Dynamic Asset Pricing in a System of Local Housing Markets

Transcription

1 Dyamc Asse Prcg a Sysem of Local Housg Markes By Parck Bayer, Brya Ellckso ad Paul B. Ellckso For mos people, buyg a house s oe of he mos sgfca vesme decsos of her lfemes. Ecoomss have maly focused o he cosumpo aspecs of hs process. For example, a ypcal model urba ecoomcs mgh frame he decso of where o lve as a dscree choce over a budle of housg ad eghborhood arbues such as locao, square fooage, schoolg opos, ad crme levels. The vesme sde of he problem has receved cosderably less aeo, a surprsg omsso sce housg asses comprse approxmaely wo-hrds of he average Amerca household s facal porfolo, serve a mpora role savg for rereme ad, as has become creasgly appare, ca be que rsky. Ths paper vews housg markes from a asse-prcg perspecve, usg face heory o relae he rsk premum of a housg asse (he dfferece bewee s expeced reur ad he reur for a rsk-free vesme) o s exposure o rsk. As usual face, wha maers for he rsk premum of a housg asse s s exposure o sysemac rsk, o dosycrac rsk. I our model, here are wo forms of sysemac rsk o whch housg asses are exposed: aoal rsk (whch s commo o houses everywhere) ad local rsk (whch affecs all houses wh a gve meropola area, bu owhere else). Houses are sad o be of he same ype h f hey are locaed he same meropola area ad have he same exposure o sysemac rsk. Our ma coclusos are ha (1) houses of every ype face a commo se of rsk prces ( for he aoal rsk ad m for he local rsk specfc o meropola area m) ha, ogeher wh approprae measures of exposure o rsk, Bayer: Duke Uversy, 213 Socal Sceces, Durham, NC, 27708, parck.bayer@duke.edu. B. Ellckso: UCLA, 8363 Buche Hall, Los Ageles, CA 90095, ellckso@eco.ucla.edu. P. Ellckso: Smo GSB, Uversy of Rocheser, Rocheser, NY, 14627, paul.ellckso@smo.rocheser.edu. 1 accou for he varao rsk-premums across housg ypes ad (2) he parameers measurg exposure o sysemac rsk facors ca be esmaed usg rasacos daa for repea sales of houses, daa ha are ow readly avalable. We also aalyze a verso of he model ha akes e res raher ha house values as he prmves. Ths specal case provdes some uo regardg he mpac of re ad rsk premums o he growh of house values. I. A Model of Housg Marke Rsk The seg s a colleco of N sgle-famly housg us locaed M meropola areas. Besdes meropola locao, houses are classfed o K caegores. We refer o a specfc parg h D.m; k/ as a housg ype, for example a large house Los Ageles. The model s formulaed as a sysem of sochasc dffereal equaos (SDE s) drve by a mul-dmesoal Weer process, usg as a framework he sadard muldmesoal marke model face (see Duffe (2001) or Shreve (2004)). We assume ha our aoal housg marke s observed over a me erval Œ0; T <, for example he 20-year perod Œ0; 20. The prce process of house of ype h D.m; k/ s assumed o be he soluo o he SDE (1) dv D V h hd C h db Equao (1) expresses he saaeous rae of prce apprecao dv =V of house a me as he sum of a expeced rae of prce apprecao hd ad a radom shock h db, where h (he drf) ad h (he volaly) are parameers ad db s he sochasc dffereal of a Weer process assocaed wh house. The sochasc dffereal db s ur assumed o be a lear combao of hree uderlyg rsk facors, (2) db h WD h dw C hm h dw h C hh h dw

2 2 PAPERS AND PROCEEDINGS MAY 2010 where dw, dw m ad dw are sochasc dffereals of Weer processes represeg aoal rsk W, local rsk W m specfc o meropola area m, ad dosycrac rsk W specfc o housg asse. The parameers h, hm ad hh are covarao parameers ha measure he sesvy of db o he aoal rsk facor, he local rsk facor for meropola area m ad he dosycrac rsk facor specfc o house. The volaly parameer h of equao (1) s lked o he covarao parameers of equao (2) by he followg dey, (3). h / 2 WD. h / 2 C. hm / 2 C. hh / 2 The prce process of every house s assumed o be govered by a SDE of he form gve by equaos (1) (3), all defed o a commo flered probably space. ; F ; F; P /. Face mposes equlbrum resrcos o hs colleco of asse-prce processes o by equag supply ad demad for each ype of asse or by some oher meas of relag asse prces o fudameals, bu sead by mposg he hypohess ha equlbrum every possble opporuy for arbrage has bee elmaed:.e., o self-facg porfolo comprsed of houses ad he rsk-free asse ca make a posve prof wh o rsk of loss uless he al vesme s srcly posve a.s. (.e., wh probably oe). The ga process G D.G / 2Œ0;T assocaed wh housg asse s defed by G D V C D where D WD R 0 d ad s he cash flow (e of expeses) receved by he ower of he asse a me. Thus G G0 s he sum of he capal ga V V0 ad he accumulaed e cash flow D accrug o a vesor holdg he asse over he erval Œ0;. For a ladlord, s smply he flow of real come less expeses for maeace, repars ad he lke, whch we wll refer o as e real flow. For a homeower, s he pued e real flow.1 The Fudameal Theorem of Asse Prcg assers ha, provded he housg marke elmaes all arbrage opporues, here exss a prcg process Z D.Z / 2Œ0;T such ha he 1 Esmag for homeowers s more dffcul ha for ladlords. As we wll see, o-arbrage heory provdes a way aroud hs problem. rsk-adjused ga process ZG for every housg asse s a margale:.e., for all s; 2 Œ0; T such ha s E.Z G j F s/ D Z s G s where F s s he formao se a me s for he flered probably space. ; F ; F; P / o whch all of he sochasc processes he model are defed. Whe he prce processes are as specfed equaos (1) (3), he he rsk-prcg process Z akes a smple form. I s he sochasc process geeraed by he SDE (4) dz D Z " dw C X m m dw m wh Z 0 D 1, where he summao s over all meropola areas. The fac ha ZG s a margale mples ha hs process, whch self s geeraed by a SDE, mus have zero drf. Assume he rao =V of e re o house value s he same for all housg asses of ype h ad ha hs e real yeld ı h remas cosa over me. 2 The (5) h C ı h D h C m hm for every housg ype h. I he face leraure, equaos (5), oe for each asse ype h, are called he marke-prce-of-rsk equaos. 3 The lef-had sde of equao (5) s he rsk premum of housg ype h, he expeced saaeous oal reur (.e., capal gas plus e real yeld) a me, e of he rsk-free rae. The rgh-had sde s he oal value of rsk exposure for a housg asse of ype h, he sum of he prce of aoal rsk mes he exposure h o ha rsk plus he prce m of local rsk mes he exposure hm o ha rsk. Raher ha mulplyg he ga G by Z, here s a equvale way o adjus for rsk by chagg he probably measure. The value Z T of he prcg process a me T s a Rado- Nkodym dervave d P Q =dp ha chages he rue probably measure P o a equvale 2 Shorly we provde a proof ha, f e res are geeraed by a geomerc Browa moo, he he e real yeld mus be cosa. 3 See Shreve (2004). #

3 VOL. 100 NO. 2 LOCAL HOUSING MARKETS 3 margale measure (EMM). Uder he EMM Q P he ga process G self, raher ha he rskadjused ga process ZG, s a margale:.e., for all s; 2 Œ0; T such ha s QE.G j F s/ D G s where he lde o he expecao sg dcaes ha he codoal expecao s ake wh respec o P. Q If probables are adjused for rsk, asses ca be prced as hough vesors are rsk eural, eve hough hey are o. Esablshg a coeco bewee housg value ad e real flow provdes a ce llusrao of a arbrage-based approach o asse prcg. As usual, s easer o esablsh a lk o fudameals f we assume a fe horzo, so for he mome we replace he me se Œ0; T wh he me se Œ0;1/. We ake he dscoued e real processes as he prmves of our model, demosrag below ha hs s equvale o a model whch he value processes V are he prmves. Alhough he aalyss ca be geeralzed o hadle me-varyg parameers, we assume drf ad volaly are cosa. I a more geeral model, hese e real processes mgh deped o he value households derve from lvg a parcular house, cludg s physcal feaures, local amees ad labor marke opporues. Whe expecaos are ake wh respec o he EMM, house values equal he expeced dscoued value of fuure res e of expeses. For hs reaso, s easer o aalyze he coeco bewee value ad e re uder he probably measure P. Q Leg deoe he flow of dscoued e re for house a me, suppose he process s a geomerc Browa moo geeraed by he SDE (6) d D ŒQh d C h d QB where QB s a Weer process uder P. Q The sochasc dffereal d QB s assumed o be a lear combao of aoal, local ad dosycrac rsk facors, (7) d QB D h h d QW C hm h d QW m C hh h d QW where WQ, WQ m ad WQ are Weer processes uder P Q (compare equaos (1) ad.2/ descrbg he SDE geerag he value process V ). I equao (6), Q h s he drf of dscoued e re uder P. Q Assume ha Q h < 0 ad defe he dscoued value process V by V D QE R 1 u du j F for 2 Œ0; 1/. I follows ha V D =Qh. Thus, uder he probably measure PQ he e real yeld of a house of ype h s he same for all houses of ype h, ad s me vara. Because P ad PQ are equvale measures, hs relaoshp also holds uder P:.e., (8) ı h WD V D Q h.p-a.s/ Leg ı h D Q h equao (5), (9) h D Q h C h C m hm whch offers a alerave perspecve o he marke-prce-of-rsk equao. If rsk-prces are zero (so vesors are fac rsk eural) he r C h D r C Q h : prce apprecao o houses of ype h equals he rsk-free rae plus he expeced rae of crease of e re uder he EMM. O he oher had, f rsk prces are posve ad he covarao parameers are posve, he h Q h D h C m hm > 0 House values apprecae a a more rapd rae ha Q h o compesae for he rsk. Wha happes o he process uder he rue probably measure P? Grsaov s Theorem, used o derve equao (5), mples ha! (10) d QB h D db C C ı h h d Subsug (10) o (6) ad usg (8) o smplfy, we oba d D Œ hd C h db : uder P he drf maches he drf V. Because V s a scalar mulple of, uder PQ dv D V ŒQh d C h d QB. Usg (10) o subsue for d QB yelds equao (1), he SDE for V uder he rue probably measure P. We coclude ha, hs specal case where e re follows a geomerc Browa moo, (1) he e re o value rao s cosa for

4 4 PAPERS AND PROCEEDINGS MAY 2010 all houses of he same ype ad (2) he growh rae dv =V of house value ad he growh rae d = of e res are drve by he same process. By resrcg hs fe horzo model o he erval Œ0; T, hese coclusos carry over mmedaely o our orgal fe-horzo model. II. Hedoc Reurs I coras o purely facal asses such as socks or bods, housg asses are heerogeeous ad rade a very low frequecy. However, daa o repea sales ca be used o overcome hese problems. Assume ha Œ0; T s dvded o N ervals. 1 ;, say mohs. Le R WD log.v =V s / deoe he logarhmc reur for a housg asse of ype h D.m; k/ ha sells a me s ad aga a me, where he sellg mes s; 2 Œ0; T are assumed rouded o he begg or ed of a moh. Defe WD s, he durao of repea sale, ad le M deoe he se of mohs covered by hs repea sale. Defe WD 1, he legh of moh. Smlarly, le W ad W m deoe he cremes over moh of he Weer processes W ad W m respecvely. Solvg he sochasc dffereal equao (1) s easy o show ha (11) R D h C X where h WD ad 2M r h C ". hh / 2 =2, " WD hh.w r h. WD Œ h h / 2 C. hm / 2 2 C h W C hm W m W s /, Le N h be he se of repea sales of houses of ype h D.m; k/ over he me erval Œ0; T. For D 1; 2; : : : ; N le I f2m g be a dcaor varable ha equals 1 f moh s covered by he h repea sale ad 0 oherwse. I regresso form equao (11) becomes (12) R D h C The coeffce r h NX r h I f2m g C " D1 hs regresso s he poro of he logarhmc reur for moh ha s commo o all housg asses of ype h. We refer o he mohly me seres.r h/n D1 geeraed by hese regressos as hedoc reurs. Equao (12) bears more ha a passg resemblace o he mehods used by Karl Case ad Rober Shller (1989) o cosruc housg prce dces. The dfferecg used o oba he logarhmc reur for he h repea sale allows us o corol for house-specfc fxed effecs: he cosa erm V0 specfc o repea sale drops ou of he expresso for he logarhmc reur. Thus, he level of housg prces s allowed o be que heerogeeous, eve for houses of he same ype. The homogeey we mpose oly requres ha log reurs (he creme o log prces over a fxed erval of me) for houses of he same ype are draw from he same dsrbuo. Because we see oly a sgle realzao, hs regresso he realzaos of W, W m ad W are fxed, bu here are N h radom varables ", oe for each repea sale. By defo of he Weer processes W, he expecao E" D 0 ad he dsurbaces are depedely dsrbued. Cosequely, he parameers of equao (11) ca be cossely esmaed usg OLS. Equao (11) hghlghs wo effecs of durao o he reur. Frs, he varace of he dsurbace erm for repea sale s. hh / 2. As Case ad Shller, hs heeroskedascy s easly hadled. Secod, durao has a drec effec o he mea reur: he regresso coeffce h o he durao of he h repea sale provdes a esmae of. hh / 2 =2 ad hece a esmae for hh, he volaly of he dosycrac rsk for a housg asse of ype h. I hs way, dervg equao (11) from a couous-me srucural model leads o a poeally mpora modfcao o he classc Case-Shller specfcao, a mea correco for durao. III. Esmag he Model The marke-prce-of-rsk equaos (5) provde H lear equaos (oe for each house ype) M C 1 ukows (he prce of aoal rsk ad M local rsk prces m ). Usg he mohly hedoc reurs of Seco II o esmae he covarao parameers h or hm ad he prce apprecao parameers h s sragh-

5 VOL. 100 NO. 2 LOCAL HOUSING MARKETS 5 forward. 4 Esmag e real yelds ı h s more dffcul, especally for houses occuped by homeowers raher ha reers. Foruaely, our srucural model of rsk prcg comes o he rescue. If we kow he rsk prces ad m, he esmaes of h, h ad hm allow us o esmae ı h. From equao (5) for house ype h (13) ı h D h C m hm h All we requre s ha rsk prces be defable. If H > M, he parameers h, h ad hm are defed. I follows from equaos (5) ha he rsk prces are defable provded we ca esmae e real yelds for M C1 housg ypes wh a leas oe locaed each meropola area. Esmag e real yelds for real properes s relavely easy. Furhermore, f houses of ype h are occuped by homeowers as well as reed, he he e real yeld mpued o homeowers mus equal he e real yeld eared by ladlords: all of he parameers of equao (5) excep ı h are he same, so he e real yelds mus also agree. Thus, wha we requre for defcao s M C 1 housg ypes for whch some houses are reed, a leas oe such ype for each meropola area. IV. Arbrage I s ofe assered ha arbrage prcg does o apply o housg markes because he majory of rasacos ake place bewee dvdual ower occupas ad he exsece of subsaal rasacos ad holdg coss lm he ably of oher vesors o ake advaage of arbrage opporues. Bu hs vew gores he fac ha a umber of sake-holders (baks, ladlords, developers, ad lad-owers) have clear facal eres he marke. The ecoomc decsos of hese sake-holders mpose dscple o he marke. The housg-relaed vesmes hey make compee wh alerave poeal vesmes ad cosequely face he same rsk prces. So he ladlord s problem dscples house prces segmes of he marke wh sgfca real acvy, owers of udeveloped lad ha mgh be developed dscple he reurs for properes already place, ad he 4 See our workg paper (2009) for deals. facal eress of baks dscple he offers buyers make o sellers. Complex owershp srucures arse may coexs. Fscher Black ad Myro Scholes (1973) ad Rober Mero (1974) proposed a smple model of corporae face whch sock s vewed as a call opo gvg equy-holders he rgh (bu o he oblgao) o ow he frm provded hey pay off he ousadg deb. The BSM model of corporae face seems a leas as releva o facg a house. Compared o corporaos, houses are raded very frequely, ad repea sales provde corol for asse heerogeey. Mos housg asses, perhaps eve hose owed by ladlords, are hghly leveraged, ad he deb s usually held by large suos. These suos are by far he larges sake-holders resdeal real esae, hey hold large porfolos of houses, ad hey have he ceve ad he power o make sure ha he asses backg hs deb are correcly prced. The fac ha hs deb has creasgly bee repackaged o morgage-backed secures ad (supposedly) hedged by cred-defaul swaps oly serves o reforce he vew ha housg markes are sophscaed asse markes. The rece marke collapse suggess ha our udersadg of how housg markes prce rsk s o as good as should be. Ths paper akes a sep oward mprovg ha udersadg. REFERENCES Bayer, Parck, Brya Ellckso ad Paul B. Ellckso Dyamc Asse Prcg a Sysem of Local Housg Markes, workg paper. Black, Fscher, ad Myro Scholes The Prcg of Opos ad Corporae Lables, Joural of Polcal Ecoomy 81(3): Case, Karl E., ad Rober J. Shller The Effcecy of he Marke for Sgle-Famly Homes, Amerca Ecoomc Revew 79(1): Duffe, Darrell Dyamc Asse Prcg Theory (Thrd Edo). Prceo Uversy Press, Prceo. Mero, Rober C O he Prcg of Corporae Deb: The Rsk Srucure of Ieres Raes, Joural of Face 29(2): Shreve, Seve E Sochasc Calculus for Face: II. Sprger-Verlag, New York.