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The method of using the Factor model is very elaborate, and many already published research has introduced many ways to use the Factor model. Out of all the research, I chose Gregory Connor and Sanjay Sehgal (2001)[1]. The reason why I chose this research was, first, the number of citations exceeded 200 times. As for research representing the Factor model, it was judged that the minimum number of citations should be above at least 100 times, so research with less than 100 citations was excluded. Second, this research was a good paper to study how to examine the Fama-French three-factor model of stock returns for other countries (here case, India), comparing the factors with the US stock market. So, later I can try my research for examining the model for the Korea stock market. Thirdly, the contents of the research are very organized and include various statistical analyses that I wanted to study practically, such as standard multivariate regression framework, seasonality, zero-beta variants[2] of the standard model, etc).
[Overall introduction]
The research empirically examines the Fama-French three-factor model for the Indian stock market, with both a one-factor linear pricing relationship by CAPM and three-factor linear pricing by Fama and French. The main goal is to analyze if the market, size, and value factors of the Fama-French model can be applied in the cross-section of random stock returns from the Indian stock market. Additionally, it examines the linear exposures of US equities to these three factors in explaining earnings growth rates, which can be tied to the equity return factors. Overall, the writers try to see if the Fama-French model can be applied to Indian equities to analyze what's similar and different from US market in applying to the Indian stock market.
[1. Data Collection]
- 364 companies’ month-end adjusted share prices data from June 1989 to March 1999
- Characteristics of the data: India is a very large emerging market with 8000 listed companies, but the top ten percent of companies take a major portion of market capitalization, whereas the remainder is thinly traded.
- Stock market index: CRISIL-500, which is a value-weighted stock market index in India, covering 97 industry groups
- Obtaining data: the share data got obtained from Capital Market Line (a financial database for India)
- The adjusted share price series got converted into return series (arithmetic returns)
- Risk-free Proxy: 91-day Treasury bills (Reserve bank of India)
- Note that before 1993, 91-day T-bills were regulated to have a 4.6% constant annual yield. Since 1993, it has been determined on an auction basis. Even in the deregulated period since 1993, the investors in the Indian stock market had higher borrowing and lending rates than the government T-bills. (There will be an analysis of the effect of the regulated T-bill rate, by using zero-beta variants of the standard model)
[2. Making Fama-French Three-Factor]
- Market factor: (market index return - risk-free return) proposed by the CAPM. CRISIL-500 return is used as a market index return, and the 91-day Treasury bill is used as a risk-free return proxy.
-Size & Value factor: Out of all the sample stocks from 1989 to 1998, based on the size of firms, two portfolios (S, B) is composed of the top 50% size and the bottom 50% (median sample size to split the sample companies). After that, these two portfolios (S, B) are again divided into the top 30% (High), the middle 40% (Medium), and the bottom 30% (Low), based on BE/ME. Overall, there can be a total of six portfolios (S/L, S/M, S/H, B/L, B/M, and B/H) from the intersection of the two sizes and three BE/ME groups. Subsequently, to find the difference in returns due to differences in size (SMB), it can be calculated by subtracting the average of returns for large portfolios (B/L, B/M, and B/H) from the average of returns for small portfolios (S/L, S/M, and S/H). Additionally, the HML can be calculated by subtracting the average of the two low BE/ME portfolios (S/Low, B/Low) returns from the average of the two high BE/ME portfolios (S/Low, B/Low)[3]. The six-size-BE/ME portfolios are constructed to be equally weighted.
[1] Connor, G., & Sehgal, S. (2001). Tests of the Fama and French model in India.
[2] A zero-beta portfolio is a portfolio constructed to have zero systematic risk (a beta of zero). Azero-beta portfolio would have the same expected return as the risk-free rate. Investopedia, https://www.investopedia.com/terms/z/zero-betaportfolio.asp
[3] 곽승주. (2021). 파이썬으로 배우는 포트폴리오. 길벗.
[3. Descriptive Statistics on the Return Series]
1) Summary statistics of the factor portfolio returns
The portfolio is composed of the six size and value sorted portfolio returns and the three-factor portfolio returns. The result shows a negative relation between the size and the average return, as mentioned in Fama and French (1993). However, interestingly, the value and the average return are not positively correlated. The small-cap stocks have a positive correlation, whereas, the larger stocks are negatively correlated with average returns. This is a different result from the strong positive correlation between the value found by Fama and French (1993) in the US stock market. Overall, in the case of the Indian market, the size effect has the same effect as the US market, but the value effect appears to be conditional. The market factor has a monthly volatility of 10.26%, and 35.54% annually. Positive skewness and positive excess kurtosis are shown in all the portfolios.
Table 2 shows the correlation coefficients between the three factors (MKT, SMB, HML). Three of the factors all have a positive correlation with each other, ranging from 0.11 to 0.27.
2) Seasonality in the Returns
For the seasonality in monthly returns in the Indian market, there are some assumptions. First, at the end of March, the financial closing occurs. According to the tax-loss selling hypothesis [1], investors would sell loss-making stocks in March and earlier months and reallocate the portfolios in April. Keim (1983) illustrated the seasonality of return, called the January effect, that the daily abnormal return distributions in January have large means relative to the remaining eleven months. The April effect in India is similar to the January effect in the USA. Second, the government financial budget in India is presented on the last day of February every year, which leads to portfolio rebalancing. This would lead to a March effect. Thirdly, in October-November of every year, the festival of Divali affects Indian consumption spending. This may push down stock prices for these two festival months with recovery in the succeeding month, December. With these three assumptions, Connor and Sehgal test for January, March, April, and October-November seasonality in mean returns. Table 3 shows the simple mean differences and t-statistics of whether mean returns are different in a given month. It is shown that there is no evidence for January, March, or April effect, but has evidence for the October-November effect (negative return differences between festival months' mean return and other months' mean return). Except SMB and HML portfolios, the Divali effect seems applied to all the portfolios including the two-size and three value-sorted portfolios (S/L, S/M, S/H, B/L, B/M, and B/H), and the market portfolio.
[3. Conduct standard multivariate regression framework]
Estimation of the multivariate regression system:
Rt=a+bMKTt+sSMBt+hHMLt+ εt
where t=1, 2, ..., T
Rt =the excess return to the portfolio in month t
a = abnormal mean return of portfolio (equal to 0 in CAPM)
b, s, h = market, size, and value factor exposures of the portfolio
: the mean-zero asset-specific return of the portfolio
MKTt = the excess return to the market portfolio
SMBt =the return to the size factor portfolio
HMLt =the return to the value factor portfolio
Connor and Sehgal estimate and test variants with some coefficients being zero, excluding the variables from the regression. For instance, Sharpe-Lintner CAPM can be estimated by imposing the restriction, ‘s=h =0’. They supposed that follows a multivariate normal distribution and is independently distributed. Table 4 shows the estimates excluding one or more factors. The market factor explains the largest fraction of variation in stock returns for the six size and value-sorted portfolios. By excluding both SMB and HML, the market factor has adjusted R squared of 70 -80%. For some portfolios, adding SMB to the market model regression (SMB and Market factor without HML) increases R squared more than the market factor itself. However, excluding market factors with only two factors (SMB and HML) has low explanatory power with R squared (-0.11 ~ 0.226), which means the market factor ranks first in explanatory power. The table is cut off from the original source, but the writer says that in the three-factor regression (MKT, SMB, and HML), the SMB factor has three significant exposures and the HML has four, which means that there is no clear rank of explanatory power in these two factors. Even if the part of the table showing the three factors regression got cut off, the writer says that it indicates that the three-factor model provides the most suitable description of risk in this two size and three value-sorted portfolios (S/L, S/M, S/H, B/L, B/M, and B/H).
[1] Keim, D. B. (1983). Size-related anomalies and stock return seasonality: Further empirical evidence. Journal of financial economics, 12(1), 13-32.
[4. Testing of zero-beta variants of the Fama-French model]
Standard multifactor pricing models, such as APT and CAPM, has an assumption that the investors can borrow and lend freely at a risk-free rate. As mentioned above, in this case, the risk-free Proxy uses the 91-day Treasury bills. For the observed risk-free rate from this 91-day T-bills, there are two possible problems in it. First, before 1993, 91-day T-bills were regulated to have a 4.6% constant annual yield. Since 1993, it has been determined on an auction basis. The rate was regulated and fixed artificially during the first 30 months of the sample period. Second, even in the deregulated period since 1993, the investors in the Indian stock market had higher borrowing and lending rates than the government T-bills.
The problems can be addressed by estimating a zero-beta version of Rt = a + bMKTt + sSMBt + hHMLt + εt, with the zero-beta rate (not observed rate, but estimated rate). The writers address the problems by supposing that the true model of expected returns has a zero-beta expected return (imposing condition of a=0, and replacing the risk-free return Rf with zero-beta expected return Rz), given:
Rt + Rf - Rz = b(MKTt + Rf - Rz) + sSMBt + hHMLt + εt
where Rf =risk-free return (observed)
Rz = zero-beta expected return (estimated)
Note) SMB and HML are unaffected by the use of zero-beta since they are portfolio return differences
which can be rearranged as:
Rt = (1-b)γ + b(MKTt) + sSMBt + hHMLt + εt
where γ = Rz- Rf (Zero-beta return premia)
Finally, the writers estimate a version of a zero-beta corrected model that reflects the difference during the regulated and unregulated period of 91-day Treasury bills, with the form:
Rt = δ1(1-b)γ1+ δ2(1-b)γ2+ b(MKTt) + sSMBt + hHMLt + εt
where δ1, δ2 = = dummy variable for the pre and post periods
γ1, γ2 = zero-beta return permia from the pre and post periods
Table 5 shows the estimates and approximate t-statistics of the coefficients. The t-value (point estimate) for γ = Rz- Rf (Zero-beta return premia) in regulated period ( γ1 ) is is higher than unregulated the period ( γ2), each point estimate is not significantly different from zero, which doesn't give accurate estimates of zero-beta returns. The writers assume the reason from the high volatility of Indian equity returns and not enough same size, which is not insufficient to estimate a zero-beta return properly.
[5. Common Risk Factor in Earnings]
Fama and French (1995) argued that the market, size, and value factors in returns can be associated with common factors in earnings shocks.[1] Fama and French examined that high BE/ME signals persistent poor earnings and low BE/ME signals strong earnings. Additionally, the market, size, and BE/ME factors in earnings are valid like those in returns. In this research, the writers examine the evidence in market, size, and BE/ME factors in earnings for the Indian stock market.
For measuring year-to-year growth in earnings, PBDT (Profit Before Depreciation and Taxes) is used, since it is a noramlly positive value, thereby having no problems for growth rate calculation. There are three factors in earnings growth like in stock returns: EGSMB, EGHML, and EGMKT. EGSMB (size factor in earnings growth) is the simple average of the percentage change in earnings for small stock portfolios (S/L, S/M, S/H) minus the average of the percentage change in earnings for three big portfolios (B/L, B/M, B/H). EGHML (value factor in earnings growth) is the simple average of the percentage change in earnings for the two high BE/ME portfolios (S/H, B/H) minus the average of the percentage change in earnings for the two low BE/ME (S/L, B/L). SMKT (market factor in earnings growth) is the average percentage change in earnings in all stocks.
Table 6 shows the result of the regression of growth in earnings for the six size and value sorted portfolios on three factors (GEMKT, GESMB, GEHML). The results, except for the SMB factor exposure of the B/M portfolio (the factor exposure coefficient would be negative, if following the Fama-French factor model, however, is positive), are in line with what Fama and French (1995) argued. Except that, the other exposures are in line, where SMB factor exposure of small-cap portfolios is positive, whereas large-cap portfolios are negative. Also, the HML factor exposure of the high and middle B/E portfolios is positive, whereas the low B/E portfolio is negative. All of the six portfolios’ adjusted R-squared is also fairly high, ranging from 0.644 to 0.967.
[1] Fama, E. F., & French, K. R. (1995). Size and book‐to‐market factors in earnings and returns. The journal of finance, 50(1), 131-155.
[6. Summary of the way this paper uses the Factor Model]
The paper "Test of the Fama and French Model in India” was such a good reference resource for applying all of the knowledge and intuition that I have gained through this semester. To finish the analysis of this paper, I want to summarize the paper with my own interpretation of how the authors were leading the research. First, the data was collected. The important part of the process of collecting data was that the analyst needs to consider the characteristics of the data. It can be not only the information that you can gain from the data itself, such as the data’s statistical information but also all the available information, such as cultural, social, and political information relevant to financial information. Here, the case was the 91-day T-bills regulated till 1993 from political issues. Second, the factor model portfolios were made. For that, all necessary data was defined, such as market factor as market index return(CRISIL-500) or 91-day Treasury bill as risk-free return proxy. Overall, the two size and three value-sorted portfolios (S/L, S/M, S/H, B/L, B/M, and B/H), and MKT, SMB, and HML portfolios were made. Thirdly, before conducting the Fama-French model regression, the authors conducted descriptive statistics on the factor portfolio returns to see the relationship between the factor returns. The result showed a negative relation between the size and the average return and the small-cap stocks have a positive correlation, whereas, the larger stocks are negatively correlated with average returns. This result was compared with Fama and French (1993) in the US stock market. Also, the correlations between the factor portfolios, skewness, and kurtosis were measured. Fourthly, the seasonality in the returns was measured. I realized that the seasonality is also important to be analyzed for catching the anomaly in the market. Seasonality can be also connected with social, cultural, and political factors, so analyzing the market is not about just analyzing market data, but the overall circumstances surrounding the market. After the descriptive statistics analysis, a multivariate regression was conducted for estimating the coefficients of each factor (MKT, SMB, HML). Connor and Sehgal estimate and test variants with some coefficients being zero, excluding the variables from the regression. From this, the explanatory power of each factor was more visible (i.e. the market factor ranks first in explanatory power, but there is no clear rank of explanatory power in SMB and HML factors), and the effect of adding certain factors with different combinations (e.g. excluding market factor with only two factors (SMB and HML) has low explanatory power. Fifthly, the zero-beta variants test was conducted from two possible problems of the observed risk-free rate (91-day T-bills); regulated and unregulated periods. The problems can be addressed by estimating a zero-beta version of the Fama-French three-factor multivariate regression model. However, the point estimates for zero-beta return premia in different periods were not significant because of the high volatility of Indian equity returns and not enough sample size. Lastly, the common risk factor in earnings was analyzed. It was following Fama and French (1995)'s argument that the market, size, and value factors in returns can be associated with common factors in earnings shocks. The three-factor factors in earnings growth (EGSMB, EGHML, and EGMKT) and PBDT (Profit Before Depreciation and Taxes) were analyzed through regression. Except for the SMB factor exposure of the B/M portfolio, the result was in line with what Fama and French (1995) argued; SMB factor exposure of small-cap portfolios is positive, whereas large-cap portfolios are negative. Also, the HML factor exposure of high and middle B/E portfoliosis positive, whereas that of the low B/E portfolio is negative.