Financial econometrics assignment

ECON334: Financial Econometrics
ASSIGNMENT – S2, 2015
Due date and time: 4pm on Friday of Week 11
23 October
Instructions
– This is an individual assignment. It should reflect your individual effort
– The assignment should be typed, with the main tables, charts and results presented
throughout the assignment to highlight your responses to the questions
– There should be no appendices (appendices will not be marked)
– Marks will be awarded for neatness, conciseness and clarity of answers
– Where answers call for explanation, a simple reporting of numerically correct results
will yield few (if any) marks
– When conducting hypothesis tests, outline all steps in your answer
– Maximum number of pages allowed: 15 (additional pages will not be marked)
– Pages should be numbered
– Be as concise as you can, while clearly addressing each question
– Total marks: 50
Submission instructions
– You are required to submit the assignment in both print and electronic copies
– Electronic submission is via a submission link on iLearn
– Print copy (with a signed assignment coversheet) must be submitted at BESS (E4B) –
ensure you know your tutor’s name, as there will be separate submission boxes for
each tutor
– A link to the FBE cover sheet is provided under the “Assignment” heading on iLearn
Fill in the details of the cover sheet and staple it to the front of your assignment
Part 1
Part 1 – Total number of marks: 15
The Eviews workfile “assignment_part1.wf1” located under “Assignment” heading on iLearn
contains four series for the period July 1963 – December 2014 (618 observations). The
following variables are included:
1. Monthly prices for
 ?? (General Electric)
2. Returns on three pricing factors from Fama and French (1993)
 ???_?? (Market Risk Premium)
 ℎ?? (High minus Low)
 ??? (Small minus Big)
3. ????_???? (the US risk free rate)
Answer the following questions based on this dataset:
A. How are the factors computed and what do they represent? (hint: read the Fama and
French 1993 paper – its on iLearn). Briefly describe major company characteristics of
General Electric. (2 marks)
B. Create log returns (in percent, e.g. 3.25%) and name them ?_??. Consider the
following model:
(?_??? − ???
) = ?0 + ?1???_??? + ?2ℎ??? + ?3???? + ??
What signs (positive or negative) would you expect to estimate for each of the factors?
Why? (3 marks)
C. Estimate the model in B and present the fitted equation. Interpret the fitted
coefficients. Which parameters are statistically significant at the 5% level? Are the
estimated parameters of the same sign as you expected in B? (4 marks)
D. Conduct a test for the validity of the CAPM. What do you find? (3 marks)
E. Conduct the basic diagnostic tests on the estimated model, i.e. autocorrelation (use 4
lags of residuals), heteroskedasticity (no cross product), non-normality,
misspecification of functional form (only quadratic term). Comment on your results
and carry out remedies for the problems you encounter. (3 marks)
Part 2
Part 1 – Total number of marks: 35
The Eviews workfile “assignment_part2.wf1” located under “Assignment” heading on iLearn
contains daily adjusted closing price for Cochlear from 02 Jan 2006 to 31 Dec 2014.
A. Plot a graph of the share price, and comment on its salient features. Conduct ADF and
KPSS unit-root tests on the price series. Be careful to properly state the null and
alternative hypotheses for the two tests. Comment on your findings.
(3 marks)
B. Generate a new variable for the daily log returns (in percentage terms). Present a
graph of the series along with summary statistics, and comment. Conduct ADF and
KPSS unit-root tests on the returns. Comment on your findings.
(3 marks)
C. Set the sample to 10 Jan 2006 – 10 June 2014. Plot the ACF and PACF functions for
Cochlear returns (include 15 lags). What optimal ????(?, ?) model would you
choose based on these? Why?
(4 marks)
D. Set the sample to 10 Jan 2006 – 10 June 2014. Select an optimal ????(?, ?) model
for the returns based on the Hannan-Quinn criterion. Select from the set of models up
to and including the largest model of ????(4,4). You may use the automatic
procedure (maximum-difference = 0) or you may undertake this task manually.
Present a table of the criterion values over all combinations of ? and ? highlighting
the minimised Hannan-Quinn value.
(5 marks)
E. Estimate the ???? model selected in D, on the same subsample. Report the fitted
equation and comment on it. Present the following for the residuals: ACF and PACF
(use 15 lags), tests for a) heteroskedasticity (no cross product), b) non-normality, c)
misspecification of functional form (only quadratic term). Comment on your findings.
(Note: you don’t need to carry out remedies if any problems are identified.)
(5 marks)
F. Using the model estimated in E, forecast the return on Cochlear over the period 11
June 2014 – 31 Dec 2014 using ?) dynamic and ??) static forecasting techniques.
Explain how these forecasts are made and how they differ, making use of equations
in your answer. Present forecast graphs and discuss the accuracy of your forecasts. Be
sure to compare the relative performance of the two forecasting strategies.
(7 marks)
G. Using the 10 Jan 2006 – 10 June 2014 subsample re-estimate the model selected in D
by adding ?) ?????(1,1) and ??) ???(1,1,1) specifications. What are these models,
and how they differ? Report the fitted equations, comment on the output, including
careful interpretation of the parameters in the volatility equation, and compare across
the two specifications. Provide graphs of the estimated conditional variances and
comment on them.
(8 marks)
References
Fama, E. F. and French, K. R. (1993). “Common risk factors in the returns on stocks
and bonds”. Journal of Financial Economics 33(3), 3-56.

Journal of Financial Economics 33 (1993) 3-56. North-Holland
Common risk factors in the returns on
stocks and bonds*
Eugene F. Fama and Kenneth R. French
Unirrrsit.v 01 Chicayo. Chiccup. I .L 60637, C;S;L
Received July 1992. final version received September 1992
This paper identities five common risk factors in the returns on stocks and bonds. There are three
stock-market factors: an overall market factor and factors related to firm size and book-to-market
equity. There are two bond-market factors. related to maturity and default risks. Stock returns have
shared variation due to the stock-market factors, and they are linked to bond returns through
shared variation in the bond-market factors. Except for low-grade corporates. the bond-market
factors capture the common variation in bond returns. Most important. the five factors seem to
explain average returns on stocks and bonds.
1. Introduction
The cross-section of average returns on U.S. common stocks shows little
relation to either the market /Is of the Sharpe (1964tLintner (1965) assetpricing
model or the consumption ps of the intertemporal asset-pricing model
of Breeden (1979) and others. [See, for example, Reinganum (198 1) and Breeden,
Gibbons, and Litzenberger (1989).] On the other hand, variables that have
no special standing in asset-pricing theory show reliable power to explain
the cross-section of average returns. The list of empirically determined averagereturn
variables includes size (ME, stock price times number of shares),
leverage, earnings/price (E/P), and book-to-market equity (the ratio of the
book value of a firm’s common stock, BE, to its market value, ME). [See
Banz (1981). Bhandari (1988). Basu (1983). and Rosenberg, Reid, and Lanstein
(19853.1
Correspondence to: Eugene F. Fama. Graduate School of Business. University of Chicago, 1101
East 58th Street. Chicago. IL 60637, USA.
*The comments of David Booth, John Cochrane. Sai-fu Chen, Wayne Ferson. Josef Lakonishok.
Mark Mitchell, G. William Schwert. Jay Shanken. and Rex Sinquefield are gratefully acknowledged.
This research is supported by the National Science Foundation (Fama) and the Center for Research
in Securities Prices (French).
030%405X.93.S05.00 C 1993-Elsevier Science Publishers B.V. Ail rights reserved
4 E.F. Fuma and K.R. French. Common risk f&run in r~ock and bond remrns
Fama and French (1992a) study the joint roles of market 8, size, E;P, leverage,
and book-to-market equity in the cross-section of average stock returns. They
find that used alone or in combination with other variables, /I (the slope in the
regression of a stock’s return on a market return) has little information about
average returns. Used alone, size, E/P, leverage, and book-to-market equity
have explanatory power. In combinations, size (ME) and book-to-market equity
(BE/ME) seem to absorb the apparent roles of leverage and E;‘P in average
returns. The bottom-line result is that two empirically determined variables, size
and book-to-market equity, do a good job explaining the cross-section of
average returns on NYSE, Amex, and NASDAQ stocks for the 1963-1990
period.
This paper extends the asset-pricing tests in Fama and French (1992a) in three
ways.
(a) We expand the set of asset returns to be explained. The only assets considered
in Fama and French (1992a) are common stocks. If markets are
integrated, a single model should also explain bond returns. The tests here
include U.S. government and corporate bonds as well as stocks.
(b) We also expand the set of variables used to explain returns. The size and
book-to-market variables in Fama and French (1992a) are directed at
stocks. We extend the list to term-structure variables that are likely to play
a role in bond returns. The goal is to examine whether variables that are
important in bond returns help to explain stock returns, and vice versa. The
notion is that if markets are integrated, there is probably some overlap
between the return processes for bonds and stocks.
(c) Perhaps most important, the approach to testing asset-pricing models is
different. Fama and French (1992a) use the cross-section regressions of
Fama and MacBeth (1973): the cross-section of stock returns is regressed on
variables hypothesized to explain average returns. It would be difficult to
add bonds to the cross-section regressions since explanatory variables like
size and book-to-market equity have no obvious meaning for government
and corporate bonds.
This paper uses the time-series regression approach of Black, Jensen, and
Scholes (1972). Monthly returns on stocks and bonds are regressed on the
returns to a market portfolio of stocks and mimicking portfolios for size,
book-to-market equity (BE/‘ME), and term-structure risk factors in returns. The
time-series regression slopes are factor loadings that, unlike size or BE/ME,
have a clear interpretation as risk-factor sensitivities for bonds as well as for
stocks.
The time-series regressions are also convenient for studying two important
asset-pricing issues.
(a) One of our central themes is that if assets are priced rationally, variables
that are related to average returns, such as size and book-to-market equity, must
proxy for sensitivity to common (shared and thus undiversiliable) risk factors in
E.F. Famu und K.R. French. Common risk factors in stock and bond returns 5
returns. The time-series regressions give direct evidence on this issue. In particular,
the slopes and R’ values show whether mimicking portfolios for risk factors
related to size and BE/lVCIE capture shared variation in stock and bond returns
not explained by other factors.
(b) The time-series regressions use excess returns (monthly stock or bond
returns minus the one-month Treasury bill rate) as dependent variables and
either excess returns or returns on zero-investment portfolios as explanatory
variables. In such regressions, a well-specified asset-pricing model produces
intercepts that are indistinguishable from 0 [Merton (1973)J The estimated
intercepts provide a simple return metric and a formal test of how well different
combinations of the common factors capture the cross-section of average
returns. Moreover, judging asset-pricing models on the basis of the intercepts in
excess-return regressions imposes a stringent standard. Competing models are
asked to explain the one-month bill rate as well as the returns on longer-term
bonds and stocks.
Our main results are easy to summarize. For stocks, portfolios constructed to
mimic risk factors related to size and BE/ME capture strong common variation
in returns, no matter what else is in the time-series regressions. This is evidence
that size and book-to-market equity indeed proxy for sensitivity to common risk
factors in stock returns. Moreover, for the stock portfolios we examine, the
intercepts from three-factor regressions that include the excess market return
and the mimicking returns for size and BE/ME factors are close to 0. Thus
a market factor and our proxies for the risk factors related to size and bookto-market
equity seem to do a good job explaining the cross-section of average
stock returns.
The interpretation of the time-series regressions for stocks is interesting. Like
the cross-section regressions of Fama and French (1992a), the time-series regressions
say that the size and book-to-market factors can explain the differences in
average returns across stocks. But these factors alone cannot explain the large
difference between the average returns on stocks and one-month bills. This job is
left to the market factor. In regressions that also include the size and bookto-market
factors, all our stock portfolios produce slopes on the market factor
that are close to 1. The risk premium for the market factor then links the average
returns on stocks and bills.
For bonds, the mimicking portfolios for the two term-structure factors (a term
premium and a default premium) capture most of the variation in the returns on
our government and corporate bond portfolios. The term-structure factors also
‘explain’ the average returns on bonds, but the average premiums for the
term-structure factors, like the average excess bond returns, are close to 0. Thus,
the hypothesis that all the corporate and government bond portfolios have the
same long-term expected returns also cannot be rejected.
The common variation in stock returns is largely captured by three stockportfolio
returns, and the common variation in bond returns is largely explained
by two bond-portfolio returns. The stock and bond markets. however, are far
from stochastically segmented. Used alone in the time-series regressions. the
term-structure factors capture strong variation in stock returns; indeed, the
slopes on the term-structure factors in the regressions for stocks are much like
those for bonds. But interestingly. when stock-market factors are also included
in the regressions, all of our stock portfolios load in about the same way on the
two term-structure factors and on the market factor in returns. As a result,
a market portfolio of stocks captures the common variation in stock returns
associated with the market factor and the two term-structure factors.
The stochastic links between the bond and stock markets do. however. seem
to come largely from the term-structure factors. Used alone. the excess market
return and the mimicking returns for the size and book-to-market equity factors
seem to capture common variation in bond returns. But when the two termstructure
factors are included in the bond regressions, the explanatory power of
the stock-market factors disappears for all but the low-grade corporate bonds.
In a nutshell, our results suggest that there are at least three stock-market
factors and two term-structure factors in returns. Stock returns have shared
variation due to the three stock-market factors, and they are linked to bond
returns through shared variation in the two term-structure factors. Except for
low-grade corporate bonds, only the two term-structure factors seem to produce
common variation in the returns on government and corporate bonds.
The story proceeds as follows. We first introduce the inputs to the time-series
regressions: the explanatory variables and the returns to be explained (sections
Z and 3). We then use the regressions to attack our two central asset-pricing
issues: how do different combinations of variables capture (a) the common
variation through time in the returns on bonds and stocks (section 4) and (b) the
cross-section of average returns (section 5).
2. The inputs to the time-series regressions
The explanatory variables in the time-series regressions include the returns on
a market portfolio of stocks and mimicking portfolios for the size. bookto-market,
and term-structure factors in returns. The returns to be explained are
for government bond portfolios in two maturity ranges, corporate bond portfolios
in five rating groups, and 25 stock portfolios formed on the basis of size
and book-to-market equity.
The explanatory variables fall into two sets, those likely to be important for
capturing variation in bond returns and those likely to be important for stocks.
Segmenting the explanatory variables in this way sets up interesting tests of
whether factors important in stock returns help to explain bond returns and vice
versa.
2.1 .I. Bond-mnrket factors
One common risk in bond returns arises from unexpected changes in interest
rates. Our proxy for this factor, TERM, is the difference between the monthly
long-term government bond return (from Ibbotson Associates) and the onemonth
Treasury bill rate measured at the end of the previous month (from the
Center for Research in Security Prices, CRSP). The bill rate is meant to proxy
for the general level of expected returns on bonds. so that TERM proxies for the
deviation of long-term bond returns from expected returns due to shifts in
interest rates.
For corporate bonds. shifts in economic conditions that change the likelihood
of default give rise to another common factor in returns. Our proxy for
this default factor, DEF, is the difference between the return on a market
portfolio of long-term corporate bonds (the Composite portfolio on the corporate
bond module of Ibbotson Associates) and the long-term government bond
return.
Chen. Roll, and Ross (1986) use TERM and a variable like DEF to help
explain the cross-section of average returns on NYSE stocks. They use the Fama
and MacBeth (1973) cross-section regression approach: the cross-section of
average stock returns is explained with the cross-section of slopes from timeseries
regressions of returns on TERM, a default factor, and other factors. In
their tests. the default factor is the most powerful factor in average.stock returns.
and TER.Cl sometimes has power. We confirm that the tracks of TER,LI and
DEF show up clearly in the time-series variation of stock returns. We also find
that the two variables dominate the common variation in government and
corporate bond returns. In contrast to the cross-section regressions of Chen.
Roll, and Ross, however, our time-series regressions say that the average
premiums for DEF and TERM risks are too small to explain much variation in
the cross-section of average stock returns. [Shanken and Weinstein (1990) make
a similar point.]
2.1.2. Stock-market fuctors
Motiuztion – Although size and book-to-market equity seem like ad hoc
variables for explaining average stock returns, we have reason to expect that
they proxy for common risk factors in returns. In Fama and French (1992b) we
document that size and book-to-market equity are related to economic fundamentals.
Not surprisingly, firms that have high BE/ME (a low stock price
relative to book value) tend to have low earnings on assets, and the low earnings
persist for at least five years before and five years after book-to-market equity is
measured. Conversely. low BE. .CfE (a high stock price relative to book value) is
associated with persistently high earnings.
Size is also related to profitability. Controlling for book-to-market equity,
small firms tend to have lower earnings on assets than big firms. The size effect in
earnings, however, is largely due to the 1980s. Until 1981. controlling for
BE;.LfE, small firms are only slightly less profitable than big firms. But for small
firms, the 198G1982 recession turns into a prolonged earnings depression. For
some reason, small firms do not participate in the economic boom of the middle
and late 1980s.
The fact that small firms can suffer a long earnings depression that bypasses
big firms suggests that size is associated with a common risk factor that might
explain the negative relation between size and average return. Similarly. the
relation between book-to-market equity and earnings suggests that relative
profitability is the source of a common risk factor in returns that might explain
the positive relation between BE:.CfE and average return. Measuring the common
variation in returns associated with size and BE,hfE is a major task of this
paper.
The Buikfiny Blocks – To study economic fundamentals, Fama and French
(1992b) use six portfolios formed from sorts of stocks on .LfE and BE ‘IlIE. We
use the same six portfolios here to form portfolios meant to mimic the underlying
risk factors in returns related to size and book-to-market equity. This
ensures a correspondence between the study of common risk factors in returns
carried out here and our complementary study of economic fundamentals.
In June of each year t from 1963 to 1991, all NYSE stocks on CRSP are
ranked on size (price times shares). The median NYSE size is then used to
split NYSE, Amex. and (after 1972) NASDAQ stocks into two groups. small and
big (S and B). Most Amex and NASDAQ stocks are smaller than the NYSE
median, so the small group contains a disproportionate number of stocks 13,616
out of 4,797 in 1991). Despite its large number of stocks, the small group
contains far less than half (about 8% in 1991) of the combined value of the two
size groups.
We also break NYSE, Amex, and NASDAQ stocks into three book-tomarket
equity groups based on the breakpoints for the bottom 30% (Loc),
middle 40% (LCfediurn). and top 30% (High) of the ranked values of BE’.fE for
NYSE stocks. We define book common equity, BE. as the COMPUSTAT book
value of stockholders’ equity, plus balance-sheet deferred taxes and investment
tax credit (if available), minus the book value of preferred stock. Depending on
availability, we use the redemption, liquidation, or par value (in that order) to
estimate the value of preferred stock. Book-to-market equity, BE,‘,CfE. is then
book common equity for the fiscal year ending in calendar year t – 1, divided by
market equity at the end of December oft – 1. We do not use negative-BE firms,
which are rare before 1980, when calculating the breakpoints for BE) .bfE
or when forming the size-BE$.LfE portfolios. Also. only firms with ordinary
E.F. Fama und K.R. Fwnch. Common rusk /Lcrorr in srock and bond renum 9
common equity (as classified by CRSP) are included in the tests. This means that
ADRs, REITs, and units of beneficial interest are excluded.
Our decision to sort firms into three groups on BE,‘ICI E and only two on ME
follows the evidence in Fama and French (1992a) that book-to-market equity
has a stronger role in average stock returns than size. The splits are arbitrary,
however, and we have not searched over alternatives. The hope is that the tests
here and in Fama and French (1992b) are not sensitive to these choices. We see
no reason to argue that they are.
We construct six portfolios (S/L, S;,V, S,!H. B,‘L, B,!M, B/H) from the intersections
of the two ,bfE and the three BE!hfE groups. For example. the S/L
portfolio contains the stocks in the small-%fE group that are also in the
low-BE/ME group, and the BI’H portfolio contains the big-.CIE stocks that also
have high BE,MEs. Monthly value-weighted returns on the six portfolios are
calculated from July of year t to June oft + 1. and the portfolios are reformed in
June of t + 1. We calculate returns beginning in July of year t to be sure that
book equity for year c – 1 is known.
To be included in the tests, a firm must have CRSP stock prices for December
of year t – 1 and June of t and COMPUSTAT book common equity for year
t – 1. Moreover, to avoid the survival bias inherent in the way COMPUSTAT
adds firms to its tapes [Banz and Breen (1986)], we do not include firms until
they have appeared on COMPUSTAT for two years. (COMPUSTAT says it
rarely includes more than two years of historical data when it adds firms).
Size- Our portfolio S,LfB (small minus big), meant to mimic the risk factor in
returns related to size, is the difference, each month, between the simple average
of the returns on the three small-stock portfolios (SjL, S/.Cf, and S,,H) and the
simple average of the returns on the three big-stock portfolios (B; L. B/&f, and
B/H). Thus, ShfB is the difference between the returns on small- and big-stock
portfolios with about the same weighted-average book-to-market equity. This
difference should be largely free of the influence of BE/ME, focusing instead on
the different return behaviors of small and big stocks.
BE//LIE – The portfolio HhfL (high minus low). meant to mimic the risk
factor in returns related to book-to-market equity, is defined similarly. HML is
the difference, each month, between the simple average of the returns on the two
high-BE/ME portfolios (S,‘H and B/H) and the average of the returns on the two
low- BE/ME portfolios (S;L and B/L). The two components of H,tlL are returns
on high- and low-BE,‘&fE portfolios with about the same weighted-average size.
Thus the difference between the two returns should be largely free of the size
factor in returns, focusing instead on the different return behaviors of high- and
low-BEllME firms. As testimony to the success of this simple procedure, the
correlation between the 1963-1991 monthly mimicking returns for the size and
book-to-market factors is only – 0.08.
True mimicking portfolios for the common risk factors in returns minimize
the variance of firm-specific factors. The six size-BEilV E portfolios in S&fB and
H,CfL are value-weighted. Using value-weighted components is in the spirit of
minimizing variance, since return vririances are negatively related to size
(table 1. below). More important, using value-weighted components results in
mimicking portfolios that capture the different return behaviors of small and big
stocks. or high- and low-BEl.VE stocks, in a way that corresponds to realistic
investment opportunities.
Market – Finally. our proxy for the market factor in stock returns is the excess
market return, R.M-RF. R&l is the return on the value-weighted portfolio of the
stocks in the six size-BE’ME portfolios, plus the negative-BE stocks excluded
from the portfolios. RF is the one-month bill rate.
22. The returns to he espkuiwd
Bonds -The set of dependent variables used in the time-series regressions
includes the excess returns on two government and five corporate bond portfolios.
The government bond portfolios (from CRSP) cover maturities from 1 to
5 years and 6 to 10 years. The five corporate bond portfolios, for Moody’s rating
groups Aaa, Aa. A, Baa. and LG (low-grade, that is, below Baa) are from the
corporate bond module of Ibbotson Associates (provided to us by Dimensional
Fund Advisors).
Stocks – For stocks. we use excess returns on 25 poitfolios, formed on size and
book-to-market equity. as dependent variables in the time-series regressions.
We use portfolios formed on size and BE/ME because we seek to determine
whether the mimicking portfolios SMB and HAIL capture common factors in
stock returns related to size and book-to-market equity. Portfolios formed on
size and BE.‘AIE will also produce a wide range of average returns to be
explained by competing asset-pricing equations [Fama and French (1992a)].
Later, however, we use portfolios formed on E.P (earnings/price) and DiP
(dividend/price). variables that are also informative about average returns [e.g..
Keim (1988)], to check the robustness of our results on the ability of our
explanatory factors to capture the cross-section of average returns.
The 25 size-BE,‘,LfE portfolios are formed much like the six size-BE;&LIE
portfolios discussed earlier. In June of each year t we sort NYSE stocks by size
and (independently) by book-to-market equity. For the size sort. .LIE is measured
at the end of June. For the book-to-market sort, ME is market equity at
the end of December of c – 1. and BE is book common equity for the fiscal year
ending in calendar year r – 1. We use NYSE breakpoints for ME and BE;.tfE to
allocate NYSE. Amex. and (after 1972) NASDAQ stocks to five size quintiles
and five book-to-market quintiles. We construct 25 portfolios from the intersections
of the size and BE, ,LfE quintiles and calculate value-weighted monthly
returns on the portfolios from July off to June of r + 1. The excess returns on
these 25 portfolios for July 1963 to December 1991 are the dependent variables
for stocks in the time-series regressions.
E.F. Fumo and R.R. French, Commor? rtsk /actors in srock und bond rerurns I1
Table I
Descriptive statistics for 25 stock portfolios formed on size and book-to-market equity: 1963-1991. 29 years.’
Book-to-market equity (BE, ME) quintiies
Size
quintile Low 1 3 4 High Low z 3 4 High
Small
2
3
4
Big
Small
4
Big
Small
1
;
4
Big
Average of annual averages of firm size Average of annual 8. E ratios for portfolio
20.6 20.8 _ ‘0 ._ ’ 19.4 15.1 0.30 0.62 0.84 1.09 1.80
89.7 89.3 89.3 89.9 88.5 0.31 0.60 0.83 1.09 1.71
209.3 211.9 210.8 214.8 210.7 0.31 0.60 0.84 1.08 1.66
535.1 537.4 545.4 551.6 538.7 0.31 0.6 1 0.84 1.09 1.67
3583.7 2885.8 2819.5 2700.5 1337.9 0.29 0.59 0.83 1.08 1.56
Average of annual percent of market Average of annual number of firms in
value in portfolio portfolio
0.69 0.49 0.46 0.48 0.64 428.0 276.6 263.8 191.5 512.7
0.92 0.71 0.65 0.6 I 0.55 121.6 94.0 86.7 79.8 71.3
1.78 1.36 1.26 1.14 0.82 102.7 78.3 73.0 64.5 45.9
3.95 3.01 2.71 2.4 I 1.50 90. I 68.9 60.7 53.1 33.4
30.13 15.87 12.85 10.44 4.61 93.6 63.7 51.7 44.0 23.6
Average of annual E’P ratios (in percent) Average of annual D’P ratios (in percent)
for portfolio for portfolio
2.42 7.24 8.26 9.06 2.66 1.00 I .94 2.60 3.13 2.82
5.2091 8.6173 10.1643 10.95 Il.61 10.78 9.2l.59 1.56 2.45 3034.04 3454.2568 4.5364
5.85 8.94 10.45 11.64 11.39 1.80 3.09 4.22 5.01 4.94
6.00 9.07 10.90 12.45 13.92 2.34 3.69 4.68 5.49 5.90
“The 25 size-BE. ME stock portfolios are formed as follows. Each year t from 1963 to 1991 NYSE quinttle
breakpoints for size (.UE. stock price times shares outstanding), measured at the end of June, are used to
allocate NYSE. Amex. and NASDAQ stocks to five size quintiles. Similarly, NYSE quintile breakpoints for
BE, ME are used to allocate NYSE. Amex. and NASDAQ stocks to five book-to-market equity quintiles. The
25 size-BE,‘.LIE portfolios are formed as the intersections of the five size and the five BE. ME groups. Book
equity. BE. is the COMPUSTAT book value of stockholders’ equity, plus balance sheet deferred taxes and
investment tax credits lif available). minus the book value of preferred stock. Depending on avjailability. we use
the redemption. liquidation. or par value (in that order) to estimate the book value of preferred stock. Bookto-market
equity. BE .ME. for a stock is BE for the fiscal year ending in calendar year r – 1. divided by ME at
the end of December oft – 1.
A portfolio’s book-to-market equity, BE,‘XfE. for the portfolio formation year c is the sum of book equity.
BE. for the firms in the portfolio for the fiscal year endmg in calendar year t – I, divided by the sum of their
market equity. ME, in December oft – I. A portfolio’s earnings/price ratio (E P) for year I is the sum ofequity
income for the firms in the portfolio for the fiscal year ending in calendar year t – 1. divided by the sum of their
market equity in December of r – 1. Equity income is income before extraordinary items, plus incomestatement
deferred taxes. minus preferred dividends. A portfolio’s dividend yield (D P) for year t is the sum
(across firms in the portfolio) of the dividends paid from July oft – 1 to June of r. divided by the sum of market
equity in June oft – I. We use the procedure described in Fama and French (1988) to estimate dividends.
The descriptive statistics are computed when the portfolio is formed in June ofeach year. 1963-1991, and are
then averaged across the 29 years.
12 E.F. Fama und K. R. French. Common risk /&tom in stock md bond returns
Table 1 shows that, because we use NYSE breakpoints to form the 25
size-BE, ,CIE portfolios, the portfolios in the smallest size quintile have the most
stocks (mostly small Amex and NASDAQ stocks). Although they contain many
stocks, each of the five portfolios in the smallest size quintile is on average less
than 0.70% of the combined value of stocks in the 25 portfolios. In contrast, the
portfolios in the largest size quintile have the fewest stocks but the largest
fractions of value. Together, the five portfolios in the largest JIE quintile
average about 74% of total value. The portfolio of stocks in both the largest size
and lowest BE/ME quintiles (big successful firms) alone accounts for more than
30% of the combined value of the 25 portfolios. And note that using all stocks,
rather than just NYSE stocks, to define the size quintiles would result in an even
more skewed distribution of value toward the biggest size quintile.
Table 1 also shows that in every size quintile but the smallest, both the
number of stocks and the proportion of total value accounted for by a portfolio
decrease from lower- to higher-BE/ME portfolios. This pattern has two causes.
First, using independent size and book-to-market sorts of NYSE stocks to form
portfolios means that the highest-BE/ME quintile is tilted toward the smallest
stocks. Second, Amex and NASDAQ stocks, mostly small, tend to have lower
book-to-market equity ratios than NYSE stocks of similar size. In other words,
NYSE stocks that are small in terms of ME are more likely to be fallen angels
(big firms with low stock prices) than small Amex and NASDAQ stocks.
3. The playing field
Table 2 summarizes the dependent and explanatory returns in the time-series
regressions. The average excess returns on the portfolios that serve as dependent
variables give perspective on the range of average returns that competing sets of
risk factors must explain. The average returns on the explanatory portfolios are
the average premiums per unit of risk (regression slope) for the candidate
common risk factors in returns.
3.1. The dependent retwxs
Stocks – The 25 stock portfolios formed on size and book-to-market equity
produce a wide range of average excess returns, from 0.32% to 1.05% per
month. The portfolios also confirm the Fama-French (1992a) evidence that
there is a negative relation between size and average return, and there is
a stronger positive relation between average return and book-to-market equity.
In all but the lowest-BE/ME quintile, average returns tend to decrease from the
small- to the big-size portfolios. The relation between average return and
book-to-market equity is more consistent. In every size quintile, average returns
tend to increase with BE/:bfE, and the differences between the average returns
E.F. Fame und K. R. French. Common risk fk!ors m slack and bond rerurm 13
for the highest- and lowest-BE;‘.CJE portfolios range from 0.19% to 0.62% per
month.
Our time-series regressions attempt to explain the cross-section of average
returns with the premiums for the common risk factors in returns. The wide
range of average returns on the 25 stock portfolios, and the size and bookto-market
effects in average returns, present interesting challenges for competing
sets of risk factors.
Most of the ten portfolios in the bottom two BEI’ME quintiles produce
average excess returns that are less than two standard errors from 0. This is an
example of a well-known problem [Merton (1980)] : because stock returns have
high standard deviations (around 6% per month for the size-BE ‘.CJE portfolios),
large average returns often are not reliably different from 0. The high
volatility of stock returns does not mean, however, that our asset-pricing tests
will lack power. The common factors in returns will absorb most of the variation
in stock returns, making the asset-pricing tests on the intercepts in the timeseries
regressions quite precise.
Borrds – In contrast to the stock portfolios, the average excess returns on the
government and corporate bond portfolios in table 2 are puny. All the average
excess bond returns are less than 0.15% per month, and only one of seven is
more than 1.5 standard errors from 0. There is little evidence in table 2 that (a)
average returns on government bonds increase with maturity, (b) long-term
corporate bonds have higher average returns than government bonds, or (c)
average returns on corporate bonds are higher for lower-rating groups.
The flat cross-section of average bond returns does not mean that bonds are
uninteresting dependent variables in the asset-pricing tests. On the contrary.
bonds are good candidates for rejecting asset-pricing equations that predict
patterns in the cross-section of average returns based on different slopes on the
common risk factors in returns.
3.2. The explanatory returns
In the time-series regression approach to asset-pricing tests, the average risk
premiums for the common factors in returns are just the average values of the
explanatory variables. The average value of RXJ-RF (the average premium per
unit of market p) is 0.43% per month. This is large from an investment
perspective (about 5% per year), but it is a marginal 1.76 standard errors from 0.
The average S,VJB return (the average premium for the size-related factor in
returns) is only 0.27% per month (t = 1.73). We shall find, however, that
the slopes on SAJB for the 25 stock portfolios cover a range in excess of 1.7, so
the estimated spread in expected returns due to the size factor is large, about
0.46% per month. The book-to-market factor HAIL. produces an average
premium of 0.40% per month (t = 2.91), that is large in both practical and
statistical terms.
i
Kill 0.97 4.52 3.97 0.05 – 0.05 0.03
7‘11 0.54 0.22 45.97 0.94 0.90 0.65
l.‘i’(; 0.60 3.03 3.66 0.05 – 0.w 0.00
(‘1) 0.62 2.24 5.10 0.20 – 0.04 0.04
RWRb 0.43 4.54 I .76
RhlO OS0 3.55 2.61
Shf0 0.27 2.89 1.73
IlhfL 0.40 2.54 2.9 I
7’1:R hl 0.06 3.02 0.38
DEb O.O:! I.60 0.2 I
0.05 – 0.04 0.03 RIII-RI.’ RMO
– 0.10 – 0.05 0.02 0.78 I.00
0.19 0.07 0.23 0.32 – 0.00
O.II( 0.06 0.07 – 0.38 – 0.00
0.05 ~ 0.00 – 0.00 0.34 0.w
~ 0.20 — 0.04 – 0.00 – 0.07 – 0.00
ari;iblcs: lixccss returns on governwznl and corporvk bonds
I SC; 0. I 2
6 IOG 0. I4
AAA 0.06
AA 0.07
A 0.0x
BAA 0.14
LG 0. I 3
Sld
1.25
2.03
2.34
2.23
2.25
2.35
2.52
Aulocorr. for lag
1 (tw1) I 2 12
Explanatory returns
Ikpendzlll
1.71
1.24
0.44
0.58
0.63
I.09
0. I 5 – 0.08
0. I ? – 0.05
0. I6 – 0.03
0. I9 – 0.04
0.2 I – 0.03
0.2 I 0.00
0.23 0.05
SMII
1.00
– 0.08
– 0.07
0. I7
0.01
0.02
0.02
0.03
0.04
0.03
0.08
I .m
– 0.05 I.00 c
0.0x – 0.69
c.
c-
?
k
2
2
:
t
Dependent variables: Excess returns on 25 stock portfolios formed on ME and BE/ME
Size
quintilc
sm:111
2
3
4
Kg
LOW
0.39
0.44
0.43
0.48
0.40
Book-to-market equity (BE/ME) quintiles
~~~ ~____~~ ~_. _
2 3 4 High Low 2 3 4 High
Means Standard deviations
0.70 0.79 0.88 1.01 7.76 6.84 6.29 5.99 6.27
0.71 0.85 0.84 I .02 7.28 6.42 5.85 5.33 6.06
0.66 0.68 0.8 I O.Y7 6.71 5.71 5.27 4.Y2 5.bY
0.35 0.57 0.17 I .05 5.97 5.44 5.03 4.95 5.75
0.36 0.32 0.56 0.59 4.95 4.70 4.38 4.27 4.85
t-statistics for means
Snxlll 0.93 I .8X 2.33 2.73 2.97
2 I.1 1 2.05 2.6’) 2.9 I 3.1 I
3 I.IX 2.12 2.39 3.04 3.15
4 I .49 1.19 2.08 LXX 3.36
I)& I .50 I .42 I .34 2.43 2.2b
‘Uhf is the value-weighted monthly percent return on the stocks in the 25 size-BE/ME portfolios, plus the negative-BE stocks excluded from the
portfolios. KF is the one-month Treasury bill rate, observed at the beginning of the month. LTG is the long-term government bond return. CB is the return
on ii proxy for the rmlrket portfolio of long-term corporate bonds. TERM is LTG-RF. DEF is CB~LTG. SMB (small minus big) is the drlTerence between
the returns on sndl-stock antI big-stock porttidios with ahout the s:mx weighted average hook-to-market equity. I/Ml. (high minus low) is the Merence
between the returns on high and low hook-to-market equity portfolios with ahuut the same weighted average size. RMO is the sum of the interccpl and
residuals l&n lhc regression (I) of KM -RF on ?‘ERM, L)EF, SMB, and IIML.
‘l‘he scvc’n hod portlidius used HIS dependcn~ vari;thles in the excess-return regressions are I – to 5-year and b- to IO-year governments (I -5G and 6 ~IW)
;IIIJ corlxnxte bonds rated AXI, A;I, A, Iklu. and below Hna (I.(;) by Moody’r. The 25 size-BE/ME stock portfoolios are formed as roollows. Each year r from
1903 to 1991 NYSE quintile hre;tkpoLits for size (ME, stock price times shares outstanding), measured al Ihe end of June, are used lo allocate NYSE.
An~x, and NASDAQ stocks to live size quintiles. Similarly, NYSE quintile breakpoints for BE/ME are used to allocate NY!%, Amex, and NASDAO
SIOC~S lo live hook-to-nl;lrkct equity quintiles. In U/:‘/ME, WB is hook comm~m equity for the liscal year ending in calendar year I – I, and ML’ is li)r the
cd ol’ Decemhcr d I – I. The 25 six &E/ME portfolios are hrmd as the intersections of the tivc size and the live HE/ME groups. Value-weighted
nlonthly prrcent returns on the pd’dios are caldad thrum July ul year I to June d I + I.
The average risk premiums for the term-structure factors are trivial relative to
those of the stock-market factors. TER.Ll (the term premium) and DEF (the
default premium) are on average 0.069/o and 0.029;b per month; both are within
0.3 standard errors of 0. Note, though, that TER,Zl and DEF are about as
volatile as the stock-market returns S.LfB and H.CIL. Low average premiums
will prevent TERM and DEF from explaining much cross-sectional variation in
average returns, but high volatility implies that the two factors can capture
substantial common variation in returns. In fact. the low means and high
volatilities of TER./I and DEF will be advantageous for explaining bond
returns. But the task of explaining the stron, 0 cross-sectional variation in
average stock returns falls on the stock-market factors. RN-RF. SMB, and
HML. which produce higher average premiums.
We turn now to the asset-pricing tests. In the time-series regression approach.
the tests have two parts. In section 4 vve establish that the two bond-market
returns, TER.tI and DEF, and the three stock-market returns, RXI-RF, SMB.
and H&IL, are risk factors in the sense that they capture common (shared and
thus undiversifiable) variation in stock and bond returns. In section 5 we use the
intercepts from the time-series regressions to test whether the average premiums
for the common risk factors in returns explain the cross-section of average
returns on bonds and stocks.
4. Common variation in returns
In the time-series regressions, the slopes and R’ values are direct evidence on
whether different risk factors capture common variation in bond and stock
returns. We first examine separately the explanatory power of bond-market
and stock-market factors. The purpose is to test for overlap between the
stochastic processes for stock and bond returns. Do bond-market factors
that are important in bond returns capture common variation in stock returns
and vice versa? We then examine the joint explanatory power of the bondand
stock-market factors, to develop an overall story for the common variation
in returns.
1. I. Bond-market fktors
Table 3 shows that, used alone as the explanatory variables in the time-series
regressions, TERM and DEF capture common variation in stock and bond
returns. The 25 stock portfolios produce slopes on TERM that are all more than
five standard errors above 0; the smallest TER.Cf slope for the seven bond
portfolios is 18 standard errors from 0. The slopes on DEF are all more than
7.8 standard errors from 0 for bonds, and more than 3.5 standard errors from 0
for stocks.
Table 3
Regressions of excess stock and bond returns (in percent) on the bond-market returns. TER./ and
DEf: July 1963 to December 1991. 342 months.”
R(t) – RF(t) = a + mTERJf(t) + dDEF(r) -t e(t)
Dependent variable: Excess returns on 25 stock portfolios formed on size and book-to-market
equity
Book-to-market equity (BE, ME) qumtiles
Size
quintile Low 2 3 4 High Low 2 3 -I High
Small
2
3
4
Big
Small
z
3
4
Big
Small
2
3
4
Big
m r(m)
0.93 0.90 0.89 0.86 0.89 5.02 5.50 5.95 6.08 6.0 1
0.99 0.96 0.99 1.01 0.98 5.71 6.32 7.29 8.3-t 6.92
0.99 0.94 0.94 0.95 0.99 6.25 7.10 7.80 8.50 7.60
0.92 0.95 0.97 1.05 1.03 6.58 7.57 a.53 9.64 7.83
0.82 0.82 0.80 0.80 0.77 7.14 7.60 8.09 8.26 6.84
d rid)
-_
1.39 1.31 1.33 1.45 1.X 3.96 4.27 4.73 5.45 5.45
1.26 1.28 1.35 1.38 1.41 3.84 4.47 5.28 6.05 5.29
1.21 1.19 I.25 1.24 1.21 4.05 4.74 5-19 5.89 4.98
0.96 1.01 1.13 1.21 1.22 3.65 1.28 5.25 5.89 4.91
0.78 0.73 0.78 0.83 0.89 3.59 3.60 4.18 4.56 4.15
R’ SW
0.06 0.08 0.09 0.10 0.10 7.50 6.57 6.00 5.68 5.95
0.08 0.10 0.13 0.17 0.12 6.97 6.09 5.45 4.87 5.69
0.10 0.12 0.15 0.17 0.14 6.38 5.35 1.86 4.48 5.2
0.11 0.14 0.17 0.21 0.15 5.63 5.04 4.57 1.39 5.31
0.13 0.15 0.16 0.17 0.12 4.61 4.33 4.00 3.89 4.55
Dependent variable: Excess returns on government and corporate bonds
I-5G 6-IOG Aaa Aa A Baa LG
m 0.45 0.72 I .02 0.99 1.00 1.01 0.81
t(m) 31.73 3880 99.94 130.44 139.80 56.24 18.05
d 0.25 0.27 0.94 0.96 I .02 1.10 1.01
t(d) 9.51 7.85 48.95 67.54 75.74 32.33 11.95
R’ 0.79 0.87 0.97 0.98 0.98 0.90 0.19
s(e) 0.57 0.75 0.41 0.30 0.29 0.72 1.80
“TERM is LTG-RF. where LX is the monthly percent long-term government bond return and
RF is the one-month Treasury bill rate. observed at the beginning of the month. DEF is C&LX,
where CB is the return on a proxy for the market portfolio of corporate bonds.
The seven bond portfolios used as dependent variables in the excess-return regressions are I- to
S-year and 6- to lo-year governments (I-5G and GlOG) and corporate bonds rated A.aa. Aa, A. Baa,
and below Baa (LG) by Moody’s, The 25 size-BE;.CfE stock portfolios are formed as follows. Each
year t from 1963 to 1991 NYSE quintile breakpoints for size (.WE, stock price times shares
outstanding). measured at the end of June. are used to allocate NYSE, Amex. and NASDAQ stocks
to five size quintiles. Similarly, NYSE quintile breakpoints for BE’.CfE are used to allocate NYSE,
Amex, and NASDAQ stocks to five book-to-market equity quintiles. In BE, ME, BE is book
common equity for the fiscal year ending in calendar year f – 1, and ME is for the end of December
oft – 1. The 25 size-BE, .1fE portfolios are formed as the intersections of the five size and the five
BE/ME groups. Value-weighted monthly percent returns on the portfolios are calculated from July
of year f to June oft + 1.
R’ and the residual standard error. s(e), are adjusted for degrees of freedom.
The slopes on TER.Cf and DEF allow direct comparisons of the common
variation in stock and bond returns tracked by the term-structure variables.
Interestingly. the common variation captured by TER.Cf and DEF is. if anything,
stronger for stocks than for bonds. Most of the DEF slopes for stocks are
bigger than those for bonds. The TER.tf slopes for stocks (all close to 1) are
similar to the largest slopes produced by bonds.
As one might expect, however, the fractions of return variance explained by
TER,M and DEF are higher for bonds. In the bond regression, R’ ranges from
0.49 for low-grade corporates to 0.97 and 0.98 for high-grade corporates. In
contrast, R’ ranges from 0.06 to 0.21 for stocks. Thus, TERM and DEF clearly
identify shared variation in stock and bond returns, but for stocks and lowgrade
bonds. there is plenty of variation left to be explained by stock-market
factors.
There is an interesting pattern in the slopes for TER.Cl. The slopes increase
from 0.45 to 0.72 for I- to S-year and 6- to lo-year governments, and then settle
at values near I for four of the five long-term corporate bond portfolios. (The
low-grade portfolio LG. with a slope of 0.81. is the exception.) As one would
expect. long-term bonds are more sensitive than short-term bonds to the shifts in
interest rates measured by TER.LI. What is striking. however, is that the 25 stock
portfolios have TER.Ll slopes like those for long-term bonds. This suggests that
the risk captured by TER,Cf results from shocks to discount rates that affect
long-term securities. bonds and stocks, in about the same way.
There are interesting parallels between the TER,Lf slopes observed here and
our earlier evidence that yield spreads predict bond and stock returns. In Fama
and French (1959), kve find that a spread of long-term minus short-term bond
yields (an ex ante version of TERXI) predicts stock and bond returns, and
captures about the same variation through time in the expected returns on
long-term bonds and stocks. We conjectured that the yield spread captures
variation in a term premium for discount-rate changes that affect all long-term
securities in about the same way. The similar slopes on TER,Lf for long-term
bonds and stocks observed here seem consistent with that conjecture.
Our earlier work also finds that the return premium predicted by the longterm
minus short-term yield spread wanders between positive and negative
values, and is on average close to 0. This parallels the evidence here (table 2) that
the average premium for the common risk associated with shifts in interest rates
(the average value of TERM) is close to 0.
The pattern in the DEF slopes in table 3 is also interesting. The returns on
small stocks are more sensitike to the risk captured by DEF than the returns on
big stocks. The DEF slopes for stocks tend to be larger than those for corporate
bonds, which are larger than those for governments. DEF thus seems to capture
a common ‘default’ risk in returns that increases from government bonds to
corporates, from bonds to stocks. and from big stocks to small stocks. Again,
there is an interesting parallel between this pattern in the DEF slopes and the
E.F. Fumu und K.R. Frewh. Common risk fucrors in srocb und bond reiurns 19
similar pattern observed in Fama and French (1989) in time-series regressions of
stock and bond returns on an ex ante version of DEF (a spread of low-grade
minus high-grade bond yields).
Using the Fama-Macbeth (1973) cross-section regression approach and stock
portfolios formed on ranked values of size, Chan, Chen. and Hsieh (1985) and
Chen, Roll, and Ross (1986) find that the cross-section of slopes on a variable
like DEF goes a long way toward explaining the negative relation between size
and average stock returns. Given the negative relation between size and the
slopes on DEF in table 3, it is easy to see why the DEf slopes work well in
cross-section return regressions for size portfolios.
Our time-series regressions suggest, however, that DEF cannot explain the
size effect in average stock returns. In the time-series regressions, the average
premium for a unit of DEF slope is the mean of DEF, a tiny 0.02% per month.
Likewise, the average TERM return is only 0.06% per month. As a result, we
shall see that the intercepts in the regressions of stock returns on TERM and
DEF leave strong size and book-to-market effects in average returns. We shall
also find that when the stock-market factors are added to the regressions, the
negative relation between size and the DEF slopes in table 3 disappears.
42. Stock-market f&ton
The role of stock-market factors in returns is developed in three steps. u’e
examine (a) regressions that use the excess market return, RAGRF, to explain
excess bond and stock returns, (b) regressions that use SMB and NML, the
mimicking returns for the size and book-to-market factors, as explanatory
variables. and (c) regressions that use RM-RF, S‘SJB. and H,VfL. The threefactor
regressions work well for stocks, but the one- and two-factor regressions
help explain why.
The Murket – Table 4 shows, not surprisingly, that the excess return on the
market portfolio of stocks, RM-RF, captures more common variation in stock
returns than the term-structure factors in table 3. For later purposes. however.
the important fact is that the market leaves much variation in stock returns that
might be explained by other factors. The only RZ values near 0.9 are for the
big-stock low-book-to-market portfolios. For small-stock and high-BE/ME
portfolios, R’ values less than 0.8 or 0.7 are the rule. These are the stock
portfolios for which the size and book-to-market factors, SMB and H.LIL, will
have their best shot at showing marginal explanatory power.
The market portfolio of stocks also captures common variation in bond
returns. Although the market fls are much smaller for bonds than for stocks.
they are 5 to 12 standard errors from 0. Consistent with intuition, /? is higher for
corporate bonds than for governments and higher for low-grade than for
high-grade bonds. The /I for low-grade bonds (LG) is 0.30, and R.V-RF explains
a tidy 19% of the variance of the LG return.
Table 1
Regressions of excess stock and bond returns (in percent) on the excess stock-market return,
R.WRF: July 1963 to December 1991. 342 months.”
R(t) – RF(t) = a + b[R.Lf(rl – RF(r)] + r(r)
_
Dependent variable: Excess returns on 25 stock portfolios formed on size and book-to-market
equity
Book-to-market equity (BE .CIE quintiles
Size
quintile Lou 2 3 1 High Low 2 3 1 Htgh
h r(h)
Small 1.40 1.26 I.11 I .06 I .08 16.33 28.12 27.01 25.03 23.01
2 I .‘I? I.15 I.12 I .02 1.13 35.76 35.56 33.12 33.1-I
3
29.04
1.36 I.15 I.04 0.96 I .oa 12.98 42.52 37.50 35.81 31.16
1 I.24 I.14 I .03 0.95 I.10 51.67 55.IZ 46.96 37.00 32.76
Btg I .03 0.99 0.89 0.84 0.89 5 I .92 61.51 13.03 35.96 27.75
R’ s(e)
Small 0.67 0.70 0.68 0.65 0.6 I -1.46 3.76 3.55 3.56 3.92
2 0.79 0.79 0.76 0.76 0.71 3.34 2.96 2.85 2.59
?
3.25
0.8-l 0.81 0.80 0.79 0.74 2.65 2.28 2.33 3.26 2.90
; 0.89 0.90 0.87 0.80 0.76 2.0 I 1.73 I.% 2.2 1 2.83
Big 0.89 0.91 0.54 0.79 0.69 1.66 1.35 1.73 1.95 2.69
Dependent variable: Excess returns on government and corporate bonds
I -5G 6-IOG Aaa -a A Baa LG
h 0.08 0.13 0.19 0.20 0.21 0.21 0.30
r(h) 5.24 5.57 7.53 8.14 8.42 8.73 II.90
RJ 0.07 0.0s 0.14 0.16 0.17 0.19 0.29
s(e) I.21 1.95 2.17 2.05 2.05 2.12 2.12
‘R.W is the value-heighted monthly percent return on all the stocks in the 25 size-BE.‘IW&
portfolios, plus the negative-BE stocks excluded from the 25 portfolios. RF is the one-month
Treasury bill rate. observed at the beginning of the month.
The seven bond portfolios used as dependent variables in the excess-return regressions are I- to
5-vear and 6- to IO-vear eovernments (I-SC and 6-IOG) and corporate bonds rated Aaa. Aa. .A. Baa.
and below Baa (LG) by Moody’s The 25 size-BE, .LfE stock portfolios are formed as follous. Each
year r from 1963 to 1991 NYSE quintile breakpoints for size (,LfE. stock price times shares
outstanding). measured at the end of June, are used to allocate NYSE. Amex. and NASDAQ stocks
to five size quintiles. Similarly. NYSE quintile breakpoints for B&ME are used to allocate NYSE,
Amex. and NASDrQ stocks to five book-to market equity quintiles. In BE ME. BE is book
common equity for the fiscal year ending in calendar year r – I, and ME is for the end of December
of r – I. The 25 size-BE .UE portfolios are formed as the intersections of the five size and the five
BE .LfE groups. Value-weighted monthly percent returns on the portfolios are calculated from July
ofyearrtoJuneofr+l.
R’ and the residual standard error. s(e), are adjusted for degrees of freedom.
E.F. Fuma and K.R. French. Common risk factors VI s[ock and bond rerurns 21
S.LfB and H.CIL -Table 5 shows that in the absence of competition from the
market portfolio. SMB and H&IL. typically capture substantial time-series
variation in stock returns; 20 of the 25 R’ values are above 0.2 and eight are
above 0.5. Especially for the portfolios in the larger-size quintile, however, SXfB
and H~ML leave common variation in stock returns that is picked up by the
market portfolio in table 4.
The Marker, S,LIB, and HML – Table 5 says that, used alone, SMB and HAIL
have little power to explain bond returns. Table 6 shows that when the excess
market return is also in the regressions, each of the three stock-market factors
captures variation in bond returns. We shall find, however, that adding the
term-structure factors to the bond regressions largely kills the explanatory
power of the stock-market factors. Thus the apparent role of the stock-market
factors in bond returns in table 6 probably results from covariation between the
term-structure and stock-market factors.
The interesting regressions in table 6 are for stocks. Not surprisingly. the three
stock-market factors capture strong common variation in stock returns. The
market ps for stocks are all more than 38 standard errors from 0. With one
exception, the t-statistics on the SMB slopes for stocks are greater than 4; most
are greater than 10. SMB, the mimicking return for the size factor, clearly
captures shared variation in stock returns that is missed by the market and by
HML. Moreover, the slopes on SMB for stocks are related to size. In every
book-to-market quintile, the slopes on SMB decrease monotonically from
smaller- to bigger-size quintiles.
Similarly, the slopes on HML, the mimicking return for the book-to-market
factor, are systematically related to BEI.Lf E. In every size quintile of stocks, the
H&IL slopes increase monotonically from strong negative values for the lowestBE,!.LIE
quintile to strong positive values for the highest-BE/.LIE quintile.
Except for the second BE/ME quintile, where the slopes pass from negative to
positive, the HML slopes are more than five standard errors from 0. HML
clearly captures shared variation in stock returns, related to book-to-market
equity. that is missed by the market and by SMB.
Given the strong slopes on SMB and H&IL for stocks, it is not surprising that
adding the two returns to the regressions results in large increases in R2. For
stocks, the market alone produces only two (of 25) R’ values greater than 0.9
(table 4); in the three-factor regressions (table 6) RZ values greater than 0.9 are
routine (21 of 25). For the five portfolios in the smallest-size quintile, R2 increases
from values between 0.61 and 0.70 in table 4 to values between 0.94 and
0.97 in table 6. Even the lowest three-factor R’ for stocks, 0.83 for the portfolio
in the largest-size and highest-BE!rLIE quintiles, is much larger than the 0.69
generated by the market alone.
Adding SMB and HML to the regressions has an interesting effect on the
market ps for stocks. In the one-factor regressions of table 4, the p for the
portfolio of stocks in the smallest-size and lowest-BE/ME quintiles is 1.40. At
Table 5
Regressions of wcess dock and bond returns (in percent) on the mimicking returns for the size (SMB) and book-lo-market euuilv (IIML) factors: Julv
.~.
I963 IO December 1991, 342 months.”
H(r) – W(r) = 0 + .sSMB(r) + IrIfM L(r) + r(r)
Size
quinlile
Smdl
2
3
4
Big
Sm;III
2
3
‘I
Big
2
3
4
Big
LWV
I .Y3
1.52
1.28
0.X6
0.28
– O.Y5
-~ 1.22
— 1.0’)
– I.11
~ 1.07
0.65
0.5’)
0.51
0.43
0.34
Depcndrn~ variable: Excess rcIurns on 25 stock podolios formed on size and book-to-market equily
Book-lo-market equity (BE/ME) quindes
2 3
1.73
1.46
1.12
0.x2
0.35
I .63
1.35
1.05
0.77
0.22
/I
1.5’) I .67 22.52
1.18 1.40 17.23
o.Y3 I.16 14.43
0.12 O.Y5 IO.16
0.2Y 0.44 3.70
_ 0.57
~ 0.66
– 0.65
~ 0.65
– 0.65
– 0.35
– 0.3x
– 0.31
– 0.36
– 0.42
K2
~ 0.18 0.0 I – Y.72
– 0.16 O.lK) ~ 12.25
~~ 0.1 I – 0.01 – IO.84
– 0.1 I ~ 0.0 I – II.43
~ 0.06 0.08 – 12.46
0.00 0.60 0.60 0. SY
0.53 0.4’) 0.42 0.44
0.43 0.37 0.31 0.35
0.30 0.24 0.18 0.23
0.18 0.08 0.04 0.06
s
4 High Low
21.38
17.68
I3.W
9.64
4.3’)
21.8X 22.30 22.16
17.08 15.47 16.42
13.42 12.13 13.45
9.29 x.57 IO.02
2.7’) 3.69 5.02
f(N
– 6.1’)
~ 7.02
– 7.07
– 6.6’)
– 7.07
– 4.10 _ 2.20 0.16
– 4.20 – 1.82 0.05
– 3.43 – 1.23 – 0.12
– 3.80 – 1.12 ~ 0.0’)
– 4.64 – 0.66 O.XI
s(e)
4.57 4.3 I 3.9x 3.7Y 4.01
4.6X 4.4 I 4.20 4.06 4.53
4.7 I 4.3 I 4.19 4.10 4.60
4.53 4.55 4.40 4.48 5.06
4.02 4.27 4.20 4.19 4.6Y
2 3 4 High
IN
E.F. Fumu and K. R. French. Common risk f&tors in SIL)L% und bond rerurns
23
Table 6
Regressions of excess stock and bond returns (in percent) on the excess market return (M-RF) and the mimicking returns for the size (SMU) .md bookto-market
equity (Ill) factors: July 1963 IO December 1991, 342 months.’
H(l) – RF(l) = ‘l + /,LKM(f) — KI$)j -t ssnlryr) + M/hlL(r) + r(/)
Dependent variable: Excess returns on 25 stock portfolios formed on size and book-lo-markel equity
Book-lo-market equity (BE/ME) quintiles
4 High Low 2
Sire
quinlile
Sm;rll
2
3
4
Big
Small
2
3
4
Big
2
3
4
Big
Low 2 3
h
1.04 1 a2 0.95 0.91 0.96
I.11 I .06 1.00 0.97 I.09
1.12 I .02 0.98 0.97 1.09
I .07 I .0x 1.04 I .05 1.18
O.Y6 1.02 0.9X O.YY I .06
–_~_s
39.37 51.80 60.44
52.49 61.18 55.88
56.X8 53.17 50.78
53.94 53.51 51.21
60.93 56.76 46.57
I .46 1.26 1.19 1.17 1.23
I.00 0.98 0.88 0.73 0.89
0.76 0.65 0.60 0.48 0.66
0.37 0.33 0.2Y 0.24 0.4 I
– 0.17 – 0.12 – 0.23 – 0.17 – 0.05
II
37.Y2 44.11
32.73 38.79
26.40 23.39
12.73 11.11
– 7.18 – 4.51
– 0.20 0.0x 0.26 0.40 0.62 – 6.47 2.35
_ 0.52 0.0 I 0.26 0.46 0.70
_ 14.57 0.4 I
– 0.31 – 0.00 0.32 0.5 I 0.6X – II.26 – 0.05
_ 0.42 0.04 0.30 0.56 0.74 – 12.51 1.04
– 0.46 0.00 0.21 0.57 0.76 – 17.03 0.09
3
I(S)
52.03
34.03
21.23
9.X I
– 7.58
f(M
__~~~
9.66
X.56
Y.75
X.X3
5.80
4
59.73 57.x’)
61.54 65.52
54.38 52.52
47.0 46. IO
53.87 3X.61
52.85
31.66
IX.62
7.38
– 6.27
50.Y7
36.7X
21.91
II.01
– 1.18
15.53 22.24
17.24 24.X0
16.88 19.39
14.84 17.OY
1x.34 t 6.24
High
R’ s(4
Small 0.94 0.96 0.97 0.97 0.96 I .94 I .44 1.16 1.12 1.22
2 0.95 0.96 0.95 0.95 0.96 1.55 1.27 1.31 1.16 1.23
3 0.95 0.94 0.93 0.93 0.93 1.45 1.41 1.43 1.32 1.52
4 0.94 0.93 0.91 0.89 0.89 I .46 I.411 I .49 1.63 1.88
Big 0.94 0.92 0.8X 0.90 0.x3 1.16 1.32 1.55 I .36 2.02
I-SC
h 0.10
l(h) 6.45
S – 0.06
l(S) – 2.70
/I 0.07
w 2.66
R2 0.10
44
I.19
Dependent variable: Excess returns on government and corporate bonds
66IOG
0.18
6.75
– 0.14
– 3.65
0.08
I .83
0.12
1.91
Aaa Aa A
~~ ~____._ ~~~~_
0.25 0.25 0.26
8.60 9.30 9.46
– 0.12 – 0.1 I – 0.09
– 2.89 – 2.72 – 2.18
0.14 0.15 0.16
2.77 3.26 3.51
0.17 0.20 0.20
2.13 2.00 2.01
Baa LG
0.27 0.34
9.58 12.22
– 0.04 0.04
– 0.91 Ott9
0.20 0.23
4.08 4.75
0.22 0.33
2.08 2.06
‘RM is the value-weighted percent monthly return on all the stocks in the 25 size-BE/ME portfolios, plus the negative-lit‘ stocks excluded from the 25
portfolios. RF is the one-month Treasury bill rate, observed at the beginning of the month. SMB(small minus big) is the return on the mimicking portfolio
for the size factor in stock returns. f/ML (high minus low) is the return on the mimicking portfolio for the book-to-market factor. (See table 5.)
The seven bond portfolios used as dependent variables are I- lo S-year and 6- IO IO-year governments (I-SC and 6-IOG) and corporate bonds rated
Aaa, An, A, Baa, and below Baa (LG) by Moody’s The 25 size-BE/ME stock portfolios are formed as lollows. Each year I from 1963 to 1991 NYSE
quintile hrcakpoints Car size, AI K, measured al the end of June, are used to allocate NYU!, Amex, and NASDAQ stocks lo live size quintiles. Similarly,
NYSIl quintile breakpoints for LIE/ME are used to allocate NYSE, Amex, and NASDAQ stocks to live book-to-market equity quintiles. In SE/ME, tlE is
book common equity for the liscal year ending in calendar year I – I, and ME is for the end ol December olt – 1. The 25 size-HE/ME portfolios are the
intersections of the live size and the tive SE/ME groups. Value-weighted monthly percent returns on the 25 portfolios are calculated from July elf to June
orI+ I.
Kz and the residual standard error, S(P), are adjusted for degrees ol freedom.
26 E.F. Fuma und K. R. Frmch. Common ruk jtictorr in srock and bond rrrurns
the other extreme, the univariate /I for the portfolio of stocks in the biggest-size
and highest-BE;.CfE quintiles is 0.89. In the three-factor regressions of table 6.
the fls for these two portfolios are 1.04 and 1.06. In general. adding .S,LfB and
H&IL to the regressions collapses the ps for stocks toward 1.0: low gs move up
toward 1.0 and high ps move down. This behavior is due. of course, to
correlation between the market and SMB or H&IL. Although S.1fB and HML
are almost uncorrelated ( – O.OS), the correlations between R.lf-RF and the
SMB and HML returns are 0.32 and – 0.38.
4.3. Stock-mnrkrt and bond-market factors
Used alone, bond-market factors capture common variation in stock returns
as well as bond returns (table 3). Used alone, stock-market factors capture
shared variation in bond returns as well as stock returns (table 6). These results
demonstrate that there is overlap between the stochastic processes for bond and
stock returns. We emphasize this point because the joint tests on the stock- and
bond-market factors that follow muddy the issue a bit.
First Pass – Table 7 shows that, used together to explain returns, the
bond-market factors continue to have a strong role in bond returns and the
stock-market factors have a strong role in stock returns. For stocks. adding
TER,Zf and DEF to the regressions has little effect on the slopes on the
stock-market factors: the slopes on R&f-RF. SXfB. and H.LfL for stocks in table
7a are strong and much like those in table 6. Similarly, adding R.Cf-RF, SMB,
and HhfL to the regressions for bonds has little effect on the slopes on TERhf
and DEF. which are strong and much like those in table 3.
The five-factor regressions in table 7 do. however. seem to contradict the
evidence in tables 3 and 6 that there is strong overlap between the return
processes for bonds and stocks. Adding the stock-market factors to the regressions
for stocks kills the strong slopes on TERM and DEF observed in the
two-factor regressions of table 3. The evidence in table 6 that bond returns
respond to stock-market factors also largely disappears in table 7b. In the
five-factor regressions, only the low-grade bond portfolio, LG. continues to
produce nontrivial slopes on the stock-market factors.
Table 7 seems to say that the only shared variation in bond and stock returns
comes through low-grade bonds. But tables 3 and 6 say there is strong common
variation in bond and stock returns when bond- and stock-market factors are
used alone to explain returns. Can we reconcile these results? We argue next that
the two term-structure factors are indeed common to bond and stock returns. In
the five-factor regressions for stocks, however, the tracks of TER.Cf and DEF are
buried in the excess market return. R.bf-RF. In contrast to the two termstructure
factors. the three stock-market factors are generally confined to stock
returns; except for low-grade bonds, these factors do not spill over into bond
returns. In short. we argue that stock returns share three stock-market factors,
E.F. Fama und K.R. French. Common ruk facrors in stuck and bond r~furns 17
and the links between stock and bond returns come largely from two shared
term-structure factors.
Second Pass: in Orthoyonali:ed itfarket Fuctor – If there are multiple common
factors in stock returns, they are all in the market return, RM, which is just
a value-weighted average of the returns on the stocks in the CRSP-COMPUSTAT
sample. The regression of RM-RF on SIVB. HAIL, TERM, and DEF for
monthly returns of July 1963 to December 1991 illustrates the point:
RIM-RF = 0.50 + 0.44SMB – 0.63 HML + 0.81 TERM
(2.55) (6.48) ( – 8.23) (9.09)
+ 0.79 DEF + e.
(4.62)
(1)
The r-statistics are in parentheses below the slopes; the R’ is 0.38. This
regression demonstrates that the market return is a hodgepodge of the common
factors in returns. The strong slopes on TER,LI and DEF produced by RM-RF
(the excess return on a proxy for the portfolio of stock-market wealth) are clear
evidence that the two term-structure factors capture common variation in stock
returns.
The sum of the intercept and the residuals in (l), call it RMO, is a zeroinvestment
portfolio return that is uncorrelated with the four explanatory
variables in (I). We can use RR/IO as an orthogonalized market factor that
captures common variation in returns left by SR;IB, HML, TERM, and DEF.
Since the stock-market returns, S&fB and HML, are largely uncorrelated
with the bond-market returns, TERM and DEF (table 2). five-factor regressions
that use R,ClO, SMB, HML, TERM. and DEF to explain bond and
stock returns will provide a clean picture of the separate roles of bond- and
stock-market factors in bond and stock returns. The regressions are in
table 8.
The story for the common variation in bond returns in table 8b is like that in
table 7b. The bond-market factors, TER.Ll and DEF, have strong roles in bond
returns. Some bond portfolios produce slopes on the stock-market factors that
are more than two standard errors from 0. But this is mostly because TERM and
DEF produce high R’ values in the bond regressions, so trivial slopes can be
reliably different from 0. As in table 7b. only the low-grade bond portfolio (LG)
produces nontrivial slopes on the stock-market factors. Otherwise, the stockmarket
factors don’t add much to the shared variation in bond returns captured
by TERM and DEF.
For the stock portfolios, the slopes on R,CIO in the five-factor regressions of
table 8a are identical (by construction) to the large slopes on RM-RF in table
7a. The slopes on the size and book-to-market returns in table 8a shift somewhat
(up for S;LIB, down for HXIL) relative to the slopes in table 7a. But the spreads
Table 7a
Kcgressions of L’XCCSS stock returns on 25 stock portfolios formed on size and book-to-market equity (in percent) on the stock-market returns, Rhf -RF,
Shff3, and IIML. and the bond-market returns, 7’EHM and DEF: July 1963 to December 1991, 342 months.”
Size
quintile
Small
2
3
4
Iiig
Small
2
3
4
Big
Smdl
2
3
4
Big
R(r) – RF(r) = LI + b[Rhl(r) – RF(rj] + sSMB(r) + hIfML(r) + mTERM(r) + dDEF(t) + e(r)
Book-to-market equity (BE/ME) quintiles
Low 2 3 4 High Low 2 3 4 High
h
l(h)
I .06 1.04 0.96 0.92 0.98 35.97 47.65 54.46 54.51 53.15
1.12 I .06 O.YX 0.94 I.10 47.1’) 54.95 49.0 I 54.19 5Y.W
1.13 I.01 0.Y7 0.95 I .0x 50.93 46.95 44.57 47.59 46.92
I .07 I .07 I.01 1.00 I.17 48.18 47.55 44.83 41.02 41.02
O.Y6 I .02 O.YX I so I.10 53.x7 51.01 41.35 4X.2Y 35.96
s r(s)
1.45 I .26 1.20 I.15 I.21 37.02 43.42 50.89 51.36 49.55
I .Ol O.YX 0.x’) 0.74 0.x’) 32.06 3x. IO 33.6X 32.12 35.7’)
0.76 0.66 0.00 0.4Y 0.6X 25.X2 22.Y7 20.x3 IX.54 22.32
0.3x 0.34 0.30 0.26 0.42 12.71 I I .36 Y.YY x.05 I I .07
– 0.17 -0.11
– 0.23
– 0. I7
– 0.06 – 7.03 – 4.07 – 7.3 I – 6.07 – 1.44
/I r(h)
– 0.27 0.10 0.27 0.40 0.63 – 5.Y5 2.90 9.82 15.47 22.27
– 0.51 0.02 0.25 0.44 0.71 – 14.01 0.69 8.1 I 16.50 24.61
– 0.37 ~ 0.00 0.31 0.50 0.69 – IO.81 – 0.1 I 9.28 16.18 19.34
– 0.42 0.04 0.29 0.53 0.75 – 12.09 I.10 8.37 14.20 16.88
– 0.46 0.01 0.21 0.58 0.78 – 16.85 0.38 5.70 18.16 16.59
E.F. Famo and K.R. French, Common risk factors in stock and bond relurns 29
zswc,* -_-
ddddd
I I I I
I I I I
J.F E.-B
30
Table 7b
Regresstons of cwess stock returns on gobernment and corporate bonds tin percent) on the
stock-market returns. R.Lf-RF. S.fB. and H.UL. and the bond-market returns. T&R.! and DEf:
July 1963 to December 1991. 342 months.”
R(t) – RF(r) = u + h[R.U(o – RF(r)] + rS.CfB(rl i hH.lfL(O + mTER.U(rt + dDEF(fI + e(t)
Bond portfoIl”
I-Xi 6-1OG Aaa .Aa A Baa LG
h
C(h)
– 0.02 – 0.04
– 2.8-I – 3 I4
0.00 – 0.02
0.30 – I.12
0.00 – 0.02
0.44 – 1.29
0.47 0.75
30.0 I 36 8-t
0.27 0.32
9.57 x.77
0.80 0.87
0.56 0.73
– 0.01
_ 2.96
– 0.0’
_ 2.28
– 0.02
– 2.46
1.03
93.30
0.97
49.X
0.97
0.40
0.00 0.00 0.02 0.18
0.06 I .05 I .99 7.39
– 0.01 0.00 0.05 0.08
_ 2.42 0.40 3.20 2.34
– 0.00 0.00 0.04 0.12
– 0.40 0.90 2.39 3.13
0.99 1.00 0.99 0.64
117.30 124.19 50.50 14.25
0.97 I .02 1.05 0.80
65.04 71.51 30.33 9.92
0.98 0.98 0.9 I 0.58
0.30 0.29 0.70 1.63
“R.Lf IS the value-weighted monthly percent return on all stocks in the 25 size-BE ME portfolios.
plus the negative-BE stocks excluded from the portfolios. RF is the one-month Treasury bill rate.
observed at the beginning “fthe month. S.fB (small minus big) is the difference each month between
the simple average of the returns on the three small-stock portfolios (S L. S .Lf. and SH) and the
sample average of the returns on the three big-stock portfolios (FL. B .I. and B H). H.rfL (high
minus loul is the ditference each month between the simple average of the returns on the two
high-BE .fE portfohos (S H and 6 H) and the average of the returns on the two lo&-BE ME
portfolios (5 L and B I!.). TER,Lf is LTG-RF, where LTG is the long-term government bond return.
DEF is CB-LTG. uhere CB is the return on a pro.xy for the market portfolio of corporate bonds.
The seven bond portfolios used as dependent variables in the excess-return regessions are I- to
j-year and 6- to IO-qear governments ( I-5G and 6IOGJ and corporate bonds rated Aaa, Aa. A, Baa.
and below Baa (LGJ by Moody’s. The 25 size-BE:.fE stock portfolios are formed as follows. Each
year r from 1963 to 1991 NYSE quintile breakpoints for size (.fE, stock price times shares
outstanding). measured at the end of June, are used to allocate NYSE. Amex. and NASDAQ stocks
to tice size quintiles. Simtlarly. NYSE quintile breakpoints for BE ‘.LfE are used to allocate NYSE.
Amex. and NASDAQ stocks to five book-to-market quintiles. In BE ./E. BE is book common
equity for the fiscal year ending in calendar year t – 1. and ,LfE is for the end of December of r – I.
The 25 size-BE .fE portfolios are the intersections of the five size and the fire BE .fE groups.
Value-weighted monthly percent returns on the portfolios are calculated from July of year f to June
ofr+ I.
R’ and the residual standard, error, s(r). are adjusted for degrees of freedom.
in the .S.LfB and H,LfL slopes across the stock portfolios in table 8a are like those
in table 7a, and .S,LfB and H,LfL again capture strong shared variation in stock
returns.
What changes dramatically in the five-factor regressions of table 8, relative to
table 7. are the slopes on the term-structure factors for stocks. The slopes on
TERM are more than 14 standard errors from 0; the DEf slopes are more than
seven standard errors from 0. The slopes on TERM and DEF for stocks are like
those for bonds. Thus unlike table 7, the five-factor regressions in table 8 say
that the term-structure factors capture strong common variation in stock and
bond returns.
How do the tracks of the term-structure variables get buried in the five-factor
regressions for stocks in table 7a? Table 8a says that stocks load strongly on
RMO, TER.CI. and DEF, but there is little cross-sectional variation in the slopes
on these factors. All the stock portfolios produce slopes on TER,V and DEF
close to 0.81 and 0.79, the slopes produced by the excess market return in (I).
And the stock portfolios all produce slopes close to 1.0 on R.CIO in table 8a, and
thus on R.Lf-RF in table 7a. Tables 7a and 8a then say that because there is little
cross-sectional variation in the slopes on RJI-RF, RJlO, TER.U, and DEF, the
excess market return in table 7a absorbs the common variation in stock returns
associated with R.LJO. TER.Vf, and DEF. In short, the common variation in
stock returns related to the term-structure factors is buried in the excess market
return in table 7a.
Is there any reason to prefer the five-factor regressions in table 8 over those in
table 7? Only to show that, in addition to the three stock-market factors. there
are two bond-market factors in stock returns. Otherwise, the two sets of
regressions produce the same R’ values and thus the same estimates of the total
common variation in returns. And the two sets of regressions produce the same
intercepts for testing the implications of five-factor models for the cross-section
of average stock returns.
5. The cross-section of average returns
The regression slopes and R’ values in tables 3 to 8 establish that the
stock-market returns. SMB, HhJL, and R.&RF’ (or R.MO), and the bondmarket
returns, TERM and DEF. proxy for risk factors. They capture common
variation in bond and stock returns. Stock returns have shared variation related
to three stock-market factors, and they are linked to bond returns through
shared variation in two term-structure factors. We next test how well the
average premiums for the five proxy risk factors explain the cross-section of
average returns on bonds and stocks.
The average-return tests center on the intercepts in the time-series regressions.
The dependent variables in the regressions are excess returns. The explanatory
variables are excess returns (R.WRF and TERXJ) or returns on zero-investment
portfolios (R,bf 0, S.CJB, HALI L. and DEF). Suppose the explanatory returns have
minimal variance due to firm-specific factors, so they are good mimicking
returns for the underlying state variables or common risk factors of concern to
investors. Then the multifactor asset-pricing models of Merton (1973) and Ross
Size
quinlile
Sn1all
2
3
4
Big
SlIldl
2
3
4
Big
Sm:~ll
2
3
4
Big
Table 8a
Regressions ofexcess stock returns on 25 stock portfolios formed on size and book-to-market equity (in percent) on the stock-market returns, KMo, SMB,
and ffML, and the bond-market relurns, TEKM and DEF: July 1963 IO December 1991, 342 months.’
K(r) – RF(f) = u + bRMO(r) + sSMB(r) + hIIML(r) + ruTERM(r) + dDEF(~) + r(f)
Book-to-market equity (BE/ME) quintiles
Low 2 3
b
4 High Low 2 3
t(b)
4 High
I .06 I.04 0.96 0.92 0.98 35.97 47.65 54.48 54.51 53.15
I.12 I .06 0.98 0.94 1.10 47.19 54.95 49.01 54.19 59.00
1.13 I.01 0.97 0.95 I .08 50.93 46.95 44.57 47.59 46.92
1.07 I .07 1.01 I.00 1.17 48.18 47.55 44.83 41.02 41.02
0.96 I .02 0.98 1.00 1.10 53.87 51.01 41.35 48.29 35.96
s
r(s)
I .92 1.72 I .62 1.56 1.64 51.96 62.88 73.21 73.72 71.32
1.50 I .45 1.33 1.16 1.3x 50.66 59.80 53.02 53.20 58.79
I .26 I.11 I .03 0.9 1 I.16 45.37 40.94 37.83 36.47 40.24
0.85 0.8 I 0.75 0.70 0.94 30.49 28.84 26.42 23.02 26.22
0.26 0.34 0.20 0.28 0.43 II.56 13.69 6.85 10.62 II.17
II t(h)
~___________
– 0.94
– 0.56
– 0.34
– 0.18 0.01 – 22.65 – 18.19 – 13.67 – 7.49 0.57
1.22
– 0.65
– 0.37 0.15 0.0 I – 36.52 – 23.89 – 13.09 – 6.22 0.51
– 1.0x
– 0.64
– 0.30
– 0.10 0.00 – 34.6X – 21.18 – 9.x2 – 3.61 0.16
– 1.09
– 0.64
– 0.35
– 0.10 0.00 – 34.85 – 20.12 – 10.93 – 2.83 0.10
– 1.07
– 0.63
– 0.4 1 – 0.05 0.09 – 42.62 – 22.46 – 12.30 – 1.75 2.06
Sn1all 0.75 0.73
2 0.85 0.82
3 0.88 0.84
4 0.85 0.87
Ijig 0.x0 0.79
Small 0.67 0.63
2 0.76 0.72
3 0.80 0.78
4 0.74 0.74
nig 0.81 0.66
Small 0.94 0.96
2 0.95 0.96
3 0.95 0.94
4 0.94 0.93
Big 0.94 0.92
_- ._
“See kmnote under table 8b.
0.73
0.86
0.84
0.90
0.79
cl
0.71 0.73 15.66
0.89 0.84 22.08
0.86 0.88 24.21
0.98 0.94 23.24
0.77 0.73 27.M)
0.66
0.81
0.83
0.84
0.75
HZ
0.97
0.95
0.93
0.91
0.87
0.78 0.79 7.25
0.19 0.79 10.23
0.84 0.69 II.53
0.91 0.80 10.56
0.72 0.68 14.56
_____
0.97 0.96 1.93
0.95 0.96 1.55
0.93 0.93 1.45
0.90 0.89 1.46
0.90 0.83 1.17
_~___
—
20.60
25.96
23.85
23.77
24.17
9.20
11.94
Il.64
10.48
10.62
—
1.43
1.27
1.41
1.47
1.31
-__
dm)
~-_ -_I
25.32 25.67 24.24
26.40 31.68 27.57
23.73 26.34 23.52
24.35 24.76 20.11
20.42 22.83 14.66
f@)
11.90
12.96
12.25
11.88
IO.15
s(e)
I.16
I.31
1.43
1.48
1.55
14.81 13.73
16.36 13.57
13.53 9.63
12.01 X.98
II.04 7.15
___–
1.11 1.20
1.13 1.23
1.31 1.50
1.59 1.88
1.36 2.00
Table 8b
Regresstons ofswess returns on government and corporate bonds (in percent1 on the stock-market
returns. R.UO. S.UB. and H.tfL. and the bond-market returns. TERJI and DEf: July 1963 to
December 1991. 3-t: months.’
I-SG 66IOG .Aaa .-a .A Baa LG
h – 0.03 – 0.04 – 0.02 0.00 0.00 0.0’ 0.18
ribI – 1.34 – 3.14 – 2.96 0.06 I .05 I .99 7.39
5 – 0.00 – 0.03 – 0.03 – 0.01 0.00 0.06 0.16
r(s) – 0.68 – 2.30 – 3.17 – 2.55 0.80 1.09 5.09
/I 0.02 – 0.00 – 0.01 – 0.00 0.00 0.03 0.00
f(lll 1.76 – 0.00 – 1.36 – 0.17 0.52 1.72 0.12
V, 0.45 0.71 1.02 0.99 1.00 1.01 0.79
t(trll 32.09 39.55 102.65 130.93 139.1 I 57.34 19.56
Cl 0.15 0.29 0.95 0.97 I .02 1.07 0.94
t(ll) 9.16 8.25 50.04 67.05 74.00 31.77 I’.09
R2 0.130 0.87 0.97 0.9s 0.98 0.9 I 0.58
S(Z) 0.56 0.73 0.40 0.30 0.29 0.70 1.63
_ _
‘R.IO, the orthogonalized market return. is the sum of intercept and residuals from the
represston of R.WRF on .S.fB. H.LIL. TER.W, and DEF. R.ll 1s the value-ueighted monthly percent
return on all stocks in the 25 size-BE .tfE portfolios. plus the negative-BE stocks excluded from the
portfoltos. RF is ths one-month Treasury btll rate, observed at the beginning of the month. S.LfB
(smdll mtnus big). the return on the mimicking portfolio for the common size factor tn stock returns,
is the dtfference each month between the simple aberage of the returns on the three small-stock
portioitos (S L. 5’ .I. and S Hl and the simple aterage of the returns on the three big-stock portfoltos
(5 L. B .If. and B Hi fl./L (htgh minus low). the return on the mimicking portfolio for the common
book-to-market equtty factor m returns, is the dtfference each month between the simple alerage of
the returns on the t&o hi!h-BE .A/& portfolios 1.5 H and B i-l) and the average of the returns on the
two low-BE .ME portfoltos (S f. and B L). TER.Lf is LTG-RF, where LX is the long-term
government bond return. DEF IS CB-LTG, where CB is the return on a proxy for the market
portfolto of corporate bonds.
The seben bond portfolios used as dependent variables m the excess-return regressions are I- to
j-year and 6- to IO-year governments (1-S and &lOG) and bonds rated .Aaa. Aa. A. Baa, and
belou Baa (LG, by Moody’s The 25 size-BE.‘.LfE stock portfolios are formed as follows. Each year
I from 1963 to 1991 XYSE quinttle breakpoints for size (LIE. stock price times shares outstandinp).
measured at the end of June. are used to allocate SYSE. Amer. and NASD.A.Q stocks to fire size
qutnttles. NYSE quinttie breakpoints for BE .CfE are also used to allocate NYSE. .Amex. and
S;SD.AQ stocks to tire-book-to-market equity quintiles. In BE .L/E. BE is book common equity
for the fiscal year endtng tn calendar year I – I. and .fE ts for the end of December oft – 1. The 25
stze-BE .5/E portfolios are the intersections of the fie size and the five BE .ME groups. Vnluewelshted
monthly percent returns on the portfolios are calculated from July of year r to June of
I – I.
R2 and the resdtual standard error. hIti), are adJusted for degrees of freedom.
Bond portfolio
E. F. Fmnu und R. R. French. Common rd fucrors in srock und bond renm~s 35
(1976) imply a simple test of whether the premiums associated with any set of
explanatory returns suffice to describe the cross-section of average returns: the
intercepts in the time-series regressions of excess returns on the mimicking
portfolio returns should be indistinguishable from 0.’
Since the stock portfolios produce a wide range of average returns, we
examine their intercepts first. We are especially interested in whether the
mimicking returns S,CIB and HML. absorb the size and book-to-market effects
in average returns, illustrated in table 2. We then examine the intercepts for
bonds. Here the issue is whether different factor models predict patterns in
average returns that are rejected by the flat average bond returns in table 2.
3.1. The cross-section oj’acerage stock returns
R&f-RF – When the excess market return is the only explanatory variable in
the time-series regressions, the intercepts for stocks (table 9a) show the size effect
of Banz (1981). Except in the lowest-BE/ME quintile. the intercepts for the
smallest-size portfolios exceed those for the biggest by 0.22% to 0.37% per
month. The intercepts are also related to book-to-market equity. In every size
quintile, the intercepts increase with BE/ME; the intercepts for the highestBE/ME
quintile exceed those for the lowest by 0.25% to 0.76% per month.
These results parallel the evidence in Fama and French (1992a) that, used alone,
market fls leave the cross-sectional variation in average stock returns that is
related to size and book-to-market equity.
In fact, as in Fama and French (1992a), the simple relation between average
return and /3 for the 25 stock portfolios used here is flat. A regression of average
return on /J yields a slo’pe of – 0.22 with a standard error of 0.31. The Sharpe
(1964)-Lintner (1965) model (/I suffices to describe the cross-section of average
returns and the simple relation between /? and average return is positive) fares no
better here than in our earlier paper.
SMB and HML – The two-factor time-series regressions of excess stock
returns on SMB and HML produce similar intercepts for the 25 stock portfolios
(table 9a). The two-factor regression intercepts are, however, large (around 0.5%
per month) and close to or more than two standard errors from 0. Intercepts
that are similar in size support the conclusion from the cross-section regressions
in Fama and French (1992a) that size and book-to-market factors explain the
strong differences in average returns across stocks. But the large intercepts also
say that S.LfB and HML. do not explain the average premium of stock returns
over one-month bill returns.
RM-RF, S.CfB, and H,LIL -Adding the excess market return to the timeseries
regressions pushes the strong positive intercepts for stocks observed in the
‘This implication is only an approximation in the Ross (19761 model. Ser. for example. Shanken
(1982).
Table Ya
lntrrceprs from excess stuck return regressions for 25 stuck purtfulios formed on size and book-to-market equity: July 1963 IO December IYYI,
342 months.”
Book-to-market equity (HE/ME) quintiles
Size
quinlile
Small
2
3
4
Uig
Low 2
0.31 0.62
0.35 0.63
0.34 0.58
0.4 1 0.27
0.34 0.30
t, I(4
3 4 High Low 2 3 4 High
~~
(i) K(r) – W’(r) = (I + OI~‘ERM(I) + dDEF(f) + t$r)
0.71 0.80 0.92 0.75 1.73 2.20 2.61 2.87
0.77 0.75 0.93 0.93 I.91 2.60 2.85 3.03
0.60 0.73 0.89 I.00 I .9Y 2.28 3.01 3.1 I
0.4’) 0.69 0.96 1.34 1.01 1.96 2.X8 3.35
0.25 0.50 0.53 1.35 1.27 1.17 2.36 2.14
(ii) R(r) – RF(r) = u + ~[RM(I) – RF(r)] + ($0
sm;III – 0.22 0.15 0.30 0.42 0.54 – O.YO 0.73 1.54 2.19 2.53
2 – 0. IX 0.17 0.36 0.3’) 0.53 – 1.00 I .05 2.35 2.7Y 3.01
3 – 0. I6 0.1s 0.23 0.39 0.50 – 1.12 1.25 1.n2 3.20 3.19
4 – 0.05 – 0.14 0.12 0.35 0.57 – 0.50 – 1.50 1.20 2.91 3.71
Big – 0.04 – 0.07 – 0.07 0.20 0.21 – 0.4Y – 0.95 – 0.70 1.89 1.41
SIll;Lll 0.24
2 0.52
3 0.52
4 0.69
Big 0.76
Smnll – 0.34
2 -0.11
3 – 0.1 I
4 0.09
Big 0.21
(iii) X(r) – RF(I) = u + sSMtqr) + hlfML&) + t(r)
0.46 0.49 0.53 0.55 0.97 I.92 2.24
0.58 0.64 0.58 0.64 2.00 2.40 2.76
0.61 0.52 0.60 0.66 2.00 2.58 2.25
0.39 0.50 0.62 0.79 2.78 1.55 2.07
0.52 0.43 0.51 0.44 3.41 2.23 1.84
(iv) K(r) – W(r) = 0 + h[RM(r) – RF(r)] + sSM&I) + h/IML(f) + V(I)
– 0.12 – 0.05 0.01 0.00 – 3.16 – 1.47 – 0.73
– 0.01 0.08 0.03 0.02 – 1.24 – 0.20 1.04
0.04 – 0.04 0.05 0.05 – 1.42 0.47 – 0.47
– 0.22 – 0.08 0.03 0.13 I .07 – 2.65 – 0.99
– 0.05 -0.13 – 0.05 – 0.16 3.27 – 0.67 – 1.46
Sm;ill – 0.35
2 – 0.1 I
3 – 0.12
4 0.0X
Big 0.21
~~_.___
‘See footnote under table 9c.
(v) H(l) – RF(f) = u + h[RM(r) – RF(r)] + sSMB(r) + hHML(r) + ntTERM(r) + dDEF(r) + &)
– 0. I3 – 0.05 0.0 I 0.00 – 3.24 – 1.58 – 0.79
– 0.02 0.0x 0.04 0.02 ~ 1.29 – 0.24 1.10
0.04 – 0.03 0.06 0.05 – 1.45 0.48 – 0.42
– 0.22 – 0.08 0.04 0. I 3 1.04 – 2.67 – 0.94
– 0.05 – 0.13 – 0.06 – 0.17 3.29 – 0.72 – 1.46
_ ~~~ ~~~~__.~~~.~.
2.52 2.49
2.61 2.56
2.66 2.61
2.51 2.85
2.20 1.70
0.22 0.14
0.51 0.34
0.7 I 0.56
0.33 I .24
0.69 – I.41
0.20 0.09
0.67 0.29
0.79 0.56
0.47 1.23
0.73 – I.51
Table Yb
portfolios. Jul> 1963 to December 1931. 343 months.’
Bond portfolIo
1-K 6-IOG ‘AU .A.l A B&I LG
II) R(r) – Rf(tl = tt 4 mTER.LIItI +

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