Cross-Sectional Regression Analysis of
Return and Beta in Japan
Jiro Hodoshima, Xavier Garza–Go´mez, and Michio Kunimura
This paper investigates the relationship between return and beta using the cross-sectional regression method. Regression of return on beta without differentiating positive and negative market excess returns produces a flat relationship between return and beta. Taking into account the difference between positive and negative market excess returns yields significant conditional relationships between return and beta. The conditional relationship between return and beta is found to be in general better fit when the market excess return is negative than positive in terms of the goodness of fit measures such as R2 and the standard error of the equation. © 2000 Elsevier Science Inc.
Keywords: Beta; Up market; Down market
JEL classification: G12
I. Introduction
A recent article by Fama and French (1992) draws a conclusion contradicting the capital asset pricing model (CAPM) which states the cross-section of expected returns is positive and linear in beta. They find virtually no relation between average return and beta. The impact of Fama and French (1992) is quite large on both academics and practitioners, causing them to reinvestigate relevance of beta. After Fama and French (1992), a number of empirical studies have provided evidence supporting the CAPM or more appropriately relevance of beta. One body of these studies conditions on the sign of the market excess return, defined by market return minus risk free rate, and investigates the relation between return and beta by taking into account whether the market excess return is positive or negative, or more simply stated, whether the market is up or down. They are, among others, Chan and Lakonishok (1993), Grundy and Malkiel (1996), Fletcher (1997), and Pettengill et al. (1995).
Nagoya City University, Nagoya 467-8501, Japan.
Address correspondence to: Jiro Hodoshima, Faculty of Economics, Nagoya City University, 1 Yamanohata, Mizuho-cho, Mizuho-ku, Nagoya 467-8501, Japan.
Although expected returns as well as expected excess returns, that is, expectation of excess returns (returns minus risk free rate), are both assumed positive in the CAPM, realized returns and realized excess returns do in fact take negative values quite often. For example, about 40% of monthly observations of the market excess return consist of negative return months in our Japanese stock market data from January 1956 to December 1995, as shown at Table 1. All of the above studies find data are better explained by drawing a distinction between positive and negative market excess returns. Differentiating up markets from down markets, all of them consistently find significant conditional relationships between return and beta with the sign of the market excess return as a condition. They imply there exists a significant positive relation between return and beta when the market excess return is positive and a significant negative relation between return and beta when the market excess return is negative. On the other hand, Fama and French (1992) do not take into account the difference between up markets and down markets when they find absence of any unconditional relation between return and beta. The results of the above studies suggest that a positive conditional relation between return and beta when the market is up is basically offsetting a negative conditional relation between return and beta when the market is down, causing absence of any unconditional relation between return and beta as seen in Fama and French (1992) and others.
This paper provides another evidence from the Japanese stock market of the conditional relation between return and beta with the sign of the market excess return as a condition. So far, most of the studies of this conditional relationship use U.S. data. The only Table 1. Summary Statistics of Market Returns and Risk Free Rate with the Difference of Up Markets and Down Markets (January 1956 –December 1995)
1. Number of up and down months
Index Total Sample Up Months Down Months
EWI (raw) 480 311 169
EWI (exc) 480 291 189
VWI (raw) 480 292 188
VWI (exc) 480 269 211
2. Average and standard deviation (SD)
Total Sample Up Months Down Months
Index Average SD Average SD Average SD
EWI (raw) 0.0153 0.053 0.0440 0.035 20.0376 0.037
EWI (exc) 0.0096 0.053 0.0412 0.035 20.0390 0.038
VWI (raw) 0.0115 0.051 0.0414 0.034 20.0349 0.035
VWI (exe) 0.0059 0.051 0.0392 0.033 20.0367 0.035
RFR (EWI exc) 0.00568 0.0021 0.00562 0.0020 0.00587 0.0023 RFR (VWI exc) 0.00568 0.0021 0.00554 0.0020 0.00594 0.0022
In the upper part of Table 1, the number of up and down months is shown. For example, EWI (raw), the EWI market return index, takes 311 positive values and 169 negative values. EWI (exc), the market excess return with EWI as the market return index, takes 291 positive values and 189 negative values.
exception is Fletcher (1997) which uses U.K. data. Evidence from other countries, particularly from non-Western countries, seems relevant and important. Because of its relative importance in the world, we consider appropriate to investigate the Japanese stock market. There exist some Japanese studies such as Hawawini (1991) and Jagannathan et al. (1998) for the unconditional return and beta relationship. Hawawini (1991) finds beta not significant except in the months of January, using monthly data from January 1955 to December 1985. Jagannathan et al. (1998) also find a flat unconditional relationship between return and (stock-index) beta, using monthly data from September 1981 to December 1993, while they find labor-beta, based on the growth rate of labor income, can explain well return. Both of the above studies use the data from the first section of the Tokyo Stock Exchange (TSE) and the cross-sectional regression method of Fama and MacBeth (1973), the same data and method as ours. So far, there exists no Japanese evidence for the conditional relationship between return and beta based on the sign of the market excess return. Therefore, our study verifies for the first time the conditional relationship and relevance of beta with the Japanese stock market data.
The main purpose of this paper is to present another evidence of the conditional relationship between return, beta, and also other idiosyncratic explanatory variables from the Japanese stock market, showing characteristics of the Japanese stock market. In addition to the simple model of return and beta, which Pettengill et al. (1995) and Fletcher (1997) investigated, we analyze a model which also contains, as explanatory variables, size, and book to market equity ratio which Fama and French (1992) studied. The size and book to market equity ratio are both well-accepted idiosyncratic explanatory variables for return [see, e.g., Chan et al. (1991) and Fama and French (1992)]. Fletcher (1997) includes the size but not the book to market equity ratio in his study of the conditional relationship. Our emphasis in this paper lies in presenting proper statistical inference of the relationships between return, beta, and other explanatory variables. In other words, we make comparisons of different relationships based on summary statistics of goodness of fit and testing results obtained from the cross-sectional regression method. Summary statistics of goodness of fit such R2and the standard error of the equation are given to all the regression results in this paper while they are not given in Pettengill et al. (1995) or Fletcher (1997). These summary statistics are essential to judge how models fit the data and also help to evaluate whether significance test results on coefficients are reliable or not. In general, they should not be omitted in every regression result. We show the conditional relation between return and beta as well as the conditional relation between return, beta, size, and book to market equity ratio are in general, based on the summary statistics of goodness of fit, better fit when the market excess return is negative than positive. This phenomenon was not observed by Pettengill et al. (1995) or Fletcher (1997) because they did not provide any summary statistics of goodness of fit.
The paper is organized as follows. In Section II, models to be analyzed and compared are presented. In Section III, data are described with summary statistics showing how returns differ when the market is up and down. In Section IV, cross-sectional regression results are presented and compared. In Section V, concluding comments are given.
II. Models
The CAPM can be written as:
where Rpt and Rmt denote, respectively, the monthly return on the portfolio p and the market for month t, Rftdenotes the risk free rate for month t, E[z] denotes the expectation operator, and bp5cov(Rpt, Rmt)/var(Rmt) where cov(Rpt, Rmt) is the covariance between Rptand Rmtand var(Rmt) is the variance of Rmt. Equation (1) is equivalent to:
E@Rpt#5g01g1bp (2)
whereg15 E[Rmt2 Rft] andg05Rft.
The CAPM assumes the expected market excess return E[Rmt2Rft] is positive. Under the positive expected market excess return, Equation (2) denotes a positive linear relation between return and beta. Empirical researches of Equation (2) such as Fama and French (1992) findg1, the expected market excess return, not significantly different from zero.
Even when the expected market excess return is positive, the realized market excess return
Rmt2Rft can and does take negative values. In fact, about 40% of our monthly obser-vations of the market excess return in Japan from January 1956 to December 1995 consist of negative months, as shown at Table 1.
Verification of the existence of the conditional relationship between return and beta with the sign of the market excess return as a condition does not imply testing whether the CAPM holds or not. In other words, empirical investigations in this paper do not aim to verify the CAPM but instead intend to verify the appropriateness of beta and the conditional information given by the sign of the market excess return. When we verify Equation (2) using data, that is, using realized portfolio return and estimated beta, Equation (2) implies a positive linear relation between realized portfolio return and estimated beta when the realized market excess return is positive and a negative linear relation between realized portfolio return and estimated beta when the realized market excess return is negative. Thus, in this paper we intend to test whether positive and negative linear relationships hold between realized portfolio return and estimated beta when the realized market excess return is positive and negative, respectively. Even when positive and negative linear conditional relationships do in fact hold, a poor unconditional linear relationship between realized return and estimated beta may well arise from mixing positive and negative conditional relationships. Therefore, the existence of the two conditional relationships between return and beta with the sign of the market excess return as a condition does not contradict poor unconditional regression results between return and estimated beta. If we can verify the existence of the conditional linear relationships, beta may be considered relevant to describe the data.
Equation (2) implies the following model of return;
Rpt5got1g1tbpt1 «pt p51, . . . , N;t51, . . . , T (3)
whereg0t,g1t, andbptcorrespond, respectively, tog0,g1, andbpat month t,«ptdenotes an error term with E[«pt]50, and N and T are the number of portfolios and observations, respectively.
In addition to Equation (3), we also study:
Rpt5gpt1g1tbpt1g2tln~MEpt!1g3tln~BE/MEpt!
1 «9ptp51, . . . , N; t51, . . . , T (4)
where ln(MEpt) denotes the logarithm of the market value of equity, or size variable, ln(BE/MEpt) the logarithm of the book to market equity ratio,g2t, andg3tcoefficients of
The conditional relationship between return and beta for Equation (3) is given by two relationships between return and beta when the market excess return is positive and negative. The cross-sectional regression estimation of the conditional relationship is defined as follows. Let us put two sets of intercept and slope parameters, (g0up,g1up) and
(g0down,g1down), as the intercept and slope parameters of the conditional relationship
between return and beta when the market excess return is positive and negative, respec-tively. Then, the cross-sectional regression estimates of (g0up,g1up) and (g0down,g1down)
are given, respectively, by the average of monthly cross-sectional regression intercept and slope estimates in the up market months when the market excess return is positive and in the down market months when the market excess return is negative, denoted as (g¯0up,g¯1up) and (g¯0down,g¯1down).
The cross-sectional regression method of Fama and MacBeth (1973) for Equation (3) first estimates beta using past observations by regressing stock returns on the constant and the market return to obtain the estimated beta ˆbpt. Substituting the estimated beta into beta, Equation (3) can be written as:
Rpt5g0t1g1tbˆpt1npt p51, . . . , N; t51, . . . , T (5)
wherenpt5 «pt2 g1tuptwith upt5 bˆpt2bptwhere uptdenotes an estimation error in beta. The cross-sectional regression method next estimates Equation (5) to obtain each month cross-sectional estimategˆ0tandgˆ1tby the least squares. Thus, we have:
Rpt5gˆ0t1gˆ1tbˆpt1nˆpt p51, . . . , N; t51, . . . , T (6)
wherenˆptdenotes the least squares residual of Equation (5). The cross-sectional regression estimates ofg0andg1given at Equation (2) are given by the average of all of the monthly
cross-sectional intercept and slope estimates, that is,g¯0andg¯1whereg¯05¥tT51gˆ0t/T and
g¯15¥tT51gˆ1t/T.g¯0andg¯1thus correspond to the cross-sectional regression estimates of
the coefficients of the unconditional relationship between return and beta. The average and standard deviation of the month-by-month regression coefficient estimates provide a t test on the coefficient of an explanatory variable in the cross-sectional regression method of Fama and MacBeth (1973). The cross-sectional regression estimate and its t test of the conditional relationship can be obtained from the monthly regression coefficient estimates in the up and down market months.
Similarly, we can also define straightforwardly the cross-sectional regression estimates of the unconditional relationship and two conditional relationships when the market excess return is positive and negative for the extended model (4).
In addition to the average and t test, we fully make use of the summary statistics of goodness of fit such as R2and the standard error of the equation to evaluate different
relationships. The summary statistics for the unconditional and two conditional relation-ships in the cross-sectional regression method are given by the average of the monthly summary statistics for the total sample and two subsamples of the up market months and down market months, respectively. Pettengill et al. (1995) consider a unique intercept and allow two different slopes in the up and down markets in their specification of the conditional relationships for Equation (2); their specification of the conditional relation-ships for Equation (2) is given by
wheredtdenotes 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:
E@Rpt#5g0updt1g0down~12dt!1g1updtbp
1g1down~12dt!bp p51, . . . , N; t51, . . . , 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 R2, adjusted R2, 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 ofg0upandg0down,
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.
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 t1 1. ME is calculated every June using the number of outstanding stocks and the stock price at the end of June. The BE/ME 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 BE/ME 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.2Thus, we totally
1Chan 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.
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 markets3when 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 t11, 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 t11 of return, beta, ln(ME), and ln(BE/ME), 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.
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 2.4Table 3 is quite similar to Table 2 as far as the return is concerned. Both the
4A 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
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;
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 R2 and standard error of the equation given, respectively, by the time series average of the month-by-month cross-sectional regression R2s 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
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 (B/M)
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.
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 (B/M)
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.
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 R2s 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
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 g0up is positive and significant in all of the sample periods while that ofg0downis 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 R2
and Average Standard Error of the Equation (SE) for the Unconditional Relationship of Equation (3) (January 1956 –December 1995)
Number of
Months g0 t-Statistic g1 t-Statistic R2 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 R2and Average Standard Error of the Equation (SE) for the Conditional Relationships of Equation (3) (January 1956 –December 1995)
Number of
Months g0 t-Statistic g1 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.
We also have the two equality test results by the t test;5the two equality hypotheses are given byg0up5g0downandg1up5 2g1down. 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 R2 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 R2and 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 condiuncondi-tional 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
5The 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 g0 t-Statistic g1 t-Statistic R2 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 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 condiuncondi-tional 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 g0 g1 g2 g3
Months (t-Stat) (t-Stat) (t-Stat) (t-Stat) R2 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.
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 of21.22, 2.22, and21.99, 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.6On 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
We find that about 40% of monthly observations of the market excess return consist of negative months in the TSE from 1956 to 1995. Mixing months of positive and negative market excess returns yields absence of any significant linear relationship between return and beta; positiveg1tappears to be simply offsetting negativeg1tin the simple model (3) with beta as the only explanatory variable and the extended model (4) with beta, size, and book to market equity ratio as the explanatory variables, making flat the relation between return and beta. Using the cross-sectional regression method, we have shown that taking into account the difference between positive and negative market excess returns produces the significant conditional relationships between return and beta. We have also made use of the goodness of fit measures such as the R2and the standard error of the equation to evaluate the regression outcomes whereas neither Pettengill et al. (1995) nor Fletcher (1997) utilize them. We have found the conditional relationship is in general better fit in the down market than in the up market in terms of the goodness of fit measures.
An earlier version of the paper was presented at Kobe University, Osaka University, and Nagoya City University and Finance Kenkyuukai in Tokyo. We thank Jonathan Fletcher, C.R. McKenzie, Glenn N. Pettengill, and Yasushi Yoshida for helpful comments and discussions. We also thank the executive editor Kenneth J. Kopecky, the editor J. Jay Choi, and two anonymous referees for very helpful comments. Financial support from Ishii Memorial Securities Research Promotion Foundation to Hodoshima and Garza–Go´mez is gratefully acknowl-edged.
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