average and standard deviation of the return do not differ much between Tables 2 and 3. On the other hand, the two tables do differ considerably in the average as well as in the
standard deviation when the beta is concerned. The average of beta at Table 3 is in general smaller than that at Table 2 by about 0.1. The standard deviation of beta at Table 3 tends
to be larger than that at Table 2. Therefore, beta tends to take smaller values but to vary more when the VWI is used as the market return index compared to when the EWI is used
as the market return index. In addition, beta is considerably different between Tables 2 and 3 in the last two subperiods; the average beta becomes larger than 1 at the 17th portfolio
in the third subperiod and at the 20th portfolio in the last subperiod at Table 3 while it occurs around the middle 10th portfolio both in the first two subperiods. This result on
beta implies the majority of 20 portfolios do not vary so much as the VWI. We consider this indicates in the last two subperiods the nonmanufacturing sector, particularly the
financial sector, dominates other sectors with respect to variation of the stock prices so that other sectors’ stock prices do not fluctuate so much as the VWI, heavily influenced
by the nonmanufacturing sector or the financial sector. Because of this phenomenon, we consider the VWI is not an appropriate proxy for the market return to be used to find the
relationship between return and beta. Thus, we omit presenting results obtained with the VWI as the market return index.
Tables 4 and 5 present, respectively, summary statistics of the 25 portfolios obtained using the EWI and VWI as the market return index from July 1962 to December 1995.
They give the time series average and standard deviation of portfolio returns, betas, sizes, and book to market equity ratios. A similar difference of the average and standard
deviation of beta between the EWI and VWI also exists in Tables 4 and 5 as in Tables 2 and 3. Therefore, we also present only results obtained with the EWI as the market return
index for the model 4.
IV. Empirical Results
Table 6 gives summary statistics in the total sample period of the 20 portfolios when the market excess return is positive and negative, using the EWI as the market return index.
Figure 2 is a scatter diagram obtained from the average portfolio return and the average
Figure 1. Relation between return and beta obtained with the EWI market return index January; 1956-December 1995.
524 J. Hodoshima et al.
portfolio beta in 20 portfolios when the market excess return is positive as well as negative, taken from Table 6. From Figure 2, we can easily recognize clear ex post
positive and negative linear relationships between return and beta when the market is up and down. Comparison of Figures 1 and 2 naturally motivates us to differentiate up
markets from down markets in empirical investigations of the relationship between return and beta.
We first show the cross-sectional regression results for the unconditional as well as conditional relationships of the simple model 3 in the total sample and the four
subsamples. Tables 7 and 8 present the time series average estimate of intercept and slope given, respectively, by the average of the month-by-month cross-sectional regression
intercept and slope estimates, the t-statistic given by the time series average estimate divided by its time series standard deviation, and the time series average estimate of R
2
and standard error of the equation given, respectively, by the time series average of the month-by-month cross-sectional regression R
2
s and standard errors of the equation. Table 7 provides the cross-sectional regression results for the unconditional relationship of the
Table 3. Summary Statistics of the VWI Based Portfolios January 1956 –December 1995
Portfolio 1
2 3–4
5–6 7–8
9–10 11–12 13–14 15–16 17–18 19
20 Returns
Average 1956–95
0.015 0.014 0.015 0.015 0.015 0.015 0.016 0.016 0.016 0.017 0.016 0.015 1956–65
0.018 0.017 0.020 0.019 0.017 0.017 0.018 0.018 0.022 0.021 0.020 0.022 1966–75
0.019 0.017 0.017 0.017 0.016 0.016 0.019 0.020 0.017 0.020 0.020 0.022 1976–85
0.014 0.013 0.014 0.013 0.015 0.015 0.016 0.016 0.015 0.015 0.016 0.012 1986–95
0.007 0.008 0.009 0.011 0.009 0.010 0.010 0.009 0.008 0.009 0.007 0.005 Standard deviation
1956–95 0.052 0.050 0.054 0.055 0.056 0.057 0.059 0.059 0.060 0.062 0.065 0.071
1956–65 0.042 0.038 0.050 0.052 0.057 0.056 0.061 0.065 0.067 0.071 0.075 0.085
1966–75 0.050 0.045 0.048 0.052 0.050 0.055 0.057 0.057 0.057 0.059 0.067 0.071
1976–85 0.031 0.028 0.027 0.029 0.029 0.032 0.033 0.034 0.035 0.036 0.039 0.041
1986–95 0.075 0.077 0.077 0.077 0.076 0.077 0.076 0.073 0.072 0.073 0.072 0.079
Beta Average
1956–95 0.49
0.59 0.68
0.74 0.77
0.85 0.92
0.95 1.01
1.07 1.22
1.25 1956–65
0.42 0.51
0.70 0.84
0.92 0.99
1.08 1.07
1.19 1.19
1.39 1.35
1966–75 0.53
0.66 0.77
0.83 0.86
1.00 1.05
1.07 1.10
1.11 1.29
1.29 1976–85
0.41 0.47
0.50 0.53
0.55 0.65
0.74 0.87
0.89 1.10
1.26 1.29
1986–95 0.62
0.75 0.77
0.79 0.80
0.79 0.81
0.83 0.86
0.91 0.93
1.10 Standard deviation
1956–95 0.31
0.30 0.28
0.29 0.29
0.31 0.29
0.29 0.27
0.29 0.31
0.34 1956–65
0.19 0.19
0.19 0.21
0.18 0.17
0.20 0.12
0.11 0.24
0.17 0.28
1966–75 0.19
0.15 0.19
0.23 0.17
0.13 0.17
0.28 0.23
0.24 0.31
0.43 1976–85
0.42 0.43
0.32 0.33
0.31 0.35
0.29 0.32
0.20 0.31
0.27 0.37
1986–95 0.35
0.29 0.28
0.27 0.31
0.34 0.29
0.28 0.30
0.26 0.23
0.18
Data for portfolios 3–4, 5–6, etc. denote the average of the two portfolios.
Cross-Sectional Analysis of Return and Beta 525
Table 4. Summary Statistics of the 25 Portfolios Beta Estimated with an EWI July 1962– December 1995
Returns Average
Standard Deviation Beta Quintile
Beta Quintile 1
2 3
4 5
1 2
3 4
5 1
1.8 1.9
1.9 1.9
1.8 1
0.060 0.066
0.073 0.076
0.081 ME
2 1.3
1.5 1.4
1.6 1.6 ME
2 0.054
0.060 0.064
0.071 0.078
Quintile 3
1.3 1.3
1.2 1.2
1.2 Quintile
3 0.052
0.058 0.061
0.067 0.075
4 1.1
1.1 1.1
1.2 1.0
4 0.050
0.053 0.055
0.063 0.067
5 1.0
1.0 1.0
1.1 0.9
5 0.047
0.050 0.057
0.059 0.061
Beta Average
Standard Deviation Beta Quintile
Beta Quintile 1
2 3
4 5
1 2
3 4
5 1
0.48 0.84
1.07 1.30
1.65 1
0.13 0.11
0.10 0.09
0.11 ME
2 0.52
0.86 1.08
1.32 1.69 ME
2 0.12
0.07 0.05
0.09 0.19
Quintile 3
0.49 0.85
1.06 1.29
1.65 Quintile
3 0.13
0.07 0.04
0.08 0.17
4 0.42
0.74 0.94
1.15 1.52
4 0.14
0.08 0.06
0.09 0.17
5 0.33
0.63 0.83
1.02 1.35
5 0.13
0.08 0.08
0.10 0.15
ln ME Average
Standard Deviation Beta Quintile
Beta Quintile 1
2 3
4 5
1 2
3 4
5 1
8.8 8.8
8.8 8.8
8.8 1
1.15 1.14
1.13 1.10
1.10 ME
2 9.5
9.5 9.5
9.5 9.5
ME 2
1.09 1.07
1.05 1.09
1.07 Quintile
3 10.2
10.2 10.2
10.2 10.1
Quintile 3
1.04 1.05
1.05 1.05
1.04 4
10.9 10.9
10.9 10.8
10.8 4
1.04 1.05
1.04 1.06
1.07 5
12.5 12.3
12.3 12.2
12.0 5
1.27 1.16
1.00 1.06
1.08 ln BM
Average Standard Deviation
Beta Quintile Beta Quintile
1 2
3 4
5 1
2 3
4 5
1 20.47
20.44 20.50
20.59 20.65
1 0.50
0.51 0.55
0.56 0.58
ME 2
20.50 20.50
20.55 20.62
20.73 ME 2
0.48 0.44
0.41 0.45
0.44 Quintile
3 20.57
20.54 20.61
20.67 20.76
Quintile 3
0.38 0.38
0.33 0.36
0.39 4
20.68 20.60
20.60 20.65
20.78 4
0.35 0.32
0.35 0.31
0.36 5
20.57 20.73
20.76 20.77
20.86 5
0.47 0.33
0.33 0.38
0.38
Sample includes all nonfinancial firms of the first section of the Tokyo Stock Exchange. Calculations of the average and standard deviation use 402 observations for return and 34 observations for beta, logarithm
of market equity ME and logarithm of book to market ratio BM.
526 J. Hodoshima et al.
Table 5. Summary Statistics of the 25 Portfolios Beta Estimated with a VWI July 1962– December 1995
Returns Average
Standard Deviation Beta Quintile
Beta Quintile 1
2 3
4 5
1 2
3 4
5 1
1.8 1.9
1.8 1.7
2.0 1
0.064 0.067
0.070 0.075
0.081 ME
2 1.2
1.5 1.5
1.6 1.6 ME
2 0.058
0.061 0.067
0.070 0.073
Quintile 3
1.2 1.3
1.3 1.2
1.1 Quintile
3 0.054
0.059 0.063
0.065 0.073
4 1.1
1.2 1.2
1.0 1.1
4 0.050
0.054 0.056
0.061 0.068
5 1.1
1.0 1.0
1.0 0.9
5 0.047
0.052 0.055
0.059 0.066
Beta Average
Standard Deviation Beta Quintile
Beta Quintile 1
2 3
4 5
1 2
3 4
5 1
0.21 0.55
0.75 0.95
1.29 1
0.34 0.35
0.36 0.36
0.35 ME
2 0.28
0.62 0.83
1.06 1.44 ME
2 0.37
0.32 0.32
0.33 0.36
Quintile 3
0.32 0.67
0.88 1.10
1.49 Quintile
3 0.34
0.32 0.32
0.32 0.33
4 0.35
0.66 0.89
1.11 1.50
4 0.26
0.23 0.20
0.23 0.29
5 0.37
0.73 1.02
1.28 1.64
5 0.20
0.11 0.11
0.17 0.29
ln ME Average
Standard Deviation Beta Quintile
Beta Quintile 1
2 3
4 5
1 2
3 4
5 1
8.8 8.8
8.8 8.8
8.9 1
1.15 1.13
1.12 1.11
1.09 ME
2 9.5
9.5 9.5
9.5 9.6
ME 2
1.08 1.07
1.07 1.07
1.07 Quintile
3 10.1
10.2 10.2
10.2 10.2
Quintile 3
1.03 1.05
1.04 1.06
1.05 4
10.8 10.8
10.9 10.8
10.8 4
1.04 1.04
1.04 1.06
1.07 5
12.2 12.1
12.2 12.3
12.5 5
1.13 1.15
1.06 1.20
1.16 ln BM
Average Standard Deviation
Beta Quintile Beta Quintile
1 2
3 4
5 1
2 3
4 5
1 20.51
20.48 20.49
20.51 20.53
1 0.53
0.52 0.52
0.54 0.56
ME 2
20.59 20.52
20.55 20.57
20.66 ME 2
0.51 0.45
0.43 0.42
0.41 Quintile
3 20.61
20.60 20.61
20.64 20.71
Quintile 3
0.41 0.35
0.36 0.36
0.38 4
20.68 20.64
20.62 20.64
20.73 4
0.31 0.33
0.35 0.34
0.34 5
20.58 20.73
20.80 20.76
20.84 5
0.47 0.31
0.33 0.36
0.40
Sample includes all nonfinancial firms of the first section of the Tokyo Stock Exchange. Calculations of the average and standard deviation use 402 observations for return and 34 observations for beta, logarithm
of market equity ME and logarithm of book to market ratio BM.
Cross-Sectional Analysis of Return and Beta 527
simple model 3. The results are similar to Hawawini 1991, showing slope or the coefficient of beta is not significant when the difference between up markets and down
markets is not taken into account. The slope estimate takes positive values as well as negative values but it is always insignificant. The intercept estimate is all positive and
significant except the last subsample. The R
2
s are around 0.2, ranging from 0.196 in the second subsample to 0.290 in the first subsample. The standard errors of the equation are
around 0.02. Table 8 gives the cross-sectional regression results for the conditional
Table 6. Relation Between Return and Beta with the Difference of Up Markets and Down Markets for the EWI Based Portfolios January 1956 –December 1995
Portfolio Beta
U-return D-return
1 0.61
0.038 20.022
2 0.63
0.037 20.021
3 0.72
0.041 20.024
4 0.77
0.042 20.026
5 0.85
0.044 20.029
6 0.86
0.043 20.029
7 0.90
0.043 20.031
8 0.94
0.046 20.033
9 0.95
0.044 20.032
10 1.02
0.047 20.033
11 1.04
0.047 20.035
12 1.07
0.050 20.036
13 1.09
0.052 20.035
14 1.09
0.049 20.036
15 1.14
0.052 20.038
16 1.20
0.052 20.036
17 1.25
0.052 20.039
18 1.25
0.054 20.041
19 1.32
0.053 20.043
20 1.37
0.056 20.045
The average return and average beta when the market excess return is positive and negative are given for 20 portfolios. U-return and D-return denote, respectively, the average return when the market is up and down.
Figure 2. Relation between return and beta obtained with the EWI market return index when the difference between up and down markets is introduced January 1956-December 1995.
528 J. Hodoshima et al.
relationships of the simple model 3. The slope estimate becomes always significant and takes positive values in up markets and negative values in down markets in all of the
sample periods. The intercept estimate of g
0up
is positive and significant in all of the sample periods while that of g
0down
is significant only in the first subsample with positive sign and the fourth subsample with negative sign and insignificant otherwise. The results
of these intercept and slope estimates for the conditional relationships of the simple model 3 imply the unique intercept assumption made by Pettengill et al. 1995 is rejected in
all but the first subsample because the two intercepts in the up market and down market are determined significantly different.
Table 7. Cross-Section Regression Average Estimates, t-Statistics. Average R
2
and Average Standard Error of the Equation SE for the Unconditional Relationship of Equation 3 January
1956 –December 1995
Number of Months
g t-Statistic
g
1
t-Statistic R
2
SE Total sample
1956–95 480
0.014 6.37
0.001 0.43
0.243 0.018
Subsamples 1956–95
120 0.019
5.51 20.000
0.02 0.290
0.021 1966–75
120 0.018
4.31 20.000
20.06 0.196
0.020 1976–85
120 0.012
3.57 0.003
0.79 0.237
0.015 1986–95
120 0.008
1.30 0.002
0.32 0.249
0.014
Significant at the 5 level. In each column of t-statistic, t-statistics are given for the test whether the coefficient on the left is 0 or not.
Table 8. Cross-Section Regression Average Estimates, t-Statistics. Average R
2
and Average Standard Error of the Equation SE for the Conditional Relationships of Equation 3 January
1956 –December 1995
Number of Months
g t-Statistic
g
1
t-Statistic R
2
SE Total sample
Up 291
0.026 8.67
0.021 6.90
0.233 0.020
Down 189
20.003 20.94
20.030 29.12
0.258 0.015
Subsamples Up
1956–95 75
0.022 4.71
0.030 5.15
0.253 0.024
1966–75 78
0.029 5.26
0.017 3.10
0.179 0.022
1976–85 75
0.017 4.04
0.013 3.03
0.261 0.016
1986–95 63
0.036 3.95
0.026 2.89
0.244 0.016
Down 1956–65
45 0.014
2.87 20.051
29.45 0.352
0.018 1966–75
42 20.002
20.32 20.033
24.75 0.228
0.016 1976–85
45 0.003
0.62 20.014
22.69 0.198
0.014 1986–95
57 20.022
23.17 20.025
23.47 0.254
0.012
Significant at the 5 level. Up and down denote, respectively, up-month and down-month periods for the total sample and subsamples.
Cross-Sectional Analysis of Return and Beta 529
We also have the two equality test results by the t test;
5
the two equality hypotheses are given by g
0up
5 g
0down
and g
1up
5 2g
1down
. The intercept equality hypothesis is rejected at the 5 significance level in all but one case with the exception of the first subsample,
which can be anticipated from Table 8. Neither Pettengill et al. 1995 nor Fletcher 1997 carry out testing the intercept equality hypothesis so that comparison with ours is not
feasible. On the other hand, the slope equality hypothesis is on the contrary accepted at the 5 significance level in all but the first subsample. Therefore, the first subsample has a
different characteristic compared to the rest of the sample. This is closer to the U.S. evidence by Pettengill et al. 1995 where the slope equality hypothesis is always accepted
and hence the symmetrical relation between return and beta in up markets and down markets is supported, compared to the U.K. evidence by Fletcher 1997 where the slope
equality hypothesis is rejected in the total sample and also in one of the two subsamples.
The summary statistics of goodness of fit such as the average R
2
and the average standard error of the equation in the cross-sectional regression approach are not much
different in the unconditional and conditional relationships of the simple model 3 because they are averages in any case. In the cross-sectional estimation of the conditional
relationships of the simple model 3, the goodness of fit measures given by the R
2
and standard error of the equation are better in down markets than in up markets in all but the
third subsample; the conditional relationship is better fit in the down market than in the up market except the third subsample. Neither Fletcher 1997 nor Pettengill et al. 1995
provide the goodness of fit measures so that such interpretation does not exist in them. We consider the goodness of fit measures are too important to be omitted in every regression
analysis. Our interpretation based on the goodness of fit measure is in contrast to Fletcher 1997, who states the conditional relationship is stronger in the down market than in the
up market based on the bigger absolute value of the slope estimate in the down market than in the up market. We consider that the strength of the relationship between return and
beta is more appropriately measured by the goodness of fit measure than the magnitude of the absolute value of the slope estimate.
Table 9 shows seasonality in the cross-sectional regression analysis for the uncondi- tional and conditional relationships of the simple model 3 in the total sample, from
January 1956 to December 1995. It shows the slope is positive and highly significant in the months of January and the unconditional relationship is better fit in the months of
5
The results are omitted for the sake of brevity.
Table 9. Seasonality in the Cross-Sectional Regression Analysis for the Unconditional and Conditional Relationships of Equation 3 January 1956 –December 1995
Number of Months
g t-Statistic
g
1
t-Statistic R
2
SE January
40 0.018
1.99 0.032
3.17 0.302
0.022 Non-January
440 0.014
6.04 20.002
20.67 0.238
0.017 Jan up
35 0.022
2.46 0.041
3.96 0.305
0.023 Jan down
5 20.012
20.37 20.024
20.76 0.283
0.014 NJ up
256 0.026
8.33 0.018
5.85 0.224
0.019 NJ down
184 20.003
20.86 20.030
29.17 0.258
0.015
Significant at the 5 level. January and non-January denote, respectively, months of January and non-January. Similarly Jan up down and NJ up
down denote, respectively, up down-months of January and non-January.
530 J. Hodoshima et al.
January than in the months of non-January. It also shows January is exceptional in the sense that 35 out of 40 months of January are up-months and that the conditional
relationship is better fit in the up market than in the down market. On the other hand, in the months of non-January the unconditional and conditional relationships are similar to
those in the total sample given at Tables 7 and 8. t values for the intercept and slope equality hypotheses are 1.01 and 0.51, respectively, in the months of January, making the
two hypotheses accepted while they are 6.20 and 22.67, respectively, in the months of non-January, making the two hypotheses rejected, but not so different from the total
sample testing results.
Table 10 presents the results of the cross-sectional regression analysis for the extended model 4 based on the EWI 25 portfolios sorted on ME and then on beta in the total
sample from July 1962 to December 1995. Beta is not significant, size is significant with a negative coefficient, and book to market equity ratio is not significant in the uncondi-
tional relationship of the extended model 4. In the conditional relationships of the extended model 4, beta becomes significantly positive and negative in the up and down
markets, respectively, the size becomes significantly negative in the up markets but insignificant in down markets, the book to market equity ratio remains insignificant in
both markets. In the months of January, all of the three explanatory variables become significant with positive coefficients for beta and book to market equity ratio and a
negative coefficient for size, which is conformable to Hawawini 1991. In other months, the result is similar to that of the total sample with one exception that the size, the only
one explanatory variable significant in the total sample, becomes insignificant. In the months of January, the up market conditional relationship does not differ much from the
unconditional relationship whereas the down market conditional relationship makes all of
Table 10. Cross-Sectional Regression Results for the Extended Model 4 Based on the 25 Portfolios Sorted on ME and Then on EWI Beta July 1962–December 1995
Number of g
g
1
g
2
g
3
Months t-Stat
t-Stat t-Stat
t-Stat R
2
SE Total
402 0.0323
0.0007 20.0017
0.0037 0.500
0.0194 3.87
0.41 22.40
1.38 Up
234 0.0771
0.0143 20.0039
0.0035 0.481
0.0216 7.12
6.28 24.23
0.92 Down
168 20.0300
20.0181 0.0015
0.0039 0.526
0.0164 22.60
28.29 1.54
1.11 January
33 0.1181
0.0249 20.0083
0.0301 0.566
0.0228 4.27
4.52 23.55
2.66 Non-January
369 0.0247
20.0014 20.0011
0.0013 0.494
0.0191 2.85
20.77 21.49
0.49 Jan up
28 0.1508
0.0277 20.0104
0.0308 0.554
0.0239 6.21
4.73 24.81
2.41 Jan down
5 20.0646
0.0094 0.0035
0.0261 0.635
0.0167 20.71
0.60 0.43
1.10 NJ up
206 0.0671
0.0125 20.0031
20.0002 0.471
0.0212 5.74
5.12 23.06
20.05 NJ down
163 20.0289
20.0190 0.0015
0.0032 0.523
0.0164 22.49
28.70 1.48
0.90
Significant at the 5 level. January and non-January denote, respectively, months of January and non-January. Similarly Jan up down and NJ up
down denote, respectively, up down-months of January and non-January.
Cross-Sectional Analysis of Return and Beta 531
the three explanatory variables insignificant. In other months, the two conditional rela- tionships become similar to those in the total sample. The slope equality test for beta is
accepted in the total sample but rejected in the months of both January and non-January with t-values of 21.22, 2.22, and 21.99, respectively, which coincides with the result for
the simple model 3 in the total sample and months of non-January but not in the months of January. The intercept equality test is rejected in all of the three samples of the total
sample and the months of January and non-January with t values of 6.77, 2.29, and 5.83, respectively, which again coincides with the result for the simple model 3 in the total
sample and months of non-January except in the months of January.
In terms of the goodness of fit measures, the conditional relationship for the extended model 4 is consistently better fit in down markets than in up markets in the total sample
and in the months of January and non-January, which is similar to the case for the simple model 3 except for the months of January. Consequently, properties of the unconditional
and conditional relationships for the extended model 4 are in general similar to those for the simple model 3 with respect to return and beta.
Overall, beta becomes quite suitable to explain return in the conditional relationships both for the simple model 3 and the extended model 4. Therefore, we consider it
appropriate to divide the sample into two parts of the up and down markets for the relevance of beta. But the same does not apply to other explanatory variables of the size
and book to market equity ratio, which seems not to be surprising at all because the distinction of the up and down markets is made to make beta, not other explanatory
variables, relevant to explain return. Although the book to market equity ratio has been recognized to be strong to explain average return [cf, e.g., Chan et al. 1991 and
Jagannathan et al. 1998], it does not appear to help explain return except January and up months of January in our data.
6
On the other hand, the size is negatively related to return in the total sample, up market, and the months of January while not significant in down
market and the months of non-January.
V. Conclusion
