THE JOURNAL OF FINANCE . VOL. LI, NO. 1 . MARCH 1996Multifactor Explanations of AssetPricing AnomaliesEUGENE F. FAMA and KENNETH R. FRENCH*ABSTRACTPrevious work shows that average returns on common stocks are related to firmcharacteristics like size, earnings/price, cash fiow/price, book-to-market equity, pastsales growth, long-term past return, and short-term past return. Because thesepatterns in average returns apparently are not explained by the CAPM, they arecalled anomalies. We find that, except for the continuation of short-term returns, theanomalies largely disappear in a three-factor model. Our results are consistent withrational ICAPM or APT asset pricing, but we also consider irrational pricing and dataproblems as possible explanations.RESEARCHERS HAVE IDENTIFIED MANY patterns in average stock returns. For ex-ample, DeBondt and Thaler (1985) find a reversal in long-term returns; stockswith low long-term past returns tend to have higher future returns. In con-trast, Jegadeesh and Titman (1993) find that short-term returns tend tocontinue; stocks with higher returns in the previous twelve months tend tohave higher future returns. Others show that a firm's average stock return isrelated to its size (ME, stock price times numher of shares), book-to-market-equity (BE/ME, the ratio of the book value of common equity to its marketvalue), earnings/price (E/P), cash flow/price (C/P), and past sales growth. (Banz(1981), Basu (1983), Rosenberg, Reid, and Lanstein (1985), and Lakonishok,Shleifer and Vishny (1994).) Because these patterns in average stock returnsare not explained by the capital asset pricing model (CAPM) of Sharpe (19)and Lintner (1965), they are typically called anomalies.This paper argues that many of the CAPM average-return anomalies arerelated, and they are captured hy the three-factor model in Fama and French(FF 1993). The model says that the expected return on a portfolio in excess ofthe risk-free rate [E(i?j) - Rj] is explained by the sensitivity of its return tothree factors: (i) the excess return on a broad market portfolio {RM ~ ^f)\\ (ii)the difference between the return on a portfolio of small stocks and the returnon a portfolio of large stocks (SMB, small minus big); and (iii) the differencebetween the return on a portfolio of high-book-to-market stocks and the returnon a portfolio of low-book-to-market stocks (HML, high minus low). Specifi-cally, the expected excess return on portfolio i is.-Rf= blEiRu) - Rf] + SiE(SMB) + /liE(HML), (1)* Fama is from the Graduate School of Business, University of Chicago, and French is from theYale School of Management, The comments of Clifford Asness, John Cochrane, Josef Lakonishok,G. William Schwert, and Rene Stulz are gratefully acknowledged.5556 The Journal of Financewhere E(i?;if) -factor sensitivities or loadings, 6j, s^, and Rf, E(SMB), and E(HML) are expected premiums, and theregression. hi, are the slopes in the time-seriesRi- Rf= a; + biiRu - Rf) + SiSMB + /i;HML -I- e^. (2)HML proxy for relative distress. Weak firms with persistently low earningsFama and French (1995) show that hook-to-market equity and slopes ontend to have high BE/ME and positive slopes on HML; strong firms withpersistently high earnings have low BE/ME and negative slopes on HML.Using HML to explain returns is thus in line with the evidence of Chan andChen (1991) that there is covariation in returns related to relative distress thatis not captured hy the market return and is compensated in average returns.Similarly, using SMB to explain returns is in line with the evidence of Huher-man and Kandel (1987) that there is covariation in the returns on small stocksthat is not captured hy the market return and is compensated in averagereturns.variation in average stock returns. FF (1993) show that the model is a goodThe three-factor model in (1) seems to capture much of the cross-sectionaldescription of returns on portfolios formed on size and BE/ME. FF (1994) usethe model to explain industry returns. Here we show that the three-factormodel captures the returns to portfolios formed on E/P, C/P, and sales growth.In a nutshell, low E/P, low C^, and high sales growth are typical of strongfirms that have negative slopes on HML. Since the average HML return isstrongly positive (ahout 6 percent per year), these negative loadings, which aresimilar to the HML slopes for low-BE/ME stocks, imply lower expected returnsin (1). Conversely, like high-BE/ME stocks, stocks with high E/P, high C/P, orlow sales growth tend to load positively on HML (they are relatively dis-tressed), and they have higher average returns. The three-factor model alsocaptures the reversal of long-term returns documented hy DeBondt and Thaler(1985). Stocks with low long-term past returns (losers) tend to have positiveSMB and HML slopes (they are smaller and relatively distressed) and higherfuture average returns. Conversely, long-term winners tend to he strong stocksthat have negative slopes on HML and low future returns.turns documented hy Jegadeesh and Titman (1993). Like long-term losers,Equation (1), however, cannot explain the continuation of short-term re-stocks that have low short-term past returns tend to load positively on HML;like long-term winners, short-term past winners load negatively on HML. As itdoes for long-term returns, this pattern in the HML slopes predicts reversalrather than continuation for future returns. The continuation of short-termreturns is thus left unexplained hy our model.in (1) and (2), with intercepts in (2) equal to 0.0, is a parsimonious descriptionAt a minimum, the available evidence suggests that the three-factor modelof returns and average returns. The model captures much of the variation inthe cross-section of average stock returns, and it ahsorhs most of the anomaliesthat have plagued the CAPM. More aggressively, we argue in FF (1993, 1994,Multifactor Explanations of Asset Pricing Anomalies 571995) that the empirical successes of (1) suggest that it is an equilibriumpricing model, a three-factor version of Merton's (1973) intertemporal CAPM(ICAPM) or Ross's (1976) arbitrage pricing theory (APT). In this view, SMBand HML mimic combinations of two underlying risk factors or state variablesof special hedging concern to investors.Our aggressive interpretation of tests of (1) has produced reasonable skep-ticism, much of it centered on the premium for distress (the average HMLreturn). Kothari, Shanken, and Sloan (1995) argue that a substantial part ofthe premium is due to survivor bias; the data source for book equity (COM-PUSTAT) contains a disproportionate number of high-BE/ME firms that sur-vive distress, so the average return for high-BE/ME firms is overstated. An-other view is that the distress premium is just data snooping; researchers tendto search for and fixate on variables that are related to average return, butonly in the sample used to identify them (Black (1993), MacKinlay (1995)). Athird view is that the distress premium is real but irrational, the result ofinvestor over-reaction that leads to underpricing of distressed stocks andoverpricing of growth stocks (Lakonishok, Shleifer, and Vishny (1994), Haugen(1995)).Section VI discusses the competing stories for the successes of the three-factor model. First, however. Sections I to V present the evidence that themodel captures most of the average-return anomalies of the CAPM.I. Tests on the 25 FF Size-BE/ME PortfoliosTo set the stage. Table I shows the average excess returns on the 25 Fama-French (1993) size-BE/ME portfolios of value-weighted NYSE, AMEX, andNASD stocks. The table shows that small stocks tend to have higher returnsthan big stocks and high-book-to-market stocks have higher returns thanlow-BE/ME stocks.Table I also reports estimates of the three-factor time-series regression (2).If the three-factor model (1) describes expected returns, the regression inter-cepts should be close to 0.0. The estimated intercepts say that the model leavesa large negative unexplained return for the portfolio of stocks in the smallestsize and lowest BE/ME quintiles, and a large positive unexplained return forthe portfolio of stocks in the largest size and lowest BE/ME quintiles. Other-wise the intercepts are close to 0.0.The F-test of Gibbons, Ross, and Shanken (GRS 19) rejects the hypothesisthat (1) explains the average returns on the 25 size-BE/ME portfolios at the0.004 level. This rejection of the three-factor model is testimony to the explan-atory power of the regressions. The average of the 25 regressioh 2?^ is 0.93, sosmall intercepts are distinguishable from zero. The model does capture most ofthe variation in the average returns on the portfolios, as witnessed by thesmall average absolute intercept, 0.093 percent (about nine basis points) permonth. We show next that the model does an even better job on most of theother sets of portfolios we consider.58 The Journal of FinanceA comment on methodology is necessary. In the time-series regression (2),variation through time in the expected premiums E(i?2Vf) - Rf, E(SMB), andE(HML) in (1) is embedded in the explanatory returns, R^ - Rf, SMB, andHML. Thus the regression intercepts are net of (they are conditional on)variation in the expected premiums. We also judge that forming portfoliosTable ISummary Statistics and Three-Factor Regressions for SimpleMonthly Percent Excess Returns on 25 Portfolios Formed on Sizeand BE/ME: 7/63-12/93, 366 MonthsRf is the one-month Treasury hill rate ohserved at the heginning ofthe month (from CRSP). Theexplanatory returns R,^, SMB, and HML are formed as follows. At the end of June of each year t(1963-1993), NYSE, AMEX, and Nasdaq stocks are allocated to two groups (small or hig, S or B)hased on whether their June market equity (ME, stock price times shares outstanding) is helow orahove the median ME for NYSE stocks. NYSE, AMEX, and Nasdaq stocks are allocated in anindependent sort to three hook-to-market equity (BE/ME) groups (low, medium, or high; L, M, orH) hased on the hreakpoints for the hottom 30 percent, middle 40 percent, and top 30 percent ofthe values of BE/ME for NYSE stocks. Six size-BE/ME portfolios (S/L, S/M, S/H, B/L, B/M, B/H) aredefined as the intersections of the two ME and the three BE/ME groups. Value-weight monthlyreturns on the portfolios are calculated from July to the following June. SMB is the difference, eachmonth, hetween the average ofthe returns on the three small-stock portfolios (S/L, S/M, and S/H)and the average of the returns on the three hig-stock portfolios (B/L, B/M, and B/H). HML is thedifference hetween the average ofthe returns on the two high-BE/ME portfolios (S/H and B/H) andthe average of the returns on the two low-BE/ME portfolios (S/L and B/L). The 25 size-BE/MEportfolios are formed like the six size-BE/ME portfolios used to construct SMB and HML, exceptthat quintile hreakpoints for ME and BE/ME for NYSE stocks are used to allocate NYSE, AMEX,and Nasdaq stocks to the portfolios.BE is the COMPUSTAT hook value of stockholders' equity, plus balance sheet deferred taxes andinvestment tax credit (if availahle), minus the book value of preferred stock. Depending onavailability, we use redemption, liquidation, or par value (in that order) to estimate the hook valueof preferred stock. The BE/ME ratio used to form portfolios in June of year t is then hook commonequity for the fiscal year ending in calendar year t — 1, divided hy market equity at the end ofDecemher of i - 1. We do not use negative BE firms, which are rare prior to 1980, when calculatingthe hreakpoints for BE/ME or when forming the size-BE/ME portfolios. Also, only firms withordinary common equity (as classified by CRSP) are included in the tests. This means that ADR's,REIT's, and units of heneficial interest are excluded.The market return R^ is the value-weight return on all stocks in the size-BE/ME portfolios, plusthe negative BE stocks excluded from the portfolios.Book-to-Market Equity (BE/ME) QuintilesSizeLow234HighLow234HighPanel A:SummaryStatistic:3MeansSmall234BigStandard Deviations0.950.930.860.800.581.081.091.051.040.717.677.136.525.8.846.746.255.535.284.616.145.715.114.974.285.855.234.794.814.186.145.945.485.674.0.310.480.440.510.370.700.710.680.390.390.820.910.750.0.36Multifactor Explanations of Asset Pricing Anomalies59Table I—ContinuedBook-to-Market Equity (BE/ME) QuintilesSizeLow234HighLow234HighPanelB: Regressions: iRi-Rf-= a,. + 6iCBM - Rf)+ s,SMB1 + hflML + e;at(a)Small-0.45-0.16-0.050.040.02-4.19-2.04-0.820.690.292-0.07-0.040.090.070.03-0.80-0.591.331.130.513-0.080.04-0.000.060.07-1.070.47-0.060.880.40.14-0.19-0.060.020.061.74-2.43-0.730.270.59Big0.20-0.04-0.10-0.08-0.143.14-0.52-1.23-1.07-1.17bt(b)Small1.031.010.940.0.9439.1050.59.9358.4757.7121.101.040.990.971.0852.9461.1458.1762.9765.5831.101.020.980.971.0757.0855.4953.1155.9652.3741.071.071.051.031.18.77.4851.7945.76.27Big0.961.020.980.991.0760.2557.7747.0353.2537.18st(s)Small1.471.271.181.171.2339.0144.4852.2653.8252.6521.010.970.880.730.9034.1039.9436.1932.9238.1730.750.630.590.470.27.0924.1322.3718.9722.0140.360.300.290.220.4112.8710.10.176.8211.26Big-0.16-0.13-0.25-0.16-0.03-6.97-5.12-8.45-6.21-0.77ht(h)Small-0.270.100.250.370.63-6.283.039.7415.1623.622-0.490.000.260.460.69-14.660.349.2118.1425.593-0.390.030.320.490.68-12.560.10.7317.4520.434-0.440.030.310.0.72-13.980.979.4514.7017.34Big-0.470.000.200.560.82-18.230.186.0418.7117.57R=s(e)Small0.930.950.960.960.961.971.491.181.131.2220.950.960.950.950.961.551.271.281.161.2330.950.940.930.930.921.441.371.381.301.5240.940.920.910.880.1.461.471.511.691.91Big0.940.920.870.0.811.191.321.551.392.15periodically on size, BE/ME, E/P, C/P, sales growth, and past returns results inloadings on the three factors that are roughly constant. Variation through timein the slopes is, however, important in other applications. For example, FF(1994) show that because industries wander between growth and distress, it is60 The Journal of Financecritical to allow for variation in SMB and HML slopes when applying (1) and(2) to industries.II. LSV DecilesLakonishok, Shleifer, and Vishny (LSV 1994) examine the returns on sets ofdeciles formed from sorts on BE/ME, E/P, C/P, and five-year sales rank. TahleII summarizes the excess returns on our versions of these portfolios. Theportfolios are formed each year as in LSV using COMPUSTAT accounting datafor the fiscal year ending in the current calendar year (see tahle footnote). Wethen calculate returns heginning in July ofthe following year. (LSV start theirreturns in April.) To reduce the influence of small stocks in these (equal-weight) portfolios, we use only NYSE stocks. (LSV use NYSE and AMEX.) Tobe included in the tests for a given year, a stock must have data on all the LSVvariables. Thus, firms must have COMPUSTAT data on sales for six yearsbefore they are included in the return tests. As in LSV, this reduces biases thatmight arise because COMPUSTAT includes historical data when it adds firms(Banz and Breen (1986), Kothari, Shanken, and Sloan (1995)).Our sorts of NYSE stocks in Tahle II produce strong positive relationsbetween average return and BE/ME, E/P, or C/P, much like those reported byLSV for NYSE and AMEX firms. Like LSV, we find that past sales growth isnegatively related to future return. The estimates of the three-factor regres-sion (2) in Table III show, however, that the three-factor model (1) capturesthese patterns in average returns. The regression intercepts are consistentlysmall. Despite the strong explanatory power ofthe regressions (mosti?^ valuesare greater than 0.92), the GRS tests never come close to rejecting the hypoth-esis that the three-factor model describes average returns. In terms of both themagnitudes of the intercepts and the GRS tests, the three-factor model does abetter job on the LSV deciles than it does on the 25 FF size-BE/ME portfolios.(Compare Tables I and III.)For perspective on why the three-factor model works so well on the LSVportfolios. Table III shows the regression slopes for the C/P deciles. Higher-C/Pportfolios produce larger slopes on SMB and especially HML. This pattern inthe slopes is also observed for the BE/ME and E/P deciles (not shown). It seemsthat dividing an accounting variable by stock price produces a characterizationof stocks that is related to their loadings on HML. Given the evidence in FF(1995) that loadings on HML proxy for relative distress, we can infer that lowBE/ME, E/P, and C/P are typical of strong stocks, while high BE/ME, E/P, andC/P are typical of stocks that are relatively distressed. The patterns in theloadings of the BE/ME, E/P, and C/P deciles on HML, and the high averagevalue of HML (0.46 percent per month, 6.33 percent per year) largely explainhow the three-factor regressions transform the strong positive relations be-tween average return and these ratios (Table II) into intercepts that are closeto 0.0.Among the sorts in Table III, the three-factor model has the hardest timewith the returns on the sales-rank portfolios. Recall that high sales-rank firmsMultifactor Explanations of Asset Pricing Anomalies 61Table IISummary Statistics for Simple Monthly Excess Returns (in Percent)on the LSV Equal-Weight Deciles: 7/63-12/93, 366 MonthsAt the end of June of each year t (1963-1993), the NYSE stocks on COMPUSTAT are allocatedto ten portfolios, based on the decile breakpoints for BE/ME (book-to-market equity), E/P(earnings/price), C/P (cashflow/price), and past five-year sales rank (5-Yr SR). Equal-weightreturns on the portfolios are calculated from July to the following June, resulting in a timeseries of 366 monthly returns for July 1963 to December 1993. To be included in the tests fora given year, a stock must have data on all of the portfolio-formation variables of this table.Thus, the sample of firms is the same for all variables.For portfolios formed in June of year t, the denominator of BE/ME, E/P, and C/P is market equity(ME, stock price times shares outstanding) for the end of December of year t — 1, and BE, E, andC are for the fiscal year ending in calendar year t — 1. Book equity BE is defined in Table I. E isearnings before extraordinary items but after interest, depreciation, taxes, and preferred divi-dends. Cash flow, C, is E plus depreciation.The five-year sales rank for June of year t, 5-Yr SR(<), is the weighted average ofthe annual salesgrowth ranks for the prior five years, that is,55-Yr SR(t) = 21 (6 -j) X Rank« -j)The sales growth for year t -j is the percentage change in sales from t-j-1 to t-j,ln[Sales(i -j)/Sa\\esit -j - 1)]. Only firms with data for all five prior years are used to determinethe annual sales growth ranks for years t — 5 to t — 1. •For each portfolio, the table shows the mean monthly return in excess ofthe one-month Treasury billrate (Mean), the standard deviation of the monthly excess returns (Std. Dev.), and the ratio of themean excess return to its standard error [<(mean) = Mean/(Std. Devy365'^)]. Ave ME is the averagesize (ME, in $millions) ofthe firms in a portfolio, averaged across the 366 sample months.Deciles1234567.10BE/MELowHighMean0.420.500.530.580.650.720.810.841.031.22Std. Dev.5.815.565.575.525.235.034.965.065.526.82((Mean)1.391.721.822.022.382.743.103.173.553.43Ave. ME225613901125103710018838730572362E/PLowHighMean0.550.450.0.630.670.770.820.900.991.03Std. Dev.6.095.625.515.355.145.184.944.885.055.87«(Mean)1.721.521.2.242.492.843.163.513.743.37Ave. ME1294136712111209141110291022909862661C/PLowHighMean0.430.450.600.670.700.760.770.860.971.16Std. Dev.5.805.675.575.395.395.195.004.884.966.36t (Mean)1.411.522.062.372.472.782.933.363.753.47Ave. ME149112661112119909949749519906525 Yr SRHighLowMean0.470.630.700.680.670.740.700.780.1.03Std. Dev.6.395.665.465.155.225.105.005.105.256.13t (Mean)1.422.142.452.522.462.782.682.913.233.21Ave. ME93712331075118212651186107588474443462The Journal of FinanceTable IIIThree-Factor Time-Series Regressions for Monthly Excess Returns(in Percent) on the LSV Equal-WeightDeciles: 7/63-12/93, 366 MonthsRi -Rf=ai + bi{RM - Rf) + SiSMB + AjHML + e,The fonnation ofthe BE/ME, E/P, C/P, and five-year-sales-rank (5-Yr SR) deciles is described inTable IL The explanatory returns, R^ - Rf, SMB, and HML are described in Table L t() \\a aregression coefficient divided by its standard error. The regression RH are adjusted for degrees offreedom. GRS is the F-statistic of Gibbons, Ross, and Shanken (19), testing the hypothesis thatthe regression intercepts for a set often portfolios are all 0.0. p(GRS) is the p-value of GRS, thatis, the probability of a GRS value as large or larger than the ohserved value if the zero-interceptshypothesis is true.Deciles123456710GRS piGRS)BE/MELowHigha0.08-0.02-0.09-0.11-0.08-0.030.01-0.040.03-0.001.19-0.26-1.25-1.39-1.16-0.400.15-0.610.43-0.020.57 0.841R^tia)0.950.950.940.930.940.940.940.940.950.E/PLowHigha-0.00-0.07-0.07-0.04-0.030.020.060.090.120.00tia)-0.07-1.07-0.94-0.52-0.430.241.011.461.490.05R20.910.84 0.5920.950.940.940.940.940.940.940.920.92C/PLowHigha0.02-0.08-0.07-0.00-0.040.000.000.050.060.0161.041.061.081.061.051.040.991.000.981.14s0.450.500.0.510.550.500.530.480.570.92h-0.39-0.180.070.110.230.310.360.500.670.79tia)0.22-1.14-1.00-0.04-0.510.000.060.720.920.14tib)51.4561.1662.490.49 0.8.1559.0461.2860.0263.3658.92tis)15.5620.3246.4922.1121.5721.4920.7222.1921.1724.13tih)-12.03-6.5226.182.5.287.8511.4013.5219.4624.8819.74i?20.930.950.950.950.940.940.940.940.940.925-Yr SRHighLowa-0.21-0.06-0.03-0.01-0.04-0.02-0.040.000.040.07b1.161.101.091.031.031.031.000.990.991.02s0.720.560.520.490.520.510.500.570.670.95h-0.090.090.210.200.240.330.330.360.470.50tia)-2.60-0.97-0.49-0.20-0.61-0.25-0.660.07tib)0.470.6059.0170.590.87 0.56367.6565.3456.6868.62.49.12tis)50.0834.25.6925.1122.5921.6520.1523.21.21.6523.6522.34tih)-2.883.558.057.988.0713.6312.8012.1314.7810.320.950.960.950.950.930.950.940.930.920.87(strong past performers) have low future returns, and low sales-rank firms(weak past performers) have high future returns (Table II). The three-factormodel of (1) captures most of this pattern in average returns, largely becauselow sales-rank stocks behave like distressed stocks (they have stronger load-Multifactor Explanations of Asset Pricing Anomalies 63ings on HML). But a hint of the pattern is left in the regression intercepts.Except for the highest sales-rank decile, however, the intercepts are close to0.0. Moreover, although the intercepts for the sales-rank deciles produce thelargest GRS F-statistic (0.87), it is close to the median of its distribution whenthe true intercepts are all 0.0 (its p-value is 0.563). This evidence that thethree-factor model describes the returns on the sales-rank deciles is importantsince sales rank is the only portfolio-formation variable (here and in LSV) thatis not a transformed version of stock price. (See also the industry tests in FF(1994).)III. LSV Double-Sort Portfoliosdistinguishes between strong and distressed stocks, and produces largerLSV argue that sorting stocks on two accounting variables more accuratelyspreads in average returns. Because accounting ratios with stock price in thedenominator tend to be correlated, LSV suggest combining sorts on sales rankwith sorts on BE/ME, E/P, or C/P. We follow their procedure and separatelysort firms each year into three groups (low 30 percent, medium 40 percent, andhigh 30 percent) on each variable. We then form sets of nine portfolios as theintersections of the sales-rank sort and the sorts on BE/ME, E/P, or C/P.Confirming their results. Table IV shows that the sales-rank sort increases thespread of average returns provided by the sorts on BE/ME, E/P, or C/P. In fact,the two double-whammy portfolios, combining low BE/ME, E/P, or C/P withhigh sales growth (portfolio 1-1), and high BE/ME, E/P, or C/P with low salesgrowth (portfolio 3-3), always have the lowest and highest post-formationaverage returns.returns on the LSV double-sort portfolios. Strong negative loadings on HMLTable V shows that the three-factor model has little trouble describing the(which has a high average premium) bring the low returns on the 1-1 portfolioscomfortably within the predictions of the three-factor model; the most extremeintercept for the 1-1 portfolios is -6 basis points (-0.06 percent) per monthand less than one standard error from 0.0. Conversely, because the 3-3 port-folios have strong positive loadings on SMB and HML (they behave likesmaller distressed stocks), their high average returns are also predicted by thethree-factor model. The intercepts for these portfolios are positive, but againquite close to (less than 8 basis points and 0.7 standard errors from) 0.0.three-factor regression (2) are 0.0; the smallest p-value is 0.284. Thus, whetherThe GRS tests in Table V support the inference that the intercepts in thethe spreads in average returns on the LSV double-sort portfolios are caused byrisk or over-reaction, the three-factor model in equation (1) describes themparsimoniously.IV. Portfolios Formed on Past Returnsterm (three- to five-year) past returns, losers (low past returns) have highDeBondt and Thaler (1985) find that when portfolios are formed on long- The Journal of FinanceTable IVSummary Statistics for Excess Returns (in Percent) on the LSVEqual-Weight Double-Sort Portfolios: 7/63-12/93, 366 MonthsAt the end of June of each year t (1963-1993), the NYSE stocks on COMPUSTAT are allocated tothree equal groups (low, medium, and high: 1, 2, and 3) based on their sorted BE/ME, E/P, or C/Pratios for year t - 1. The NYSE stocks on COMPUSTAT are also allocated to three equal groups(high, medium, and low: 1, 2, and 3) based on their five-year sales rank. The intersections of thesales-rank sort with the BE/ME, E/P, or E/P sorts are then used to create three sets of nineportfolios (BE/ME & Sales Rank, E/P & Sales Rank, C/P & Sales Rank). Equal-weight returns onthe portfolios are calculated from July to the following June. To be included in the tests for a givenyear, a stock must have data on all of the portfolio-formation variables. The sample of firms is thusthe same for all variables. BE/ME (book-to-market equity), E/P (earnings/price), C/P (cashflow/price), and five-year sales rank are defined in Table II. The 1-1 portfolios contain strong firms (highsales growth and low BE/ME, E/P, or C/P), while the 3-3 portfolios contain weak firms (low salesgrowth and high BE/ME, E/P, or C/P).For each portfolio, the table shows the mean monthly return in excess of the one-month Treasurybill rate (Mean), the standard deviation of the monthly excess returns (Std. Dev.), and the ratio ofthe mean excess return to its standard error [<(mean) = Mean/(Std. Dev./365^'^)]. Ave. ME is theaverage size (ME, in $millions) of the firms in a portfolio, averaged across the 366 sample months.Count is the average across months of the number of firms in a portfolio.1-11-21-32-12-22-33-13-23-3BE/ME and Sales RankMean0.470.49Std. Dev.5.955.19«(Mean)1.521.81Count151109Ave. ME15301867E/P andSales RankMean0.410.47Std. Dev.6.025.44<(Mean)1.311.66Count11498Ave. ME13941524C/P andSales RankMean0.440.45Std. Dev.6.035.261.401.rtMean)Count122107Ave. ME136515270.525.631.77410.5.752.111067230.694.972.661800.745.022.831168660.936.452.794820.945.593.201186551.115.993.55144510610.775.762.576873911100.724.942.801630.635.762.101050.824.963.161049280.806.082.51876510.865.333.0814571.065.903.4313150611030.625.802.0310613550.715.012.701660.705.762.336280.835.093.101157960.856.132.786150.915.343.271348811.065.903.4412561610671187future returns and winners (high past returns) have low future returns. Incontrast, Jegadeesh and Titman (1993) and Asness (1994) find that whenportfolios are formed on short-term (up to a year of) past returns, past loserstend to be future losers and past winners are future winners.Table VI shows average returns on sets often equal-weight portfolios formedmonthly on short-term (11 months) and long-term (up to five years of) pastreturns. The results for July 1963 to December 1993 confirm the strongcontinuation of short-term returns. The average excess return for the monthMultifactor Explanations of Asset Pricing Anomalies65Table VThree-Factor Regressions for Monthly Excess Returns (in Percent)on the LSV Equal-Weight Douhle-Sort Portfolios:7/63-12/93, 366 MonthsRi -Rf=ai + biiRu - Rj) + s.SMB +The formation of the double-sort portfolios is described in Table IV. BE/ME (book-to-marketequity), E/P (earnings/price), C/P (cashflow/price), and five-year sales rank are described in TableII. The 1-1 portfolios contain strong firms (high sales growth and low BE/ME, E/P, or C/P), whilethe 3-3 portfolios contain weak firms (low sales growth and high BE/ME, E/P, or C/P). «() is aregression coefficient divided by its standard error. The regression R'^ are adjusted for degrees offreedom. GRS is the F-statistic of Gibbons, Ross, and Shanken (19), testing the hypothesis thatthe nine regression intercepts for a set of double-sort portfolios are all 0.0. p(GRS) is the p-valueof GRS.1-11-21-32-12-22-33-13-23-3 GRS p (GRS)BE/ME & Sales Ranka-0.000.00-0.06-0.19-0.000.00-0.19-0.070.07b1.101.031.001.121.000.991.171.061.01s0.490.310.550.630.480.500.870.740.97h-0.33-0.14-0.040.310.250.320.750.700.68t{a)-0.100.12-0.57-2.59-0.070.12-1.-0.940.69 1.22 0.284tib)71.6767.8535.6561.8167.3651.0041.29.4538.46t(s)22.3014.3213.7724.4222.4418.1821.3626.6225.76tih)-13.19-5.74-0.9410.5710.3310.1716.3022.3115.91R^0.960.950.860.940.950.920.0.930.E/P& SalesRanka-0.06-0.060.02-0.090.030.06-0.19-0.060.06b1.111.041.021.111.010.991.131.041.00s0.480.450.740.580.430.480.820.650.92h-0.34-0.120.180.140.250.390.530.580.61t(a)-0.-0.870.24-1.230.530.81-2.10-0.820.59 1.06 0.394t(.b)62.1256.0941.5258.9767.4853.8051.3259.0537.61tis)18.6117.0421.0721.3020.1818.1326.0825.6623.98tih)-11.56-3.8.414.5010.4612.8814.9220.4914.19R^0.950.940.900.940.950.920.930.940.C/P& SalesRanka-0.02-0.06-0.02-0.140.000.07-0.17-0.020.0461.111.011.021.121.021.001.131.041.00s0.460.420.720.630.460.530.800.0.92h-0.36-0.120.140.170.260.340.620.620.68t(a)-0.27-1.03-0.24-1.930.080.95-1.73-0.340.34 1.04 0.405tib).0465.8240.2063.3167.9652.2845.5558.4836.63tis)18.3719.1219.8624.7721.3419.4722.5725.3223.47tih)-12.71-4.903.425.8210.6110.8415.2121.15.40R^0.950.950.0.950.950.920.910.940.88after portfolio formation ranges from -0.00 percent for the decile of stocks withthe worst short-term past returns (measured from 12 to 2 months heforeportfolio formation) to 1.31 percent for the decile with the best short-term past66The Journal of FinanceTable VINYSE Deciles Formed Monthly Based on Continuously CompoundedAverage Monthly Excess Returns (in Percent) on Equal-WeightPast ReturnsAt the beginning of each month t, all NYSE firms on CRSP with returns for months t - xiot -yare allocated to deciles based on their continuously compounded returns between t - x and t - y.For example, firms are allocated to the 12-2 portfolios for January 1931 based on their continu-ously compounded returns for January 1930 through November 1930. Decile 1 contains the NYSEstocks with the lowest continuously compounded past returns. The portfolios are reformedmonthly, and equal-weight simple returns in excess of the one-month bill rate are calculated forJanuary 1931 (3101) to December 1993 (9312). The table shows the averages of these excessreturns for 6307 to 9312 (366 months) and 3101 to 6306 (390 months).PortfolioAverage Excess ReturnsPeriodMonths1234567106307-931212-2-0.000.460.610.550.720.686307-931224-20.360.600.590.850.660.900.711.081.310.816307-931236-20.460.600.770.730.690.800.730.931.050.810.696307-931248-20.660.700.770.780.740.840.710.970.710.726307-931260-20.860.760.710.730.720.750.700.0.716307-931260-131.160.810.770.740.700.760.740.660.720.730.720.730.0.423101-630612-21.491.521.321.493101-630624-22.241.391.601.451.571.451.551.701.411.583101-630636-22.311.741.311.871.651.321.241.461.401.263101-6308-22.341.811.621.401.321.461.601.231.371.271.301.363101-630660-22.491.781.741.331.221.241.501.391.331.263101-630660-132.621.851.271.631.181.611.431.281.141.241.341.281.081.01returns. (Skipping the portfolio formation month in ranking stocks reducesbias from bid-ask bounce.)Table VI also confirms that average returns tend to reverse when portfoliosare formed using returns for the four years from 60 to 13 months prior toportfolio formation. For these portfolios, the average return in the month afterportfolio formation ranges from 1.16 percent for the decile of stocks with theworst long-term past returns to 0.42 percent for stocks with the best pastreturns. In the 1963-1993 results, however, long-term return reversal isobserved only when the year prior to portfolio formation is skipped in rankingstocks. When the preceding year is included, short-term continuation offsetslong-term reversal, and past losers have lower future returns than past win-ners for portfolios formed with up to four years of past returns.Can our three-factor model explain the patterns in the future returns for1963-1993 on portfolios formed on past returns? Table VII shows that theanswer is yes for the reversal of long-term returns observed when portfoliosare formed using returns from 60 to 13 months prior to portfolio formation. Theregressions of the post-formation returns on these portfolios oni?^ - R^, SMB,and HML produce intercepts that are close to 0.0 both in absolute terms andon the GRS test. The three-factor model works because long-term past losersMultifactor Explanations of Asset Pricing AnomaliesTable VII67Three-Factor Regressions for Monthly Excess Returns (in Percent)on Equal-Weight NYSE Portfolios Formed on Past Returns:7/63-12/93, 366 MonthsRi-Rf= a, + biiRu - Rf)The formation of the past-return deciles is described in Table VI. Decile 1 contains the NYSEstocks with the lowest continuously compounded returns during the portfolio-formation period(12-2, 48-2, or 60-13 months before the return month), ti) is a regression coefficient divided by itsstandard error. The regression R^s are adjusted for degrees of freedom. GRS is the F-statistic ofGibbons, Ross, and Shanken (19), testing the hypothesis that the regression intercepts for a setof ten portfolios are all 0.0. p(GRS) is the p-value of GRS.123456710GRS p(GRS)Portfolio formation months areM2 to t-2absh-1.15-0.39-0.21-0.22-0.04-0.051.021.041.021.021.141.060.480.590.531.350.770.660.320.300.330.0.350.350.121.040.470.290.211.030.450.230.331.100.510.230.591.130.680.04R^tia)-5.34-3.05-2.05-2.81-0.-0.931.943.083.884.5.45 0.000tib)21.3133.32.0351.4861.0373.6268.9662.6751.7535.25tis)17.16.9618.5920.8722.0623.9621.5319.0316.14.84tih)6.216.728.7410.1811.8613.1611.888.506.680.700.750.850.0.920.940.960.950.940.920.86Portfolio formation months aret-48 to t-2a-0.73-0.32-0.09-0.08-0.05-0.001.021.011.121.061.05b1.160.420.0.520.480.87s1.590.310.440.440.360.900.60h0.071.000.410.180.100.150.370.991.041.110.400.420.490.11-0.05-0.26tia)-2.91tib)18.61tis)17.91tih)8.91R^abs-2.79-0.96-0.99-0.67-0.011.081.462.093.602.02 0.03139.2246.5553.1957.8263.78.7258.6257.0243.3721.3619.6818.6119.1718.5118.5216.6116.2213.4012.9411.9313.7812.6111.877.344.19-1.55-6.350.920.930.940.950.930.940.900.910.730.880.000.990.470.340.021.000.380.290.061.000.350.230.10-0.07-0.121.011.061.150.400.450.500.13-0.00-0.26Portfolio formation months aret-60 to M3-0.18-0.16-0.13-0.071.071.041.131.090.670.591.500.830.420.870.0.50htia)-0.80-1.-1.69-0.990.020.400.961.43-0.92-1.361.29 0.235tib)20.2444.4055.0361.0963.7965.6862.5858.2660.4953.04tis)18.7723.6324.0924.0621.2117.4415.4316.1818.0616.33tih)9.5913.6715.9415.3113.4611.828.984.46-0.14-7.50R^0.750.910.930.940.940.940.940.930.940.93load more on SMB and HML. Since they behave more like small distressedstocks, the model predicts that the long-term past losers will have higheraverage returns. Thus, the reversal of long-term returns, which has producedso much controversy (DeBondt and Thaler (1985,1987), Chan (1988), Ball and68 The Journal of FinanceKothari (19), Chopra, Lakonishok, and Ritter (1992)), falls neatly within thepredictions of our three-factor model. Moreover, since the model captures theeconomic essence of long-term winners (strong stocks) and losers (smallerdistressed stocks), we speculate that it can explain the stronger reversal oflong-term returns observed in the 1931-1963 period (Table VI).In contrast. Table VII shows that the three-factor model misses the contin-uation of returns for portfolios formed on short-term past returns. In thethree-factor regressions for these portfolios, the intercepts are strongly nega-tive for short-term-losers (low-past-returns) and strongly positive for short-term winners. The problem is that losers load more on SMB and HML (theybehave more like small distressed stocks) than winners. Thus, as for theportfolios formed on long-term past returns, the three-factor model predictsreversal for the post-formation returns of short-term losers and winners, andso misses the observed continuation.As noted earlier, when portfolios are formed on long-term past returns thatinclude the year prior to portfolio formation, short-term continuation offsetslong-term reversal, leaving either continuation or little pattern in futurereturns. Again, however, future returns on long-term losers load more on SMBand HML, so the three-factor model (1) incorrectly predicts return reversal.The regressions in table VII for portfolios formed using returns from two to 48months prior to portfolio formation are an example.V. Exploring Three-Factor ModelsThe tests above suggest that many patterns in average stock returns, so-called anomalies of the CAPM, are captured by the three-factor model of (1). Inthis section we show that the explanatory returns of the model are not unique.Many other combinations of three portfolios describe returns as well as R^ -Rf, SMB, and HML. These results support our conclusion that a three-factormodel is a good description of average returns.We first provide some background. Fama (1994) shows that a generalizedportfolio-efficiency concept drives Merton's (1973) ICAPM. Because ICAPMinvestors are risk averse, they are concerned with the mean and variance oftheir portfolio return. ICAPM investors are, however, also concerned withhedging more specific state-variable (consumption-investment) risks. As aresult, optimal portfolios are multifactor-minimum-variance (MMV): theyhave the smallest possible return variances, given their expected returns andsensitivities to the state-variables.In a two-state-variable ICAPM, MMV portfolios are spanned by (they can begenerated from) the risk-free security and any three linearly independentMMV portfolios. (With two state variables and a finite number of risky secu-rities, a third MMV portfolio is needed to capture the tradeoff of expectedreturn for return variance that is unrelated to the state variables.) Thisspanning result has two implications that we test below.(SI) The expected excess returns on any three MMV portfolios describe theexpected excess returns on all securities and portfolios. In other words, theMultifactor Explanations of Asset Pricing Anomalies 69intercepts in regressions of excess returns on the excess returns on anythree MMV portfolios are equal to 0.0.(S2) The realized excess returns on any three MMV portfolios perfectlydescribe (intercepts equal to 0.0 and R^ equal to 1.0) the excess returns onother MMV portfolios.In the usual representation of a three-factor ICAPM, the three explanatoryportfolios are the value-weight market and MMV portfolios that mimic the twostate variables of special hedging concern to investors. (SI) and (S2) say,however, that any three MMV portfolios can be used to generate MMV port-folios and describe returns.The tests that follow can also be interpreted in terms of a model in the spiritof Ross' (1976) APT. Suppose (i) investors are risk averse, (ii) there are twocommon factors in returns, and (iii) the number of risky securities is finite.Fama's (1994) analysis again implies that optimal portfolios are MMV: theyhave the smallest possible variances given their expected returns and theirloadings on the two common factors. With a finite number of securities,however, the returns on MMV portfolios in general are not perfectly explainedby the two common factors in returns. As a result, as in the ICAPM, therisk-free security and three MMV portfolios are needed to span MMV portfoliosand describe expected returns. Again, (SI) and (S2) hold.A. Spanning TestsIn principle, the explanatory variables in the ICAPM (or the APT) are theexpected returns on MMV portfolios in excess of the risk-free rate. SMB andHML in (1) are, however, each the difference between two portfolio returns.Equation (1) is still a legitimate three-factor risk-return relation as long as thetwo components of SMB (S and B) and the two components of HML (H and L)are MMV. Rg - i?^and R^ - Rf are then exact linear combinations of iJ^f ~ ^f>Rs - Rf, and Rjj - Rf, so subtracting RB from Rg (to get SMB) and R^ from RH(HML) has no effect on the intercepts or the explanatory power of the three-factor regressions.Obviously, we do not presume that our ad hoc size and book-to-marketportfolios are truly MMV. We suggest, however, that if R^ - Rf, SMB, andHML do a good job describing average returns, then M, S, B, H, and L are closeto MMV. (SI) and (S2) say that this hypothesis has two testable implications.(i) All combinations of three of the portfolios M, S, B, H, and L should providesimilar descriptions of average returns (SI), (ii) Realized excess returns on anythree of the candidate MMV portfolios should almost perfectly describe theexcess returns on other candidate MMV portfolios (S2).Table VIII tests (S2) with regressions that use the four different triplets ofRuf - Rf, Rg - Rf, RH - Rf> and R^ - Rf to explain the excess return on theexcluded MMV proxy. (We drop the big-stock portfolio B from the list of MMVproxies because the correlation between R^ and Rg is 0.99.) The results areconsistent with (S2). Excess returns on any three of M, S, H, and L almost70The Journal of FinanceTable VIIIRegressions to Explain Monthly Excess Returns (in Percent) on M,S, L, H, SMB and HML: 7/63-12/93, 366 MonthsThe portfolios (described in Table I) include the market (M), the small-stock portfolio (S), thelow-book-to-market portfolio (L), the high-book-to-market portfolio (H), the difference between Hand L (HML), and the difference between S and the return on the big-stock portfolio B (SMB). Tosimplify the notation, the table uses the portfolio labels, rather than explicit notation for theirexcess returns. The regression iJ^ and the residual standard error, s(e), are adjusted for degrees offreedom. The numbers in parentheses are ^-statistics (regression coefficients divided by theirstandard errors).R^s(e)s0.28+ 1.17 M+ e(1.99)(36.95)0.792.68L-0.10+ 1.20 M+ e(-1.15)(62.84)0.921.62H0.46+0.99 M+ e(4.08)(38.73)0.802.16SMB0.19+0.21 M+ e(1.32)(6.)0.102.74HML0.56-0.21 M+ e(4.42)(-7.53)0.132.41S0.00-0.83 M+ 1.00 L(0.17)+0.81 H(-29.12)+ e(46.81)0.99(50.12)0.65L-0.03+0.86 M(-0.90)+0.86 S(51.83)-0.67 H(46.81)+ e0.99(-29.30)0.60H0.06+0.98 M(1.36)+ 1.09 S(31.38)-1.05 L(50.12)+ e0.98(-29.30)0.75M0.00-0.85 S+ 1.03 L(0.08)(-29.12)+0.75 H(51.83)+ e0.980.66(31.38)perfectly describe the excess return on the fourth. The regression interceptsare close to 0.0, and the R^different tripletsTable IX summarizes the intercepts from regressions that use the four values are close to 1.0 (0.98 and 0.99).excess returns on the different sets of portfolios examined in previous sections. O^RM - Rf, Rs - Rf, RH - Rf, and R^ - Rf to describe theAs predicted by (SI), different triplets of M, S, L, and H provide equivalentdescriptions of returns. Specifically, different three-factor regressions producemuch the same GRS tests, mean absolute and squared intercepts, and averagevalues ofidentical for different triplets of explanatory returns. Suhstantively, Table EK R^. Moreover, the regression intercepts (not shown) are nearlysays that different three-factor regressions all miss the continuation of returnsMultifactor Explanations of Asset Pricing AnomaliesTable K71Summary of Intercepts from One-Factor CAPM Excess-ReturnRegressions and Different Versions of the Three-Factor ICAPMRegressions: 7/63-12/93, 366 MonthsThe alternative sets of dependent excess returns (and the tables that describe them) include the 25size-BE/ME portfolios (Table I), the E/P and five-year sales-rank deciles (Table II), the nine portfoliosdouhled-sorted on C/P and five-year sales rank (Table IV), the long-term and short-term past returndeciles (60-13 and 12-2) (Table VI). The explanatory variables (described in Table I) include the excessreturns on the market portfolio (M), the small-stock portfolio (S), the low- and high-book-to-marketportfolios (L and H), SMB (the return on S minus the return on the big-stock portfolio B) and HML (Hminus L). GRS is the F-statistic of Gibbons, Ross, and Shanken (19), testing the hypothesis that theregression intercepts for a set of dependent portfolios are all 0.0. jo(GRS) is the p-value of GRS. Ave | a \\and Ave o^ are the average absolute and squared values of the intercepts for a set of dependentportfolios, and Ave R^ is the average of the regression R^ (adjusted for degrees of freedom).Dependent Ports.25 Size-BE/ME25 Size-BE/ME25 Size-BE/ME25 Size-BE/ME25 Size-BE/ME25 Size-BE/MEE/PE/PE/PE/PE/PE/PExplanatory Ports.MMMMMSMMMMMSMMMMMSMMSMBGRS2.761.972.062.161.872.062.850.840.951.020.860.862.510.871.010.960.920.932.931.041.131.141.031.052.511.291.381.191.291.305.134.4.4.584.514.46p(GRS)0.0000.0040.0020.0010.0080.0020.0020.5920.4880.4270.5750.5710.0060.5630.4370.4740.5140.5090.0020.4050.3380.3330.4160.3960.0060.2350.1860.2990.2340.2300.0000.0000.0000.0000.0000.000Ave 1 a 1Ave a^0.2860.0930.0970.1020.0940.0940.2600.0510.0590.00.0520.0510.2560.0530.0550.0520.0520.0520.2680.0620.0670.0630.0610.0610.2680.0920.0940.0770.00.0900.3370.3310.3220.3290.3260.3280.11400.010.01700.01830.01590.01620.10590.00390.00510.00570.00410.00400.08210.00580.00680.00590.00570.00570.10070.00680.00680.000.000.00650.090.01140.01120.00740.01020.01070.170.20970.20270.20400.20470.20690.770.930.930.920.920.920.830.930.940.940.930.930.820.930.940.940.930.930.800.930.930.930.920.930.800.920.920.920.910.910.790.900.900.900.900.90SMBSLLSMBSSLLSMBSsHMLHLHHHMLHLHHHMLHLHHHMLHLHHHMLHLHHHMLHLHSales RankSales RankSales RankSales RankSales RankSales RankC/P & Sales RankC/P & Sales RankC/P & Sales RankC/P & Sales RankC/P & Sales RankC/P & Sales Rank60-1360-1360-1360-1360-1360-1312-212-212-212-212-212-2sLLMMSMSMMMMMSMMMMMSsLLSMBSsLLSMBSSLLH72 The Journal of Financefor portfolios formed on short-term past returns. On the other hand, everytriplet of M, S, L, and H does a similar and excellent job describing the returnson the LSV deciles formed on E/P and sales rank, and the LSV portfoliosdouble-sorted on C/P and sales rank. In results not shown in Table IX, excel-lent three-factor descriptions of returns are also obtained for the LSV BE/MEand C/P deciles, and for portfolios double-sorted on sales rank and BE/ME orE/P. Finally, Table EX shows that all triplets of M, S, L, and H capture thereversal of returns for portfolios formed on long-term past returns.Table IX says that our original (FF 1993) combination of the market, SMB,and HML fares no better or worse than triplets of M, S, H, and L. But theoriginal set of portfolios has one advantage. Table X shows thati?^ - Rf, SMB,and HML are much less correlated with one another than R^ - Rf, Rs - Rf,RB ~ Rf, RH ~ Rf, and i?^ - Rf. This makes three-factor regression slopeseasier to interpret, and it is why we use RM - Rf, SMB, and HML in theregressions of Tables I, III, V, and VII.B. Additional MMV ProxiesM, S, H, and L are not the only portfolios that give equivalent descriptionsof returns. We construct explanatory portfohos (MMV proxies) that are simpleaverages of the returns for the bottom and top three deciles of each of the LSV(BE/ME, E/P, C/P, and sales-rank) sorts and the short- and long-term past-return sorts. For example, the high E/P return (HE/P) is the average of the topthree E/P decile returns.The MMV proxies formed from the LSV BE/ME, E/P, and C/P deciles workmuch like our L and H (low- and high-BE/ME) portfohos in describing returns.The reason is clear from Table X. Excess returns on the LSV low BE/ME, E/P,and C/P portfolios are correlated 0.99 with each other, and they are correlated0.98 with our L (low-BE/ME) portfolio. Excess returns on the LSV high BE/ME,E/P, and C/P portfolios are correlated 0.98 and 0.99 with each other, and theircorrelations with our H portfolio are 0.97 and 0.98. The \"high\" portfolios aremuch more correlated with one another than with the \"low\" portfolios. TheMMV proxies produced by the LSV BE/ME, E/P, and C/P sorts also havesimilar average excess returns, 0.48 to 0.51 for the three \"low\" portfolios and0.97 to 1.03 for the three \"high\" portfolios. These returns are a bit higher thanthose of our L and H portfolios, 0.44 and 0.90, probably because L and H areconstructed from value-weight components.In short, the \"low\" and \"high\" MJVTV proxies from the LSV BE/ME, E/P, andC/P sorts mimic our L and H portfolios. Thus it is not surprising that they canreplace L and H in the three-factor model. Without showing the details,combining the market portfoho M with LBE/ME and HBE/ME, or LE/P andHE/P, or LC/P and HC/P produces three-factor descriptions of returns hkethose in Table IX.Ball (1978) argues that scaling stock prices with accounting variables, likeearnings, cash flow, or book equity, is a good way to extract the information instock prices about expected returns. Our tests suggest, more precisely, thatMultifactor Explanations of Asset Pricing Anomalies73COOi(MdCOlilt280.2.1.88inCOCOOi3.||co*^, a, *^ 1-1rHCJi •* 1-H (Mt>d CO (MO O 00d in COO p OS O(N CD-.COrH in COt> 00 00dOi T- «i COH in^ I—I a j2 .—0) - 03 'C SCO CO l>S 1 (I) ^ op CO •*A m 11- \"1-i in COfcia) in •* ;o [>*; d in^.- £to i> pd in (N05 00 O5TJ\" in CDd in t-J^K1-1in CO CD l>;•*d in 1-iPO00 in t>.^ in COd in >HCO rHin inC o C o
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