Journal of Financial Economics 33 (1993) 3-56. North-Holland
Common risk factors in the returns on stocks and bonds*
Eugene F. Fama and Kenneth R. French Unirrrsit.v 01 Chicayo. Chiccup. I .L 60637, C;S;L
Received July 1992. final version received September 1992
This paper identities five common risk factors in the returns on stocks and bonds. There are three stock-market factors: an overall market factor and factors related to firm size and book-to-market equity. There are two bond-market factors. related to maturity and default risks. Stock returns have shared variation due to the stock-market factors, and they are linked to bond returns through shared variation in the bond-market factors. Except for low-grade corporates. the bond-market factors capture the common variation in bond returns. Most important. the five factors seem to explain average returns on stocks and bonds.
1. Introduction
The cross-section of average returns on U.S. common stocks shows little relation to either the market /Is of the Sharpe (1964tLintner (1965) asset- pricing model or the consumption ps of the intertemporal asset-pricing model of Breeden (1979) and others. [See, for example, Reinganum (198 1) and Breeden, Gibbons, and Litzenberger (1989).] On the other hand, variables that have no special standing in asset-pricing theory show reliable power to explain the cross-section of average returns. The list of empirically determined average- return variables includes size (ME, stock price times number of shares), leverage, earnings/price (E/P), and book-to-market equity (the ratio of the book value of a firm¡¯s common stock, BE, to its market value, ME). [See Banz (1981). Bhandari (1988). Basu (1983). and Rosenberg, Reid, and Lanstein (19853.1
Correspondence to: Eugene F. Fama. Graduate School of Business. University of Chicago, 1101 East 58th Street. Chicago. IL 60637, USA.
*The comments of David Booth, John Cochrane. Sai-fu Chen, Wayne Ferson. Josef Lakonishok. Mark Mitchell, G. William Schwert. Jay Shanken. and Rex Sinquefield are gratefully acknowledged. This research is supported by the National Science Foundation (Fama) and the Center for Research in Securities Prices (French).
030%405X.93.S05.00 C 1993-Elsevier Science Publishers B.V. Ail rights reserved
4 E.F. Fuma and K.R. French. Common risk f&run in r~ock and bond remrns
Fama and French (1992a) study the joint roles of market 8, size, E;P, leverage, and book-to-market equity in the cross-section of average stock returns. They find that used alone or in combination with other variables, /I (the slope in the regression of a stock¡¯s return on a market return) has little information about average returns. Used alone, size, E/P, leverage, and book-to-market equity have explanatory power. In combinations, size (ME) and book-to-market equity (BE/ME) seem to absorb the apparent roles of leverage and E;¡®P in average returns. The bottom-line result is that two empirically determined variables, size and book-to-market equity, do a good job explaining the cross-section of average returns on NYSE, Amex, and NASDAQ stocks for the 1963-1990 period.
This paper extends the asset-pricing tests in Fama and French (1992a) in three ways.
(a) We expand the set of asset returns to be explained. The only assets con- sidered in Fama and French (1992a) are common stocks. If markets are integrated, a single model should also explain bond returns. The tests here include U.S. government and corporate bonds as well as stocks.
(b) We also expand the set of variables used to explain returns. The size and book-to-market variables in Fama and French (1992a) are directed at stocks. We extend the list to term-structure variables that are likely to play a role in bond returns. The goal is to examine whether variables that are important in bond returns help to explain stock returns, and vice versa. The notion is that if markets are integrated, there is probably some overlap between the return processes for bonds and stocks.
(c) Perhaps most important, the approach to testing asset-pricing models is different. Fama and French (1992a) use the cross-section regressions of Fama and MacBeth (1973): the cross-section of stock returns is regressed on variables hypothesized to explain average returns. It would be difficult to add bonds to the cross-section regressions since explanatory variables like size and book-to-market equity have no obvious meaning for government and corporate bonds.
This paper uses the time-series regression approach of Black, Jensen, and Scholes (1972). Monthly returns on stocks and bonds are regressed on the returns to a market portfolio of stocks and mimicking portfolios for size, book-to-market equity (BE/¡®ME), and term-structure risk factors in returns. The time-series regression slopes are factor loadings that, unlike size or BE/ME, have a clear interpretation as risk-factor sensitivities for bonds as well as for stocks.
The time-series regressions are also convenient for studying two important asset-pricing issues.
(a) One of our central themes is that if assets are priced rationally, variables that are related to average returns, such as size and book-to-market equity, must proxy for sensitivity to common (shared and thus undiversiliable) risk factors in
E.F. Famu und K.R. French. Common risk factorsin stock and bond returns 5
returns. The time-series regressions give direct evidence on this issue. In particu- lar, the slopes and R¡¯ values show whether mimicking portfolios for risk factors related to size and BE/lVCIEcapture shared variation in stock and bond returns not explained by other factors.
(b) The time-series regressions use excess returns (monthly stock or bond returns minus the one-month Treasury bill rate) as dependent variables and either excess returns or returns on zero-investment portfolios as explanatory variables. In such regressions, a well-specified asset-pricing model produces intercepts that are indistinguishable from 0 [Merton (1973)J The estimated intercepts provide a simple return metric and a formal test of how well different combinations of the common factors capture the cross-section of average returns. Moreover, judging asset-pricing models on the basis of the intercepts in excess-return regressions imposes a stringent standard. Competing models are asked to explain the one-month bill rate as well as the returns on longer-term bonds and stocks.
Our main results are easy to summarize. For stocks, portfolios constructed to mimic risk factors related to size and BE/ME capture strong common variation in returns, no matter what else is in the time-series regressions. This is evidence that size and book-to-market equity indeed proxy for sensitivity to common risk factors in stock returns. Moreover, for the stock portfolios we examine, the intercepts from three-factor regressions that include the excess market return and the mimicking returns for size and BE/ME factors are close to 0. Thus a market factor and our proxies for the risk factors related to size and book- to-market equity seem to do a good job explaining the cross-section of average stock returns.
The interpretation of the time-series regressions for stocks is interesting. Like the cross-section regressions of Fama and French (1992a), the time-series regres- sions say that the size and book-to-market factors can explain the differences in average returns across stocks. But these factors alone cannot explain the large difference between the average returns on stocks and one-month bills. This job is left to the market factor. In regressions that also include the size and book- to-market factors, all our stock portfolios produce slopes on the market factor that are close to 1.The risk premium for the market factor then links the average returns on stocks and bills.
For bonds, the mimicking portfolios for the two term-structure factors (a term premium and a default premium) capture most of the variation in the returns on our government and corporate bond portfolios. The term-structure factors also ¡®explain¡¯ the average returns on bonds, but the average premiums for the term-structure factors, like the average excess bond returns, are close to 0. Thus, the hypothesis that all the corporate and government bond portfolios have the same long-term expected returns also cannot be rejected.
The common variation in stock returns is largely captured by three stock- portfolio returns, and the common variation in bond returns is largely explained
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by two bond-portfolio returns. The stock and bond markets. however, are far from stochastically segmented. Used alone in the time-series regressions. the term-structure factors capture strong variation in stock returns; indeed, the slopes on the term-structure factors in the regressions for stocks are much like those for bonds. But interestingly. when stock-market factors are also included in the regressions, all of our stock portfolios load in about the same way on the two term-structure factors and on the market factor in returns. As a result, a market portfolio of stocks captures the common variation in stock returns associated with the market factor and the two term-structure factors.
The stochastic links between the bond and stock markets do. however. seem to come largely from the term-structure factors. Used alone. the excess market return and the mimicking returns for the size and book-to-market equity factors seem to capture common variation in bond returns. But when the two term- structure factors are included in the bond regressions, the explanatory power of the stock-market factors disappears for all but the low-grade corporate bonds.
In a nutshell, our results suggest that there are at least three stock-market factors and two term-structure factors in returns. Stock returns have shared variation due to the three stock-market factors, and they are linked to bond returns through shared variation in the two term-structure factors. Except for low-grade corporate bonds, only the two term-structure factors seem to produce common variation in the returns on government and corporate bonds.
The story proceeds as follows. We first introduce the inputs to the time-series regressions: the explanatory variables and the returns to be explained (sections Z and 3). We then use the regressions to attack our two central asset-pricing issues: how do different combinations of variables capture (a) the common variation through time in the returns on bonds and stocks (section 4) and (b) the cross-section of average returns (section 5).
2. The inputs to the time-series regressions
The explanatory variables in the time-series regressions include the returns on a market portfolio of stocks and mimicking portfolios for the size. book- to-market, and term-structure factors in returns. The returns to be explained are for government bond portfolios in two maturity ranges, corporate bond port- folios in five rating groups, and 25 stock portfolios formed on the basis of size and book-to-market equity.
The explanatory variables fall into two sets, those likely to be important for capturing variation in bond returns and those likely to be important for stocks. Segmenting the explanatory variables in this way sets up interesting tests of
whether factors important in stock returns help to explain bond returns and vice versa.
2.1 .I. Bond-mnrket factors
One common risk in bond returns arises from unexpected changes in interest rates. Our proxy for this factor, TERM, is the difference between the monthly long-term government bond return (from Ibbotson Associates) and the one- 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 returns due to shifts in interest rates.
For corporate bonds. shifts in economic conditions that change the likelihood of default give rise to another common factor in returns. Our proxy for this default factor, DEF, is the difference between the return on a market portfolio of long-term corporate bonds (the Composite portfolio on the corpo- rate bond module of Ibbotson Associates) and the long-term government bond return.
Chen. Roll, and Ross (1986) use TERM and a variable like DEF to help explain the cross-section of average returns on NYSE stocks. They use the Fama and MacBeth (1973) cross-section regression approach: the cross-section of average stock returns is explained with the cross-section of slopes from time- 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 returns. and TER.Cl sometimes has power. We confirm that the tracks of TER,LI and DEF show up clearly in the time-series variation of stock returns. We also find that the two variables dominate the common variation in government and corporate bond returns. In contrast to the cross-section regressions of Chen. Roll, and Ross, however, our time-series regressions say that the average premiums for DEF and TERM risks are too small to explain much variation in the cross-section of average stock returns. [Shanken and Weinstein (1990) make a similar point.]
2.1.2. Stock-market fuctors
Motiuztion – Although size and book-to-market equity seem like ad hoc variables for explaining average stock returns, we have reason to expect that they proxy for common risk factors in returns. In Fama and French (1992b) we document that size and book-to-market equity are related to economic funda- mentals. Not surprisingly, firms that have high BE/ME (a low stock price relative to book value) tend to have low earnings on assets, and the low earnings persist for at least five years before and five years after book-to-market equity is
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measured. Conversely. low BE. .CfE (a high stock price relative to book value) is associated with persistently high earnings.
Size is also related to profitability. Controlling for book-to-market equity, small firms tend to have lower earnings on assets than big firms. The size effect in earnings, however, is largely due to the 1980s. Until 1981. controlling for BE;.LfE, small firms are only slightly less profitable than big firms. But for small firms, the 198G1982 recession turns into a prolonged earnings depression. For some reason, small firms do not participate in the economic boom of the middle and late 1980s.
The fact that small firms can suffer a long earnings depression that bypasses big firms suggests that size is associated with a common risk factor that might explain the negative relation between size and average return. Similarly. the relation between book-to-market equity and earnings suggests that relative profitability is the source of a common risk factor in returns that might explain the positive relation between BE:.CfE and average return. Measuring the com- mon variation in returns associated with size and BE,hfE is a major task of this paper.
The Buikfiny Blocks – To study economic fundamentals, Fama and French (1992b) use six portfolios formed from sorts of stocks on .LfE and BE ¡®IlIE. We use the same six portfolios here to form portfolios meant to mimic the underly- ing risk factors in returns related to size and book-to-market equity. This ensures a correspondence between the study of common risk factors in returns carried out here and our complementary study of economic fundamentals.
In June of each year t from 1963 to 1991, all NYSE stocks on CRSP are ranked on size (price times shares). The median NYSE size is then used to split NYSE, Amex. and (after 1972) NASDAQ stocks into two groups. small and big (S and B). Most Amex and NASDAQ stocks are smaller than the NYSE median, so the small group contains a disproportionate number of stocks 13,616 out of 4,797 in 1991). Despite its large number of stocks, the small group contains far less than half (about 8% in 1991) of the combined value of the two size groups.
We also break NYSE, Amex, and NASDAQ stocks into three book-to- market equity groups based on the breakpoints for the bottom 30% (Lo\c), middle 40% (LCfediurn).and top 30% (High) of the ranked values of BE¡¯.\fE for NYSE stocks. We define book common equity, BE. as the COMPUSTAT book value of stockholders¡¯ equity, plus balance-sheet deferred taxes and investment tax credit (if available), minus the book value of preferred stock. Depending on availability, we use the redemption, liquidation, or par value (in that order) to estimate the value of preferred stock. Book-to-market equity, BE,¡®,CfE. is then book common equity for the fiscal year ending in calendar year t – 1,divided by market equity at the end of December oft – 1. We do not use negative-BE firms, which are rare before 1980, when calculating the breakpoints for BE) .bfE or when forming the size-BE$.LfE portfolios. Also. only firms with ordinary
E.F. Fama und K.R. Fwnch. Common rusk /Lcrorr in srock and bond renum 9
common equity (as classified by CRSP) are included in the tests. This means that ADRs, REITs, and units of beneficial interest are excluded.
Our decision to sort firms into three groups on BE,¡®ICIE and only two on ME follows the evidence in Fama and French (1992a) that book-to-market equity has a stronger role in average stock returns than size. The splits are arbitrary, however, and we have not searched over alternatives. The hope is that the tests here and in Fama and French (1992b) are not sensitive to these choices. We see no reason to argue that they are.
We construct six portfolios (S/L, S;,V, S,!H. B,¡®L, B,!M, B/H) from the intersec- tions of the two ,bfE and the three BE!hfE groups. For example. the S/L portfolio contains the stocks in the small-%fE group that are also in the low-BE/ME group, and the BI¡¯H portfolio contains the big-.CIE stocks that also have high BE,MEs. Monthly value-weighted returns on the six portfolios are calculated from July of year t to June oft + 1. and the portfolios are reformed in June of t + 1. We calculate returns beginning in July of year t to be sure that book equity for year c – 1 is known.
To be included in the tests, a firm must have CRSP stock prices for December of year t – 1 and June of t and COMPUSTAT book common equity for year t – 1. Moreover, to avoid the survival bias inherent in the way COMPUSTAT adds firms to its tapes [Banz and Breen (1986)], we do not include firms until they have appeared on COMPUSTAT for two years. (COMPUSTAT says it rarely includes more than two years of historical data when it adds firms).
Size- Our portfolio S,LfB (small minus big), meant to mimic the risk factor in returns related to size, is the difference, each month, between the simple average of the returns on the three small-stock portfolios (SjL, S/.Cf, and S,,H) and the simple average of the returns on the three big-stock portfolios (B; L. B/&f, and B/H). Thus, ShfB is the difference between the returns on small- and big-stock portfolios with about the same weighted-average book-to-market equity. This difference should be largely free of the influence of BE/ME, focusing instead on the different return behaviors of small and big stocks.
BE//LIE – The portfolio HhfL (high minus low). meant to mimic the risk factor in returns related to book-to-market equity, is defined similarly. HML is the difference, each month, between the simple average of the returns on the two high-BE/ME portfolios (S,¡®H and B/H) and the average of the returns on the two low- BE/ME portfolios (S;L and B/L). The two components of H,tlL are returns on high- and low-BE,¡®&fE portfolios with about the same weighted-average size. Thus the difference between the two returns should be largely free of the size factor in returns, focusing instead on the different return behaviors of high- and low-BEllME firms. As testimony to the success of this simple procedure, the correlation between the 1963-1991 monthly mimicking returns for the size and book-to-market factors is only – 0.08.
True mimicking portfolios for the common risk factors in returns minimize the variance of firm-specific factors. The six size-BEilV E portfolios in S&fB and
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H,CfL are value-weighted. Using value-weighted components is in the spirit of minimizing variance, since return vririances are negatively related to size (table 1. below). More important, using value-weighted components results in mimicking portfolios that capture the different return behaviors of small and big stocks. or high- and low-BEl.VE stocks, in a way that corresponds to realistic investment opportunities.
Market – Finally. our proxy for the market factor in stock returns is the excess market return, R.M-RF. R&l is the return on the value-weighted portfolio of the stocks in the six size-BE¡¯ME portfolios, plus the negative-BE stocks excluded from the portfolios. RF is the one-month bill rate.
22. The returns to he espkuiwd
Bonds -The set of dependent variables used in the time-series regressions includes the excess returns on two government and five corporate bond port- folios. The government bond portfolios (from CRSP) cover maturities from 1to 5 years and 6 to 10 years. The five corporate bond portfolios, for Moody¡¯s rating groups Aaa, Aa. A, Baa. and LG (low-grade, that is, below Baa) are from the corporate bond module of Ibbotson Associates (provided to us by Dimensional Fund Advisors).
Stocks – For stocks. we use excess returns on 25 poitfolios, formed on size and book-to-market equity. as dependent variables in the time-series regressions. We use portfolios formed on size and BE/ME because we seek to determine whether the mimicking portfolios SMB and HAIL capture common factors in stock returns related to size and book-to-market equity. Portfolios formed on size and BE.¡®AIE will also produce a wide range of average returns to be explained by competing asset-pricing equations [Fama and French (1992a)]. Later, however, we use portfolios formed on E.P (earnings/price) and DiP (dividend/price). variables that are also informative about average returns [e.g.. Keim (1988)], to check the robustness of our results on the ability of our explanatory factors to capture the cross-section of average returns.
The 25 size-BE,¡®,LfE portfolios are formed much like the six size-BE;&LIE portfolios discussed earlier. In June of each year t we sort NYSE stocks by size and (independently) by book-to-market equity. For the size sort. .LIE is mea- sured at the end of June. For the book-to-market sort, ME is market equity at the end of December of c – 1. and BE is book common equity for the fiscal year ending in calendar year r – 1. We use NYSE breakpoints for ME and BE;.tfE to allocate NYSE. Amex. and (after 1972) NASDAQ stocks to five size quintiles and five book-to-market quintiles. We construct 25 portfolios from the intersec- tions of the size and BE, ,LfE quintiles and calculate value-weighted monthly returns on the portfolios from July off to June of r + 1. The excess returns on these 25 portfolios for July 1963 to December 1991 are the dependent variables for stocks in the time-series regressions.
E.F. Fumo and R.R. French, Commor? rtsk /actors in srock und bond rerurns I1
Descriptive statistics for 25 stock portfolios formed on size and book-to-market equity: 1963-1991. 29 years.¡¯
Book-to-market equity (BE, ME) quintiies
quintile Low 1 3 4 High
Low z 3 4 High
89.7 89.3 209.3 211.9 535.1 537.4
3583.7 2885.8
_¡®0 ._¡¯ 19.4 15.1
89.3 89.9 88.5 210.8 214.8 210.7 545.4 551.6 538.7
2819.5 2700.5 1337.9
0.30 0.62 0.84 0.31 0.60 0.83 0.31 0.60 0.84 0.31 0.6 1 0.84 0.29 0.59 0.83
1.09 1.80 1.09 1.71
Small 2.42 7.24 1
; 5.2910 8.7631 4 5.85 8.94 Big 6.00 9.07
10.4163 I1l0.6.915 10.45 11.64 10.90 12.45
190.278 11.39 13.92
l1..596 1.80 2.34
I.94 2.60 23.4053 43.0445
3.09 4.22 3.69 4.68
4.6285 4.6543 5.01 4.94 5.49 5.90
Average of annual averages of firm size
Average of annual
8. E ratios for portfolio
Average of annual percent of market value in portfolio
Average of annual number of firms in portfolio
0.69 0.49 0.92 0.71 1.78 1.36 3.95 3.01 30.13 15.87
0.46 0.65 1.26 2.71 12.85
0.48 0.64 0.6 I 0.55 1.14 0.82 2.4 I 1.50 10.44 4.61
428.0 121.6 102.7
90. I 93.6
276.6 263.8 191.5 512.7 94.0 86.7 79.8 71.3 78.3 73.0 64.5 45.9
Average of annual E¡¯P ratios (in percent) for portfolio
Average of annual D¡¯P ratios (in percent) for portfolio
68.9 60.7 53.1 63.7 51.7 44.0
¡°The 25 size-BE. ME stock portfolios are formed as follows. Each year t from 1963 to 1991 NYSE quinttle breakpoints for size (.UE. stock price times shares outstanding), measured at the end of June, are used to allocate NYSE. Amex. and NASDAQ stocks to five size quintiles. Similarly, NYSE quintile breakpoints for BE, ME are used to allocate NYSE. Amex. and NASDAQ stocks to five book-to-market equity quintiles. The 25 size-BE,¡®.LIE portfolios are formed as the intersections of the five size and the five BE. ME groups. Book equity. BE. is the COMPUSTAT book value of stockholders¡¯ equity, plus balance sheet deferred taxes and investment tax credits lif available). minus the book value of preferred stock. Depending on avjailability. we use the redemption. liquidation. or par value (in that order) to estimate the book value of preferred stock. Book- to-market equity. BE .ME. for a stock is BE for the fiscal year ending in calendar year r – 1. divided by ME at the end of December oft – 1.
A portfolio¡¯s book-to-market equity, BE,¡®XfE. for the portfolio formation year c is the sum of book equity. BE. for the firms in the portfolio for the fiscal year endmg in calendar year t – I, divided by the sum of their market equity. ME, in December oft – I. A portfolio¡¯s earnings/price ratio (E P) for year I is the sum ofequity income for the firms in the portfolio for the fiscal year ending in calendar year t – 1. divided by the sum of their market equity in December of r – 1. Equity income is income before extraordinary items, plus income- statement deferred taxes. minus preferred dividends. A portfolio¡¯s dividend yield (D P) for year t is the sum (across firms in the portfolio) of the dividends paid from July oft – 1to June of r. divided by the sum of market equity in June oft – I. We use the procedure described in Fama and French (1988) to estimate dividends.
The descriptive statistics are computed when the portfolio is formed in June ofeach year. 1963-1991, and are then averaged across the 29 years.
12 E.F. Fama und K.R. French. Common risk /&tom in stock md bond returns
Table 1 shows that, because we use NYSE breakpoints to form the 25 size-BE, ,CIE portfolios, the portfolios in the smallest size quintile have the most stocks (mostly small Amex and NASDAQ stocks). Although they contain many stocks, each of the five portfolios in the smallest size quintile is on average less than 0.70% of the combined value of stocks in the 25 portfolios. In contrast, the portfolios in the largest size quintile have the fewest stocks but the largest fractions of value. Together, the five portfolios in the largest JIE quintile average about 74% of total value. The portfolio of stocks in both the largest size and lowest BE/ME quintiles (big successful firms) alone accounts for more than 30% of the combined value of the 25 portfolios. And note that using all stocks, rather than just NYSE stocks, to define the size quintiles would result in an even more skewed distribution of value toward the biggest size quintile.
Table 1 also shows that in every size quintile but the smallest, both the number of stocks and the proportion of total value accounted for by a portfolio decrease from lower- to higher-BE/ME portfolios. This pattern has two causes. First, using independent size and book-to-market sorts of NYSE stocks to form portfolios means that the highest-BE/ME quintile is tilted toward the smallest stocks. Second, Amex and NASDAQ stocks, mostly small, tend to have lower book-to-market equity ratios than NYSE stocks of similar size. In other words, NYSE stocks that are small in terms of ME are more likely to be fallen angels (big firms with low stock prices) than small Amex and NASDAQ stocks.
3. The playing field
Table 2 summarizes the dependent and explanatory returns in the time-series regressions. The average excess returns on the portfolios that serve as dependent variables give perspective on the range of average returns that competing sets of risk factors must explain. The average returns on the explanatory portfolios are the average premiums per unit of risk (regression slope) for the candidate common risk factors in returns.
3.1. The dependent retwxs
Stocks – The 25 stock portfolios formed on size and book-to-market equity produce a wide range of average excess returns, from 0.32% to 1.05% per month. The portfolios also confirm the Fama-French (1992a) evidence that there is a negative relation between size and average return, and there is a stronger positive relation between average return and book-to-market equity. In all but the lowest-BE/ME quintile, average returns tend to decrease from the small- to the big-size portfolios. The relation between average return and book-to-market equity is more consistent. In every size quintile, average returns tend to increase with BE/:bfE, and the differences between the average returns
E.F. Fame und K. R. French. Common risk fk!ors m slack and bond rerurm 13
for the highest- and lowest-BE;¡®.CJE portfolios range from 0