Forgotten Books...UNDER ASYMMETRIC INFORMATION J iang Wang Alfred P . S loan School of Management...
Transcript of Forgotten Books...UNDER ASYMMETRIC INFORMATION J iang Wang Alfred P . S loan School of Management...
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A MODEL OF INTERTEMP ORAL AS S ET PRI CE SUNDER ASYMMETRI C INFORMATI ON
J i ang Wang
Al fr ed P . S l oan S cho o l o f Manag ement
Mas sachus e t t s In s t i tute o f T e chno l ogy50 Memor i a l Dr ive
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A MODEL OF INTERTEMP ORAL AS S ET PRICE SUNDER ASYMMETRIC INFORMATION
J i ang Wang
Al fr ed P . S l oan S cho o l o f Manag ement
Ma s s a chus e t t s I n s t i tut e o f T e chno l ogy
50 Memor ia l Dr ive
Cambr id e , MA 0 2 13 9gLino B lé lj
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UNDER ASYMMETRIC INFORMAT ION
J i ang Wang
Al fr e d P . S l o an S cho o l o f Management
Mas s achus e t t s I ns t i tu t e o f Te chno l o gy50 Memo r i a l Dr ive
C amb r i dge , MA 02 139
F i r s t D r a f t : Novemb e r 198 9Th i s r ev i s i on : De c emb e r 1990
AB STRACT
Th i s p ap e r p r e s ent s a dynam i c a s s e t p r i c ing mode l unde r asymme tr i c
i nfo rma t i on . I nve s t o r s have d i f f e r ent info rma t i on c onc e rn ing the futuregr owth ra te o f d iv i dends . They r a t i ona l ly ex t rac t informat i on fr om p r i c e s a s
we l l as d iv i dends and max im i z e th e i r exp e c t e d ut i l i ty . The mode l ha s ac l o s e d - form s o lut i on to the r a t i o na l exp e c ta t i ons e qu i l ib r ium . We f ind tha t
e x i s t ence o f un info rme d inve s t o r s inc r e a s e s the r i sk pr em ium . S upp ly sh o cksc an a f fe c t the r i sk p r em ium only unde r as ymme tr i c info rma t i on . I nfo rma t i onas ymme try among inve s t o r s c an inc r e a s e p r i c e vo l a t i l i ty and ne ga t iveaut o c o rre l a t i on in r e turns . Le s s i nfo rme d inve s to r s may r a t i ona l ly b ehavel ike p r i c e C has e r s .
I am ve ry gr a te ful t o S anfo rd C r o s sman fo r h i s enc ouragement and many
s ugge s t i ons . Many thanks go t o Da r r e l l Duf f i e fo r va l uab l e c omment s . He l p fulc ommen t s and sugge s t i ons we r e ma de by Ana t Ama t i
,M i chae l Brennen
,J ohn
Campb e l l , Ge o rge Cons tantinides , Doug D i amond , Be rnard Dumas , Ph i l Dybvig ,Ch i - fu Huang , R i chard K ih l s t r om , Al an K l e i don , Alb e r t Ky l e , Andr ew Lo , G e o r ge
Pennac ch i , Paul P fl e i de re r , Kr i shna Rama swamy , Ri chard Ro l l , S tephen Ro s s , OiShen , Kenne th S ingl e t on , Sher iden T i tman and par t i c ip ants o f workshop s a t
Ch i c ago , Carne g i e - Me l l on , C o l umb i a , Duke , Harvard , London S cho o l o f
Ec onom i c s , M i ch i gan , M i nne s o t a , No r thwe s te rn , Oh i o S ta t e
P r inc e t on , Roche s t e r , S tanfo rd , Te l - Av iv , UCLA , Wha r t on and Yal e . P au lVa l ent in pr ov ided ab l e re s e arch a s s i s t anc e . Th i s pap e r i s a rev i s e d ve r s i ono f Chap te r 2 and 3 o f my Un ive r s i ty o f P ennsy lvan i a Ph . D . d i s s e r ta t i on and
wa s c i r cul a t e d e ar l i e r unde r th e t i t l e,
"As s e t Pr i c e s,S tock Re turns
,P r i c e
Vo l a t i l i ty , R i sk P r em ium , and Tr ad ing S tr a t e g i e s unde r Asymme tr i c
I nfo rma t i on . " Any e r r o r s ar e o f c our s e my own .
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I . INTRODUCT ION
In th i s p ap e r , we p r e s ent a gene ra l e qu i l ib r i um mo de l o f dynam i c as s e t
p r i c ing unde r a symme tr i c info rma t i on . We c ons i de r an e c onomy endowe d w i th a
g iven quant i ty o f e qu i ty . The d ividends on the e qu i ty gr ow a t a s t o chas t i c
r a t e . The inve s t o r s in the e c onomy can b e e i th e r i nf o rme d o r un informe d : the
info rmed inve s t o r s know the future d iv i dend gr ow th r a t e,wh i l e the un informed
inve s to r s do no t . Al l the inve s to rs ob s e rve c ur r en t d iv i dend p ayment s and
s t ock p r i c e s . S i nc e the gr owth ra te o f d iv i de nds de t e rm ine s the r a t e o f
appr e c i a t i on o f s t o ck p r i c e s , change s o f p r i c e s p r ov i de s i gna l s ab ou t the
fu ture gr ow th o f d iv idends . The un info rme d inve s t o r s r a t i ona l ly ex tr ac t
informa t i on ab out the s ta te o f the e c onomy fr om th e p r i c e s a s we l l a s
d iv i dends . Be c aus e we as s ume an inc omp l e te ma rke t s t ruc tur e , the s i gna l s do
no t ful ly r eve a l the t rue va lue s o f a l l the s ta t e va r i ab l e s . I n e qu i l ib r ium ,
i nve s to r s w i th a c c e s s t o d i f fe r ent info rma t i on w i l l an t i c ip a te d i f fe r ent
expe c t e d r e turns from inve s t ing in the s t o cks
We exp l o re the imp l i c a t i ons o f our mode l fo r the b ehav i o r o f s t o ck p r i c e s ,
r i sk pr em i a,p r i c e vo l a t i l i ty , autoc o r re l a t i on in s t o ck r e turns and inve s t o r s
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t rad ing s t r at e g i e s .
Unde r a symme t r i c info rma t i on,the un info rme d inve s t o r s l e arn ab out the
fu ture d iv idend gr owth ra te fr om d ivi dends and p r i c e s . The i r e s t ima t e o f the
d ivi dend growth ra t e w i l l a ffe c t the i r demand o f th e s t o cks , henc e the
e qu i l ib r ium p r i c e s . The e r r or s o f the un info rme d inve s t o r s in e s t ima t ing the
d iv i dend gr owth ra t e c aus e s p r i c e s t o dev i a te fr om th e i r fundamen ta l va lue s .
Howeve r,the fl ow o f d iv i dends p rov i de s a f l ow o f new info rma t i on ab out the
unde r ly ing growth ra te . The un info rme d inve s t o r s w i l l c ont inuous ly upda te
the i r e s t ima te b as e d on the newly arr ive d info rma t i on and c or r e c t the e rr o r s
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made in p r ev i ous e s t imat i on . The r e fo r e , the e s t ima t i on e r r or s o f the
un info rmed inve s to r s are only t emp o ra ry .
One app l i c a t i on o f our mo de l r evo lve s a round the p r ob l em o f e qu i ty p r em i a .
Fo r examp l e , Mehra and P re s c o t t ( 198 5 ) a r gue tha t ex i s t ing mode l s o f a s s e t
p r i c ing w i th i dent i cal inve s t o r s,p e r fe c t info rma t i on and t ime - add i t ive
p re fe rence s are inadequa t e i n exp l a i n ing the h i gh e qu i ty p rem i a ob s e rve d in
the U . S . s to ck marke t .2Ab e l ( 1 9 8 8 ) sugge s t s tha t he t e r o gene i ty i n i nve s t o r s
'
b e l i e f s may he l p us to unde r s t and the h i gh r i s k pr em i a . P r e s umab ly , i nve s t or s'
beb ief s are gene ra ted by the i n fo rma t i on they have . Ou r mode l p r ov i de s a
r a t i onal framewo rk to s tudy th e impac t o f d i ffe rent i a l i nfo rma t i on on r i sk
p rem i a . We show tha t the ex i s t enc e o f un info rmed inve s t o r s c an l e ad t o r i sk
p rem ia much h i ghe r than tha t unde r symme tr i c and pe r fe c t info rma t i on . When
mor e inve s to r s ar e l e s s info rme d in the e c onomy,p r i c e s w i l l c on ta in l e s s
info rma t i on ab out the fundame n t a l s o f the s to ck . Henc e , the re w i l l b e mo r e
unc e r ta inty c onc ern ing the fu tur e payo ffs fr om the s t o cks . H i ghe r p r em i a w i l l
b e demanded by the l e s s info rme d inve s t or s t o inve s t in the s t o cks . As the
frac t i on o f the un informe d inve s t o r s in the e c onomy inc r e a s e s the r i sk p r em i a
on s tocks a l s o inc rease s .
De Long et . al . ( 1990 ) have sugge s t ed tha t" no i s e t rad ing in th e marke t
( o r" supp ly shocks “ in the t e rm ino l o gy o f th i s pap e r ) c oul d inc r e as e th e pr i c e
vo l a t i l i ty,henc e the r i sk o f i nve s t ing in the s tock marke t and the r i s k
p r em i a . The ba s i c a s sump t i on tha t dr ive s the i r re sul t i s tha t s ome i nve s t o r s
have f ini te inve s tment ho r i z o n s . The cur rent mode l a s s ume s inf in i t e ho r i z on
fo r a l l inve s t or s . I t i s , the n , shown tha t the unc ond i t i ona l exp e c t e d e xc e s s
1S e e Gro s sman and Sh i l l e r Mehra and Pr e s c o t t
More gene ra l pre fe r enc e s have b e en s ugge s ted i n exp l a in ing the ob s e rvedp r em ium . S e e , for examp l e , Ep s t e in and Z in Cons tant inidesHe a t on ( 19 90 )
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r e turn from the s tocks on ly de p ends on the r i s k c onc e rn ing the s t o cks ' fu tur e
c a sh fl ows , wh ich we w i l l c a l l the fundament a l r i s k o f the s t o cks . Even though
no i s e tr ading doe s move p r i c e s and inceas e th e p r i c e vo l a t i l i ty,w i thou t
info rma t i on asymme try i t do e s no t change the fundamenta l r i s k o f th e s t o cks .
Howeve r,when informa t i on as ymme t ry i s p r e s en t in the e c onomy
,no i s e t r ad ing
w i l l a ffe c t the informa t i on qua l i ty o f p r i c e s in r eve a l ing p r iva t e i nfo rma t i on
ab ou t future cash fl ows . Mo r e no i s e t rad ing make s p r i c e l e s s info rma t ive ab out
the future d ividend growth . Th i s may inc r e a s e the unc e r ta inty o f futur e c a sh
f l ows to the l es s info rme d inve s t o r s,henc e th e r e qu i r e d the re turn
The r e fore,in our mode l only unde r info rma t i on a symme try , no i s e t r ad ing c an
a f fe c t the r isk prem i a
Many author s have ar gue d tha t a typ i c a l a s s e t p r i c ing mode l w i th i dent i c al
i nve s t o r s and pe rfe c t info rma t i on c anno t r e c onc i l e th e l ar ge vo l a t i l i ty o f
s t o ck p r i c es obs e rve d in the marke t w i th the h i s t o ry o f smo o th d iv i dends .3
Campb e l l and Kyl e ( 19 8 8 ) sugge s t tha t the ex i s t enc e o f no i s e t rad ing i n the
marke t c an help in exp l a in ing the h i gh vo l a t i l i ty i n s t o ck p r i c e s . I n our
mode l,we show tha t the imp e r fe c t i nfo rma t i on o f s ome inve s to r s w i l l c aus e
s t o ck p r i c es to be mor e vo l a t i l e,c ompar e w i th the c a s e when a l l inve s t o r s
have pe r fe c t informa t i on . Unab l e t o ob s e rve th e t rue gr owth r a t e o f d iv i dends ,
the un informed inve s to r s us e d iv i dends and p r i c e s t o make infe renc e s . They
w i l l r at i onal ly at tr ibute a change i n d iv i dend t o r e f l e c t par t i a l ly a change
in d ivi dend growth ra t e even when th e r e i s no change i n i t . Mo r e p r e c i s e ly ,
the innova t i ons to the un info rme d inve s t o r s ' e s t ima t e o f the d iv i dend gr ow th
ra t e w i l l be pos i t ive ly c o r r e l a t e d w i th d iv i dend sh o cks . Th i s o f c our s e w i l l
fe e d back to the pr i c e s thr ough the i r demand . Henc e , s ma l l change s in
3S e e LeRoy and P o r t e r ( 19 8 1 ) and Sh i l l e r S e e a l s o Fl av in
Kl e i don Marsh and Me r t on Mank iw , Rome r and Shap i r o Fo r
mor e re c ent work , s e e C ampb e l l and Sh i l l e r We s t ( 19 8 8 )
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d iv i dends c an c aus e l ar ge change s in p r i c e s . Howeve r , we have al s o found tha t
the innova t i on var ianc e o f p r i c e s doe s no t a lway s monotonocally inc re as e wi th
the frac t i on o f uninforme d inve s tors . I n o the r wo r ds,add ing mo re informed
inve s to rs t o the e c onomy c an s ome t ime s des tab lize the p r i c e s and inc re as e the
p r i c e vo l a t i l i ty . Th i s i s due t o the e f fe c t o f a s ymme t r i c info rma t i on . As
info rmed inve s t o r s are in t r oduc ed into the e c onomy,mo r e informa t i on ab out the
fundamental s w i l l be ava i l ab l e t o the marke t th r ough th e demand o f the
info rmed inve s to r s and th e p r i c e . Th i s shou l d h ave th e e f fe c t o f reduc ing the
p r i c e vo l a t i l i ty . At the s ame t ime , info rma t i on a s ymm e t ry int roduc e s
heterogeity among the inve s t o r s . Les s info rma t i on w i l l b e r eve al e d by the
p r i c e s ab out non - fundamen t a l s , such as the l eve l o f s up p ly shocks . Th i s c an
c aus e p r i c e t o ac tual ly b e c ome more vo l a t i l e
Rec ent emp i r i c al s tud i e s s ugge s t tha t the r e may e x i s t s i gn i f i c ant ne ga t ive
s e r i a l co r re l a t i on in l ong ho r i z on s to ck r e turns .4
A s s um ing c e r ta in
s ta t i onar i ty o f the e c onomy , we can have s t o ck r e turns ne ga t ive ly s e r i a l ly
c o rr e la ted due t o mean r eve r s i on in s ome underlving s t a t e var i ab l e s tha t
a f fe c t s the expe c ted exc e s s re turns . Howeve r , whe n i nf o rma t i on asymme try i s
p r e s ent,the l e s s info rme d inve s tors c an on ly l e a rn ab ou t the s e s ta te
var i abl e s fr om re al i z e d r e turns . Th i s w i l l inc r e a s e th e dep endenc e o f the i r
e xp e c ted futur e re turn on p a s t re turns and gene r a t e a dd i t iona l ne ga t ive s e r i al
c o r r e la t i on in s tock re turns
We al s o app ly our mode l t o ana lyz e the t rad i ng s t r a t e g i e s tha t are us e d by
inve s tor s w i th d i ffe rent info rma t i on s e t s . When i nf o rma t i on a symme try i s
p r e s ent,the i nfo rme d inve s t o r s hedge aga ins t no t on l y p o s s ib l e change s in the
unde r ly ing s ta t e var i ab l e s but a l s o the r e a c t i o n o f the un info rme d inve s t o r s .
AS e e Fama and Fr ench Poterba and S umm e r s Lo and
MacKinlay Al s o s e e R i chards on ( 19 8 8 )
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I n o th e r wo r ds , they take advantage o f the e r r o r s made by the l e s s i nfo rmed
inve s t o r s t o make p r o f i t s . The un informe d inve s t o r s t rade b a s e d on the i r
info rma t i on ex tr ac t e d fr om p r i c e s and d iv i dends . We f ind tha t i n s ome c as e s,
the un info rmed inve s t o r s r a t i ona l ly adop t tr ad i ng s t r a t e g i e s tha t l o ok l ike
p r i c e cha s i ng : they buy s t o cks when p r i c e s go up and s e l l when p r i c e s go down .
A numbe r o f s tud i e s p r e s e n t ev i denc e c ons i s t en t w i th th i s typ e o f t r e nd
cha s ing s t r a t e gy .5
Our r e s u l t s show tha t t r end - chas ing l ike b ehav i o r c an be
r a t i ona l fo r the l e s s info rme d inve s t o r s unde r a symme t r i c inf o rma t i on wh i l e
the inf o rme d inve s t o r s b ehave as c ontra r i ans .
The r e a r e two maj o r ob s ta c l e s in tack l ing the p rob l em o f dynam i c a s s e t
p r i c ing unde r a symme t r i c info rma t i on . One i s the no t i on o f " no t r ad i ng " and/o r
ful ly r eve a l i ng ,6
and th e o the r i s the ma thema t i c a l d i f f i c u l ty invo lve d in
s o lv ing th e s ys tem . The no - t r ad ing the o r em s ta t e s tha t i f a s ymme t r i c
informa t i on i s the only mo t iva t i on fo r t r ad ing , then by h i s w i l l i ngne s s t o
t rade an inve s t o r r eve a l s h i s info rma t i on to the marke t . Henc e , i nf o rma t i on
a symme t ry w i l l b e e l im ina t e d i n e qu i l ib r ium and no tr ade w i l l a c tua l ly o c cur
a s new inf o rma t i on c ome s in . Th i s r e sul t c ruc i a l ly dep ends on th e marke t
s t ruc tur e o f the e c onomy ( s e e , e . g . , Gr o s sman ( 19 7 7 , Unde r i nc omp l e t e
marke t s,the r e c an b e mo t iva t i ons o the r than the ar r iva l o f new i nf o rma t i on
tha t c aus e inve s t o r s t o t r ade and the ar gument for the i r r e l evancy o f d ive r s e
i nforma t i on b r e aks down . When info rma t i on a s ymme try i s c oup l e d w i th an
inc omp l e t e marke t s t ruc tur e,the p r ob l em b e c ome s ma th ema t i c a l l y ve ry
i nvo lve d .7
The curr ent mode l,t o our knowl e dge , i s the f i r s t dynam i c a s s e t
5S e e
, Anderas s en and Kraus C a s e and Sh i l l e r Fr e nke l and
Froot ( 19 8 8 ) and Sh i l l e r S e e a l s o S o r o s
6S e e G r o s sman Milgrom and S t oky ( 19 8 5 )
7S e e Duf f i e and Huang ( 19 8 6 ) and He and P e ar s on ( 19 8 8 )
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p r i c ing mode l unde r as ymme t r i c info rma t i on wh i ch pr ov i de s a c l o s e d - fo rm
s o lu t ion
The fo rmal mode l i s sp e l l e d out i n s e c t i on I I . I n s e c t i on I I I , we c ons ide r
the trad i t i ona l s i tua t i on i n wh i ch i nve s t o r s have homogene ous and pe r fe c t
info rma t i on about the e c onomy . Th i s p r ov i de s a benchmark c as e fo r our e c onomy
A ra t iona l expe c ta t i ons e qu i l ib r i um o f the ful l mode l i s ob ta i ne d in s e c t i on
IV by s e quent i a l ly s o lv ing the p r ob l ems o f inve s to r s ' ra t i ona l l e arn ing,
op t im i za t i on and marke t e qu i l ib r ium . I n s e c t i on V,we d i s cus s the e qu i l ib r ium
p r i c e func t i on and how , unde r a s ymme t r i c i nfo rma t i on , pr i c e s dev i a t e fr om
the i r pe r fe c t informa t i on b e nchmark va lue . The p rob l ems o f r i sk p r emia and
p r i c e vo l a t i l i ty ar e addre s s e d in s e c t i on V I and VI I r e sp e c t ive ly . I n s e c t i on
V I I I,we s tudy the aut oc o r r e l a t i on i n s t o ck re turns . I n s e c t i on IX , we ana lyz e
the op t ima l inve s tment s t ra t e g i e s fo l l owe d by inve s t o r s w i th d i f fe r ent
info rma t i on . S ome fur the r c omment s a r e p r ov i de d in s e c t i on X. S e c t i on XI
c onc lude s
I I . THE ECONOMY
We c ons i de r a s imp l e e c onomy w i th a s ingl e phys i c a l good . The e c onomy i s
de f ined a s fo l l ows
c e r ta in amount o f r i sky cap i tal .
The informa t i on s t ruc tur e in th i s p ap e r i s s im i l ar to the onec ons i der e d by Towns end S ing l e t on Car i no ( 198 6 ) c ons i de re dmode l s in wh i ch al l p r iva te i nf o rma t i on b e c ome s pub l i c a fte r a sho r t p e r i o d
( one or two pe r i od ) s o tha t e f fe c t ive l y the l e s s info rmed inve s t o r s' l earn ing
p r ob l em b e c ome s s ta t i c . I n th i s p ap e r , we al l ow pr iva t e informa t i on to rema inp r iva te . There fore , the l e s s i nfo rme d i nve s t o r s
' l e a rn ing pr ob l em b e c ome sdynam i c wh i ch w i l l gene r a t e in t e r e s t ing r e s ul t s conserning op t ima l inve s tmen ts t ra t e gi e s and equi l ib r ium p r i c e s
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Ea ch un i t o f the ri s ky c ap i ta l gene r a t e s a f l ow o f ou tput ( d iv idend ) a t an
ins tant ane ous ra te D . D i s gove rned by th e d i f fus i on p r o c e s s
dD aD(p -kD ) d t+a dwD D
whe r e p i s a s ta te var i ab l e fo l l ow ing an Orns t e in - Uh l e nb e ck (O - U ) p roc e s s
du a (b -p ) d t+o dw71 P 1“
He r e , wD
,w”
are two indep ende n t s t anda rd W i ene r p r o c e s s e s and aD
,a b
,k
,
0D ’
a
“ar e p o s i t ive c ons tant s .
C l e ar ly , aD(p -kD ) g ive s the expe c t e d gr ow th r a t e o f d iv i dends . When k>0 ,
u/k c an b e inte rp re t e d as the shor t - run s t e ady s ta t e l eve l fo r the d iv idend
r a t e D,and i t f luc tua te s a r ound a l ong - run s t e ady s t a t e l eve l b/k . (D , p ) i s
a Gaus s i an Markov p r oc e s s .
0 s a t i s f i e s
the s t o chas t i c d i ffe r ent i a l e qua t i on
d 9 - a00dt+a
odw9
,
W i th th i s fo rm o f the d iv i dend p r o c e s s , D c an take ne ga t ive value s .
Howeve r,one c an make the p r ob ab i l i ty o f D r e ach ing ne ga t ive va lue s be as
sma l l a s one l ike s by cho o s ing the p arame t e r va lue s . One inte rp re ta t i on forne ga t ive d ividends i s tha t i nve s to r s have t o put b a c k s ome inve s tments toma in t a in the future c a sh f l ow .
10An e qu ival ent inte rp r e ta t i on to the s t o cha s t i c supp ly i s the ex is tenc e
o f no i s e t rade rs w i th the i r demand o f th e s t o ck b e in g - 0 .
-
whe re ao
, 00
are p o s i t ive c ons tants and we
i s a s tanda rd W i ene r p ro c e s s . For
s imp l i c i ty , we as s ume tha t we
i s unc orre la te d w i th wD
and w .11
By A2 , the to ta l s upp ly o f the r i sky c ap i ta l i s a s s ume d t o b e s to chas t i c .
I t ha s a l ong - run s ta t i ona ry l eve l wh i ch i s no rma l i z e d t o l . 9 g ive s the
dev ia t i on o f the cur r ent s upp ly from i t s l ong - run s ta t i onary l eve l.
p e r fe c t ly d ivi s ib l e shar e o f the s t ock he l d by the inve s t o r s in the e c onomy
Shar e s are trade d in a c omp e t i t ive s t o ck marke t w i th no trans ac t i ons c o s t s .
The s t o ck i s the only s e cur i ty that i s traded in the ma rke t . Le t S b e the
equ i l ib r ium p r i c e o f the s t o ck .
e c onomy w i th a c ons tant ra t e o f re turn 1+r Al l i nve s t o r s have ac c e s s t o
the s to rage t e chno l o gy a t no c o s t .13
I n orde r to have inc omp l e t e marke t s,we have intr o duc e d an add i t i ona l s ta te
var iab l e by as s um ing the aggr e ga te quant i ty o f the r i s ky c ap i ta l to b e
s t o cha s t i c . Th i s i s only fo r th e s imp l i c i ty in expo s i t i on . The inc omp l e t ene s s
o f the marke t can b e mode l e d i n a ra t i ona l,s e l f - c ont a i ne d fr amework w i thout
11The indep endenc e among al l the sho cks t o the e c onomy i s as s ume d
pur e ly for a l geb ra i c s imp l i c i ty . One c an e a s i ly ex t end the mode l to the c as eo f mor e gene ra l c o r r e l a t i on s t ruc ture s .
12We w i l l no t a l l ow fr e e d i sp o s a l o f the s t ocks . The re fo r e , the share s
in th i s mode l do no t have l im i t e d l i ab i l i ty . Th i s woul d a l l ow the p r i c e o f thes t o ck to take ne ga t ive va lue s .
1 3A gene ral e qu i l ib r ium J us t i f i c a t ion fo r a c ons t ant inte re s t ra t e in an
e c onomy wi th CARA p r e fe r enc e s i s g iven by Sunda re s an ( 19 8 3 )
-
mak ing ad ho c a s sump t i ons ab out the s t o ck s upp ly o r inve s to r s ' b ehav i o r . In
p a r t i cul ar , Wang ( 1 9 8 9 , 19 9 0 ) has gene r a t e d the s ame re sul t by a s s um ing
ra t i ona l inve s t o r s , f ix e d s t o ck s upp ly and p o s s ib i l i t i e s o f r i s ky p r oduc t i on .
We have a l s o r e s tr i c te d th e shar e s o f e qu i ty to be the only s e cur i ty tr ade d in
th e marke t .14
i nfo rme d and the un info rme d . The info rme d inve s to r s have pe r fe c t p r iva t e
i nfo rma t i on ab out the s t a t e var i ab l e s y and 0 , wh i l e the unin fo rme d inve s t o r s
have no p r iva t e info rma t i on ab out them . Al l inve s t o r s ob s e rve the p r i c e o f the
s t o ck and d iv i dends . Henc e , i s the info rme d inve s t o r s'
in fo rma t i on s e t a t t ime t and i s the un informe d inve s t o r s '
i nfo rma t i on s e t . Le t the f r a c t i on o f the un informed inve s to r s b e w .
inve s t o r s in the e c onomy .
By AS , we have as sume d tha t inve s t o r s' in fo rma t i on ab out the s t a t e
15var i ab l e s c an b e imp e r fe c t . I n a dd i t i on , we a l l ow inve s to r s t o d i f fe r in
the i r informa t i on ab out the e c onomy . G iven the s truc tur e o f the marke t s and
th e e c onomy, 8 and D w i l l no t b e s uf f i c i e nt to reve al the va lue s o f b o th u and
0 t o the un info rme d inve s t o r s . I nfo rma t i on asymme try w i l l p e r s i s t i n the
e qu i l ibr ium .
14Th i s a s s ump t i on make s the c ap i ta l marke t dy nam ic al l y inc omp l e t e in the
s ens e o f Harr i s on and Kr e p s
15Fo r e ar l i e r wo rk on the op t ima l p o r t fo l i o cho i c e and as s e t p r i c e s
unde r imp e r fe c t info rma t i on,s e e
, e . g . , Me r ton Do than and Fe l dman
De temp l e Gennot te
-
o rde r t o maximi z e exp e c ted ut i l i t i e s c ond i t i one d on the i r informa t i on s e t s
Al l inve s to r s have p r e fe r enc e s exh ib i t ing c ons tant ab s o lute
r i sk ave r s i on ( CARA )'16
' eXP {-P t
whe r e p i s the t ime impa t i enc e pa rame t e r and c ( t ) i s the c ons ump t i on r a t e a t
t ime t .
The CARA pre ferenc e s ar e as s ume d s o tha t a c l o s e d - fo rm s o lut i on t o the
mode l c an be ob ta ined . Wi th CARA p r e fe r enc e s,an inve s t o r 's a s s e t demand w i l l
b e i ndep endent of h i s wea l th .17
Th i s imp l i e s tha t the e qu i l ib r ium p r i c e o f the
s t o ck we l l be indep endent o f th e we a l th d i s tr ibu t i on o f the inve s to r s a s we l l
a s the l eve l o f aggr e ga t e wea l th . Th i s indep endence gr e a t ly s imp l i f i e s our
p r ob l em .
I I I . THE BENCHMARK CAS E : UNDER P ERFECT INFORMAT I ON
Be fore we s o lve the ful l mode l s pe c i f i e d i n the p r ev i ous s e c t i on , l e t us
f i r s t c ons ider the sp e c ia l cas e in wh i ch a l l inve s to r s ar e info rmed ( i . e .
16He re
,we have as sume d a l l inve s t o r s to have th e s ame r i sk ave r s i on and
the Arrow - Pratt measure to be 1 . Th i s i s b e c aus e we want to pur e ly fo cus onthe e f fe c t of informa t i on on s t o ck p r i c e s . Th i s a s sump t i on can b e e as i lyr e l axe d as long as we rema in in the CARA c l a s s . See the d i s cus s i on l a t e r .
1 7W i th CARA ut i l i ty
,ne ga t ive cons ump t i on and ne ga t ive we a l th ar e
p o s s ib l e . In th is pap e r , we do no t impo s e non - ne ga t iv i ty c ons t ra int t o rul eou t ne ga t ive weal th .
-
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-
12
the d i s c ount r ate . Inde e d, 03
004 6490
r ep r e s e nt s th e d i s c ount on the
p r i c e o f the s tock to c omp e ns a te for the r i sk in i t s fu ture c ash f l ow . I t
i nc r e as e s w i th 0 , the supp ly sh ock o f the r i sky cap i t a l , b e c aus e 6 de t e rm ine s
the t o ta l amount o f aggr e ga t e r i sk the e c onomy i s e xp o s e d to . As 0 inc r e a s e s ,
e ach inve s t o r has to b e ar mo r e o f the marke t r i sk i n e qu i l ib r i um by ho l d i ng
mo r e s tocks . The pr i c e o f the s t ock has t o adj us t t o g ive h i ghe r exp e c t e d
re turn in orde r to induc e i nve s t or s t o inve s t mor e i n th e s t o ck
G iven the e qui l ibr i um p r i c e o f the s t o ck,the c ond i t i ona l exp e c t e d e xc e s s
r e turn to a share o f s t o ck a t t ime t i s
Et[ dS+Ddt - rSdt ] /dt - rs
OO( 3 2 )
The unc ond i t i onal exp e c t e d e xc e s s re turn i s e E - rs e00 00
dep ends on ly on a00
and a”
wh i ch charac te r i z e the fundamen ta l r i sk a s s o c i a t e d w i th fu tur e c ash
D
fl ows o f the s tock . No t e tha t e i s indep endent o f a00
The r e fo r e,the s i z e o f
9.
the s upp ly shocks doe s no t a f f e c t the unc ond i t i ona l e xp e c te d exc e s s r e turn ,
even though i t doe s a ffe c t the p r i c e vo l a t i l i ty . Th i s r e s ul t c ont r a s t s w i th
the r e sul t o f De Long et . al . I t i s a s sumed in D e Long et . al . tha t
r a t i onal inve s t or s have f in i t e ho r i z ons and have to l i qu i da te th e i r p o s i t i on
a t the end o f the i r l i fe t ime . S inc e supp ly shocks w i l l c aus e t emp o r ary sh i f t s
in the pr i c e from i ts fundame n tal value " , inve s t o r s w i th f in i t e l i fe t ime w i l l
fac e addi t i ona l r i sk in p r i c e due to supp ly shocks whe n they have t o l i qu i da t e
the i r s tock ho l d ing . Henc e , th ey wi l l demand ex tra p r em ium fo r th e add t ional
r i sk in l i qu i da t i on pr i c e s due t o supp ly shocks . Fo r i nve s t o r s w i th l onge r
ho r i z ons,the e ffec t o f the add i t i ona l r i s k o f l i qu i da t i on p r i c e s due t o
supp ly shocks w i l l be sma l l e r . When inve s t or s have inf i n i t e ho r i z ons , the
unc ond i t i ona l expec ted exc e s s r e turn i s de te rm ine d only by the fundamen ta l
-
r i sk o f the s tock wh i ch i s th e r i sk in i t s futur e c a sh f l ows . Mo re vo l a t i l e
p r i c e s caus ed by t emp o ra ry sh o cks in supp ly do e s no t change the r i sk o f
inve s t ing in the s to ck and th e r i s k p r em ium . As we w i l l s e e l a t e r , th i s i n no
l onge r the cas e unde r asymme t r i c i nfo rma t i on .
The innova t i on var i anc e o f th e s tock p r i c e the var i anc e o f
ins tantaneous s t o ck r e turn ) i s a c ons tant g iven by
2 2 2 2 2 2 2
S OsDO
aD+s +5 0apo
op ao o
A c ons tant var i anc e o f p r i c e s imp l i e s tha t the var i anc e o f the p e r c e nt a ge
r e turns increas e s a s the p r i c e o f the s t o ck de c r e a s e s and vi c e ve r s a ( s e e
Campb e l l and Ky l e Th i s phenomenon has b e en no t e d in the U . S . s t o ck
marke t da ta ( s e e B l ack Ne l s on
When both the info rme d and th e un inf o rme d inve s t o r s ar e p r e s en t i n th e
e c onomy , the s i tua t i on b e c ome s mo r e c omp l i c a t e d . Howeve r , we may exp e c t tha t
the e qui l ibr ium p r i c e w i l l s omewha t r e s emb l e the func t i onal fo rm o f the ab ove
b enchmark cas e due t o the unde r ly ing l ine ar i ty in th e mode l .
IV . EQ UILIBRIUM OF THE ECONOMY
I n th i s s e c t i on , we s o lve fo r the e qu i l ib r ium o f th e e c onomy de s c r ib e d i n
s e c t i on I I . The e qu i l ib r ium c onc ep t i s tha t o f the r a t i ona l e xp e c t a t i o ns
deve l ope d by Luc as G r e e n Gr o s sman and Krep s In
s uch an equi l ibr ium,i nve s to r s max im i z e the i r e xpe c t e d ut i l i t i e s ba s e d on
th e i r informat i on,the marke t s c l e a r
,and inve s t o r s ' exp e c ta t i ons ab ou t th e
d i s tr ibut i ons o f a l l the random var i ab l e s a re s e l f - fu l f i l l ing
The equ i l ibr ium s to ck p r i c e dep ends on a l l the s ta t e var i ab l e s o f th e
e c onomy . The pr imary s ta te va r i ab l e s a r e the cur rent d iv i dend ra t e D , th e
-
shor t - te rm s ta t i onary l eve l o f the d iv i dend p and th e aggr ega te supp ly shock
o f the s t ock 9 . Howeve r , when informa t i on a s ymme t ry i s p r e s ent in e qu i l ib r ium
the s e p r imary s ta te var i ab l e s ar e no t suf f i c i e n t t o ful ly de s c r ibe the s ta te
o f the e c onomy . The un informe d inve s t o r do no t d i r e c t l y ob s e rve p and 0 . They
c an only us e d ividends and p r i c e s to infe r the t rue va l ue s o f u and 0 . The i r
demand o f the s t ock w i l l dep end on the infe r r e d va l ue s ( e s t ima te s ) o f y and 9
The re fo r e,the ex i s tenc e o f the un info rme d inve s t o r s i nt r oduc e s new s ta te
var iab l e s into the e c onomy,such as the e s t ima t o r s they us e . We c a l l the s e new
s ta te var i ab le s induc e d s ta te var iab l e s
The way we ob ta in an e qu i l ib r ium o f the e c onomy i s s im i l ar in e s s ence t o
tha t o f Gro s sman ( 19 7 6 ) and o ther s in a s ta t i c s e t t i ng .2 1
We f i r s t c onj e c ture
an e qui l ib r ium p r i ce func t i on . Bas ed on the as s ume d p r i c e func t i on , we s o lve
th e inve s t o r s ' l ea rn ing and op t im iz a t i on p r ob l em . Ma rke t c l e ar ing i s then
imp o s e d to ver i fy the c onj e c ture d pr i c e func t i on .2 2
We c an wr i te whe re " o " deno t e s the i nduc e d s ta t e var i ab l e s .
The ob s e rva t ions o f the p e r fe c t informa t i on c a s e i n th e p rev i ous s e c t i on
sugge s t the fol l ow ing p r op o s i t i on .
exp e c ta t i ons equ i l ib r ium . The e qu i l ibr ium p r i c e func t i o n has the fo l l owing
l ine ar form
AS ( 5 +5 0 ) s
n(u- p ) ,
S e e,for examp l e
,He l lw i g D i amond and Ve r r e c ch i a ( 198 1 ) Amat i
The uniquene s s o f e qu i l ib r ium w i l l no t b e d i s c us s e d in th i s pap e r .
-
Awhe r e y i s the Ka lman f i l te r fo r u us e d by th e un info rme d inve s t o r s
bas e d on the i r i nfo rma t i on s e t It
’
Th e r e s t o f th i s s e c t i on w i l l p rov ide a p r o o f o f th e PROPO S I T I ON fo l l ow ing
the app r o ach s p e c i f i e d e ar l i e r
Be c aus e an un info rmed inve s t o r 's demand o f the s t o ck dep ends on h i s
e s t ima t e s o f th e unob s e rve d s t a te var i ab l e s, ai s in th e p r i c e func t i on a s an
induc e d s ta te var i ab l e . By the s ame token , one m i gh t exp e c t 9 , the f i l t e r for
9 , t o b e a l s o i n the p r i c e func t i on . Howeve r , th i s i s unne c e s s a ry . D e f ine
s =s - s and §=s p+s77 flu go
9 . We have the fo l l ow ing l emma :9
LEMMA G iven the ob s e rva t i on s o f D and S,the f i l t e r s o f an un in fo rme d
inve s t o r fo r p and 9 , Q and 9 r e s p e c t ive ly , s a t i s fy th e c ond i t i o n
9 s up+ sa9 , o r s
“(u-fi) - s
e( 9 (a 3 )
u 9
Pr o o f : Ob s e rv ing D , S and know ing fl, an un info rme d i nve s t o r
c an de duc e 5 . Henc e , 5 i s in the informa t i o n s e t It
and
The r e fo r e, 9 i s func t i ona l ly r e l a te d t o th e s t a te va r i ab l e s
in The e s t ima t i on e r r o r s o f the two va r iab l e s a re
Fo r 5 , we have
d § [ a s (b -p ) - a s 9 ] d t a dw
Q . E . D
a l r e ady inc luded
l i ne a r ly r e l a te d .
-
16
2 2 2 2whe r e and dw
6i s a s t andard W i ene r p ro c e s s de f ine d i n App end ix A .
An un informe d inve s t o r l e a r ns ab ou t the va lue s o f p and 9 fr om h i s
ob s e rva t i ons o f D and S . H i s O p t ima l e s t ima tor s fo r u and 9 b a s e d on h i s
info rma t i on s e t a r e de r ived in Append i x A . The r e s ul t s a re
summar i z e d in the the o r em b e l ow :
th e f i l t e r s 2 and 9 s a t i s fy th e f o l l ow ing
s tochas t i c d i ffe rent i a l e qua t i o n 5 °
hpD ufi
9 9Dh9 §
dD -aD(fi-kD ) d t
A Adg- a
fls
u(b -p ) d t+a
9599 d t
whe r e hpD
hpfi
’h9D
h9§
ar e c ons tant s g iven in Append i x A . w , the
innova t i on p r o c e s s o f the f i l t e r s , i s a s tandard W i ene r p r o c e s s w i th r e sp e c t
D,S
to F ( t ) , the a - al geb r a gene r a t e d by It
' Fur the rmo r e , the info rma t i on
s trucuture gene rate d by Fw
( t ) i s e qu iva l ent to the one gene ra t e d by FD ’ S
( t )
As sp e c i f i e d in p de t e rm ine s the growth r a t e o f d iv i dends . No t
ob s e rv ing p,the un info rme d inve s t o r s ra t i ona l ly draw infe renc e ab out u from
d iv i dends and p r i c e s . They w i l l r a t i ona l ly a t tr ibut e a shock in d iv i dend t o
r e f l e c t par t i a l ly a sh i f t i n th e grow th ra te o f d iv i dends . Th i s g ive s r i s e to
the p o s i t ive c ontr ibut i on o f d iv i dend shocks t o innova t i ons o f Q , a s shown in
Thus, 0 eve n though dw and dw
yar e i ndep e ndent .
D
-
Fr om we ob s erve th a t 9 a l s o p o s i t ive ly r e s p onds to d iv idend shocks
Fo r the un informe d inve s t o r s , a p o s i t ive i nnova t i on i n D sugge s t s an inc r e as e
i n p . Condi t i one d the inc r e a s e i n u mus t b e o ffs e t by an
inc r e as e in 9 ( s inc e 50
i s ne ga t ive ) . Henc e , 0 a l though 9 and
D a r e indep enden t . The j o i n t e s t ima t i on o f p and 9 b a s e d on b o th D and S
gene r a te s the induce d c o r r e l a t i on b e twe en the two f i l t e r s and d iv i dends and
p r i c e s . The induc e d c o r r e l a t i o n be twe e n the un info rme d inve s t o r s ' e s t ima te s
and d iv idends and pr i c e s a r e ve ry imp o r tant in unde r s t and ing the b ehavi o r o f
s t o c k p r i ce s and re turns .
Fo r future c onven i enc e , de f i ne n=Q -p t o b e the un i n fo rmed inve s tor s ' e r ro r
i n e s t ima t ing p . As de r ive d i n Append ix A,
dn - annd t
- a
udwp} ’ (a 7 )
wh e r e a,7
i s a pos itve c ons t an t . Henc e , n f o l l ows a s t andard O -U p ro c e s s . The
fa c t tha t n i s me an - reve r t ing t o z e r o imp l i e s tha t th e e s t ima t i on e rro r o f the
un informe d inve s t o r s ar e on ly t emp orary . As a ma t t e r o f fac t , the c ont inuous
f l ow o f div idends as we l l a s change s in p r i c e s p r ov i de s a fl ow o f new
i nf o rma t i on about the unde r ly i ng growth r a t e o f d iv i de nds . The un informed
i nve s t or s c ons tant ly up da t e th e i r e s t ima t e s ab ou t u ( and 9 ) b a s e d on the newly
a r r ive d informa t i on and c o r r e c t the e r r o r s made in the i r p rev i ous e s t ima t i on .
S upp o s e tha t ther e has b e en a p o s i t ive sho ck in D wh i l e the re has be en no
sh o ck to p . No t ob s e rv ing u , th e un info rme d inve s t o r s w i l l ra t i ona l ly
a t t r ibute the d iv i dend sh o ck t o par t i a l ly r e f l e c t a h i gh p , henc e inc re as e
the i r e s t imate 3. Howeve r , th e futur e d iv i dend w i l l no t grow a s exp ec te d s inc e
u ha s no t inc reas e d . When new l eve l s o f D ar e r e a l i z e d , the un informed
-
Ainve s to r s W i l l r ev i s e the ir e s t ima te and l owe r p
,e l im ina t ing the e rror in
p rev i ous e s t ima t i on .
2 . Inve s tment oppo r tun i t i e s
AG iven th e p r o c e s s o f f i l t e r p , the s t o ck p r i c e f o l l ows the p r o c e s s
d S [ sDO
aD(p -p ) - s
€a09 - a
na ]d t a ws ,
2 2 2 2 2swhe r e as( sDO
huD
s
n) 0
D( 1 h
flfs
n) 05
’ and dwS
i s a s t andard W i ene r p r o c e s s
de f ine d in Append ix A . I n orde r to charac t e r i z e th e i nve s tment oppo r tun i t ie s
in the e c onomy,we c ons i de r a z e ro - weal th p o r t f o l i o l ong one share o f s to ck
ful ly f inanc e d by b or r ow ing a t the r i sk - fr e e r a t e . The ins tantane ous re turn to
th i s z e ro - wea l th p o r t fo l i o g ive s the exc e s s r e turn t o one share o f s t ock .
c a sh f l ow fr om the z e r o -we al th p or tfol io,deno t e d by Q , s a t i s f i e s the
fo l l ow ing s t o cha s t i c d i f fe rent i a l e quat i on
dQ (D - rS ) dt+dS [ eo
whe r e e - rs0 0
Henc e,the exp e c t e d exc e s s r e turn on the s t o ck onl y dep ends on 9 and q . The
l eve l o f aggre ga t e s to ck supp ly affec ts the ex c e s s r e turn on the s t o ck becaus e
i t de te rmine s the t o ta l r i sk expo sure o f the e c onomy . The sho r t - run s ta t i onary
l eve l o f d iv idend p doe s no t a ffe c t the exc e s s r e turn p e r s e , but 0 , the
un info rmed inve s t o r s ' e r ro r in e s t imat ing u , c an change the expe c te d exce s s
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-
x £0
£99 f
un,
2 2W he r e
2 24fn= [ - a
savon
- 05nvflnl/ ( ra
s
P r o o f : S e e App end ix B .
As we po inte d out e ar l i e r,th e s ta t e o f the e c onomy i s char ac t e r i z e d by four
s ta t e var iab l e s , H oweve r , the va lue func t i on o f an inf o rmed
inve s t or only dep ends on 9 and n s inc e they ar e the only s ta t e var i ab l e s tha t
a f fe c t h i s inve s tment opp o r tun i t i e s .
For the un informed inve s t o r , the opt imazat ion p r ob l em can , i n gene r al , be
mo r e c omp l ic a t e d . The un info rme d inve s t o r s ' comsumpt ion - inve s tmen t p o l i cy i s
gene ra l ly a func t i on o f the i r informa t i on s e t,wh ich i s the who l e h i s to ry o f
d iv i dends and p r i c e s . Howeve r , g iven the p r o c e s s e s a s sumed for th e s ta te o f
0
the e c onomy as we l l as th e p r i c e the info rma t i on s t ruc ture gene r a t e d by Ithas
an e qu ival ent rep re s ent a t i on wh i ch i s one gene ra ted by the W i ene r p r o c e s s w
n C O u 0 a 0 0
tha t innovate s the f i l te r s . The f i l t e r s a re s uff i c i en t s ta t i s t i c fo r It
' Us ing
th i s equival ent rep r e s ent a t i on o f the info rma t i on s t ruc ture , one c an re s ta t e
the un info rme d inve s tor s ' op t im i z a t i on p r ob l em as a s tandard Markov i an one
w i th the f i l te r s be ing th e e f fe c t ive s ta t e var iabl e s and the innova t i on
p r o c e s s gene ra t ing the dynam i c s . I t then formallv l o oks s im i l ar t o the
24In th i s pape r , we us e the fo l l ow ing no ta t i on : a =E [ dzdz
'It] ’ whe r e
z and z ' are two random va r i ab l e s .
-
2 1
info rme d i nve s to r s ' p rob l em . Thus , we have a s i tua t i on in wh i ch the S epara t i on
2 5P r inc ip l e app l i e s
We c an the re for e r e fo rmul a t e the un info rme d inve s t o r s ' op t im i z a t i on p r ob l em
a s fo l l ows
Ao Max -exp < - pc (w I J /d t (a 12 )
c* , x*t
it 9:
s . t . dW [ rW +x* (D - rS )
l im o,t-Nn
whe r e W i s h i s we a l th , x* h i s h o l d ing in the s t o ck , c* h i s c onsump t i on
A AJ (W h i s va lue func t i on . S inc e E [ n
tIt] =0 , f r om the
e xp e c t e d exce s s re turn o f an un informe d inve s t o r only dep ends on 9 . The
s o lu t i on t o program i s g iven in The o r em
12 ) has a s o lu t i on o f the form
- exp <-pc
- tu*
(4 13 1 )
whe r e
1 2v
5veg? +v
69+v
0. 2 )
The op t imal demand o f the s t o ck i s
f*
f*
9 a 14
2 5S e e Fl em ing and R i she l ( 19 7 5 ) S e e a l s o Fe l dman De temp l eGennotte
-
2 a 2whe r e f
0= [ eO
and fo= [
P r o o f : S e e Append ix C
5 . Ma rke t c l ear ing
Unde r the as s ume d fo rm o f the pr i c e func t i on,the demand o f s to ck by
i nd iv i dua l informe d and un informed inve s t o r s a r e g ive n r e s p e c t ive ly by
and When marke t c l e ar s , they mus t sum t o Thus
( l ( 4 1 5 )9
Us ing we have the fo l l owing e qua t i on 5 °
ws f - 0 .p
S o lv ing we c an ob ta in the c o e f f i c i ent s so
, 59
’
func t i on .2 6
Th i s c omp l e te s our proo f o f the PRO PO S I T I ON .
2 6i s a s e t o f non - l ine ar a l geb ra i c e qua t i ons . We are no t ab l e to
ob ta in the exp l i c i t fo rm o f the s o lut i on . Howeve r , nume r i c a l s o lut i on cane a s i ly be ob ta ine d fo r the c o e ff i c i ent s . In the two l im i t ing c as e s , i e . , m=oo r w=1 , exp l i c i t fo rm o f the s o lut i on c an b e c a l cul a t e d . Pe r turba t i onme thodo l ogy c an b e us e d in the vi c in i ty o f m=o and e xp l i c i t s o lut i on o fr e cur s ive form c an b e de r ive d .
-
I n wha t fo l l ows , we want t o ana lyz e the e f fe c t o f i nfo rma t i on on s t o ck
p r i c e s , r i sk p r em ium , p r i c e vo l a t i l i ty and s e r i a l c o r r e l a t i on in r e turns . I n
the current mode l , the. parame ter w char ac te r i z e s th e inf o rma t i on s t ruc tur e o f
the e c onomy . Mo s t o f the c omp ar a t ive s ta t i c analys i s a r e c onc e rne d w i th the
e f fe c t o f chang ing w . I t i s imp o r tant t o re al i z e tha t w c ap ture s two a s p e c t s
o f the informa t i on s truc tur e o f th i s mo de l . One asp e c t i s the imp e r fe c t i on in
s ome inve s to r s ’ i nfo rma t i on . As w inc re as e s mo r e inve s t o r s have imp e r fe c t
info rma t i on and the t o tal amount o f info rma t i on in the marke t ab out th e
fundamental s the d iv i dend gr owth ) i s l e s s in s ome s ens e . The o the r
a s p e c t o f the info rma t i on s t ruc tur e i s informa t i on a s ymme try . As w change s ,
the extent to wh i ch info rma t i on i s a symme tr i c among inve s t o r s a l s o ch ange s .
Fo r examp l e,when m= l a l l inve s t o r s are un info rmed and the re i s no i n f o rma t i on
a symme try wh i l e when w i s l i gh t l y l e s s than 1 the i nve s t o r s ar e no l onge r
homo gene ous and info rma t i on i s a symme t r i c . I nf orma t i on a s ymme t ry c an a c tua l ly
c aus e pr i c e to be l e s s info rma t ive a s we w i l l s e e l a t e r . Henc e , a s we ch ange u
we ar e chang ing the s e two a s p e c t s o f the info rma t i on s t ruc tur e a t the s ame
t ime and the e f fe c t we ob ta in ar e o f c our s e th e ne t o f the two .
The e f fe c t o f imp e r fe c t inf o rma t i on i s b e s t i l lus t r a t e d by c omp ar i ng the
two extreme c as e s : m=o and m=l . The fo rme r c o r r e sp onds th e c a s e o f p e r fe c t
info rma t i on wh i l e the l a t e r th e c a s e o f pur e impe r fe c t in fo rma t i on . C omp ar ing
the r e sul t fo r w i n the v i c i n i ty o f the two ex t reme s and the e x t r eme c a s e
thems e lve s may demons t r a te the e f fe c t o f info rma t i on a s ymme t ry .
V . S TOCK PRI C ES
As s ta ted in the PROPOS I T I ON,the e qu i l ib r i um p r i c e o f the s t o ck i s
S Q ( SO+ S90 ) s
qq . ( 5 1 )
-
We c an show tha t when the informe d i nve s t o r s dom ina t e in the marke t ( i . e .
2 7we l) , s
”i s a lways p o s i t ive . Fo r r e a s ons d i s cus s e d in s e c t i on I I I
, 89i s
ne ga t ive . Examp l e s a re sh own in F i g . l .
The ex i s tenc e o f the un info rme d inve s t o r s c aus e s dev i a t i ons in the s t o ck
p r i c e fr om i t s benchmark va lue S0
g iven in s e c t i on I I I
S - SO
s
na
The d i ffe renc e be twe en S and SO
c an b e de c omp o s e d i n t o two p ar t s . The f i r s t
par t , sqq , i s d ire c t ly c aus e d by the e s t ima t i on e rr o r s o f the un info rme d
inve s tor s . Wi th sqb e ing p o s i t ive
,the e qu i l ib r ium p r i c e re sponds p o s i t ive ly
to the e s t ima t i on e r r or s o f the un in fo rme d inve s to r s . The un info rmed
inve s t or s ' op t imi sm dr ive s the p r i c e up and th e i r p e s s im i sm dr ive s the p r i c e
down . Al though the informe d inve s t o r s have p e r fe c t i nfo rma t i on about the
e rr or s o f the uninfo rmed inve s t o r s,the i r risk s avers ion p reven t s them fr om
tak ing l ar ge p o s i t i ons to e l im ina t e th e s e dev i a t i ons . S inc e n fo l l ows an O -U
p ro c e s s and reve rt s t o a z e ro me an, 5
00 r ep r e s ent s a tempo rary c omponent in
p r i c e devi at i ons . As d i s c us s e d e a r l i e r , th i s i s be caus e the un informe d
inve s to r s c ont inuous ly upda te th e i r e s t ima t e s b a s e d on the new info rma t i on
from the d ividend fl ow and the p r i c e s and the e r ro r s made in p revi ous
e s t ima t i on are c orre c te d ba s e d on the newly ar r ive d i nfo rma t i on . The s e c ond
par t i s given by the d i f fe renc e b e twe en ( so+so9 ) and ( s We p o inte d
oo+seo
out ear l i e r tha t g ive s the d i s c ount i n p r i c e in o rde r t o c omp ens a t e
for the r i sk o f inve s t ing in the s t o ck . I t r ep r e s e n t s a pe rmanent dev i a t i on in
2 7Th i s r e sul t i s ob ta ine d by us ing a p e r turba t i on me thodo l ogy . The p ro o f
i s ava i l ab le from the autho r up on r e que s t .
-
2 5
th e s t ock p r i c e from i t s b enchmark va lue . I nde e d,th i s s e c ond c omp onent has a
non - z e ro l ong - run s ta t i onary l eve l o f ( so
As we w i l l s e e l a te r,the
p e rmanent c omp onent in the p r i c e dev i a t i o n ha s t o do w i th the fac t tha t
imp e r fe c t i nfo rmat ion o f s ome inve s t o r s a f fe c t s th e r i s k o f inve s t ing in the
s t o ck for th e s e inve s to r s .
V I . RI SK PREM IUM
From the unc ond i t i onal exp e c t e d e xc e s s r e turn fr om the s t ock i s
g iven by eo
. As di s cus s e d in S e c t i on I I I,
eO
i s inde p e ndent o f 09unde r
p e r fe c t info rma t i on S im i l ar ly , one c an show th a t th e s ame re sul t ho l ds
when w= l , i . a . , al l inve s to r s ar e un inf o rme d .2 8
Henc e,even though the supp ly
sh o cks do c aus e the p r i c e to vary and t o b e mo r e vo l a t i l e,they w i l l no t
a f fe c t the unc ondi t i onal exp e c t ed exc e s s r e turn w i th ou t i nfo rma t i on asymme try
Howeve r,a s shown by F i g . 2 , when info rma t i on a s ymme t ry i s p r e s ent 0
0w i l l
a f fe c t the unc ondi t i onal expe c t e d exc e s s r e turn . Th i s i s b e c aus e unde r
a symme tr i c i nfo rma t i on,l e s s informe d i nve s t o r s r e l y o n p r i c e s and d iv i dends
t o l e arn ab ou t the fu ture d iv i dend gr owth . As 0 i nc r e a s e s,the pr i c e may
9
b e c ome l e s s info rmat ive ab out p . Th i s w i l l inc r e a s e th e unc e r ta inty in futur e
c a sh fl ows p erc e ived by the un info rme d i nve s t o r s . C on s e quent ly , they w i l l
demand a h i ghe r re turn .
The r i s k p r em ium on the s t o ck i s g iven by 6 s e/S wh e r e we only c ons i de r the
s i tua t i on whe n S 0 . e i s the expe c t e d e xc e s s re turn on the s t o ck g iven by
e e r+a9) so9
0
2 8Ac tua l ly , when a l l inve s t o r s a r e un info rme d the e qu i l ib r ium p r i c e wi l l
ful ly reve a l 9 .
-
C l ea r l y , 8 i s no t c ons tant ove r t ime . I t s ins tantane ous l eve l dep ends on a l l
the s t a te var i ab l e s . Le t us l o ok a t the " l ong - run l eve l “ 6 de f ine d by 6=e/S ,
where e=e0
and S ar e , r e sp e c t ive ly , the l ong - run s ta t i ona ry l eve l o f e and S .
Then,
The i nfo rma t i on s t ruc tur e a f f e c t s the l ong - run s ta t i o na ry l eve l o f r i s k
p r em ium only through the l ong - run l eve l o f p r i c e , S . G iven the va lue s o f the
unde r ly ing s ta te var iab l e s , wh i ch gove rn the future c a sh f l ows o f the s t o ck ,
the p r i c e o f the s to ck de c r e a s e s a s the frac t i on o f un info rme d inve s t o r s
i nc re a s e s ( s e e F i g . l ) . The s ame r e sul t ho l ds fo r the l ong - run l eve l o f the
p r i c e . The r e for e , we c onc lude tha t the r i sk p rem ium on the s t o ck inc r e a s e s
w i th the fr ac t i on o f un informe d inve s to r s . As shown in F i g . 3 , 5 i s an
inc r e a s ing func t i on o f w .
Th i s r e sul t i s qu i t e intu i t ive . As we d i s cus s e d e a r l i e r , th e r i s k p r em ium
only dep ends on the fundamen t a l r i sk o f the s t o ck p e r c e ive d by the inve s t o r s .
As the frac t i on o f un info rme d i nve s t o r s inc re a s e s , the p r i c e w i l l c on ta i n l e s s
info rma t i on ab out futur e d iv i dend growth . Th i s inc re a s e s the p e rc e ive d
unc e r t a inty ab out futur e c ash f l ows by the un info rme d inve s t o r s , henc e the
r i sk o f inve s t ing in the s t o c k . I n the e qu i l ib r ium , a h i ghe r p r em ium fo r the
s to ck w i l l b e demanded . I t sh ou l d b e c l e ar tha t th i s r e sul t i s ma i nly dr iven
by the imp e r fe c t informa t i on the marke t has ab out the fundamenta l s . The
informa t i on asymme try p l ays no imp o r tan t ro l e he re .
V I I . PR I CE VOLATILITY
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-
inc r e a s e mono toni c a l ly . As shown in F i g . 4 , 02
3may p e ak a t s ome w in the
int e r i o r o f Th i s r e f l e c t s the imp a c t o f info rma t i on asymme try.
I t has o f ten b een argued in the l i t e r a tur e tha t r a t i onal inve s to r s w i th
s up e r i o r info rma t i on wi l l a lways s tab lize p r i c e s .2 9
The bas ic r eas on i s tha t
b e t t e r i nformed inve s t or s w i l l take p r o f i t ab l e p o s i t i ons wheneve r p r i c e i s out
o f l ine w i th the fundamenta l s . By do ing s o , th e y pa r t i a l ly revea l the i r
p r iva t e informa t i on to the marke t and move the p r i c e in the di re c t i on o f
fundamental s . More o f the s e informed inve s t o r s ex i s t , l a r ger impac t they w i l l
have on the p r i c e and l e s s i t w i l l dev i a t e f r om i t s fundamental value . Henc e ,
inc re a s i ng the fra c t i on o f info rmed inve s t o r s sh oul d r e duc e the pr i c e
vo l a t i l i ty .
G iven the info rma t i on s t ruc ture in th i s p ap e r , i t i s no t the cas e tha t mo re
info rme d inve s tor s wi l l a lway s s tab lize th e p r i c e s . To s e e why , l e t us l o ok a t
the v i c in i ty of m=l . When w=l , inve s t o r s a r e h omogene ous and al l uninfo rmed
The ab s enc e o f info rma t i on asymme try enab l e s i nve s t o r s to pe r fe c t ly infe r the
l eve l o f supp ly shock fr om the e qu i l ib r i um p r i c e . I n o the r words , the fac t
tha t " no b ody knows any th ing enabl e s everyb odv t o know s ome th ing . Cons ide r
now a sma l l frac t i on o f informed inve s t o r s a r e int r o duc e d into the marke t
( i . e w i s s l i gh t ly l e s s than On the one h and , they wi l l b r ing in more
info rma t i on about the d iv i dend growth thr ough the i r demand and p r i c e s b e caus e
they know p . Thi s has the e ffe c t o f r e duc i ng th e p r i c e vo l at i l i ty s inc e now
the un informe d inve s to rs have b e t te r informa t i o n ab ou t u and are l e s s re l i ab l e
on d iv i dends in l earning ab ou t p . On the o the r hand , the informa t i on asymme try
2 9I n a re c ent pape r , De Long et . al . ( 19 9 0 ) a r gue tha t i f the re ex i s t
inve s t o r s in the marke t who fo l l ows , by a s s ump t i on , c e r ta in po s i t ive fe e dbacks tra t e g i e s
,the introduc t i on o f ra t i ona l s p e cu l a to r s c an ac tual ly destab lize
p r i c e s . Our mode l d i ffe r fr om the i r s in tha t h e r e a l l inve s tor s ' inve s tments t ra t e gy are di re c t ly de r ived from ut i l i ty max im i z a t i on . Informa t i on asymme tryi s exp l i c i t ly mode l e d to gene r a te the r e s u l t .
-
2 9
w i l l make p r i c e l e s s info rma t ive about the l eve l o f s upp ly sho cks . P r i c e w i l l
no l onge r fu l ly r evea l the supp ly sho cks in e qu i l ib r ium . The un informe d
inve s t o r s have t o us e pas t a s we l l a s p re s ent p r i c e s and d iv i dends t o infe r
the cur r en t l eve l o f 9 . The Baye s i an upda t ing o f the un info rme d i nve s t or s
enhanc e s the c o rr e l a t i on b e twe en innova t i ons i n 9 and p r i c e change s .
App r e c i a t i on in the s tock p r i c e (mo re p re c i s e ly , an inc re a s e in 5 ) w i l l b e
ra t i ona l l y in t e rp re te d as par t i a l ly r e fl e c t ing an de c re a s e in 9 . Th i s imp l i e s
a h i ghe r fu tur e re turn , henc e induc e s h i ghe r s t o ck demand by the un info rmed
inve s t o r s wh i ch fe e ds b ack in t o the p r i c e s . Henc e,the info rma t i on a symme try
w i l l make p r i c e mor e vo l a t i l e . The ne t e f fe c t o f inc r e as i ng the fra c t i on o f
info rme d inve s t o r s w i l l b e the s um o f the two , wh i ch i s gene r a l ly amb i guous .
Fo r s ome p a rame t e r value s , the e ffe c t o f inf o rma t i on a symme try c oul d dom ina te .
When mo r e i nfo rme d inve s t o r s a r e intr oduc e d in the marke t,th e p r i c e c an
dev i a t e mo r e fr om the fundament al s and be c ome mo re vo l a t i l e
Whe n aD
,a“
,89and k a r e s t r i c t ly p o s i t ive , the p r i c e of the s t o ck has a
f in i t e unc ond i t i onal var i anc e . The unc ond i t i ona l var i anc e o f th e s t o ck p r i c e ,
i s c a l cul a ted in App end ix E . F i g . 5 . 2 shows tha t Var [ S ] c an b ehave
in the s i m i l a r fash i on as w change s .30
I n var i ous p r i c e vo la t i l i ty te s t s o f the s t o ck marke t e f f i c i e ncy ( e . g .
LeRoy and P o r t e r Sh i l l e r the e qu i l ib r ium s t o ck p r i c e s are
ob ta ine d by a s sum ing a rep r e s en ta t ive inve s t o r w i th p e r fe c t i nfo rma t i on . The
e ffe c t o f d ive r s e and/or imp e r f e c t info rma t i on have no t b e en taken int o
c ons i de ra t i on . Our mode l sh ows tha t the ex i s t enc e o f i nve s t o r s w i th imp e r fe c t
30I t sh ou l d be po in t e d ou t tha t Var [ S ] c an behave in var i ous ways as w
change s,de p end ing on the par ame te r va lue s . Fo r examp l e , i t i s even p o s s ib l e
t o have th e s i tua t i on tha t the unc ond i t i ona l var ianc e i s l owe r unde r imp e r fe c t
info rma t i on (m=l ) than unde r p e r fe c t info rma t i on Th i s r e s ul t should no tpuz z l e us b e c aus e the unc ond i t i ona l var i anc e o f pr i c e no t only dep ends on theins tant ane ous var iance - c ovar i anc e o f i t s c omp onent s but a l s o dep ends on a l lthe adj us tmen t c o e ff i c i ent s s uch as a
n, a
“, a
eand
a n
-
i nfo rma t i on c an c aus e s t o ck p r i c e s t o dev i a te from the i r fundamenta l va lue s
and to b e c ome mo r e vo l a t i l e . Mo r e ove r , informa t i on a s ymme try c an a l s o c aus e
p r i c e t o b e mo r e vari ab l e . The t r ad i t i ona l mode l s o f i dent i c a l i nve s to r s and
p e r fe c t inf orma t i on can unde r s t a t e the p r i c e vo lat i l i ty3 1
V I I I . AUTOCORRELAT ION IN S TOCK RETURNS
Re c ent emp i r i c a l s tud ies '( Fama and French Poterba and S umme r s
Lo and Mck inl ay have found s i gn i f i cant ne ga t ive s e r i a l
c o r r e l a t i on in s t o ck re turns ove r l ong ho r i z ons of 3 - 5 ye ar s . Ne ga t ive
au t o c orr e l a t i on in re turns s ugge s t s the ex i s tenc e o f s ta t i onary c omp onent s in
s t o ck p r i c e s ( S umme rs I n th i s s e c t i on , we emp l oy our mode l t o s tudy
the b ehav i o r o f s t o ck re turns . Fo r c onven i ence,we cons ider
‘
the r e turns to the
z e r o - wea l th p or t fo l io de f ine d i n S e c t i on IV .32
Unde r asymme tr i c info rma t i o n , the re turn to the z e r o -weal th p o r t fo l i o i s
de r ived in THEOREM
dQ (D - rS ) dt+dS [ eo
Q ha s the fo l l owing s o lu t i on ( s e e , e . g . , Arno l d
Q ( t ) Q ( to)e0( t - t
o) [ a
sdws( s ) - 7
ns
no
ndwn( s )
7950[ 9 ( t ) 7 S [ n( t )
31S e e
,for e xamp l e , Mank iw and De Long ( 19 8 9 )
32Or equiva l ent ly , the e xc e s s r e turn to a shar e o f the s t o ck .
33Q g ive s the und i s c ount e d cumul a t ive c ash f low fr om the z e r o we a l th
p o r t fo l i o
-
whe r e 19
De f ine A( t ) =v s 9 ( t ) . x( t ) =7 s n( t ) and Z ( t )
by
dZ e d t [ asdw d d
0w 7
05
00
”W
")
s’
losoao a
Q c an b e exp re s s e d as the s um o f the thr e e c omp onent s Z , A and x
Q ( t ) Z ( t ) A( t ) x( t )
C l e ar ly, Z ( t ) fol l ows a r andom wa lk w i th a c ons t an t
Ax( t ) fo l l ow s tandard O - U p r o c e s s e s
3
I n o rder t o l o ok a t the au t o c o rre l a t i on i n s t o ck
th e fo l l owing rat i o
- r ) 1
fl( r ) i s c al cul a te d in App end ix D .
The nume ra to r in i s the unc ond i t i ona l c ova r i anc e b e twe en the r e turns
fr om two c ons e cut ive h o l d ing p e r i o ds o f l e ng th r
- Q ( t - f ) ] 75560Za§< 1 3 5— 13 5 0by , [ c< r a
9 n
35The c ovar ianc e is ne ga t ive i f r a and p o s i t ive i f r a
9 9'
34I t should b e p o in t e d out tha t Z ( t ) , A( t ) and n( t ) are no t indep enden t
( s e e Append ix D ) .
3 5When r a the f inanc ing c o s t o f th e z e r o -we a l th p or t fo l i o dom ina te s
th e r e turn , wh ich le ads t o p o s i t ive s e r i a l c o r r e l a t i on in the re turns
( 8 3 )
dr i f t and b o th A( t ) and
r e turns , l e t us c ons i de r
-
32
C ons i de r the case when r 39
’ c ovar i anc e i s ne ga t ive . The s t r ong
me an - reve r s i on in 9 w i l l gener ate ne gat ive au t o c o r r e l a t i on in s to ck re turns
even unde r symme tr i c info rmat i on . Unde r as ymme tr i c i nfo rma t i on,howeve r
,the
ne ga t ive au to c o rre l a t i on w i l l b e enhanc e d . I n th i s c a s e,the un info rme d
inve s to r s us e p as t re turns to f i l te r the cur r e n t l eve l o f supp ly shock . A l ow
r e turn in the pas t p e r i od wi l l be a t tr ibu t e d p a r t i a l ly to an inc r ea s e in 9 ,
even when i t was caus ed by a drop in the futur e d ivi dend gr ow th . The r e fo r e ,
they wi l l exp e c t a fu tur e de c re as e in 9 due t o i t s me an - reve r t ing na tur e henc e
a h i gh re turn . Th i s enhanc e d c o rre l at i on b e twe e n the un informe d inve s t o r s
up da t ing o f 9 and the rea l i z e d re turns amp l i f i e s th e s e r i a l c o r re l a t i on in the
r e turns . As shown in F i g . 6 , informat i on a symme try c an s i gn i f i c ant ly inc r e as e
the ne ga t ive autoc orre la t i on in re turns .
IX . OPT IMAL INVESTMENT STRATEG I ES
I n th i s s e c t i on,we exam ine the trad ing s t r a te g i e s o f b o th the info rme d and
the un info rme d inve s to rs . As we have s e en in the p r ev i ous s e c t i ons , inve s to r s
w i th d i f fe r en t informa t i on vi ew the as s e t s d i f f e rent ly even in e qu i l ib r i um .
Henc e , they w i l l adop t d i ffe rent inve s tment p o l i c i e s
G iven the e qu i l ib r ium p r i c e,THEOREM g ive s th e op t ima l s t o ck demand o f
an informed inve s t or
9 10 9 r)
"
As shown in s e c t i on IV,the exc e s s re turn ar e de t e rm ine d by two fac t o r s
aggr e ga te supp ly o f the r i sky c ap i tal 9 and th e un inf o rme d inve s t o r s '
Ae s t imat i on e rr or n=u-u . They g ive r i s e t o the two he dg ing c omp onent s in the
info rmed inve s to rs ' demand fo r s tock,as exp r e s s e d by the l as t two te rms in
-
Knowl edge ab out e xp e c t ed futur e r evi s i ons i n the un info rme d
inve s t o r s ' e s t ima te s do e s no t p r ovide the informe d i nve s t o r s w i th arb i t rage
opp o r tun i t i e s s inc e 0 i s s t o cha s t ic . Howeve r , the i nf o rme d inve s t o r s do take
advantage o f the i r s up e r i o r inf ormat i on to make pr o f i t s by hedg ing a ga ins t
change s in 0 . Due t o r i sk - ave r s i on , the h e dg ing p o s i t i on o f the inf o rme d
inve s t o r s w i l l no t b e l ar ge enough to e l i m ina t e th e p r i c e dev i a t i ons c aus ed by
the un informe d inve s t o r s
The op t ima l inve s tme nt p o l i cy o f the un info rme d inve s t o r s i s g ive n by
in THEOREM We r ewr i t e i t a s
x* f*
3 9 . ( 9 2 )0 9
The s e c ond par t o f the demand r epre s en t s an un info rme d inve s t o r 's h e dg ing
demand . 9 i s the only s ta t e var i ab l e tha t may af fe c t the exc e s s r e turn on the
s t o ck ant i c i pa ted by an un info rmed inve s t o r , henc e th e var i ab l e tha t he wants
t o he dge aga ins t .
S eve r a l empi r i c a l s tud i e s s ugge s t tha t s ome inve s t o r s in th e ma rke t b ehave
l ike p r i c e C has er s : buy when p r i c es r i s e and s e ll whe n p r i c e s dr op .3 6
I n wha t
fo l l ows,we show tha t t r end - cha s ing c an b e the ra t iona l b ehav i o r o f l e s s
info rme d inve s to r s unde r a s ymme t r i c info rma t i on , as a c ons e quenc e o f our
a s s ump t i ons ab out the d iv i dend growth ra t e . I n orde r t o s tudy how an
un informed inve s to r ’ s demand f o r s to ck r e s p onds to change s i n p r i c e s , we
c ons i de r the c orre l a t i on b e twe e n dx* and d S c ond i t i one d on It, E [ dx*d S It] .
G iven the demand s che dul e in
S e e the di s cus s i on in D e Long, Shleifer , S umme r s and Wa l dmann ( 19 90 )
and the r e fe renc e s the r e i n .
-
E d *dS 1*
f*
£ d fi 1*
f*
xt ] a c ]
When £9
i s p o s i t ive,the s i gn o f i s the s ame as tha t o f
”k
intui t ive ly,f9
sh ou l d b e p o s i t ive . When the info rme d inve s t o r s
infe r tha t 9 i s h i gh , the i r h o l d ing shoul d a l s o inc re as e . O th e rw i s e , i t mus t
b e the info rme d inve s t o r s who a r e ho l d ing mo r e , wh i ch c anno t b e an equ i l ib r ium
s i tua t i on for the un informe d inve s t o r s
When the un info rmed inve s t o r s dom ina t e the marke t we l ) , the
e qui l ib r ium pr i c e i s a lmo s t a p e r fe c t s i gna l fo r the supp ly o f th e s t o ck .
Ac tua l ly , when a l l the inve s t o r s a re un info rme d the p r i c e ful ly r eve al s
9 . As d i s cus s ed b e for e , when th e aggr e ga t e supp ly o f th e s t o ck inc r e a s e s , e ach
inve s to r has to ho l d mor e s t o ck in e qu i l ib r ium . The p r i c e o f th e s t o ck ha s t o
de c re as e s o tha t the exp e c t e d ex c e s s r e turn o f the s t o ck inc r e a s e s and
inve s t o r s are wi l l ing to h o l d mo r e s t o cks . Th i s ne ga t ive c o r r e l a t i on b e twe en
d 9 and dS make s the un info rme d i nve s t o r s ra t i ona l ly in te rp r e t an ob s e rve d
de c re as e in the p r i c e a s p a r t i a l ly due to an inc re a s e in 9 . Th ey , the r e fo re
ant i c ipa te a h i ghe r exc e s s r e turn from the s to ck and inc re a s e the i r h o l d ing o f
s t o ck . Henc e , when the pr i c e p r ov i de s a go o d s i gna l fo r the s t o ck supp ly ,
i s ne gat ive and th e un info rmed inve s to r s b ehave l ike c ont r ar i ans .
Howeve r , when the un info rme d i nve s to r s do no t dom ina te th e marke t , the
p r i c e doe s no t p r ov i de a go o d s i gna l fo r the s upp ly b e c aus e o f heterogeity
among the inve s t o rs . The s i gn o f E [ d 9 | Iz] i s amb i guous . As sh own by Eq . ( 4 6 )in S e c t i on IV
,E [ dDd9 | It] i s p o s i t ive wh i l e i s ne ga t ive . A p o s i t ive
E [ dDd9 | IZ] can be unde r s t o o d a s f o l l ows . Supp o s e tha t the r e i s a p o s i t ive
innova t i on in D and no innova t i on in 5 . The un informe d inve s t o r s w i l l
ra t i ona l ly infer a po s i t ive sho ck in u from the d iv i dend change . Al s o
ob s e rv ing no innova t i on in 5=s p+ $09 w i th S
9b e ing ne ga t ive
,the un info rme d
p
-
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-
Fo r examp l e , we c an a l l ow inve s tor s to have d i f fe r e n t de gr e e o f r i sk
ave r s i on as l ong a s they r ema in in the CARA c l a s s . Th e a ggr e ga t i on the or ems
( s e e , e . g . Rub i ns te in ( l9 74 ) ) enab l e us t o re duc e th e e c onomy t o an e f fe c t ive
two - p e r s on s e tup us e d in th i s p ap e r . We shoul d p o in t ou t tha t the r e sul t s in
th i s p ap er dep ends on the a s s ump t i on o f r i sk - ave r s i o n o f al l i nve s t o r s . I t i s
the r i s k ave r s i on o f the info rme d inve s t o r s tha t p r eve nt them fr om tak ing ve ry
l a r ge p o s i t i ons t o take advantage o f the un info rme d i nve s t o r s ' e s t ima t i on
e r r o r s . I f the informed inve s t o r s we re r i sk neu tra l , th e only e qu i l ib r i um in
the curr ent c omp e t i t ive s e t - up woul d b e a ful ly reve a l i ng one . I n c oul d b e
int e re s t ing to l o ok a t how the r i s k ave r s i on o f the two typ e s o f inve s t o r s
a f fe c t s ome o f the r e sul t s .
As fo r the info rma t i on s t ruc tur e,one c an a s s ume tha t in
‘
add i t ion t o
d iv i dends and p r i c e s the un info rme d inve s to r s r e c e ive s ome o the r exo genous
s i gna l s,wh i ch a l s o have the s ame l inear i ty a s ment i o ne d e arl i e r . I n add i t i on ,
we may ex tend the mode l t o i nc lude mor e than two c l a s s e s o f d i f fe r ent l y
info rme d inve s t o r s . When the i nforma t i on s e ts o f d i f f e r en t c l a s s e s o f
inve s to r s have a c omp l e t e rank ing in te rms o f s t ‘ t i s t ical dom inanc e , the
ex t ens i on woul d b e s t ra i gh t - fo rward . Howeve r , i f the r ank ing i s no t c omp l e te
the s i tua t i on b e c ome s mo r e c omp l i c a t e d
We have mode l l e d the info rma t i on s truc tur e o f th e e c onomy in a d i s c r e t e
fa sh i on . The r e a re a f in i t e numbe r o f s i gna l s ava i l ab l e t o the inve s t o r s
D , S , u , For e ach s i gna l , an inve s t o r e i th e r has a c c e s s t o i t o r
doe s no t have any ac c e s s . Th i s may intr oduc e c e r ta in i ns tab i l i t i e s int o the
e qu i l ib r ium o f the e c onomy in t e rms o f the info rma t i o n s t ruc tur e . To
i l lus t ra te th i s,l e t us c ons i de r the c as e in wh i ch th e p opul a t i on o f the
informe d inve s t o r s i s ve ry sma l l . The pr i c e w i l l c on t a i n c e r ta in info rma t i on
tha t only the info rmed inve s t o r s have and henc e pr ov i de s a va luab l e s our c e o f
-
3 7
i nforma t i on t o the un inf o rme d inve s t o r s . Th i s i s t ru e no ma t t e r h ow sma l l the
info rmed inve s t o r s ' p opu l a t i on i s as l ong as i t i s no t exa c t ly z e r o . The
s t ruc tur e o f the e c onomy a s we l l a s the va lue s o f a l l th e p ar ame t e r s a r e
c ommon knowl e dge . Eve n th ough th e info rme d inve s t o r s only have a ve ry sma l l
impac t on the p r i c e ,3 8
th e un inf orme d inve s to r s wou l d have no t r oub l e i n
i dent i fy ing th e e xac t p e r turb a t i on on th e p r i c e c aus e d by the in fo rme d
inve s t o r s . The ab s o lut e ma gn i tude o f th e s i gna l has no th ing t o do w i th i t s
informa t i on c ontent . Henc e , th e un info rme d inve s t o r s c an s t i l l e x tr ac t th e
s ame amount o f info rma t i o n f r om the p r i c e . Howeve r , when the p opul a t i on o f the
informed inve s tor s i s e xa c t l y z e r o , the p r i c e do e s no t c onvey any inf o rma t i on
he l d by the info rmed inve s t o r s . The re fo r e , the info rma t i on c on t e n t may change
ab rup t ly when w re ache s 1 . Th i s imp l i e s tha t the l im i t ing e qu i l ib r i um a s we l
c an b e dra s t i c a l ly d i f fe r e n t f r om th e e qu i l ib r ium wh en m=l
I n the curr ent s e tup,h oweve r
,the i ns tab i l i ty s ugge s t e d ab ove i s no t
p r e s ent . The r e a s on i s th a t whe n m= l , p r i c e r ema ins a va luab l e s our c e o f
info rmat i on t o the un inf o rme d inve s t o r s b e c aus e i t fu l ly r eve a l s the s upp ly o f
the s to ck . The info rma t i o n c o ntent i s c ont inuous in s ome s ens e a s w app r oa che s
1 . But we have found examp l e s o f ins tab i l i t i e s in a var i ant s e tup in wh i ch
p r i c e s be c ome i nforma t i ona l l y va lue l e s s when w i s l .3 9
The ex i s t e nc e o f
ins tab i l i t i e s in info rma t i o n s t ruc tur e w i l l b e c ome an imp o rtan t i s s ue i f we
try t o endogenize the in f o rma t i on s t ruc tur e .
XI . CONCLUS ION
38Th i s i s the re s ul t o f the c omp e t i t ive as sumpt i on and may no t b e t rue
i f inve s to r s a r e a l l owe d t o b ehave s t r a t e g i c a l ly .
39In the s e tup deve l op e d by Wang in wh i ch hedg ing a ga ins t
change s in p r iva te inve s tm e n t opp or tun i t i e s p l ay s th e ro l e o f s upp ly sh o cks ,one c an show the d i s rup t i ve change in e qu i l ib r ium as w app roach e s 1
-
Th i s pap er p re s ent s a gene r a l e qu i l ib r ium mode l o f a s s e t p r i c ing unde r
a symme tr i c info rma t i on . The mode l invo lve s inve s t o r s who are d i ffe rent ly
info rme d ab out the s ta t e o f the e c onomy . I nve s t o r s r a t i onal ly extr ac t
info rma t ion from a l l the ava i l ab l e s i gna l s inc lud ing p r i c e s and c omp e t i t ive ly
tr ade in the marke t t o max im i z e exp e c t e d ut i l i ty . We ob ta in a c l o s e d - fo rm
s o lut i on to the ra t i o na l exp e c t a t i ons e qu i l ib r ium o f the ec onomy
We fur the r emp l oy the mode l t o inve s t i ga t e the imp ac t of asymme t r i c
info rma t i on on e qu i l ib r ium as s e t p r i c e s,the r i sk p r em ium , pr i c e vo l a t i l i ty ,
aut o c o rre l a t i on i s re turns and op t ima l t r ad i ng s t r a t e g i e s . We i l lus tr a te tha t
the ex i s tenc e o f inve s to r s w i th imp e r f e c t i nf o rma t i on i nc reas e s r i s k p r em i a on
s t o cks . We show tha t the ex i s t enc e o f un i nfo rme d inve s t o r s c rea te s t emp o rary
dev i a t i ons in p r i c e s from the i r fundamen t a l va lue s . Th i s caus e s s t ock p r i c e s
t o b e more vo l a t i l e . We a l s o f i nd tha t th e i nfo rme d inve s to r s can des tab lize
the p r i c e and info rma t i on asymme try c an i nc r e a s e the p r i c e vo l at i l i ty and
ne ga t ive auto c o rr e l a t i on i n r e turns . Ano th e r imp l i c a t i on o f our mode l i s tha t
a l though supp ly shocks do make p r i c e mo r e vo l a t i l e,only unde r as ymme tr i c
info rma t i on the s i z e o f supp ly shocks c an a f fe c t the unc ondi t i ona l exp e c te d
exc e s s re turn . We a l s o show tha t unde r a s ymme t r i c inf o rma t i on , inve s t o r s w i th
d i f fe r ent informa t i on adop t d i f fe rent i nve s tment s t r a t e g i e s . For l e s s informed
inve s t o rs,they may ra t i ona l ly a c t l ike p r i c e cha s e r s o r p o s i t ive fe e dback
tr ade r s
I n a s epara te work,we us e a var i ant fo rm o f the cur r ent mode l t o s tudy the
b ehav i o r o f t rad ing vo lume and i t s r e l a t i onsh ip w i th p r i c e behavior .A0
Examp l e s o f s ome re l a t e d emp i r i c a l wo rk ar e Fr ench and Ro l lB arc l ay , Li tz enb e r ge r and Warne r ( 19 8 8 )
-
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"TheEc onom i c C ons e quenc e s o f No i s e Tr ade r s "
,J ourna l o f P o l i t i c a l Ec onomy 9 8 , 70 3
7 3 8
D e Long , J . B . , A . Shleifer , L . H . S umme r s and R . J . Wa l dmann , 1990 ," P o s i t ive
Fe e db ack I nve s tment S t ra te g i e s and De s tab i l i z ing Ra t i ona l S p e cul at i on" ,
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40
D e t emp l e , J . B . , 19 8 6 , As s e t Pr i c ing in a Pr oduc t i o n Ec onomy w i th Incomp l e te
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" E qu i l ib r ium Inte re s t Ra t e s and Mult iperiod Bonds
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"Mult iperiod Se cur i ty Marke t s w i th
Ep s t e in,L . G . and S . E . Zin , l 9 8 9a .
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,and the
Temp o r al B ehavi or o f Comsumpt ion and As s e t Re turns : A The or e t i c a l Fr amework
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Temp o ra l B ehav ior o f Comsumpt ion and As s e t Re turns : An Emp i r i c a l Ana lys i s"
,
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5 - 2 6
Gennotte , G . , 19 8 6 ," Op t ima l P o r t fo l i o Cho ic e Unde r I nc omp l e t e I nfo rma t i on
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4 1
Gr o s sman , S . J and R . J . Sh i l l e r , 19 8 1 ,“ C onsump t i on Correlatednes s and R i sk
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Har r i s on , M . , and D . Kr ep s , 1 9 7 9 ,"Mar t inga l e s and Mult iperiod S e cur i t i e s
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,Trade and Common Knowl e dge
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4 2
Ne l s on , D . B . , 198 7 ,"The T ime S e r i e s B ehav i o r o f S t o ck Marke t Vo l a t i l i ty and
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Advanc e d Appl ied Pr ob ab i l i ty 13 , 7 64 - 7 7 7
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