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

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

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 .

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

'

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

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 )

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 )

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 )

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 )

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

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 .

https://www.forgottenbooks.com/join

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

https://www.forgottenbooks.com/join

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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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" ,

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

Do than,and Fe l dman , 19 8 6 ,

" E qu i l ib r ium Inte re s t Ra t e s and Mult iperiod Bonds

Duf f i e,D . and C . - F . Huang , 19 8 6 ,

"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 .

" Sub s t i tut i on,R i sk Ave r s i on

,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

Ep s t e in,L . G . and S . E . Z in , 198 9b . Sub s t i tu t i on , R i sk Ave r s i on , and the

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"

,

fo r thc om ing,J ourna l o f P o l i t i c a l Ec onomy

Fama,E . F . and K . R . Fr e nch , 19 8 8 ,

" Pe rmanent and Tr ans i to ry C omp onent s o fS t o ck P r i c e s "

,J ourna l o f P o l i t i c a l Ec onomy 9 6 (Ap r i l ) , 246 - 2 7 3

Fl av in,M . A . , 19 8 3 ,

" Exc e s s Vo l a t i l i ty in the F inanc i a l Marke t s : ARe a s s e s sment o f the Emp i r i c a l Evidenc e " , J ourna l o f P o l i t i c a l Ec onomy 9 1

( De c emb e r ) , 9 2 9 - 95 6

Fl em ing W . H . , and R . W . R i sh e l , 19 7 5 , De term i ni s t i c and S t o cha s t i c Opt ima l

C ontr o l, NY : S p r inge r -Ve r l a g

Fr anke l,J . , and K . Froot , 1 9 8 8 ,

" Exp l a ining the Demand fo r Do l l ar s :I nte rna t i ona l Ra te s o f Re turn and the Expec ta t i ons o f Chari s ts andFundament a l i s t s "

,in R . Chamb e r s and P . Paarlberg , e ds Agr i cul tur e ,

Mac r oe c onom i c s,and Exchange Ra te s

French,K . R . , and R . Ro l l , 19 8 6 ,

" S to ck Re turn Var i anc e s : The Ar r iva l o f

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

Gr e en, J . , 19 7 3 ,

" Info rma t i o na l Eff i c i ency and Equ i l ib r ium Harvard Ins t i tute

o f Ec onom i c s Re se arch D i s c us s i on Pap e r , # 2 84

G ros sman , S . J . , 197 6 ," On th e Eff i c i ency of C ompe t i t ive S to ck Marke t s whe re

G r o s sman , S . J . , 19 7 7 , Th e Ex i s tenc e o f Futur e s Ma rke t s , No i sy Ra t i ona l

S tud i e s 44 (No . 4 3 1 - 4 4 9

Gr o s sman,S . J . , 198 1 ,

"An I n t r oduc t i on to the The o ry o f Ra t i ona l Expe c ta t i ons

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

Me asur ement I n Ec onom i e s w i th Non - Traded As s e t s and H e t e rogene ous

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

Marke t s ",J ourna l o f Ec onom i c Th e o ry 20 , 3 8 1 - 4 08

He,H and N . D . P e ar s on , 1 9 8 8 ,

" C ons ump t i on and P o r t fo l i o Po l i c i e s w i thI nc omp l e te Marke t s and Sho r t - S a l e Cons t r a int s : the I nf ini t e D imens i on C as eS l oan S ch oo l o f Management , m ime o

He a t on,J . , 19 90 ,

“ The I n t e r a c t i on b e twe en T ime - nons ep arab l e Pr e f e r enc e s andT ime Aggre ga t i on " , Wo rk ing P ap e r # 3 1 8 l - 90 - EFA , S l oan S cho o l o f Mana gement ,

He l lw i g,M . , 19 8 0 ,

" On the Aggr e ga t i on o f Info rma t i on in C omp e t i t ive Marke t s

J ourna l o f Ec onom i c The o ry 2 6 , 2 7 9 - 312 .

Kl e i don,A . W . , 19 8 6 ,

"Var i anc e B ounds Te s t s and S t o c k Pr i c e Va l ua t i on

Mode l s ",J ourna l o f P o l i t i c a l Ec onomy 94 ( Oc t ob e r ) , 9 5 3 - 1001 .

LeRoy , S . F and R . D . P o r t e r , 19 8 1 ," The Pr e s ent - Va lue Re l a t i on : Te s t s B a s e d

_I , Sp r inger -Ve r l ag .

Lo,A . W . , and A . C . Mack in l ay , 19 8 8 ,

" S t o ck Marke t P r i c e s Do No t Fo l l ow

S tud i e s 1 , 4 1 - 6 6 .

40 (March ) , 14 2 9 - 144a

Mankiw , N . G . , D . Rome r , and M . D . Shap i r o , 19 8 5 ,"An unb i a s ed Re exam ina t i on

Mar sh,T . A . , and R . C . Me r t o n , 19 8 6 ,

" D iv i dend Var i ab i l i ty and Var i anc e

Rev i ew 7 6 (June ) , 4 8 3 - 4 9 8

Mehr a,R . , and E . C . Pr e s c o t t , 1 9 8 5 ,

"The Equ i ty P rem ium : A Puz z l e " , J ourna l

o f Mone tary Ec onom i c s 15 , 14 5 - 16 1

Me r ton,R . C . , 19 7 1 , Op t imum C ons ump t i on and P o r t fo l i o Rul e s in a C on t inuous

Ti me Mode l ",J ourna l o f Ec onom i c The o ry ( De c emb e r ) , 3 7 3 - 4 13

Milgrom , P . and N . S t oky , 1 9 8 2 ," I nfo rma t i on

,Trade and Common Knowl e dge

J ournal o f Ec onom i c The ory 2 6 , 17 - 2 7 .

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

Re turns unpub l i she d pap e r , MIT

Poterba , J . M . , and L . H . Summe r s , 19 8 8 ,"Me an Reve r s i on in S t o ck Pr i c e s

R i chards on , M . , 198 9 ,"Tempo rary Comp onent s o f S t o ck P r i ce s : A Skep t i c 's

V i ew ",Whar ton S choo l , Unive r s i ty o f P ennsy lvan i a , m ime o

Rub ins t e in , M . , 1974 ,"An Aggregat i on The o r em fo r S e cur i t i e s Marke t s J ourna l

Sh i l l e r,R . J . , 19 8 1 , Do S to ck Pr i c e s Move To o Much t o be Jus t i f i ed by

Sh i l l e r,R . J . , 198 8 ,

" P or t fo l i o Insur anc e and O the r I nve s to r Fash i ons a sFac to r s in the 198 7 S to ck Marke t Crash " , NB ER Mac r o Ec onomi c s Annual , 2 8 7 - 2 9 6 .

S ingl e t on,K . J . , 198 7 ,

"As s e t Pr ic e s in a T ime - S e r i e s Mode l w i th D i spar a t e ly

I nfo rme d,Compe t i t ive Tr ade r s " , New Apporoaches t o Mone tary Ec onom i c s

( P ro c e e d ings o f the S e c ond I nte rna t i ona l S ymp o s ium in Ec onom i c The o ry and

Econome tr i c s , ed i te d by W . A . Barne t t and K . J . S ing l e t on , Cambr i dge

Un ive r s i ty P re s s

S undar e s an, M . , 198 3 ,

" Cons tant Ab s o lut e R i sk Ave r s i on Prefe r enc e s and

Summe r s,L . H . , 19 8 6 , Do e s the S to ck Marke t Ra t i ona l ly Re fl e c t Fundamenta l

P o l i t i c a l Ec onomy 91 (No . 546 - 58 8

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"A Mode l o f Dynami c As s e t P r i c i ng Unde r Asymme tr i c

I nfo rma t i on ",The Whar ton S cho o l , Unive r s i ty o f P enns y lvani a , m ime o .

Wang,J . , 1990 ,

" Inte r t emp oral As s e t P r i c e s unde r Asymme tr i c I nfo rma t i onPh . D . d i s s e r ta t i on , Unive r s i ty of Pennsy lvan i a .

We s t,K . D . , 19 8 8 ,

" D iv idend Innova t i ons and S t o ck P r i c e Vo l a t i l i ty

Wh i t t l e, P . , 19 8 1 ,

"Ri sk - S ens i t ive Linear/Q uad rat ic/Gaus s ian C ontr o l

Advanc e d Appl ied Pr ob ab i l i ty 13 , 7 64 - 7 7 7

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