Working Paper Optimal housing, consumption, and investment decisions over the life-cycle

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1 econsor Der Open-Access-Publikaionsserver der ZBW Leibniz-Informaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Kraf, Holger; Munk, Claus Working Paper Opimal housing, consumpion, and invesmen decisions over he life-cycle Working paper series // Johann-Wolfgang-Goehe-Universiä Frankfur am Main, Fachbereich Wirschafswissenschafen Finance & accouning, No. 197 Provided in Cooperaion wih: Faculy of Economics and Business Adminisraion, Goehe Universiy Frankfur Suggesed Ciaion: Kraf, Holger; Munk, Claus 29 : Opimal housing, consumpion, and invesmen decisions over he life-cycle, Working paper series // Johann-Wolfgang-Goehe- Universiä Frankfur am Main, Fachbereich Wirschafswissenschafen Finance & accouning, No. 197 This Version is available a: hp://hdl.handle.ne/1419/3946 Nuzungsbedingungen: Die ZBW räum Ihnen als Nuzerin/Nuzer das unengelliche, räumlich unbeschränke und zeilich auf die Dauer des Schuzrechs beschränke einfache Rech ein, das ausgewähle Werk im Rahmen der uner hp:// nachzulesenden vollsändigen Nuzungsbedingungen zu vervielfäligen, mi denen die Nuzerin/der Nuzer sich durch die erse Nuzung einversanden erklär. Terms of use: The ZBW grans you, he user, he non-exclusive righ o use he seleced work free of charge, erriorially unresriced and wihin he ime limi of he erm of he propery righs according o he erms specified a hp:// By he firs use of he seleced work he user agrees and declares o comply wih hese erms of use. zbw Leibniz-Informaionszenrum Wirschaf Leibniz Informaion Cenre for Economics

2 JOHANN WOLFGANG GOETHE-UNIVERSITÄT FRANKFURT AM MAIN FACHBEREICH WIRTSCHAFTSWISSENSCHAFTEN Holger Kraf / Claus Munk Opimal Housing, Consumpion, and Invesmen Decisions Over he Life-Cycle No. 197 February 29 WORKING PAPER SERIES: FINANCE & ACCOUNTING

3 HOLGER KRAFT * / CLAUS MUNK OPTIMAL HOUSING, CONSUMPTION, AND INVESTMENT DECISIONS OVER THE LIFE-CYCLE The paper conains graphs in color. Use color priner for bes resul. No. 197 February 29 ISSN * Deparmen of Finance, Goehe-Universiy, Grüneburgplaz 1, 6323 Frankfur am Main, Germany, holgerkraf@finance.uni-frankfur.de School of Economics and Managemen & Dep. of Mahemaical Sciences, Aarhus Universiy Danish Cener for Accouning and Finance D-CAF, Barholin's Alle 1, Building 1322, DK-8 Aarhus C, Denmark, cmunk@econ.au.dk We appreciae commens from paricipans a presenaions a he European Finance Associaion meeing in Ahens, he Copenhagen Business School, he Universiy of Souhern Denmark, and Aarhus Universiy. Björn Bick provided excellen research assisance. Holger Kraf graefully acknowledges nancial suppor by Deusche Forschungsgemeinschaf DFG. Claus Munk graefully acknowledges nancial suppor from he Danish Research Council for Social Sciences. The working papers in he series Finance and Accouning are inended o make research findings available o oher researchers in preliminary form, o encourage discussion and suggesions for revision before final publicaion. Opinions are solely hose of he auhors

4 Absrac We provide explici soluions o life-cycle uiliy maximizaion problems simulaneously involving dynamic decisions on invesmens in socks and bonds, consumpion of perishable goods, and he renal and he ownership of residenial real esae. House prices, sock prices, ineres raes, and he labor income of he decision-maker follow correlaed sochasic processes. The preferences of he individual are of he Epsein-Zin recursive srucure and depend on consumpion of boh perishable goods and housing services. The explici consumpion and invesmen sraegies are simple and inuiive and are horoughly discussed and illusraed in he paper. For a calibraed version of he model we find, among oher hings, ha he fairly high correlaion beween labor income and house prices imply much larger life-cycle variaions in he desired exposure o house price risks han in he exposure o he sock and bond markes. We demonsrae ha he derived closed-form sraegies are sill very useful if he housing posiions are only rese infrequenly and if he invesor is resriced from borrowing agains fuure income. Our resuls sugges ha markes for REITs or oher financial conracs faciliaing he hedging of house price risks will lead o non-negligible bu moderae improvemens of welfare. Keywords: Housing, labor income, porfolio choice, life-cycle decisions, recursive uiliy, REITs JEL-Classificaion: G11, D14, D91, C6

5 Opimal Housing, Consumpion, and Invesmen Decisions over he Life-Cycle 1 Inroducion The wo larges asses for many individuals are he human capial and he residenial propery owned and occupied by he individual. The financial decisions of individuals over he life-cycle are bound o be affeced by he characerisics of hese asses. While he early lieraure on dynamic consumpion and porfolio decisions Samuelson 1969; Meron 1969, 1971 ignored such non-financial asses, progress has recenly been made wih respec o incorporaing and undersanding housing decisions and labor income in a life-cycle framework of consumpion and porfolio choice. Due o he complexiy of such decision problems, almos all of hese sudies resor o raher coarse and compuaionally very inensive numerical soluion echniques wih an unknown precision. In conras, his paper provides closedform soluions for coninuous-ime problems involving boh consumpion, housing, and invesmen decisions when sock prices, ineres raes, labor income, and house prices vary sochasically over ime. Preferences are modeled by a wo-good exension of Epsein-Zin recursive uiliy ha allows for a separaion of he risk aversion and he elasiciy of ineremporal subsiuion, wih exac closedform soluions given for he wo special cases of power uiliy and a uni elasiciy of subsiuion and an approximae closed-form soluion for he general case. These closed-form soluions lead o a deeper undersanding of he economic forces driving individual decisions in such a complex seing. For a calibraed version of he model we show ha he soluions from he model imply fairly realisic consumpion and invesmen paerns over he life-cycle. Our model has he following feaures. The individual derives uiliy from consumpion of perishable goods and of housing services and maximizes life-ime uiliy of he Epsein-Zin ype. The individual receives an exogenous sochasic sream of labor income unil a fixed reiremen dae afer which he individual lives for anoher fixed period of ime. Our specificaion of he income process encompasses life-cycle variaions in he expeced growh rae and volailiy and also allows for variaions in expeced income growh relaed o he shor-erm ineres rae in order o reflec dependence on he business cycle. The pure financial asses available are a sock, a bond, and shor-erm deposis cash. The shor-erm ineres rae and he reurns on he bond are modeled by he classical Vasicek model, and for he sock price we assume a consan expeced excess reurn, a consan volailiy, and a consan correlaion wih he bond price. The individual can buy and sell houses 2 a a uni price ha varies sochasically wih a consan expeced growh rae in excess of he shor-erm ineres rae, a consan volailiy, and consan correlaions wih labor income and financial asse prices. The purchase of a house serves a dual role by boh generaing consumpion services and by consiuing an invesmen affecing fuure wealh and consumpion opporuniies. We allow he individual o disenangle he wo dimensions of housing by rening he house insead of owning i he ren is proporional o he price 2 In order o keep he erminology simple we use house insead of he more general erm residenial propery. 1

6 of he house rened and/or by invesing in a financial asse linked o house prices. In curren financial markes, shares in REITs Real Esae Invesmen Truss and he S&P/Case-Shiller Home Price Indices CSI fuures and opions raded a he Chicago Mercanile Exchange offer such opporuniies; more informaion on hese conracs is provided in Secion 2. In order o derive closed-form soluions our main model exhibis marke compleeness cf., e.g., Liu 27 so, in paricular, he labor income sream has o be spanned by he raded asses. The correlaions beween an individual s labor income and he reurns on socks and bonds are probably quie low. 3 However, labor income ends o be highly correlaed wih house prices e.g. Cocco 25 repors a correlaion of.55 so ha he income spanning assumpion is less unrealisic in our model wih housing han in he models wih labor income, bu no housing, sudied in he exising lieraure references given below. Sill i may no be possible o find a rading sraegy in socks, bonds, deposis, and houses ha perfecly replicaes he income risk. Wihou perfec spanning i seems impossible o derive he opimal invesmen sraegy in closed-form or even wih a precise, numerical soluion echnique. While he invesmen sraegy we derive in his paper will hen be sub-opimal, he resuls presened in Bick, Kraf, and Munk 28 for a similar, hough slighly simpler, model indicae ha i will be near-opimal in he sense ha he invesor will a mos suffer a loss corresponding o a few percen of his iniial wealh by following he closed-form sub-opimal sraegy insead of he unknown opimal sraegy. The resuls we presen below will herefore be highly relevan even wihou perfec spanning. The high correlaion beween labor income and house prices implies he following disinc life-cycle paern in he invesmen exposure o house price risk. When human wealh is big relaive o financial wealh e.g. early in life, he individual should inves very lile in housing so ha he desired housing consumpion is mainly achieved by rening. When human wealh is low relaive o financial wealh e.g. lae in life, he opimal housing invesmen is quie big due o is fairly aracive risk-reurn rade-off. We find ha he opimal housing invesmen varies much more over he life-cycle han he opimal invesmens in bonds and socks. In our main model he individual can coninuously and coslessly adjus boh he housing consumpion and he housing invesmen, bu we also consider problems wih limied flexibiliy in housing decisions. Changes in physical ownership of housing generae subsanial ransacion coss no included in our model, so coninuous adjusmens of housing invesmen mus be implemened by rebalancing he posiion in he house-price linked financial asse. We have o assume a perfec correlaion beween he reurns on ha asse and house prices, which may be unaainable in acual markes bu carefully seleced REITs or CSI housing conracs will come close. 4 The case where boh housing consumpion and housing invesmen are coninuously adjusable can be seen as an upper bound on he life-ime uiliy ha he individual can realisically obain. We invesigae he imporance of he frequency 3 The correlaion beween average labor income and he general sock marke is usually esimaed o be close o zero see, e.g., Cocco, Gomes, and Maenhou 25, bu i should be possible o find single socks highly correlaed wih he labor income of a paricular individual. 4 Tsai, Chen, and Sing 27 repor ha REITs behave more and more like real esae and less and less like ordinary socks. 2

7 of adjusmens of he housing consumpion and housing invesmen in wo ways. Firs, we derive an explici soluion o he problem where he individual consumes a consan level of housing services hrough life, adjuss he housing invesmen posiion coninuously, and has ime-addiive power uiliy of consumpion wih no uiliy from erminal wealh. We find ha he uiliy decrease due o he fixed housing consumpion is equivalen o less han 1% of oal wealh. Second, we implemen a Mone Carlo simulaion procedure o compue he expeced uiliy of an individual resriced o infrequen adjusmens of a housing consumpion, b housing invesmen, or c boh. Again, we find ha he wealh-equivalen loss is fairly small even when he housing posiions are only adjused every five years. Our resuls indicae ha i is more imporan o adjus he housing invesmen posiion frequenly han he housing consumpion posiion, suggesing ha a well-funcioning marke for REITs or oher financial conracs relaed o house prices can lead o a non-negligible improvemen in he welfare of individual invesors. Anoher quesionable feaure of our main model is he possibiliy of he invesor o borrow agains fuure labor income. The opimal unconsrained consumpion and invesmen sraegy will lead o cases where he angible wealh he sum of financial wealh and he value of he housing sock owned will be negaive bu he human wealh more han ouweighs ha so ha oal wealh is posiive. Such a sraegy may no be feasible in real life due o moral hazard and asymmeric informaion issues in he valuaion of human wealh. We inroduce a minor ransformaion of he opimal unconsrained consumpion and invesmen sraegy ha ensures non-negaive angible wealh a all poins in ime. We evaluae he expeced uiliy generaed by his ransformed sraegy by Mone Carlo simulaion and find ha i corresponds o a wealh-equivalen loss of only abou one percen compared o he maximum expeced uiliy in he unconsrained case, using our benchmark parameer values. Hence, he ransformaion of he closed-form soluion mus be a leas near-opimal in he borrowing consrained case, emphasizing he relevance of our closed-form soluion in more realisic seings. Nex we briefly compare our seing and findings o some recen relaed papers. Cocco 25 considers a model feauring sochasic house prices and labor income wih an assumed perfec correlaion beween house prices and aggregae income shocks. Ineres raes are assumed consan. Rening is no possible. The individual is allowed o borrow only up o a percenage of he curren value of he house. There is a minimum choice of house size, and house ransacions carry a proporional cos. The individual has o pay a one-ime fixed fee o paricipae in he sock marke. Yao and Zhang 25a add moraliy risk and he possibiliy of rening o Cocco s framework and do no impose a perfec correlaion beween house prices and income. Van Hemer 27 generalizes he seing furher by allowing for sochasic variaions in ineres raes and hereby inroducing a role for bonds, and he also addresses he choice beween an adjusable-rae morgage and a fixed-rae morgage ignoring he imporan prepaymen opion, however. The laer wo papers disregard he sock marke enry fee in Cocco s model. All hese hree papers apply numerical soluion echniques based on a discreizaion of ime and he sae space. Yao and Zhang 25a and Cocco 25 solve he dynamic programming equaion relaed o he problem by applying a very coarse discreizaion, e.g. using binomial processes and large ime inervals beween revisions of decisions. Van Hemer 27 is able o handle a finer dis- 3

8 creizaion by relying on 6 parallel compuers. I is difficul o assess he precision of such numerical echniques and, in any case, he compuaional procedures are highly ime-consuming and cumbersome. The closed-form soluions derived in his paper are much easier o analyze, inerpre, and implemen and hus faciliae an undersanding and a quanificaion of he economic forces a play. Moreover, he hree above-menioned papers assume preferences of he ime-addiive Cobb-Douglas syle. We build he insananeous Cobb-Douglas uiliy of perishable consumpion and housing services ino an Epsein-Zin recursive uiliy formulaion allowing us o disenangle he risk aversion and he elasiciy of ineremporal subsiuion ψ, as has been shown o be valuable boh for consumpion-porfolio choice wih one consumpion good see, e.g., Campbell and Viceira 1999, Campbell, Cocco, Gomes, Maenhou, and Viceira 21, and Chacko and Viceira 25 and for equilibrium asse prices see Bansal and Yaron 24. We provide exac closed-form soluions for he special case of ime-addiive Cobb-Douglas uiliy, corresponding o ψ = 1/, and for he more reasonable case where ψ = 1 and > 1. Exending he log-linearizaion echnique of Campbell 1993 and Chacko and Viceira 25, we derive an approximae closed-form soluion for general combinaions of ψ and. Damgaard, Fuglsbjerg, and Munk 23 do provide a closed-form soluion for a relaed bu much simpler problem of an individual maximizing ime-addiive Cobb-Douglas uiliy over consumpion and owner-occupied housing, when he size of he house occupied can be coninuously and coslessly rebalanced. They ignore he possibiliy of rening as well as labor income and variaions in ineres raes. They provide a mahemaical and numerical analysis of he case wih a proporional cos on house ransacions. Some more marginally relaed papers deserve o be menioned. Campbell and Cocco 23 sudy he morgage choice in a life-cycle framework wih sochasic house price, labor income, and ineres raes. By fixing he house, however, hey are no able o address he ineracion beween housing decisions and porfolio decisions. Moreover, heir soluion relies on a very coarse discreizaion of he model, e.g. wih wo year ime inervals where decisions canno be revised. Munk and Sørensen 28 solve he life-cycle consumpion and invesmen problem wih sochasic labor income and ineres raes, bu do no incorporae houses in neiher consumpion nor invesmen decisions. They find a closed-form soluion for a complee marke version of heir model, which is generalized o include housing decisions and recursive uiliy in our paper. They also repor resuls from a numerical soluion for he case where labor income risk is no spanned by raded financial asses. While we invesigae individual decision making in he presence of housing wealh and human capial on individual decisions, he role of hese wo facors in equilibrium asse pricing have also been subjec o recen heoreical and empirical research. Papers on he impac of housing decisions and prices on financial asse prices include Piazzesi, Schneider, and Tuzel 27, Lusig and van Nieuwerburgh 25, and Yogo 26, while papers such as Consaninides, Donaldson, and Mehra 22, Sanos and Veronesi 26, and Soresleen, Telmer, and Yaron 24, 27 focus on he ineracion of labor income risk and asse prices. To summarize our conribuion, we derive explici expressions for he opimal life-cycle housing, consumpion, and invesmen decisions of an invesor having Epsein-Zin uiliy in a rich model aking ino accoun variabiliy in labor income, ineres raes, and he prices of houses, socks, and bonds. 4

9 We discuss he srucure of he soluion and show in a numerical example ha our soluion generaes a life-cycle behavior wih many realisic feaures. We also provide resuls suggesing ha our soluion will sill be close o opimal if some of he unpleasan model assumpions are relaxed. The remainder of he paper is organized as follows. Secion 2 formulaes and discusses he ingrediens of our model and he uiliy maximizaion problem faced by he individual. Secion 3 saes, explains, and illusraes he opimal housing, consumpion, and invesmen sraegies in he case when housing decisions can be conrolled coninuously. Secion 4 invesigaes he effec of limiing he flexibiliy in revising housing decisions and provides esimaes of he value of being able o make coninuous revisions for example via rade in financial conracs linked o house prices. Secion 5 summarizes and concludes. All proofs are colleced in he appendices a he end of he paper. 2 The problem The main elemens of our modeling framework are specified as follows. Consumpion goods. The individual can consume wo goods: perishable consumpion and housing. The perishable consumpion good is aken as he numeraire so ha he prices of he housing good and of all financial asses are measured in unis of he perishable consumpion good. Financial asses. The individual can inves in hree purely financial asses: a money marke accoun cash, a bond, and a sock represening he sock marke index. The reurn on he money marke accoun equals he coninuously compounded shor-erm real ineres rae r, which is assumed o have Vasicek dynamics dr = κ[ r r ] d σ r dw r, 2.1 where W r = W r is a sandard Brownian moion. The price of any bond or any oher ineres rae derivaive is hen of he form B = Br, wih dynamics db = B [r + λ B σ B r, d + σ B r, dw r ], 2.2 where σ B r, = σ r B r r, /Br, is he volailiy and λ B he Sharpe raio of he bond, which is idenical o he marke price of ineres rae risk. In paricular, if we inroduce he noaion B m τ = 1 m 1 e mτ for any posiive consan m, he ime price of a real zero-coupon bond mauring a some dae T > can be wrien as aτ = B T = e at BκT r, 2.3 r λ Bσ r σ2 r κ 2κ 2 τ B κ τ + σ2 r 4κ B κτ

10 An invesor will no gain from rading in more han one bond in addiion o he money marke accoun. The sock price S has dynamics [ ] ds = S r + λ S σ S d + σ S ρ SB dw r + 1 ρ 2 SB dw S, 2.5 where W S = W S is a sandard Brownian moion independen of W r, σ S is he consan volailiy and λ S he consan Sharpe raio of he sock, and ρ SB is he consan correlaion beween he sock and he bond reurns. Houses. The individual can also buy or ren houses. A given house is assumed o be fully characerized by a number of housing unis, where a uni is some one-dimensional represenaion of he size, qualiy, and locaion. Prices of all houses move in parallel. The purchase of a unis of housing coss ah ; here are no ransacion coss. The uni house price H is assumed o have dynamics dh = H [ r + λ H σ H r imp d + σ H ρ HB dw r + ˆρ HS dw S + ˆρ H dw H ], 2.6 where W H = W H is a sandard Brownian moion independen of W r and W S, σ H is he consan price volailiy and λ H he consan Sharpe raio of houses, ρ HB is he consan correlaion beween house and bond prices, and ˆρ HS = ρ SH ρ SB ρ HB, ˆρ 1 ρ 2 H = SB 1 ρ 2 HB ˆρ2 HS where ρ SH is he consan correlaion beween house and sock prices. Finally, r imp is he impued ren, i.e. he marke value associaed wih he ne benefis offered by a house similar o he convenience yield of commodiies, which is assumed o be consan as, e.g., in Van Hemer 27. The uni renal cos of houses is assumed o be proporional o he curren uni house price, i.e. νh for some consan ν. For laer use, define λ H = λ H + ν r imp /σ H. By rening insead of owning he house, he individual can isolae he consumpion role of housing. We assume ha he individual can inves in a financial asse wih a price ha follows he movemens in house prices. In a number of counries, shares in REITs are publicly raded. A REIT Real Esae Invesmen Trus is an invesmen company ha invess in and ofen operaes real esae generaing renal income and hopefully capial gains so, by consrucion, he prices of REIT shares will be closely relaed o real esae prices. 5 While REITs in general may be ineresing as an asse class improving he overall risk-reurn radeoff, REITs specializing in residenial real esae are paricularly ineresing for exising or prospecive individual homeowners as a vehicle o manage exposure o house price risk wihou having o physically rade houses frequenly. 5 REITs were inroduced in he U.S. in he 196s and he REIT indusry has experienced subsanial growh since he early 199s. According o he websie of he Naional Associaion of Real Esae Invesmen Truss on November 7, 27 see shares in 19 U.S. REITs are publicly raded wih a oal marke capializaion of more han $4 billion, and as of Sepember 28, of he companies in S&P5 index are REITs. Well-esablished REIT markes also exis in counries such as Japan, Canada, France, and he Neherlands, and are under developmen in many oher counries, e.g. in Germany. 6

11 If we le R denoe he value of he REIT per uni of housing and assume ha he REIT passes on renal income o shareholders as a dividend, we will have R = H and he oal insananeous reurn from a REIT is dh + νh d = H [r + λ Hσ H d + σ H ρ HB dw r + ˆρ HS dw S + ˆρ H dw H ]. 2.7 An alernaive o REITs is he housing fuures and opions on housing fuures raded since 26 a he Chicago Mercanile Exchange. The payoff of such a conrac is deermined by eiher a U.S. naional home price index or by a home price index for one of 1 major U.S. ciies; he indices were developed by Case and Shiller, hence he conracs are also referred o as CSI fuures and opions. See de Jong, Driessen, and Van Hemer 28 for a parial analysis of he economic benefis of having access o such housing fuures. Noe ha when an individual physically owns a house, a negaive posiion in he money marke accoun can be inerpreed as an adjusable-rae morgage, whereas a negaive posiion in he longerm bond resembles a fixed-rae morgage. In order o obain closed-form soluions we do no limi borrowing o some fracion of he marke value of he house owned. Labor income. The individual is assumed o reire from working life a ime T and live unil ime T T. During working life he individual receives a coninuous and exogenously given sream of income from non-financial sources e.g. labor a a rae of Y which has he dynamics dy = Y [µ Y r, d + σ Y r, ρ Y B dw r + ˆρ Y S dw S + ˆρ Y dw H ]. 2.8 For analyical racabiliy here is no idiosyncraic shock o he income process, hence he marke is complee, bu as discussed in he inroducion his is no a crucial assumpion for he relevance of our resuls. The expeced percenage income growh µ Y and volailiy σ Y are allowed o depend on ime age of he individual and he ineres rae level o reflec flucuaions of labor income over he life- and business cycle, cf., e.g., Cocco, Gomes, and Maenhou 25 and Munk and Sørensen 28. ρ Y B is he consan correlaion beween income growh and bond reurns, and ˆρ Y S = ρ SY ρ SB ρ Y B, ˆρ 1 ρ 2 Y = 1 ρ 2 Y B ˆρ2 Y S SB where ρ SY is he consan correlaion beween house and sock prices. Due o he compleeness assumpion he correlaion beween income growh and house prices follow from he oher pairwise correlaions, ρ Y H = ρ HB ρ Y B + ˆρ HS ˆρ Y S + ˆρ H ˆρ Y. In he reiremen period [ T, T ], he individual is assumed o have no income from non-financial sources. The human capial of he individual is he presen value of he enire fuure labor income sream. In a complee marke wih risk-neural probabiliy measure Q, he human capial is unique and given by L = L, r, y = E Q [ T e s ru du y s ds ] = y E Q [ T e s ru du y s ds y ] y F, r, 7

12 using he fac ha he disribuion of ys y is independen of y. For general funcions µ Y and σ Y, F can be found by solving a PDE. We specialize o he case µ Y r, = µ Y + br, σ Y r, = σ Y 2.9 for deerminisic funcions µ Y and σ Y, where he nex heorem gives a closed-form soluion for F, r. This specificaion allows for he well-known life-cycle paern in expeced income growh and income volailiy, see e.g. Cocco, Gomes, and Maenhou 25, and also for a business-cycle variaion in he expeced income growh via he relaion o he real ineres rae, see e.g. Munk and Sørensen 28. Theorem 2.1 Human capial When labor income is given by 2.8 and 2.9, he human capial is L, r, y = y F, r wih where Ã, s = κ r + σ r λ B 1 b s B κs κ T F, r = 1 { T } e Ã,s 1 bbκs r ds, 2.1 ρ Y B σ r 1 b 1 2 σ2 r1 b 2 1 κ 2 [s 2B κs + B 2κ s ] s s σ Y ub κ s u du s µ Y u du + λ Y σ Y u du, and λ Y is defined in A.3 in Appendix A. The expeced fuure income rae is { E [Y ] = Y exp µ Y u du + br + b r r + bσ2 r 2κ 2 B κ b2 σr 2 } 4κ B κ 2 bσ r ρ Y B σ Y ub κ u du, 2.11 and expeced fuure human capial is E [L, r, Y ] = F E [Y ] F, re [Y ], where F is given in A.1. For a proof, we refer he reader o Appendix A. Wealh dynamics. The individual s angible wealh a any ime is denoed by X and defined as he value of his curren posiion in he money marke accoun, he bond, he sock, and REITs, plus he value of he house owned by he individual. Le π S and π B denoe he fracion of angible wealh invesed in he sock and he bond, respecively, a ime. Le ϕ o and ϕ r denoe he unis of housing owned and rened, respecively, a ime. Le ϕ R denoe he number of shares in REITs owned a ime. The wealh invesed in he money marke accoun is hen X 1 π S π B ϕ o + ϕ R H. Finally, le c denoe he rae a which he perishable good is consumed a ime. The dynamics of 8

13 angible wealh is hen dx = π S X ds S + π B X db B + [X 1 π S π B ϕ o + ϕ R H ] r d + ϕ o dh + ϕ R dh + νh d ϕ r νh d c d + 1 { T } Y d ] = [X r + π S λ S σ S + π B λ B σ B + ϕ I λ Hσ H H ϕ C νh c + 1 { T } Y d π S X ρ SB σ S + π B X σ B + ϕ I H ρ HB σ H dw r + π S X σ S 1 ρ 2SB + ϕ IH ˆρ HS σ H dw S + ϕ I H ˆρ H σ H dw H, where ϕ C ϕ o + ϕ r, ϕ I ϕ o + ϕ R, 2.13 so ha ϕ C is he oal unis of housing occupied by and hus providing housing services o he individual and ϕ I is he oal unis of housing invesed in eiher physically or indirecly hrough REITs. The wealh dynamics and he welfare of he individual are hus only affeced by ϕ C and ϕ I so ha, in general, here will be one degree of freedom. To obain a unique soluion we will have o fix one of he hree conrol variables ϕ o, ϕ r, and ϕ I. Preferences. We use a sochasic differenial uiliy or recursive uiliy specificaion for he preferences of he individual so ha he uiliy index V ω associaed a ime wih a given conrol process ω = c, ϕ o, ϕ r, ϕ R, π S, π B over he remaining lifeime [, T ] is recursively given by ] V ω [ T = E f zu ω, Vu ω du + V T ω Here zu ω = c β uϕ 1 β Cu is he weighed composie consumpion a ime u wih β, 1 defining he relaive imporance of he wo consumpion goods, where ϕ C = ϕ o + ϕ r as in A uni of housing is assumed o conribue idenically o he direc uiliy wheher owned or rened. We assume ha he so-called normalized aggregaor f is defined by δ 1 1/ψ fz, V = z1 1/ψ [1 ]V 1 1/θ δθv, for ψ 1 1 δv ln z δv ln [1 ]V, for ψ = where θ = 1 /1 1 ψ. The preferences are characerized by he hree parameers δ,, ψ. I is well-known ha δ is a ime preference parameer, > 1 reflecs he degree of relaive risk aversion owards aemporal bes on he composie consumpion level z in our case, and ψ > reflecs he elasiciy of ineremporal subsiuion EIS owards deerminisic consumpion plans. 6 The erm V T ω represens erminal uiliy and we assume ha V T ω = ε 1 Xω T 1, where ε and XT ω is he erminal wealh induced by he conrol process ω. The special case where ψ = 1/ so ha 6 I is also possible o define a normalized aggregaor for = 1 and for < < 1 bu we focus on he empirically more reasonable case of > 1. 9

14 θ = 1 corresponds o he classic ime-addiive uiliy wih he Cobb-Douglas-syle insananeous 1. uiliy funcion c β uϕ 1 β Cu Le A denoe he se of admissible conrol processes ω over he remaining lifeime [, T ]. Consrains on he conrols are refleced by A. A any poin in ime < T, he individual maximizes V ω over all admissible conrol processes given he values of he sae variables a ime. The value funcion associaed wih he problem is defined as J, x, r, h, y = sup { V ω ω u u [,T ] A, X = x, r = r, H = h, Y = y } 2.16 ignoring y in he reiremen phase [ T, T ]. Throughou he analysis we solve he relevan uiliy maximizaion problems applying he dynamic programming principle; see Duffie and Epsein 1992 on he validiy of his soluion echnique in he case of sochasic differenial uiliy. The above uiliy specificaion is he coninuous-ime analogue of he Kreps-Poreus-Epsein-Zin recursive uiliy defined in a discree-ime seing. Boh he discree-ime and he coninuous-ime versions have been applied in a few recen sudies of uiliy maximizaion problems involving a single consumpion good, cf. Campbell and Viceira 1999, Campbell, Cocco, Gomes, Maenhou, and Viceira 21, and Chacko and Viceira 25, and was also applied in a wo-good seing relaed o ours by Yao and Zhang 25b. Oher recen papers modeling relaed wo-good uiliy maximizaion problems apply he classic ime-addiive uiliy wih a Cobb-Douglas-syle insananeous uiliy funcion, cf. Cocco 25, Yao and Zhang 25a, and Van Hemer 27. Benchmark parameer values. When we illusrae our findings in he following secions, we will use he parameer values lised in Table 1 unless oherwise noed. Our benchmark parameer values are roughly in line wih hose used in similar sudies referred o in he inroducion. In our illusraions we assume consan µ Y and σ Y. This allows us o focus on undersanding he impac of he sae variables and heir ineracions on he life-cycle behavior and disregards he more mechanical ime-dependence, which is of secondary imporance. Whenever we need o specify he bond ha he individual invess in, we ake i o be a 2-year zero-coupon bond. Unless menioned oherwise, he resuls repored presume ha he curren value of he shor-erm ineres rae is idenical o he long-erm average, r = r. Whenever we need o use levels of curren or fuure house prices, wealh, labor income ec., we use a uni of USD 1 scaled by one plus he inflaion rae in he perishable consumpion good. For concreeness we hink of houses as being fully represened by he number of square fee of average qualiy and locaion and will laer use an iniial value of h = 2 corresponding o USD 2, for a house of 1, square fee. When he shor rae is a is long-erm average, he expeced growh rae 7 Wih ψ = 1/, he recursion 2.14 is saisfied by V ω [ T = δ E e δu 1 1 z1 u du + 1 ] δ e δt ε 1 Xω T 1, which is a posiive muliple of he radiional ime-addiive power uiliy specificaion. Noe ha ε = δ would correspond o he case where uiliy of a erminal wealh of X will coun roughly as much as he uiliy of consuming X over he final year. 1

15 of house prices is a modes.7% per year. This value may seem low given he house price inflaion in mos developed counries over he las decade, bu i is in fac very reasonable considering house price movemens over a longer period, cf. he discussions in Cocco 25 who assumes an expeced growh rae of 1% and Yao and Zhang 25a who use %. [Table 1 abou here.] 3 Soluion wih fully flexible housing decisions Assume for now ha he individual can coninuously and coslessly adjus boh he number of housing unis consumed and he number of housing unis invesed in. We shall refer o his siuaion as fully flexible housing decisions. Due o 2.13, we can assume ha he individual never has any direc ownership of housing unis bu coninuously adjuss he invesmen in REITS o obain he desired housing invesmen level and coninuously adjus he number of housing unis rened o achieve he desired housing consumpion level. Alernaively, we can disregard REITs and assume a coninuously adjused direc ownership of housing unis admiedly, ha may involve subsanial ransacions coss excluded from he heoreical framework of his paper, as well as a coninuously adjused rening posiion. In Appendix B we demonsrae ha he value funcion under fully flexible housing decisions can be separaed as J, x, r, h, y = 1 1 g, r, h x + yf, r 1, 3.1 where g solves a parial differenial equaion PDE. This form of he value funcion has also been found in many simpler cases. The oal iniial wealh of he individual is he sum of he angible wealh x and he human capial which, according o Theorem 2.1, equals yf, r wih F given by 2.1. As in he exising soluions o similar, bu simpler, problems sudied in he lieraure, he g funcion is deermined by he assumed asse price dynamics and will generally depend on variables sufficien o describe relevan variaions in he invesmen opporuniy se; see, e.g., Liu 27. Long-erm invesors will generally wan o hedge variaions in invesmen opporuniies as capured by he shor-erm ineres rae and he maximum Sharpe raio, which ogeher define he locaion of he insananeous mean-variance efficien fronier, cf. Nielsen and Vassalou 26. Since λ B, λ S, and λ H are assumed consan, here are no variaions in he maximum Sharpe raio, so he shor-erm ineres rae alone drives invesmen opporuniies. In addiion, a long-erm invesor who can conrol her consumpion of muliple goods affecing her uiliy will wan o hedge variaions in he relaive prices of hose consumpion goods. In our model, he relaive price of he wo consumpion goods is given by H. This explains why g is a funcion of r and h in our seing. In erms of he funcions g and F, he opimal fracions of angible wealh invesed in he sock 11

16 and he bond are π S = 1 ξ S x + yf σ Y ζ S yf σ S x σ S x, 3.2 π B = 1 ξ B x + yf σy ζ B yf σ B x x σ r yf F r σ r g r x + yf, 3.3 σ B x F σ B g x σ B respecively, while he opimal unis of housing invesed in physically or hrough REITs are ϕ I = 1 ξ I x + yf σ H h σ Y ζ I σ H yf h + x + yf g h g. 3.4 The consans ξ B, ξ S, ξ I are defined in B.12-B.14 in Appendix B in erms of he marke prices of risk λ B, λ S, λ H and he pairwise correlaions beween prices on he bond, he sock, and he house. The consans ζ B, ζ S, ζ I are defined in A.4-A.6 in Appendix A in erms of he pairwise correlaions beween he bond, he sock, he house, and he labor income. The firs erms in 3.2, 3.3, and 3.4 reflec he speculaive demand well-known from he saic mean-variance analysis and are deermined by wealh, relaive risk aversion, variances and covariances, and he marke prices of risk. The second erms in he equaions reflec an adjusmen of he invesmens o he risk profile of human wealh. We can hink of he individual firs deermining he desired exposure o all he exogenous shocks i.e. he sandard Brownian moions W r, W S, and W H and hen adjusing for he exposure implici in he human wealh in order o obain he desired exposure of he explici invesmens owards he shocks. The appropriae adjusmen is deermined by he insananeous correlaions beween he asses and he labor income hrough he consans ζ B, ζ S, ζ I. In addiion, human wealh is discouned fuure labor income and herefore ineres rae dependen. From 2.1, i follows ha T F r, r = 1 { T } 1 b B κ s e Ã,s 1 bbκs r ds. Hence, as long as he ineres rae sensiiviy of he expeced income growh rae b is below 1, human wealh is decreasing in he ineres rae level and is hus similar o an invesmen in he bond. If he expeced income growh rae is srongly pro-cyclical, i.e. b > 1, human wealh is increasing in he ineres rae corresponding o an implici shor posiion in he bond, which is correced for by a posiive explici demand for he bond. For furher discussion of his poin, see Munk and Sørensen 28. The ime-dependence of human wealh, as refleced by he funcion F, r, induces a non-consan opimal sock porfolio weigh. To be consisen wih he popular advice of having more socks when you have a long invesmen horizon, we need ξ S > σ Y ζ S, which obviously depends on he level of risk aversion and he income volailiy, bu also on he marke prices of risk and numerous correlaions embedded in ξ S and ζ S. The las erm in 3.3 hedges agains variaions in fuure invesmen opporuniies which are summarized by he shor-erm ineres rae and hus hedgeable hrough a bond invesmen. A leas in he wo cases below wih a closed-form soluion for g, r, we find g r /g < so ha he ineremporal hedge demand for he bond is posiive consisen wih inuiion and he exising lieraure. Finally, he 12

17 las erm in 3.4 represens a hedge agains variaions in he house price. When house prices increase, he coss of fuure housing increase. To compensae for ha, he individual can inves more in houses so ha an increase in house prices will also increase her wealh. Consisen wih ha inerpreaion, g h /g is posiive in he closed-form soluions below. An invesmen in a house is a hedge agains fuure coss of housing consumpion. The opimal consumpion rae and he opimal unis of housing consumed are given by c = η βν 1 β hk x + yf g ψ 1 1, 3.5 ϕ C = ηh k 1 x + yf g ψ 1 1, 3.6 where k = 1 ψ 1 β and η = δβ ψ βν 1 β k 1. This implies ha he opimal oal expendiure on he wo consumpion goods is k ν c + νhϕ C = δ ψ β βψ 1 h k x + yf g ψ β The individual disribues he oal consumpion expendiure o perishable consumpion and housing consumpion according o he relaive weighs β and 1 β of he goods in he preference specificaion. The opimal spending on each good is a ime- and sae-dependen fracion of he oal wealh x + yf. I can be shown ha subsiue he above expression for oal consumpion ino B.15, using he opimal sraegies, he dynamics of oal wealh W = X + Y F, r will be dw = r + 1 W λ λ g r σr λ B g + σ Hλ g h HH g ην 1 β Hk g where λ = λ B, + 1 λ dw g r g σ r dw r + H g h g σ H ρ H dw, λ S ρ SB λ B 1 ρ 2 SB, 1 ˆρ H [ λ H ρ SH ρ SB ρ HB 1 ρ 2 SB ψ 1 1 d λ S ρ ] HB ρ SH ρ SB 1 ρ 2 λ B SB is he vecor of marke prices of risk associaed wih he sandard Brownian moion W = W r, W S, W H, and ρ H = ρ HB, ˆρ HS, ˆρ H. The erm 1 λ dw reflecs he opimal risk aking in a seing wih consan invesmen opporuniies and he erm 1 λ λ in he drif gives he compensaion in erms of excess expeced reurns for ha risk. The shock erms gr g σ r dw r and H g hg σ H ρ H dw are he opimal adjusmens of he exposure o ineres rae risk and house price risk, respecively, due o ineremporal hedging of shifs in invesmen opporuniies, again wih appropriae compensaion in he drif of wealh. The raios g r /g and g h /g involve he risk aversion and he elasiciy of ineremporal subsiuion EIS of he individual. The specificaion of he funcion g, r, h depends on he EIS parameer ψ. When ψ is differen from 1, g has o saisfy he non-linear PDE B.22. However, i is apparenly only possible o solve ha PDE in closed form in he special case of power uiliy where ψ = 1/ since he PDE is hen linear. We presen and discuss ha soluion nex. When ψ = 1 and ε >, so ha he individual

18 has some uiliy from erminal wealh, g has o saisfy he PDE B.31. A closed-form soluion and he resuling opimal sraegies in ha case are discussed in Secion 3.2. For he case where he EIS parameer ψ is differen from 1 and 1/, we presen in Secion 3.3 a closed-form approximae soluion following he approach of Chacko and Viceira Power uiliy Wih ime-addiive power uiliy so ha ψ = 1/, he opimal consumpion sraegies simplify o c = η βν x + yf hk, 1 β g 3.8 k 1 x + yf ϕ C = ηh. g 3.9 The nex heorem saes he g funcion and summarizes he full soluion o he problem for power uiliy. Theorem 3.1 Soluion, power uiliy For he case where ψ = 1/, he value funcion is given by 3.1, where F is defined in 2.1 and where g, r, h = ε 1 e D T 1 BκT r + ην T 1 1 β hk d1u β e Bκu r du, 3.1 δ D τ = + 1 λ λ 2 2 τ + r + 1 σ r λ B 1 τ B κ τ κ 1 σ 2 2 r 1 2 κ 2 τ B κ τ κ 2 B κτ 2, 3.11 δ d 1 τ = λ λ 1 k 2 σ Hλ H ν k 1σ2 H τ + β r + 1 σ r λ B kσ rσ H ρ HB 1 τ B κ τ κ κ 1 β 2 σ 2 2 r 1 2 κ 2 τ B κ τ κ 2 B κτ 2, 3.12 wih λ λ = λ B ξ B + λ S ξ S + λ H ξ I. The opimal conrols are given by and In he following, we will discuss and illusrae he opimal sraegies. Figure 1 shows how he raio of opimal perishable consumpion o oal wealh, c/x + yf, r = ηβν 1 β hk /g, r, h, varies wih he lengh of he remaining life-ime. The benchmark parameers in Table 1 are applied ogeher wih an iniial uni house price of h = 2 USD per square foo and an iniial shor-erm ineres rae of r = r =.2. The four curves differ wih respec o he value of ε, which indicaes he relaive preference weighing of erminal wealh and inermediae consumpion: a erminal wealh of X will roughly conribue o life-ime uiliy ε/δ imes as much as a consumpion of X in he final year, cf. foonoe 7. For ε =, he consumpion o wealh raio goes o infiniy as he ime horizon goes 14

19 o since in ha case he individual will wan o spend everyhing before he end. The individual will annually spend around 4-5% of oal wealh on perishable consumpion goods when young, almos independenly of he value of ε. This propensiy o consume ou of oal wealh will hen gradually increase as he individual grows older. The consumpion-wealh raio is only lile sensiive o he ineres rae level and he house price level. The opimal spending on housing consumpion equals he perishable consumpion muliplied by he facor 1 β/β, which is.25 wih our benchmark parameers. [Figure 1 abou here.] Nex, we derive he expeced consumpion over he life-cycle. Assume for simpliciy ha he individual has no uiliy from erminal wealh, i.e. ε =. In his case g, r, h = ην 1 β hk G, r, where G, r = T e d1u β1 1 B κu r du, and he opimal spending on consumpion goods will be W c = β G, r, ϕ CνH = 1 β G, r, 3.13 and, in paricular, independen of he curren house price. The firs-order derivaives of g ha ener he opimal porfolio weighs are hen g h g = k h = h 1 1 β 1 g r, g = ˆβD, r, where ˆβ T = β 1/ and D, r = B κ u e d1u ˆβB κu r du /G, r. The dynamics of oal wealh in 3.7 simplifies o dw = r + 1 W λ λ + σr λ B ˆβD, r + kσ H λ H G, r 1 d λ dw + ˆβσ r D, r dw r + kσ H ρ H dw. In Appendix B.4 we compue he ime expecaion of W /G, r, which leads o he expeced spending on he wo goods over he life-cycle. Appendix B.4 also conains similar resuls for ε >. Figure 2 illusraes he expeced consumpion paern over he life-cycle. In addiion o he benchmark parameers, we have assumed an iniial angible wealh of X = 2, and an iniial income rae of Y = 2,. The figure shows he expeced expendiure on each of he wo consumpion goods on he lef scale. The expeced perishable consumpion grows from around 13, o 36, USD per year over he assumed 4 year horizon. The expeced expendiure on housing consumpion is again jus a 1 β/β muliple of he expeced perishable consumpion. The expecaion of he house price on he lef-hand side is 8 E [H ] = H exp { r + λ H σ H r imp + W r r + σ2 r 2κ 2 σ rσ H ρ HB B κ σ2 r κ } 4κ B κ The house price dynamics 2.6 implies ha H = H exp{ ru du + λ Hσ H r imp 1 2 σ2 H + σ H ρ H dwu}. Subsiuing A.9 and aking expecaions, we find he expeced house price saed in he ex. 15

20 Now we can esimae he expeced number of housing unis consumed as 1 βe [W /G, r ]/νe [H ], cf This is illusraed by he blue curve using he righ scale in Figure 2. The expeced number of housing unis consumed grows from abou 3 o 685 over he 4 year life-ime. Recall ha a housing uni can be hough o represen one square foo of housing of average qualiy so he above numbers square fee per person are of a reasonable magniude. Finally noe ha for ε > a lile consumpion over he life-cycle is given up o generae posiive erminal wealh. [Figure 2 abou here.] Concerning he opimal invesmens, noe ha g h /g > so ha he risk of higher fuure housing coss is hedged by an increased invesmen in houses, and g r /g < so ha he ineremporal hedging of shifs in invesmen opporuniies leads o a posiive bond demand. Figure 3 shows how he opimal invesmens as fracions of oal wealh vary wih he human wealh o oal wealh raio. The fracion of oal wealh invesed in he sock consiss of a consan speculaive posiion of 23.% wih an adjusmen for labor income which increases linearly from o 4.4% wih he relaive imporance of human wealh; since he auxiliary parameer ζ S is negaive, he income-moivaed adjusmen of he sock posiion is posiive. The fracion of oal wealh invesed in he bond consiss of four erms: i a consan speculaive posiion of -42.3%, ii an adjusmen due o he insananeous correlaion of income wih financial asses, which increases linearly from o 65.8% wih he human wealh o oal wealh raio, iii an adjusmen due o he dependence of human wealh on he ineres rae varying from o -35.8% as he human/oal wealh raio goes from o 1, and iv an ineremporal hedge agains ineres rae risk which amouns o 47.6% no maer how he oal wealh is decomposed. 9 The oal bond demand varies from 5.3% o 35.4% as he human/oal wealh is varied from o 1. Here, he componen iii depends on F r /F, he relaive sensiiviy of human wealh wih respec o he ineres rae. The numbers jus repored and used o generae he figure assume 2 years o reiremen in which case F r /F 1.8, bu he raio goes o as reiremen is approaching which will slighly increase he fracion of oal wealh invesed in he bond. The componen iv depends on he raio g r /g, which is approximaely -2.3 for a remaining life-ime of 4 years. The raio approaches zero relaively slowly as ime passes, which leads o a lower hedge-moivaed bond posiion. The fracion of oal wealh invesed in houses physically or financially consiss of a consan speculaive demand of 86%, an income-adjusmen erm varying from o approximaely -1% as he human/oal wealh raio goes from o 1, and an ineremporal hedge agains house price risk equal o 15% independen of wealh composiion. The large negaive income-adjusmen is due o he large posiive correlaion beween labor income and house prices. The oal invesmen in houses varies from roughly 1% wih no human wealh o roughly % wih only human wealh. [Figure 3 abou here.] 9 The hedge demands repored for he bond here and for he house invesmen below are compued assuming no uiliy of erminal wealh, bu hey are only lile sensiive o he value of ε. 16

21 For a fixed raio of human wealh o oal wealh and ε =, he fracions of oal wealh invesed in he sock and houses are independen of he remaining lifeime, whereas he fracion invesed in he bond varies quie slowly due o he ime-dependence of he raios F r /F and g r /g. The main source of variaions in porfolio weighs over he life-cycle is ha he human/oal wealh raio will decrease o zero as reiremen is approaching. According o Figure 3, he individual should herefore hrough his life increase he fracion of oal wealh invesed in he house and decrease he fracion of oal wealh invesed in he sock and in he bond. Figure 4 shows how he expeced oal wealh, human wealh, and financial wealh vary over he life-cycle again assuming an iniial financial wealh of X = 2, and an iniial labor income rae of Y = 2,. The graph is produced using an approximaion of he expeced oal wealh E [W ] as given in B.27 and he approximaion F, re [Y ] of expeced human capial, where he expeced income is given by 2.11 in Theorem 2.1. The expeced financial wealh is compued residually. When ε is assumed o be zero, all wealh is opimally consumed before he end. Human wealh dominaes iniially bu drops o zero a reiremen of course. Financial wealh is hump-shaped since saving is necessary when working in order o finance consumpion during reiremen. [Figure 4 abou here.] Figure 5 illusraes how he invesmens in he sock, he bond, and housing unis physical or hrough REITs are expeced o evolve over he life of he invesor. Early in life human wealh is he major par of oal wealh and, in accordance wih Figure 3, i is opimal o inves close o nohing in houses and subsanial amouns in socks and long-erm bonds, financed in par by shor-erm borrowing. Due o he large posiive correlaion beween house prices and labor income, he human wealh crowds ou housing invesmens. As human wealh decreases, he housing invesmen will increase. This rend coninues unil reiremen. A and afer reiremen, he housing invesmen is dominaed by he large speculaive demand which will fall owards zero as he invesor consumes ou of wealh. The expeced sock invesmen falls seadily wih age in line wih he sandard more socks when you are young advice. The bond demand is more sensiive o he composiion of wealh han he sock demand as seen from Figure 3, and his is refleced by variaion of he expeced bond invesmen over he life-cycle. Noe ha we assume ha a any dae he individual rades in a zero-coupon bond mauring 2 years laer. If we had chosen a differen bond or anoher ineres rae dependen asse, e.g. a bond fuure, he opimal invesmen in ha asse would have been a muliple of he opimal invesmen in he 2-year bond in order o obain he same overall exposure o he shocks o he shor-erm ineres rae. [Figure 5 abou here.] 17