Dynamic portfolio and mortgage choice for homeowners

Transcription

1 Oo van Hemer Frank de Jong Joos Driessen Dynamic porfolio and morgage choice for homeowners Discussion Papers May 2005

2 Dynamic porfolio and morgage choice for homeowners Oo van Hemer Frank de Jong Joos Driessen Universiy of Amserdam May 2, 2005 Absrac We invesigae he impac of owner-occupied housing on financial porfolio and morgage choice under sochasic inflaion and real ineres raes. To his end we develop a dynamic framework in which invesors can inves in socks and bonds wih differen mauriies. We use a coninuous-ime model wih CRRA preferences and calibrae he model parameers using daa on inflaion raes and equiy, bond, and house prices. For he case of no shor-sale consrains, we derive an implici soluion and idenify he main channels hrough which he housing o oal wealh raio and he horizon affec financial porfolio choice. This soluion is used o inerpre numerical resuls ha we provide when he invesor has shor-sale consrains. We also use our framework o invesigae opimal morgage size and ype. A moderaely risk-averse invesor prefers an adjusable-rae morgage (ARM), while a more risk-averse invesor prefers a fixedrae morgage (FRM). A combinaion of an ARM and an FRM furher improves welfare. Choosing a subopimal morgage leads o uiliy losses up o 6%. The auhors would like o hank he Case-Shiller-Weiss company for providing he house price daa. This paper has benefied from discussions wih Francisco Gomes, Alex Michaelides, Yihong Xia and seminar paricipans a he Universiei van Amserdam, he conference on Porfolio Choice and Invesor Behavior organised by he Sockholm Insiue for Financial Research, and London School of Economics. Conac address for all auhors: Universiei van Amserdam, Finance Group, Faculy of Economics and Economerics, Roeerssraa 11, 1018 WB Amserdam, The Neherlands. hemer@inbergen.nl; fdejong@uva.nl; J.J.A.G.Driessen@uva.nl.

3 1 Inroducion For homeowners a house is no only a place o live in, bu also a risky financial invesmen. Ofen he owner-occupied house is a large, if no he larges, asse for a household. I can herefore have a major impac on financial porfolio choice. Moreover, homeowners face he decision how o finance he house. The financial porfolio and morgage choice are boh imporan elemens of overall financial planning and are herefore closely relaed. Even hough many invesors own he house hey live in, mos lieraure on financial porfolio choice ignores housing and morgages alogeher. This paper is he firs (o he bes of our knowledge) o provide an inegraed analysis of a homeowner s opimal financial porfolio and morgage choice. We ake a long-erm invesmen perspecive, where he invesor derives uiliy from he services supplied by he house and from he consumpion of oher goods which is equal o oal erminal wealh. We inerpre curren invesor s wealh as including human capial, i.e. we assume ha fuure labor income is capialized and invesed in he house and he available menu of financial asses. The housing invesmen is aken as fixed and given, while posiions in financial asses are rebalanced dynamically. Our seup incorporaes ha housing differs from financial asses in a leas four crucial respecs. Firs, he oal amoun of housing is ofen dicaed by consumpion moives raher han invesmen moives. Second, he housing invesmen is far less liquid han financial invesmens because of high moneary and effor coss involved wih moving. Third, he house can serve as collaeral for a morgage loan up o he marke value of he house. Fourh, he expeced housing reurn will be lower han on a hypoheical pure financial asse wih comparable risk characerisics, because he marke will recognize ha a house also provides housing services. Anoher imporan par of our seup is a realisic model for he erm srucure of ineres raes, wih expeced inflaion and real ineres rae as facors. Our model hus exends he Brennan and Xia (2002) model by including housing. The model gives a raionale for holding nominal bonds wih differen mauriies. Imporanly, i also allows us o invesigae he implicaions of differen ypes of morgage loans. In addiion o he wo erm srucure facors, we model unexpeced inflaion, house risk and sock marke risk, leading o a oal of five sources of uncerainy. This srucure enables us o realisically examine he ineracion of financial asse prices and he house price. We also invesigae morgage choice. Welinkheopimalmorgagesizeandypeohecoefficien of relaive risk aversion, house 1

4 size and horizon, and sudy he inerplay of financial porfolio choice and morgage choice. This paper relaes o several srands in he porfolio choice lieraure. Firs, Brennan and Xia (2002) and Campbell and Viceira (2001) illusrae he imporance of bonds for a long-erm invesor. Boh use a wo-facor model similar o ours for he nominal ineres rae. A long-erm invesor holds bonds no only o exploi he risk premium, bu also o hedge changes in he invesmen opporuniy se. Our paper exends his work by aking ino accoun he impac of an owner-occupied house on porfolio choice and by opimally choosing he morgage size and ype. Anoher srand of lieraure focuses on he housing and financial porfolio choice in a saic one-period mean-variance seing (Brueckner (1997) and Flavin and Yamashia (2002)). These aricles focus on he so-called housing consrain : he invesmen in housing mus be a leas as large as he consumpion of housing. The classical muual fund separaion heorem is no longer valid in he presence of he housing consrain. In conras o he saic one-period mean-variance seing, we use CRRA preferences, allow for dynamic sraegies, and focus on long horizons. More imporanly, hese papers ignore he long erm real ineres rae and inflaionrisks,andhereforehavelileroleforbondinvesmens.in addiion, he one period framework gives no advice for morgage choice. A number of papers have exended he lieraure on porfolio choice over he life cycle by examining he opimal house size decision. Examples are Cocco (2004), Hu (2003), and Yao and Zhang (2003). The laer wo papers also model he house enure decision. Our paper can be seen as complemening his lieraure. We have a much richer asse menu, we sudy he choice for morgage ype, and implemen more sophisicaed modelling of he ineracion of he reurn on he house wih financial asse reurns. We do no, however, model life cycle feaures like he labor income profile or labor income risk. Finally, his paper relaes o Campbell and Cocco (2003) who examine he choice beween a Fixed-Rae Morgage (FRM) and an Adjusable-Rae Morgage (ARM). They consider a model which has persisen shocks o he expeced inflaion rae only. In his case an FRM has a variable real value and herefore has wealh risk. An ARM has a fixed real value, bu has income risk ha is imporan in heir life-cycle seup wih an explici borrowing consrain. A posiive shock o ineres raes may cause a emporary reducion in consumpion, which can be very cosly in uiliy erms. Our morgage analysis differs from Campbell and Cocco (2003) in wo imporan ways. Firs, in our model seup, here is no income risk bu wealh risk is more complex since we incorporae persisen shocks o 2

5 boh real ineres raes and inflaion. This makes he choice beween an ARM and FRM nonrivial. Second, we model a much richer asse menu wih socks and bonds of various mauriy, and provide an inegraed analysis of porfolio and morgage choice, where he morgage is modeled as a shor posiion in one or several bonds. Our analysis herefore sheds ligh on a differen, bu imporan, deerminan of morgage choice. The main resuls of he paper can be summarized as follows. In he absence of shorsale consrains, we are able o derive an implici analyical expression for he invesor s opimal financial porfolio. This porfolio is composed of (i) he nominal mean-variance angency porfolio; (ii) a porfolio ha mos closely resembles an inflaion-indexed bond; and (iii) a porfolio ha bes offses he risk of he illiquid house. The firs wo porfolios were also derived by Brennan and Xia (2002). The size of he posiion in he hree porfolios depends on he coefficien of relaive risk aversion, he financial wealh raio and he effecive housing o oal wealh raio. Because he provided housing services make a house no a pure financial asse, and because he posiion in he house is fixed, i is no he curren marke value of he house ha appears in he expressions for opimal porfolio choice, bu a value we refer o as effecive housing wealh. The hird componen can be seen as a posiion offseing he effecive housing exposure. Our analysis shows ha house ownership affecs he opimal financial porfolio in several ways. Firs, i gives rise o a porfolio hedging he house risk. Second, i deermines a facor o leverage he financial porfolio weighs in order o mainain an appropriae absolue financial risk exposure. Third, i deermines he effecive o oal wealh raio. We show ha horizon effecs in he porfolio choice arise due o a horizon dependen hedge agains real ineres changes and due o he effecive o oal wealh raio which decreases wih he invesmen horizon. In he more realisic case ha he invesor can only hold nonnegaive posiions, we use numerical echniques o calculae he opimal porfolio. The inerpreaion of he numerical resuls is grealy enhanced by he lessons learned from he no-consrains case. We esimae he model parameers using daa on equiy, bond, and house prices, and sudy he opimal financial porfolios for differen invesor horizons and house sizes. As he house size increases, wo effecs deermine he change in financial porfolio weighs. Firs, in order o mainain an approximaely consan absolue sock marke exposure, he financial porfolio weighs are levered up. Second, since he risk-averse invesor is exposed o undiversified risk of he fixed house posiion, he will decrease his exposure o sock and bond marke risk. In case of shor-sale consrains, an addiional effecishasockand bond posiions compee in erms of heir hedging and reurn benefis. As a resul, he 3

6 weighed average duraion of he wo bonds and cash posiion increases wih he housing o oal wealh raio, which is consisen wih a desire for a leveraged exposure o real ineres rae and expeced inflaion rae shocks. The horizon effecs can be undersood from he fac ha he effecive o oal wealh raio decreases subsanially wih horizon. For moderaely risk-averse invesors his resuls in a decrease of he fracion invesed in socks and for more risk-averse invesors in a decrease of he fracion invesed in long-erm bonds wih horizon. We show ha neglecing house ownership in porfolio choice can lead o uiliy losses up o 2%, which illusraes he imporance of incorporaing house ownership in calculaing opimal financial porfolios. Finally, we allow for morgage loans, and invesigae he choice beween fixed-rae and adjusable-rae morgages. We find ha a moderaely risk-averse invesor prefers an ARM, in order o avoid paying he risk premium on long-erm bonds. A more risk-averse invesor is more concerned abou hedging inflaion and ineres rae risk, and raher chooses an FRM. An even beer morgage for his invesor is a hybrid morgage, which is a combinaion of an FRM and an ARM. Choosing a subopimal morgage leads o uiliy losses up o 6% for long horizons. This illusraes ha he morgage choice should play a cenral role in a household s financial planning. The srucure of he paper is as follows. Secion 2 presens he invesor s porfolio allocaion problem, describes he price processes of he available asses, and discusses he opimal porfolio choice. In secion 3 we discuss he esimaion of he model parameers. Secion 4 conains our main resuls for unconsrained invesors as well as for invesors wih shor-sale consrains. We also invesigae how sensiive our resuls are o alernaive parameers for he house price process. In secion 5 we inroduce morgage loans. Secion 6 concludes. 2 Opimal asse allocaion In his secion we presen he invesor s porfolio allocaion problem. We describe he economy and he price processes of he available asses, and discuss he opimal porfolio choice. For he case wih no consrains on he size of posiions in financial asses, we are able o provide an analyical expression for he opimal invesmen in socks, bonds wih differen mauriies and cash. In he special case ha housing risk is perfecly hedgeable we can solve for he opimal invesmen in closed form. Finally, we discuss he numerical echniques o analyze he model wih shor sale consrains. 4

7 2.1 The invesor s opimizaion problem We consider opimal financial porfolio choice for an invesor from ime 0 unil ime T,his horizon. We assume ha besides financial asses, he invesor owns he house he lives in, 1 which has given size H. The house size is inerpreed as a one-dimensional represenaion of he qualiy of he house. The invesor sells his house a ime T, which could be inerpreed as he momen of reiremen. The possibiliy o sell his house or buy a second house before ime T is ignored. The nominal price of a uni of housing a ime is denoed Q. We normalize Q 0 o 1. Nominal housing wealh is denoed W H Q H. We define W F as nominal financial wealh. W0 F includes human capial, bu excludes housing wealh. Labor income risk and moral hazard issues involved in capializing labor income are ignored. We make he simplifying assumpion ha mainenance coss are capialized and paid in advance, which means hey do no play an explici role in our analysis. Taking ino accoun ha labor income is capialized, we like o hink of he housing o oal wealh raio, h, as being in he order of magniude of 0.2, and in our ables i ypically ranges from 0 o Toal nominal wealh is denoed W W F +W H W F +Q H.AimeT, he invesor uses his oal wealh for consumpion of oher goods. The real price of hese consumpion goods is chosen o be he numeraire. The nominal price level a ime is denoed as Π and we normalize Π 0 =1. We use uppercase leers for nominal variables and he corresponding lowercase leer for heir real counerpar, so oal real wealh is w = w F + w H.Following Cocco (2004), Hu (2003) and Yao and Zhang (2003) we represen preferences over housing consumpion o oher goods by he Cobb-Douglas funcion u (w T,H)= ³w ψ T H1 ψ 1 γ 1 γ = w1 γ T 1 γ ν H (1) wih ν H ψh (1 ψ)(1 γ), γ 1 ψ (1 γ). We have γ = w u ww /u w,whichishe coefficien of relaive risk aversion given a fixedposiioninhousing. 1 Owning migh be preferred o rening a house because of favorable ax reamen or fricions in he renal marke. For example, Henderson and Ioannides (1983) focus on a fricion in he renal marke which hey refer o as rener exernaliy. This exernaliy is due o moral hazard issues, like propery abuse by reners. 2 Heaon and Lucas (2000, Table V) repor housing o oal wealh raios in he range 0.1 o 0.3 while including capialised labor income, social securiy and pension benefis in he oal wealh measure. In conras, Flavin and Yamashia (2002) ignore human capial as par of oal wealh. Their housing o oal wealh raio ranges from 0.65 o

8 A [0,T] he invesor solves he porfolio allocaion problem max x(τ) A, τ T E u w F T (x)+w H T,H (2) subjec o he consrain ha wt F is financed by he sraegy for financial porfolio weighs x (τ), wihwealhaime equal o w F,andwhereA is he se of admissible financial porfolio weighs. Throughou he paper, we assume ha his se is independen of oal wealh w and he real ineres rae. We consider hree specific casesforhesea: no consrains, shor sale consrains on all asses, and he case wih a morgage up o he iniial value of he house. Four remarks regarding he invesors objecive are in place here. Firs, he model can easily be embedded ino a model where jus before ime =0he invesor also opimizes over house size. In his case he preference parameer ψ is crucial for deermining he opimal house size. Because our focus is on financial porfolio choice, we choose o condiion on a given house size (or equivalenly, housing wealh a =0) direcly. Second, defining uiliy over inerim consumpion insead of over erminal wealh is no likely o change he resuls qualiaively, bu basically reduces he effecive horizon. 3 Third, we can ignore he muliplicaive facor ν H of he uiliy funcion when solving for opimal porfolio choice x. So given he house size, uiliy is of he power uiliy form over erminal wealh. This also meanshaweonlyhaveoknowγ and no ψ and γ separaely. Fourh, hroughou we will assume ha W0 F 0, implyinghaiisalwayspossibleohavew F 0 for all [0,T], which ensures ha he invesor s problem is well defined. 2.2 Asse price dynamics We consider an economy wih five sources of uncerainy represened by innovaions in five Brownian moions. This is similar o Brennan and Xia (2002, henceforh BX), excep ha we have an exra source of uncerainy o capure idiosyncraic risk in house price changes. As saed earlier, we focus on he financial porfolio choice of a single invesor who akes 3 This is a well-known resul in he absence of an owner occupied house (see e.g. Brennan and Xia (2002) for a furher discussion). We argue ha his resul also holds in our se-up, which enhances he comparison beween his paper and e.g. Cocco (2004), Hu (2003) and Yao and Zhang (2003) who use inerim consumpion. This relies on he observaion ha also for inerim consumpion he facor ν H can be separaed in he uiliy funcion and on he asserion ha he difference in house size beween he inerim consumpion and he erminal wealh formulaion will be small. To expand on he laer, in a world wihou uncerainy i is easy o show ha he Cobb-Douglas preference srucure implies ha he opimal house size is he same in he inerim consumpion and erminal wealh cases. 6

9 price processes as given. Furhermore, we assume ha he risk premia on he sources of uncerainy are consan. The nominal sock price follows a geomeric Brownian moion ds S =(R f + σ S λ S ) d + σ S dz S, (3) where R f is he nominal ineres rae and λ S is he nominal risk premium. Boh he insananeous real riskless rae r and he insananeous expeced rae of inflaion π follow Ornsein-Uhlenbeck processes dr = κ ( r r) d + σ r dz r, (4) dπ = α ( π π) d + σ π dz π. (5) The oal expeced reurn on owner-occupied housing is he expeced house price appreciaion plus a convenience yield represening he benefis from he housing services. We refer o his convenience yield as impued ren and denoe i by r imp. We assume ha he impued ren is a consan fracion of he house value. 4 Thenominalpriceprocessforhe house price follows a geomeric Brownian moion dq Q + rimp d =(R f + σ Q λ Q ) d + σ Q dz Q. (6) where σ Q λ Q is he premium for he financial risk of exposure o he σ Q dz Q shocks. We can rewrie and orhogonalize his equaion as dq Q = R f + θ 0 λ r imp d + θ 0 dz, (7) where z =(z S,z r,z π,z v ) and z v is orhogonal o z S, z r and z π. In addiion, θ =(θ S,θ r,θ π,θ v ) 0 are he loadings on he various Brownian moions and λ =(λ S,λ r,λ π,λ v ) 0 is he vecor of nominal risk premia on he sources of uncerainy. The price level follows dπ Π = πd + σ Πdz Π = πd + ξ 0 dz + ξ u dz u, (8) where ξ =(ξ S,ξ r,ξ π,ξ v ) 0 and dz u is orhogonal o dz. 4 Flavin and Yamashia (2002) also specify he impued ren as a consan fracion of he house value, buihasaslighlydifferen inerpreaion. In heir mean-variance se-up i reflecs he moneary value of he uiliy an individual derives from he housing services. In conras, in our case i represens he reurn differenial beween he house and a (hypoheical) pure financial asse wih comparable risk characerisics, as deermined by he marke. 7

10 Defining he covariance marix of dz by à ρs,r,π 0 ρ = 0 1!, we have σ 2 Q = θ0 ρθ and σ 2 Π = ξ0 ρξ + ξ 2 u. We will someimes use he real risk premia given by λ = λ ρξ and λ u = λ u ξ u. For noaional convenience we inroduce he noaion φ =(φ S,φ r,φ π,φ v ) 0 ρ 1 λ and φ u = λ u. For an inerpreaion of hese parameers, recall from BX ha he real pricing kernel, M, evolvesas dm M = rd + φ0 dz + φ u dz u. (9) We assume ha he available asses are nominal bonds wih differen mauriies (including an insananeous bond which will be referred o as cash), socks and he nonradable house. Also noice ha we assume ha here are no asses available whose nominal reurn have nonzero loading on dz u, and herefore ha here are no inflaion-indexed asses available. We also assume ha here are no radable financial asses available whose nominal reurn have nonzero loading on dz v, i.e. no house price dependen conracs. This means ha we do no need informaion on λ v and r imp separaely because hey are only relevan for he expeced house price appreciaion. Tha is, informaion on θ v λ v r imp is sufficien. Only in Theorem III where we consider he special θ v =0case, we can illusrae he implicaions of he impued ren in isolaion. BX show ha he nominal price a ime of a discoun bond mauring a ime T, denoed as P T,saisfies dp P =(R f Bσ r λ r Cσ π λ π ) d Bσ r dz r Cσ π dz π, (10) where B T = κ 1 1 e κ(t ), C T = α 1 1 e α(t ),andr f = r + π ξ 0 λ ξ u λ u is he reurn on he insananeous nominal risk free asse (cash). Using Io s lemma one can show ha he real reurn on a nominal bond is given by dp p = r Bσ r λ r Cσ π λ π ξ 0 λ ξ u λ u d Bσr dz r Cσ π dz π ξ 0 dz ξ u dz u. (11) A firs poin o noice is ha he real reurn on a nominal bond of a given mauriy has a fixed real risk premium. In paricular i does no depend on he expeced inflaion rae, π. I is sraighforward o show ha he same holds for he real risk premia on socks and he 8

11 house. Since uiliy is defined over real wealh his implies ha he invesor s indirec uiliy funcion will depend on he real riskless ineres rae, r, bu no on he curren expeced rae of inflaion, π. A second poin o noice is ha he reurn processes of bonds wih differen mauriies differ only in heir loadings on dz r and dz π. When here are no consrains on posiion size, any desired combinaion of loadings on dz r and dz π can be accomplished by posiions in any wo bonds wih differen mauriies. In he unconsrained case we herefore firs characerize opimal porfolio choice by opimal allocaion o facor asses, whose nominal reurn has a nonzero loading on exacly one facor (source of uncerainy). Only hereafer we choose wo paricular bond mauriies and describe he opimal porfolio choice in erms of he weighs on hese bonds. 2.3 Opimal porfolio choice Since uiliy is defined over real wealh and all available asses have consan real risk premia over he real riskless asse, we can choose w, h, r, and as sae variables, where h w H / w F + w H is he housing o oal wealh raio. Here we use ha w F,w H maps one-o-one o (w, h), where as before w w F + w H is oal real wealh. Noice also ha here are no financial asses available whose nominal reurn has nonzero loading on dz u. Given hese resricions on he menu of asses, he evoluion of real wealh is given by dw/w = [r + µ e w (x, h) ξ u λ u] d + σ 0 w (x, h) dz ξ u dz u, (12) µ e w (x, h) (1 h) µ e F (x)+hµ e q, σ 0 w (x, h) (1 h) σ 0 F (x)+hσ 0 q, where x are fracions of financial wealh invesed in available asses. The µ e variables denoe expeced excess reurns, for example µ e q =(θ ξ) 0 λ r imp,andσ q = θ ξ. µ e F and σ F are simple linear funcions of x. We can now prove a heorem ha will urn ou o be very useful. 9

12 Theorem I If he se of admissible porfolio weighs, A, is independen of w and r, hen he indirec uiliy funcion can be wrien as J (w, h, r, ) w1 γ 1 γ ν H exp {(1 γ)(r r) B T } n ³ exp (1 γ) ξ u λ u γ o 2 ξ2 u (T ) I (h,), (13a) where I saisfies I (h,) = min x A E s.. " µ ½Z T exp µ r s + µ ew 12 Z T ¾ 1 γ # σ0wρσ w ds + σ 0 wdz (13b) dh = h µ e q µ e w d + σ 0 q σ 0 w dz hσ 0 w ρ (σ q σ w ) d (13c) d r = κ ( r r) d + σ r dz r (13d) r = r (13e) wih I (h, T )=1for all h and where B T was defined before. Proof: see Appendix A. The fac ha indirec uiliy is separable in wealh is a well-known consequence of power uiliy. I is more surprising ha i is also separable in he real ineres rae. The assumpion ha he variance of incremens in r is independen of he level of r is key for his o hold. Noice ha we have no ye specified wheher shor posiions in available asses are possible or no. We only assumed ha resricions do no depend on w and r. Theorem I has wo imporan implicaions for financial porfolio choice. Firs, financial porfolio choice is independen of he curren value of real wealh, w, and he curren value of he real ineres rae, r. Second, i implies ha he degree of marke incompleeness caused by he lack of financial asses wih nominal reurns wih nonzero loading on dz u,as measured by ξ u, has no impac on he financial asse allocaion. The reason is ha dz u is orhogonal o dz and ha he financial asse allocaion does no influence he fuure degree of marke incompleeness due o he lack of inflaion-indexed asses. We now derive he expressions for he asse allocaions. We assume ha here are no asses wih non-zero loading on dz v and dz u,i.e. noinflaion-indexed asses and no house price dependen conracs. Wih he available asses we can ge any combinaion of loadings 10

13 on dz S, dz r and dz π. More precisely, we assume ha here are four financial asses available whose insananeous variance-covariance marix has rank hree. Socks, wo bonds wih differen mauriies and cash would be obvious choices o achieve his. If we assume ha he invesor can unconsrained allocae fracions x S, x r and x π of his financial wealh o hree facor asses whose nominal reurns have sochasic componens σ S dz S, σ r dz r and σ π dz π respecively, and allocae a fracion 1 x S x r x π o cash, hen we can derive an implici expression for he opimal asse allocaion. 5 Theorem II Le he se of admissible porfolio weighs, A, be such ha effecively he invesor can allocae unconsrained fracions x S, x r and x π of his financial wealh o hree facor asses whose nominal reurns have sochasic componens σ S dz S, σ r dz r and σ π dz π respecively, and allocae a fracion 1 x S x r x π o cash. The opimal fracions are given by x S = J µ w F 1 ξ w F γ S φ S J w F w F γ σ S γ x r = J µ w F 1 ξ w F γ r φ r J w F w F γ σ r γ x π = J µ w F 1 ξ w F γ π φ π J w F w F γ σ π γ ξs σ S µ ξr ξπ σ π J w F w H J w F w F B T σ r J w F w H J w F w F θ S h, 1 h σ S J w F w H J w F w F h θ π, 1 h σ π θ r (14a) h,(14b) 1 h σ r (14c) where parial derivaives are deermined using w = w F + w H, h = w H / w F + w H and applying he chain rule. Proof: see Appendix B. The inuiion for hese porfolio weighs is fairly simple. The asse posiions consis of wo componens. The expression in square brackes is exacly he same as he long-erm invesmen porfolio derived by Brennan and Xia (2002). The firs erm of his BX porfolio can be seen as a posiion in he nominal mean-variance angency porfolio. The second erm is he projecion of an inflaion-indexed bond wih mauriy T on dz, whichishe bes possible hedge agains unexpeced inflaion plus a hedge agains real ineres changes, capured by B T. This BX porfolio is pre-muliplied by he raio of he coefficien of relaive risk aversion associaed wih oal wealh changes γ, which equals wj ww /J w,ohe coefficien of relaive risk aversion associaed wih financial wealh changes w F J w F w F /J w F. 5 The auhors hank Yihong Xia for poining ou o us ha solving for an implici expression is informaive and ha i boils down o solving a sysem of hree linear equaions in hree unknowns. 11

14 The correcion facor akes ino accoun ha increases in oal and financial wealh have differen consequences for h and herefore differen uiliy implicaions. 6 For our parameer values his correcion facor will generally be smaller han one. I capures he well-known effec ha in he presence of an illiquid asse an invesor effecively behaves more risk averse in his liquid asse allocaion. 7 The second componen is a hedge erm, which arises when he financial asse reurn and he housing reurn are correlaed. Taking ino accoun he relaive size of financial and housing wealh, a one-for-one hedge agains non-idiosyncraic house risk would give (h/ (1 h)) θ i /σ i. However, changes in financial and housing wealh have differen consequences for h, and herefore differen uiliy implicaions. This gives rise o he correcion facor J w F w H /J w F w F. The correcion facors for he long-erm porfolio and he hedge porfolio are relaed and can be furher worked ou. This is shown in Corollary I. Corollary I From equaions (14a)-(14c) in Theorem II we can derive he following expression µ 1 ξ (1 h) x S = (1 h + ωh) S φ S γ σ S γ µ 1 ξ (1 h) x r = (1 h + ωh) r φ r γ σ r γ µ 1 ξ (1 h) x π = (1 h + ωh) π φ π γ σ π γ ξs σ S µ ξr ξπ σ π ωh θ S, σ S B T σ r ωh θ π, σ π ωh θ r σ r, (15a) (15b) (15c) where ω (h, τ) =1+ Proof: see Appendix B. γi h +hi hh γ(1 γ)i 2γhI h h 2 I hh. These equaions show ha he invesor behaves as if he value of he house is differen from he prevailing price in he marke. The invesor acs as if his house is worh ωhw. We refer o ωh as he effecive housing o oal wealh raio. This means ha an invesor acs as if he value of his asses is effecively only a fracion 1 h + ωh of oal wealh. We will refer o his raio as he effecive o oal wealh raio. Anoher poin o noice is ha 6 Here a change in oal wealh means a wealh change leaving he housing o oal wealh raio, h, he same. Tha is, a $1 increase in w corresponds o a $h increase w H and a $1 h increase w F. In conras, a change in financial wealh does affec h. 7 See e.g. Grossman and Laroque (1990). 12

15 financial wealh is a fracion 1 h of oal wealh, so ha he financial porfolio should beleveragedupbyafacor1/ (1 h) o ge he desired exposure for he oal porfolio. For exposiional easy he leverage facor is pu on he lef-hand side of equaions (15a)- (15c). Observe ha here are wo disinc horizon effecs. Firs, B T capures he horizon dependen hedge agains changes in he real ineres rae. Second, as we will show below, he effecive housing o oal wealh raio changes subsanially wih horizon. Boh effecs make he asse allocaion change over ime. In addiion, wih a fixedposiioninhehouse, he housing o oal wealh raio h is sochasic and generaes ime-varying asse allocaions. For he parameer choice presened in secion 3, we find ha ω is beween zero and one, and declining wih he invesmen horizon. In Figure I we plo ω as a funcion of horizon for a γ =3invesor (Panel A) and a γ =7invesor (Panel B) for various housing o oal wealh raios. Comparing panel A and B, noice he inuiive resul ha he effecive housing wealh, and herefore ω, is lower for he more risk-averse γ =7invesor (ceeris paribus). To furher illusrae he effecive housing o oal wealh raio we provide an explici closed-form soluion for ω in he special case ha he house price risk is spanned by available asses, i.e. θ v =0. Theorem III If (i) he nominal housing reurn is perfecly hedgeable, i.e. θ v =0and (ii) he invesmen opporuniy se, A, is such ha effecively he invesor can allocae unconsrained fracions x S, x r and x π of his financial wealh o hree facor asses whose nominal reurns have sochasic componens σ S dz S, σ r dz r and σ π dz π respecively, and allocae a fracion 1 x S x r x π o cash, hen I (h, ) = [1 h + ωh] 1 γ exp{(1 γ)[ rτ γ φ0 ρφτ + (16) µ σ r φ 0 ρe 2 (τ B) 1 µ κ γ 4κ σ 2 r 2κτ 3+4e κτ e 2κτ ]} γ and ω = e rimpτ. Proof: see Appendix C. In Theorem III he only marke incompleeness is he absence of inflaion-indexed asses. Noicehaherealime- value of a uni of housing o be delivered a ime T is 13

16 Figure I: ω (h, τ) for unconsrained financial asse allocaion ( γ =3and γ =7). The figure shows ω (h, τ) as a funcion of horizon for various housing o oal wealh raios. We use he parameer values presened in secion 3. Panel A: he invesor has risk aversion γ = ω(h,τ) h=0.1 h=0.2 h=0.3 h=0.4 h= Horizon Panel B: he invesor has risk aversion γ = ω(h,τ) h=0.1 h=0.2 h=0.3 h=0.4 h= Horizon 14

17 e rimp (T ) q <q for a posiive convenience yield, r imp. This means ha ω = e rimp (T ). Consisen wih Figure I, he effecive o oal wealh raio is smaller han one and declining in horizon. This makes sense since he house has a lower expeced reurn han a porfolio of pure financial asses wih he same risk characerisics. The longer he horizon, he lower he effecive financial value of he house. In case θ v 6=0, i is easy o see ha he funcional form ω = e rimpτ again obains if a house price dependen asse is available, ha has a nonzero loading on dz v, zero loading on dz u, arbirary loadings on dz S, dz r, dz π,anda consan real risk premium. In Secion 4 and 5 we will use numerical echniques o evaluae opimal asse allocaion for he general case where no house price dependen asse exiss and θ v 6=0.Theneffecive housing wealh differs from he marke value of he house no only because of he impued ren, bu also because housing involves an exposure o unhedgeable, idiosyncraic house risk. Assuming unconsrained allocaions o available asses are possible, we decompose he numerical soluion ino he hree componens of Theorem II. Subsequenly, we also invesigae he opimal asse allocaion when here are shor-sale consrains and when he invesor can ake a morgage loan. For he numerical resuls we coninue o assume ha he invesmen opporuniy se, A, is independen of w and r. In his case, we can use Theorem I o see ha he only par of he indirec uiliy funcion ha is no known in closed form is I (h, ). We know ha I (h, T )=1for all h. Agridoverh and is chosen and we solve for I (h, ) and he opimal asse allocaion backwards in ime. More precisely, wihou loss of generaliy a node (h, ) we normalize w =1and r = r, andseξ u = λ u =0. The laer reduces he number of Brownian moions from five o four in he numerical procedure. Thus we deermine I (h, ) by solving I (h,)=max x A E " w 1 γ +d (x) # e( r +d r)b +d,t I (h +d (x),+ d) w =1, r = r, h 1 γ where d is he sep size of he grid over ime. 8 8 To deermine I (h, + d) for values of h ha are no on he grid, we use cubic spline inerpolaion. The expecaion is evaluaed using Gaussian quadraure wih 5-poins for he unconsrained porfolio choice and 3-poins for he consrained porfolio choice. Increasing he number of poins did no aler resuls in he presened precision. For he opimizaion over x we use a search algorihm ha does no use any derivaive informaion and is robus o differen saring values. The grid on h and is chosen fine enough o ensure precision up o he presened number of decimals. (17) 15

18 3 Calibraion To illusrae he impac of an owner-occupied house on he financial porfolio and morgage choice for differen horizons and housing o oal wealh raios, we calibrae he model parameers o quarerly daa on sock reurns, inflaion, T-bill raes, long bond yields, and house price reurns. We firs esimae a erm srucure model on quarerly daa on nominal ineres raes and inflaion from 1973Q1 o 2003Q4. We use a Kalman filer o exrac he real ineres rae and expeced inflaion rae from he daa, and esimae he model by Quasi Maximum Likelihood. 9 This procedure provides esimaes of he mean reversion parameers and provides ime series of innovaions in he real ineres rae and expeced inflaion, and a ime series of unexpeced inflaion. The values for he mean reversion parameers of real ineres and expeced inflaion rae, κ = and α =0.0548, imply half-lives of 1.1 and 12.6 years respecively. 10 In he second sep of he calibraion, we fi he means, sandard deviaions and correlaions of sock reurns, real ineres raes, expeced and unexpeced inflaion and house prices, and he marke prices of risk. The sample period for his second sep is limied due o he availabiliy of house price daa, and runs from 1980Q2 unil 2003Q4. The reason o esimae he mean reversion parameers over a longer sample period han he oher parameers is ha we need a long sample o obain good esimaes of he mean reversions; all he oher parameers are bes fied o he more recen common sample period, aking he esimaed mean reversions from he firs sep as given. Table I provides all he (annualized) parameer esimaes. We now give some more deail on he second sep of he calibraion process. To esimae he sock reurn process we use quarerly sock reurns on an index comprising all NYSE, AMEX and NASDAQ firms. 11 Following Fama and French (2002), amongs ohers, we believe ha he equiy premium, σ S λ S in our model, is lower han he realized premium when measured over he pas few decades. While he realized excess reurn in our daa is 6.4%, weseλ S such ha he equiy premium is 4.0%. 9 Deails on he procedure are provided in Appendix D. 10 Using differen sample periods, Brennan and Xia (2002) and Campbell and Viceira (2001) also find a half-life of around 1 year for innovaions in he real rae and a much longer half-life for expeced inflaion. 11 The auhors would like o hank Kenneh R. French for making his daa available a his websie. 16

19 Table I. Choice of model parameers. The able repors calibraed parameer values for he dynamics of he asse prices, he inflaion rae and real ineres rae (as described secion 2.2). The parameer values are obained using quarerly daa. Parameer Esimae Sock reurn process: ds/s =(R f + σ S λ S ) d + σ S dz S σ S λ S Real riskless ineres rae process: dr = κ ( r r) d + σ r dz r r κ σ r λ r Expeced inflaion process: dπ = α ( π π) d + σ π dz π π α σ π λ π House price process: dq/q = R f + σ Q λ Q r imp d + σ Q dz Q = R f + θ 0 λ r imp d + θ 0 dz θ S θ r θ π θ v θ v λ v r imp σ Q σ Q λ Q r imp Realized inflaion process: dπ/π = πd + σ Π dz Π = πd + ξ 0 dz + ξ u dz u ξ S ξ r ξ π ξ v ξ u σ Π Correlaions: ρ Sr ρ Sπ ρ rπ Table II Correlaion marix for (dz S,dz r,dz π,dz Q,dz Π ) 0 dz S dz r dz π dz Q dz Π dz s 1 dz r dz π dz Q dz Π

20 For he ineres rae and inflaion par of he model, he firs sep already provides us wih he mean reversion parameers. The oher parameers are esimaed as follows. The mean expeced inflaion is esimaed by he mean increase of he CPI. For he mean real ineresraeweakehedifference beween he means of he T-bill rae and he expeced inflaion, minus a 0.5% correcion o reflec he premium on unexpeced inflaion. 12 The sandard deviaions of he real ineres rae, expeced inflaion and unexpeced inflaion are deermined using he ime series generaed by he Kalman filer. 13 We esimae he risk premia λ r and λ π by maching he average yields of wo bond porfolios wih a consan ime o mauriy of 3.4 and 10.4 years. For his we use formulas derived by Brennan and Xia (2002), Appendix A. We esimae he house price process using repeaed sales daa a he ciy level for Alana, Boson, Chicago and San Francisco from 1980Q2 o 2003Q4. There are quie some differences in he price processes for he four ciies. We choose o focus on general house price characerisics and consruced a naion-wide reurn index by weighing he ciies equally. Case and Shiller (1989) argue ha he sandard deviaion of individual house price changes are close o 15%, like individual socks. Because price changes of differen houses are far from perfecly correlaed, aggregaion leads o a considerable reducion of he variabiliy. In our naion-wide index we find a sandard deviaion of 2.67%. Since we are ineresed in he dynamics of an individual house, we correc his series by simply scaling house price shocks wih a facor 15.00%/2.67% = 5.6 around is mean. Finally, we esimae he correlaion marix ρ and he coefficien vecors ξ and θ using quarerly sock reurns, house price reurns, he innovaions in he real ineres rae, expeced inflaion, and unexpeced inflaion. We have daa on house prices, bu no on impued ren. Therefore we calibrae θ v λ v r imp and no λ v and r imp separaely. As discussed insecion2hisissufficien o deermine he opimal asse allocaion. Table II provides he implied correlaion marix of he sochasic vecor (dz S,dz r,dz π,dz Q,dz Π ) The 50 basis poins unexpeced inflaion risk premium is based on he esimae of Campbell and Viceira (2002, p.72)). Wih his assumpion here is no furher need o esimae he marke price of risk for unexpeced inflaion, λ u, because i does no influence he asse allocaion in our se-up wih only nominal securiies (i does however influence indirec uiliy). 13 The discree-ime sandard deviaions are convered o he coninuous-ime counerpars, incorporaing he effec of mean reversion in he processes. 18

21 4 Porfolio Choice wihou Morgage In his secion we firs presen he unconsrained opimal porfolio choice for a moderaely risk-averse invesor (γ =3). We do his firs in erms of facor asses and spli he soluion ino he hree componens discussed in secion 2. Thereafer we ranslae his o porfolio choice in erms of available asses. Equipped wih he inuiion of he unconsrained case, we ackle he porfolio allocaion problem for an invesor who is consrained o holding nonnegaive posiions in available asses. Here we consider a moderaely risk-averse invesor (γ =3) as well as a more risk-averse invesor (γ =7). Finally we invesigae how sensiive our resuls are o alernaive parameers for he house price process. 4.1 Unconsrained porfolio choice We presen he opimal porfolio for he siuaion where here are no financial asses whose nominal reurn has nonzero loading on dz v and dz u. Tha is, here is no house price dependen conrac nor an inflaion-indexed securiy. Table III shows he opimal allocaion o socks, real ineres and expeced inflaion facor asses, whose nominal reurns have sochasic componens σ S dz S, σ r dz r and σ π dz π respecively. We use he parameer values presened in Table I. Panel A shows he allocaion as fracion of financial wealh, i.e. he sum corresponds o x S, x r and x π respecively. Panel B shows he allocaion as fracion of oal wealh, i.e. he sum corresponds o (1 h) x S, (1 h) x r and (1 h) x π respecively. The able also shows he opimal allocaion spli ino he hree componens given in Theorem II. The firs componen comprises he posiions in he mean-variance angency porfolio. The fracion allocaed o he real ineres facor asse and he expeced inflaion facor asse are much larger in absolue erms han he fracion allocaed o he socks facor asse. The main reason is ha he reurns on he real ineres and expeced inflaion facor asse boh have a relaively low sandard deviaion, resuling in large invesmens o obain he opimal risk exposure. As a fracion of oal wealh a he 1-monh horizon he allocaion is he same for all housing o oal wealh raios, bu as a fracion of financial wealh here is a leverage effec. All posiion sizes decrease in horizon because he effecive housing o oal wealh raio and herefore he effecive o oal wealh raio decreases in horizon. This horizon effec is more profound for a larger housing o oal wealh raio, as ω is smaller for larger housing o oal wealh raios (see Figure 1). This can be undersood from he fac ha he risk of he fixed house posiion can only be hedged parially. As he house size 19

22 Table III. Unconsrained facor asse allocaion (γ =3). The able presens he opimal allocaion o socks, real ineres and expeced inflaion facor asses, whose nominal reurns have sochasic componens σ S dz S, σ r dz r and σ π dz π respecively. Panel A shows he allocaion as fracion of financial wealh, i.e. he sum corresponds o x S, x r and x π respecively. Panel B shows he allocaion as fracion of oal wealh, i.e. he sum corresponds o (1 h) x S, (1 h) x r and (1 h) x π respecively. The able also shows he opimal allocaion spli ino he hree componens given in Theorem II. The invesor has risk aversion γ =3. PanelA:asfracionoffinancial wealh h =0.2 h =0.4 Facor Componens Componens Horizon Asse 1s 2nd 3rd Sum 1s 2nd 3rd Sum Socks monh Real ineres Exp. inflaion Socks years Real ineres Exp. inflaion Socks years Real ineres Exp. inflaion Panel B: as fracion of oal wealh h =0.2 h =0.4 Facor Componens Componens Horizon Asse 1s 2nd 3rd Sum 1s 2nd 3rd Sum Socks monh Real ineres Exp. inflaion Socks years Real ineres Exp. inflaion Socks years Real ineres Exp. inflaion Table IV. Unconsrained financial porfolio choice (γ =3). The able presens opimal financial porfolio weighs for socks, bonds wih mauriies of 5 and 20 years, and cash. The invesor has risk aversion γ =3. Horizon Asse h =0 h =0.1 h =0.2 h =0.3 h =0.4 h =0.5 Socks monh 5yearbond year bond Cash Socks years 5yearbond year bond Cash Socks years 5yearbond year bond Cash

23 increases, he presence of his undiversifiable risk makes a risk-averse invesor decrease his exposure o risky financial asses. The second componen comprises posiions in he hedge porfolio agains real ineres changes and unexpeced inflaion. A he 1-monh horizon here is mainly hedging agains unexpeced inflaion and posiions are limied. Only real ineres rae changes are considerably (posiively) correlaed wih unexpeced inflaion leading o a posiive hedge demand for he real ineres facor asse. A longer horizons he hedge agains real ineres changes becomes imporan, resuling in a subsanially negaive allocaion o he real ineres facor asse. If he real ineres rae shocks would be more persisen han in our calibraion (where he half life is jus 1.1 year) he value would become even more negaive (see e.g. Brennan and Xia (2002)). The sign for he real ineres facor asse a longer horizons in his second componen is negaive, because a shor erm gain compensaes for deerioraing invesmen opporuniies when confroned wih a downward shock o he real ineres rae. The same remarks regarding he horizon and housing o oal wealh raio effec as in he analysis of he firs componen apply. In our calibraion he reurn on housing is posiively correlaed o changes in he real ineres rae and expeced inflaion rae. The hird componen, he posiion in he porfolio ha hedges he effecive housing wealh, herefore involves negaive values for he real ineres and expeced inflaion facor asses. A he 1-monh horizon magniudes increase abou linear in h as a fracion of oal wealh and more han linear in h as fracion of financial wealh. The posiion size decreases in horizon because he effecive housing wealh raio decreases in horizon. Again, his is more profound for a larger housing o oal wealh raio. Summed over all hree componens, Table III shows ha in paricular he allocaion o he real ineres facor asse (as fracion of financial wealh denoed by x r ) is large and negaive. Noe ha Theorem II implies ha for a more risk-averse invesor (say γ =7 insead of γ =3), he posiions in he firs componen become (a facor 7/3) smallerand he posiions in he second componen become (a facor 9/7) larger. Inhiscasehesizeof x r would sand ou even more. In Table IV, we ranslae he facor asse posiions o porfolio weighs in financial asses. If we assume ha he invesor can inves unconsrained in socks, wo bonds wih differen mauriies and cash, any combinaion of loadings on dz S, dz r and dz π can be accomplished. Table IV repors he opimal oal porfolio choice for various values for h, 21

24 when bonds of 5 and 20 years mauriy are available. To inerpre he bond posiions, firs noe ha in our calibraion we have κ>α.tha is, he mean reversion in he real ineres rae is quicker han he mean reversion in he expeced inflaion rae. Remember ha he sochasic componen of he nominal reurn on a nominal bond is given by dp T P T =[...] d B T σ r dz r C T σ π dz π. Since κ>α, for any horizon T we have 0 <B T <C T. Moreover, we have ha B T C T is decreasing in τ T. 14 This implies ha o obain a negaive value for x r in he same order of magniude (or even larger, in size) han x π, one needs a long posiion in a shor-erm bond and a shor posiion in a long-erm bond. Because B and C are larger for longer horizons, he size of he shor posiion will be smaller han he size of he long posiion. This is exacly wha we see in Table IV. We also observe in Table IV ha he opimal bond posiions are very large in size. 4.2 Consrained porfolio choice The unconsrained resuls in Table IV exhibi large shor posiions in he 20-year bond and cash. In pracice, such posiions can no be easily achieved for a ypical invesor who faces shor sale consrains. Table V herefore shows he resuls when we consrain he fracion invesed in socks, he wo bonds and cash o be posiive. 15 Panel A shows he opimal porfolio for a moderaely risk averse invesor (γ =3) and Panel B for a fairly risk averse invesor (γ =7). For he moderaely risk-averse invesor (γ =3), he consrained allocaion o socks approximaely equals he unconsrained allocaion o socks for h 0.3. Sincealmos all sockholdings in he unconsrained case originae from he firs componen (i.e. he 14 d To see his noice ha BT dτ C T = e κτ C T e ατ B T. We have e κτ C (C T ) 2 T = e κτ τ 0 e αs ds and e ατ B T = e ατ τ 0 e κs ds. Because for s ]0,[ we have e αs κτ <e κs ατ if and only if κ>α,i follows ha d dτ BT C T < 0. For a mauriy of 5 years we have: B =1.48 and C =4.37. For a 20 year mauriy we have: B =1.54 and C = In he unconsrained case he available bond mauriies have no impac on indirec uiliy as long as here are a leas wo differen mauriies available a any ime. In he consrained case available mauriies do maer for indirec uiliy. This in urn makes fuure available bond mauriies relevan for curren porfolio choice. In he remainder of his paper we assume ha he mauriies of he available bonds are consan. In pracice his would mean ha he invesor can inves in wo bond porfolios ha are rebalanced in such a way ha he duraion is always 5 and 20 years. 22

25 Table V. Consrained porfolio choice (γ =3and γ =7). The able presens opimal financial porfolio weighs for socks, bonds wih mauriies of 5 and 20 years, and cash in he presence of shor-sale consrains, using he base case parameer se in Table I. The invesor has risk aversion γ =3. The bond mauriies are assumed o be consan over he invesmen period. Panel A: he invesor has risk aversion γ =3 Horizon Asse h =0 h =0.1 h =0.2 h =0.3 h =0.4 h =0.5 Socks monh 5yearbond year bond Cash Socks years 5yearbond year bond Cash Socks years 5yearbond year bond Cash Panel B: he invesor has risk aversion γ =7 Horizon Asse h =0 h =0.1 h =0.2 h =0.3 h =0.4 h =0.5 Socks monh 5yearbond year bond Cash Socks years 5yearbond year bond Cash Socks years 5yearbond year bond Cash