Illiquidity and Pricing Biases in the Real Estate Market

Transcription

1 Illiquidiy and ricing Biases in he Real Esae arke Zhenguo Lin Fannie ae 39 Wisconsin Avenue Washingon DC 16 Kerry D. Vandell School of Business Universiy of Wisconsin adison 975 Universiy Avenue adison, WI 5376 Absrac his aer addresses he micro-analyic foundaions of illiquidiy and rice dynamics in he real esae marke by inegraing modern orfolio heory wih models describing he real esae ransacion rocess. Based on he noion ha real esae is a heerogeneous good ha is raded in decenralized markes and ha ransacions in hese markes are ofen characerized by cosly searches, we argue ha he mos imoran asecs defining real esae illiquidiy in boh residenial and commercial markes are he ime required for sale and he uncerainy of he markeing eriod. hese asecs rovide wo sources of bias in he commonly adoed mehods of real esae valuaion, which are based solely on he rices of sold roeries and imlicily assume immediae execuion. We demonsrae ha esimaed reurns mus be biased uward and risks downward. hese biases can be significan, esecially when he markeing eriod is highly uncerain relaive o he holding eriod. We find also ha real esae risk is closely relaed o invesors ime horizons, secifically ha real esae risk decreases when he holding eriod increases. hese resuls are consisen wih he convenional wisdom ha real esae is more favorable o long-erm invesors han o shor-erm invesors. hey also rovide a heoreical foundaion for he recen economeric lieraure (e.g., Gazlaff and aurin (1997, 1998), Fisher, Gazlaff, Gelner, and aurin (3), and Goezmann and eng (3)) which finds evidence of smoohing of real esae reurns. Our findings hel exlain he aaren risk-remium uzzle in real esae -- i.e., ha ex-os reurns aear oo high, given heir aaren low volailiy and can lead o he formal derivaion of adjusmens ha can define real esae s roer role in he mixed-asse orfolio. resened before he Gumann Cener Symosium on Real Asses and orfolio anagemen, Universiy of Vienna, arch 7, 6. No for quoaion wihou ermission of he auhors. he auhors wish o hank hilie Jioron, Nai-Fu Chen, and aricians in a worksho a he Universiy of California-Irvine, anonymous referees a Real Esae Economics and edior David Ling, and Shaun Bond a Cambridge Universiy for heir commens on an earlier draf. All errors remain our own.

2 Illiquidiy and ricing Biases in he Real Esae arke I. Inroducion: he Risk remium uzzle in Real Esae roer ricing, evaluaion of invesmen erformance, and allocaion of real esae in a mixed-asse orfolio has roven o be a vexing quesion. Real esae is highly heerogeneous, is hinly raded over relaively-long holding eriods, and is raded hrough a ransacions rocess ha is yically no a simulaneous-bid aucion bu insead is a sequenial bid rocess wihou recall which may involve significan ransacion coss. hus, i dislays characerisics of illiquidiy, bu a ye of illiquidiy ha may dear from ha dislayed by hinly-raded securiies. A subsanial lieraure has evolved over ime, summarized in Secion II, ha aems o correc for various of hese idiosyncraic characerisics of real esae as an asse class. he resen aer conribues o ha lieraure by exloring in a formal framework he micro-analyic foundaions ha underlie he divergen rading mechanism in real esae inegraing he radiional lieraure in modern orfolio heory wih more recen models describing he real esae ransacions rocess. Secion III will firs describe formally how real esae reurn and risk were radiionally measured and esimaed. hen we will briefly examine sources of bias in esimaion ha have been discussed in he lieraure. Finally, we will urn o he rimary conribuions of his aer: consideraion of wo sources of bias in he measuremen of risk and reurn originaing from wo disinc yes of risk (which we erm markeing eriod risk and liquidaion risk) aribuable o lack of recogniion of he fac ha (1) i is he ex ane risk and reurn execaions ha are he relevan consideraions for invesor behavior, no ex os realizaions of ransacion rices a he oin of sale, and () observed ransacion rices wih osiive markeing eriods do no reflec he underlying disribuion of roery valuaions. hese biases, reflecing he heerogeneiy of real esae asses, a ransacion rocess characerized by sequenial bidding wihou recall, and an exended and uncerain markeing eriod, could under lausible condiions accoun for much of wha has been considered o be he anomalous ricing characerisics of real esae. II. he Evoluion of Valuaion Aroaches o he easuremen and Esimaion of Real Esae Reurn and Risk Early emirical sudies in he real esae lieraure comaring real esae reurns o hose of oher asse classes consisenly concluded ha real esae as an asse class no only had exremely low volailiy, bu also exremely high risk-adjused reurns, relaive o wha would be execed from conemorary asse valuaion heory. 1 ore recen 1 During he eriod 1978 o 1998, for examle, he sandard deviaion of he Naional Council of Real Esae Invesmen Fiduciaries (NCREIF) oal reurn series was 3.66%, which is less han one-fifh ha for common socks (.8% for large ca socks and 4.4% for small ca socks), and less han half ha for long-erm bonds (8.3% for cororae bonds and 8% for governmen bonds). In erms of riskadjused reurn, NCREIF s Share raio (1.47) was more han six imes ha for bonds (beween.17 and.8) and a leas hree imes ha for boh large ca socks (.41) and small ca socks (.35) (Ibboson

3 sudies have argued ha hese esimaed reurns o real esae are less volaile han hose of common socks because real esae reurns, such as he NCREIF index, are esimaed by using boh araisal daa and ransacion rices, and araisals end o smooh real esae values and make hem less volaile (e.g. Gelner (1991), Clayon, Gelner and amilon (1), Gelner, acgregor and Schwann (3)). Afer excluding araisal daa in he NCREIF roery samle, however, Gelner and Goezman () consruced a ransacion-based NCREIF index and found ha he sandard deviaion of he ransacion-based NCREIF reurn index increased only slighly, from 3.66% o 4.6% in he eriod 1978 o 1998, bu was sill very low comared o ha for socks and bonds. In addiion, he annual reurn of he ransacion-based NCREIF index was 9.%, and is Share raio (1.7) remained exremely high. Such resuls ersisenly resened he quesion: was here a risk remium uzzle in real esae? In oher words, did conemorary real esae valuaion mehods misrice real esae s risk-adjused reurn? radiional emirical esimaes of risk and reurn for real esae and financial asses were based on observed hisorical rices. For examle, in a given ime eriod, suose he rice of an asse a ime is (,1,,..., ). hen he simles and mos commonly used formulae for esimaing he reurn and risk of his asse were rˆ σˆ i 1 i 1 i i 1 i i ( i 1 i rˆ) (1) Associaes, 5). his aern was even more ronounced for earlier eriods when common socks erformed less well. For examle, Firsenberg, Ross and Zisler (1988) found ha he sandard deviaion of sock reurns was over five imes greaer han ha of real esae reurns, and he sandard deviaion of bond reurns was hree imes ha of real esae reurns, while he average reurns of real esae were slighly higher han ha of boh socks and bonds. oag (198) develoed a roery index based on a samle of 463 unleveraged roeries and found is reurn o be 14.%, comared o common sock reurns of 3.7% reored by Ibboson and Sinquefield (198) for he corresonding ime eriod. Zerbs and Cambon (1984) found he reurn of Commingled Real Esae Funds (CREF) o be 14.%, comared o 6.5% for common socks over he same eriod. By examining he sandard deviaion of reurns, hey also found ha real esae reurns aeared far less volaile han hose for common socks and cororae bonds. For examle, he sandard deviaion of CREF reurns for was 4.7%, comared o 1.% for common socks and 7.8% for cororae bonds in he same eriod. Oher sudies ha found similar resuls for he comarable ime eriod include Ibboson and Fall (1979), who found, in he eriod 1947 o 1978, he sandard deviaion of real esae (3.5%) was abou one-fifh ha of socks (18%) and real esae s Share raio (1.31) was abou hree imes ha of socks (.38). During he eriod , Kelleher (1976) found real esae s Share raio o be even higher, a abou eigh imes ha of socks (1.75 vs..1). Some also argued ha real esae has such a high risk-adjused reurn because i ofen involves high ransacion coss. his argumen has been challenged by Kallberg, Liu and Greig (1996). 3

4 where curren rices included any curren dividends as well as caial areciaion. Due o he heerogeneiy and infrequency of rading of real esae asses, i became aaren ha we could no direcly aly he formulae above o he rices of each sold roery over ime o esimae real esae reurn and volailiy. he simles aroach adoed o correc his roblem was o comose a real esae rice index as he average of he values of he roeries being sudied. his was he aroach used in he comosiion of he commonly-referenced NCREIF index of roery erformance in he commercial marke and he Naional Associaion of Realors (NAR) Index in he residenial marke. A more sohisicaed aroach was reresened by economerically-esimaed indices of roery values over ime. he commonly-used aroaches here include he reea-sales model (firs roosed by Bailey, uh, and Nourse (1963) and oularized by Case and Shiller (1987) and currenly he source of he OFEO Residenial rice Index), he hedonic model (he source of he UD Residenial rice Index), he hybrid model (see Quigley (1995), and he araisal-based model (see Quan and Quigley (1991)). We refer o Lin (4) for a deailed discussion of he srenghs and weaknesses of each of hese mehodologies. Suffice i o say ha each is subjec under cerain condiions o resening cerain biases in he esimaion of rue reurns. hese biases originae from four sources, which have been recognized in he lieraure: (1) araisal bias, which can originae from a bias in he holding-eriod reurn, even if he araised value is an unbiased esimaor (Gilibero, 1988), or from a bias caused by he smoohing ha is inheren in using araisal-based reurns (Gelner (1989b, 1991), Ross and Zisler (1991)); () samle selecion bias, which is creaed by he fac ha ransacion rices used for esimaion by necessiy could reresen a biased samle of he enire sock (e.g., Gazlaff and aurin (1997, 1998) for he residenial marke and unneke and Slade () for he commercial marke), ofen addressed by using eckman s wo-se rocedure and he inclusion of he inverse ills raio; (3) ransformaion bias, creaed in he reea-sales model under cerain condiions, when reea-sales esimaors are essenially equal-weighed cross-secional averages, while he reurns of equal-weighed orfolios are arihmeic averages of cross-secional individual asse reurns (Considered in Shiller (1991), Gelner and Goezmann (), and Goezman and eng ()); and (4) aggregaion bias, caused eiher by saial aggregaion bias (see homas and Sekler (1983) and Goodman (1998)) or emoral aggregaion bias (see Gelner (1993) and Dombrow, Knigh, and Sirmans (1997)), reresening uneven clusering of ransacions across sace or ime. A final air of recen aers (Fisher, Gazlaff, Gelner, and aurin (3), Goezmann and eng (3)) idenify an addiional source of bias, namely he bias associaed wih differences in he ease of selling a roery over ime, which hey relae o differences in he markeing eriod or ime-on-marke (O). In his scenario, when he marke is ho, here are many bidders in he marke comeing o urchase roeries, which ends o resul in a shorer O a which he reservaion rice of he buyer mees or exceeds he reservaion rice of he buyer. Commensurae wih his henomenon is an increase in he volume of sales because he disribuion of buyer 4

5 reservaion rices is shifed u relaive o he disribuion of seller reservaion rices. In heir view, hese variaions over ime are direcly relaed o each oher and reresen roxies for variaions in he degree of liquidiy resen in he real esae marke a any eriod of ime. hus, observed rices (conrolling for all he sources of bias idenified above) are biased esimaes of value unless he degree of liquidiy, as hey define i (consan O or consan sales volume) is conrolled for. he resuling consan liquidiy rice index hen reresens an unbiased rue measure of rice erformance over ime. 3 he above wo aers, in aricular, are imoran recursors o ours. hey recognize he endogenous relaionshi beween markeing eriod, rading volume, and observed selling rice and he fac ha his has o do wih variaions in he degree of liquidiy, as oeraionally defined in some way, in he sense of how easy or quick one is able o sell a roery a he oimal sales rice under he circumsances in he marke, and how ha oimal (observed) rice may be observed under differen liquidiy condiions. he curren aer differs from hem in he following ways: (1) Boh aers are rimarily inended o be economeric aers ha derive roer economeric correcions for biases creaed by differen rading volumes or O s over ime (essenially exending he eckman samle-selecion bias correcions earlier o conrol also for varying O s and/or rading volumes). hey do no include formal heoreical derivaions of wha consiues he fundamenal essence of he risk characerizing he ransacion rocess in real esae ha roduces he observed bias. () Boh aers oeraionally define liquidiy a a consan (average or yical) level using he roxy of a consan (average) O or rading volume. hey imlicily assume in heir oeraional definiion ha such condiions (i.e. consan O or rading volume) are boh necessary and sufficien o define consan liquidiy. hey do no recognize eiher he ossibiliy ha rading volumes and O could vary under a broader definiion of consan liquidiy (for examle, consider ha rading volumes vary considerably over ime in he urely liquid sock marke). Nor do hey recognize ha under a broader definiion of liquidiy, a consan O may no be consisen wih a consan rading volume (for examle, hese are correlaed, bu by no means direcly relaed in he real esae marke). Finally, hey do no recognize ha even if hey boh (or even one) were consan, liquidiy, by a broader definiion could vary. 4 3 he Goezmann and eng model differs from he Fisher e al. model rimarily in ha i resens a simler economeric aroach o idenifying he bias and incororaes direc, raher han imlied measures of rading volume. 4 For examle, conceivably, sabiliy in boh ime on marke and ransacion volume could occur as he resul of he righ confluence of changes in sellers and buyers reservaion rice disribuions over ime in 5

6 (3) Finally, boh aers consider he relevan correcion for liquidiy bias o be one ha brings he rice index o reresen an average or yical degree of liquidiy in he marke (by heir definiion). his means an average or yical O or rading volume. Raher, he roer correcion o comare he erformance of real esae as an asse class in a mixed asse orfolio agains he erformance of oher asse classes and o esimae roer allocaion of real esae in he orfolio, is o adjus real esae rice movemens o reresen full liquidiy. While heir oeraional definiion of liquidiy as O is meaningful when he noion of comlee liquidiy is considered (O), his does no seem o be he case if heir oeraional definiion is rading volume, since such volume would aroach infiniy (or consan rading of all roeries a every oin in ime). he resen aer does no deal secifically wih he economeric correcion of observed sales rices. Raher we delve heoreically ino he fundamenal rocesses driving rice risk and markeing-eriod risk over ime. his formal analyical framework allows us o define a coninuum of liquidiy condiions for he individual asse, from fully liquid, when i is oimal for he seller o acce he firs offered bid a ime eriod zero, o a low degree of liquidiy, when he markeing eriod exerienced before he acceance of a bid becomes oimal may be quie long (or in he limi even infinie, in which case he sale is no observed). Aggregaing hese individual siuaions u, given roery, buyer, and seller heerogeneiy, hen rovides a broader se of observed condiions in he marke over ime wih resec o rice, O, and rading volume ha reflec a more comlex disribuion of degrees of liquidiy. Allowing hese relaionshis o vary in a broader framework as he oimal O is moved oward zero (full liquidiy), hen races ou he correced rice dynamics for real esae as a liquid asse, on an ales-o-ales basis wih oher liquid asses in he orfolio. he nex secion develos his framework. III. he Real Esae ransacion rocess Inuiion. We begin wih he formalizaion of he real esae ransacion rocess as disinc from ha in financial markes. Inuiively, due o he uncerainy of he markeing eriod in he real esae marke, he formaion of ransacion rices in he real esae marke is very differen from ha in he financial marke. Firs, he classical (erfecly comeiive, fricionless, and comlee) financial marke is a homogeneous and hickly raded marke. A any insan, here is unresriced availabiliy of buyers and sellers a he marke rice, and rices are deermined by marke clearing. On he oher hand, he real esae marke is a heerogeneous and hinly raded he face of macro-marke dynamics. 6

7 marke, and is rices are formed by sellers sequenial search and heir oimal soing rule: acceing he firs rice above heir reservaion rice (e.g. Arnold (1999) and Yavas (199)). Second, rices exis in he financial marke a any ime. Sellers can sell heir financial asses a he marke rice a any ime wihou waiing. In conras, real esae rices exis only when here is a curren buyer wih an offer rice ha is a leas as high as he seller s reservaion rice. When here is eiher no buyer or an offer rice is below he seller s reservaion rice, he real esae rice does no exis and he sellers have o coninue o search for he nex buyer. ence, sellers canno sell heir real esae asses a any given rice wihou waiing. Due o he naure of sochasic arrival of oenial buyers and he uncerainy of heir offer rices, ime on marke canno be fully conrolled by he sellers. herefore, unlike he financial marke, real esae invesors face no only rice risk, as in he financial marke, bu also markeing eriod risk. he odel. 5 Exhibi 1 illusraes hese rice dynamics in he conex of he ransacion rocess in he real esae marke. Suose an invesor urchases a real esae asse a ime and laces i on he marke a ime, and suose is a ossible (discree) markeing eriod wih a sale rice, where ( 1,,...) 1 is he waiing ime of he firs buyer, is he waiing ime of he second buyer, and i is he waiing ime of i h buyer. 6 In each assing eriod, he seller faces random arrivals of oenial buyers. here are wo ossibiliies: (1) a successful sale when a oenial buyer is resen and his/or her asking rice equals or exceeds seller s minimum acceed rice; or () no successful sale when here is no buyer resen or an asking rice is oo low for he seller o acce. Assume he seller s reservaion rice a he ime of he ih buyer s arrival is reservaion bid i and he buyer s bidding rice is i. Noe ha he reservaion rice is oimally, no arbirarily, se by he seller. By he oimal soing rule, he robabiliy of a successful sale a ime i given ha he real esae has no been sold is i bid reservaion r ob( i ) () i herefore, we can deduce ha he real esae asse will be sold a he firs offer wih robabiliy 1, a he second offer wih robabiliy ( 1 1), a he hird offer wih robabiliy 1 (1 )), and so on. Since i ( i 1,,...) is largely 3( he analysis ha follows is consisen wih ha develoed by ohers (e.g., see Fisher, Gazlaff, Gelner, and aurin (3)) who recognize he sochasic naure of he ransacion rocess for real esae as i affecs ransacion rices and ime on marke. We develo i exlicily only o se he sage for laer develomen of our revised risk/reurn measures. 6 For mahemaical simliciy, we consider a discree-ime model. owever, he analysis can be readily exended o a coninuous-ime model wih no subsanive change in our findings. 7

8 deermined by he marke, he robabiliy of a successful sale a each offer canno be fully conrolled by he seller. 7 Since he seller does no know when a oenial buyer will arrive afer when he asse is u on he marke, he arrival ime i is also sochasic. We hus denoe i as i o emhasize is randomness. Exhibi 1 illusraes he ossibiliy of sale a ime i ( i 1,,... i,...), where is he seller s holding ime and i is he seller s markeing eriod. ence, he markeing ime i of a successful sale no only deends on he arrival disribuion of oenial buyers bu also on wheher buyers bidding rices are higher han he seller s reservaion rice. Since boh he arrival disribuion and seller s reservaion rice are closely relaed o marke condiions, he uncerainy of he markeing eriod is beyond he seller s conrol. If he asse is successfully sold a markeing eriod (,1,,... ), he seller receives ex-os reurn. he robabiliy of sale in markeing eriod is denoed by R (,1,, 3,... ), which characerizes real esae illiquidiy and saisfies, 1. A ransacion rice can be observed if and only if a bid rice equals or exceeds he reservaion rice : Where: if unobserved if <,. V ε, ε N(, σ ) (4) ε (3) and V o reresens he disribuion of marke valuaions. Equaions (3) and (4) can be combined as V [ ε ε ] (5) V Similar roeries wih differen sellers may ransac very differenly. For a variey of reasons, some sellers may have o lower heir reservaion rices in order o sell heir houses more quickly han ohers. Equaion (5) imlies ha he disribuion of ransacion rices varies over reservaion rice levels. herefore, sricly seaking he reurn and risk in he real esae marke are seller-secific. 7 I is ossible, indeed likely, ha sellers can influence he rae of arrival of oenial buyers, and ossibly he likelihood and level of buyers bids by sraegic seing of heir asking rice. We shall ignore his ossibiliy for he resen and assume realisically he seller lacks a leas some degree of conrol. 8

9 As in Goezmann and eng (3), unneke and Slade (), Englund, Quigley and Redfearn (1999), and Gazlaff and aruin (1998), we assume a oenial buyer, on he oher hand, offers a bid rice based on some figure reresening he marke valuaion. ence, he bid rice is a rice from a single disribuion of marke valuaion, which is no seller-secific. As we know, he only rices ha can be observed in he real esae marke are ransacion rices, and bid rices below he seller s reservaion rice are unobservable direcly hrough observed ransacion rices. hus, he marke valuaion disribuion canno be direcly observed in he real esae marke. We will revisi his issue when we discuss he valuaion bias resen from he use of observed real esae ransacions as an esimae of he underlying valuaion disribuion. he Disribuion of Buyers Arrivals. Le us now solve he above roblem for secific disribuions for buyers arrivals and bid rices. he yical assumion of buyers sochasic arrival is he oisson rocess, wih a consan arrival rae in each oin of ime. Following Gazlaff and aurin (1998), Salan (1991), aurin (1988), and Liman and ccall (1986), we assume ha a oenial seller receives bid rices from oenial buyers a a rae of one er eriod (unis of ime can be made arbirarily small). 8 he Disribuion of Bid rices. Regarding he disribuion of bid rices, Arnold (1999) and Sirmans, urnbull and Dombrow (1995) assume ha he bid disribuion is bid over [, ] wih densiy funcion f ( ), where ( ) is he minimum (maximum) bid rice. 9 he assumion of a ime-invarian disribuion of execed offer rices is robably oo simle for wo reasons. Firs, he underlying value of a secific roery is yically execed o increase over ime, esecially for residenial roery in nominal erms. According o OFEO s reea-ransacions home rice index (esimaed using daa from Fannie ae and Freddie ac), home rices rose in excess of he rae of inflaion over he las years in all bu a few markes and in nominal erms in all markes. ence, a leas for he residenial marke, we yically would exec he disribuion of underlying home values o shif uward over ime. Second, afer a relaively long holding eriod, buyers end o have weaker informaion regarding he rue marke value of a secific roery, and hus are more likely o agree on a rice ha may differ subsanially from he marke value. 1 herefore, he underlying risk should increase as ime asses beween ransacions (Case and Shiller 1987). Consisen wih 8 he assumion is made for echnical simliciy. Anglin (3), Arnold (1999), Glower, aruin and endersho (1998) and iceli (1989) assume ha an arrival rae of λ er eriod. A more comlicaed model would allow buyers o resond o sellers asking rice, i.e., a higher asking rice imlies a lower arrival rae, bu ha is beyond he scoe of his analysis. 9 can be regarded as he seller s asking rice. We recognize ha in unique circumsances in which he marke is aricularly ho, bidding by oenial urchasers raise above he asking rice. Inclusion of his ossibiliy would no change he essenial resuls of our analysis. 1 his is because of he increasing remoeness of he anchoring rovided by he revious ransacion rice. 9

10 hese wo facs, we assume ha he disribuion of bid rices varies over ime. In aricular, we assume ha i is disribued over τ, τ ] a ime τ, where is he original urchase rice a ime. 11 [ Yavas (199) and Read (1988) assume he bid densiy funcion f ( ) is uniformly disribued. For echnical simliciy, we ado he same assumion here. 1 ence, he bid buyers bid rice τ a ime τ is disribued as, 1 bid, [ τ, τ ] bid f ( ) ( ) τ τ (6), oherwise he seller decides wheher or no o acce an offer based on he reservaion rice for he roery. 13 A each oin in ime, he oimal markeing sraegy for he seller is o acce he firs bid above he reservaion rice, and o rejec all bids below. Like he buyers bid rice, we allow a ime-varying reservaion rice and denoe his as τ. herefore, in any eriod, here exis wo ossibiliies: Firs, a ransacion occurs when a buyer arrives wih a bid rice bid and he seller acces ha offer and sells he τ τ roery for ha rice. Second, no ransacion occurs if an offer rice bid τ < τ. If here is no ransacion, no deal is reached and he seller will coninue o search for he nex buyer. hus, observable real esae ransacion rices (denoed by ) mus be in he range of [ τ, τ ], and in order o rade a an observable ransacion rice, he real esae seller exeriences he uncerainy of sale in each eriod. 14 olding oher hings equal, he higher he underlying marke value of a roery, he higher he seller s reservaion rice. Accordingly, we assume ha he seller s τ 11 Noe ha i is sill ossible for he bid disribuion o reresen a ossibiliy of declining rices using his disribuion if is negaive. Of course negaive bid rices are no ossible; our assumion is consisen, however, wih hose sochasic ineres rae models ha ermi negaive ineres raes o allow fiing of observed ineres rae drifs. In a more generalized model, we could also add a consan (or consans) o he τ erm which could bes fi aniciaed drifs in roery values over ime. he essenial resuls of our model would be reserved regardless. 1 In fac, our essenial resuls would hold under a wide variey of more comlex disribuion funcion assumions. 13 o evaluae ossible disribuions of reservaion rices is no he focus of his analysis. We rea he disribuion as given. Liman and ccall (1986, 1976) and DeGroo (197) have some discussion on his. 14 In he real esae marke, mos sellers sell heir asses by exercising he oimal soing rule: o acce all bids above he reservaion rice and o rejec all bids below. enceforh we assume observable ransacion rices are all rices ha are a leas as high as sellers reservaion rices. 1

11 reservaion rice also increases wih ime. For analyical racabiliy and wihou loss of generaliy, we furher assume ha, for a aricular seller, a ime τ his reservaion rice τ is τ. 15 robabiliy of Sale: Consan azard Rae. Given he discussion above, in any eriod here are wo ossibiliies: a ransacion or no ransacion. If a buyer s offer is oo low for he seller o acce, he seller has o wai and coninue o search for he nex buyer. In his model, he robabiliy of having a ransacion in each eriod given he roery is no sold can be obained as, which for he ime being we shall assume is consan over ime. ence, he robabiliy of sale a markeing eriod (,1,, 3... ) (denoed by ) is a geomeric disribuion, Where, rob( ) ) π (1 π (7) π (8) and π is he (consan) hazard rae of sale. Equaions (7) and (8) highligh wo ineresing facs. Firs, he robabiliy of a successful sale in each eriod no only deends on he seller s reservaion rice ( ), bu also deends on he disersion of buyers valuaion, ( ), and a lower reservaion rice indicaes a higher robabiliy of sale. herefore, he reservaion rice lays an imoran role in he deerminaion of maching robabiliy beween sellers and buyers. Second, sellers canno sell heir asse a a redeermined ime wih cerainy. Since rarely occurs in he real esae marke, rob ( ) < 1 holds for all. In oher words, he ime required for sale in he real esae marke is a random variable (e.g. rii 1977). Execed arkeing eriod: Consan azard Rae. aving derived he robabiliy of a successful sale in each eriod, we nex sudy how long he seller is execed o wai on he marke. By definiion, he execed markeing eriod can be exressed as follows, i E[ ] π (1 π ) (9) 15 here could of course be circumsances, once a roery is laced on he marke and a seller has received insufficien ineres given her moivaion o sell, she my revise her reservaion rice downward based uon a revised ersecive of he underlying bid disribuion. his comlexiy could cerainly be inroduced in a more elaborae model, bu i is beyond he scoe of he resen effor. 11

12 Simlifying (9) yields, 1 E [ ] 1 (1) π Equaion (1) indicaes ha he robabiliy of sale is uniquely relaed o he execed markeing eriod hrough he (consan) hazard rae π. aking Equaions (1) and (8) ogeher, we can conclude ha real esae sellers sell heir asses immediaely only when hey decide o acce whaever rice he buyer can offer (, hence π 1 and E [ ] ). IV. Real Esae Illiquidiy and arkeing eriod Bias Given he above develomen of he formal srucure of he real esae ransacions rocess, we can now roceed o formally defining real esae illiquidiy and markeing eriod bias. We ado as an oeraional measure of real esae illiquidiy he firs and second momen of he random variable reresening he markeing eriod i, or he execed ime required for sale and he uncerainy in he markeing eriod σ m E[ ] (11) σ Var[ ] [ E[ ]] (1) 16 In Aendix 1, o beer undersand how he disribuion of robabiliy of a ossible sale over he markeing eriod affecs he execed markeing eriod and is volailiy, we consider hree secial cases ha in which he disribuion of he robabiliy of sale over ime is consan, uniform, and exonenial. Consisen resuls are obained in each case: A higher level of illiquidiy imlies a longer execed markeing eriod and a greaer uncerainy in markeing eriod, yically an increase of he same order magniude for boh measures. Ex os vs. Ex Ane easures of Reurn: As discussed reviously, associaed wih each random markeing eriod variable and is firs and second momens, as defined above, is also a random reurn variable wih is firs and second momens he execed 16 Noe ha we are now ermiing he robabiliy of sale over ime o vary. Since Var ( ) [ E[ ]] Var( ), and is he ime of sale, we use he erms uncerainy of markeing eriod and uncerainy of ime of sale inerchangeably. Our oeraional definiion of illiquidiy we feel ges a he essence of illiquidiy in he real esae marke, having o do wih he aniciaed markeing ime for an individual asse and he uncerainy in ha measure. igher order reamen of he sochasic markeing ime variable is cerainly ossible bu beyond he scoe of he curren analysis. 1

13 reurn and risk. A seller lacing his real esae asse on he marke afer holding i for eriods receives reurn R uon successfully selling i a ime ( 1,,... ). We define he ex-ane measure of reurn as he forward-looking measure uncondiional uon a successful sale a a secific oin in ime. he ex-ane execed reurn and risk can be defined as follows, Var E ex ane R ] E[ E[ R ]] (13) ex ane [ R [ R ex ane ] E[ E[ [ ]] R E R ]] (14) R Noe ha ex-ane execed reurn and risk are closely relaed o he uncerainy of he markeing eriod ( ). Concurrenly, we define he ex-os measure of reurn as he observed reurn based uon he observed sales rice a he ime a which he roery is sold. hose esimaion mehodologies for reurn discussed above ha acce he sales rice as an unbiased esimae of he marke value a he ime of sale are all ex-os reurn measures. hey ignore ossible underlying variaions in he markeing eriod, hence variaions in he degree of liquidiy resen. If reurns generaed in his manner are comared o reurns generaed for fully liquid asses such as socks, hen imlicily his measure is assuming real esae ossesses full liquidiy (i.e., is a measure condiional uon immediae execuion ( )). In his siuaion, ex-os execed reurn and risk are defined as follows, Var ex os E [ R ] E[ R R ] E[ R E[ R ]] ex os [ ] (15) (16) Since he ex-os execed reurn and risk are indeenden of he uncerainy of he markeing ime, he ex-os measure of reurn only involves rice risk. Wha is he racical imorance of hese ex ane vs. ex os measures of reurn and risk? As we know, hisorical rices are he daa recorded on an ex-os, or afer he fac basis of successful sales. In realiy, a real esae seller who ries o sell his asse does no know when i will acually be sold; i.e., acually ossesses an ex-ane view of he risk he faces. In order o disinguish our alernaive aroach from radiional valuaion aroaches o real esae asse erformance, we label our new measure he ex-ane measure and he radiional measure he ex-os measure. arkeing eriod bias in his framework is herefore naurally defined as he ricing difference beween hese wo measures. hus, markeing eriod bias essenially caures he effec of he uncerainy of markeing eriod on real esae ricing. 13

14 Equaions (13)-(16) indicae ha ex-ane execed reurn and risk are idenical o ex-os execed reurn and risk if and only if asses can be sold immediaely. We can rove ha in he resence of uncerainy in he markeing eriod, however, and under realisic assumions for he disribuion of reurns, he ex ane execed reurn remains equivalen o he ex os reurn, bu he ex ane variance is higher han he ex os variance. his is demonsraed in he following heorem: heorem 1: Suose an invesor urchases a real esae asse a ime and markes i in eriod. Assume he robabiliy of being sold a markeing eriod (,1,,..., ) is ( 1), wih ex-os (oal) reurn R, which is disribued wih mean ( ) u and variance ( ) σ, where u he eriodic mean ex-os reurn and σ he eriodic mean ex-os variance. hen (1) he eriodic ex-ane execed reurn is he same as he eriodic execed ex-os reurn: and u ex ane u ; (17) () he eriodic ex-ane variance is higher han he eriodic ex-os variance, secifically, Var( ) ( u E[ ] ex ane σ ) σ (18) [roof rovided in Aendix ] he assumion ha he ex-os reurn R is disribued wih mean ( ) u and variance ( ) σ is jusified because, as we noed above, emirical evidence from he residenial and commercial marke confirm ha he oal ex-os reurn ends o increase secularly wih he holding eriod (since rices end o increase, a leas in nominal erms), and ex-os risk would be execed o increase over ime as one looks furher in he fuure. 17 Case and Shiller (1987) noe ha he risk associaed wih a secific roery should be osiively relaed o he lengh of ime elased beween ransacions of ha roery. One exlanaion for his is ha boh buyers and he seller end o have beer informaion regarding he rue marke value of a given roery if he ime afer he las ransacion of ha roery is relaively shor. On he oher hand, afer a relaively long holding eriod, boh buyers and he seller end o have weaker informaion regarding he rue marke value of he roery, and hus are more likely o agree on a rice ha 17 Again, i would be ossible o generalize his secificaion, bu over a wide range of ossibiliies, our fundamenal resuls would no be affeced. We ado hese assumions rimarily for racabiliy uroses. 14

15 differs significanly from he rue marke value. herefore, he ex-os risk should increase as ime asses beween ransacions. Four conclusions can be drawn from heorem 1. Firs, based on our alernaive ex ane ersecive, real esae risk can be decomosed ino wo elemens: rice risk and markeing eriod risk. If we confine real esae risk only o rice risk, we always undersae real esae risk. his underesimaion becomes more serious when he uncerainy of markeing eriod Var( ) is relaively large and/or when he execed annualized ex-os reurn u is high. As a resul, using risk esimaed from hisorical rices and he ime of ransacions, ignoring he resence and uncerainy of markeing eriod, will underesimae real esae risk, and hus lead o markeing eriod bias. Second, we recognize ha we imlicily are assuming in our analysis ha he seller faces neiher a liquidiy shock nor borrowing consrains; hence he can always wai for he bes buyer. uang (3), however, considers an invesor who holds an illiquid asse having o liquidae his asse immediaely by a discoun rice when a liquidiy shock occurs. If his is he case, higher ex-ane risk and lower ex-ane reurn may lead o an even higher markeing eriod bias. We will discuss his siuaion laer when we consider liquidaion risk. hird, he ex-ane reurn is he same as he reurn esimae from radiional aroaches ha ignore markeing eriod risk. owever, his resul holds only when he ex-os reurn execaions are realized. If he marke faces an unaniciaed downurn (or uurn), his assumion may be violaed, and we can show ha he annualized ex-os reurn should decrease (increase) over ime relaive o a riori execaions; hence he exane reurn will be greaer (less) han he reurn esimaed from he radiional aroach. Finally, a longer holding eriod imlies lower markeing eriod risk, hence lower real esae risk and lower markeing eriod bias, ceeris aribus. his is because he relaive magniudes of he holding eriod and markeing eriod change o such a degree ha he order of magniude affec of markeing eriod risk in comarison wih rice risk becomes increasingly small. his resul rovides a formal jusificaion for he convenional wisdom ha real esae is more favorable o long-erm invesors han o shor-erm invesors. Exhibi summarizes how he alernaive disribuions of markeing eriod considered in Aendix 1 affec real esae risk. For examle, when he markeing eriod follows a consan condiional robabiliy of sale, he higher he condiional robabiliy, he less he markeing eriod risk, and herefore he less he real esae risk. If he markeing eriod is disribued as an exonenial disribuion, hen he longer he execed markeing eriod (noice ha E [ ] η ), he higher he real esae risk, given oher hings equal. Emirical Esimaions of arkeing eriod Risk. heorem 1 demonsraes ha he curren aroach of using ex-os variance o measure real esae risk has a bias roblem. 15

16 he ex-os variance always underesimaes real esae risk. owever, heorem 1 also rovides a formula o correc for his bias, i.e. Var( ) ( u E[ ] ex ane σ ) σ (19) Equaion (19) ells us ha, in order o correc for markeing eriod bias, besides ex-os reurn, ex-os risk and holding eriod, we need o know he disribuion of he markeing eriod random variable. Bond, wang, Lin and Vandell (5) invesigae a number of assumions abou he disribuion of imes o sale, such as he normal, chisquare, gamma and Weibull disribuions, and find ha he exonenial densiy funcion exlains he U.K. commercial real esae daa emloyed beer han he ohers. By assuming he markeing eriod is disribued as he exonenial disribuion, we nex esimae he degree o which markeing eriod risk ogeher wih invesmen ime horizon affec markeing eriod bias and real esae risk in boh he residenial and commercial roery markes, using U.S. daa. 18 Case I: he U.S. Residenial roery arke We firs consider he U.S. residenial roery marke. We assume an esimaed average annual reurn of 5.% and sandard deviaion of 1.67%, which are derived from he OFEO U.S. home rice index during he eriod 198Q1 o 4Q4. Exhibi 3 illusraes by how much he ex-ane variance would exceed he ex-os variance under various scenarios of execed markeing eriod and holding eriod. From his able, we can readily see firs ha if he execed ime-on-marke is zero and real esae is aniciaed o be a liquid asse, hen real esae risk is comleely comosed of rice risk and he ex-ane variance is he same as he ex-os variance. Second, he degree of underesimaion obained by using he ex-os variance for real esae risk increases wih he execed markeing eriod and decreases wih he holding eriod. For examle, if he execed markeing eriod is eigh monhs, he ex-ane variance will be abou wo and a half imes higher han he ex-os variance if he holding eriod is only one year, bu only 38% higher if he holding eriod exends o 1 years. herefore, he radiional racice by OFEO and oher relaed indices of using he ex-os variance as a roxy for he ex-ane variance can seriously underesimae real esae risk, esecially when he execed markeing eriod is high and he holding eriod is relaively shor. Case II: he U.S. Commercial roery arke 18 We assume in he following simulaions ha he OFEO and NCREIF indices reresen eiher a diversified orfolio of roeries or a single roery being evaluaed by he individual invesor whose erformance is idenical o he averages reresened by he indices. We do no adjus for oher biases ha have been recognized in he lieraure, such as samle selecion bias, smoohing hrough araisal bias, ec. 16

17 For our emirical evaluaion of he U.S. commercial roery marke, we choose an average annual reurn of 8.63% and a sandard deviaion of 3.%, which are based on he Naional Council of Real Esae Invesmen Fiduciaries (NCREIF) roery index during he eriod 198Q1 o 4Q4. Exhibi 4 summarizes our findings. Similar conclusions o hose from Exhibi 3 are obained, and of a similar magniude. Firs, if he execed ime-on-marke is zero, he ex-ane variance is he same as he ex-os variance. Second, ex-ane variance increases wih he execed markeing eriod and decreases wih he holding eriod. For examle, if he execed markeing eriod is eigh monhs, he ex-ane variance will be abou wo imes higher han he ex-os variance if he holding eriod is only one year, bu is only 3% higher if he holding eriod exends o 1 years. Conclusion wih Resec o arkeing eriod Bias. A common view in he academic communiy has been ha risk due o real esae illiquidiy is rivial when he invesmen ime horizon is long. Bu lile has been sudied on he magniude of he risk associaed wih markeing eriod and is uncerainy. he resuls above srongly sugges ha real esae illiquidiy risk can be subsanial even when he invesmen ime horizon is relaively long, yical of holding eriods observed in he marke. herefore, he uncerainy of he markeing eriod should cerainly be considered in he esimaion of rue reurn and risk in he real esae marke. V. Real Esae Illiquidiy and Liquidaion Bias Inuiion. In addiion o markeing eriod bias (markeing eriod risk ha comounds he effecs of rice risk), here exiss a second oenial source of bias in he esimaion of marke reurns and volailiy ha is relaed o real esae illiquidiy. his we will erm liquidaion bias. In he financial marke, since invesors can sell heir asses a observed marke rices almos immediaely, a hisorical rice a ime reresens he marke rice of his asse a ha ime, meaning ha a seller could rade a ha rice a ha ime. owever, wo differences exis beween his siuaion and ha in he real esae marke due o he uncerainy of he markeing eriod in he real esae marke: Firs, in each ime eriod, only a small orion of roeries is sold successfully, while a large orion of he roeries being offered for sale are sill siing on he marke. he ransacion rices a ime may reflec rices of he sold roeries a ha ime; however, hey may no be reflecive of he (unobserved) rices of similar roeries ha are sill waiing on he marke. Second, sellers offering heir roeries for sale have ofen waied on he marke for a long eriod of ime before selling. Said anoher way, curren real esae ransacion rices end o reflec he rices of hose asses ha have been on he marke for some ime. Unlike he financial marke, in which he rice a ime is he rice of hose asses recenly u on he marke, here is a subsanial ime lag beween he ime when real esae is laced on he marke and when i is sold. ence, he rice of sold roeries a 17

18 ime may no reresen he rice of oher roeries recenly laced on he marke. Anoher way of hinking of his is ha sellers would be unlikely o be willing o rade a observed ransacion rices a he ime a which hey lace heir asses on he marke. he risk of a subsanial ime lag beween he even of sale and even of lacing he roery on he marke is imoran o real esae invesors. For examle, a household may exerience a surrise liquidiy shock, such as a job loss or a divorce, and facing a borrowing consrain, mus sell is illiquid real esae in a shor ime eriod; firms may have sudden invesmen ooruniies, bu face an imerfec and cosly exernal caial marke, and hence have o sell heir real esae asse immediaely when such an ooruniy arrives; fund managers may face an unexeced increase in wihdrawals, resuling in a need o liquidae a orion of heir real esae asse orfolio. In all hese siuaions, when such a shock occurs, real esae invesors are forced o sell real esae asses immediaely. ow does his ossibiliy of he necessiy of a quick sale have an imac on real esae ricing, given ha such a significan ime lag may be resen? In order o answer his quesion, we need o look again a how ransacion rices are formed in he real esae marke. As observed formally above when we discussed he real esae ransacion rocess, a real esae ransacion rice can be observed if and only if a buyer s bid rice equals or exceeds he seller s reservaion rice. ence, ransacion rices reflec he rices when a bid rice is above he seller s reservaion rice; i.e., he rices from a runcaed disribuion of bid rices. his imlies ha he execed ransacion rice is likely o be higher han he execed bid rice a each oin of ime. he yical seller has o face he uncerainy of markeing eriod o receive a bid reflecing he disribuion of ransacion rices. owever, an invesor who exeriences a surrise liquidiy shock and has o sell his real esae asse wihin a shor eriod of ime mus acce a rice from he disribuion of bid rices raher han he disribuion of ransacion rices. In oher words, a bias may resul if his reurn and risk is direcly esimaed from observed ransacion rices, ignoring he ossibiliy of a need for immediae liquidaion. We define his bias o be liquidaion bias. Because of he close relaionshi beween he exisence and magniude of liquidaion bias and he observed markeing eriod, we may regard liquidaion bias as he value of he oenially consrained ime on marke. We emhasize ha, while markeing eriod bias as we have described i above, would be execed o affec all sellers o he exen ha he markeing eriod and acual sales rice remain uncerain ex ane, he degree of liquidaion bias resen is deenden uon each individual seller s circumsance wih resec o he degree of likelihood of exeriencing a forced sale. Formal Definiion of Liquidaion Bias. As we saw above in Secion III, a real esae ransacion rice will be observed if and only if a bidding rice equals or exceeds he reservaion rice : 18

19 unobserved if if <,. (3 ) Where: V ε, ε N(, σ ) (4 ) ε Equaions (3 )and (4 ) can be combined as, V [ ε ε ] (5 ) V As we know, he only rices ha can be observed in he real esae marke are ransacion rices, and bid rices below he seller s reservaion rice canno be observed. As a resul, he marke valuaion disribuion canno be direcly observed in he real esae marke. ow can we imue he unobservable marke valuaion disribuion from available marke informaion? o answer his quesion, we rewrie equaion (5 ) as follows, V ε V ε () Equaion () can be regarded as defining he liquidaion bias beween he underlying, unobservable marke values and he observed ransacion rices. Since he reservaion rice is closely relaed o he markeing eriod (O), we can rewrie () as: V [ ε ε f ( O )] (1) olding oher facors consan, inuiion suggess a higher reservaion rice no only dislays a greaer liquidaion bias beween he observable ransacion rice and he marke valuaion, bu also resuls in a longer markeing eriod. herefore, we should exec a correlaion beween he lengh of he markeing eriod and he degree of liquidaion bias. Normally, we canno observe he reservaion rice; however, informaion on markeing eriod is readily available. Equaion (1) suggess ha we may use he available informaion on ransacion rices and ime-on-marke o imue an esimae of marke reurn and volailiy. We underake his exercise in he following secion for he secific case of a uniform disribuion of bid rices. 19

20 Esimaion of Liquidaion Bias under a Uniform Bid Disribuion. Suose r and σ are he eriod reurn and volailiy from he marke valuaion disribuion and r and σ are he eriod reurn and volailiy based on observed ransacion rices. We formally define liquidaion bias as, reurn bias r r () volailiy bias σ σ (3) As before, we assume ha an invesor urchases a real esae asse a ime a a rice, holds i for eriods, and hen offers i on he marke for sale. In each markeing eriod (,1,,...), condiional uon a ransacion being observed, he oal reurn from his ransacion can be esimaed as, R (4) and, bid bid, if ( ) (5) bid unobserved, if < ( ) ence, is uniformly disribued over [( ),( ) ]. As a resul, he average eriod reurn from his ransacion is, r (6) ere, is uniformly disribued over [, ]. Similarly, we can obain he average eriod reurn from he underlying marke valuaion as follows, r (7) Where: is uniformly disribued over [, ]. Unforunaely, we can only observe marke values when and canno observe hem when [, ). In oher words, we canno esimae marke reurn and risk solely based on he ransacion rices. owever, he relaionshi beween he marke valuaion and ransacion rices can be

21 esablished by he use of markeing eriod informaion. We summarize his resul in he following heorem. heorem : arke Valuaion, ransacion rices, and arkeing eriods. Under he assumion above ha ransacion rices are uniformly disribued over [, ] and he underlying marke valuaion is uniformly disribued over [, ], he liquidaion bias for a random markeing eriod can be reresened as 1. r r 3E[ ] σ (8). σ σ E[ ] σ (9) where: E ] is he execed markeing eriod. [ [roof rovided in Aendix 3] Equaions (8) and (9) reresen he liquidaion reurn and risk bias, resecively. heorem also illusraes he relaionshi beween liquidaion bias and he execed markeing eriod. hrough his relaionshi, we can use he available observed ransacion reurn ( r ), observed ransacion volailiy ( σ ) and he execed markeing eriod ( E [ ] ) o imue he underlying marke reurn ( r ) and volailiy ( σ ) as follows, r r 3E[ ] σ (3) σ ( 1 E[ ]) σ (31) Several conclusions can be drawn from heorem. Firs, noe ha as execed, liquidaion bias disaears when asses are able o be raded a observable ransacion rices wih immediae execuion (i.e., when E [ ] ). In oher words, here is no liquidaion bias in he financial marke when observed ransacion rices reresen marke valuaion. Second, unlike in he case of markeing eriod bias, he holding eriod lays no role in liquidaion bias in he real esae marke. owever, he execed markeing eriod ( E [ ] ) lays an imoran role in he deerminaion of liquidaion bias. In aricular, liquidaion bias increases when he execed markeing eriod increases, which is consisen wih he common erceion ha a longer execed markeing eriod imlies a higher reservaion rice, ceeris aribus, hence a greaer deviaion of ransacion rices from he underlying marke valuaion. 1

22 hird, he rue marke reurn is always lower han he reurn esimaed from observed ransacion rices, and he rue marke volailiy is always higher han ha esimaed from observed ransacion rices. ence, evaluaing real esae erformance based solely on he ransacion samle is likely o resul in an oversaed risk-adjused reurn o hose invesors who face he oenial necessiy of a quick sale. Fourh, liquidaion bias increases wih he disersion of observed ransacion reurns, σ. When σ, liquidaion bias disaears. he inuiion behind his is sraighforward. σ imlies ha all ossible bid rices collase o a single rice a each oin in ime. u differenly, he disribuion of ossible bidding rices, which becomes a single rice, is he same as ha of ransacion rices, so no bias resuls. A final imlicaion of heorem is ha we canno conclude ha real esae submarke A is beer han real esae submarke B simly by looking a heir comaraive reurn and risk esimaed from ransacion rices. arkeing eriod informaion mus be also considered. For examle, suose he reurn and volailiy based on he ransacion samle are he same in boh submarke A and submarke B (i.e., A B A B r r and σ σ ), bu assume he execed markeing eriod in submarke A is A B much higher han ha in submarke B (i.e., E [ ] > > E[ ]). By heorem, we readily A B A B see ha σ > > σ and r < < r. herefore, he real esae marke in submarke B is acually beer han ha in submarke A in he sense of dominaing in boh reurn and risk. Naively looking a he reurn and volailiy based on he ransacion rices alone, however, would lead us o conclude ha hey were idenical. In he financial markes for hickly-raded securiies, he markeing eriod is rivial, and consequenly, he marke reurn and volailiy are he same as he reurn and volailiy esimaed from ransacion rices. In he real esae marke, however, he markeing eriod is subsanial, and he marke reurn and volailiy can be very differen from he reurn and volailiy based on he ransacion rices. his rovides an addiional raionale for he noion ha reurns risk-adjused reurns for real esae obained hrough observed ransacions daa require subsanial adjusmen o lace hem in a common framework for analysis wih liquid securiies, esecially for hose real esae asses in which he invesors oenially face he necessiy of quick sale. Seller eerogeneiy. hus far, we have assumed all sellers are idenical in erms of heir reservaion rice levels over ime. owever, holding everyhing else consan, sellers wih differen financial siuaions may have very differen selling sraegies. Sellers who are recenly divorced, or face a job loss likely have o sell heir houses more quickly han ohers. Examles of seller moivaions ha have been sudied in he lieraure include: roeries owned by relocaed sellers (urnbull, Sirmans and Benjamin, 199); vacan roeries (e.g. Zuelke, 1987); foreclosure roeries (e.g. Forgey, Ruherford and Vanbuskirk, 1994); and ime-consrains on sale (e.g. Glower, endersho, and aurin 1998).

23 We nex discuss how sellers moivaions affec heir execed reurn and volailiy. Le us ignore markeing eriod risk and focus only uon liquidaion risk under seller heerogeneiy. 19 Consider wo yes of sellers: ye A has a liquidiy consrain wih a shorer execed markeing eriod, denoed by, and ye B does no have any financial disress wih a longer execed markeing eriod, denoed by ( > ). Suose ha he execed eriod reurn and volailiy ha yes A and B receive are denoed by ( ra, σ A ) and ( rb, σ B ), resecively. We summarize our findings in he following heorem. heorem 3: Risk remium and markeing eriod. If wo sellers of ye A and B differ in heir execed markeing eriods (hence reservaion rices) and, wih <, seller A will suffer a lower eriod reurn and a higher volailiy in reurn han borrower B, as demonsraed by he relaionshis where: 1. rb ra 3 ( ) σ > (3) 1 1. σ A σ B σ > (33) (1 )(1 ) σ is he volailiy from marke valuaion. [roof rovided in Aendix 4] wo oins are worh noing from his resul. Firs, ye B sellers are execed o exerience a longer ime on he marke and hus have a higher liquidaion risk. owever, hey are also execed o receive a higher execed reurn and a lower volailiy in comensaion. he inuiion behind his resul is quie clear. ye A sellers who are moivaed o sell quickly have a lower reservaion rice and acce earlier, hence receive lower offers. hose sellers who are no moivaed o sell quickly will have a higher reservaion rice and will only acce offers ha are relaively high, even if his means an exended wai. his resul is also consisen wih recen findings by uang (3). uang sudies an equilibrium in which agens face surrise liquidiy shocks and inves in liquid and illiquid asses. e finds ha he illiquid asse generaes a higher execed reurn o comensae is holders for he liquidaion risk. A second imlicaion of heorem 3 is ha ye A sellers wih financial disress have o sell more quickly han ye B sellers. owever, hey have o give u a higher execed reurn and a lower volailiy. In oher words, frequency and immediacy of ransacion is closely relaed o he reurn and volailiy of he ransacion roeries. he sysemaic differences in reurn and volailiy among differen sellers in he real esae marke sugges ha roeries wih owners subjec o differen selling moivaions, hence differen markeing eriods before sale, should exhibi differen aerns of rice behavior. hese differen submarkes, disinguished by breviy of markeing eriod, 19 ence, risk and reurn essenially reresen ex os and no ex ane measures. 3

24 should be analyzed searaely, or a leas he analyical mehodology describing heir rice behavior should ake ino accoun he sysemaic differences in he frequency and immediacy of ransacions (see eerogeneous Sellers in Aendix 5). Emirical Alicaions. In his secion, we discuss how we can emirically correc for liquidaion bias. he required informaion in our correcion mehod includes ransacion rices and he markeing eriod for each ransacion roery, boh of which are readily available from he marke. For simliciy, we assume ha sellers in he real esae marke are homogeneous. ence, here is only one uniform reservaion rice in he marke, and every seller acces an offer only when he offer rice is a leas as high as his rice. Furhermore, we assume ha sellers are 1 ercen likely o exerience he necessiy of forced sale a a secific oin in ime during he markeing eriod. 1 Suose he esimaed reurn and volailiy from he ransacion rices are rˆ and σ, resecively, and he observed average markeing eriod is ˆ. From equaions (3) and (31), we can back ino an esimae of he unobservable marke reurn and volailiy as follows, rˆ rˆ 3ˆ σˆ (3 ) σˆ ˆ ˆ (31 ) (1 ) σ Equaions (3 ) and (31 ) rove ha, given he same reurn and volailiy esimaed from ransacion rices, he underlying marke reurn and volailiy can be quie divergen. We nex look emirically a how he average markeing eriod affecs valuaion bias in boh he residenial and commercial roery markes. Again, we assume 1 ercen likelihood of forced sale a a secific oin in ime during he markeing eriod. Case I: he US Residenial roery arke. As before, we assume an observed average annual ransacion-based reurn of 5.% and a sandard deviaion of 1.67%, as aken from OFEO s U.S. ouse rice Index during he eriod 198Q1 o 4Q4. Exhibi 5 illusraes how he marke reurn and volailiy correced for liquidaion bias change wih he average markeing eriod. As discussed earlier, we have ignored markeing eriod bias o focus soley on liquidaion bias; marke reurn and volailiy are essenially he reurn and volailiy absen real esae illiquidiy, i.e., wih a markeing eriod. From Exhibi 5, we can see ha a higher average markeing eriod imlies a lower effecive marke reurn and higher effecive marke volailiy, hence a higher he case of heerogeneous sellers wih differen reservaion rices is discussed in Aendix 4. 1 his assumion will be relaxed in fuure work. 4

25 liquidaion bias, given he same ransacion reurn (5.%) and volailiy (1.67%). When he acual markeing eriod is six monhs, he effecive marke reurn is almos 9 ercen lower han he ransacion reurn, and he effecive marke volailiy is over 5 ercen higher han he ransacion volailiy. herefore, in he resence of a high average markeing eriod, here is a significan bias roblem for he reurn and volailiy esimaed from a ransacion samle. As we know, in he residenial marke, he average markeing eriod is abou eigh o en monhs when he marke is cold, and abou four o five monhs when he marke is ho. Case II: he U.S. Commercial roery arke. he annual reurn and volailiy of he Naional Council of Real Esae Invesmen Fiduciaries (NCREIF) roery index during he 198Q1-4Q4 eriod was 8.6% and 3.%, resecively. herefore, as in he residenial case, we choose an observed average annual reurn of 8.6% and sandard deviaion of 3.% as he ransacion-based reurn and volailiy in he commercial marke over his eriod. Exhibi 6 illusraes he relaionshi beween average markeing eriod and valuaion bias in he commercial marke. As in Exhibi 5, we can see ha a higher average markeing eriod imlies a lower marke reurn and higher marke volailiy, given he same ransacion reurn (8.6%) and volailiy (3.%). When he acual markeing eriod is eigh monhs, he effecive marke reurn is over 43 ercen lower han he ransacion reurn, and he effecive marke volailiy is 67 ercen higher han he ransacion volailiy. Given he fac ha average markeing eriods in commercial markes are ofen longer han hose in residenial markes, valuaion bias in he commercial marke could be an even more serious roblem han in he residenial marke when using he reurn and volailiy esimaed from ransacion rices. VI. Conclusions his aer documens he oenial ricing biases resen in radiional real esae valuaion mehodologies ha imlicily assume real esae asses can be sold immediaely wihou waiing. he assumion of immediae execuion may be reasonable in he financial marke where he ime o rade an asse is rivial; however, i is cerainly no valid in he real esae marke where markeing eriod is no only uncerain bu also subsanial. his large magniude and uncerainy of markeing eriod exoses an invesor addiional risk; however, lile work has been done o sudy formally he naure and magniude of his risk. Over he as hree years (u unil only recenly), he residenial real esae marke in U.S. could be characerized as a ho marke, in which areciaion was high and markeing eriods were shor (4.5 monhs). In he early 199s, however, he marke was deressed, and he average markeing eriod was abou 9 monhs. Krainer (1) has a good discussion on ho and cold real esae markes. 5

26 We formally define and measure markeing eriod risk by roosing a new measure of ex-ane reurn and variance, which relaces he radiional ex-os ransacionbased reurn and variance measure, o caure boh rice risk and one comonen of liquidiy risk faced by real esae invesors. Our findings sugges ha he risk associaed wih ex ane reurns is subsanially higher han ha esimaed under radiional mehodologies. For examle, in he case of U.S. residenial marke, for an execed markeing eriod of 8 monhs, he acual risk o shor-erm invesors (one year holding eriod) can be as high as 3.4 imes ha found under radiional esimaes. o long-erm invesors (1 years holding eriod), alhough he risk of markeing eriod can be amorized over he longer holding eriod, he acual risk can sill be almos fory ercen higher han ha esimaed using radiional rocedures. A similar resul is found for he commercial marke: Under an execed markeing eriod of 8 monhs, shor-erm invesors incur an effecive risk almos 3 ercen ha assumed under radiional rocedures; long-erm invesors incur an effecive risk 3 ercen higher. herefore, radiional real esae valuaion mehodologies, which assume real esae can be sold immediaely and ignore he uncerainy of markeing eriod, can seriously underesimae real esae risk. A by-roduc of his finding is ha invesors wih longer invesmen ime horizons will be less affeced by real esae illiquidiy due o markeing eriod risk, no because hey do no rade frequenly bu because he addiional risk caused by he uncerainy of markeing eriod is amorized over he longer holding eriod. his resul is consisen wih he common erceion ha real esae is more favorable o long-erm invesors. Afer correcing for markeing eriod risk, anoher bias which we erm liquidaion bias, is also found o exis in radiional real esae valuaion mehodologies, esecially for cerain classes of invesors who face a high robabiliy of immediae forced sale. his is because ransacion rices are he rices from a runcaed disribuion of he underlying disribuion of marke valuaion by oenial buyers. olding oher hings equal, a higher reservaion rice imlies a longer execed markeing eriod and a larger deviaion of ransacion rices from he rue marke valuaion. We rovide evidence o suor he noion ha here is a osiive relaionshi beween he execed markeing eriod and his liquidaion bias. We formally examine how he execed markeing eriod of real esae affecs he magniude of liquidaion bias by considering a model in which he buyers bid rices are based on marke valuaion and a ransacion rice occurs if and only if a bid rice equals or exceeds he seller s reservaion rice. We derive a closed-form relaionshi among marke reurn/risk, ransacion reurn/risk, and he execed markeing eriod. We find ha a longer execed markeing eriod imlies a larger liquidaion bias, ceeris aribus. Consisen wih invesmen heory, invesors wih a higher execed markeing eriod are execed o receive a higher reurn and lower risk in comensaion. Looked a in anoher way, our resuls sugges ha ignoring he exisence of a finie and someimes significan markeing eriod in real esae can cause he rue underlying marke reurn o be much lower han he observed ransacion-based reurn and he underlying marke risk o be much higher han he observed ransacion-based risk. 6

27 We conclude ha radiional mehods of esimaion of real esae reurn and risk, which borrow in a naïve fashion from finance heory by ignoring real esae illiquidiy, no only undersae real esae risk bu also oversae real esae reurns. Our findings exend he emirical resuls of Fisher, Gazlaff, Gelner, and aurin (3) and Goezmann and eng (3) who seek a consan liquidiy index o correc for he neglec of he imorance of ime on marke and rading volume on rue reurns. We rovide a heoreical foundaion for esimaing he liquidiy bias ha resuls from his henomenon and underake simulaions ha obain reliminary esimaes of is magniude. he resen effor can hel us o undersand he aaren risk remium uzzle in real esae and aid in he develomen of correcive measures o rea real esae aroriaely in a mixed-asse orfolio. We have made cerain simlifying assumions hroughou our analysis for he sake of racabiliy which bear furher scruiny. Firs, we assume ha ex-os execed nominal rices and oal reurns increase linearly over he holding eriod. While his has hisorically been rue emirically over he long run, i is likely o be violaed in individual circumsances over he shor run when he marke faces a downurn. Second, we rea he holding eriod, rae of bid arrivals, and reservaion rices and bids as given, whereas a more generalized model would rea hem as endogenous. ore fundamenally, our model reas boh he ex-os execed reurn and volailiy and rice levels as exogenously given and hence indeenden of he robabiliy of sale. In a more general equilibrium framework, he ex-os execed reurn and volailiy should be saedeenden (Clarke R. and Silva. (1998)), as should he robabiliy of sale (Krainer (1)). Anglin (3) discusses how boh sale rice and robabiliy of sale are correlaed and vary over marke condiions. Finally, we have considered he single real esae asse in isolaion and no as a ar of a orfolio in which nonsysemaic risk comonens, oenially including elemens associaed wih liquidiy and valuaion bias, can be diversified away or minimized over long-erm holding eriods during which invesors have discreion as o he iming of markeing he asse. here also exiss he quesion as o he relaive relevance of our reamen in he residenial marke, in which a single residenial asse lacks diversificaion oenial, vs. he commercial marke, in which he insiuional invesor has many asses which are held in a diversified mixed-asse orfolio. In a subsequen working aer wih Bond and wang (5) we address a few of hese orfolio issues. 7

28 Exhibi 1 he ransacion rocess for Real Esae arkeing eriod (random) 1 1 Reurn uon A Successful Sale (random) Exhibi 8

29 Exhibi 3 Liquidiy Bias beween Ex-ane and Ex-os Variance Esimaes: he U.S. Residenial roery arke (Source: OFEO house rice index, 198Q1 o 4Q4) he able above shows by how much he variance of he reurns, afer aking he uncerainy of markeing eriod ino accoun, is greaer han ha given by radiional esimaion mehods using he unadjused sales rice as an unbiased esimae of marke value. For examle, if a homeowner holds a roery for 1 years and he execed markeing eriod is 8 monhs, he ex ane risk faced by he homeowner is 38% higher han ha given by he radiional ex os esimaion. 9

30 Exhibi 4 Liquidiy Bias beween Ex-ane and Ex-os Reurn Esimaes: he US Commercial roery arke (Source: NCREIF roery erformance index, 198Q1 o 4Q4) he able above shows by how much he variance of he reurns, afer aking he uncerainy of markeing eriod ino accoun, is greaer han ha given by radiional esimaion mehods using he unadjused sales rice as an unbiased esimae of marke value. For examle, if an invesor holds a roery for 1 years and he execed markeing eriod is 8 monhs, he ex ane risk faced by he invesor is 3% higher han ha given by radiional ex os esimaion. 3

31 Exhibi 5 Liquidaion Bias: he US Residenial roery arke (Source: OFEO house rice index, 198Q1 o 4Q4) arke reurn and volailiy reresen he reurn and volailiy received by sellers who mus sell heir real esae asses immediaely. he able above shows how we can imue he rue marke reurn and volailiy from he reurn and volailiy esimaed from a ransacion-based samle, given he execed markeing eriod. For examle, suose he observed ransacion reurn and volailiy are 5.% and 1.67%, resecively, and assume he average execed markeing eriod is 8 monhs. hen he rue marke reurn and volailiy (i.e., assuming insananeous sale, wih liquidaion risk bu wihou markeing eriod risk) are 3.9% and.78%, resecively. he liquidaion bias is simly he difference beween he marke reurn/volailiy and he ransacion reurn/volailiy. Exhibi 6 Liquidaion Bias: he US Commercial roery arke (Source: NCREIF roery erformance index, 198Q1 o 4Q4) arke reurn and volailiy reresen he reurn and volailiy received by he sellers who mus sell heir real esae asses immediaely. he able above shows how we can imue he rue marke reurn and volailiy from he reurn and volailiy esimaed from a ransacion-based samle, given he execed markeing eriod. For examle, suose he ransacion reurn and volailiy are 8.63% and 3.%, resecively, and assume he execed markeing eriod is 1 monhs. hen he rue marke reurn and volailiy are 4.1% and 5.87%, resecively. he liquidaion bias is simly he difference beween he marke reurn/volailiy and he ransacion reurn/volailiy. 31

32 Aendix 1 Execed arkeing eriod and Is Uncerainy under Alernaive Assumions abou he Disribuion of he robabiliy of Sale over ime Case 1: Consan Condiional robabiliy of Sale In his case, he condiional robabiliy (or hazard rae) of selling a similar roery in each marke eriod is consan. Suose he hazard rae is λ ( λ 1), hen he robabiliy of selling he roery in each markeing eriod (,1,,...) is a geomeric disribuion, λ ( 1 λ ) (A1.1) Insering he equaion above ino equaions (11) and (1), we can obain, 1 E [ ] 1 λ (A1.) 1 Var ( ) (1 λ ) λ (A1.3) herefore, a lower hazard rae (i.e., a less liquid marke) indicaes a higher execed markeing eriod ( E [ ] λ < ) and higher uncerainy of ime of sale ( Var ( ) λ < ). When λ 1 (i.e., a erfecly liquid asse), boh E [ ] and Var( ) become zero. Case : Uniform robabiliy of Sale Suose he robabiliy of sale is consan across all markeing eriods. Assume ha N 1 is he maximum markeing eriod. hen he robabiliy of sale in each markeing eriod (,1,,..., N 1) is 1 (A1.4) N We hus have, 1 E [ ] N (A1.5) 1 Var ( ) N (A1.6) 1 Equaions (A1.5) and (A1.6) demonsrae again ha a larger N (less liquidiy) imlies a longer execed markeing eriod and higher uncerainy of ime of sale. In 3

33 aricular, he execed markeing eriod and he volailiy (sandard deviaion) of sale ime will boh increase a abou he same order of magniude as N ; when N doubles, he execed markeing eriod and he uncerainy of sale ime almos double. Case 3: Exonenial Disribuion of Sale Suose real esae sales follow he oisson disribuion. hen he markeing eriod follows an exonenial densiy funcion wih arameer η, 1 1 η e, η (A1.7) We hus have, E [ ] η (A1.8) Var ( ) η (A1.9) herefore, he arameer η reresens he execed markeing eriod. Again we have, for a higher η (less liquidiy), boh he execed markeing ime and is volailiy are execed o increase a he same order of magniude. 33

34 Aendix roof of heorem 1 By he definiions of ex ane execed reurn and variance (i.e., equaions (13) and (14)), and he assumed disribuion of ex os reurn wih mean u ) ( and variance ) ( σ, we have ]) [ ( ) ( ] [ ] [ E u u R E R E ane ex (A.1) ] [ ) ] [ ] ([ ] [ ] [ ]] [ [ ) ( ane ex ane ex ane ex ane ex R E u R E R E R E R E R Var σ (A.) Insering equaion (A.1) ino equaion (A.) yields, ) ( ]) [ ( ) ( u Var E R Var ane ex σ (A.3) ence, he annualized ex-ane reurn and variance are, ] [ ) ( ) ( u E Var u u ane ex ane ex σ σ (A.4) Q.E.D. 34

35 Aendix 3 roof of heorem Based on equaions (6) and (7), he ransacion reurn (annualized) and he marke reurn (annualized) can be exressed as: r (A3.1) where, is uniformly disribued over ], [ r (A3.) where, is uniformly disribued over ], [. ence, we have, ) ( ] [ E r (A3.3) ] [ E r (A3.4) 1 ) ( ] [ Var σ (A3.5) 1 ) ( ] [ Var σ (A3.6) Equaions (8) and (1) yield, ] [ E (A3.7) Given, 1 ) ( ) 3( ) ( ) ( (A3.8) and equaions (A.4) o (A.6), we herefore have, 35

36 r r 3E[ ] σ (A3.9) Similarly, we can rewrie ( 1 ) [1 ] ( ) 1 (A3.1) From (A.1) and equaions (A.5), (A.6) and (A.7), we herefore have σ ( 1 E[ ]) σ (A3.11) Q.E.D. 36

37 Aendix 4 roof of heorem 3 Given he execed markeing ime for ye A sellers and for ye B sellers, and equaions (A3.9) and (A3.11), we have, ra r 3σ A (A4.1) σ σ σ (A4.) A A rb r 3σ B (A4.3) σ σ σ (A4.4) B B Rewriing equaions (A4.) and (A4.4) yields, σ A 1 σ 1 σ B 1 σ 1 (A4.5) (A4.6) From equaions (A4.5) and (A4.6), we have, σ A σ B σ (1 )(1 ) > (A4.7) From equaions (A4.1), (A4.3), (A4.5) and (A4.6), we have, herefore, r r B A r r 3 σ 1 3 σ 1 (A4.8) (A4.9) rb ra 3 ( ) σ > (A4.1) 1 1 Q.E.D. 37

38 Aendix 5 eerogeneous Sellers We have hereofore assumed all sellers are idenical in erms of heir reservaion rice characerisics. owever in realiy, differeniaed sellers coexis in he real esae marke. Sellers wih differen financial siuaions, as discussed earlier, receive differen ransacion reurns and volailiies. Suose here are N differen yes of sellers in he i marke, he average markeing eriod for sellers of ye i ( i 1,,..., N) is ˆ. Assume he roorional reresenaion of sellers of ye i is given by g, and denoe he reurn i i and volailiy for sellers of ye i based on he ransacion samle as rˆ and σˆ, resecively. hen, we can esimae he marke reurn and volailiy using one of wo aroaches. he firs aroach is o aly equaions (3) and (431 o each ye of seller and esimae marke reurn and volailiy searaely. he second aroach is o ake ino accoun he differences in he roorional reresenaion of each seller ye and esimae marke reurn and volailiy as follows, N i σˆ g i i 1 σ ˆ N (A5.1) 1 g i i 1 1 ˆ i 3ˆ uˆ σ g (A5.) N N i i rˆ g ˆ i i i 1 i 1 1 ˆ i i 38

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