where d
t
denotes a dummy variable which takes 1 and 0 when the market is up and down, respectively. On the other hand, our specification of the conditional relationships for
Equation 2 can be written as: ER
pt
5 g
0up
d
t
1 g
0down
~1 2 d
t
1 g
1up
d
t
b
p
1 g
1down
~1 2 d
t
b
p
p 5 1, . . . , N; t 5 1, . . . , T. 8
The difference between 7 and 8 lies in the intercept specification; the intercept is allowed to differ in Equation 8. The slope specification difference when the market is up
and down is apparently the most important in the conditional relationships and the intercept specification difference between 7 and 8 may not matter much. However, we
prefer Equation 8 to Equation 7 by the following two reasons. The first reason is because summary statistics of goodness of fit such as R
2
, adjusted R
2
, and the standard error of the equation cannot be unambiguously defined in the cross-sectional regression
estimation of Equation 7. Neither summary statistics for the total sample nor two sets of summary statistics for the up-market months and the down-market months are appropriate
in the cross-sectional regression estimation of Equation 7. On the other hand, in the cross-sectional regression estimation of Equation 8 two sets of summary statistics of
goodness of fit, obtained from averaging the month-by-month regression summary sta- tistics in the up-market months and the down-market months, are quite relevant. The
second reason is because we consider Equation 8 is a more flexible and natural model than Equation 7; intercept in the up market months may or may not be the same as that
in the down market months and we can decide, by testing equality of g
0up
and g
0down
, which equation, 7 or 8, fits the data better. The same reasoning applies to the
specification of the conditional relationships for Equation 4. Thus, we also allow the intercept to differ in the up market months and down market months in the specification
of the conditional relationships for Equation 4.
III. Data
In this paper we use data from the Japanese stock market. We use monthly returns for stocks listed on the first section of the Tokyo Stock Exchange TSE. The return data come
from a standardized database, similar to the Center for Research in Securities Prices at the University of Chicago, obtained from the Japanese Securities Research Institute JSRI.
Returns are adjusted for dividends and capital modifications. The TSE consists of the first and second sections and the first section typically lists bigger firms. The return data
include all the companies listed on the first section of the TSE. Information about stock prices is taken from the Toyo Keizai database. Accounting information for nonfinancial
companies listed on the first section of the TSE comes from the Japan Development Bank Kaigin database. As the risk free rate, we use monthly average of the next day call
money rates with collateral, obtained from the Nikkei NEEDS database supplied by Nihon Keizai Shimbunsha, mainly for continuity of its observations during the total sample
period for the returns.
To verify the simple model 3 of return and beta, we only use the monthly return data from the JSRI and the risk free rate. On the other hand, to verify the extended model 4
we use all nonfinancial companies listed on the first section of the TSE that have data on the three databases mentioned above. We match accounting and market data information
520 J. Hodoshima et al.
in the following way. The Kaigin file for year t contains accounting data for fiscal year-ends mostly from April of year t - 1 to March of year t. We assume that three months
is enough time for financial information to be disseminated to investors, and relate accounting data for year t with returns from July of year t to June of year t 1 1. ME is
calculated every June using the number of outstanding stocks and the stock price at the end of June. The BEME ratio is also calculated at the end of June’s stock prices. Monthly
cross-sectional regressions are run from July of a year to June of the next year in the extended model 4. ME and the ratio BEME that use stock prices are held constant for
a year.
To verify unconditional as well as conditional relationships of return and beta of Equation 3, our source of data are just return data for all the companies listed on the first
section of the TSE obtained from the JSRI from January 1952 to December 1995. On the other hand, to investigate unconditional as well as conditional relationships of the
extended model of Equation 4, we use all nonfinancial companies listed on the first section of the TSE that have data on the three databases explained above. As a result, our
source of data starts in July 1960 and ends in December 1995 for the extended model of Equation 4. Our data cover a relatively long period compared to other studies on the
Japanese stock returns such as Chan et al. 1991
1
, which cover January 1971 to December 1985, Jagannathan et al. 1998, which cover from September 1981 to December 1993,
and Hawawini 1991, which covers January 1955 to December 1985. Therefore, the present study reveals characteristics of the Japanese stock market during this relatively
long period.
We use two proxies of the market return, a value weighted index VWI, provided by the JSRI for all the firms listed on the first section of the TSE, and an equally weighted
index EWI of all the firms in the sample. An EWI is known to explain small stocks better than a VWI so that the use of the EWI as the market return index in the CAPM tests
usually results in a better description of the data. Because the nonmanufacturing sector, particularly financial sector, is very large in Japan, the VWI is heavily influenced by the
nonmanufacturing sector or the financial sector. Therefore, the VWI may not be a good proxy for the market return in the Japanese stock market.
The portfolio returns and betas for the simple model 3 are constructed as follows in three steps. In the first step, we estimate, using two years of data, beta for each individual
stock by regressing the stock return on the constant term and the market return and then construct 20 portfolios of stocks based on the ranking of the estimated betas. In the second
step, we re-estimate, using the next two years of data, beta for each portfolio by the average of re-estimated betas of the stocks assigned to that portfolio. In the last step, using
another two years of new data we assign stocks to the portfolios formed in the first step and obtain the portfolio return by averaging returns of the stocks belonged to each
portfolio. Then we use the portfolio returns in the last step and the portfolio betas in the second step for estimation, testing, and comparison of the unconditional and conditional
relationships of the simple model 3. Discarding the earliest two years of data and adding new two years of data, we repeat this three step procedure of six years.
2
Thus, we totally
1
Chan et al. 1991 also use firms listed on the second section of the TSE so that proportion of smaller firms is larger in their study.
2
In the first version of the paper, we used another three step procedure of six years with two years in the first step, three years in the second step, and one year in the last step, following Hawawini 1991 which tests
the CAPM and other modified models of the CAPM using monthly returns for stocks listed on the first section
Cross-Sectional Analysis of Return and Beta 521
obtain 40 years or 480 months of data from January 1956 to December 1995 for the inference of the model 3. We use two years of data to construct portfolios and two years
of data to estimate betas of the portfolios. Consequently four years of data are lost on the initial preparation. But the four years’ data lost are a short period compared to other
studies so that we can still reserve for the inference relatively large observations of 40 years as the Japanese data of monthly returns. We remark that we do not need to take into
account the difference of up markets and down markets
3
when we estimate beta for each stock in the first and second steps because beta is treated as a parameter to be estimated
in these steps instead of an explanatory variable in the inference of the model 3. The portfolios for the extended model 4 are constructed as follows. In the first step,
we estimate, using two to five years of previous data up to June of year t, beta for each individual stock by regressing the stock return on the constant term and the market return
and then construct 25 portfolios of stocks of one year from July of year t to June of year t 1 1, sorted on the value of ME and then on the value of beta; first construct 5 portfolios
based on ME and within each portfolio sorted by ME construct 5 portfolios based on beta. In the next step, we estimate, test, and compare the unconditional and conditional
relationships of the model 4 using one year 25 portfolio data from July of year t to June of year t 1 1 of return, beta, lnME, and lnBEME, constructed in the first step; each
portfolio of every variable is obtained by averaging values of the variable assigned to each portfolio. Discarding the earliest one year of data and adding new one year of data, we
repeat this two step procedure to obtain 402 months of data from July 1962 to December 1995 for the inference of the extended model 4. Therefore, we do not have a step of
re-estimating beta in the portfolio formation for the extended model 4. This is mainly to make the explanatory variables of size and book to market equity ratio relevant to explain
return in the last step; the size and book to market equity ratio become less relevant to explain return as the time gap between return and the two variables increases. Fama and
French 1992 also used the size-beta portfolios in their analysis.
Table 1 presents summary statistics of the market return and the market excess return from January 1956 to December 1995. Summary statistics of the market return and the
market excess return from July 1962 to December 1995 for the model 4 are quite similar to Table 1 and hence omitted. It shows the market return and the market excess return are
negative in substantial proportions of the sample; more than one third of observations of the market return are negative and even larger observations of the market excess return are
negative. For example, the EWI market return index takes 311 positive values and 169 negative values and the market excess return with the EWI market return index takes 291
positive values and 189 negative values. It also shows negative returns are canceling out positive returns considerably, resulting in small positive average returns of above one
of the TSE from January 1955 to December 1985, obtained also from the JSRI. Repeating this procedure by discarding the earliest one year of data and adding a new one year of data, we obtained 39 years of data from
January 1957 to December 1995. As we replace the earliest one year data with a new one year data each time, however, we use the same one year data twice in the first step and three times in the second step, which may
well distort the inference of Equation 3. Therefore, we adopt the current procedure that does not use the same data more than once in any step. But comparison results and conclusions under the current procedure change
little as compared to those under the former procedure.
3
Wiggins 1992 does take into account the difference of up markets and down markets when he estimates beta and obtains a better description of data. As a test for robustness of their results, Pettengill et al. 1995 also
try, stated in a footnote, to estimate betas separately in up and down market periods and then regress returns on up or down betas depending on the sign of the market excess return, resulting in a significant relationship
between return and beta.
522 J. Hodoshima et al.
percentage for the market return and less than one percentage of the market excess return. Standard deviations of the market return and the market excess return also become smaller
after differentiating up markets from down markets. Table 2 presents summary statistics of the 20 portfolios obtained using the EWI as the
market return index in the total sample and four equally divided subsamples, from January 1956 to December 1965, from January 1966 to December 1975, from January 1976 to
December 1985, and from January 1986 to December 1995. It gives the time series average and standard deviation of portfolio betas and portfolio returns. Figure 1 is a scatter
diagram obtained from the average portfolio return and the average portfolio beta in 20 portfolios in the total sample period given at Table 2. This shows a flat relation between
average return and average beta in the 20 portfolios. Table 3 presents summary statistics of the portfolios obtained using the VWI as the market return index in the same periods
as Table 2.
4
Table 3 is quite similar to Table 2 as far as the return is concerned. Both the
4
A figure for Table 3, equivalent to Figure 1 for Table 2, is similar to Figure 1 and, hence, omitted.
Table 2. Summary Statistics of the EWI 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
0.014 0.015 0.015 0.015 0.014 0.015 0.015 0.016 0.017 0.016 0.016 0.016 1956–65
0.019 0.019 0.019 0.017 0.018 0.018 0.018 0.019 0.022 0.019 0.021 0.027 1966–75
0.019 0.016 0.018 0.016 0.015 0.017 0.019 0.019 0.019 0.020 0.019 0.020 1976–85
0.011 0.013 0.013 0.014 0.013 0.014 0.014 0.016 0.018 0.017 0.015 0.016 1986–95
0.007 0.008 0.009 0.009 0.009 0.009 0.010 0.010 0.009 0.009 0.007 0.007 Standard deviation
1956–95 0.048 0.047 0.051 0.052 0.054 0.055 0.058 0.059 0.063 0.064 0.067 0.074
1956–65 0.039 0.040 0.049 0.050 0.060 0.056 0.062 0.062 0.072 0.071 0.075 0.085
1966–75 0.045 0.042 0.048 0.049 0.049 0.052 0.057 0.058 0.059 0.062 0.067 0.075
1976–85 0.031 0.026 0.025 0.029 0.029 0.030 0.031 0.034 0.034 0.038 0.039 0.042
1986–95 0.069 0.069 0.072 0.072 0.071 0.075 0.074 0.075 0.078 0.079 0.079 0.085
Beta Average
1956–95 0.61
0.63 0.75
0.85 0.92
0.99 1.05
1.09 1.17
1.25 1.32
1.37 1956–65
0.39 0.54
0.76 0.82
0.93 0.95
1.14 1.05
1.17 1.28
1.39 1.46
1966–75 0.56
0.60 0.75
0.91 0.89
1.00 1.04
1.10 1.15
1.21 1.38
1.40 1976–85
0.74 0.58
0.60 0.78
0.91 1.00
1.04 1.15
1.23 1.37
1.34 1.39
1986–95 0.77
0.79 0.90
0.92 0.96
0.99 1.00
1.06 1.11
1.13 1.15
1.22 Standard deviation
1956–95 0.24
0.17 0.20
0.13 0.13
0.11 0.11
0.14 0.10
0.15 0.19
0.22 1956–65
0.19 0.12
0.16 0.08
0.09 0.10
0.10 0.13
0.11 0.11
0.13 0.24
1966–75 0.24
0.17 0.17
0.18 0.18
0.08 0.06
0.17 0.08
0.18 0.27
0.29 1976–85
0.18 0.16
0.17 0.06
0.12 0.15
0.09 0.11
0.08 0.10
0.14 0.09
1986–95 0.10
0.04 0.15
0.10 0.08
0.07 0.06
0.06 0.09
0.08 0.10
0.15
Data for portfolios 3–4, 5–6, etc. denote the average of the two portfolios.
Cross-Sectional Analysis of Return and Beta 523
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
