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Comparative tests of Fama-French Three and
Five-Factor models using Principal Component
Analysis on the Chinese Stock Market
Name: Eric Lai
Degree: BSc Investment and Financial Risk Management
Cass Business School
Supervisor Name: John Hatgioannides
Submission Date: 8th April 2016
"I certify that I have complied with the guidelines on plagiarism
outlined in the Course Handbook in the production of this
dissertation and that it is my own, unaided work".
Signature...................................................................................................
1
Acknowledgements
I offer my gratitude and thanks to my supervisor John Hatgioannides who has
piqued my interest in financial engineering for asset pricing models and guided me
throughout the project.
I also thank my friends and family for their continued support with helping me
make this project possible.
2
Table of Contents
Acknowledgements .................................................................................................................. 1
List of Tables............................................................................................................................ 4
List of Figures .......................................................................................................................... 5
Abstract .................................................................................................................................... 6
1. Introduction ......................................................................................................................... 7
1.1. Chinese Stock Market Overview .................................................................................. 7
1.2. Outline of the paper....................................................................................................... 9
2. Literature Review ............................................................................................................. 10
2.1. Harry Markowitz Portfolio Theory............................................................................. 10
2.2. Capital Asset Pricing Model ....................................................................................... 12
2.3. Time-varying beta ....................................................................................................... 13
2.4. Early Tests ................................................................................................................... 13
2.5. Alternative variables and higher moment models .................................................... 14
3. Empirical Framework ....................................................................................................... 17
3.1. Fama and French Three-Factor Model ...................................................................... 17
3.2. Principal Component Analysis ................................................................................... 18
3.3. Fama and French Five-Factor Model ........................................................................ 22
4. Data .................................................................................................................................... 24
5. Methodology ....................................................................................................................... 27
5.1. Fama-French Three-Factor Model ............................................................................. 27
5.2. Principal Component Regression ............................................................................... 28
5.3. Fama and French Five-Factor Model ........................................................................ 31
6. Results ................................................................................................................................ 34
7. Diagnostics ......................................................................................................................... 40
7.1. Multicollinearity .......................................................................................................... 40
7.2. Normality ..................................................................................................................... 41
7.3. Heteroscedasticity ....................................................................................................... 42
3
7.4. Autocorrelation ............................................................................................................ 43
7.5. Stationarity .................................................................................................................. 44
8. Empirical Analysis ............................................................................................................ 46
8.1. Summary Statistics for factor returns ....................................................................... 46
8.2. Regression Analysis ..................................................................................................... 48
8.3. Principal Component Analysis ................................................................................... 50
8.4. Model Evaluation ......................................................................................................... 51
9. Conclusion .......................................................................................................................... 53
References .............................................................................................................................. 55
4
List of Tables
Table 1: Total index weight of the top 100 market constituents listed in the Shanghai
Stock Exchange from the period June 2004 to June 2015 ................................................. 25
Table 2: Construction of Size and BE/ME factors ............................................................. 28
Table 3: Construction of Size, BE/ME, profitability, and investment factors ................. 33
Table 4: Average monthly percent returns for portfolios formed on Size and BE/ME, Size
and OP, Size and Inv; June 2004 to June 2015, 132 months ........................................... 34
Table 5: Summary statistics for average monthly factor returns .................................... 35
Table 6: Fama and French 3-Factor model regressions for 9 Size-BE/ME portfolios; June
2004 to June 2015 ................................................................................................................. 36
Table 7: Principal component regressions for 9 Size-BE/ME portfolios; June 2004 to June
2015 ....................................................................................................................................... 37
Table 8: Fama and French 5-factor 2 x 2 sort regressions for 9 Size-BE/ME portfolios;
June 2004 to June 2015 ....................................................................................................... 38
Table 9: Fama and French 5-factor 2 x 2 x 2 x 2 sort regressions for 9 Size-BE/ME
portfolios; June 2004 to June 2015 ..................................................................................... 39
Table 10: Correlation matrix of Fama and French 3-factor model factors ...................... 40
Table 11: Correlation matrix of Fama and French 5-factor 2 x 2 sort model factors ..... 40
Table 12: Correlation matrix of Fama and French 5-factor 2 x 2 x 2 x 2 sort model factors
................................................................................................................................................. 41
Table 13: Jarque-Bera test of Fama and French 3-factor model ...................................... 41
Table 14: Jarque-Bera test of Fama and French 5-factor 2 x 2 sort model ..................... 42
Table 15: Jarque-Bera test of Fama and French 5-factor 2 x 2 x 2 x 2 sort model ......... 42
Table 16: White's Heteroscedastic test of Fama and French 3-factor model .................. 43
Table 17: White's Heteroscedastic test of Fama and French 5-factor 2 x 2 sort model .. 43
Table 18: White's Heteroscedastic test of Fama and French 5-factor 2 x 2 sort model .. 43
Table 19: Breusch-Godfrey test of Fama and French 3-factor model .............................. 44
Table 20: Breusch-Godfrey test of Fama and French 5-factor 2 x 2 sort model .............. 44
Table 21: Breusch-Godfrey test of Fama and French 5-factor 2 x 2 x 2 x 2 sort model . 44
5
Table 22: Augmented Dickey-Fuller test of all nine portfolio log returns series (SL, SN,
SH, ML, MN, MH, BL, BN, and BH) ................................................................................. 45
Table 23: Correlation of PCA factors and factors of the Fama-French three-factor model
................................................................................................................................................. 50
Table 24: Weighting coefficients of principal components of average monthly returns for
nine portfolios sorted on Size-BE/ME from June 2004 to June 2015, 132 months ......... 50
Table 25: Summarised alpha and R2 of regressions on nine portfolios sorted on Size-
BE/ME from June 2004 to June 2015, 132 months ........................................................... 51
List of Figures
Figure 1: Graphical orthogonal transformation of principal component analysis .......... 19
Figure 2: Scree plot of ordered 9 eigenvalues and Kaiser-Guttman rule ........... 30
6
Abstract
The Fama-French (1993) three-factor model directed at capturing size and value
patterns in average stock returns is comparatively tested using principal component
analysis. Motivated by the missed variations in average returns of the three-factor
model. The three-factor model is augmented with Fama-French (2015) 2 x 2 and 2 x 2 x
2 x 2 joint controls for profitability and investment factors. Size, value, and profitability
effects are found similar to the findings of Fama and French (2015). Furthermore,
opposite investment effects are observed in which firms that invest aggressively yield
higher average returns than firms that invest conservatively. Moreover, the value factor
is not found to be redundant or absorbed by the slopes of profitability and investment
factors.
7
1. Introduction
Following the central role of Markowitz's (1952) portfolio theory, an abundant
amount research of fundamental factors and tests within asset pricing models has been
widely debated among practitioners and academics. Embodied by the Markowitz (1952)
mean-variance model of optimal portfolio construction, the Capital Asset Pricing Model
(CAPM) of Sharpe (1964), Lintner (1965), and Black (1972), has long shaped the
perception of risk and return. With growing recognition of asset pricing modelling,
researchers (see, Ross, 1976; Stattman, 1980; Banz, 1981; Basu, 1983; Rosenberg, Reid
and Lanstein, 1985 and Chan, Hamao, and Lakonishikok, 1991) have since incorporated
alternative theories and factors such as: macroeconomic indices, debt-to-equity,
earnings-to-price, book-to-market and size, to more accurately model the cross-sectional
excess portfolio returns.
Tests within the study follow the Fama and French (1993, 2015) three-factor and
five-factor empirical model, which is directed at capturing the size, value, profitability
and investment patterns in average stock returns. Evaluations of three-factor model are
made against a constructed principal component regression up to the third component.
Evidence shows that the three-factor model does not fully capture average cross-
sectional returns of the Chinese stock market, which further prompts the performance
assessment and augmentation of the model to accommodate for profitability and
investment factors.
Since the economic reforms of China in 1978, emerging markets have gradually
repealed government controls for a more open economy to domestic markets and foreign
investments. Spurred by the historic large returns of emerging markets, investors
undertake the inherent sources of higher risk associated with the higher returns in
these markets. The motivation of the study lies in these additional sources of risks in
emerging markets, and tests whether these risks and premiums are sufficiently priced
within the Fama-French three-factor and five-factor model.
1.1. Chinese Stock Market Overview
Stocks listed on the Shanghai Stock Exchange (SSE) and Shenzhen Stock
Exchange (SZSE) are listed as A-shares and B-shares. Due to government restrictions,
A-shares are not available for purchase through foreign investment except through
strict regulations are known as Qualified Foreign Institutional Investor (QFII) system.
On the contrary, B-shares are eligible for foreign investment. The focus of the paper will
8
be the constituents of Shanghai Stock Exchange Composite Index which track daily
performance of all A-shares and B-shares listed on the Shanghai Stock Exchange.
Through long-term efforts of reformation, the People's Republic of China plan for a more
open economy and uniform investment by the process of combining the two stock
classes. This transition, albeit slowly, of blending the two classes has seen the
conversion of listed B-shares into H-shares, the neighbouring market asset class of Hong
Kong. As a result of persistent tight regulations, foreign listed B-shares experienced
poor liquidity, thus making sizeable investments difficult. The conversion to H-shares
into an environment with overall more standardised regulations served to attract higher
foreign investment.
Under China's stringent political system, the government plays an important
role in the economy by means of control and ownership of the country's largest
monopolies, known as the China state-owned enterprise (SOE) system. These state-
controlled enterprises extend throughout sectors such as finance, chemical, transport,
construction, etc. SOEs dominate sectors such as the oil industry where the state-owned
portion of shares exceed 70 percent, whereas the tobacco industry is completely state-
owned. However, the transition to a more open economy has seen progress in late 2014
which saw Sinopec, one of China's largest monopolies alongside PetroChina co., sell a
$17.4 billion stake to domestic investors. Despite these efforts, the political and
regulatory environment in China still operates as a heavy hand under a transition
economy. Additionally, more market does not necessarily mean less government, it
advocates different government where enterprises such as banks must be separated to
become more competitive in a free market.
Asset pricing models assume normality in returns which generally performs well
in developed markets, even with small departures from the over-simplified assumption
of normal returns. On the contrary, emerging markets such as China are more likely to
experience shocks induced by political, regulatory, and many more issues. These
complications indicate that stock returns may have different skewness, kurtosis, and
higher volatility. Thus, treating emerging markets in the same manner as other
developed markets could be erroneous. This study aims to provide insight on how asset
pricing models perform in emerging markets and what implications emerging markets
have on financial theory.
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1.2. Outline of the paper
The remainder of the study is organised as follows. Section 2 provides the
background literature review. Section 3 defines the empirical framework. Section 4
discusses the data used. Section 5 demonstrates the methodology, and the results and
diagnostic tests are presented in Section 6 and 7 respectively. Section 8 analyses the
empirical results and Section 9 concludes the evidence and findings.
10
2. Literature Review
The paradigm of portfolio theory, in particular, has been heavily researched and
debated over the past five decades. The purpose of the literature review is to outline and
follow the landmarks in the key issues concerning wealth allocation and discuss the
pivotal implications and contributions to financial theory.
2.1. Harry Markowitz Portfolio Theory
Harry Markowitz’s (1952) research on portfolio theory was the first to clearly
outline and pave the way for modern portfolio theory. Markowitz’s paper “Portfolio
Selection” which has proven to be one of the most pivotal contributions to financial
theory, introduces the “mean-variance” (E-V rule) model which constructs the optimal
portfolio for a number of risky assets according to the risk-adverse nature of an investor.
The prevailing theory defines two parameters, expected return and variance, in which
Markowitz asserts that there is a rate at which investors can gain by taking on
variance, or reduce by giving up expected return. Markowitz builds the model on the
following assumptions:
I. Investors are risk adverse such that given an expected return, they will minimise
portfolio variance.
II. Investors will maximise portfolio expected return given a portfolio variance.
III. Investors maximise the expected utility curve that demonstrates the diminishing
marginal utility of wealth.
IV. Investors make decisions and estimates solely on the basis of expected return
and risk.
V. Markets are perfectly efficient, such that all assets are liquid and there are no
transaction costs or taxes.
VI. Quantities of all assets are restricted on short selling such that for all
securities i.
Under these assumptions, the efficient portfolio is defined by considering the
portfolio expected return and variance parameters. These parameters are derived using
the following formulas,
expected portfolio return,
11
the variance of portfolio returns,
and the covariance between ith and jth assets in the portfolio,
.
The is the expected return of the portfolio, and are percentage
weights of securities i and j where is the expected return of security i. is the
variance of the portfolio calculated using the term which is the covariance between
the ith and jth assets in the portfolio.
Markowitz theorem emphasised the importance of portfolio selection with
consideration to imperfect correlations and co-movements with all other securities in the
portfolio. Due to these factors, diversification can be achieved such that two imperfectly
correlated securities can yield higher returns with a lower weighted variance of both
securities. The ability to construct these portfolios resulted in a portfolio that has same
expected return, but reduced risk than a portfolio constructed ignoring the interactions
between securities (Elton, Gruber 1997). Under these assumptions, Markowitz
developed the concept of the efficient frontier which is a set of portfolios in which there
is no other portfolio that has a higher expected return given a standard deviation of
return (E-V rule).
Furthermore, there have been many alternative portfolio theories such as Kraus
and Litzenberger (1976) which incorporated the effect of skewness on valuation to more
realistically explain the distribution of returns. Since then, researchers (see, Ross, 1976;
Stattman, 1980; Banz, 1981; Basu, 1983; Rosenberg, Reid and Lanstein, 1985 and Chan,
Hamao, and Lakonishikok, 1991) consider higher moment factors such as:
macroeconomic indices, debt-to-equity, earnings-to-price, book-to-market and size, to
more accurately model the cross-sectional excess portfolio returns. Much of the research
have been milestones in the journey of portfolio theory which will be further discussed.
Nevertheless, Markowitz’s mean-variance has remained the cornerstone of modern
portfolio theory despite the abundant research and alternative theories in the past five
decades.
12
2.2. Capital Asset Pricing Model
The problem remained concerning how to evaluate the relationship between
variance and expected return, and how to discern what proportion of risk is from the
market. Subsequently, a tremendous amount of research was dedicated to estimate the
inputs in Markowitz’s portfolio theory, variance, correlation and covariance. The main
tool to estimate covariance was found to be derived from single index models, in which
Sharpe (1964) developed and popularised.
Where is the return of the market at time t and the error term is assumed
to have a mean of zero and variance is the sensitivity of the security to movements
in the market, a indicates that the security is more sensitive than the market, and
shows that the security is less sensitive than the market.
Following the portfolio theory developed by Markowitz (1952), William Sharpe
(1964), John Lintner (1965) and Jan Mossin (1966) formed the Sharpe Lintner Mossin
Capital Asset Pricing Model more commonly known as the CAPM. CAPM asserts that
the equilibrium portfolio return of risky assets is a linear function of the portfolio
covariance with the market. The proxy for the market beta is denoted as , CAPM
postulates that the market m is mean-variance efficient according to the similar
assumptions made by Markowitz (1952). The CAPM is formed under the following
assumptions:
I. Perfect markets in which all securities are priced correctly reflecting freely
available perfect information with a large number of buyers and sellers in the
market.
II. There are no transaction costs or taxes.
III. Investors are all risk adverse and desire to maximise their expected utility that
demonstrates the diminishing marginal utility of wealth.
IV. Investors hold diversified portfolios requiring return for the systematic risk of
the portfolio, idiosyncratic risk is therefore ignored.
V. Investors can freely borrow and lend at the risk-free rate of return.
VI. Investors are risk adverse and wish to aim to maximise portfolio return.
The CAPM equation is written as follows:
13
Where is the risk-free rate of lending and borrowing. The coefficient of
CAPM is assumed to be constant over all risky securities in the portfolio and stable over
time. Thus, the CAPM models a linear positively sloped relationship between expected
portfolio returns and beta. Because of the simplistic nature and ease of use, the CAPM
still remains a widely used and is the most popular model by US companies to estimate
the cost of capital (Graham and Harvey 2001). Black, Jensen and Scholes (1972) and
also Fama and Macbeth (1973) support the CAPM in their paper and find that during
the sample period prior to 1969, there exist a simple positive relationship between beta
and expected portfolio returns. However, in response to the over-simplistic assumptions
made by CAPM, many authors (see, Fama and French, 1992; Jagannathan and Wang,
1996; Fabozzi and Francis, 1978; Lakonishok and Shapiro, 1986; among others) argue
that the standard CAPM that uses the market index as a proxy performs poorly in
explaining cross-sectional stock returns.
2.3. Time-varying beta
Jagannathan and Wang (1996) assert that the systematic risk measured by
CAPM vary over time. Empirical evidence showed that during recessionary periods,
leverage of troubled firms increased which also caused the beta to increase. Thus, the
constant CAPM assumed beta does not capture accurately the real world returns which
are inherently dynamic and not static. Fabozzi and Francis (1978) classify the factors
that lead to a time-varying beta into four categories, firstly, microeconomic variables
such as leverage, dividend and management changes. Second, macroeconomic influences
such as inflation and business cycles. Third, political factors such as war, labour
legislation, elections and pollutions. And finally, market factors such as market
conditions, disintermediation and credit crunches.
2.4. Early Tests
Many of the early tests of CAPM (see, Lintner, 1965; Douglas, 1968; Black,
Jensen and Scholes, 1972) employed time series and cross-sectional regressions to test
the performance of CAPM. However, Lintner (1965) found that the variance of the error
term of the time series regression to be significant, which suggests that the CAPM is
inadequate in explanatory power. Similar results were also found by Douglas (1968) in a
different study by using the same method of Lintner.
14
2.5. Alternative variables and higher moment models
In response to the early tests and poor performance of the standard CAPM,
alternative variables were examined beyond the mean and variance parameters which
allowed models to take higher moments for financial modelling. The abundance of early
tests and investigations of CAPM were a pivotal point in financial theory as this
subsequently led to an enormous amount of research, where a variety of variables were
tested to explain portfolio characteristics and asset returns. Of these numerous
variables studied, some of the most prominent factors will be explored in depth such as
size, book-to-equity, earnings-price, debt-to-equity and macroeconomic variables.
Banz (1981) tested the size effect of firms in explaining the residual variation of
average cross-sectional returns. Banz finds that average return for small stocks are too
high given their beta estimates, and average returns for large stocks are too low. Banz
further asserts that there is a size effect premium which is an empirical finding that
firms with small market capitalisation exhibit higher returns than those of large firms.
This initial evidence showed that size effect is not linear to market value, and the effect
is greater observed for small firms. Chan et al. (1985) also found that small size firms
are more sensitive to macroeconomic factors such as production risks and changes in
risk premium than larger firms, which may explain why the size factor is significant in
capturing cross-sectional returns. Researchers have since tried to explain the size
premium anomaly due to illiquidity, greater transaction costs, less information available
and unreliable beta estimates. However, these explanations do not explain with
certainty if size factor really is insignificant. Thus, the simplistic standard CAPM does
not seem to fully capture the size effect factor.
Additional early evidence of alternative relevant factors was found in the book-
to-market equity (BE/ME) effect of firms. Stattman (1980) and Rosenberg et al. (1985)
found a value premium in US stocks whereby high book value firms were undervalued
by the market and would become profitable if held long as prices increase. This positive
relation between average returns and ratio of a firm’s book-to-market value were also
confirmed in the study of Chan, et al. (1991) in the Japanese market.
Prior to the 1990’s, the additional factors that were researched were only used to
identify the insufficient explanatory power of beta and anomalies of the CAPM. Fama
and French (1992) evaluated the significance of market beta, size, book-to-market,
earnings/price (E/P) and debt-to-equity on NYS, AMEX and NASDAQ stocks and found
15
that beta alone was not a comprehensive measure and statistically insignificant in
explaining the cross-section of average returns. Basu (1983) showed that the earnings-
price (E/P) of US stocks help explain the cross-section of average returns. Stocks that
exhibited higher risks and return were more likely to have a higher earnings-price ratio.
Bhandari (1988) found that common stock returns were positively related to debt-to-
equity (DER) but stocks with higher DER did not necessarily signal higher risk.
However, it still posed as a natural proxy for risk of common equity and can be used as
an additional variable to explain cross-sectional returns.
Fama and French (1992) in their 1963 to 1990 sample period, evaluated each of
the variables and found that each factor had significant power in explaining the cross-
section of average returns. However, when jointly tested it was observed that size and
book-to-market factors seem to absorb the explanatory roles of E/P and DER. Thus,
Fama and French proposed their three-factor model with variables SMB (small-minus-
big) and HML (high-minus-low) such that systematic risk is not only characterised by
market beta alone but also the size and value premiums. The research asserted that in
order to justify a price of an asset, the multidimensional risks should be considered.
Lakonishok and Shapiro (1986) investigated the single factor CAPM model beta
over the 50 year period from 1941-1990 to be statistically weak. Additional tests from
Fama and French (1996) test their three-factor model to the CAPM and find that the
CAPM beta becomes insignificant and intercept coefficients of the three-factor model
were closer to zero when the two extra factors are accounted for. Hence, the combination
of size and BE/ME in addition to the CAPM has superior performance and greater
explanatory power in cross-sectional average returns.
At the stage of financial theory in the 1970s, there existed a gap between the
theoretical importance of systematic variables and their exogenous influences. Ross
(1976) first developed and introduced the arbitrage pricing theory (APT) to explain the
pricing of assets. The author asserts that the mean-variance efficiency of the market
portfolio and the underlying theory and assumptions such as normality in returns are
difficult to justify. The arbitrage pricing model estimates the expected returns as a
linear function of various fundamental macroeconomic variables, where the sensitivity
to each factor has its own beta specific coefficient. The arbitrage pricing model is
expressed as,
16
where is a constant for asset i, is the systematic kth factor, is the factor loading
for factor k and is the idiosyncratic component of the securities return. If the return
structure holds, then according to arbitrage pricing theory the expected return of ith
security is,
where is the risk premium associated with factor k and is the risk-free rate. The
main assumptions made by arbitrage pricing theory are that markets are perfectly
competitive, idiosyncratic risk is assumed to be diversified away and the factor model
explains the relationship between risk and return of a security. Chen, Roll, Ross (1986)
propose a set of relevant “state variables” and employ their method in identifying
important macroeconomic variables and whether exposure to these systematic variables
explains expected returns. The paper tests these macroeconomic variables and
individual systematic effect on market returns and finds that: the spread of long and
short interest rates, inflation, industrial production and spread of bonds are influential
sources that are significantly priced in stock returns.
17
3. Empirical Framework
3.1. Fama and French Three-Factor Model
The Fama and French (1993) three-factor model was developed in response to the
increasing empirical evidence that the CAPM beta proxy was not sufficient in explaining
variations in returns. Leaning on the research of Banz (1981), Stattman (1980),
Rosenberg, Reid and Lanstein (1985), Fama and French found empirical evidence that
small size and high BE/ME firms on average exhibit higher returns than of large firms
and low BE/ME stocks. In addition to the CAPM beta, the size and value factors seemed
to significantly capture the majority of returns of non-financial firms in the NYSE,
AMEX, and NASDAQ market. Thus, Fama and French (1993) create diversified
portfolios sorted on size and book-to-market equity using NYSE median breakpoints as
intersections of portfolios. The diversified portfolios are exposed to combinations of state
variables, SMB (small minus big, is the difference in average returns between small
stocks and big stocks) and HML (high minus low, is the difference in average return
between high BE/ME stocks and low BE/ME stocks) factors, which are used to augment
the CAPM to identify the size and BE/ME effects. The established three-factor model
explained approximately 95% of the variation of excess returns from 1963 to 1990. The
model is as follows,
where is the return of a security or portfolio i over period t, is the return of the
market portfolio, is the risk-free rate of return, and is the zero-mean residual.
Intercept is equal to zero if , , and completely capture the variation of cross-
sectional returns.
There have been various studies regarding the validity and power of the size and
value factors. Because the unknown state variables are explained through mimicking
portfolios, and the justification of state variables are not identified. Black (1993) and
Kothari et al. (1995) argue that a substantial part of premiums may be the result of data
snooping and survivor bias. Sorting portfolios on book-to-market equity also includes
firms that are financially distressed which typically exhibit higher returns and
disproportionately high BE/ME ratios. Studies suggest that the distress risk is priced
into the premiums due to fixation of variables which include the distressed firms that
survive, but exclude the distressed firms that fail. Other arguments include the
18
illiquidity of small stocks which will incur greater transaction costs, and given the lower
traded frequency of stocks, the parameters and beta coefficients may be unreliable.
Moreover, further research of Fama and French (1995) and Lakonishok et al.
(1994) find positive empirical evidence which supports the validity of size and BE/ME
factors in capturing the variation of cross-sectional returns to be significant. Barber and
Lyon (1997) analyse the survivor bias effects of the Fama and French (1992) sample and
include those firms outside of the holdout sample. Evidence showed that the inclusion of
stocks excluded from the sample had no significant effect on the SMB and HML
estimates.
3.2. Principal Component Analysis
Principal component analysis (PCA) was first defined by Pearson (1901) and
later developed by Hotelling (1933). With the increase of computational power, principal
component analysis has seen application in many fields such as finance, biology, ecology,
health and architecture. Principal component analysis is a statistical procedure that
reduces the dimensionality of the original dataset of linear correlated variables through
orthogonal transformation, these variables retain the highest variance possible and are
called principal components. Principal components are decomposed linear combinations
of statistically independent random variables, each with a mean of zero and unit
variance. Thus, the number of principal components are less than or equal to the
number of original variables. Figure 1 illustrates the orthogonal transformation of the
original dataset to the independent principal components.
19
Figure 1:
Graphical orthogonal transformation of principal component analysis for a two-
dimensional data set.
The reduced dimensions of the x1 and x2 data set are rotated by onto the first
principal component axis z1 and second principal component axis z2. The variance of the
first eigenvalue dominates the fractional contribution of explained variances by
maximising the variability of returns for z1 and z2, where
.
Mathematically, the two orthogonal transformed projections in terms of the return
vector can be expressed as,
which can be expressed in matrix form,
2
1
2
1
)cos()sin(
)sin()cos(
x
x
z
z
denoted as , where
2221
1211
)cos()sin(
)sin()cos(
aa
aaA
and is the return vector.
20
The orthogonal transformation is demonstrated by the zero values of the anti-diagonal
of the identity matrix,
.
Moreover, principal component analysis used in this study is defined by higher
dimensional vectors. Thus, suppose for p portfolios or variables, the principal
components for are expressed as,
which is written in matrix form as,
ppppp
p
p
p x
x
x
aaa
aaa
aaa
z
z
z
2
1
21
22221
11211
2
1
where and . Generally,
where j is the jth principal component and is the vector.
Thus the variance of the principal components its expected value for the sample of p is,
where denotes the sample covariance matrix and is the expected sample return.
Assuming that the returns are normally distributed, it follows that,
.
The variability of PC1 is maximised using the Lagrangian method of constrained
optimisation for the following function,
21
such that the first principle component is constrained subject to . Additionally,
subsequent preceding principal components are maximised subject to and
for , this constraint shows the orthogonal relationship of
principal components. Taking the first derivative of equation with respect to equal
to zero derives,
which solves for the eigenvectors and the determinant, solves for the
eigenvalues. If S is a p x p matrix, then the above is the pth order polynomial where the
p roots are,
.
Choosing the number of factors is generally agreed to be one of the most
important decision in factor analysis (Jackson, 1993). Accordingly, there is an
abundance of literature with various methods for determining the number of factors for
analysis. Peres-Neto, Jackson and Somers (2005) investigate individual methods for
determining the number of principal components which total to 20 stopping rules. The
authors conclude that the Bartlett's (1954) test of sphericity, which determines whether
the data exhibits multivariate normal distribution with zero covariance, seemed to be
the best method as it maximised the detection of non-trivial components.
One commonly used method is to retain as many factors required such that the
sum of eigenvalues exceed a certain threshold fraction of the total variance,
where is the number of retained eigenvalues and is the total number of eigenvalues.
Researchers typically retain principal components comprising 95% of total variance
(Jackson, 1993).
By far the most popular method is the Kaiser-Guttman rule (Kaiser, 1960),
where the significant principal components retained are defined with eigenvalues
greater than 1 .
22
3.3. Fama and French Five-Factor Model
Extending from the evidence that average stock returns are related to book-to-
market equity, BE/ME. The dividend discount model illustrates that profitability and
investment add to the average returns of BE/ME. The dividend discount model defines a
stocks market price by the discounted value of future dividends per share,
.
In the dividend discount model, is the share price at time t, is the
expected dividend per share at period , and r is the internal rate of return of
expected dividends. If prices of two firms at time t have same expected dividends but
different prices, the firm with a lower price has a higher return. Assuming that the
pricing is rational, the firm with a lower price and higher return must therefore have a
higher risk. However, the models drawn from the dividend discount model and below,
are the same whether pricing is rational or irrational.
Miller and Modigliani (1961) manipulate the dividend discount model to show
the relations of expected returns and expected profitability, expected investment, and
BE/ME. The market value of a stock at time t is given by,
.
In this equation, is the total equity earnings for the period and
is the change in total equity. Dividing the whole equation by book equity gives,
Three statements are made about the expected stock return in relation to
BE/ME, earnings, and investment. Firstly, keeping all else constant except the market
value of a stock, , and the expected returns, r. Lower market value of , or
equivalently a higher, , implies a higher return. Fixing market value, , and all
other values except future earnings, , and expected returns, r. Higher expected
earnings imply a higher expected return. Lastly, fixing market value, , and all other
values except investment, , and expected returns, r. Higher growth in book equity,
or investment, indicated a lower expected return.
Motivated by the empirical evidence of Novy-Marx (2013), Titman, Wei, and Xie
(2004), which shows that the three-factor model fails to capture the variation of average
23
returns related to profitability and investment. Fama and French (2015) add
profitability and investment factors to the three-factor model,
.
In the augmented model, the profitability premium (robust minus weak)
mimics the risk factors related to profitability and is the difference between the average
returns of robust profitability portfolios minus the average returns of weak profitability
portfolios. The profitability measure (OP) is measured as revenues minus cost of goods
sold, minus selling, general, and administrative expenses, minus interest expense all
divided by book equity.
Additionally, the investment premium (conservative minus aggressive)
mimics the risk-returns factors related to investment. The is calculated as the
average returns of conservative investment portfolios minus the average returns of
aggressive investment portfolios. The investment premium (Inv) is measured as the
change in total assets from the fiscal year ending to , divided by total assets.
24
4. Data
Stock return data used in this research ranges over the period of June 2004 to
June 2015. These returns are collected from constituents belonging to the Shanghai
Stock Exchange (SSE) market and sourced from Bloomberg. Hence, the monthly market
portfolio returns used in this study is calculated from the Shanghai Stock Exchange
Composite Index1. Such data includes the bond prices, earnings-before-tax, book-equity,
total assets and the risk-free rate. The maturity profile of Chinese government bond
issuance ranges from 3 months to 50 years. Over time, many securities were added and
also discontinued. Other issuances such as the 15 year were issued only from 2001 to
2009, and the 8 year was only issued once in 1999. Thus, to accurately reflect the
Chinese monthly risk-free rate, a short term government bond with a life spanning over
the data period will be used. However, as China 1 month government bond issue is not
reported by Bloomberg, the shortest next available government bond will serve as a
suitable replacement. The risk-free rate used in the report is the 3 Month Central Bank
Bill obtained from DataStream, expressed as a decomposed effective monthly rate.
Constituents of the Shanghai Stock Exchange are chosen to create portfolios,
which follow a similar stock sample selection criteria similar to that of Fama and French
(1993). Firstly, stocks must be traded and have monthly returns within the last 12
months preceding July of year t, where June of year t is the size cut off point for
portfolio construction. Secondly, stocks must report a positive book value at fiscal year-
end in order to achieve a positive book-to-market ratio. Lastly, stocks that do not satisfy
these conditions will be excluded and replaced by the next successive stock.
Fama and French (1993) allocate 10 size portfolios of NYSE, AMEX, and
NASDAQ stocks based on the NYSE breakpoints. This process, however, is difficult to
replicate in China's emerging and relatively new market environment. In June 2004, the
constituents of the SSE numbered approximately 857 and rapidly grew to 1108
constituents in June 2015. Therefore, the SSE constituents belonging to the bottom size
group referred to as 'microcaps' by Fama and French (2015), would be highly dynamic
and may be unrepresentative of microcap portfolio performance. Additionally, these
newly listed firms that are actively traded within the last 12 months are inherently
more complicated due to unreported values and missing data.
1 Bloomberg Ticker: SHCOMP:IND. The Shanghai Stock Exchange Composite Index is a capitalisation-
weighted index that tracks the daily price performance of all A-shares and B-shares listed on the Shanghai
Stock Exchange developed on 19th December 1990.
25
With consideration to China's oversized monopolies that are majority state-
owned, or state-controlled enterprises throughout sectors such as finance, chemical,
transport, construction, etc. A prime example being the state-owned largest oil and gas
integrated company, China Petroleum and Chemical Corporation, which encompasses
11.9% of the entire Shanghai Stock Exchange market capitalisation in June 2004 and
16.84% in June 2008. The portfolios constructed in the research will consist of the top
100 market capital firms of each year. Table 1 shows the total index weight of the top
100 stocks in each year.
Table 1:
Total index weight of the top 100 market constituents listed in the Shanghai Stock
Exchange from the period June 2004 to June 2015.
2004 2005 2006 2007 2008 2009 2010
57.71% 46.71% 62.00% 76.03% 81.41% 78.44% 72.74%
2011 2012 2013 2014 2015 Average
69.21% 69.50% 69.09% 65.04% 58.79% 66.69%
Due to the monopolistic nature of China's market, selecting a sample of the top
100 stocks will subsequently capture the majority of the market. On average during the
period from June 2004 to June 2015, the top 100 market capital stocks capture 66.69%
of the market index. Thus, the sample is convincingly large enough to represent the
market and also avoids data complications of dynamic microcaps.
By construction, the process of selecting the top 100 firms in each year may be a
logical error by concentrating the sample to firms that exhibit survivor bias and could
possibly yield inaccurate conclusions. However, in extension to the sample used in Fama
and French (1993), Barber and Lyon (1997) analyse the returns of financial firms which
Fama and French excluded from their analysis and show that the "survivor bias"
element does not have a significant effect on the size and value premium estimates.
Thus, the portfolio samples used in the research for factor estimation leans on the
findings of Barber and Lyon (1997) and assumes that the data outside of the holding
sample does not significantly affect the factor premium estimates.
Under the pricing model assumption of normally distributed returns, monthly
returns data is used instead of daily returns to reduce the fluctuations by averaging out
the "noise". Whether these noise-induced biases become insignificant when returns are
observed at a monthly frequency depends on the simplifying assumption of normality in
26
returns. However, if the pricing model captures the significant factors of cross-sectional
returns and is overall representative of the market. The model is still useful for making
rational portfolio decisions even with the small departures from the over-simplified
assumption of normally distributed monthly returns.
Following these normality assumptions, the prices of stocks are also assumed to
be log normally distributed which enables the use of logarithmic returns. Logarithmic
returns allow factors to be compared in a comparable metric, the compound returns over
n periods is additive such that the return over period n is the logarithmic difference
between the final and initial period.
Thus, the return of asset i at period t is expressed as,
where is the return of asset i for month t, denotes the price of a stock at time t
and is the price of the stock one month before period t.
27
5. Methodology
5.1. Fama-French Three-Factor Model
To study the economic fundamentals highlighted in the Fama and French (1993)
model and its performance in the Chinese stock market, six portfolios are formed from
sorts of size and book-to-market equity (BE/ME). The six portfolios formed are exposed
to different levels of size and book-to-market equity effects and therefore mimic the
underlying risk factors in returns related to size and book-to-market. In June of each
year t from 2004 to 2015, constituents of the Shanghai Stock Exchange are ranked
according to size (price times shares or constituent weight) and the top 100 firms are
then chosen. Stocks are then assigned into two size groups (Small (S) and Big (B)) where
the size breakpoint is defined as the size median of the top 100 stocks.
Within the small and big size groups, portfolios are further sorted into three
book-to-market equity groups based on the breakpoints for the top 30% (High), middle
40% (Neutral), and bottom 30% (Low) book-to-market equity (BE/ME) stocks. Moreover,
the decision to sort portfolios into three groups on BE/ME follows the findings of Fama
and French (1992) where book-to-market equity plays a stronger role than size in
capturing the variation of returns. Portfolios created are reformed at June for each year
from 2005 to 2015.
The intersections of two size groups and three book-to-market equity groups form
six portfolios denoted as BH, BN, BL, SH, SN, and SL. The first letter categorises the
portfolios into small (S) or big (B) size groups, and the second letter denotes the
portfolios in different BE/ME groups, high (H), neutral (N), and low (L). For example,
the group denoted as SL represents the portfolio that contains small stocks that have a
low book-to-market equity.
By construction, the six portfolios formed are components used to calculate the
fundamental size and value premiums. The size premium SMB (small minus big)
mimics the risk factors related to size and is the average returns of small stocks (SH,
SN, and SL) minus the average returns of big stocks (BH, BN, and BL). Additionally,
the value premium HML (high minus low) mimics the risk-return factors related to
book-to-market equity. The HML premium is defined similarly, calculated as the
average returns of two high-BE/ME portfolios (SH and BH) minus the average returns
of two low-BE/ME portfolios (SL and BL). Further descriptions of Fama-French factors
28
can be found online at Kenneth French's data library2 and factor constructions are
summarised in Table 2.
Table 2:
Construction of Size and BE/ME factors. Independent sorts are used to assign two Size groups
and three BE/ME groups. The factors are SMB (small minus big) and HML (high minus low).
Factor Breakpoints Factor Construction
SMB Size: Median of top 100 size stocks
HML BE/ME: top 30%, middle 40%, and
bottom 30% of the respective size group
It should be noted that SMB and HML are not themselves state variables,
instead, the factors are just diversified portfolios with different combination of
exposures to state variables. Thus, this allows observations of size and value factors and
their significance in capturing returns without specifically identifying the state
variables.
The market risk premium, SMB and HML estimates are analysed through
regression upon 9 sorted portfolios on size and BE/ME formed from the top 100 size
firms of the SSE, which will be used as the dependant variables. The breakpoints of size
are top 30%, middle 40%, and bottom 30%. Within each size group, portfolios are then
sorted according to the BE/ME breakpoints, top 30%, middle 40%, and bottom 30%.
Fama and French (1993) create 25 portfolios sorted on size and BE/ME based upon three
market indices, NYSE, AMEX and NASDAQ which are well developed and highly
traded markets. However, with consideration to the smaller Shanghai Stock Exchange
index and sample size of 100 firms, the number of dependent portfolios in this study will
be reduced to 9.
5.2. Principal Component Regression
One of the major advantages of principal component regression (PCR) is
overcoming the problem of multicollinearity, which is a result of explanatory variables
being collinear, or highly correlated. As the portfolios and estimates are calculated using
the same underlying data, sorted in various combinations, normal regression of such
portfolios would be statistically erroneous due to multicollinearity. Principal components
2 Description of Fama/French Factors:
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/Data_Library/f-f_factors.html
29
of the 9 Size and BE/ME sort portfolios are used as the explanatory variables. Thus, the
orthogonal nature of principal components curbs the effects of multicollinearity within
the dataset.
Firstly, the portfolio return matrix of nine portfolios is constructed where each
column represents the time series return of one portfolio, and each row contains the
monthly returns of each respective portfolio. From the period of June 2004 to June
2015, a 132 by 9 return matrix X is formed,
9,1323,1322,1321,132
9,33,32,31,3
9,23,22,21,2
9,13,12,11,1
xxxx
xxxx
xxxx
xxxx
where denotes the first-month return of portfolio 1 of 9 portfolios. Secondly, the 9 by
9 standardised variance-covariance matrix S is formulated on the 9 portfolios through
Excel,
1
1
1
1
3,92,91,9
9,32,31,3
9,23,21,2
9,13,12,1
such that covariance between portfolio 1 and 2 for example is
. To
obtain the eigenvalues and eigenvectors, PC1 is maximised using the Lagrangian
method for constrained optimisation. The Lagrangian equation to maximise is as
follows,
under constraints:
where ,
is the vector and is the time vector of portfolio returns.
Subsequent maximisations of following principal components are subject to the
additional constraint which represents the orthogonal relationship
between and .
30
Taking the first derivative of equation with respect to equal to zero derives,
which solves for the eigenvectors, and the determinant,
solves for the eigenvalues.
Given the points discussed 3.2 in relation to the stopping point of principal
components, in addition to the motivation of using PCR as a comparative tool for the
Fama and French three-factor model. The number of principal components to be
analysed in PCR will be the first three components, resulting in a 9 by 3 principal
component matrix denoted as A. The scree plot of eigenvalues is illustrated in Figure 2,
where the horizontal red line depicts the Kaiser-Guttman rule .
0
1
2
3
4
5
6
7
8
1 2 3 4 5 6 7 8 9
Scree Plot (Ordered Eigenvalues)
Figure 2:
Scree plot of ordered 9 eigenvalues, the horizontal red line where , depicts the
Kaiser-Guttman rule .
According to the Kaiser-Guttman rule, only the first component should be retained.
However, the first principal component only explains 83.24% of the total variance.
Whereas the retention of the first three principal components captures 93.66% of the
total variance. Thus, the analysis of the first three components provides a satisfactory
model where the principal components explain almost 95% of total variance.
31
The principal components used as the explanatory variables denoted as Z, a 132
by 3 matrix, is calculated through the matrix multiplication of X and A . Where
each column of matrix Z are the explanatory variables PC1, PC2, and PC3. The PCR is
written,
By construction, the chosen three principal components regressed against the
excess returns of 9 created portfolios sorted on size and BE/ME should produce the best
three factor model by capturing the maximum variance explained in each principal
component.
5.3. Fama and French Five-Factor Model
The evidence of Novy-Marx (2013), Titman, Wei, and Xie (2004) shows that the
three-factor model was incomplete and overlook much of the variation of returns related
to profitability and investment. Fama and French (2015) motivated by this, design the
five-factor model directed at capturing size, value (BE/ME), profitability (OP) and
investment (Inv) patterns of average stock returns.
Similar to the Fama and French (1993) three-factor model, definitions of the
factors are calculated by various sorts based upon the fundamental variables studied.
The factor constructions closely follow the approach used in Fama and French (2015)
where the three-factor model is augmented with profitability and investment factors,
.
In this augmented model, the profitability premium (robust minus weak) mimics
the risk factors related to profitability and is the difference between the average returns
of robust profitability portfolios minus the average returns of weak profitability
portfolios. The profitability measure (OP) is observed in June for each year and is
revenues minus cost of goods sold, minus selling, general, and administrative expenses,
minus interest expense all divided by book equity.
Additionally, the investment premium (conservative minus aggressive)
mimics the risk-returns factors related to investment. The is calculated as the
average returns of conservative investment portfolios minus the average returns of
aggressive investment portfolios. The investment premium (Inv) is likewise sorted in
June of each year and is the change in total assets from the fiscal year ending to ,
32
divided by total assets. Portfolios created are reformed at June for each year from
2005 to 2015.
For the purpose of evaluation and accommodation of the additional factors, 2 x 3,
2 x 2, and 2 x 2 x 2 x 2 sorts are joint controls to isolate estimates of factor premiums.
Concluded from the evidence of Fama and French (2015), the 2 x 2 sorts Size-BE/ME,
Size-Op, and Size-Inv is preferred over the 2 x 3 sorts. Since the 2 x 3 sort excludes the
middle 40% of the sample whereas the 2 x 2 version produces better diversified
portfolios by using all stocks. Estimates of factors (SMB, HML, RMW, and CMA) are
calculated differently for 2 x 2 and 2 x 2 x 2 x 2 sorts under the breakpoints and
constructions summarised in Table 3.
The 2 x 2 sort portfolios are labelled with two letters, the first letter defines the
Size group, small (S) or big (B). In the 2 x 2 sort, the second letter describes the BE/ME
group, high (H), neutral (N), or low (L), the profitability group, robust (R), neutral (N),
or weak (W), and the investment group, conservative (C), neutral (N), or aggressive (A).
For example, the portfolio denoted as BR contains stocks with big size and robust
profitability.
Notation becomes more intricate in the joint control 2 x 2 x 2 x 2 sort portfolios.
The first letter represents the size group, the second letter is the BE/ME group, the
third is the profitability group, and the fourth is the investment group. For example, the
portfolio denoted as SHRA contains stocks with small size, high BE/ME, robust
profitability and aggressive investment.
It should be noted that for the 2 x 2 sort, 12 portfolios are created containing 50
stocks each as the sorts carried out on two intersections, Size-BE/ME, Size-OP, or Size-
Inv. However, 16 portfolios are created in the 2 x 2 x 2 x 2 sort within the top 100 size
stocks. Thus, the portfolios in the 2 x 2 x 2 x 2 are poorly diversified and must be
considered in the inference of results.
Similar to the Fama and French (1993) three-factor model, the factors SMB,
HML, RMW, and CMA are not themselves state variables, but are instead portfolios
with different combinations of exposures to the state variables. Thus, the significance of
size, value, profitability and investment are captured in the returns without specifically
identifying the state variables.
33
Table 3:
Construction of Size, BE/ME, profitability, and investment factors.
Portfolios are labelled with two or four letters. The first letter describes the Size group small (S) or big (B). In the 2 x 2 sort, the second letter
describes the BE/ME group, high (H), neutral (N), or low (L), the profitability group, robust (R), neutral (N), or weak (W), and the investment group,
conservative (C), neutral (N), or aggressive (A). In the 2 x 2 x 2 x 2 sort, the second letter is the BE/ME group, the third is the profitability group, and
the fourth is the investment group. The factors are SMB (small minus big), HML (high minus low BE/ME), RMW (robust minus weak OP), and CMA
(conservative minus aggressive Inv).
Sort Factor Breakpoints Component Construction
2 x 2 sorts on
Size and BE/ME,
or Size and OP,
or Size and Inv
SMB Size: Median of top 100 size stocks
HML BE/ME: Median of top 100 size stocks
RMW OP: Median of top 100 size stocks
CMA Inv: Median of top 100 size stocks
2 x 2 x 2 x 2 sorts
on Size, BE/ME,
OP, and Inv
SMB Size: Median of respective size portfolio
HML BE/ME: Median of respective BE/ME
portfolio
RMW OP: Median of respective OP portfolio
CMA Inv: Median of respective Inv portfolio
34
6. Results
Table 4:
Average monthly percent returns for portfolios formed on Size and BE/ME, Size and OP, Size and
Inv; June 2004 to June 2015, 132 months.
At the end of each June, stocks are allocated into three small to big size groups (top
30%,middle 40%, and bottom 30%), stocks are then allocated into three BE/ME groups (Low to
High), again using the same percent groups (top 30%, middle 40%, and bottom 30%). The Size-OP
and Size-Inv portfolios are formed in the same way, except that the second sort variable is
operating profitability or investment. Operative profitability, OP, in the sort for June of year t is
measured with accounting data for the fiscal year ending in year t-1 and is revenues minus cost
of goods sold, minus selling, general and administrative expenses, minus interest expense all
divided by book equity. Investment, Inv, is the change in total assets from the fiscal year ending
t-1 to the fiscal year ending t, divided by t-1 total assets. The table shows averages of monthly
returns of the 9 portfolios formed on Size and BE/ME, Size and OP, Size and Inv.
Low Neutral High
Panel A: Size-BE/ME portfolios
Small 0.34% 1.93% 3.23%
Medium -0.07% 1.43% 3.89%
Big 0.11% 0.90% 2.57%
Panel B: Size-OP portfolios
Small 0.96% 1.04% 1.30%
Medium 1.68% 1.66% 1.45%
Big 0.95% 1.33% 1.77%
Panel C: Size-Inv portfolios Small 1.04% 0.95% 1.32%
Medium 0.91% 1.93% 1.86%
Big 0.54% 1.61% 1.93%
35
Table 5:
Summary statistics for average monthly factor returns; June 2004 to June 2015, 132 months. Panel A of the table shows the average monthly returns
(Mean) and the standard deviation of monthly returns (Std dev.) for the average returns. Panel B shows the correlations of the same factor from
different sorts and Panel C shows the correlations for each set of factors. The Fama and French three factor model is notated as FF3, Fama and
French 5 Factor 2 x 2 sort is notated as FF5 2x2, and the Fama and French 5 Factor 2 x 2 x 2 x 2 sort is notated as FF5 2x2x2x2.
Panel A: Averages and standard deviations for monthly returns
Fama-French 3 Factors
Fama-French 5 Factor 2 x 2 Factors
Fama-French 5 Factor 2 x 2 x 2 x 2 Factors
Rm-Rf SMB HML Rm-Rf SMB HML RMW CMA Rm-Rf SMB HML RMW CMA
Mean 0.644 0.428 -3.156
0.644 -0.472 -2.121 0.108 0.498
0.644 -0.392 -2.160 -0.277 -0.029
Std dev. 8.708 3.847 6.078
8.708 4.216 4.109 3.386 3.443
8.708 4.215 4.204 3.084 2.782
Panel B: Correlation between different version of the same factor
SMB
HML
RMW
CMA
FF3 FF5 2x2 FF5 2x2x2x2 FF3 FF5 2x2 FF5 2x2x2x2
FF5
2x2
FF5
2x2x2x2
FF5
2x2
FF5
2x2x2x2
FF3 1.00 0.32 0.35
1.00 0.80 0.81
FF5 2x2 1.00 0.82
1.00 0.84
FF5 2x2 0.32 1.00 0.98
0.80 1.00 0.99
FF5 2x2x2x2 0.82 1.00
0.84 1.00
FF5 2x2x2x2 0.35 0.98 1.00
0.81 0.99 1.00
Panel C: Correlation between different factors
Fama and French 3 Factor Model
Fama and French 5 Factor Model 2 x 2
Fama and French 5 Factor Model 2 x 2 x 2 x 2
Rm-Rf SMB HML Rm-Rf SMB HML RMW CMA Rm-Rf SMB HML RMW CMA
Rm-Rf 1.00 -0.14 0.07
1.00 -0.32 -0.22 -0.05 0.25
1.00 -0.30 -0.19 0.10 0.14
SMB -0.14 1.00 -0.40
-0.32 1.00 0.33 -0.02 -0.23
-0.30 1.00 0.35 -0.20 -0.17
HML 0.07 -0.40 1.00
-0.22 0.33 1.00 0.12 -0.51
-0.19 0.35 1.00 -0.10 -0.24
RMW - - -
-0.05 -0.02 0.12 1.00 -0.33
0.10 -0.20 -0.10 1.00 -0.26
CMA - - - 0.25 -0.23 -0.51 -0.33 1.00 0.14 -0.17 -0.24 -0.26 1.00
36
Table 6: Fama and French 3-Factor model regressions for 9 Size-BE/ME portfolios; June 2004 to June 2015, 132 months.
At the end of June each year, stocks are allocated to three size groups (top 30%, middle 40%, and bottom 30%). Stocks are then independently
sorted into three BE/ME groups (Low, Neutral, and High) again using the top 30%, middle 40%, and bottom 30% sort. The intersection of these two
sorts produce 9 Size-BE/ME portfolios. The RHS variables are excess market return, RM-RF, the Size factor, SMB, and the value factor, HML. Each
variable is uniquely constructed according to the model factor definition.
Dependant variable: excess returns on 9 stock portfolios formed on size and book-to-market equity
Size Book-to-market equity (BE/ME) 30%, 40% and 30% groups
Low Neutral High
Low Neutral High
a
t(a)
Small 0.006 0.004 0.010
Small 1.952 1.198 2.510
Medium 0.004 0.005 0.011
Medium 1.109 1.476 2.579
Big 0.006 0.006 0.004
Big 2.139 0.047 0.976
b
t(b)
Small 1.053 1.198 0.954
Small 32.230 31.515 23.644
Medium 1.104 1.195 1.079
Medium 29.600 31.826 24.066
Big 0.986 0.981 1.068
Big 35.093 29.707 26.343
s
t(s)
Small 0.679 1.141 0.883
Small 8.448 12.187 8.892
Medium 0.362 0.319 0.356
Medium 3.942 3.445 3.224
Big -0.280 -0.517 -0.574
Big -4.039 -6.365 -5.756
h
t(h)
Small 0.462 -0.009 -0.333
Small 9.146 -0.146 -5.334
Medium 0.490 0.071 -0.542
Medium 8.491 1.226 -7.822
Big 0.378 0.113 -0.488
Big 8.685 2.214 -7.784
R2
S(e)
Small 0.898 0.893 0.838
Small 0.032 0.038 0.040
Medium 0.883 0.888 0.835
Medium 0.037 0.037 0.044
Big 0.921 0.890 0.861 Big 0.028 0.033 0.040
37
Table 7:
Principal component regressions for 9 Size-BE/ME portfolios; June 2004 to June 2015, 132 months. Time series principal components are the columns
of matrix where is the 132 by 3 matrix.
Dependant variable: excess returns on 9 stock portfolios formed on size and book-to-market equity
Size Book-to-market equity (BE/ME) 30%, 40% and 30% groups
Low Neutral High
Low Neutral HIgh
a
t(a)
Small 0.001 -0.003 -0.003 Small 0.288 -1.236 -1.164
Medium -0.001 -0.003 -0.005 Medium -0.367 -1.207 -1.881
Big -0.005 0.001 0.001 Big -2.360 0.228 0.288
B1
t(B1)
Small 0.280 0.353 0.332 Small 30.027 42.609 41.366
Medium 0.328 0.374 0.356 Medium 37.198 45.690 48.521
Big 0.340 0.320 0.314 Big 48.062 44.777 36.975
B2
t(B2)
Small 0.515 0.376 -0.124 Small 15.539 12.761 -4.348
Medium 0.420 -0.025 -0.237 Medium 13.390 -0.874 -9.063
Big -0.036 -0.387 -0.444 Big -1.449 -15.235 -14.672
B3
t(B3)
Small -0.066 -0.406 -0.318 Small -1.495 -10.310 -8.338
Medium 0.372 -0.055 -0.313 Medium 8.866 -1.410 -8.965
Big 0.665 0.221 -0.053 Big 19.766 6.500 -1.322
R2
S(e)
Small 0.906 0.945 0.934 Small 0.030 0.027 0.026
Medium 0.931 0.943 0.951 Medium 0.029 0.027 0.024
Big 0.954 0.944 0.922 Big 0.023 0.023 0.028
38
Table 8:
Fama and French 5-factor 2 x 2 sort regressions for 9 Size-BE/ME portfolios; June 2004 to June
2015, 132 months.
At the end of June each year, stocks are allocated to three size groups (top 30%, middle
40%, and bottom 30%). Stocks are then independently sorted into three BE/ME groups (Low,
Neutral, and High) again using the top 30%, middle 40%, and bottom 30% sort. The intersection
of these two sorts produce 9 Size-BE/ME portfolios. The RHS variables are excess market return,
RM-RF, the Size factor, SMB, the value factor, HML, the profitability factor, RMW, and the
investment factor, CMA. Each variable is uniquely constructed according to the model factor
definition.
Dependant variable: excess returns on 9 stock portfolios formed on size and book-to-market equity
Size Book-to-market equity (BE/ME) 30%, 40% and 30% groups
Low Neutral High
Low Neutral High
a
t(a)
Small 0.004 0.003 0.008
Small 1.085 0.581 1.504
Medium 0.002 0.004 0.009
Medium 0.600 1.146 1.860
Big 0.004 0.006 0.005
Big 1.167 1.694 1.037
b
t(b)
Small 0.958 1.140 0.932
Small 26.025 19.583 16.461
Medium 0.980 1.100 1.008
Medium 27.606 29.833 19.356
Big 0.942 0.953 1.042
Big 27.901 25.036 22.397
s
t(s)
Small 0.072 0.067 -0.152
Small 0.924 0.546 -1.270
Medium -0.262 -0.427 -0.572
Medium -3.478 -5.464 -5.179
Big -0.162 -0.265 -0.514
Big -2.264 -3.279 -5.209
h
t(h)
Small 0.252 -0.438 -0.749
Small 2.893 -3.179 -5.590
Medium 0.509 0.014 -0.957
Medium 6.059 0.159 -7.766
Big 0.549 0.331 -0.496
Big 6.874 3.680 -4.505
r
t(r)
Small -0.279 -0.375 -0.460
Small -2.967 -2.526 -3.190
Medium -0.201 -0.129 0.053
Medium -2.220 -1.377 0.398
Big 0.232 0.211 0.221
Big 2.695 2.175 1.865
c
t(c)
Small 0.530 0.360 -0.027
Small 4.927 2.117 -0.160
Medium 0.263 0.089 0.371
Medium 2.539 0.822 2.439
Big 0.012 -0.063 0.159
Big 0.120 -0.564 1.171
R2
S(e)
Small 0.888 0.784 0.723
Small 0.034 0.054 0.052
Medium 0.908 0.907 0.807
Medium 0.033 0.034 0.048
Big 0.901 0.874 0.842
Big 0.031 0.035 0.043
39
Table 9:
Fama and French 5-factor 2 x 2 x 2 x 2 sort regressions for 9 Size-BE/ME portfolios; June 2004 to
June 2015, 132 months.
At the end of June each year, stocks are allocated to three size groups (top 30%, middle
40%, and bottom 30%). Stocks are then independently sorted into three BE/ME groups (Low,
Neutral, and High) again using the top 30%, middle 40%, and bottom 30% sort. The intersection
of these two sorts produce 9 Size-BE/ME portfolios. The RHS variables are excess market return,
RM-RF, the Size factor, SMB, the value factor, HML, the profitability factor, RMW, and the
investment factor, CMA. Each variable is uniquely constructed according to the model factor
definition.
Dependant variable: excess returns on 9 stock portfolios formed on size and book-to-market equity
Size Book-to-market equity (BE/ME) 30%, 40% and 30% groups
Low Neutral High
Low Neutral High
a
t(a)
Small 0.002 0.003 0.007 Small 0.631 0.560 1.316
Medium 0.001 0.003 0.009 Medium 0.157 0.866 1.854
Big 0.004 0.006 0.005 Big 1.318 1.721 1.047
b
t(b)
Small 0.995 1.162 0.938 Small 26.198 19.794 16.745
Medium 1.008 1.118 1.015 Medium 28.412 30.778 20.099
Big 0.952 0.958 1.037 Big 28.617 25.712 23.132
s
t(s)
Small 0.050 0.123 -0.135 Small 0.599 0.950 -1.091
Medium -0.303 -0.449 -0.559 Medium -3.872 -5.616 -5.022
Big -0.162 -0.297 -0.540 Big -2.216 -3.618 -5.462
h
t(h)
Small 0.368 -0.315 -0.734 Small 4.512 -2.499 -6.096
Medium 0.533 0.013 -0.907 Medium 6.992 0.161 -8.356
Big 0.537 0.298 -0.509 Big 7.508 3.725 -5.285
r
t(r)
Small -0.332 -0.168 -0.297 Small -3.029 -0.993 -1.841
Medium -0.226 -0.190 0.125 Medium -2.211 -1.820 0.861
Big 0.116 0.047 0.225 Big 1.209 0.436 1.740
c
t(c)
Small 0.510 0.375 0.053 Small 4.136 1.969 0.291
Medium 0.292 0.046 0.277 Medium 2.538 0.386 1.691
Big -0.128 -0.253 0.059 Big -1.188 -2.091 0.408
R^2
S( e)
Small 0.877 0.772 0.719 Small 0.036 0.055 0.053
Medium 0.905 0.906 0.812 Medium 0.033 0.034 0.048
Big 0.901 0.875 0.848 Big 0.031 0.035 0.042
40
7. Diagnostics
Econometric theory suggests that information obtained from statistical models
can be improved, by identifying their strengths and weaknesses through systematic
testing. Inferences from the empirical analysis must therefore be tested econometrically,
also known as diagnostic model testing. The empirical results are examined for,
multicollinearity, normality, heteroscedasticity, autocorrelation, and stationarity. (The
p-values in the tables which are annotated with an asterisk (*) indicate the rejection of
the null hypothesis.)
7.1. Multicollinearity
Multicollinearity occurs when two or more explanatory variables are highly
correlated. Regressions containing multicollinearity exhibit high R2, inflated standard
errors, and individual effects between two correlated variables become difficult to
distinguish. Multicollinearity can be detected in the matrix of correlations, the
correlation matrices of Fama and French 3 factor model, 2 x 2 Fama and French 5-
factor, and 2 x 2 x 2 x 2 Fama and French 5 factor model, are presented below in Table
10, 11, and 12.
Table 10:
Correlation matrix of Fama and French 3-factor model factors.
RM-RF SMB HML
RM-RF 1 -0.1416 0.0677
SMB -0.1416 1 -0.3988
SMB 0.0677 -0.3988 1
Table 11:
Correlation matrix of Fama and French 5-factor 2 x 2 sort model factors.
RM-RF SMB HML RMW CMA
RM-RF 1 -0.3195 -0.2242 -0.0515 0.2521
SMB -0.3195 1 0.3339 -0.0207 -0.2315
HML -0.2242 0.3339 1 0.1181 -0.5063
RMW -0.0515 -0.0207 0.1181 1 -0.3341
CMA 0.2521 -0.2315 -0.5063 -0.3341 1
41
Table 12:
Correlation matrix of Fama and French 5-factor 2 x 2 x 2 x 2 sort model factors.
RM-RF SMB HML RMW CMA
RM-RF 1 -0.2982 -0.1862 0.1043 0.1423
SMB -0.2982 1 0.3541 -0.1988 -0.1700
HML -0.1862 0.3541 1 -0.0979 -0.2423
RMW 0.1043 -0.1988 -0.0979 1 -0.2629
CMA 0.1423 -0.1700 -0.2423 -0.2629 1
By observing all three matrices, the correlations between explanatory variables
in each model are below 0.40, except for the correlation of CMA and HML factors in the
five factor 2 x 2 sort that has a correlation of 0.5063. The correlation of 0.5063 indicates
a moderate positive relationship, nonetheless, the overall correlations of variables are
weak with values around 0.20 to 0.30. Thus, multicollinearity is assumed to be excluded
from the data.
7.2. Normality
The Jarque-Bera test is employed to test whether the sample returns data
exhibits skewness and kurtosis of a normal distribution. The null of normality is
rejected if the p-value is less than 5%, indicating that the returns are not well-modelled
by a normal distribution. The Jarque-Bera tests of Fama and French 3 factor model, 2 x
2 Fama and French 5 factor, and 2 x 2 x 2 x 2 Fama and French 5-factor model, are
presented below in Table 13, 14, and 15.
Table 13:
Jarque-Bera test of Fama and French 3-factor model.
L N H
Jarque-Bera S
112.8774 3.2765 9.5389
p-value 0* 0.1943 0.0085*
Jarque-Bera M
91.7681 0.5130 31.6161
p-value 0* 0.7738 0*
Jarque-Bera B
54.0863 148.4156 52.7935
p-value 0* 0* 0*
42
Table 14:
Jarque-Bera test of Fama and French 5-factor 2 x 2 sort model.
L N H
Jarque-Bera S
45.0937 25.0976 64.9976
p-value 0* 0* 0*
Jarque-Bera M
73.7371 2.9461 30.7317
p-value 0* 0.2292 0*
Jarque-Bera B
17.2563 55.4682 1.2040
p-value 0.0002* 0* 0.5477
Table 15:
Jarque-Bera test of Fama and French 5-factor 2 x 2 x 2 x 2 sort model.
L N H
Jarque-Bera S
127.7486 35.3914 97.4646
p-value 0* 0* 0*
Jarque-Bera M
117.5979 4.2246 20.7802
p-value 0* 0.1210 0*
Jarque-Bera B
28.2211 53.7411 1.9471
p-value 0* 0* 0.3777
The null hypothesis for residual normality is rejected for all Fama-French three
factor portfolios except SN, MN, and MN, BV, in the Fama-French five factor model.
This indicates that the returns of the portfolios constructed of Chinese stocks mostly
reject the null of a normal distribution.
7.3. Heteroscedasticity
The test of heteroscedasticity is crucial in regression analysis as it evaluates if
the variance of error terms are constant. Consequences of heteroscedasticity include
unbiased estimates of OLS estimation and estimates that are no longer the best linear
unbiased estimators (BLUE). A White's test of heteroscedasticity with no cross terms is
applied to the pricing models. Variance of error terms are constant if the p-value is less
than 0.50, the White's test of Fama and French 3 factor model, 2 x 2 Fama and French 5
factor, and 2 x 2 x 2 x 2 Fama and French 5 factor model, are presented below in Table
16, 17, and 18.
43
Table 16:
White's Heteroscedastic test of Fama and French 3-factor model.
L N H
F-Statistic S
2.2032 2.7221 8.4241
p-value 0.0909 0.0471 0
F-Statistic M
15.8387 17.9736 26.5585
p-value 0 0 0
F-Statistic B
0.6784 7.4666 3.0464
p-value 0.5668 0.0001 0.0312
Table 17:
White's Heteroscedastic test of Fama and French 5-factor 2 x 2 sort model.
L N H
F-Statistic S
2.3793 1.7257 2.9104
p-value 0.0423 0.1334 0.0160
F-Statistic M
2.5320 1.9541 1.2234
p-value 0.0321 0.0900 0.3020
F-Statistic B
1.4565 4.8947 2.8992
p-value 0.2088 0.0040 0.0164
Table 18:
White's Heteroscedastic test of Fama and French 5-factor 2 x 2 sort model.
L N H
F-Statistic S
1.4040 2.5272 3.2895
p-value 0.2273 0.0324 0.0079
F-Statistic M
1.7699 2.3705 2.2878
p-value 0.1237 0.0430 0.0499
F-Statistic B
2.2016 7.2965 2.4047
p-value 0.0582 0.0000 0.0404
The null hypothesis of constant variance of error terms is
accepted as p-values for all models are less than 0.50, except BL of Fama and French
three-factor model. Thus, the variance of error terms in all other portfolios are constant.
7.4. Autocorrelation
Autocorrelation is found in datasets that repeat similar patterns when exposed to
shocks or experience overlapping effects of shocks in a given time period. The
observation in error terms are then found to be correlated, which causes inflated R2
values, inefficient coefficient estimates, and misleading standard errors. The models are
tested according to the Breusch-Godfrey test with 10 lagged residuals. The Breusch-
Godfrey test of Fama and French 3 factor model, 2 x 2 Fama and French 5-factor, and 2
x 2 x 2 x 2 Fama and French 5 factor model, are presented below in Table 19, 20, and 21.
44
Table 19:
Breusch-Godfrey test of Fama and French 3-factor model.
L N H
F-Statistic S
2.4776 1.5008 1.2654
p-value 0.0099* 0.1474 0.2579
F-Statistic M
1.9524 0.4085 1.6412
p-value 0.0447* 0.9403 0.1032
F-Statistic B
0.5617 1.6804 1.3788
p-value 0.8421 0.0932 0.1983
Table 20:
Breusch-Godfrey test of Fama and French 5-factor 2 x 2 sort model.
L N H
F-Statistic S
0.6488 2.3041 1.4276
p-value 0.7691 0.0166* 0.1767
F-Statistic M
1.4856 2.8480 2.3133
p-value 0.1533 0.0033* 0.1610
F-Statistic B
0.8848 2.4680 0.9555
p-value 0.5496 0.0103* 0.4859
Table 21:
Breusch-Godfrey test of Fama and French 5-factor 2 x 2 x 2 x 2 sort model.
L N H
F-Statistic S
0.9726 2.5082 1.6586
p-value 0.4711 0.0091* 0.0989
F-Statistic M
1.9226 2.6843 2.8557
p-value 0.0487* 0.0054* 0.0033*
F-Statistic B
1.0296 3.3373 0.8274
p-value 0.4232 0.0008* 0.6031
The null hypothesis is rejected for, SG, and MG in the three
factor model, SN, MN, and BN in the 2 x 2 sort five factor model, and SN, MG, MN, MV,
and SN in the 2 x 2 x 2 x 2 sort five-factor model. To correct for these autocorrelated
error terms, standard errors are adjusted for using Newey-West standard errors which
allow for autocorrelated residuals.
7.5. Stationarity
Stationarity is a stochastic process such that any subset of the time series data
has a distribution function identical to any other subset. If a regression is non-
stationary, the persistence of shocks will be infinite, the subsequent mean and variance
estimates are then not well defined and inferences from their coefficients are unreliable.
The Augmented Dickey-Fuller (ADF) test is utilised to examine for a unit root with 12
45
lagged differences. The ADF test is carried on the series of each portfolio, illustrated in
Table 22.
Table 22:
Augmented Dickey-Fuller test of all nine portfolio log returns series (SL, SN, SH, ML,
MN, MH, BL, BN, and BH).
L N H
t-Statistic S
-10.3186 -9.9959 -10.3780
p-value 0* 0* 0*
t-Statistic M
-3.3533 -3.1407 -5.8497
p-value 0.0145* 0.0261* 0*
t-Statistic B
-11.3839 -10.3113 -9.7500
p-value 0* 0* 0*
As all p-values are less than 0.05, the null hypothesis that the series contains a
unit root is rejected and the return series for all nine portfolios are stationary.
46
8. Empirical Analysis
8.1. Summary Statistics for factor returns
The empirical tests investigate whether portfolios formed on Size, BE/ME,
profitability, and investment capture the variation in average returns that are exposed
to these factors. Firstly, each factor is examined for the pattern in average returns.
Table 4 shows the average returns of nine portfolios sorted on Size-BE/ME, Size-OP, and
Size-Inv.
In each of the BE/ME columns under Panel A , average returns fall from small
stocks to big stocks in Low and Neutral BE/ME stocks which show the size effect.
However, the Medium size stocks with high BE/ME have a higher average return of
3.89% than that for similar small stocks (3.23%). The relation of stock returns and
BE/ME, known as the value effect is more consistently observed as average returns in
each column from Low to High increases monotonically with higher BE/ME. The results
are similar to those of Fama and French (1993) where the value effect is stronger among
small stocks than in large stocks. The returns of small stocks in the first row increase
from 0.34% per month to 3.23% per month for increasing BE/ME, compared to the
smaller degree of increase for big stocks in the third row, from 0.11% to 2.57%.
Panel B of Table 4 shows the average returns of nine portfolios sorted on Size-
OP. In contrast to the findings of Fama and French (2015), the effect of size on average
returns is not as prevalent in comparison to the size effect of the BE/ME sort. The size
effect in the Size-OP sort is not observed as there is not a pattern of higher premiums
for small size firms. However, higher profitability sorts are associated with higher
returns within small and large stocks, average returns of small stocks increase from
0.96% to 1.30%, and 0.95% to 1.77% for large stocks. The reverse case is seen for
medium size stocks where Low profitability stocks yield higher average returns (1.68%)
than High profitability stocks (1.45%).
The size effect in Panel C of Table 4 is detected only in stocks with Low
investment (conservative stocks) with average returns increasing from big to small
stocks, 0.54% to 1.04%. Similarly, the size effect of Size-OP sort portfolio of small stock
premiums is not observed in neutral and higher investment stocks. Big stocks in
Neutral and High investment have higher returns than those of small stocks. For
example, the returns of medium and big stocks are 1.93% and 1.61% respectively, in
comparison to the small stock average return of 0.95%. The average returns of High
47
investment (aggressive) stocks exceed the average returns of Low investment
(conservative) stocks. This result is interesting. The results diverge from the findings of
Fama and French (2015) such that average returns on portfolios with Low investment
are higher portfolios of High investment. These results show that in the Chinese stock
market, average returns are typically higher for firms that investment more
aggressively.
Building from the findings of Fama and French (1995), high BE/ME stocks tend
to have low profitability and investment, and low BE/ME stocks tend to be profitable
and invest aggressively. Table 4 however does not isolate the value, profitability and
investment effects in average returns. To more cleanly investigate the effects of value,
profitability and investment, the factors in the regressions of the Fama and French
three-factor and five-factor model are examined in section 8.2.
Table 5 shows the average summary statistics for individual factors of each
model. In Panel A, the average SMB returns are 0.428%, -0.472%, and -0.392% for the
three versions of the factor. The SMB correlations between different versions of the
same factor are quite low, 0.32 and 0.35, which explains the differences in SMB average
returns and standard deviations. This is quite surprising since the breakpoints for the
size sort is defined in the same way, and all three versions use utilise all top 100 stocks
in the construction.
The summary statistics for HML, RMW and CMA depend more on the factor
construction as each model constructs these factors in a different way. The results from
the FF5 2x2 and FF5 2x2x2x2 are the easiest to compare. By construction, the factors
from the 2 x 2 sort controls for size and one other variable, whilst the 2 x 2 x 2 x 2 sort
jointly controls for all four variables. In Panel A and B of Table 5, the joint controls have
little effect on HML as the average returns of the FF5 2x2 and FF5 2x2x2x2 are -2.121%
and -2.160% respectively. Moreover, the correlations of the HML between FF5 2x2 and
FF5 2x2x2x2 are high at 0.99, which also accounts for the similarity in standard
deviations of 4.109 and 4.215.
However, the correlations between RMW and CMA in the two five-factor models
are lower, with correlations of 0.82 for RMW, and 0.84 for CMA. The comparison
between these two factors for each model is noteworthy. Firstly, the 2 x 2 sort model
produces positive RMW and CMA of 0.108% and 0.498% respectively, whereas the 2 x 2
x 2 x 2 sort model yields negative returns of RMW and CMA, -0.277 and -0.029
48
respectively. By construction, the 2 x 2 x 2 x 2 sort portfolios are poorly diversified (as
mentioned in 5.3), therefore it is logical to expect the standard deviation of the model to
be higher than that of the 2 x 2 sort. However, despite the similarities in standard
deviation in SMB and HML, the 2 x 2 RMW and CMA has higher standard deviation
(3.386 and 3.443) in contrast to the 2 x 2 x 2 x 2 portfolios, although inherently
possessing better diversification than the 2 x 2 x 2 x 2 model (3.084 and 2.782).
Small stocks normally exhibit high market betas than those of big stocks, such
that increase in excess market return is further positively reflected in higher beta
stocks. Thus, the negative relationship in all models between SMB and Rm-Rf is
counterintuitive. This indicates that in China's emerging market, big stocks perform
better than small stocks when the market performs well.
Most notably from the correlations, the divergence from the findings of Fama and
French (2015) that low investment firms produce higher average returns of high
investment firms (CMA), is further supported by the correlations of RMW and CMA.
The negative correlations between RMW and CMA (-0.33 in FF5 2x2, and -0.26 in FF5
2x2x2x2) shows that as firms with high profitability outperform firms with low
profitability, aggressive investment firms yield higher returns than those of
conservative firms. This empirical result is also illustrated in Panel C of Table 4 where
aggressive investment firms produce higher average returns than conservative firms.
8.2. Regression Analysis
Through examining the R2 of the Fama and French three-factor model in Table 6.
The pricing model captures 83.5% to 92.1% in the variation of average returns, this
result is significantly lower to the 95% R2 observed in Fama and French (1993). In
comparison to the R2 of the principal component regressions defined in Table 7, which
explains 90.6% to 95.4%. The empirical results show that there are variations in average
returns are not sufficiently captured by the three-factor model. These results motivate
the augmentation of the three factor model with profitability and investment factors,
using the Fama and French five-factor model.
Tables 6, 8, and 9 shows the Fama-French intercepts from the 9 Size-BE/ME
portfolios. The Fama and French (2015) "fatal" problem of extreme microcap growth
stocks with large t-statistics that reject the model are not reflected in the results shown
in Table of 8 and 9. The portfolios of small growth stocks produce positive three-factor
and five-factor intercepts. Moreover, a recurring pattern of negative coefficients is
49
observed in high BE/ME (values) stocks and small neutral BE/ME stocks, with the
highest negative coefficients in the medium size stocks (-0.542, -0.957, and -0.907) in the
three-factor and the two five-factor models.
Fama and French (2015) drop the HML factor and find that the five-factor model
never improves the description of average returns from the four-factor model that drops
HML factor. On to contrary, HML is not recognised as a redundant factor within the
SSE market returns from 2004 to 2015. The HML t-statistic is strongly significant in all
models, especially at high and low extremes of BE/ME sort portfolios. Furthermore, the
t-statistic is most significant in SL portfolio of Fama-French three-factor model (9.146),
and in MH portfolio of both Fama-French five-factor models (-7.766 and -8.356).
Analysing the profitability factors in Table 8 and 9, microcap stocks returns
behave like those of unprofitable firms but grow rapidly. Small unprofitable firms are
reflected in the negative coefficients of all small stocks. The most extreme negative
profitability intercept lies in the top right of the matrix (-0.460) for the 2 x 2 sort, and
top left of the matrix (-0.332) for the 2 x 2 x 2 x 2 sort. RMW intercept gradually
becomes positive from small to big stocks. It can be inferred that big stocks are more
profitable than small stocks, especially within high BE/ME stocks. This result is not
surprising as the majority of large size firms are state owned by China and generally
have high BE/ME characteristics.
Aside from the high BE/ME stocks in the 2 x 2 and 2 x 2 x 2 x 2 portfolios which
are majority classified as state-owned firm, investment intercepts (CMA) are largely
positive for small stocks, and decreasing or negative for big stocks. This adds an extra
size dimension to the previous findings that aggressive firms have higher average
returns than conservative firms. Negative CMA coefficients for aggressive big firms
confirms that higher investment firms outperform lower investment firms
Additionally, the extremely high t-statistic (t(b)) of the excess market return
coefficients in all Fama and French models is similar to the results found in Fama and
French (1993, Table 6). The high t-statistic is due to the high explanatory power in these
regressions since the regressors and regressands are both constructed from the same
underlying data, the market constituents of the index. As the nine diversified Size-
BE/ME portfolios are constructed within the constituents of the SSE market index, this
consequently results in high t-statistic for the coefficient of excess market returns.
50
8.3. Principal Component Analysis
The variations in average returns can be analysed through principal components.
The components allow for interpretation of average returns of different portfolios and
the individual effects on portfolios sorted on Size-BE/ME.
Table 23:
Correlation of PCA factors and factors of the Fama-French three-factor model.
RM-RF PC1 PC2 PC3 SMB HML
RM-RF 1 0.966 -0.010 0.015 -0.142 0.068
PC1 0.966 1 0.084 -0.022 -0.039 0.040
PC2 -0.010 0.084 1 -0.002 0.791 -0.804
PC3 0.015 -0.022 -0.002 1 -0.455 -0.508
SMB -0.142 -0.039 0.791 -0.455 1 -0.399
HML 0.068 0.040 -0.804 -0.508 -0.399 1
Table 24:
Weighting coefficients of principal components of average monthly returns for nine
portfolios sorted on Size-BE/ME from June 2004 to June 2015, 132 months.
Portfolio Principal Components
PC 1 PC 2 PC 3 PC 4 PC 5 PC 6 PC 7 PC 8 PC 9
BL 0.330 -0.028 0.662 -0.133 -0.492 0.140 0.290 -0.227 0.192
BN 0.333 -0.398 0.243 0.214 -0.151 -0.239 -0.621 0.369 -0.155
BH 0.325 -0.456 -0.063 0.453 0.357 -0.123 0.560 0.059 0.133
ML 0.324 0.400 0.361 -0.281 0.633 0.061 -0.044 0.331 0.100
MN 0.354 -0.017 -0.055 -0.259 0.145 -0.303 0.035 -0.506 -0.659
MH 0.344 -0.217 -0.316 -0.233 0.128 0.152 -0.365 -0.423 0.573
SL 0.311 0.545 -0.077 0.681 -0.065 0.224 -0.167 -0.224 -0.063
SN 0.332 0.338 -0.374 -0.152 -0.359 -0.561 0.184 0.294 0.223
SH 0.344 -0.118 -0.346 -0.212 -0.190 0.652 0.137 0.357 -0.308
Examining the Table 23 correlation matrix of principal components, it can be
seen that the first principal component is strongly related to the excess market return
(RM-RF) with a correlation of 0.966. This level effect approximately influences all
portfolios in a similar magnitude shown in the column of the first principal component
in Table 24.
The second principal component is strongly correlated to SMB (0.791) and also
strongly negatively related to HML (-0.804). Moreover, the third principal component
seems to be simultaneously affected by both SMB and HML factors, with negative
correlations of -0.455 and -0.508 respectively.
51
8.4. Model Evaluation
Given that the exposures to the three and five-factor, bi, si, hi, ri, and ci, fully
capture all variations in the average expected return, the intercept a i is zero for all
portfolios. Therefore, by definition, effective pricing models should have alphas close to
zero. The Fama and French three-factor model, 2 x 2 five-factor model, 2 x 2 x 2 x 2 five
factor model, and principal component regressions are evaluated by comparison of alpha
values and R2 which are summarised from Tables 6, 7, 8, and 9 in Table 25.
The Fama and French three-factor model have alphas that are close to zero
which range from 0.004 to 0.011. However, small and medium high BE/ME portfolios
(SH and MH) alphas are significant at 2.510 and 2.579 t-statistic respectively. By itself,
the three-factor intercept is sufficient to reject the three-factor model at a 99% one-tail
confidence interval. Not surprisingly the portfolios that reject the three-factor model
have the lowest R2 of 83.8% and 83.5%. Aside from these portfolios, the Fama and
French three-factor model explains the variation in average returns well with R2 of
86.1% to 92.1%.
Table 25:
Summarised alpha and R2 of regressions on nine portfolios sorted on Size-BE/ME from
June 2004 to June 2015, 132 months.
a
R2
Low Neutral High Low Neutral High
Fama and French 3 Factor Model
Small 0.006 0.004 0.010
0.898 0.893 0.838
Medium 0.004 0.005 0.011
0.883 0.888 0.835
Big 0.006 0.006 0.004
0.921 0.890 0.861
Principal Component Regression
Small 0.001 -0.003 -0.003
0.906 0.945 0.934
Medium -0.001 -0.003 -0.005
0.931 0.943 0.951
Big -0.005 0.001 0.001
0.954 0.944 0.922
Fama and French 5 Factor 2x2 Model
Small 0.004 0.003 0.008
0.888 0.784 0.723
Medium 0.002 0.004 0.009
0.908 0.907 0.807
Big 0.004 0.006 0.005
0.901 0.874 0.842
Fama and French 5 Factor 2x2x2x2 Model
Small 0.002 0.003 0.007
0.877 0.772 0.719
Medium 0.001 0.003 0.009
0.905 0.906 0.812
Big 0.004 0.006 0.005
0.901 0.875 0.848
52
As the first three principal components are constructed to maximise the variation
of returns upon increasing dimensions. Principal component regression inevitably
produces the best three factor model. Consequently, alpha intercepts are individually
smaller in all principal component regressions than the Fama and French three-factor
model. Likewise, the average R2 of PCR is 93.7% which exceeds the highest R2 of the
three-factor model.
In terms of R2, both the five-factor models do not improve the explanatory power
of the three-factor model except the ML portfolio (90.8% and 90.5%). This may be due to
the difference in how factors are defined in the augmented models which do not fully
reflect the political, regulatory, and complex legal environment under a transition
economy. Nonetheless, alpha intercepts of the five factor model are all in a lower degree
(except BH) than those of the three factor model. Moreover, the t-statistics of all alphas
for the five factor model do not reject the model.
In summary, the empirical results show that both the three-factor and five-factor
model are useful tools in asset pricing. The inferences drawn from the results suggest
that different factors have contrasting effects and implications upon particular markets.
Such inconsistencies are found in the average returns of aggressive stocks being higher
than the average returns of conservative stocks, in addition to the non-redundant HML
factor for the Chinese stock market.
53
9. Conclusion
The Fama-French three and five-factor model is studied to examine whether
risks associated with emerging markets are identified by the factors and sufficiently
priced within the model. Firstly, patterns of small size premiums are identified in Size-
BE/ME sort portfolios which are in line with the findings of Banz (1981) and Fama and
French (1993). Additionally, the empirical results clearly show the value effect similar to
Fama and French (1993), where these effects are stronger among small stocks than in
large stocks.
Secondly, contrary to the Fama and French (2015), small firms that invest a lot
despite low profitability are not found to be a problem. Such that all portfolio alpha
intercepts in the five-factor sorts do not reject the model. Furthermore, opposite
investment effects are found in which firms that invest aggressively yield higher
average returns than firms that invest conservatively. The investment premiums are
observed to be stronger in big size firms that are characterised by low or negative CMA
intercepts.
Thirdly, Fama and French (2015) suggest that HML is a redundant factor and its
average returns are captured by exposures to RM-RF, SMB, RMW, and CMA. However,
the results show that the HML factor is significant in all models, especially in low and
high BE/ME portfolios. The inferences drawn from the results indicate that different
factors and the way factors are defined have contrasting effects and implications in an
emerging market. Such inconsistencies are found in the average returns of aggressive
stocks being higher than the average returns of conservative stocks, in addition to the
non-redundant HML factor for the Chinese stock market.
By construction, the orthogonal nature of principal components that maximise
variability along each transformed dimension produces the best three-factor model.
Overall, the Fama-French five-factor model is not prominently better than the Fama-
French three-factor model. The three-factor model captures higher average cross-
sectional returns defined by the higher R2. On the other hand, the alpha intercepts of
the five-factor model are all in a lower degree (except BH) than the three-factor model.
Thus, both the three-factor and the two sort five-factor models are useful for making
rational portfolio decisions even with small departure from the over-simplified
assumption of normally distributed returns in the Chinese stock market.
54
Furthermore, the joint controls of the 2 x 2 and 2 x 2 x 2 x 2 sorts are attractive
for isolating and identifying the estimates for factor premiums. However, given the top
100 weight sample, it's not clear whether the poorly diversified 2 x 2 x 2 x 2 sort
effectively isolate size, value, profitability, and investment patterns.
Finally, future studies may consider a larger sample size to accommodate for 16
well-diversified portfolios in the 2 x 2 x 2 x 2 joint control Fama-French five-factor
model. Additional investigations of profitability and investment factors in alternative
emerging markets can be explored to confirm or refute the disparities found contrary to
the findings of Fama and French (2015).
55
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