Commodities in Dynamic Asset Allocation: Implications of Mean Reverting Commodity Prices

Transcription

1 Commodiies in Dynamic Asse Allocaion: Implicaions of Mean Revering Commodiy Prices Renxiang Dai Finance Group, CenER, and Nespar Tilburg Universiy P.O. Box 9153, 5 LE Tilburg, The Neherlands r.dai@uv.nl This version: March 17, 29 Absrac This paper sudies commodiy invesmen in he conex of dynamic asse allocaion, wih a focus on he implicaions of he commodiy reurn predicabiliy arising from mean revering commodiy prices. The model of financial markes consiss of hree asse classes: socks, bonds, and commodiies, which generalizes he benchmark seing of Meron (1969). The risk premium in he commodiy marke is assumed o be dependen on he mean-revering spo commodiy price, and his assumpion is suppored by he empirical findings of he paper. I solve, in closed form, he opimal porfolio and consumpion sraegies. The sudy suggess ha allocaion o commodiies is needed o opimize he insananeous risk-reurn profile (myopic purposes), as well as o hedge he sochasic changes of he invesmen opporuniy se (ineremporal purposes). The welfare cos of excluding he commodiy from financial decision making is also solved in closed form. A simple numerical exercise shows ha here is subsanial marke iming in he opimal financial policy, and ha excluding he asse class of commodiies may incur subsanial welfare coss, especially for long-erm and less risk-averse invesors. I hank Frank de Jong, Rober Kozarski, Berrand Melenberg, Juan Carlos Rodriguez, Zaifu Yang, and Leon Zolooy for helpful commens. I am especially graeful o Hans Schumacher for many simulaing discussions and insighful suggesions. 1

2 1 Inroducion Commodiies have been emerging as an increasingly imporan class of asses for insiuional and individual invesors in recen years. Sysemaic invesigaion of commodiies as an invesable asse class goes back a leas some 3 years ago Greer, 1978, Bodie and Rosansky, 198]. However, he growh of commodiy markes o a major alernaive invesmen vehicle is a more recen developmen. Around 27, he size of he global commodiies derivaives marke is esimaed o be abou 75 billion US dollars Till and Eagleeye, 27]. As he markes have grown, more invesors have been araced o commodiies. Increased exposure o commodiies has been acquired by insiuional invesors, wih pension funds as a noable example, and o a less exen by individual invesors as well (see e.g. Mongars and Marchal-Dombra 26] and Doyle e al. 27]). In he lieraure of commodiy invesmen, i remains an open quesion wheher and how o include commodiies in mainsream porfolios. Sudies on commodiy invesmen have generally been based on he performance of invesmen in commodiy fuures. The reason is ha invesmen in commodiies is mosly by means of derivaive producs, especially commodiy fuures, while spo ransacions of commodiies play lile role in commodiy invesmen. Mos exising sudies on commodiy invesmen apply he one-period mean-variance opimizaion framework of Markowiz 1952]. In he saic mean-variance framework, he key issues invesigaed by hese sudies have been wheher invesmen in commodiies fuures yields a posiive risk premium, how such invesmen covaries wih bonds and socks, and how i hedges agains inflaion (see e.g. Erb and Harvey 26], Gordon and Rouwenhors 26], Ka and Oomen 27] and references herein). In his lieraure, he mos conroversial issue has presumably been wheher or no commodiy invesmen offers a posiive risk premium, and if i does, wha drives he risk premium. The absence of an appreciably posiive risk premium does no necessarily make mean-variance invesors refrain from allocaing o an asse class, bu i will surely make i less aracive or of lile pracical relevance in mos sudies. As such, he ongoing debae over he risk premium of commodiy fuures invesmen has lef i an open quesion wheher or no commodiies are an appealing asse class. Empirical evidence has documened ha risk premia in commodiy fuures markes are iming-varying and predicable. For example, Bessembinder and Chan 1992] show ha prices in commodiy fuures markes can be forecas on he basis of insrumenal variables known o possess forecas power in equiy and bond markes. Some sudies have found ha risk premia of commodiy invesmen vary in differen saes, like he phase of he business cycle, he sance of moneary policy, marke senimen, and he hisory of invesmen reurns (see, for example, Jensen e al. 2, 22], Wang and Yu 24], Erb and Harvey 26], Miffre and Rallis 27], Nijman and Swinkels 27], and Vrug e al. 27]). Similarly, ime-variaion and predicabiliy of asse reurns have been well documened in he asse classes of socks and bonds. Wha are he implicaions of ime-variaion and predicabiliy of asse reurns for porfolio choice? For he mainsream asse classes, heir implicaions for porfolio choice have been explored in deph, for example, Kim and Omberg 1996] and Wacher 22] in he case of socks. In he case of commodiies, however, much less research effors have been devoed o he implicaions of ime-varying and predicable commodiy reurns for porfolio decision making. In view of his, his paper presens a sudy of he asse class of commodiies in an ineremporal framework, wih an explici focus on he ime-varying and predicable reurns in commodiy markes. In he lieraure of commodiy invesmen, he closes o his sudy is presumably Hoevenaars e al. 28], who address he opimal porfolio policy in he conex where expeced reurn of alernaive asses, including commodiy reurns, are ime-varying and 2

3 predicable. For echnical reasons, however, Hoevenaars e al. 28] consider only consan-proporion porfolio sraegies, and hence absrac from marke iming ha in principle will arise from reurn predicabiliy. This sudy aims o furher he undersanding of commodiy invesmen, especially in exploiing commodiy reurn predicabiliy by marke iming. This paper invesigaes he asse class of commodiies in he dynamic opimizaion framework ha was inroduced ino finance by Meron 1969]. The so-called Meron s problem has been analyzed and exended in various conexs, reflecing differen aribues of people s preferences and of financial markes (see, for example, Chaper 9 of Duffie 21] for a exbook reamen). The lieraure of dynamic asse allocaion, however, has focused predominanly on such radiional asse classes as socks and bonds. Owing o he growing imporance of commodiies, i is perinen o ask, in his esablished framework, how invesors should opimally make heir porfolio and consumpion decisions when commodiies are available in addiion o socks and bonds. To his end, I inroduce ino he classical Black-Scholes economy a commodiy marke. Consisen wih he fac ha commodiy fuures are he major commodiy invesmen vehicle, he commodiy marke is modeled as a fuures marke. Wih he addiion of he commodiy marke, he Black-Scholes economy consising of a riskless bond and a risky sock is augmened o an economy equipped wih hree asse classes. This hree-asse economy, referred o as he Bond-Sock-Commodiy economy in he following, enables one o capure he richer invesmen opporuniies semming from he presence of commodiies. The commodiy fuures marke is characerized by a generalized version of he single-facor model in Schwarz 1997]. Following Schwarz 1997], he non-radable spo commodiy price follows a mean-revering process. Raher han assuming a consan risk premium in he fuures marke as in Schwarz 1997], I generalize his model by assuming ha he risk premium is dependen on he spo commodiy price. This generalizaion can be jusified by hree reasons. Firs, as menioned above, empirical evidence has shown ha risk premia in commodiy markes are ime-varying, and can be prediced by insrumenal variables characerisic of he business cycle. Second, empirical sudies, for example Fama and French 1988], have idenified a srong business cycle componen in he variaion of spo commodiy prices. I suggess ha spo commodiy prices migh have forecas power for risk premia in commodiy fuures markes. Las bu no leas, he esimaion resuls of his exended model presened in his paper has provided srong evidence ha he effec of he spo price on he risk premium is significan. In his simple characerizaion of he asse class of commodiies, he risk premium in he commodiy marke is prediced by he mean-revering spo commodiy price. As is well known, mean reversion is an imporan propery of commodiy prices, and mean reversion of prices has become a prevailing assumpion in he lieraure relaed o he sochasic behavior of commodiy prices, for insance, Gabillon 1995], Schwarz 1997], Geman 25], o name bu a few. Moreover, empirical sudies of commodiy prices have found evidence of mean reversion o various degrees (for example Bessembinder e al. 1995], Pindyck 21], and Andersson 27]). This sudy, by relaing commodiy marke reurns o spo commodiy prices, underscores he implicaions of he mean-revering naure of he commodiy price for commodiy invesmen and porfolio decisions. The dynamic framework adoped here makes his sudy disinc from ones ha use a saic perspecive. In he saic one-period paradigm, people are assumed o make a one-off invesmen decision a he beginning of he period in order o maximize heir uiliy over he invesmen oucome a he end of he period. In comparison, he dynamic framework buil on an ineremporal seing allows people o make inermediae rebalancing. Undoubedly he dynamic framework offers 3

4 a richer srucure han he saic one does, and arguably i is closer o financial decision-making in pracice. I has long been known ha unless (i) invesors have logarihmic uiliy, or (ii) he financial marke offers a consan invesmen opporuniy se in he sense ha boh he riskfree rae and he marke price of risk are consan, he opimal financial policy derived from he dynamic framework is differen from he so-called myopic policy based solely on one-period analysis. As will be shown, he ime variaion and predicabiliy of expeced reurns in commodiy markes, once invesigaed in he framework of dynamic asse allocaion, has profound implicaions for commodiy invesmen and porfolio decisions, which he saic mean-variance analysis is unable o accommodae. Therefore, his sudy, by virue of a richer framework, will shed new ligh on he debae on commodiy invesmen, and enable us o expound on some conenious issues arising from saic analysis in ligh of he findings from a dynamic perspecive. This sudy conribues o he discussion of commodiy invesmen by aking a novel roue o approach he issue. Differen from focusing on indices of commodiy fuures in exan lieraure, I model commodiies ino he economy as commodiy fuures underlying hose indices. And his approach may have an advanage in comparison wih ha based on commodiy fuures indices. I has been recognized ha commodiy fuures indices embed rading sraegies of commodiy fuures (Gordon and Rouwenhors 25], Erb and Harvey 26]). Owing o differing ways of composiion, weighing and rebalancing, differen commodiy indices imply differen rading sraegies, and hence may well give divergen picures of commodiy invesmen reurns, even in a common ime period. In addiion o bringing an elemen of arbirariness because of varying ways of index building, he applicaion of indices may blur some imporan characerisics of commodiy invesmen, like he implicaions of mean reversion in commodiy prices. In conras, his sudy direcly specifies he underlying commodiy fuures as such, in he hope of achieving a sharp focus on implicaions of his propery. I shall consider he opimal financial sraegy for an invesor in wo classical cases. In he firs, he invesor is concerned wih maximizing he expeced uiliy over wealh on some fixed horizon dae. The second case I consider is ha of an invesor who derives uiliies over life-ime consumpion. Of hese wo cases, he firs, erminal wealh case only involves porfolio decisions, and is concepually easier. The inermediae consumpion case, being slighly more complicaed, involves boh porfolio and consumpion decisions. In boh cases, he invesor is assumed o have consan relaive risk aversion, and o be more risk averse han a logarihmic invesor. By he specificaion of he commodiy fuures marke in his aricle, he risk premium of commodiy invesmen urns ou o follow an Ornsein-Uhlenbeck process. This propery allows us o approach he dynamic opimizaion problem in a roue similar o ha developed by Kim and Omberg 1996] and Wacher 22], who address he dynamic opimizaion problem in he conex of predicable equiy premia. Thanks o he simple srucure of he model, he opimal policy and he uiliy cos of excluding he commodiy are solved in closed form. The opimal policy dicaes ha allocaion o commodiies is made boh for myopic purposes and for ineremporal purposes, whereas sock allocaion is made solely ou of myopic consideraions. The opimal financial sraegy involves iming on he spo commodiy price, and hus pus forward a heoreical case for he acical iming sraegies sudied in some empirical invesigaion. The remainder of he aricle is organized as follows. I presen he basic model of he Bond-Sock-Commodiy economy in he nex secion. Secion 3 and 4 are devoed o he opimal sraegy in he erminal wealh case and in he inermediae consumpion case, respecively. In secion 5, I esimae he model of commodiy fuures, and offer some represenaive numerical examples and discussion. Secion 4

5 6 concludes. 2 The economy In he Bond-Sock-Commodiy economy, people can inves in hree asse classes: bonds, socks, and commodiies. I op o characerize he radiional asse classes of bonds and socks by he sandard Black and Scholes 1973] model in order o isolae he effec of he inroducion of commodiies, alhough i is possible o follow oher formulaions which were developed in recen years o reflec sochasic ineres raes, and he documened predicabiliy of sock reurns. For he riskless bond, he consan ineres rae is denoed by r. The sock price S follows a geomeric Brownian moion ds = µ 1 S d + σ 1 S dz 1,, (1) where µ 1 and σ 1 are posiive consans, and Z 1, is a sandard Wiener process. Following Schwarz 1997], he spo commodiy price M is specified by he following single-facor model wih mean-revering propery: dm = θ(µ 2 lnm )M d + σ 2 M dz M,, (2) where θ, µ 2, and σ 2 are posiive consans, and Z M, is anoher sandard Wiener process joinly normally disribued wih Z 1,. Defining m = lnm, we have ( ) dm = θ µ 2 σ2 2 2θ m d + σ 2 dz M,. (3) As in Schwarz 1997], he spo commodiy price is assumed o be non-radable. Assuming he marke price of risk associaed wih he Wiener process driving he spo commodiy price, Z M,, is given by λ M, = α + βm. (4) This assumpion is moivaed by wo empirical findings: (i) expeced reurns in commodiy markes are ime-varying and can be prediced by some insrumenal variables characerisic of he business cycle; and (ii) here is a srong business cycle componen in he variaion of spo commodiy prices. Noe ha his model is reduced o he one-facor model of Schwarz 1997] when β =. Then under he risk-neural measure dm = ( µ 2 θm ) d + σ 2 d Z M,, where θ := θ + σ 2 β, µ 2 := θµ 2 σ 2 α σ 2 2 /2, and Z M, is a sandard Wiener process under he risk-neural measure. From he above equaion, The disribuion of m T condiioning on m ( < T) under he risk-neural measure is normal wih mean and variance: E Q m T ] = e θ m + 1 e θ(t ) ] µ 2 θ Var Q m T ] = σ2 2 1 ] e 2 θ(t ) 2 θ From he maringale propery of fuures prices under he risk-neural measure, i follows ha he fuures price of he commodiy wih mauriy T a ime is ( ) F (T) = E Q M T ] = exp E Q m T] VarQ m T]. 5

6 Then, { ] F (T) = exp e θ(t ) m + 1 e θ(t ) µ ]} + 2 θ σ2 2 1 e 2 θ(t ). (5) 4 θ To faciliae he soluion of Meron s problem, I characerize he asse class of commodiies by he self-financing porfolio M : dm = M (r + σ 2 λ M, )d + M σ 2 dz M,. (6) This porfolio is formed by a porfolio sraegy of he riskless bond and he commodiy fuures as follows. Have a long posiion in he commodiy fuures, and he fuures holding is consanly rolled over o keep he ime-o-mauriy of fuures conracs consan, say equal o l where l is a posiive consan. Moreover, he fuures holding is such ha he noional value of he fuures conrac (he number of fuures conracs imes he fuures price) a ime is equal o M e θl. Because fuures conracs have zero value, he enire porfolio value M is invesed in he riskless bond. For more deail of he porfolio sraegy of M, see Appendix A.1. As such, he specificaion of he Bond-Sock-Commodiy marke has been compleed. To ease he soluion o Meron s problem, however, I reformulae his model by convering he wo possibly correlaed driving Wiener processes o a sandard wo-dimensional Wiener process (i. e. is wo componens are independen). Denoing he correlaion coefficien beween Z 1, and Z M, by ρ where ρ < 1, he price dynamics of he wo classes of risky asses given by (1) and (6) can be rewrien as ds = µ 1 S d + σ 1 S 1 ] dz, dm = M (r + σ 2 λ M, )d + M σ 2 ρ ρ ] dz, (7) where ρ := 1 ρ 2, and Z := ] Z 1, Z 2, is a sandard wo-dimensional Wiener process. 1 In erms of he sandard vecor Wiener process, m can be wrien as ( ) dm = θ µ 2 σ2 2 2θ m ] d + σ 2 ρ ρ dz. Given he specificaion of he financial marke as in (7) and he consan ineres rae r, he marke price of risk associaed wih he sandard vecor Wiener process Z is ] λ1 λ =, λ 2, where λ 1 = µ 1 r σ 1, λ 2, = λ 2 (m ) := λ M, ρ ρ ρ λ 1. (8) Tha is o say, he marke price of risk wih respec o Z 1, is consan, whereas ha wih respec o Z 2, is sochasic and dependen on he commodiy price. As such, he invesmen opporuniy se is sochasic in he Bond-Sock-Commodiy economy, and he opimal sraegy should be differen from he myopic one unless he uiliy funcion of he invesor is logarihmic. Moreover, being a linear ransform of m, λ 2, also follows an Ornsein-Uhlenbeck process: dλ 2, = θ ( λ2 λ 2, ) d + βσ 2 ρ 1 In his aricle, boldface noaion is used o denoe vecors and marices. ρ ρ ] dz, (9) 6

7 where λ 2 = β ρ ( ) µ 2 σ2 2 + ᾱ 2θ ρ ρ ρ λ 1. For a marke o exclude arbirage, i suffices ha he Novikov condiion holds (see, for example, Chaper 6 in Duffie 21]): ( )] 1 T E exp λ λ d <. 2 I can be verified ha he Novikov condiion holds indeed in our model, 2 so he Bond-Sock-Commodiy economy is free of arbirage. Moreover, his economy can be shown o be a complee marke (see e.g. Kreps and Pliska 1981]), wih a unique sae-price densiy ξ given by dξ ξ = rd λ dz, and ξ = 1. (1) Now urn o an invesor wih iniial wealh W. Her consumpion plan is characerized by an consumpion-rae process c, and her porfolio plan is a process of porfolio weighs in he wo risky asses x = ], x S, x M, where xs, and x M, denoe he porfolio weighs in he sock, S, and in he invesable represenaive commodiy, M, respecively. The residual, 1 x S, x M,, is allocaed o he riskless bond. From he self-financing propery of he consumpion-porfolio plan, i follows ha he wealh process W is given by where dw = W r + x σλ ] d c d + W x σdz, (11) σ1 σ = σ 2 ρ σ 2 ρ Noe ha given x M,, we can calculae he corresponding holding of he underlying commodiy fuures conracs as follows. Recall ha in one uni of he represenaive commodiy M, he holding of he fuures conrac wih consan ime o mauriy l has a noional value of M e θl. When x M, weigh of oal wealh is allocaed o M, i implies he raio of he noional value of he underlying fuure conrac o he wealh value W is x e θl M,. In he erminal wealh case, he invesor solves he following dynamic opimizaion problem: sup E x ] W 1 T 1 ]. s.. dw =W r + x σλ ] d + W x σdz, where is he consan rae of relaive risk aversion. For reasons ha will become clear laer, is assumed o be larger han one hroughou he paper o ensure he exisence of a well-behaved soluion. This assumpion is empirically relevan as i is generally suppored by empirical sudies of people s risk aversion (see e.g. Friend and Blume 1975], Pindyck 1988], and Szpiro 1986]), and by he lieraure 2 As λ 1 is consan and hence saisfies he Novikov condiion, we only have o verify ha i is also rue for he Ornsein-Uhlenbeck λ 2,. Dokuchaev 27] has proved ha a marke price of risk following an Ornsein-Uhlenbeck process saisfies he Novikov condiion. Following he same reasoning as in Dokuchaev 27], we can prove ha his condiion applies in our model. (12) 7

8 on he equiy premium puzzle. In he inermediae consumpion case, he dynamic opimizaion is ] T η c1 sup E e x,c 1 d s.. dw =W r + x σλ ] d c d + W x σdz, W T, where η denoes he subjecive discoun rae. 3 Pure porfolio opimizaion I sar wih he erminal wealh case, in which here are only porfolio decisions o make. Using he maringale mehod, he dynamic opimizaion problem (12) is equivalen o he following saic variaional problem Cox and Huang, 1991]: sup E x W 1 T 1 ] s.. W = E ξ T W T ] In paricular, he budge consrain in (12) is equivalen o he saic one in (14) ha is formulaed in erms of he unique sae price densiy. For a soluion o his saic opimizaion problem o exis, i suffices ha E ξ 1 ] T is finie Cox and Huang, 1991], namely he growh-opimal porfolio has a finie expecaion. Under he assumpion ha his condiion applies, he opimal erminal wealh is deermined by WT = (kξ T ) 1/, where k is a Lagrange muliplier deermined by subsiuing he opimal erminal wealh ino he saic budge consrain. Following Cox and Huang 1989], I define a new variable By Io s formula, N = (kξ ) 1. (13) (14) dn = ( r + λ λ 2 ) ] 2, d + λ1 λ 2, dz. (15) N From he definiion of N and he fac ha ξ W is a maringale, we have W = 1 E ξ T WT ξ ] = 1 ] ] E kξ T (kξ T ) 1 1 = N E N 1 T λ 2,, N, kξ where he las equaliy follows from he fac ha λ 2, and N ogeher form a srong Markov process, and hence λ 2, and N are all he invesor needs o know o evaluae momens of N T a ime. Therefore, we can define W := F(N, λ 2,, ; T). 8

9 3.1 Opimal wealh To simplify noaion, I define he following parameers ( 1 a 1 = 2 q = ) βσ 2 θ, a 2 = 1 ( ) 2 βσ2, ρ a a 2, λ = θ λ ρ ρ βσ 2λ 1. (16) Then he opimal wealh can be presened as follows. Lemma 3.1 For an invesor concerned wih maximizing he expeced uiliy over wealh a ime T as described in (12), he opimal wealh is given by where and A 1 (τ) = 1 A 2 (τ) = 1 A 3 (τ) = 1 W = F(N, λ 2,, ; T) = N H(λ 2,, T ), (17) { ]} 1 1 H(λ 2,, τ) = exp 2 A 1(τ)λ 2 2, + A 2 (τ)λ 2, + A 3 (τ), (18) τ 2 (1 e qτ ) 2q (q + a 1 )(1 e qτ ), 4λ ( 1 e qτ/2) 2 q 2q (q + a 1 )(1 e qτ )], a2 2 A2 2 (x) + λ A 2 (x) + a 2 2 A 1(x) + (1 )r λ2 1 ] dx. Proof Given ha W = F(N, λ 2,, ; T) and he sochasic differenial equaions (9) and (15) for λ 2, and N, we can wrie he wealh process in he form of a sochasic differenial equaion by applying Io s formula: where µ W = F + F N N ( r + λ λ 2 F 2,) + σ W = F N N λ 1 (19) dw = µ W d + σ W dz, (2) λ 2 θ ( λ2 λ 2, ) F λ 2 2 ( ) 2 βσ2 ρ F 2 N 2 N2 (λ λ 2 2,) 2 F λ 2 N N βσ 2 ρ (ρλ 1 + ρλ 2, ), λ 2, ] + F βσ 2 ] ρ ρ. λ 2 ρ Because W is a self-financing wealh process, no arbirage requires µ W rf = σ W λ. Wriing i explicily leads o he following parial differenial equaion (PDE) F + r F ( N N + θ λ 2 θλ 2, βσ ) 2 F ρ ρλ 1 βσ 2 λ 2, + 1 λ 2 2 ( ) 2 βσ2 2 F ρ λ F 2 N 2 N2 (λ λ 2 2,) + βσ 2 2 F ρ λ 2 N N (ρλ 1 + ρλ 2, ) = rf. (21) 9

10 F also saisfies he boundary condiion, F(N T, λ 2,T, T; T) = W T. I is noeworhy ha his PDE bears a close resemblance o he PDE for he opimal wealh process in Wacher, 22, Eq. (2)], where he opimal porfolio choice problem is addressed in he conex of mean-revering sock risk premia. The resemblance arises from he fac ha he marke price of risk in Wacher s model is characerized by an Ornsein-Uhlenbeck process, as is λ 2, in he Bond-Sock- Commodiy model. The PDE can be solved by firs guessing a general form for he soluion. Enlighened by he soluion o he PDE in Wacher 22], I guess he form given by (17) and (18). Subsiuing hem back ino (21) yields a quadraic equaion for λ 2, ; from he fac ha boh he consan erm and he coefficiens on λ 2 2, and λ 2, mus be zero, one obains a sysem of hree ordinary differenial equaions: da 1 dτ (τ) = a 2A 2 1 (τ) + a 1A 1 (τ) + 1, da ( 2 dτ (τ) = a a1 ) 2A 1 (τ)a 2 (τ) + A 2 (τ) + λ A 1 (τ), (22) 2 da 3 dτ (τ) = a 2 2 A2 2 (τ) + λ A 2 (τ) + a 2 2 A 1(τ) + (1 )r λ2 1. Equaions of he same form appear in Kim and Omberg 1996] and Wacher 22], and he soluion mehod is sandard. Following Wacher 22], I assume ha > 1 o ensure he exisence of a well-behaved soluion. Under his assumpion, he soluion is given by (19). For he validiy of he opimal soluion (17), some echnical condiions need o be saisfied Cox and Huang, 1989]. Appendix A.2 verifies ha hese condiions hold. Therefore he opimal wealh is given by (17). 3.2 Opimal porfolio plan Turn o he opimal porfolio plan, a plan ha secures he opimal wealh. In he maringale soluion, he opimal porfolio plan can be obained by equaing he diffusion erms in he wo characerizaions of he opimal wealh given by (11) and (2). So he opimal sraegy can be summarized as follows. Theorem 3.2 For an invesor facing he problem (12) in he Bond-Sock-Commodiy economy, he opimal porfolio plan x := ] x S, x M, is given by x = 1 ( σ ) 1 σ W W = 1 λ 1 σ 1 ρ σ 1 ρ λ 2, 1 σ λ 2 ρ 2, } {{ } myopic par + 1 β } ρ A 1(T )λ 2, + A 2 (T )] {{ } ineremporal par. (23) As is sandard in he lieraure, he opimal sraegy in he above presenaion is decomposed ino wo pars: a myopic par, and an ineremporal par. The myopic par, independen of invesmen horizon, is he allocaion ha an invesor would choose if she ignored changes in he invesmen opporuniy se or if her uiliy funcion is logarihmic. The inerpreaion from he perspecive of logarihmic uiliy can be seen direcly by seing o one: when is one, A 1 and A 2 are zero, and he second par disappears. The ineremporal allocaion, he concep 1

11 of which was firs inroduced by Meron 1971] and repeaed in many subsequen sudies, depends on he invesmen horizon and sems from sochasic variaions in he invesmen opporuniy se. Le us look a he allocaion o he wo risky asses in more deail. Given he empirical evidence presened in Secion 5 ha he risk premium in he commodiy marke is decreasing in he spo commodiy price, namely a significanly negaive esimae of β in (4), i is assumed ha β < in he following discussion. The sock allocaion, x S,, consiss solely of a myopic par. I should no come as a surprise, considering ha he sochasic changes in he invesmen opporuniy se in he Bond-Sock-Commodiy economy are caused by he variaions of he commodiy price as shown in (8), and hence he commodiy should be in a beer posiion o deal wih hem. The sock allocaion, wrien as a combinaion of wo ρ λ 1 σ 1, erms λ1 σ 1 σ λ 1 ρ 2,, has a naural economic inerpreaion. The firs erm, is he classical sock allocaion in he Black-Scholes economy Meron, 1969]. The second erm, ρ σ λ 1 ρ 2,, is more ineresing for our purposes, as i arises from he inroducion of he commodiy. Because of he relaionship beween λ 2, and he commodiy price as given in (4) and (8), he second erm implies ha he sock allocaion is dependen on he commodiy price. And he dependence may ake hree forms, according o he way he commodiy and he sock covary wih each oher: (i) when he sock price is posiively correlaed wih he commodiy price (ρ > ), he sock weigh is increasing wih he commodiy price; (ii) when he sock price is negaively correlaed wih he commodiy price (ρ < ), he sock weigh is decreasing wih he commodiy price; and (iii) when hey are independen from each oher (ρ = ), he sock weigh is immune o he commodiy price variaion, and consan a λ1 σ 1. Differen from he case for he sock, he commodiy allocaion x M, is made boh for myopic purposes and for ineremporal purposes. Firs consider he myopic demand for he commodiy, λ 2, /σ 2 ρ. Wih β <, i is a decreasing funcion of he commodiy price, and he myopic demand requires o sell he commodiy when is price rises, and o buy when is price drops. This propery follows from he fac ha he insananeous expeced reurn on he commodiy is negaively relaed o he curren commodiy price. Wheher he invesor should be long or shor he commodiy depends on he sign of λ 2,. Seing λ 2, =, we can solve he hreshold value of he commodiy price for he myopic demand ( ) M mpc ρλ1 α = exp. β Wih β <, we can disinguish hree cases: (i) when he commodiy price is lower han M mpc, and hen λ 2, is posiive, he myopic demand is a long posiion; (ii) when he commodiy price is greaer han M mpc, and hen λ 2, is negaive, he invesor is shor he commodiy; and (iii) when he commodiy price is equal o M mpc, and hen λ 2, is zero, he opimal policy dicaes no exposure o he commodiy for myopic purposes. These are naural resuls if recalling ha he myopic allocaion is concerned only wih insananeous reurns of asses. Thus, he myopic demand boh for he sock and for he commodiy is dependen on he commodiy price. This dependence can be accouned for more inuiively by analogy wih mean-variance opimizaion. As is well known (see, for example, Chaper 13 in Ingersoll 1987]), opimal myopic allocaion o risky asses, i.e. he porfolio of risky asses opimally chosen by log invesors, can be inerpreed as he angency porfolio in he insananeous sandard deviaion-expecaion graph as illusraed in Figure 1. In he Bond-Sock-Commodiy economy, he expecaion and variance of he reurn on he sock is fixed, so he sock is characerized by a fixed poin in he figure. However, he expecd reurn on he commodiy is 11

12 Figure 1: The dependence of he myopic allocaion on he commodiy price: an illusraion The figure shows he insananeous mean-variance opimizaion for wo differen commodiy price M 1 and M 2. The corresponding angency porfolios are P1 and P2. condiional on he curren commodiy price, so he locus of he commodiy in he sandard deviaion-expecaion graph is ime-varying. Suppose ha he spo commodiy price is M 1 a a cerain ime, a myopic invesor would find he opimal allocaion by looking for he angency porfolio (labeled as P1 in he figure), a porfolio based on he fixed locus of he sock and he curren locus of he commodiy. If he spo commodiy price changes o M 2, say, and he commodiy changes o a corresponding new locus, hen he corresponding new angency porfolio (labeled as P2 ) will be formed according o he new commodiy. As such, he myopic allocaion changes wih he variaion of he commodiy price. Now urn o he ineremporal demand for he commodiy. From is expression β ρ A 1(T )λ 2, + A 2 (T )], (24) i follows ha he ineremporal demand changes wih he spo commodiy price, hrough he presence of λ 2,, and wih he invesmen horizon, hrough he presence of A 1 (T ) and A 2 (T ). Firs look a he impac of he spo commodiy price. Because β < and A 1 (τ) is negaive (see Appendix A.3), he ineremporal allocaion is a decreasing funcion of he commodiy price. In oher words, as in he myopic allocaion o commodiy, he ineremporal allocaion is o buy he commodiy when is price falls, and o sell when is price rises. By definiion, he ineremporal allocaion is concerned wih reurns on asses beyond he nex period (an infiniesimal period in coninuous ime). For he mean-revering commodiy, a price hike implies ha no only he reurn on he commodiy over he nex infiniesimal period is geing worse 3, bu he reurns beyond he nex period are deerioraing as well. And i calls for, as a response, lowering he exposure o he commodiy beyond wha is done is he myopic allocaion. Conversely, a price slump implies improved prospecs of fuure reurns, and requires an increased exposure in he ineremporal allocaion. These observaions may help o undersand why he ineremporal demand is decreasing in he commodiy price. Consider now he dependence of he ineremporal allocaion on he invesmen horizon. In paricular, should a long-erm invesor allocae more o he commodiy 3 This implicaion has been capured by he myopic allocaion. 12

13 han a shor-erm one? The horizon effec of he ineremporal allocaion o he commodiy also represens he enire horizon effec of he oal risky allocaion, as i is he sole elemen dependen on he horizon. The horizon effec can be characerized by β ρ (A 1 (T )λ 2, + A 2 (T )), (25) which follows from differeniaing (24) wih respec o he horizon. As shown in Appendix A.3, A 1 (τ) is negaive, whereas A 2 (τ) can be posiive or negaive, depending on he sign of λ, where λ, as defined in (16), includes he parameers characerizing he financial marke and he risk aversion of he invesor. Thus, wihou imposing furher consrains on he parameer values, we canno decide he sign of (25), or he horizon effec. Furher discussion of horizon effec will be given in he numerical example in Secion 5. Afer looking ino is wo componens, we are ready o consider he oal commodiy allocaion. Firs of all, he oal commodiy allocaion is decreasing in he commodiy price since boh componens are decreasing in he price. Anoher quesion ha one may ask is when he invesor should be long or shor he commodiy as a whole. To answer his quesion, one can derive he following hreshold commodiy price for deciding he long/shor posiion of he oal commodiy allocaion M l ρλ1 α = exp σ ] 2 ρa 2 (T ). (26) β 1 + σ 2 βa 1 (T ) Therefore, when he commodiy price is lower, or higher han his hreshold price, he invesor should be long, or shor he commodiy, respecively. The hreshold M l depends on he invesmen horizon, so i is possible ha oher hings being equal, one invesor is shor he commodiy and anoher is long simply because hey have differen invesmen horizons. Unless λ =, he hreshold in he oal commodiy allocaion, Ml, is differen from ha in he myopic commodiy allocaion, M mpc. This highlighs he difference beween he dynamic framework and he saic one: while a myopic invesor would be shor he commodiy, he opimal allocaion may be a long posiion if ineremporal rebalancing is allowed. This difference is aribued o he ineremporal allocaion o he commodiy, in which he following hreshold of deciding a long/shor posiion is used: M in ρλ1 α = exp β ρa 2(T ) βa 1 (T ) ]. (27) This hreshold is differen from M mpc unless λ =, and i may be less or greaer han M mpc, depending on he sign of λ. 3.3 The imporance of commodiies as an asse class: welfare analysis For an emerging asse class like commodiies, i is naural o ask how imporan i is o ake i ino accoun when making invesmen decisions. In oher words, how cosly is i if he new asse class is lef ou in invesmen decision-making? To address his quesion, welfare analysis is applied, as is sandard in he lieraure. In his paper, he imporance of incorporaing he new asse class, or he uiliy cos of omiing i, is measured by he percenage exra iniial wealh ha is necessary o bring he invesor o he same expeced uiliy as is obained by following he opimal sraegy. The uiliy cos can be solved in closed form. Proposiion 3.3 Suppose ha in he Bond-Sock-Commodiy economy, an invesor is concerned wih maximizing he expeced uiliy over wealh a ime T as described 13

14 in (12). If he commodiy is excluded in porfolio decisions, hen L percen of exra iniial wealh is needed o achieve he same expeced uiliy level as is obained by following he opimal sraegy (23), and { L 1 = exp A 1(T)λ 2 2, + A 2 (T)λ 2, + T ( a2 2 A2 2 (x) + λ A 2 (x) + a ) 2 2 A 1(x) dx ]} 1. (28) Proof Firs, we need o know he expeced uiliy from he opimal sraegy, namely he indirec uiliy funcion. Cox and Huang 1989] show ha he indirec uiliy funcion J(W, λ 2,, ) saisfies he differenial equaion From (17), i follows ha Then he differenial equaion becomes J W = 1 N. 1 N = W H(λ 2,, T ). J W = W H(λ 2,, T ). Therefore, he boundary condiion J(W T, λ 2,T, T) = W 1 T /(1 ) implies ha he indirec uiliy funcion is J(W, λ 2,, ) = W 1 1 H(λ 2,, T ). When he commodiy is lef ou, he Bond-Sock-Commodiy economy is reduced o he sandard Black-Scholes economy. I is well known ha in his specificaion, he indirec uiliy is J BS (W, ) = W 1 ( 1 exp r(1 ) + 1 ) ] 1 2 λ2 1 (T ). From he definiion of L ( ) 1 + L J BS W, 1 (28) follows immediaely. = J(W, λ 2,, ), The exponen in he righ hand side of (28) is a quadraic funcion of he iniial commodiy price. From he quadraic form, i follows ha wih eiher very high or very low commodiy prices, he welfare loss is relaively large, whereas wih inermediae commodiy prices, he loss is relaively small. The effec on he welfare loss of oher facors, like he invesmen horizon and he risk aversion of he invesor, will be discussed in he numerical illusraions in Secion 5. 14

15 4 Opimal porfolio and consumpion decisions Now consider he case where he invesor derives uiliy from inermediae consumpion. In his case, apar from deciding wha asse mix o hold, he invesor needs o decide wha fracion of wealh o consume. Thus, assuming uiliy over consumpion allows Meron s problem o be relaed o people s financial decisions in a way ha he previous erminal wealh case does no. On he oher hand, he inermediae-consumpion case has a close link wih he erminal wealh case, in ha he he single opimizaion problem in he former case can be hough of as a series of opimizaion problems for a coninuum of fuure daes Wacher, 22]. In paricular, he invesor wih uiliy over consumpion decides he opimal series of consumpion evens, and hen applies he erminal wealh analysis o each fuure consumpion even. This is analogous o he equivalence beween a bond ha pays coupon coninuously and a coninuum of zero-coupon bonds. In he following maringale soluion, I shall use his insigh and follow a procedure similar o Wacher 22]. The maringale approach ransforms he budge consrain in (13) ino a saic one, ] T W = E ξ c d. (29) The opimal consumpion plan follows from he firs order condiion of he opimizaion problem in saic form, 4 c = (Kξ ) 1 e 1 η, (3) where K is a Lagrange muliplier deermined by insering c ino (29). The opimal porfolio plan is deermined so as o mee he need o finance he consumpion plan (3). The wealh a ime, denoed by W, is he discouned value of fuure consumpion ill ime T, ] W = 1 T E ξ s c ξ sds. As in he erminal wealh case, define a new variable I follows from Io s formula ha N = (Kξ ) 1. dn = ( r + λ 2 1 N + ) ] λ2 2, d + λ1 λ 2, dz. (31) Tha is, N has he same dynamics as N, bu wih a differen iniial value. From he inroducion of N, and he srong Markov propery of ], N λ 2, i follows ha ] T 1 W = N E N 1 e 1 ηs ds λ 2,, N. s Therefore one can define W := G(N, λ 2,, ; T). 4 Here we work under he same echnical assumpion ha E(ξ 1 T ) is finie as in he erminal wealh case. The oher echnical condiions for he soluion s validiy are proved in Appendix A.2. 15

16 4.1 Opimal wealh The opimal wealh assuming inerim consumpion can be characerized as follows. Lemma 4.1 For an invesor concerned wih maximizing he expeced uiliy over life-ime consumpion as described in (13), he opimal wealh is given by where and W = G(N, λ 2,, ; T) = N 1 e η T H(λ 2,, s ) = exp H(λ 2,, s )ds, (32) { A 1(s )λ 2 2, + A 2 (s )λ 2, + A 3 (s ) ]}, (33) A 1 (τ) =A 1 (τ), A 2 (τ) =A 2 (τ), (34) A 3 (τ) =A 3 (τ) ητ Proof Applying Io s formula o W = G(N, λ 2,, ; T), one has where µ W = G + G N N ( r + λ λ 2 G 2,) + σ W = G N N λ 1 dw = µ W d + σ W dz, (35) λ 2 θ ( λ2 λ 2, ) G λ 2 2 ( ) 2 βσ2 ρ G 2 N 2 N 2 (λ λ 2 2,) 2 G λ 2 N N βσ 2 ρ (ρλ 1 + ρλ 2, ), λ 2, ] + G βσ 2 ] ρ ρ. λ 2 ρ Differen from he erminal wealh case, he porfolio process W is no self-financing, since a coninuous consumpion flow c is wihdrawn. Thus G(N, λ 2,, ) iself does no saisfy he generalized Black-Scholes equaion. Insead, in his case no arbirage requires µ W + c rg = σ W λ. Wriing i explicily, we have he following PDE for G, G + r G ( N N + θ λ 2 θλ 2, βσ ) 2 G ρ ρλ 1 βσ 2 λ 2, + 1 λ 2 2 ( ) 2 βσ2 2 G ρ λ G 2 N 2 N 2 (λ λ 2 2,) + βσ 2 2 G ρ λ 2 N N (ρλ 1 + ρλ 2, )N 1 e η = rg, (36) wih he boundary condiion, G(N T, λ 2,T, T) =. Because a PDE of similar form has been solved by Wacher 22], I ake (32) and (33) as he guessed form of soluion here. Subsiuing hem ino (36) and maching he coefficiens of λ 2 2,, λ 2, and he consan erm produces a sysem of hree differenial equaions very similar o (22). And heir soluion is (34). 16

17 A firs glance, i seems hard o undersand why he differenial equaion (36) should have a soluion in he inegral form as in (32). This guessed soluion, however, may follow naurally when uilizing he link beween he inermediae consumpion analysis and he erminal wealh analysis. Consider a series of auxiliary invesors deriving uiliy from erminal wealh a ime i, T], and each invesor is indexed by her fixed horizon dae i. Suppose ha invesor i has iniial wealh W i, = W ξ 1 1 i ] e η i E T E 1 ξ1 e 1 η d ]. (37) Applying he erminal wealh analysis o invesor i, we can wrie her opimal wealh a, i] as ] 1 W i, := F(N, λ 2,, ; i) = e η i N E N 1 λ 2,, N Wih he inroducion of F(N, λ 2,, ; i), he opimal wealh of he invesor wih uiliy over consumpion G(N, λ 2,, ) can be characerized as he sum of he opimal wealh of he auxiliary invesors: G(N, λ 2,, ) = T i F(N, λ 2,, ; s)ds. (38) The erminal wealh analysis can yield a soluion of F ha is similar o F given by (17). So he soluion (32) follows immediaely. The derivaion hrough a series of auxiliary invesors has an ineresing economic inerpreaion. W i, given in (37) is he value a ime zero of he opimal erminal wealh a ime i. From (29) and (3), i follows ha W i, is he value a ime zero of he opimal consumpion even a period i for he invesor concerned wih inermediae consumpion. The fracion a he righ hand side of (37) is he raio of period-i consumpion o her life-ime consumpion in erms of he presen value. Therefore, i is correc o hink of he invesor as holding separae accouns for each fuure consumpion even, disribuing her iniial wealh ino each accoun according o (37) o achieve he opimal consumpion plan, and hen invesing each accoun so ha he consumpion needs are me. 4.2 Opimal porfolio and consumpion policy In he inermediae consumpion case, he opimal financial sraegy is characerized by he following heorem. Theorem 4.2 Suppose ha in he Bond-Sock-Commodiy economy, an invesor seeks o maximize he expeced uiliy over life-ime consumpion by choosing consumpion and invesmen plans, as formalized in (13). Then he opimal consumpion plan can be characerized by he following consumpion-wealh raio, c W = The opimal porfolio plan, denoed by x := x S, x = 1 λ 1 σ 1 ρ σ 1 ρ λ 2, 1 σ λ 2 ρ 2, } {{ } myopic par 1 T H(λ 2,, s )ds]. (39) + 1 β ρ T x M, ], is H(λ2,,s )A1(s )λ2,+a2(s )]ds T H(λ2,,s )ds }{{} ineremporal par. (4) 17

18 Proof The opimal consumpion-wealh raio follows from (3) and (32). The opimal porfolio plan can be obained by equaing he diffusion erms in (11) and (35), he wo characerizaions of W. For he opimal consumpion-wealh raio, noe ha i changes wih he commodiy price, bu no wih he sock price. The myopic allocaion in he porfolio plan is he same as ha in he erminal wealh case. I is a naural oucome when considering ha in he analogy of he invesor wih uiliy over consumpion o a series of invesors concerned wih erminal wealh, each of he auxiliary invesors has idenical myopic allocaion. The only new elemen arising from assuming inermediae consumpion is conained in he ineremporal allocaion o he commodiy. Comparing (4) and (23), i is clear ha he ineremporal allocaion in he inermediae consumpion case is a weighed average of ha in he erminal wealh case, using H(λ 2,, τ) as he weigh. For H, (39) implies W c = T H(λ 2,, s )ds. Hence, H(λ 2,, τ) can be inerpreed as he ime- value of fuure consumpion in τ periods normalized by he opimal consumpion rae a ime-. In all, he ineremporal allocaion assuming inermediae consumpion is an average of hose assuming erminal wealh, and he average is weighed by he value of fuure consumpion in each period. 4.3 Welfare analysis When people are concerned wih inerim consumpion, he uiliy loss of leaving ou he commodiy is as follows. Proposiion 4.3 Suppose ha in he Bond-Sock-Commodiy economy, an invesor is concerned wih maximizing he expeced uiliy over life-ime consumpion as described in (13). If he commodiy is excluded in consumpion and porfolio decisions, hen L percen of exra iniial wealh is needed o achieve he same expeced uiliy level as is obained by following he opimal sraegy given by Theorem 4.2, and where L 1 = ω := ω T H(λ 2,, s)ds 1 e ωt η r(1 ) ] 1 1. (41) λ 2 1. Proof By reasoning similar o ha in he erminal wealh case, he indirec uiliy funcion assuming inermediae consumpion, denoed by J (W, λ 2,, ), is given by T J (W, λ 2,, ) = W1 1 e η H(λ 2,, s )ds]. In he sandard Black-Scholes economy afer dropping he commodiy, he corresponding indirec uiliy is J BS (W, ) = W1 1 (1 e ω(t ))], 1 ω Then L as given by (41) follows from is definiion. 18

19 5 Calibraion and discussion In his secion, I shall firs esimae he commodiy fuures price model. Then some represenaive numerical illusraions and discussions will be presened. 5.1 Esimaion of he commodiy fuures model The parameers ha characerize he asse class of commodiies are esimaed using he GSCI Commodiies Index fuures prices. The GSCI index underlying he fuures conrac racks he price levels of major commodiies, and he fuures conrac can be viewed as being wrien on a baske of commodiies. Therefore, he GSCI Commodiies Index is aken o be he non-radable spo commodiy price, and he GSCI Commodiies Index fuures prices are he radable fuures prices. The esimaion is carried ou in wo seps. In he firs, he hree parameers ha characerize he spo commodiy price, θ, µ 2, and σ 2, are esimaed. From (3), he logarihm of he spo commodiy price follows a firs-order auoregressive process, and he maximum likelihood mehod is used o ge he esimaes (Table 1). I use he monhly daa of GSCI Commodiy Index from December 1969, he sar dae of he index, o November 28. The daa are deflaed by he US CPI-U index, for he reason ha he asse prices are assumed o be measured in real erms in he Bond-Sock-Commodiy economy. The deflaed daa are normalized in such a way ha he value was 1 in July 1992 when Chicago Mercanile Exchange (CME) inroduced fuures on his index. The second sep is o esimae α and β, which specify he risk premium in he fuures marke (4). The fuures price (5) can, in log form, be rewrien as lnf (T) = e θt m + (1 e θt ) µ 2 θ + σ2 2 4 θ (1 e 2 θt ). Then he sysem of esimaion equaions is y = am + d + ε, = 1, 2,..., K (42) where K is he number of he observaions. In he above, ε is an N 1 vecor of disurbance, lnf (T 1 ) e θt 1 y =., a =. lnf (T N ) and d = ( 1 e θt 1 ) µ2 θ + σ θ ( 1 e θt N ) µ2 θ + σ2 2 4 θ e θt 1 (1 e 2 θt 1 ) (1 e 2 θt N ) where T 1,...,T N are he ime o mauriy of he fuures conracs and N is he number of conracs. In view of he liquidiy of fuures conracs, he firs hree nearby fuures conracs are used for he esimaion. In paricular, he daa, obained from Bloomberg, consis of he monhly observaions of he prices of hese hree conracs a he end of each monh from July 1992 o November 28. Since he fuures rading erminaes on he elevenh business day of he conrac monh, he ime o mauriy for each of hese hree conracs does no change wih he observaions. As wih he underlying index, he fuures prices are deflaed by he CPI-U index, and normalized correspondingly. 19

20 The esimaes of α and β, presened in Table 1, are obained by applying he ieraive leas squares mehods o (42). The esimae of β is significanly negaive, suggesing ha he risk premium of he commodiy fuures is decreasing in he spo commodiy price. Esimaed parameer values Assumed parameer values ˆθ ˆµ 2 ˆσ 2 ˆα ˆβ µ1 σ 1 r ρ η (.82) (.66) (.12) (.323) (.69) Table 1: The parameer values used for he numerical exercise The parameers characerizing he asse class of commodiies are esimaed, and he sandard errors are in parenhesis. The oher parameer values are aken o be consisen wih many exising sudies. For he purpose of numerical illusraion, I assume he oher parameer values as given in Table 1. The parameer values for he sock and he riskless bond, µ 1, σ 1, and r, are aken o be consisen wih many empirical sudies, e.g. Campbell 23]. The value of he correlaion coefficien is chosen o be -.1, a level corresponding o he finding in many sudies ha he commodiy and sock reurns have a moderae negaive correlaion. 5.2 Opimal sraegy and uiliy of commodiy invesmen For he parameer values as given in Table 1, Opimal financial sraegies and uiliy losses are deermined. The focus of his numerical exercise is on he influence of he commodiy price, of he horizon of he invesor, and of he risk aversion of he invesor. From he propery of Ornsein-Uhlenbeck processes, i follows ha he logarihm of commodiy price is asympoically saionary, and he asympoically saionary disribuion is normal: Φ ( µ 2 σ2 2 2θ, σ2 2 2θ ). To have an inuiive idea of he level of curren commodiy price, I shall locae i wih respec o his asympoical disribuion. Figure 2 shows he opimal sraegies and uiliy losses for a range of curren commodiy prices from 66 hrough 251. Wih respec o he asympoically saionary disribuion of he commodiy price, his commodiy price range corresponds o he one from he 5h-percenile o he 95h-percenile. In his example, he opimal sock weigh is decreasing wih he spo commodiy price, owing o he assumed negaive correlaion coefficien beween he wo driving Brownian moions, and he decreasing relaionship beween he risk premium in he commodiy fuures marke and he spo commodiy price. Noably, he commodiy allocaion varies considerably wih he changes of he commodiy price. The commodiy allocaion decreases from a long posiion of abou 1% for he commodiy price a 5h-percenile, o a shor posiion of around 5% for he commodiy price a 95h-percenile. In he opimal porfolio sraegy, here is subsanial marke iming. We have learned ha he uiliy losses of leaving ou he commodiy depend on he curren commodiy price (Proposiions 3.3 and 4.3). In his numerical example, he curren commodiy price has a significan impac on he magniude of uiliy loss. For he given range of commodiy price, he uiliy loss varies from 1% o 47% in he erminal wealh case, and from 5% o 2% in he inermediae consumpion case. I suggess ha excluding commodiies in financial decisions is much more cosly when commodiy prices are very low or very high han when hey are moderae. Figure 3 shows ha he uiliy loss of excluding he commodiy is increasing in he horizon, T, and decreasing in he degree of risk aversion,. The decreasing 2

21 Panel A: he erminal wealh case sock weigh commodiy weigh: myopic commodiy weigh: ineremporal commodiy weigh: oal welfare loss (L%) 1.8 Panel B: he inermediae consumpion case commodiy weigh: ineremporal commodiy weigh: oal welfare loss (L%) consumpion wealh raio Figure 2: Opimal policy and he commodiy price This figure shows how he opimal sraegy and he uiliy loss changes wih he curren commodiy price in he erminal wealh case (Panel A), and in he inermediae consumpion case (Panel B). I is assumed ha T = 1, and = 5. The sock allocaion and he myopic allocaion o he commodiy are he same in boh cases, and hey are no repeaed in Panel B for ease of reading. Figure 3: The effec on uiliy loss of invesmen horizon and risk aversion This figure shows how he uiliy loss changes wih he invesmen horizon and he degree of risk aversion in he erminal wealh case and he inermediae consumpion case. The spo commodiy price is assumed o be 129, he median value of he asympoic disribuion. 21