Indebted Sellers, Housing Liquidity and Mortgage Standards

Transcription

1 Indebed Sellers, Housing Liquidiy and Morgage Sandards Hongfei Sun Queen s Universiy Chenggang Zhou Universiy of Waerloo Absrac We develop a racable dynamic general-equilibrium model of housing and lending o sudy he effec of morgage deb on house-selling decisions, as well as he impac of housing marke liquidiy on morgage sandards. Our model feaures direced search, long-erm morgages and limied commimen. There arise equilibrium defaul and equilibrium disribuions of housing prices and morgage deb. The model generaes endogenous responses of housing prices, liquidiy and morgage sandards upon shocks. We calibrae our model o he U.S. economy. Key findings are: Firs, seller behavior varies significanly according o one s indebedness. A seller s asking price is firs decreasing and raher fla-sloped in leverage for lower leverage levels, bu hen urns increasing and raher seep-sloped in leverage for higher leverage levels. This is consisen wih he empirical findings. Moreover, he iniial responses of houseselling decisions upon shocks vary qualiaively across households wih differen deb balances. Second, housing marke liquidiy maers for seing morgage sandards. As is observed from daa, our model generaes negaive co-movemens beween housing prices and down-paymen raios. We also explore wo oher alernaive seings respecively wih fricionless housing markes and direced search wih middlemen. These alernaives offer qualiaively differen predicions over equilibrium dynamics in many aspecs. JEL Classificaion: E3, G21, R1, R31 Keywords: Housing; Morgage; Search; Block Recursiviy Conac informaion: Sun, hfsun@econ.queensu.ca; Zhou, cmzhou@uwaerloo.ca. Earlier drafs of his paper were circulaed under he ile Housing Liquidiy and Lending Sandards by C. Zhou. We hank Allen Head and Huw Lloyd-Ellis for insighful commens and suggesions. Sun graefully acknowledges financial suppor from he Bank of Canada Governor s Award. The opinion expressed in he paper is our own and does no reflec he view of he Bank of Canada. All errors are ours. 1

2 1 Inroducion We consruc a dynamic general-equilibrium model of housing and lending o sudy he influence of household morgage deb posiion on house-selling decisions and he effec of housing marke liquidiy on morgage sandards. Two empirical observaions moivae our paper: Firs, according o??, sellers wih high leverage pos higher asking prices, wai longer on he marke, and generally ge o sell a higher prices. Second, here are negaive co-movemens beween housing prices and down-paymen raios. For example, Figure 1 illusraes ha housing prices are negaively correlaed wih he down-paymen raios for firs-ime home buyers, marginal ones among whom are mos likely affeced by he down-paymen consrain. This relaionship is paricularly prominen for he period leading up o he U.S. sub-prime morgage crisis. In his ime period, he U.S. housing marke was fas booming ye accompanied by much relaxed lending sandards, 1 e.g., lower down-paymen requiremens. 2 To help explain hese phenomena, we develop a dynamic model of fricional housing marke and long-erm morgage deb. The model gives rise o equilibrium disribuions of housing prices and morgage deb, as well as equilibrium defaul. I generaes endogenous responses of housing marke liquidiy, housing prices and morgage sandards, in addiion o consrucion and enry of new buyers. Our baseline model demonsraes ha (i) houseselling decisions criically depend on he level of home equiy involved, and (ii) liquidiy in he housing marke is an imporan facor in deermining he ease wih which households can borrow. When calibraed o he U.S. economy, our baseline model gives rise o he aforemenioned heerogeneiy in seller behaviors and he negaive correlaion. To accenuae he imporance of properly-modelled house rading, we also explore wo oher alernaive ways of modelling housing ransacions, one wih fricionless markes and he 1 Lending sandards may include he level of morgage approval raes, down-paymen raios, documen requiremens, morgage raes, ec. We focus on down-paymen raios. Many believe ha morgages wih low down-paymen requiremens are more risky because hey are more likely o be underwaer in down housing markes. 2? show ha many subprime loans in ha period were characerized by high leverage a originaion and non-radiional amorizaion schedules.? consruc a series for loan-o-value raios (LTVs) faced by firs-ime home buyers and show ha he overall LTV raio increased from 2 o 25.? use daa compiled for over 2 U.S. ciies beween 2 and 28 o find ha ineres-only (IO) morgages were used sparingly in ciies in which an elasic housing supply kep housing prices in check, bu were common in ciies wih an inelasic supply in which housing prices rose sharply and hen crashed.? documen and show ha lending sandards (denying raes) declined more in areas ha experienced larger credi booms (more applicans) and greaer price appreciaion.? find ha regions wih high laen demand from 21 o 25 experienced large relaive decreases in denial raes, increases in morgages originaed, and increases in housing price appreciaion, despie he fac ha he same regions experienced significanly negaive relaive income and employmen growh over his ime period. 2

3 oher wih fricional markes bu wihou direc maching beween house buyers and sellers. We show ha hese alernaives have drasically differen predicions over equilibrium dynamics on many aspecs. % of home purchase price 21% 19% 17% 15% 13% 11% 9 % 7 % Home Price Index (1 in 2Q1) 5 % Year % 2 % 1 % % 1% 2% Percenage change (year ago) 3 % 2 % 1 % % 1% 2% 3% Year Downpaymen 1s ime home buyers (lef axis) Case Shiller Home Price Index (righ axis) 3% Figure 1: Values and percenage changes (from one year earlier) in average firs-ime home buyers down-paymen raios and S&P/Case-Shiller U.S. Naional Home Price Index. Source: American Housing Survey (AHS) 27, 29, 211 naional daa. The key feaures of our model are direced search, long-erm morgages and limied commimen. The model focuses on households living in an area resembling a ciy. Wihin he ciy, houses are idenical and a household can own a mos one house. The housing marke consiss of a variey of submarkes, inside each of which raders are randomly mached. Each submarke is characerized by a paricular combinaion of ransacion price and maching probabiliies (respecively for buyers and sellers). Search is direced in he sense ha rading decisions of buyers and sellers are based on he rade-off beween he ransacion price and he maching probabiliy. For example, a buying household undersands ha i can enjoy a higher probabiliy of being mached wih a seller by visiing a submarke wih a higher rading price. And vice versa for a seller. Households ake on morgages o finance house purchases. A morgage is a finiehorizon long-erm conrac wih fixed ineres raes. The morgage indusry is compeiive and morgage companies earn zero expeced profi on each individual conrac. Aggregae 3

4 income flucuaions govern he in-flow of households and hus newly-enering house buyers. Households are also subjec o shocks ha force hem ino financial disress and shocks ha cause hem o relocae elsewhere ouside he ciy. Based on heir specific siuaions, households make decisions on house purchasing/selling and on wheher o defaul on morgage deb. If a household defauls, is house ges seized by he morgage company, a foreclosure flag is placed on he household s record and he household is prohibied from paricipaing in he housing marke. Laer on, he flag ges lifed randomly. In equilibrium, households defaul eiher because hey have failed o sell heir houses o obain funds o pay off due deb or because hey choose o do so sraegically due o negaive home equiy (which may occur upon a negaive income shock o he ciy). There is ex ane heerogeneiy among house sellers, which consis of home owners, consrucion firms and morgage companies (o unload invenory of foreclosed houses), and ex pos heerogeneiy among household sellers in ha hese home owners differ in heir ousanding morgage deb. Endogenous disribuions of sellers over morgage deb give rise o equilibrium disribuions of housing prices. To mainain model racabiliy, we assume ha goods are non-sorable and hus households canno save across periods. Given his assumpion, all house buyers are rendered homogeneous. Free-enry of hese homogeneous buyers ino he housing marke gives rise o he aforemenioned rade-off beween house prices and maching probabiliies. Heerogeneous sellers opimally separae hemselves ino various submarkes based on heir individual saes. As a resul, he individual decision problem is independen of he disribuion of sellers. Therefore, he model is block recursive as in? and?. This grealy improves racabiliy, which is paricularly imporan for our ask because we examine no only he seady sae bu also shor-run dynamics of he economy. By calibraing he model o he U.S. economy, we have wo general findings: (i) Seller behavior varies significanly according o one s indebedness; (ii) Housing marke liquidiy is an imporan facor in deermining he ease wih which households can borrow. Deailed summary of key resuls is given by he following: Firs, a seller s asking price is firs decreasing and raher fla-sloped in leverage for lower leverage levels, bu hen urns increasing and raher seep sloped in leverage for higher leverage levels. This resul maches o a grea exen he general curvaure of he leverage-price relaionship suggesed by empirical findings of Genesove and Mayer (1997). Second, upon ciy-level income shocks, he iniial response of house-selling decisions varies qualiaively across households a differen sages of he morgage repaymen process. Consider posiive shocks for insance. For heavily indebed households, heir choices of 4

5 selling probabiliy jump up a he impac of shocks. In conras, such choices fall relaive o he seady-sae level for less indebed sellers. On he choice of selling probabiliy (and rading price), indebed households face a rade-off beween foreclosure coss and capial gains. On one hand, choosing a high selling probabiliy implies a beer chance of avoiding foreclosure. On he oher hand, choosing a low selling probabiliy is associaed wih a high selling price, which implies higher capial gains. Households a earlier sages of repaymen have higher LTVs. The rise in home equiy induced by posiive shocks and housing price appreciaion generally reduces he levels of household LTVs. This leads o a rise in he choice of selling probabiliies as explained in he previous paragraph. Moreover, also as previously menioned, he choice of selling probabiliy (and rading price) is more sensiive o changes in LTVs for higher LTV levels while i is virually unresponsive for lower ones. Thus he increase in selling probabiliies is more significan for households wih high LTVs han hose wih low LTVs. For his reason, he consideraion of foreclosure coss dominaes for more indebed households while he consideraion of capial gains dominaes for hose less indebed. Third, housing marke liquidiy impacs lending sandards hrough affecing he expeced defaul rae of a morgage and he morgage company s expeced loss upon defaul. The inuiion is as follows. If he housing marke is more liquid, indebed households ge o avoid foreclosure more easily because hey can sell heir houses faser and use he proceeds o pay off deb. Thus, he expeced defaul rae of a morgage ends o be lower. Similarly, since morgage companies also ge o sell off foreclosed houses more quickly, he expeced carrying cos of a foreclosed house becomes lower as well. Moreover, since a hoer marke is ypically associaed wih higher rading prices, rise in he marke value of houses means a foreclosed house can be sold a a higher price, which also conribues o lower expeced carrying coss. Alogeher, higher housing marke liquidiy reduces boh he lender s exposure o defaul risk and he expeced loss on a foreclosed house. Given hese wo facors, he morgage company is willing o offer a higher volume of morgages wih more relaxed erms when he housing marke is more liquid. Fourh, here is a negaive correlaion beween housing prices and down-paymen raios. In paricular, upon a posiive ciy-wide income shock, more households are moving ino he ciy. Such posiive housing demand shock resuls in an immediae rise in marke ighness. In effec, housing prices rise iniially upon shock and exhibi shor-erm momenum (i.e., serially correlaed growh raes). In he meanwhile, he down-paymen raio falls iniially and gradually reurns o he long-run rend hereafer. Inuiively, he posiive housing demand shock resuls in an immediae increase in marke ighness. Neverheless, due 5

6 o rading fricions, he marke only absorbs he rise in ighness gradually. As a resul, he marke remains ho for a prolonged period before evenually cooling down. As is explained before, hoer marke condiions encourage morgage companies o offer loans a more lenien erms. The effecs of a negaive income shock are jus he opposie. Finally, he wo oher alernaive seings have qualiaively differen predicions over he co-movemen beween housing prices and down-paymen raios. In he seing wih fricionless housing markes, houses ge sold immediaely and hus here is no gradual adjusmen in marke liquidiy. Defaul on deb is purely due o exogenous shock causing a household o defaul, insead of any endogenous decision. Therefore, none of he aforemenioned effecs of marke liquidiy on lending sandards in he baseline model ake place in his seing. Insead, his seing leads o posiive co-movemens beween housing prices and down-paymen raios. The inuiion is he following: Wih fricionless markes, upon a posiive income shock he housing price adjuss quickly back o he long-run rend afer he iniial increase. Thus he morgage company s expeced loss upon defaul of a morgage is higher a originaion because he housing price is expeced o fall very soon. Given ha he defaul rae is exogenous, he morgage company mus require a higher down-paymen rae o compensae for he rise in expeced cos of defaul, in order o issue larger loans a he spike of housing demand. The oher alernaive seing has fricional markes bu no direc maching of buyers and sellers. This paricular search process follows?, which we name direced search wih middlemen. The housing marke consiss of a buying marke and a selling marke, where house buyers and sellers respecively are mached and rade wih real esae agens. The moivaion for? o propose his novel arrangemen is o ackle he echnical diffi culy of wo-sided heerogeneiy in a direced search environmen. We show, however, ha his heoreical shorcu does come a he cos of quaniaive performance. Two model specifics maer for he resuls: Firsly, he sizes of he agen cohor respecively sen o he wo markes are chosen by a profi-maximizing real-esae firm (RE henceforh), and here is no limi on he measure of agens ha a RE firm can send. Thus he wo cohors do generally differ in size. Such discrepancy causes housing marke liquidiy o be differen from ha generaed by direc maching of buyers and sellers. In oher words, marke liquidiy no longer direcly reflecs he fundamenal marke ighness (he acual buyer-seller raio) as in he baseline model. Secondly, he RE firm can acquire new houses from consrucion firms in a perfecly compeiive marke. This differs from our baseline model where consrucion firms also supply new housing ino he fricional marke. In oher words, he housing supply in he 6

7 Hedlund seup is much more elasic han our baseline. Upon a posiive income shock, for example, he RE firm can send more agens ino he buying marke o ake advanage of he rise in housing demand, and a he same ime, acquire more houses o mee he demand from he compeiive consrucion marke and by sending more agens o he fricional selling marke. Wih a direc cos o sending more agens o he marke and wih he cos of fricional maching, much of he rise in housing demand is me hrough acquiring more unis from he compeiive consrucion marke. As a resul, he responses of he maching probabiliy faced by a house seller o shocks are drasically differen from hose in he baseline model. Calibraion of his alernaive seing suggess he following for he co-movemen beween housing prices and down-paymen raios: Following posiive income shocks, he co-movemen is firs negaive bu hen urns posiive in laer periods afer he shock. The co-movemen is posiive given negaive income shocks. Relaed lieraure. Our model environmen is based on? (HLS henceforh), which sudies housing marke dynamics of house prices, sales, consrucion, and populaion growh. Our model inroduces long-erm morgage conracs and seller heerogeneiy ino he HLS framework. We focus on heerogeneiy in sellers rading behaviors and he housing-lending linkage. In conras, here is no heerogeneiy in house raders in? and hence no equilibrium disribuions of house raders. Moreover, he analysis in? is focused on random search, alhough direced search is considered as a robusness check. We, however, adop direced search for racabiliy reasons. Our paper is closely relaed o heoreical work on he ineracion beween housing marke liquidiy and lending sandards, such as? and?, wih he laer being he mos relaed.? quaniaively accouns for he cyclical dynamics of he housing and morgage markes. Modelling-wise, boh? and our baseline are feaured by direced search, long-erm morgages and limied commimen. Bu here are some key differences. In erms of search fricions, as we explained previously, he Hedlund seup is a direced search arrangemen using real-esae agens as middlemen o bridge ransacions and mainain model racabiliy. In conras, our model has one-sided heerogeneiy, where direced search is beween heerogeneous house sellers and homogeneous buyers. Boh seups render he model block recursive and racable. 3 Neverheless, we do find he quaniaive implicaions differ dras- 3 Direced search and soring wih wo-sided heerogeneiy is a challenging problem o ackle. There are a handful of papers ha characerize he seady sae of such an economy under cerain condiions (see?,?,?, and?).? furher shows ha dynamics of soring wih wo-sided heerogeneiy can be racable in some seings. Neverheless, our model would be inracable if allowing for endogenous saving decisions along wih endogenous morgage choices. 7

8 ically by comparing resuls from our baseline and he alernaive rading arrangemen of wo-sided direced search similar o ha in?. In erms of endogenous credi consrains,? analyzes infinie-horizon morgage conracs wih flexible ineres raes. To he conrary, our model looks a finie-horizon conracs wih fixed raes. Our choice of he conrac form is quie realisic in ha convenional U.S. morgages ypically have a 3-year erm and abou 7% of hese morgages have fixed raes alhough his percenage has decreased in recen years. Even hough i adds o complexiy of he model o work wih finie-horizon conracs, i does allow us o obain insighs on how he opimal house-rading decisions vary across households a differen sages of morgage repaymen. Anoher imporan difference beween our model and? is ha we focus on housing and morgage markes ha are local in naure while? addresses business cycle dynamics of a domesic economy wih housing and morgage markes.? shows ha expansionary moneary policy leads o higher leverage, in a New Keynesian model wih credi-consrained consumers and housing marke fricions. In his model, a decrease in morgage raes booss demand for housing purchases. Wih more buyers in he fricional marke, lenders can liquidae foreclosed houses more quickly, effecively reducing he expeced carrying cos of a foreclosed house and making lender more willing o finance a larger fracion of he house purchase. Our paper conrass wih? in several aspecs: Firs,? considers one-period deb, whereas we sudy long-erm morgages. Second, here is no equilibrium defaul and hus no equilibrium house liquidaion in?. Financial fricions ake he form of Kiyoaki-Moore ype of collaeral consrain. Lenders opimally offer deb o he exen ha defaul is prevened in equilibrium. In conras, we devoe aenion on endogenous defaul choices of indebed households. Defaul does ake place in equilibrium, where houses acually ge seized, become par of he foreclosure invenory and conribue o housing supply. As a resul, marke liquidiy no only affecs he expeced carrying cos of a foreclosed house bu also he expeced defaul rae of a morgage. Boh facors conribue o he negaive correlaion beween housing prices and down-paymen raios. Furhermore, considering long-erm morgage conracs allows us o race ou he defaul decision of indebed households a every sage of heir morgage repaymen process. Third, in erms of model deails, in? houses are divisible, housing sock is fixed and here is no consrucion secor. Our model is he exac conrary in all hese aspecs.? is a DSGE model as in?. The environmen is populaed by wo ypes of represenaive households, paien and impaien. In conras, our model feaures ex pos heerogeneiy among households. Moreover, he search process is random search in? and direced search in ours. The laer offers a grea deal of racabiliy for our environmen 8

9 wih equilibrium disribuions of house sellers. Our paper conribues o he general lieraure on housing (see? and? for deailed reviews on his lieraure). This paper expands on he growing lieraure on search fricions in he housing marke. To name a few (in addiion o? and?),?,?,?,?,?. None of hese papers, however, models morgage conracs and examines he housing-lending relaionship in focus here in our paper. The remainder of his paper is organized as follows. Secion 2 inroduces he environmen of our baseline search economy. Secion 3 formalizes he equilibrium for he baseline. Secion 4 describes he firs alernaive seing wih fricionless housing markes. Secion 5 inroduces he second alernaive seing of direced search wih middlemen. Secion 6 provides jusificaions for our calibraion choices. Secion 7 presens seady-sae resuls. Secion 8 discusses he dynamic implicaions of our model. Secion 9 concludes. 2 Model Environmen Time is infinie and discree. Each ime period is indexed by. The economy consiss of a single housing marke, considered a ciy, and he res of he world, where households decide wheher o ener he ciy a he beginning of each period. The aggregae economy is populaed by a measure Q of ex ane idenical households, which grows exogenously a ne rae µ. Each household lives infiniely and supplies one uni of labor in-elasically every period. In period, he uni of labor supplied earns income y, which is in erms of consumpion goods and follows a saionary sochasic process in log-levels. A household s period-by-period uiliy from consumpion c and housing z is given by U = u(c ) + z, (1) where z = z H if he household is he residen owner of a house, and z = oherwise. The funcion u( ) is sricly increasing, sricly concave and wice coninuously differeniable, wih he boundary properies: u() =, u( ) = and u() being suffi cienly large. All households discoun fuure wih facor β (, 1). Consumpion goods are non-sorable and here is no echnology for households o save across periods. 4 A he beginning of each period, a measure µq of new households arrive in he economy. 4 This assumpion is used o mainain racabiliy. Given his assumpion, buyers in he housing marke are homogenous because hey earn idenical labor income and do no have any savings from he previous period. As a resul, search and maching in his model is beween homogeneous buyers and heerogeneous sellers. Having homogeneous buyers significanly simplifies our analysis. 9

10 Each of hese households has a bes alernaive o enering he ciy, denoed as ε. The variable ε is independenly and idenically disribued across new households according o a saionary disribuion funcion G(ε), wih suppor [, ε]. There is a criical alernaive value ε c, a which a new household is indifferen beween wheher o ener he ciy or no. Upon enering he ciy, households are randomly and permanenly separaed ino wo ypes: buyers who value owning a house, and perpeual reners who do no value owning a house. The criical value is given by ε c = ψu + (1 ψ)w p, (2) where ψ is he exogenous fracion of buyers among newly-enering households. The respecive values U and W p are of being a buyer and a perpeual rener in he ciy. In effec, he populaion inflow depends on he value of living in he ciy, eiher as a buyer or as a perpeual rener. Houses are indivisible and idenical. Each household can own a mos one house. In he ciy, here are a large number of compeiive consrucion firms ha build new housing unis. Houses consruced in period become available a he beginning of period + 1. Each new house requires one uni of land, which can be purchased in a compeiive marke a uni price q = Q(N ) and incurs a consrucion cos k = K(N ), where N denoes he amoun of new houses buil in period and available a + 1. Purchased houses are occupied immediaely o provide housing services and owner premiums. Mainenance coss are also incurred immediaely. Houses depreciae over ime, regardless of wheher hey are occupied or no. Any depreciaion will be offse by mainenance. Le m denoe he cos of mainenance incurred by he owner o offse depreciaion every period. Non-owner households in he ciy also need dwellings. They receive housing services from landlords. To simplify analysis and focus on house rading, we assume ha landlords sell housing services compeiively a a price equal o a fixed fracion of he average concurren house price. Tha is, ren is given by R = ςp. Renal houses are no couned ino he ciy s housing sock. A he end of each period, perpeual reners experience a shock wih probabiliy π p (, 1) ha induces hem o leave he ciy. On receiving his shock, perpeual reners move ou immediaely and receive a coninuaion value, L. Oherwise hey remain as perpeual reners ino he nex period. Similarly, wih probabiliy π h (, 1) residen owners and house buyers receive shocks ha cause hem o leave he ciy a he end of each period. Upon receiving his shock, residen owners and buyers move ou immediaely and receive 1

11 he coninuaion value, L. Moving owners have vacan houses ha hey may wan o sell. In addiion, moving owners wih ousanding deb mus also decide wheher o coninue heir morgage conracs, which will be specified laer. Sellers in he housing marke consis of household sellers, consrucion firms and he morgage company (will be specified laer). Households may sell houses hey own. Consrucion firms sell houses hey have buil and he morgage company sells houses seized from defauling borrowers. The housing marke is characerized by direced search. In paricular, here are a variey of submarkes indexed by (θ, p), where θ denoes he marke ighness (i.e., he raio of he measure of buyers [B] o ha of sellers [S]) and p denoes he ransacion price. Buyers and sellers ake (θ, p) of all submarkes as given and decide which submarke o ener and search for a rade. The maching process is governed by he funcion M (B, S), which is increasing in boh argumens and has consan reurns o scale. The respecive maching probabiliies of a buyer γ(θ) and a seller ρ(θ) are given by γ(θ) = ρ(θ) = ( M (B, S) = M 1, 1 ) B θ (3) M (B, S) = M (θ, 1) = θγ(θ). S (4) There is no cos o enering a submarke. Free-enry renders a rade-off beween he house price and he maching probabiliy across submarkes. Inuiively, higher-price submarkes have lower levels of ighness so ha a buyer who is willing o pay a higher price ges compensaed by enjoying a higher chance o rade wih a seller. Households mus obain morgages o help finance a house purchase because hey canno save and he periodic labor income is no suffi cien o pay he full purchase price in any acive submarke. The morgage secor is perfecly compeiive and represened by one morgage company ha issues long-erm morgage conracs o borrowers. The morgage company is owned by risk-neural invesors who consume all profis and losses ex pos. 5 To finance is loans, he morgage company rades one-period risk-free bonds a an exogenous ineres rae i in he inernaional bond marke. In addiion, he morgage company incurs service coss (premium) φ per period as he opporuniy cos of funds loaned o households for housing purposes. The morgage conrac is a sandard fixed-rae morgage wih finie mauriy T. Le m and r m represen he size of a loan a originaion and he morgage rae, respecively. 5 An alernaive mehod is o assume all ex pos profis and losses are ransferred in a lump sum o households. This assumpion complicaes he compuaion alhough no significanly changing he qualiaive resuls. 11

12 Boh T and r m are exogenously given. For a conrac (m ; r m, T ), he consan periodic paymen x( ) is given by x(m ) = r m 1 (1 + r m ) T m. (5) Given he number of compleed paymens, he principle balance d afer n {, T 1} paymens is d(m, n + 1) = (1 + r m ) d(m, r m, n) x (m, r m ) (6) and d(m, ) = m. Since boh x( ) and d(, ) are unrelaed o afer originaion, we use (m, n) o represen he sae of an ongoing morgage. A borrower can erminae his/her morgage conrac a any ime. A erminaion is a foreclosure if he borrower canno pay off her ousanding morgage balance. Adverse shocks, such as accidens or unexpeced illness, may hi households a he beginning of each period. A residen owner wih morgage deb has probabiliy π d of experiencing his financial disress shock. A disressed residen owner mus erminae her curren morgage conrac wihin he same period, eiher by paying off he ousanding deb or by defauling. In he even of defaul, a borrower s morgage balance is se o zero and a foreclosure flag is placed on his/her credi record, i.e., f = 1. The morgage company repossesses he borrower s house, pus i in real-esae-owned (REO) invenory, and ses i up for sale in he following period. The defauling borrower receives any posiive difference beween he value of a foreclosed house and her ousanding morgage balance. Upon a successful sale, he morgage company loses a fracion χ (, 1) of he revenue as coss, e.g., legal fees. As a penaly for defauling, buyers wih f = 1 lose access o he morgage marke and are hus excluded from he housing marke. Saring he nex period, he foreclosure flag says on a buyer s record wih probabiliy π f (, 1). 6 Finally, he morgage rae is given by r m = i + φ + ϱ, where ϱ represens a risk premium. Wihou ϱ, a morgage conrac earns sricly negaive expeced profi due o he posiive probabiliy of defaul. Timing. Each period consiss of wo sub-periods. A he beginning of sub-period 1, new households wih ε ε c ener he ciy. Income shocks, financial disress shocks and shocks on he foreclosure flag are all revealed. Then he housing marke opens. Buyers and sellers decide on which submarke (p, θ) o search and rade in. The morgage secor becomes acive afer he housing marke closes. Borrowers decide wheher o defaul on heir cur- 6 According o he policies of Fannie Mae and Freddie Mac, foreclosure filings say on a borrower s credi record for a finie number of years. 12

13 Sub-period 1 Sub-period 2 +1 New enrans, Shocks on (y,d,f) Housing marke opens (buying/selling decisions) Credi marke opens (new morgages, defaul) Consumpion, morgage paymens Moving shock Figure 2: Time Line ren morgage conracs. New owners ake on morgages o finance heir purchases. In sub-period 2, households receive income, make paymens (mainenance, down paymens, morgage paymens or rens), and consume whaever is lef. A he end of he period, moving shocks are revealed for all households and hose who are hi leave he ciy immediaely. Figure 2 provides a more deailed illusraion of he iming of decisions. 3 Equilibrium in he Baseline Economy 3.1 Household value funcions We now proceed o describe household value funcions, sequenially for sub-periods 1 and 2 of a ypical ime period Households in sub-period 1 House rading decisions and morgage defaul decisions are all made in sub-period 1. We specify he value funcions for he following individuals: buyers wihou he foreclosure flag U, residen owners wih deb bu no in financial disress S (m, n), disressed residen owners S f (m, n), residen owners wihou deb S, owners ha have lef he ciy and are in deb V L (m, n), and owners ha have lef he ciy and are no in deb V L, ogeher wih he value of a vacan house in he consrucion firm s invenory V c, and he value of owning a foreclosed house V REO. In sub-period 1, a buyer wihou he foreclosure flag on her credi record will search o 13

14 buy a house. The value of such a buyer enering submarke (p, θ) is given by U = γ(θ)v o (p, m ) + (1 γ(θ))w (). (7) In he submarke, he buyer ges mached wih a seller wih probabiliy γ(θ), in which case she will proceed as a new owner wih value V o (p, m ). The loan volume m is deermined by he morgage company and will be specified laer. Wih probabiliy 1 γ(θ), he buyer fails o ge a mach and will proceed as a buyer wihou he foreclosure flag who has value W (). Wih free-enry of buyers, any acive submarke mus offer he exac same level of U o he buyers. Rewriing (7) yields ( ) θ = γ 1 U W () Ω (p), (8) V o (p, m, ) W () which is he free-enry condiion of buyers ha pins down he relaionship beween he ransacion price and he marke ighness. A residen owner who is currenly in deb (m, n) bu no hi by adverse shocks has he following value: ρ(θ s )W (max[, p s d(m, n)]) + (1 ρ(θ s )) { } S (m, n) = max (1 D )V (m, n) (p s,θ s ) max D {,1} +D W f (max[, βe V+1 REO d(m, n)]) s.. θ s = Ω (p s ). (9) Firs of all, his residen owner decides on wheher o pu her house up for sale a his poin. If yes, hen he decision is on which submarke (p s, θ s ) o rade he house in. This paricular decision is subjec o he buyers free-enry condiion (8). Wih probabiliy ρ(θ s ), he owner successfully sells her house. In his case, his individual pays off he ousanding deb wih sale proceeds, keeps any remaining profi a = max[, p s d(m, n)], and moves on as a buyer wihou he foreclosure flag who has value W (a). If she chooses no o sell her house, or has failed o sell i, he owner makes a decision on wheher o defaul on her curren morgage conrac. The respecive coninuaion values are V (m, n) (value of an indebed residen owner a he beginning of sub-period 2; will be specified laer) if no defaul and W f (max[, βe V REO +1 d(m, n)]) if defaul, where V REO +1 is he value of a vacan house in he morgage company s REO invenory and he expecaion is aken over shocks such as income shocks, adverse shocks, foreclosure flag shocks and moving shocks. In he case of defaul, his owner ges o keep any residual value of he house afer paying 14

15 off he ousanding deb d(m, n). Consider a residen owner who is jus hi by an adverse shock. 7 This owner mus erminae her morgage conrac wihin he same period. If he house ges sold, he owner receives he residual value ne of deb and hen becomes a buyer wihou he foreclosure flag W (max[, p sd d(m, n)]). Oherwise, he owner is forced o defaul and he foreclosure flag is placed on her credi record W f (max[, βe V REO +1 d(m, n)]). In his case, she will receive any residual value of he house afer he ousanding balance is covered. 8 Thus he value of a disressed residen owner wih deb d(m, n) is given by: S f (m, n) = max (p sd,θ sd ) { ρ(θ sd )W (max[, p sd d(m, n)]) + ( 1 ρ(θ sd ) ) W f (max[, βe V REO +1 d(m, n)]) s.. θ sd = Ω ( p sd). } (1) Similarly, a residen owner wihou any morgage deb decides on wheher and how o sell her house. If he house ges sold, he owner moves on as a buyer wihou he foreclosure flag of value W (p sw ). Oherwise, she moves ono he nex sub-period as an owner wihou deb V. Alogeher, he value of his owner is S = max (p sw,θ sw ) {ρ(θsw )W (p sw ) + (1 ρ(θ sw )) V } (11) s.. θ sw = Ω (p sw ). If an owner is hi by he moving shock, she mus relocae o anoher ciy immediaely and receive coninuaion value L saring he nex period. A he new locaion, she will earn income y, pay ren R, and also pay mainenance cos on he house she owns in he ciy m. If such an owner is in morgage deb when leaving he ciy, hen she mus make decisions on wheher o sell he house and o defaul on deb. The respecive scenarios of house sale and defaul are similar o wha has been described before. The value of an 7 In he even of financial disress, i is always in an owner s bes ineres o sell before going ino foreclosure if hey have posiive equiy. In an economy wih perfecly liquid housing markes, disressed owners wih posiive equiy would never defaul because hey can immediaely sell heir houses and pay off morgage deb. However, according o he RealyTrac repor, less han 5% of homeowners who go ino foreclosure have negaive equiy. Time-consuming search and maching is responsible for his observaion. The appraisal value of a house is no equal o is rue liquidaion value in decenralized housing markes. Appraisal value is ypically esimaed based on he mos recen sale prices of houses wih similar characerisics, whereas liquidaion value akes no only he price of a house bu also he probabiliy of sale ino consideraion. 8 In he curren seup, disressed residen owners can use proceeds from sales, bu no labor income, o pay off ousanding morgage deb. Relaxing his consrain would complicae he model and does no change he resuls in any significan way. 15

16 owner who has lef he ciy bu sill in deb is given by V L (m, n) ρ(θ L ) { u(max [, p L d(m, n) ] + y L R L ) + βl } + (1 ρ(θ L )) { = max (1 D L (p L,θ L ) max ) { u(y L R L x(m ) m) + βe V+1(m L, n + 1) } D L { [ +D L u(max, βe V+1 REO d(m, n) ] + y L R L ) + βl } s.. θ L = Ω ( p L). } (12) If a relocaed owner is no in deb prior o moving, her only decision is on wheher o sell he house: V L = max (p Lw,θ Lw ) { ρ(θ Lw ) { u(p LW + y L R L ) + βl } + (1 ρ(θ Lw )) { u(y L R L m) + βe V L +1 } s.. θ Lw = Ω ( p Lw). (13) Finally, he value of a vacan house in a consrucion firm s invenory V c, and he value of a foreclosed house V REO are respecively given by { V c = max ρ(θ c )p c + (1 ρ(θ c ))[ m + βe V+1] } c (14) (p c,θ c ) s.. θ c = Ω (p c ). } { V REO = max ρ(θ REO )(1 χ)p REO + (1 ρ(θ REO ))[ m + βe V (p REO,θ REO +1 REO ] } ) s.. θ REO = Ω ( p REO). (15) Noe ha he morgage company loses a fracion χ of he sale proceeds as a foreclosure cos Households in sub-period 2 We now esablish he values a he beginning of sub-period 2 of period, respecively for a perpeual rener W p, a buyer wih he foreclosure flag W f (a), a buyer wihou he foreclosure flag W (a), a residen owner wih morgage deb V (m, n), a new owner who jus purchased a house in he previous sub-period V o (p, m ), and an owner wihou deb V. There is no paricular decision o be made in his sub-period excep for consumpion. A perpeual rener remains a rener (and never seeks o purchase a house) he enire 16

17 ime she says in he ciy. The corresponding value is given by W p = u(c ) + π p βl + (1 π p )βe W p +1 (16) s.. c = y R Wih probabiliy π p, he perpeual rener is hi by he moving shock, leaves he ciy immediaely and receives he coninuaion value L. Oherwise, she moves ono he nex period as a rener. Consumpion is simply income ne of ren. A buyer wih he foreclosure flag on her credi record has no access o credi, nor he housing marke. She will remain rening unil she moves ou of he ciy or he foreclosure flag is lifed from her record. Le W f (a) be he value of such a buyer wih asse a a he beginning of sub-period 2. The balance a represens his buyer s inra-period asse holding. This balance is sricly posiive if (i) he residen owner has defauled on her morgage in he preceding sub-period 1; and (ii) he value of a foreclosed house in he REO invenory is above he ousanding morgage deb. Oherwise, a =. Thus we have [ ] W f (a) = u(c ) + π h βl + (1 π h )β π f E W+1() f + (1 π f )E U +1 (17) s.. c = y + a R Condiional on saying in he ciy, wih probabiliy π f he foreclosure flag says on his individual s record and she moves ono he following period wih expeced value W+1(). f Wih probabiliy 1 π f, he foreclosure flag is lifed and his household will coninue on wih value U +1 as a buyer searching for a house in he sub-period 1 of period + 1. A buyer wihou he foreclosure flag a he beginning of sub-period 2 is eiher a residen owner who jus successfully sold her house or a buyer who has failed o purchase a house in sub-period 1. Such a buyer may hold sricly posiive inra-period asse a, which came from sale proceeds ne of he ousanding morgage deb. She will move on wih value U +1 and paricipae in he housing marke in he nex period if no hi by he moving shock a he end of he curren period. The value of such a buyer is given by W (a) = u(c ) + π h βl + (1 π h )βe U +1 (18) s.. c = y + a R In sub-period 2 of period, a residen owner wih a morgage has he principle balance d(m, n), where n [, T 1]. The owner s periodic income is used o cover repaymen, 17

18 mainenance cos and consumpion. Le V (m, n) denoe he value of such an owner. I follows ha for n [, T 2], V (m, n) = u(c ) + z H + π h βe V+1(m L, n + 1) + (1 π h ) [ ] π d βe S+1(m f, n + 1) + (1 π d )βe S +1 (m, n + 1) s.. c = y x(m ) m (19) If he owner is hi by he moving shock, hen she will exi he ciy righ away and coninue on wih value V L +1(m, n + 1). Noe ha her morgage deb does no vanish because she has relocaed. Condiional on no relocaing, in he nex period he owner ges hi by he adverse shock wih probabiliy π d. In his case, she will coninue on as a disressed residen owner wih deb S f +1(m, n + 1). Oherwise, she eners he nex period as a non-disressed owner wih value S +1 (m, n + 1). For n = T 1, we have V (m, T 1) = u(c ) + z H + π h β V L +1 + (1 π h )βe S+1 (2) s.. c = y x(m ) m In his case, he curren period is he las ime ha he owner makes he repaymen and he deb will be fulfilled afer his. Thus she will coninue on wih value V L +1 if hi by he moving shock and wih value S +1 oherwise. A new owner who has jus purchased a house in he preceding sub-period 1 pays he difference beween he purchase price and oal deb m, i.e., he down paymen. The periodic morgage paymen sars from he following period. Thus he amoun of deb going ino he nex period is d(m, ). Le V (p, m ) denoe he value of a new owner. I follows ha V o (p, m ) = u(c ) + z H + π h βv+1(m L, ) + (1 π h ) [ ] π d βe S+1(m f, ) + (1 π d )βe S +1 (m, ) s.. c = y (p m ) m (21) Finally, owners wihou morgage deb do no suffer from adverse shocks. They say in he ciy unil hey experience moving shocks. The value of such an owner is given by V = u(c ) + z H + π h β V L +1 + (1 π h )βe S+1 (22) s.. c = y m 18

19 3.2 Consrucion firm The consrucion indusry has a large number of compeiive firms. Building a new house requires one uni of land a he price q = Q(N ) and also incurring a cos k = K(N ), where N is he measure of new houses buil in period and available a + 1. A represenaive firm s zero-profi condiion requires ha N is such ha he cos of building a house is equal o he expeced value of a vacan house for sale in period + 1. Tha is, Q(N ) + K(N ) = βe V c +1. (23) 3.3 Morgage company No arbirage implies ha he morgage company earns zero profi on each loan conrac. In paricular, he expeced reurn ne of expeced foreclosure coss should be exacly equal o he opporuniy cos of funds, which is he ineres rae i of he inernaional bonds plus he servicing premium φ. In his model, households canno save over ime. Moreover, he firs regular repaymen of all new morgages sars from he period afer he one when he morgage is iniiaed. In oher words, all new borrowers are idenical o he morgage company a he poin of iniiaing he loan. Therefore, he represenaive morgage company issues he same loan m, o all new borrowers in period, regardless of he house price hey pay. Le ι = (m ; r m ) denoe he morgage held by a residen owner. Le P ι (m, n) be he morgage s presen value a he beginning of sub-period 2 of period afer n paymens have been made up o period 1. Correspondingly, P Lι (m, n) is he presen value of he morgage held by an owner ha has relocaed elsewhere. Noe ha here n 1 for a relocaed owner because one repaymen has been made by he beginning of he firs 19

20 sub-period 2 afer he household relocaed. I follows ha for all n {, 1,, T 1}, P ι (m, n) = x(m )I {n } + I {n T 1} 1 + i + φ E π h ρ(θ+1) L min [ p+1, L d(m, n + 1) ] { D L +1(m, n) min [ βv REO +2, d(m, n + 1) ] +(1 ρ(θ L +1)) + (1 D+1(m L, n))p+1(m Lι, n + 1)] +(1 π h ) { ρ(θ sd π +1) min [ p sd +1, d(m, n + 1) ] } d + (1 ρ(θ+1)) sd min [ βv+2 REO, d(m, n + 1) ] ρ(θ s +1) min [ p s +1, d(m, n + 1) ] +(1 ρ(θ s +(1 π d ) { +1)) D +1 (m, n) min [ βv+2 REO, d(m, n + 1) ] + (1 D+1(m, n))p+1(m ι, n + 1) } } (24) and for all n {1,, T 1}, P Lι (m, n) = x(m ) + I {n T 1} 1 + i + φ ρ(θ L E +1) min [ p L +1, d(m, n + 1) ] +(1 ρ(θ L +1)) { D L +1 min [ βe V REO +2, d(m, n + 1) ] +(1 D L +1)P Lι +1(m, n + 1) } (25) where p sd +1, θ sd +1, p s +1, θ s +1, p L +1, θ L +1, D +1, D L +1 are policies ha households will follow in period +1 given morgage balance (m, n+1). Moreover, I {n } and I {n T 1} are index funcions such ha I {n } = I {n T 1} = {, if n = 1, oherwise {, if n = T 1 1, oherwise. The firs index funcion I {n } is such ha a borrower sars making regular repaymens 2

21 from he period nex o originaion. The second one I {n T 1} indicaes ha he morgage maures afer he curren repaymen is made. The presen value P ι (m, n) is equal o he period- repaymen x(m ) plus he discouned expeced value of period + 1. The laer is affeced by wheher he borrower is hi by he moving shock or he financial disress shock, and wheher he borrower decides o sell he house or o defaul given he paricular siuaion she is in. 9 If he borrower sells he house in period + 1, he amoun ha he morgage company will receive is he minimum of he sale proceeds and he ousanding deb d(m, n + 1). The value P Lι (m, n) is deermined in a similar way excep ha a relocaed borrower does no experience any moving or disress shock. Finally, he morgage company s zero-profi condiion is given by P ι (m, ) m =. (26) Thus he problem solved by he morgage company is o find he value of m, ha saisfies (26) in period. 3.4 Laws of moion We now proceed o describe he evoluion of he socks of households. We normalize he sock variables by he oal populaion Q. of period. Consider he ime poin a he beginning In per capia values, he aforemenioned sock variables are he (relaive) measures of perpeual reners F, buyers wihou he foreclosure flag B, buyers wih he foreclosure flag (and are hus excluded from he housing marke) B R, indebed owners Φ, indebed and relocaed owners Φ L, he consrucion firm s invenory Φ c, and foreclosed houses Φ REO. Period- perpeual reners consis of perpeual reners from he previous period who were no hi by he moving shock, and newly-enering perpeual reners. The measure of perpeual reners F evolves according o (1 + µ)f = (1 π f )F 1 + (1 ψ)g(ε c )µ. (27) Poenial buyers wih he foreclosure flag a he beginning of period consiss of he following households who were no hi by he moving shock a he end of period 1: 9 To compue he presen value of a morgage conrac ι a originaion, we firs compue he value for n = T 1, and hen use backward inducion o obain P ι (, ) for n [, T 2]. 21

22 buyers whose foreclosure flag sill remains on record in period and residen owners who defauled in period 1. The laer are eiher residen owners who were no hi by he financial disress shock bu chose o defaul sraegically or hose who were hi by he disress shock and forced o defaul because of failing o sell off he house. Noe in paricular ha owners wihou morgage debs, n = T, are no subjec o he defaul shock. Thus we have π f B 1 R + (1 π d ) (1 + µ)b R = (1 π h ) T 1 n= { (1 ρ(θ s 1 (m, 1 n, n)))d 1 (m, 1 n, n) Φ 1 (m, 1 n, n) +π T 1 d n= (1 ρ(θsd 1 (m, 1 n, n)))φ 1 (m, 1 n, n)} } (28) where Φ 1 (m, 1 n, n) is he measure of indebed owners who had made n periodic paymens by he beginning of 1 on a morgage wih volume m, 1 n a originaion. The measure of buyers wihou foreclosure flags a he beginning of period consiss of newly-enering buyers, previously flagged buyers whose flag jus go lifed, and nonrelocaing buyers from he previous period who paricipaed in he housing marke bu did no ge o rade. Noe ha he measure of buyers who successfully purchased a house in period 1 is calculaed as he respecive sellers in period 1, i.e., { Φ L 1(, ), Φ c, Φ REO}, imes he corresponding maching probabiliies of a seller in heir choices of submarkes. I follows ha (1 + µ)b = ψg(ε c )µ + (1 π f )B R + (1 π h ) B 1 ρ(θ Lw 1)Φ L 1 (, ) ρ(θ c 1)Φ c ρ(θ REO 1 )Φ REO T 1 n= ρ ( θ L 1 (m, 1 n, n) ) Φ L 1(m, 1 n, n) (29) Le Φ (m, n) be he measure of indebed owners who have made n periodic paymens by he beginning of period on a morgage wih volume m, n a originaion. For n >, his measure evolves according o: (1 + µ)φ (m, n) = (1 π h )(1 π d )(1 ρ(θ s 1 (m, n, n 1))) (1 D 1 (m, n, n 1))Φ 1 (m, n, n 1). (3) 22

23 Tha is, he curren period owners wih an ongoing morgage are hose indebed owners from he previous period who were hi by neiher he moving shock nor he financial disress shock, eiher did no choose o sell he house or did ry o sell he house bu failed, and did no choose o defaul sraegically. For n =, Φ (m, ) represens he measure of residen owners who jus purchased a house in he las period, have sayed in he ciy and are no hi by any disress shock. Noe ha he measure of new owners can be accouned by he measure of sellers geing mached for a rade. Thus we have (1 + µ)φ (m, ) = (1 π h ) (1 π d ) T 1 n= ρ(θs 1 (m, 1 n, n))φ 1 (m, 1 n, n) +π d T 1 n= ρ(θsd 1 (m, 1 n, n))φ 1 (m, 1 n, n) + T 1 n= ρ(θl 1 (m, 1 n, n))φ L 1 (m, 1 n, n) +ρ(θ sw 1)Φ sw 1 + ρ(θ Lw 1)Φ Lw 1 +ρ(θ c 1)Φ c + ρ(θ REO 1 )Φ REO Finally, he measure of residen owners wihou a morgage Φ (, ) is given by (1 π d )(1 ρ(θ s 1 (m, T, T 1))) (1 + µ)φ (, ) = (1 π h ) (1 D 1 (m, T, T 1))Φ 1 (m, T, T 1) +(1 ρ(θ sw 1))Φ 1 (, ). (31), (32) where residen owners who jus made he las paymen in period 1 are couned ino Φ (, ). Similarly, Φ L (m, n) is he measure of relocaed owners who have made n periodic paymens a he beginning of period. Again, he loan volume a originaion is m, n. Moreover, Φ L (, ) represens he measure of relocaed owners who are no in deb. I 23

24 follows ha (1 + µ)φ L (m, n) = (1 ρ(θ sl 1 (m, n, n 1)))(1 D L 1 (m, n, n 1)) (1 + µ)φ L (m, ) = π h Φ L 1 (m, n, n 1) +π h (1 π d )(1 ρ(θ s 1 (m, n, n))) (1 D 1 (m, n, n))φ 1 (m, n, n) (33) (1 π d ) T 1 n= ρ(θs 1 (m, 1 n, n))φ 1 (m, 1 n, n) +π T 1 d n= ρ(θsd 1 (m, 1 n, n))φ 1 (m, 1 n, n) + T 1 n= ρ(θl 1 (m, 1 n, n))φ L 1 (m, 1 n, n) +ρ(θ sw 1)Φ sw 1 + ρ(θ Lw 1)Φ Lw 1 +ρ(θ c 1)Φ c + ρ(θ REO 1 )Φ REO (1 ρ(θ sw 1))Φ 1 (, ) +(1 π d )(1 ρ(θp s 1 (m, T, T 1))) (1 + µ)φ L (, ) = π h (1 D 1 (m, T, T 1))Φ 1 (m, T, T 1) +(1 ρ(θ sl 1 (m, T, T 1)))(1 D L 1( ))Φ L 1 (m, T, T 1) (34) +(1 ρ(θ slw 1 ))Φ L 1(, ). (35) Le Φ c be he sock of houses in he represenaive consrucion firm invenory a he beginning of period. This sock includes he invenory ha did no ge sold in he previous period and he newly buil houses ha are made available in period, he laer of which is measured by N. Thus Φ c evolves according o (1 + µ)φ c = ( 1 ρ(θ c 1) ) Φ c 1 + N. (36) The sock of houses in he REO invenory a he beginning of period, Φ REO, evolves 24