**Journal**of Financial Economics 33 (1993) 3-56. North-Holland Common stocks and bonds* risk factors in the returns on Eugene F. Fama and Kenneth Unirrrsit.v 01 Chicayo. Chiccup. I .L 60637, C;S;L R. French Received July 1992. final version received September 1992 This paper identities five common 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 shared variation due to the stock-market factors, shared variation in the bond-market factors. Except for low-grade factors capture the common variation in bond returns. explain average returns on stocks and bonds. risk factors in the returns on stocks and bonds. There are three and default risks. Stock returns have and they are linked to bond returns corporates. Most important. the five factors seem to through the bond-market 1. Introduction The cross-section relation to either the market pricing model or the consumption of Breeden (1979) and others. [See, for example, Reinganum Gibbons, and Litzenberger no special standing in asset-pricing the cross-section of average returns. The list of empirically return variables includes size (ME, stock leverage, earnings/price (E/P), and book-to-market book value of a firm’s common Banz (1981). Bhandari (1988). Basu (1983). and Rosenberg, (19853.1 of average returns /Is of the Sharpe ps of the intertemporal on U.S. common stocks shows little (1965) asset- asset-pricing (198 1) and Breeden, (1964tLintner model (1989).] On the other hand, variables theory show reliable that have power to explain determined number of shares), equity (the ratio of the value, ME). [See Reid, and Lanstein average- price times stock, BE, to its market Correspondence East 58th Street. Chicago. to: Eugene F. Fama. Graduate IL 60637, USA. School of Business. University of Chicago, 1101 *The comments Mark Mitchell, G. William Schwert. Jay Shanken. and Rex Sinquefield are gratefully This research is supported by the National in Securities Prices (French). of David Booth, John Cochrane. Sai-fu Chen, Wayne Ferson. Josef Lakonishok. acknowledged. Science Foundation (Fama) and the Center for Research 030%405X.93.S05.00 C 1993-Elsevier Science Publishers B.V. Ail rights reservedE.F. Fuma and K.R. French. Common risk f&run in r~ock and bond remrns 4 Fama and French (1992a) study the joint roles of market 8, size, E;P, leverage, and book-to-market equity in the cross-section find that used alone or in combination regression of a stock’s return on a market average returns. Used alone, size, E/P, leverage, have explanatory power. In combinations, (BE/ME) seem to absorb the apparent returns. The bottom-line result is that two empirically and book-to-market equity, do a good job explaining average returns on NYSE, Amex, and NASDAQ period. This paper extends the asset-pricing tests in Fama and French (1992a) in three ways. (a) We expand the set of asset returns sidered in Fama and French (1992a) are common integrated, a single model should also explain include U.S. government and corporate (b) We also expand the set of variables book-to-market variables in Fama stocks. We extend the list to term-structure a role in bond returns. The goal is to examine important in bond returns help to explain stock returns, and vice versa. The notion is that if markets are integrated, between the return processes for bonds and stocks. (c) Perhaps most important, the approach different. Fama and French (1992a) use the cross-section Fama and MacBeth (1973): the cross-section variables hypothesized to explain average returns. add bonds to the cross-section regressions size and book-to-market equity have no obvious and corporate bonds. This paper uses the time-series regression Scholes (1972). Monthly returns on stocks and bonds returns to a market portfolio of stocks book-to-market equity (BE/‘ME), and term-structure time-series regression slopes are factor loadings have a clear interpretation as risk-factor stocks. The time-series regressions are also convenient asset-pricing issues. (a) One of our central themes is that if assets are priced rationally, that are related to average returns, such as size and book-to-market proxy for sensitivity to common (shared and thus undiversiliable) of average stock returns. with other variables, return) has little information and book-to-market size (ME) and book-to-market roles of leverage determined They /I (the slope in the about equity equity and E;‘P in average variables, the cross-section stocks for the 1963-1990 size of to be explained. The only assets con- stocks. If markets bond returns. The tests here bonds as well as stocks. used to explain returns. and French (1992a) are directed variables that are likely to play whether are The size and at variables that are there is probably some overlap to testing asset-pricing models is of regressions of stock returns is regressed on It would be difficult to since explanatory meaning variables for government like of Black, Jensen, are regressed portfolios risk factors in returns. The that, unlike size or BE/ME, sensitivities for bonds as well as for approach and on the for size, and mimicking for studying two important variables equity, must risk factors in

E.F. Famu und K.R. French. Common risk factors 5 in stock and bond returns returns. The time-series lar, the slopes and R’ values show whether mimicking related to size and BE/lVCIE capture shared variation not explained by other factors. (b) The time-series regressions returns minus the one-month either excess returns or returns variables. In such regressions, intercepts that are indistinguishable intercepts provide a simple return metric and a formal test of how well different combinations of the common factors returns. Moreover, judging asset-pricing excess-return regressions imposes a stringent asked to explain the one-month bill rate as well as the returns bonds and stocks. Our main results are easy to summarize. mimic risk factors related to size and BE/ME capture strong common in returns, no matter what else is in the time-series that size and book-to-market equity indeed proxy for sensitivity factors in stock returns. Moreover, for the stock portfolios intercepts from three-factor regressions and the mimicking returns for size and BE/ME a market factor and our proxies for the risk factors related to size and book- to-market equity seem to do a good job explaining stock returns. The interpretation of the time-series regressions the cross-section regressions of Fama and French (1992a), the time-series sions say that the size and book-to-market average returns across stocks. But these factors alone cannot difference between the average returns on stocks and one-month left to the market factor. In regressions to-market factors, all our stock portfolios 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 premium and a default premium) capture most of the variation our government and corporate bond portfolios. ‘explain’ the average returns on bonds, term-structure factors, like the average excess bond returns, are close to 0. Thus, the hypothesis that all the corporate and government same long-term expected returns also cannot The common variation in stock returns portfolio returns, and the common variation regressions give direct evidence on this issue. In particu- portfolios in stock and bond returns for risk factors use excess returns Treasury bill rate) as dependent on zero-investment a well-specified from 0 [Merton (monthly stock or bond variables as explanatory model produces (1973)J The estimated and portfolios asset-pricing capture models on the basis of the intercepts standard. Competing the cross-section of average in models are on longer-term For stocks, portfolios constructed to variation regressions. This is evidence to common we examine, risk the that include the excess market factors are close to 0. Thus return the cross-section of average for stocks is interesting. Like regres- factors can explain the differences in explain the large bills. This job is the size and book- that also include produce slopes on the market factor factors (a term in the returns on factors also premiums The term-structure but the average for the bond portfolios have the be rejected. is largely captured in bond returns is largely explained by three stock-

by two bond-portfolio from stochastically term-structure slopes on the term-structure those for bonds. But interestingly. in the regressions, two term-structure a market portfolio associated with the market The stochastic to come largely from the term-structure return and the mimicking seem to capture common structure factors are included the stock-market factors disappears In a nutshell, our results suggest that there are at least three stock-market factors and two term-structure factors in returns. variation due to the three stock-market returns through shared variation in the two term-structure low-grade corporate bonds, only the two term-structure common variation in the returns on government The story proceeds as follows. We first introduce regressions: the explanatory variables and the returns to be explained Z and 3). We then use the regressions issues: how do different combinations variation through time in the returns on bonds and stocks (section 4) and (b) the cross-section of average returns (section 5). The stock and bond markets. Used alone in the time-series strong variation factors in the regressions when stock-market all of our stock portfolios factors and on the market of stocks captures the common factor and the two term-structure links between the bond and stock markets do. however. seem factors. Used alone. the excess market returns for the size and book-to-market variation in bond returns. in the bond regressions, for all but the low-grade returns. however, are far regressions. indeed, for stocks are much like factors are also included load in about the same way on the factor in returns. variation segmented. the the factors capture in stock returns; As a result, in stock returns factors. equity factors But when the two term- the explanatory corporate power of bonds. Stock returns have shared factors, and they are linked to bond factors. Except for factors seem to produce and corporate the inputs to the time-series bonds. (sections to attack our two central of variables asset-pricing (a) the common capture 2. The inputs to the time-series regressions The explanatory a market to-market, for government folios in five rating groups, and 25 stock portfolios and book-to-market equity. in the time-series and mimicking factors in returns. The returns to be explained bond portfolios in two maturity variables of stocks regressions portfolios include the returns on for the size. book- portfolio and term-structure are ranges, corporate formed on the basis of size bond port- The explanatory capturing Segmenting variables in bond returns and those likely to be important the explanatory variables fall into two sets, those likely to be important for variation for stocks. tests of in this way sets up interesting

whether factors important versa. in stock returns help to explain bond returns and vice 2.1 .I. Bond-mnrket factors One common rates. Our proxy for this factor, TERM, is the difference between the monthly long-term government bond return (from Ibbotson month 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 interest rates. For corporate bonds. shifts in economic of default give rise to another common this default factor, DEF, is the difference portfolio of long-term corporate bonds (the Composite rate bond module of Ibbotson Associates) and the long-term return. Chen. Roll, and Ross (1986) use TERM and a variable explain the cross-section of average returns on NYSE stocks. They use the Fama and MacBeth (1973) cross-section regression average stock returns is explained with the cross-section series 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 and TER.Cl sometimes has power. We confirm DEF show up clearly in the time-series that the two variables dominate the common corporate bond returns. In contrast to the cross-section Roll, and Ross, however, our time-series premiums for DEF and TERM risks are too small to explain much variation the cross-section of average stock returns. [Shanken a similar point.] risk in bond returns arises from unexpected changes in interest Associates) and the one- returns due to shifts in conditions factor between that change the likelihood in returns. Our the return portfolio government proxy for on a market on the corpo- bond like DEF to help approach: the cross-section of slopes from time- of returns. that the tracks of TER,LI and of stock returns. We also find variation in government regressions regressions say that the average variation and of Chen. in and Weinstein (1990) make 2.1.2. Stock-market fuctors Motiuztion variables they proxy for common document that size and book-to-market mentals. Not surprisingly, relative to book value) tend to have low earnings persist for at least five years before and five years after book-to-market - Although for explaining size and book-to-market average stock returns, risk factors in returns. In Fama and French (1992b) we equity are related to economic firms that have high BE/ME on assets, and the low earnings equity seem like ad hoc we have reason to expect that funda- (a low stock price equity is

measured. associated Size is also related 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 BE;.LfE, small firms are only slightly less profitable firms, the 198G1982 recession turns into a prolonged some reason, small firms do not participate and late 1980s. The fact that small firms can suffer a long earnings big firms suggests that size is associated explain the negative relation between relation between book-to-market equity profitability is the source of a common risk factor in returns that might explain the positive relation between BE:.CfE and average return. Measuring mon variation in returns associated with size and BE,hfE is a major task of this paper. The Buikfiny Blocks - To study economic (1992b) use six portfolios formed from sorts of stocks on .LfE and BE ‘IlIE. We use the same six portfolios here to form portfolios ing risk factors in returns related to size and book-to-market ensures a correspondence between the study of common carried out here and our complementary In June of each year t from 1963 to 1991, all NYSE stocks on CRSP are ranked on size (price times shares). The median split NYSE, Amex. and (after 1972) NASDAQ big (S and B). Most Amex and NASDAQ median, so the small group contains a disproportionate out of 4,797 in 1991). Despite its large number contains far less than half (about 8% in 1991) of the combined size groups. We also break NYSE, Amex, and NASDAQ market equity groups based on the breakpoints middle 40% (LCfediurn). and top 30% (High) of the ranked values of BE’.\ fE for NYSE stocks. We define book common value of stockholders’ equity, plus balance-sheet tax credit (if available), minus the book value of preferred stock. Depending availability, we use the redemption, liquidation, estimate the value of preferred stock. Book-to-market book common equity for the fiscal year ending in calendar market equity at the end of December oft - 1. We do not use negative-BE which are rare before 1980, when calculating or when forming the size-BE$.LfE portfolios. Conversely. with persistently low BE. .CfE (a high stock price relative to book value) is high earnings. to profitability. Controlling for book-to-market equity, 1981. controlling than big firms. But for small earnings depression. in the economic boom of the middle for For depression risk factor that might return. Similarly. and earnings suggests that bypasses with a common size and average the that relative the com- fundamentals, Fama and French meant to mimic the underly- equity. This risk factors in returns fundamentals. study of economic NYSE size is then used to stocks into two groups. small and stocks are smaller than the NYSE number of stocks 13,616 of stocks, the small group value of the two stocks into three book-to- for the bottom 30% (Lo\c), equity, BE. as the COMPUSTAT deferred taxes and investment book on or par value (in that order) to equity, BE,‘,CfE. is then year t - 1, divided by firms, for BE) .bfE the breakpoints Also. only firms with ordinary

9 E.F. Fama und K.R. Fwnch. Common rusk /Lcrorr in srock and bond renum 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 has a stronger role in average stock returns than size. The splits are arbitrary, however, and we have not searched over alternatives. 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 intersec- tions of the two ,bfE and the three BE!hfE portfolio contains the stocks in the small-%fE low-BE/ME group, and the BI’H portfolio contains have high BE,MEs. Monthly value-weighted calculated from July of year t to June oft + 1. and the portfolios are reformed in June of t + 1. We calculate returns beginning 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 t - 1. Moreover, to avoid the survival bias inherent adds firms to its tapes [Banz and Breen (1986)], we do not include firms until they have appeared on COMPUSTAT rarely includes more than two years of historical 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 simple average of the returns on the three big-stock B/H). Thus, ShfB is the difference between the returns on small- and big-stock portfolios with about the same weighted-average 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 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 on high- and low-BE,‘&fE portfolios with about the same weighted-average Thus the difference between the two returns factor in returns, focusing instead on the different return behaviors low-BEllME firms. As testimony to the success of this simple procedure, correlation between the 1963-1991 monthly book-to-market factors is only - 0.08. True mimicking portfolios for the common the variance of firm-specific factors. The six size-BEilV E portfolios equity The hope is that the tests the S/L groups. group the big-.CIE stocks that also returns on the six portfolios For example. that are also in the are in July of year t to be sure that book common in the way COMPUSTAT equity for year for two years. (COMPUSTAT data when it adds firms). says it (SjL, S/.Cf, and S,,H) and the portfolios (B; L. B/&f, and portfolios book-to-market equity. This HML is equity, is defined similarly. of H,tlL are returns size. should be largely free of the size of high- and the mimicking returns for the size and risk factors in returns minimize in S&fB and

H,CfL are value-weighted. minimizing (table 1. below). More important, mimicking portfolios stocks. or high- and low-BEl.VE 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 stocks in the six size-BE’ME portfolios, from the portfolios. RF is the one-month Using value-weighted since return using value-weighted that capture the different return behaviors stocks, in a way that corresponds components are negatively is in the spirit of related components of small and big variance, vririances to size results in to realistic portfolio of the stocks excluded plus the negative-BE bill rate. The returns 22. to he espkuiwd set of dependent the excess returns variables used in the time-series and five corporate (from CRSP) cover maturities bond portfolios, for Moody’s rating that is, below Baa) are from the Associates (provided to us by Dimensional regressions bond port- from 1 to Bonds -The includes folios. The government 5 years and 6 to 10 years. The five corporate groups Aaa, Aa. A, Baa. and LG (low-grade, corporate bond module of Ibbotson Fund Advisors). Stocks - For stocks. we use excess returns on 25 poitfolios, book-to-market equity. as dependent We use portfolios formed on size and BE/ME whether the mimicking portfolios stock returns related to size and book-to-market size and BE.‘AIE will also produce explained by competing asset-pricing Later, however, we use portfolios (dividend/price). variables that are also informative Keim (1988)], to check the robustness explanatory factors to capture The 25 size-BE,‘,LfE portfolios portfolios discussed earlier. In June of each year t we sort NYSE stocks by size and (independently) by book-to-market sured at the end of June. For the book-to-market the end of December of c - 1. and BE is book common ending in calendar year r - 1. We use NYSE breakpoints allocate NYSE. Amex. and (after 1972) NASDAQ and five book-to-market quintiles. We construct tions of the size and BE, ,LfE quintiles returns on the portfolios from July off to June of r + 1. The excess returns on these 25 portfolios for July 1963 to December for stocks in the time-series regressions. on two government bond portfolios formed on size and regressions. in the time-series because we seek to determine SMB and HAIL capture equity. Portfolios a wide range of average equations [Fama formed on E.P (earnings/price) about average returns of our results on the ability the cross-section of average returns. are formed much like the six size-BE;&LIE variables common factors in formed on returns (1992a)]. and DiP to be and French [e.g.. of our For the size sort. .LIE is mea- sort, ME is market equity at equity for the fiscal year for ME and BE;.tfE to stocks to five size quintiles 25 portfolios from the intersec- and calculate value-weighted equity. monthly 1991 are the dependent variables

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.’ equity (BE, ME) quintiies Book-to-market Size quintile 4 Low 1 3 4 High Low z 3 High Average of annual averages of firm size Average of annual 8. E ratios for portfolio Small 2 3 4 Big 20.6 89.7 209.3 535.1 3583.7 20.8 89.3 211.9 537.4 2885.8 _ ‘0 ._ ’ 89.3 210.8 545.4 2819.5 19.4 89.9 214.8 551.6 2700.5 15.1 88.5 210.7 538.7 1337.9 0.30 0.31 0.31 0.31 0.29 0.62 0.60 0.60 0.6 1 0.59 0.84 0.83 0.84 0.84 0.83 1.09 1.09 1.08 1.09 1.08 1.80 1.71 1.66 1.67 1.56 Average of annual percent of market Average of annual number of firms in value in portfolio portfolio Small 0.69 0.92 1.78 3.95 30.13 0.49 0.71 1.36 3.01 15.87 0.46 0.65 1.26 2.71 12.85 0.48 0.6 I 1.14 2.4 I 10.44 0.64 0.55 0.82 1.50 4.61 428.0 121.6 102.7 90. I 93.6 276.6 94.0 78.3 68.9 63.7 263.8 86.7 73.0 60.7 51.7 191.5 79.8 64.5 53.1 44.0 512.7 71.3 45.9 33.4 23.6 4 Big E’P ratios (in percent) for portfolio Average of annual Average of annual D’P ratios (in percent) for portfolio I .94 Small 1 2.42 7.24 8.26 9.06 2.66 1.00 2.60 3.13 2.82 ; 4 Big 5.20 5.91 5.85 6.00 8.61 8.73 8.94 9.07 10.16 10.43 10.45 10.90 10.95 Il.61 11.64 12.45 10.78 9.28 11.39 13.92 l.59 1.56 1.80 2.34 2.45 3.03 3.09 3.69 4.04 3.45 4.22 4.68 4.25 4.68 5.01 5.49 4.53 4.64 4.94 5.90 are formed as follows. Each year t from 1963 to 1991 NYSE quinttle measured stocks to five size quintiles. Similarly, stocks to five book-to-market of the five size and the five BE. ME groups. book value of stockholders’ equity, plus balance “The 25 size-BE. ME stock portfolios breakpoints for size (.UE. stock price times shares outstanding), allocate NYSE. Amex. and NASDAQ BE, ME are used to allocate NYSE. Amex. and NASDAQ 25 size-BE,‘.LIE portfolios are formed as the intersections equity. BE. is the COMPUSTAT investment tax credits lif available). minus the book value of preferred stock. Depending the redemption. liquidation. or par value (in that order) to estimate to-market equity. BE .ME. for a stock is BE for the fiscal year ending in calendar the end of December oft - 1. A portfolio’s book-to-market equity, BE,‘XfE. for the portfolio BE. for the firms in the portfolio for the fiscal year endmg in calendar market equity. ME, in December oft - I. A portfolio’s earnings/price income for the firms in the portfolio for the fiscal year ending in calendar market equity in December of r - 1. Equity statement deferred taxes. minus preferred dividends. (across firms in the portfolio) of the dividends equity in June oft - I. We use the procedure The descriptive statistics are computed when the portfolio then averaged across the 29 years. at the end of June, are used to NYSE quintile breakpoints equity quintiles. The for Book sheet deferred on avjailability. we use taxes and the book value of preferred stock. Book- year r - 1. divided by ME at formation year c is the sum of book equity. year t - I, divided by the sum of their ratio (E P) for year I is the sum ofequity year t - 1. divided by the sum of their before extraordinary dividend yield (D P) for year t is the sum income is income A portfolio’s items, plus income- paid from July oft - 1 to June of r. divided by the sum of market described in Fama and French (1988) to estimate dividends. is formed in June ofeach year. 1963-1991, and are

12 E.F. Fama und K. R. French. Common risk /&tom in stock md bond returns 1 shows Table size-BE, ,CIE portfolios, stocks (mostly small Amex and NASDAQ stocks, each of the five portfolios than 0.70% of the combined portfolios in the largest fractions of value. Together, average about 74% of total value. The portfolio and lowest BE/ME quintiles 30% of the combined value of the 25 portfolios. rather than just NYSE stocks, to define the size quintiles more skewed distribution of value toward the biggest size quintile. Table 1 also shows that in every size quintile number of stocks and the proportion decrease from lower- to higher-BE/ME First, using independent size and book-to-market portfolios means that the highest-BE/ME stocks. Second, Amex and NASDAQ 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 that, because the portfolios we use NYSE in the smallest size quintile stocks). Although in the smallest size quintile value of stocks in the 25 portfolios. size quintile have the fewest stocks but the largest the five portfolios in the largest of stocks in both the largest size (big successful firms) alone accounts And note that using all stocks, breakpoints to form the 25 have the most they contain is on average less In contrast, many the JIE quintile for more than would result in an even but the smallest, both the of total value accounted portfolios. for by a portfolio This pattern has two causes. sorts of NYSE stocks to form is tilted toward the smallest stocks, mostly small, tend to have lower quintile stocks. 3. The playing field Table 2 summarizes regressions. variables give perspective risk factors must explain. The average returns on the explanatory the average premiums per unit of risk (regression common risk factors in returns. the dependent and explanatory returns in the time-series that serve as dependent The average excess returns on the portfolios on the range of average returns that competing sets of are portfolios slope) for the candidate 3.1. The dependent retwxs Stocks - The 25 stock portfolios produce a wide range of average month. The portfolios there is a negative a stronger positive relation In all but the lowest-BE/ME small- to the big-size book-to-market equity is more consistent. tend to increase with BE/:bfE, formed on size and book-to-market excess returns, from 0.32% to 1.05% per also confirm the Fama-French relation between size and average between average return and book-to-market quintile, average returns tend to decrease from the portfolios. The relation between In every size quintile, average returns and the differences equity (1992a) evidence return, that and there is equity. average return and between the average returns

13 E.F. Fame und K. R. French. Common risk fk!ors m slack and bond rerurm for the highest- and lowest-BE;‘.CJE portfolios month. Our time-series regressions returns with the premiums range of average returns to-market effects in average returns, present interesting sets of risk factors. Most of the ten portfolios average excess returns that are less than two standard example of a well-known problem [Merton high standard deviations folios), large average returns volatility of stock returns will lack power. The common in stock returns, making the asset-pricing series regressions quite precise. Borrds - In contrast to the stock portfolios, government and corporate excess bond returns are less than 0.15% per month, more than 1.5 standard errors from 0. There is little evidence in table 2 that (a) average returns on government bonds corporate bonds have higher average returns average returns on corporate bonds are higher for lower-rating The flat cross-section of average bond returns does not mean that bonds are uninteresting dependent variables in the asset-pricing bonds are good candidates for rejecting patterns in the cross-section of average returns based on different slopes on the common risk factors in returns. range from 0.19% to 0.62% per attempt for the common on the 25 stock portfolios, to explain the cross-section risk factors in returns. and the size and book- challenges for competing of average The wide two BEI’ME quintiles errors from 0. This is an (1980)] : because stock returns have 6% per month for the size-BE ‘.CJE port- often are not reliably different from 0. The high does not mean, however, that our asset-pricing factors in returns will absorb most of the variation tests on the intercepts in the bottom produce (around tests in the time- the average excess returns on the in table 2 are puny. All the average and only one of seven is bond portfolios increase with maturity, than government (b) long-term bonds, or (c) groups. tests. On the contrary. equations asset-pricing that predict 3.2. The explanatory returns In the time-series premiums explanatory unit of market perspective The average S,VJB return returns) is only 0.27% the slopes on SAJB for the 25 stock portfolios the estimated spread in expected returns 0.46% per month. The book-to-market premium of 0.40% per month statistical terms. approach factors in returns are just the average values of the The average value of RXJ-RF p) is 0.43% per month. This is large from an investment (about 5% per year), but it is a marginal (the average premium per month (t = 1.73). We shall find, however, regression to asset-pricing tests, the average risk for the common variables. (the average premium per 1.76 standard for the size-related errors from 0. factor in that cover a range in excess of 1.7, so due to the size factor is large, about factor HAIL. produces (t = 2.91), that is large in both practical an average and

c. c- ? k 2 2 c t : i - 0.69 I.00 I .m 0.0x - 0.05 1.00 SMII 0. I7 - 0.07 - 0.08 bonds RMO - 0.00 - 0.00 - 0.00 I.00 0.w and corporvk RIII-RI.’ 0.78 0.32 - 0.38 - 0.07 0.34 on governwznl 0.02 0.04 0.03 0.08 returns 0.02 0.03 0.23 - 0.00 - 0.00 0.01 0.02 0.03 0.07 0.03 0.65 0.00 0.04 12 for lag Explanatory - 0.03 - 0.08 - 0.05 lixccss returns - 0.04 - 0.05 - 0.04 - 0.03 0.00 0.05 0.07 ~ 0.00 -- 0.04 - 0.04 0.06 - 0.05 0.90 - 0.w 2 Aulocorr. 0. I 5 0. I ? 0. I6 0.2 I 0.05 - 0.10 0.2 I 0. I9 0.23 O.II( ~ 0.20 0.19 0.05 0.05 0.20 0.05 0.94 ari;iblcs: \ I 1 (tw1) 1.71 1.24 2.9 I 0.2 I I .76 1.73 0.63 I.09 Ikpendzlll 2.61 0.44 0.58 3.66 5.10 3.97 0.38 45.97 1.25 2.35 2.52 2.34 2.23 2.25 3.02 2.03 3.03 3.55 2.89 2.54 2.24 4.54 I.60 4.52 0.22 S ld 0. I 3 0. I 2 O.O:! 0. I4 0.14 0.06 0.07 0.0x OS0 0.60 0.97 0.54 0.62 0.27 0.40 0.06 0.43 7’1:R hl I SC; 6 IOG RWRb RhlO BAA AAA l.‘i’(; 7‘11 Shf0 IlhfL DEb LG Kill AA (‘1) A

for the liscal year ending in calendar year I - I, and ML’ is li)r the SMB (small minus big) is the drlTerence between plus the negative-BE stocks excluded from the is the sum of the interccpl and stock portfoolios are formed as roollows. Each year r from bill rate, observed at the beginning of the month. LTG is the long-term government bond return. CB is the return 5.bY groups. Value-weighted 5.75 High 4.85 (high minus low) is the Merence 6.06 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) 6.27 for size (ME, stock price times shares outstanding), measured al Ihe end of J une, are used lo allocate NYSE. Amex, and NASDAO 5.99 5.33 4.Y2 4.95 4.27 4 ~_. _ Standard deviations are used to allocate NY!%, 5.27 5.03 the tivc size and the live HE/ME 4.38 5.85 6.29 3 ~____~~ with ahuut the same weighted average size. RMO formed on ME and BE/ME antI big-stock porttidios with ahout the s:mx weighted average hook-to-market equity. I/Ml. ~~~ 5.71 5.44 4.70 6.84 6.42 2 portfolios, DEF is CB~LTG. quintile breakpoints for BE/ME 7.28 5.97 6.71 4.95 Low 7.76 quintiles d I + I. ;IIIJ corlxnxte bonds rated AXI, A;I, A, Iklu. and below Hna (I.(;) by Moody’r. The 25 size-BE/ME of Dependent variables: Excess returns on 25 stock portfolios is the value-weighted monthly percent return on the stocks in the 25 size-BE/ME as the intersections equity (BE/ME) I .02 I .05 thrum J uly ul year I to J une O.Y7 equity 1.01 is LTG-RF. 3.15 3.36 2.97 0.59 High 3.1 I 2.2b and IIML. equity quintiles. In U/:‘/ME, WB is hook comm~m for the rmlrket portfolio of long-term corporate bonds. TERM between the returns on high and low hook-to-market equity portfolios 0.8 I Book-to-market 2.73 3.04 2.43 2.9 I 0.84 0.17 0.88 LXX 0.56 4 lhc regression (I) of KM -RF on ?‘ERM, L)EF, SMB, hrmd NYSE t-statistics for means are stocks to live size quintiles. Similarly, Means I .34 2.33 2.39 2.08 0.85 0.32 0.79 0.68 0.57 portfolios 2.6’) 3 are caldad &E/ME I .8X 1.19 I .42 2.05 2.12 0.71 0.35 0.70 0.36 0.66 quintile hre;tkpoLits 2 nlonthly prrcent returns on the pd’dios portfolios. KF is the one-month Treasury cd ol’ Decemhcr d I - I. The 25 six SIOC~S lo live hook-to-nl;lrkct I.1 1 I .49 I .50 LOW ‘l‘he scvc’n hod portlidius 0.44 0.48 0.40 0.93 0.39 0.43 I.IX the returns on sndl-stock An~x, and NASDAQ 1903 to 1991 NYSE residuals l&n ii proxy quintilc sm:111 ‘Uhf Size Snxlll I)& Kg on 3 2 3 4 4 2

The average risk premiums those of the stock-market default premium) 0.3 standard errors of 0. Note, though, volatile as the stock-market will prevent TERM and DEF from explaining average returns, but high volatility substantial common volatilities of TER.\ /I and DEF will be advantageous returns. But the task of explaining average stock returns falls on the stock-market HML. which produce higher average premiums. We turn now to the asset-pricing the tests have two parts. In section returns, TER.tI and DEF, and the three stock-market and H&IL, are risk factors in the sense that they capture common thus undiversifiable) variation in stock and bond returns. In section 5 we use the intercepts from the time-series regressions for the common risk factors in returns returns on bonds and stocks. for the term-structure factors. TER.Ll (the term premium) are on average 0.069/o and 0.029;b per month; both are within that TER,Zl and DEF are about returns S.LfB and H.CIL. Low average much cross-sectional implies that the two factors can capture variation in returns. In fact. the low means factors are trivial relative to and DEF (the as premiums variation in and high bond variation SMB, and for explaining the stron, 0 cross-sectional factors. in RN-RF. tests. In the time-series 4 vve establish regression approach. that the two bond-market returns, RXI-RF, SMB. (shared and to test whether the average premiums explain the cross-section of average 4. Common variation in returns the slopes and R’ values are direct evidence on common variation the explanatory The purpose is to test for overlap and bond returns. Do bond-market common the joint explanatory factors, to develop an overall story for the common In the time-series whether different returns. We first examine and stock-market stochastic processes that are important and vice versa? We then examine and stock-market in returns. regressions, risk factors capture separately factors. for stock in bond returns capture in bond and stock power of bond-market between the factors variation in stock returns power of the bond- variation 1. I. Bond-market fktors Table 3 shows that, used alone as the explanatory regressions, TERM and DEF capture returns. The 25 stock portfolios five standard errors above 0; the smallest portfolios is 18 standard 7.8 standard errors from 0 for bonds, and more than 3.5 standard for stocks. variables variation in the time-series in stock and bond common produce slopes on TERM that are all more than TER.Cf slope for the seven bond errors from 0. The slopes on DEF are all more than errors from 0

Table 3 TER.\/ Regressions of excess stock and bond returns (in percent) on the bond-market DEf: July 1963 to December returns. and 1991. 342 months.” R(t) - RF(t) = a + mTERJf(t) + dDEF(r) -t e(t) Dependent variable: Excess returns on 25 stock portfolios equity formed on size and book-to-market Book-to-market equity (BE, ME) qumtiles Size quintile 4 -I 3 High Low 2 3 Low 2 High m r(m) 6.0 1 6.92 7.60 7.83 6.84 0.93 0.99 0.99 0.92 0.82 0.90 0.96 0.94 0.95 0.82 0.89 0.99 0.94 0.97 0.80 0.86 1.01 0.95 1.05 0.80 0.89 0.98 0.99 1.03 0.77 5.02 5.71 6.25 6.58 7.14 5.50 6.32 7.10 7.57 7.60 5.95 7.29 7.80 a.53 8.09 6.08 8.3-t 8.50 9.64 8.26 Small 2 3 4 Big d rid) -_ 1.39 1.26 1.21 0.96 0.78 1.31 1.28 1.19 1.01 0.73 1.33 1.35 I.25 1.13 0.78 1.45 1.38 1.24 1.21 0.83 1.X 1.41 1.21 1.22 0.89 3.96 3.84 4.05 3.65 3.59 4.27 4.47 4.74 1.28 3.60 4.73 5.28 5-19 5.25 4.18 5.45 6.05 5.89 5.89 4.56 5.45 5.29 4.98 4.91 4.15 Small z 3 4 Big R’ SW 0.06 0.08 0.10 0.11 0.13 0.08 0.10 0.12 0.14 0.15 0.09 0.13 0.15 0.17 0.16 0.10 0.17 0.17 0.21 0.17 0.10 0.12 0.14 0.15 0.12 7.50 6.97 6.38 5.63 4.61 6.57 6.09 5.35 5.04 4.33 6.00 5.45 1.86 4.57 4.00 5.68 4.87 4.48 1.39 3.89 5.95 5.69 5.2 5.31 4.55 Small 2 3 4 Big Dependent variable: Excess returns on government and corporate bonds I-5G Aaa Aa A Baa 6-IOG LG I .02 99.94 0.72 3880 0.99 1.00 1.01 56.24 0.81 18.05 0. 45 31.73 m t(m) 130.44 139.80 I .02 75.74 d t(d) 0.25 9.51 0.27 7.85 0.94 48.95 0.96 67.54 1.10 32.33 1.01 11.95 R’ s(e) 0.79 0.57 0.87 0.75 0.97 0.41 0.98 0.30 0.98 0.29 0.90 0.72 0.19 1.80 “TERM RF is the one-month where CB is the return on a proxy for the market The seven bond portfolios S-year and 6- to lo-year governments and below Baa (LG) by Moody’s, The 25 size-BE;.CfE year t from 1963 to 1991 NYSE quintile outstanding). measured at the end of June. are used to allocate NYSE, Amex. and NASDAQ to five size quintiles. Similarly, NYSE quintile breakpoints Amex, and NASDAQ stocks to five book-to-market common equity for the fiscal year ending in calendar oft - 1. The 25 size-BE, .1fE portfolios are formed as the intersections BE/ME groups. Value-weighted monthly percent returns on the portfolios of year f to June oft + 1. R’ and the residual standard error. s(e), are adjusted is LTG-RF. where LX Treasury is the monthly bill rate. observed government of the month. DEF is C&LX, of corporate bonds. in the excess-return bonds rated A.aa. Aa, A. Baa, stock portfolios are formed as follows. Each breakpoints for size (.WE, stock percent long-term at the beginning portfolio variables bond return and used as dependent (I-5G and GlOG) and corporate regressions are I- to price times shares stocks NYSE, for BE’.CfE are used to allocate equity quintiles. year f - 1, and ME is for the end of December of the five size and the five are calculated In BE, ME, BE is book from July for degrees of freedom.

The slopes on TER.Cf and DEF allow direct comparisons variation in stock and bond returns Interestingly. the common variation thing, 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 TER,M and DEF are higher for bonds. In the bond regression, 0.49 for low-grade corporates to 0.97 and 0.98 for high-grade contrast, R’ ranges from 0.06 to 0.21 for stocks. Thus, TERM and DEF clearly identify shared variation in stock and bond returns, grade bonds. there is plenty of variation 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, at values near I for four of the five long-term low-grade portfolio LG. with a slope of 0.81. is the exception.) expect. long-term bonds are more sensitive than short-term 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 the risk captured by TER,Cf results from shocks to discount long-term securities. bonds and stocks, in about the same way. There are interesting parallels between the TER,Lf slopes observed our earlier evidence that yield spreads predict bond and stock returns. and French (1959), kve find that a spread of long-term yields (an ex ante version of TERXI) predicts captures about the same variation through long-term bonds and stocks. We conjectured variation in a term premium for discount-rate securities in about the same way. The similar slopes on TER,Lf for long-term bonds and stocks observed here seem consistent Our earlier work also finds that the return term minus short-term yield spread wanders values, and is on average close to 0. This parallels the evidence here (table 2) that the average premium for the common risk associated (the average value of TERM) is close to 0. The pattern in the DEF slopes in table 3 is also interesting. small stocks are more sensitike to the risk captured big stocks. The DEF slopes for stocks tend to be larger than those for corporate bonds, which are larger than those for governments. a common ‘default’ risk in returns that increases corporates, from bonds to stocks. and from big stocks to small stocks. Again, there is an interesting parallel between this pattern of the common variables. tracked captured by the term-structure by TER.Cf and DEF is. if any- of return variance explained R’ ranges from corporates. by In but for stocks and low- by stock-market left to be explained and then settle bond portfolios. As one would bonds to the shifts in corporate (The bonds. This suggests that rates that affect here and In Fama bond returns, returns captures minus short-term stock and bond time in the expected that the yield spread changes that affect all long-term and on with that conjecture. premium predicted between positive by the long- and negative with shifts in interest rates The returns on by DEF than the returns on DEF thus seems to capture from government bonds to in the DEF slopes and the

19 E.F. Fumu und K.R. Frewh. Common risk fucrors in srocb und bond reiurns similar pattern observed in Fama and French (1989) in time-series 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 portfolios formed on ranked values of size, Chan, Chen. and Hsieh (1985) and Chen, Roll, and Ross (1986) find that the cross-section like DEF goes a long way toward explaining and average stock returns. Given the negative 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 size effect in average stock returns. In the time-series 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. shall see that the intercepts in the regressions DEF leave strong size and book-to-market also find that when the stock-market factors are added to the regressions, negative relation between size and the DEF slopes in table 3 disappears. regressions of regression approach and stock of slopes on a variable the negative relation relation between between size size and the explain the average the regressions, As a result, we on TERM and We shall of stock returns effects in average returns. the 42. Stock-market f&ton The role of stock-market examine (a) regressions excess bond and stock returns, mimicking returns variables. and (c) regressions factor regressions help explain why. The Murket - Table 4 shows, not surprisingly, market portfolio of stocks, RM-RF, returns than the term-structure the important fact is that the market leaves much variation might be explained by other factors. The only RZ values near 0.9 are for the big-stock low-book-to-market portfolios. 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 have their best shot at showing marginal The market portfolio of stocks also captures returns. Although the market fls are much smaller for bonds than for stocks. they are 5 to 12 standard errors from 0. Consistent corporate bonds than for governments high-grade bonds. The /I for low-grade bonds (LG) is 0.30, and R.V-RF a tidy 19% of the variance of the LG return. factors in returns that use the excess market return, RAGRF, (b) regressions for the size and book-to-market that use RM-RF, work well for stocks, but the one- and two-factor is developed in three steps. u’e to explain that use SMB and NML, the factors, as explanatory S‘SJB. and H,VfL. The three- regressions that the excess return on the more common variation factors in table 3. For later purposes. captures in stock however. in stock returns that For small-stock and high-BE/ME factors, SMB and H.LIL, will explanatory power. common variation in bond with intuition, for low-grade /? is higher for than explains and higher for

Table 1 Regressions of excess stock and bond July 1963 to December returns (in percent) on the excess stock-market 1991. 342 months.” return, R.WRF: 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 equity (BE .CIE quintiles Book-to-market Size quintile Lou 2 1 3 1 High Low 2 3 Htgh h r(h) Small 2 3 1 Btg 1.40 I .‘I? 1.36 I.24 I .03 25.03 33.1-I 35.81 37.00 35.96 23.01 29.04 31.16 32.76 27.75 1.26 I.15 I.15 I.14 0.99 I.11 I.12 I.04 I .03 0.89 I .08 1.13 I .oa I.10 0.89 16.33 35.76 12.98 51.67 5 I .92 27.01 33.12 37.50 46.96 13.03 I .06 I .02 0.96 0.95 0.84 28.12 35.56 42.52 55.IZ 61.51 R’ s(e) 0.6 I 0.71 0.74 0.76 0.69 0.67 0.79 0.8-l 0.89 0.89 0.70 0.79 0.81 0.90 0.91 Small 2 ? 0.68 0.76 0.80 0.87 0.54 -1.46 3.34 2.65 2.0 I 1.66 3.56 2.59 3.26 2.2 1 1.95 3.92 3.25 2.90 2.83 2.69 0.65 0.76 0.79 0.80 0.79 3.76 2.96 2.28 1.73 1.35 3.55 2.85 2.33 I.% 1.73 ; Big Dependent variable: Excess returns on government and corporate bonds Baa I -5G 6-IOG Aaa -\a A LG h r(h) 0.08 5.24 0.21 8.73 0.30 II.90 0.13 5.57 0.19 7.53 0.20 8.14 0.21 8.42 RJ s(e) 0.19 2.12 0.07 I.21 0.0s 1.95 0.16 2.05 0.17 2.05 0.29 2.12 0.14 2.17 ‘R.W is the value-heighted portfolios, plus the negative-BE Treasury bill rate. observed The seven bond portfolios 5-vear and 6- to IO-vear eovernments and below Baa (LG) by Moody’s year r from 1963 to 1991 NYSE outstanding). measured at the end of June, are used to allocate to five size quintiles. Similarly. Amex. and NASDr\Q stocks common equity for the fiscal year ending in calendar of r - I. The 25 size-BE .UE portfolios BE .LfE groups. Value-weighted ofyearrtoJuneofr+l. R’ and the residual standard monthly stocks at the beginning used as dependent (I-SC The 25 size-BE, .LfE stock portfolios quintile breakpoints percent excluded return from on all the stocks the 25 portfolios. in the 25 size-BE.‘IW& RF is the one-month of the month. variables and 6-IOG) in the excess-return and corporate regressions are I- to bonds rated Aaa. Aa. .A. Baa. are formed as follous. for size (,LfE. stock NYSE. Amex. and NASDAQ for B&ME are used to allocate equity quintiles. In BE ME. BE is book year r - I, and ME is for the end of December as the intersections of the five size and the five percent returns on the portfolios Each price times shares stocks NYSE, NYSE quintile to five book-to breakpoints market are formed monthly are calculated from July error. s(e), are adjusted for degrees of freedom.

21 E.F. Fuma and K.R. French. Common risk factors VI s[ock and bond rerurns S.LfB and H.CIL -Table market portfolio. variation in stock returns; above 0.5. Especially for the portfolios and H~ML leave common 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, captures variation in bond returns. We shall find, however, term-structure factors to the bond regressions power of the stock-market factors. Thus the apparent factors in bond returns in table 6 probably term-structure and stock-market factors. The interesting regressions in table 6 are for stocks. Not surprisingly. stock-market factors capture strong common market ps for stocks are all more than 38 standard exception, the t-statistics on the SMB slopes for stocks are greater than 4; most are greater than 10. SMB, the mimicking 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 smaller- to bigger-size quintiles. Similarly, the slopes on HML, the mimicking factor, are systematically related to BEI.Lf E. In every size quintile H&IL slopes increase monotonically from strong negative values for the lowest- BE,!.LIE quintile to strong positive values Except for the second BE/ME quintile, positive, the HML slopes are more than five standard clearly captures shared variation in stock returns, 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 adding the two returns to the regressions stocks, the market alone produces only two (of 25) R’ values greater than 0.9 (table 4); in the three-factor regressions routine (21 of 25). For the five portfolios creases 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 in the largest-size and highest-BE!rLIE generated by the market alone. Adding SMB and HML to the regressions market ps for stocks. In the one-factor portfolio of stocks in the smallest-size and lowest-BE/ME 5 shows that in the absence of competition SMB and H&IL. typically 20 of the 25 R’ values are above 0.2 and eight are in the larger-size variation in stock returns from the time-series capture substantial quintile, however, SXfB that is picked up by the each of the three stock-market factors that adding kills the explanatory role of the stock-market the largely results from covariation between the the three variation in stock returns. errors from 0. With one The return for the size factor, clearly monotonically from return for the book-to-market of stocks, the for the highest-BE/.LIE where the slopes pass from negative quintile. to errors from 0. HML related to book-to-market that in R2. For results in large increases (table 6) RZ values greater than 0.9 are in the smallest-size R2 in- quintile, R’ for stocks, 0.83 for the portfolio quintiles, is much larger than the 0.69 has an interesting regressions effect on the of table 4, the p for the quintiles is 1.40. At

IO.02 factors: Julv High 4.6Y O.XI 5.06 ~ 0.0’) 4.60 - 0.12 4.01 4.53 0.05 5.02 0.16 13.45 16.42 22.16 euuilv (IIML) 3.7Y - 1.82 - 1.23 - 1.12 4.10 - 0.66 4.06 4.48 4.19 3.69 _ 2.20 12.13 x.57 15.47 22.30 4 .~. equily returns for the size (SMB) and book-lo-market - 3.43 3.9x 4.19 4.40 4.20 2.7’) - 3.80 - 4.64 4.20 - 4.20 17.08 13.42 9.29 - 4.10 21.8X s(e) IN f(N 3 formed on size and book-to-market 4.3 I 4.4 I 4.3 I - 6.6’) - 7.07 - 6.1’) ~ 7.02 - 7.07 4.55 4.27 4.3’) 17.68 9.64 21.38 I3.W 2 quindes = 0 + .sSMB(r) + IrIfM L(r) + r(r) 4.7 I 4.6X - Y.72 ~ 12.25 - IO.84 - 12.46 1991, 342 months.” 4.57 4.53 4.02 IO.16 14.43 3.70 17.23 22.52 - II.43 Low equity (BE/ME) Excess rcIurns on 25 stock podolios 0. SY O.lK) O.Y5 Table 5 ~ 0.0 I 0.0 I I .67 1.40 0.44 0.35 0.23 0.06 0.08 - 0.01 0.44 I.16 High Regressions of wcess dock and bond returns (in percent) on the mimicking Book-lo-market I963 IO December 0.60 0.42 1.5’) ~~ 0.1 I - 0.1 I 1.18 o.Y3 0.31 0.18 0.04 0.2Y ~ 0.06 - 0.16 ~ 0.18 0.12 H(r) - W(r) 4 0.4’) I .63 1.35 1.05 0.24 0.08 - 0.42 0.60 0.37 - 0.35 - 0.3x - 0.31 - 0.36 0.77 0.22 K2 /I s 3 variable: 1.46 1.73 1.12 - 0.65 0.00 0.53 0.43 0.30 0.18 ~ 0.66 - 0.65 ~ 0.65 _ 0.57 0.35 0.x2 2 Depcndrn~ - O.Y5 -- 1.0’) ~ 1.07 -~ 1.22 I .Y3 0.5’) 1.28 1.52 0.65 0.43 0.34 0.X6 0.51 - I.11 0.28 LWV quinlile Sm;III Smdl Size Big Big Big 2 3 2 3 4 ‘I 3 2 4

23 E.F. Fumu and K. R. French. Common risk f&tors in SIL)L% und bond rerurns

17.OY 46. IO - 1.18 57.x’) 50.Y7 36.7X 19.39 returns for the size (SMU) .md book- 3X.61 t 6.24 24.X0 52.52 21.91 22.24 65.52 High II.01 7.38 - 6.27 16.88 14.84 15.53 17.24 1x.34 54.38 52.85 31.66 IX.62 59.73 53.87 61.54 47.0 4 equity 9.X I X.X3 X.56 Y.75 - 7.58 5.80 9.66 50.78 51.21 52.03 34.03 55.88 21.23 60.44 46.57 3 I(S) __~~~ f(M variable: Excess returns on 25 stock portfolios formed on size and book-lo-markel 1991, 342 months.’ and the mimicking 0.4 I 1.04 - 4.51 2.35 - 0.05 11.11 0.09 51.80 53.17 53.51 56.76 38.79 44.11 23.39 61.18 + r(/) 2 quintiles + M/hlL(r) _ 14.57 - 12.51 56.X8 37.Y2 - 7.18 - 6.47 - 17.03 12.73 53.94 32.73 39.37 52.49 60.93 26.40 - II.26 Low factors: July 1963 I O December equity (BE/ME) Regressions of excess stock and bond returns (in percent) on the excess market return (M-RF) -t ssnlryr) 1.23 0.4 I 1.18 I .06 1.09 0.6X - 0.05 0.74 0.76 0.89 0.66 0.62 0.70 0.96 High I.09 Table 6 -- KI$)j Book-lo-market H(l) - RF(l) = ‘l + /,LKM(f) 0.5 I O.YY 1.17 I .05 0.56 0.57 0.73 0.48 - 0.17 0.40 0.46 0.91 0.97 0.97 0.24 4 equity (Ill) 1.00 0.98 1.04 1.19 0.9X 0.2Y - 0.23 0.26 0.26 0.32 0.30 0.21 0.95 0.88 0.60 3 h II --_~_s to-market I .02 I .06 I .0x 1 a2 0.0 I 1.26 1.02 - 0.12 - 0.00 0.04 0.00 0.98 0.65 0.33 0.0x 2 Dependent I .46 O.Y6 1.04 1.12 I .07 - 0.31 _ 0.42 - 0.46 - 0.17 - 0.20 _ 0.52 I.00 0.76 0.37 I.11 Low quinlile Sm;rll Small Sire Big Big Big 2 3 3 2 3 2 4 4 4

1.23 portfolios are the plus the negative-lit‘ stocks excluded from the 25 tlE is groups. Value-weighted monthly percent returns on the 25 portfolios are calculated from J uly elf to J une used as dependent variables are I- lo S-year and 6- IO IO-year governments (I-SC and 6-IOG) and corporate bonds rated 1.22 1.52 1.88 2.02 12.22 2.06 Ott9 4.75 0.04 0.23 0.33 0.34 minus big) is the return on the mimicking portfolio stocks lo live size quintiles. Similarly, are formed as lollows. Each year I from 1963 to 1991 NYSE LG factor. (See table 5.) stocks to live book-to-market equity quintiles. In SE/ME, 1.12 1.16 1.32 1.63 I .36 2.08 4.08 9.58 - 0.04 - 0.91 0.20 0.22 0.27 Baa olt - 1. The 25 size-HE/ME 1.31 1.43 1.55 1.16 I .49 (high minus low) is the return on the mimicking portfolio for the book-to-market s(4 - 0.09 3.51 - 2.18 2.01 0.26 9.46 0.16 0.20 A Dependent variable: Excess returns on government and corporate bonds Amex, and NASDAQ I.411 1.27 1.41 1.32 I .44 ~~~~_ book common equity for the liscal year ending in calendar year I - I, and ME is for the end ol December portfolios, bill rate, observed at the beginning of the month. SMB(small 1.55 1.45 1.16 I .94 I .46 - 0.1 I 3.26 - 2.72 2.00 0.25 9.30 0.15 0.20 Aa quintile hrcakpoints Car size, AI K, measured al the end of J une, are used to allocate NYU!, ‘RM is the value-weighted percent monthly return on all the stocks in the 25 size-BE/ME stock portfolios ~____._ Amex, and NASDAQ 0.96 0.96 0.93 0.89 0.x3 Kz and the residual standard error, S(P), are adjusted for degrees ol freedom. ~~ 2.77 8.60 - 0.12 - 2.89 2.13 0.14 0.17 0.25 Aaa The 25 size-BE/ME 0.93 0.97 0.95 0.89 0.90 are used to allocate NYSE, 66IOG 1.91 I .83 0.8X - 3.65 6.75 0.95 0.93 - 0.14 0.08 0.12 0.97 0.91 0.18 Aaa, An, A, Baa, and below Baa (LG) by Moody’s R’ intersections of the live size and the tive SE/ME for the size factor in stock returns. f/ML portfolios. RF is the one-month Treasury 0.94 0.96 0.93 0.92 0.96 quintile breakpoints for LIE/ME 0.10 - 2.70 2.66 6.45 - 0.06 0.07 0.10 I-SC I.19 The seven bond portfolios 0.95 0.94 0.95 0.94 0.94 I. Small NYSIl orI+ l(h) l(S) 44 w Big R2 S h /I 2 3 4

26 E.F. Fuma und K. R. Frmch. Common ruk jtictorr in srock and bond rrrurns the other extreme, the univariate and highest-BE;.CfE the fls for these two portfolios H&IL to the regressions toward 1.0 and correlation between the market and SMB or H&IL. Although are almost uncorrelated ( - O.OS), the correlations SMB and HML returns are 0.32 and /I for the portfolio is 0.89. In the three-factor are 1.04 and 1.06. In general. adding .S,LfB and collapses the ps for stocks toward high ps move down. This behavior of stocks in the biggest-size regressions quintiles of table 6. 1.0: low gs move up is due. of course, S.1fB and HML between R.lf-RF to and the - 0.38. 4.3. Stock-mnrkrt and bond-market factors Used alone, bond-market as well as bond shared variation demonstrate stock returns. We emphasize bond-market First Pass - Table bond-market stock-market TER,Zf and DEF to the regressions 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, 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. evidence in tables 3 and 6 that there is strong processes for bonds and stocks. Adding the stock-market sions for stocks kills the strong slopes on TERM and DEF observed two-factor regressions of table 3. The evidence respond to stock-market factors also largely five-factor regressions, only the low-grade produce nontrivial slopes on the stock-market Table 7 seems to say that the only shared variation 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 used alone to explain returns. Can we reconcile these results? We argue next that the two term-structure factors are indeed common the five-factor regressions for stocks, however, the tracks of TER.Cf and DEF are buried in the excess market return. R.bf-RF. structure factors. the three stock-market returns; except for low-grade bonds, these factors do not spill over into bond returns. In short. we argue that stock returns factors capture (table 3). Used alone, stock-market in bond returns as well as stock returns (table 6). These results that there is overlap between the stochastic this point because the joint tests on the stock- and factors that follow muddy the issue a bit. 7 shows that, used together factors continue to have a strong factors have a strong role in stock returns. has little effect on the slopes on the common variation in stock returns factors capture returns processes for bond and to explain returns, the role in bond returns and the For stocks. adding R.Cf-RF, SMB, adding seem to contradict overlap between factors to the regres- the the return in the in table 6 that bond returns disappears in table 7b. In the bond portfolio, LG. continues factors. in bond and stock returns to factors are to bond and stock returns. In In contrast to the two term- confined factors are generally to stock share three stock-market factors,

17 E.F. Fama und K.R. French. Common ruk facrors in stuck and bond r~furns and the links between term-structure Second Pass: in Orthoyonali:ed mon factors in stock returns, they are all in the market return, RM, which is just a value-weighted average of the returns STAT sample. The regression of RM-RF monthly returns of July 1963 to December stock and bond returns come largely from two shared factors. itfarket Fuctor - If there are multiple com- on the stocks in the CRSP-COMPU- on SIVB. HAIL, TERM, and DEF for 1991 illustrates the point: RIM-RF = 0.50 + 0.44SMB - 0.63 HML + 0.81 TERM (9.09) ( - 8.23) (2.55) (6.48) + 0.79 DEF + e. (1) (4.62) below the slopes; the R’ is 0.38. This The r-statistics regression demonstrates factors in returns. The strong slopes on TER,LI and DEF produced (the excess return on a proxy for the portfolio evidence that the two term-structure returns. The sum of the intercept and the residuals investment portfolio return that is uncorrelated variables in (I). We can use RR/IO as an orthogonalized captures common variation in returns Since the stock-market returns, with the bond-market returns, TERM and DEF (table 2). five-factor sions that use R,ClO, SMB, HML, stock returns will provide a clean picture stock-market factors in bond and stock table 8. The story for the common variation 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 are more than two standard errors from 0. But this is mostly because TERM and DEF produce high R’ values in the bond regressions, reliably different from 0. As in table 7b. only the low-grade produces nontrivial slopes on the stock-market market factors don’t add much to the shared variation by TERM and DEF. For the stock portfolios, the slopes on R,CIO in the five-factor table 8a are identical (by construction) 7a. The slopes on the size and book-to-market (up for S;LIB, down for HXIL) relative to the slopes in table 7a. But the spreads are in parentheses that the market return is a hodgepodge of the common by RM-RF wealth) are clear variation of stock-market factors capture common in stock in (l), call it RMO, is a zero- with the four explanatory market left by SR;IB, HML, TERM, and DEF. S&fB and HML, are largely factor that uncorrelated regres- TERM. and DEF to explain of the separate returns. The regressions bond and roles of bond- and are in in bond returns in table 8b is like that in factors that so trivial slopes can be bond portfolio (LG) factors. Otherwise, in bond returns captured the stock- regressions of to the large slopes on RM-RF returns in table 8a shift somewhat in table

I I .07 - 1 .44 Rhf -RF, 46.92 35.7’) 35.96 49.55 22.32 1 9.34 5Y.W 24.61 22.27 1 6.88 1 6.59 53.1 5 41 .02 H igh retu rn s, 4X.2Y x.05 - 6.07 IX.54 47.59 54.51 1 5.47 1 4.20 54.1 9 1 6.50 41 .02 32.1 2 51 .36 1 6.1 8 1 8.1 6 4 o n th e sto ck-m arket 342 m o n th s.” 49.0 I - 7.3 I + e(r) 8.1 I Y.YY 5.70 54.46 9.82 9.28 8.37 44.57 44.83 33.6X 50.89 20.x3 41 .35 3 r(s) r(h ) l(h ) 1 991 , + dDEF(t) (in p ercen t) - 0.1 I 1 963 to D ecem b er 3x. IO I I .36 - 4.07 2.90 0.69 0.38 54.95 43.42 47.65 46.95 47.55 22.Y7 I.1 0 51 .01 2 + mTERM(r) q u in tiles eq u ity - 5.Y5 - 7.03 25.X2 35.97 50.93 53.x7 32.06 37.02 - 1 2.09 - 1 6.85 - 1 4.01 47.1 ’) 48.1 8 1 2.71 - IO .81 Lo w Ju ly (B E /M E ) + hIfML(r) o n size an d b o o k-to -m arket D E F: eq u ity 7a an d I .0x 0.6X 0.42 0.98 - 0.06 0.x’) 0.63 0.69 0.75 0.78 I.21 0.71 I.1 0 I.1 7 I.1 0 H igh + sSMB(r) Tab le 7’E H M B o o k-to -m arket retu rn s, - RF(rj] - 0. I7 0.4Y 0.44 0.94 0.74 0.40 1 .00 0.92 0.95 I so 0.26 0.50 0.53 0.58 I.1 5 4 fo rm ed R(r) - RF(r) = LI + b[Rhl(r) and th e b o n d -m arket o n 25 sto ck p o rtfo lio s O .YX O .YX 0.Y7 1 .20 0.96 0.x’) 0.00 0.30 - 0.23 0.27 0.25 0.29 I.01 0.31 0.21 3 /I s h I .06 I .07 I .02 I .26 Shff3, and IIML. O .YX 1 .04 0.34 0.04 ~ 0.00 0.66 0.02 0.01 I.01 0.1 0 2 -0.1 1 o f L’XC C SS sto ck retu rn s I .06 I .07 1 .45 O .Y6 - 0.42 - 0.46 I .O l 0.76 0.3x - 0.27 - 0.37 1 .1 2 1 .1 3 - 0.51 - 0.1 7 Lo w Kcgressio n s q u in tile Sm all Sm all Sm d l Size B ig Iiig B ig 4 2 3 3 4 2 2 3 4

29 E.F. Famo and K.R. French, Common risk factors in stock and bond relurns zswc,* -_- ddddd I I I I I I I I J.F E.-B

30 Table 7b Regresstons stock-market of cwess returns. stock R.Lf-RF. returns S.\ fB. and H.UL. July 1963 to December on gobernment and corporate and the bond-market 1991. 342 months.” bonds returns. tin percent) T&R.\ ! and DEf: on the 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 - 0.02 - 2.8-I - 0.04 - 3 I4 - 0.01 _ 2.96 h C(h) 0.02 I .99 0. 00 0.06 0. 00 I .05 0. 18 7.39 0.00 0.30 - 0.02 - I.12 - 0.01 _ 2.42 0.05 3.20 0. 00 0.40 - 0.0’ _ 2.28 0.08 2.34 - 0.02 - 1.29 0.04 2.39 - 0.02 - 2.46 - 0.00 - 0.40 0.12 3.13 0.00 0. 44 0.00 0.90 1.00 0.47 30.0 I 0.75 36 8-t 1.03 93.30 0.99 50.50 0.99 0.64 14.25 117.30 124.19 0.27 9.57 0.97 49.X 1.05 30.33 0.32 x.77 0.97 65.04 I .02 71.51 0.80 9.92 0.97 0.40 0.80 0.56 0.87 0.73 0.98 0.30 0.98 0.29 0.9 I 0.70 0.58 1.63 “R.Lf IS the value-weighted plus the negative-BE observed at the beginning the simple average of the returns sample average of the returns minus loul is the ditference high-BE .\fE portfohos portfolios (5 L and B I!.). TER,Lf is LTG-RF, DEF is CB-LTG. uhere The seven bond portfolios j-year and 6- to IO-qear governments and below Baa (LGJ by Moody’s. year r from 1963 to 1991 NYSE outstanding). measured at the end of June, are used to allocate to tice size quintiles. Simtlarly. Amex. and NASDAQ stocks to five book-to-market equity for the fiscal year ending in calendar The 25 size-BE .\fE portfolios Value-weighted monthly percent returns on the portfolios ofr+ I. R’ and the residual standard, percent return on all stocks in the 25 size-BE from the portfolios. RF is the one-month S.\fB (small minus big) is the difference on the three small-stock on the three big-stock portfolios each month between the simple average of the returns (S H and 6 H) and the average of the returns where LTG is the long-term CB is the return on a pro.xy for the market used as dependent variables ( I-5G and 6IOGJ The 25 size-BE:.\fE stock portfolios quintile breakpoints monthly ME portfolios. Treasury each month between (S L. S .Lf. and SH) and the B .\I. and B H). H.rfL (high stocks excluded “fthe bill rate. month. portfolios (FL. on the two on the two lo&-BE government portfolio of corporate in the excess-return and corporate bonds rated Aaa, Aa. A, Baa. are formed as follows. for size (.\fE, stock NYSE. Amex. and NASDAQ breakpoints for BE ‘.LfE are used to allocate quintiles. In BE .\/E. year t - 1. and ,LfE is for the end of December are the intersections of the five size and the fire BE .\ fE groups. are calculated ME bond return. bonds. are I- to regessions Each price times shares stocks NYSE. NYSE quintile BE is book common of r - I. from July of year f to June error, s(r). are adjusted for degrees of freedom. in the .S.LfB and H,LfL in table 7a, and .S,LfB and H,LfL returns. What changes dramatically table 7. are the slopes on the term-structure slopes across the stock portfolios again capture strong shared variation in table 8a are like those in stock in the five-factor regressions factors for stocks. The slopes on of table 8, relative to

TERM are more than 14 standard 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 that the term-structure factors capture bond returns. How do the tracks of the term-structure regressions for stocks in table 7a? Table 8a says that stocks load strongly RMO, TER.CI. and DEF, but there is little cross-sectional on these factors. All the stock portfolios close to 0.81 and 0.79, the slopes produced 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, excess market return in table 7a absorbs the common associated with R.LJO. TER.Vf, and DEF. In short, the common stock returns related to the term-structure return in table 7a. Is there any reason to prefer the five-factor regressions table 7? Only to show that, in addition are two bond-market factors in stock returns. regressions produce the same R’ values and thus the same estimates of the total common variation in returns. And the two sets of regressions intercepts for testing the implications of five-factor of average stock returns. errors from 0; the DEf slopes are more than regressions variation in table 8 say in stock and strong common variables get buried in the five-factor on variation in the slopes slopes on TER,V and DEF produce by the excess market return in (I). RJlO, TER.U, and DEF, the variation in stock returns variation in factors is buried in the excess market in table 8 over those in factors. there Otherwise, the two sets of to the three stock-market produce the same models for the cross-section 5. The cross-section of average returns slopes and R’ values in tables returns. SMB, HhJL, TERM and DEF. proxy for risk factors. They capture common in bond and stock returns. Stock returns have shared variation factors, and they are linked in two term-structure factors. for the five proxy risk factors explain The regression stock-market market returns, variation to three stock-market shared variation average premiums average returns on bonds and stocks. The average-return The dependent variables variables are excess returns (R.WRF portfolios (R,bf 0, S.CJB, HALI L. and DEF). Suppose the explanatory minimal variance due to firm-specific returns for the underlying state variables investors. Then the multifactor asset-pricing 3 to 8 establish (or R.MO), and the bond- that the and R.&RF’ related through to bond returns We next test how well the the cross-section of tests center on the intercepts in the regressions in the time-series are excess returns. The explanatory and TERXJ) or returns on zero-investment regressions. returns have factors, or common models of Merton (1973) and Ross so they are good mimicking risk factors of concern to

SMB, 2.06 53.15 59.00 35.96 58.79 0.57 0.51 0.16 0.10 71.32 26.22 46.92 41.02 40.24 High II.17 ~___________ returns, KMo, - 1.75 - 7.49 10.62 - 6.22 - 3.61 - 2.83 53.20 36.47 54.51 54.19 73.72 23.02 47.59 41.02 48.29 4 equity (in percent) on the stock-market - 13.67 - 13.09 6.85 - 9.x2 - 10.93 - 12.30 54.48 37.83 73.21 53.02 26.42 49.01 44.57 44.83 41.35 t(h) 3 t(b) r(s) + dDEF(~) + r(f) - 18.19 13.69 54.95 51.01 59.80 - 23.89 - 21.18 - 20.12 - 22.46 47.65 62.88 28.84 46.95 47.55 40.94 2 1991, 342 months.’ quintiles + ruTERM(r) - 34.6X 35.97 50.93 53.87 51.96 50.66 - 36.52 - 34.85 30.49 - 22.65 47.19 48.18 45.37 - 42.62 Low II.56 equity (BE/ME) Regressions ofexcess stock returns on 25 stock portfolios formed on size and book-to-market + hIIML(r) and DEF: July 1963 IO December Table 8a 0.0 I 1.17 1.10 1.64 1.3x 1.10 I .08 0.98 0.94 0.43 0.01 0.00 0.00 0.09 I.16 High Book-to-market + sSMB(r) 0.9 1 1.00 1.56 1.16 0.92 0.94 0.95 - 0.18 0.15 - 0.10 - 0.10 - 0.05 I.00 0.70 0.28 4 K(r) - RF(f) = u + bRMO(r) - 0.4 1 1.33 1.01 I .62 I .03 0.96 0.98 0.97 0.98 0.75 0.20 - 0.34 - 0.37 - 0.30 - 0.35 3 b s II relurns, TEKM 0.8 I I .06 I .07 I .02 1.72 I .45 0.34 - 0.56 - 0.65 - 0.64 - 0.64 - 0.63 I.01 I.11 I.04 2 and the bond-market 1.07 1.22 - 1.0x - 1.09 - 1.07 I .06 1.13 I .92 1.50 I .26 0.96 - 0.94 0.85 0.26 I.12 Low and ffML, quinlile Sm:~ll SlIldl S n1all Size Big Big Big 2 3 2 3 3 4 2 4 4

1. 88 1. 20 1. 23 1. 50 13. 57 14. 66 13. 73 X. 98 7. 15 2. 00 24. 24 23. 52 20. 11 27. 57 9.63 -_I _ _ _ - - 1. 59 1. 36 14. 81 16. 36 13. 53 12. 01 I I . 04 1. 11 1. 13 1. 31 25. 67 24. 76 31. 68 22. 83 26.34 I . 31 1. 43 1. 48 12. 25 11. 88 I O. 15 1. 55 11. 90 12. 96 25. 32 26. 40 23. 73 24. 35 20. 42 I.16 s( e) dm) ~-_ f @ ) -- 11. 94 I l . 64 10. 48 1. 43 1. 27 1. 47 1. 31 10. 62 1. 41 20. 60 25. 96 23. 85 23. 77 24. 17 9. 20 -__ -- 1. 46 14. 56 1. 93 1. 55 1. 45 1. 17 10. 23 I I . 53 7. 25 15.66 10.56 27.M) 24.21 23.24 22.08 0. 93 0. 89 0. 73 0. 79 0. 79 0. 69 0. 80 0. 68 0. 96 0. 96 0. 83 0.84 0.88 0.94 0.73 _ _ _ _ _ _ ~_ _ _ 0. 91 0. 90 0. 71 0. 78 0. 19 0. 84 0. 72 0. 97 0. 95 0. 93 0. 90 0.89 0.86 0.98 0.77 0. 73 0. 66 0. 81 0. 83 0. 84 0. 93 0. 91 0. 75 0. 97 0. 95 0. 87 0.84 0.90 0.86 0.79 HZ cl . _ 0. 73 0. 84 0. 87 0. 72 0. 78 0. 74 0. 93 0. 92 0. 82 0. 79 0. 96 0. 94 0.63 0.96 0.66 under t abl e 8b. 0. 75 0. 85 0. 74 0. 85 0. 88 0. x0 0. 80 0. 81 0. 94 0. 95 0. 94 0. 94 0. 95 0.67 0.76 " See km not e _ - Sn1all Sm al l Sm al l I j i g ni g Bi g 2 3 2 4 3 4 3 4 2

Table 8b Regresstons ofswess returns. R.UO. S.UB. and H.tfL. returns on government and corporate bonds (in percent1 on the stock-market returns. TERJI and DEf: 1991. 3-t: months.’ and the bond-market December July 1963 to Bond portfolio I-SG 66IOG .-\a .A Baa .Aaa LG h ribI - 0.03 - 1.34 - 0.04 - 3.14 0.00 I .05 0.0’ I .99 - 0.02 - 2.96 0.00 0.06 0.18 7.39 5 r(s) - 0.00 - 0.68 - 0.03 - 2.30 - 0.01 - 2.55 0.06 1.09 - 0.03 - 3.17 0.00 0.80 0.16 5.09 /I f(lll 0.02 1.76 - 0.00 - 0.00 - 0.00 - 0.17 - 0.01 - 1.36 1.02 102.65 0.00 0.52 0.03 1.72 0.00 0.12 0.99 V, t(trll 0.45 32.09 0.71 39.55 1.00 1.01 57.34 0.79 19.56 139.1 I I .02 74.00 130.93 0.97 67.05 Cl t(ll) 0.15 9.16 0.29 8.25 0.95 50.04 1.07 31.77 0.94 I’.09 R2 S(Z) 0.130 0.56 0.9s 0.30 _ 0.98 0.29 0.87 0.73 0.97 0.40 0.9 I 0.70 0.58 1.63 _ ‘R.\ IO, the orthogonalized represston of R.WRF return on all stocks in the 25 size-BE portfoltos. RF is ths one-month (smdll mtnus big). the return on the mimicking is the dtfference each month portioitos (S L. 5’ .\I. and S Hl and the simple aterage of the returns on the three big-stock (5 L. B .If. and B Hi fl.\ /L (htgh minus low). the return on the mimicking book-to-market equtty factor m returns, is the dtfference the returns on the t&o hi!h-BE .A/& portfolios two low-BE .ME portfoltos (S f. and B L). TER.Lf government bond return. DEF IS CB-LTG, portfolto of corporate bonds. The seben bond portfolios used as dependent j-year and 6- to IO-year governments (1-S and &lOG) belou Baa (LG, by Moody’s The 25 size-BE.‘.LfE I from 1963 to 1991 XYSE quinttle breakpoints measured at the end of June. are used to allocate qutnttles. NYSE quinttie breakpoints for BE .CfE are also used to allocate S;\SD.AQ stocks to tire-book-to-market equity for the fiscal year endtng tn calendar year I - I. and .\fE ts for the end of December stze-BE .5/E portfolios are the intersections welshted monthly percent returns on the portfolios I - I. R2 and the resdtual standard error. hIti), are adJusted for degrees of freedom. market return. is the sum of intercept and residuals from percent the on .S.\fB. H.LIL. TER.W, and DEF. R.ll 1s the value-ueighted .tfE portfolios. Treasury btll rate, observed portfolio between the simple aberage of the returns monthly plus the negative-BE at the beginning for the common stocks excluded from the of the month. size factor tn stock returns, on the three small-stock S.LfB portfoltos portfolio for the common each month between the simple alerage of 1.5 H and B i-l) and the average of the returns on the is LTG-RF, where where CB is the return on a proxy LX is the long-term for the market m the excess-return and bonds rated .Aaa. Aa. A. Baa, and stock portfolios are formed as follows. for size (LIE. stock price times shares outstandinp). SYSE. Amer. and NASD.A.Q variables regressions are I- to Each year stocks to fire size NYSE. .Amex. and quintiles. In BE .L/E. BE is book common equity oft - 1. The 25 of the fi\e size and the five BE .ME groups. are calculated Vnlue- from July of year r to June of

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 explanatory returns suffice to describe the cross-section intercepts in the time-series portfolio returns should be indistinguishable Since the stock portfolios examine their intercepts first. We are especially mimicking returns S,CIB and HML. absorb the size and book-to-market in average returns, illustrated in table 2. We then examine bonds. Here the issue is whether average returns that are rejected by the flat average bond returns associated of average returns: the of excess returns from 0.’ a wide range of average interested with any set of regressions on the mimicking produce returns, we the in whether effects the intercepts predict for in different factor models patterns in table 2. 3.1. The cross-section oj’acerage stock returns R&f-RF - When the excess market return is the only explanatory the time-series regressions, the intercepts of Banz (1981). Except in the lowest-BE/ME 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 quintile, the intercepts increase with BE/ME; BE/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 related to size and book-to-market equity. In fact, as in Fama and French (1992a), the simple relation return and /3 for the 25 stock portfolios return on /J yields a slo’pe of - 0.22 with a standard (1964)-Lintner (1965) model (/I suffices to describe the cross-section 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 returns on SMB and HML produce similar intercepts (table 9a). The two-factor regression intercepts per month) and close to or more than two standard that are similar in size support the conclusion in Fama and French (1992a) that size and book-to-market strong differences in average returns across stocks. But the large intercepts say that S.LfB and HML. do not explain the average premium over one-month bill returns. RM-RF, S.CfB, and H,LIL -Adding series regressions pushes the strong positive intercepts variable in for stocks (table 9a) show the size effect quintile. the intercepts for the equity. In every size for the highest- the intercepts in average stock returns that is between average of average used here is flat. A regression error of 0.31. The Sharpe of average regressions for the 25 stock portfolios are, however, large (around 0.5% errors from 0. Intercepts from the cross-section factors explain of excess stock regressions the also of stock returns the excess market return to the time- for stocks observed in the ‘This implication (1982). is only an approximation in the Ross (19761 model. Ser. for example. Shanken

equity: J uly 1963 IO December IYYI, High 3.1 I 2.87 3.03 3.35 2.53 1 .41 2.1 4 3.71 3.01 3.1 9 ~~ 2.X8 2.7Y 2.36 3.20 2.85 1 .89 3.01 2.61 2.91 2.1 9 4 1 .54 2.20 2.60 2.28 1 .96 2.35 I(4 1 .20 - 0.70 1 .n 2 1 .1 7 3 formed on size and book-to-market I .9Y I .05 - 1 .50 1 .27 0.73 1 .25 - 0.95 1 .73 1 .01 I.91 2 = (I + OI~‘ERM(I) + dDEF(f) + t$r) quintiles (ii) R(r) - RF(r) = u + ~[RM(I) - RF(r)] + ($0 equity (HE/ME) - 0.4Y - O .YO 1 .34 - 1 .00 - 0.50 0.75 0.93 1 .35 I.00 - 1 .1 2 Low 342 months.” Table Ya High 0.54 0.92 0.93 0.96 0.53 0.53 0.57 0.89 0.50 0.21 Book-to-market from excess stuck return regressions for 25 stuck purtfulios K(r) - W ’(r) 0.42 0.3’) 0.39 0.35 0.20 0.80 0.75 0.73 0.69 0.50 4 (i) 0.4’) - 0.07 0.77 0.60 0.25 0.30 0.36 0.23 t, 0.71 0.1 2 3 - 0.07 0.62 0.63 0.58 0.27 0.30 - 0.1 4 0.1 5 0.1 7 0.1 s 2 0.4 1 - 0.04 - 0. IX - 0. I6 0.34 0.34 - 0.22 - 0.05 0.35 0.31 Low lntrrceprs quinlile Size Sm all sm ;III Uig Big 3 4 2 3 4 2

1.70 I .24 1.23 2.56 2.61 0.56 2.49 2.85 0.09 0.29 - I.51 0.14 0.34 0.56 - I.41 0.7 I 2.61 2.20 0.69 0.47 0.73 2.52 2.66 2.51 0.33 0.20 0.67 0.79 0.22 0.51 + dDEF(r) + &) 1.10 - 1.46 - 1.46 1.84 1.04 2.76 - 0.42 - 0.94 2.24 2.25 2.07 - 0.73 - 0.47 - 0.99 - 0.79 + V(I) - 1.47 - 1.58 H(l) - RF(f) = u + h[RM(r) - RF(r)] + sSMB(r) + hHML(r) + ntTERM(r) 1.55 - 2.65 - 2.67 2.40 2.23 - 0.67 - 0.72 2.58 - 0.20 0.47 - 0.24 0.48 I.92 + h/IML(f) + t(r) X(r) - RF(I) = u + sSMtqr) + hlfML&) - RF(r)] + sSM&I) ~ 1.29 - 1.45 - 1.42 I .07 1.04 - 1.24 - 3.16 3.27 - 3.24 3.29 2.00 3.41 2.00 2.78 0.97 0.00 0. I 3 - 0.16 - 0.17 0.00 0.13 0.02 0.05 0.55 0.64 0.66 0.79 0.44 0.02 0.05 ~~~~__.~~~.~. = 0 + h[RM(r) 0.0 I - 0.05 0.06 - 0.06 0.53 0.58 0.62 0.51 0.01 0.03 0.05 0.03 0.04 0.04 0.60 K(r) - W(r) ~~~ (iii) - 0.05 - 0.04 - 0.08 - 0.05 - 0.03 - 0.08 - 0.13 0.49 0.64 0.52 0.50 0.43 0.08 0.0x -0.13 _ (iv) - 0. I3 - 0.12 - 0.22 - 0.05 - 0.22 - 0.05 0.58 0.52 - 0.01 0.04 - 0.02 0.04 0.46 0.61 0.39 ‘See footnote under table 9c. (v) - 0.1 I - 0.1 I 0.0X - 0.34 - 0.12 0.52 0.69 0.76 0.09 0.21 - 0.35 0.21 0.24 0.52 -0.11 ~~_.___ Smnll Sm;ill S Ill;Lll Big Big Big 2 3 2 3 3 4 2 4 4

Table Yb 1931. 343 months.’ 1963 to December portfolios. Jul> Bond portfolIo 1-K 6-IOG ‘AU .A.l A B&I LG II) R(r) - Rf(tl = tt 4 m TER.LIItI + <IfIEf-ltl + ettt - 0.00 - 0.2Y 0.06 I .42 - 0.00 - 0.55 0.06 0.67 0.09 7.16 - 0.02 - 1.10 0.08 2.70 Ilil Rlr) - RF(t) = (I 4 h[R.\!~tl - RF(t)] + fir1 0.M) 0.03 - 0.0 I - 0.1 I 0.0-l 0.37 - 002 - 0.15 0.08 0.76 - 0.03 - 0.21 0.08 I.27 luil R(rl - RF(t) = LI + sS.tlB(t) + hH.LfLlt) + e(t) 0.08 0.58 0.0’ 0.55 0.1 I 0.82 0.07 0.58 0. I6 I .47 0.07 0.52 0. I1 1.70 (1~) R(r) - RF(t) = LI + h[RM(t) - RF‘ltl] + s.S.\IBltl + ItH.VLltl + CICI - 0.1 I - 1.00 - 0.08 - 0.69 - 0.05 - 0.41 - 0.07 - 0.64 0.07 0.62 - 0.07 - 0.62 0.06 0.8Y R~ri - RF(t) = (1 + h[R.\l(t) - RFltl] t sS.\/51t1 + hff.\IL(t~ (\I + ntTER.Lf(t) + riDEflt) + p(t) - 0.07 _ 0.77 - 0.00 - 0.57 0.02 0.52 - 0.00 - 0.25 0.1 I 1.77 - o.Oil - 0.17 0.09 2.8-l ‘See footnote under table Yc. tuo-factor intercepts month: use R.11-RF. returns do a good job explaining There is an interesting excess market return is added to the two-factor the three-factor regressions. clos2 to I. The average market risk premium the similar strong positive intercepts on .S.\fB and H.LJL. In short, the size and book-to-markzt the differences in averagz returns across stocks. but the markzt factor is needed to explain why stock returns are on average above the one-month TER.lf cud DEF - Table 9a shows that adding TER.11 and DEF. to the tim2-series regressions the intercepts produced by the three stock-market the strong slopes on TER.tf and DEF when they are used alone to explain stock (.S.LfB and H.CIL) regressions in the three-factor 16 are wtthtn 0.144 o f 0. Intercepts S.lIB. and HAIL to absorb to values close to 0. Only thr22 of the 25 differ from 0 by more than O.Z’?,b per close to 0 say that the regressions common time-series the cross-section of average stock returns. story for the smaller intercepts (S.CfB and H,CIL) regressions. the stock portfolios produce (0.33% per month) observed in the regressions regressions that variation in obtained when the In slopes on R.&RF thzn absorbs of stock returns factors can sxplain bill rate. factors, the term-structure for stocks has almost no effect on factors. Likewise. in spite of

39 E.F. Funu md R. R. Frmch. Conrnwn rd lircrors in .uuck und bond rr~urrrs Table 9c F-startstics testing the intercepts in the excess-return levels of bootstrap regressions and F-dtstrtbutions.” against 0 and matching probability Regression (from tables 9a and 9b) (iii) (ii) (iv) (i) (v) !.91 1.75 F-statistic Probability Bootstrap F-distribution 2.09 1.56 1.66 level 0.985 0.990 0.998 0.999 0.996 0.996 0.95 1 0.96 I 0.971 0.975 --__ percent return on all stocks in the 25 size-BE .lfE portfolios. stocks excluded from the 25 portfolios. of the month. S.LfB (small minus big), the return size factor in stock returns. is the difference each month between the simple portfolios (5 L, S’XI. and S H) and the simple average portfolios IB L. B’.LI, and B H). H,ML (high minus low). the portfolio for the common book-to-market “R.11 is the value-weighted plus the negative-BE observed at the beginning portfolio for the common average of the returns on the three small-stock of the returns on the three big-stock return on the mimicking difference each month between the simple average of the returns on the two high-BE .WE portfolios (S,‘H and B H) and the average of the returns on the two low-BE TER.W is LX-RF. where LX is the long-term CB is the return on a proxy for the market portfoho The seven bond portfolios used as dependent j-year And 6- to IO-year governments (I-5G and CLOG) and corporate and below Baa (LG) by Moody’s, The 25 size-BE year f from 1963 to I991 NYSE quintile breakpoints outstanding). measured at the end of June, are used to allocate NYSE. Amex. and N.ISDAQ to live size quintiles. NYSE quintile breakpoints for BE .tfE are also used to allocate NYSE. Amex, and N.ASDAQ stocks to five book-to-market equity quintiles. equity for the tiscal year ending in calendar year I - I. and :1fE is for the end of December The 25 size-BE .tfE portfolios are the intersections Value-vveightsd monthly percent returns on the portfoltos oft+ I. Regressions (I~+v) in table 9c correspond to the regressions monthly RF 15 the one-month Treasury on the mimicking bill rate. equity factor in returns. is the .bfE portfolios (S L and B LI. bond return. DEF is CB-LK. of corporate bonds. variables in the excess-return government where regressions are I- to bonds rated Aaa, Aa. A, Baa. are formed as follows. Each for stze (.WE. stock price times shares .%I E stock portfolios stocks In BE .LfE. BE is book common oft - I. of the five size and the five BE .LfE groups. are calculated from July of year c to June tn tables 9a and 9b. The F-stattstic is F = (.-l’_r-‘?t)(.v - K - f + I),(L*(.V - K)*W,,,). K is 1 plus the number ofexplanatory intercepts. and o,.r Ross. and Shanken (19891 show that this statistic where .V = 342 observations. the regression. covariance matrix of the residuals (.Y’X)-r corresponding has an F-distribution returns and explanatory In the bootstrap residuals from the regressions 342 monthly excess returns for the 25 stock and seven bond portfolios These model returns and the exrlanatory 1963 to December 1991, are the population sample. with replncement. of 342 paired observations five regression models) on the model returns regressicns. For each model. the table shows the proportion F-Xdttstic is smaller than the empirical drawn from an F-distribution is smaller than the empirical L = 31 regressions, vector of the 32 regression from the 32 regressions. to the intercept. Gibbons, with L and N - K - f. i I degrees of freedom under the assumption variables are normal and the true intercepts simulations, the slopes (with intercepts for July 1963 to December variables in A is the (column) Z (L x L) is the unbiased is the diagonal element of that the are 0. set to 0). explanatory 1991 in tables 3 to 7 are used to generate for each regression S.LfB. H.CfL, TER.Lf. and DEF, for July for the stmulations. Each simulation (the same set of observations and the explanatory variables. of 10.000 simulations estimate. The table also shows the probability estimate. variables. and model. returns. R.Lf-RF. takes a random for each of the and estimates rn which the that a value the

returns (table 3). the two variables returns for the 25 stock portfolios The reason for these results is straightforward. returns (the average risk premiums 0.06% and 0.02% per month. The high volatility allows them to capture substantial in the two-factor regressions But the low average TER,LI and DEF returns imply that the two term-structure factors can’t explain much of the cross-sectional returns. produce intercepts in table 2. close to the average excess The average TER.Ll and DEF for the term-structure of TER.Ll and DEF (table 2) common variation in bond and stock returns of table 3 and the five-factor factors) are puny, regressions of table 8. variation in average stock 5.2. The cross-section of average hod retwrrs Tables dominated bond portfolio TERM and DEF are in the bond regressions. and DEF. the average (table 2), so it is not surprising for bonds (table 9b) are close to 0. Do low average TERM and DEF premiums factors are irrelevant in a well-specified DEF are the dominant Moreover, Fama and-French values of variables like TER;LI and DEF vary through business conditions. The expected discount-rate risks, is positive around peaks. The expected value of the default premium conditions are weak and default conditions are strong. Thus, the common TERM and DEF implies interesting and bond returns. 3. 7b and 8b say that the common by the bond-market (LG) has nontrivial variation in bond returns is factors, TERM and DEF. Oniy the low-grade slopes on the stock-market Like the average values of TERM excess returns on the bond that the intercepts in the time-series factors when portfolios are close to 0 regressions imply that the term-structure asset-pricing model? Hardly. TER.Ll and in the common variation (1989) and Chen (1991) find that the expected variables in bond returns. time and are related to value of TER.CJ, the term premium business cycle troughs and negative near in DEF is high when economic risks are high, and it is low when business sensitivity of stocks and bonds intertemporal variation for to in expected stock j.3. Joiut tests on the regression intercepts We use the F-statistic the hypothesis for the 32 bond and stock portfolios bootstrap probability explanatory variables The F-tests support hypothesis that the term-structure the average returns Ross. and Shanken variables that are all equal to 0. The F-statistics, levels, for the five sets of intercepts in tables 3 to 8 are in table 9c. the analysis of the intercepts returns, TER,Ll and DEF. suffice to explain on bonds and stocks at the 0.99 level. This confirms of Gibbons, (1989) to formally test regression that a set of explanatory produces intercepts and produced by the above. The tests reject the the

conclusion, average stock returns. The F-test rejects the hypothesis average returns at the 0.99 level. This confirms cannot explain the size and book-to-market large positive intercepts explanatory variables produce hypothesis at the 0.98 level. In terms of the F-test. the three stock-market HML, produce the best-behaved intercepts for the seven bond and 25 stock portfolios 0.95 level. The rejection stocks. Among stocks with the lowest ratios of book-to-market stocks), the smallest stocks have returns that are too low ( - 0.34% per month, t = - 3.16) relative to the predictions of the three-factor stocks have returns that are too high (0.21% per month, differently, the rejection of a three-factor of a size effect in the lowest-BE,‘ME quintile. BE/ME quintile produce slopes on the size factor SMB that are strongly negatively related to size (table 6). But unlike average returns in the lowest-BE/ME quintile show no relation to size (table 2). Despite its marginal rejection in the F-tests, our view is that the three-factor model does a good job on the cross-section rejection of the model simply says that because absorb most of the variation in the returns on the 25 stock portfolios (the typical R’ values in table 6 are above 0.93). even small abnormal to show that the three-factor model is just a model, that is, it is false. To answer the important question of whether the model can be useful in applications, interesting result is that only one of the 25 three-factor stocks (for the portfolio in both the smallest-size quintiles) is much different from 0 in practical Indeed, our view is that the three-factor and H,bfL to explain average returns do surprisingly the mimicking returns SMB and HML for the size and book-to-market are constructed. The regressions produce intercepts even though SMB and HML surely contain for the risk factors in returns related to size and book-to-market Adding the term-structure returns. TER.Cf and DEF, to regressions use R.11-RF, S.bfE, and H.ClL as explanatory F comes from bonds. The five-factor regression stocks are close to those produced by the three stock-market for bonds, adding TER.Cf and DEF results in much lower residual errors. and the increased precision pushes the five-factor obvious from the regression intercepts explain in table 9a, that the low the cross-section that RAWRF suffices to explain that the excess market return effects in average stock returns. The TERM and DEF returns cannot of average for stocks observed when SicIB and HAlL are the only an F-statistic that rejects the zero-intercepts factors, R&I-RF, Nevertheless, are 0 rejects at about the from the lowest-BE’JIE SMB, and intercepts. the joint test that all comes largely quintile of equity (growth model, and the biggest t = 3.27). Put a bit model in table 9c is due to the absence The five portfolios in the lowest- the other BE/ME quintiles, of average R,WRF, stock returns. S,LIB, and HML The average returns suffice the for regression and the lowest-BE/ME intercepts terms. that use RM-RF, well, given the simple way SMB, regressions factors for stocks that are close to 0, some firm-specific noise as proxies equity. that also increases F. The larger intercepts and R’ values for variables factors. standard for the two But intercepts

government intercepts The three stock-market regressions provide the best model for returns and average returns on bonds and stocks. TER.Lf and DEF dominate the variation variation in the expected values of TERJf an interesting part of the variation through stocks and bonds that is missed by the F-test, which is concerned long-term average returns. bond portfolios beyond two standard errors from 0. The two rather small, 0.09O 0 and 0. I 1% per month. factors produce a lower F, but we think the five-factor are. however, in bond returns. And the and DEF with business conditions time in the expected is returns only with on 6. Diagnostics In this section we check the robustness factors explain the cross-section the residuals from the five-factor regressions capture returns. We then examine seasonals in stock and bond returns. use one set of stocks in the explanatory dependent returns. These tests address the concern book-to-market factors in the regressions cause vve use size and book-to-market explanatory returns. The last and most interesting stock-market factors that capture the average returns on size-BE,‘.LfE work as well on portfolios formed on other variables about av’erage returns. in particular, of our inference that five common of expected stock and bond returns. We first use time-series regressions through time in the cross-section whether our live risk factors capture Next come split-sample returns and another. that the evidence of size and above is spurious, portfolios for both our dependent tests examine risk to check that the of expected the January regressions disjoint, set in the the variation that arising only be- and the whether portfolios known to be informative earnings, price and dividend,‘price ratios. There is evidence dividend (default spreads, (term spreads. references therein.] returns. the predictability explanatory regressions. hypothesis. that stock and bond returns can be predicted over high-grade over short-term using (a) bond yields bond yields yields (D,,P), (b) spreads DFS), (c) spreads TS), and (d) short-term If our five risk factors capture of stock and bond returns should be embodied returns (the month-by-month The regression residuals should we estimate the 32 time-series of low-grade of long-term interest rates. [See Fama (1991) and the the cross-section of expected in the risk premiums) be unpredictable. regressions, in the five-factor To test this e,(t + 1) = k,, + k,D(r) P(r) -I- k2DFS(r) + k,TS(t) + k,RF(r) (2) + r/Jr + 1).

The r,(t + I)in (2) are the time series of residuals for our 25 stock and seven bond portfolios from the five-factor regressions D(t). P(t), is dividends on the value-weighted year ending in month t divided by the value of the portfolio default spread, DFS(t), is the difference at the end of month r between the yield on a market portfolio of corporate bonds and the long-term yield (from Ibbotson Associates). The term spread. between the long-term government bond yield at the end of month one-month bill rate, RF(r). The estimates of (2) produce no evidence that the residuals from the five-factor time-series regressions are predictable. tive values of R’ (adjusted for degrees of freedom). Only four of the 32 R’ values exceed 0.01; the largest is 0.03. Out of 1X (32 x 4) slopes in the residual regressions. ten are more than two standard between positive and negative values. and they are scattered the 32 regressions and the four explanatory The fact that variables known to predict predict the residuals from our live-factor the five risk factors capture the cross-section The residual tests are also interesting information tion. Since we estimate regression slopes on returns period, we implicitly assume that the sensitivities risk factors are constant. If the true slopes vary through residuals may be spuriously predictable. that the assumption of constant slopes is reasonable, used here. of table 7. The dividend portfolio of NYSE stocks for the at the end oft. The yield, government bond TS(r), is the difference c and the In the 31 regressions, 15 produce nega- errors from 0; they are split evenly randomly across variables. stock and bond regressions ofexpected on a key regression for the entire of the dependent returns our inference that stock and bond returns. do not supports specifica- 1963-1991 returns to the time, the regression The absence of predictability at least for the portfolios suggests Since the work Roll (1983) and Keim (1983). documenting especially returns on small stocks, tend to be higher in January, tests of asset-pricing models to look for unexplained leery of judging models on their ability seasonals are, in whole or in part, sampling snooping bias toward rejection test for January seasonals in the residuals Despite our fears. we find that, except for the smallest stocks. residual January seasonals are weak at best. The strong stocks and bonds are largely absorbed Table 10 shows regressions of returns January and 0 in other months. The regression non-January months, and the slopes on the dummy measure differences between average January returns and average returns that stock returns, it is standard January effects. We are January seasonals. error, the tests can contain [Lo and MacKinlay (1990)]. Nevertheless, from our five-factor in to explain If the a data- we regressions. January by strong seasonals on a dummy intercepts seasonals in the returns in our risk factors. variable that is 1 in are average returns for on in other months.

1991, 0.04 0.00 - 0.00 - 0.00 0.03 ~ 0.00 - 0.00 0.06 0.01 0.02 0.00 0.00 0.02 0.00 0.00 R2 regressions: July 1963 IO December ((4 ~ 1.74 - 1.48 - 1.80 - 1.15 3.06 3.57 - 2.16 - 2.90 2.01 4.09 4.94 - 0.74 - 0.15 - 0.97 0.49 regression residuals - 1.17 - 1.02 - 0.57 - 0.81 - I.41 0.21 0.04 0.62 0.2x - 0.14 0.50 0.42 0.52 0.33 0.x3 I(4 Five-factor 1.13 - 0.23 - 0.04 - 0.22 - 0.49 I.51 0.56 0.69 0.76 - 0.55 0.12 - 0.41 - 0.x0 - 0.46 - 0.34 h returns, and residuals from the five-factor 0.02 - 0.12 - 0.05 - 0.06 - 0.06 - 0.09 0.02 - 0.01 0.04 0.03 0.07 0.04 0.03 0.00 0.04 u + r 0.10 quintitc - 0.00 0.00 0.00 0.06 0.06 0.03 0.05 0.06 0.05 0.08 0.00 0.03 0.03 0.05 0.07 0.00 0.02 0.02 0.04 0.04 Size quinlite 3 Size quintite 2 R2 342 months.’ K(r) = u + b JAN (r) Table 10 Smattrst-size 6.3 I I .67 I .70 3.6X I .78 3.56 5.55 - 0.69 2.04 3.34 3.48 4.96 4.70 4.23 4.27 4.93 4.22 2.57 2.99 4.99 4.12 returns l(b ) returns, explanatory Excess stock returns exptanalory 1.66 - 0.8 I I .22 I .53 1.14 1.20 I .63 t .47 I .65 1.31 2.03 0.30 0.56 - 0.30 0.03 0.90 0.48 0.62 I.04 I.92 t.9t w Five-factor I .49 1.19 2.74 2.29 5.62 5.91 7.39 5.76 2.35 3.06 3.51 0.41 6.31 6.29 2.92 3.95 2.87 Tests for January seasonals in the dependent I.10 4.17 4.32 4.53 b _ 0.10 0.2 I 0.20 - 0. I3 - 0.07 0.31 0.40 0.05 0.24 0.31 0.37 0.40 0.37 0.53 0.4x 0.55 0.24 0.42 0.43 0.52 0.60 u t~t~/Ar t: Low Stock portfolio High High nEjnlE High Low Low tlE/ME4 Mr/Are ? NT/MI: 3 E 3 2 UE/M E 4 2 3 I: 4 HM-RF TERM t/ML ICaclor BE/ME BE/ME BE/ME KM0 DEF SMB B E /M E H E /M E H E /M E B E /M E H E/At B E/M

equity quinliles in is a dummy variable that is I if month I is January and 0 otherwise. RMO is the sum of the intercept and residuals from the regression of where CB is the return on a proxy for the market portrolio of and bonds rated Aaa, Aa, A. Baa, and below Baa (LG) by return. RF is the one-month Treasury bill rate, observed at -- O.(K) O.lNJ equity factors in stock returns. - 0.00 0.0 I 0.0 I 0.00 - 0.00 0.00 - 0.00 - 0.00 - 0.00 - 0.00 0.02 0.00 - 0.00 0.03 0.00 __~~ - 1.60 I .67 - 2.17 2.0x - 2.54 - 3.27 - 0.40 - 0.79 - I.85 0.58 - 0.09 - 0.02 - 0.57 0.63 0.31 I.01 - I.17 regression residuals portfolios are formed as the intersections of independent sorts of stocks into size and book-to-market - 0.18 0.23 0.62 0.53 - 0.60 - 0.29 - 0.17 0.73 - 0.48 - 0.09 0.12 0.46 0.93 0.34 0.03 0.00 0.16 are the returns on the mimicking portfolios for the size and book-to-market Five-factor - 0.1 I - 0.1 I - 0.17 0.12 0.19 - 0.46 - 0.73 - 0.04 0.14 - 0.93 - 0.37 - 0.03 0.38 - 0.00 - 0.17 0.08 0.25 0.06 0.00 0.08 0.03 0.04 and GIOG) Tl:‘Rhf, and DEP. KM is the value-weighted monthly stock-market The variables are described in more detail in table 8. 7’k’RM is LTG-RF, where LTG is the long-term government bond return. DEF is Cn-LTG, Biggest-size quintile - 0.00 0.00 0.03 0.04 0.0 I - 0.00 0.00 0.00 0.02 0.03 Size quintile 4 The seven bond portfolios are I- to 5-year and 6- to IO-year governments (I-5G 1.65 3.24 0.95 4.00 1.28 1.19 2.1 I 3.59 2.85 0.35 Excess bond returns I .02 1.16 1.88 I .40 2.15 0.68 1.34 1.54 1.17 0.92 1.12 1.77 2.0X 3.12 4.45 1.11 1.11 2.31 3.38 0.34 the beginning ol’the month. SMU and f/ML June of each year from 1963-1991. 0.21 0.39 0.40 0.52 0.68 0.37 0.37 0.32 0.27 0.23 Moody’s, The 25 size--BE/ME _ RI%-RI’ w Shfn, //bfl.. corporate bonds. Bond portfolio BE/ME High High LIE/ME Low BE/ME Low BE/ME 3 BEfME 2 LX/ME 3 NEjhfE 2 BE/ME 4 SE/ME 4 ‘JAN(r) 6-I o<; nE/ME I S C Aa;l Baa A;1 LG A

The table confirms and the seasonals more than 1.91% per month and more than two standard portfolios in the two smallest size quintiles. January return declines monotonically the January seasonal in stock returns In every size quintile. the slopes on the January BE ‘!Cf E. The extra January return for the two highest-BE quintile is always at least 2.38% per month and 2.85 standard January season& are not limited to stock returns. The slopes on the January dummy for corporate bonds increase monotonically portfolio. The extra January returns are 0.86’5b. 1.13%. and 1.56% per month for the A, Baa, and LG portfolios, and these extra average returns are at least 1.94 standard errors from 0. If our five-factor time-series regressions in stock and bond returns. there must be January Table 10 shows that, except for TERJI. returns in excess of I % per month and at least I .67 standard season& in the size and book-to-market average S,LIB and H,tfL returns in January greater than in other months, and the extra January standard errors from 0. Indeed, like the excess returns on the 25 stock portfolios and the five corporate bond portfolios five-factor regressions. the extra January much larger and more reliably different non-January months. Finally, table 10 shows that the January absorb the seasonals in stock and bond five-factor residuals on the January dummy. smallest-size quintile produce systematically are only one-quarter to one-tenth the positive excess returns on the portfolios. If anything. remaining size quintiles show negative January January dummy for these stock portfolios. and mostly within two standard errors of 0. In short, whether spurious the January seasonals in the returns on stocks and corporate largely explained by the correspondin, five-factor model. that there are January are related to size. The slopes on the January season& in excess stock returns, dummy errors from 0 for the for BE .ifE. the extra size. More interesting, are all Controlling with increasing is also related to book-to-market dummy equity. tend to increase with .\JE portfolios in a size errors from 0. from the Aaa to the LG are to explain the January seasonals the risk factors have extra January seasonals in the risk factors. errors from 0. The are especially are 2.73% and 2.29% per month returns are 3.96 and 4.70 factors strong. The that are the dependent returns on the risk factors are generally from 0 than the average variables in the returns for seasonals returns. only the stock portfolios posttive slopes: even these slopes January seasonals the five-factor seasonals, but the slopes on the and for the bond portfolios, in our risk factors largely fn the regressions of the in the in the raw for the residuals are small or real, bonds seem to be u seasonals in the risk factors of our In the time-series tvvo explanatory regressions returns for stocks. the dependent returns and the S:LIB and H.LlL are portfolios formed on size and

E.F. Fuamu umf K. R. French. Common risk /&torr WI stock md bond returns 17 book-to-market power of SLVB and N.CIf. is spurious, this is unlikely. given that the dependent and BE:.CfE sorts (25 portfolios) unlikely that we have stumbled factors that (a) measure strong common when really there is none, and (b) produce exactly the patterns slopes on S,bfB and HML needed to explain the size and book-to-market in the average returns on the 25 portfolios. interest. We split the stocks in each of the 25 size-BE groups. One group is used to form the 25 dependent returns for the time-series regressions. versions of the explanatory returns. RIM-RF. SMB. and HhfL. The roles of the two groups are then reversed. and another we have two sets of regressions. In each set, the explanatory returns are from disjoint groups of stocks. Without showing all the details, we can report that the results for the two sets of regressions of excess returns for 25 size-BEj’,LfE versions of RM-RF. Slbff3, and H.bfL are similar to the full-sample tables 6 and 9. The slopes on RXf-RF, regressions are close to those in table 6, and the intercepts. full-sample three-factor regressions in table 9. are close to 0. In short, the split-sample regressions confirm that there are common related to size and book-to-market equity. They also confirm that market, size, and book-to-market factors seem to capture returns. If anything, the split-sample regressions hypothesis that R.Lf-RF, SMB, and average stock returns than the full-sample dependent portfolio returns in the split-sample available stocks. the portfolios are less diversified Although the three-factor split-sample R’ (mostly greater than 0.88), they are a bit lower than (mostly greater than 0.9). As a result. hypothesis are weaker for the split-sample regressions. equity. Many readers worry that the apparent induced by the regression setup. We think returns are based on much finer size than the S.IIB and H.LIL returns. It also seems on two mimicking returns for size and BE/ME variation in the returns on 25 portfolios explanatory in the regression effects Still, an independent test is of ,LfE portfolios value-weighted into two equal portfolio The other is used to form half-sample set of regressions is run. In this way and dependent portfolios on disjoint results in S&f& and H.CfL in the split-sample like those for the risk factors in returns the cross-section of average stock show less power H&IL capture regressions. to reject the of 25 the cross-section Since regressions than those produce those in table 6 the F-tests of the zero-intercepts regressions than for the full-sample the use half the in table high values 6. of regressions 6.4. Portfolios fbrtwd on E, P The most interesting book-to-market explain the returns check on our inferences risk factors in returns on portfolios about the role of size and whether these variables variables known is to examine formed on other to be

informative one-factor regressions (D.‘P) ratios. The average returns on the E/P portfolios Gaffe, Keim, and Westerfield of firms with negative quintile have the highest average return increases pattern is an interesting about average returns. Table 11 shows summary (RM-RF) and three-factor for portfolios formed on earningsprice statistics, as well as SMB, and (E, P) and dividend/price (R.LJ-RF. HAIL) have the U-shape documented in (1989) and Fama and French (1992a). The portfolio earnings and the portfolio average returns. For the positive-E/P from the lowest- to the highest-E/P challenge for our risk factors. of firms in the highest-E/P portfolios, quintile. This Table I I Summary statistics for value-weighted dividend price (D P 1 and earnings, price (E, P), and regressions of excess portfolio returns on (i) the excess market return (R.WRF) and (ii) the excess market returns for the size ISCUB) and book-to-market equity (H.VL) 342 months.” monthly excess returns (in percent) on portfolios formed on return (R.Lf-RF) factors: July 1963 to December and the mimicking 1991. R(t) - RF(r) = a + b[RM(t) - RF(t)] + e(t) Ii) R(r) - RF(t) = tz + b[Rhf(r) - RF(r)] + sS.CfB(r) (ii) + hH.VfQt) + e(t) Portfolios formed on E P Portfolios formed on D P _ r(mn) Purrfolio Mean Std. Mean Std. t(mn) 1.72 0.96 0.48 0.39 7.36 5.48 1.20 I .30 0.72 0.27 7.77 5.13 GO Low 7 1.53 I .82 2.27 3.30 0.44 0.47 0.57 0.56 4.53 4.65 4.33 3.86 1.68 1.87 1.42 2.67 ; 4 Htgh 0.17 0.16 0.55 0.86 4.76 4.68 4.4s -I.%+ Portfolios formed on E’P Regression (i) Regression (ii) ll b 5 h R2 R? Portfolio ‘1 h EPGO 0.13 (0.50) - 0.30 ( - 1.68) I.21 I.13 0.16 (6.10) 0.82 1.37 0.64 (‘4.70) (27.82) (17.42) 0.9 I Low - 0.10 ( - 2.35, 0.04 (0.70) 0.99 (66.75) - 0.01 ( - 0.55) - 0.50 t - 19.73) 0.96 1.10 (57.42) 2 0.03 (0.46) I.01 0.03 (OAO, 1.01 0.02 (1.01) - 0.00 ( - 0.081 0.94 0.91 (70.24) (61.17) - 0.00 ( - 0.11) 1 .oo (55.16) 3 0.04 (0.50) 0.99 161.62) 0.91 001 (0.40) 009 (3.86, 0.92 0.9 I 1 0.15 t 1.761 0.93 (49.78) - 0.02 ( - 0.28) 0.98 (53.57) I .03 (51.56) 0.05 ( 1.95) 0.33 ( lO.UI 0.58 High 0.46 (3.69) 0.91 (34.733 0.08 11.01) 0.24 (8.34) 0.67 (19.62) 0.91 0.78

Table I1 (continued) formed on D P Portfolios Regression (i) Regression (ii) h R’ b s R’ U b 0 Portfolio D* P=O - 0.15 I - 0.86) - 0.1 I ( - 1.29) - 0.01 ( - 0.19) 1.45 0.80 - 0.23 ( - 2.30) 0.99 (35.09) - 0.21 ( - 5.17) 0.94 I.20 (37.18) I.15 (59.15) (49.45) I .03 (65.09) 0.9 I - 0.48 ( - 17.92) 0.11 (1.64) 0.09 (3.92) 0.95 Low 1.04 0.96 0.06 (1.17) - 0.14 ( - 6.49) 2 1.01 - 0.01 ( - 0.66) 0.02 (0.72) 0.96 (85.34) (77.07) I .02 (64.43) 0.99 (69.14) 0.9 I (58.42) 0.93 - 0.03 ( - 0.44 0.14 (5.09) 3 0.04 (0.64) 0.94 4 0.17 (2.45) 0.9 1 0.04 (0.59) 0.98 (66.51) - 0.06 ( - 2.80) - 0.05 ( - 1.77) 0.30 (12.00) 0.91 0.24 (2.22) 0.72 (30.16) 0.73 - 0.01 (0.16) 0.85 (40.08) 0.54 (15.04) 0.8-t High yield (D P) for year f is the described in Fama and price ratio (E/P) for year t is year t - 1. divided by market items. plus income-statement for D.‘P or E’P are determined Regression t-statistics “Portfolios dividends paid from July of I - 1 to June oft [measured French (1988)]. divided by market equity in June of r - 1. The earnings the equity income for the fiscal year ending in calendar December of t - I. Equity income is income before extraordinary deferred taxes, minus preferred dividends. The quintile breakpoints using only NYSE firms with positive dividends or earnings. See table 7 for definitions of R,Lf-RF. are formed in June of year r, 1963-1991. The dividend using the procedure equity in are in parentheses. S.Wf3. and H&IL. Table Sharpe-Lintner largely unexplained. factor regressions (t = - 2.35) for the lowest-E/P failure of the one-factor positive-E/P portfolios explain the positive relation In contrast, explain returns leaves no residual E/P effect in average returns. The three-factor intercepts for the five positive-E P portfolios to 1.01). Interestingly, the three-factor regressions in the average returns on the positive-E/P the book-to-market factor HML. The lowest positive-E slope, - 0.50, like those produced by portfolios in the lowest-BEIME the three-factor regressions in table 6. The highest-E/P slope, 0.67, like those for portfolios in the highest-BE Table 1 confirms that there is also a positive relation for our 25 portfolios formed on size and BE/ME. It confirms model leaves the relation For the positive-E/P increase the evidence in Basu between portfolios. (1983) that the return one-factor and E,‘P in the one- per month average the intercepts - 0.20% monotonically, quintile from to 0.46% (t = 3.69) for the highest. The model has a simple explanation. are all close to 1.0, so the one-factor between E/P and average return. the three-factor model that uses RM-RF, The market j?s for the model cannot SMB. and HML to are within 0.1 of0 ( C’S from - 0.12 say that the increasing portfolios is due to their loadings P quintile has an HML pattern on quintile in has an HML in table 6. quintile .UE quintile between E’P and BE/ME

Fama and French stocks. that is. stocks vvith persistently in high stock prices relative to book equity. High BE .\fE. on the other hand, is associated with distress. that is. persistently result in low stock prices. The loadings on H.1fL in the three-factor of table II then say that low-E P stocks have the low average returns typical of (low-BE.‘.CfE) growth stocks, while high-E P stocks returns associated with distress (high-BE’.!IE). The negative-E. P portfolio produces three-factor model. In spite of the portfolio’s per month). the three-factor model says that its average month too low, given its strong loadings portfolios in table 6) and H.LfL (0.16. like the higher-BE 6). In other words, according to the three-factor this portfolio should be higher because relatively depressed. stocks. The three-factor portfolio is, however. only I.65 standard In short. E/P portfolios produce a strong spread in average returns, seems to be absorbed by the three common portfolios are thus interesting corroboration common risk factors in stock returns related to size and book-to-market and (bj R!tf-RF. S.LfB. and H.CfL. the mimicking BE .tfE risk factors. capture the cross-section (1991bl find that lovv BE .tfE is characteristic high earnings of growth on book equity that result low earnings on book equity that regressions have the high average the only hint of evidence high average excess return (0.72% return on .S.!fB (1.13. like the smallest-size .!fE portfolios model. the average return on its return behaves like those of small. intercept for the negative-E errors from 0. against the is O.?‘!/o per in table P which risk factors in stock returns. The E,P of our inferences that (a) there are equity. returns for market. size, and of average stock returns. Table I 1 shows that. as in Keim ( 3983). average returns on portfolios on D P are also U-shaped; they drop from the zero-dividend lowest positive-D P portfolio. and then increase across the positive-D, P port- folios. The U-shaped pattern. and the overall spread hovvever. much weaker for the D P portfolios Table I1 also confirms Keim’s (1983) finding Lintner model leaves a pattern in average returns on dividends. The one-factor intercepts to the highest-D P portfolios. This suggests that pre-tax returns stocks must be higher to equalize after-tax But the apparent tax effect in average three-factor regressions that use R.Lf-RF, The three-factor intercepts for the five positive-D show no relation to D P. The three-factor pattern in the average returns on the positive-D increasing pattern in their loadings on the book-to-market lovvest-(positive:)-D P quintile has a strong negative H.CfL slope, - 0.18, and the formed to the portfolio in average returns, are, than for the E/P portfolios. that the one-factor that looks like a tax penalty increase monotonically Sharpe- from the lowest- on higher-D,P returns. does not survive S.LfB, and H.LfL to explain P portfolios regressions say that the increasing P portfolios risk-adjusted returns in the rc,urns. are close to 0 and is due to the factor H.bfL. The

highest-D. P portfolio model says that low-D P stocks have the low average returns typical of growth stocks, whereas high-D P stocks have the high average returns relative distress. Table 1 confirms that there is also a positive relation D P and BE’ME for our 25 portfolios The zero-dividend portfolio produces three-factor model. The three-factor model says that the high average excess return on this portfolio (0.48% per month) is 0.13% too low (r = - 2.30). given its strong loading (0.99) on SSIB, the mimicking other words, because the return on the zero-dividend return on a portfolio of small stocks. the three-factor return on this portfolio is not high enough. But the three-factor zero-dividend portfolio is small in practical model produces intercepts for the five positive-D,‘P to 0. both statistically and practically. portfolios are consistent with our inference that the three stock-market R,tf-RF, .SICIB, and H,LlL. capture the cross-section has a strong positive slope. 0.54. Again. the three-factor associated with between formed on size and BE,‘JIE. the strongest evidence against the return for the size factor. In portfolio model says that the high intercept terms. Moreover, portfolios We conclude that. overall. varies like the for the the three-factor that are all close the Dif factors, of average stock returns. 7. Interpretation and applications This paper studies the common tests whether these shared risks capture There are at least five common produce common variation bonds, the stock-market corporate bonds. The stock and bond markets are linked, however. through two shared term-structure factors. risk factors in stock and bond returns the cross-section factors in returns. Three stock-market in stock returns. Except for low-grade factors have little role in returns and of average returns. factors corporate on government and 7.1. Interprrttrtion factors, R,LfO, SJIB. and H.LlL. and with the two term-structure in table 8 that use RXIO. .S.!lB. H.ML, stock and bond returns Table 2 shows that the three stock-market are largely uncorrelated factors, TER.Ll and DEF. The regressions TER.Ll, and DEF to explain summary of the separate roles of the five factors in the volatility the cross-section of average returns. The 25 stock portfolios produce slopes on the orthogonalized R.CfO. that are all around 1. Thus R,LiO. which has a standard 3.55% per month, accounts for similar common the stock portfolios. The average RJlO return. 0.50% per month also a common part of the average excess returns slopes for stocks are all around 1. we can interpret with one another thus provide of returns and in a good market return, deviation in the returns on all (r = 2.61) is on stocks. Since the RMO the average R.ifO return as of variation

the premium general stock-market For stocks, the slopes on the two term-structure around 0.8. The standard month (table 2). then say that TERAI accounts returns on all the stock portfolios, DEF captures less common returns are only 0.06% and 0.02% per month. so they explain almost none of the average excess returns on stocks. But the expected TERM and DEF returns vary through time with business conditions (1991)]. Thus TER:bI and DEF produce expected bond and stock returns. Except for low-grade corporate bonds. the common variation in bond returns identified table 8. Thus the low average excess returns average TERM and DEF returns. R’ values near I in tables 3 and 8 say that TERM and DEF explain almost all the variation corporate returns. Since the TERM and DEF slopes (around I) are similar to the slopes for stocks (around stocks share almost all the variation Stocks, however, have substantial additional market factors. In the five-factor regressions of table 8, the slopes on RMO. TER.\ I, and DEF do not vary much across the 25 stock portfolios. TER,V, and DEF in stock returns are captured RM-RF, in table 7. The slopes on RM-RF the slopes on R.tIO in table 8. Thus, like RMO, TERM, and DEF, the excess market return does not explain the strong cross-sectional stock returns and their volatilities (table 2). That job is left to S,LIB and HML, the mimicking returns for the risk factors related to size and book-to-market equity. The slopes on .S.CfB in table 8 exceed 1.5 for portfolios quintile, and they drop to around 0.3 for portfolios The standard deviation of S,LIB is large. 2.89% per month. size-related factor in returns is thus important returns are much more variable than big-stock SMB return is only 0.2746 per month range from 1.92 to 0.20, however, so the predicted across the 25 stock portfolios due to the size-related per month. The slopes on H.UL in table 8 range from about lowest-book-to-market quintile to values near 0 in the highest-BE;.LIE HML thus tends to increase the volatility for being a stock (rather risk. than a one-month bill) and sharing returns in table 8 are all of TER,CI and DEF. 3.01% and 1.60% per for similar on the order of that captured variation in returns. The average TER:CI and DEF deviations variation by R&IO, while in the [Fama interesting and French (1989) and Chen time-series variation in TERM and DEF capture in the five-factor regressions on bonds fit nicely with the low almost all of in high-grade for corporate 0.8), we can infer that corporate bond common volatility (Aaa. Aa, A) bonds in high-grade returns. due to stock- As a result, the roles of RMO, well by the excess market return, in table 7 are, however, the same as differences in average in the smallest-size in the biggest-size quintile. The common why small-stock (table 2). The average in explaining returns (t = 1.73). The S.CIB slopes in table 8 spread in average risk factor is large, 0.46% returns - 1 for portfolios in the quintile. of low-BE/ME stock returns. Table 2