a reexamination of factors
TRANSCRIPT
A RE-EXAMINATION OF FACTORS AFFECTING
RETURNS IN INDIAN STOCK MARKET
Rahul Kumar 1
Dr. Chandra Prakash Gupta2
1 The author is a member of The Institute of Chartered Accountants of India and a Research Scholar, Management Development Institute, Gurgaon (India). He can be reached at [email protected]. Mobile +919911007220
2 The author is Professor in Finance, Management Development Institute, Gurgaon (India). He can be reached at [email protected]. Mobile +919818041308
Abstract
The paper evaluates the return generating process for the Indian stock market implied by
general multi-factor model and the Fama-French three-factors model in specific. It tests
systematically and robustly the relevance of Fama-French three-factor model in
explaining the cross sectional differences in returns in Indian using a large sample data
pooled from wide range of companies and periods. The empirical results show that the
Indian equity market exhibits a strong size effect and value effect which are consistent
with the findings of Fama and French (1996) for US portfolios and Sehgal (2003) for
Indian stocks. Thus, it provides an evidence of the pervasiveness of the Fama-French
three-factor model in explaining the cross sectional differences of stock returns. This
study may provide a strong support for a broader and generalized asset pricing model in
which there are multiple risk factors.
Key words: Multi-Factor Model; Asset Pricing; Size effect; Value effect.
II
Introduction
The primary implication of the Single Index Model3 developed by Sharpe (1964) is that
the portfolio risk can be divided into two parts, namely, diversifiable (unsystematic), and
non-diversifiable (systematic). Unsystematic risk is the component of the portfolio risk
that can be eliminated by increasing the portfolio size, the reason being that risks that are
specific to an individual security can be eliminated by constructing a well-diversified
portfolio. Systematic risk is associated with overall movements in the general market or
economy and therefore is often referred to as the market risk. The market risk is the
component of the total risk that cannot be eliminated through portfolio diversification
(Elton and Gruber, 1996). Based on the Single Index Model, Sharpe (1964), Lintner
(1965) and Mossin (1966) independently developed what has come to be known as the
Capital Asset Pricing Model (CAPM).The Capital Asset Pricing Model (CAPM) relates
the expected rate of return of an individual security to a measure of its systematic risk
(Galagedera, 2004). The oldest complete model of asset pricing, the model explains the
differences in riskiness between assets. Miller (1999) stated that CAPM not only
expressed new and powerful insight into the nature of risk but also through its empirical
investigation contributed to the development of finance and to major innovation in the
field of econometrics. The CAPM is an ex-ante equilibrium model based on expectations
and a string of assumptions (Malin & Veeraghavan, 2004). The CAPM model states that
(a) expected returns on security is a positive linear function of its market (the slope in
the regression of a security’s return on the market’s return), and (b) the market suffices
to describe the cross section of expected return.3 Details of the Single Index Model is given in Appendix - I
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Following the publication of Single Index Model and CAPM, there were many empirical
studies that tested whether the model adequately describes the way stock market prices
behave in practice. Many empirical researchers have found that there are influences
beyond the market that cause stocks prices to move together (Elton & Gruber, 1996) and
this laid to the development of multi-index4 (multifactor5) models. Specifically, these
studies have found through their empirical researches that single factor (market) is not
sufficient in explaining security returns, as postulated by single index model and CAPM.
Further, these studies found that the company variables, that do not find a place in one
factor asset pricing framework of CAPM, tend to empirically exhibit a relationship with
stock returns. Prominent amongst these company characteristics are Firm size (measured
in terms of market capitalization) [(Banz, 1981), (Cook and Roseff, 1982)], earning- yield
(E/P Ratio) [(Ball, 1978), (Basu, 1983)], Leverage (Bhandari, 1988), Cash flow to price
(C/P ratio) [(Hawawini,1991), (Chan, Hamao, and Lakonishok, 1991)] and the firm’s
book-to-market equity (BE/ME) ratio (Chan, Hamao, and Lakonishok, 1991). These
company characteristics were found to provide a better explanation than market factor
alone for the cross-section of average stock returns. These evidences suggest that
multifactor models should be considered in research applications that require estimates of
expected returns (Fama, 1996).
It is now well accepted that multi-factor model is a natural representation of the real
world, but a central empirical issue is which factors best account for differences in the 4 For the purpose of this study, the terms ‘multi-index’ and ‘multifactor’ are used synonymously, unless stated otherwise.5 Multifactor models are an attempt to capture some of the non market influences that cause securities to move together (Elton & Gruber, 1996?).
2
returns and thus, finally providing the systematic risks in the market for a multi-factor
asset pricing model. In an empirical attempt to operationalise multifactor model, Fama
and French developed three-factor model. Using a multifactor approach, they, in their
landmark paper in 1992, empirically examined the joint role of market beta, firm’s size,
firm’s book-to-market equity (BE/ME) ratio, earning yield (E/P ratio) and leverage in the
cross-section of average stock returns. They found that (a) the excess market return has
some information about average returns; and (b) the combination of size (market
capitalization) and book-to-market (BE/ME) absorbs the role of leverage and earning
yield (E/P) in average stock returns. Based on their empirical findings in Fama and
French (1992), Fama and French (1993) propounded a three-factor asset pricing model,
comprising of the market factor and two mimicking portfolios that proxy for common
factors in returns relating to size and book to market equity (often called a “value”). They
showed that their three-factor model captures much of the variations in the cross-section
of average stock returns in a portfolio, which is missed by Sharpe’s Single Index Model.
The conclusions reached by the Fama and French (1993) could be consistent with a
multifactor version of Merton’s (1973) intertemporal capital asset pricing model
(ICAPM) or the arbitrage pricing model (APT) of Ross (1976), in suggesting that the
higher average returns on value stocks are compensation for risk missed by the CAPM.
The contribution of Fama-French (1996), particularly relating to the interpretation of the
results, ignited a flurry of responses from the academy. The first wave of responses was
based on result of DeBondt and Thaler (1987). Based on the empirical evidences of
overreaction hypothesis of DeBondt and Thaler (1987), Lakonishok, Shleifer, and
3
Vishny (1994) and Haugen (1995), suggests that the high returns associated with value
stocks are generated by investors who incorrectly extrapolate the past earnings growth
rates of firms. As a result, the market tends to undervalue low BE/ME stocks and
overvalue low size stocks.
A second area of controversy relates to the controversial findings of Daniel and Titman
(1997), suggesting that the return premia on small capitalization and value stocks do not
arise because of the co-movement of these stocks with pervasive factors. They argued
that it is characteristics of the portfolios constructed by sorting rather than the covariance
structure of returns that appear to explain the cross-section variation in stock returns.
Moreover, they further claim that the value premia in the three-factor model are
determined by value characteristics, not by underlying risk characteristics.
In replying to the critique of the Fama and French mulitfactor model, Davis, Fama and
French (2000) extend data back to 1926 (a 68-year sample period) and expand the sample
coverage to all NYSE industrial firms. The results provide no evidence of a sample-
specific explanation for the value premium, with the value premium in pre-1963 returns
close to that observed for the subsequent period in earlier work. These results led Davis et
al (2000) to conclude that the model of Fama and French (1996) explains the value
premium better than the characteristic based model of Daniel and Titman (1997).
Moreover, Davis et al suggest that the evidence in favor of the characteristic model,
provided by Daniel and Titman (1997), appears to be a feature of the sample period.
Davis et al (2000) note that “the acid test of a multifactor model is whether it explains
differences in returns.” and not a particular type of covariance structure.
4
Presently there is considerable evidence from other world markets in support of the three-
factor model. [Chan, Hano, and Lakonishok (1991), Capasul, Rowley and Sharpe (1993),
Fama and French (1998), Chui and Wei (1998)]. However much of the empirical support
is limited to developed capital markets. Kothari, Shanken and Sloan (1995) asserted that
any robust multi-factor model must be tested to work under a variety of conditions and
not for a limited set of portfolios. Hence, there is a need for more sample tests, especially
relating to emerging markets. It is commonly observed that emerging market returns have
unusual features6 [Harvey C (1995)]. Hence they pose a greater challenge to the universal
applicability of rational asset pricing theory. Indian capital market is grossly under-
researched in the area of CAPM and factors model (Manjunatha and Mallikakarjunappa,
2006).
The result of empirical studies in Indian context (Vaidyanathan and Chava, 1997;
Marisetty and Vedpurishwar 2002; Mohanty, 1998, 2002; Sehgal, 2003; Connor and
Sehgal, 2003) supports the Fama and French multi-factor model. However, a recent study
carried out by Manjunatha and Mallikakarjunappa (2006) reveal confounding relationship
among factors viz., market, size, and book-to-market (BE/ME) ratio and portfolio return
(dependent variable). Their study is based on daily data of 66 companies and is
susceptible to suffer from selection bias. Further, various empirical researches suggest
that the fat tails7 are more commonly observable in daily data than monthly data.
6 Emerging markets differ from developed markets as they are expected to exhibit lower level of market efficiency, a less evolved institutional and regulatory framework, and less mature investor behavior. Their economic parameters are relatively unstable, and hence less predictable, and their stock markets are fairly volatile.7 A distribution is said to be a ‘fat-tailed’ or ‘heavy tailed distribution’ if the tails of a non-normal distribution are heavier than the normal distribution. Fat-tailed distributions are also leptokurtic
5
(Agrawal, Rao, and Hiraki (1989) Hence, taking daily data for the regression analysis
may lead to spurious results. The diverse findings in Indian context demands systematic
investigation of the effectiveness of the model by taking a large sample of data pooled
from wide range of companies and periods. And, it is this demand that provides the
necessary motivation for the present study. Therefore, the objective of the current study is
to empirically examine the effectiveness of three-factor model, by testing its validity in
different time periods and on different data sets, in explaining the cross section of stock
returns in Indian market.
The remainder of the study proceeds as follows: Section 2 outlines the objective of the
current study. Section 3 describes the underlying data, date source and data definition.
Section 4 describes the methodology followed for examining factors affecting returns.
Section 5 presents and discusses empirical results and finally, the article ends with
conclusion and directions for future research.
2. Objectives of the study
The confounding empirical results in the literature on Fama and French three-factor
model demands systematic investigation of the effectiveness of the model by taking a
large sample of data pooled from wide range of companies and periods, especially in
emerging economies like India. As discussed earlier, the empirical results related to
multi-factor models in India are showing diverse results so as to reach at some
conclusion. Therefore, there is a need to re-examine the whole issues related to the
distribution.
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effectiveness of three-factor model in India in explaining differences in returns on various
stocks. Consequently, the current study is aimed at examining empirically the
effectiveness of three-factor model – that is, by testing its validity in different time
periods and on different data sets, in explaining the cross section of stock returns in
Indian market. With this objective the study specifically examines the following
questions:-
Is there a significant size effect in Indian stock returns?
Is there a significant value (BE/ME) effect in Indian stock returns?
Is the Fama-French three-factor model a better descriptor of return generating
process in Indian context as compared to Single Factor Model?
3. Data source and data definition
The data comprises of the month end adjusted share price for companies from August
1990 to March 2006. The sample companies form part of S&P CNX 5008, a broad based
stock market index that gives representation to companies of varying level of size and
trading activity. The sample companies account for a major portion of the market
capitalization and average trading volume in India, and hence are presumed to be fairly
representative of market performance. Moreover, bulk of the out of sample companies are
either very thinly traded or do not provide a long and continuous track record of financial
and accounting information. Their inclusion would have posed a serious estimation
problem.
8 Earlier known as CRISIL 500 Equity Index
7
The adjusted share price data has been obtained from CMIE database “Prowess”, the
leading financial database in India. The share price series has been used to construct
monthly return series. The National Index of the Bombay Stock Exchange (BSE, India)
has been used as a surrogate for market index. The BSE National Index is a broad based
and value weighted stock market proxy, constructed on the lines of Standard and Poor,
USA.
Implicit yield on 91 Treasury bill (T- Bills) have been used as proxy for risk free return.
The data source is Reserve Bank of India Bulletin. Since the auction of 91 days Treasury
bill was introduced in India from January 1, 1993, for the period August, 1990 to
December, 1992 the monthly running yield on Government of India securities are taken
as proxy for risk free return. The data source is Report on Currency and Finance, an
annual publication of the Reserve Bank of India.
The necessary accounting information have been obtained for the sample companies from
1990 to 2006 for March end each year, which is the financial closing month in India and
is followed by the sample companies under study. The data source is CMIE Prowess,
which is extensively used by academic researchers as well as practitioners in India.
3.1. Data Definition
i. Size: Market equity (ME) stands as the proxy for the size. ME is also termed as
Market Capitalization which is market price per share times number of shares
outstanding. Market capitalization is calculated in the beginning of July of each year
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t. We have assumed a time lag of one quarter (three months) from the end of the
financial year for the release of financial information to the public by companies.
ii. BE/ME: BE/ME is the ratio of Book value per share of equity to market value per
share of equity. BE/ME is also termed as “value”. BE/ME has been calculated as
book value per share in March end of year t, divided by the market value per share in
beginning July of year t.
Table- I presents the summary statistics of the sample data for the annual period of 1990
to 2006. Column (2) of table (1) represents the number of observations in each year
available for the study. Column (3) to (6) shows the summary statistics of Size (measured
in terms of market capitalization of firms). Column (6) to (8) represents the summary
statistics of BE/ME ratio of firms.
Table I : Summary Statistics of the Sample Data for the annual period of 1990 TO 2006.Year No. of Obs. Size (Rs. In Crores) Book to Market Ratio
Median Mean Std. Dev Median Mean Std. Dev(1) (2) (3) (4) (5) (6) (7) (8)
1990 190 188.38 59.62 399.23 15.65 1.74 100.161991 205 246.81 81.38 528.48 13.73 1.22 105.861992 232 416.16 142.95 986.33 5.75 0.35 60.531993 259 480.27 125.11 1048.45 5.83 0.94 38.911994 307 975.73 283.20 2325.32 2.60 0.58 8.951995 339 726.28 190.51 1663.70 3.29 0.70 23.091996 365 940.46 150.56 2916.24 4.86 1.13 32.831997 375 1145.51 138.00 4057.26 5.40 1.97 12.721998 387 859.39 115.37 2963.14 6.71 2.12 17.271999 394 1173.31 180.05 4191.53 6.08 2.25 13.132000 408 1338.17 136.47 5216.42 4.92 1.85 11.462001 430 985.81 103.29 3971.57 7.21 2.61 16.382002 439 1105.37 155.18 3934.95 6.24 2.42 14.012003 449 1515.87 227.55 5189.35 5.34 2.06 11.342004 455 2258.32 372.36 7460.70 2.07 0.97 4.992005 471 3748.47 826.60 11038.45 1.04 0.60 1.742006 487 4826.81 1045.05 14535.88 0.54 0.39 0.50
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4. Methodology for examining factor affecting returns
This study primarily adopts the methodology suggested in Fama and French (1993) to
capture the cross sectional variation in returns. For the purpose of this study, return at
time t, R(t), is defined as shown in equation (1).
)1(
Where:-
:Return of portfolio p at time t
P (t) : Price of a security at the end of month t
P (t-1) : Price of a security at the end of month t-1
Figure I presents graphically the process followed for examining the factors affecting
returns in the Indian stock market.
10
Figure I: Graphical representation of the methodology for examining common factors affecting
returns
Now, we describe the procedure followed for examining factors affecting return
generating process:-
4.1. Create Size Portfolios
We focus on size portfolio and for the purpose in the beginning of July of each year t
from 1990 to 2006, on the basis of ME, all the sample stocks are sorted in descending
order on the basis of size. The sample is then divided into three groups based on the
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breakpoints for the bottom 30% (Small), middle 40% (Central), and top 30% (Big). This
leads to creation of three size portfolios of stocks falling under each group viz. Small,
Central, and Big named as S, C, and B respectively.
4.2. Create Value portfolio
In the next stage, beginning July of each year t, on the basis of BE/ME we sorted the
sample stocks in descending order and created three value portfolios based on the
breakpoints for the bottom 30% (Low), middle 40% (Medium) and top 30% (High) of the
ranked values of BE/ME of sample companies. This leads to creation of three value
portfolios of stocks falling under each group viz. Low, Medium, and High named as L,
M, and H respectively.
The split of the sample stocks into different categories (3 ME groups and 3 BE/ME
groups) is arbitrary and Fama and French argued that there is no reason that the tests
should be sensitive to this choice.
4.3. Calculate size and value portfolio return
We then calculate returns on nine portfolios constructed from the intersection of three
sizes and three BE/ME groups named as S/L, S/M, S/H, C/L, C/M, C/H, B/L, B/M, and
B/H. S/L portfolio contains stocks of small ME and low BE/ME companies, while B/H
portfolio represents big ME companies with high BE/ME ratio. Monthly equally
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weighted returns on the nine portfolios are calculated from July of year t to June of year
t+1, and the portfolios are reformed in July of year t+1. We calculate returns beginning in
July of year t to ensure that the book equity for year t is known to the investors by that
time for decision making purposes.
The nine size BE/ME portfolios have been consciously constructed to be equally
weighted as suggested by Lakonishok, Shliefer and Vishny (1994), since they contain
less estimation errors compared to the value weighted portfolios. Fama and French
(1996) documented that the three-factor model does a better job in explaining equally
weighted portfolios than value weighted portfolios.
4.4. Size and Value factors return
The Fama- French methodology involves use of three-factors for explaining returns. The
market factor (Rm – Rf) and other two factors relating to size (SMB) and BE/ME ratio
(HML). Given the market factor, we next construct the SMB and HML factors return. To
calculate SMB and HML returns, as explained earlier, companies are divided into nine
groups based on size and the BE/ME ratio. The intersections of the three sizes and three
BE/ME groups produce nine portfolios of stocks which are used to compute the SMB and
HML factor returns.
The SMB factor return is the average of return on three small size portfolio i.e. {(average
of (S/L, S/M, & S/H))} minus the average of return on three big size portfolios i.e.
13
{(average of (B/L, B/M, B/H))}. These returns are calculated for each month over the 12
months following portfolio formation. The process was repeated until the portfolios were
reconstructed. Similarly, the HML factor return is the average of return on three high
BE/ME portfolios i.e. {(average of (H/S, H/C, & H/B))} minus the average of return on
three low BE/ME portfolios i.e. {average of (L/S, L/C, & L/B))}. This way monthly
factor returns are calculated for the each month over the 12 months following portfolio
formation. The process was repeated until the portfolios were reconstructed. In the next
stage, we investigate the relationship between portfolio returns and explanatory variables
viz., market factor, size (ME), and value (BE/ME) factors.
4.5. Examination of explanatory factors of returns
We run time series regressions, to examine whether different risk factors, individually
and/or collectively, capture variations in returns. The time series regression equations to
examine the relationship between portfolio returns and overall market factor, size (ME),
and value (BE/ME) factors, separately and/or collectively are listed in equation (2) to (6)
as follows:-
a. Regression using only the market factor (RM – RF) as explanatory variable (the
Single Index Model). Symbolically,
)2(
b. Regression using SMB and HML as explanatory factors. Symbolically,
)3(
c. Regression using market and SMB as explanatory factors. Symbolically,
14
)4(
d. Regression using market and SMB as explanatory factors. Symbolically,
)5(
e. Regression using market, SMB and HML factors (the Fama French Model)
)6(
Where:
is the monthly return of a certain portfolio (S/L, S/M, S/H, C/L, C/M,C/H, B/L, B/M,
B/H). is the monthly risk free rate. is the monthly return on market. For the
purpose of this study, Bombay stock exchange (BSE, India) national index has been used
as a surrogate for market.
(Small minus Big) represents the size factor. HML (High minus Low) represents
the BE/ME (value) factor. The loadings are the slopes in the time series
regression.
Thereafter, before making any final assertion about the risk factors in explaining returns,
we examined the significance of the intercept of the regression equation listed in equation
(2) to (6).
5. Data analysis and discussion
15
Table II shows the mean monthly returns over the risk free return (excess return) on the
Size & BE/ME sorted portfolios. The nine size BE/ME portfolios exhibit an excess return
ranging from -0.31% to 1.45% per month. The portfolio returns confirm the Fama (1993,
1995) evidence that there is a negative relation between size and return. Various theories
have been put forth to explain the size effect. One of the most frequently mentioned
explanations holds that small stocks contain some systematic risks that are not adequately
measured by empirical researchers. Small firms are small because the market uses a high
discount rate to capitalize its future cash flows, or because they have lost market values
due to poor past performance. They are more likely to have cash flow problems and less
likely to survive adverse economic conditions. Since these risks cannot be easily captured
by empirical models, small stocks tend to exhibit a higher risk-adjusted return. (See, for
example, Chan and Chen (1991), Fama and French (1996), Berk (1995), Vassalou and
Xing (2004), Berk, Green, and Naik (1999), Gomes, Kogan, and Zhang (2003)]. Chan
and Chen (1991) reported that the firm size proxy a distress effect in expected stock
returns. Other explanation for the size effect, dating back to Stoll and Whaley (1983), is
based on liquidity. Larger stocks are generally more liquid, and investors are willing to
compromise returns for higher liquidity. Therefore equilibrium returns of larger stocks
are lower. Another possible explanation of the size effect can be the transaction cost.
Since the large stocks attract lesser transaction costs such as search cost, monitoring cost
etc vis-à-vis that of small stocks, size premium for large stocks are negative.
Furthermore as shown in table II, the relation between BE/ME and return is positive.
Various researchers attempted to explain the value premium in stock returns. Chan and
16
Chen (1991) and Fama and French (1992) suggested that it is possible that the risk
captured by BE/ME is the relative distress factor. They postulate that the earning
prospects of the firms are associated with a risk factor in returns. Firms that the market
judges to have poor prospects signaled here by low stock price and high ratios of book to
market equity have higher expected returns (they are penalized with higher cost of
capital) than firms with high prospects.
Hence, the Indian equity market seems to exhibit a strong size effect and value effect.
This finding contrast with Fama and French (1995) findings where they show a strong
value effect and a conditional size effect for US data.
Table II : Mean monthly excess returns on the Size (ME)– Value (BE/ME) sorted portfolios
Value (BE/ME)
Size
(ME)
Low Medium HighSmall 0.76% 0.85% 1.45%
Central -0.30% 0.07% 1.17%
Big -0.31% 0.03% 0.62%
Table III provides the summary statistics of risk premiums of the factors viz., market,
size, and value factors. The average value of RM - RF (average premium per unit of
market factor) is 0.64% per month. The annual excess return of about 7.71% is low from
the investors’ perspective compared to the risks posed by emerging market such as India
and the risk free rate of return available on investment in 91 days Treasury bill. The
average SMB factor return (the average premium for the size factor in returns) is about
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0.90% per. The size premium is about 1.14 times the market premium. The average HML
factor return (the average premium for the value factor in return) is 1.03% per month.
Table III : Summary statistics of Market, Size, and Book to Market
Mean Standard Deviation
RM-RF 0.0064 0.087
SMB 0.0090 0.047
HML 0.0103 0.033
Table IV provides the Pearson’s correlation coefficient between factor returns. The
correlation is significant between market, SMB, and HML suggesting some overlapping
amongst factors.
Table IV: Pearson’s Correlation Coeffecient between Market, SMB and HML FactorsMarket SMB HML
Market 1 -.216(**) .145(*)SMB -.358(**)HML** Correlation is significant at the 0.01 level (2-tailed).* Correlation is significant at the 0.05 level (2-tailed).
In the subsequent sub-section, we report and discuss the empirical results of our time
series regressions.
a. The market factor as explanatory variable
Table V reports the result of the regression using market factor as explanatory variable of
the stock returns. The slope of the market factor (b) indicates that beta risk does not vary
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significantly between smaller companies and companies with lower BE/ME ratio, and big
companies and companies with high BE/ME ratio. The empirical result is not consistent
with Fama and French (1993). Further, our result (see Table V) shows that the market
factor coefficient (b) is positive and highly significant in all nine portfolios. The t
statistics of all the beta (b) values are more than 7 implying statistical significance of beta
in explaining cross section of expected returns. The adjusted R2 value ranges from 0.20 to
0.62 for the sample portfolios. The adjusted R2 values are relatively lower for small size
stock portfolios (S/L, S/M, and S/H) showing the inability of single index model’s market
factor in explaining the size effects in return. The average of adjusted R2 is 40.63%. It
implies that market factor does explain a proportion of the common variation in stock
returns.
Table V : Excess Portfolio Returns Regressed on Excess Returns for the Market Factor
Portfolio b t(b) Adjusted R2
S/L 0.554 7.23 0.204S/M 0.426 7.05 0.197S/H 0.541 9.84 0.324C/L 0.607 11.70 0.405C/M 0.692 12.37 0.433C/H 0.647 13.92 0.492B/L 0.564 14.17 0.501B/M 0.604 13.49 0.476B/H 0.704 18.22 0.624
b. SMB and HML factors as explanatory variable:
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Table VI shows the power of size and value factors in explaining portfolio returns. The
SMB slopes (s) are statistically significant at the 5% level for the small stock and central
stock portfolios (S/L, S/M, S/H and C/L, C/M, C/H). But for big size portfolios (B/L,
B/M, and B/H) slopes (s) are not statistically significant. The HML slopes (h) indicate
strong significance for low and high value stocks, but for the central value stock, it does
not indicate statistically significant effect. The average of the adjusted R2 is 13%
indicating that that in the absence of the market factor, the SMB and HML fail to capture
common variations in returns.
Table VI : Excess Portfolio Returns Regressed on Mimicking returns for Size (SMB) and Value (HML) factors
Portfolio s h t(s) t(h) Adjusted R2
S/L 1.063 -0.92 8.04 -4.91 0.399S/M 0.881 0.09 7.50 0.55 0.230S/H 0.933 0.582 8.10 3.56 0.243C/L 0.313 -0.387 2.42 -2.12 0.06C/M 0.362 0.151 2.48 0.73 0.201C/H 0.341 0.208 2.67 1.15 0.025B/L -0.097 -0.543 -0.89 -3.52 0.050B/M -0.029 -0.063 -0.24 -0.36 -0.009B/H 0.004 0.361 0.033 2.05 0.013
c. Market and SMB factors as explanatory variable:
Table VII shows that the coefficients of the market factor (b), and size factor (s) is
positive and highly significant in the case of all of the nine portfolios. While t statistics of
all the market slopes are more than 14 standard errors from 0, the size slopes are
statistically significant at the 5% level for small stock portfolios. The improved adjusted
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R2 values for the small stock portfolios also confirm that the size factor in association
with market factors contributes noticeably towards the explanation of stock returns. The
average of the adjusted R2 is 58.37% which indicate that the market and SMB factors
capture a greater proportion of common variations in returns compared to one factor
single index model.
Table VII : Excess portfolio returns regressed on excess returns for the market factor and mimicking returns for the size (SMB) factors
Portfolio b s t(b) t(s) Adjusted R2
S/L 0.7398 1.591 14.91 17.34 0.683S/M 0.553 1.078 11.96 12.62 0.553S/H 0.663 1.051 16.88 14.46 0.671C/L 0.688 0.686 14.85 8.01 0.549C/M 0.765 0.629 14.69 6.54 0.532C/H 0.715 0.574 16.87 7.33 0.600B/L 0.596 0.278 15.13 3.82 0.533B/M 0.632 0.238 14.02 2.87 0.495B/H 0.728 0.204 18.718 2.83 0.637
d. Market and HML factors as explanatory variable:
Table VIII presents the impact of market factor and value factor in explaining returns.
We observe from Table VIII that market slopes are statistically significant at 5% level in
all cases. The value factor coefficient (h) is negative for all portfolios except (B/H)
portfolio, and statistically significant for the (S/L), (S/M), (C/L), (C/M), (B/L), and (B/M)
portfolios. The value factor coefficient (h) increases monotonically for the low value
portfolio to high value portfolio. The result suggests pervasiveness of value effect in
return. The average of the adjusted R2 is 46.87% and show a marginal improvement in
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explanation of the cross sectional differences in returns as compared to the regression
model using market factor alone (table- V)
Table VIII : Excess portfolio returns regressed on excess returns for the market factor and mimicking returns for the Book to market equity (HML) factors
Portfolio b h t(b) t(h) Adjusted R2
S/L 0.649 -1.706 10.39 -10.42 0.485S/M 0.456 -0.529 7.65 -3.39 0.237S/H 0.546 -0.100 9.817 -0.69 0.323C/L 0.652 -0.796 13.59 -6.336 0.504C/M 0.709 -0.304 12.637 -2.068 0.442C/H 0.659 -0.217 14.102 -1.774 0.497B/L 0.604 -0.724 17.182 -7.866 0.618B/M 0.619 -0.285 13.864 -2.438 0.489B/H 0.699 0.091 17.88 0.892 0.624
e. Market, SMB and HML factors as explanatory variable
Table IX shows the regression results using all the three-factors viz. market, size, and
value to explain the portfolio returns. We observe from Table IX that the market slope (b)
and SMB slope (s) are positive and statistically significant in all the cases. The SMB
slope (s) is decreasing from small to big size portfolio. The HML slope (h) is statistically
significant except in the case of S/M, C/M and C/H stock portfolios. The HML slope (h)
is monotonically increasing from low to high value stocks. The mean adjusted R2 of the
regression model is 61.2% indicating that the three-factor model provides the best
description of portfolio returns.
Table IX : Excess portfolio returns regressed on excess returns for the market factor and mimicking returns for the size (SMB) and Book to market equity (HML) factors
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Portfolio b s H t(b) t(s) t(h) Adjusted R2
S/L 0.769 1.330 -1.077 18.668 16.467 -9.544 0.782S/M 0.553 1.073 -0.022 11.912 11.786 -0.174 0.551S/H 0.651 1.159 0.447 17.249 15.677 4.329 0.698C/L 0.702 0.557 -0.533 15.805 6.398 -4.375 0.587C/M 0.765 0.628 -0.007 14.624 6.119 -0.049 0.529C/H 0.713 0.588 0.060 16.755 7.058 0.521 0.597B/L 0.614 0.116 -0.669 17.282 1.668 -6.872 0.622B/M 0.637 0.192 -0.195 14.159 2.175 -1.580 0.498B/H 0.722 0.255 0.211 18.655 3.364 1.999 0.643
f. Testing of significance of intercept
The regression results in Table IX suggest that the market, SMB and HML proxy for
common risk factors in returns. We next verify if the proxy risk factors suffice to explain
the returns on portfolio. If the explanatory factors are suitable and sufficient proxies for
underlying common risk factors, the intercept of the time series regression of excess
returns on the mimicking portfolios should not be significantly different from 0. Table X
lays out the intercept results for regression in Table V to Table IX.
While the intercepts of regressions (b) (c)and (d), relating to Table V to VII, are
important, the intercept of the regression (a) and (f) are of great interest as they pertain to
the one factor single index model and three-factor model respectively. Using the single
factor model, the intercept values are statistically significant for S/H portfolio at 5%
level. The sample intercepts however sober down and none of them are statistically
significant in the three-factor model framework. These intercept term confirm that the
three-factor model does capture most of the variations in stock returns, that is missed by
single factor model.
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Table X: Intercepts for Excess returns on nine portfolios formed on Size and BE/ME(a)
Portfolio a t (a)S/L 0.004 0.605S/M 0.005 1.094S/H 0.0109 2.286C/L -0.006 -1.522C/M -0.003 -0.757C/H 0.007 1.856B/L -0.006 -1.944B/M -0.003 -0.919B/H 0.001 0.500
(b)
Portfolio a t(a)S/L 0.007 1.166S/M -0.0004 -0.073S/H -1.93E-5 -0.003C/L -0.001 -0.298C/M -0.004 -0.583C/H 0.006 1.050B/L 0.003 0.636B/M 0.001 0.202B/H 0.002 0.410
( c)
Portfolio a t(a)S/L -0.011 -2.675S/M -0.004 -1.195S/H 0.0006 0.187C/L -0.013 -3.38C/M -0.009 -2.178C/H 0.002 0.5155B/L -0.009 -2.762B/M -0.005 -1.513B/H -0.0003 -0.092
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(d)
Portfolio a t(a)S/L 0.021 3.728S/M 0.011 2.053S/H 0.011 2.385C/L 0.001 0.232C/M -0.0006 -0.135C/H 0.009 2.299B/L 0.0004 0.139B/M -0.0007 -0.187B/H 0.007 0.222
(e) Portfolio a t(a)
S/L 0.002 0.441S/M -0.004 -1.048S/H -0.004 -1.383C/L -0.007 -1.710C/M -0.009 -2.007C/H 0.001 0.291B/L -0.001 -0.375B/M -0.003 -0.844B/H -0.002 -0.810
5. Summary and conclusion
Having systematically and robustly tested the relevance of Fama- French three-factor
model in explaining the cross sectional differences in portfolio returns, in the Indian
context using a hitherto large sample data pooled from wide range of companies and
periods, this section summarizes the findings herein below:
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1. There is a strong size and value effects in Indian stock market. The difference
between the mean returns for small and big size stocks (SMB) is about 11% on an
annualized basis. Whereas, the difference between the mean returns for high and
low value stocks (HML) is about 12% on an annualized basis. The evidence
shows that the value stocks outperform the size stock. The result is inconsistent
with the findings of Sehgal (2003).
2. The pearson’s correlation between market and SMB and SMB and HML is
significant at 1% level. Whereas, the correlation between Market and HML is
significant at 5% level. The correlation result suggests some overlapping amongst
the factors and hence, explanatory factors are not orthogonal to each other.
3. The Fama-French three-factor model explains the cross section of average stock
returns that is missed by single Index model. The mean adjusted R2 improves
from 40.63% for single factor model to 61.2% for the Fama French model. The
mean absolute alpha (extra-normal return) declines from 0.0051 per month for
one factor model to 0.0036 per month for the multi-factor model.
We observe that overall the FF three-factor model explains the variation in cross section
of stock returns in a meaningful manner. We also note that the three-factor model is
robust in the sense that the value of alpha intercept is statistically insignificant in the case
of most of the portfolios. Our results are consistent with the findings of FF (1996) for US
portfolios and Sehgal (2003) for Indian stock market and provide evidence of the
pervasiveness of the FF three-factor model in explaining the return generating process.
Consequently, we reject the data snooping hypothesis, arguing that if the FF (1996) were
26
involved in data snooping, our empirical findings would have contradicted their results.
Our empirical findings are based on different time periods as well as on different sample
of securities than in Sehgal (2003). In summary, this study provides support for a
broader, rational asset pricing model in which there are multiple risk factors.
27
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Appendix I
Single Index Model
Sharpe (1963) develop single index model. The single index model states that co-
movement between stocks is due to single factor. The basis equation underlying the
single index model is:
------------------------------------------------------------------ (i)
Where:
= Return on the jth stock
= Component of security j that is independent of factor performance
= Coefficient that measures change in given a change in
= Rate of change in factor
The term in equation (i) comprises of two elements which is the expected value of
and which is the random element of
The single index model equation (i), therefore, becomes:
------------------------------------------------------------------ (ii)
It is assumed that and are uncorrelated as are and for .
31