**2019 | 37 views | 29 Pages | 473.23 KB**

Permanent and Temporary Components of Stock Prices Eugene F. Fama and Kenneth R. French University of Chicago A slowly mean-reverting component of stock prices tends to induce negative autocorrelation in returns. The autocorrelation is weak for the daily and weekly holding periods common in market efficiency

Permanent and Temporary Components of

Stock Prices

Eugene F Fama and Kenneth R French

Universityof Chicago

A slowly mean reverting component of stock prices tends to induce

negative autocorrelation in returns The autocorrelation is weak for

the daily and weekly holding periods common in market efficiency

tests but stronger for long horizon returns In tests for the 1926 85

period large negative autocorrelations for return horizons beyond a

year suggest that predictable price variation due to mean reversion

accounts for large fractions of 3 5 year return variances Predict

able variation is estimated to be about 40 percent of 3 5 year return

variances for portfolios of small firms The percentage falls to

around 25 percent for portfolios of large firms

I Introduction

Early tests of market efficiency examined autocorrelations of daily

and weekly stock returns Sample sizes for such short return horizons

are typically large and reliable evidence of nonzero autocorrelation is

common Since the estimated autocorrelations are usually close to 0 0

however most studies conclude that the implied predictability of re

turns is not economically significant Fama 1970 summarizes this

early work which largely concludes that the stock market is efficient

Summers 1986 challenges this interpretation of the autocorrela

tion of short horizon returns He argues that the claim in common

The comments of Craig Ansley David Booth John Cochrane John Huizinga

Shmuel Kandel Robert Kohn Richard Leftwich Merton Miller Sam Peltzman

Charles Plosser Rex Sinquefield and especially G William Schwert are gratefully

acknowledged This research is supported by the National Science Foundation Fama

the Center for Research in Security Prices French and Batterymarch Financial Man

agement French

Journal of Political Economy 1988 vol 96 no 21

1988 by The University of Chicago All rights reserved 0022 3808 88 9602 0005 01 50

COMPONENTS OF STOCK PRICES 247

models of an inefficient market is that prices take long temporary

swings away from fundamental values which he translates into the

statistical hypothesis that prices have slowly decaying stationary com

ponents He shows that autocorrelations of short horizon returns can

give the impression that such mean reverting components of prices

are of no consequence when in fact they account for a substantial

fraction of the variation of returns

Our tests are based on the converse proposition that the behavior of

long horizon returns can give a clearer impression of the importance

of mean reverting price components Specifically a slowly decaying

component of prices induces negative autocorrelation in returns that

is weak for the daily and weekly holding periods common in market

efficiency tests But such a temporary component of prices can induce

strong negative autocorrelation in long horizon returns

We examine autocorrelations of stock returns for increasing hold

ing periods In the results for the 1926 85 sample period large nega

tive autocorrelations for return horizons beyond a year are consistent

with the hypothesis that mean reverting price components are impor

tant in the variation of returns The estimates for industry portfolios

suggest that predictable variation due to mean reversion is about 35

percent of 3 5 year return variances Returns are more predictable

for portfolios of small firms Predictable variation is estimated to be

about 40 percent of 3 5 year return variances for small firm port

folios The percentage falls to around 25 percent for portfolios of

large firms

Our results add to mounting evidence that stock returns are pre

dictable see e g Bodie 1976 Jaffe and Mandelker 1976 Nelson

1976 Fama and Schwert 1977 Fama 1981 Campbell 1987 French

Schwert and Stambaugh 1987 Again this work focuses on short

return horizons De Bondt and Thaler 1985 are an exception and

the common conclusion is that predictable variation is a small part

usually less than 3 percent of the variation of returns There is little

in the literature that foreshadows our estimates that 25 45 percent of

the variation of 3 5 year stock returns is predictable from past re

There are two competing economic stories for strong predictability

of long horizon returns due to slowly decaying price components

Such price behavior is consistent with common models of an irrational

market in which stock prices take long temporary swings away from

fundamental values But the predictability of long horizon returns

can also result from time varying equilibrium expected returns gen

erated by rational pricing in an efficient market Poterba and Sum

mers 1987 show formally how these opposite views can imply the

same price behavior The intuition is straightforward

248 JOURNAL OF POLITICAL ECONOMY

Expected returns correspond roughly to the discount rates that

relate a current stock price to expected future dividends Suppose

that investor tastes for current versus risky future consumption and

the stochastic evolution of the investment opportunities of firms re

sult in time varying equilibrium expected returns that are highly

autocorrelated but mean reverting Suppose that shocks to expected

returns are uncorrelated with shocks to rational forecasts of divi

dends Then a shock to expected returns has no effect on expected

dividends or expected returns in the distant future Thus the shock

has no long term effect on expected prices The cumulative effect of a

shock on expected returns must be exactly offset by an opposite ad

justment in the current price

In this scenario autocorrelated equilibrium expected returns lead

to slowly decaying components of prices that are indistinguishable

from the temporary price components of an inefficient market at

least with univariate tests like those considered here More informed

choices between the competing explanations of return predictability

will require models that restrict the variation of expected returns in

plausible ways for example models that restrict the relations between

the behavior of macroeconomic driving variables and equilibrium ex

pected returns

Finally tests on long horizon returns can provide a better impres

sion of the importance of slowly decaying stationary price compo

nents but the cost is statistical imprecision The temporary compo

nent of prices must account for a large fraction of return variation to

be identified in the univariate properties of long horizon returns We

find reliable evidence of negative autocorrelation only in tests on

the entire 1926 85 sample period and the evidence is clouded by the

statistical issues changing parameters heteroscedasticity etc that

such a long time period raises

II A Simple Model for Stock Prices

Let p t be the natural log of a stock price at time t We model p t as

the sum of a random walk q t and a stationary component z t

p t q t z t 1

q t q t 1 X t 2

where p is expected drift and mq t is white noise Summers 1986

argues that the long temporary price swings assumed in models of an

inefficient market imply a slowly decaying stationary price compo

COMPONENTS OF STOCK PRICES 249

nent As an example he suggests a first order autoregression ARI

z t 4 z t 1 E t 3

where E t is white noise and is close to but less than 1 0

The model 1 3 is just one way to represent a mix of random

walk and stationary price components The general hypothesis is that

stock prices are nonstationary processes in which the permanent gain

from each month s price shock is less than 1 0 Our tests are relevant

for the general class of models in which part of each month s shock is

permanent and the rest is gradually eliminated The tests center on

the fact that the temporary part of the shock implies predictability

negative autocorrelation of returns

A The Implicationsof a StationaryPrice Component

Since p t is the natural log of the stock price the continuously com

pounded return from t to t T is

r t t T p t T p t

q t T q t z t T z t

The random walk price component produces white noise in re

turns We show next that the mean reversion of the stationary price

component z t causes negative autocorrelation in returns

The slope in the regression of z t T z t on z t z t T the

first order autocorrelation of T period changes in z t is

p T cov z t T

z t z t z t T 5

o 2 Z t T z t

The numerator covariance is

cov z t T z t z t z t T u2 z 2 cov z t z t T

cov z t z t 2T

The stationarity of z t implies that the covariances on the right of 6

approach 0 0 as T increases so the covariance on the left approaches

cr2 z The variance in the denominator of the slope

U2 z t T z t 2U2 z 2 cov z t T z t 7

approaches 2 T

2 z We can infer from 6 and 7 that the slope in the

regression of z t T z t on z t z t T approaches 0 5 for

250 JOURNAL OF POLITICAL ECONOMY

The slope p T has an interesting interpretation used often in the

empirical work of later sections If z t is an AR 1 the expected change

from t to T is

E z t T z t 4T 1 Z t 8

and the covariance in the numerator of p T is

cov z t T z t z t z t T 1 2 2T 2 z

1 XT 2UF2 Z

With 8 and 9 we can infer that the covariance is minus the variance

of the T period expected change cr2 Etz t T z t Thus when

z t is an ARI the slope in the regression of z t T z t on z t

z t T is minus the ratio of the variance of the expected change in

z t to the variance of the actual change This interpretation of the

slope is a valid approximation for any slowly decaying stationary pro

Equation 8 shows that when is close to 1 0 the expected change

in an ARI slowly approaches z t as T increases Likewise the slope

p T is close to 0 0 for short return horizons and slowly approaches

0 5 This illustrates Summers s 1986 point that slow mean rever

sion can be missed with the short return horizons common in market

efficiency tests Our tests are based on the converse insight that slow

mean reversion can be more evident in long horizon returns

B The Propertiesof Returns

Since we do not observe z t we infer its existence and properties

from the behavior of returns Let P T be the slope in the regression

of the return r t t T on r t T t If changes in the random walk

and stationary components of stock prices are uncorrelated

T cov r t t T r t T t 10

p T uf2 z t T z t

u2 z t T z t u2 q t T q t ld

For long return horizons the interpretation of the slope as the proportion of the

variance of the change in z t due to the expected change is valid for any stationary

process If z t is a stationary process with a zero mean the expected change from t to T ap

proaches z t as T increases and the variance of the expected change approaches

r2 z The ratio of the long horizon variance of the expected change in z t Cr z to the

long horizon variance of the actual change 2 2 z is thus 0 5 the negative of the long

horizon value of p T

COMPONENTS OF STOCK PRICES 251

r2 Etz t T z t lOb

cr2 r t T t

Expression lOb highlights the result that 3 T measures the propor

tion of the variance of T period returns explained by or predictable

from the mean reversion of a slowly decaying price component z t

Expression 10a helps predict the behavior of the slopes for increas

ing values of T If the price does not have a stationary component the

slopes are 0 0 for all T If the price does not have a random walk

component f T p T and the slopes approach 0 5 for large

values of T

Predictions about the slope f T are more complicated if the stock

price has both random walk and stationary components The mean

reversion of the stationary component tends to push the slopes to

ward 0 5 for long return horizons while the variance of the white

noise component q t T q t pushes the slopes toward O O Since

the variance of z t T z t approaches 2cr2 z as the return horizon

increases and the white noise variance grows like T the white noise

component eventually dominates Thus if stock prices have both

random walk and slowly decaying stationary components the slopes

in regressions of r t t T on r T t t might form a U shaped

pattern starting around 0 0 for short horizons becoming more nega

tive as T increases and then moving back toward 0 0 as the white

noise variance begins to dominate at long horizons

Finally existing evidence e g Fama and Schwert 1977 Keim and

Stambaugh 1986 Fama and French 1987 French et al 1987 sug

gests that expected returns are positively autocorrelated The nega

tive autocorrelation of long horizon returns due to a stationary com

ponent of prices is consistent with positively autocorrelated expected

returns For example the model 1 3 implies negatively autocor

related returns Poterba and Summers 1987 show however that if

the stationary price component z t in 3 is an ARI with parameter

0 0 the expected return is an ARI with parameter and so is

positively autocorrelated The economic intuition is that shocks to

expected returns discount rates can generate opposite shocks to

current prices and returns can be negatively autocorrelated when

expected returns are positively autocorrelated

III The Autocorrelation of Industry and Decile

Portfolio Returns

A The Data

The mix of random walk and stationary components in stock prices

can differ across stocks Firm size and industry are dimensions known

252 JOURNAL OF POLITICAL ECONOMY

to capture differences in return behavior see King 1966 Banz 1981

Huberman and Kandel 1985 We examine results for industry port

folios and for portfolios formed on the basis of size

The basic data are 1 month returns for all New York Stock Ex

change NYSE stocks for the 1926 85 period from the Center for

Research in Security Prices At the end of each year stocks are ranked

on the basis of size shares outstanding times price per share and

grouped into ten decile portfolios One month portfolio returns

with equal weighting of securities are calculated and transformed

into continuously compounded returns These nominal returns are

adjusted for the inflation rate of the U S Consumer Price Index CPI

and then summed to get overlapping monthly observations on

longer horizon returns Unless otherwise noted return henceforth

implies a continuously compounded real return

There is a problem with the decile portfolios Stocks with unusually

high or low returns tend to move across deciles from one year to the

next If unusual returns are caused by temporary price swings subse

quent reversals may be missed the tests may understate the impor

tance of stationary price components because of the movement of

stocks across deciles Since the problem is less severe for portfolios

that include all stocks we also show results for the equal and value

weighted portfolios of all NYSE stocks The value weighted market

portfolio summarizes the return behavior of large stocks while the

equal weighted portfolio is tilted more toward small stocks

Using Standard Industrial Classification codes we also form 17

industry portfolios with equal weighting of the stocks in a portfolio

One criterion in defining an industry is that it contains firms in similar

activities The other criterion is that the industry produces diversified

portfolios during the 1926 85 period Each of the 17 industries al

ways has at least seven firms 15 after 1929 and the number of firms

per industry is usually greater than 30 Within industries there is

little concentration of firms by size For example the average of the

decile ranks of the firms in an industry is typically between 4 0 and

7 0 Thus size and industry are not proxies and size and industry

portfolios can provide independent evidence on the behavior of long

horizon returns Details on the industry portfolios are available from

the authors

The tests center on slopes in regressions of r t t T on r t T t

The slopes are first order autocorrelations of T year returns Ordi

nary least squares OLS estimates have a bias that depends on the

true slopes sample sizes and the overlap of monthly data on long

horizon returns see Kendall 1954 Marriot and Pope 1954 Huizinga

1984 Proper bias adjustments when the true slopes are 0 0 prices do

not have stationary components are difficult to determine analyt

COMPONENTS OF STOCK PRICES 253

ically We use simulations constructed to mimic properties of stock

returns to estimate the bias adjustments see the Appendix The

simulations also show that when prices have stationary components

that generate negative autocorrelations on the order of those ob

served here simple OLS slopes have little bias We examine both OLS

and bias adjusted slopes

B Regression Slopesfor the 1926 85 Sample Period

Industries

Table 1 shows slopes in regressions of r t t T on r t T t for

return horizons from 1 to 10 years using the industry portfolio data

for the 1926 85 sample period As predicted by the hypothesis that

prices have stationary components negative slopes are the rule The

bias adjusted slopes are uniformly negative for return horizons from

2 to 5 years The unadjusted slopes are almost always negative for all

horizons The slopes reach minimum values for 3 5 year returns

and they become less negative for return horizons beyond 5 years

This U shaped pattern is consistent with the hypothesis that stock

prices also have random walk components that eventually dominate

long horizon returns Estimated slopes not shown for nominal re

turns are usually within 0 04 of those for real returns

The slopes for 3 4 and 5 year returns are large in magnitude and

relative to their standard errors The average values of the bias

adjusted slopes for 3 4 and 5 year returns are 0 30 0 34 and

0 32 the averages of the unadjusted slopes are 0 38 0 45 and

0 45 Expression lOb says that the slope measures the proportion

of the variance of T year returns due to time varying expected re

turns generated by slowly decaying stationary price components The

slopes for the industry portfolios thus suggest that these time varying

expected returns average between 30 percent and 45 percent of the

variances of 3 5 year returns

Moreover the limiting argument for the slopes in Section II says

that the variance of the expected change in the stationary price com

ponent z t approaches half the variance of the long horizon change

in z t Thus regression slopes that average between 0 30 and 0 45

estimate that on average between 60 percent and 90 percent of the

variances of 3 5 year industry returns are due to the stationary price

component z t

A caveat is in order The hypothesis that prices contain both

random walk and slowly decaying stationary components predicts a

U shaped pattern of slopes for increasing return horizons This pro

vides some justification for leaning toward extreme slopes to estimate

tn in r cn o o 1 C4 c C4

I I I I I I I I I I I I I I I I I

c s s O C1 00 0 t 00 t O

00 C4 l n n C4c C O O4 C

O K C1 n O G O Cq

I I I I I II I I I I I I I II I

n cn C4 n C14C4 VK O n O cnan Ioc cn n

I I I I I I I I I I I I I I I I I

o t Cj C j cn C4 in Ln C

c Xc Cj in

O IIA ItI II In In I

It I In I I In In II

v n GCq n C U c

X XI 1 IA CC

I I11I nI nI I nI I In It I I1 I I

00 00 00CA

a t r v cM N r r v r r rt n C 1 E00

U U V U fC

a C W M c r cc v cld cM

No cn n n v e ono i on I I

O II I I I I I I II IlIlI lI IIl

on n t cq qn a 1 nt n Q

m II I I I I I I I I I I I I I I I c

0II I I I I 0I I I I I i I I I I I z

z Cl GM Nd C13 CfNG sCTJ

O O4 O O4O0O O4O O O O OOOO245 O

e V Y m S S S i s

I I D I I I I I IE I I I I 0 o

v n n c v S DH m U S S O

256 JOURNAL OF POLITICAL ECONOMY

proportions of return variances due to the two components of prices

Since we do not predict the return horizons likely to produce extreme

slopes however using the observed extremes to estimate proportions

of variance probably overstates the importance of stationary compo

nents of prices

Moreover a pervasive characteristic of the tests is that small effec

tive sample sizes imply imprecise slope estimates for long horizon

returns The large standard errors of the industry slopes averaging

0 11 for 1 year returns and 0 26 for 10 year returns leave much

uncertainty about the true slopes and thus about the proportions of

variance due to the random walk and stationary components of

prices See the Appendix for pertinent details

There is no obvious pattern in the variation of the regression slopes

across industries There is a clearer pattern in the slopes for the decile

portfolios in table 2 Like the industry slopes the decile slopes are

negative and large for 2 5 year returns However the minimum

values of the slopes tend to be more extreme for lower smaller firm

deciles All the bias adjusted slopes less than 0 30 and all the unad

justed slopes less than 0 37 are generated by the equal weighted

market portfolio and deciles 1 7 Most of the 4 and 5 year bias

adjusted slopes for these portfolios are more than 2 0 standard errors

below 0 0 The value weighted market and the larger firm deciles 9

and 10 produce no bias adjusted slopes more than 2 0 standard er

rors below 0 0

Again perspective is in order The large standard errors of the

decile slopes between 0 13 and 0 20 for 3 5 year returns mean

that if stock prices have stationary components they must generate

large negative slopes and account for large fractions of variance to

be identified reliably even when the estimates use the entire 1926 85

sample period Nevertheless every decile produces a simple OLS

slope for 3 4 or 5 year returns more than 2 0 standard errors below

0 0 And the U shaped pattern of the slopes across return horizons

predicted by the hypothesis that prices have both random walk and

slowly decaying stationary components is observed for all the deciles

the industry portfolios and the two market portfolios

We conclude that the tests for 1926 85 are consistent with the

hypothesis that stock prices have both random walk and stationary

components The estimates suggest that stationary price components

account for large fractions of the variation of returns and that they

are relatively more important for small stock portfolios We recog

nize however that the imprecision of the tests implies substantial

COMPONENTS OF STOCK PRICES 257

uncertainty about any interpretation of the results The relevance of

this caveat is obvious in the subperiod results that follow

C SubperiodAutocorrelations

Because the regression slopes are not estimated precisely the results

for the 1926 85 period are in principle the strongest test of the

hypothesis that stock prices have stationary components There are

however reasons to examine subperiods First return variances drop

substantially after 1940 see Officer 1973 French et al 1987 The

variance changes make inference less precise even if the autocorrela

tions of returns are stationary Moreover the high variances of the

early years are associated with large price swings It is possible that the

large negative autocorrelations estimated for 1926 85 are a conse

quence of the early years

We have estimated the slopes in the regression of r t t T on

r t T t for the 30 year splits 1926 55 and 1956 85 and for the

longer 1946 85 and 1941 85 periods The estimates for 1941 85 are

in tables 3 and 4 We choose 1941 85 because it is the longest period

of roughly constant return variances The regression slopes it pro

duces are similar in magnitude and pattern to those for 1946 85 and

Like 1926 85 the 1941 85 period produces a general pattern of

negative autocorrelation of returns that is consistent with the hy

pothesis that prices have stationary components However the 194 1

85 bias adjusted slopes are typically closer to 0 0 and they do not

produce the strong U shaped pattern across return horizons observed

for 1926 85 Moreover large standard errors averaging 0 13 for

1 year industry portfolio returns and 0 27 for 8 year returns make

the hypothesis that prices contain no stationary components the true

slopes are 0 0 difficult to reject

Large standard errors make most hypotheses about subperiods dif

ficult to reject For example slope estimates for 1926 55 not shown

have an even stronger U shaped pattern than those for 1926 85

while estimates for 1956 85 also not shown are much like those for

1941 85 However the hypothesis that the slopes for 1926 55 and

1956 85 are equal cannot be rejected indeed large standard errors

make the hypothesis essentially untestable

In short the preponderance of negative slopes observed for all

periods shown and not shown is consistent with the hypothesis that

stock prices have stationary components that generate negative auto

correlation in long horizon returns Subperiod slopes suggest that the

negative autocorrelation is weaker stationary price components are

less important in the variation of returns after 1940 But reliable

O00 C 1crS cn Cu O 0

Lr C CA c 00 0 Ln Cz C

t cl c cn CA u O

o s o z I I I I I I I I I I I I

b n xC oo cq s in tc I oo t 1t

Ln I t 4 1 n cn CI cq cn

t 00 04 t O in Cq

tr cn C1 G

O0M MG MG AG MG MG

aS0 J 1 0 1

0 H c Nbt1

cn cin in in v

if i nM in rXC DttOsx

Ct cn cn C c cosGMG

I In I I I I I I CI I

n n cXc C X O in in C tss cn

Gt ac in if C

cn 1 o x t ckr

X cr cr c s

CGACr in C C r C r on X tin S

A t in C r X C t z