Dynamic Asset Allocation with Commodities and Stochastic Interest Rates

Transcription

1 World Review of Business Research Vol.. No. 4. July 0. Pp. 5 9 Dynamic Asse Allocaion wih Commodiies and Sochasic Ineres Raes Sakkakom Maneenop* his research aims a finding an explici invesmen policy wih hedged variaions of mixed bond-sock-commodiy dynamic porfolio problems under a simple ineres rae model and mean-revering commodiy prices. he findings sugges ha he opimal allocaion o a zero-coupon bond and a commodiy is a combinaion of speculaive erms and hedge erms as proecion of change in ineres raes and change in commodiy marke price of risk, respecively. he allocaion o socks, however, depends only on he speculaive porfolio as here is no need for risk proecion. he policy recommends a negaive relaionship beween risk-aversion facor and riskier asses, socks and commodiies, while i proposes a posiive relaionship o ha of zero-coupon bonds. his is consisen wih he professional advice ha invesors who can olerae more risk should inves more in riskier asses such as socks and commodiies. he paper also finds inverse relaionships beween commodiy prices and posiions in he sock and he commodiy bu no conclusions can be made regarding he direcion of zero-coupon bond invesmen from a rise in commodiy prices. he welfare loss due o neglec in commodiy invesmen is also solved in explici form. Noneheless, he resul should be verified wih numerical examples so ha one can deermine how a bond-sock-commodiy porfolio differs from a pure bond and sock porfolio. Field of Research: Finance. Inroducion Commodiies have played an imporan role as an alernaive asse class for invesors in recen years. Commodiies have been emerging as an increasingly imporan class of asses and are claimed o have value-added effeciveness due o heir diversificaion benefis. Ibboson Associaes (006), exhibiing he correlaion coefficiens of annual oal reurns ( ), provide inuiive evidence of he low correlaion of commodiies wih radiional asse classes. Of he seven asse classes, reasury bills and commodiies are he only wo asse classes wih negaive average correlaions o he oher asse classes. In addiion, commodiies are posiively correlaed wih inflaion (see, for example, Erb and Harvey, 006; Goron and Rouwenhors, 006). here are some sudies concerned wih he diversificaion and inflaion hedge effecs of commodiies especially research of shor-erm invesmen using he framework of Markowiz (95). Despie he rise of commodiy invesmen, porfolios of mos invesors are sill comprised mainly of radiional asses as socks and bonds, so i is imporan o keep hem in invesmen decisions. In fac, given he growing imporance of commodiies, i is ineresing o se up a porfolio consised of bonds, socks, and commodiies. o accomplish such an objecive, his sudy combines characerisics from he models *Sakkakom Maneenop, Hiosubashi Universiy, Japan. ed0005@g.hi-u.ac.jp

2 including erm srucure of ineres raes as in Sørensen (999) and Korn and Kraf (00); he models of mean-revering excess reurns as in Kim and Omberg (996), Wacher (00), and Munk, Sørensen, and Vinher (004); and he models including commodiy asan alernaive asse class as in Dai (009). wo sandard mehods of invesigaion, dynamic programming and maringale mehod, can be used o deermine opimal sraegies. We will use he dynamic programming approach o solve he problem of allocaing bonds, socks, and commodiies. In conras o Dai (009) who focuses only on sock and commodiy selecion, we presen bonds ino he porfolio selecion which complicaes he sudy. Ye, since bonds are one of he mos imporan asse classes for boh insiuional and individual invesors, inroducing bonds increases pracical applicabiliy. he paper aims a finding soluions of dynamic bond-sock-commodiy allocaion under a sochasic opporuniy se of invesors holding consan relaive risk-aversion (CRRA) uiliy. I will also compare radiional advice from financial markes o resuls from he lieraure. For example, as debaed in Campbell and Viceira (00), wo invesmen rules of humb claim ha () aggressive invesors should hold socks while conservaive invesors should hold only bonds; and () long-erm invesors inves more in socks han shor-erm invesors. his sudy will discuss wheher hose wo beliefs hold up well when approached academically and how inroducion of commodiies affecs hese rules of humb. Moreover, he paper considers relaionships beween movemens in commodiy prices and changes in invesmen sraegies. In shor, we discuss wheher an increase in commodiy prices leads o long or shor posiions in oher asses. Finally, o undersand he imporance of having commodiy asses in he porfolio, we examine wealh loss from excluding commodiies from he porfolio. he resul is an explici invesmen sraegy wih hedge variaions in ineres raes and he commodiy marke price of risk. In he opimal policy, he allocaion o zerocoupon bonds and commodiies is made for speculaive or myopic purposes as well as for ineremporal hedging purposes. he opimal allocaion o he sock depends on he spo commodiy price and does no conain he hedge erm compared o previous wo asses. Posiions in socks and commodiies, he riskier asses, have negaive relaionships wih invesor s risk olerance, while posiions in zero-coupon bonds, he less risky asse, have he opposie resul. Despie finding he exac soluion o he problem, here are a leas hree limiaions in his sudy. he dynamic commodiy price model is a one-facor model alhough recen sudies have shown more opions of wo or muli-facor models ha are more effecive in illusraing commodiy price movemens. Moreover, he paper does no consider he correlaion among commodiy price reurns, inflaion, and he value of US dollars which are all of he debaed opics in recen years. Finally, as we mainly discussed he resul based on mahemaical models, here are some inconclusive pars ha require verificaion wih numerically. he res of he paper is organized as follows. Secion discusses relevan lieraure. Secion 3 ses up invesmen asse dynamics and examines an opimal asse allocaion sraegy. Secion 4 analyzes such an opimal soluion and invesigaes more in horizon effec. Welfare analysis will also be discussed in his secion. Finally, secion 5 concludes he paper. 6

3 . Lieraure Review Dynamic asse allocaion wihin coninuous-ime economics environmen has been sudied inensively since Meron (969, 97). he sudy helps an invesor find coninuous-ime porfolio sraegies. he sandard mehod of invesigaion is based on he sochasic conrol framework and he Hamilon-Jacobi-Bellman (HJB) equaion, resuling in nonlinear equaions which are ypically hard o solve. Meron ses up he framework of dynamic porfolio problems for an invesor who maximizes expeced wealh uiliy a a given invesmen horizon. his groundbreaking work, however, has an impracical assumpion of consan ineres raes while i is well-known in financial markes ha ineres raes are no deerminisic. As o relax he above assumpion and o inroduce bonds as one of he invesmen choices, i is necessary o include he model of erm srucure of ineres rae dynamics ino porfolio problems. Sørensen (999), Brennan and Xia (000), and Korn and Kraf (00) consider invesmen problems under sochasic ineres raes of Vasicek (977) ype an invesor wih CRRA uiliy can inves in a bank accoun, socks, and bonds. hey argue ha he opimal invesmen sraegy is a simple combinaion of a speculaive erm and a hedge erm. While he former explains he need o opimize an immediae risk-reurn profile in a mean-variance framework, he laer describes how he invesor proecs sochasic behaviors of ineres raes. Paricularly, Brennan and Xia (000) show ha, consisen wih popular recommendaions, he bond-sock raio has a posiive relaionship wih he degree of risk-aversion. Using more advanced ineres rae model assumpions, Munk and Sørensen (004) invesigae invesmen sraegies similar o hose assuming simple ineres rae models. However, one of heir key resuls is ha he hedge porfolio is more sensiive o he curren form of he erm srucure han o he specific dynamics of ineres raes. hus, no only is i easier o apply basic models such as he Vasicek model in our research, bu i is also possibly sufficien in erms of correcness and pracicaliy. Anoher ype of allocaion problems relaed o his paper is he opimal porfolio wih he sochasic marke price of risk. As some empirical sudies sugges evidence of mean reversion in sock reurns, Kim and Omberg (996) and Wacher (00) achieve exac opimal invesmen sraegies in a se-up wih a consan risk-free ineres rae r and socks he marke price of risk is idenical o he Sharpe raio of he sock. Munk, Sørensen, and Vinher (004) also find he exac opimal asse allocaion sraegy for a porfolio of bonds and socks in a model feauring mean reversion in sock prices and inflaion uncerainy. heir main resul is he combinaion of speculaive and hedge erms in he opimal invesmen policy as in aforemenioned papers. Insead of assuming mean reversion in sock prices, we will apply he sochasic characerisics of marke price of risk ino commodiy prices as in Schwarz (997). As menioned earlier abou he imporance and populariy of commodiies, in he lieraure, i is sill debaable wheher one should include commodiies in he porfolio and, if included, how commodiies affec he oal wealh. Research in commodiy invesmen has commonly been founded on he performance of commodiy fuures since mos commodiies raded in financial markes are in derivaive forms. Mos exising sudies on commodiy invesmen apply he one-period mean-variance opimizaion framework of Markowiz (95). In ha kind of myopic framework, he 7

4 main hypoheses examined by hese sudies are wheher commodiy invesmen gives a posiive risk premium, correlaes wih oher asses, and is capable of hedging agains inflaion. For example, Erb and Harvey (006) find ha some securiy characerisics and porfolio sraegies provides a posiive risk premiums. Conover, Jensen, Johnson, and Mercer (00) also show ha he oal porfolio can be benefied by inclusion of commodiies. Specifically, an equiy porfolio wih commodiy exposures performs beer during periods of high inflaion. However, much less research effors have been devoed o long-erm allocaion sraegy. he closes lieraure o he presen paper is he work by Dai (009), sudying dynamic asse allocaion using he Maringale approach wih a focus on inroducing commodiies ino porfolio managemen. As he same objecives of hose who sudy allocaion wih sochasic ineres raes, he solves for he exac soluion of opimal porfolio and consumpion sraegies wih he adven of commodiies as a new asse class. he conclusion is ha such policies are a combinaion of a speculaive erm and a hedge erm. He also finds he wealh loss due o disregard in commodiy invesmen. he main difference beween Dai (009) and our work is ha while Dai uses he Maringale mehod as a means of finding he resul, he presen research paper applies he radiional dynamic programming approach o invesigae he sraegy. Moreover, Dai s porfolio includes socks and commodiies while his sudy includes bonds, socks, and commodiies; hus adding more complicaion in a search for an opimal invesmen sraegy. o provide horough deails of he presen aricle, he nex secion will se up relaed mahemaical models and solve he problem of dynamic asse allocaion. 3. Invesmen Asse Dynamics and Opimal Asse Allocaion In his secion we inroduce he invesmen asse dynamics and follow up by solving for he opimal asse allocaion. 3. Invesmen Asse Dynamics he ineres rae dynamics are explained by Ornsein-Uhlenbeck process as in Vasicek (977). d r ( r - r )d - dz () r r r indicaes he long-run mean of he ineres rae, denoes he degree of mean reversion, r is he volailiy of he ineres rae, and Z r is a sandard Brownian moion. Such a process leads o a zero-coupon bond price wih mauriy given by exp B a b r () r r r a( ) r b( ) b( ), b( ) e 4 8

5 wih he consan parameer,, is he premium on ineres rae risk. Using Io s lemma, he dynamics of he bond price, sochasic differenial equaion in he form B, can be described by a d B B r B( r, ) d B( r, )dzr (3) B( r, ) rb is he sensiiviy erm of he zero-coupon bond price. he dynamics of a sock price, differenial equaion S, are assumed as he following sochasic ds S r d dz dz s rs s r rs s s (4) s describes he sock volailiy, denoes he sock marke price of risk, and he produc of hese wo parameers expresses expeced excess reurn from equiy invesmen. he correlaion beween reurns in socks and ineres raes is denoed by rs. Z s is anoher sandard Brownian moion and independen of Z r. he spo commodiy price, C, is assumed o follow he one-facor model in Schwarz (997) given by ˆ dc lnc C d C dz (5) c c c c is he long run mean of he spo price, denoes he degree of mean reversion, c explains he commodiy volailiy, and Z ˆc is a sandard Brownian moion. he model is popular for modeling energy and agriculural commodiies and aims a inroducing mean reversion o he long-run mean,. Under risk-neural measure, i is possible o find he fuures price of such a commodiy wih he following marke price of risk c c c X, X lnc (6) As discussed in Dai (009), he above equaion is inspired by empirical evidence ha expeced reurns are ime-varying and can be prediced by some insrumenal variables such as spo commodiy prices hemselves. Noe ha his model is reduced o he one-facor model of Schwarz (997) if c 0. Also, following Dai (009), he commodiy price process may be adaped o he following process c ˆ dv V r d dz c c c c (7) 9

6 V is he self-financing porfolio characerized as a new asse class of commodiies. Wih his process, i is possible o se up he dynamics of hree asses, bonds, socks, and commodiies consisen wih asse allocaion problems of Meron (969, 97). he dynamics of hree asses can be expressed in erms of he following marix db dp ds diag( P) r r, 3, ( 3 ) d r, 3, dz dv 3 3 ( ) (8) B( r, ) 0 0 cr sc s r,, 0,, 3 rs s rs cr sc \and parameers wih one underline are vecors and parameers wih wo underlines denoes marices. he vecor of marke price of risk is composed of wo consans, and, wih respec o Z r and Z s respecively; and anoher sae variable, 3, wih respec o Z c. Noe ha he Brownian moion Z c is independen of Z r and Z s. he parameer 3 is sochasic and dependen on he commodiy price. and 3 can be inferred from he above price dynamics as and rs (9) 3 c cr sc (0) As c depends on he commodiy price, he marke price of risk, 3, can be expressed in he following dynamics d d dz () c cr sc 3 c c cr sc he sochasic characerisics of he commodiy marke price of risk and he ineres rae as menioned earlier generaes a sochasic invesmen opporuniy se effec in 0

7 hedging erms. he opimal invesmen sraegy will be differen from he case wih saic mean-variance framework. wo sae variables, r and 3, can be wrien in he following marix or r 0 0 dr r r d d c Z c d ccr csc c c d x m r, d v r, dz 3 3 () 3. Opimal Asse Allocaion Following Munk (00), he wealh process, W, can be expressed as (3) d W W r r, 3, ( 3 ) d W r, 3, dz is he hree-dimensional vecor of invesmen porion a ime in he porfolio consised of bonds, socks, and commodiies. he remains of his wealh, B S V, is invesed in he risk-free asse. An invesor is assumed o maximize uiliy from he erminal wealh, o a power uiliy funcion. he indirec uiliy funcion is given by W, wih respec W J W, r, 3, sup EW, r, 3, (4) he parameer 0 describes he risk olerance level of an invesor. he Hamilon-Jacobi-Bellman equaion associaed wih he above dynamic opimizaion problem has he form 0 WJW r, 3, ( 3 ) JWWW r, 3, r, 3, J W r v r J rwj J m r J r Where,,,, r, 3 3 Wx W x 3 xx 3 (5) r, vr, vr, he firs order condiion wih respec o he invesmen sraegy provides he following form

8 JWx JW r,, ( ) r,, v r, (6) WJ WW WJWW he opimal asse allocaion of a power uiliy invesor is saed in he following heorem. heorem : he indirec uiliy of wealh funcion of a CRRA invesor is given by A 0 ( ) A ( ) ra ( ) 3 A 3 ( ),, 3, e 3 J W r W (7) A0 ( ) r r A ( s)ds g A ( s)d s A ( s)d s A ( s)ds 0 3 * 3 r c c r cr * 0 0 A ( s) A ( s)d s A ( s)ds d ae d d af e A ( ) e, A ( ) A3 ( ), A3 ( ), d d e b d e d wih a, b c c, c *, d b 4 ac, c c * c cr c sc c c, c c f 3 g c rcr A ( ), g c cr sc he vecor of opimal risk asse allocaions a ime is given by rs 3 r A ( ) cr rssc B( r, ) B( r, ) B 3 sc S S V 3 c A ( ) A3 ( ) 3 C (8)

9 From he opimal sraegy, here exis speculaive erms and hedge erms as in previous sudies. When risk-aversion facor,, increases, posiions in he sock decreases while he posiion in a zero-coupon bond increases. his is he same as he professional advice ha invesors who can olerae less risk should inves more in less riskier asses such as he zero-coupon bond. Posiions in he commodiy, however, do no necessarily have an exac movemen relaed o he risk-aversion facor. Also, we noice ha allocaion o he sock is independen from he invesmen horizon. his conradics wih radiional advice ha he sock weigh should increase wih invesmen horizon. Basically, hedge erms in he zero-coupon bond and he commodiy explain he hedge agains change in ineres raes and change in commodiy marke price of risk, respecively. Nex secion will discuss he heorem in more deails including how invesmen horizon affecs he porion of commodiies in he porfolio. 4. Findings and Discussions his secion analyzes he sraegy in heorem and considers wealh loss due o excluding commodiies from he porfolio. 4. Allocaion Analysis and Invesmen Horizon Effec In his subsecion we invesigae he resul from he opimal invesmen sraegy obained from he previous secion. For simpliciy, each componen of wealh fracion will be discussed following wih an argumen in horizon effec in commodiies. Zero-coupon bond allocaion rs ( ) 3 r A B cr rssc B( r, ) B( r, ) (9) Wih he adven of commodiy as an asse choice, here exiss he erm 3 cr rssc which could be eiher posiive or negaive depending on correlaion facors, cr, rs, and sc. Regularly, empirical sudies as in Ibboson Associaes (006) sugges posiive and negaive in. Also, Dai s rs sc and cr (009) finding ha c is significanly less han zero, along wih he empirical es of Schwarz (997) concluding ha he spo commodiy price has significanly negaive effec on he risk premium, suggess an inverse relaionship beween he commodiy price, C, and he marke price of risk, 3. However, we sill canno ascerain he direcion of zero-coupon bond caused by arise in he commodiy price since he sign of he erm cr rssc also depends on magniudes of each consan. he hedge erm suggess ha, in he long run, invesmen in he zero-coupon bond depends on he ineres rae volailiy and he erm A () bu no on he commodiy. If he allocaion in bonds is he zero-coupon mauring a he end of he invesmen horizon,, we will obain he equaliy A ( ) (, ) B r and he hedge erm will reduce o which no longer depends on ime. By his assumpion, i is clear 3

10 ha he hedge posiion of a more risk-averse invesor ( ) is posiive as a less risk-averse invesor ( ) is negaive. Sock Allocaion S S 3 sc (0) As shown in (6) ha par of he hedge erm is creaed by he derivaive J Wx while here are no sae variables relaed o he sock compared o hose of oher asses. Sock allocaion, hus, resrics only in he speculaive erm as here are no risks o hedge as in bonds as explained earlier or commodiies which will be discussed laer. Noe ha Munk e al. (004) assume he mean reversion in socks and find he hedge erm in sock allocaion. By examining (0), i is clear from he equaion ha if sc is greaer han zero, he sock invesmen increases when commodiy price increases. However, if sc is less han zero, he invesmen in sock has negaive relaionship wih he commodiy price; his case is more suggesed empirically. I may be inerpreed ha when he commodiy price reduces, his suggess in he increase in he sock price; and invesors, herefore, should inves in sock. Commodiy Allocaion () 3 c V A ( ) A3 ( ) 3 C he resul is similar o Munk e al. (004) and Dai (009) ha he hedge erm includes boh A () and A ( ). 3 Focusing only on he firs erm, he speculaive porfolio, i can be implied ha, wih c less han zero, here is a negaive relaionship beween he commodiy price and he speculaive erm. Nex, we concenrae on he hedge erm and recall he value of A () 3. he fac ha d is greaer han b in heorem indicaes he posiiviy of his erm. his, herefore, leads o an inverse relaionship beween he commodiy price and he hedge porfolio. All in all, i is cerain o sele ha he invesor should inves in commodiies when he commodiy price decreases and reduce he porion when he commodiy price increases. Horizon Effec in Commodiies As described above ha allocaion o he sock is independen from he invesmen horizon. However, i remains unclear how invesmen horizon affecs posiions in he commodiy. By differeniaing he commodiy wealh fracion wih respec o ime horizon, we obain he following derivaive 4

11 V c A( ) A( ) 3 3 () I can be shown ha A () 3 is always posiive while A () can be eiher posiive or negaive depending on parameers. hus, i is inconclusive wheher he invesmen horizon has posiive or negaive effec in commodiy invesmen. However, if 3 is high enough, he derivaive will be posiive and lead o a higher commodiy allocaion for a longer erm invesor. Furher empirical sudies are suggesed o examine he horizon effec in commodiies and o invesigae invesmen rules of humb. 4. Welfare Analysis his subsecion sudies he wealh loss due o disregard in commodiy invesmen. he loss is assumed as he percenage L of exra iniial wealh ha is necessary o bring he invesor o he same uiliy level as he invesor considering invesmen in a commodiy. NC H( r, 3, ) J W ( L), r, 3, J W, r, 3, We (3) herefore, we obain 3 NC L exp H( r,, ) H ( r, ) (4) H( r,, ) A ( ) A ( ) r A ( ) A ( ) NC NC NC H ( r, ) A ( ) A ( ) r is he funcion wih no commodiy consideraion. and 0 NC From Munk (00), i follows ha A ( ) A( ) and NC A0 ( ) r r A ( s)d s r A ( s)ds (5) 0 0 NC Hr (,, ) and H ( r, ) ino he wealh loss equaion, we obain Insering 3 L exp A ( ) 3 A3 ( ) 3 3 A ( s)d s * A3 ( s)ds 0 0 c c r cr A ( s) A ( s)d s * A ( s)ds 0 0 (6) I can be seen from he equaion ha wih very high or very low commodiy prices, he square of marke price of risk will be high, leading o an increase in he welfare loss. On he oher hand, he sandard level of commodiy values can reduce wealh loss. his may be reckoned as he opporuniy loss from exreme movemen in commodiy prices. 5

12 5. Conclusions his sudy has aimed a examining he bond-sock-commodiy porfolio in he framework of dynamic asse allocaion as in Meron (969). he commodiy marke price of risk is sochasic and dependen on a spo commodiy price, while he spo price iself has he propery of mean reversion. As i is well-documened ha ineres raes are sochasic, he paper assumes he simple Vasicek ineres rae model along wih he mean reversion in he spo commodiy price. he sudy has aimed a finding an opimal invesmen sraegy for a porfolio including zero-coupon bonds, socks, and commodiies using he dynamic programming approach. he opimal invesmen policy was derived for an invesor who is concerned wih erminal wealh. Closed-form expressions were obained for he opimal sraegies and he uiliy losses of excluding he commodiy from he financial decisions. In he opimal policy, he allocaion o zero-coupon bonds and commodiies were made for speculaive or myopic purposes, as well as for ineremporal hedging purposes. he opimal allocaion o he sock is solely speculaive and dependen on he spo commodiy price. Posiions in socks and commodiies, he riskier asses, have negaive relaionships wih invesor s risk olerance, while posiions in zero-coupon bonds, he less risky asses, have he opposie resul. his is consisen wih he professional advice ha invesors who can accep more risk should inves more in riskier asses. Assuming ha he spo commodiy price has a significan negaive effec on he risk premium as in Schwarz (997) and Dai (009), no conclusions could be made regarding he direcion of zero-coupon bond invesmen resuling from a rise in commodiy prices. However, i may be concluded ha here are inverse relaionships beween commodiy prices and posiions in socks and commodiies in he porfolio. In brief, one should inves in socks and commodiies when commodiy prices decrease and dives such posiions when commodiy prices increase. he welfare loss due o neglec in commodiy invesmen is also solved in explici form. he resul implies ha exreme movemen in commodiy prices may lead o more wealh loss compared o sable commodiy prices. In our furher empirical sudies will deermine he relaion beween commodiy prices and invesmen in zero-coupon bonds as well as examine wheher he wealh loss creaes a huge difference beween bond-sock-commodiy porfolio and pure bond and sock porfolio in our furher sudies. References Björk, 009, Arbirage heory in Coninuous ime, Oxford Universiy Press, USA. Brennan, MJ & Xia, Y 000, Sochasic ineres raes and he bond-sock mix, European Finance Review, vol. 4, pp Campbell, JY & Viceira, LM 00, Sraegic Asse Allocaion, Oxford Universiy Press. USA. Conover, CM, Jensen, GR, Johnson, RR & Mercer JM 00, Is now he ime o add commodiies o your porfolio?, he Journal of Invesing, vol. 9, no. 3, pp Dai, R 009, Commodiies in dynamic asse allocaion: Implicaions of mean revering commodiy prices, Working Paper, ilburg Universiy. 6

13 Erb, CB & Harvey, CR 006, he sraegic and acical value of commodiy fuures, Financial Analyss Journal, vol. 6, no., pp Goron, GB & Rouwenhors, GK 006, Facs and fanasies abou commodiy fuures, Financial Analyss Journal, vol. 6, pp Ibboson Associaes 006, Sraegic asse allocaion and commodiies, Ibboson Associaes, Chicago. Kim, S & Omberg, E 996, Dynamic nonmyopic porfolio behavior, Review of Financial Sudies, vol. 9, no., pp Korn, R & Kraf, H 00, A sochasic conrol approach o porfolio problems wih sochasic ineres raes, SIAM Journal on Conrol and Opimizaion, vol. 40, no. 4, pp Markowiz, H 95. Porfolio selecion, Journal of Finance, vol. 7, no., pp Meron, RC 969, Lifeime porfolio selecion under uncerainy: he coninuous-ime case, Review of Economics and Saisics, vol. 5, no. 3, pp Meron, RC 97, Opimum consumpion and porfolio rules in a coninuous-ime model, Journal of Economic heory, vol. 3, pp Munk, C 00, Dynamic asse allocaion, Lecure Noe. Munk, C & Sørensen, C 004, Opimal consumpion and invesmen sraegies wih sochasic ineres raes. Journal of Banking and Finance, vol. 8, pp Munk, C, Sørensen, C & Vinher, N 004, Dynamic asse allocaion under meanrevering reurns, sochasic ineres raes, and inflaion uncerainy: Are popular recommendaions consisen wih raional behavior?, Inernaional Review of Economics and Finance, vol. 3, pp Øksendal, B 003, Sochasic Differenial Equaions: An Inroducion wih Applicaions, Springer. Sørensen, C 999, Dynamic asse allocaion and fixed income managemen, Journal of Financial and Quaniaive Analysis, vol. 34, no. 4, pp Schwarz, ES 997, he sochasic behavior of commodiy prices: Implicaions for valuaion and hedging, Journal of Finance, vol. 5, no. 3, pp Vasicek, OA 977, An equilibrium characerizaion of he erm srucure, Journal of Financial Economics, vol. 5, pp Wacher, JA 00, Porfolio and consumpion decisions under mean-revering reurns: An exac soluion for complee markes, Journal of Financial and Quaniaive Analysis, vol. 37, no., pp

14 Appendix: Proof of heorem Maneenop Subsiuing he candidae opimal value of invesmen sraegy ino (5) and gahering erms, we obain he second order PDE JW W 3 x 3 xx 3 JWW 0 J rwj J m r, r J r, J J ( ) v r, J v r, v r, J W Wx 3 3 Wx 3 3 Wx JWW JWW (7), such ha he sraegy is feasible, hen is equal o he indirec If his PDE has a soluion, J W, r, 3, he sraegy is opimal and ha he funcion J W, r,, uiliy funcion. Ineresed readers may consul Øksendal (003) and Björk (009) for more deails. A guess soluion o he PDE is given by A 0 ( ) A ( ) ra ( ) 3 A 3 ( ),, 3, e 3 J W r W (8) Subsiuing he relevan derivaives of J W, r,, obain ha J will be a soluion if he funcion H solves he PDE 3 3 ino (7) and simplifying, we 0 r H r,, mr, vr, H r r, 3 H xx H x r, 3 H x x (9) Guess ha equaion is in he form,, ( ) ( ) ( ) ( ) H r A A r A A (30) Finding he relevan derivaives and insering hem ino (9), we obain an equaion linear in he ineres rae and quadraic in he commodiy marke price of risk. Is four coefficiens mus be zero, resuling in he following sysem of ODEs. A( ) A ( ), A (0) 0 (3) A ( ) a ba ( ) ca ( ), A (0) 0 (3) A ( ) fa3 ( ) b ca3 ( ) A ( ), A (0) 0 (33) 8

15 a, b, c, d, and f are presened in heorem. Solving he above PDEs, we A( ), A ( ), and A () 3 as also shown in he heorem. Plugging hem ino obain he opimal sraegy, we achieve he opimal invesmen sraegy. 9