Frank de Jong and Yang Zhou Portfolio and Consumption Choice with Habit Formation under Inflation

Transcription

1 Frank de Jong and Yang Zhou Porfolio and Consumpion Choice wih Habi Formaion under Inflaion DP 08/

2 Porfolio and Consumpion Choice wih Habi formaion under inflaion Preliminary Version: Commens Welcome Frank de Jong Yang Zhou Augus 20, 2013 Absrac We invesigae he opimal porfolio and consumpion policies for a finiehorizon invesor in a life-cycle model wih habi formaion and inflaion risk. We consider wo ypes of habi invesors: one forms habi based on real pas consumpion, while he oher on nominal pas consumpion, which is moivaed by money illusion. The opimal sraegy is expressed explicily in erms of he soluion o a linear parial differenial equaion. We find ha he effecs of inflaion on he opimal sraegy depend on he ype of habi invesor, because i deermines he risk profile of he hedge porfolio and subsisence porfolio. This dependence is robus o he incompleeness of he financial marke. Keywords: Porfolio and consumpion choice; Habi formaion; Inflaion risk; Money illusion JEL Codes: D91; G11 This research has been parly funded by he Nework of Pensions, Aging and Reiremen Nespar). Deparmen of Finance, Tilburg Universiy, P.O.Box 90153, 5000 LE Tilburg, The Neherlands, f.dejong@uv.nl Deparmen of Finance, Tilburg Universiy, P.O.Box 90153, 5000 LE Tilburg, The Neherlands, y.zhou@uv.nl 1

3 1 Inroducion Time separable uiliy funcions, such as power uiliy, are common in he opimal porfolio and consumpion choice lieraure. Noneheless, his ime-separabiliy has long been criicized because i is a odds wih he empirical evidence ha households consumpion depends on heir own pas consumpion and/or he consumpion of a reference group 1. To his end, uiliy funcions wih habi formaion have been proposed, which prescribe ha invesors form habi on he basis of heir own previous consumpion inernal habi formaion) or he pas hisory of aggregae consumpion exernal habi formaion) and derive uiliy only from he consumpion in excess of he habi levels. Alhough some sudies 2 on he opimal porfolio choice already employ preferences wih inernal habi formaion, o he bes of our knowledge none of hem considers inflaion uncerainy. However, hedging inflaion risk is imporan for long-erm invesors, as i subsanially increases he volailiy of he households wealh. In paricular, he ineracion beween he need o susain fuure minimum consumpion and ha o hedge inflaion risk may have subsanial influence on he composiion of he opimal porfolio. Moreover, he under-developmen of he inflaion-indexed bonds marke in pracice makes i difficul for he habi-households o ensure fuure habi consumpion due o he unspanned residual inflaion risk. Therefore, incorporaing inflaion risk in he habi-based life-cycle models will add realism o he analysis of household decision making and enables us o obain relevan policy implicaions. The inroducion of inflaion risk may have differen effecs on differen ypes of habi formaion. Specifically, we consider wo habi formaion models, namely real habi formaion and nominal habi formaion. habi level is generaed direcly by pas real consumpion raes. Under real habi formaion, he real In conras, under nominal habi formaion, invesors form heir nominal habi on he basis of previous nominal consumpion bu derive uiliy sill from consumpion in excess of real habi. This mismach can be referred o as money illusion, because invesors misake nominal consumpion for real consumpion in forming habi levels. More imporanly, i produces wo disincive feaures of real habi dynamics under nominal habi formaion relaive o hose under real habi formaion: he evoluion of he real habi level becomes sochasic 1 See, for example, Heien and Durham 1991), Ferson and Consaninides 1991) Ravina 2005), Korniois 2010). 2 See, for example, Deemple and Karazas 2003), Bodie, Deemple, Oruba, and Waler 2004), Munk 2008). 2

4 and subjec o he erosion of inflaion. These disincions reveal he bigger role of inflaion under nominal habi formaion. Therefore, i is of ineres o compare he opimal porfolio and consumpion sraegy under differen ypes of habi formaion. In his paper, we inroduce inflaion risk o a life-cycle model wih habi formaion and sudy he effecs of inflaion risk on he opimal porfolio and consumpion sraegy of a represenaive habi invesor. In paricular, we compare boh he qualiaive and quaniaive properies of he opimal porfolio and consumpion sraegy under differen habi formaion and link he differences o he differen roles of inflaion. We begin by invesigaing a complee marke case as he benchmark and proceed o a case wih a single nominal bond. The analysis of boh complee and incomplee marke cases is performed under real habi formaion and nominal habi formaion, respecively. Specifically, he moivaions for he incomplee marke case are wofold. Firs, he perfec hedge agains boh expeced inflaion risk and ineres rae risk by linear combinaion of wo nominal bonds always requires a shor posiion in one of he bonds, which is unrealisic from a pracical poin of view. Second, i is emping o exclude he inflaion-indexed bond from he asse menu due o he lack of a well-developed inflaion-indexed bonds marke. Our main resuls are as follows: Firs, consisen wih Munk 2008) 3, he opimal porfolio can be explicily expressed in erms of he soluion o a linear parial differenial equaion and is a combinaion of hree porfolios: 1) a myopic mean-variance porfolio, 2) a hedge porfolio agains variaion of fuure invesmen opporuniies in he economy wih adjusmen of habi formaion and 3) a subsisence porfolio ensuring fuure minimum consumpion. Habi formaion affecs he opimal porfolio sraegy hrough wo channels: on he one hand, i induces a subsisence demand and hus reduces he free wealh. This channel is referred o as leverage effec. On he oher hand, habi persisence eners he pricing kernel and is a deerminan of fuure invesmen opporuniies. As a resul, he hedge porfolio depends on habi srengh as well as he risk profile of fuure habi levels. Second, he effecs of inflaion risk on he opimal porfolio sraegy differ subsanially beween he wo cases. Under real habi formaion, he imporance of inflaion risk is raher limied because here is no ineracion beween habi persisence and expeced inflaion. A direc consequence is he absence of influence of expeced inflaion on he 3 Munk 2008) sudy a life-cycle model of consumpion and invesmen wih boh habi formaion and sochasic invesmen opporuniies. 3

5 wo habi formaion channels; boh he hedge porfolio and he subsisence porfolio are affeced by inflaion risk merely hrough he hedge agains unexpeced inflaion. In conras, inflaion risk plays a much bigger role in he case of nominal habi formaion, because he expeced inflaion exposure borne by he real habi levels carries over o he hedge porfolio and subsisence porfolio. On he one hand, expeced inflaion eners he pricing kernel, leading o a sharp increase in he inflaion risk exposure of he hedge porfolio. On he oher hand, he erosion of he subsisence porfolio caused by inflaion leads o weaker leverage effec. Furhermore, he subsisence porfolio becomes subsanially exposed o inflaion risk. Anoher disincion beween he wo cases arises wih respec o unexpeced inflaion hedging: he opimal porfolio akes full insurance agains unexpeced inflaion risk in he case of real habi formaion bu leaves he subsisence porfolio uninsured in he alernaive case because of he perfecly negaive correlaion beween he real habi level and realized inflaion. Third, comparison wih Brennan and Xia 2002) 4 idenifies he effecs of habi formaion on he opimal porfolio in he presence of inflaion risk: Firs of all, a new porfolio for susaining subsisence consumpion shows up. Moreover, he opimal porfolio akes lower equiy exposure and ineres rae risk exposure and he decline is more pronounced for higher habi srengh. The inflaion exposure is lower in he case of real habi formaion, bu higher in he case of nominal habi formaion. Third, boh he equiy exposure and inflaion risk exposure become dependen on invesmen horizon and he horizon effec on he ineres rae exposure srenghens. Fourh, in he incomplee marke case, he opimal sock invesmen, opimal bond invesmen and sock-o-bond raio decrease wih habi srengh and iniial habi level. The horizon effec is negaive for he sock invesmen and posiive for he bond invesmen. Boh he opimal sock invesmen and bond invesmen are higher under nominal habi formaion han hose under real habi formaion, bu he opimal porfolio leans more owards he bond. Finally, we examine he expeced wealh and expeced consumpion for hree ypes of invesors, namely non-habi invesor, real habi invesor and nominal habi invesor. The wealh decumulaion is slowes for he nominal habi invesor, modes for he real habi invesor and fases for he non-habi invesor. The non-habi invesor sars wih 4 Brennan and Xia 2002) sudy he opimal porfolio and consumpion choice of a finie-horizon invesor in he presence of inflaion risk. 4

6 higher consumpion bu has much lower consumpion growh han her counerpars. Wihin he habi invesors, he nominal one has higher wealh and consumpion over he whole life-cycle han he real one. Based on he heoreical analysis, some policy implicaions can be drawn for longerm invesors, paricularly for pension funds. There is some empirical evidence ha households exhibi srong demand for guaranees for heir pension income 5. A possible explanaion for such a demand is habi formaion. As pension funds inves on behalf of heir members, i is imporan o ake ino accoun he habi persisence of pension paricipans in making invesmen decisions: Firs, o ensure fuure guaraneed pension payou, here should be a clear separaion beween he subsisence porfolio and oher radiional porfolios proposed in he porfolio choice lieraure. Moreover, he composiion of he hedge porfolio and subsisence porfolio should depend on he ype of guaranees offered real v.s. nominal). This aricle builds on he srand of papers on dynamic asse allocaion wih inflaion risk. See, for example, Campbell and Viceira 2001), Brennan and Xia 2002), Sangvinasos and Wacher 2005), Munk and Sørensen 2004), De Jong 2008), Koijen, Nijman, and Werker 2010) and Van Hemer 2010). In paricular, we follow Brennan and Xia 2002) in modeling he asse price dynamics in he presence of inflaion risk. On he oher hand, his paper also relaes o he lieraure on he opimal porfolio and consumpion choice wih habi formaion in preferences. See, for example, Consaninides 1990), Deemple and Zapaero 1992), Deemple and Karazas 2003), Bodie, Deemple, Oruba, and Waler 2004) and Munk 2008). Consaninides 1990) derives he opimal porfolio and consumpion sraegy for an infiniely-lived invesor under he assumpion of consan invesmen opporuniies. Based on he insighful observaion of Schroder and Skiadas 2002) ha he model wih linear habi formaion can be mechanically ransformed ino an equivalen model wihou habi formaion, Bodie, Deemple, Oruba, and Waler 2004) provide an analysis of opimal porfolio and consumpion decision in a more general seing wih endogenous labor supply and sochasic wages and Munk 2008) inroduces sochasic invesmen opporuniies o he habi-based lifecycle model, which is closes o his paper. We exend Munk s model by incorporaing inflaion risk and sudy how inflaion influences he opimal sraegy of he habi invesor and how hese effecs depend on he ype of habi formaion. 5 See, for example, Van Rooij, Kool, and Pras 2007) and Anolín, Paye, Whiehouse, and Yermo 2011). 5

7 The remainder of he paper is organized as follows. Secion 2 ses up he model by describing he financial markes and preferences. Secion 3 presens he soluion o he opimizaion problem. Secion 4 calibraes he model and carries ou some numerical experimens. Secion 5 concludes he paper and Appendix shows all proofs. 2 The Model 2.1 Financial Markes We follow Brennan and Xia 2002) in modeling he asse price dynamics. There are four variables deermining asse prices in he Brennan-Xia model: he nominal sock price S, he insananeous real ineres rae r, he insananeous expeced inflaion π and he commodiy price level Π. The erm srucure is characerized wih he real ineres rae and expeced inflaion. For simpliciy, we assume ha he risk premia on sources of uncerainy are consan 6. The sock price follows a geomeric Brownian moion as in he Black and Scholes 1973) model. The real ineres rae and expeced inflaion follow Ornsein-Uhlenbeck processes as in he Vasicek 1977) model. The realized inflaion equals he expeced inflaion plus a random shock. The equaions driving he sae variables are given by, ds S = R + σ S λ S )d + σ S dz S, 1) dr = κ r r )d + σ r dz r, 2) dπ = θ π π )d + σ π dz π, 3) dπ Π = π d + σ Π dz Π, 4) where R is he nominal ineres rae, σs capure he volailiy, λ S is he nominal price of equiy risk, dzs are changes in sandard Brownian moions z, θ and κ are mean reversion parameers, and r and π are uncondiional means. Noe ha hroughou his paper, we use uppercase leers for nominal variables and he corresponding lowercase leers for heir real counerpars. 6 Sangvinasos and Wacher 2005) and Koijen, Nijman, and Werker 2010) allow for ime variaion in risk premia, bu absrac from habi formaion in preferences. 6

8 We can orhogonalize Equaion 4) for unexpeced inflaion: dπ Π = π d + ξ S dz S + ξ r dz r + ξ π dz π + ξ u dz u = π d + ξ dz, 5) where dz = dz S, dz r, dz π, dz u ) denoes he vecor of innovaions in sandard Brownian moions wih dz u 7 orhogonal o dz S, dz r, and dz π. The correlaion marix of dz herefore is ρ = {ρ S,r,π } ). 6) The real pricing kernel of he economy m, follows a diffusion process: dm m = r d + φ S dz S + φ r dz r + φ π dz π + φ u dz u = r d + φ dz, 7) where, φs represens he consan loadings on he sochasic innovaions in he economy and deermines he marke prices of risk, λ S, λ r, λ π and λ u, which are associaed wih innovaions dz S, dz r, dz π and dz u, respecively. Brennan and Xia 2002) show ha he nominal shor-erm risk-free rae R and he vecor of nominal marke price of risk λ = λ S, λ r, λ π, λ u ) are given by λ = ρξ φ), 8) R = r + π ξ λ. 9) The ime nominal price of a nominal zero-coupon bond mauring a T, denoed by P, T ), evolves as, dp, T ) P, T ) = [R B r, T )σ r λ r B π, T )σ π λ π ]d B r, T )σ r dz r B π, T )σ π dz π, 10) 7 Noe ha in Brennan and Xia 2002), he subscrip "u" means unhedgeable. However, as explained below, we consider a complee marke in he benchmark model and hus here is no unhedgeable componen of inflaion risk. We follow his noaion for he purpose of comparison. 7

9 where B r, T ) = κ 1 1 e κt ) ), 11) B π, T ) = θ 1 1 e θt ) ). 12) In conras, he ime real price of an inflaion-indexed bond mauring a ime T evolves as dp, T ) p, T ) = [r B r, T )σ r λr ]d B r, T )σ r dz r, 13) where λ r = φ ρe 2 and e 2 = 0, 1, 0, 0). Applying Iô s Lemma o is nominal value, P = Π p, yields is nominal reurn, dp, T ) P, T ) = [r + π B r, T )σ r λ r ]d B r, T )σ r dz r + ξ dz. 14) Equaion 10) shows ha nominal bonds have loadings on dz r and dz π, bu no loading on dz u. Thus, in an economy wih only socks and nominal bonds, he inflaion process can no be fully spanned and he marke is incomplee, which corresponds o he seing of Brennan and Xia 2002). In his paper, however, we add an inflaion-indexed bond o he asse menu in order o complee marke, because, as shown in 14), inflaionindexed bonds have non-zero loading on dz u, which allows he invesor o hedge agains unexpeced inflaion risk. I is imporan o noe ha he reurn processes of nominal bonds wih differen mauriies only differ in heir loadings on dz r and dz π. Hence, any desired combinaion of loadings on dz r and dz π can be achieved by posiions in any wo bonds wih differen mauriies. In wha follows, we consider wo financial marke seings. In he benchmark model, we assume ha he invesor can inves in five securiies: a nominal insananeously riskless asse, a sock, wo nominal bonds wih mauriies T 1 and T 2 and an inflaionindexed bond wih mauriy T 3. Le σ be he facor loadings marix of he sock and hree bonds and Λ be he vecor of he nominal risk premia, which are given by, σ S σ = 0 B r 0, T 1 )σ r B π 0, T 1 )σ π 0 0 B r 0, T 2 )σ r B π 0, T 2 )σ π 0, 15) ξ S ξ r B r 0, T 3 )σ r ξ π ξ u 8

10 and Λ = σλ = σ S λ S, B r 0, T 1 )σ r λ r B π 0, T 1 )σ π λ π, B r 0, T 2 )σ r λ r B π 0, T 2 )σ π λ π, B r 0, T 3 )σ r λ r + ξ λ). 16) In he alernaive seing, he invesor has access o only one nominal bond and socks and herefore he marke is incomplee. The moivaions for he incomplee marke case are wofold. Firs, he perfec hedge agains boh expeced inflaion risk and ineres rae risk by linear combinaion of wo nominal bonds always requires a shor posiion in one of he bonds 8. However, borrowing consrains prevail for mos of marke paricipans, making his combinaion largely infeasible in pracice. Second, i is of ineres o exclude he inflaion-indexed bond from he asse menu because he inflaion-indexed bonds marke is less developed. Le σ I be he facor loadings marix of he sock and he nominal bond wih mauriy of T 4 and Λ I be he vecor of he nominal risk premia, which are given by, ) σ S σ I =, 17) 0 B r 0, T 4 )σ r B π 0, T 4 )σ π 0 and Λ I = σ I λ = σ S λ S, B r 0, T 4 )σ r λ r B π 0, T 4 )σ π λ π ). 18) 2.2 Preferences We consider an invesor wih a fixed invesmen horizon T. The objecive of he invesor is o maximize over her life-cycle he expeced discouned sum of all finie uiliy which are generaed by he difference beween real consumpion c and real habi level h. In line wih mos of he lieraure, he uiliy funcion is assumed o be of he isoelasic form wih risk aversion parameer γ. The individual s porfolio and consumpion opimizaion problem can be formulaed as max C,x) A E [ˆ T 0 C δ Π e h ) 1 γ 1 γ 8 See, for example, Brennan and Xia 2002) and Sangvinasos and Wacher 2005). d ] 19) 9

11 where δ is he subjecive discoun facor, C is he nominal consumpion rae, h is he real habi level and A is he se of admissible consumpion and porfolio sraegy. x is he vecor of he porfolio weighs on he risky asses and 1 x ι is he weigh on he nominally riskless asse. The invesor maximizes her uiliy by appropriaely choosing a nominal consumpion process C = C ) and a porfolio sraegy x = x ). The nominal wealh dynamics can be wrien as, dw = [W R + x Λ) C ] d + W x σdz. 20) The requiremen ha he fuure consumpion sreams mus be financeable by he iniial wealh of he invesor implies a saic budge consrain, [ˆ T E 0 ] m C d W 0. 21) m 0 Π Π 0 where W 0 is he nominal iniial wealh, Π 0 is he iniial price level. Choosing C τ and x τ over he period τ [, T ] o maximize uiliy in he remaining lifeime yields he indirec uiliy: J = max C,x) A E [ˆ T Cs δs ) Π e s h s ) 1 γ 1 γ ds ]. 22) As shown in 19), he habi level can be regarded as a subsisence consumpion rae, since he consumpion rae mus exceed he habi level. Noe ha γ is no he acual level of relaive risk aversion, bu sill an imporan deerminan of i: c RRA = γ. 23) c h Obviously, he relaive risk aversion is no longer consan, bu decreasing in he raio of consumpion o habi. In oher words, for any given habi level, higher consumpion rae leads o lower risk aversion. We consider wo ypes of inernal habi formaion. The firs one is real habi formaion, in which he real habi level is generaed by previous real consumpion raes, h = h 0 e β + α ˆ 0 e β s) c s ds, 24) 10

12 and evolves as, dh = βh αc )d. 25) Here c is he real consumpion, α is he scaling parameer, β is he persisence parameer and h 0 is he iniial real habi level. The real habi level is a weighed average of pas consumpion raes. The weighs are exponenially decreasing so ha he recen consumpion raes are given higher weighs. Following Munk 2008), we require ha β > α o ensure ha he real habi level will decline when her consumpion rae coincides wih he habi level. Noe ha when c = h, dh = β α)h d. Thus, β α) can be inerpreed as he decay rae of habi level a he minimum consumpion and capures habi srengh 9. The alernaive is nominal habi formaion, in which he nominal habi level is generaed by previous nominal consumpion raes: H = H 0 e β + α ˆ 0 e β s) C s ds. 26) As he invesor derives uiliy from consumpion on op of real habi level, bu forms habi on he basis of previous nominal consumpion, here is a mismach beween uiliy funcion and habi formaion process. This can considered money illusion: he invesor misakes nominal consumpion sream for real consumpion sream in forming habi levels. Applying Iô s lemma o he relaionship h = H /Π yields he dynamics of h, dh = βh + π ξ ρξ αc )d h ξ dz. 27) Comparison wih 25) reveals wo noeworhy feaures of real habi dynamics under nominal habi formaion: Firs, he evoluion of he real habi level becomes sochasic because of he uncerainy inheried from unexpeced inflaion. Second, expeced inflaion eners he drif erm, which implies ha he real habi level in his case is eroded by inflaion and herefore decays faser han ha in he case of real habi formaion. 9 In wha follows, we refer o β α) as habi srengh. Bu, i is imporan o noe ha he smaller β α), he sronger he habi formaion preference. 11

13 3 Soluions 3.1 Real Habi Formaion Solving he porfolio and consumpion opimizaion problems formulaed in Secion 2 is far from rivial, because linear habi formaion produces srong pas dependence and renders he uiliy funcion no ime separable. We follow Schroder and Skiadas 2002) and Munk 2008) in finding he soluions. Schroder and Skiadas 2002) show ha he opimal porfolio choice models wih habi formaion in a given financial markes is closed linked o he corresponding models wihou habi formaion in a financial marke wih a habi-adjused price kernel. Applying his relaion, Munk 2008) derives a general characerizaion of he opimal porfolio and consumpion sraegy and sudies he quaniaive effecs of habi formaion in some concree seings. Under real habi formaion, we exend Munk 2008) by incorporaing inflaion risk and examining how i affecs he opimal porfolio sraegy in boh complee and incomplee marke seings. We firs presen wo auxiliary processes, f and g, which are used o characerize he soluions under real habi formaion. The process f is defined by f = E [ˆ T e β α)s ) m ] ˆ T s ds = e β α)s ) p, s)ds. 28) m If c s = h s for all s, fuure real habi levels depreciae a a rae of β α). Hence, f can be hough of as he ime marke price of a bond paying coninuous real coupons which are declining a he decay rae of real habi levels and h f is he cos of ensuring ha fuure real consumpion never falls below he curren real habi. The process g is defined by, g = E [ˆ T ) ] 1 1 e δ/γ)s ) ms γ 1 + αfs ) 1 1 γ ds. 29) m As 1 + αf) can inerpreed as he shadow price of one uni of consumpion oday, g capures he effecs of boh he habi formaion via f) and he fuure invesmen opporuniies via m) on he expeced uiliy. I should be noed ha for γ > 1, boh f and g decrease wih β α). 12

14 We wrie he dynamics of f and g as df = f [ µf d + σ fdz ], 30) dg = g [ µg d + σ gdz ], 31) where 0, g/ rr, ) σ r, 0, 0), 32) gr, ) σ g = σ f = 0, T B, s)e β α)s ) p, s)ds σ r, 0, 0), 33) e β α)s ) p, s)ds T and µ f and µ g are some adaped processes. Equaion 33) shows ha under real habi formaion, he volailiies of f and g are driven solely by he ineres rae risk. This sems from he fac ha real zero-coupon bonds, which consiue f, only carry exposure o ineres rae risk and his exposure is passed on o g hrough f. Theorem 1 characerizes he opimal sraegy in erms of he soluion of a one dimensional, second order PDE for g. Theorem 1. Assume ha w 0 h 0 f 0. The indirec uiliy is and gr, ) solves he PDE, J = g γ w h f ) 1 γ 1 γ g r, ) + κ r r ) ) ) g σ r φ r γ r r, ) g 2 σ2 r r, ) r αfr, )) 1 1 γ δ = γ ) r + γ 1 ) γ 2γ 2 φ ρφ gr, ) 34) 35) wih he erminal condiion gr T, T ) = 0. The opimal real consumpion sraegy is c = h αf ) 1 γ w h f g. 36) 13

15 The opimal porfolio sraegy, x = x S, xt 1 N, xt 2 x = w h f w = w h f w N, xt 3 I ), is given by 1 γ σ ) 1 φ) + w h f σ ) 1 σ w g + h f σ ) 1 σ w f + σ ) 1 ξ 1 γ Σ 1 Λ + w h f 1 1 ) w γ Σ 1 σρˆσ g + ξ) + h f Σ 1 σρσ w f + ξ), 37) where w is he real wealh process induced by he opimal sraegy, and ) h is he real habi level induced by he opimal real consumpion sraegy. ˆσ g = σ g. Σ = σρσ is he variance-covariance marix of he nominal asse reurns and σ ) 1 ξ represens he vecor of covariances beween he asse reurns and inflaion. γ γ 1 The condiion w 0 h 0 f 0 ensures ha he iniial wealh of he invesor can susain he minimum consumpion level in he fuure. As shown in Appendix A, we firs derive he soluion of he dual model wihou habi formaion, which is closely relaed o he model of Brennan and Xia 2002) and hen ransform i o he soluion of he primal model wih habi formaion by applying he resuls of Schroder and Skiadas 2002) o he case wih inflaion risk. The opimal consumpion in 36) conains wo componens: he curren habi level and a ime and sae-dependen fracion of he free wealh w h f. Since boh f and g decrease wih β α) for γ > 1, he marginal propensiy o consume 1+αf ) 1/γ /g and he consumpion rae increase wih β α), implying ha as he habi srengh declines he invesor ends o consume more ou of her wealh. As f and g have no loadings on boh expeced and unexpeced inflaion risk facors, he opimal consumpion sraegy is unaffeced by inflaion risk. Equaion 37) expresses he opimal porfolio as he sum of hree porfolios: a myopic porfolio ha invess in he nominal mean-variance angency porfolio represened by Σ 1 Λ, a hedge porfolio ha provides hedge agains variaion of fuure invesmen opporuniies in he economy modified by he presence of habi formaion, and a subsisence porfolio ha ensures fuure minimum consumpion. As he presence of habi formaion induces he invesor o se aside a fracion of wealh for fuure minimum consumpion sream, he free wealh is reduced o w h f, which dampens boh he myopic demand and he hedge demand. In addiion o his leverage effec, habi formaion affecs he hedge demand also hrough σ g. Equaion 64) in Appendix A shows 14

16 ha he habi-adjused pricing kernel, which deermines he invesmen opporuniies in he presence of habi formaion, involves f. Therefore, he opimal hedge agains variaions in fuure invesmen opporuniies mus ake ino accoun he changes in he cos of ensuring he minimum consumpion level. Comparison wih Munk 2008) reveals ha under real habi formaion, he effecs of inflaion risk on he opimal porfolio sraegy are very small: i only induces a hedge agains unexpeced inflaion, which corresponds o he erm σ ) 1 ξ. This is a direc consequence of no ineracion beween habi persisence and expeced inflaion risk under real habi formaion: since σ f and σ g are unaffeced by inflaion risk, boh he hedge porfolio and he subsisence porfolio carry exposure o inflaion risk only hrough he hedge agains unexpeced inflaion, which is consisen wih Brennan and Xia 2002). Turning o he incomplee case wih only one nominal bond, we follow he approach aken in De Jong 2008) o derive he opimal porfolio sraegy. The raionale behind he approach is o minimize a pre-specified norm of he difference beween opimal and feasible wealh dynamics. Theorem 2 characerizes he soluion. Theorem 2. The opimal porfolio sraegy in he case of one nominal bond x = x S, xt 4 N ), is given by x = w h f w = w h f w 1 γ Σ 1 I 1 γ Σ 1 I σ I ρ φ) + w h f Λ I + w h f w w Σ 1 I 1 1 γ σ I ρσ g + h f w ) Σ 1 I Σ 1 I σ I ρˆσ g + ξ) + h f w σ I ρσ f + Σ 1 σ I ρξ I Σ 1 I σ I ρσ f + ξ), 38) where Σ I = σ I ρσ I is he variance-covariance marix of he nominal asse reurns. 3.2 Nominal Habi Formaion In his subsecion, we urn o nominal habi persisence, which is formed based on he households previous nominal consumpion. The individual s porfolio and consumpion opimizaion problem can be reformulaed as max C,x) A [ˆ T E e 0 C H δ Π ) 1 γ 1 γ d ] 39) 15

17 where H is he nominal habi level defined by, H = H 0 e β + α ˆ 0 e β s) C s ds 40) Once again, we presen he soluion in erms of wo auxiliary processes denoed by ˆf and ĝ, respecively. The process ˆf is defined by ˆf = E [ˆ T e β α)s ) m ] ˆ T s/m ds = e β α)s ) P, s)ds. 41) Π s /Π If C s = H s for all s, fuure nominal habi levels depreciae a a rae of β α). Hence, ˆf can be hough of as he ime marke price of a bond paying coninuous nominal coupons which are declining a he decay rae of nominal habi levels and H ˆf is he cos of ensuring ha fuure nominal consumpion never falls below he curren nominal habi. Comparison beween 28) and 41) shows ha he habi bond under nominal habi formaion is comprised of nominal zero-coupon bonds raher han inflaion-indexed zero-coupon bonds. I is worh noing ha under he calibraed parameer values shown below, f > ˆf for any < T, which implies ha he nominal habi bond is cheaper han he real habi bond. This can be explained by he fac ha in he case of nominal habi formaion, he real habi level is allowed o be eroded by inflaion and depreciaes faser. Since he values of fuure coupons decline, he price of he habi bond drops. The process ĝ is defined by, ĝ = E [ˆ T ) ] 1 1 e δ/γ)s ) ms γ 1 + α m ˆf ) 1 1 γ s ds. 42) ĝ capures he effecs of boh he habi formaion via ˆf) and he fuure invesmen opporuniies via m) on he expeced uiliy. I should be noed ha for γ > 1, boh ˆf and ĝ decrease wih β α) and ĝ < g. We define he dynamics of ˆf and ĝ as d ˆf = ˆf [ µ ˆf d + σ ˆf dz ] dĝ = ĝ [ µĝ d + σ ĝdz ] 43) 44) 16

18 where σĝ = 0, ĝ/ rr, π, ) σ r, ĝr, π, ) ĝ/ πr, π, ) σ π, 0), 45) ĝr, π, ) σ ˆf = 0, T B, s)e β α)s ) P, s)ds σ r, T D, s)e β α)s ) P, s)ds σ π, 0). e β α)s ) P, s)ds e β α)s ) P, s)ds T and µ ˆf and µĝ are some adaped processes. Equaion 46) shows ha under nominal habi formaion, he volailiies of ˆf and ĝ are driven by boh he ineres rae risk and expeced inflaion risk. This is because nominal zero-coupon bonds, which consiue ˆf, are exposed o boh risk facors and ĝ inheri hese exposures from ˆf. Theorem 3 characerizes he opimal sraegy in erms of he soluion of a wo dimensional, second order PDE for ĝ. Theorem 3. Assume ha w 0 h 0 ˆf0. The indirec uiliy is T 46) and ĝr, π, ) solves he PDE, δ γ ) γ J = ĝγ w h ˆf ) 1 γ 1 γ r + γ 1 ) 2γ 2 φ ρφ ĝr, π, ) = ĝ + κ r r ) ) γ + θ π π ) γ r, π, ) α ˆfr, π, )) 1 1 γ ) ĝ σ r φ r r r, π, ) ĝ 2 σ2 r r, π, ) r2 π r, π, ) ĝ 2 σ2 π r, π, ) π2 ) σ π φ π ) ĝ 47) 48) wih he erminal condiion ĝr, π, T ) = 0. The opimal real consumpion sraegy is c = h α ˆf ) 1 γ w h ˆf ĝ. 49) 17

19 The opimal porfolio sraegy, x = x S, xt 1 N, xt 2 x = W H ˆf 1 W γ σ ) 1 φ) + W + H ˆf σ ) 1 σ W ˆf + W = w h ˆf w N, xt 3 I H ˆf σ ) 1 σĝ ) W H ˆf σ ) 1 ξ W 1 γ Σ 1 Λ + w h ˆf w ), is given by 1 1 ) Σ 1 σρˆσĝ + ξ) + h ˆf γ w Σ 1 σρσ ˆf. 50) We assume ha γ > 1 and focus on he comparison beween he opimal sraegy in wo cases. The relaion ˆf < f implies ha he value of he habi bond declines. This is a resul of erosion by inflaion: as he inflaion drives he real habi level o decay faser, he habi bond price goes down and herefore less money is needed o ensure fuure subsisence consumpion. The relaion ĝ < g, ogeher wih ˆf < f implies ha he marginal propensiy o consume 1 + αf ) 1/γ /g and he consumpion rae increase. On he oher hand, here are some major changes o he opimal porfolio sraegy. Firs, he reducion in he value of he habi bond leads o weaker leverage effecs and lower subsisence demand. Therefore, he speculaive porfolio expands while he subsisence porfolio shrinks. However, i is no possible o deermine analyically how he hedge porfolio changes beween he wo cases, because habi persisence influences he hedge porfolio no only hrough he leverage effec bu also hrough is effec on invesmen opporuniies and he laer effec has o be evaluaed numerically. Second, he inflaion risk has much larger impac on he opimal porfolio han i does under he real habi formaion. The explanaion for his bigger effec is ha he habi bond ˆf, which deermines no only he risk profile of he subsisence porfolio bu also fuure invesmen opporuniies, is comprised of nominal zero-coupon bonds and herefore bears expeced inflaion risk. As a resul, boh σ f and σ g become subjec o expeced inflaion risk, hereby subsanially increasing he inflaion risk exposures of boh he hedge porfolio and he subsisence porfolio. Third, he opimal porfolio no longer akes full insurance agains unexpeced inflaion risk; he subsisence porfolio is lef uninsured. This is because under nominal habi formaion he real habi level is permied o be reduced by inflaion and herefore has a perfecly negaive correlaion wih realized inflaion, which is clearly shown in 27). Turning o he incomplee case wih only one nominal bond, we once again follow 18

20 he approach aken in De Jong 2008) o find he opimal porfolio sraegy. Theorem 4 characerizes he soluion. Theorem 4. The opimal porfolio sraegy in he case of one nominal bond x = x S, xt 4 N ), is given by x = w h ˆf w + h ˆf w = w h ˆf w 1 γ Σ 1 I Σ 1 I 1 σ I ρ φ) + w h ˆf w σ I ρσ ˆf + w h ˆf γ Σ 1 I w Λ I + w h ˆf w Σ 1 I σ I ρξ ) 1 1 γ Σ 1 I σ I ρσĝ Σ 1 I σ I ρˆσĝ + ξ) + h ˆf w Σ 1 I σ I ρσ ˆf. 51) 4 Numerical Illusraions In his secion, we carry ou some numerical experimens o compare he effecs of inflaion risk and habi persisence on he opimal consumpion and porfolio sraegy under differen ypes of habi formaion. In he benchmark case, we consider an invesor wih risk aversion parameer γ = 3, a 30-year horizon, and a ime preference rae δ = Iniial wealh, iniial habi level and iniial price level are se o W 0 = 10000, h 0 = 400 and Π 0 = 1, respecively. Habi parameers are aken o be α = 0.3 and β = 0.4. To calibrae he model, we follow he parameer esimaes repored in Brennan and Xia 2002), which are shown in Table 1. Noe ha we assume ha unexpeced inflaion is uncorrelaed wih sock reurns, real ineres rae and expeced inflaion, so ha only inflaion-index bonds can be used o hedge agains unexpeced inflaion. In he cases wih complee marke, we assume ha he here are hree bonds available o he invesor, namely an 1-year nominal bond T 1 = 1), an 10-year nominal bond T 2 = 10) and an 1-year inflaion-indexed bond T 3 = 1). Resuls under real habi formaion are obained by solving he one dimensional PDE 35) for g using a Crank- Nicolson finie difference scheme, wih 500 real ineres rae subinervals and 1000 ime seps. In conras, resuls under nominal habi formaion are obained by solving he wo dimensional PDE 48) for g using an explici finie difference scheme, wih 50 real ineres rae subinervals, 50 expeced inflaion subinervals and 1000 ime seps. We can calculae he loadings on he innovaions in differen risk facors o decompose 19

21 Table 1: Parameer values Parameer Value Sock reurn process: ds/s = R f + λ S σ S )d + σ S dz S σ S λ S Real ineres rae process: dr = κ r r)d + σ r dz r r κ σ r λ r Expeced inflaion process: dπ = θ π π)d + σ π dz π π θ σ π λ π dπ Realized inflaion process: Π = πd + ξ Sdz S + ξ r dz r + ξ π dz π + ξ u dz u ξ S 0 ξ r 0 ξπ 0 ξ u dm Pricing kernel process: m = rd + φ Sdz S + φ r dz r + φ π dz π + φ u dz u φ S φ r φ π φ u 0 Correlaions ρ Sr ρ Sπ ρ rπ This able shows he parameer values aken from Brennan and Xia 2002). he risk exposure of he opimal porfolio: L S = x S, L r = x T 1 N B r0, T 1 ) x T 2 N B r0, T 2 ) x T 3 I L π = x T 1 N B π0, T 1 ) x T 2 N B π0, T 2 ) + x T 3 L u = x T 3 I. I B π 0, T 3 ) ξ r ξ r σ r 52) ), 53) σ r, 54) The loadings on he innovaions in he equiy risk and unexpeced inflaion risk coincide 55) 20

22 Table 2: Opimal porfolio sraegy in complee marke under real habi formaion a) For differen habi srengh β α) β α L r L π x S x 1 N x 10 N x 1 I No habi b) For differen iniial habi level h 0 h 0 L r L π x S x 1 N x 10 N x 1 I c) For differen invesmen horizon T T L r L π x S x 1 N x 10 N x 1 I The able shows he opimal porfolio sraegy in complee marke under real habi formaion. L r L π) is he sensiiviy of he opimal porfolio o innovaions in r π). x S, x 1 N x10 N and x1 I are he fracions of wealh invesed in he sock, he 1-year nominal bond, he 10-year nominal bond and he 1-year inflaion-indexed bond respecively. The parameer values are as follows: α = 0.3, β = 0.4 varying in panel a)), h 0 = 400 varying in panel b)) and T = 30 varying in panel c)). The curren ineres rae and curren expeced inflaion are se a he uncondiional means r and π, respecively. Oher parameers are shown in Table 1. wih he opimal sock allocaion and opimal inflaion-indexed bond allocaion, because hese wo risks are borne solely by he sock and inflaion-indexed bond, respecively. Hence, in wha follows, we don repor L S and L u. I should be noed ha he sock is only conained in he myopic porfolio, because i is appropriaed neiher for hedging purpose nor for ensuring he fuure subsisence consumpion. Table 2 summarizes he opimal porfolio sraegy in complee marke under real habi formaion. As shown in panel a), he inroducion of habi formaion remarkably reduces he equiy exposure and expeced inflaion risk exposure and his effec is more pronounced for sronger habi formaion, which is associaed wih smaller β α). These 21

23 lower risk exposures can be aribued o he reducion of he free wealh, because under real habi formaion he equiy risk and expeced inflaion risk are only aken by he myopic porfolio. In conras, habi srengh has differen effecs on he ineres risk exposure of differen porfolios. While he leverage effec reduces he myopic demand and hedge demand, he expansion of he subsisence demand driven by larger habi srengh leads o higher ineres rae loadings. Moreover, habi srengh can affec he hedge porfolio also by changing he volailiy of habi-adjused invesmen opporuniies. The observaion ha he ineres rae sensiiviy is decreasing in habi srengh indicaes ha he leverage effec dominaes. The lower ineres rae and inflaion risk exposures associaed wih weaker habi persisence reduce he absolue demand for boh nominal bonds. Panel b) shows ha as iniial habi level rises, he opimal porfolio akes less ineres rae risk exposure and inflaion risk exposure and reduces he holdings of he sock and he wo nominal bonds because of he pure leverage effec. Panel c) illusraes he imporance of invesmen horizon. Equiy exposure and inflaion exposure are decreasing in he invesmen horizon, since longer horizon subsanially increases he price of he habi bond and generaes sronger leverage effec. On he conrary, he opimal ineres rae loadings rise wih invesmen horizon. The reason is ha he volailiies of boh he habi bond σ f and fuure invesmen opporuniies σ g increase sharply, which induces much larger subsisence demand and hedge demand and offses he leverage effec. As a resul, he absolue porfolio shares in boh nominal bonds are higher for longer horizon. These observaions sand in sark conras o Brennan and Xia 2002), who find limied horizon effec on he opimal ineres rae risk exposure abou five years) and no horizon effecs on he opimal equiy exposure and opimal inflaion risk exposure. Finally, he opimal inflaion-indexed bond holding is independen of habi parameers and invesmen horizon, because he opimal porfolio simply akes a full insurance agains he unexpeced inflaion risk, which corresponds o he erm σ ) 1 ξ. Table 3 repors he opimal porfolio sraegy wih one nominal bond under real habi formaion. From panel a) we can see ha he presence of habi persisence in preference drives down he demand for boh risky asses because of he leverage effec. While boh he sock holding and bond holding increase wih habi srengh, he whole porfolio ils owards he bond. This is a resul of higher hedge demand and subsisence demand induced by sronger habi persisence. Panel b) shows ha higher iniial habi level dampens he risky invesmen because of he reducion in free wealh and makes 22

24 Table 3: Opimal porfolio sraegy wih one nominal bond under real habi formaion a) For differen habi srengh β α) Bond mauriy No habi 1 year x S x N x S x N year x S x N x S x N b) For differen iniial habi level h 0 Bond mauriy year x S x N x S x N year x S x N x S x N c) For differen invesmen horizon T Bond mauriy 1 year 5 years 10 years 20 years 30 years 1 year x S x N x S x N year x S x N x S x N The able shows he opimal porfolio sraegy wih one nominal bond under real habi formaion. x S /x N is he socko-bond raio. We consider 1-year nominal bond and 10-year nominal bond, respecively. x S x N ) is he demand for he sock he nominal bond). The parameer values are as follows: α = 0.3, β = 0.4 varying in panel a)), h 0 = 400 varying in panel b)) and T = 30 varying in panel c)). The curren ineres rae and curren expeced inflaion are se a he uncondiional means r and π, respecively. Oher parameers are shown in Table 1. 23

25 he opimal porfolio lean owards he bond because he sock can be used neiher for hedging purpose nor ensuring fuure minimum consumpion. Panel c) illusraes he horizon effec. I urns ou ha while he sock demand decreases wih invesmen horizon, he bond demand increases, since longer horizon generaes larger value of he habi bond and higher volailiy of fuure invesmen opporuniies. Comparison beween Table 2 and Table 3 shows ha for given parameer values, he opimal porfolio share in he sock is higher in he incomplee marke han i is in he complee marke and he difference is larger for he case wih long-erm bond. The higher demand for he sock sems from he fac ha dz S is calibraed o be negaively correlaed wih boh dz r and dz π. As he bonds have negaive loadings on dz r and dz π, he correlaion beween he nominal reurns beween he sock and he bonds is posiive, which dampens he sock invesmen in he myopic porfolio. Unrepored resuls show ha in he case of one bond, he opimal porfolio akes lower ineres risk exposure bu higher inflaion risk exposure han does he opimal porfolio in he complee marke. Because he correlaion beween dz S and dz r is much higher han ha beween dz S and dz π, he decreased correlaion effec associaed wih lower ineres rae exposure ouweighs he increased correlaion effec associaed wih higher inflaion exposure. Moreover, since he ineres risk exposure decreases wih he bond mauriy in he incomplee marke case, he correlaion effec diminishes accordingly. Now we urn o he opimal porfolio sraegy under nominal habi formaion. Table 4 shows he resuls for differen habi parameers and invesmen horizon. Some ineresing changes emerge as compared o he opimal porfolio sraegy under real habi formaion shown in Table 2. Firs, as shown in panel a), he presence of habi persisence induces larger inflaion risk exposure and his effec inensifies wih habi srengh, which is in sharp conras o he decreasing inflaion risk exposure in he real habi case. Moreover, for any given habi srengh, he opimal porfolio under nominal habi formaion has much larger loadings on he inflaion risk han i does under real habi formaion. These disincions are consequences of differen risk profiles of he hedge porfolio and he subsisence porfolio under differen ypes of habi formaion: while hese wo porfolios under real habi formaion are only subjec o he ineres rae risk, hose under nominal habi formaion carry he expeced inflaion risk hrough he habi bond ˆf, because ˆf is comprised of nominal zero-coupon bonds raher han inflaion-indexed zero-coupon bonds. As a resul, he impac of he inflaion risk on he opimal porfolio is subsanially amplified. Second, alhough he equiy exposure 24

26 Table 4: Opimal porfolio sraegy in complee marke under nominal habi formaion a) For differen habi srengh β α) β α L r L π x S x 1 N x 10 N x 1 I No habi b) For differen iniial habi level h 0 h 0 L r L π x S x 1 N x 10 N x 1 I c) For differen invesmen horizon T T L r L π x S x 1 N x 10 N x 1 I The able shows he opimal porfolio sraegy in complee marke under nominal habi formaion. L r L π) is he sensiiviy of he opimal porfolio o innovaions in r π). x S, x 1 N x10 N and x1 I are he fracions of wealh invesed in he sock, he 1-year nominal bond, he 10-year nominal bond and he 1-year inflaion-indexed bond respecively. The parameer values are as follows: α = 0.3, β = 0.4 varying in panel a)), h 0 = 400 varying in panel b)) and T = 30 varying in panel c)). The curren ineres rae and curren expeced inflaion are se a he uncondiional means r and π, respecively. Oher parameers are shown in Table 1. and ineres risk exposure remains decreasing in habi srengh, hey ge higher as compared o he real habi case due o he sronger leverage effec: inflaion erodes he habi bond price, hereby leaving more free wealh. Third, he opimal demand for he inflaion-index bond becomes dependen on he habi parameers and invesmen horizon, because he subsisence porfolio is lef uninsured agains unexpeced inflaion. As he habi bond price increases, which is associaed wih sronger habi persisence and longer horizon, he opimal inflaion-index bond holding declines. Fourh, panel c) shows ha he horizon effec on he inflaion risk sensiiviy is reversed. This is also due o he bigger impac of he inflaion risk on he opimal porfolio sraegy. 25

27 Table 5: Opimal porfolio sraegy wih one nominal bond under nominal habi formaion a) For differen habi srengh β α) Bond mauriy No habi 1 year x S x N x S x N year x S x N x S x N b) For differen iniial habi level h 0 Bond mauriy year x S x N x S x N year x S x N x S x N c) For differen invesmen horizon T Bond mauriy 1 year 5 years 10 years 20 years 30 years 1 year x S x N x S x N year x S x N x S x N The able shows he opimal porfolio sraegy wih one nominal bond under nominal habi formaion. We consider 1-year nominal bond and 10-year nominal bond, respecively. x S x N ) is he demand for he sock he nominal bond). x S /x N is he sock-o-bond raio. The parameer values are as follows: α = 0.3, β = 0.4 varying in panel a)), h 0 = 400 varying in panel b)) and T = 30 varying in panel c)). The curren ineres rae and curren expeced inflaion are se a he uncondiional means r and π, respecively. Oher parameers are shown in Table 1. 26

28 Table 5 repors he opimal porfolio sraegy wih one nominal bond under nominal habi formaion. Comparison beween Table 3 and Table 5 reveals ha he demand for boh risky asses grows because of he increase in free wealh. The sock-o-bond raio is lower under nominal habi formaion han i is under real habi formaion, implying ha he composiion of he porfolio leans more owards he bond in he former seing. This il sems from he higher hedge demand and subsisence demand induced by he inflaion risk. Finally, we invesigae he expeced wealh and expeced consumpion under differen ypes of habi formaions, which are illusraed in Figure 1. Panel a) shows ha all hree ypes of invesors accumulae wealh in he early periods and decumulae wealh in he lae periods. The accumulaion is slowes for he non-habi invesor, modes for he real habi invesor and fases for he nominal habi invesor. Compared wih he habi invesors, he non-habi invesor does no have o reserve a fracion of wealh for ensuring fuure subsisence consumpion and enjoy higher consumpion in he early periods, which is clearly displayed in he righ graph. In he lae periods, however, he consumpion of he habi invesors exceeds ha of he non-habi invesor because of he higher saving rae generaed by habi formaion. The nominal habi invesor has higher wealh and consumpion han he real habi invesor over he whole life-cycle, boh because he nominal habi invesor has more free wealh o inves in socks and benefi more from equiy risk premium and because she has a higher marginal propensiy o consume on average han he real habi invesor. The wealh accumulaion phase arises from he equiy risk premium implied by he esimaes in Brennan and Xia 2002), which seems unrealisically high in he curren marke circumsances. Therefore, i is of ineres o sudy he case wih lower equiy risk premium, which is shown in panel b). When equiy risk premium is se a a lower level, he wealh of hree ypes of invesors decumulaes over he whole life-cycle. Ineresingly, in face of worse marke condiions, he habi invesors begin wih higher consumpion han heir counerpar, bu reduce spending for some periods, because hey have o drive down he habi level and increase saving o ensure ha fuure habi consumpion can be susained. 27