Optimal consumption and investment strategies with stochastic interest rates

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1 Journal of Banking & Finance 28 (2004) Opimal consumpion and invesmen sraegies wih sochasic ineres raes Claus Munk a, Carsen Sørensen b, * a Deparmen of Accouning and Finance, Universiy of Souhern Denmark, Odense, Denmark b Deparmen of Finance, Copenhagen Business School, Solbjerg Plads 3, DK-2000 Frederiksberg, Denmark Received 24 December 2001; acceped 15 July 2003 Available online 18 December 2003 Absrac We characerize he soluion o he consumpion and invesmen problem of a power uiliy invesor in a coninuous-ime dynamically complee marke wih sochasic changes in he opporuniy se. Under sochasic ineres raes he invesor opimally hedges agains changes in he erm srucure of ineres raes by invesing in a coupon bond, or porfolio of bonds, wih a paymen schedule ha equals he forward-expeced (i.e. cerainy equivalen) consumpion paern. Numerical experimens wih wo differen specificaions of he erm srucure dynamics (he Vasicek model and a hree-facor non-markovian Heah Jarrow Moron model) sugges ha he hedge porfolio is more sensiive o he form of he erm srucure han o he dynamics of ineres raes. Ó 2003 Elsevier B.V. All righs reserved. JEL classificaion: G11 Keywords: Dynamic asse allocaion; Hedging; Term srucure of ineres raes 1. Inroducion Since he pahbreaking papers of Meron (1969, 1971, 1973) i has been recognized ha long-erm invesors wan o hedge sochasic changes in invesmen opporuniies, such as changes in ineres raes, excess reurns, volailiies, and inflaion raes. The main conribuion of his paper is o enhance he undersanding of how invesors wih * Corresponding auhor. Tel.: ; fax: address: cs.fi@cbs.dk (C. Sørensen) /$ - see fron maer Ó 2003 Elsevier B.V. All righs reserved. doi: /j.jbankfin

2 1988 C. Munk, C. Sørensen / Journal of Banking & Finance 28 (2004) consan relaive risk aversion (CRRA) preferences for consumpion (and, possibly, erminal wealh) should opimally hedge ineres rae risk. We demonsrae ha he opimal hedge agains changes in ineres raes is obained by invesing in a coupon bond, or porfolio of bonds, wih a paymen schedule ha precisely equals he cerainy equivalens of he fuure opimal consumpion raes. Furhermore, we sudy he imporance for ineres rae hedging of boh he curren form and he dynamics of he erm srucure. In a numerical example we compare he soluions for a sandard one-facor Vasicek and a hree-facor model where he erm srucure can exhibi hree kinds of changes: a parallel shif, a slope change, and a curvaure change. Our findings sugges ha he form of he iniial erm srucure is of crucial imporance for he opimal fuure consumpion plan and, hence, imporan for he relevan ineres rae hedge, while he specific dynamics of he erm srucure is of minor imporance. As shown by Heah e al. (1992), any dynamic ineres rae model is fully specified by he curren erm srucure and he forward rae volailiies. Therefore, he Heah Jarrow Moron (HJM) modeling framework is naural for he purpose of comparing he separae effecs of he curren erm srucure and he dynamics of he erm srucure on he opimal ineres rae hedging sraegy. The HJM class ness all Markovian ineres rae models, such as he Vasicek model. However, models ouside his Markovian class also frequenly arise wihin he HJM modeling framework. This is, for example, he case for he hree-facor model considered in our numerical example. Given ha we wan o compue opimal invesmen sraegies in possibly non- Markovian models, we firs derive a general, exac characerizaion of boh opimal consumpion and porfolio choice in a framework ha also allows for non-markovian dynamics of asse prices and he erm srucure of ineres raes, bu requires dynamically complee markes. This characerizaion generalizes recen resuls in specialized Markovian seings (Liu, 1999; Wacher, 2002a). For he special case where ineres raes have Gaussian, bu sill poenially non-markovian, HJM dynamics, we obain he explici soluion for he opimal consumpion and invesmen sraegies ha we use for sudying he impac of he curren form and he dynamics of he erm srucure on hedging demand. To our knowledge, his paper provides he firs explici soluion o an ineremporal consumpion and invesmen problem where he dynamics of he opporuniy se is non-markovian and he invesor has non-logarihmic uiliy. There has recenly been a number of sudies of opimal invesmen sraegies wih specific assumpions on he dynamics of ineres raes. Brennan and Xia (2000) and Sørensen (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 a Vasicek-ype model and marke prices on risk (and expeced excess reurns) are consan, he opimal hedge porfolio is he zero-coupon bond ha expires a he invesmen horizon. This paricular resul is also obained as a special case wihin he framework of his paper. Liu (1999) provides similar insigh using he one-facor square-roo model of Cox e al. (1985). A few papers have addressed he porfolio problem under sochasic ineres raes for invesors wih uiliy over consumpion. In a general complee-marke seing,

3 C. Munk, C. Sørensen / Journal of Banking & Finance 28 (2004) Wacher (2002b) shows ha an infiniely risk-averse agen will only inves in a coupon bond. This resul is also a special case of our findings, bu we solve for he opimal porfolio for CRRA invesors wih any level of risk aversion. Boh Campbell and Viceira (2001) and Brennan and Xia (2002) sudy consumpion and porfolio choice problems in seings wih uncerain inflaion, where real ineres raes follow a one-facor Vasicek model. While we ignore inflaion risk, we allow for more general dynamics of ineres raes. 1 The general modeling of he invesmen opporuniy se in his paper ness he Markovian models sudied in he above-menioned papers. Furhermore, we explicily link he opimal hedge porfolio o he opimal consumpion paern of he invesor. In addiion, we sudy how sensiive he opimal hedge agains ineres rae risk is o he curren form of he erm srucure and o he dynamics of he erm srucure of ineres raes. The res of he paper is organized as follows. In Secion 2 we se up he general coninuous-ime 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 showing how o hedge agains changes in he erm srucure of ineres raes using coupon bonds in a specialized HJM muli-facor Gaussian erm srucure seing. In Secion 4 we consider wo specific numerical examples based on he Vasicek model and an HJM hree-facor model. We compare he hedge bonds in he wo examples for differen levels of risk aversion and differen forms of he iniial erm srucure of ineres raes. Secion 5 concludes Porfolio choice wih general dynamics in invesmen opporuniies We consider a fricionless economy where he dynamics is generaed by a d-dimensional Wiener process, w ¼ðw 1 ;...; w d Þ, defined on a probabiliy space ðx; F; PÞ. F ¼fF : P 0g denoes he sandard filraion of w and, formally, ðx; F; F ; PÞ is he basic model for uncerainy and informaion arrival in he following Preferences We will consider he invesmen problem of an expeced uiliy maximizing invesor wih a ime-separable consan relaive risk aversion uiliy funcion given by Z T K E 0 U 1 ðc ; Þd þð1 KÞE 0 ½U 2 ðw T ÞŠ; ð1þ 0 1 Inflaion can be inroduced along he same lines in he se-up of his paper in which case he relevan bond for hedging purposes would be an indexed bond wih paymens ha in real erms mach he forwardexpeced consumpion paern. 2 Proofs and deailed derivaions are conained in an appendix which is available from he auhors by reques.

4 1990 C. Munk, C. Sørensen / Journal of Banking & Finance 28 (2004) where U 1 ðc; Þ ¼e b C1 c 1 ; 1 c U 2 ðw Þ¼e bt W 1 c 1 ; 1 c and where b is a consan subjecive ime discoun rae and c is a consan relaive risk aversion 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 c ¼ 1 is he logarihmic uiliy case: U 1 ðc; Þ ¼e b log C and U 2 ðw Þ¼e bt log W 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 ¼ðV 1 ;...; V d Þ 0. The price dynamics of he risky asses (cum dividend) is given by dv ¼ diagðv Þ½ðr 1 d þ r k Þd þ r dw Š; ð2þ where k is an R d -valued sochasic process of marke prices of risk, r is an R dd - valued sochasic process of volailiies, 1 d is a d-dimensional vecor of ones, and diagðv Þ is a ðd dþ-dimensional marix wih V in he diagonal and zeros off he diagonal. I is assumed ha r has full rank d implying ha markes are dynamically complee (cf. Duffie and Huang, 1985). As a consequence of markes being dynamically complee, he pricing kernel (or sae-price deflaor) is uniquely deermined and given by (see, e.g., Duffie, 1996, Chaper 6) Z Z f ¼ exp r s ds k 0 s dw s 1 Z kk s k 2 ds ; P 0; ð3þ or, equivalenly, in differenial form, df ¼ f ½ r d k dw Š; f 0 ¼ 1: ð4þ 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 f PV ½X Š¼E s X ¼ P ðsþbe s ½X Š; ð5þ f where P ðsþ 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, be s ½X Š; see, e.g., Jamshidian (1987, 1989) and Geman (1989) who inroduce he noion of he forward risk-neural maringale measure, as being disinc from he usual riskneural maringale measure in he conex of ineres rae models.

5 C. Munk, C. Sørensen / Journal of Banking & Finance 28 (2004) The problem and he general soluion Le p be an R d -valued process describing he fracions of wealh ha he agen allocaes ino he d differen risky asses. The wealh of he agen hen evolves according o dw ¼½ðr þ p 0 r k ÞW C Šd þ W p 0 r dw : ð6þ The agen s problem is o choose a dynamic consumpion sraegy, C, and porfolio policy, p, in order o maximize he expeced uiliy in (1). This problem has radiionally been addressed and solved by using a dynamic programming approach, cf. Meron (1969, 1971, 1973). The main idea of he maringale soluion approach suggesed and formalized by Cox and Huang (1989, 1991) and Karazas e al. (1987) is o alernaively consider he saic problem sup K E 0 fc ;W T g subjec o E 0 Z T 0 f f 0 Z T 0 C d þ U 1 ðc ; Þd þð1 KÞE 0 ½U 2 ðw T ÞŠ f T W T f 0 ð7þ 6 W 0 : ð8þ 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. As shown by Cox and Huang (1989, 1991) and Karazas e al. (1987), he soluion o his problem also provides he soluion o he dynamic problem of choosing he opimal consumpion sraegy and porfolio policy. 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, 06 6 T. 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 ) o deermine he opimal consumpion process, C, and erminal wealh, W T. Thus, under he specific CRRA uiliy assumpion in (1), he opimal consumpion plan given informaion available a ime akes he form 3 C s ¼ W K 1 b c e Q c ðs Þ 1 c; s 6 T ; ð9þ f s f 3 The deails in his derivaion are conained in he appendix which is available from he auhors by reques.

6 1992 C. Munk, C. Sørensen / Journal of Banking & Finance 28 (2004) where he (invesor-specific) sochasic process Q is defined by Z " T # " c 1 # c 1 Q ¼ K 1 c e b c ðs Þ c f E s ds þð1 KÞ 1 c e b c ðt Þ c f E T : ð10þ f f Noe ha he curren consumpion rae a ime is given by C ¼ðW =Q ÞK 1 c and, hence, ha Q describes he wealh-o-consumpion raio. As formalized in Proposiion 1 below, Q is also crucial for deermining how o hedge agains changes in he opporuniy se. While he consumpion policy is usually given explicily by solving (7) and (8), as in (9), 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). The exisence and uniqueness of such a porfolio policy follow from he Maringale Represenaion Theorem (see, e.g., Duffie, 1996). 4 For log-invesors (c ¼ 1) i is well known ha he opimal porfolio is he growhopimal porfolio, bu in order o derive an explici expression for he opimal porfolio for oher invesors i is generally recognized ha he price dynamics mus be specialized. Cox and Huang (1989) show ha when he sae-price deflaor 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, he following proposiion provides a closed-form expression for he opimal invesmen sraegy for a power uiliy invesor in a general possibly non-markovian complee marke seing for a CRRA invesor. Since Q, as defined in (10), is a posiive sochasic process adaped o he filraion generaed by w, i follows from he Maringale Represenaion Theorem, ha he dynamics of Q can be described by dq ¼ Q ½l Q d þ r Q dw Š ð11þ for some drif process l Q and some volailiy process r Q. The precise forms of l Q and r Q depend on he specific assumpions on he pricing kernel and, subsequenly, we will consider such specific examples and apply he following general proposiion. Proposiion 1. The value funcion of he general problem in (7) and (8) has he form J ¼ Qc W 1 c AðÞ ; ð12þ 1 c where AðÞ ¼ K b ð1 e bðt Þ bðt Þ Þþð1 KÞe and Q is defined in Eq. (10). The opimal consumpion choice and he opimal porfolio policy a ime are given by 4 The 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).

7 C. Munk, C. Sørensen / Journal of Banking & Finance 28 (2004) and C ¼ K 1 c W Q p ¼ 1 ðr 0 c Þ 1 k þðr 0 Þ 1 r Q : ð13þ ð14þ Proof. The proof is available from he auhors upon reques. h Proposiion 1 saes he opimal consumpion and invesmen sraegies in our seing ha allows for general, possibly non-markovian, shifs in he invesmen opporuniy se. However, a special case is he Markovian seing where he shifs in invesmen opporuniies are governed by a Markov diffusion process x wih dynamics dx ¼ l x ðx ; Þd þ r x ðx ; Þdw : In his case, he basic opimizaion problem considered in his paper could alernaively be solved using a radiional dynamic programming approach, and i is well known ha some (unknown) funcion Qðx ; Þ exiss such ha J is given as in (12) wih Q replaced by Qðx ; Þ (see, e.g., Ingersoll, 1987). In his case he characerizaion of opimal consumpion in (13) follows from he so-called envelope condiion and, furhermore, i follows direcly by Io s lemma ha r Q in (14) can be characerized on he form r x ðx ; Þ oq =Qðx ox ; Þ. Proposiion 1 provides an explici characerizaion of he funcion Q and, in paricular, exends he resul so ha i also applies for non-markovian marke seings where a dynamic programming approach does no direcly apply. As in Meron (1971), he porfolio policy can be decomposed ino a speculaive porfolio (he firs erm in (14)) and a hedge porfolio ha describes how he invesor should opimally hedge agains changes in he invesmen opporuniy se (he las erm in (14)). The invesor mus hus form a hedge porfolio ha basically mimics he dynamics of Q and, hence, Q reflecs everyhing of imporance for how o hedge agains changes in he invesmen opporuniy se. For a given invesor i can be inferred from (10) 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 hedging agains 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. I is insrucive o consider wo special cases: he log-uiliy case (c ¼ 1) and he case of an infiniely risk-averse invesor (c ¼1). 5 The log-uiliy invesor does no hedge agains changes in he opporuniy se a all (he las erm in (14) vanishes as c! 1) and he opimal consumpion rae is C ¼ KW =AðÞ, i.e. a ime-varying, bu 5 Formally, he resuls for an infiniely risk averse invesor are defined as he limiing resuls of Proposiion 1 as c!1.

8 1994 C. Munk, C. Sørensen / Journal of Banking & Finance 28 (2004) deerminisic, fracion of wealh. The infiniely risk-averse invesor has no speculaive demand for securiies a all (he firs erm in (14) vanishes as c!1). If his invesor has uiliy from boh consumpion and erminal wealh, he Q-process reduces o Q ¼ Z T P ðsþds þ P ðt Þ: Hence, he hedge porfolio is an annuiy bond. (In he special case where he invesor has uiliy from erminal wealh only, i.e. K ¼ 0, he hedge porfolio is a zero-coupon bond ha expires a he invesmen horizon.) According o (9), he opimal consumpion sraegy is in his case consan, C ¼ W =Q ¼ W 0 =Q 0, and he opimal consumpion sraegy is hus basically implemened by using he cerain paymens on he annuiy bond for consumpion. 6 ð15þ 3. Hedging changes in ineres raes In he res of he paper we focus on how o hedge changes in ineres raes. In his secion we will provide an explici soluion o he consumpion and invesmen choice problem when ineres raes evolve according o a HJM model. 7 This is an applicaion of Proposiion 1. Furhermore, we demonsrae a close link beween he hedging demand and he opimal consumpion sream. For convenience and clariy we separae he invesmen asses ino socks and bonds in he following. Formally, we spli he d-dimensional Wiener process generaing he financial asse reurns as w ¼ðw B ; w S Þ, where w B is of dimension k and w S is of dimension l ¼ 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 sock prices may depend on boh w B and w S which allow 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 is given by db ¼ diagðb Þ½ðr 1 k þ r B k B Þd þ r B dw B Š ð16þ and ds ¼ diagðs Þ½ðr 1 l þ u S Þd þ r S1 dw B þ r S2 dw S Š; where r B, r S1,andr S2 are marix valued processes of dimension k k, l k, and l l, respecively. Again, r B and r S2 are assumed non-singular so ha markes are complee. Changes in he reurns of he erm srucure derivaives and he socks are ð17þ 6 An annuiy bond is a coupon bond where he cerain cash flows (coupon + principal repaymen) from he bond are he same hroughou he finie life of he bond. 7 The HJM approach is, o our knowledge, he mos general ineres rae modeling framework, and any erm srucure model ha does no allow for arbirage can be represened in a HJM seing.

9 correlaed wih k l covariance marix r B r 0 S1. The marke price of risk process k (which is no dependen on he paricular se of asses chosen) has he form where k ¼ C. Munk, C. Sørensen / Journal of Banking & Finance 28 (2004) k B k S ; k S ¼ r 1 S2 u S r 1 S2 r S1k B : Noe ha we have inroduced he R l -valued sochasic process u S ð¼ r S1 k B þ r S2 k S Þ, which can be inerpreed as he expeced excess reurn on he socks. More specifically, we assume ha he dynamics of he erm srucure of ineres raes can be described by a k-facor model of he HJM class inroduced by Heah e al. (1992). For any mauriy dae s he dynamics of he s-mauriy insananeous forward rae is f ðsþ ¼f 0 ðsþþ Z 0 aðs; sþds þ Z 0 r f ðs; sþ 0 dw Bs ; where r f ð; sþ is an R k -valued deerminisic funcion and f 0 ðsþ is he s-mauriy forward rae observed iniially a ime 0. The shor-erm ineres rae is r ¼ f ðþ. As a no-arbirage drif resricion we have ha Z s að; sþ ¼r f ð; sþ 0 k B ðþ þ r f ð; uþdu ; so ha one only has o specify he iniial erm srucure of forward raes and he volailiy srucure r f ð; sþ. Among he many erm-srucure derivaives, we focus on defaul-free bonds. The dynamics of he price P ðsþ ¼expð R s f ðsþdsþ of he zero-coupon bond mauring a ime s is given by ð18þ dp ðsþ ¼P ðsþ½ðr þ r P ð; sþ 0 k B ðþþd þ r P ð; sþ 0 dw B Š; ð19þ where r P ð; sþ ¼ R s r f ð; uþdu. For laer use we will also consider a bond paying a coninuous coupon of kðþ up o ime T and a lump sum paymen of kðt Þ a ime T. The ime price of such a bond is B cpn ¼ Z T kðsþp ðsþds þ kðt ÞP ðt Þ: Applying a Leibniz-ype rule for sochasic processes (which in he specific conex is formally saed and proved in he appendix which is available from he auhors by reques), i is seen ha he coupon bond price mus evolve according o db cpn ¼ kðþd þ B cpn h ðr þ r 0 B cpn k B ðþþd þ r 0 B cpn dw B i;

10 1996 C. Munk, C. Sørensen / Journal of Banking & Finance 28 (2004) where r B cpn ¼ R T kðsþp ðsþr P ð; sþds þ kðt ÞP ðt Þr P ð; T Þ : ð20þ kðsþp ðsþds þ kðt ÞP ðt Þ R T Our specific resuls on how o hedge agains changes in ineres raes, as saed in Proposiion 2, are based on he following assumpion. Assumpion 1. The marke price of risk process k kðþ and he forward rae volailiies r f ð; sþ are deerminisic funcions of ime. The implicaions of he assumpion ha marke prices of risk and forward rae volailiies are deerminisic are imporan since we only allow ineres raes o change sochasically and, hence, here are no reasons o hedge agains sochasic changes in marke prices of risk or 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 r B, r S1, and r S2 of he invesmen asses are deerminisic and, in fac, hey may be described by non-markovian processes. Muli-facor Gaussian models are in many respecs flexible and hus ofen used for derivaive pricing since hey allow closed-form soluion for mos European-ype erm srucure coningen claims (e.g., Amin and Jarrow, 1992; Brace and Musiela, 1994). A shorcoming of Gaussian erm srucure models, hough, is ha hey are no able o rule ou negaive ineres raes. The Gaussian assumpion also allows closed-form expressions for opimal invesmen sraegies, as we shall see in he following. Furhermore, i is imporan o poin ou ha also in Gaussian HJM models, he shor rae process is no necessarily Markovian (as is he case in he HJM hreefacor example considered in a subsequen secion). 8 From he assumpion ha prices of risk and forward rae volailiies are deerminisic, i follows ha he shor-erm ineres rae is normally disribued (Gaussian) and ha he pricing kernel f, as saed in (3), is lognormally disribued. I is hus possible o compue in closed form he expecaions in he definiion of Q in (10) and, hence, obain an analyical expression for Q. The proof of he following proposiion is based on his feaure. Proposiion 2. Under Assumpion 1, he value funcion and he opimal consumpion sraegy are given by (12) and (13) in Proposiion 1, where in his case Q ¼ Z T Z ðsþds þ Z ðt Þ ð21þ 8 In fac, he shor rae is only Markovian if r f ð; sþ can be separaed as r f ð; sþ ¼GðÞHðsÞ, where H is a real-valued coninuously differeniable funcion ha never changes sign and G is an R k -valued coninuously differeniable funcion (cf. Carverhill, 1994).

11 wih and Z ðsþ ¼K 1 c ðp ðsþþ c 1 c exp b c Z ðt Þ¼ð1 KÞ 1 c ðp ðt ÞÞ c 1 c exp b c gð; sþ ¼ C. Munk, C. Sørensen / Journal of Banking & Finance 28 (2004) Z s kkðuþk 2 du þ Z s c ðs Þþ1 gð; sþ ; s < T ; ð22þ 2c 2 c ðt Þþ1 2c gð; T Þ 2 kr P ðu; sþk 2 du 2 Z s k B ðuþ 0 r P ðu; sþdu: The opimal porfolio policy a ime is described by p ¼ 1 ðr 0 c Þ 1 kðþþ c 1 ðr 0 r cpn B c Þ 1 ; ð25þ 0 where r cpn B is he volailiy vecor of a bond, as defined in Eq. (20), which pays coninuous coupon according o kðsþ ¼bE s ½C sš ¼ K 1 c W Q ðp ðsþþ 1 c exp b c and has a erminal lump sum paymen a ime T of kðt Þ¼bE T ½W T Š ¼ð1 KÞ 1 c W Q ðp ðt ÞÞ 1 c exp ð23þ ð24þ c ðs Þþ1 gð; sþ ; s < T ; ð26þ 2c 2 b c c ðt Þþ1 2c gð; T Þ : ð27þ 2 Proof. The proof is available from he auhors by reques. h Proposiion 2 provides an explici expression for he opimal invesmen sraegy wih possibly non-markovian and muli-facor dynamics of ineres raes. The opimal porfolio policy in (25) is described by wo erms: he firs erm describes he usual speculaive demand for risky asses while he second erm describes he hedging demand for risky asses. The form of he hedging erm is such ha by choosing risky asse weighs according o his erm (and he residual invesed in he risk-free bank accoun), one obains a bond porfolio ha basically replicaes a specific coupon bond. This specific coupon bond will be referred o as he hedge bond in he following. In paricular, Proposiion 2 shows ha wih uiliy from inermediae consumpion his hedge bond is equivalen o a coupon bond wih coupon raes equal o he cerainy equivalens of opimally planned fuure consumpion raes. In he special case of uiliy from erminal wealh only (corresponding o K ¼ 0), he relevan bond for hedging reduces o a zero-coupon bond ha expires a he invesmen horizon. This is similar o he insigh obained in specialized Vasicek seings by Brennan and Xia (2000) and Sørensen (1999). A zero-coupon bond seems an

12 1998 C. Munk, C. Sørensen / Journal of Banking & Finance 28 (2004) inuiively appealing insrumen for hedging changes in ineres raes in he case of uiliy from erminal wealh only since his securiy has a cerain paymen a he invesmen horizon irrespecively of how ineres raes evolve. Likewise, in he case including uiliy of inermediae consumpion he suggesed coupon bond seems a reasonable insrumen for hedging shifs in ineres raes in he sense ha he cerain paymens on he bond mach he currenly planned fuure consumpion expendiure profile irrespecively of how ineres raes evolve. The opimal porfolio policy described in (25) can in fac be implemened by allocaing a fracion of wealh (1=c) ino he speculaive porfolio and a fracion of wealh (1 1=c) ino he appropriae hedge bond. In order o see his, le p be he R dþ1 -valued vecor process describing he augmened opimal porfolio weighs where he fracion of wealh invesed in he risky asses, p, are included as he firs d-enries while he fracion of wealh invesed in he risk-free bank accoun is included as he ðd þ 1Þh enry. Noe ha by insering he opimal risky asse porfolio weighs in (25), he opimal augmened porfolio policy can be obained in he form p ¼ ¼ 1 c p d p ðr 0 Þ 1 kðþ d ðr0 Þ 1 kðþ 0 þ 1 1 ðr0 r B cpn Þ 1 B r B cpn d ðr0 Þ C A : The firs erm in (28) describes he augmened opimal porfolio weighs in he loguiliy case where c ¼ 1. This porfolio is usually referred o as he growh-opimal porfolio or, equivalenly, he speculaive porfolio. On he oher hand, he las erm in (28) describes he augmened porfolio weighs needed o implemen he appropriae hedge bond. According o Proposiion 2, he specific dynamics of he erm srucure of ineres raes is of imporance for how o hedge agains 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 forward-expeced consumpion paerns and, in paricular, our focus is on wheher he curren form of he erm srucure or he dynamics of he erm srucure is of crucial imporance for he opimal forward-expeced consumpion paern. In his conex i can be noed ha, even in he general seing of Secion 2, only he form of he erm srucure maers for he opimal forward-expeced consumpion paerns for he benchmark cases of log-invesors and infiniely risk averse invesors, while he erm srucure dynamics is irrelevan. For infinie risk aversion, his follows from he fac ha he opimal consumpion rae is consan and equal o W =Q where Q describes he price of an annuiy bond which is fully deermined by he prevailing erm srucure a ime, cf. he descripion of Q in (15) for his special case. For log uiliy, i can be shown ha he forward-expeced opimal consumpion rae is be s ½C K sš¼w AðÞ ðp ðsþþ 1 e bðs Þ ; s < T ; ð29þ ð28þ

13 C. Munk, C. Sørensen / Journal of Banking & Finance 28 (2004) wih AðÞ being defined in Proposiion 1. The forward-expeced erminal wealh a ime T is given by a similar expression which is also fully deermined by he curren erm srucure of ineres raes, as refleced in zero-coupon bond prices P ðsþ, and no influenced by he parameers describing he erm srucure dynamics. 4. Specific examples In his secion we consider wo specific examples of ineres rae dynamics in he se-up of he previous secion. The firs example is based on he erm srucure dynamics from he Vasicek (1977) model while he second example is based on a flexible 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. As shown by Heah e al. (1992), any dynamic ineres rae model is fully specified by he curren form of he erm srucure and he forward rae volailiies. Hence, he HJM framework is naural for he purpose of comparing he separae effecs of he curren form of he erm srucure and he dynamics of he erm srucure on he opimal ineres rae hedging sraegy. In our specific examples he iniial erm srucures are hus chosen o be idenical across he wo examples, i.e. he iniial form of he erm srucure curve in he hree-facor HJM erm srucure model is adoped from he Vasicek example. We compue he opimal sraegies in boh examples using empirically reasonable parameer values. For various degrees of risk aversion and for differen forms of he iniial erm srucure we compare he relevan hedge bond under Vasicek dynamics and he relevan hedge bond under he hreefacor HJM dynamics. Our resuls below sugges ha he opimal paymen schedule on he hedge bond is very sensiive o he form of he iniial erm srucure of ineres raes while he opimal paymen schedule on he hedge bond is insensiive o he dynamics of ineres raes over ime when he curren erm srucure is held fixed Vasicek example In he following example we allow for uiliy from boh inermediae consumpion and erminal wealh by seing he preference parameer K equal o 1 in he specificaion of he uiliy funcion in (1). This implies ha uiliy from inermediae con- 2 sumpion and uiliy from erminal wealh are weighed equally. The se-up for invesmen asses in he following example is basically as in Brennan and Xia (2000) and Sørensen (1999), bu hey only consider uiliy from erminal wealh. The agen can inves in a single sock and a single bond as well as he insananeously risk-free bank accoun. The erm srucure dynamics is 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 ¼ jðh r Þd r r dw B ; ð30þ

14 2000 C. Munk, C. Sørensen / Journal of Banking & Finance 28 (2004) where he parameer h describes he long-run level for he shor-erm ineres rae, j is a mean-reversion parameer ha deermines he srengh of endency o he longrun level, and he parameer r r describes he ineres rae volailiy. Besides he parameers describing he ineres rae dynamics, he parameer denoed k 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 s-mauriy forward rae a ime in he Vasicek model is given by f ðsþ ¼e jðs Þ r þ r 1 ð1 e jðs Þ Þþ r2 r 2j e jðs Þ bðs Þ; where ð31þ r 1 ¼ h þ k Br r j r2 r 2j 2 ; bðsþ ¼ 1 j ð1 e js Þ: The dynamics of he s-mauriy forward rae can be deermined from (31) and (30) and an applicaion of Io s lemma. In paricular, i is seen ha he forward rae volailiy srucure in his example has he form r f ð; sþ ¼ r r e jðs Þ. Wihin he HJM framework of Secion 3, his volailiy srucure and an iniial erm srucure of forward raes of he form in (31) provide a complee specificaion of he Vasicek (1977) erm srucure model. The agen can inves in a single sock as well as bonds and he bank accoun. In he specific case of a one-facor ineres rae model i is sufficien ha he agen can inves in a single bond besides he sock and he bank accoun in order o implemen he complee-marke opimal soluion. The price process of he single sock is described in Eq. (17) where in his case r S1 and r S2 are scalars (i.e. of dimension 1 1). The specific parameer values used in he following numerical example are chosen as follows: h ¼ 0:04; j ¼ 0:15; r r ¼ 0:015; r S1 ¼ 0:0625; r S2 ¼ 0:2421; u S ¼ 0:05; k S ¼ 0:19365; k B ¼ 0:05: ð32þ In paricular, he parameers j, h, and r r, which describe he ineres rae dynamics, are chosen so ha hey are close o hose obained by Chan e al. (1992) for he Vasicek ineres rae process. The parameers for he sock process are chosen so ha he expeced excess reurn on he sock is u S ¼ 5%, he volailiy of he sock is consan 25% ð¼ ðr 2 S1 þ r2 S2 Þ1=2 Þ, 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 hisorical reurns daa on socks

15 C. Munk, C. Sørensen / Journal of Banking & Finance 28 (2004) (see, e.g., Brealey and Myers, 1996, Chaper 7, Table 7-1). Though, as poined ou by Brown e al. (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 he 20.2% hisorical volailiy esimae on he S&P 500 index based on he Ibboson 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, k B ¼ 0:05, implies ha e.g. he expeced excess reurn on a 10-year zero-coupon bond in he Vasicek model is 0.39%. 9 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. Hence, he speculaive porfolio under he specific parameer values involve no speculaive demand for bonds. Agens wih nonlogarihmic uiliy, however, wan o inves in a bond, or bond porfolio, ha has payoffs ha equal heir forward-expeced consumpion paern in order o hedge agains changes in he opporuniy se, as described in Proposiion 2. In line wih he discussion afer Proposiion 2, he appropriae invesor specific bond in his respec is referred o as he hedge bond. As formalized in (28), he infiniely risk averse invesors inves 100% in he hedge bond while e.g. an invesor wih consan relaive risk aversion, c, equal o 2 will inves 50% (¼ 1=c) of wealh in he speculaive porfolio and 50% (¼ 1 1=c) 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. The opimal asse allocaions of invesors wih risk aversion parameers 1, 4/3, 2, 4, and infiniy are abulaed in Table 1 in accordance wih (28). I can be noed ha he asse allocaions abulaed in Table 1 do no depend on he ime horizons of he invesors; however, he appropriae hedge bonds differ across invesors ha are heerogeneous wih respec o boh risk aversion and ime horizon. Moreover, he asse allocaion choices in Table 1 do no depend on he form of he curren erm srucure, bu he opimal paymen schedules on he hedge bonds do. Finally, i may be noed ha if he relevan coupon bonds for hedging are no explicily available in he marke, hey can always be replicaed by rading in any single bond and he bank accoun wihin he Vasicek model. We will consider he opimal paymen schedules on he relevan hedge bonds in 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 Fig 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 Associaes (1995) reurns daa.

16 2002 C. Munk, C. Sørensen / Journal of Banking & Finance 28 (2004) Table 1 Opimal asse allocaions for invesors wih differen degrees of relaive risk aversion Relaive risk aversion c ¼ 1 c ¼ 4=3 c ¼ 2 c ¼ 4 c ¼1 Sock 80% 60% 40% 20% 0% Bank accoun 20% 15% 10% 5% 0% Hedge bond 0% 25% 50% 75% 100% The opimal allocaions are in accordance wih (28), and relevan Vasicek model parameer values and sock price parameer values are given in (32). Noe: The opimal asse allocaions are idenical for invesors wih differen ime horizons (T ) and ime preference parameers (b). The appropriae hedge bonds, however, differ across invesor ypes. Also, he relevan hedge bonds for he differen invesors in his able, which are heerogeneous wih respec o degree of relaive risk aversion, are no idenical Forward raes Time o mauriy r = 0.01 r = 0.04 r = 0.07 Fig. 1. Term srucures of forward ineres raes. The figure displays forward raes as a funcion of ime o mauriy for differen Vasicek erm srucures described by shor ineres rae levels of 0.01, 0.04, and 0.07, respecively. As formalized in Proposiion 2, he forward-expeced consumpion paern of he agen is crucial for how he agen should opimally hedge agains changes in ineres raes. The forward-expeced consumpion paern and he forward-expeced erminal wealh of he agen are described by he expressions in (26) and (27). In paricular, he consumpion paern over ime depends on he erm srucure of forward raes hrough he occurrence of he zero-coupon price Pð; sþ ¼expð R s f ð; sþdsþ in he expressions. Also, he consumpion paern over ime depends on he prices on risk in he economy hrough he expression for he variance of he log-pricing kernel, gð; sþ as saed in Eq. (24). Using ha he zero-coupon bond volailiy is

17 C. Munk, C. Sørensen / Journal of Banking & Finance 28 (2004) r P ð; sþ ¼ R s r f ð; uþdu ¼ r r bðs Þ and by evaluaing he inegrals in (24), one obains gð; sþ ¼ðk 2 B þ k2 S Þðs Þþ2ðr 1 hþðbðs Þ ðs ÞÞ r2 r 2j ðbðs ÞÞ2 : ð33þ The forward-expeced consumpion paerns are displayed in Fig. 2 for differen degrees of relaive risk aversion, a subjecive ime discoun rae of b ¼ 0:03, and a ime horizon of T ¼ 25 (years). The invesors have iniial wealh of W 0 ¼ 100. The consumpion paerns in he figure describe he specific paymen schedules for he relevan coupon bonds ha he differen invesors should use in order o hedge agains 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 Forward-expeced consumpion rae (a) Time Forward-expeced consumpion rae (b) Time Forward-expeced consumpion rae (c) Time Forward-expeced consumpion rae (d) Time Fig. 2. Expeced consumpion paerns wih iniial wealh W 0 ¼ 100 and ime horizon T ¼ 25. The figure displays he forward-expeced consumpion sreams for four differen levels of he relaive risk aversion coefficien c; panel (a): c ¼ 1, panel (b): c ¼ 2, panel (c): c ¼ 4, panel (d): c ¼ 1. The hree curves in each panel correspond o he differen iniial forward rae curves displayed in Fig. 1. The dashed curve is for he upward sloping erm srucure (r ¼ 0:01), he hick solid curve is for he nearly fla erm srucure (r ¼ 0:04), and he hin solid curve is for he downward sloping erm srucure (r ¼ 0:07). The presen value of he consumpion policy mus equal curren wealh, and he discouned value of he forward-expeced consumpion sream is hus in all cases W 0 ¼ 100. Moreover, he curren consumpion-o-wealh raios in percen are described by he curren, ime ¼ 0, consumpion raes.

18 2004 C. Munk, C. Sørensen / Journal of Banking & Finance 28 (2004) bonds in order o hedge agains changes in he opporuniy se. For example, he illusraed invesors in Fig. 2(b) and (c) wih relaive risk aversion of 2 and 4 should opimally inves 50% and 75% in heir specific hedge bonds, cf. Table 1. For a log-uiliy invesor and for an invesor wih c ¼1, he forward-expeced consumpion and erminal wealh paerns only depend on he iniial erm srucure of ineres raes, as described in he discussion following Proposiion 2. In paricular, for c ¼1he forward-expeced consumpion paern is always fla, as displayed in Fig. 2(d), while he forward-expeced consumpion paern for a log-uiliy invesor in Fig. 2(a) depends on he subjecive discoun rae b and he specific form of he curren erm srucure. From (29) i follows ha in he logarihmic uiliy case, c ¼ 1, he forward-expeced consumpion rae kðsþ mus saisfy k 0 ðsþ ¼ðf ðsþ bþkðsþ 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 b ¼ 0:03, and vice versa. Furhermore, he consumpion-o-wealh raios are described by he curren (ime ¼ 0) consumpion raes, and according o Fig. 2(a) he curren consumpion-o-wealh raios are idenical across he hree erm srucure cases for he log-uiliy invesors wih C 0 =W 0 ¼ 5:686=100 ¼ 5:686%. On he oher hand, he opimal consan consumpion raes ha can be susained by he infiniely risk-averse invesors are deermined enirely by he curren annuiy bond price which differs across he hree erm srucure cases. The consumpion paerns for he invesors in Fig. 2(b) and (c) are basically in beween he wo polar benchmark cases of invesors wih logarihmic uiliy and infiniely risk-averse invesors 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 as inpu in he HJM modeling approach. The relevan curren erm srucures are adoped from he above Vasicek example, as displayed in Fig. 1; he enire erm srucures of forward raes in Fig. 1 are hus used as an inpu in he invesmen/consumpion decision problem. The erm srucure can basically exhibi hree kinds of changes: a parallel level change, a slope change, and a curvaure change. Specifically, he forward rae volailiy srucure is assumed o have he form r f ð; sþ 0 ¼ r 1 ; r 2 e j2ðs Þ ; r 3 ðs Þe j 3ðs Þ ; s 6 T : ð34þ The dynamics of he forward rae curve is described by insering he volailiy srucure (34) in (18). 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, and his hus causes a slope change of he forward curve.

19 C. Munk, C. Sørensen / Journal of Banking & Finance 28 (2004) 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, and 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 (1987) 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 of any zero-coupon bond is described by r P ð; sþ ¼ R s r f ð; uþdu and under he above specificaion of forward curve volailiy we have r P ð; sþ 0 ¼ r 1 ðs Þ; r 2 b 2 ðs Þ; r 3 ðb 3 ðs Þ ðs Þe j3ðs Þ Þ ; ð35þ j 3 where b j ðsþ ¼ 1 j j ð1 e jjs Þ for j ¼ 2; 3. As in he Vasicek example above, i is possible o deermine he opimal forwardexpeced consumpion paern and, hence, he relevan coupon bond for hedging agains 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, gð; sþ, deermines he relevan consumpion paerns in (26) and (27). Sraighforward calculaions using (24) show ha he analogy o (33) in he Vasicek example is now given by gð; sþ ¼ðk 2 B1 þ k2 B2 þ k2 B3 þ k2 S Þðs Þ k B1r 1 ðs Þ 2 þ 1 3 r2 1ðs Þ3 þ 2k B2r 2 r2 2 ðb j 2 j 2 2 ðs Þ ðs ÞÞ r2 2 ðb 2 ðs ÞÞ 2 2 2j 2 4k B3r 3 3 r 2 3 ðs Þ 1 r 2 3 ðs Þ 2 j j j 3 3 þ 4k B3r 3 3 r 2 3 þ 2k B3r 3 þ r2 3 ðs Þþ r2 3 ðs Þ b 2 j j 4 3 j 3 j 3 3 j 2 3 ðs Þ 3 r2 3 5 þ 3 j 2 3 4j 3 2 ðs Þþ1 2 j 3ðs Þ ðb 2 3 ðs ÞÞ 2 : ð36þ 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 is chosen such ha he volailiies 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 his is achieved bu, specifically, he parameer values in he base case are: j 2 ¼ 1:00; j 3 ¼ 0:50; r 1 ¼ 0:00325; r 2 ¼ 0:01184; r 3 ¼ 0:00869; r S1 ¼ð0:03187; 0:02305; 0:04857Þ 0 ; r S2 ¼ 0:24206; u S ¼ 0:05; k S ¼ 0:19365; k B ¼ð0:02549; 0:01844; 0:03886Þ 0 : ð37þ

20 2006 C. Munk, C. Sørensen / Journal of Banking & Finance 28 (2004) In choosing he parameers in (37) we firs fixed j 2 and j 3, which deermine he slope effec and he curvaure effec in he dynamics of he forward rae curve in (18). Our raionale for choosing he specific parameer values is given below, bu he following numerical resuls are no sensiive o he specific parameer values used for j 2 and j In he presen conex, he innovaions in he forward curve are generaed by a hree-dimensional Wiener process, w B ¼ðw B1 ; w B2 ; w B3 Þ 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, j 2 ¼ 1:00 implies ha if an innovaion in w B2 increases he spo rae wih 100 basis poin, he 1-year forward rae is only increased by ð100 e j 21 ¼Þ 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 j 3 ¼ 0:50 he maximum ampliude occurs for he ð1=j 3 ¼Þ 2-year forward rae. Hence, he specific parameer values chosen for j 2 and j 3 are reasonable in order o empirically capure wha is usually referred o as a slope change and a curvaure change in he erm srucure, and his is he raionale for he specific parameer choices. While he parameers j 2 and j 3 are specified exogenously, he forward rae volailiy parameers r 1, r 2, and r 3 are calibraed in order o ensure ha he volailiies of zero-coupon bonds wih imes o mauriy equal o 0.25, 2, and 10 years, respecively, are idenical o hose in he Vasicek example. 11 Nex, r S1 and r S2 are chosen so ha he volailiy on he sock is 25% and so 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 ineres rae in he Vasicek example. Finally, risk premia are also calibraed o be comparable o hose in he Vasicek example. In paricular, he expeced excess reurn on he sock is 5% while he risk premia on bonds, as refleced in k B, are calibraed so ha here is no speculaive demand for bonds (also, kk B k¼0:05, as in he Vasicek example). 12 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 he Vasicek example; cf. Table Numerical resuls based on alernaive parameer values for j 2 and j 3 (varied separaely in inervals from 0.10 o 10) are available from he auhors by reques. 11 This is done by equaing he relevan zero-coupon bond volailiies in (35) o hose in he Vasicek example, and hus basically solving hree equaions wih respec o he hree unknowns: r 1, r 2, and r This is achieved by choosing he hree parameers in k B so ha here is no speculaive demand for bonds exposed alone o innovaions in w B1, w B2, and w B3, respecively. In pracice, he speculaive demand for hese bonds, as described by he firs erm in (25), are equaed o zero, and he hree parameers in k B are hus basically obained by solving hree equaions wih respec o hese hree parameers.

21 C. Munk, C. Sørensen / Journal of Banking & Finance 28 (2004) The speculaive demand for securiies in his example is by consrucion exacly similar o he speculaive demand in he above Vasicek example. The way he invesors wan o hedge agains changes in he opporuniy se, however, may be quie differen compared o he Vasicek case 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 (37) is relevan for addressing quesions such as: (i) is he presen form of he erm srucure of ineres raes imporan for how o hedge agains changes in he opporuniy se? and (ii) is he flexibiliy and dynamics of he erm srucure of ineres raes imporan for how o hedge agains changes in he opporuniy se when he curren erm srucure is kep fixed? 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 agains 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 differen scenarios. The firs quesion, (i), can be addressed by looking a he diversiy of consumpion paerns under differen curren erm srucures, bu under he same fundamenal model of ineres rae dynamics. (Such resuls have already been presened in he above numerical analysis under Vasicek ineres rae dynamics.) The second quesion, (ii), can be addressed by looking a he diversiy of consumpion paerns under differen erm srucure dynamics, bu by using he same curren erm srucure as inpu in he analysis. This is done below where we make numerical comparisons across he Vasicek model and he HJM hree-facor model when he same curren erm srucure of forward raes applies; in paricular, his analysis is based on using he enire Vasicek erm srucures of forward raes exhibied in Fig. 1 as inpu curren erm srucures in he HJM hree-facor model. The opimal consumpion paerns are abulaed in Table 2 for invesors wih degrees of relaive risk aversion equal o 1, 4/3, 2, 4, and infiniy so ha he differen invesors inves 0%, 25%, 50%, 75%, and 100%, respecively, in heir appropriae hedge bonds; cf. Table 1. As in he Vasicek example, he invesors have an invesmen horizon of 25 years, a subjecive ime discoun rae of b ¼ 0:03, and hey equally weigh uiliy from inermediae consumpion and final wealh, i.e. K ¼ 1 in he general uiliy funcion specificaion in (1). The invesors have iniial wealh W 0 ¼ The forward-expeced consumpion paerns for he Vasicek dynamics are exacly idenical o hose displayed in Fig. 2 in he Vasicek example above. The forwardexpeced consumpion paerns for he HJM hree-facor model are for he benchmark parameers in (37) and by using he Vasicek forward rae erm srucures in Fig. 1 as separae curren erm srucure inpus. 13 Also, as discussed in relaion o 13 The relevan Vasicek inpu forward rae erm srucures are given in analyical form by (31). The abulaed forward raes in Table 2 basically represen single poins on he enire erm srucure of forward raes which consiues he inpu o he analysis.