Commonality In The Determinants Of Expected Stock
Returns*
by Robert A. Haugen** and Nardin L. Baker***
Journal of Financial Economics, Summer 1996
** Professor of Finance, University of California,
Irvine[1]
*** Grantham, Mayo, Van Otterloo, Boston, MA
* The authors would like to thank Michael Adler, Michael Brennan, Naifu Chen, John Comerford, Joe Dada, Eugene Fama, Dan Givoly, Campbell Harvey, Philippe Jorion, Josef Lakonishok, Harry Markowitz, Jay Ritter, Richard Roll, Mark Rubinstein, Jay Shanken, Neal Stoughton, and the participants of workshops at the University of Southern California, the University of California at Los Angeles and the Berkeley Program in Finance, as well as the referee for insightful comments and Darius Miller for computational assistance.
Commonality
in the Determinants of Expected Stock Returns
--
Abstract --
Evidence is presented that the
determinants of the cross-section of expected stock returns are stable in their
identity and influence from period to period and from country to country. The determinants are related to risk,
liquidity, price-level, growth potential, and stock price history. Out-of-sample predictions of expected
return, using moving average values for the payoffs to these firm
characteristics, are strongly and consistently accurate. Two findings, however, distinguish this
paper from others in the contemporary literature. First, the stocks with higher expected and realized rates of
return are unambiguously of lower risk than the stocks with lower returns. Second, we find that the important
determinants of expected stock returns are strikingly common to the major
equity markets of the world. Given the
nature of the tests, it is highly unlikely that these results may be attributed
to bias or data snooping. Consequently,
the results seem to reveal a major failure in the Efficient Markets Hypothesis.
Commonality In The Determinants Of Expected Stock
Returns
Evidence
is mounting that relative stock returns can be predictable with factors that
are inconsistent with the accepted paradigms of Modern Finance. DeBondt and Thaler (1985), Jegadeesh and
Titman (1993), Chopra, Lakonishok and Ritter (1992), and Jegadeesh (1990) show
that the return history of a stock contains useful information in predicting relative
returns. In addition, Fama and French
(1992), Lakonishok, Shleifer, and Vishny (1994) and Davis (1994) show that
future returns can be predicted by the relative sizes of (a) the current market
price of a stock and (b) the current values of accounting numbers such as book
value or earnings-per-share. The
reaction to this evidence has been strong; three interpretations have been
offered to explain the results.
Some
believe that the evidence is flawed and results, at least in part, from
bias. Kothari, Shanken, and Sloan
(1995), Brown and Goetzmann, (1995) and Brown, Goetzmann, and Ross (1995) cite
survival bias as a problem that can exaggerate predictive power. Black (1993), Merton (1988) and Lo and
MacKinlay (1990) suggest that the results can be the result of snooping the
data in some fashion prior to testing.
Others
take the view that, while success in prediction may be exaggerated to some
degree by the influence of various biases, the fundamental nature of the
results still stands, and it deserves the close attention of the field. We can divide those who take this position
into two groups. One group believes
that the differences are related to relative risk, while the other attributes
them to bias in pricing by the market.
Those
in the first group believe that the differentials in expected stock returns are
expected and required by investors. (See Fama and French, 1992 and 1993 and
Ball, Kothari, and Shanken 1995.) They
believe that the differentials are risk premiums. While they argue that the nature of the risk premiums seem
inconsistent with the predictions of the Capital Asset Pricing Model, they
claim that they may be consistent with other, multi-factor models. Thus, while this group believes that the new
results can lead to a rejection of the CAPM, in their view the efficient
markets theory remains intact.
The
second group, on the other hand, believes that the differentials in predicted
returns come as a surprise to investors. (See Chopra, Lakonishok and Ritter
(1992), Lakonishok, Shleifer, and Vishny (1994), and Haugen (1995)) The
differential returns are said to derive from market over or under reactions to
various events. Distortions in the patterns of realized
returns, caused by bias in the pricing of stocks, can mask the true
nature of the relation between expected return and risk, whatever its
nature. This group sees the results as
a major setback for the efficient markets hypothesis.
The
tests of forecasting power conducted in this paper minimize the various sources
of bias discussed in the literature.
Given our procedures and the size of the predictable return
differentials found, it seems unlikely that these differentials are merely
artifacts of bias in methodology. Since
the differences in realized returns are too large to be credibly called risk
premiums and since the high return deciles are not relatively risky, the
results also strongly favor the pricing bias hypothesis.
The
determinants of differential stock returns are surprisingly stable over time,
and the forecasting power of our expected return factor model is also
surprisingly high. We also find high
power in other countries.
Interestingly, there seems to be a great deal of commonality across
markets in firm characteristics that explain differences in expected
returns. This is true in spite of the
fact that the monthly “payoffs” to these characteristics are not significantly
correlated across the five countries examined.
Thus, the determinants of expected stock returns appear to be common
across different time periods and across different markets.
In
the next section we identify the sources of bias that can distort the results
of studies of predictive power in stock markets. Following this discussion, we move to a discussion of the nature
of the firm characteristics (factors) used to predict return. We then discuss our methodology, and finally
we present our results.
1. Sources of bias in predicting stock returns
As
discussed above, some have argued that reports of success in predicting
relative stock returns are flawed by several sources of bias. Our objective is to design a test, where the
effects of the problems discussed below are minimized to so that we are confident
that our results are real.
If
significant numbers of firms that have become individually inactive are
systematically excluded from a data base, the data can be said to suffer from survival
bias. To illustrate the bias,
consider studies of the performance of mutual funds. For simplicity, suppose all mutual funds have identical expected
rates of return (equal to that of the market index), but they have different
variability in return. Assume also
that, if performance falls below some threshold, a fund becomes inactive. Then the probability of reaching that
threshold will increase with the risk of the fund. If we observe the performance of only those funds that remain
active, we will tend to find that the performance of survivors exceeds the market's. We will also tend to find that performance
increases with the level of variability in return. Thus, it will appear that one can predict performance on the
basis of fund risk. For studies of
portfolios of individual firms, the nature of the bias is less clear because
many firms can disappear because of merger as well as failure. In either case, however, it is likely that
the overall returns to nonsurvivors will be abnormal. This being the case, if factors used in prediction are somehow
related to the probability of going inactive, failure to include inactive firms
in the data base will result in misleading estimates of predictive power. Survival bias is exacerbated by the nature
of firms that tend to be back filled in commercial data bases. Providers tend to add companies that have
significant market positions when the records are back filled. Thus, given two firms of identical size five
years prior to the back fill, the larger (and more successful) firm at the time
of the back fill is more likely to be added to the data base.
Look
ahead bias occurs when data items are
used as predictive factors, the values of which were unknown when the
predictions are assumed to be made.
Suppose, for example, that the earnings-to-price ratio (earnings yield) is
used as the predictive factor. If the
ratio is calculated with an earnings number that was not reported as of the
date of the prediction, the predictive power of the factor will be exaggerated. This is because the set of firms with
relatively high (low) earnings yields will include those with unexpectedly high
(low) last quarter numbers. Market
reactions to these numbers are likely to be positive (negative). Thus, high (low) earnings yields will be
associated with high (low) subsequent returns, even though there may be no true
predictive information in the number whatsoever.
A
phenomenon called bid-asked bounce can also instill bias in tests of
predictive power in equity markets.
Stocks trade at bid or asked prices, and returns are usually measured close-to-close. Suppose that the underlying market value of
a stock does not change during a month, t, but that the last trade of the month
was at the bid. Assume also that the
stock continues to remain constant in price during month t+1. There is roughly an even chance that it will
close at the asked price at the end of t-1 or t+1. Thus, assuming no change in the bid-asked spread, the measured
return will either be zero, or negative, for t, and either zero, or positive,
for t+1. Thus, returns measured over
closing prices may appear to be negatively auto correlated, even when they are
not. Thus, the existence of bid-asked
bounce can lead a researcher to falsely conclude that last period's return has
predictive power, even when successive stock returns are completely
uncorrelated.
Bias
associated with data snooping occurs when researchers (a) examine the
properties of a data base or the results of other studies of a data base, then
(b) build predictive models employing promising factors based in the previous
results, and then (c) test the power of their models on the same data
base. Since most researchers currently
employ the same data base of U.S. firms and publish and discuss their results,
this is both an important concern and a difficult problem to address. Nevertheless, the problem can be addressed
by employing data from markets that have not been studied extensively or by
attempting to predict over time periods that are new to analysis.
2. Firm characteristics (factors) that may
induce differentials in expected returns
Factor
models that employ firm characteristics to predict the second moment (the
variance) of stock returns (or statistics related to the second moment such as
volatility relative to a benchmark portfolio, market beta and residual
variance), have been applied by practicing analysts for decades.[2] In this study
we shall employ such a model to predict the first moment (the expected value)
of stock returns. Our model will employ
a variety of factors similar in number and nature to those employed in second
moment factor models.
In
more traditional tests of the determinants of the cross-section of expected
stock returns, empiricists have chosen factors based on theoretical models of
asset pricing (Fama and MacBeth, (1973)), or variables that have power in
explaining the covariances between stocks (Chen, 1983). If stock markets are perfectly liquid and
highly efficient, differences in risk should be the sole determinant of
differences in expected return.
However,
if stocks are heterogeneous in their liquidity and if pricing is biased
relative to the available information set, many nonrisk related variables can
be important in predicting the cross-section.
In light of this possibility, our predictions of expected stock returns
will be based on five classes of factors: risk, liquidity, price-level, growth
potential, and price-history.
Given
the price reactions to unexpected changes in market risk reported in
longitudinal studies (French, Schwert and Stambaugh (1987) and Haugen, Talmor
and Torous (1991)), differences in the risk of stocks are likely to have
predictive power in the cross-section.
Accepted paradigms point to specific risk variables, such as CAPM and
APT betas, that are theoretically appropriate variables for forecasting returns. However, as discussed by Haugen (1995), it
is becoming increasingly apparent that these models can have low power. In spite of this, we include the standard
market related beta[3] and betas related macro-economic variables. These include monthly percentage changes in
industrial production and inflation, the rate of return on 30 day Treasury
bills, as well as the differences in return between (a) a 30 year Treasury bond
and a 30 day Treasury bill, and (b) the Salomon Brothers composite corporate
bond index. and a two security portfolio of Treasury bonds with the same
duration.
In addition, we shall also include a stock's own variance, its residual
variance, the ratio of total debt to stockholders’ equity, income available for
the payment of interest relative to total interest charges, and the standard
error of preceding five year, time trended, quarterly earnings-per-share scaled
by average earnings-per-share over the trailing period. Collectively, we expect to find that the
payoffs to the risk variables is positive.
Differences
in the liquidity of stocks are also potentially important. In rebalancing their portfolios, traders
must buy at asked prices and sell at bid prices. The bid-asked spread serves as part of the cost of trading. The market impact of a trade is also
important. Individual stocks have
widely differing degrees of liquidity.
To keep the expected rates of return, net of trading costs,
commensurate, stocks must have gross expected returns that reflect the
relative cost of trading (See Stoll and Whaley (1983) and Amihud and Mendelson
(1986)). Factors associated with
liquidity include price-per-share, the annual average volume of daily trading
relative to annual average total market capitalization (price-per-share times
the total number of shares outstanding), the five year time trend in this
variable, and contemporary total market capitalization. Overall, an investor should expect the
payoffs to the various factors that represent differentials in liquidity to be
negative, with the liquid stocks having the lower expected returns.
Factors
related to price-level indicate the level of current market price
relative to various accounting numbers.
These measures indicate whether a stock is selling cheap or dear. Factors representing cheapness in price
include contemporary market price relative to earnings-per-share, cash
flow-per-share, dividends-per-share, book value-per-share, and sales-per-share. The trailing five year time trends and
variability about trend in these variables are also included as factors. We include the time trends to differentiate
firms that are declining in their profitability, from those that are emerging
or recovering. Recent research has
shown that stocks with low ratios of price to current cash flows have earned
relatively high rates of return in recent decades. The source of these higher returns is the subject of much
controversy.
Some
(Chan and Chen (1991) and Fama and French (1992)) believe that value stocks are
"fallen angels" and therefore are more risky. They believe the premium returns to these
stocks are expected and required. Given
that this is true, factors showing cheapness in price actually belong in the
risk category discussed above.
Others
(Chopra, Ritter and Lakonishok (1992), Lakonishok, Shleifer and Vishny (1994),
and Haugen (1995)), believe that the premium returns to value stocks are
unexpected and systematically surprise investors. They believe that investors over react to the past records of
success and failure by firms.
Proponents of over-reactive markets believe that the forces of
competition in a line of business tend to quickly drive profits to normal
levels. By projecting prolonged rapid
growth, investors in growth stocks can drive prices too high. As the forces of competition come into play
faster than these investors believe, they tend to be disappointed by the
earnings reports of growth stocks. The
future dividends and capital gains on these stocks tend to be smaller than
expected and returns tend to be relatively low. The converse tends to be true of value stocks.
Irrespective
of whether these payoffs spring from risk or over reaction, they should be
positive, with the stocks having the highest current cash flows in relation to
market price having the greatest expected rates of return.
Factors
related to growth potential indicate the probability for faster (or
slower) than average future growth in a stock's earnings and dividends. Within the cross-section, relatively
profitable firms will tend to grow faster, at least until competitive entry
into their lines of business forces profits to normal levels. Based on the assumption that firms that are
currently relatively profitable have greater potential for future growth, we
include several measures of profitability as predictive factors. They include the ratios of net earnings to
book equity, operating income to total assets, operating income to total sales,
total sales to total assets, and the trailing, five year time trends in these
variables. We also include the
trailing, five year time trend in earnings-per-share, expressed as a percentage
of average earnings over the five year period.[4] Given the
size of the factors that reflect the price-level of a stock, the greater the growth potential for profits
and dividends, the greater the expected future rate of return. If the market mistakenly assigns identical
prices to stocks with differing growth potential, one would expect the payoffs
to the growth potential factors to be collectively positive.
Technical
factors describe the price history of a stock. Recent research shows three relations between the history of
return and future expected return.
First, there appears to be very short term (one to two months) reversal
patterns in returns. If a stock went up
significantly in price last month, this seems to signal a reversal for the next
month (see Jegadeesh, (1990)). These
short term reversal patterns can be caused by price pressure induced by
investors actively attempting to buy or sell large amounts of a particular
stock quickly. An investor attempting
to sell can drive the price of the stock below its fair value. This being the case, the stock can be
expected to recover and return to fair value shortly thereafter. As discussed above, it is also possible that
short term negative serial correlation can be induced by "bid-asked
bounce." Jegadeesh (1992) argues
that this bias is likely to be small.
He finds that trading strategies that attempt to exploit short term
reversals remain successful even when returns for the previous month do not
reflect the last day of trading. On the
other hand, Ball, Kothari, and Wasley (1995) find that bid-asked problems can
be very troublesome in simulations of short term contrarian strategies that
seek to exploit short term reversal patterns.
Bid-asked problems have little impact on the tests reported in this
paper. The results remain fundamentally
intact with a one month gap separating the point in time when our expected
return deciles are formed and the period over which performance is measured. In addition, our deciles are not distinguished
by the short term performance of their stocks.
Second,
there are intermediate term inertia patterns in stock returns, with stocks that
have done well (poorly) in the previous six to 12 months having good (poor)
future prospects. These intermediate
term, inertia patterns in stock returns can be due to the market's tendency to
(a) exhibit lagged reactions to individual earnings reports and (b) to under
react to initial reports of unusually high or low rates of profitability
by firms. An initial good (bad)
quarterly earnings report tends to be followed by one or two more. Failing to recognize this, the market under
reacts to the first report and then completes its reaction as the next one or
two are reported in the six months that follow (see Jegadeesh and Titman (1993)
and Bernard and Thomas (1990)).
Finally,
Jegadeesh and Titman (1993) show that there
are long term (three to five years) reversal patterns in stock returns. This can be due to the fact that the market
over reacts to a chain of positive (negative) reports of good (bad)
earnings numbers. Believing that the
chain will continue into the future for an extended period, investors drive the
price up (down) too high (low).
Consistent with the discussion above, as competitive forces come into
play, the stocks that went up or down in price in the past tend to reverse
their performance in the future.
Proponents
of efficient markets contend that these technical patterns are not the product
of market under and over reaction (see, for example, Chen, (1991)). They believe,
instead, that risk premiums on stocks are time varying. Risk premiums in expected returns become
larger and smaller as the risk of stocks becomes larger or smaller, or as
investors' sensitivity to risk grows or declines. Both the levels of risk and risk aversion can change with the
business cycle. As we move into a
recession, the risk of common stocks can increase; we also become poorer so our
aversion to taking on risk can become stronger. Given this, the expected returns to stocks can be higher in
recessions and lower in booms. To the
extent that changes in prosperity occur in roughly regular time patterns, the
systematic patterns that we see in the
history of stock returns can be induced by time varying risk premiums.
Given
the patterns stemming from pricing bias, we will expect the payoffs to be (a)
negative, (b) positive, and (c) negative with respect to a stock's performance
in the past (a) one to two months (b) six to 12 months, and (c) two to five
years respectively. A comprehensive
list of all the factors used in the model is provided in the Appendix to the
paper.
3. Estimating and projecting factor payoffs
In
building an expected return factor model, one needs to estimate the tendency
for stocks with differing exposures to different factors to produce differing
returns. In a given month, we will
simultaneously estimate the monthly payoffs (cross-sectional regression
coefficients) to the variety of factor characteristics using an ordinary least
squares, cross-sectional, multiple regression analysis:
Where:
rj,t = rate of return to stock j in month t,
= regression
coefficient or payoff to factor i in month t,
Fj,i,t-1 = exposure
(firm characteristics such as APT betas, size, measures of profitability, etc.)
to factor i for stock j at the end of month t-1,[5]
uj,t = unexplained
component of return for stock j in month t.
Equation
(1) is estimated over a sequence of months to obtain a history of the payoffs
to the various factors.
One
can use the information embodied in the payoff histories to make out-of-sample
projections of the sizes of the future payoffs in future periods. The experiments reported in this paper for
U.S. stocks employ simple averages of the payoffs observed in the 12 months
prior to the month in which expected return is to be estimated.[6] Expected
return for month t is then projected as:
E(rj,t) = 
Where:
E(rj,t) = expected
rate of return to stock j in month t,
E(Pi,t) = expected
payoff to factor i in month t, (the arithmetic mean of the estimated payoff
over the trailing 12 months)
Fj,i,t-1 = exposure
to factor i for stock j based on information available at the end of month t-1.
As
stated above, models similar to the one employed here are used by practicing
portfolio analysts to predict statistics related to the second moment of the
return distribution for equity portfolios for decades. These risk models interface the covariance
matrix of factor payoffs with portfolio exposures (differential exposures from
an index) to obtain estimates of portfolio volatility (or perhaps tracking
error relative to an index).
4. Results of the monthly regressions
In
fitting the factor model in each month, we begin with the actual (as of each date) monthly lists of the stocks in the
Russell 3000 Stock Index. The indexes
consist collectively of roughly the 3000 largest stocks in the United
States. Subject to data availability,[7] the sample includes all stocks that were actually
represented in the index, as it existed, from 1979 through 1993.[8] In addition,
if a particular stock's record is incomplete, so that the set of data required
to compute its exposure to a particular factor is unavailable in a given month,
the stock remains in the sample and is assigned the population mean value for
the exposure. This procedure can bias
our results, because numbers unavailable in the current record might have been
available at the time the forecasts are to be made. It is our opinion that filling the missing records with
population average exposure numbers instills less of a bias than removing the
stock from the population. We have run
the tests both ways, however, with little difference in the results.
We
estimate the payoffs in all months from 1979 through 1993. We employ factors related to (a) risk, (b)
liquidity, (c) price-level, (d) growth potential, and (e) the technical history
of stock returns. For accounting
numbers, such as earnings-per-share, we assume a reporting lag of three
months. However, beginning in 1988, the
set of data files that were actually commercially available in the forecast
month, are used to calculate all factor exposures. Thus, "look ahead" bias should not seriously affect our
results prior to 1988, and it should have no impact whatsoever on the results
after 1988.
Over the period 1979 through
1993, 180 multiple regressions are run to explain the differential monthly
return to the individual stocks in the Russell 3000 population. The time series of multiple, adjusted
coefficients of determination for these regressions is presented in figure
1. In interpreting the numbers, the
reader should keep in mind that we are attempting to explain the monthly
differentials in the returns to individual securities and not well diversified
portfolios. In any given month, the
great majority of the return differentials are caused by the receipt of
unexpected information, causing relative realized returns to deviate from
expectations.
Across the 90 regressions for
the first half of the overall period, the regression coefficients, or payoffs,
associated with the various factors, are averaged. The factors are then ranked on the basis of the absolute values
for the "T" statistics associated with the means across the first
half. The Fama and MacBeth (1973) means
and “T” statistics for these factor payoffs are presented in the first panel of
table 1. The mean coefficient values
and “T” statistics for the second half of the period for the 12 most important
factors of the first half are presented in the second panel of table 1.
Note that all the factors continue to have consistent signs, and the
sizes of the mean payoffs are remarkably similar.
As reported by others, we find
evidence of short term reversal patterns and intermediate term inertia patterns
in the technical history. We also find
that the payoffs to the variables showing cheapness in price (the ratios of
book, earnings and cash flow to price) all payoff positively and are important
in explaining the cross-section. It
also appears that, given the level of market price relative to current cash
flows, the more profitable firms tend to have the greater expected
returns. Liquidity also appears to be
important, with stocks characterized by high and growing levels of trading
volume selling at prices to produce lower levels of expected return. Since, to invest in these stocks, you must
buy now and then sell in the future, the signs for the coefficients related to
the current level of trading volume and its trend are as expected.
Interestingly, only one of the
12 most important factors (variability in ratio of cash flow to market price)
seems to be related to risk in the distributions of monthly returns, and its
payoff seems to be of the wrong sign. We
report that none of the market or APT related beta coefficients have
significant "T" scores.
Ironically, much of the previous work in explaining the cross-section
has concentrated exclusively on these variables. The average payoff to
market beta, volatility of total return and residual variance is .01%, .03%,
and .15% respectively.
Comparing the two periods, we
see a high degree of commonality in the signs and sizes of the
coefficients. All the important factors
maintain their signs, and the sizes of the coefficients are remarkably similar.
To test the null hypothesis that
the mean payoffs to all the factors across the entire period are all zero, we
run a Hotelling-T2 test of the joint significance of the mean values
of the payoffs to all of the factors except those relating to sector. Since the majority of our factors are not
statistical estimates and are measured without error, we do not adjust the
Hoteling-T2 for errors-in-variables. The value for the unadjusted Hotelling-T2 is 8.206
(p=.000). Thus, we conclude that the
payoffs are jointly nonzero at an extremely high level of confidence.
5. A test of the out-of-sample accuracy of the predictions of
expected returns
To test the accuracy of our predictions, we first
estimate the payoffs to the factors in the 12 months prior to 1979. The payoffs
are then averaged and these mean values are used as projections for the first
month of 1979. We employ a 12 month
trailing mean to take advantage of the possibility that the expected values of
the payoffs are time varying. Given the
exposures of each stock, (based on
information available at the end of the previous month) and the projected
factor payoffs for the next month, we can calculate each stock's relative
expected rate of return. We then rank
on the relative expected returns, and we form the stocks into ten equally
weighted deciles, with decile one containing the stocks with the lowest
expected rates of return.
The
process is repeated through December 1993, with the 12 month trailing period,
over which the payoffs are observed and averaged, moving with the process. We then calculate the actual linked,
realized rates of return to the deciles after they have been so ranked.
The
results are shown in table 2. Over the
entire period, the spread between decile ten and decile one is approximately
35%. The slopes reported in table 2 are
derived from a regression of realized annual return on decile ranking. They can be interpreted as the expected
increase in realized return when moving from one decile to the next. The coefficients of determination are also
reported; they are surprisingly high.
To test for the reliability of the factor model, we also separate the
realized returns by year in table 2. In
each year, as we go from decile one to decile ten, the realized return tends to
become larger, and the spreads are surprisingly large.[9]
To
test for the potential effect of the "bid-asked bounce," we reran the
tests, where we attempted to predict returns in month t+2 on the basis of
information available at t. The effect
of separating the forecasts from the exposures by a month is to slightly reduce
the overall slope of table 2, while slightly increasing the coefficient of
determination. This effect seems
consistent with staleness in the factor exposure estimates, and we conclude
that our principle results are largely unaffected by bid-asked problems.
The
results of table 2 are dramatically different from those reported by
others. Two differences in methodology
account for the improvement in
predictive power. First, the model
simultaneously employs a variety of predictive variables, rather a than one or
two at a time. Second, and importantly,
the deciles of table 2 are reformed monthly rather than annually as in other
studies. Many of the factor exposures,
such as excess return in the previous month or quarter, tend to mean revert
rather quickly. As a result, their
power in predicting return is much greater over a one month horizon than over a
one year horizon.
To
determine whether the results reported in table 2 are primarily driven by
market behaviors reported previously by others, we ran the tests excluding all
but selected factors. First, we
replicated the tests using factors related only to the intermediate momentum
patterns in stock returns. We include
only excess returns measured over the previous three, six, and 12 months. Using only these factors, we find a
significant deterioration in predictive power.
The overall spread drops from 35% to 15% and we find negative slopes
relating predicted return to decile number in four of the 15 years. To determine whether variables related to
cheapness in price are the primary drivers, we reran the tests using
book-to-price and earnings-to-price as lone factors. The spread drops to 12% for book-to-price, and we find four years
with negative slopes. The spread drops
to 14% for earnings-to-price with three negative slope years. We conclude that it is the collective power
of many of the factors in the group that accounts for the high level of
accuracy in the predictions of return.[10]
In
interpreting the table 2, it is important to realize that each stock has a term
structure of expected return, with some components of expected return more
persistent than others. For example,
the exposures of a particular stock to factors relating to recent stock
performance can be expected to mean revert very quickly. Exposure to size, on the other hand, mean
reverts very slowly for small firms and shows little or no tendency to mean
revert for large firms. Thus, the
numbers in table 2 actually reflect annualized differences in the
rates of return between the deciles for the first month following
projection. As an aside, we have done
the analysis based on monthly arithmetic mean returns across the deciles,
(which is consistent with an assumed one month investor horizon). The result over the 1979 to 1983 interval
shows a slope of 3.0% with an R2 of 91.7%.
At
the bottom of table 2 we report the annualized volatilities of the monthly
rates of return to the deciles. Note
that volatility tends to decrease as we move from decile one to decile
ten. This provides us with initial
evidence that investors might not regard the stocks in decile ten as being
highly risky. Figure 2 plots the
frequency distributions of monthly returns for deciles one and ten. Once again, there is nothing apparently
alarming in the nature of the distribution of returns for decile ten relative
to decile one that might induce investors to expect and require a large
differential in expected return. Of
course it is possible that the returns to the deciles have highly differing
sensitivities to macroeconomic variables that are of great concern to portfolio
investors. This seems unlikely, because
none of the APT betas surface as important determinants of expected return.
Table
2 also gives some important initial information on the influence of data
snooping on tests of the predictive accuracy of stock forecasting models. When this particular test was conducted, the
authors were unaware of any results extending past 1990. Thus, the post 1990 period was not
"premined". Nevertheless, the
results for the period 1991 to 1993 do not differ materially from the preceding
periods. We will more fully address the
data snooping issue later, when we take the model to other, international
markets.
6. Average characteristics of the stocks within
the deciles
Table
3 shows selected (equally weighted) average fundamental characteristics of the
stocks within each decile. The numbers
are averages taken over the 1979 to 1993 period. The results are rather striking.
As we move from decile one (low return) to decile ten (high return), the
stocks exhibit lower degrees of financial leverage, higher levels of interest
coverage, lower market betas, lower volatility of total return, higher rates of
earnings growth, and higher rates of profitability in all dimensions considered
(profit margin, asset turnover, return on assets and equity and on trailing
rates of growth in earnings per share).
Moreover, the trailing upward trend in the profitability numbers becomes
more pronounced as we move toward decile ten.
Decile ten stocks also tend to be larger companies, selling at higher
levels of price-per-share. To assure
that these results are not driven by small firms and bid-asked problems, we
provide some additional results related to the distributions of price and size
within the deciles. The median values
for price for Deciles one and ten are $12.21 and $26.94 respectively. The median values for market capitalization
are $167.85M and $439.66M respectively.
We also computed the mean value for the highest and lowest 10% of the
stocks within each decile. For price,
in decile one, the values are $36.81 and $3.35. For price in decile ten the values are $65.53 and $11.33. For size in decile one, the values are
$1,925.25M and $34.26M. For size in
decile ten, the values are $3,793.51M and $69.30M. Thus, there doesn’t appear to be anything unusual about the
distributions for either price or market capitalization. Decile ten stocks have also exhibited good
relative performance during the past year.
It is interesting to note that the high return deciles contain more
liquid stocks, even though the payoff to liquidity is negative. This is because liquidity is positively
correlated with profitability, and the negative payoff to liquidity is
overwhelmed by the collective positive payoffs to the profitability factors, resulting
in the inclusion of liquid stocks in the high return deciles. The profitability factors (profit margin,
asset turnover, return on assets, return on equity, and trailing growth in
earnings-per-share) have a collective average payoff of 1.57% over the 1979 to
1983 period. It is also interesting to
note that the deciles do not distinguish themselves in terms of last month’s
excess return, in spite of the fact that this is the single most powerful
factor. This is because exposures to
this particular factor tend to be uncorrelated in the cross-section with
exposures to other factors. In this
regard, last month’s excess return is different from exposures to the momentum
factors, which tend to be reinforced by their positive correlation with
exposures to the profitability factors.
Based
on these characteristics, and the nature of the distributions of monthly
returns, it seems extremely difficult to make a case for the notion that the
stocks of decile ten, with relatively high expected returns, are distressed
companies that are perceived to be more risky relative to the stocks of decile
one. Indeed, the very opposite is
almost certainly true. It would
also be difficult to make a case for the notion that the relatively high
returns of decile ten stocks are an artifact of survivorship bias, given the
fact that (a) our coverage of the Russell 3000 populations is very high and (b)
that the attrition rate for the type of stocks that populate decile ten is
likely to be very low.
In
spite of the rather impressive fundamental characteristics of the high expected
return stocks, table 3 shows that they tend to sell at inexpensive prices
relative to their earnings, cash flow, and dividends-per-share. This result seems counter intuitive in the
context of an efficient market.
However, in a market characterized by serious biases and inefficiencies,
is it really surprising that we can to build a "high quality"
portfolio at a "bargain" price?
There is, in fact, an acronym in the investment business for the type of
stocks that resemble those that collectively appear to populate decile
ten. These are called GARP stocks
(Growth At a Reasonable Price).
Actually, the stocks of decile ten might be better named GAIP stocks
(Growth At an Inexpensive Price), in light of their relatively high earnings,
cash flow, and dividend yields.
It
is important to note that, while of decile ten stocks collectively have the
characteristics reported in table 3, individual members of the decile do not
have the complete profile.
Individually, by construction, their expected returns will always be
relatively high, but they will be high because the stocks are outstanding in
terms of selected characteristics.
Since managers typically screen stocks individually to form “buy lists,”
populated by stocks with homogeneous, desired characteristics, they should have
a difficult time screening into the type of stocks that populate our decile
ten. Indeed, if you were to screen,
requiring each to have decile ten characteristics, you might well find an empty
set, with no stocks at once exhibiting, low risk, high liquidity, low
price-level, and high profitability.
7. Risk-adjusting the realized returns
Fama
and French (1993) claim that they can explain the cross-section of expected
stock returns on the basis of the loadings of stocks with respect to three
factors: (a) the excess return to the capweighted market index, (b) the
difference between the rates of return to small stocks and large stocks, and
(c) the difference between the rates of return to stocks with high
book-to-price ratios and stocks with low book-to-price ratios. After regressing the monthly returns to
ranked groupings of stocks on the three factors, they find that the intercepts
of their regressions are generally not significantly different from zero. They conclude from this that the
cross-section can be explained by differences in relative risk, as given by
differences in the factor loadings.
To
see if this same result holds for our deciles, we shall employ a similar three
factor model. In our model the factors
are defined as follows:
MKTPREM: The monthly excess return to the
capweighted Russell 3000 stock index (using the monthly return to 90 day
Treasury bills).
SML: The Russell 3000 stock population is ranked monthly
(in accord with the procedures used to create the decile returns) on the basis
of market capitalization. Equally
weighted quintiles are formed. SML is
the monthly difference in return between the smallest and largest quintile.
HML: Assuming a six month reporting lag, the Russell 3000
stock population is ranked monthly on the basis of the ratio of the most
recently reported book value-per-share to market price-per-share. Equally weighted quintiles are formed. HML is the difference in monthly return
between the highest and lowest quintile.
For
each of the deciles of table 2, the monthly excess return is regressed on the
three factors over the period 1979 through 1993. The results of these regressions are presented in table 4. Note that the intercepts for the low
expected return deciles are negative and highly significant; the intercepts for
the high expected return deciles are positive and also highly significant. In fact, when annualized, the differences
between the risk adjusted intercepts are larger than the differences between
the raw realized returns of table 2.
This is because risk tends to decrease across the deciles even as
raw returns increase.
The
patterns in the factor loadings are also interesting. Note that the loadings on SML for the high expected return
deciles are smaller, because these deciles are populated by large cap stocks. The loadings on HML are also smaller for
these deciles. While there is no
tendency for the high return deciles to contain stocks with high ratios of
book-to-price, they do contain stocks with strong growth characteristics
(stocks that are highly profitable and stocks with rapid trailing rates of
growth in earnings-per-share). The high
return deciles also have smaller loadings on MKTPREM, reflecting their lower
levels of (market-related) risk. Note
in table 3 that there is no tendency for book to price to become larger in
moving from decile one to decile ten.
However, profitability tends to increase significantly for the upper
deciles. This accounts for the smaller
loadings on HML. We note that if Fama
and French believe that value stocks, that have high loadings on SML and HML
are more risky than average, then, presumably, they must also believe that the
high expected return stocks found here (with low loadings on SML and HML) are
below average in their risk.
8. Simulating the investment performance of the
expected return factor model
There
is considerable turnover of names within these deciles, as stocks migrate from
one decile to another. Given the costs
of trading these stocks, the return differentials actually experienced,
relative to a buy and hold strategy, will be considerably less than those
depicted in table 2. To get a more
accurate picture of attainable
performance, we now move to a simulation in which the factor model is
employed to import expected returns to a Markowitz type optimization. In the simulation, trading is controlled,
and transaction costs are accounted for.
To
minimize any residual survival bias, we employ a Markowitz optimization on the
largest 1000 stocks in the population at the beginning of each quarter from
1979(q1) to 1993(q4). The simulation is
based on the Russell 1000 stock index, as it existed in the Frank Russell
Company’s records at the beginning of each quarter. Estimates of portfolio volatility are based on the full
covariance matrix of returns to the 1000 stocks in the previous 24 months. Estimates of expected returns to the 1000
stocks are based on the factor model discussed above. The following constraints are applied to portfolio weights for
each quarterly optimization:
(1) The maximum weight in a portfolio that
can be assigned to a single stock is limited to 5%. The minimum is
0% (Short selling is not permitted).
(2) No more than three times its percentage of the Russell 1000
total market capitalization can be invested in any one stock in the portfolio.
(3) The portfolio industry weight is restricted to be within 3%
of the market capitalization weight of that industry. (Based on the two-digit SIC code.)
(4) Turnover
in the portfolio is constrained from 20% to 40% annually, depending on the
emphasis in the optimization toward higher expected return.
These
constraints are designed to merely keep the portfolios diversified. Reasonable changes in the constraints do not
affect the results materially.
Numerical search procedures are used to find the lowest volatility
portfolio, given expected return. Thus,
we do not have to invert the covariance matrix. Given the constraints imposed on the optimization, exact unique
solutions to the problems exist.
In
each quarterly optimization, three portfolios are constructed. One is designed to have the lowest possible
volatility, irrespective of expected rate of return. This is the Global Minimum Variance Portfolio (G), or the
portfolio at the nose of the set of constrained, minimum variance
portfolios. The other two portfolios
are designed to emphasize return vs. risk to different degrees. We shall call
them the Intermediate Return Portfolio (I) and the High Return Portfolio
(H). To show the spread in achievable
returns within the cross-section of the Russell 1000 population, we also
construct a low return portfolio (L), that can be taken to be the converse of
the H portfolio. In the optimization,
we minimize the function, s2 - lE(r). For
portfolio G, l takes a value of zero. The
coefficient l is assigned
progressively higher values for the I and H portfolios. The value for l for the L portfolio is the negative of the value used
for the H portfolio. For all
simulations in both the U.S. and in other countries, the values for l are identical over countries and constant over time
for all portfolios. The returns of these four portfolios are then
observed, on a buy and hold basis, over each quarter following the quarterly
optimization. A conservative (for the 1
thousand largest U.S. stocks) 2% round trip transactions cost is assumed.
The
results are shown in figure 3. Note
that the global minimum variance portfolio has both lower risk and higher
return than the capweighted Russell 1000 stock index. This result is consistent with the evidence provided in Haugen
(1995), as well as with the results, presented in table 6 of this paper,
showing that the average payoff to volatility of return is negative in each of
the five largest markets of the world.
The I and H portfolios also both dominate the capweighted Russell 1000
stock index. The H portfolio has
approximately four hundred basis points greater return than the index, while
achieving the same overall level of volatility.[11] The portfolio
denoted by an L in the graph is built with the opposite signed l from the H portfolio. In the
case of the L portfolio, transactions costs are added to the returns because
investors would presumably be short selling this portfolio in which case the
cost of raising the funds would be the returns of the stocks plus the costs of
trading the portfolio. Note that the
spread between L and H is nearly nine hundred basis points. Thus, the return
differentials of table 2 appear to be realizable to an economically meaningful
degree even to an investor who must bear significant trading costs.
As
with the deciles, we regress the excess returns on H and L on the market’s
excess return, SML, and HML over the period 1979 through 1993. The regression yields the following results:
rj,t
- rf,t = a +
s SMLt + h HMLt + m MKTPREMt + et
|
Portfolio |
a |
T-stat |
s |
T-stat |
h |
T-stat |
m |
T-stat |
R2 |
|
H |
.0041 |
3.923 |
-.0508 |
-2.608 |
-.0546 |
-1.728 |
.9558 |
39.35 |
.921 |
|
L |
-.0060 |
-5.006 |
.0508 |
2.283 |
.2129 |
5.914 |
1.111 |
40.13 |
.910 |
Note
that H and L have statistically significant positive and negative intercepts
respectively, with H loading negatively (but insignificantly) on HML and L
loading positively. The annualized,
risk adjusted spread between H and L is approximately 12%.
9. The accuracy of factor models in other
countries: A test of the size of the
data snooping bias in the U.S. results
Bias
associated with data snooping is a very difficult problem. As stated previously, we gain some comfort
in the fact that the model holds up over a period (1991 through 1993) that had
not been snooped as of the date of our tests.
In this section we shall attempt to gain additional comfort by taking
the same procedure to four other countries.
Tests
with the same set of factors discussed above are run over a total of 208 stocks
in France, 195 in Germany, 715 in Japan and 406 in The United Kingdom. Because of limitations in coverage
associated with commercially available data files, the tests must be run over
the period 1985 through midyear 1994.
Our data is taken from Compustat's Global Vantage, and we employ their
research data base, that includes inactive companies. A reporting lag for accounting variables of three months is again
assumed. Global Vantage does not
include quarterly accounting numbers, so that the assumed reporting lag can be
as long as 15 months. In the tests,
payoffs are projected, based on the basis of their trailing average
values. We stress that these payoffs
are estimated individually across each country. To economize on data, the payoff to all factors is presumed to be
zero in the first month of 1985; in the second month of 1985 the projected
payoffs are assumed to take on the values for January of 1985. Payoffs in subsequent months are then based
on the available trailing histories, up to a limit of 12 months. All returns are in local currency. Based on the projected expected returns, the
firms are again formed into deciles within each country, and the actual monthly
returns for the deciles are then observed.
The process is repeated for all months of the year, and the monthly
decile returns are then linked. The
results for each country are shown in the four panels of table 5. Once again, the factor model proves to be
very powerful. Note that in nearly all
cases, the slopes, relating realized return to decile ranking, are positive and
the coefficients of determination continue to be high.
As
in the U.S., an examination of the annualized volatility of return at
the bottom of each panel shows no evidence of an increase in risk as we move
from decile one (low return) to decile ten (high return) in any of the
countries examined.
10. Simulation of investment performance for
global markets
The
same optimization constraints are employed as in the U.S., except in those
cases where a few stocks dominate a country’s total market capitalization. In this case the maximum weight assigned to each stock is the lowest of (a) three
times the stock's market capitalization weight, (b) its capitalization weight
plus 2%, or (c) 10%. As in the U.S.
simulation, the portfolios are reoptimized quarterly. Portfolio volatility is again estimated on the basis of the full
covariance matrix of the 24 monthly returns trailing each quarter. Three portfolios are again constructed in
the optimization process: the global
minimum variance portfolio (G), an intermediate return portfolio (I), and a
high return portfolio (H). Following
the optimization, the subsequent quarter’s returns are observed and
linked. A 2% round trip transaction
cost is assumed.
The
results of the simulations are presented in figure 4. Note that, in every country, realized returns as well as
volatilities increase monotonically as we go from the global minimum variance
portfolio to the high return portfolio.
In addition, the intermediate and high return portfolios dominate the
capitalization weighted FTA equity index for every country.
Also
in figure 4, we show the results of a combined optimization across the largest
two thousand stocks in the five largest countries (including the U.S.), where
returns are denominated in U.S. dollars.
Since the factor models project relative expected returns within markets
only, the country weightings were constrained to approximate the capitalization
weightings of each country. Note that,
through international diversification, one is able to appreciably lower
volatility, while enhancing the spread in realized return, relative to the
capitalization weighted FTA five country equity index.