Optimal Portfolio Choices and the Determination of. Housing Rents in the Context of Housing Price Uncertainty

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1 Opimal Porfolio Choices and he Deerminaion of Housing Rens in he Conex of Housing Price Uncerainy Gang-Zhi FAN * Ming PU Xiaoying DENG Seow Eng ONG Nov 015 Absrac This paper develops a uiliy indifference-based model o invesigae he pricing issue of house rens under housing price uncerainy. Our model no only allows for he crucial feaures in he housing marke, such as illiquidiy, marke incompleeness, and idiosyncraic propery risks, bu also he ineracion of invesors house enure choices wih heir financial asse holdings. Our model provides ineresing insighs ino he hedging of house resale risk and deerminaion of housing renal prices. In addiion o he parameers describing he expeced changes and volailiy on sock and house prices, we also show ha he invesors precauionary savings moive, idiosyncraic propery risks, and he correlaion beween sock and housing price have imporan implicaion for he deerminaion issue of housing renals. We empirically es he model predicions using he daa from major Asian markes and he resuls overall suppor he model predicions. Key Words: Tenure Choice, Resale Risk, Reservaion Renal Prices, Uiliy Maximizaion, Incomplee Markes * Deparmen of Real Esae Sudies, Konkuk Universiy, 10 Neungdong-ro, Gwangjin-gu, Seoul , Korea. fan10@konkuk.ac.kr. School of Insurance & Collaboraive Innovaion Cener of Financial Securiy, Souhwesern Universiy of Finance and Economics, 55 Guanghuacun Sree, Chengdu, China. puming@swufe.edu.cn. Corresponding Auhor. Economics and Managemen School, Wuhan Universiy, 99 Bayi Road, Wuhan, China, xdeng@whu.edu.cn. Deparmen of Real Esae, Naional Universiy of Singapore, 4 Archiecure Drive, Singapore, rsongse@nus.edu.sg. 1

2 Opimal Porfolio Choices and he Deerminaion of Housing Rens in he Conex of Housing Price Uncerainy 1. Inroducion Owner-occupied housing has been widely hough of as one of he mos imporan componens of household asse porfolios for mos homeowners (e.g., Flavin and Yamashia, 00), and homeownership is also herefore ofen viewed as one imporan channel o creae household wealh (e.g., Beracha and Johnson, 01). However, he recen house crisis in he U.S. have aggregaed our concern abou he effec of housing price uncerainy on household housing decisions due o he high concenraion of homeowners household wealh on residenial real esae in his counry. According o he 010 Survey of Consumer Finances (SCF), abou 67.3% of U.S. households possess heir primary residences, bu he average value of he primary residences of hese U.S. homeowners, in effec, dropped 17.6% during he period from 007 o 010. Figure 1 furher demonsraes ha alhough he average housing price in he U.S. had experienced a coninuously increasing process prior o 005, he price flucuaed obviously and showed a downward rend afer 005. However, i is noeworhy ha he average housing ren did no go down wih he housing price afer 005 and, insead, sill mainain a seady growh. This suggess he imporance and necessiy of invesigaing he role of house rening in hedging agains housing price uncerainy and of exploring he implicaion of housing price uncerainy for he deerminaion of housing renal prices in he conex of household asse porfolios. As a resul, our sudy exends Meron s (1969) opimal porfolio model o examine he deerminaion of housing renal prices in his conex. Theoreically, one migh hink ha homeowners can deermine heir housing rens according o he user coss of heir homes based on he renal equivalen approach, whereby he renal price of a home can usually be considered o be equivalen o is user coss. However, recen research has noiced he marked divergence beween

3 housing rens and user coss (Verbrugge (008) and Díaz and Luengo-Prado, 008), implying he lack of usefulness of his approach. The increasing uncerainy of real esae marke faciliaes improving our undersanding for he usefulness lack and also poses a new challenge for incorporaing he uncerainy of real esae marke ino he evaluaion of housing renal prices. On he oher hand, recenly Beracha and Johnson (01) demonsrae ha he invesmen performance of house rening is acually superior over house buying during mos of he period. Given ha housing is usually one of he mos imporan componens of household asse porfolios, his suggess ha i is possible o improve he performance of household asse porfolio by aking rening housing services ino consideraion. For hese reasons, our sudy exends he real opions based approach o look a he deerminaion of housing renal prices under housing price uncerainy. In examining he crucial role of housing in deermining opimal household porfolios, exising sudies usually pay aenion o he ineracion of owner-occupied housing wih raded financial asse holdings [see, e.g., Brueckner (1997), Flavin and Yamashia. (00), Cocco (005), Hu (005), and Yao and Zhang (005)]. Our sudy also aemps o invesigae he implicaion of house enure choice of an invesor who is exposed o subsanial housing price risk for his household asse holdings, while we develop a dynamic asse porfolio model by allowing for he sochasic evoluion of boh sock and housing price. Alhough homeownership allows he invesor o lock in fuure housing coss and hedge agains flucuaions in fuure ren paymens, he also has o be faced wih housing price risk. Higher house price risk probably lowers his invesor s willingness o own a house and increase his rening likelihood. So far, however, lile research has looked a he hedging of housing price risk and is ineracion wih house enure choice in he conex of sochasic evoluion of boh sock and housing price. Such model specificaion can make us beer capure he crucial feaure of housing price risk in order o shed new ligh on is imporan implicaion for household asse porfolios. 3

4 Homeownership can bring benefis o households in numerous counries in erms of axes, and is also found o play an imporan role in hedging agains ren flucuaions (e.g., Englund e al. 00; Sinai and Souleles, 005). However, how o hedge agains house price risk has also been one key research issue ha canno be ignored in he exising real esae lieraure. Oralo-Magné and Rady (00) show ha financial securiies such as real esae socks are a poor financial device hedging he idiosyncraic risk associaed wih residenial real esae, while household home probably plays an imporan role in hedging agains adverse flucuaions in housing prices and rens [see also Díaz and Luengo-Prado (008)]. Englund e al. (00) sress he necessiy of incorporaing home price derivaives as a caegory of poenial hedging insrumens ino real esae risk-managemen sraegies, while hey also demonsrae he imporance of financial asses such as socks, bonds and -bills in hedging residenial propery price risk over longer holding periods. Meyer and Wieand (1996) find ha idiosyncraic risk in he housing marke is an imporan deerminan for diversifying and hedging housing marke risk, and ha housing renal sraegies can exer a noiceable role in hedging agains housing price risk. More recenly, Yao and Zhang (005) emphasize ha since invesors can pariion heir housing consumpion from heir housing invesmen, rening housing services can be viewed as an imporan sraegy agains house price risks [see also Meyer and Wieand (1996) Voicu and Seiler (013)]. ** The recen real esae crisis also has furher exacerbaed our concerns on developing and using housing renal sraegies o avoid house price risks. Of course, oher hedging sraegies migh be useful in hedging agains housing price uncerainy, bu here is lile evidence supporing he usefulness in hedging agains housing price uncerainy. Rening house services provides a hedge agains subsanial housing price risk a he ime of resale. Convenionally, he opions-based pricing approach has been ** In sharp conras, Sinai and Souleles (005) also provide evidence ha housing ren volailiy has a significan posiive impac of he demand for homeownership so ha home owning can reduce a household s exposure o fuure ren flucuaions. The 007 Survey of Consumer Finances (SCF) demonsraes ha abou 31.4% of US households consume housing services via rening whereas for he res households heir consumpion of housing services are realized 4

5 employed for pricing various lease conracs due o is poenial in allowing for uncerainies involved in he leasing aciviies. However, while asse lease conracs can be viewed as compound opions, sandard opion pricing models are sill subjec o he following problems in examining renal choices in he housing marke. Firs, suppose ha capial markes are complee and fricionless, and financial asses are raded coninuously, such ha any coningen claims buil on hese asses can mee he spanning condiion. We can readily price hese claims based on non-arbirage argumens. However, housing markes are largely differen from he capial markes, and characerized by he marke fricions menioned above. Consequenly, i is difficul o find a rading sraegy o compleely replicae he payoffs of coningen claims buil on residenial real esae, and no-arbirage argumens will no longer hold rue. Second, in he sandard models coningen claims are evaluaed in he risk-neural world where all he invesors are risk preference-free. As a resul, he models do no explicily consider he effec of invesors subjecive degree of risk aversion, whereas Shilling (003) provides supporive evidence ha real esae invesors are exremely risk averse. Third, he opion-based models usually only calculae and predic he fair marke value of he coningen claims, and i is difficul in he complee marke seing o analyze he bid and ask prices of real esae lease rades, while hese rades are ypically reached hrough a bargaining process. Suppose ha a house rener or invesor seek o maximize heir expeced uiliy of erminal wealh, and he presen model evaluaes various housing renal choices based on he principle of expeced wealh uiliy equivalence insead of no arbirage argumens due o he reasons discussed above. Similar o Yao and Zhang (005), we incorporae he rening-versus-owning decision ino opimal porfolio choices, and are concerned abou which of he wo enure opions yields a higher expeced wealh uiliy. Specifically, if he rener chooses o ren a house raher han own, he expeced hrough homeownership. See, e.g., Smih (1979), McConnell and Schallheim (1983), Grenadier (1995,1996, 005), Ambrose, Hendersho and Klosek (00), Hendersho and Ward (003), Sanon and Wallace (00), Buimer and O (007), and Cho and Shilling (007). 5

6 uiliy he can obain from his opimal wealh porfolio under his choice should be a leas no lower han ha he can derive condiional on purchasing he house or anoher comparable house; vice versa. When he is indifferen, in he sense of expeced wealh uiliy, beween rening and purchasing his house, we can derive his housing reservaion renal price. The conribuion of his sudy o he relevan lieraure is apparen. I develops a uiliy indifference-based framework o value reservaion housing renal prices by allowing for a homeowner s and rener s opimal wealh porfolio choice and he effecive hedging of house resale price risk. Rosen e al. (1984) find ha housing price uncerainy plays an imporan role in a household s housing enure choice decision, and higher uncerainy in housing prices relaive o rens could lead o he reducion of he proporion of homeownership. Our model furher explores ha he effec of housing price uncerainy on he deerminaion of several housing renal prices. Alhough he convenional presen value model implies ha housing price movemens should be inerpreed by changes in housing rens, empirical resuls usually rejec he implicaion (e.g., Poerba, 1991). Our model is also parallel o he indifference pricing heory associaed wih he valuaion of coningen claims on non-radeable or illiquid asses (Musiela and Zariphopoulou, 004a, 004b). I no only explicily akes accoun of invesors precauionary savings moive, bu also invesigaes he impac of he price correlaion beween he raded risky asse and he nonradable propery. The expeced change and volailiy of boh he raded asse price and he housing price are also idenified o be imporan deerminans for he agen s housing reservaion renal prices. *** Idiosyncraic risk in he housing marke is also shown o play an imporan role in hedging agains house price risk and deermining housing renal choices. Our model firs invesigaes he opimal choice problem by focusing on he A large body of economic lieraure, boh heoreical and empirical, has also paid aenion o he imporan implicaion of uncerainies in oher influenial facors on housing enure saus and consumpion. See, e.g., Haurin (1991), Fu (1995), Robs e al. (1999), Chung and Haurin (00), Oralo-Magné and Rady (00), and Díaz and Luengo-Prado (008). *** Buimer and O (007) have shown ha propery reservaion lease prices play an imporan role in he valuaion of real esae and real esae leases. 6

7 non-insiuional economic aspecs, and hen incorporae he consideraion of oher facors such he ax sysem, as in Henderson and Ioannides (1983). The remainder of his aricle is organized as follows. Secion develops a heoreical model for valuing housing renal conracs, by considering he deerminaion of marke clearing renal prices. Secion 3 describes he daa and empirical designs. Secion 4 presens he empirical resuls. Secion 5 summarizes his aricle and draws relevan conclusions.. Uiliy Indifference Models.1 Model Seup This secion develops a uiliy indifference-based model framework for pricing house renal conracs based on he equivalen principle proposed above. Since a risk-neural pricing approach buil on he assumpion of marke compleeness is inappropriae o be uilized for he purpose of examining our research problem, we resor o a sochasic dynamic opimizaion approach. Suppose ha a risk-averse represenaive agen could be a house buyer or rener, whose opimizaion problem is o maximize his expeced uiliy of erminal wealh. We also assume ha he agen faces a house choice problem on wo muually exclusive modes of eiher purchasing a given house or rening his space in order o saisfy his consumpion or invesmen demand. As a resul we can define he reservaion housing renal price as he amoun which he agen is willing o pay for rening a piece of available housing space for a pre-specified period of ime so ha he is indifferen in he sense of expeced wealh uiliy owards buying and rening his propery. While purchasing his given house can be regarded as a ypical good consumpion behavior, i likewise is also a real esae invesmen choice. Suppose ha in addiion o he house invesmen opporuniy, he agen s wealh is held in he form of financial 7

8 asses consising of a risk-free bond and a risky raded sock. Wihou loss of generaliy, he agen s uiliy funcion is assumed o saisfy he consan absolue risk aversion (CARA) form 1 u( x) exp x, (1) where 0 represens his absolue risk aversion level. Such a uiliy specificaion allows us o derive closed-form soluions o he agen s opimizaion problem in order o deeply invesigae he effecs of uncerainy arising from capial and real esae markes on he pricing of housing renal conracs. Suppose ha he price process of he risk-free bond is governed by dr rr d, R 1 T, () 0 where r is he risk-free rae. Tha is, he risk-free bond price changes wih ime a he rae rr. However, he price of he raded sock evolves in a geomeric Brownian moion ds S d S db, S0 s 0 T, (3) where boh coefficiens and are given posiive consans, and he process B sands for a sandard Brownian moion. Under he specificaions above, he agen wih an iniial endowmen x, given a ime 0, faces an opimizaion problem of choosing a house invesmen opporuniy and allocaing he remaining endowmen beween he risk-free bond and he raded sock in order o maximize his expeced uiliy of erminal wealh. Meron (1969) considers a similar bu simplified opimal invesmen problem, where a risk averse individual needs o choose an opimal invesmen allocaion sraegy beween a riskless bond and a risky raded asse for maximizing his expeced wealh uiliy. Following his 0 is also he efficien of absolue prudence, and herefore measures he precauionary savings moive (Miao and Wang, 007). 8

9 framework, le ( s) s T denoe he agen s invesing sraegy, where s is he amoun he inves in he raded sock a ime s. Given ha he agen seeks o maximize his expeced uiliy of wealh a some fuure imet, we define his value funcion as V ( x, ) max E[ u( X ) X x], (4) ( ) s s T T where X : s T represens he wealh process. In he absence of he real esae s invesmen opporuniy, by a sandard argumen V ( x, ) saisfies he following Hamilon-Jacobi-Bellman (HJB) equaion 1 V rxv max rv V 0 x x xx, (5) subjec o equaion V ( x, T) u x. We can derive he following analyic soluion o his 1 r T r T V x, exp xe ( ). (6) Correspondingly, he opimal invesmen sraegy can be idenified hrough he firs-order condiion for (5) * rt r e. (7) Given ha in he exponenial uiliy funcion he absolue risk aversion is consan and independen of he wealh process, he sraegy * is readily found o be a deerminisic decreasing funcion of he rading horizon T, he volailiy of he raded sock and he risk aversion coefficien, and does no depend on he wealh or sock dynamics. Given ha invesing in real esae is also a possible invesmen choice for his agen, we exend Meron s (1969) analyical framework o incorporae his addiional choice ino his invesmen porfolio. Since uncerainy on housing prices is a key deerminan See, e.g., Young and Zariphopoulou (00). 9

10 for ren-versus-buy decisions as discussed above, le he value of he given dwelling evolve in he following Brownian process dp a( P, ) d b( P, ) dw (8) where a, and, b are he drif and diffusion coefficiens for he value process, and W is a new Brownian moion correlaed o B wih coefficien 1,1. As a resul, he wealh dynamics of he agen saisfies he following conrolled diffusion process ds dy r( Y ) d. (9) S Under he consideraion, he value funcion of his agen can herefore be rewrien as U( y, p, ) max E[ u( Y P ) Y y, P p]. (10) ( ) s s T T T Alernaively, he agen can ren he house for meeing his consumpion demand raher han buy. Le L be he payou rae of rens and be he amoun allocaed o he raded sock a ime. The wealh dynamics a any ime saisfy he following conrolled diffusion process L ds L dx r( X ) d Ld (11) S Correspondingly, wih an iniial endowmen x, he value funcion can be specified as L L V ( x, L, ) max E[ u( X ) X x]. (1) ( ) s s T T To build a uiliy indifference-based model, we give he following definiion Definiion 1: The renal reservaion price for he agen a ime is defined as he amoun L( x, p, ) such ha V( x, L( x, p, ), ) U( x p, p, ) (13) Tha is, he agen is willing o pay L( x, p, ) for rening he residenial space a ime such ha he is indifferen owards rening i or purchasing i wih he cos p a ime. By his definiion, if he expecs ha he lef hand side of equaion (13) will be greaer 10

11 han he righ side, he agen will choose o ren in ha rening helps him realize greaer wealh uiliy han buying. On he conrary, if he hinks ha purchase can bring him a greaer expeced wealh uiliy, he agen will choose o buy raher han ren. A. Complee Markes For exposiional convenience, we firs ake ino accoun a special case where here is an insananeous perfec posiive correlaion beween he wo Brownian moions B and db dw W, implying. Based on he principle of dynamic programming, he HJB equaion wih regard o he value funcion V can be wrien as follows: 1 V LV rxv max rv V 0 x x x xx subjec o V ( x, T) condiion for his equaion (14) u x. We can obain he opimal sraegy based on he firs-order * ( rv ) x. (15) V xx Subsiuing (15) ino equaion (14) produces he following HJB equaion V ( r) V LV rxv 0. (16) x x x V xx Since he value funcion V is smooh, i is he unique smooh soluion of he HJB equaion. As a resul, we can give he follow proposiion abou his unique soluion. Proposiion 1: Given a perfec posiive correlaion beween he sock price and housing price, under he opimal invesmen sraegy (15) and he house rening choice, he value funcion, namely he soluion o HJB equaion (16), is given by 1 L r( T ) ( r) ( T ) L V ( x, L, ) exp ( x ) e exp( ). (17) r r Proof: See Appendix A. 11

12 Compared wih he soluion (6) o Meron s (1969) problem, his resul implies ha if he agen chooses o ren and make renal paymens wih he rae L per uni ime, he needs o se aside an amoun of L r for renal paymens a ime, which can be explained as he capialized presen value of he payable ren sream. As a resul, he agen s value funcion is differeniaed from (6) derived from he Meron problem, L because i is also deermined by he remaining amoun ( x ) insead of x and r L adjused by a risk aversion-relaed erm exp( ). r On he oher hand, he perfec correlaion implies he compleeness of he house marke. In oher words, he price risk in his house marke can be compleely hedged wih he raded sock. Consequenly, one can easily derive he cerainy equivalen value for he given house a ime by discouning is expeced value under he risk neural probabiliy CE( P, ) E P, (18) T T where is usually referred o as he sae price densiy and can be expressed as ( ) 1 exp ( r r T r e ) ( T ) ( )( BT B ) (19) Based on he derived cerainy equivalen value, we can conver he agen s opimizaion problem wih he house invesmen ino he classic Meron (1969) problem. Accordingly, his value funcion in his case is rewrien as U( y, p, ) V y CE( P, ), (0) T wherev is he sandard value funcion of he Meron (1969) problem defined in (6). In he end, seing 0as he beginning ime we have he following proposiion. Proposiion : Under a perfec posiive correlaion beween he sock price and Under he risk neural probabiliy P, all he financial asses has a fixed rae of reurn equal o r such ha dp dp CE( P, ) E [ P ] E P. For more deails, see Shreve (003, chaper 5). and T T T 1

13 housing price and he opimal invesmen sraegies (15) and (7), he renal reservaion price for he agen is given by r p CE( PT,0) L rt 1 e. (1) Proof: Subsiuing (18) ino (0) and hen boh he resulan relaionship and (17) ino (13), i is sraighforward o derive he analyical soluion wih regard o L hrough simple manipulaion. This proposiion clearly shows ha he renal reservaion price is independen of he risk aversion coefficien. This implies ha under he perfec correlaion, he housing renal price is unique and independen of individual risk preference, and all he poenial users in he housing marke are willing o offer he same price for rening he residenial space. Furhermore, he price change of he given house, p CE( P T,0), is found o be a crucial deerminan for he housing renal price over he ime period (0, T). The economic implicaion is more apparen if equaion (1) is rearranged as follows L 1 e r rt p CE( P,0). () T The lef hand side of equaion () represens he presen value of he payable renal reservaion price sream over he ime period (0, T), while he righ side reflecs he price change of he given house during his period. When p CE( P T,0), he agen is exposed o higher house resale price risk, and rening house services is herefore picked as a feasible hedging sraegy agains he risk. As a consequence, he housing renal conrac can be priced according o he renal paymen rae L. However, if p CE( P T,0), equaion () will no longer hold rue, and he agen will prefer o direcly own his house. B. Incomplee Markes 13

14 Owner-occupied housing is a ypical class of differeniaed commodiies such ha i is difficul, in pracice, o find a raded financial asse whose value is compleely correlaed wih he underlying house value. As a resul, we relax he assumpion of he perfec relaedness specified above and allow he raded sock o be imperfecly correlaed wih he given house. Based on he principle of dynamic programming, he value funcion (10) saisfies he following nonlinear HJB equaion 1 1 U ryu a p, U b p, U max ru U b p, U 0 y p pp y yy yp (3) Differeniaing his equaion wih respec o produces ** ( r) U y b y, U yp (4) U U yy yy The firs erm of he righ side of (4) is of he same form as he opimal invesmen sraegy in he Meron (1969) problem, while heir value funcions may be differen. However, he second erm in he opimal sraegy represens he sraegy hedging he. price risk of he house marke. If he correlaion coefficien beween B and W,, is zero implying ha he capial and house markes are independen of each oher, hen he hedging erm vanishes because house price risk is idiosyncraic and canno be hedged using he raded sock. If 0 1, he house price risk can be parially hedged using he raded sock. This erm herefore suggess ha he volailiy of he housing price is anoher imporan deerminan for his hedging sraegy in addiion o. GivenU yy 0, a decrease in he volailiy leads o a less hedging demand for offseing he price risk in he house marke. Under he opimal sraegy (4), HJB equaion (3) can be reduced o 1 ( r) U y b p, U yp U ryu y a p, U p b p, U pp 0 (5) U yy 14

15 subjec o 1 U( y, p, T) ( y p) e. Then we have he following proposiion. Proposiion 3: Given a less perfec correlaion beween he housing price and sock price and he opimal invesmen sraegy (4), he value funcion soluion o HJB equaion (5) can be expressed as ( r) ( T ) r T P 1 T U( y, p, ) exp ye e E e 1 ( ) (1 ) 1 (6) where 1 ( r) ( r). exp ( ) ( T ) ( ZT Z ) Proof: See Appendix B. Based on he resul of Proposiion 3, we can furher derive he following proposiion. Proposiion 4: Given a less perfec correlaion beween he house price and sock price and he opimal invesmen sraegies (15) and (4), he renal reservaion price for he agen can be found o be L p e (1 ) P ln T r E rt 0e rt 1 e (1 ). (7) Proof: Subsiuing (6) and (17) ino (13), we can readily derive he analyical soluion wih regard o L hrough simple manipulaion. Given ha he erm e rt ln E 0e (1 ) P T (1 ) is he cerainy equivalen of he resale price P T of he house a ime 0, we can inerpre he crucial role of he price change of he given house over he ime period (0, T) in deermining he renal reservaion price L as in Proposiion. However, compared wih Proposiion, we find ha he housing renal price is no longer independen of he risk aversion degree or precauionary savings moive, and is affeced by he price correlaion beween he house and sock. Neverheless, when 1 and 0, he cerainy equivalen 15

16 value approaches ha under he perfec correlaion such ha he renal reservaion price is also close o he one given in (1). Equaion (7) can be also rearranged as L 1 rt (1 ) P e ln E T rt 0e pe r (1 ). (8) This suggess ha he value of housing renal conracs is no only associaed wih he parameers describing he ineres rae, renal erm, expeced reurn and volailiy of he raded sock, expeced growh and volailiy of he housing price, bu also wih he agen s precauionary savings moive and he price correlaion beween he raded sock and house. C. A Special Case For exposiional convenience, his subsecion assumes ha he housing price evolves in an arihmeic Brownian moion. **** We also firs ake ino accoun ha simplified siuaion where here is an insananeous perfec price correlaion beween he housing and sock. As a resul, equaion (8) is rewrien as dp d db (9) According o (18), he ime-0 cerainy equivalen of he resale price P T of he house is given by CE( P,0) E ( p T ( B B ). (30) T 0 T 0 We can evaluae (30) based on he risk neural pricing argumen in ha he housing price risk can be hedged compleely by rading he sock and bond. Under he risk r neural probabiliy, B B B0 is a Brownian moion, and herefore **** Sochasic differenial equaion (8) includes several special processes for asse prices, such as he geomeric Brownian moion, arihmeic Brownian moion and mean-revering process. These processes have been widely uilized in he economic lieraure. We keep o he arihmeic Brownian moion o highligh he conribuions in our paper wihou causing unnecessary complicaions ha hrow no addiional insighs. 16

17 rt r CE( PT,0) e E p T BT ( ) T rt r e p T ( ) T (31) where E denoes he expeced operaor under he risk neural probabiliy. By proposiion, we can derive he following proposiion. Proposiion 5: Given ha he housing price follow an arihmeic Brownian moion process and here is a perfec posiive correlaion beween he sock price and housing price, under he opimal invesmen sraegy (15) he renal reservaion price of he agen is given by r rt rt r L (1 e ) p e ( ) T T rt 1 e. (3) Proof: Subsiuing (31) ino (1), i is sraighforward o derive he analyical soluion wih regard o L. This proposiion demonsraes ha he renal reservaion price is dependen on he parameers describing he ineres rae, renal erm, expeced reurn and volailiy of he raded sock, expeced growh and volailiy of he housing price. Differeniaing equaion (3) wih respec o he key parameers produces he following comparaive saic resuls: L L L L L 0, 0, 0, 0, 0. p The exac comparaive saic derivaives are given in Appendix C. Holding oher hings unchanged, i is shown ha he renal reservaion price L decreases in he expeced reurn on he propery, bu increases in he volailiy of he propery reurn. The resuls are no surprising in ha higher expeced reurns on he propery sugges housing invesmen becoming more aracive han leasing, while increasing volailiy of he housing price implies a higher propery resale risk and herefore increases L. 17

18 On he oher hand, he resuls also show ha a higher expeced reurn on he raded sock leads o he higher renal reservaion price, while increasing volailiy of he sock price has a negaive effec on he renal reservaion price. This is because higher expeced sock reurns make invesing in he raded sock more aracive han invesing in he house and herefore raises he agen s willingness o ren, while increasing sock volailiy decreases his willingness. One can also readily find ha L is an increasing funcion of he iniial price p of he house, as higher iniial prices discourage he agen from housing invesmen. Also we noice ha he renal reservaion price is no direcly dependen on P T, which seems o be differen from our inuiion. Since he housing price follows a Markov process, he fuure housing price is only deermined by p, T and WT. Now we urn o he incomplee marke scenario, and allow for an imperfec correlaion beween B and W. In his scenario, he cerainy equivalen value of he house a ime 0 is given by CE( P,0) e T rt ln E 0e (1 ) P T (1 ) where 1 ( r) ( r) 0 exp ( ) T ( BT B0) and P p T W. T T, (33) Given he correlaion coefficien, we have db dw 1 dz, where Z is a new Brownian moion independen of reduced o r rt CE( PT,0) e p T T (1 ) T W. As a consequence, equaion (33) can be 1 (34) 18

19 where r is he price of marke risk in he financial marke. According o proposiion 4, we have he following resul. Proposiion 6: Given he less perfec correlaion beween he housing price and sock price and he opimal invesmen sraegies (15) and (4), if he housing price evolves in an arihmeic Brownian moion, hen he renal reservaion price for he agen can be found o be r r 1 1 e. (35) rt rt L (1 e ) p e T (1 ) T T rt Proof: Subsiuing (34) ino (7), i is sraighforward o derive he analyical soluion wih regard o L. Compared wih (3), i is shown ha he renal reservaion price is also deermined by he agen s precauionary savings moive and he price correlaion beween he sock and house as well as hose parameers idenified from (3). However, when 1and 0, (34) can be reduced o (3). Differeniaing (35) wih respec o he key parameers, we have he following comparaive saic resuls: L L L L L L L 0, 0, 0, 0, 0, 0, 0 p. The exac parial derivaives are repored in Appendix D. These resuls show ha he effecs of changes in,,,and, on he renal reservaion price L are consisen wih hose idenified above. In addiion, our resuls also show ha an increase in he iniial house price decreases he agen s willingness o buy, and hence raises L. Higher precauionary savings moive discourages he agen from invesing in he residenial propery, and herefore increases L. However, he effec of varying correlaion beween he wo risky asses on he lease reservaion price is ambiguous, and an increase in he correlaion is likely o produce opposie impacs on he lease reservaion price. More specifically, i is shown ha he lease reservaion price 19

20 r increases in he correlaion when 1, where r can be explained as he marke price of risk. Under his condiion, Figure 1 clearly shows ha he lease reservaion price rises as he correlaion increases. [Inser Figure 1] In addiion, if 0, we can direcly obain he risk-neural house renal price from (35) r r 1 e. rt rt L (1 e ) p e T T rt Define he idiosyncraic risk premium of he house marke as he difference beween a renal reservaion price and he risk-neural renal price. Then he risk premium can be derived as follow r 1 (1 ) 1 e. (36) rt I e T rt This implies ha even hough invesors chooses o ren housing services, he risk premium is sill a remarkable facor deermining heir house renal reservaion prices due o he effec of heir exposure o resale price risk in he housing marke in owning a house. Equaion (36) also shows ha he risk premium increases wih he agen s precauionary savings moive, and volailiy of housing price, bu decreases wih he price correlaion beween he house and raded sock.. Marke Clearing Renal Price Since a house renal conrac is usually reached hrough a bilaeral bargaining process, his secion invesigaes he deerminaion of he marke clearing renal price. To address his issue, we require allowing for he second marke agen in his model, namely, he owner of he given house. Suppose ha he owner is also likewise facing A similar definiion can also be found in Miao and Wang (007). 0

21 wo differen opions a iniial ime 0: eiher selling or rening ou her house. If she chooses o ren ou, he owner would receive regular renal incomes derived from her propery enan, namely, he firs agen. On he oher hand, he owner probably decides o sell up her propery and receives a lump sum paymen a ime 0. As a consequence, he renal reservaion price for he owner is ha amoun which she is willing o regularly accep for rening ou her propery over a pre-specified period of ime so ha she is indifferen in he sense of expeced wealh uiliy owards rening or selling his house. We assume ha he agen s uiliy funcion also saisfies he consan absolue risk aversion (CARA) form 1 u ( x) exp x, (37) where 0represens her absolue risk aversion level. To evaluae he owner s renal reservaion price, suppose ha if he owner sells her residenial propery a iniial ime 0, she receives a lump sum paymen P0 p; if she chooses o ren ou her propery, her payoff will be composed of wo componens: he renal incomes derived from he enan over he ime period from 0 o T, and he resale price PT of he given propery a ime T. Therefore, we have he following relaionship: T r p Le d CE P T 0 (,0) (38) where L denoes he owner s renal reservaion price, and CE (,0) PT represens he ime-0 cerainy equivalen of he propery resale price. As a resul, we give he following proposiion Proposiion 7: Given he definiion of he owner s renal reservaion price, he renal reservaion price is given by L r p CE( PT,0) rt 1 e ; (39) 1

22 when here is a less perfec correlaion beween he housing price and sock price, he renal reservaion price can be found o be L p e ln E e (1 r ) rt 0 rt 1 e (1 ) P T. (40) Proof: We can direcly obain (39) via a simple manipulaion for (38); since CE (,0) PT can be wrien as he form of (33), we can derive (40) by subsiuing CE (,0) PT ino (39). One can readily find ha he owner s renal reservaion price have he same funcion form as ha of he enan. For simpliciy, suppose ha he wo agens have differen precauionary savings moives, bu have common beliefs abou he price evoluions of he risk-free bond, and raded sock and house. As a resul, le L1 be he enan s renal reservaion price, and if he renal ransacion is carried ou we migh expec he following inequaliy holds: L L L1, (41) where L is he marke clearing renal price. This inequaliy suggess a range of renal price wihin which boh he agens are willing o ener ino a renal conrac. If L L, he owner would prefer o direcly sell ou her residenial propery raher han ener ino a renal ransacion. If L1 house insead of rening. L, he enan would choose o purchase his Since inequaliy (41) specifies he upper and lower bounds for he marke clearing renal price, his implies ha he marke power of hese wo agens play a crucial role Alernaively, we can also direcly solve he HJB equaions analyically in order o price he owner s renal reservaion prices. A similar assumpion can also be found in Wang (1996).

23 in deermining he marke clearing renal price. Le Q0,1 represen he owner s deerminisic marke power, ha is, her abiliy o raise he marke renal price independenly. Consequenly, he marke clearing renal price can be expressed as follows L QL (1 Q) L. (41) 1 When he marke power Q is equal o uniy, he owner has a full marke power such ha he highes renal price ha he enan is willing o accep is he marke clearing renal price. Reversely, when Q is equal o zero, he full marke power is possessed by he enan raher han he owner, and he marke clearing renal price is herefore he lowes renal price ha he owner is willing o accep. If 0 < Q < 1, he marke clearing renal price is deermined by he ineracion beween supply and demand forces in he house renal marke..3 Oher Consideraions The previous secions examine he choice problem by only focusing on he idenical residenial uni available o he buyer or rener and he effec of he non-insiuional economic aspecs. We can exend he examinaions wihin he framework developed above by allowing he housing unis wih differen characerisics and he impac of he mainenance coss and insiuional facors such as he ax sysem. Since individuals migh have differen invesmen and consumpion demands for housing (Henderson and Ioannides, 1983; Fu, 1991), we incorporae he difference ino he presen dynamic porfolio model. Suppose ha if he represenaive agen chooses o consume house services hrough homeownership, hen he will buy one housing uni from his owner. However, if he prefers o acquire housing services only by rening, his agen will lease a -uni house due o his differen demand for housing in he scenario. ***** Under his consideraion, his renal paymen will be ***** For simpliciy, while a house uni can be described by many housing characerisics, we make use of o represen he consumpion demand for housing, which can also be a vecor of housing characerisics. 3

24 L according o he previous discussion. As a resul, his wealh dynamics now becomes L ds L dx r( X ) d Ld. (45) S Following he same proving procedure of proposiion 1, we may derive he value funcion for his agen under rening housing services 1 L r( T ) ( r) ( T ) L V ( x, L, ) exp ( x ) e exp( ). (46) r r Subsiuing (6) and (46) ino (13), we find ha he agen s renal reservaion price can be rewrien as ln E e L p e (1 ) r rt 0 rt (1 e ) (1 ) P T. (47) This suggess ha holding oher hings unchanged, he renal reservaion price per uni is no necessarily increasing in he demand for rener-occupied housing, which usually reflecs individuals real consumpion demand for housing. In addiion, we also invesigae he effecs of he mainenance coss and propery ax facors, which usually disor house choice problem. If he agen chooses o ren a house, he does no require allowing for hese effecs. However, if he decides o purchase a house uni, hen his wealh process and value funcion will differ from hose discussed above due o hese impacs. Suppose ha boh he propery ax and mainenance coss be proporional o he price of he house, which are paid coninuously for 0 T. Le c 1 and c represen he propery ax rae and mainenance cos, respecively, and hen he wealh process follows ds dy r( Y ) d ( c c ) Pd. (48) 1 S Correspondingly, he agen s value funcion can be defined as 4

25 U( y, p, ) max E[ u( Y (1 c c ) P ) Y y, P p]. (49) ( s ) s T T 1 T I can be shown ha he value funcion saisfies he following expression where ( r) ( T ) r T c1 c P 1 T U( y, p, ) exp ye e E e exp ( ) ( T ) ( ZT Z ) 1 ( ) (1 )(1 ) 1 1 ( r) ( r). Subsiuing (50) and (17) ino (13) yields he following renal reservaion price ln E r e L 1 c c p e rt (1 e ) (1 ) (1 ) 1 0 rt 1 c1 c P T I can be shown ha he effecs of changes in he propery ax rae, c 1, and mainenance cos, c, on he renal reservaion price are unexpeced. An increase in hese wo parameers discourages he agen from buying his house, and herefore enhances his renal reservaion price. However, when he expeced price growh on his propery is high enough o compensae he adverse impac, he increase canno lower his renal reservaion price.. (50).4 The Model Predicions The model and is exensions generae ineresing implicaions for he relaions among housing ren, he housing risk, he co-movemen of he asses and invesmen. In he complee marke, equaion (17) shows ha in he presence of house invesmen, he L agen s value is a funcion of renal paymens (decreasing funcion) and furher r L adjused by a risk aversion-relaed erm exp( ). Boh equaion (1) and () show r ha he renal reservaion price is a funcion of he ineres rae, renal erm, expeced reurn and volailiy of he raded sock, expeced growh and volailiy of he housing 5

26 price. These equaions predic he following. The renal reservaion price L decreases in he expeced reurn on he propery and he volailiy of he sock price, bu increases in he volailiy of he propery reurn and expeced reurn on he raded sock. Higher expeced sock reurns make invesing in he raded sock more aracive han invesing in he house and herefore raises he agen s willingness o ren, while increasing sock volailiy decreases his willingness. The above relaions generae he following hypoheses for household invesmen: H1: The renal reservaion price is negaively associaed wih expeced reurn on he propery and posiively associaed wih he volailiy of he propery reurn. H: The renal reservaion price is negaively associaed wih he volailiy of he sock price and posiively associaed wih he expeced reurn on he raded sock. In he incomplee markes, boh equaion (7) and (8) sugges ha he value of housing renal conracs is furher dependen on he agen s precauionary savings moive and he co-movemen beween he raded sock and house. Equaion (36) shows ha he idiosyncraic risk premium increases wih he agen s precauionary savings moive, hence he renal price. Alhough invesors chooses o ren housing services, he idiosyncraic risk premium is sill a remarkable facor deermining heir house renal reservaion prices due o he effec of heir exposure o resale price risk in he housing marke in owning a house. However, he effec of varying correlaion beween he wo risky asses on he lease reservaion price seems ambiguous, i is shown ha he lease reservaion price increases in he correlaion when he marke price of risk r saisfies r 1. Hence empirically we hypohesize: H3: The renal reservaion price is posiively associaed wih he idiosyncraic risk premium. H4: The renal reservaion price is posiively associaed wih he correlaion beween he wo risky asses. 6

27 3 The sample We empirically es he model predicions using he daa from several major Asian markes. We focus on he following Asian markes: Taiwan, Hong Kong, Korea, and Singapore, given he ransparency and compleeness of hese real esae markes. Also, unlike he U.S., hese counries or regions have made less preferenial ax policies associaed wih homeownership. The sample o be used is from 1994 o 010. The housing daa are colleced from heir respecive saisics bureaus, and macro daa are rerieved from DaaSream. To es he model s implicaions on he renal price, we regress he renal price on he asses aribues and macro characerisics in he panel: (51) where RenalPrice is he renal price colleced from Saisics Bureau of respecive region repored on quarerly basis (in logarihm), is region fixed effec, and AsseAribues akes Propery Volailiy, Propery Reurn, Sock Volailiy, Sock Reurn, Asse Co-movemen, and Idiosyncraic Risk Premium respecively. We measure Propery Volailiy as he sandard deviaion of he housing price wihin he quarer, a ime-series measure for he volailiy of he propery reurn. In similar vein, we calculae he sandard deviaion of he sock price index wihin he quarer and define i as he volailiy of he raded sock reurn (Sock Volailiy). We use he realized reurn in he nex quarer as he proxy for he expeced reurn on he propery (Propery Reurn) and he raded sock (Sock Reurn). We include he conemporary housing price (Propery Price) as he proxy for he iniial housing price in he model, 7

28 since he model shows ha he renal reservaion price is no direcly dependen on he curren housing price P T, bu deermined by he iniial housing price. To measure he varying correlaion beween differen asses, we calculae he correlaion on a quarerly basis beween he housing price index reurns and sock index reurns for each region using monhly observaions, named Asse Co-movemen. To capure he idiosyncraic risk premium, we use quarer-specific unexpeced flucuaion o capure he ime-series variaion in he idiosyncraic risk of households. To do so, we firs orhogonalize he excess reurns of housing price index reurns o he excess marke reurns: where is he reurn on he housing price index for each region. is he sock price index reurn for each region. Boh index reurns are measured in excess of he (0) risk-free rae on he 1-monh Treasuries for he respecive region. The regression is conduced wih monhly daa over he full sample period, 1994 o 010. The esimaed residuals are he housing marke specific reurns orhogonal o he sock marke reurns, which we define as he idiosyncraic risk premium (Idiosyncraic Risk Premium). The esimaed coefficien on he idiosyncraic risk premium from his regression, measures he change of renal prices in response o he increase in he idiosyncraic risk premium of he households. Macro conrol variables include GDP, Ineres Rae, CPI, and Consumpion. GDP is he log difference of gross domesic produc (GDP), Ineres Rae is he 3-monh deposi rae, CPI is he log difference of CPI, and Consumpion is he log difference of privae consumpion. 8

29 In he robusness check, we analyse he impac of asses aribues on he renal price using he vecor auoregressive model for each region and esimae he impulse response funcion of differen asse aribue o he renal prices. 4 Empirical Resuls This secion repors he empirical evidence on renal price, he resuls of which overall suppor he model s predicions on he deerminans of he renal price. In Figure 4, we plo he housing ren and housing price for HongKong, Korea, Singapore and Taiwan. All figures show a salien shif around he Asian financial crisis. Boh housing price volailiy and renal price volailiy before 000 are relaively smaller han i is aferwards in Korea and Taiwan, vice versa for Hong Kong and Singapore. The house price in all hese regions flucuaed obviously. However, i is noeworhy ha he average housing ren did no flucuae as much as he housing price. This reinforces he imporance of house rening in hedging agains housing price uncerainy. [Inser Figure 4] Table 1 repors he resuls on how he propery aribues affec he renal prices. Column (1) repors he resuls wih he simples esimaion wih only conrol variables, and hey explain abou 11% of renal prices. Column () shows ha he conemporary propery prices, as a proxy for he iniial housing price, is significanly and posiively associaed wih he renal price. For each 1% increase in he housing price, he renal price is furher increased by 41.6%. Column (3) shows ha he volailiy of he propery reurn (Propery Volailiy) is significanly and posiively associaed wih he renal prices. When he volailiy of he propery reurn increases by 1%, he renal prices are shrunk by.75%. In column (4), he expeced propery reurn (Propery Reurn) is significanly and negaively associaed wih he renal prices. For each 1% of increase of he expeced propery reurn, he renal prices decreases by 13.7%. 9

30 Overall, he resuls in Table 1 suppor he hypoheses ha he renal reservaion price is negaively associaed wih expeced reurn on he propery and posiively associaed wih he volailiy of he propery reurn. (H1). [Inser Table 1] Table repors he resuls on how he asse aribues affec he renal prices. Column (1) and Column () show ha he renal reservaion price is negaively associaed wih he volailiy of he sock price and posiively associaed wih he expeced reurn on he raded sock (H). The coefficiens are all significan a 95% confidence level. For each 1% of increase of he volailiy of raded sock reurn, he renal prices decreases by 10%. For each 1% of increase of he raded sock s expeced reurn, he renal prices increases by.9%. If he invesor decides o ren house services raher han purchase, his wealh porfolio is composed of hose liquid financial asses, and he is herefore no exposed o he house price risk. The resuls show ha he invesor spo renal reservaion price is an increasing funcion of he expeced reurn on raded risky asses, bu a decreasing funcion of raded asse volailiy. This is possibly because of he effecive hedging of house resale price risk. In Column (3), we furher add he correlaion beween house and raded sock ino he specificaion, which shows a significan and posiive impac on he renal prices. Finally, in Column (4), Idiosyncraic Risk Premium is found o be significanly and posiively associaed wih he renal prices. For each 1% of increase of he idiosyncraic risk premium, he renal prices increases by.6%. This is consisen wih he agen s precauionary savings moive, as he idiosyncraic risk premium increases wih he agen s precauionary savings moive, hence he renal price. Overall, he resuls in Table suppor he hypoheses ha he renal reservaion price is dependen on he asse aribues like he raded sock volailiy, he varying correlaion beween he asses class and he idiosyncraic risk (H, H3, H4). 30

31 [Inser Table ] The robusness ess discussed in he previous Secion generae similar resuls o hose repored, which are available upon reques. 5 Conclusions This paper proposes a uiliy indifference-based model for analyzing he join decisions of household porfolio selecion, house price risk hedging and housing renal behavior under asse price uncerainies. We obain closed-form soluions o he opimal problem and carry ou comparaive saic analysis based on he soluions. Our model can provide ineresing insighs ino he join decisions and esable predicions on he deerminaion of housing renal prices. Our resuls show ha he invesor renal reservaion price is an increasing funcion of he expeced reurn on raded risky asses, bu a decreasing funcion of raded asse volailiy. However, he varying expeced reurn on residenial real esae is found o have a reverse impac on he renal reservaion price, while he effec of increasing real esae volailiy is shown o be posiive. In paricular, we find ha higher precauionary savings moive makes his invesor enhance his renal reservaion price, while he effecs of changes in he correlaion beween he raded asse and residenial real esae on he reservaion price are ambiguous. In addiion, i is also shown ha idiosyncraic risk premium in he housing marke is also a major consideraion for he invesor o deermine renal reservaion price, while under he siuaion of rening housing services he is no faced wih idiosyncraic propery risk. Moreover, our model is also exended o allow for he impacs of he mainenance coss and insiuional facors such as he ax sysem, which can disor he above findings. Neglecing he impacs can resul in an inaccurae forecas for he housing renal prices. Finally, we empirically verify he model s predicions using he panel daa from several Asians 31