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multifactor explanations

Rf is the one-month Treasury bill rate observed at the beginning of the month (from CRSP). The explanatory returns RM, 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 big, S or B) based on whether their June market equity (ME, stock price times shares outstanding) is below or above the median ME for NYSE stocks. NYSE, AMEX, and Nasdaq stocks are allocated in an independent sort to three book-to-market equity (BE/ME) groups (low, medium, or high; L, M, or H) based on the breakpoints for the bottom 30 percent, middle 40 percent, and top 30 percent of the values of BE/ME for NYSE stocks. Six size-BE/ME portfolios (S/L, S/M, S/H, B/L, B/M, B/H) are defined as the intersections of the two ME and the three BE/ME groups. Value-weight monthly returns on the portfolios are calculated from July to the following June. SMB is the difference, each month, between the average of the returns on the three small-stock portfolios (S/L, S/M, and S/H) and the average of the returns on the three big-stock portfolios (B/L, B/M, and B/H). HML is the difference between the 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 25 size-BE/ME portfolios are formed like the six size-BE/ME portfolios used to construct SMB and HML, except that quintile breakpoints for ME and BE/ME for NYSE stocks are used to allocate NYSE, AMEX, and Nasdaq stocks to the portfolios.

BE is 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 redemption, liquidation, or par value (in that order) to estimate the book value of preferred stock. The BE/ME ratio used to form portfolios in June of year t is then book common equity for the fiscal year ending in calendar year t - 1, divided by market equity at the end of December of t - 1. We do not use negative BE firms, which are rare prior to 1980, when calculating the breakpoints for BE/ME or when forming the size-BE/ME portfolios. Also, only firms with ordinary common equity (as classified by CRSP) are included in the tests. This means that ADRs, REITs, and units of beneficial interest are excluded.

The market return RM is the value-weight return on all stocks in the size-BE/ME portfolios, plus the negative BE stocks excluded from the portfolios.

Book-to-Market Equity (BE/ME) Quintiles Size Low 2 3 4 High Low 2 3 4 High

Panel A: Summary Statistics Means Standard Deviations

Small

0.31

0.70

0.82

0.95

1.08

7.67

6.74

6.14

5.85

6.14

0.48

0.71

0.91

0.93

1.09

7.13

6.25

5.71

5.23

5.94

0.44

0.68

0.75

0.86

1.05

6.52

5.53

5.11

4.79

5.48

0.51

0.39

0.64

0.80

1.04

5.86

5.28

4.97

4.81

5.67

0.37

0.39

0.36

0.58

0.71

4.84

4.61

4.28

4.18

4.89



Book-to-Market Equity (BE/ME) Quintiles

Size

High

High

Panel B: Regressions: Rt

-Rf =

a, + bt{RM - Rf)

+ stSMB + hflML + et

t(a)

Small

-0.45

-0.16

-0.05

0.04

0.02

-4.19

-2.04

-0.82

0.69

0.29

-0.07

-0.04

0.09

0.07

0.03

-0.80

-0.59

1.33

1.13

0.51

-0.08

0.04

-0.00

0.06

0.07

-1.07

0.47

-0.06

0.88

0.89

0.14

-0.19

-0.06

0.02

0.06

1.74

-2.43

-0.73

0.27

0.59

0.20

-0.04

-0.10 -

-0.08

-0.14

3.14

-0.52

-1.23

-1.07

-1.17

t(b)

Small

1.03

1.01

0.94

0.89

0.94

39.10

50.89

59.93

58.47

57.71

1.10

1.04

0.99

0.97

1.08

52.94

61.14

58.17

62.97

65.58

1.10

1.02

0.98

0.97

1.07

57.08

55.49

53.11

55.96

52.37

1.07

1.07

1.05

1.03

1.18

54.77

54.48

51.79

45.76

46.27

0.96

1.02

0.98

0.99

1.07

60.25

57.77

47.03

53.25

37.18

t(s)

Small

1.47

1.27

1.18

1.17

1.23

39.01

44.48

52.26

53.82

52.65

1.01

0.97

0.88

0.73

0.90

34.10

39.94

36.19

32.92

38.17

0.75

0.63

0.59

0.47

0.64

27.09

24.13

22.37

18.97

22.01

0.36

0.30

0.29

0.22

0.41

12.87

10.64

10.17

6.82

11.26

-0.16

-0.13

-0.25 -

-0.16

-0.03

-6.97

-5.12

-8.45

-6.21

-0.77

t(h)

Small

-0.27

0.10

0.25

0.37

0.63

-6.28

3.03

9.74

15.16

23.62

-0.49

0.00

0.26

0.46

0.69 -

-14.66

0.34

9.21

18.14

25.59

-0.39

0.03

0.32

0.49

0.68 -

-12.56

0.89

10.73

17.45

20.43

-0.44

0.03

0.31

0.54

0.72 -

-13.98

0.97

9.45

14.70

17.34

-0.47

0.00

0.20

0.56

0.82 -

-18.23

0.18

6.04

18.71

17.57

s(e)

Small

0.93

0.95

0.96

0.96

0.96

1.97

1.49

1.18

1.13

1.22

0.95

0.96

0.95

0.95

0.96

1.55

1.27

1.28

1.16

1.23

0.95

0.94

0.93

0.93

0.92

1.44

1.37

1.38

1.30

1.52

0.94

0.92

0.91

0.88

0.89

1.46

1.47

1.51

1.69

1.91

0.94

0.92

0.87

0.89

0.81

1.19

1.32

1.55

1.39

2.15

periodically on size, BE/ME, E/P, C/P, sales growth, and past returns results in loadings on the three factors that are roughly constant. Variation through time in the slopes is, however, important in other applications. For example, FF (1994) show that because industries wander between growth and distress, it is

Table I-Continued



critical to allow for variation in SMB and HML slopes when applying (1) and (2) to industries.

II. LSV Deciles

Lakonishok, Shleifer, and Vishny (LSV 1994) examine the returns on sets of deciles formed from sorts on BE/ME, E/P, C/P, and five-year sales rank. Table II summarizes the excess returns on our versions of these portfolios. The portfolios are formed each year as in LSV using COMPUSTAT accounting data for the fiscal year ending in the current calendar year (see table footnote). We then calculate returns beginning in July of the following year. (LSV start their returns in April.) To reduce the influence of small stocks in these (equal-weight) portfolios, we use only NYSE stocks. (LSV use NYSE and AMEX.) To be included in the tests for a given year, a stock must have data on all the LSV variables. Thus, firms must have COMPUSTAT data on sales for six years before they are included in the return tests. As in LSV, this reduces biases that might 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 Table II produce strong positive relations between average return and BE/ME, E/P, or C/P, much like those reported by LSV for NYSE and AMEX firms. Like LSV, we find that past sales growth is negatively related to future return. The estimates of the three-factor regression (2) in Table III show, however, that the three-factor model (1) captures these patterns in average returns. The regression intercepts are consistently small. Despite the strong explanatory power of the regressions (most R2 values are greater than 0.92), the GRS tests never come close to rejecting the hypothesis that the three-factor model describes average returns. In terms of both the magnitudes of the intercepts and the GRS tests, the three-factor model does a better 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 LSV portfolios, Table III shows the regression slopes for the C/P deciles. Higher-C/P portfolios produce larger slopes on SMB and especially HML. This pattern in the slopes is also observed for the BE/ME and E/P deciles (not shown). It seems that dividing an accounting variable by stock price produces a characterization of 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 low BE/ME, E/P, and C/P are typical of strong stocks, while high BE/ME, E/P, and C/P are typical of stocks that are relatively distressed. The patterns in the loadings of the BE/ME, E/P, and C/P deciles on HML, and the high average value of HML (0.46 percent per month, 6.33 percent per year) largely explain how the three-factor regressions transform the strong positive relations between average return and these ratios (Table II) into intercepts that are close to 0.0.

Among the sorts in Table III, the three-factor model has the hardest time with the returns on the sales-rank portfolios. Recall that high sales-rank firms



Table ii

Summary Statistics for Simple Monthly Excess Returns (in Percent) on the LSV Equal-Weight Deciles: 7/63-12/93, 366 Months

At the end of June of each year t (1963-1993), the NYSE stocks on COMPUSTAT are allocated to 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-weight returns on the portfolios are calculated from July to the following June, resulting in a time series of 366 monthly returns for July 1963 to December 1993. To be included in the tests for a 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, and С are for the fiscal year ending in calendar year t - 1. Book equity BE is defined in Table I. E is earnings before extraordinary items but after interest, depreciation, taxes, and preferred dividends. Cash flow, C, is E plus depreciation.

The five-year sales rank for June of year t, 5-Yr SR(£), is the weighted average of the annual sales growth ranks for the prior five years, that is,

5-Yr SR(*) = X (6 -j) X Rank(* -j)

7 = 1

The sales growth for year t - j is the percentage change in sales from t - j - 1 to t - j, ln[Sales(£ -j)/Sa\es(t - j - 1)]. Only firms with data for all five prior years are used to determine the 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 bill rate (Mean), the standard deviation of the monthly excess returns (Std. Dev.), and the ratio of the mean excess return to its standard error [£(mean) = Mean/(Std. Dev./3651/2)]. Ave ME is the average size (ME, in $millions) of the firms in a portfolio, averaged across the 366 sample months.

Deciles

BE/ME

High

Mean

0.42

0.50

0.53

0.58

0.65

0.72

0.81

0.84

1.03

1.22

Std. Dev.

5.81

5.56

5.57

5.52

5.23

5.03

4.96

5.06

5.52

6.82

£(Mean)

1.39

1.72

1.82

2.02

2.38

2.74

3.10

3.17

3.55

3.43

Ave. ME

2256

1390

1125

1037

1001

High

Mean

0.55

0.45

0.54

0.63

0.67

0.77

0.82

0.90

0.99

1.03

Std. Dev.

6.09

5.62

5.51

5.35

5.14

5.18

4.94

4.88

5.05

5.87

£(Mean)

1.72

1.52

1.89

2.24

2.49

2.84

3.16

3.51

3.74

3.37

Ave. ME

1294

1367

1211

1209

1411

1029

1022

High

Mean

0.43

0.45

0.60

0.67

0.70

0.76

0.77

0.86

0.97

1.16

Std. Dev.

5.80

5.67

5.57

5.39

5.39

5.19

5.00

4.88

4.96

6.36

t (Mean)

1.41

1.52

2.06

2.37

2.47

2.78

2.93

3.36

3.75

3.47

Ave. ME

1491

1266

1112

1198

5-Yr SR

High

Mean

0.47

0.63

0.70

0.68

0.67

0.74

0.70

0.78

0.89

1.03

Std. Dev.

6.39

5.66

5.46

5.15

5.22

5.10

5.00

5.10

5.25

6.13

t (Mean)

1.42

2.14

2.45

2.52

2.46

2.78

2.68

2.91

3.23

3.21

Ave. ME

1233

1075

1182

1265

1186

1075



Table III

Three-Factor Time-Series Regressions for Monthly Excess Returns (in Percent) on the LSV Equal-Weight Deciles: 7/63-12/93, 366 Months

Ri - Rf= at + bt(RM - Rf) + sSMB + ЛДШЬ + et

The formation of the BE/ME, E/P, C/P, and five-year-sales-rank (5-Yr SR) deciles is described in Table II. The explanatory returns, RM - Rf, SMB, and HML are described in Table I. t{ ) is a regression coefficient divided by its standard error. The regression R2s are adjusted for degrees of freedom. GRS is the F-statistic of Gibbons, Ross, and Shanken (1989), testing the hypothesis that the regression intercepts for a set of ten portfolios are all 0.0. p(GRS) is the p-value of GRS, that is, the probability of a GRS value as large or larger than the observed value if the zero-intercepts hypothesis is true.

Deciles

123456789 10 GRS p(GRS)

BE/ME

0.08

-0.02

-0.09

-0.11

-0.08

t(a)

1.19

-0.26

-1.25

-1.39

-1.16

0.95

0.95

0.94

0.93

0.94

-0.00

-0.07

-0.07

-0.04

-0.03

t(a)

-0.07

-1.07

-0.94

-0.52

-0.43

0.91

0.95

0.94

0.94

0.94

0.02

-0.08

-0.07

-0.00

-0.04

1.04

1.06

1.08

1.06

1.05

0.45

0.50

0.54

0.51

0.55

-0.39

-0.18

0.07

0.11

0.23

t(a)

0.22

-1.14

-1.00

-0.04

-0.51

51.45

61.16

62.49

64.15

59.04

t(s)

15.56

20.32

22.11

21.57

21.49

t{h)

-12.03

-6.52

2.56

4.28

7.85

0.93

0.95

0.95

0.95

0.94

5-Yr SR

High

-0.21

-0.06

-0.03

-0.01

-0.04

1.16

1.10

1.09

1.03

1.03

0.72

0.56

0.52

0.49

0.52

-0.09

0.09

0.21

0.20

0.24

t(a)

-2.60

-0.97

-0.49

-0.20

-0.61

t(b)

59.01

70.59

67.65

65.34

56.68

t(s)

25.69

25.11

22.59

21.65

20.15

t(h)

-2.88

3.55

8.05

7.98

8.07

0.95

0.96

0.95

0.95

0.93

High

-0.03

0.01

-0.04

0.03

-0.00

-0.40

0.15

-0.61

0.43

-0.02

0.57

0.841

0.94

0.94

0.94

0.95

0.89

High

0.02

0.06

0.09

0.12

0.00

0.24

1.01

1.46

1.49

0.05

0.84

0.592

0.94

0.94

0.94

0.92

0.92

High

0.00

0.00

0.05

0.06

0.01

1.04

0.99

1.00

0.98

1.14

0.50

0.53

0.48

0.57

0.92

0.31

0.36

0.50

0.67

0.79

0.00

0.06

0.72

0.92

0.14

0.49

0.898

61.28

60.02

63.36

58.92

46.49

20.72

22.19

21.17

24.13

26.18

11.40

13.52

19.46

24.88

19.74

0.94

0.94

0.94

0.94

0.92

-0.02

-0.04

0.00

0.04

0.07

1.03

1.00

0.99

0.99

1.02

0.51

0.50

0.57

0.67

0.95

0.33

0.33

0.36

0.47

0.50

-0.25

-0.66

0.07

0.47

0.60

0.87

0.563

68.89

62.49

54.12

50.08

34.54

23.64

21.89

21.65

23.65

22.34

13.63

12.80

12.13

14.78

10.32

0.95

0.94

0.93

0.92

0.87

(strong past performers) have low future returns, and low sales-rank firms (weak past performers) have high future returns (Table II). The three-factor model of (1) captures most of this pattern in average returns, largely because low sales-rank stocks behave like distressed stocks (they have stronger load-



ings 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 to 0.0. Moreover, although the intercepts for the sales-rank deciles produce the largest GRS F-statistic (0.87), it is close to the median of its distribution when the true intercepts are all 0.0 (its p-value is 0.563). This evidence that the three-factor model describes the returns on the sales-rank deciles is important since sales rank is the only portfolio-formation variable (here and in LSV) that is not a transformed version of stock price. (See also the industry tests in FF (1994).)

III. LSV Double-Sort Portfolios

LSV argue that sorting stocks on two accounting variables more accurately distinguishes between strong and distressed stocks, and produces larger spreads in average returns. Because accounting ratios with stock price in the denominator tend to be correlated, LSV suggest combining sorts on sales rank with sorts on BE/ME, E/P, or C/P. We follow their procedure and separately sort firms each year into three groups (low 30 percent, medium 40 percent, and high 30 percent) on each variable. We then form sets of nine portfolios as the intersections 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 the spread 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 with high sales growth (portfolio 1-1), and high BE/ME, E/P, or C/P with low sales growth (portfolio 3-3), always have the lowest and highest post-formation average returns.

Table V shows that the three-factor model has little trouble describing the returns on the LSV double-sort portfolios. Strong negative loadings on HML (which has a high average premium) bring the low returns on the 1-1 portfolios comfortably within the predictions of the three-factor model; the most extreme intercept for the 1-1 portfolios is -6 basis points (-0.06 percent) per month and less than one standard error from 0.0. Conversely, because the 3-3 portfolios have strong positive loadings on SMB and HML (they behave like smaller distressed stocks), their high average returns are also predicted by the three-factor model. The intercepts for these portfolios are positive, but again quite close to (less than 8 basis points and 0.7 standard errors from) 0.0.

The GRS tests in Table V support the inference that the intercepts in the three-factor regression (2) are 0.0; the smallestp-value is 0.284. Thus, whether the spreads in average returns on the LSV double-sort portfolios are caused by risk or over-reaction, the three-factor model in equation (1) describes them parsimoniously.

IV. Portfolios Formed on Past Returns

DeBondt and Thaler (1985) find that when portfolios are formed on long-term (three- to five-year) past returns, losers (low past returns) have high



Table IV

Summary Statistics for Excess Returns (in Percent) on the LSV Equal-Weight Double-Sort Portfolios: 7/63-12/93, 366 Months

At the end of June of each year t (1963-1993), the NYSE stocks on COMPUSTAT are allocated to three equal groups (low, medium, and high: 1, 2, and 3) based on their sorted BE/ME, E/P, or C/P ratios 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 the sales-rank sort with the BE/ME, E/P, or E/P sorts are then used to create three sets of nine portfolios (BE/ME & Sales Rank, E/P & Sales Rank, C/P & Sales Rank). Equal-weight returns on the portfolios are calculated from July to the following June. To be included in the tests for a given year, a stock must have data on all of the portfolio-formation variables. The sample of firms is thus the 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 (high sales growth and low BE/ME, E/P, or C/P), while the 3-3 portfolios contain weak firms (low sales growth 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 Treasury bill rate (Mean), the standard deviation of the monthly excess returns (Std. Dev.), and the ratio of the mean excess return to its standard error Y(mean) = Mean/(Std. Dev./3651/2)]. Ave. ME is the average 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.

BE/ME and Sales Rank

Mean

0.47

0.49

0.52

0.64

0.69

0.74

0.93

0.94

1.11

Std. Dev.

5.95

5.19

5.63

5.75

4.97

5.02

6.45

5.59

5.99

£(Mean)

1.52

1.81

1.77

2.11

2.66

2.83

2.76

3.20

3.55

Count

Ave. ME

1530

1867

1061

1110

E/P and Sales Rank

Mean

0.41

0.47

0.77

0.63

0.72

0.82

0.80

0.86

1.06

Std. Dev.

6.02

5.44

5.76

5.76

4.94

4.96

6.08

5.33

5.90

£(Mean)

1.31

1.66

2.57

2.10

2.80

3.16

2.51

3.08

3.43

Count

Ave. ME

1394

1524

1103

1355

C/P and Sales Rank

Mean

0.44

0.45

0.70

0.62

0.71

0.83

0.85

0.91

1.06

Std. Dev.

6.03

5.26

5.76

5.80

5.01

5.09

6.13

5.34

5.90

£(Mean)

1.40

1.64

2.33

2.03

2.70

3.10

2.64

3.27

3.44

Count

Ave. ME

1365

1527

1067

1187

future returns and winners (high past returns) have low future returns. In contrast, Jegadeesh and Titman (1993) and Asness (1994) find that when portfolios are formed on short-term (up to a year of) past returns, past losers tend to be future losers and past winners are future winners.

Table VI shows average returns on sets of ten equal-weight portfolios formed monthly on short-term (11 months) and long-term (up to five years of) past returns. The results for July 1963 to December 1993 confirm the strong continuation of short-term returns. The average excess return for the month



Table V

Three-Factor Regressions for Monthly Excess Returns (in Percent) on the LSV Equal-Weight Double-Sort Portfolios: 7/63-12/93, 366 Months

Rt - Rf = at + bt(RM ~ Rf) + stSMB + HML + et

The formation of the double-sort portfolios is described in Table IV. BE/ME (book-to-market equity), E/P (earnings/price), C/P (cashflow/price), and five-year sales rank are described in Table II. The 1-1 portfolios contain strong firms (high sales growth and low BE/ME, E/P, or C/P), while the 3-3 portfolios contain weak firms (low sales growth and high BE/ME, E/P, or C/P). tO is a regression coefficient divided by its standard error. The regression R2 are adjusted for degrees of freedom. GRS is the F-statistic of Gibbons, Ross, and Shanken (1989), testing the hypothesis that the nine regression intercepts for a set of double-sort portfolios are all 0.0. p(GRS) is the p-value of GRS.

1-1 1-2 1-3 2-1 2-2 2-3 3-1 3-2 3-3 GRS p (GRS)

BE/ME & Sales Rank

-0.00

0.00

-0.06

-0.19

-0.00

0.00

-0.19

-0.07

0.07

1.10

1.03

1.00

1.12

1.00

0.99

1.17

1.06

1.01

0.49

0.31

0.55

0.63

0.48

0.50

0.87

0.74

0.97

-0.33

-0.14

-0.04

0.31

0.25

0.32

0.75

0.70

0.68

t(a)

-0.10

0.12

-0.57

-2.59

-0.07

0.12

-1.64

-0.94

0.69

t(b)

71.67

67.85

35.65

61.81

67.36

51.00

41.29

54.45

38.46

t(s)

22.30

14.32

13.77

24.42

22.44

18.18

21.36

26.62

25.76

t(h)

-13.19

-5.74

-0.94

10.57

10.33

10.17

16.30

22.31

15.91

0.96

0.95

0.86

0.94

0.95

0.92

0.89

0.93

0.89

E/P <

& Sales

Rank

-0.06

-0.06

0.02

-0.09

0.03

0.06

-0.19

-0.06

0.06

1.11

1.04

1.02

1.11

1.01

0.99

1.13

1.04

1.00

0.48

0.45

0.74

0.58

0.43

0.48

0.82

0.65

0.92

-0.34

-0.12

0.18

0.14

0.25

0.39

0.53

0.58

0.61

t(a)

-0.89

-0.87

0.24

-1.23

0.53

0.81

-2.10

-0.82

0.59

t(b)

62.12

56.09

41.52

58.97

67.48

53.80

51.32

59.05

37.61

t(s)

18.61

17.04

21.07

21.30

20.18

18.13

26.08

25.66

23.98

t(h)

-11.56

-3.86

4.41

4.50

10.46

12.88

14.92

20.49

14.19

0.95

0.94

0.90

0.94

0.95

0.92

0.93

0.94

0.89

C/P & Sales

Rank

-0.02

-0.06

-0.02

-0.14

0.00

0.07

-0.17

-0.02

0.04

1.11

1.01

1.02

1.12

1.02

1.00

1.13

1.04

1.00

0.46

0.42

0.72

0.63

0.46

0.53

0.80

0.64

0.92

-0.36

-0.12

0.14

0.17

0.26

0.34

0.62

0.62

0.68

t(a)

-0.27

-1.03

-0.24

-1.93

0.08

0.95

-1.73

-0.34

0.34

t(b)

64.04

65.82

40.20

63.31

67.96

52.28

45.55

58.48

36.63

t(s)

18.37

19.12

19.86

24.77

21.34

19.47

22.57

25.32

23.47

t(h)

-12.71

-4.90

3.42

5.82

10.61

10.84

15.21

21.64

15.40

0.95

0.95

0.89

0.95

0.95

0.92

0.91

0.94

0.88

1.22 0.284

1.06 0.394

1.04 0.405

after portfolio formation ranges from -0.00 percent for the decile of stocks with the worst short-term past returns (measured from 12 to 2 months before portfolio formation) to 1.31 percent for the decile with the best short-term past



Table VI

Average Monthly Excess Returns (in Percent) on Equal-Weight NYSE Deciles Formed Monthly Based on Continuously Compounded

Past Returns

At the beginning of each month t, all NYSE firms on CRSP with returns for months t - x to t - у are 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 continuously compounded returns for January 1930 through November 1930. Decile 1 contains the NYSE stocks with the lowest continuously compounded past returns. The portfolios are reformed monthly, and equal-weight simple returns in excess of the one-month bill rate are calculated for January 1931 (3101) to December 1993 (9312). The table shows the averages of these excess returns for 6307 to 9312 (366 months) and 3101 to 6306 (390 months).

Portfolio Average Excess Returns Formation -

Period

Months

6307-9312

12-2

-0.00

0.46

0.61

0.55

0.72

0.68

0.85

0.90

1.08

1.31

6307-9312

24-2

0.36

0.60

0.59

0.66

0.71

0.81

0.73

0.80

0.93

1.05

6307-9312

36-2

0.46

0.60

0.77

0.69

0.73

0.81

0.69

0.78

0.84

0.97

6307-9312

48-2

0.66

0.70

0.77

0.74

0.71

0.71

0.72

0.71

0.72

0.89

6307-9312

60-2

0.86

0.76

0.73

0.75

0.70

0.71

0.74

0.70

0.66

0.73

6307-9312

60-13

1.16

0.81

0.77

0.76

0.74

0.72

0.72

0.73

0.54

0.42

3101-6306

12-2

1.49

1.52

1.32

1.49

1.39

1.45

1.45

1.55

1.58

1.87

3101-6306

24-2

2.24

1.60

1.57

1.70

1.41

1.31

1.32

1.24

1.26

1.46

3101-6306

36-2

2.31

1.74

1.65

1.46

1.40

1.40

1.32

1.23

1.27

1.36

3101-6306

48-2

2.34

1.81

1.62

1.60

1.37

1.30

1.33

1.22

1.24

1.26

3101-6306

60-2

2.49

1.78

1.74

1.50

1.39

1.33

1.27

1.18

1.28

1.14

3101-6306

60-13

2.62

1.85

1.63

1.61

1.43

1.24

1.34

1.28

1.08

1.01

returns. (Skipping the portfolio formation month in ranking stocks reduces bias from bid-ask bounce.)

Table VI also confirms that average returns tend to reverse when portfolios are formed using returns for the four years from 60 to 13 months prior to portfolio formation. For these portfolios, the average return in the month after portfolio formation ranges from 1.16 percent for the decile of stocks with the worst long-term past returns to 0.42 percent for stocks with the best past returns. In the 1963-1993 results, however, long-term return reversal is observed only when the year prior to portfolio formation is skipped in ranking stocks. When the preceding year is included, short-term continuation offsets long-term reversal, and past losers have lower future returns than past winners for portfolios formed with up to four years of past returns.

Can our three-factor model explain the patterns in the future returns for 1963-1993 on portfolios formed on past returns? Table VTI shows that the answer is yes for the reversal of long-term returns observed when portfolios are formed using returns from 60 to 13 months prior to portfolio formation. The regressions ofthe post-formation returns on these portfolios on RM - Rf, SMB, and HML produce intercepts that are close to 0.0 both in absolute terms and on the GRS test. The three-factor model works because long-term past losers



Table VII

Three-Factor Regressions for Monthly Excess Returns (in Percent) on Equal-Weight NYSE Portfolios Formed on Past Returns: 7/63-12/93, 366 Months

Rt- Rf= at + bt(RM - Rf) + sSMB + ЛДШЬ + et

The formation of the past-return deciles is described in Table VI. Decile 1 contains the NYSE stocks with the lowest continuously compounded returns during the portfolio-formation period (12-2, 48-2, or 60-13 months before the return month). t() is a regression coefficient divided by its standard error. The regression R2s are adjusted for degrees of freedom. GRS is the F-statistic of Gibbons, Ross, and Shanken (1989), testing the hypothesis that the regression intercepts for a set of ten portfolios are all 0.0. p(GRS) is the p-value of GRS.

123456789 10 GRS p(GRS)

Portfolio formation months are

M2 to t-

-1.15

-0.39

-0.21

-0.22

-0.04

-0.05

0.12

0.21

0.33

0.59

1.14

1.06

1.04

1.02

1.02

1.02

1.04

1.03

1.10

1.13

1.35

0.77

0.66

0.59

0.53

0.48

0.47

0.45

0.51

0.68

0.54

0.35

0.35

0.33

0.32

0.30

0.29

0.23

0.23

0.04

t(a)

-5.34

-3.05

-2.05

-2.81

-0.54

-0.93

1.94

3.08

3.88

4.56

t(b)

21.31

33.36

42.03

51.48

61.03

73.62

68.96

62.67

51.75

35.25

t(s)

17.64

16.96

18.59

20.87

22.06

23.96

21.53

19.03

16.89

14.84

t(h)

6.21

6.72

8.74

10.18

11.86

13.16

11.88

8.50

6.68

0.70

0.75

0.85

0.89

0.92

0.94

0.96

0.95

0.94

0.92

0.86

Portfolio formation months are

J-48 to t-

-0.73

-0.32

-0.09

-0.08

-0.05

-0.00

0.07

0.10

0.15

0.37

1.16

1.12

1.06

1.05

1.02

1.01

1.00

0.99

1.04

1.11

1.59

0.87

0.64

0.52

0.48

0.42

0.41

0.40

0.42

0.49

0.90

0.60

0.44

0.44

0.36

0.31

0.18

0.11

-0.05

-0.26

t(a)

-2.91

-2.79

-0.96

-0.99

-0.67

-0.01

1.08

1.46

2.09

3.60

18.61

39.22

46.55

53.19

57.82

63.78

64.72

58.62

57.02

43.37

t(s)

17.91

21.36

19.68

18.61

19.17

18.51

18.52

16.61

16.22

13.40

t(h)

8.91

12.94

11.93

13.78

12.61

11.87

7.34

4.19

-1.55

-6.35

0.73

0.88

0.91

0.92

0.93

0.94

0.95

0.93

0.94

0.90

Portfolio formation months are

J-60 to M3

-0.18

-0.16

-0.13

-0.07

0.00

0.02

0.06

0.10

-0.07

-0.12

1.13

1.09

1.07

1.04

0.99

1.00

1.00

1.01

1.06

1.15

1.50

0.83

0.67

0.59

0.47

0.38

0.35

0.40

0.45

0.50

0.87

0.54

0.50

0.42

0.34

0.29

0.23

0.13

-0.00

-0.26

t(a)

-0.80

-1.64

-1.69

-0.99

0.02

0.40

0.96

1.43

-0.92

-1.36

20.24

44.40

55.03

61.09

63.79

65.68

62.58

58.26

60.49

53.04

t(s)

18.77

23.63

24.09

24.06

21.21

17.44

15.43

16.18

18.06

16.33

t(h)

9.59

13.67

15.94

15.31

13.46

11.82

8.98

4.46

-0.14

-7.50

0.75

0.91

0.93

0.94

0.94

0.94

0.94

0.93

0.94

0.93

4.45 0.000

2.02 0.031

1.29 0.235

load more on SMB and HML. Since they behave more like small distressed stocks, the model predicts that the long-term past losers will have higher average returns. Thus, the reversal of long-term returns, which has produced so much controversy (DeBondt and Thaler (1985, 1987), Chan (1988), Ball and



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