X-CAPM: An Exapolaiv Capial Ass Picing Modl Nicholas Babis*, Robin Gnwood**, Lawnc Jin*, and Andi Shlif** *Yal Univsiy and **Havad Univsiy Absac Suvy vidnc suggss ha many invsos fom blifs abou fuu sock mak uns by xapolaing pas uns: hy xpc h sock mak o pfom wll (pooly) in h na fuu if i pfomd wll (pooly) in h cn pas. 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, bu is also consisn wih h suvy vidnc on invso xpcaions. This suggss ha h suvy vidnc dos no nd o b sn as an inconvnin obsacl o undsanding h sock mak; on h conay, i is consisn wih h facs abou pics and uns, and may b h ky o undsanding hm. Jun 4, 3 W a gaful o smina paicipans a Damouh Collg, h London Businss School, h London School of Economics, Oxfod Univsiy, h Univsiy of Souhn Califonia, and Yal Univsiy fo usful fdback; and also o Jonahan Ingsoll fo many hlpful convsaions.
. 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 (3) 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 pvious 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, xpcd uns a low whn sock pics a high: in hs modls, sock pics a high whn invsos a lss isk avs o pciv lss 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 (3) analyz daa fom six diffn suvys; som 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), Amomin and Shap (8), and Baccha, Mns, and Wincoop (9).
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, h invsos hy dscib xpc high cash flows, bu no 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 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 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 3
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. This is bcaus hy 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, hy 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 simulaions, w hn xamin ou modl s pdicions abou pics and uns. W find ha hs pdicions a consisn wih sval of h 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 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 4
of consumpion laiv o pas consumpion, is colad wih h valu of h sock mak; moov, i pdics h mak s subsqun pfomanc. 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 (3) and ohs o paamiz h funcional fom of xapolaion in ou modl. Fo his paamizaion, w find, fo xampl, 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 momnum in pic changs: h psnc of fully aional ads mans ha pic changs a ngaivly auocolad a all lags. 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 xapolaiv invsos 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 al-wold invsos. This suggss ha w do no nd o hink of h suvy vidnc as a nuisanc, o as an impdimn o undsanding h facs abou pics and uns. On h conay, h xapolaion ha is psn in h suvy daa is pfcly 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 5
quaniis, such as h pic-dividnd aio, ha a commonly sudid by mpiiciss. Scion 6 concluds. All poofs, as wll as 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 o inp, 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. 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 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. W dno h valu of h sock mak a im by P. 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 un of h isky ass, whil h oh yp dos no. D D W discuss h consan ins a assumpion a h nd of Scion. 6
Th modling of xapolaos is moivad by h suvy vidnc analyzd by Vissing-Jognsn (4), Amomin and Shap (8), Baccha, Mns, and Wincoop (9), and Gnwood and Shlif (3). 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. 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 xpcd pic 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 (3): af good sock mak uns, xapolaos xpc h sock mak o coninu o pfom wll; and af poo sock mak uns, hy xpc coninud wak pfomanc. Whil w lav and unspcifid fo now, naual valus a and, and hs a indd h valus ha w us la. W do no ak a song sand on h undlying souc of h xapolaiv xpcaions in (3). Howv, on possibl souc is h psnaivnss huisic, o h closly-lad blif in h law of small numbs (Babis, Shlif, and Vishny 998; Rabin ). Fo xampl, und h law of small numbs, popl hink ha vn sho sampls will smbl h pan populaion fom which hy a dawn. As a sul, 7
whn hy s good cn uns in h sock mak, hy inf ha h sock mak mus cunly hav a high avag un and ha i will hfo coninu o do wll. 3 Th scond yp of invso, h aional ad, has coc blifs abou h voluion of fuu sock pics. By cocly conjcuing h quilibium pic pocss, h aional invsos ak full accoun of xapolaos ndognous sponss o pic movmns a all fuu ims. 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. 4 A im, ach xapolao maximizs C d (4) subjc o his budg consain dw W d W ( W Cd N P)( d) N Dd N P d W (5) W d C d N Pd N dp N D d, wh N is h p-capia numb of shas h invss in h isky ass a im. Similaly, a im, ach aional ad maximizs C d (6) subjc o his budg consain dw W d W ( W C d N P)( d) N Dd N P d W (7) W d C d N Pd N dp N D d, wh N is h p-capia numb of shas h invss in h isky ass a im, 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. 3 W hink of h ffc ha w a modling as disinc fom h xpinc ffc pod by Malmndi and Nagl (). Som vidnc ha hs a indd disinc ffcs is ha, as w show la, h invso xpcaions documnd in suvys dpnd only on cn pas uns, no h disan pas uns ha play a ol in Malmndi and Nagl s () suls. 4 Th modl mains analyically acabl vn if h wo yps of invsos hav diffn valus of o. 8
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 ( ) N, (8) Q wh 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. 5 Using h sochasic dynamic pogamming appoach dvlopd in Mon (97), w obain h following poposiion. Poposiion (Modl soluion). In h hognous-agn modl dscibd abov, h quilibium pic of h isky ass is D P ABS. (9) Th pic of h isky ass P and h snimn vaiabl S volv accoding o B g D dp D S d d, () B ( B) ( B) gd D ds S d d. () B ( B) A im, h valu funcions fo h xapolaos and h aional ads a s Cs J ( W, S, ) max xp, ds W a S b S c { Cs, Ns} s J max xp. () s Cs ( W, S, ) ds W a S b S c { Cs, Ns} s Th opimal p-capia sha dmands fo h isky ass fom h xapolaos and fom h aional ads a 5 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
Q N S, N N, (3) and h opimal consumpion flows of h wo yps a log( ) C W a S b S c, log( ) C W a S b S c, wh h opimal walh lvls, W and W, volv as in (5) and (7), spcivly. Th cofficins A, B, a, b, c, a, b,, c and a dmind hough a sysm of (4) simulanous quaions. 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. 6 Coollay (Raional bnchmak). If all ads in h conomy a aional ( ), h quilibium pic of h isky ass saisfis D gd D P Q. (5) I hfo volvs accoding o gd D dp d. d (6) Th valu funcion fo h aional ads is Th opimal consumpion flow is J ( W, ) W Q xp D. (7) C DQ W, (8) 6 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 p uni im pic chang of h isky ass.
wh h opimal walh lvl, W, volvs as dw Q Q d. (9) 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 h basic popis of ou famwok. Fom quaion (), w s ha h snimn lvl S follows a man-ving pocss wih long-un man g D /. Equaion (9) shows ha, whn snimn is high, sock mak pics a pushd up -- h cofficin B is posiiv fo all valus of h basic paams ha w hav considd. Inuiivly, if h snimn lvl is high, indicaing ha pas pic changs hav bn high, xapolaos xpc h sock mak o pfom wll in h fuu and hfo push is cun pic high. Whil xapolaos blifs a, by dfiniion, xapolaiv, aional ads blifs a conaian: hi blifs a basd on h u pic pocss () whos dif dpnds ngaivly on S. Compaing quaions () and (6), w also 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 (6) -- quaion () shows ha hy a pdicabl in h psnc of xapolaos. Spcifically, 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. In h analysis w conduc la, w find ha, fo asonabl valus of h basic modl paams, h divd paam in quaion (3) is posiiv. In oh wods, af a piod of good sock mak pfomanc, on ha gnas a high lvl of
snimn S, xapolaos 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. 7 W also find ha, fo asonabl valus of h basic modl paams, h divd paams a, a, b, and b in quaion (4) ypically saisfy a, a, and b b. Th fac ha 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, cognizing ha xapolaos bullinc has causd h sock mak o bcom ovvalud, cocly pciv low fuu uns; hy hfo do no ais hi consumpion as much. Equaions (9) and () indica ha h mispicing cad by xapolaos is vnually cocd, and mo quickly so fo high valus of β. To undsand his in oh wods, o undsand why, in ou famwok, bubbls vnually bus call ha an ovpicing occus whn good cash-flow nws gnas a pic incas ha hn fds ino xapolaos blifs, lading hm o 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 bubbl dflas. This happns mo apidly whn β is high bcaus, in his cas, xapolaos quickly fog all bu h mos cn pic changs. 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 7 Sinc h supply of h isky ass is fixd and h a only wo goups of ads, h sha dmand of aional ads mus vay ngaivly wih h snimn lvl. In a sippd-down vsion of ou famwok, w hav also analyzd wha happns whn h a h yps of ads: h wo yps w xamin h, bu also a goup of paially-aional invsos who buy (sll) h isky ass whn is pic is low (high) laiv o fundamnals. W find ha, in his conomy, h sha dmand of h fully aional ads is posiivly lad o h snimn lvl. In oh wods, consisn wih h findings of Bunnmi and Nagl (4), hs ads id h bubbl gnad by xapolaos.
of walh. In oh wods, h 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. 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 cun pics 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, () 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 3
fom h oh yp: whn snimn S is high, aional ads incas hi dmand fo 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. 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 asonabl 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 ; h popoion μ of aional ads in h conomy;, which govns h laiv wighing of cn and disan pas pic changs in h dfiniion of h snimn vaiabl; and, finally, and, which govn h laionship bwn h snimn vaiabl and xapolaos blifs. 8 W s =.5%, consisn wih h low hisoical isk-f a. W s h iniial dividnd lvl D o, and givn his, w choos.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 la. W s gd.5 o mach, appoximaly, h mpiical aio of g / in h daa. Finally, w s h isky ass supply o Q 5. D D D 8 Th vaiabls,, and a lisd a h boom of Tabl bcaus, fo much of h analysis, w do no nd o spcify hi valus; hi valus a ndd only fo h simulaions in Scion 5. 4
W now un o h invso-lvl paams. W s h iniial walh lvls o W W 5. Givn his, w s isk avsion qual o. so ha laiv isk WJWW avsion, compud fom h valu funcion as RRA W, is.5 a h iniial J walh lvls. W choos a low im discoun a of =.5%, consisn wih mos oh ass picing famwoks; and, as nod ali, w s and in quaion (3) o and. Ths valus imply ha, 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. This lavs jus wo paams: μ, h facion of aional invsos in h conomy; and, h wighing paam in quaion (). Bcaus hs wo paams play an impoan ol in ou famwok, w consid a ang of valus fo ach on in h numical analysis ha follows. Spcifically, w consid fou diffn 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. Fo, w consid h possibl valus:.5,.5, and.75. Rcall ha, fo high valus of, xapolaos wigh cn uns mo havily whn focasing fuu uns. Fo xampl, whn.5, h alizd annual pic chang on h sock mak saing fou yas ago is wighd 86% as much as h mos cn annual pic chang; whn.5, i is wighd % as much; and whn.75, i is wighd only % as much. Whil w consid fou diffn valus of μ, w focus on h lows of h fou valus, namly.5. Th fac ha h avag invso in h suvys sudid by Gnwood and Shlif (3) suvys ha includ boh sophisicad and lss sophisicad spondns xhibis xapolaiv xpcaions suggss ha many if no mos invsos in acual financial maks may b xapolaos. W can also us h suvy vidnc o g a b sns of a asonabl valu of. Th ida is simpl: if invsos xpcaions of fuu sock mak uns dpnd pimaily on vy cn mak uns on uns ov h pas ya, o pas wo yas W 5
hn is high. Convsly, if invsos xpcaions of fuu uns dpnd o a significan xn on uns in h disan pas, hn his poins o a low valu fo. In h Appndix, w dscib in dail how w us h suvy daa o sima. Th simaion maks us of Poposiion blow, and spcifically, quaion (), which dscibs h chang in h valu of h sock mak 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 D gd P P S s B s, () whil xapolaos xpcaion of h sam quaniy is: wh m ( m ) P P S s s ms, m () k B and m ( ). Whn and, () ducs o P. P S s s (3) Equaions () and () confim ha h xpcaions of xapolaos load posiivly on h snimn lvl, whil h xpcaions of aional ads load ngaivly. Whn w us h pocdu dscibd in h Appndix o sima fom h suvy daa, w obain a valu of appoximaly.5. Consqunly, whil w psn suls fo h diffn valus of, w pay mos anion o h cas of.5. 9 Fo a givn s of valus of h basic paams in Tabl, w us h pocdu oulind in h Appndix o compu h divd paams: and, which 9 Whn w sima fom h suvy daa, w assum, fo simpliciy, ha all h suvyd invsos a xapolaos. If w insad allowd som of hm o fall ino h cagoy of aional invsos in oh wods, if w insad id o sima boh and μ fom h suvy daa -- w migh obain a diffn valu of. Howv, w would no xpc h sima of o chang vy much. 6
dmin xapolaos, and hnc aional ads, opimal sha dmand (s quaion (3)); a, b, c, a, b and c, which dmin invsos opimal consumpion policis (s quaion (4)); 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 (s quaion (9)); and finally P, h volailiy of pic changs in h sock mak (s quaion ()). 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, 9.75, A7.4, B.99, P 3 3 3 a., a.8, b 7.3, b.4, c.63, c 3.47. (4) 4. Empiical Implicaions In his scion, w psn a daild analysis of h mpiical pdicions of h modl. Und h assumpions ha h dividnd lvl follows an aihmic Bownian moion and ha invsos hav xponnial uiliy, i is mo naual, in ou analysis, 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 ovall 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 7
psn blow, w always includ, as a bnchmak, h cas of, in oh wods, h cas wh h conomy 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, bu no subsqun cash flows. I is hlpful o xpss his fac in h mo sucud way suggsd by Cochan (), among ohs. If w un h univaia gssions a gssion of fuu uns on h cun dividndpic 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 and his is h basic fac ha nds o b xplaind 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 cash flows, and no is own fuu valu. W can xpss 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) fuu dividnd chang, and h fuu dividnd-pic diffnc, on h cun dividndpic diffnc h h cofficins w obain mus sum o on, a any hoizon. To mach h mpiical facs, ou modl nds o pdic a gssion cofficin in h fis gssion ha is clos o on, paiculaly a long hoizons. Th nx poposiion shows ha his is xacly h cas. (5) Poposiion 3 (Th 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 8
sa of h hoizon. In populaion, h cofficin on h indpndn vaiabl in h wh k. B cov D/ P, P P gssion is (6) k DP ( ), va D/ P Tabl 3 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. Th abl shows ha, consisn wih h mpiical facs, D/ P dos indd pdic fuu pic changs wih a posiiv sign. Moov, h gssion cofficin, ( ), is incasing in h lngh of h im hoizon. Fo long im DP hoizons, h gssion cofficin convgs o, as i dos mpiically in a gssion of long-hoizon uns on h dividnd yild. In h bnchmak aional conomy, h quaniy D/ P is consan; h gssion cofficin w compu 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 sock pics vn high. A his poin, h sock mak is ovvalud; h valu of D/ P is hfo low. Pcisly bcaus h sock mak is 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 is divn by 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 also wak fo low : whn is low, xapolaos blifs a mo psisn; as a sul, i aks long fo an 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 im-sis avags will convg o ou analyical suls in h long un. 9
ovvaluaion o coc, ducing h pdiciv pow of D/ P fo pic changs a any fixd hoizon. 4.. Auocolaions of P D/ In h daa, pic-dividnd aios a highly auocolad a sho lags, and w would lik o know if ou modl can capu his. Th naual analog of h picdividnd 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 (5), w nod ha, if w un gssions of h fuu chang in h sock pic, h fuu chang in dividnds, and h fuu dividnd-pic diffnc on h cun dividnd-pic diffnc, hn h h gssion cofficins w obain mus sum o on. Sinc dividnds follow 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 auocolaion of h dividnd-pic k diffnc D/ P, mus hfo qual. Th nx poposiion confims his. Poposiion 4 (Auocolaions of P D/). In populaion, h auocolaion of P D/ a a im lag of is D D k PD ( ) co P, P, (7) and k. B In Tabl 4, w compu h auocolaions in Poposiion 4 fo sval pais of valus of and, and fo lags of on qua, on ya, wo yas, h yas, and fou yas. 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: a long hoizons, h pic-dividnd diffnc focass pic changs, no is own fuu valu. Th abl shows ha h auocolaions a
high fo low valus of : whn is low, xapolaos blifs a vy psisn, which, in un, impas psisnc o h pic-dividnd diffnc. 4.3. 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 naual 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 dscibd in Coollay, an conomy wh pics a s only by aional focass of fuu cash flows. Poposiion 5 (Excss volailiy). In h conomy of Scion, h volailiy of pic changs ov a fini im hoizon (, ) is ( S B D k D P ) va P P SB, k (8) whil h volailiy of P D/ ov (, ) is wh k D D P D( ) va ( P ) ( P ) SB, k k B and D S. (B) (9) 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 is considably high: 3% high whn h a an qual numb of xapolaos and aional ads in h conomy, a figu ha, as w xplain blow, dpnds lil on h paam. Similaly, whil in h fully aional conomy h pic-dividnd diffnc is consan, in h psnc of xapolaos, i vais significanly.
Th suls in Poposiion 5 and in Tabl 5 confim h inuiion w dscibd in h Inoducion, namly ha h psnc of xapolaos amplifis h volailiy of sock pics. A good cash flow shock pushs sock pics up. Howv, his incas in sock pics immdialy lads xapolaos o xpc high fuu pic changs in h sock mak, which, in un, lads hm o push h valu of h sock mak 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, which, in un, mans ha subsqun pic changs, whil low han avag, will no b vy low. As a consqunc, aional invsos do no push back songly agains h ovvaluaion causd by h xapolaos. Pu diffnly, vn if h facion of xapolaos in h ovall populaion is low, his can b nough o significanly amplify h volailiy of h sock mak. Th abl shows ha, as xpcd, h ga h facion of xapolaos in h conomy, h mo xcss volailiy h is in pic changs and in h pic-dividnd diffnc. Mo insing, i also shows ha 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, h is anoh foc ha 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 sul, whn is high, aional ads ad mo aggssivly agains h xapolaos, dampning volailiy. Ovall, hn, 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 ad incass in valu. In oh wods, whil h high pic volailiy lows aional ads uiliy, his is mo han compnsad fo by h high pofis hy mak by xploiing h xapolaos.
4.4. Auocolaions of pic changs Empiically, uns on h sock mak a posiivly auocolad a sho lags; a long lags, hy a ngaivly auocolad. 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 (Auocolaions of pic changs). In populaion, h auocolaion of pic changs bwn (, ) and (, 3 ), wh, is P 3 3 P P, P P 3 P P va P P cov (,, ) co P P, P P, (3) va 3 wh S B D k3 k k cov P P, P P, 3 SB k va k P P SB B, k S D D S B D k( 3 ) D va P P ( 3 S B 3 ), k and k. B (3) 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. To s why, suppos ha h is good cash flow nws a im. Th sock mak gos up in spons o his nws; bu sinc his pic is causs xapolaos o xpc high fuu pic changs, h sock mak is pushd vn fuh up. Now ha h sock mak is ovvalud, h pic chang is low, on avag, going fowad. In oh wods, pas pic changs hav ngaiv pdiciv pow fo fuu pic changs. Ngaiv auocolaions a also obsvd in h daa, a sval lags; o som xn, hn, ou modl machs h daa. Howv, h is also a way in which ou 3
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. Som ali modls of un xapolaion fo xampl, Cul, Poba, and Summs (99), D Long al. (99b), Hong and Sin (999), and Babis and Shlif (3) do gna posiiv sho-m auocolaion, o momnum, fo sho. Babis and Shlif (3), fo xampl, consid an conomy wih wo goups of invsos. Th fis goup s dmand fo h isky ass a im dpnds on h ass s pas pic changs up o im ; h scond goup buys (slls) h isky ass whn is pic is low (high) laiv o fundamnals, bu dos no know h xac sucu of xapolao dmand. This modl gnas posiiv sho-m auocolaion and ngaiv long-m auocolaion in pic changs. In D Long al. (99b) and Hong and Sin (999), h im isky ass dmand of som invsos dpnds posiivly on h pic chang bwn im and im ; hs famwoks also gna posiiv sho-m auocolaion in pic changs, and ngaiv long-m auocolaion. Givn ha hs ali xapolaion-basd modls gna momnum, why dos ou modl no do so? Th a wo diffncs bwn h ali modls and h cun on, ach of which, akn alon, suffics o mov any momnum. Fis, in conas o h ali modls, ou conomy conains fully aional invsos who undsand how xapolaos fom blifs. Scond, in h cun modl, xapolaos dmand fo h isky ass dpnds on vn h mos cn pic chang P, whil P d in h ali modls, i dpnds only on pic changs up o, bu no including, h mos cn pic chang. This las assumpion is impoan fo gnaing momnum: if xapolaos dmand a im dpnds on h pic chang bwn im and im, a posiiv pic chang bwn and is likly o gna a posiiv pic chang bwn and. Ou claim ha ih h psnc of aional invsos o an xapolao dmand funcion ha dpnds on h mos cn pic chang is nough o mov momnum is basd on a -xaminaion of h Babis and Shlif (3) modl. W find ha, if w ih plac h paially aional invsos in ha modl wih fully aional invsos, o mak xapolao dmand a funcion of h mos cn pic chang, h modl no long gnas momnum. 4
4.5. 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: 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 wh C C P P cov( C C, P P ) c o,, va( C C ) va( P P ) (3) and cov( C C, P P ) ag b B B k k k ( k ) W S D S ( k) ( k) S k k g D S S DaW DbW k k k k (33) k g DS k S DW ad bd( ), k k B va( ) S ( D k P )( ) D P SB, (34) k 5
va( C C ) a W k 4 k k 4gDS S bws ( ) ( 3 ) ( ) k k 4k k k k a b g g k k k k k k 4 k W W S D k 4 D S S( ) (3 ) W a b k S ( 4 ab g g k k k k k S D k D S ) ( ) aww ( ) g 4 3 b b a k k k k k k k S S D k W W ( ) W ( ) 4 a g k W DS k k S ( ) Wb( ). k (35) No ha a a ( ) a, b b ( ) b, a a/, b W DQ, k and (B) B D S. ( B) W W b BQ BQ, B Panls A and B of Tabl 7 us Poposiion 7 o compu h colaion of consumpion changs and pic changs a a qualy and annual fquncy, spcivly, and fo sval (, ) pais. Th wo panls show ha, whil h psnc of xapolaos slighly ducs his colaion laiv o is valu in h fully aional conomy, h colaion is nonhlss high. As is h cas fo viually all consumpionbasd picing modls, hn, ou modl fails o mach h low colaion of consumpion gowh and uns in h daa. 4.6. Pdiciv pow of h suplus consumpion aio Pio mpiical sach has shown ha a vaiabl calld h suplus consumpion aio a masu of consumpion laiv o pas consumpion lvls, is conmpoanously colad wih h pic-dividnd aio on h ovall sock mak; and fuhmo, ha i pdics subsqun uns wih a ngaiv sign. 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. 6
As w hav don houghou his scion, w look a a suplus consumpion diffnc ah han a 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 conmpoanous pic-dividnd diffnc P D/. Poposiion 8: (Colaion bwn consumpion chang and P D/). In populaion, h colaion bwn h chang in consumpion ov a fini im hoizon (, ) and P D/ is D co( C C, P ) D cov( C C, P ), (36) D va( C C ) va( P ) wh D k a WgD S bw S W S agd S b S cov( C C, P ) ( ) B, (37) k k k k k va( P D ) S, B k b b ( ) b, a a/, b W and va( C C ) is as in (35). Also, a a ( ) a, W b BQ 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 (Th pdiciv pow of changs 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 7
C (, cov C C, P P ) va C C wh cov( C C, P P ) k( ) k a WgDS bws WS agds b S ( )( ) B k k k k k,, a and va( C C ) is as in (35). Also, a a ( ) a, b b ( ) b, aw, (38) (39) 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 and h pic-dividnd diffnc. H, h suplus consumpion diffnc is compud as h cun lvl of aggga consumpion minus h lvl of aggga consumpion a qua ago. 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 hoizon on qua, on ya, wo yas, h yas, o fou yas on h suplus consumpion diffnc. I shows ha h suplus consumpion diffnc has significan ngaiv pdiciv pow fo pic changs, and ha h pdiciv pow is paiculaly song fo low μ and high β. Takn ogh, hn, Panl C of Tabl 7 and Tabl 8 show ha h suplus consumpion diffnc can b colad wih h valuaion lvl of h sock mak and wih h subsqun sock pic chang vn in a famwok ha dos no involv habi-yp pfncs in any way. Th inuiion fo hs suls is saighfowad. Af a squnc of good cash flow nws, xapolaos caus h sock mak o bcom ovvalud and hnc h pic-dividnd diffnc o b high. Howv, a h sam im, xapolaos opimisic blifs abou h fuu lad hm o ais hi consumpion; whil h aional ads do no ais hi consumpion as much, aggga consumpion nonhlss incass ovall, pushing h suplus consumpion diffnc up. This gnas a posiiv 8
colaion bwn h pic-dividnd diffnc and h suplus consumpion diffnc. Sinc h sock mak is ovvalud a his poin, h subsqun pic chang in h sock mak is low, on avag. As a consqunc, h suplus consumpion diffnc pdics fuu pic changs wih a ngaiv sign. 4.7. Equiy pmia and Shap aios Poposiion blow compus h quiy pmium and Shap aio of h sock mak. Poposiion (Equiy Pmium and Shap Raio). In h conomy of Scion, h quiy pmium, dfind as h p uni im xpcaion of h xcss pic chang and dividnd, can b win as gd dp Dd Pd ( B) A. (4) d Th Shap aio is d ( dp D dpd) Va d ( dp Dd Pd ) / D B D gd B ( B) A. (B) B ( B) (4) Panl A of Tabl 9 uss h poposiion o compu h avag xcss pic chang and dividnd a an annual hoizon fo sval (, ) pais. Th panl shows ha h quiy pmium iss as h facion of xapolaos in h conomy gos up: h mo xapolaos h a, h mo volail h sock mak is; h quiy pmium hfo nds o b high o compnsa fo h high isk. Panl B of h abl shows ha i is no jus h quiy pmium ha gos up as falls, bu also h Shap aio. 9
5. Fuh Analysis: Raio-basd Quaniis In Scion 4, w focusd on quaniis dfind in ms of diffncs: on pic changs, and on h pic-dividnd diffnc P D/. Givn h addiiv sucu of ou modl, hs a h naual quaniis o sudy. Howv, mos mpiical sach woks wih aio-basd quaniis such as uns and pic-dividnd aios. Whil hs a no h mos naual quaniis o look a in h conx of ou modl, w can nonhlss xamin wha ou modl pdics abou hm. This is wha w do in his scion. Sinc analyical suls a no availabl fo aio-basd quaniis, w us numical simulaions o sudy hi popis. In Scion 5., w xplain h mhodology bhind hs simulaions. In Scion 5., w psn ou suls. In bif, h suls fo h aio-basd quaniis a boadly consisn wih hos fo h diffncbasd quaniis in Scion 4. Howv, w also inp hs suls cauiously: pcisly bcaus hy a no h naual objcs of sudy in ou modl, h aio-basd quaniis a no as wll-bhavd as h diffnc-basd quaniis w xamind in Scion 4. 5.. Simulaion Mhodology To conduc h simulaions, w fis disciz h modl. In his discizd vsion, w us a im-sp of ¼, in oh wods, of on qua. As indicad in Scion 3, h iniial lvl of h dividnd is D and h iniial walh lvls a W W 5. W fuh s h iniial snimn lvl, S, o h sady-sa man of g D. W know fom Poposiion ha, a im, D N S, P ABS, i i i i i C W a S b S c log( ), i{, }. (4) Th poposiion also lls us ha, fom im n o (n+), w hav: 3
D D g ( n) n D D ( n), g S S S B (B) D D ( n) n n ( n) D P( n) ABS( n) N W S ( n) ( n) ( n) ( n) W N ( P i i i i ( n) n n ( n) n n, N, P Q ( ) N ) C, ( n), (43) C W N P N D, W i i i n n n n ( n) a S bs i i i i i ( n) ( n) ( n) ( n) c lo g ( ), wih i {, }, and wh { ( n), n } a i.i.d. sandad nomal andom vaiabls wih man and a sandad dviaion of. W mak h convnional assumpions ha h lvl of h consumpion sam fo h piod bwn (n, (n+)) is dmind a h bginning of h piod; and ha h lvl of h dividnd paid ov his piod is dmind a h nd of h piod. Fo a givn s of valus of h basic modl paams in Tabl, w us h pocdu dscibd in h poof of Poposiion o compu h paams ha dmin h opimal pofolio holdings and consumpion choic vaiabls such as, fo xampl. W hn us h abov quaions o simula a sampl pah fo ou conomy ha is piods long, in oh wods, 5 yas long. W compu quaniis of ins fom his -piod im sis h auocolaion of sock mak uns, say. W hn pa his pocss, ims. In h nx scion, w po h avag un auocolaion ha w obain acoss hs, simulad pahs. H, w a assuming ha h valus of h divd paams, such as, ha dmin invsos opimal policis in h coninuous-im famwok a a good appoximaion o h valus of hs paams in h disc-im analog of ou modl. On indicaion ha his is a asonabl assumpion is ha ou numical suls a obus o changing fom /4 o /48, say. 3
5.. Rsuls Tabl psns h modl s pdicions fo aio-basd quaniis fo.5 and fo h diffn valus of. Fo ach (, ) pai, w simula, pahs, ach of which is piods long. Fo ach of h, pahs, w compu vaious quaniis of ins spcifically, h quaniis lisd in h lf column of Tabl. Th abl pos h avag valu of ach quaniy acoss h, pahs. Th las column of h abl pos h mpiical valu of ach quaniy ov h pos-wa piod fom 947 o. 3 W now discuss ach of hs quaniis in un. Mos of hm a simply h aio-basd analogs of h quaniis w sudid in Scion 4: fo xampl, insad of compuing h sandad dviaion of pic changs, w compu h sandad dviaion of uns. Howv, w a also abl o addss som qusions ha w did no discuss in any fom in Scion 4, such as whh h consumpion-walh aio o mo complx fomulaions of h suplus consumpion aio hav any pdiciv pow fo uns. Row : W po h cofficin on h indpndn vaiabl in a gssion of oal log xcss uns masud ov a on-ya hoizon on h log dividndpic aio a h sa of h ya. To b cla, as dscibd abov, w un his gssion in ach of h, pahs w simula; h abl pos h avag cofficin acoss all pahs, as wll as h avag R-squad, in panhss. Consisn wih h findings of Scion 4., h abl shows ha h dividnd-pic aio pdics subsqun uns wih a posiiv sign. Row : W po h auocolaion of h pic-dividnd aio a a onya lag. Consisn wih h suls of Scion 4., h aio is highly psisn. Row 3: W compu h xcss volailiy of uns -- spcifically, h sandad dviaion of sock uns in h hognous-agn conomy laiv o h sandad dviaion of uns in h aional bnchmak conomy. Consisn wih h findings of Scion 4.3, w s ha sock uns xhibi xcss volailiy. 3 Fo h nonduabl consumpion daa, h sampl piod sas in 95. Runs a basd on h CRSP valu-wighd indx. Fo h consumpion-walh aio, walh is compud in wo diffn ways: h fis way uss h mak capializaion of h CRSP sock mak, and h scond uss aggga houshold walh fom h Flow of Funds accouns, following Lau and Ludvigson (). 3
Row 4: W compu h xcss volailiy of pic-dividnd aios: h sandad dviaion of h pic-dividnd aio in h hognous-agn conomy laiv o is sandad dviaion in h aional bnchmak conomy. Consisn wih Scion 4.3, h sandad dviaion of h pic-dividnd aio gos up in h psnc of xapolaos. Row 5: W compu h auocolaion of qualy log xcss sock uns a lags of on qua and wo yas. As in Scion 4.4, uns a ngaivly auocolad. Row 6: W compu h colaion of annual log xcss sock uns wih annual changs in qualy log consumpion. As in Scion 4.5, his colaion is high han h colaion obsvd in h daa. Row 7: W compu h colaion bwn h suplus consumpion aio and h pic-dividnd aio, wh boh quaniis a masud a a qualy fquncy. Givn h ga flxibiliy affodd by numical simulaions, w us a mo sophisicad dfiniion of suplus consumpion han in Scion 4.6, on ha is sill simpl han in Campbll and Cochan (999) bu ha nonhlss psvs h spii of hi calculaion. Spcifically, w dfin h suplus consumpion aio as: S a C X, (44) a C a wh h supscip a sands fo aggga, and wh h habi lvl X adjuss slowly o changs in consumpion: k n X wkn ( ;, ) C, wh ( ;, ). k k wk n n j In simpl ms, X is a wighd sum of pas consumpion lvls, wh cn consumpion lvls a wighd mo havily. Fo a givn, w choos n so ha n j j j ( n) 9%; j ha is, w choos so ha vn consumpion j (45) 33
changs in h disan pas civ a las som wigh in h compuaion of h habi lvl. In ou calculaions, w s.95 and n. 4 Row 7 of Tabl shows ha, as in Scion 4.6, h suplus consumpion aio and pic-dividnd aio a posiivly colad, consisn wih h acual daa. Row 8: W po h cofficin on h indpndn vaiabl in a gssion of oal log xcss uns ov a ya on h suplus consumpion aio a h sa of h ya. Consisn wih ou suls in Scion 4.6 using a simpl masu of suplus consumpion, h suplus consumpion aio pdics subsqun uns wih a ngaiv sign, as i dos in acual daa. Row 9: Empiically, h consumpion-walh aio has pdiciv pow fo subsqun uns. H, w xamin whh ou modl can gna his pan. W compu h cofficin on h indpndn vaiabl in a gssion of oal log xcss uns ov a ya on h log consumpion-walh aio a h sa of h ya. Th abl shows ha h aio dos indd hav som pdiciv pow. Wha is h inuiion fo his pdiciv pow? Af a squnc of good cash flow nws, xapolaos caus h sock mak o bcom ovvalud. This, in un, incass aggga walh in h conomy; i also incass aggga consumpion, bu no o h sam xn: aional ads, in paicula, do no incas hi consumpion vy much bcaus hy aliz ha fuu uns on h sock mak a likly o b low. Ovall, h consumpion-walh aio falls. Sinc h sock mak is ovvalud, is subsqun un is low han avag. Th consumpion-walh aio hfo pdics subsqun uns wih a posiiv sign. Row : W compu h quiy pmium and Shap aio in ou conomy. In summay, whil i is naual, in ou famwok, o sudy diffnc-basd quaniis ah han aio-basd quaniis, Tabl shows ha h aio-basd quaniis xhibi pans ha a boadly simila o hos ha w obaind in Scion 4 fo h diffnc-basd quaniis. 4 Whn =.95, qualy consumpion on ya ago is wighd abou 4% as much as cun consumpion. 34
6. Conclusion Suvy vidnc suggss ha many invsos fom blifs abou fuu sock mak uns by xapolaing pas uns: hy xpc h sock mak o pfom wll (pooly) in h na fuu if i has cnly pfomd wll (pooly). Such blifs a had o concil wih xising modls of h aggga sock mak. W sudy a hognous-agn modl in which som invsos fom blifs abou fuu sock mak pic changs by xapolaing pas pic changs, whil oh invsos hav fully aional blifs. W find ha h modl capus many faus of acual uns and pics. Impoanly, howv, i is also consisn wih h suvy vidnc on invso xpcaions. This suggss ha h suvy vidnc dos no nd o b sn as a nuisanc; on h conay, i is consisn wih h facs abou pics and uns and may b h ky o undsanding hm. 35
Appndics A. Poof of Poposiion In od o solv h sochasic dynamic pogamming poblm, w nd h diffnial foms fo h voluion of h sa vaiabls. Fom h dfiniion of h ( s) snimn vaiabl, S dps d, is diffnial fom is ds Sd dp. (A) Th m S d capus h fac ha, whn w mov fom im o im d, all h ali pic changs ha conibu o S nd o b associad wih small wighs sinc hy a fuh away fom im d han hy w fom im ; h m dp capus h fac ha h las pic chang pushs S in h sam dicion; and h paam capus h sickinss of his blif updaing ul. Also, h walh of ach yp of ad volvs as i i i i i i W ( W C d N P )( d) N D d N P d d (A) i i i i i i dw W d Cd NPd N dp N D d, i {, }, consisn wih h budg consains in (5) and (7). As nod in h main x, h divd valu funcions fo h xapolaos and h aional ads a i scs i i i J ( W, S, ) max, {, }. i i ds i { Cs, Ns} s (A3) Th assumpions ha ads hav CARA pfncs, ha D follows an aihmic Bownian moion, ha S volvs in a Makovian fashion as in (A), and ha xapolaos biasd blifs in (3) a linaly lad o S joinly guaan ha h divd valu funcions a only funcions of im, of h lvl of walh, and of h lvl of snimn, bu of nohing ls (such as D o P ). W vify his and discuss i fuh af solving h modl. If w dfin i C i i i i i i ( C, N ; W, S, ) [ dj ], d i{, }, (A4) hn, fom h hoy of sochasic conol, w hav ha 5 i i i i ma x ( C, N ; W, S, ), i {, }. (A5) i i { C, N } By Io s lmma, (A5) lads o h sochasic Bllman quaions which sa ha, along h opimal pah of consumpion and ass allocaion, i C i i i i i i i i i i J JW W C N PN gp N D JWWP( N ) i i i i i JS SgP JSSP JWSN P, i{,}, (A6) 5 S Kushn (967) fo a daild discussion of his opic. 36
wh g P and g P a h p uni im pic chang of h sock mak xpcd by xapolaos and aional ads, spcivly, and wh P is h p uni im volailiy of h sock pic. No ha, as sad in (3), gp S, and ha g P coms fom aional ads conjcu abou h sock pic pocss, which is y o b dmind. No also ha, in coninuous im, h volailiy P is ssnially obsvabl by compuing h quadaic vaiaion; as a sul, h wo yps of ads ag on is valu. W assum, and la vify, ha P is an ndognously dmind consan ha i dos no dpnd on S o. Finally, fom h voluion of S in (A), w know ha dw and S a locally pfcly colad fo boh yps of ad. Sinc h infini-hoizon modl is ppual, and sinc, as vifid la, h voluions of W andw do no dpnd xplicily on h lvl of h dividnd o h sock pic, w know ha h passag of im only affcs h valu funcions hough im discouning. W can hfo wi, fo i {, }, i ( ) ), wh ma s Cs i i i i i i i J ( W, S, ) I ( W, S I ( W, S) x. i i ds (A7) { Cs, Ns} s Subsiuing (A7) ino (A6) givs h ducd Bllman quaions i C i i i i i i i i i i I IW W C N PN gp N D IWWP( N ) (A8) i i i i i ISS gp ISSP IWSN P, i{, }. Th fis-od condiions of (A8) wih spc oc i i and W a i i C I W (A9) and i i i i IW gp P D IWS N, i{, }. i i (A) IWW P IWW Th fis m on h igh-hand sid of (A) is h sha dmand du o man-vaianc considaions; h scond m is h hdging dmand du o snimn-lad isk. W now conjcu, and la vify, ha h u quilibium sock pic saisfis D P ABS. (A) Th cofficins A and B a y o b dmind. Assuming ha h aional ads know his pic quaion and h u pocss fo D, hy can obain h u voluion of h sock pic as B gd D dp S d d (A) B ( B) ( B) by combining (), (A), and (A). Subsiuing (A) ino (A) yilds gd D ds S d d. (A3) B ( B) 37
Fom (A) and (A3) i is cla ha whn B, h snimn vaiabl S follows an Onsin-Uhlnbck pocss wih a sady-sa disibuion ha is Nomal wih man g D D and vaianc, and ha h xpcd p uni im pic chang, ( B) [ dp] g, D also flucuas aound is long-un man of wih long-un vaianc of d 3 D B. In addiion 3 (B) B gd D gp S, P. (A4) B ( B ) ( B) Tha is, aional ads fuu xpcd pic chang is ngaivly and linaly lad o h snimn lvl, and P is a consan if h conjcu in (A) is valid. Givn h imposd blif sucu ha gp S, h xapolaos subjcivly bliv ha h sock pic volvs as D dp Sd d, (A5) ( B ) wh d is xapolaos pcivd innovaion m fom h dividnd pocss, which islf follows dd gd D Dd, (A6) wh g D is xapolaos pcivd xpcd p uni im dividnd chang. 6 Diffniaing (A) and subsiuing in (A) and (A6), xapolaos obain B gd D dp S d d, (A7) B ( B) ( B ) in conas wih h pic pocss (A) obaind by h aional ads. Compaing (A5) and (A7) suggss ha g ( S ) ( B) ( B) B S. (A8) D Tha is, xapolaos pcivd xpcd dividnd chang p uni im dpnds xplicily on S. (W no ha his is qui diffn fom dicly xapolaing pas dividnd changs.) Pic-agmn acoss h wo yps of ads, in oh wods, dp gpd PdgPd Pd (A9) pvns xapolaos fom sing, hough ospcion, ha hi blif sucu is biasd, and povids a dic laion bwn d and d. Equaions (A), (A7), and (A9) joinly confim dividnd-agmn acoss ads: dd g d dg d d. (A) D D D D 6 If insad, h xapolaos know h u pocss of D, hy will bliv ha dp ( S )d + P d, a pic pocss ha, givn ha B/( B), claly dvias fom h u pocss in (A). In oh wods, vn af a im inval of lngh d, xapolaos will, in pincipl, b abl o lan ha hi blifs a wong. 38
W guss ha h soluions of I ( W, S) and I ( W, S) a I i ( W i, S) x p W i as i b i S c i, i{, }. (A) Subsiuing (A) ino h opimal consumpion ul in (A9) and h opimal sha dmand of h sock in (A) yilds i i i i i C W a S b S c log( ), (A) and i i i i gp P D ( a Sb ) N, i{, }. (A3) P Fo h xapolaos, subsiuing gp S and h pic quaion (A) ino (A3) givs Ab P B a P N S, wh and. (A4) P P Subsiuing h pic quaion (A), h fom of I in (A), h opimal consumpion C in (A), and h opimal sha dmand N in (A4) ino h ducd Bllman quaion (A8) fo h xapolaos, w obain h following quadaic quaion in S: ( Ab P) ( B a P) S ( ) ( SABS) P as bsc log ( ) [( Ab P) ( B a P) S] P (A5) ( asb)[ S( S)] a P P( asb) ( asb)[( Ab ) ( B a ) S], P which is quivaln o h simulanous quaions: ( B a P) a a ( ) ( a P), (A6) P ( B a P)( Ab P) b a b ( ) Pa b, (A7) P ( Ab P) ( ) log( ) ( c b a P b P ). (A8) P Ths h quaions dmin h cofficins a, b, c,, and as funcions of h cofficins A and B. If, as w assum, xapolaos know h blif sucu of h aional ads as wll as h paams and Q, i follows ha hy can go hough h inmpoal maximizaion poblm fo h aional invsos (spcifid blow) and figu ou h pic quaion (A). As a sul, xapolaos know h cofficins A and P 39
B, and hough quaions (A6), (A7), and (A8), hy can solv fo hi opimal sha i dmand N, as wll as fo hi valu funcion J. W now un o h aional ads. Using g P and P fom (A4), h fom of I in (A), N fom (A3), h opimal sha dmand of h sock fom xapolaos in (A4), and h mak claing condiion N ( ) N Q, w obain a and b as funcions of A and B, ( B) B ( ) D a, B D B ( B) ( B) D gd () D b Q A. D ( B) ( B) ( B) Subsiuing h pic quaion (A), g P fom (A4), h fom of I in (A), h opimal Q consumpion C in (A), and h opimal sha dmand N ( S) ino h ducd Bllman quaion (A8) fo h aional ads, w obain anoh quadaic quaion in S Q( )( S) B gd ( ) S ABS B (B ) as bsc log( ) P[ Q( )( S)] (a Sb) gd S B P( a Sb )[ Q( ) ( ] a P P( a Sb ) S), which is quivaln o h simulanous quaions: ( ) B ( ) B [( ) P] a a ( a ) P ( B) B a ( ) P, Q ( ) B ( ) gd B Ab B (B) P( ) [ Q( ) ] b a gd Pab B ( B) P a [ Q( ) ] b ( ), (A9) (A3) (A3) (A3) 4
[ Q( ) ] g c ( B ) P[ Q( ) ] bgd [ ( ) ] ( b P ) P Q a P b. ( B ) ( ) D A log( ) Ths h quaions dmin h cofficins A, B, and c. Equaions (A6)-(A8) and (A3)-(A33) a h mahmaical chaacizaion of h ndognous inacion bwn aional ads and h xapolaos. Th pocdu fo solving hs simulanous quaions is lf o h nx scion of h Appndix. Th fac ha h conjcud foms of P, I, and I in (A) and (A) saisfy h Bllman quaions in (A8) fo all W and S vifis hs conjcus, condiional on h validiy of h assumpion ha W and S a h only wo sochasic sa vaiabls. To vify h la, no ha h pic quaion in (A), h opimal consumpion uls in (A), and h fac ha h soluions of N and N a linaly lad o S joinly guaan ha h voluions of W andw in (A) dpnd xplicily only on S. Lasly, h divd voluion of h sock pic in (A) vifis h assumpion ha P is an ndognously dmind paam. This compls h vificaion pocdu. Equaions (A), (A) and (A3), (A4), and (A) confim quaions (9), (), (), (3), and (4) in h main x, spcivly, and quaions (A7) and (A) ogh confim (). This compls h poof of Poposiion. (A33) B. Solving h Simulanous Equaions To solv quaions (A6), (A7), (A8), (A3), (A3), and (A33), w goup hm ino h pais of quaions and solv ach pai in squnc. Fis, w us (A6) and (A3) o dmin a and B, wh, in un, w us (A4), (A4), and (A9) o xpss, P, and a as funcions of a and B. Scond, w us (A7) and (A3) o dmin b and A, wh, in un, w us (A4) and (A9) o xpss and b as funcions of b, A, and B. Lasly, w solv ach of (A8) and (A33) o obain c and c, spcivly. i Th fac ha h valu funcion J ( W, S, ) is muliplicaivly spaabl in W, S, and simplifis h modl and nsus acabiliy. Fo insanc, ou modl has h fau ha h discoun faco only affcs opimal consumpion and opimal walh, bu no h quilibium pic: fo boh yps of invso, opimal sha dmand is unlad o. C. Poof of Coollay Whn all h ads in h conomy a fully aional, (A) ducs o W I ( W ) K, (A34) wh K is a consan o b dmind. Subsiuing (A34) ino (A) and using N Q, w know ha h quilibium sock pic is 4
P D gd D Q. (A35) This hid m on h igh-hand sid of his quaion shows ha P is pggd o h cun lvl of h dividnd; h oh wo ms capu dividnd gowh and compnsaion fo isk. Subsiuing (A34) and (A35) ino h Bllman quaion (A8) dmins h cofficin K as D K x p Q. (A36) Fom (A9), opimal consumpion is DQ CW log( K) W. (A37) Fom (A), (A35), and (A37), opimal walh volvs accoding o DQ QD dw d d. (A38) This compls h poof of Coollay. D: Poof of Coollay Diffniaing boh sids of (A) givs gdd Dd dp B( Sd dp). (A39) If h is posiiv cash-flow nws ha incass h sock pic by, hn, fom (A39), h psnc of snimn in h quilibium pic will push h pic up by a fuh amoun B, and hn by a fuh amoun B, and so on. Th oal pic incas du o a shock of siz is hfo B B ( B B ). (A4) B This gomic sis convgs if and only if B. Tha is, h pic quaion (A) is an quilibium pic quaion if and only if B. Whn all invsos a xapolaos ( ), h mak claing condiion implis Ab P Q, (A4) P B a P B a P. (A4) P Subsiuing (A4) ino (A6), w obain a [ ( ) a ] a [ ( B) ( )]. (A43) P Und h condiion ha, (A43) implis a. Givn his, (A4) hn implis ha B /. (A44) 4
Sinc h ncssay and sufficin condiion fo xisnc of h conjcud quilibium is B, (A44) now mans ha a ncssay condiion fo xisnc is. (A45) W hav no y shown h sufficincy of his condiion; o do so, w nd o chck (A7) and (A8) o s whh w can dmin A, b, and c. Subsiuing a and (A44) ino (A7), w obain b unlss ( ). (A46) Wih b, w hn obain, fom (A4), ha QP QD ( ) A. (A47) D Now subsiuing a, b, (A4), (A44), and P ino (A8) givs ( B) Q c log( ). (A48) ( ) D Qui gnally, hn, w can solv fo A, b, and c if condiion (A45) holds. Thfo, w can claim ha (A45) is boh a ncssay and sufficin condiion. W no ha his poof dos no ul ou any nonlina quilibia. E: Poofs of Poposiions o Th saisical popis of h snimn pocss S can b divd by sudying a k lad pocss, Z S, which volvs accoding o k k gd k dz d Sd. (A49) Unlik h snimn pocss, h Z pocss has a non-sochasic dif m, and is hfo asi o analyz. W us his pocss padly in ou poofs of Poposiions o 6. E.. Poof of Poposiion I is saighfowad o calcula h pic chang xpcaions of aional ads. Combining xapolaos blif abou h insananous pic chang, (A5), and h diffnial dfiniion of h snimn vaiabl, (A), w find ha xapolaos subjciv blif abou h voluion of S is ds [ ( ) S] d Sd. (A5) Exapolaos bliv ha is a sandad Win pocss. This mans ha, fom h m pspciv of xapolaos, h voluion of Z S, wh m ( ), is m m d Sd. dz (A5) Using h saisical popis of h Z pocss, as pcivd by xapolaos, w obain (). Whn m, applying L'Hôpial's ul o () givs (3). 43
E.. Poof of Poposiion 3 Fom (A), w know ha cov D / P, P P B cov S, S S B cov S, D D. (A5) I is obvious ha cov S, D D. show W also obain Using h popis of h Z pocss, w can k ( ) S cv o S, S S. (A53) k S va D/ P B va S B. (A54) k Equaions (A5), (A53), and (A54) hn joinly giv (6). E.3. Poof of Poposiion 4 Fo h auocolaion sucu of P D/, w know fom (A) ha PD( ) co S, S. (A55) W can show ha covs, S s S s S S s S k k g (A56) D S s s. k I is saighfowad o show ha va( S) va ( S ) /. S k Puing hs suls ogh, w obain quaion (7) in h main x. E.4. Poof of Poposiion 5 Fom h pic quaion (A), w know ha h vaianc of pic changs is givn by va P P B va S S B cov S S, D D va D D. (A57) Th quaniy va S S can b xpssd as S S s S S s s S S s S S va va, (A58) wh h subscip s mans ha w a condiioning on S s. W can show k k k k ( ) S va S S s va Z Z s. Sd (A59) k Using h popis of h Z pocss, w also find ha k g D S S s ( ) s, (A6) 44
and k ( ) S D cov S S, D D. (A6) k Subsiuing (A59) and (A6) ino (A58) givs k ( ) S va S S. (A6) k Subsiuing (A6), (A6), and va( D D ) Dino (A57) givs quaion (8) in h main x. Combining h pic quaion (A) wih (A6) lads o (9). E.5. Poof of Poposiion 6 Fom (A), w know ha cov P P, P P B cov S S, S S cov D D, D 3 3 D 3 B cov S S, D D B cov S S, D D. 3 3 (A63) Using h popis of h Z pocss, w obain S k3 k k cov S S, S S, 3 (A64) k and SD k3 k k cv o D D, S S. 3 (A65) k In addiion, sinc h incmns in fuu dividnds a indpndn of any andom vaiabl ha is masuabl wih spc o h infomaion s a h cun im, cov D D, D D cov S S, D D. (A66) 3 3 Subsiuing (A64), (A65), and (A66) ino (A63) yilds h fis quaion in (3). Th scond quaion in (3) is divd in Poposiion 5, and h hid quaion can b divd in a simila way. E.6. Poof of Poposiions 7 o 9 Fom h budg consains (A), h pic quaion (A), and h opimal consumpions (A), w know ha aggga walh volvs as dw ( a W S b W S c W ) d W d. (A67) Subsiuing his ino (A) yilds C C ( W W ) [ a( S S ) b( S S )] (A68) ( a S bsc) d d [ as ( S) bs ( S)]. W W W W 45
To compu cov ( C C, P P ) and va( C C ), w fis nd o compu h covaianc of vy combinaion of wo ms in h las lin of (A68). Fo xampl, on of hs covaiancs is 7 s d s s k ( ) S c ov Sd, S SS s S s S s d. k Th oh covaianc ms can b compud in a simila way. Raanging and simplifying ms, w obain (33), (35), (37), and (39). Equaion (34) has bn divd in Poposiion 5. (A69) E.7. Poof of Poposiions Subsiuing h quilibium pic quaion (A) and is voluion (A) ino ou dfiniion of h quiy pmium, dp Dd Pd, givs (4) in h main x. Fo d h Shap aio, by h law of oal vaianc, Va d ( dp Dd Pd ) s Va d ( dp Dd Pd ) s S Va s d ( dp Dd Pd ) s S (A7) D B D B (B) B ( B) Combining (4) wih (A7) givs (4).. F: Esimaing Esimaing Equaions Ou objciv is o sima h modl paams,, and using h suvy daa. Suppos w hav a im-sis of aggga sock mak pics wih sampl fquncy (w us ¼ fo qualy daa). Thn, a im, h pop discizaion of (A) is n S (, n) w( j;, n) ( P P ), (A7) j j j j wh w( j;, n). H h wighing funcions a paamizd by and n k k by n, which masus how fa back invsos look whn foming hi blifs. Ths wighs mus sum o. 7 Th divaion of (A69) maks us of Fubini s hom. W hav chckd ha h condiions ha allow h us of Fubini s hom hold in ou conx. Fo mo on hs condiions, s Thom.9 in Lips and Shiyav (). 46
Th ky assumpion of ou modl is ha xapolaos xpcd pic chang (no xpcd un) is [ dp ]/ d S ( ). (A7) Th xpcaion in (A7) is compud ov h nx insan of im, fom o d, no ov a fini im hoizon. In h suvys, howv, invsos a ypically askd o sa hi blifs abou sock mak pfomanc ov h nx ya. I is hfo no fully coc o sima (,, ) using (A7). W mus insad compu wha h modl implis fo h pic chang xapolaos xpc ov a fini hoizon. W do his in Poposiion of h pap, and find: m ( ) m ( ) [ P P ] ( S s s)( ) ( ms), (A73) m wh m ( ). Th fis m on h igh-hand sid of (A73) is xapolaos xpcd pic chang a im, s, muliplid by h im hoizon,. (Fo xampl,.5 fo a six-monh hoizon). Th scond m capus xapolaos subjciv blifs abou how h snimn lvl will volv ov h im hoizon,, Th paams (,, ) n h in a non-lina fashion. To dmin (,, ), w hfo sima boh and [ P P ] [ ˆ ˆ S ( ˆ)]( ), (A74) ˆ ˆ ( ) ( ) ( ˆ ˆ (ˆ ˆ ( ˆ m m P P S )] ms ) [ ] [ ( ) ), (A75) m wih m( ˆ, ˆ ) ˆ (ˆ ) and S ( ˆ) consucd as dscibd abov. W also sima quaion (A75) fo h spcial cas wh is fixd a. In his cas, quaion (A75) bcoms: ˆ ˆ ( ) [ ] [ ˆ P P S ( ˆ)] ( ). (A76) Suvy Daa W sima quaions (A74), (A75), and (A76) using h Gallup suvy daa sudid by Gnwood and Shlif (3) and ohs. W sa wih h scald vsion of h sis dscibd in ha pap. Af h scaling, h pod xpcaions a in unis of pcnag xpcd uns on h aggga sock mak ov h following monhs. W hn conv his sis ino xpcd pic changs by muliplying by h lvl of h S&P 5 pic indx a h nd of h monh in which paicipans hav bn suvyd. Tha is, 47
P P P P P [ ] [ ] P Suvy Th suling Gallup sis compiss 35 daapoins bwn Ocob 996 and Novmb. Th daa a monhly bu h a also som gaps. W sima quaions (A74), (A75), and (A76) using nonlina las squas gssion. W us 6 quas of laggd pic changs in h S&P 5 pic indx whn consucing S abov. W po cofficins and Nwy Ws sandad os using a lag lngh of 6 monhs. (A77) Cofficin Equaion (A74) Equaion (A75) Equaion (A76) β.49.44.68 [-sa] [6.5] [5.77] [.73] λ.9.7.7 [-sa] [3.4] [35.4] [36.8] λ.35.3 [-sa] [8.7] [9.48] R-squad.77.74.75 Rfncs Ali, Aydogan, and Paul Tlock. 3. Biasd Blifs, Ass Pics, and Invsmn: A Sucual Appoach. Jounal of Financ, fohcoming. Amomin, Gn, and Svn A. Shap. 8. Expcaion of Risk and Run Among Houshold Invsos: A hi Shap Raios Councyclical? Financ and Economics Discussion Sis Woking Pap, 8-7, Fdal Rsv Boad, Washingon DC. Baccha, Philipp, Elma Mns, and Eic van Wincoop. 9. Pdicabiliy in Financial Maks: Wha Do Suvy Expcaions Tll Us? Jounal of Innaional Mony and Financ, 46-46. Bansal, Ravi, and Ami Yaon. 4. Risks fo h Long Run: A Ponial Rsoluion of Ass Picing Puzzls. Jounal of Financ 59, 48 59. Bansal, Ravi, Dana Kiku, and Ami Yaon.. An Empiical Evaluaion of h Long- Run Risks Modl fo Ass Pics. Ciical Financ Rviw, 83. Babis, Nicholas, Ming Huang, and Tano Sanos.. Pospc Thoy and Ass Pics. Qualy Jounal of Economics 6, 53. Babis, Nicholas, and Andi Shlif. 3. Syl Invsing. Jounal of Financial Economics 68, 6 99. 48
Babis, Nicholas, Andi Shlif, and Rob Vishny. 998. A Modl of Invso Snimn. Jounal of Financial Economics 49, 37 343. Bao, Rob. 6. Ra Disass and Ass Maks in h Twnih Cnuy. Qualy Jounal of Economics, 83 866. Bunnmi, Makus, and Sfan Nagl. 4. Hdg Funds and h Tchnology Bubbl. Jounal of Financ 59, 3-4. Campbll, John, and John Cochan. 999. By Foc of Habi: A Consumpion-Basd Explanaion of Aggga Sock Mak Bhavio. Jounal of Poliical Economy 7, 5 5. Campbll, John, and Rob Shill. 988. Th Dividnd-Pic Raio and Expcaions of Fuu Dividnds and Discoun Facos. Rviw of Financial Sudis, 95 8. Choi, Jams. 6. Exapolaiv Expcaions and h Equiy Pmium. Woking pap, Yal Univsiy. Cochan, John.. Psidnial Addss: Discoun Ras. Jounal of Financ 66, 47 8. Cul, David, Jams Poba, and Lawnc Summs. 99. Spculaiv Dynamics and h Rol of Fdback Tads. Amican Economic Rviw Paps and Pocdings 8, 63 68. D Long, Bad, Andi Shlif, Lawnc Summs, and Rob Waldman. 99a. Nois Tad Risk in Financial Maks. Jounal of Poliical Economy 98, 73 738. D Long, Bad, Andi Shlif, Lawnc Summs, and Rob Waldman. 99b. Posiiv Fdback Invsmn Sagis and Dsabilizing Raional Spculaion. Jounal of Financ 45, 379 395. Fama, Eugn, and Knnh Fnch. 988. Dividnd Yilds and Expcd Sock Runs. Jounal of Financial Economics, 3 5. Fus, Andas, Bnjamin Hb, and David Laibson.. Naual Expcaions, Macoconomic Dynamics, and Ass Picing. NBER Macoconomics Annual 6, 48. Gabaix, Xavi.. Vaiabl Ra Disass: An Exacly Solvd Famwok fo Tn Puzzls in Maco-Financ. Qualy Jounal of Economics 7, 645 7. Gnwood, Robin, and Andi Shlif. 3. Expcaions of Runs and Expcd Runs. Woking pap, Havad Univsiy. Hansn, Las, and Knnh Singlon. 98. Gnalizd Insumnal Vaiabls Esimaion of Nonlina Raional Expcaions Modls. Economica 5, 69-86. 49
Hansn, Las, and Knnh Singlon. 983. Sochasic Consumpion, Risk Avsion, and h Tmpoal Bhavio of Ass Runs. Jounal of Poliical Economy 9, 49-65. Hishlif, David, and Jianfng Yu. 3. Ass Picing wih Exapolaiv Expcaions and Poducion. Woking pap, Univsiy of Califonia and Univsiy of Minnsoa. Hong, Haison, and Jmy Sin. 999. A Unifid Thoy of Undacion, Momnum Tading, and Ovacion in Ass Maks. Jounal of Financ 54, 43 84. Ju, Nngjiu, and Jianjun Miao.. Ambiguiy, Laning, and Ass Runs. Economica 8, 559 59. Kushn, Haold. 967. Sochasic Sabiliy and Conol (Acadmic Pss, Nw Yok). LRoy, Sphn, and Richad Po. 98. Th Psn-Valu Rlaion: Tss Basd on Implid Vaianc Bounds. Economica 49, 555574. Lau, Main, and Sydny Ludvigson.. Consumpion, Aggga Walh, and Expcd Sock Runs. Jounal of Financ 56, 85 849. Lips, Rob, and Alb Shiyav.. Saisics of Random Pocsss I, Gnal Thoy, nd diion (Sping-Vlag, Nw Yok). Malmndi, Ulik, and Sfan Nagl.. Dpssion-babis: Do Macoconomic Expincs Affc Risk-aking? Qualy Jounal of Economics 6, 373-46. Mha, Rajnish, and Edwad Psco. Th Equiy Pmium: A Puzzl. Jounal of Monay Economics 5, 456. Mon, Rob. 97. Opimum Consumpion and Pofolio Ruls in a Coninuous- Tim Modl. Jounal of Economic Thoy 3, 373 43. Rabin, Mahw.. Infnc by Blivs in h Law of Small Numbs. Qualy Jounal of Economics 7, 775 86. Riz, Thomas. 988. Th Equiy Risk Pmium: A Soluion. Jounal of Monay Economics, 7 3. Shill, Rob. 98. Do Sock Pics Mov Too Much o b Jusifid by Subsqun Changs in Dividnds? Amican Economic Rviw 7, 4436. Timmmann, Alan. 993. How Laning in Financial Maks Gnas Excss Volailiy and Pdicabiliy in Sock Pics. Qualy Jounal of Economics 8, 35 45. Vissing-Jognsn, Ann. 4. Pspcivs on Bhavioal Financ: Dos 5
Iaionaliy Disappa wih Walh? Evidnc fom Expcaions and Acions, in NBER Macoconomics Annual, Volum 8, ds. Mak Gl and Knnh Rogoff, Cambidg, MA: MIT Pss. Wach, Jssica. 3. Can Tim-Vaying Risk of Ra Disass Explain Aggga Sock Mak Volailiy? Jounal of Financ 68, 987 35. 5
Tabl : Slcd Modls of h Aggga Sock Mak TRADITIONAL Modl allows fo inmdia consumpion D/P pdics uns Accouns fo volailiy Accouns fo quiy pmium Accouns fo suvy vidnc Habi Campbll and Cochan (999) Ys Ys Ys Ys No Long-un isk Ra disass Bansal and Yaon (4) Ys No Ys Ys No Bansal, Kiku, and Yaon () Ys Ys Ys Ys No Riz (988), Bao (6) Ys No No Ys No Gabaix (), Wach (3) Ys Ys Ys Ys No LEARNING Timmman (993) Ys Ys Ys No No BEHAVIORAL Pfnc-basd Pospc hoy Babis, Huang, and Sanos () Ys Ys Ys Ys No Ambiguiy avsion Ju and Miao () Ys Ys Ys Ys No Blif-basd Nois ad isk D Long al. (99a) Ys Ys Ys No No Exapolaion of fundamnals Exapolaion of uns Babis, Shlif, Vishny (998) No Ys Ys No No Choi (6) Ys Ys Ys No No Fus, Hb, Laibson () Ys Ys Ys No No Ali and Tlock (3) No Ys Ys No No Hishlif and Yu (3) Ys Ys Ys No No Cul, Poba, Summs (989) No Ys Ys No Ys D Long al. (99b) No Ys Ys No Ys Hong and Sin (999) No Ys Ys No Ys Babis and Shlif (3) No Ys Ys No Ys Babis, Gnwood, Jin, and Shlif (3) Ys Ys Ys No Ys 5
Tabl : Paam Valus Th abl pos h valus w assign o h isk-f a ; h p uni im man g D and sandad dviaion D of dividnd changs; h isky ass p-capia supply Q; absolu isk avsion ; h discoun a ; h popoion of aional ads in h conomy; h paams, and which govn h blifs of xapolaos; h iniial lvl of h dividnd D ; and h iniial walh lvls, and, of xapolaos and aional ads, spcivly. Paam Valu.5% g D.5 σ D.5 Q 5. δ.5% {.5,.5,.75, } β {.5,.5,.75} λ λ D 5 5 53
Tabl 3: Pdiciv Pow of D/ P fo Fuu Sock Pic Changs Th abl pos h populaion sima of h gssion cofficin whn gssing h pic chang fom im o im k (in quas) on h im lvl of D/ P fo k =, 4, 8,, and 6, and fo vaious pais of valus of h paams μ and β: P P ab( D / P) ε. k k Th calculaions mak us of Poposiion 3 in h main x. β k.75.5.5 -.4.6. 4 -.55.64.85.5 8 -.6.4.6 -.55.8.33 6 -..33.98 -.34.6.9 4 -.438.54.68.5 8 -.684.754.86 -.8.878.948 6 -.9.94.98 -.94.3.3 4 -.579.65.774.75 8 -.8.879.949 -.95.958.988 6 -.968.985.997 54
Tabl 4: Auocolaions of P D/ Th abl pos h auocolaion of P D/ a vaious lags k (in quas) and fo vaious pais of valus of h paams and. Th calculaions mak us of Poposiion 4 in h main x. β k.75.5.5 -.986.984.978 4 -.945.936.95.5 8 -.894.876.838 -.845.8.767 6 -.799.767.7 -.866.839.78 4 -.56.496.37.5 8 -.36.46.39 -.78..5 6 -..6.9 -.86.768.689 4 -.4.348.6.75 8 -.78..5 -.75.4. 6 -.3.5.3 55
Tabl 5: Volailiy of Pic Changs and Volailiy of P D/ Panl A pos h volailiy of annual pic changs fo vaious pais of valus of h paams μ and β; Panl B pos h volailiy of P D/, masud a an annual fquncy, fo vaious pais of μ and β. Th calculaions mak us of Poposiion 5 in h main x. Panl A: Volailiy of Annual Pic Changs.75.5.5.5. 3.5 7.43.5.7 3.3 6.86.75.4.67 5.9 Panl B: Volailiy of Annual P D/.75.5.5.5. 3.9 7.53.5.3 3.4 7.77.75.5 3. 7.9 56
Tabl 6: Auocolaions of Pic Changs Th abl pos h auocolaions of qualy and cumulaiv sock pic changs a vaious lags k (in quas) and fo vaious pais of valus of h paams μ and β. Th calculaions mak us of Poposiion 6 in h main x. Auocolaions a Hoizon k Cumulaiv Auocolaions o Hoizon k k.75.5.5.75.5.5 -. -.3 -.7 -. -.3 -.7 -. -.3 -.7 -. -.5 -..5 3 -. -.3 -.7 -. -.6 -.3 4 -. -.3 -.7 -.3 -.7 -.5 8 -. -.3 -.6 -.4 -. -..5.75 -. -.3 -.6 -.5 -. -.4 -.6 -.38 -.79 -.6 -.38 -.79 -.3 -.3 -.6 -. -.5 -.3 3 -. -.7 -.48 -.4 -.58 -.8 4 -. -. -.38 -.6 -.63 -.7 8 -.6 -. -.4 -.9 -.7 -.38 -.3 -.6 -.5 -.9 -.69 -.34 -. -.54 -. -. -.54 -. -.8 -.4 -.76 -.9 -.69 -.4 3 -.4 -.3 -.53 -.3 -.77 -.54 4 -. -.4 -.36 -.34 -.8 -.6 8 -.5 -.8 -.8 -.35 -.83 -.59 -. -.3 -. -.33 -.77 -.47 57
Tabl 7: Consumpion, P D/, and Pic Changs Panl A shows h colaion bwn qualy changs in consumpion and qualy changs in pic; Panl B shows h colaion bwn annual changs in consumpion and annual changs in pic; Panl C shows h colaion bwn h annual chang in consumpion and P D/. Th calculaions mak us of Poposiions 7 and 8 in h main x. Panl A: Colaion bwn qualy consumpion changs and qualy pic changs.75.5.5.5.994.985.984.5.99.84.84.75.93.794.79 Panl B: Colaion bwn annual consumpion changs and annual pic changs.75.5.5.5.994.985.984.5.947.878.876.75.935.853.849 Panl C: Colaion bwn qualy consumpion changs and P D/.75.5.5.5 -.5.48.48.5 -.436.398.49.75 -.54.446.456 58
Tabl 8: Pdiciv Pow of Changs in Consumpion fo Fuu Pic Changs Th abl pos h populaion sima of h gssion cofficin whn gssing h pic chang fom im o im k (in quas) on h mos cn qualy consumpion chang fo k, 4, 8,, 6, and fo vaious pais of valus of h paams and : P P ab ( C C ) ε. k k Th calculaions mak us of Poposiion 9 in h main x. k.75.5.5 -. -.6 -.53 4 -.43 -. -.5.5 8 -.84 -.95 -.393 -.3 -.84 -.565 6 -.59 -.366 -.7 -.7 -.4 -.44 4 -.35 -.67 -.66.5 8 -.547 -.3 -.738 -.658 -.68 -.94 6 -.7 -.5 -.979 -.44 -.7 -.55 4 -.49 -.759 -.375.75 8 -.69 -.3 -.686 -.686 -.5 -.756 6 -.78 -.47 -.77 59
Tabl 9: Equiy Pmia and Shap Raios Panl A pos quiy pmia fo vaious pais of valus of h paams and ; Panl B pos Shap aios fo vaious pais of and. Th calculaions mak us of Poposiion in h main x. Panl A: Equiy Pmia.75.5.5.5.5.58.9 3.9.5.5.65.46 4.88.75.5.66.48 4.9 Panl B: Shap Raios.75.5.5.5.5.4.65..5.5.43.73.33.75.5.43.7.7 6
Tabl : Modl Pdicions fo Raio-basd Quaniis Th abl summaizs h modl s pdicions fo aio-basd quaniis. A full dscipion of hs quaniis can b found in Scion 5. of h main x. Th valus of h basic modl paams a givn in Tabl, and (h facion of aional ads) is.5. Fo.5,.5, and.75, w po simas of ach quaniy avagd ov, simulad pahs. In ows (), (8), and (9), w po boh a gssion cofficin and, in panhss, an R- squad. Th igh column shows h mpiical simas fo h pos-wa piod fom 947- (95- fo consumpion-lad quaniis bcaus nonduabl consumpion daa a availabl only fom 95). quaniy of ins () pdiciv pow of log(d/p).5.5.75.9 (.).46 (.).45 (.) pos-wa U.S. sock mak daa. (.8) () auocolaion of P/D.93.84.85.94 (3) xcss volailiy of uns.3.9.97 - (4) xcss volailiy of P/D 7. 4.85 4.55 - (5) auocolaion of log xcss un (k = ) -. -.9 -.4. auocolaion of log xcss un (k = 8) -. -. -. -. (6) colaion bwn Δ 4 c and.7.54.47.3 (7) colaion bwn suplus consumpion and P/D.6.33.7. (8) pdiciv pow of h suplus consumpion aio (9) pdiciv pow of log(c/w) -.7 (.5).5 (.5) -.89 (.8). (.5) -.77 (.7). (.5). (.3) -.77 (.9) () quiy pmium.6%.6%.64% 7.97% Shap aio..4.3.44.33 (.5) 6