A NEW APPROACH OF VALUING ILLIQUID ASSET PORTFOLIOS

Transcription

1 Yale SOM Workig Paper No. ICF A NEW APPROACH OF VALUING ILLIQUID ASSET PORTFOLIOS Liag Peg Yale School of Ecoomic JANUARY 9, 00 Thi paper ca be dowloaded wihou charge from he Social Sciece Reearch Nework Elecroic Paper Collecio: hp://paper.r.com/paper.af?abrac_id=58599

2 A New Approach of Valuig Illiquid Ae Porfolio * Liag Peg Yale Ecoomic Deparme Box 0868 New Have Coecicu liag.peg@yale.edu Fir Draf: March 5, 000 Curre Draf: Jauary 9, 00 I hak William N. Goezma, Roger G. Ibboo, Rober J. Shiller, ad Mahew Spiegel for heir ecourageme ad umerou helpful comme ad dicuio. I have alo grealy beefied from he comme of Doald Adrew, Sefao Ahaaouli, George Hall, Sefa Krieger, Giueppe Mocarii, Peer C. B. Phillip, Raier Schulz, ad paricipa i he workhop a Yale Uiveriy. All error are mie aloe.

3 Abrac: Thi paper propoe a ew approach of valuig porfolio ha coai illiquid ae. The approach ha hree major advaage. Fir, he eimaor are arihmeic average of idividual ae reur or heir proxie, o hey ricly correpod o acual porfolio reur. Secod, he approach i able o value porfolio i which ae are arbirarily weighed, icludig equal-weighed, price-weighed, ad value-weighed porfolio. Third, he model i eay o exed o icorporae ae characeriic daa o improve he accuracy. Simulaio wih acual daa of Dow Joe Idurial how ha hi ew approach provide uperior eimaor ha ome currely available aleraive.

4 A New Approach of Valuig Illiquid Ae Porfolio May impora ae raac ifrequely. For example, he real eae, ar, ad bod marke are geerally coidered illiquid. I real eae ad ar marke, ae ed o be held for year or eve decade bewee ale. I Uied Sae bod marke, le ha 0% of bod raac daily. A he ame ime, while more high-frequecy daa of ock rade ad quoe are available, reearcher ecouer ifreque radig problem more ofe, for example whe calculaig high frequecy ock price idex. The global equiy marke, if coidered a a iegraed marke, i alo illiquid: while he global equiy marke i coidered ope, regioal exchage may be cloed hu heir lied equiie may o be radable. No oly may exiig ae marke are illiquid; may o-be-eablihed marke migh be o. For example, for he ew macro marke origially propoed by Shiller (993a), uch a aioal icome ad labor icome marke, he uderlyig cah marke price may be obervable oly ifrequely. Clearly illiquid ae are widely pread wihi he ecoomy, which herefore raie he queio how o value porfolio ha coai illiquid ae, i pie of a poeial pauciy of raacio daa. Oe well-kow mehod for eimaig he reur of illiquid ae porfolio i Repea ale regreio (RSR). Thi echique eimae he ime erie reur uig he oberved raacio price for a ube of ae. Fir uggeed by Bailey, Muh, ad Noure (963), he RSR ha bee he ubjec of a grea deal of dicuio ice he. The origial RSR model ha wo eriou limiaio. Fir, i eimaor are geomeric average of cro-ecio For example, Cae ad Shiller (987, 989), Clapp ad Giaccoo (99, 999), Geler (997), Goezma (989, 99), Goezma ad Peg (000), Goezma ad Spiegel (995, 997), Shiller (99, 993b). All ugge variou way o modify ad improve he echique.

5 idividual ae reur, while he rue reur for a porfolio, o maer a equal-weighed or a value-weighed oe, are alway arihmeic average of idividual ae reur. Jee' iequaliy implie ha he geomeric average of ay e of poiive umber o all equal i le ha he arihmeic average of hem. Thu he RSR eimaor ed o be biaed away from acual porfolio reur. Eve whe all raacio are oberved ad he acual porfolio reur are already kow, he RSR eimaor ill do' equal he acual reur. Goezma (99) propoe a correcio mehod ha approximae he arihmeic average give he geomeric average, uder he aumpio ha he ae reur i each period are ideically logormally diribued. Thi mehod work well i imulaio. However, i eed o eimae uoberved cro-ecioal variace, which may o be eay i ome ceario uch a whe ime erie daa are heerokedaic. A a aleraive, Goezma ad Peg (000) propoe a mehod ha direcly provide arihmeic average eimaor of he equal-weighed porfolio reur. The ecod limiaio of he origial RSR mehod i ha i acually provide eimaor for equalweighed porfolio reur oly, while oe may be more iereed i price-weighed, valueweighed, or oher pecial-weighed porfolio. Shiller (99) propoe eimaor, eiher priceweighed or equal-weighed, ha are aalogou o he origial RSR eimaor bu are arihmeic average, which i called arihmeic repea ale eimaor (ARS). However, more flexible approache ha are able o value arbirary-weighed porfolio would be deirable. Reearcher have propoed mehod uig boh raacio daa ad daa of ae characeriic o eimae reur of ifreque-raded ae porfolio. For example, Cae ad Quigley (99), Cae e al (99), Clapp ad Giaccoo (99), hedoic repeaed meaure mehod (HRM) by Shiller (993b), ad diace-weighed repea-ale procedure (DWRS) by Goezma ad Spiegel (997). The primary mehodological advaage of uig boh

6 raacio ad characeriic daa lie i i abiliy o exploi he relaio bewee ae reur wih i characeriic. Limiaio of hee mehod are ha hey may o provide reur eimaor for arbirarily weighed porfolio, ad heir eimaor may o have aural ierpreaio, uch a beig arihmeic mea of idividual ae reur. Thi paper propoe a ew approach of valuig porfolio ha coai illiquid ae baed o he mehod of mome, which i called he GMM approach hroughou. The GMM approach ha hree major advaage. Fir, i i capable of valuig arbirarily weighed porfolio a log a ae weigh are kow or derivable. Few previou mehod claim o be able o do o. Secod, all GMM eimaor of porfolio reur are cro-ecioal arihmeic average of idividual ae reur (or proxie of hem), o he eimaor ricly correpod o acual porfolio ad o correcio i eeded, which i a impora improveme over he currely broadly ued RSR mehod. Third, he GMM approach i poeially exedable o icorporae ae characeriic daa o improve he accuracy. O oe had, he characeriic daa help o differeiae oe ae from aoher, which faciliae he correcio for he biaed ample problem ha raacio more likely ake place upo a ube of ae i he porfolio (hi will be how i ecio four). O he oher had, ice ae wih differe characeriic may have differe reur procee, characeriic daa would help o proxy idividual ae' reur more accuraely. Therefore he GMM eimaor of porfolio reur would be more accurae ice hey are average of proxie of idividual ae reur. Thi paper oice a fiie ample problem of he ew approach ha ifrequely raded ae ed o be over-weighed i he eimaio. A correcio mehod i propoed o provide he fiie ample verio of he GMM approach. To compare i performace wih ha of he RSR ad he ARS ad a imple mehod ha eimae a porfolio' reur for a period by 3

7 averagig all available idividual reur for ha period, acual fiacial daa are ued o do imulaio. Each imulaio firly coruc ifrequely raacio daa e by drawig ome of acual daily price for he Dow Joe Idurial Idex (DJII) ock over Sepember o December 999, he eimae he acual DJII daily reur wih he limied daa e. The accuracy of each mehod i meaured wih four differe aiic. They are he quared error of he geomeric mea of reur eimaor, he adard deviaio of he reur eimaor, he R reulig from regreio of he acual DJII daily reur upo he eimaed ime erie reur, ad he mea quared differece bewee he acual ad he eimaed reur. The imulaio how ha he GMM approach i uperior o oher mehod o meaurig he overall performace of porfolio ad o capurig period o period reur evoluio a well. The uperioriy of he GMM i more obviou o valuig porfolio coaiig boh liquid ad illiquid ae. The paper i orgaized a follow. Secio pree he mahemaical model of reur proce. Secio dicue he baic model eimaio ad illurae he eimaor by eimaig a exremely mall daa e. I alo dicue he fiie ample problem ad propoe a correcio mehod. Secio 3 decribe he procedure of he imulaio e ad repor reul. Secio 4 dicue poeial exeio of he model, icludig icorporaig characeriic daa io eimaio ad uig hem o correc he biaed ample problem. Secio 5 coclude. A appedix pree deail of eimaio algorihm for he equal-weighed ad he price-weighed porfolio. 4

8 I. Mahemaical Model of Reur Proce I.. Ae reur Defie capial appreciaio of ae a i ime period, ae a he ed of ime period over i price a he ed of period. r P P. a, a, / a, r a,, a he raio of he price of he Aume ha he r a, i deermied a followig: r a, E( ra, m, ca, ) εa, =. () The erm m i a e of porfolio-wide commo facor ha affec all ae reur i ime period. The erm c, i a e of characeriic of ae a i ime. The error erm a ε a, capure ae-pecific eve ha are repoible for uexpeced chage of price. Aume E( ε a, m, ca, ) = ad ε a, i idepede. Baed o hee aumpio, commo facor ad ae characeriic joily deermie a ae expeced reur i a ime period. The commo facor could be macroecoomic variable like he rik-free iere rae, iflaio rae, uemployme rae, ad o o. For houe, ae characeriic could be hedoic variable uch a locaio or quare-fee of floor pace. For equiie, hey could be P/E raio, B/M raio, capializaio ad o o. For bod, hey could be bod mauriy, raig, coupo rae or oher characeriic. The aumpio abou he ae reur proce are coie wih ha ae wih differe characeriic may have differe reur procee. 5

9 I.. Porfolio reur A porfolio coi of ui of value, ay dollar, ha are iveed i differe ae. The reur of a porfolio i ime period, r, equal he raio of he porfolio value a he ed of ime period over i value a he ed of ime period. Suppoe a porfolio coi of N dollar iveed i A differe ae a he ed of ime period, ad a dollar d become r d, a he ed of ime. The he reur of hi porfolio i ime period i r N r d, d= N. () Sice he reur of a dollar equal he reur of he ae i which hi dollar i iveed, all dollar iveed i he ame ae have ame value a he ed of period. Thu he porfolio reur i ime ca be expreed a average of ae reur. r ( w ra ) a, = A a=,. (3) The erm w a, i he weigh of ae a i hi porfolio. I equal he proporio of he dollar value of he porfolio iveed i ae a a he ed of. I.3. Reur of a radom dollar i a porfolio Dollar i a porfolio are diiguihed from each oher by he characeriic of ae i which hey are iveed. The probabiliy for a radomly eleced dollar o have pecific characeriic equal he weigh of he ae havig hee characeriic i he porfolio. Obviouly differely weighed porfolio have differe probabiliy diribuio of dollar characeriic. For example, coider porfolio of wo ae: oe riky bod ad oe rik-free 6

10 bod. For he equal-weighed porfolio, he probabiliy of a radom dollar beig rik-free i 0.5. For a price-weighed porfolio, if he rik-free bod ha higher price, he probabiliy of a radom dollar beig rik-free i larger ha 0.5. Le f (c) deoe he probabiliy of a radomly eleced dollar i ime period havig characeriic e c. Deoe by γ he expeced reur of he radom dollar i period codiioal upo he e of commo facor, he γ E ( rd m ) = E( E( rd, m, c ) = E( rd, m, c) f ( c) A ( ) = E( r m c ), a,, c a= a, w a,, (4) which equal he expeced porfolio reur. The ecod equaliy hold becaue of he law of ieraed expecaio. The la equaliy hold ice he probabiliy for he radom dollar o have pecific characeriic equal he weigh of he ae i he porfolio ha ha hee characeriic. Uig equaio () ad (4) ad he fac ha a dollar' reur equal he reur of he ae i which hi dollar i iveed, oe ca alway wrie he reur of a radomly eleced dollar d i he porfolio a r d, γηd, εd, = (5) wih η E r m, c ) / E( r m ), which i a fucio of he dollar characeriic. Sice d, ( d, d, d, he dollar i radomly eleced, i ha radom characeriic ad he erm η d, i a radom calar wih E( η, γ ) = by he law of ieraed expecaio. d Equaio (5) coec he reur of a radomly eleced dollar wih he expeced reur of he porfolio. I ha a iuiive ierpreaio. The reur of a radomly eleced dollar i a porfolio coi of hree par. The fir par i he expeced reur of he porfolio. The 7

11 ecod par i he expeced deviaio from he porfolio expeced reur due o he ae' characeriic. The hird par i a radom hock. II. Model Eimaio II.. GMM eimaor Now aume ae characeriic are o obervable ad he daa coi of raacio price ad ime. Aume ha a raacio alway ake place a he ed of a ime period. A repea-ale obervaio coi of he fir raacio price, he ime of he fir raacio, he ecod raacio price, ad he ime of ecod raacio. For obervaio, deoe by B he fir raacio price (he purchae or buy price), by S he ecod raacio price (he ale price), by b he ime of fir raacio, by ha of he ecod raacio. The holdig ierval of obervaio, deoed by H, coi of all ime period laer ha b ad o laer ha, i.e., H { b + }. The legh of H i deoed by T, o T b. Aume here are N repea-ale obervaio ad T + ime period i he ample, umbered from 0 o T. For ime period, deoe by O H } he e of all obervaio ha have { hi ime period i heir holdig ierval. Defie he ize of ha belog o O, by N. O, i.e. he umber of obervaio Le y equal he compoud reur from he oberved buy o ell, y S / B. The, 8

12 y ) = r = (, γ η, ε, H H H y γ ). / = ( η ε,, H H Sice E ( η, ε, ) =, he mome codiio E( y / γ ) 0 = for =,..., N, yield a H H parameer-defiig mappig uder uiable regulariy codiio, which are aumed o hold. Sample couerpar o he mome codiio defie he eimaor of γ : O w / ˆ y γ = 0, for =,..., T. (6) H The erm w, idicae how may dollar ample he repea-ale obervaio provide, o i equal he weigh of he ae ha correpod o he repea-ale obervaio. A obervaio provide more dollar ample of he porfolio if i ae ha heavier weigh i he porfolio. Thu by chooig differe w,, oe could ue he ame daa e o eimae reur of differe porfolio. For example, each obervaio provide he ame amou of dollar ample for he equal-weighed porfolio, o i w, hould equal o each oher o eimae he reur of he equalweighed porfolio. A he ame ime, a obervaio provide he amou of dollar ample proporioal o i ae price for he price-weighed porfolio, o w, hould be proporioal o i i ae price o eimae he price-weighed porfolio reur. Rearragig equaio (6), he eimaor of porfolio reur i ime period i γ ˆ = w, y / γˆ. (7) O { / H, } 9

13 Sice he y erm i a compoud reur, he y / { / H, } γ ˆ erm i he compoud reur from which all expeced reur for ime period wihi holdig ierval excep are ubraced. Thu hi erm i a proxy of a ae reur i ime period. The obviouly he eimaor γˆ i a arihmeic average of reur (or proxie of reur) of idividual ae i ime period. I iclude he reur (or proxie of reur) of all ae i he porfolio, a log a he ae are raded a lea oce before ad oce afer curre ime period. Alo, he reur of each ae, o maer he ae i raded frequely or ifrequely, i direcly icluded i he eimaor oly oce. Thu, hi eimaor doe' direcly over cou frequely raded or ifrequely raded ae. A very ice propery of he eimaor i ha: whe all ae are frequely raded, he eimaor are average of idividual ae reur, which exacly equal he acual porfolio reur. w, y = w, O O ( S B ) ˆγ =. II.. Reur of Equal-weighed ad Price-weighed Porfolio From equaio (6), hi approach i capable of valuig arbirarily weighed porfolio, a log a ae weigh are kow or derivable. The equal-weighed ad he price-weighed porfolio may be he mo widely ued porfolio i reearch. (Here he value-weighed porfolio i coidered a a pecial cae of price-weighed porfolio, i he ee ha he ae price i a value-weighed porfolio i price for he whole ae iead of for ju oe hare of he ae.) Here he deail of valuig hee wo kid of porfolio are preeed. 0

14 The eimaor of he equal-weighed porfolio reur ca be eaily obaied by leig N w, =, where N i defied earlier, a he umber of obervaio ha iclude ime period i heir holdig ierval. Deoe by e γ he reur of equal-weighed porfolio i ime period. The eimaor-defiig equaio are = O H e e y N }, / { ˆ / ˆ γ γ, for T,..., =. (8) For he price-weighed porfolio, a ae weigh i ime period i proporioal o i price a he ed of ime period. Deoe by p γ he reur of price-weighed porfolio i ime period. For illiquid ae, price are o obervable for all ime period, or are correpodig weigh. However, he model ielf provide eimaor for all uoberved price. For he ae correpodig o he repea-ale obervaio, a eimaor for i price a he ed of ime period - i + = =, ˆ ˆ b p B P γ. The a eimaor of he weigh of ae i ime period i + = + = = = O b p b p O i i B B P P w,,, ˆ ˆ ˆ ˆ ˆ γ γ. Wih he eimaed weigh, he reur eimaor of he price-weighed porfolio are defied a = + = = O p O b p S B γ γ ˆ / ˆ, for T,..., =. (9) Rearrage equaio (9), he eimaor of he price-weighed porfolio reur i ime period i

15 p S / γˆ p O = + γ ˆ =. (0) p B γˆ O = b II.3. A Illuraio of Eimaor For a example of he eimaor of equal-weighed ad price-weighed porfolio, coider a very mall daa e coiig of wo ae ad hree ime period umbered from 0 o. The fir ae wa old a he ed of each ime period, while he ecod oe wa old oly a he ed of he ime period 0 ad ime period. Deoe by P,0, P,, P, he price of he fir ae, by P,0,P, he price of he ecod ae. Thu here are hree repea-ale obervaio, he fir wo are for he fir ae ad he la oe i for he ecod ae. P, / P,0 Y = P, / P,. P, / P,0 I hi example, he eimaor of equal-weighed porfolio reur i ime period ad are e P, P, = γ ˆ + e, P,0 P ˆ,0 γ e P, P, = γ ˆ + e. () P, P ˆ,0 γ The eimaor of price-weighed porfolio reur are P + P / γˆ p p,, p,, γ ˆ =, γˆ p P,0 + P,0 P, + P,0γˆ P + P =. () Obviouly he reur eimaor of boh he equal-weighed ad he price-weighed porfolio have aural ierpreaio. The reur eimaor of equal-weighed porfolio are average of

16 idividual ae reur or proxie of hem; he eimaor of price-weighed porfolio equal he raio of porfolio value or proxie of hem. A he ame ime, calculaig he eimaor i eay becaue here are wo equaio for wo reur eimaor of each porfolio. II.4. A Fiie Sample Problem ad i Correcio Equaio (6) how ha a obervaio wih holdig ierval H direcly appear i he eimaor-defiig equaio for all ime period ha belog o H. For example, a obervaio whoe holdig ierval coiig of period ad would be ued o eimae ˆγ ad ˆγ, ad herefore appear i he defiig equaio for boh wo period. A he ame ime, from equaio (6), he ˆγ appear i he defiig equaio of ˆγ ad vice vera. Thu hi obervaio i acually ued wice i he eimaio of ˆγ : oe ime i direcly appear i he defiig equaio of ˆγ ad aoher ime i i icluded i ˆγ ad ˆγ appear i he defiig equaio of ˆγ. A fiie ample problem would rie whe he porfolio coi of mall amou of ae ad he legh of he repea ale obervaio holdig ierval varie a lo. The obervaio ha have log holdig ierval may domiae hoe wih hor holdig ierval whe eimaig he porfolio reur becaue hey are ued for much more ime. Uig he mall daa e from la ubecio a example, oe ca eaily how ha he compouded eimaed reur of he price-weighed porfolio for period ad i p p γ γˆ = ( P + P ) /( P + ). The acual compoud reur i already kow from he ˆ,,,0 P, 0 daa, which i P + P )/( P + ). Clearly he ecod ae are over-weighed i he GMM (,,,0 P, 0 eimaor, imply becaue i i le frequely raded ad he correpodig repea ale obervaio ha loger holdig ierval. 3

17 Dow-weighig repea ale obervaio wih loger holdig ierval ca olve he fiie ample problem. Dividig each repea ale obervaio wih i legh of holdig ierval, equaio (6) chage o O w / ˆ y γ = 0, for =,..., T. (3) T H Uig equaio (3) o eimae he ame daa e ued earlier, he compouded eimaed reur p p of he wo-ae price-weighed porfolio for wo period i γ γˆ = ( P + P ) /( P + ), ˆ,,,0 P, 0 which exacly equal he acual oe. Afer he correcio, he equal-weighed GMM eimaor i acually equivale o he arihmeic-average equal-weighed eimaor by Goezma ad Peg (000). III. Simulaio Te III.. Aleraive Mehod ad Accuracy Meaureme The imulaio e he performace of four aleraive mehod i he eimaig of ime erie reur of he price-weighed idex, i.e. he acual DJII. The fir oe i he fiie ample verio of he GMM mehod propoed i hi paper (GMM). The ecod oe i he verio of he repea ale regreio (RSR) ha ca be juified a maximum likelihood eimaor accordig o Goezma (99). The hird oe i he irumeal variable verio of he arihmeic repea ale regreio (ARS) propoed by Shiller (99). The fourh oe i a imple mehod ha eimae a daily DJII reur by averagig all available idividual daily reur for ha day (weighed by price). Though he repea ale regreio eeially provide eimaor of equal-weighed porfolio reur, I ill ue i a a bechmark becaue i i well 4

18 kow ad widely ued. The ARS i a aural bechmark ice i provide eimaor of priceweighed porfolio reur. The imple mehod would be a hady choice whe he problem of daa pauciy i o eriou, hu i i iereig o pu he mehod i he imulaio ad e i uefule. I i impora o make ure i i fair o pu hee four mehod ogeher i he imulaio ice each of hem could have may varia. For example, he RSR ha i varia like hreeage RSR ad Baye RSR; he ARS ha i ierval-weighed verio ad hedoic verio. The GMM i alo exedable o have imilar varia. I coider i i fair o pu hee four mehod ogeher i a hore race fir becaue hey are all oe-ep mehod while heir varia ypically ivolve more ha oe ep ad exra regreio, ad alo becaue hey provide eimaor wih obviou ecoomic meaig while heir varia geerally do'. Table provide a imple compario of he properie of hee four mehod. Amog hem, he GMM mehod, he ARS mehod, ad he imple mehod provide eimaor ha are arihmeic mea of idividual ae reur, while he RSR provide geomeric mea eimaor. The GMM, he ARS, ad he RSR eimae porfolio reur wih regreio, while he imple mehod doe' ue regreio. All mehod excep he RSR ue he aural price iead of he logarihmic oe. The RSR i able o eimae reur for equal-weighed porfolio oly ad he ARS i able o value boh equal-weighed ad price-weighed porfolio, while he GMM ad he imple mehod are able o value equal-weighed, price-weighed, ad oher weighed porfolio. The RSR ad he GMM mehod boh dow-weigh obervaio wih loger holdig ierval, while he ARS doe'. Shiller (99) propoe oher varia ha akig accou of error heerokedaiciy for differe obervaio bu require exra regreio 5

19 There are four differe meaureme for he accuracy of a mehod. Specifically, he fir meaureme evaluae he overall accuracy of a mehod. I i he quared differece of he geomeric mea of he eimaed reur ad ha of he acual reur, which i calculaed by T = T T T γ ˆ γ. = The maller i he quared differece, he more accurae i he mehod o valuig he porfolio' log erm performace. The ecod meaureme i he adard deviaio of he eimaed reur becaue a good mehod i expeced o provide eimaor whoe adard deviaio i cloer o he acual oe. The hird meaureme i he R reulig from regreio of acual reur erie upo eimaed oe, which capure he correlaio bewee he acual ad eimaed reur. A good mehod i expeced o have a higher R. The fourh meaureme i he mea of quared error (MSE) of eimaed reur for all period. I alo help o capure a mehod' period o period performace, ad i calculaed a T ( ˆ γ ) T = γ. Amog he four meaureme, he fir oe, which evaluae a mehod' abiliy o meaure a porfolio' overall performace, i coidered he mo impora. III.. Simulaio Procedure The daa are acual daily price of 30 Dow Joe Idurial Idex ock from Sepember o December 999. There are 85 radig day (o 84 daily reur for DJII ice he fir day i ad he eimaor may o loger be arihmeic average of idividual ae reur. 6

20 he bae period) ad oally,550 daily price for he 30 ock. The baic approach of a imulaio i o radomly elec ome price from all,550 daily price o coruc a ifreque-raacio daa e, he eimae he ime erie of DJII daily reur over he hree moh wih differe mehod. Baed o eimaor, oe i able o calculae he four differe accuracy meaureme for each mehod. The accuracy of differe mehod may deped o he frequecy of radig (he legh of holdig ierval) ad maybe he perceage of liquid ae i he porfolio a well. Therefore he e procedure carefully corol he umber of price draw from he acual daa ad he perceage of liquid ae i he porfolio. Specifically, here are wo group of imulaio. I he fir group, here i o liquid ae, ad all "oberved price" are radomly draw from acual price. Thi group coai welve ceario i which he umber of "oberved price" i 00, 400, 600, 800, 000, 00, 400, 600, 800, 000, 00, ad 400 repecively, repreeig ceario i which illiquid ae rade wih differe frequecy. Smaller umber of oberved price correpod o ceario i which ae rade le frequely ad vice vera. I he ecod group of imulaio, firly radomly elec ome ock a "liquid ae", whoe price are oberved over all ample period. There are 4 ceario i which he porfolio ha differe perceage of "liquid ae": 0%, 0%, 30%, ad 40%. I each ceario, he umber of price draw for oher ock, he "illiquid ae", are alo corolled, beig 0%, 0%, ad 30% repecively. Coequely, here are oal differe ceario i he ecod group of imulaio: perceage of "liquid ae" rage over 0%, 0%, 30%, ad 40%; ad i each cae, he umber of oberved price for "illiquid ae", rage over 0%, 0%, ad 30%. The ecod group of imulaio e performace of differe mehod i valuig porfolio coiig of boh liquid ad illiquid ae. For each ceario i he fir ad he ecod group, 7

21 imulaio i repeaed for 00 ime. Thu he repored aiic of accuracy meaureme are average over 00 imulaio. III.3. Simulaio Reul Table repor he imulaio reul for he four mehod o valuig porfolio orely coiig of "illiquid ae" porfolio. From he quared error of he geomeric mea of reur eimaor, he GMM mehod periely provide he mo accurae meaureme of porfolio' overall performace. For all welve ceario, he quared error of i geomeric mea i alway maller ha ha of oher mehod. The ruig up i he ARS, followed by he RSR ad he he imple mehod. The imple mehod perform poorly whe ae rade o very frequely, ad fially over-perform he RSR whe he daa miig i le ha 7%. The adard deviaio of eimaor for all four mehod geerally decreae ad coverge o he acual adard deviaio whe more ad more price are oberved. However, o mehod i obviouly uperior i he ee ha havig adard deviaio much cloer o he acual oe. No a urprie, for all four mehod he average R icreae wih he umber of "oberved price", ad he average MSE decreae wih i, which cofirm ha all mehod beer capure acual reur' evoluio whe more price are oberved. A he ame ime, he GMM mehod periely ad obviouly perform beer ha he ARS ad lighly beer ha he RSR i erm of higher R. The imple mehod ha high R whe ae rade very ifrequely, which i acually mileadig becaue here may be may period for which he imple mehod i imply o able o provide eimaor. The RSR i acually doig well whe ae rade very ifrequely. I average MSE i maller ha ha of GMM ad ARS whe he umber of oberved price i le ha 00. A a cocluio, he GMM work well i capurig he 8

22 evoluio of daily reur for price-weighed porfolio coiig of illiquid ae oly, a lea whe ae rade reaoably frequely. Table repor he imulaio reul o valuig porfolio coiig of boh "liquid ae" ad "illiquid ae". All mehod eem more accurae ha whe eimaig porfolio orely coiig of illiquid ae, afer he umber of oberved price i corolled. For example, he average R for he GMM o value porfolio coaiig 0% liquid ae ad wih 0% of illiquid ae' price oberved (correpodig o abou 500 oberved price) i 49.85%, while he average R for i o value porfolio orely coaiig illiquid ae ha have 600 oberved price i 37.46%. I cocluio, all mehod are more accurae if ome of ae i he porfolio rade very frequely, eve hough he oal umber of oberved price i low. The GMM mehod i clearly uperior o all oher mehod o meaurig he overall performace of porfolio. I quared error of geomeric mea of reur i obviouly maller ha ha of oher mehod i all ceario. A he ame ime, he GMM alway ha higher R ha oher hree mehod, ad i ha maller MSE ha oher mehod excep i everal ceario uch a valuig porfolio coiig of 0% liquid ae ad wih 0% or 30% price oberved for illiquid ae. I cocluio, he imulaio cofirm ha he GMM mehod i clearly uperior o oher hree mehod: he ARS, he RSR, ad he imple mehod, o meaurig porfolio' overall performace ad period o period evoluio a well. I uperior i more obviou whe valuig price-weighed porfolio coaiig boh liquid ad illiquid ae. Whe valuig he porfolio orely coiig of illiquid ae, he GMM mehod i obviouly more accurae ha he oher mehod a lea whe ae rade reaoably frequely. 9

23 IV. Poible Exeio Whe ae characeriic are obervable, he accuracy of porfolio reur eimaor may be improved. Fir, ice ae characeriic are aumed o help o deermie reur, kowig characeriic help o obai beer proxie for each ae' igle period reur, hu help o obai more accurae porfolio reur eimaor, ay, hedoic GMM eimaor. Secod, kowig characeriic help o differeiae ae from each oher, which make i poible o correc he biaed ample problem. The fir ubecio dicue he hedoic GMM eimaor, ad he ecod oe dicue he correcio of biaed ample. IV.. Hedoic GMM eimaor Equaio (7), (), ad () how ha he eimaor of γ i a arihmeic average of idividual igle-period reur or proxie of hem. If all proxie exacly equal acual idividual igle-period reur, he eimaor would exacly equal he acual porfolio reur. Clearly he accuracy of he porfolio reur eimaor deped o he accuracy of he proxie of idividual igle-period reur. Whe ae characeriic are o obervable, ae are o differeiaed from each oher, i which cae he be proxy of a ae' reur i he expeced porfolio reur. However, whe ae characeriic are obervable, i i poible o ge beer proxie becaue he characeriic may yemaically affec a ae expeced reur. The model aume ha expeced deviaio of a ae' reur from he expeced porfolio reur i a fucio of he ae' characeriic. A hree-age procedure may be ued o eimae he fucioal form ad he provide more accurae eimaor of he porfolio reur. I he fir age, eimaig he model a if characeriic were o obervable, which 0

24 provide coie eimaor of he expeced porfolio reur, deoed by γ ˆ ice hi eimaio i he fir age. The differece of a oberved ae reur (or compoud reur) from he eimaed porfolio reur i. y γ ). (4) / ˆ = y / γ eˆ = ( η ε eˆ,, H H H The erm ê i eimaio error i he fir age. For he purpoe of impliciy, aume ae characeriic remai he ame wihi each holdig ierval. The for repea-ale obervaio, he deviaio of i reur from he expeced porfolio reur, i.e. η, for T, are coa over he whole holdig ierval. The implify he oaio o η. Clearly he erm wih error coaied. T ~ η y ˆ / γ i a meaureme of η H Give he η ~ ad obervable characeriic, he ecod age eimae he fucioal form how he expeced deviaio of a ae' reur deped o i characeriic. η ~ = g( ) ε, wih E [ ε g ( )] =. (5) c c Here boh parameric ad o-parameric approache may be ued o eimae he fucioal form of g (.). Deoe by ηˆ he eimaor of η, i.e. η ˆ = gˆ( c ). The hird age defie T y = / ˆ y η, ad eimae he porfolio reur wih y iead of y.

25 3 w, /( ˆ ) y γ = 0, for =,..., T. H O The eimaor of porfolio reur i 3 y y γ ˆ = w, = w,. (6) 3 3 O γˆ O γˆ ηˆ { / H, } { / H, } Clearly η ˆ i more accurae ha γ a he proxy for he ae' reur i period becaue i γ 3 ˆ ˆ exploi he fac ha ae characeriic help o deermie a ae' reur. Coequely he erm y γˆ 3 ηˆ { / H, } i a beer proxy for he ae' reur i ime period, ad he 3 γ ˆ, a average of idividual igle-period reur or heir proxie, i herefore more accurae. Thi procedure alo provide eimaor of expeced reur for a ub-e of ae wih pecific characeriic. For example, i real eae reearch, hi procedure i able o eimae houig idex for o oly a broad meropolia bu alo a pecific eighborhood wihi. Suppoe a idicaor variable equal o if a houe i i he eighborhood ad 0 oherwie. The value of hi idicaor variable ca be rea a a characeriic for a houe. The hree age procedure propoed here i able o eimae he meropolia idex ad he impac of he idicaor variable o a houe' reur a well, which i he expeced reur deviaio from he meropolia idex for houe i he eighborhood. The he eimaor of houig idex for he eighborhood equal o he meropolia idex adjued by he expeced deviaio from i for houe i he eighborhood.

26 IV.. Correcio for biaed ample The biaed ample problem exi if he probabiliy for raacio o ake place i higher for oe ube of ae i he porfolio ha oher. Thi problem may be correced if he ae characeriic ha help o differeiae he ube of ae from oher are obervable. For example, uppoe oe i iereed i a porfolio coiig of all houe i ow A ad ow B. Durig he ample period, a ew compay headquarer ielf i ow A bu ohig imilar happe i ow B, which may caue much more houe raacio i ow A ha ow B. Therefore while mo houe i ow A are icluded i he repea ale daa, oly ome houe i ow B are icluded becaue of much le raacio akig place here wihi he ample period. Thu he raacio daa are biaed. If houe locaio i uobervable, houe i ow A ad ow B ca' be differeiaed from each oher. The here may o be ay way o correc for he biaed ample. The eimaed porfolio reur are acually for a porfolio ha coi of more houe i ow A ha wha i deired. However, if houe locaio i obervable, oe ca ell how may houe i each ow are icluded i he repea ale daa. Suppoe he acual umber of houe i each ow are roughly he ame, bu here are wo ime of houe i ow A icluded i he repea ale daa ha houe i ow B. Baed o he belief ha houe i he ame ow follow he ame reur proce, oe ca double he weigh of he houe i ow B ha are icluded i he daa durig eimaio, which make he daa provide equal umber of ample for houe i each ow. Therefore he eimaed porfolio reur are for he porfolio ha coi of ame proporio of houe i boh ow. 3

27 V. Cocluio Thi paper propoe a ew approach o value porfolio coaiig illiquid ae baed o mehod of mome. The model of reur proce i meaigful ad he GMM eimaor have aural ierpreaio. All he eimaor are arihmeic average of idividual ae reur (or heir proxie) ad ricly correpod o porfolio reur, which i a impora improveme over he currely broadly ued RSR mehod. Thi ew approach provide eimaor for reur of ay arbirary-weighed porfolio, icludig equal-weighed, priceweighed, ad value-weighed porfolio, which few model claim o be able o do. Thi model accommodae he arihmeic-rsr propoed by Goezma ad Peg (000). Alo, hi model i flexible ad very eay o exed. For example, i i able o eimae he porfolio reur wih or wihou ae characeriic daa, while he eimaor could be more efficie if boh price daa ad characeriic daa are available. A he ame ime, he model may be able o provide more accurae eimaor by correcig he ample bia problem ha raacio may ake place more likely o over-valued ae. Simulaio are ued o e he accuracy of he GMM eimaor propoed i hi paper. The daa are acual fiacial daa:,550 daily price of Dow Joe Idurial Idex ock over Sepember 999 o December 999. The baic approach of a imulaio i o radomly elec ome price from all daily price o coruc a ifreque-raacio daa e, he eimae he acual DJII daily reur wih differe mehod. The accuracy of a mehod i meaured wih four aiic. They are he quared error of he geomeric mea of reur eimaor, he adard deviaio of he reur eimaor, he R reulig from regreio of he acual DJII daily reur upo he eimaed ime erie reur, ad he mea quared differece bewee 4

28 he acual ad he eimaed reur. The imulaio cofirm ha he GMM mehod i clearly uperior o he RSR, he ARS, ad he imple mehod ha eimae a daily reur by averagig all available idividual daily reur for ha day. The uperioriy of he GMM i more obviou o valuig price-weighed porfolio coaiig boh liquid ad illiquid ae. 5

29 Appedix: Eimaio Algorihm To eimae he reur of equal-weighed porfolio, defie marix X, Y, W ad I a followig. The X i a N by T dummy marix. I row correpod o repea-ale obervaio, ad colum correpod o ime period. For row, he fir ozero dummy appear i he poiio ha correpod o he ime period b +, he ime period immediaely afer he fir ale of h obervaio, ad he la ozero dummy appear i he poiio correpodig o, he ime period of he ecod ale. All eleme bewee hee wo ozero dummie alo equal oe, while oher eleme i hi row are zero. A a example, if a ae wa purchaed a he ed of ime period ad old a he ed of ime period 4, ad T=5, i correpodig row i X i ( 0,0,,,0). The Y i defied a a N by vecor whoe h eleme i y. The W i a N by N diagoal marix whoe h eleme i marix form a /T. The I i a N by vecor of. Now, he eimaor-defiig equaio for he equal-weighed porfolio ca be wrie i [ log( Y ) X log( γˆ )] X WI = X W exp, or W { exp [ log( Y ) X log( ˆ )] I } = 0 X γ. I i clear ha here are T equaio for T eimaor. Though hee equaio are o liear, olvig hem wih earchig echique may o be very difficul. The price-weighed porfolio reur are eve eaier o eimae. Defie a T by vecor β whoe h eleme i a reciprocal price idex for ime, β / ˆ. γ = 6

30 I i obviou ha kowig β i equivale o kowig γˆ. Defie by Z a N by T + marix whoe row correpod o repea-ale obervaio, ad colum correpod o ime period bu ar wih ime period 0. The b h eleme i row equal B, ad h eleme equal all oher eleme are 0. For example, for he daa e ued i earlier ecio o illurae he eimaor, he Z i P,0 P, 0 Z = 0 P, P,. P,0 0 P, Wih marix X, Z, ad vecor β, he eimaor-defiig equaio for price-weighed porfolio reur ca be wrie i marix form a X WZ = 0. β They are liear equaio ad i i rivial o olve ou β. Oce β i kow, γˆ i kow. For example, γ ˆ = β, γˆ = β β for >. S, 7

31 Referece Bailey, M.J., R.F. Muh ad H.O. Noure A Regreio Mehod for Real Eimae Price Idex Corucio. Joural of he America Saiical Aociaio 58: Cae, B., H.O. Pollakowki ad S.M. Wacher. 99. O Chooig amog Houe Price Idex Mehodologie. AREUEA Joural 9(3): Cae, B. ad J.M. Quigley. 99. The dyamic of Real Eae Price. The Review of Ecoomic ad Saiic 73(): Cae, K.E. ad R.J. Shiller Price of Sigle Family Home ice 970: New Idexe for Four Ciie. New Eglad Ecoomic Review: Cae, K.E. ad R.J. Shiller The Efficiecy of he Marke for Sigle Family Home. America Ecoomic Review 79: Clapp, J.M. ad C. Giaccoo. 99. Eimaig Price Tred for Reideial Propery: A Compario of Repea Sale ad Aeed Value Mehod. Joural of Real Eae Fiace ad Ecoomic 5(4): Clapp, J.M. ad C. Giaccoo Reviio i Repea-Sale Idexe: Here Today, Goe Tomorrow? Real Eae Ecoomic 7(): Geler, D Bia ad Preciio of Eimae of Houig Iveme Rik Baed o Repea- Sale Idice: A Simulaio Aalyi. Joural of Real Eae Fiace ad Ecoomic 4(): Goezma, W.N. 99. The Accuracy of Real Eae Idice: Repea Sale Eimaor. Joural of Real Eae Fiace ad Ecoomic 5(): Goezma, W. N., ad L. Peg The Bia of he RSR eimaor ad he Accuracy of Some Aleraive. Workig Paper, Ieraioal Ceer for Fiace, Yale Uiveriy. Goezma, W.N. ad M. Spiegel No-emporal Compoe of Reideial Real Eae Appreciaio. Review of Ecoomic ad Saiic 77(): Goezma, W.N. ad M. Spiegel A Spaial Model of Houig Reur ad Neighborhood Subiuabiliy. Joural of Real Eae Fiace ad Ecoomic 4(): -3. Shiller, R.J. 99. Arihmeic Repea Sale Price Eimaor. Joural of Houig Ecoomic ():

32 Shiller, R. J. 993a. Macro Marke: Creaig Iiuio for Maagig Sociey' Large Ecoomic Rik. Oxford Uiveriy Pre: New York. Shiller, R. J. 993b. Meaurig Ae Value for Cah Seleme i Derivaive Marke: Hedoic Repeaed Meaure Idexe ad Perpeual Fuure. Joural of Fiace 48(3):

33 Table. Propery Compario of Aleraive Mehod Thi able compare he properie of four aleraive mehod. The fir propery i if he eimaor are arihmeic average of idividual ae reur. The ecod oe i if he mehod ru regreio. The hird oe i if he mehod ue aural price iead of logarihmic price. The fourh o he ixh are if he mehod i able o value equal-weighed, price-weighed, or oher-weighed porfolio repecively. The eveh propery i if he mehod dow weigh he obervaio wih log holdig ierval. The "Y" repree "ye", ad he "N" repree "o". Arihmeic average Regreio Naural Price Equalweighed Priceweighed Oherweighed Time Dow weigh GMM Y Y Y Y Y Y Y ARS Y Y Y Y Y N N RSR N Y N Y N N Y Simple Y N Y Y Y Y NA 30

34 Table. Simulaio Reul: Valuig Porfolio Sorely Coiig of Illiquid Ae Thi able repor imulaio reul for four eimaio mehod: he GMM, he ARS, he RSR, ad he imple mehod. The baic procedure of a imulaio i o radomly elec N price from 550 daily price for all 30 ock i DJII over Sepember o December 999 o coruc a ifreque-raacio daa e, he eimae he ime erie of DJII daily reur wih differe mehod. The imulaio i ru for 00 ime for N equal o 00, 400, 600, 800, 000, 00, 400, 600, 800, 000, 00, ad 400 repecively. All accuracy meaureme umber are average for 00 imulaio. The R reul from regreio of acual daily reur upo eimaed erie of eimaor. The MSE of reur erie i he average quared differece bewee acual ad eimaed reur. N Squared Error of he Geomeric Mea of Eimaed Reur Serie (i %) GMM ARS RSR Simple Sadard Deviaio of Reur Serie (i 0.0%) Acual GMM ARS RSR Simple R (i perceage) GMM ARS RSR Simple MSE of Reur Serie (i 0.00%) GMM ARS RSR Simple

35 Table 3. Simulaio Reul: Valuig Porfolio Coiig of Liquid ad Illiquid Ae Thi able repor imulaio reul for four eimaio mehod: he GMM, he ARS, he RSR, ad he imple mehod. A imulaio coi of wo ep. Fir, coruc a ifreque-raacio daa e by radomly elecig x% ock a "liquid ae", which mea all heir price are obervable, ad y% daily price of he re of ock. The eimae he acual DJII daily reur. The x equal 0, 0, 30, ad 40, ad y equal 0, 0, ad 30 repecively. All accuracy meaureme umber are average for 00 imulaio. The R reul from regreio of acual daily reur upo eimaed erie of eimaor. The MSE of reur erie i he average quared differece bewee acual ad eimaed reur. (x, y) (0, 0) (0, 0) (0, 30) (0, 0) (0, 0) (0, 30) (30, 0) (30, 0) (30, 30) (40, 0) (40, 0) (40, 30) Squared Error of he Geomeric Mea of Eimaed Reur Serie (i %) GMM ARS RSR Simple Sadard Deviaio of Eimaed Reur Serie (i 0.0%) GMM ARS RSR Simple R (i perceage) GMM ARS RSR Simple MSE of Reur Serie (i 0.00%) GMM ARS RSR Simple