X-CAPM: An Extrapolative Capital Asset Pricing Model

Transcription

1 X-CAPM: An Exapolaiv Capial Ass Picing Modl Nicholas Babis*, Robin Gnwood**, Lawnc Jin*, and Andi Shlif** *Yal Univsiy and **Havad Univsiy Mach 6, 4 Absac Suvy vidnc suggss ha many invsos fom blifs abou fuu sock mak uns by xapolaing pas uns. Such blifs a had o concil wih xising modls of h aggga sock mak. W sudy a consumpion-basd ass picing modl in which som invsos fom blifs abou fuu pic changs in h sock mak by xapolaing pas pic changs, whil oh invsos hold fully aional blifs. W find ha h modl capus many faus of acual pics and uns; impoanly, howv, i is also consisn wih h suvy vidnc on invso xpcaions. W a gaful o Makus Bunnmi, John Campbll, Juhani Linnainmaa, David Solomon, Lawnc Summs, and smina paicipans a Calch, Damouh Collg, Havad Univsiy, h London Businss School, h London School of Economics, h Massachuss Insiu of Tchnology, Oxfod Univsiy, h Univsiy of Califonia a Bkly, h Univsiy of Chicago, h Univsiy of Souhn Califonia, Yal Univsiy, h NBER Summ Insiu, and h Sanfod Insiu fo Thoical Economics fo usful fdback; and o Jonahan Ingsoll fo many hlpful convsaions.

2 . Inoducion Rcn hoical wok on h bhavio of aggga sock mak pics has id o accoun fo sval mpiical gulaiis. Ths includ h xcss volailiy puzzl of LRoy and Po (98) and Shill (98), h quiy pmium puzzl of Mha and Psco (985), h low colaion of sock uns and consumpion gowh nod by Hansn and Singlon (98, 983), and, mos impoanly, h vidnc on pdicabiliy of sock mak uns using h aggga dividnd-pic aio (Campbll and Shill 988, Fama and Fnch 988). Boh adiional and bhavioal modls hav id o accoun fo his vidnc. Y his sach has lagly nglcd anoh s of lvan daa, namly hos on acual invso xpcaions of sock mak uns. As cnly summaizd by Gnwood and Shlif (4) using daa fom mulipl invso suvys, many invsos hold xapolaiv xpcaions, bliving ha sock pics will coninu ising af hy hav pviously isn, and falling af hy hav pviously falln. This vidnc is inconsisn wih h pdicions of many of h modls usd o accoun fo h oh facs abou aggga sock mak pics. Indd, in mos adiional modls, invsos xpc low uns, no high uns, if sock pics hav bn ising: in hs modls, ising sock pics a a sign of low invso isk avsion o low pcivd isk. Cochan () finds h suvy vidnc uncomfoabl, and commnds discading i. In his pap, w psn a nw modl of aggga sock mak pics which amps o boh incopoa xapolaiv xpcaions hld by a significan subs of invsos, and addss h vidnc ha oh modls hav sough o xplain. Th modl includs boh aional invsos and pic xapolaos, and xamins scuiy pics whn boh yps a aciv in h mak. Moov, i is a consumpion-basd ass picing modl wih infinily livd consums opimizing hi dcisions in ligh of hi blifs and mak pics. As such, i can b dicly compad o som of h xising Gnwood and Shlif (4) analyz daa fom six diffn suvys. Som of h suvys a of individual invsos, whil ohs cov insiuions. Mos of h suvys ask abou xpcaions fo h nx ya s sock mak pfomanc, bu som also includ qusions abou h long m. Th avag invso xpcaions compud fom ach of h six suvys a highly colad wih on anoh and a all xapolaiv. Eali sudis of sock mak invso xpcaions includ Vissing-Jognsn (4), Baccha, Mns, and Wincoop (9), and Amomin and Shap (3).

3 sach. W suggs ha ou modl can concil h vidnc on xpcaions wih h vidnc on volailiy and pdicabiliy ha has animad cn wok in his aa. Why is a nw modl ndd? As Tabl indicas, adiional modls of financial maks hav bn abl o addss pics of h xising vidnc, bu no h daa on xpcaions. Th sam holds u fo pfnc-basd bhavioal financ modls, as wll as fo h fis gnaion blif-basd bhavioal modls ha focusd on andom nois ads wihou imposing a spcific sucu on blifs. Sval paps lisd in Tabl hav sudid xapolaion of fundamnals. Howv, hs modls also suggl o mach h suvy vidnc: af good sock mak uns divn by song cash flows, h invsos hy dscib xpc high cash flows, bu, bcaus hs xpcaions a flcd in h cun pic, hy do no xpc high uns. Finally, a small liau, saing wih Cul, Poba, and Summs (99) and DLong al. (99b), focuss on modls in which som invsos xapola pics. Ou goal is o wi down a mo modn modl of pic xapolaion ha includs infini hoizon invsos, som of whom a fully aional, who mak opimal consumpion dcisions givn hi blifs, so ha h pdicions can b dicly compad o hos of h mo adiional modls. Ou infini hoizon coninuous-im conomy conains wo asss: a isk-f ass wih a fixd un; and a isky ass, h sock mak, which is a claim o a sam of dividnds and whos pic is dmind in quilibium. Th a wo yps of ads. Boh yps maximiz xpcd lifim consumpion uiliy. Thy diff only in hi xpcaions abou h fuu. Tads of h fis yp, xapolaos, bliv ha h xpcd pic chang of h sock mak is a wighd avag of pas pic changs, wh mo cn pic changs a wighd mo havily. Tads of h scond yp, aional ads, a fully aional: hy know how h xapolaos fom Fo xampl, in h cash-flow xapolaion modl of Babis, Shlif, and Vishny (998), invsos xpcaions of uns main consan ov im, vn hough hi xpcaions of cash flows vay significanly. Mo laboa modls of cash-flow xapolaion fo xampl, modls wih boh xapolaos and aional ads may, as a bypoduc, com clos o maching h suvy vidnc; h, w psn an alnaiv appoach ha may b simpl and mo dic. Modls in which invsos y o lan an unknown cash-flow gowh a fac simila challngs o modls of cash-flow xapolaion. Moov, Gnwood and Shlif (4) find ha suvy xpcaions of uns a ngaivly colad wih subsqun alizd uns, a pan ha is had o susain in a modl of aional laning. 3

4 hi blifs and ad accodingly. Th modl is simpl nough o allow fo a closdfom soluion. W fis us h modl o undsand how xapolaos and aional ads inac. Suppos ha, a im, h is a posiiv shock o dividnds. Th sock mak gos up in spons o his good cash-flow nws. Howv, h xapolaos caus h pic jump o b amplifid: sinc hi xpcaions a basd on pas pic changs, h sock pic incas gnad by h good cash-flow nws lads hm o focas a high fuu pic chang on h sock mak; his, in un, causs hm o push h im sock pic vn high. Mo insing is aional ads spons o his dvlopmn. W find ha h aional ads do no aggssivly counac h ovvaluaion causd by h xapolaos. Thy ason as follows. Th is in h sock mak causd by h good cash-flow nws -- and by xapolaos acion o i -- mans ha, in h na fuu, xapolaos will coninu o hav bullish xpcaions fo h sock mak: af all, hi xpcaions a basd on pas pic changs, which, in ou xampl, a high. As a consqunc, xapolaos will coninu o xhibi song dmand fo h sock mak in h na m. This mans ha, vn hough h sock mak is ovvalud a im, is uns in h na fuu will no b paiculaly low hy will b bolsd by h ongoing dmand fom xapolaos. Rcognizing his, h aional ads do no shaply dcas hi dmand a im ; hy only mildly duc hi dmand. Pu diffnly, hy only paially counac h ovpicing causd by h xapolaos. Using a combinaion of fomal poposiions and numical analysis, w hn xamin ou modl s pdicions abou pics and uns. W find ha hs pdicions a consisn wih sval ky facs abou h aggga mak and, in paicula, wih h basic fac ha whn pics a high (low) laiv o dividnds, h sock mak subsqunly pfoms pooly (wll). Whn good cash-flow nws is lasd, h sock pic in ou modl jumps up mo han i would in an conomy mad up of aional invsos alon: as dscibd abov, h pic jump causd by h good cash-flow nws fds ino xapolaos xpcaions, which, in un, gnas an addiional pic incas. A his poin, h sock mak is ovvalud and pics a high laiv o dividnds. Sinc, subsqun o h ovvaluaion, h sock mak pfoms pooly on 4

5 avag, h lvl of pics laiv o dividnds pdics subsqun pic changs in ou modl, jus as i dos in acual daa. Th sam mchanism also gnas xcss volailiy -- sock mak pics a mo volail han can b xplaind by aional focass of fuu cash flows as wll as ngaiv auocolaions in pic changs a all hoizons, capuing h ngaiv auocolaions w s a long hoizons in acual daa. Th modl also machs som mpiical facs ha, hus fa, hav bn akn as vidnc fo oh modls. Fo xampl, in acual daa, suplus consumpion, a masu of how cun consumpion compas o pas consumpion, is colad wih h valu of h sock mak and pdics h mak s subsqun un. Ths facs hav bn akn as suppo fo habi-basd modls. Howv, hy also mg naually in ou famwok. Ou numical analysis allows us o quanify h ffcs dscibd abov. Spcifically, w us h suvy daa sudid by Gnwood and Shlif (4) and ohs o paamiz h funcional fom of xapolaion in ou modl. Fo on asonabl paamizaion, w find ha if 5% of invsos a xapolaos whil 5% a aional ads, h sandad dviaion of annual pic changs is 3% high han in an conomy consising of aional ads alon. Th a aspcs of h daa ha ou modl dos no addss. Fo xampl, vn hough som of h invsos in h conomy a pic xapolaos, h modl dos no pdic h posiiv auocolaion in pic changs obsvd in h daa a vy sho hoizons. Also, h is no mchanism in ou modl, oh han high isk avsion, ha can gna a lag quiy pmium. And whil h psnc of xapolaos ducs h colaion of consumpion changs and pic changs, his colaion is sill much high in ou modl han in acual daa. In summay, ou analysis suggss ha, simply by inoducing som invsos wih xapolaiv xpcaions ino an ohwis adiional consumpion-basd modl of ass pics, w can mak sns no only of som impoan facs abou pics and uns, bu also, by consucion, of h availabl vidnc on h xpcaions of alwold invsos. This suggss ha h suvy vidnc nd no b sn as a nuisanc, o as an impdimn o undsanding h facs abou pics and uns. On h conay, h 5

6 xapolaion obsvd in h suvy daa is consisn wih h facs abou pics and uns, and may b h ky o undsanding hm. In Scion, w psn ou modl and is soluion, and discuss som of h basic insighs ha mg fom i. In Scion 3, w assign valus o h modl paams. In Scion 4, w show analyically ha h modl poducs sval ky faus of sock pics. Ou focus h is on quaniis dfind in ms of diffncs pic changs, fo xampl; givn h sucu of h modl, hs a h naual objcs of sudy. In Scion 5, w us simulaions o documn h modl s pdicions fo aio-basd quaniis, such as h pic-dividnd aio, which a mo commonly sudid by mpiiciss. Scion 6 concluds. All poofs and som discussion of chnical issus a in h Appndix.. Th Modl In his scion, w popos a hognous-agn, consumpion-basd modl in which som invsos xapola pas pic changs whn making focass abou fuu pic changs. Consucing such a modl psns significan challngs, boh bcaus of h hogniy acoss agns, bu also bcaus i is h chang in pic, an ndognous quaniy, ha is bing xapolad. By conas, consucing a modl basd on xapolaion of xognous fundamnals is somwha simpl. To pvn ou modl fom bcoming oo complx, w mak som simplifying assumpions abou h dividnd pocss (a andom walk in lvls), abou invso pfncs (xponnial uiliy), and abou h isk-f a (an xognous consan). W xpc h inuiions of h modl o cay ov o mo complx fomulaions. 3 W consid an conomy wih wo asss: a isk-f ass in pfcly lasic supply wih a consan ins a ; and a isky ass, which w hink of as h aggga sock mak, and which has a fixd p-capia supply of Q. Th isky ass is a claim o a coninuous dividnd sam whos lvl p uni im volvs as an aihmic Bownian moion dd g d d, () D D 3 Sval oh modls of h aggga sock mak mak simila assumpions; s, fo xampl, Campbll and Kyl (993) and Wang (993). W discuss h consan ins a assumpion a h nd of Scion. 6

7 wh g and a h xpcd valu and sandad dviaion of dividnd changs, D D spcivly, and wh is a sandad on-dimnsional Win pocss. Boh g and a consan in ou modl. Th valu of h sock mak a im is dnod by P and is dmind ndognously in quilibium. Th a wo yps of infinily-livd ads in h conomy: xapolaos and aional ads. Boh yps maximiz xpcd lifim consumpion uiliy. Th only diffnc bwn hm is ha on yp has coc blifs abou h xpcd pic chang of h isky ass, whil h oh yp dos no. Th modling of xapolaos is moivad by h suvy vidnc analyzd by Vissing-Jognsn (4), Baccha, Mns, and Wincoop (9), Amomin and Shap (3), and Gnwood and Shlif (4). Ths invsos fom blifs abou h fuu pic chang of h sock mak by xapolaing h mak s pas pic changs. To fomaliz his, w inoduc a masu of snimn, dfind as: S () s dps d,, () wh s is h unning vaiabl fo h ingal and wh dp sd P s P sd. S is simply a wighd avag of pas pic changs on h sock mak wh h wighs dcas xponnially h fuh back w go ino h pas. Th dfiniion of S includs vn h mos cn pic chang, dp d = P P d. Th paam plays an impoan ol in ou modl. Whn i is high, snimn is dmind pimaily by h mos cn pic changs; whn i is low, vn pic changs in h disan pas hav a significan ffc on cun snimn. In Scion 3, w us suvy daa o sima. W assum ha xapolaos xpcaion of h chang, p uni im, in h valu of h sock mak, is g [ dp ] / d S, (3) P, wh h supscip is an abbviaion fo xapolao, and wh, fo now, h only quimn w impos on h consan paams and is ha. Takn ogh, quaions () and (3) capu h ssnc of h suvy suls in Gnwood and Shlif (4): if h sock mak has bn ising, xapolaos xpc i o kp ising; and if i has bn falling, hy xpc i o kp falling. Whil w lav and D D 7

8 unspcifid fo now, w s and in ou numical analysis; w xplain in Scion 3 why hs a naual valus o us. W do no ak a song sand on h undlying souc of h xapolaiv xpcaions in (3). On possibl souc is a judgmn huisic such as psnaivnss, o h closly-lad blif in h law of small numbs (Babis, Shlif, and Vishny 998; Rabin ). Fo xampl, popl who bliv in h law of small numbs hink ha vn sho sampls will smbl h pan populaion fom which hy a dawn. As a consqunc, whn hy s good cn uns in h sock mak, hy inf ha h mak mus cunly hav a high avag un and will hfo coninu o pfom wll. 4 To compu hi opimal consumpion-pofolio dcision a ach momn of im, xapolaos nd o fom blifs no only abou h xpcd insananous pic chang, bu also abou h voluion of fuu pics. W assum ha, in xapolaos minds, pics volv accoding o dp S d (4) () d P, wh, again fom h xapolaos pspciv, is a on-dimnsional Win pocss. Th dif m simply flcs h xpcaions in (3), whil h insananous volailiy is h acual insananous volailiy ha is ndognously dmind in quilibium and ha w assum, and la vify, is a consan. Sinc volailiy can asily b simad fom pas daa, w assum ha xapolaos know is acual valu. Th scond yp of invso, h aional ad, has coc blifs, boh abou h dividnd pocss in () and abou h voluion of fuu sock pics. Th is a coninuum of boh aional ads and xapolaos in h conomy. Each invso, whh a aional ad o an xapolao, aks h isky ass pic as givn whn making his ading dcision, and has CARA pfncs wih absolu isk avsion and im discoun faco. 5 A im, ach xapolao maximizs P 4 Anoh possibl souc of xapolaiv xpcaions is h xpinc ffc analyzd by Malmndi and Nagl (). On cava is ha, as w show la, h invso xpcaions documnd in suvys dpnd pimaily on cn pas uns, whil in Malmndi and Nagl s () suls, disan pas uns also play a significan ol. 5 Th modl mains analyically acabl vn if h wo yps of invsos hav diffn valus of o. 8

9 subjc o his budg consain C d (5) dw W W ()() W C d N P d N D d N P W d d W d C d N Pd N dp N D d, wh W, C, and N a his im walh, consumpion, and p-capia numb of shas h invss in h isky ass, spcivly. Similaly, a im, ach aional ad maximizs subjc o his budg consain (6) C d (7) dw W W ()() W C d N P d N D d N P W d d W d C d N Pd N dp N D d, wh W, C, and N a his im walh, consumpion, and p-capia numb of shas h invss in h isky ass, spcivly, and wh h supscip is an abbviaion fo aional ad. Sinc aional ads cocly conjcu h pic pocss P, hi xpcaion is consisn wih ha of an ousid conomician. W assum ha aional ads mak up a facion, and xapolaos, of h oal invso populaion. Th mak claing condiion ha mus hold a ach im is N (8) ( ) N, (9) Q wh, as nod abov, Q is h p-capia supply of h isky ass. W assum ha boh xapolaos and aional ads obsv D and P on a coninuous basis. Moov, hy know h valus of μ and Q, and ads of on yp undsand how ads of h oh yp fom blifs abou h fuu. 6 Using h sochasic dynamic pogamming appoach dvlopd in Mon (97), w obain h following poposiion. 6 As in any famwok wih lss han fully aional ads, h xapolaos could, in pincipl, com o lan ha hi blifs abou h fuu a inaccua. W do no sudy his laning pocss; ah, w sudy h bhavio of ass pics whn xapolaos a unawa of h bias in hi blifs. 9

10 Poposiion (Modl soluion). In h hognous-agn modl dscibd abov, h quilibium pic of h isky ass is D P A BS. () Th pic of h isky ass P and h snimn vaiabl S volv accoding o B gd dp S d Pd, () () B B gd ds S d Pd, B () D wh P. A im, h valu funcions fo h xapolaos and h aional () B ads a J ds W J scs ( W, S,) max xp(), a S b S c { Cs, Ns } s scs ( W, S,) max xp(). ds W a S b S c { Cs, Ns } s (3) Th opimal p-capia sha dmands fo h isky ass fom h xapolaos and fom h aional ads a N S, N S, (4) and h opimal consumpion flows of h wo yps a C C a S b S c log( ), W W a S b S c log( ), (5) wh h opimal walh lvls, W and W, volv as in (6) and (8), spcivly. Th cofficins A, B, a, b, c, a, b, c,,,, and a dmind hough a sysm of simulanous quaions.

11 Compaing xapolaos blifs abou fuu pics in (4) wih h acual pic pocss in (), w s ha xapolaos blifs abou fuu pics a incoc. Whil xapolaos hink ha h xpcd insananous pic chang dpnds posiivly on h snimn lvl S, quaion () shows ha i acually dpnds ngaivly on S. Subsiuing xapolaos blifs abou fuu pics (4) ino h diffnial fom of (), namly, ds S d dp, (6) w obain xapolaos blifs abou h voluion of snimn, ds ( ( )) S d d. (7) P Compaing () o (7), w s ha xapolaos also hav incoc blifs abou h voluion of snimn. Fo xampl, whn and, hy hink ha snimn follows a andom walk, whil, in aliy, i is man-ving. Whil h xapolaos hav incoc blifs abou h voluion of pics and snimn, hy a nonhlss fully im-consisn. A im, an xapolao s consumpion-pofolio dcision is a funcion of wo sa vaiabls: his walh W and snimn S. Whn compuing his im dcision, h xapolao maks a plan of wha h will do in all fuu sas {(W, S + )}. If, a im, h aivs in sa (W, S + ), h is im-consisn: h aks h acion ha h had pviously plannd o ak in ha sa. His only o is ha, sinc his blifs abou h voluion of pics and snimn a incoc, h missimas h pobabiliy of moving fom sa (W, S ) a im o sa (W, S + ) a im. To undsand h ol ha xapolaos play in ou modl, w compa h modl s pdicions o hos of a bnchmak aional conomy, in oh wods, an conomy wh all ads a of h fully aional yp, so ha. 7 Coollay (Raional bnchmak). If all ads in h conomy a aional ( ), h quilibium pic of h isky ass is 7 Anoh way of ducing ou modl o a fully aional conomy is o s and, h paams in (3), o g D / and, spcivly. In his cas, boh h aional ads and h xapolaos hav h sam, coc blifs abou h xpcd pic chang of h isky ass p uni im.

12 P and hfo volvs accoding o Th valu funcion fo h aional ads is Th opimal consumpion flow is D gd D Q, (8) gd D dp d. d (9) D ( W,) xp Q J W. () C DQ W, () wh h opimal walh lvl, W, volvs as dw DQ Q D. d d ().. Discussion In Scions 4 and 5, w discuss h modl s implicaions in dail. Howv, h closd-fom soluion in Poposiion alady maks appan is basic popis. Compaing quaions () and (8), w s ha, up o a consan, h ffc of xapolaos on h isky ass pic is givn by h m BS in quaion (), wh, fo all of h basic paam valus w hav considd, h cofficin B is posiiv. Inuiivly, if h snimn lvl S is high, indicaing ha pas pic changs hav bn high, xapolaos xpc h sock mak o coninu o pfom wll and hfo push is cun pic high. Equaion () shows ha, in quilibium, snimn S follows a man-ving pocss, on ha vs mo apidly o is man as β incass. Pu diffnly, h mispicing BS gnad by xapolaos is vnually cocd, and mo quickly so fo high valus of β. To s why, call ha an ovpicing occus whn good cash-flow nws gnas a pic incas ha hn fds ino xapolaos blifs, lading hm o

13 push pics sill high. Th fom of xapolaion in quaion (), howv, mans ha, as im passs, h pic incas causd by h good cash-flow nws plays a small and small ol in dmining xapolaos blifs. As a sul, hs invsos bcom lss bullish ov im, and h mispicing cocs. This happns mo apidly whn β is high bcaus, in his cas, xapolaos quickly fog all bu h mos cn pic changs. Compaing quaions () and (9), w s ha, as nod in h Inoducion, h psnc of xapolaos amplifis h volailiy of pic changs spcifically, by a faco of /( B) >. And whil in an conomy mad up of aional invsos alon, pic changs a no pdicabl -- s quaion (9) -- quaion () shows ha hy a pdicabl in h psnc of xapolaos. If h sock mak has cnly xpincd good uns, so ha h snimn vaiabl S has a high valu, h subsqun sock mak un is low on avag: h cofficin on S in quaion () is ngaiv. In sho, high valuaions in h sock mak a followd by low uns, and low valuaions a followd by high uns. This anicipas som of ou suls on sock mak pdicabiliy in Scions 4 and 5. Equaion (4) shows ha xapolaos sha dmand is a posiiv lina funcion of h snimn lvl S : fo all valus of h basic modl paams w hav considd, h divd paam is posiiv. In oh wods, af a piod of good sock mak pfomanc, on ha gnas a high snimn lvl S, xapolaos fom mo bullish xpcaions of fuu pic changs and incas h numb of shas of h sock mak ha hy hold. Wih a fixd supply of hs shas, his auomaically mans ha h sha dmand of aional ads vais ngaivly wih h snimn vaiabl S : aional ads absob h shocks in xapolaos dmand. Whil xapolaos blifs a, by assumpion, xapolaiv, aional ads blifs a conaian: hi blifs a basd on h u pic pocss () whos dif dpnds ngaivly on S. Equaion () shows ha, in h fully aional conomy, opimal consumpion is a consan plus h poduc of walh and h ins a. Equaion (5) shows ha, whn xapolaos a psn in h conomy, h consumpion policy of ach yp of agn also dpnds on lina and quadaic ms in S. W find ha h divd paams a, b and b in quaion (5) ypically saisfy a, a, and b b. Th fac ha a,, 3

14 b b indicas ha xapolaos incas hi consumpion mo han aional ads do af song sock mak uns. Af song uns, xapolaos xpc h sock mak o coninu o is; an incom ffc hfo lads hm o consum mo. Raional ads, on h oh hand, cocly pciv low fuu uns and hfo do no ais hi consumpion as much. Th fac ha a and a a boh ngaiv indicas ha, as snimn incass in absolu magniud, boh yps incas hi consumpion. Whn S aks ih a vy high o a vy low valu, boh yps pciv h sock mak o b svly misvalud and hfo xpc hi spciv invsmn sagis o pfom wll in h fuu. This, in un, lads hm o ais hi consumpion. Sinc xapolaos hav incoc blifs abou fuu pic changs, i is likly ha, in h long un, hi walh will dclin laiv o ha of aional ads. Howv, h pic pocss in () is unaffcd by h laiv walh of h wo ad yps: und xponnial uiliy, h sha dmand of ach yp, and hnc also pics, a indpndn of walh. Th xponnial uiliy assumpion allows us o absac fom h ffc of suvival on pics, and o focus on wha happns whn boh yps of ad play a ol in sing pics. Th ida, inhn in ou modl, ha uninfomd invsos will affc pics vn in h long un, is no complly unalisic. In aliy, popl an labo incom which can susain a losing invsmn sagy fo many yas; and whil som uninfomd invsos may b focd o xi h financial maks bcaus of poo pfomanc, hy a likly o b placd, a las in pa, by a nw coho of naïv ads wih lil pio xpinc. In addiion, vn if h walh of an uninfomd invso dclins ov im, his pocss can ak a long im (Yan, 8). W hav usd numical simulaions o confim his in h conx of ou modl. W find ha, if, a im, h xapolaos and aional ads ach hold 5% of aggga walh, hn, af 5 yas, fo h bnchmak paam valus ha w lay ou in Scion 3, h xapolaos on avag sill hold 4% of aggga walh. 8 8 Nonhlss, i would b usful o consuc a modl of pic xapolaion in which h laiv walh of infomd and uninfomd ads affcs pics fo xampl, a modl wih pow uiliy o Epsin-Zin pfncs. Howv, such a modl would b fa lss acabl han h on w psn h; and w 4

15 A h ha of ou modl is an amplificaion mchanism: if good cash-flow nws pushs h sock mak up, his pic incas fds ino xapolaos xpcaions abou fuu pic changs, which hn lads hm o push h cun pic up vn high. Howv, his hn fuh incass xapolaos xpcaions abou fuu pic changs, lading hm o push h cun pic sill high, and so on. Givn his infini fdback loop, i is impoan o ask whh h hognous-agn quilibium w dscibd abov xiss. Th following coollay povids a condiion fo xisnc of quilibium. Coollay (Exisnc of quilibium). Th quilibium dscibd in Poposiion xiss if and only if B. Whn (all invsos a xapolaos), h quilibium dscibd in Poposiion xiss if and only if, (3) assuming ha. Coollay shows ha, whn all invsos in h conomy a xapolaos, h may b no quilibium vn fo asonabl paam valus; loosly pu, h fdback loop dscibd abov may fail o convg. Fo xampl, if = and =.5, h is no quilibium in h cas of if h ins a is lss han 5%. Howv, if vn a small facion of invsos a aional ads, h quilibium is vy likly o xis. Indd, fo.5, w hav found an quilibium fo all h paam valus w hav id. On of h assumpions of ou modl is ha h isk-f a is consan. To valua his assumpion, w compu h aggga dmand fo h isk-f ass acoss h wo yps of ad. W find ha his aggga dmand is vy sabl ov im and, in paicula, ha i is uncolad wih h snimn lvl S. This is bcaus h dmand fo h isk-f ass fom on yp of ad is lagly offs by h dmand fom h oh yp: whn snimn S is high, aional ads incas hi dmand fo conjcu ha, if i allows fo labo incom and fo cohos of nw uninfomd invsos o n h maks on a gula basis, is pdicions will b simila o hos of h cun modl. 5

16 h isk-f ass (and mov ou of h sock mak), whil xapolaos duc hi dmand fo h isk-f ass (and mov ino h sock mak). Whn snimn is low, h vs occus. This suggss ha, vn if h isk-f a w ndognously dmind, i would no flucua wildly, no would is flucuaions significanly anua h ffcs w dscib h. Ou modl is simila in som ways o ha of Campbll and Kyl (993) a modl in which, as in ou famwok, h isk-f a is consan, h lvl of h dividnd on h isky ass follows an aihmic Bownian moion, and infinily-livd aional invsos wih xponnial uiliy inac wih lss aional invsos. Th diffnc bwn h wo modls and i is an impoan diffnc is ha, in Campbll and Kyl (993), h sha dmand of h lss aional invsos is xognously assumd o follow a man-ving pocss, whil, in ou modl, xapolaos sha dmand is divd fom hi blifs. I is also usful o compa ou famwok o modls of aional laning. Wang (993) consids an conomy wih infomd invsos, uninfomd invsos, and nois ads wih xognous man-ving dmand fo a isky ass. Th dividnd sam on h isky ass has a im-vaying dif ha is known o h infomd invsos bu no o h uninfomd invsos, who insad y o sima h dif fom pas dividnds and pics. Th modl of Wang (993) capus a numb of facs abou ass pics, bu is lss consisn wih h suvy vidnc on xpcaions. Gnwood and Shlif (4) show ha, in a gssion of h avag invso xpcaion of fuu uns on pas uns and pas changs in fundamnals, h cofficin on pas uns is songly posiiv whil h cofficin on pas changs in fundamnals is insignifican. In h conomy dscibd by Wang (993), howv, h xpcaions of boh h infomd and uninfomd invsos abou fuu pic changs ypically dpnd ngaivly on pas pic changs, af conolling fo pas fundamnals: if pics go up wihou a conmpoanous incas in dividnds, boh invso yps inf ha nois ad dmand has gon up; sinc his dmand is man-ving, boh invso yps focas low, no high, pic changs in h fuu. 6

17 Th ida ha invsos xapola pas pic changs faus pominnly in classic qualiaiv accouns of ass bubbls (Kindlbg 978, Minsky 986, Shill ). Ou modl shows fomally how, vn in h psnc of fully aional ads, pic xapolaos can gna h mos fundamnal fau of a bubbl, namly a subsanial and long-livd ovvaluaion of an ass class. Ou focus on xapolaion as h souc of h ovvaluaion diffnias ou famwok fom xising modls of bubbls, such as h aional bubbl modl of Blanchad and Wason (98) and h hognous-blifs modls of Haison and Kps (978) and Schinkman and Xiong (3); in hs oh modls, invsos do no hold xapolaiv xpcaions. Whil ou modl shows how xapolaion can lad o ovvaluaion, h goal of ou analysis is no o undsand bubbls, bu ah, o undsand h bhavio of h aggga sock mak, and, in paicula, h join bhavio of consumpion, dividnds, and pics. Sinc w buil ou modl wih his goal in mind, i is no supising ha h a sval faus of bubbls ha i dos no capu fo xampl, h psisn momnum in pics whil a bubbl is foming, h high ading volum a h bubbl s pak, and h iding of h bubbl by aional invsos (Bunnmi and Nagl 4). An xapolaion-basd modl of bubbls ha capus his ich s of facs has y o b dvlopd. 3. Paam Valus In his scion, w assign bnchmak valus o h basic modl paams. W us hs valus in h numical simulaions of Scion 5. Howv, w also us hm in Scion 4. Whil h co of ha scion consiss of analyical poposiions, w can g mo ou of h poposiions by valuaing h xpssions hy conain fo spcific paam valus. Fo asy fnc, w lis h modl paams in Tabl. Th ass-lvl paams a h isk-f a ; h iniial lvl of h dividnd D ; h man gd and D sandad dviaion of dividnd changs; and h isky ass supply Q. Th invso- lvl paams a h iniial walh lvls fo h wo yps of agns, W and W ; absolu isk avsion and h im discoun a ; and, which link h snimn 7

18 vaiabl o xapolaos blifs;, which govns h laiv wighing of cn and disan pas pic changs in h dfiniion of snimn; and finally, μ, h popoion of aional ads in h invso populaion. 9 W s =.5%, consisn wih h low hisoical isk-f a. W s h iniial dividnd lvl D o, and givn his, w choos D.5; in oh wods, w choos a volailiy of dividnd changs small nough o nsu ha w only aly ncoun ngaiv dividnds and pics in h simulaions w conduc in Scion 5. W s g.5 o mach, appoximaly, h valu of g / in acual daa. Finally, w s h D isky ass supply o Q 5. W now un o h invso-lvl paams. W s h iniial walh lvls o W W 5; hs valus imply ha, a im, h valu of h sock mak consius appoximaly half of aggga walh. W s isk avsion qual o. so WJWW ha laiv isk avsion, compud fom h valu funcion as RRA W, is J.5 a h iniial walh lvls. And w choos a low im discoun a of =.5%, consisn wih mos oh ass picing famwoks. This lavs fou paams:,,, and μ. As shown in quaion (3), and dmin h link bwn h snimn vaiabl S and xapolaos blifs. W us and as ou bnchmak valus. Th ingal sum of h wighs on pas pic changs in h dfiniion of snimn in () is qual o on; infomally, S psns on uni of pic chang. I is hfo naual fo xapolaos o scal S by whn focasing a uni pic chang in h fuu. Givn his valu of, w s bcaus his nsus ha xapolaos blifs a coc on avag : whil xapolaos ovsima h subsqun pic chang of h sock mak af good pas pic changs and undsima i af poo pas pic changs, h os in hi focass of fuu pic changs ov any fini hoizon will, in h long un, avag ou o zo. D D W 9 Fo much of h analysis, w do no nd o assign spcific valus o D,, W and W ; h valus of hs vaiabls a quid only fo h simulaions in Scion 5. W hav also usd h suvy vidnc o sima and and find h simad valus o b clos o zo and on; s h Appndix fo mo dails. Th suls w psn in Scions 4 and 5 a simila whh w us h simad valus o h valus zo and on. 8

19 Th paam dmins h laiv wigh xapolaos pu on cn as opposd o disan pas pic changs whn foming xpcaions abou h fuu; a high valu of mans a high laiv wigh on cn pic changs. To sima, w us h im sis of invso xpcaions fom h Gallup suvys sudid by Gnwood and Shlif (4). W dscib h simaion pocdu in dail in h Appndix. In bif, w un a gssion of h avag invso xpcaion of h pic chang in h sock mak ov h nx ya, as codd in h suvys, on wha ou modl says xapolaos xpcaion of his quaniy should b a ha im as a funcion of h snimn lvl and h modl paams. If h avag invso xpcaion of h fuu pic chang ha w obsv in h suvys dpnds pimaily on cn pas pic changs, h simad will b high. Convsly, if i dpnds o a significan xn on pic changs in h disan pas, h simad will b low. Th simaion maks us of Poposiion blow, and spcifically, quaion (5), which dscibs h pic chang xpcd by xapolaos ov any fuu hoizon. Poposiion (Pic chang xpcaions of aional ads and xapolaos). Condiional on an iniial snimn lvl S s, aional ads xpcaion of h pic chang in h sock mak ov h fini im hoizon (, ) is () k g g P D, D P S s B s (4) whil xapolaos xpcaion of h sam quaniy is wh m P P S s () s () ms, m m k B and m ( ). Whn and, (5) ducs o (5) P. P S s s (6) Equaions (4) and (5) confim ha h xpcaions of xapolaos load posiivly on h snimn lvl, whil h xpcaions of aional ads load ngaivly. 9

20 Whn w us h pocdu dscibd in h Appndix o sima fom h suvy daa, w obain a valu of appoximaly.5. Fo his valu of, xapolaos xpcaions dpnd pimaily on cn pas pic changs; spcifically, whn foming hi xpcaions, xapolaos wigh h alizd annual pic chang in h sock mak saing fou yas ago only % as much as h mos cn annual pic chang. Whil w pay mos anion o h cas of.5, w also psn suls fo.5 and.75. Whn.5, h annual pic chang fou yas ago is wighd 86% as much as h mos cn annual pic chang, and whn.75, only % as much. Th final paam is μ, h facion of aional ads in h invso populaion. W do no ak a song sand on is valu. Whil h avag invso xpcaion in h suvy daa is obusly xapolaiv, i is had o know how psnaiv h suvyd invsos a of h full invso populaion. In ou analysis, w hfo consid a ang of valus of μ: (an conomy wh all invsos a fully aional),.75,.5, and.5. W do no consid h cas of bcaus Coollay indicas ha, whn all invsos a xapolaos, h quilibium dos no xis fo asonabl valus of and. Whil w consid fou diffn valus of μ, w focus on h low wo valus, namly.5 and.5. Th fac ha h avag invso in h suvys sudid by Gnwood and Shlif (4) suvys ha includ boh sophisicad and lss sophisicad spondns xhibis xapolaiv xpcaions suggss ha many invsos in acual financial maks a xapolaos. Fo a givn s of valus of h basic paams in Tabl, w solv a sysm of simulanous quaions, as oulind in h Appndix, o compu h divd paams:,,, and, which dmin xapolaos and aional ads opimal sha dmand (s quaion (4)); a, b, c, a, b and c, which dmin invsos opimal consumpion policis (s quaion (5)); A and B, which spcify how h pic lvl P dpnds on h lvl of h snimn S and h lvl of h dividnd D Whn w sima, w assum ha h suvyd invsos cospond o h xapolaos in ou modl: af all, h psnc of xapolaos in ou conomy is moivad pcisly by h suvy vidnc. If w insad assumd ha h suvyd invsos cospond o all invsos in ou modl, w would likly obain a simila valu of. Sinc aional ads blifs a h mio imag of xapolaos blifs, aional ads and xapolaos wigh pas pic changs in a simila way whn foming hi xpcaions.

21 (s quaion ()); and finally, P, h volailiy of pic changs in h sock mak. Fo xampl, if.5,.5, and h oh basic paams hav h valus shown in Tabl, h valus of h divd paams a:.54, 5.39,.5,.54, a a b b ,.8, 7.3,.4, c.63, c 3.47, A 7.4, B.99, Bfo w un o h mpiical implicaions of h modl, w mak on mo obsvaion abou invso xpcaions. Whn w say ha ou modl can mach h suvy vidnc, o ha i is consisn wih i, w man ha, in ou modl, a significan facion of h invso populaion is compisd of ads spcifically, xapolaos whos xpcaion of fuu pic changs dpnds posiivly on pas pic changs. Howv, w could also dfin maching h suvy vidnc in a sic sns, namly o man ha h avag xpcaion of fuu pic changs acoss all invsos in h modl is xapolaiv. Insingly, w find ha ou modl can mach h suvy daa vn in his sic sns: h xpcaion of h fuu chang in pic avagd acoss all invsos in h modl -- spcifically, h populaion-wighd avag of h xpssions in quaions (4) and (5) dpnds posiivly on h snimn lvl fo any. In oh wods, if h a any xapolaos a all in h conomy, h avag invso xpcaion is xapolaiv. Th ason is ha, whil xapolaos hold xapolaiv blifs and aional ads hold conaian blifs, aional ads a always lss conaian han xapolaos a xapolaiv. Af all, h aional ads a conaian only bcaus h xapolaos a xapolaiv; hy canno, hfo, b mo conaian han h xapolaos a xapolaiv. Whil ou modl can mach h suvy vidnc vn in his sic sns, w do no focus on his inpaion. Sinc w do no know how psnaiv h suvyd invsos a of h full invso populaion, i is uncla whh h avag xpcaion of fuu pic changs acoss all al-wold invsos is xapolaiv. P (7) 4. Empiical Implicaions

22 In his scion, w psn a daild analysis of h mpiical pdicions of h modl. A consqunc of ou assumpions ha h dividnd lvl follows an aihmic Bownian moion and ha invsos hav xponnial uiliy is ha i is mo naual o wok wih quaniis dfind in ms of diffncs ah han aios fo xampl, o wok wih pic changs P P ah han uns, and wih h pic-dividnd diffnc P D/ ah han h pic-dividnd aio. Fo xampl, Coollay shows ha, in h bnchmak aional conomy, i is P D/ ha is consan ov im, no P/D. In his scion, hn, w sudy h pdicions of pic xapolaion fo hs diffncbasd quaniis. In Scion 5, w also consid h aio-basd quaniis. W sudy h implicaions of h modl fo h diffnc-basd quaniis wih h hlp of fomal poposiions. Fo xampl, if w a insd in h auocolaion of pic changs, w fis compu his auocolaion analyically, and hn po is valu fo h paam valus in Tabl. Fo wo cucial paams, μ and β, w consid a ang of possibl valus. Rcall ha μ is h facion of aional ads in h invso populaion, whil β conols h laiv wighing of na-pas and disan-pas pic changs in xapolaos focas of fuu pic changs. W a insd in how h psnc of xapolaos in h conomy affcs h bhavio of h sock mak. To undsand his mo claly, in h suls ha w psn blow, w always includ, as a bnchmak, h cas of, in oh wods, h cas wh h invso populaion consiss nily of aional ads. 4.. Pdiciv pow of D/ P fo fuu pic changs A basic fac abou h sock mak is ha h dividnd-pic aio of h sock mak pdics subsqun uns wih a posiiv sign; moov, h aio s pdiciv pow is ga a long hoizons. In ou modl, h naual analogs of h dividndpic aio and of uns a h dividnd-pic diffnc D/ P and pic changs, spcivly. W hfo xamin whh, in ou conomy, h dividnd-pic diffnc pdics subsqun pic changs wih a posiiv sign, and whh his pdiciv pow is ga a long hoizons. I is hlpful o xpss h long-hoizon vidnc in h mo sucud way suggsd by Cochan (), among ohs. If w un h univaia gssions a

23 gssion of fuu uns on h cun dividnd-pic aio; a gssion of fuu dividnd gowh on h cun dividnd-pic aio; and a gssion of h fuu dividnd-pic aio on h cun dividnd-pic aio hn, as a ma of accouning, h h gssion cofficins mus (appoximaly) sum o on. Empiically, h h gssion cofficins a oughly,, and, spcivly, a long hoizons. In oh wods, a long hoizons, h dividnd-pic aio focass fuu uns no fuu dividnd gowh, and no is own fuu valu. W can sa his poin in a way ha fis mo naually wih ou modl, using quaniis dfind as diffncs, ah han aios. Givn h accouning idniy D D D D P P P P, i is immdia ha if w un h gssions of h fuu pic chang, h (ngaiv and scald) fuu dividnd chang, and h fuu dividnd-pic diffnc, on h cun dividnd-pic diffnc h valus of h h cofficins will sum o on in ou conomy, a any hoizon. To mach h mpiical facs, ou modl nds o pdic a cofficin in h fis gssion ha, a long hoizons, is appoximaly qual o on. Th nx poposiion shows ha his is indd h cas. (8) Poposiion 3 (Pdiciv pow of D/ P). Consid a gssion of h pic chang in h sock mak ov som im hoizon (, ) on h lvl of D/ P a h sa of h hoizon. In populaion, h cofficin on h indpndn vaiabl in h gssion is 3 wh k. B cov( D / P,) P P k DP ( ), va( D /) P (9) If h cofficin in h fis gssion is appoximaly on, his immdialy implis ha h cofficins in h scond and hid gssions a appoximaly zo, consisn wih h vidnc. Th cofficin in h scond gssion is xacly zo bcaus dividnd changs a unpdicabl in ou conomy. Th cofficin in h hid gssion is hn on minus h cofficin in h fis gssion; if h la is appoximaly on, h fom is appoximaly zo. 3 Th xpcaions ha w compu in h poposiions in Scion 4 a akn ov h sady-sa disibuion of h snimn lvl S. Egodiciy of h sochasic pocss S guaans ha sampl saisics will convg o ou analyical suls fo vy lag sampls. 3

24 Poposiion 3 shows ha, in ou modl, and consisn wih h mpiical facs, h cofficin in a gssion of h pic chang in h sock mak on h dividndpic diffnc is posiiv and incass a long hoizons, ising in valu asympoically owad on. Ths pans a claly visibl in Tabl 3, which pos h valu of h gssion cofficin in Poposiion 3 fo vaious valus of μ and β, and fo fiv diffn im hoizons: a qua, a ya, wo yas, h yas, and fou yas. In h bnchmak aional conomy ( ), h quaniy D/ P is consan; h gssion cofficin in Poposiion 3 is hfo undfind. Th inuiion fo why D/ P pdics subsqun pic changs is saighfowad. A squnc of good cash-flow nws pushs up sock pics, which hn aiss xapolaos xpcaions abou h fuu pic chang of h sock mak and causs hm o push h cun sock pic vn high, lowing h valu of D/ P. Sinc h sock mak is now ovvalud, h subsqun pic chang is low, on avag. Th quaniy D/ P hfo focass pic changs wih a posiiv sign. Th abl shows ha, fo a fixd hoizon, h pdiciv pow of D/ P is song fo low : sinc h pdicabiliy of pic changs sms fom h psnc of xapolaos, i is naual ha his pdicabiliy is song whn h a mo xapolaos in h conomy. Th pdiciv pow of D/ P is wak fo low : whn is low, xapolaos blifs a mo psisn; as a sul, i aks long fo an ovvaluaion o coc, ducing h pdiciv pow of D/ P fo pic changs ov any fixd hoizon. 4.. Auocolaion of P D/ In h daa, pic-dividnd aios a highly auocolad a sho lags. W would lik o know if ou modl can capu his. Th naual analog of h pic-dividnd aio in ou modl is h diffnc-basd quaniy P D/. W hfo xamin h auocolaion sucu of his quaniy. In ou discussion of h accouning idniy in quaion (8), w nod ha, if w un gssions of h fuu chang in h sock pic, h (ngaiv, scald) fuu 4

25 chang in dividnds, and h fuu dividnd-pic diffnc on h cun dividndpic diffnc, h h gssion cofficins w obain mus sum o on. Sinc h dividnd lvl follows a andom walk in ou modl, w know ha h cofficin in h scond gssion is zo. W also know, fom Poposiion 3, ha h cofficin in h fis gssion is k. Th cofficin in h hid gssion, which is also h k auocolaion of h pic-dividnd diffnc P D/, mus hfo qual. Th nx poposiion confims his. Poposiion 4 (Auocolaion of P D/). In populaion, h auocolaion of P D/ a a im lag of is D D k PD ( ) co P, P, (3) wh k. B W us Poposiion 4 o compu h auocolaion of h pic-dividnd diffnc fo sval pais of valus of and, and fo lags of on qua, on ya, wo yas, h yas, and fou yas. Tabl 4 pos h suls. Th abl shows ha, in ou modl, and consisn wih h mpiical facs, h pic-dividnd diffnc is highly psisn a sho hoizons, whil a long hoizons, h auocolaion dops o zo. Th abl shows ha h auocolaion is high fo low valus of : whn is low, xapolaos blifs a vy psisn, which, in un, impas psisnc o h picdividnd diffnc Volailiy of pic changs and of P D/ Empiically obsvd sock mak uns and pic-dividnd aios a hough o xhibi xcss volailiy, in oh wods, o b mo volail han can b xplaind puly by flucuaions in aional xpcaions abou fuu cash flows. W now show ha, in ou modl, pic changs and h pic-dividnd diffnc h analogs of uns and of h pic-dividnd aio in ou famwok also xhibi such xcss volailiy. In paicula, hy a mo volail han in h bnchmak aional conomy 5

26 dscibd in Coollay, an conomy wh pic changs a du only o changs in aional focass of fuu cash flows. Poposiion 5 (Excss volailiy). Th sandad dviaion of pic changs ov a fini im hoizon (, ) is BS D k D P (()) va( P P ) BS, k (3) whil h sandad dviaion of P D/ is wh D BS PD va P, k (3) k B D S. ( B) Tabl 5 pos h sandad dviaion of annual pic changs and of h annual pic-dividnd diffnc P D/ fo sval (, ) pais. Panl A shows ha, in h fully aional conomy ( ), h sandad dviaion of annual pic changs is, in oh wods, D /. Whn xapolaos a psn, howv, h sandad dviaion can b much high: fo xampl, 3% high whn h a an qual numb of xapolaos and aional ads in h conomy, a figu ha, as can b sn in h abl and as w xplain blow, dpnds lil on h paam. Similaly, Panl B shows ha whil h pic-dividnd diffnc is consan in h fully aional conomy, i vais significanly in h psnc of xapolaos. Th suls in Poposiion 5 and in Tabl 5 confim h inuiion w gav in h Inoducion fo why xapolaos amplify h volailiy of sock pics. A good cashflow shock pushs sock pics up. This lads xapolaos o xpc high fuu pic changs, causing hm o bid cun sock pics up vn fuh. Raional invsos counac his ovvaluaion, bu only mildly so: sinc hy undsand how xapolaos fom blifs, hy know ha xapolaos will coninu o hav opimisic blifs abou h sock mak in h na fuu. This mans ha, whil subsqun pic changs will b low han avag, hy will no b vy low. As a consqunc, aional invsos do no push back songly agains h ovvaluaion causd by h xapolaos. 6

27 Tabl 5 shows ha, h lag h facion of xapolaos in h conomy, h mo xcss volailiy h is in pic changs. Mo insing, h amoun of xcss volailiy is lagly insnsiiv o h paam. This may sm supising a fis: sinc xapolaos blifs a mo vaiabl whn is high, on migh hav hough ha a high would cospond o high pic volailiy. Howv, anoh foc pushs in h opposi dicion: aional ads know ha, pcisly bcaus xapolaos chang hi blifs mo quickly whn is high, any mispicing causd by h xapolaos will coc mo quickly in his cas. As a consqunc, whn is high, aional ads ad mo aggssivly agains h xapolaos, dampning volailiy. Ovall, h lvl of has lil ffc on volailiy. Dos h high pic volailiy gnad by xapolaos lav h aional ads wos off? I dos no. Spcifically, w find ha, if w sa wih an conomy consising of only aional ads and hn gadually add mo xapolaos whil kping h p-capia supply of h isky ass consan, h valu funcion of h aional ads incass in valu. In oh wods, whil h high pic volailiy svs o low aional ads uiliy, his is mo han compnsad fo by h high pofis h aional ads xpc o mak by xploiing h xapolaos Auocolaion of pic changs Empiically, sock mak uns a posiivly auocolad a sho lags; a long lags, hy a ngaivly auocolad (Cul, Poba, and Summs 99). W now xamin wha ou modl pdics abou h auocolaion sucu of h analogous quaniy o uns in ou famwok, namly pic changs. Poposiion 6 (Auocolaion of pic changs). In populaion, h auocolaion of pic changs ov h invals (, ) and (, 3 ), wh, is 4 DLong al. (99a) obain a simila sul: hy find ha h inoducion of nois ads ino h conomy aiss h uiliy of aional ads. Th ason is ha h psnc of nois ads xpands h invsmn oppouniy s. In ou modl, h xapolaos do no xpand h oppouniy s; hy chang i, by aling h bhavio of h isky ass. I is hfo lss obvious, in ou conx, ha h psnc of xapolaos will ais aional ads uiliy, bu ou calculaions indica ha i dos. DLong al. (989) poin ou ha, whn h supply of capial is ndognous, nois ads may low aional ads uiliy by dpssing h capial sock. Sinc h supply of shas is fixd in ou modl, w canno valua h impoanc of his channl. 7

28 cov( P P,) P P 3 (,,) co() P P, P P, (33) va( P P ) va() P P P wh wih B cov()()( P P P P B k k and B S D k3 k k, 3 S BS D k D va()() P P BS, k B k S D k ( 3 ) D va() P P ()( ) 3 B S 3, S D. ( B) ), (34) In Tabl 6, w us Poposiion 6 o compu h auocolaion of pic changs fo sval pais of valus of and, and a lags of on, wo, h, fou, igh, and wlv quas. Th abl shows ha pic changs a ngaivly auocolad a all lags, wih h auocolaion nding o zo a long lags. I is asy o s why, in ou modl, pic changs a ngaivly auocolad a long lags. Suppos ha h is good cash-flow nws a im. Th sock mak gos up in spons o his nws, which causs xapolaos o xpc high fuu pic changs; hy hfo push h im sock pic vn fuh up. Now ha h sock mak is ovvalud, h long-m fuu pic chang is low, on avag. I is hfo inuiiv ha pas pic changs would hav ngaiv pdiciv pow fo pic changs ha a som way ino h fuu. Ngaiv auocolaions a indd obsvd in h daa, a long lags; o som xn, hn, ou modl machs h daa. Howv, h is also a way in which ou modl dos no mach h daa: acual uns a posiivly auocolad a h fis qualy lag, whil h pic changs gnad by ou modl a no. I may iniially b supising ha ou modl gnas ngaiv auocolaions in pic changs vn a h shos lags. Th ason fo his pdicion is ha, as laid ou in quaions () and (3), h wighs xapolaos pu on pas pic changs whn hy fom xpcaions dclin h fuh back w go ino h pas. Consid again a good cashflow shock a im ha, as dscibd abov, fds ino xapolaos xpcaions and 8

29 amplifis h conmpoanous pic chang. Th wighing schm in quaion () mans ha, vn an insan la, h posiiv im pic chang ha causd xapolaos o bcom mo bullish plays a small ol in dmining hi xpcaions; xapolaos hfo bcom a lil lss bullish, and h is a pic vsal. Th abov discussion claifis why som ali modls of un xapolaion fo xampl, Cul, Poba, and Summs (99), DLong al. (99b), Hong and Sin (999), and Babis and Shlif (3) do gna posiiv sho-m auocolaion in uns. In hs modls, h wighs xapolaos pu on pas pic changs whn dciding on hi sha dmand ypically do no dclin monoonically, h fuh back w go ino h pas. In paicula, in hs modls, xapolaos sha dmand a im dpnds on h laggd pic chang fom im o im ; h laggd pic chang hfo mas mo han h mos cn pic chang fom o in dmining sha dmand. This assumpion lads o posiiv sho-m auocolaion: a pic incas a im fds ino xapolaos sha dmand only a im, gnaing anoh pic incas a ha im. This suggss ha an xnsion of ou modl in which xapolaos ac o pas pic changs wih som dlay whn foming hi xpcaions may gna boh ngaiv long-m and posiiv sho-m auocolaions in pic changs. W do no pusu his appoach h, howv: doing so would galy complica h analysis whil impoving h modl s xplanaoy pow in only a mino way Colaion of consumpion changs and pic changs Anoh quaniy of ins is h colaion of consumpion gowh and uns. In h daa, his colaion is low. W now look a wha ou modl pdics abou h analogous quaniy in ou conx: h colaion of consumpion changs and pic changs. Poposiion 7 (Colaion bwn consumpion changs and pic changs). In populaion, h colaion bwn h chang in consumpion and h chang in pic ov a fini im hoizon (, ) is 9

30 wh co( C C,) P P cov( C C,) P P, va() Cva( C P P ) (35) cov( C C,) P P ()() ag D b BS D BW k () a g b k W D W () k, S D k S () BS D k D va() P P BS ( ), (37) k and va() C a W C S D k W 4g b k k 4k k k k k 4 k k D S S W S ()()() 3 k k 4 aw bw S g D k 4() g DS S (3 ) W a k k k k b k ( 4 ab g g k k k k k S D k D S )) (() aw W k k k S S g D 4 3 bw W () bw a () k k k k 4 k aw g DS k k S ( )( W b ). k k k (36) (38) a b BQ No ha a a ( ) a, b b ( ) b, aw, bw BQ, B DQ W, k ( B) B and D S. ( B) Panls A and B of Tabl 7 po h colaion of consumpion changs and pic changs a a qualy and annual fquncy, spcivly, and fo sval (, ) pais. Th numbs a compud using Poposiion 7. Th panls show ha, whil h psnc of xapolaos slighly ducs h colaion of consumpion changs and pic changs laiv o is valu in h fully aional conomy, h colaion is 3

31 nonhlss high. As is h cas fo viually all consumpion-basd ass picing modls, hn, ou modl fails o mach h low colaion of consumpion gowh and uns in h daa Pdiciv pow of h suplus consumpion aio Pio mpiical sach has shown ha a vaiabl calld h suplus consumpion aio a masu of how cun consumpion compas o pas consumpion -- is conmpoanously colad wih h pic-dividnd aio on h ovall sock mak, and, fuhmo, pdics subsqun uns wih a ngaiv sign (Campbll and Cochan 999, Cochan ). Ths findings hav bn akn as suppo fo habi-basd modls of h aggga sock mak. W show, howv, ha hs pans also mg fom ou modl. As w hav don houghou his scion, w sudy diffnc-basd quaniis: h suplus consumpion diffnc ah han h suplus consumpion aio. Moov, w focus on h simpls possibl suplus consumpion diffnc, namly h cun lvl of aggga consumpion minus h lvl of aggga consumpion a som poin in h pas. Poposiion 8 compus h colaion bwn his vaiabl and h cun pic-dividnd diffnc P D/. Poposiion 8: (Colaion bwn h chang in consumpion and P D/). In populaion, h colaion bwn h chang in consumpion ov a fini im hoizon (, ) and P D/ masud a im is co( C C, P D /) cov( C C, P D /), va() C C va( P D / ) (39) wh ()() aw g D bw ag D b k B S cov( C C, P D /) (), S W k k k (4) D B va P k S, and va() C C is as in (38). Also, a a ( ) a, 3

32 a b BQ b b ( ) b, aw, bw BQ DQ, W, k and B ( B) B D S. ( B) Poposiion 9 xamins whh h suplus consumpion diffnc can pdic fuu pic changs. Poposiion 9 (Pdiciv pow of h chang in consumpion). Consid a gssion of h pic chang in h sock mak fom o on h chang in consumpion ov h fini im hoizon (, ). In populaion, h cofficin on h indpndn vaiabl is wh C (, cov C C, P P ) va C C cov( C C,) P P ()() aw gd bw agd b k BS S W ( )() k k, k () k a and va() C C is as in (38). Also, a a ( ) a, b b ( ) b, aw, (4) (4), b W b BQ BQ DQ, W, k, and B ( B) B S D. ( B) Panl C of Tabl 7 uss Poposiion 8 o compu, fo sval (, ) pais, h colaion bwn h suplus consumpion diffnc h cun lvl of aggga consumpion minus h lvl of aggga consumpion a qua ago and h cun pic-dividnd diffnc. Th panl shows ha h wo quaniis a significanly colad. Tabl 8 uss Poposiion 9 o compu h cofficin on h indpndn vaiabl in a gssion of h pic chang in h sock mak ov som inval on qua, on ya, wo yas, h yas, o fou yas on h suplus consumpion diffnc masud a h bginning of h inval. I shows ha h suplus consumpion diffnc pdics subsqun pic changs wih a ngaiv sign and in an conomically significan way, and ha his pdiciv pow is paiculaly song fo low 3