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Chan, Hamao & Lakonishok (1991), Fama & French (1998, 2012), Griffin (2002), Hou, Karolyi & Kho (2011) and others have determined the relationship between international stock return and company size and book-to-market value ratio. This article is not the first to study how international stock returns are related to profitability and investment. Titman, Wei & Xie (2013) pointed out that in many markets, high investment is often accompanied by a low average rate of return. Sun, Wei & Xie (2013) and Watanabe, Yu, Yao & Yu (2013) confirm this result and show that higher profitability is associated with higher stock returns. These papers did not study in detail how the average rate of return of each company's portfolio varies with profitability and investment level, and they did not try to capture these factors in the average rate of return in the asset pricing model.
The asset pricing model test in this article will explore whether the five-factor model and its variants explain the relationship between international stock returns and company size, book-to-market value ratio, profitability, and investment level. Therefore, as in FF (2015), the stocks will be sorted reasonably to construct the variables on the right side of the model equation.
This article also examines the local version of the model, that is, the return rate and factor data to be explained come from the same region. The local version of the three-factor model of FF (1993) largely ignores the relationship between the average return on stocks and profitability and investment level. The model only includes the three variables of Mkt, SMB and HML in the equation. . The null hypothesis in the test was rejected, and the local version of the five-factor model explained the relationship between the average rate of return and profitability and investment level. We also provide evidence on the global version of the five-factor model.
The portfolio we constructed reveals new results about the return on stocks in the international market. The most interesting of these is that the average return on stocks of small companies in Europe and the Asia-Pacific region is relatively low. The factor loads on these stocks are similar to those of low-profit companies that invest heavily. For example, when we rank profitability and investment, for European small company stock portfolios with low profitability and high investment, the average excess return rate from 1990 to 2015 is -0.65%, while the Asia-Pacific region is -0.71 %. The average excess return on the North American portfolio is low, but not too extreme, at 0.12% per month. In the test of the US rate of return, FF (2015) found that despite the low profitability, the average return of the stock portfolio of small companies is usually much lower than the predicted value of the five-factor model. These companies generally have low profit margins. A lot of investment. Although the average returns on these stocks are lower in Europe and Asia Pacific, to some extent, the five-factor model can capture them.
Section 2 of this article describes the portfolio and factors used in model estimation. Section 3 tests whether asset pricing in four different regions conforms to the global version of the model. The rest of this article focuses on testing the model. In these tests, this article uses regional factors to capture changes in portfolio returns in the same region. Sections 4 and 5 introduce the descriptive statistics of regional factors and portfolio returns. Sections 6-8 test the asset pricing model. Section 6 tests whether the regional factor is redundant. Section 7 introduces the statistics constructed by the regression intercept. Section 8 details the intercept improvement produced by adding RMW and CMA to the FF (1993) three-factor model.
Two, data and variables
The international stock returns and accounting data in this article mainly come from databases such as Bloomberg, Datastream and Worldscope. The sample period is from July 1990 to December 2015. The selection of the sample is limited by the availability of data and the extensive coverage of the stocks of large and small companies in the researched market. In order to increase the effectiveness of the test, this article uses a diversified portfolio in the regression. The diversified investment portfolio improves the effect of regression fitting, thereby improving the accuracy of the test intercept of the asset pricing model. Like FF (2012), in order to construct a diversified investment portfolio, this article merges 23 developed markets into four regions: (1) North America (US and Canada); (2) Japan; (3) Asia Pacific (Australia, New Zealand, Hong Kong and Singapore); (4) Europe (Austria, Belgium, Denmark, Finland, France, Germany, Greece, Ireland, Italy, Netherlands, Norway, Portugal, Spain, Sweden, Switzerland and the United Kingdom). This article also studies the portfolio that combines these four regions.
In terms of region selection, this article believes that it is reasonable to assume that the countries within the region are consistent. For example, we have reason to believe that the commodities and securities traded in the United States and Canada are very similar. In the European region, this article mainly selects EU countries. For non-EU countries, they should also participate in more EU market treaties. As for the integrity of the Asia-Pacific region, the author thinks it is doubtful, but the market value of the stock market in this region is small, only 4%. The other three regions account for 48% in North America, 30% in Europe, and 18% in Japan.
In each region, the author sorted the stocks according to the combination of company size and book-to-market value ratio, profitability, and investment level. For American stocks, this article uses the discontinuity points of the New York Stock Exchange to segment the company size and other variables to avoid the large but smaller American Stock Exchange and Nasdaq stocks from dominating the ranking. For the same reason, the discontinuous point of book-to-market value ratio, profitability, and investment level is also based on the stocks of large companies, and the discontinuity point of company size is also based on the percentage of the company's market value, so as to avoid excessive weight on the stocks of small companies.
In order to construct the variables on the right side of the model (1), this paper double-orders the company size and book-to-market value ratio, profitability, and investment. Each group of variables constructs a total of 23 investment portfolios. The company size is divided into two groups, large companies The stock of the company is in the top 10% of the company's market value, and the stock of the small company is the stock in the bottom 10% of the company's market value. For the three variables of book-to-market value ratio, profitability and investment level, in 23's portfolio, the break points are 30% and 70%. Take the portfolio structure constructed by company size and book-to-market value ratio as an example. The division of company size is as described above. Given the size of a group of companies, the stocks in the group are divided into 3 groups according to the book-to-market value ratio. The first 30%, the middle 40%, and the last 30% form a 2*3 total of 6 investment portfolios, and calculate the monthly rate of return of each investment portfolio in a value-weighted manner. In order to construct the HML variables, calculate the high book-to-market value ratio of large company stocks and small company stocks to the investment group
The three variables of HML, RMW, and CMA are generally correlated, that is, stocks with high book-to-market value ratios generally have lower profitability and investment levels, while companies with lower book-to-market value ratios tend to have higher profitability. And aggressive investment strategies. These correlations mean that the three variables of HML, RMW, and CMA are the interaction of the three effects of company value, profitability, and investment level.
This paper constructs 25 Size-B/M portfolios, 25 Size-OP portfolios, and 25 Size-Inv portfolios as the explained variables of each regression, and each portfolio will calculate the rate of return weighted by value. The size of the company is divided into four division points of 3%, 7%, 13%, and 25% of the total market value of each region, and the division of book-to-market value ratio, profitability, and investment level is divided into every 20%. Since the book-to-market value ratio, profitability, and investment level are interrelated, the above three sets of investment portfolios do not divide the effect of company value, profitability, and investment level. In order to clarify the relationship between them, we need to use company size , Book-to-market value ratio, profitability, and investment level to jointly construct an investment portfolio. Since the 3333 investment portfolio cannot be well diversified, this article constructs a 24*4 investment portfolio, in which the company size is divided into the top 10% and the bottom 10%, and the other three variables are divided according to quartiles. This results in three sets of investment portfolios, namely 32 Size-B/M-OP portfolios, 32 Size-B/M-Inv portfolios, and 32 Size-OP-Inv portfolios.
3. Test of the global model
This article first tests the international version of the model, that is, the factor variables on the right of the model are constructed based on global stock data, and the variables on the left are the portfolio returns of each region. The test results show that the factor variables based on the global stock structure perform poorly and cannot explain the expected return of each regional investment portfolio.
According to the model constructed in this article, if the global version of the model is valid, the intercept term should not be significantly different from 0. However, this article has performed 20 regressions on different portfolios in four regions, and the intercept term of 16 regressions is in It is different from 0 at the 5% significance level. The author guessed that the possible reason for this situation is that the global stock market cannot be regarded as a whole, or the model proposed in this article is wrong, so the focus of the author's test is still on the regional version of the model, that is, the left and right sides of the model equal sign are both Regional data structure.
Fourth, the factor of the rate of return descriptive statistics
in Table 1, Panel A shows that Japan 1990 - - During 2015, equity risk premium (Mkt) close to zero, while the three other parts of the equity risk premium is relatively large. From 1990 to 2015, the company size premium (SMB) was close to zero in all four regions, with the largest being 0.17% in North America. In terms of value premium (HML), only the value premium in North America is not significantly different from 0, while in Japan, only the value premium is not close to 0. This result is very important for the next test. The profitability premium (RMW) is very significant in North America and Europe, but not significant in the Asia-Pacific region. The investment level premium (CMA) in North America, Europe and the Asia-Pacific region is concentrated between 0.2% and 0.39%, and the standard deviation is between 1.86 and 2.60.
Panel B of Table 1 confirms the results in FF (2012). For North America, Europe, and Asia Pacific, the value premium is relatively higher for small company stocks, but the opposite is true for Japan, where the value premium is higher for large company stocks. The value premium of large company stocks is 0.11% higher than that of small company stocks, but the standard deviation is only 0.51. New evidence shows that, with the exception of Japan, the RMW and CMA variables in other regions are also higher for small company stocks.
Panel C shows the correlation coefficients of the five factors in different regions. The market factor has the highest correlation, but the correlation between Japan and several other regions is relatively low. The correlation coefficients of several other factors are relatively low. Since Europe and North America account for 80% of the total market value of the sample, the relationship between them requires additional attention. Among the four non-market factors, HML has the highest correlation in North America and Europe, followed by CMA, SMB, and RMW. The profitability factor RMW has the lowest correlation among different regions.
5. Excess average return rate of the investment portfolio
Panel A of Table 2 shows the 5*5 investment portfolio return rate constructed as described above, and shows the size of the spread caused by changes in the company's size of B/M, OP, and Inv. . For the Size-B/M portfolio, within a given Size group, the average rate of return is positively correlated with the book-to-market ratio. During the period from 1990 to 2015, for the stocks of large companies in North America, the correlation between book-to-market value ratio and yield was weak. For Europe and the Asia-Pacific region, the return on stocks of large companies is also positively correlated with the book-to-market value ratio. For Japan, when the company's size increases, the spread caused by the book-to-market ratio continues to increase.
It is worth noting that for different regions, the market value of each group in the 5 groups divided by Size accounts for the same proportion of the total local market value, but the average market value of the company varies greatly. For example, for the group with ultra-high market value, The average market value of Europe is slightly lower than that of North America, Japan is only half of that of North America, and the Asia-Pacific region is only 1/3 of that of North America. Therefore, it is possible to compare the rate of return models in different regions, but it is misleading to directly compare the size of the rate of return.
For the Size-OP portfolios in North America, Europe, and Asia Pacific, the average return increases with the increase in OP in all the Size quantiles, while in the larger size quantile, the degree of attenuation of the spread will be greater Less than the Size-B/M portfolio. In Japan, the average return in the highest OP quantile is greater than the lowest quantile, but the gap is small, and the relationship between profitability and average return is weak.
Fama & French (2015) found that for the US data from 1963 to 2013, the average return of the 5×5 Size-Inv portfolio in the super-large market capitalization group decreased with the increase of Inv, but for the other four groups, the return Will fall rapidly between the fourth and highest investment quantile. The Size-Inv portfolio in Panel A also proves this pattern, and Europe also has the same pattern. For the Asia-Pacific region, this feature has also extended to a group of super-large market capitalization. There is no obvious rule between the rate of return and investment in Japan.
Panel B shows the return rate of the 244 portfolio. Although there is one more variable than the 55 group, it only strengthens the conclusion in the 55 group. In North America, Europe, and the Asia-Pacific region, the rate of return has risen steadily as the book-to-market ratio and profitability increase, but for investment levels, the rate of return will drop rapidly at the highest quantile. For Japan, there is a strong correlation between portfolio yield and book-to-market value ratio, while the correlation between profitability and investment level and yield is weak.
In FF (2015), the author found that the five-factor pricing model is not applicable to the stock portfolio of small companies with the lowest profitability and the highest investment level. This is also a problem in this article. The results in Table 2 show that in North America, for stocks with low profitability and high investment levels, the average yields of small company stocks and large company stocks are significantly lower than other investment portfolios. For Europe and the Asia-Pacific region, these two investments The portfolio yield is lower, and Japan does not have this problem.
6. Factor Generation Test
FF (2015) The author found that in the regression of the HML factor to the other four factors, the intercept term is very close to 0, indicating that the HML factor is redundant and can be fully explained by the other four factors. This section will test this question to see if each factor in the four regions is redundant. Table 3 shows the regression results of each factor in the four regions against the other four factors.
In North America, Europe, and the Asia-Pacific region, the Mkt factor is not redundant for the four regions, and the intercept term of the regression is significantly different from zero. The value factor HML is very important to explain the changes in the rate of return. In the regression, the intercept terms in the four regions are all significant. Among them, the intercept terms in Europe, Asia Pacific, and Japan are positive, and the intercept terms in North America are negative. Profitability RMW is important in North America, Europe, and Asia Pacific. The coefficients in the regression are all positive and very significant. In Japan, the intercept term is relatively low, but still significant. The investment factor CMA is important in North America and the Asia-Pacific region, but it is not significant in Europe and Japan.
In general, the four factors are important in North America, but in Europe and the Asia-Pacific region, only the three factors of Mkt, HML, and RMW are important. The Asia-Pacific region also needs the assistance of the CMA factor, but the CMA factor is Europe plays a smaller role. For Japan, the HML factor is important, coupled with the assistance of the CMA factor. It should be noted that whether a factor is redundant may also depend on the time interval of the sample data. The above results only indicate that some factors are redundant in this specific time period.
7. Summary of Asset Pricing Tests
This section will test the five-factor model. For each group of 5*5 portfolios, 25 regressions will be performed, and then the GRS statistics will be used to perform a joint significance test to determine the intercepts of the 25 regressions. Whether at least one of the items is different from 0. This article tried a three-factor model constructed by Mkt, SMB, and HML, a four-factor model constructed by Mkt, SMB, HML, and RMW, and a five-factor model including all factors. The test results show that in North America, Europe, and Asia-Pacific, the performance of the five-factor model is always better than that of the four-factor model that ignores the RMW or CMP factors.
8. Asset pricing details.
Since we are more concerned about how the performance of the model changes from three factors to five factors, this section only retains these two models. At the same time, in order to save space, only the results of North America and Europe, and the results of the Asia-Pacific region are shown here. Very similar to European ones. Tables 5 to 10 show the intercept terms and their t values of 6 different portfolios. Since the coefficients of Mkt and SMB in the three-factor and five-factor models are very close, the explanation for the changes in the intercept term is mainly focused on HML , RMW, CMA three factors.
(1) Size-B/M portfolio
In North America and Europe, the small company stock portfolio with the lowest B/M did not pass the test of the three-factor model, the smallest B/M quantile in North America, the smallest and the second-smallest company The portfolio intercept items composed of stocks are -0.34% and -0.45%, respectively, and the European counterparts are -0.47% and -0.15%. In the five-factor model, the intercept items of these two portfolios are both very Close to 0. In North America and Europe, small companies and low B/M stocks that have caused problems with the three-factor model all have negative HML, RMW, and CMA coefficients in the regression, indicating that the companies corresponding to these stocks are like those with low profit and high investment. In FF (2015), this feature also caused the five-factor model to be affected. Here, a shorter sample data is used, and this situation has improved.
The GRS test in Table 4 shows that the five-factor model has other problems. For example, for the three-factor model, both small companies and large company stock portfolios with high B/M in North America have significant intercept terms. The five-factor model alleviates the large The company’s stock issue did not help small company stocks.
(2) In
the three-factor model of the Size-OP portfolio in North America, with the increase in profitability, the intercept term of the regression is also rising, and this feature is more obvious in the European region. In the five-factor model, the rising intercept term is explained by the newly added profitability factor RMW. When the OP is low, the coefficient of RMW is negative, and when the OP is high, the coefficient of RMW is positive. Nevertheless, the GRS test rejects the null hypothesis. The main problem may be that many small company stock portfolios have large positive intercept terms.
(3) Size-Inv Portfolio
In a portfolio with a high investment level, the three-factor model has a very significant intercept term. The five-factor model largely solves this problem, and the GRS test proves the five-factor model in Europe All intercept terms of is close to zero. In North America, the three-factor model provides a higher intercept term in the low Inv group. This problem still exists in the five-factor model, but in the other four Inv quantile portfolios, only one regressed The intercept term is significant.
In the highest Inv quintile, the improvement in the European portfolio's intercept is due to the negative coefficients of RMW and CMA. Since the HML coefficients of these investment portfolios are also negative, this article believes that the yields of these companies behave like high-investment low-profit companies. The improvement of the intercept term in North America is also due to negative RMW and CMA, but the coefficient of HML is positive.
(4) Size-B/M-OP investment portfolio
For small company stocks in North America, the intercept term in the three-factor model will increase with the increase in OP. The same pattern applies to large company stocks, but the growth rate is even greater. small. In the three-factor model in North America, the regression intercept term of the small company stock portfolio with the lowest B/M and OP is -0.52% and the t value is -2.90, while the problem with the large company portfolio lies in the lowest OP and the highest B For the grouping of /M, the regression intercept term is -0.46% and is significantly different from 0. After using the five-factor model, the intercept terms of these two portfolios are closer to 0, and this is also the only portfolio with two significant intercept terms in the five-factor model in North America. In addition, the five-factor model is used After that, the law that the intercept term increases with OP in the three-factor model no longer exists.
In Europe, small company stocks with low B/M and OP pose a greater challenge to the three-factor model. The intercept term of this portfolio is -0.99% and the t value is -6.01, which also does not reflect The intercept term increases with the increase in OP. For a large company stock portfolio, the set of intercept terms with the lowest OP is negative, and the set of intercept terms with the highest OP is positive. Similar to North America, the five-factor model in the European region also alleviates the above problems. The intercept item of the small company stock portfolio with low B/M and OP has risen from -0.99% to -0.37%, and the cutoff of other small company stocks The distance term is also closer to 0, although the intercept term of some portfolios is still significant. 
(5) Size-B/M-Inv Portfolio
According to the results in Table 9, for large company stocks in North America and Europe, the problem with the three-factor model lies in stocks with high B/M and Inv. For this portfolio, North America The intercept terms in Europe and Europe are -0.62% (t=-4.13) and -0.41% (t=-2.60), and the five-factor model improves this problem. The reason for the improvement may be that both RMW and CMA in the regression There are negative coefficients and large absolute values.
For small company stock portfolios, the problem of the three-factor model is high Inv and low B/M. The five-factor model in Europe solves this problem well, making the intercept term closer to 0 and no longer significant . In North America, although the five-factor model has also improved this problem, the intercept term is still significant.
(6) Size-OP-Inv portfolio
Although the GRS test rejects both the three-factor model and the five-factor model in the Size-OP-Inv portfolio, in detail, the overall performance of the five-factor model is better. For the stocks of large and small companies in North America and Europe, within each Inv quantile, the intercept term of the three-factor model increases with the increase of OP. At the same time, in all rankings, it has low OP and high Inv’s small company stocks pose the most serious problems.
For the stocks of large companies, the five-factor model has significantly improved the above problems. The law that the intercept item increases with the increase in OP no longer exists in the five-factor model, and the stocks of large companies with low OP and high Inv are no longer one. problem. Although the five-factor model has also improved the situation of small company stocks, the problem still exists. The intercept term is still significant. The law of positive correlation between the intercept term and OP no longer exists in North America, but it still has this problem in Europe. As a rule, in addition to this, for small company stocks in North America and Europe, in the lower OP and higher Inv groups, some portfolio coefficients are very significant.
In general, the five-factor model explains most of the changes in portfolio returns, but for small company stocks with low profitability and high investment levels, the five-factor model cannot provide a good explanation for them.
9. Conclusion
In North America, Europe and the Asia-Pacific region, the average return on stocks of small companies increases with the rise of B/M and OP, but this rule is not obvious for stocks of large companies. When sorting the Invs, the stock returns of small companies in the three regions will drop rapidly at the quantile of the highest Inv. This is no longer applicable to the stocks of large companies, and the stock returns of other regions have no more than North America. Discover other patterns of change. In general, the investment factors in the five-factor model may largely absorb changes in the stock returns of small companies with high investment levels.
The global version of the five-factor model performed poorly in the test, so this article focuses on the regional version of the five-factor model, that is, the factors and portfolios are from the same region. For Japan, B/M is the only factor with a clear relationship, and the three-factor model also performs well. For North America, Europe and the Asia-Pacific region, profitability and investment levels are not explained by the three-factor model. Although the five-factor model has not passed the GRS test, it has absorbed the portfolio’s return on book-to-market ratio, profitability and The law of changes in investment levels.
The factor generation test results show that all factors play a role in explaining the stock returns in North America, but the investment level factor CMA is redundant in Europe and Japan. Therefore, for these two regions, ignore the explanatory impact of the CMA factor on the return. Not big.
The author’s recent research results show that stock portfolios of small companies with high investment and low profitability generally have lower yields and will pose challenges to the three-factor model and the five-factor model. This article also found the same result. And this problem is more serious in North America and Europe.
The attached
table in the appendix shows the coefficients and t statistics of HML, RMW, and CMA in the regression of the five-factor model.
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