se seem close o he relevan duraions implied by he oher parameer ses considered in Table 2. 5 Conclusion In his paper we have derived opimal sraegies for invesmens in socks and erm-srucure derivaives for a CRRA invesor in a complee marke. We provided explici resul on how o hedge changes in he invesmen opporuniy se in he case of muli-facor Gaussian HJM ineres raes and deerminisic marke prices of risk. In paricular, we have demonsraed how changes in he invesmen opporuniy se can be hedged by a single bond: A zero-coupon bond for he case of uiliy from erminal wealh only and a coninuous-coupon bond in he case 21