Permanent And Temporary Components Of Stock Prices-Books Download

Permanent and Temporary Components of Stock Prices

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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


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