1981 Do Stock Prices Move Too Much to Be Justified by Subsequent Changes in Dividends

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merican Economic ssociation

Do Stock Prices Move Too Much to be Justified by Subsequent Changes in Dividends?Author(s): Robert J. ShillerSource: The American Economic Review, Vol. 71, No. 3 (Jun., 1981), pp. 421-436Published by: American Economic AssociationStable URL: http://www.jstor.org/stable/1802789

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Do Stock

PricesMove

Too Much

to

be

Justified

y

Subsequent

hanges

n

Dividends?

By ROBERTJ.SHILLER*

A simple

model hat

s commonly

sed

to

interpret

ovements

n corporate

ommon

stock.

rice

ndexes sserts

hat

real stock

prices

qual thepresent

alue of

rationally

expected

r

optimally

orecasted

uture

eal

dividends

iscounted

y

a

constant eal

dis-

count

rate.

This

valuation

model or

varia-

tions

n it

n which hereal

discount ate

s

not constant

ut

fairlytable)

s often

sed

byeconomistsndmarketnalystslike s a

plausible

model o

describe

he

behavior

f

aggregate

arket

ndexes nd

is viewed

s

providing

reasonable

tory o

tell

when

people

ask what

accounts

for

a

sudden

movement

n

stock price

indexes. Such

movements

re then

ttributed

o

"new

n-

formation"

bout

future

ividends.

will

refero this

model

s the

efficient

arkets

model" lthough

t should

e recognized

hat

this

namehas also

been

applied

to other

models.It hasoften eenclaimedn popular is-

cussions

hat tockprice

ndexes

eem too

"volatile,"

hat

s, that

the movements

n

stock rice

ndexes

ould

notrealistically

e

attributed

o

anyobjective

ew nformation,

since

movements

n the

rice

ndexes

eem

o

be

"too

big"

relative

o

actual subsequent

events.Recently,

he

notion

hat financial

assetprices

re too

volatile

o accord

with

efficient

arkets

as received

ome

econo-

metric

upport

n

papers

yStephen

eRoy

and

Richard orter

n the tock

market,

nd

by

myselfn the

bond

market.

To

illustrate

raphically

hy tseems

hat

stock rices

retoovolatile,

have

plotted

n

Figure

1

a stockprice

ndex

,

with

ts

ex

post

rational

counterpart

* (data

set

1).'

The

stockprice

ndex

pt

s the real

Standard

and

Poor's Composite

tock

Price ndex

de-

trended y

dividing

y

a factor roportional

to the ong-runxponential rowth ath)and

p*

is the

present

discounted

value of

the

actual subsequent

eal dividendsalso

as a

proportion

f the ame

ong-runrowth

ac-

tor).2The

analogous

eries

fora

modified

Dow Jonesndustrial

verageppear

n

Fig-

ure

2

(data

set

2).

One

is

struck

y

the

smoothness

nd stability

f the

expost ra-

tionalprice

eries

* when

ompared

with

the ctual rice

eries.

hisbehavior

f

p*

is

due

to

the

act hat

he

resent

alue elation

relates *

to

a

long-weighted

ovingverage

ofdividendswithweightsorrespondingo

discount actors)

nd

moving

verages

end

to smooth

he

series veraged.

Moreover,

while eal

dividends

id

vary ver

his am-

ple period,

hey id

notvary

ong nough

r

far

nough

o

causemajor

movements

n

p*.

For example,

hile

nenormally

hinks

f

the GreatDepression

s

a time

whenbusi-

nesswas

bad,

real dividends

ere

ubstan-

tially

below

their

long

run

exponential

growthath

i.e.,

10-25 percent

elow

the

*Associaterofessor,

niversityfPennsylvania,

nd

researchssociate,

ationalBureau f Economic

e-

search.

am gratefulo Christine msler or

esearch

assistance,

nd to her as

well

as

Benjaminriedman,

Irwin Friend, anford

Grossman, tephen

LeRoy,

Stephen

oss,

nd

Jeremyiegel

or

elpful

omments.

Thisresearch as supported

ytheNational

ureau f

Economic

esearchs part

f

the

Research roject

n

theChanging

oles of Debt and

Equity n Financing

U.S.

CapitalFormationponsored

y the American

Council

f Life

nsurance

nd by

the

National

cience

Foundation nder

grant SOC-7907561.

The views

expressed

ere re

olely

my wn

nddo notnecessarily

representhe iews f the upportinggencies.

'The

stock

rice ndex

may

ookunfamiliar

ecause

it sdeflated

y

price

ndex,

xpressed

s a proportion

of the

ong-run

rowth

ath

ndonly

January

igures

are shown.

ne

might

ote, or xample,

hat

he

tock

marketecline

f 1929-32

ooks

maller

han he

ecent

decline.

n

realterms,

t was.

The Januaryigures

lso

miss oth

he1929

eak

nd 1932

rough.

2Theprice

nd

dividenderies

s a proportion

f

the

long-run

rowth

ath re

definedelow

t

the eginning

of Section

.

Assumptions

boutpublicknowledge

r

lack

of knowledge

f the ong-run

rowthath

are

important,

s

shall e

discussed

elow.

he series

* is

computed

ubject

o

an

assumption

bout

dividends

after 978. eetext ndFigure below.

421






422

THE

AMERICAN ECONOMIC REVIEW

JUNE

1981

300-

Index

225

p

150-

*

75-

0

I

year

1870

1890 1910

1930

1950

1970

FIGURE

1

Note:

Real Standard

nd

Poor's

Compositetock

rice

Index solid ine

p)

and

ex

post

rational

rice

dotted

line *), 1871-

979, oth etrended

y

dividing long-

run

xponential

rowth actor.

he

variable

*

is

the

present

alue of actual

ubsequent

eal

detrended

i-

vidends,ubject o an

assumptionbout

the

present

value n 1979of

dividends

hereafter.ata are from

Data

Set 1,Appendix.

growth

ath for the

Standard nd Poor's

series, 6-38

percentelow

hegrowth

ath

for heDowSeries) nly or few epression

years:

933,

934, 935,

nd 1938.

The

mov-

ing

verage hich

etermines*

will mooth

out such

hort-run

luctuations.

learly

he

stockmarket

ecline

eginning

n

1929 nd

ending

n

1932 ould

not be rationalizedn

terms f

subsequentividends or

could

t

be

rationalizedn

terms f

subsequentarn-

ings, ince

arningsre

relevant

n

this

model

only

as

indicators f

later

dividends. f

course, he

fficient arkets odel

oes not

sayp=p*.

Might ne still

uppose hat his

kindof stockmarket rashwas a rational

mistake,

forecastrror hat

ational

eople

might

ake?

his

paper

will

xplore ere

he

notion hat

he

very

olatility

f

p (i.e.,

the

tendency

f

big movements

n

p

to

occur

again

and

again) mplies

hat he

answer

s

no.

To

give n idea

of the kind

of

volatility

comparisonshat

will

be

made

here,

et

us

considert this

oint

he

implest

nequality

which

uts

imits n

one measure f

volatil-

ity: he tandardeviationfp. The efficient

markets odel

an be

describeds

asserting

Index

2000-

1500

1000-

500-

yeor

0

I

I

I I

1

1928 1938 1948 1958

1968 1978

FIGURE

2

Note:Realmodifiedow Jonesndustrial

veragesolid

line

p)

and ex

post

rational

rice dotted

ine

p*),

1928-1979,othdetrended

y dividing y

a

long-run

exponentialrowthactor.

he

variable

*

is

the

resent

value of actual subsequent eal

detrended ividends,

subject

o an

assumption

bout the

present

alue

n

1979 fdividendshereafter.

ata arefrom ata Set2,

Appendix.

that

p, =E,(

p*), i.e.,

p,

is

the

mathematical

expectation

onditional

n all

information

available t time ofp*. Inother ords,, s

the

optimal

orecast f

p*.

One

can define

the

forecast

rror

s

u,=

p*

-pt.

A

funda-

mental rinciplefoptimal orecastss that

the

forecast rror , mustbe uncorrelated

with heforecast;hat

s,

the

ovariance e-

tween

,

and

u,

must

e

zero.

f

a forecast

error

howed consistent

orrelation

ith

the

forecasttself, hen hatwould

n

itself

imply

hat

he

forecastould be

improved.

Mathematically,

t can be

shown

rom he

theory

f

conditional

xpectations

hat

u,

must euncorrelatedith ,.

If one

uses the

principle

rom

lementary

statistics

hat hevariance f the um f two

uncorrelatedariables s the sum

of

their

variances,

ne then

has

var(p*) var(u)+

var(p). Sincevariances

annot e

negative,

this

means

var(p)

?var(p*) or, converting

to more

easily nterpreted

tandard evia-

tions,

(1)

(p

or(P*)

This

inequality

employed efore n the






VOL.

71

NO.

3

SHILLER:

STOCK

PRICES

423

papers

by LeRoy

and

Porter

nd myself)

s

violated

dramatically

y

the

data

in Figures

1

and

2 as is

immediately

bvious

n

looking

at the

figures.3

This paper will develop the efficient

markets

model

in Section

I

to

clarify ome

theoretical

uestions

that may

arise in

con-

nection

with the

inequality

1)

and some

similar

nequalities

will

be derived

that

put

limitson

the standard

deviation

of the

in-

novation

n

price

and the

standard

deviation

of the change

n

price.

The model

s restated

in innovation

orm

which

allows

betterun-

derstanding

f the

limits

on stock

price

volatility

mposed

by the

model.

In

particu-

lar, this

will enable

us to see (Section

I) that

the standard eviation ftvp s highestwhen

information

bout

dividends

is revealed

smoothly

nd that

f

nformation

s revealed

in big

lumps occasionally

the price

series

may

have higher

kurtosis fatter

ails) but

will have lower

variance.

The notion

ex-

pressed

by some

that

earnings

rather

han

dividend

data should

be used

is discussed

n

Section III,

and

a

way

of

assessing

the

im-

portance

of

time variation

n

real

discount

rates s shown

n Section

V. The inequalities

are comparedwith hedata in SectionV.

This

paper

takes as

its starting oint

the

approach

used earlier 1979)

which

howed

evidence suggesting

that

long-term

bond

yields

re too

volatile

to accord

with imple

expectations

models of the

term

tructure

f

interest

ates.4

n that paper,

it

was shown

how restrictions

mpliedby

efficient

arkets

on

the cross-covariance

unction f

short-

term

nd long-term

nterest ates

imply

n-

equality

restrictions

n the spectra

of

the

long-termnterestrate series which char-

acterize

the smoothness

hat the long

rate

should display.

n this paper, analogous

im-

plications

are

derived for

the volatility

f

stock prices,

although

here

a

simpler

and

more ntuitively

ppealing

discussion

of

the

model

in

terms f

its

innovation epresenta-

tion

is used.

This

paper

also

has

benefited

from

the earlier

discussion

by

LeRoy and

Porter

which ndependently

erived ome

re-

strictions

n security rice

volatility

mplied

by

the

efficient

markets

model and

con-

cluded that common stock prices are too

volatile

to

accord

with

the model. They ap-

plied

a methodology

n

some ways

similar

o

that used

here to

study

stock price

ndex

and individual

stocks

in

a

sample

period

starting

fterWorld

War II.

It

is somewhat

naccurate

o say

that this

paper

attempts

o contradict

he extensive

literature

f efficient

markets as,

for

exam-

ple,

Paul Cootner's

volume

on the

random

character f stock prices,

r

Eugene

Fama's

survey).5Most of this literature eally ex-

aminesdifferentroperties

f security

rices.

Very

ittle f

the efficient

markets iterature

bears directly

n the

characteristic

eature f

the model

considered

here:

that

expected

real returns

or the

aggregate

tock market

are constant hrough

ime or approximately

so).

Much of the

literature

on efficient

markets oncerns

he investigation

f

nomi-

nal "profit

pportunities"

variously

efined)

and whether ransactions

osts

prohibit

heir

exploitation.

Of

course,

f real stock prices

are "too volatile"as it is definedhere,then

there

may

well be

a sort of

real

profit

p-

portunity.

ime variation

n

expected

real

interest

ates

does not itself

mply

that any

3Some

people

will

object

to thisderivation

f

I)

and

say

thatone might

s well have said

that

E,(p,)

=p,*

i.e.,that

orecasts

re correct

on average,"

whichwould

lead

to a reversal

f the

nequality

1). This

objection

stems, owever, rom misinterpretationf conditional

expectations.

he subscript

on the expectations

pera-

tor

E means "taking

as given (i.e., nonrandom)

all

variablesknown

t

time ." Clearly,

t

s known

t time

and p*

is not.

n

practical

erms,

f a forecaster

ives s

his forecast

nythingther

han

Et(

p*), then

highfore-

cast is

not optimal

n the sense

of expected

squared

forecast

rror.

fhe gives forecast

hich

quals

E( p,*)

only

on average, hen

he is adding

random

noise to the

optimal

forecast.

he

amountof noise apparent

n

Fig-

ures I or

2 is extraordinary.

magine

what

we would

think

f our ocal

weather

orecaster

f,

ay,

actual

ocal

temperatures

ollowed

he dotted

ine and

his forecasts

followed

he

solid ine

4This analysis

was extended

to

yieldson

preferred

stocksbyChristine msler.

5

t should not

be inferred hat

the literature

n

efficient

markets niformly

upports he

notion of

ef-

ficiencyut

forth here, or

xample, hat

no assetsare

dominated

r thatno trading

ule dominates buy

and

hold

strategy,for

recent

papers see S. Basu;

Franco

Modigliani

nd RichardCohn;

William Brainard,

John

Shoven

and

Lawrence

Weiss;

and

the

papers

in

the

symposium

on market efficiency

dited

by Michael

Jensen).






424 THEAMERICAN

ECONOMIC

REVIEW

JUNE

1981

trading

ule

dominates buy and hold

strategy,

ut

really arge

variationsn

ex-

pectedreturnsmight eem to suggest hat

such a trading uleexists. hispaperdoes

not nvestigatehis,

r

whetherransactions

costsprohibit

ts

exploitation.

his

paper

s

concerned,owever,nstead

with

more

n-

terestingfrom

an economic

tandpoint)

question:

what ccounts

or

movements

n

real stock rices

nd can

they e explained

by new nformationbout subsequent

eal

dividends?

f

themodel ails ue

to

excessive

volatility,henwe

will

have een new har-

acterization

f how

the

simple

modelfails.

The characterizations not equivalent o

other haracterizationsf tsfailure,uch s

that one-period olding

eturns re fore-

castable,

r

that tocks avenot been good

inflationedges ecently.

The volatilityomparisons hat

will

be

madeherehavethe

dvantage

hat

hey

re

insensitive o misalignment

f

price and

dividend

eries,

s

mayhappen

with

arlier

data when

collection

rocedures ere not

ideal.The tests re

also

not affected

y

the

practice,

n

the

constructionf

stockprice

and dividend

ndexes,

f

dropping

ertain

stocks rom he ample ccasionallynd re-

placing hemwith ther tocks, o long as

the

volatility

f

the series

s not

misstated.

These comparisonsre thuswell suited

o

existingong-term

ata

in

stock

rice

ver-

ages.

The

robustnesshat he

volatility

om-

parisons ave, oupled

with heir

implicity,

may account ortheir opularityn casual

discourse.

I.

The

implefficientarkets

odel

Accordingo the imple fficient arkets

model,

he real

price

P,

of

a

share

t the

beginning

f

the ime

eriod

s

given y

00

(2)

Pt

Yk+

EtDt+k

O<Y<

I

k=O

where

,

is

therealdividend aid at (let us

say,

the end

of) time ,

Et

denotesmathe-

matical xpectationonditionaln informa-

tion vailable ttime , nd y s the onstant

real

discount actor. define he constant

real nterest

ate

r

so that

y=

7/(1

4r).

In-

formationt time

includes

t

and

Dt

and

their agged

values,

nd

will

generally

n-

clude ther ariabless well.

The one-period

holdingreturn

Ht

(APt

+Dt)/Pt

is the return rom uying

the stock t time and

selling t at time +

1.

The firstermn thenumerator

sthe apital

gain,

the

second term

s

the dividend e-

ceived t the ndof time

.They re divided

by

P,

to provide rate f return.

he model

(2) has thepropertyhat

t(Ht)

r.

The model

2) can be restated

n

terms

f

series s a proportion

f

the

ong-runrowth

factor:

pt

= Pt

/kA

dt

=Dt/Xt?

T

where

thegrowthactors -

T

=(l

+ g)-

T9,

gis the

rate

f

growth,

nd

T

is the ase year. ivid-

ing 2) byAt- and substituting

ne finds6

00

(3)

Pt=

2

(Xy)k

Etdt+k

k=O

00

=

k

'

Etdt+k

k=O

The growth

ate g mustbe less than the

discount ate if 2) is to give finite rice,

and hence

y-

AXy1,

and

definingby

y

7/(

+

r),

the discount ate ppropriate

or

the

pt

and

dt

series

s

r>

0. This discount

rate

is,

it turns ut, ust

the mean

divi-

dend

divided

by

the meanprice, .e, r=

E(d)/E(

p).7

6No

assumptions

re

ntroduced

n

going

rom

2)

to

(3), since 3)

is ust

n

algebraic

ransformation

f

2).

I

shall,

owever,ntroducehe

ssumptionhat

,

s

ointly

stationary

ith

nformation,

hich

means hat he

un-

conditional)ovarianceetween,and

t-k,where

t

is

any nformation

ariablewhich

might

e

d, tself

rp,),

depends

nly

n

k,

not

t.

It

follows hat

we

can

write

expressions

ike var(p) without

time

ubscript.

n

contrast,

realizationf the

andom ariable he ondi-

tional

xpectation

,(d1+k)

is a function f time

ince t

depends n

information

t

time .

Some

stationarity

assumption

s

necessary

f

we are

toproceedwith

ny

statistical

nalysis.

7Taking nconditional

xpectations

f

both ides f

(3)

we

find

E(p)= l

E(d)

using = I/I

+?

and

solving

e

find

=

E(d)/E(p).






VOL. 71

NO. 3

SHILLER: STOCK PRICES 425

Index

300-

225-

150

75-

year

0-l

1870

1890 1910

1930 1950 1970

FIGURE

3

Note:

Alternativemeasures f the expostrationalprice

p*,

obtainedby

alternativessumptions

bout

the

pre-

sent

value in

1979 of dividends

hereafter.

he middle

curve s

the

p*

series

plotted

n

Figure

1. The series

re

computed

recursively

rom

terminal onditions

using

dividend

eriesd of Data Set

1.

We

may also

write hemodel

as noted

above

n

terms

f

the

x

postrational rice

series

1*

analogous

o

the x

post rational

interest

ate eries

hatJeremy

iegel nd

I

used o

study

heFisher ffect,r

that used

tostudyhe xpectationsheoryfthe erm

structure).

hat s,

p1*

s

the

present

alue f

actual ubsequent

ividends:

(4) Pt

=Et(

Pt*)

00

where

P

k=

+

ldtk

k=O

Since he

ummation

xtends

o infinity,

e

never

bserve

*

without

omeerror.

ow-

ever,

with

long

nough

ividend eries

we

mayobserve

n approximate

. If

we choose

an arbitraryaluefor heterminalalueof

p*

(inFigures

and

2,p*

for

979

was set

t

the average

detrended

eal

price

over

the

sample)

henwe may

determine

1*

recur-

sively

by

p*

=Y(p*?I

+dt)

working

ack-

ward

from he terminal

ate.

As

we

move

back

from

he

erminalate,

he

mportance

of

the erminal

alue hosen

eclines.

n

data

set

1)

as shown

n

Figure ,

y

is

.954

and

Y'08

=.0063

so that

at the

beginning

f

the

sample

the

terminal

alue

chosen

has a

negligible

eight

n

thedetermination

f

pt*.Ifwehad chosen differenterminalondi-

TABLE

1- DEFINITIONS

OF

PRINCIPAL

SYMBOLS

y real discount

actor or

eries efore etrending;

y= 1/(1 r)

y= real

discount

actor

or

etrendederies; _ Ay

D,= realdividendccruingo stockndexbefore e-

trending)

d,

=

real

detrendedividend;,

D,/Xt+

-

T

A first ifference

perator

x,

_x,-x,

St

innovation

perator;

,x,X+

-E,X,+k

E,t

IX,+k;

E= unconditional

athematical

xpectations

perator.

E(x)

is

the rue

population)

ean f

x.

Et

=

mathematicalxpectations

perator

onditional

n

information

t time ; E,x,

_E(x,II,)

where

,

is

the

vector f

information

ariables

nown

t

time

.

A=

trend

actor or rice

nd

dividend

eries;

-

I

+g

where is the ong-runrowth

ate f

price

nd

dividends.

P,=

real tock rice

ndex

before

etrending)

pi

=

realdetrended

tock

rice

ndex;

r

P/AtT

p,

=

ex

post

ationaltock

rice

ndex

expression

)

r=

one-period

ealdiscount

atefor

eries efore e-

trending

r=

real iscount

ate or etrended

eries;

=

1

-y )y

r2

=

two-period

ealdiscount

ate or etrended

eries;

r2=(I

+r_)2-I

t=

time

year)

T=base

year

for

detrending

nd

for

wholesale

rice

index;

PT=PT

-nominal

tock

price

ndex

at

time

tion,

he

resultwouldbe to add

or

subtract

an

exponential

rend

rom

he

p*

shown

n

Figure . This

s shown raphically

n Figure

3, in which

*

is showncomputed

rom

alternativeerminal alues.

Since the only

thing

we need

know

to

compute *

about

dividendsfter

978 s

p*

for

1979,

t does

not matter hether

ividendsre

"smooth"

or

not after

978.

Thus,

Figure represents

our

uncertainty

bout *.

There s yet anotherway to write he

model,which

willbe usefuln the

analysis

which ollows.

or

this

purpose,

t

is con-

venient

o adoptnotation

or he nnovation

in

a variable. et

us

define

he nnovation

operator

-Et -Et-1

where

t

is the on-

ditional

xpectations

perator.

hen

for

ny

variable

t

he erm

tXt+k

equals

tXt+k

-

Et

IXt+k

which s

thechange

n

the

condi-

tional

xpectation

f

Xt+k

that s made

n

response

o new information

rriving e-

tween t-

1

and

t. The time subscript may

be droppedo that

Xk

denotes

tXt+k

and






426

THE AMERICAN ECONOMIC

REVIEW

JUNE

1981

8X

denotes X0

or

,X,.

Since

conditional

expectations

operators

satisfy

EjEk

=

Emin(j

k) it follows

that

E-m,aX,+k

=Et-m

(Et Xt+

k

Et-

IXt+

k)

=

Et-m

Xt+ k

-

Et-mXt+k

0, m 0.Thismeans hat

t

Xt+k

must e

uncorrelated

or ll k

with ll infor-

mation

nown t

time

- 1

and

must,

ince

lagged

nnovations

re nformation

t

time

,

be

uncorrelatedith

t,Xt+j

t'<t,

allj,

i.e.,

innovationsn

variables

re

serially

ncorre-

lated.

The

modelmplies hat he nnovation

n

price

6tpt

s

observable.

ince

3)

can be

written

t

=

(dt

+

Etpt+I),

we

know,

solv-

ing,

that

Etpt+1p

t/-dt.

Hence

StPtj

Etpt -Et- Pt = pt + dt- l-Pt- Y Apt

+dt_l-rpt_1.

The

variable

which

we

call

St

t

or ust

Sp)

is the

variable

which

live

Granger

nd

Paul

Samuelson

mphasized

should,

n

contrast o A

Xpt-Pt-p, by

ef-

ficient

markets,e

unforecastable.

n

prac-

tice,

with

our

data,

6tpt

so

measured

will

approximatelyqual

Apt.

The

model lso

mplies

hat he

nnovation

in

price

s related

o

the

nnovations

n di-

vidends

y

00

(5) stPt = yk Stdt+k

k=O

This xpressions

identical

o

3)

except

hat

St

replaces

t.

Unfortunately,

hile

tpt

s

observable n this

model, the

Stdt+k

erms

are

not

directly

bservable,

hat

s,

we do

not

knowwhen he

ublic ets

nformation

bout

a

particular

ividend.

hus,

n

deriving

n-

equalities

elow,

ne s

obliged

o

assume he

"worst

ossible"

attern

f

nformation

c-

crual.

Expressions2)-(5) constituteour iffer-

ent

representations

f

the same efficient

marketsmodel.

Expressions

4)

and

(5)

are

particularly

seful or

deriving

ur

nequali-

ties

on

measures f

volatility.

e have

al-

ready

sed

4)

to

derive he imit

1)

on the

standard

eviation

f

p given

he standard

deviation

f

p*,

and we

will

use

(5)

to

derive

a limit

n the

tandard eviation

f

Sp

given

the tandard eviationfd.

One issue hat elates o

the

derivation

f

(1)

can

now

be

clarified. he

inequality1)

was derived sing heassumptionhat he

forecastrror

t =P*

-Pt is

uncorrelatedith

Pt.

However,

the forecast

error

ut

is

not

serially

ncorrelated.

t

is

uncorrelatedith

all informationnown t timet, but the

lagged forecast rror

ut_1

is not known at

time since

'*I

is notdiscoveredt time

.

In fact, t=

lz3k= +kpt+k

as can be seen

by substitutinghe xpressions

or

t

nd

pt'

from 3) and (4) into

ut

=p*

-Pt,

and re-

arranging.

ince

the series

8tp,

s

serially

uncorrelated,

t

has first-order

utoregressive

serial

orrelation.8or

this eason,

t

s

inap-

propriate o test the model by regressing

Pt*

pt

on variables

nown t

timet and

using

he

ordinary-statistics

f

the

coeffi-

cients fthese ariables. owever, gener-

alized least

squares

transformation

f

the

variables ouldyield n appropriateegres-

sion test.We might husregress he

trans-

formed

variable

ut -Yu+

I

on

variables

known

at

time

t.

Since

ut

yuti

y

,

this amounts to

testing

whether

the

nnovationn

price

an be

forecasted.

will

erform

nd

discuss uch

egression

ests

in

Section below.

To

find

limit n

the tandard eviation

of

Sp for given tandard eviation

f

dt,

first otethatd, equals its unconditional

expectationlus

the umof ts nnovations:

00

(6)

dt=E(d)+ 2

St-kdt

k=0

If

we regard (d) as

E-0(dt),

thenthis

expression

s

just

a

tautology.

t

tells us,

though,

hat t

=0, 1,2,.... re ustdifferent

linear ombinationsf

the ame nnovations

individendshat nternto he inear ombi-

nation

in

(5)

which

determine

tpt

t=

0, 1,

,....

We

can

thus ask

how

large

var

(8p) mightbe

for

given var(d).

Since

innovations

re seriallyuncorrelated,

e

know rom

6)

that

he ariance

f

the

um

s

81t

follows hat

var(u)=var(8p)/(l

y2)

as

LeRoy

andPorter

oted.

hey

ase

their

olatility

ests

n

our

inequality

1) (which

hey all theorem

) and an

equal-

ity

restriction

2(p)

+a2(8p)/(l

I-2)=a2(p*)

(their

theorem

). They

oundhat,

ith

ostwartandardnd

Poor

earnings ata, both relations

ere

violated

y

sample tatistics.






VOL. 71 NO. 3

SHILLER:

STOCK

PRICES

427

the

um f the

variances:

00

00

(7)

var(d)= 2 var(dk)=

2

k=O k=O

Our

assumption

f

stationarity

or

,

mplies

that

ar(8t_kd)

-var(Sdk)

u.2 s

indepen-

dent

f t.

In expression

5)

we

have

no

information

that he

ariance

fthe

um

s

the

um

fthe

variances

ince

ll the

nnovations

re

time

innovations,

hichmay

e correlated.

n

fact,

forgiven

a6,

,

themaximum

ariance

ofthe

um

n

5)

occurs

when

he lements

n

thesum

are perfectly

ositively

orrelated.

Thismeans henhat olong s

var(Sd)#O0,

td,+k

=akS(d(,

where

k =Gk/GO-

Substitut-

ing

his

nto

6)

implies

00

(8)

dt

akEt-k

k=O

where

hat denotes

variable

minus

ts

mean:

dt

d -E(d)

and

E-t=dt.

Thus,

f

var(Sp)

is

to be

maximized

or

given

vO2,

21

the

dividend

process

must be

a

moving verage rocessn termsf tsown

innovations.9

have

hus

hown,

ather

han

assumed,

hat

f

the

variance

f

Sp

is to be

maximized,

he

orecast

f

d,+k

willhave

he

usual

ARIMA

form as

in the

forecast

popularized

y

Box

and

Jenkins.

We can

now

find he

maximum

ossible

variance

or p

for iven

ariance

fd.

Since

the

nnovations

n

5)

are

perfectly

ositively

correlated,

var(Sp)

=

(2oYkk+

k)2.

To

maximize

his subject

to

the

constraint

var(d)

=

=oAu2

with

respect

o 0,

*,

onemay etuptheLagrangean:

,c

2

1

i

,,X

where

is the

Lagrangean

multiplier.

he

first-order

onditions

or

aj,

0=

,

..

0

are

(10)

a-

=2 2

0

ok

)7

2paj 0

which

n

turn

means

hat

.

is

proportional

to

j. The second-order

onditions

or

a

maximum

re

satisfied,

nd

the

maximum

can be

viewed

s a

tangency

f an

isoquant

for

var(op),

which

is

a

hyperplane

n

(Jo,

91,

'g2'...

space,

with he

hypersphere

ep-

resented

y

the

onstraint.

t the

maximum

(u2

=

(1-y2)var(d

)y2k

and

var(Sp)

y2var(d)/(1-y2)

and so, converting

o

standard eviations orease of interpreta-

tion,we

have

(11)

u(Sp)<af(dl

)/2

where

r2

-(1

+r)21

Here,

F2

s the

wo-period

nterest

ate,

which

is

roughly

wice

the

one-period

ate.

The

maximum

ccurs,

hen,

when

dt

is a

first-

order

autoregressive

rocess,

dt

=

Ydt

1

+

et,

and

E,dt+k =Ykdt,

whered-d-E(d) as

before.

The

variance

f the

nnovation

n

price

s

thus

maximized

when

information

bout

dividends

s revealed

n a

smooth

ashion

o

that

he

tandard

eviation

f the

new

nfor-

mation

t

time

about

a

future

ividend

d,+k

is

proportional

o

its weight

n the

present

alue

formula

n the

model

5).

In

contrast,

uppose

ll

dividends

omehow

e-

came

knownyears

before

hey

werepaid.

Then

the

nnovations

n dividends

ould

be

so heavily iscountedn 5) that heywould

contribute

ittle

o the

tandard

eviation

f

the

nnovation

n

price.

Alternatively,

up-

pose

nothing

ere

known bout

dividends

until

heyear

hey

re paid.

Here,

lthough

the

innovation

ould

not

be

heavily

is-

counted

n

5),

the

mpact

f the

nnovation

would

be

confined

o only

one term

n

(5),

and the tandard

eviation

n the

nnovation

in

price

would be

limited

o

the

standard

deviation

n the

ingle

ividend.

Other nequalities

nalogous

o

(11)

can

alsobe derivedn the ameway.Forexam-

90f

course,

all indeterministic

tationary

rocesses

can

be given

inear

moving

verage representations,

s

Hermann

Wold

showed.

However,

t does

not

follow

that

heprocess

an

be

given

movingverage

epresen-

tation

n terms

f

itsown

innovations.

he

true

process

may be

generated

onlinearly

r

other

nformation

e-

sides its

own lagged

values

may

be

used

in forecasting.

These

will

generally

esult

n

a less

thanperfect

orrela-

tionof thetermsn (5).






428

THE AMERICAN

ECONOMIC

REVIEW

JUNE

1981

ple,

we

can

put

an

upper bound to

the

standard

deviation of

the

change

in

price

(rather

han

he

nnovationn

price)

for

given

standard

eviation n

dividend.

The

only

dif-

ference nduced in the above procedure s

that

/p,

is a

different

inear

combination

f

innovations n

dividends.

Using

the

fact

that

Apt

=8tpt

+

-

t-I

-dt-

,

we

find

00

(12)

Apt

=

I

ykaitdt+k

k=O

00 00

00

+r

',b_

k

dt+k-

I

2

at-idt-I

1=1I

k=O

j=1l

As above, themaximization f thevariance

of

Sp

for

given

variance

f

d

requires

hat

he

time t

innovations

n d

be

perfectly

or-

related

(innovations

at

different

imes

are

necessarily

ncorrelated)

o

that

again

the

dividend

process

must

be

forecasted

s an

ARIMA

process.

However,

the

parameters

of

theARIMA

process

for

d

which

maximize

the

variance of

lp

will

be

different.

ne

finds,

fter

maximizing he

Lagrangean ex-

pression

(analogous to

(9))

an

inequality

slightly ifferentrom1 1),

(13)

a(/\p)<a(d

)/~

The

upper bound

is

attained

f

the

optimal

dividend

forecast s

first-order

utoregres-

sive, but

with

an

autoregressive

oefficient

slightly ifferent

rom

that which

induced

the

upper

bound

to

11). The

upper

bound

to

(13)

is

attained

if

d

=(1-

r))

-

e and

Etdt+k

=(1-r)kdt,

where, s

before,

t

dt

-E(d).

II.

High

Kurtosis

nd

nfrequent

mportant

Breaks n

nformation

It

has

been

repeatedly

noted

that

stock

price

change

distributions

how

high

kurtosis

or

"fat

tails."

This

means

that,

f

one

looks

at

a

time-series

f

observations

n

Sp

or

Ap,

one

sees

long

stretches f

time when

their

(absolute)

values

are all

rather

small

and

then n occasionalextremelyarge absolute)

value.

This

phenomenon

s

commonly

ttri-

buted o

a

tendency

or

new

nformation

o

come n

big

umps

nfrequently.

here

eems

to

be a

common

resumption

hat

his

nfor-

mation umpingmight ause stockprice

changes o

have

high

or

infinite

ariance,

which

would eem

o

contradicthe

conclu-

sion

n

the

preceding

ection

hat

he

vari-

ance

of

price s

limited

nd is

maximized

f

forecasts

ave a

simple

utoregressive

truc-

ture.

High

sample

kurtosis

oes

not

indicate

infinite

ariancef

we

do

not

ssume,

s

did

Fama

(1965)

and

others,

hat

rice

hanges

are

drawn rom

he

table

Paretian

lass of

distributions.'0hemodeldoesnotsuggestthat rice hanges ave distributionnthis

class.

The

model

nstead

uggests

hat

the

existencef

moments

or he

price

eries s

implied

ythe

xistencef

momentsor

he

dividends

eries.

As

long

as d

is

jointly

tationary

ith

information

nd

hasa

finite

ariance,

hen ,

p*,

Sp, and

Apwill

be

stationary

nd

have

finite

ariance."

f

d is

normally

istributed,

however,t

does

not

follow

hat

the

price

variables ill

e

normally

istributed.n

fact,

theymayyet how igh urtosis.

To

see this

possibility,

uppose

he

div-

idends

re

serially

ndependent

nd

identi-

cally

normally

istributed.

he

kurtosis f

the

price

series s

defined

y

K=

E(

)4/

(E(fp)2)2,

where

_p-E(p).

Suppose, s

an

example, hatwith

probability

f

1/n

'0The

empiricalfact

about

the

unconditional

istri-

bution of

stock

price

changes

in not

that

they have

infinite

ariance

which

an

never

e

demonstrated

ith

any

finite

ample), but

that

theyhave

high

kurtosis n

thesample.

1

"With

ny

stationary

rocess

X, the

existence f

a

finite

ar(X,)

implies, y

Schwartz's

nequality,

finite

value of

cov(X,,

X?+k)

for

anyk,

and

hence

the

entire

autocovariance

unction f

X, and the

spectrum, xists.

Moreover,

he

variance

of

E,(X,)

must

also be

finite,

since

the

variance of

X

equals

the

variance

of

E,(X,)

plus

the

varianceof

the

forecast

rror.

While

we

may

regard

real

dividends

s

having

finite

ariance,

nnova-

tions n

dividends

may

how

high

kurtosis. he

residuals

in

a

second-order

utoregression

or

d,

have a

student-

ized

rangeof

6.29

for

the

Standard nd

Poor

series

nd

5.37 for

the Dow

series.

According to

the

David-

Hartley-Pearson

est,

normality

an be

rejected

t the

5

percent

evel

(but not

at

the

1

percent

evel) with

a

one-tailed estforbothdata sets.






VOL.

71 NO.

3

SHILLER: STOCK

PRICES

429

the public

s told

d,

at

the beginning

f time

t, but

with probability

n

- 1)/n has no

in-

formation bout

current

or future

divi-

dends."2

n timeperiods

when they re

told

dt,

pf

equals Yq, otherwise

i

=0.

Then

E()

E((Td1)4)/n

and

E(f

Pt)

E

((

ydt)2

)In

so

that kurtosis

equals

nE(

d1)4)/E((Yd1)2)

which

equals n

times

the kurtosis

of

the normal

distribution.

Hence, by

choosing

n

high

enough one

can

achieve

an

arbitrarilyigh

kurtosis, nd

yet

the

variance

f

price

will

always

exist.

More-

over,

hedistribution

f

A

conditional n

the

information

hat

the dividend

has been

re-

vealed

is also

normal,

n spite of

highkurto-

sis of theunconditional istribution.

If information

s revealed

in big lumps

occasionally

so

as to induce high

kurtosis

s

suggested

n the above example)

var(3p)

or

var( \p)

are

not

especially

arge.

The vari-

ance loses

more

from the long

interval

of

time

when nformation

s

not revealed

han t

gains

from

he nfrequent

vents

when t

is.

The highest

ossible

variance

forgiven

vari-

ance

of d

indeed comes

when

nformation

s

revealed

smoothly

s noted

in

the previous

section.

n the

above

example,

where

nfor-

mation bout dividends s revealed ne time

in

n,

a(3p)

=

n

/2a(d)

and a(Ap)

=

Y(2/n)1/2a(d).

The values

of a(3p)

and

a( \p)

impliedby

this

example

are for

all n

strictly

elow

the

upper

bounds

of the

in-

equalities

1 1)

and

13).13

III. DividendsrEarnings?

It has been

argued

hat he

model

2)

does

not

capture

what

s

generally

meant

by

effi-

cient

markets,

nd

thatthe

model should

be

replaced by a modelwhichmakespricethe

present

value

of

expected

earnings

rather

than

dividends.

n the

model

(2)

earnings

may be

relevant

o the

pricing

f shares

but

only

insofar

as

earnings

are indicators

of

future ividends.

arnings

re

thus

no differ-

ent

from

ny other

conomic

variable

which

may ndicatefuture ividends. he model 2)

is consistent

ith he usual

notion

n

finance

that

ndividuals

re

concerned

with

returns,

that

is,

capital

gains

plus dividends.

The

model

mplies

hat

xpected

otal

returns

re

constant

nd that

the capital gains

compo-

nentof

returns

s ust a

reflection

f

nforma-

tion about

future

dividends.

Earnings,

n

contrast,

re

statistics

onceived

by accoun-

tants

which

re supposed

to provide

n

indi-

cator

of

how

well a company

s doing,

and

there

s a

great

deal

of atitude

or he defini-

tion of earnings, s the recent iteraturen

inflation ccounting

will attest.

There is

no reason

why price

per

share

ought

to

be the

present

value

of expected

earnings

per

share

if some

earnings re

re-

tained.

n

fact,

s

Merton

Miller nd

Franco

Modigliani

rgued,

uch

a present

alue

for-

mula

would

entail

a fundamental

ort

of

double

counting.

t

is incorrect

o include

n

the present

value

formula

both

earnings

t

time t

and the

later

earnings

that

accrue

whentime earnings re reinvested.14 iller

and Modigliani

howed

a

formula y

which

price

might

e regarded

s

the present

alue

of

earnings

corrected

for

investments,

ut

that

formula

can

be shown,

using

an

accounting

dentity

o be

identical

o

2).

Some

people

seem to

feel

thatone

cannot

claim price

as

present

value

of expected

dividends

ince

firms outinely

ay

out

only

a fraction

of earnings

and

also attempt

somewhat

to stabilize

dividends. They

are

right

n

the

case

where

firms

paid

out no

dividends, or hen hepricep1wouldhaveto

grow

at the

discount

rate

r,

and the

model

(2)

would

not be

the solution

to

the dif-

ference

quation

implied

by

the

condition

E,(H,)=r.

On

the otherhand,

if firms

ay

out

a fraction

f

dividends

r

smooth

hort-

run fluctuations

n

dividends,

hen

the

price

of the

firm

will

grow

at

a rate ess

than

the

12For

simplicity,

n this example, the

assumption

elsewhere

n this rticle hat ,

is always

known t time

has been

dropped. t follows

hat

n

this xample

8,p,

#-

Apt

+dt

-rp,

1

but nstead

8,p,

=pt.

13 or

another illustrative

xample,

consider

d,

jd,1

+

E,

as

with the upper

bound

for the

inequality

(11)

but where

he

dividends

re announced

or henext

n yearsevery

/n years.

Here, even though

, has the

autoregressive

tructure, ,

s not the nnovation

n

d,.

As n goesto infinity,

(8p)

approaches ero.

14LeRoy and

Porter o assume

price s

present alue

of earnings

but employ

a correction

o theprice

and

earnings

series

which is, under

additional

theoretical

assumptions

ot employed

y Miller

and

Modigliani,

correctionorthedoublecounting.






430

THEAMERICAN

ECONOMIC

REVIEW

JUNE

1981

discount

rate and

(2) is

the

solution to

the

difference

quation."5

With

urStandard

nd

Poor data,

the

growth ate

of real

price is

only about 1.5 percent,while the discount

rate is

about

4.8%+1.5%=6.3%. At

these

rates,

the value

of the firm

few

decades

hence

s so

heavily

discounted

elative o

its

size

that

t

contributes

ery

ittle o

the

value

of

the stock

today;

by far the

most of

the

value

comes from

he

ntervening

ividends.

Hence

(2) and

the

implied

p* ought

to be

useful

characterizationsf the

value of

the

firm.

The

crucial thing

o

recognize n

thiscon-

text s

thatonce

we know

the

terminal

rice

and interveningividends,we have specified

all

that

investors

are

about. It

would not

make

senseto

define n

ex post

rational

rice

from

terminal

ondition n

price,

using he

same

formula with

earnings

in

place of

dividends.

IV.

Time-Varyingeal

Discount

ates

If

we

modify he

model

2) to allow

real

discount rates

to

vary

without

restriction

through

ime,

then the

model

becomes un-

testable. We do not observe real discount

rates

directly.

egardless f

the

behaviorof

P1

and

D1, there

will

always be a

discount

rate

serieswhich

makes

2) hold

identically.

We

might sk,

though,

whether

he move-

ments

n

the

real

discount ate

thatwould

be

required aren't

larger

than

we

might

have

expected.

Or is it

possible that small

move-

ments

n

the

current

ne-period

iscount

ate

coupled

with

new

information

bout

such

movements n

future

discount rates

could

accountfor

high

tock

price

volatility?16

The natural xtension

f

2)

to the case

of

time

varying

eal

discount ates s

(14)

Pt =Et

(Dt+ll

lk+r,,)

which

has the

propertythat

E,((1

+H1)/

(1

+

r)) -1. If

we set

1 +

r

=

(aU/aCt)/

(aU/aC+ l),

i.e.,

to the

marginal ate of

sub-

stitutionbetween

present and

futurecon-

sumptionwhere

U

is

the

additively eparable

utility

f

consumption,

hen

this

property

s

the

first-order

onditionfor

a

maximum f

expected

utility

ubject

to a

stock

market

budgetconstraint,nd equation 14) is con-

sistentwith

such

expected

utility

maximiza-

tion

at all

times.Note

that

while

r,

is a sort

of

ex

post real interest

ate

not

necessarily

known

until

time +

1,

only

the

conditional

distribution t time

t

or

earlier

nfluences

price

n

the

formula

14).

As

before,we can

rewrite he

model in

terms f

detrended eries:

(15)

Pt

-Et(Pt*)

00 k

where

p

2

d

j

+

1

kP

O

t+k

j0

1+

1

A-#+j

_(1 +rt)/X

This

model then

mplies

that

u(Pt)

?f(p')

as

before.

ince the

model s

nonlinear,

ow-

ever,

t

does not

allow

us to

derive

nequali-

ties

like

(11)

or

(13).

On the

other

hand,

if

movements

n

real

interest

ates are

not

too

large,

hen

we can

use the

inearization f

p*(i.e., Taylor expansion truncated fter the

linear

term)

around

d=E(d) and

r-=E(r-);

i.e.,

00

~E(d)

00

(16)

Pt

-

-Y

dt+k

(F)r

Y

rt+k

k=-O

) ~k=O

where

y=1/(1+E(rF)), and

a hat

over

a

variable

denotes he

variable

minus

ts

mean.

The

first erm n

the

above

expression s

ust

theexpression or * in (4) (demeaned). The

second term

represents

he

effect

n

p*

of

15To

understand his

point, t helps

to consider

a

traditional

ontinuous ime

growth

model, o

insteadof

(2) we

have

PO=1?D,e

-r'dt. In

such a

model,

a firm

has a constant

earnings

tream . If it

pays

out all

earnings,

henD= I

and

PO

fj'

e

-rfdt= I/r.

If

it

pays

out

only

s of its

earnings, hen

the

firm

rows at rate

(I

-s)r,

Dt

s=e('

-s)rt

which

s less than

at t=O,

but

higher han I

later on.

Then

Po=

fosIe('

s)rte-

rdt-

fO'sle

-srtdt=sI/(rs).

If

s#O

(so

thatwe're

not

divid-

ing by

zero)

PO

=

J/r.

'6James

Pesando

has

discussed the

analogous

ques-

tion:

how

argemust he

variance n

liquidity

remiabe

in order to justify he volatility f long-termnterest

rates?






VOL. 71

NO.

3

SHILLER:

STOCK

PRICES

431

movements

n

real

discount

rates.

This sec-

ond

term s

identical

o

the

expression or

*

in

(4)

except

that

dt+k

is

replaced

by rt+k

and

the

expression

is

premultiplied

by

-E(d)/E(r)-

It is

possible

to

offer

simple

intuitive

interpretationor

his

inearization.

irst

note

that the

derivative f 1(1 +

rt+k),

with

re-

spect to r

evaluated t

E(r)

is

-y

Thus,

a

one

percentage

oint ncrease n

-t+k auses

17(1

+r,+k)

to

drop

by

y2 times

1

percent,

r

slightly

ess

than 1

percent.Note

that all

terms

n

(15)

dated

t+k or

higher re

pre-

multiplied

by

17(1

'+k).

Thus,

if

rt+k

is

increased

by one

percentage

point,

all

else

constant,then all of these terms will be

reducedby

about

y2 times

1

percent.

We can

approximate

he

sum of

all

these

terms s

yk-lE(d)/E(r),

where

E(d

)/E(F)

is

the

value at

the

beginning f

time t

+

k of

a

constant

dividend

stream

E(d) discounted

by

E(F), and

yk-

1

discounts t

to

the

pres-

ent.

So,

we

see

that a

one

percentage oint

increase

n

-t+k,

all

else

constant, ecreases

p'

by

about

yk+

'E(d)/E(rF),

which

corre-

sponds

to

the kth

term n

expression

16).

There

re two

sources f

naccuracywith

his

linearization. irst, thepresentvalue of all

future

ividends

starting

with

time

t+k is

not

exactly

k- 'E(d

)/E(rF).

Second,

ncreas-

ing

?k

by one

percentage

point does

not

cause

1/(1

+rt+k)

to fall

by

exactly

2

times

1

percent.

To

some

extent,

however, hese

errors n

the

effects n

p*

of

i-,r-+

't+2'

should

average

out,

and

one can

use

(16)

to

get an

idea of

the

effects f

changes

in

discount ates.

To

give an

impression

s

to the

accuracy

of

the

linearization

16),

I

computed

* for

data set2 in twoways: first sing 15) and

then

using

16),

with

the

same

terminal on-

dition

p1*979I

n place

of

the

unobserved

r

series,

I

used

the

actual

four-

ix-month

prime

ommercial

aper

rate

plus a

constant

to

give

t

the

mean

r of

Table 2.

The com-

mercial

paper

rate s

a

nominal

nterest

ate,

and

thus

one

would

expect

ts

fluctuations

represent

hanges in

inflationary

xpecta-

tions

s well

as

real

interest

ate

movements.

I

chose

t

nonetheless,

ather

rbitrarily,

s a

serieswhichshows muchmore fluctuation

than

one

would

normally

xpect

o

see

in

an

TABLE

2-

SAMPLE

STATISTICSFOR

PRICE

AND

DIVIDEND

SERIES

Data

Set 1:

Data

Set2:

Standard Modified

and

Dow

Poor's

Industrial

Sample Period:

1871-1979

1928-1979

1)

E(p)

145.5

982.6

E(d)

6.989

44.76

2) r

.0480

0.456

r2

.0984

.0932

3) b=lnX

.0148

.0188

o(b)

(.0011)

(1.0035)

4)

cor(p,

p*)

.3918

.1626

a(d)

1.481

9.828

Elements f

nequalities:

Inequality1)

5)

a(p)

50.12

355.9

6)

a(p*) 8.968

26.80

Inequality11)

7)

a(Ap+d1-ip

1)

25.57

242.1

min(a)

23.01

209.0

8)

a(d)/Irj

4.721

32.20

Inequality

13)

9)

a(Lp)

25.24

239.5

min(a)

22.71

206.4

10)

a(d)/1/r

4.777

32.56

Note: In

this

table,

E

denotes

sample

mean,

a

denotes

standard eviationnd6 denotes tandard rror.Min a)

is the

ower

bound

on

a

computed

s

a

one-sided

x2

95

percent onfidence

nterval. he

symbols

,

d, r,

F2,

b,

and

p*

are

defined

n

the

text.

Data

sets are

described

n

the

Appendix.

nequality1)

in

the text

sserts

hatthe

standard eviation

n

row

5

shouldbe less

thanor

equal

to

that

n

row

6,

inequality

11)

that

in

row 7

should

be

less than

or

equal to

that

n

row

8,

and

inequality

(13) that

in

row 9

should

be

less

than that n

row

10.

expected real

rate.

The

commercial

paper

rate

ranges,

n

this

ample,

from

.53 to

9.87

percent. t

stayedbelow

1

percent oroveradecade (1935-46) and, at

the

end of

the

sample,

stayed

generallywell

above

5 per-

cent for

ver a

decade.

In

spite

of

this

rratic

behavior,

he

correlation

oefficient

etween

p*

computed

from

15)

and

p*

computed

from

16) was

.996, and

a(p *)

was

250.5 and

268.0

by

15)

and

(16),

respectively.

hus the

linearization

16) can

be

quite

accurate.

Note

also

that

while

these

arge

movements n

i-

cause

p*

to

move

much

more

than

was

observed

n

Figure

2,

a(

p*)

is

still

ess

than

halfof a( p). This suggests hatthevariabil-

ity

i-

that

is

needed

to

save

the

efficient






432

THE

AMERICAN

ECONOMIC

REVIEW

JUNE

1981

markets

odel s

much

arger

et, s we

shall

see.

To

put a

formal

ower

bound

on

a(r)

given he

variabilityf

Ap, note

that

16)

makes

fl*

the present value of

zt,

z

where

t-d

-

PE(d)/E(Q).

We

thus now

from

(13)

that

2E(F)var(A

p)<var(z).

Moreover,rom he

definitionf z

we

know

that

var(z)<var(d)+2a(d)a(F)E(d)/E(r)

+

var(Q)E(d)2/E(F)2

where

the

equality

holds

f

dt

and

Ft

re

perfectly

egatively

correlated.

ombining

hese wo

nequalities

and

solving or

(r) one

finds

(17)

(r)YE~(r)a(Ap)-(d)

)E(r)/E(d

)

This

nequality

uts lower

ound

on

a(r)

proportional

o

the

discrepancy

etween he

left-handide

and

right-hand

ide of

the

inequality

13).'7

t will

be

used to

examine

the

ata

n

the

next

ection.

V.

Empirical

vidence

Theelementsfthe nequalities1), (11),

and

13)

are

displayed

or

he

wo

data sets

(described

n

the

Appendix)

n

Table 2. In

both

data

sets,the

long-run

xponential

growth

ath

was

estimated

y

regressing

ln(P1) on a

constant nd

time.

Then

A

n

(3)

was

set qual

to b

where

is the

oefficient

oftime

Table

2). The

discount

ate

used o

compute

*

from

4) is

estimated

s

the

average divided

y

the

average .l8 The

terminal

alue f

p*

is

taken

s

average .

With

data set

1, the

nominal rice

and

dividend eries re thereal Standard nd

Poor's

Composite

tock

ricendex

nd

the

associated

ividend

eries.The

earlier

b-

servationsor

his

eries re

due

to

Alfred

Cowles

who

aid that he ndex

s

intendedo

represent,

gnoringhe

le-

ments f

brokerage

harges nd

taxes,

whatwould avehappened oaninves-

tor's

funds

f

he

had

bought, t

the

beginning

f

1871,

ll

stocks

uoted

n

the

NewYork

Stock

xchange,

llocat-

ing

his

purchases

mong he

ndividual

stocks n

proportion

o

their

total

monetary

alue

nd

each month

p

to

1937

had

by the

same

criterion

edis-

tributed

is

holdings

mong

ll

quoted

stocks.

[p. 2]

In

updating

is

series,

tandard

nd Poor

later

estricted

he

ample

o

500

stocks, ut

theseries ontinues o be valueweighted.

The

advantage

o

this

eries s

its

compre-

hensiveness. he

disadvantage

s

that

the

dividends

ccruing

o the

portfolio

t one

point

of

time

may

not

correspond

o

the

dividends

orecasted

y holders f

theStan-

dard

nd Poor's

portfoliot an

earlier

ime,

due

to the

hange

n

weighting

f the

tocks.

There s

no

way

o

correct

his

isadvantage

without

osing

omprehensiveness.

he

origi-

nal

portfoliof

1871

s

bound o

become

relatively

mallernd

smaller

ample

f

U.S.

commontocks s time oeson.

With

data

set

2, the

nominal

eries

re

a

modified ow

Jones

ndustrial

verage

nd

associated

ividend

eries.

With

his

ata

set,

the

dvantagesnd

disadvantages

f

data

et

1

are

reversed.

y

modificationsn

theDow

Jones

ndustrial

verage

ssure

that

this

series

reflects

he

performance

f

a

single

unchanging ortfolio.

he

disadvantage

s

that

the

performancef

only

30

stocks s

recorded.

Table

2

reveals

hat all

inequalities

re

dramaticallyiolated ythe ample tatistics

for

both

data

sets.

The

eft-hand

ideof

the

inequality

s

always

t

least

five

times s

great

s

the

right-hand

ide, nd as

much

s

thirteen

imes s

great.

The

violation

f

the

nequalities

mplies

that

innovations"n

price s

we

measure

them

an

be

forecasted.n

fact,

f

we

regress

t+

Pt+l

onto

a

constant

nd)

pt,

we

get

significant

esults:

coefficient

f

pt

of

-.1521

(t=

-3.218,

R2

=.0890) for

data set

1

and a coefficient f -.2421 (t= -2.631,

R2=.1238) for

data

set 2.

These

resultsre

'7In derivinghe

nequality

13) t

was

assumed

hat

d,

was

known

t time

,

so

by

analogy his

nequality

would

e

based

on

the

ssumption

hat

,

s

known

t

time .

However,

ithout

his

assumption

he

same

inequality

ould e

derived

nyway.

he

maximum

on-

tribution

f

t

to

the

variance f

A

P

occurs

when

t

s

known

t

time .

18JThis

s not

quivalent

o the

verage

ividend

rice

ratio,whichwas slightlyigher.0514fordata set 1,

.0484

for

ata et

2).






VOL.

71 NO. 3

SHILLER:

STOCK PRICES

433

notdueto the epresentation

fthe ata as a

proportion

f the ong-runrowth

ath. n

fact,

f the

holding

period return

H,

is

regressed

n a constant nd the

dividend

price atio ,/P,,weget esultshat reonly

slightly

ess

ignificant:

coefficient

f

3.533

(t=2.672,

R2

=.0631) fordata set

1 and a

coefficient

f4.491

t=

1.795,

R2 =

.0617) or

data set

2.

Theseregression

ests,while technically

valid,may

not

be

as generallyseful

for

appraising

he

validity

f the

model s are

thesimple olatility

omparisons.irst,

s

noted above,

the regression

estsare not

insensitiveo data

misalignment.uch ow

R2

mighte theresult f dividendr com-modity rice ndexdata errors.econd, l-

though

he

model

s

rejected

n

these

very

long amples,

he estsmaynot

be powerful

if

we confined

urselveso shorteramples,

forwhich hedata

are more ccurate,s

do

most

researchers

n

finance,

hile olatility

comparisons

ay

be muchmorerevealing.

To see this, onsider

stylizedworld

n

which for

the sake

of argument)he di-

vidend eries

,

is absolutelyonstant

hile

the

price

eries ehaves s

in

our data set.

Since the actual dividend eries s fairly

smooth,

ur

tylized

orld

s

nottoo

remote

from ur

own.

f

dividends

,

are

absolutely

constant,

owever,

t shouldbe

obvious

o

themost asual

nd

unsophisticated

bserver

by volatilityrguments

ike

hose

madehere

that the

efficient arketsmodel

mustbe

wrong.

ricemovementsannot

eflect ew

informationbout

dividends

f

dividends

never

hange. et regressions

ikethose un

above

will

have imited

ower

o

reject

he

model.

f

the alternative

ypothesis

s, say,

that

p

pfl11

+E ,

wherep is close to but

less than ne,

then

hepower

f thetest

n

short

amples

ill

e

very

ow.

n

this

tylized

worldwe are testing

or he stationarityf

the

,

series,

or

which,

s

we

know, ower

s

low

n

short

amples.'9

or example,

f

post-

wardata

from,ay,1950-65were hosen

a

period ften

sed

n recent inancial arkets

studies)

when he tockmarket as

drifting

up, then

learly heregression

ests

will

not

reject. ven nperiods howing reversalf

upward

rifthe

ejection

aynotbesignifi-

cant.

Usingnequality17),

wecan computeow

big

the standard eviation f

real

discount

rateswouldhave to

he

to possibly ccount

for the discrepancy

(/?p)-a(d)/(2r)'/2

between able

2

results

rows

and 10) and

the nequality13).

Assuming

able

2 r

row

2) equals

E(r) and that sample

variances

equal populationariances,

e find hat he

standard eviationf

,

wouldhavetobe atleast 4.36 percentage oints ordata set 1

and 7.36 percentage oints

fordata set 2.

These re very argenumbers.

f we take, s

a normal

ange

or

r,

mplied y

these

ig-

ures,

+?2 standard eviation ange round

the

real nterestate

given

n

Table

2,

then

thereal nterest

ate

r,

wouldhave to range

from 3.91

to 13.52percent ordata

set

1

and

-8.16

to

17.27

percent

or

data set

2

And these

ranges reflectowest possible

standard eviations

hich re onsistentith

themodel nly ftherealratehas thefirst-

order

autoregressive

tructure

nd

perfect

negativeorrelation

ith ividends

These stimatedtandard

eviationsf ex

ante

eal nterest

ates

re

roughlyonsistent

with the results f the simpleregressions

noted bove.

n

a regression

f

H,

on

D,/P,

and a

constant,

he tandard

eviationfthe

fitted alue f

H,

s 4.42 and 5.71percent

or

data sets

1

and

2, respectively.

hese

arge

standard

eviationsre consistent ith he

low

R2

because he tandard

eviationf

H,

is so muchhigher17.60and 23.00percent,

respectively).

he regressions

f

51p,

n

p,

suggest

higher

standard

deviations of

expected

eal interest ates.

The

standard

deviation f thefitted

alue divided

y

the

average

detrended

rice

s

5.24

and 8.67

percent

or ata sets

1

and

2, respectively.

VI.

Summary

ndConclusions

We have een hat

measuresf stock

rice

volatility

ver he

past

centuryppear

o

be

far too high-five to thirteen imestoo

'9If

dividends

reconstant

letus

say

dt

0) then

testof

the model

by a regression

f 8,+

pt+I

on

pt

amounts

o

a regression

f

pt+

on

pt

with

he null

hypothesis

hat

the coefficient

f

pt

is

(1

+

r).

This

appears

o be

an explosive

odel orwhich

-statistics

arenot alidyet ur ruemodel, hichneffectssumes

a(d)#=O,

s

nonexplosive.






434

THE

AMERICAN

ECONOMIC

REVIEW

JUNE

1981

high-

to be

attributedo

new

nformation

about

future

eal

dividendsf

uncertainty

about

future

ividends

s

measured

y the

sample

tandard

eviationsf realdividendsaround their ong-runxponential rowth

path.The

ower

ound

f a

95

percent ne-

sided

x2

onfidence

nterval

or

he

tandard

deviation f

annual

changes

n

real

stock

prices s

over

five

times

higher

han

the

upper

ound

llowed

y our

measure f

the

observed

ariability

f real

dividends.

he

failure f

the

fficient

arkets

odel s

thus

so

dramatic

hat

t would

eem

mpossibleo

attribute

he

failure

o such

things s

data

errors,

rice

ndex

problems,r

changes

n

tax aws.

One

way of

saving he

general

otion

f

efficient

arkets

ould

be to

attributehe

movementsn

stock

prices

to

changes n

expected

eal

nterest ates.

ince

expected

real

nterest

ates

re

not

directly

bserved,

such a

theoryan

not be

evaluated

tatisti-

cally nless

ome

ther

ndicator

freal

rates

is

found.

have

shown,

owever,

hat

he

movementsn

expected

eal

interest

ates

that

would

ustifyhe

variability

n

stock

prices re

very

arge-

much

arger

han he

movementsnnominalnterestates ver he

sample eriod.

Another

ay

of

saving

he

general

otion

of

efficient

arketss to

say

that ur

mea-

sure

of the

uncertainty

egarding

uture

i-

vidends-

the

sample tandard

eviation f

the

movementsf

real

dividends

round heir

long- un

exponential

rowth

path un-

derstates

he

true

uncertainty

bout

future

dividends.

erhaps

he

market as

rightfully

fearful

f

much

argermovementshan

ctu-

allymaterialized.

ne s

led to

doubt

his,

f

after centuryfobservationsothingap-

penedwhich

ould

remotely

ustify

he

tock

price

movements.he

movements

n

real

dividends

he

market

earedmust

avebeen

many

imes

arger

han hose

bservedn

the

Great

Depression

f

the

1930's,

s was

noted

above.

Since

the

market id

not know n

advance

with

ertainty

he

growth

ath

nd

distributionf

dividends

hatwas

ultimately

observed,

owever,

ne

cannot e

sure

hat

they

were

wrong

o consider

ossible

major

events hich idnotoccur. uch nexplana-

tion

f

the

olatility

f tock

rices,

owever,

is

"academic,"

n

thatt

relies

undamentally

on

unobservables

nd

cannot

be

evaluated

statistically.

APPENDIX

A.

Data

Set 1:

Standard

nd Poor

Series

Annual

1871-

979.

The

price

eries

,

is

Standard nd

Poor's

Monthly

omposite

Stock

rice ndex

or

January

ivided

y

the

Bureau

of

Labor

Statistics

holesale

rice

index

JanuaryWPI

startingn

1900,

nnual

average

WPI

before

900

scaled

to

1.00 n

the

base

year

1979).

Standard

nd

Poor's

Monthly omposite tockPrice ndex s a

continuation

f

the

Cowles

Commission

Common

tock

ndex

developed

y

Alfred

Cowles nd

Associates

nd

currently

s

based

on

500

stocks.

The

Dividend

eries

D,

is

total

dividends

for he

alendar

ear

ccruing

o

the

ortfolio

represented

y

the

stocks n

the

ndex di-

vided

by

the

averagewholesale

rice

ndex

for

he

year

annual

verage

WPI

scaled

to

1.00

n

thebase

year

1979).

tarting

n

1926

these

total dividends

re

the series Div-

idendsper share... 2monthsmovingotal

adjusted

o

ndex"

rom

tandard

nd

Poor's

statistical

ervice.

or 1871

to

1925,

total

dividends

re

Cowles

eries a-1

multiplied

by

1264

to

correct or

hange

n

base

year.

B. Data

Set

2:

Modified

ow

Jones

Industrial

verage

Annual

928-1979. ere

P,

and

D,

refer

o

real

price

nd

dividends

f the

portfolio

f

30

stocks

omprising

he

ample

or

heDowJonesndustrialverage hentwascreated

in

1928.

Dow

Jones

verages

efore

1928

exist,

utthe

30

industrials

eries

was

begun

in

that

year.The

published

ow

Jones n-

dustrial

Average,

owever,

s

not

ideal

in

that

tocks

re

dropped

nd

replaced

nd n

that

he

weightingiven

n

individualtock

is

affected

y plits.

f

the

riginal 0

stocks,

only

17

were

till

ncluded

n

theDow

Jones

Industrial

verage t the

ndof

our

ample.

The

published

ow

Jones

ndustrial

verage

is thesimple umofthepricepershare f

the

30

companies

ivided

y

a

divisor

hich






VOL.

71 NO.

3

SHILLER:

STOCK

PRICES

435

changes

hrough

ime. hus,

f

a

stock

plits

twofor

one,

thenDow

Jones ontinues

o

include

nly

one

share

but changes

he

di-

visor oprevent sudden rop n theDow

Jones

verage.

To produce

he series

sed

n this

paper,

the

Capital

Changes

Reporter

as

used

to

trace

hanges

n the ompanies

rom

928

o

1979.

Of

the

original

0

companies

f

the

Dow

Jones

ndustrial

verage,

t the

nd

of

our

ample

1979),

hadthe dentical

ames,

12 had

changed

nly

heir

ames,

nd

9

had

been

acquired,

merged

r consolidated.

or

these

atter ,

theprice

nd

dividend

eries

are

continued

as the price

and

div-

idend f the haresxchangedy the cquir-

ing

orporation.

n only

ne

case

was a

cash

payment,

long

with

hares

f the

cquiring

corporation,

xchanged

or he

hares

f

the

acquired

orporation.

n

this ase,

the

price

and

dividend

eries

were

continued

s

the

price

nddividend

f

the

shares

xchanged

by

the cquiring

orporation.

n four

ases,

preferred

hares

f

the cquiring

orporation

were

among

shares

exchanged.

Common

shares

f

equal

value

were substituted

or

these

n

our

series.

he number

f shares

f

eachfirmncludednthe otal s determined

by

the

plits,

nd

effective

plits

ffected

y

stock

ividends

nd

merger.

he

price

eries

is the

value

of

all

these

hares

n

the

ast

trading

ay

of

the

preceding

ear,

s

shown

on the

Wharton

chool's

Rodney

White

Center

Common

tock

tape.

The

dividend

series

s the

total

for

he

year

of

dividends

and

the ash

value

of other istributions

or

all

these

hares.

he

price

nd

dividend

eries

were

eflated

sing

he

ame

wholesale

rice

indexess indataset1.

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C.

Amsler,

An American

onsol:

A

Reex-

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heory

f

the

Term

Structure

f Interest

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ichigan

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Univ.

1980.

S. Basu,

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ime

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Dividend

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the

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J. Bus.,

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Oct. 1961,

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nd

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Inflation,

ational

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1979,

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J.

Pesando,

Time Varying

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Univ.

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July

1979.

P.

A. Samuelson,

Proof

that

Properly

Dis-

counted

Present

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Assets

Vibrate

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1977.

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1981

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Historical

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1979,

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1981 Do Stock Prices Move Too Much to Be Justified by Subsequent Changes in Dividends

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Transcript of 1981 Do Stock Prices Move Too Much to Be Justified by Subsequent Changes in Dividends