TESTING THE THREE FACTOR MODEL IN TURKEY
MSc International Money, Finance and Investment
Brunel University
The School of Social Sciences
September 2012
Student Number: 1129154
Supervisor: Dr. Aris KARTSAKLAS
CONTENTS
1. INTRODUCTION ...................................................................................................... 1
2. LITERATURE REVIEW .......................................................................................... 3
2.1. The Single (Index) Factor Asset Pricing Model..................................................................... 3
2.2. Testing The Single Factor Asset Pricing Model .................................................................... 6
2.3. The Multifactor Asset Pricing Model .................................................................................... 9
2.4. Testing Multifactor Asset Pricing Model ............................................................................. 11
3. THE THREE FACTOR MODEL AND ISTANBUL STOCK EXCHANGE ..... 11
3.1. The Three Factor Model....................................................................................................... 11
3.2. Istanbul Stock Exchange ...................................................................................................... 12
4. DATA ......................................................................................................................... 17
4.1. The Sample Stocks and Returns ........................................................................................... 17
4.2. The Market Proxy ................................................................................................................ 19
4.3. The Risk Free Interest Rate Proxy ....................................................................................... 19
5. METHODOLOGY ................................................................................................... 20
5.1. The Return Period ................................................................................................................ 20
5.2. SMB and HML Portfolios .................................................................................................... 21
5.3. Returns on Twelve Portfolios as Independent Variables ..................................................... 22
6. EMPIRICAL ANALYSIS ........................................................................................ 26
6.1. Regressions of excess stock portfolios returns on the excess-market proxy return ............. 27
6.2. Regressions of excess stock portfolios returns on the returns of the SMB and HML factors
..................................................................................................................................................... 28
6.3. Regressions of excess stock portfolios returns on the excess return on the market proxy and
returns on SMB and HML factors. .............................................................................................. 29
6.4. Intercepts from regressions of excess returns on stock portfolios........................................ 30
7. CONCLUSION ......................................................................................................... 33
8. REFERENCES.......................................................................................................... 35
LIST OF TABLES
Table 1 - Number of companies traded on the ISE equity markets ............................................ 14
Table 2 - Daily average of traded volume (USD) of the ISE equity markets ............................. 15
Table 3 - Daily average of traded volume of the ISE equity markets ......................................... 16
Table 4 - The capitalisation of chosen firms and the ISE equity markets ................................... 18
Table 5 - Descriptive statistics for 12 stock portfolios................................................................ 23
Table 6 - Summary statistics of dependent and explanatory variables in the regressions .......... 25
Table 7 – The root test statistics of explanatory variables .......................................................... 26
Table 8 - Regressions of excess returns of 12 portfolios on market factor ................................. 27
Table 9 - Regressions of excess returns of 12 portfolios on SMB and HML factors.................. 28
Table 10 - Regressions of excess returns of 12 portfolios on market, SMB and HML factors .. 29
Table 11- Intercepts from regressions of the excess return on the 12 stock portfolios ............... 31
Table 12 – The GRS statistics for all alternative models ............................................................ 32
ACKNOWLEDGMENTS
I would like to thank everyone who made it possible for me to study at Brunel
University and complete this thesis. They include EU and the Ministry for EU Affairs of
Turkey who granted Jean Monnet Scholarship to me, Dr. Saim KILIÇ, the Chief
Regulatory Officer at Istanbul Stock Exchange, for his stimulating support, the
Chairman/CEO and the Board of Director of Istanbul Stock Exchange for extending
leave of absence;
My supervisor at Brunel, Dr. Aris KARTSAKLAS for his supervision and guidance, I
could not have completed this study;
My wife, Berna Dibo and my family for their patient support.
Any errors and shortcomings remaining in the text are, of course, entirely my own.
ÖZET
Sharpe (1964), Lintner (1965) and Mossin (1966) tarafından literature kazandırılan
Finansal
Varlık
Fiyatlama
Modeli
(FVFM)
finansal
varlıkların
getirilerinin
açıklanmasında son derece popular olmuştur. Ancak yıllar içerisinde Finansal Varlik
Fiyatlama Modeli’ni (FVFM) destekleyen çalışmalarla birlikte, söz konusu modeli
geçersiz kılan ve yeni modeller öneren çalışmalar olmuştur. Fama and French ilk olarak
1993 ve sonrasında 1996 yılındaki çalışmalarında, standard finansal varlık fiyatlama
modeline, finansal varlık getirilerine olan etkileri bir çok calışmada test edilen, piyasa
değeri ve PD/DD faktörlerini ekleyerek Üç Faktörlü Finansal Varlık Fiyatlama modelini
geliştirmislerdir.
Bu çalışmada, öncelikli olarak İstanbul Menkul Kıymet Borsasında işlem gören hisse
senetlerinden oluşan portföyler üzerinde piyasa değeri ve PD/DD faktorlerinin etkilerini
inceleyip, Üç Faktörlü Finansal Fiyatlama Modelinin gerçeliliğini araştırdık. Çalışmada,
modelin İMKB için etkinliğinin incelenmesinde zaman serisi regresyon analizinden
yararlanılmış, GRS istatistiği kullanılarak, fiyatlama hatası olarak kabul edilen
regresyon sabitlerinin sıfırdan farklı olup olmadığı test edilmiştir.
Çalışma sonucunda piyasa değeri ve PD/DD faktorlerinin İMKB’de etkili olduğu, Üç
Faktörlü Modelin İMKB için uygulanabilir olduğu ve getirileri açıklama gücü açısından
FVFM’den daha anlamlı sonuçlar verdiği belirlenmiş olup, bulgularımız Fama ve
French’ in istatistiksel bulgularıyla uyumludur.
ABSTRACT
Sharpe (1964), Lintner (1965) and Mossin (1966) introduced the CAPM, which has
been so popular in pricing the risky financial assets. But over years, there has been
many studies supporting or contracting the CAPM and suggesting new models. Fama
and French, in 1993, suggested the “Three-Factor Asset Pricing Model” by inserting
size and BE/ME factors into CAPM.
In this study, we investigated size and BE/ME factors effects as well as the viability of
the three-factor model for Istanbul Stock Exchange (ISE) markets. Only time series
regressions have been carried out to analyse the effectiveness of the three-factor model
on the ISE markets.
We find evidence for size, and book-to-market factors in Istanbul Stock Exchange.
Considering time series regressions result, the three-factor model is found applicable for
ISE and has superior to CAPM in term of explaining the common variation in returns.
Consequently, our empirical results are reasonably consistent with the Fama and French
1993 findings.
1. INTRODUCTION
Asset Pricing literature abounds with applications of the Sharpe (1964), Lintner (1965),
and Mossin (1966) Capital Asset Pricing Model (the CAPM or the standard CAPM) to
various market data such as Canadian, US, Japan, Indian, European markets data, which
are sometimes contradicting or confirming, even criticizing, and extending the CAPM.
Fama and French are two other prominent scholars who have produced studies
contradicting, criticizing, and extending the CAPM. Fama and French (1992) find that
the CAPM is violated for US stock market, and other two factor, size (ME, price times
number of shares) and book-to-market value (BE/ME, the ratio of the book value to its
market value), explain a major part of the cross sectional returns. Fama and French
(1993) develop a three-factor model defining the combination of market, size and bookto-market model which is better than the single factor linear pricing model in terms of
explaining returns.
In this paper we examine the Fama and French three-factor model for Istanbul Stock
Exchange (ISE). We first investigate the size and BE/ME effects in the ISE markets and
find that the Fama-French (1993) evidence that there is an positive relationship between
BE/ME and average return is confirmed for ISE. But the other Fama French (1993)
evidence that there is a negative relationship between size and average return is not as
much apparent as BE/ME effect. It seems that the ISE markets display a strong BE/ME
effect but not an apparent size effect. Second, we examine the relationship between the
returns on 12 portfolios based on size and book-to-market value and the market, size
and book-to-market factors, and finally test whether the market, size and book-tomarket value are the risk factors explaining the returns in Turkey’s markets.
We use the multivariate time-series regression (OLS) approach to investigate the threefactor model. Daily excess returns on 12 portfolios, created on size and BE/ME, as
dependent variable are regressed on the daily excess returns to a market, size and
BE/ME factors as explanatory variables. The daily return on 3-month Government Debt
Security index (calculated at ISE) is used as risk-free rate proxy to calculate excess
returns.
1
In time-series regression, ISE All index is assumed to be the best proxy for the market
factor; and SMB (small minus big) and HML (high minus low) portfolios are created to
mimic size and BE/ME factors. We created SMB and HML portfolios in a similar
portfolio formation approach to that developed by Fama and French (1993, 1996)
In empirical analysis, first, the market proxy is used as explanatory variable alone to test
the CAPM. Then, we used SMB and HMB alone to examine the explanatory power of
size and BE/ME effects whether they capture common variation in portfolio returns.
Finally, we added the size and BE/ME factors into the CAPM. Fama and French (1993)
suggest that their three-factor model explain common variation in returns better than
CAPM. Our empirical results confirm that the SMB and HML portfolios capture the
risk factor related to size and BE/ME ratio, and when these two factors are added to the
CAPM, the significance of the model increases. We find that the three-factor model
captures common variation in returns better than CAPM in ISE markets.
Intercepts in the CAPM and the three-factor model are considered as pricing errors.
Firstly, we test all intercepts individually whether they are indistinguishable from zero
or not. Then we use GRS test statistic suggested by Gibbons, Ross and Shanken (1989)
to test whether all intercepts are jointly indistinguishable from zero for each model.
Since we find that all intercepts in the three-factor model are jointly indistinguishable
from zero, GRS test confirm that three-factor model is superior to the CAPM in term of
explaining returns on ISE markets.
In the following section, we summarize remarkable studies related to asset pricing
theory literature commencing with Markowitz’s portfolio selection and ending up with
multifactor asset pricing models. In section 3, we describe the three-factor model and
give basic information about the Istanbul Stock Exchange. We delineate our data used
in the model and its sources in section 4. In Section 5 we explain how we obtained and
used the data to calculate stock returns and formed 12 stock portfolios, SMB and HML
portfolios. We analyse and test the model and give the result in section 6. Summary and
concluding findings are provided in section 7.
2
2. LITERATURE REVIEW
2.1. The Single (Index) Factor Asset Pricing Model
For decades, finance professionals, researchers, and practitioners have been studying
possible ways to explain the relationship between the expected return on an asset and its
risk factors, and to find an answer to the question as to what determines assets’ prices
and to identify the best model explaining the expected return on risky assets. These
studies, such as the portfolio selection of Markowitz (1952), the Capital Asset Pricing
Model (CAPM) of Sharpe (1964), Lintner (1965), and Mossin (1966), the Arbitrage
Pricing Theory (APT) of Ross (1976), and the three-factor model of Fama and French
(1993), consist of Asset Pricing Theory.
One may undoubtedly say that Markowitz (1952, 1959) is the father of modern portfolio
theory, the first to use mean-variance analysis to emphasize the risk factor of expected
returns (Elton, Gruber 1997). In his paper “Portfolio Selection”, Markowitz (1952)
developed his model using mean-variance analysis. Markowitz defines the expected
return and variance of returns on a portfolio as the basic criteria for portfolio selection.
These two parameters, the expected return and the variance of returns, are crucial for its
model. As with every model, Markowitz’s model is based on some assumptions:
• Investors attempt to maximize the expected return.
• All investors have the same investment time horizon.
• All investors are risk-averse.
• Investors make their investment decisions on the basis of expected return and risk, as
measured by the mean and variance of the returns on varies assets.
• All markets are perfectly efficient.
Under these assumptions, Markowitz formulated the portfolio problem as a choice of
the mean and variance of a portfolio of assets. He proved the fundamental theorem of
mean variance portfolio theory, specifically given a constant variance, maximization of
the expected return, and, given a constant expected return, minimized variance. These
two principles led to the formulation of an efficient frontier from which the investor
3
could choose his or her preferred portfolio, depending on individual risk return
preferences. The important message of the theory was that the assets could not be
selected only on characteristics that were unique to the security. Rather, an investor had
to consider how each security co-moved with all other securities. Furthermore, by
taking these co-movements into account an ability to construct a portfolio was produced
that had the same expected return and less risk than a portfolio constructed by ignoring
the interactions between securities (Elton, Gruber 1997).
Although no one use Markowitz’s model anymore, however it has remained as the
central to modern portfolio theory for decades despite its alternatives, and there has
been a great deal of study implementing this theory for the last few decades.
As the mean variance theory of Markowitz requires large data inputs, variance and
covariance of returns on security in the portfolio (2n+n*(n+1)/2), the inputs to the
portfolio analysis have been the most important and crucial problem for implementing
the theory. Because of this input drawback, new models are developed to simplify the
inputs to portfolio analysis.
The foremost models developed to estimate covariance was index models. These
models have enjoyed widespread application beyond estimating covariance structures,
and are worth reviewing on their own (Elton, Gruber 1997). The prominent one is the
market model which was developed by Sharpe (1964). The Sharpe’s market model is
Rit = αi + βi * Rmt + eit ,
(1)
where Rit is the return of stock i in period t, αi is the unique expected return of security i,
βi is the sensitivity of stock i to market movements, Rmt is the return on the market in
period t, and eit is the unique risky return of security i in period t and has a mean of zero
and finite variance σ2ei, uncorrelated with the market return, pairwise and serially
uncorrelated. Namely, Sharpe’s market model relates the return on asset i to the return
on a stock market index. β in equation (1) is a risk measure arising from the relationship
between the return on a stock and the market’s return. Using the market model’s
assumption, β is given by
4
In terms of inputs to portfolio analyse, the most important characteristic of the market
model was that the number of inputs required for portfolio analysis was reduced to
3n+2.
Besides index models, the equilibrium models have been developed relating asset
returns to excess return on a market portfolio in capital markets. The first and basic
form of the general equilibrium model was developed by Sharpe (1964), Lintner (1965),
and Mossin (1966), which is why this model is called the Sharpe Lintner Mossin Capital
Asset Pricing Model (the SLM CAPM, the standard CAPM or the CAPM).
The CAPM builds on the model of portfolio selection developed by Harry Markowitz
(1959). The CAPM is considered one of the fundamental contributions to the exercise
of finance, and it has long been a guide for academics and practitioners of about the
relationship between average returns and risk. The CAPM postulates that the market
portfolio is mean-variance efficient in the sense of Markowitz (1959), and the model
simply assumes that the equilibrium rates of return on all risky assets (or portfolios) are
a linear function of their covariance with the market proxy. In this model, the risk of
asset is measured by β coefficient (market beta, β) and the β is assumed to be constant
across the entire assets (Gokgoz 2007). Namely, the standard CAPM predicts a linear,
single factor (beta), positively sloped relationship between expected stock return and
beta (Ed Vos and Byron Pepper 1997). The standard CAPM equation can be written as
follows:
Rit = Rf + βi *( Rmt – Rf ) + eit ,
(2)
where Rit is the return of stock i in period t, Rf is risk free rate, βi is the sensitivity of
stock i to excess return on a market portfolio, Rmt is the return on the market in period t,
and eit is the unique risky return of security i in period t and has a mean of zero and
variance σ2ei. Since the CAMP contains some unrealistic assumptions, this model has
evolved over time, and several forms of the CAPM were developed on relaxed and
mode realistic assumptions. One of the foremost and best-known derivations of the
CAPM is zero-beta CAPM of Black (1972). Black (1972) derived a new model of the
CAPM by relaxing the assumption of risk-free lending and borrowing.
5
2.2. Testing The Single Factor Asset Pricing Model
A model explaining the relationship between expected return on assets and its risk
factors has to be tested in terms of how well they describe the real world. There has
been a great deal of empirical testing of the standard form of CAPM.
The first and simple test of CAPM was held by Sharpe and Cooper (1972). Sharpe and
Cooper (1972) find that stocks with higher βs have produced higher future returns, and
there is a positive relationship between return and β. They simply regressed βs of
returns on US stocks (1931-1967) by using the equation as follows, Ri = a1 + a2 βi + ui
and found a1 as 5.54, a2 as 12.75 and R2 as 0.95. Their findings can be most easily
interpreted as a positive, strong and linear relationship between βs and returns. But the
intercept of 5.54 is much higher than plausible risk-free rate, which the results support
the zero-beta CAPM.
Most of the early empirical tests of the CAPM such as Lintner (1965) and Douglas
(1968), Black, Jensen and Scholes (1972) and Fama and MacBeth (1973) used a time
series (first pass) regression to estimate βs and a cross-sectional (second pass)
regression to test the hypothesis derived from the CAPM (Elton et al, 2011, p.332-334).
Lintner (1965) found the residual variance from the first-pass regression (Rit = ai + bi
RMt + eit) statistically significant and positive, which means that the CAPM does not
hold. Douglas (1968) found similar results to those of Lintner by using the same
methodology.
Black, Jensen, and Scholes (1972) consider a different time series model which is
written in terms of returns in excess of the risk-free rate Rf; and show that returns are
positively and linearly related to β, as follows:
Rit – Rft = αi + βi (Rmt – Rft ) + eit
(3)
Fama and MacBeth (1973) consider an interesting methodology to test the CAPM and
find that CAPM holds but zero-Beta model of CAPM is more consistent in terms of
equilibrium conditions than standard CAPM. However, Fama and French (1992) and
Fama and French (1996) tested CAPM and found a non-linear relationship between
return and the beta.
6
There are also several empirical tests related to CAPM resulting in contradictions of the
Sharpe-Lintner-Black model (see, for example, Reinganum (1981), Breeden, Gibbons,
and Litzenberger (1989), Fama and French (1992)) These tests reveal that not only
market portfolio but also some other variables which can be manifested as (1) size, ME
(stock price times number of shares), (2) book-to market equity ratio, BE/ME (3)
earnings-price ratio, E/P and (4) leverage ratio have a strong role in explaining average
returns.
Size (ME): Banz (1981) finds that market equity (ME) adds to the explanation of the
cross-section of average returns provided by market βs. Average returns on small stocks
are too high given their β estimates, and average returns on large stocks are too low.
Banz (1981) finds that the relationship between size and return is not linear. The main
excess return effect occurs for very small firms. The smallest firms have on average
very large unexplained mean returns. There is not much difference in return between the
middle and upper firms. The difference in return terms is about .4% per month.
Leverage: Another contradiction of the CAPM is the positive relation between leverage
and average return documented by Bhandari (1988). It is plausible that leverage is
related to risk and expected return, but in the CAPM, leverage as a risk factor should be
absorbed by market β. Bhandari finds that leverage helps explain the cross-section of
average stock returns in tests which include size (ME) as well as β.
Book-to-Market Equity (BE/ME, the ratio of the book value of a firm’s stock to its
market value): Stattman (1980), and Rosenberg, Reid, and Lanstein (1985) find that
average returns on US stocks are positively related to the ratio of a firms’s book value
of common equity to its market value, BE/ME. Chan, Hamao, and Lakonishkok (1991)
find that book-to-market equity (BE/ME), also has a strong role in explaining the crosssection of average returns on Japanese stocks.
Earnings-Price Ratios (E/P): Finally, Basu (1983) shows that earnings-price ratios
(E/P) help explain the cross-section of average returns on U.S. stocks in tests that also
include size and market β. Ball (1978) argues that E/P is a catch-all proxy for unnamed
7
factors in expected returns; E/P is likely to be higher (prices are lower relative to
earnings) for stocks with higher risks and expected returns, whatever the unnamed
sources of risk.
Shefrin and Statman (1995) suggest an explanation for the size and the BE/ME effects.
They propose a behaviour-based approach to security valuation. The essence of their
null hypothesis is that investors make decisions based on the wrong criteria when they
choose stocks, and that this is what causes the size and book to market effect.
All these variables, size (ME), leverage, BE/ME, and P/E can be regarded as different
ways to scale stock prices, to extract information in prices about risk and expected
returns (Keim 1988). Moreover, since E/P, ME, leverage, and BE/ME are all scaled
versions of price, it is reasonable to expect that some of them are redundant for
describing average returns.
Fama and French (1992) try to evaluate the joint role of market β, size, E/P, and BE/ME
in the cross-section of average returns on NYSE, AMEX, and NASDAQ stocks, and
they find that stock returns are negatively related to size (ME) and positively related to
book to market ratios, and the relationship between stock returns and beta is not
statistically significant and β has little information average returns (Fama and French
1993)
Although Black, Jensen, and Scholes (1972) and Fama and MacBeth (1973) find that, as
predicted by the CAPM, there is a positive simple relation between average stock
returns and β during the pre-1969 period, but Reinganum (1981) and Lakonishok and
Shapiro (1986) and Fama and French (1992) find that the relationship between β and
average return disappears during 1963-1990 period, even when β is used alone to
explain average returns, and the simple relation between β and average return is weak in
the 50-year 1941-1990 period. And conversely, Fama and French’s (1992) tests do not
support the most basic prediction of the CAPM, i.e. that average stock returns are
positively related to market βs.
8
Fama and French (1992) suggest that unlike the simple relation between β and average
return, the univariate relations between average return and size (ME), leverage, E/P, and
book-to-market equity (BE/ME) are strong, and their results are (a) β does not seem to
help explain the cross-section of average stock returns and (b) the combination of size
and book-to-market equity seems to absorb the roles of leverage and E/P in average
stock returns, at least during the 1963-1990 sample period.
Fama and French (1992) say that if assets are priced rationally, their results suggest that
stocks risks are multidimensional. One dimension of risk is captured by size, ME.
Another dimension of risk is proxied by the ratio of the book value of common equity to
its market value, BE/ME.
And Fama and French (1992) find that whatever the underlying economic causes, their
main result is straightforward: two easily measured variables, size (ME) and book-tomarket equity (BE/ME), provide a simple and powerful characterization of the crosssection of average stock returns for the 1963-1990 period.
Fama and French (1992) divide the set of theories within which the empirical results
can be viewed into two contexts. In one context, the results can be seen as rational
valuation, and the other as overreaction. The essence of the rational valuation theory is
that size and book to market are indicators of risk, through their relationship with the
economic prospects of companies. The essence of overreaction, as described by De
Bondt and Thaler (1985) is that investors overreact to recent stock returns, thereby
causing the stocks of the losers to become undervalued and the stocks of the winners to
become overvalued.
2.3. The Multifactor Asset Pricing Model
The standard CAPM and other versions of it have basis in mean-variance analysis and
include a single risk factor, the beta (β) of a market portfolio. However, it is postulated
that the market return is not only the factor determining the risk of a stock but also some
accounting data related to the firm. One of the first pieces of research to link the beta of
a single stock to firm’s fundamental variables was carried out by Beaver, Kettler and
9
Scholes (1970). They use seven accounting variables to investigate the relationship
between the beta (β) on a firm’s stock and these seven accounting variables such as
dividend pay-out, asset growth, leverage, liquidity, asset size, earning variability,
accounting beta (Elton et al, 2011, p.149). This and similar empirical studies (see also
Ball and Brown (1969), Beaver and Manegold (1975), Gonedes (1973), and Hill and
Stone (1980)) do not have strong theoretical roots and they present problems of serious
potential model misspecifications.
Shortly after the market model was developed, a number of researchers started to
explore whether multi-index models better explained reality (Elton, Gruber 1997). The
multi-index model is simply as follows:
Rit = αi + ∑ βij * Ijt + eit ,
(4)
where Rit is the return of stock i in period t, αi is the unique expected return of security
i, βij is the sensitivity of stock i to index j, Ij is the return on the jth index in period t, J is
the number of indexes and eit is the unique risky return of security i in period t and has a
mean of zero and variance σ2ei. The prominent theory that uses the multi-index model
which was developed and first proposed by Ross (1976) is arbitrage pricing theory
(APT). Ross has suggested a new and different approach to explaining the pricing of
assets, and postulates that the returns on any stock be linearly related to a set of indexes.
It is based on multi-index model, but differs from it in terms of the contribution in
showing how to go from a multi-index model to a description of equilibrium (Elton et
al, 2011, p.358-362). APT simply suggests that the expected return on an asset can be
explained as a linear function of various fundamental factors which are macroeconomic
aggregates rather than firm-specific characteristics. Sensitivity to changes in each factor
is represented by a factor-specific beta coefficient. APT is formulated as follows:
Rit = ai + bi1δ1t + bi2δ2t + ……………+ bikδkt + εit
(5)
Where Rit is the return on asset i on time t, ai is the expected return on asset i, δj is the
common factor affecting returns, bi is the sensitivity of asset i against common factor j,
and finally εit is non-systematic risk of asset i. The assumptions of the ART is as follows
E(δj) = 0 for all j; E(εi) = 0 for all i; E(εiεj) = 0 for all i and j where i ≠ j ;
E(εi2) = σi2 < ∞.
10
2.4. Testing Multifactor Asset Pricing Model
Roll and Ross (1980) is the first testing APT by using Fama and Macbeth (1973)
technique, first and second pass tests. Their second-pass test’s result suggests that at
least three factors are significant in explaining equilibrium prices (Elton et al, 2011,
p.366). There are numerous studies that strongly suggest that there are more than two
factors driving returns. For example Elton and Gruber (1982,1988) find that a five
factor APT model is superior to a single factor model in explaining and predicting
expected returns. (see, for more examples, Cho, Elton, and Gruber (1984), Dhrymes,
Friend, and Gultekin (1984),Connor and Korajczyk (1986), Chen, Roll, and Ross
(1986))
3. THE THREE FACTOR MODEL and ISTANBUL STOCK
EXCHANGE
3.1. The Three Factor Model
The present research has been aimed at discovering from asset pricing literature which
model is the best in terms of explaining the variations in portfolio or stock returns.
Fama and French (1993) argue that CAPM is insufficient to explain the returns on
stocks and they suggest a multifactor model. Fama and French (1996) test the CAPM
and the three-factor model by using 30 year-time data, and found that the three-factor
model is superior to CAPM as the interception values are nearly zero.
Fama and French (1993) develop the three-factor model to explain the relationship
between returns on portfolios/stocks and size, book to market value and return on
market portfolio. Fama and French (1993) find that their portfolios mimicking risk
factors related to size and book-to-market equity explain common variation in returns,
which is the evidence that these two variables (size and BE/ME) are factor loadings to
common risk factors in stock returns. The Fama and French three-factor model can be
written as follows:
E(Ri) - Rf = βi [ E(Rm) - Rf ] + βis (SMB) + βih E(HML)
11
(6)
Where E(Ri) - Rf is the expected return of the stock over the risk free rate, E(Rm) - Rf is
the expected return of the market portfolio over the risk free interest rate, SMB (small
minus big) is the difference between the returns on portfolios of small and big stocks
and HML (high minus low) is the difference between returns on portfolios of stocks
with high and low book-to-market value, βi, βis and βih are the risk factor of respectively
market portfolio, SMB returns and HML returns. Fama and French (1996) postulate that
if assets are priced rationally, size and book-to-market ratio must be a proxy for risk.
Lam (2005) states that SMB captures the risk factor related to size, HML captures the
risk factor related to the book-to-market equity and the excess market return, excess
return on market portfolio (Rm-Rf) captures the market factor in stock returns.
There are several empirical studies which test the tree factor model of Fama and French
in emerging market and Turkey. For example, Connor and Sehgal (2001) test the threefactor model in India. They use similar mimicking portfolio formation procedure that
Fama and French 1993 use. But instead of 25, they use six size-BE/ME based portfolios
as dependent variables and market factor, size factor, BE/ME factor as independent
variables in the regression. Their results confirm the Fama-French three-factor model in
Indian equity market. In another study, Gokgoz (2007) test the CAPM and the threefactor model in Turkish Stock Markets for the period of 2001-2006. In his study,
Gokgoz (2007) tests the three-factor model on the basic five indices of the Istanbul
Stock Exchange, and uses daily returns on real estate, securities, industrials, services
and technology indices as dependent variables and size, BE/ME and market factors as
explanatory variables. Gokgoz (2007) finds that the three-factor model is applicable for
Turkish Stock Markets.
3.2. Istanbul Stock Exchange
In contrast to Fama and French who improved and tested the three-factor model in
developed USA markets using 30-year period data, we investigate the viability of the
three-factor model for the Istanbul Stock Exchange (ISE or IMKB) for the period of
May 2004 – April 2011 (investigation period).
12
The Stock Exchange was established on December 26, 1985 and the first transaction
was made on January 3, 1986. Although ISE is seen an emerging market in terms of the
number of listed companies, volume, volatility, but it has grown so fast in terms of
infrastructure in both technological and operational terms and in the context of
international financial centre project, that it is a candidate to be one of leading stock
exchanges in the region and ultimately in the global arena.
The ISE offers local and international investors an opportunity to invest in various
products in an organized, transparent and reliable trading floor with modern
technological capabilities. On all markets of the ISE, transactions are conducted
electronically, and market information is announced in real-time. ISE markets are
classified under four main categories: Equity Markets, Emerging Companies Markets,
Debt Securities Markets, Foreign Securities Markets.
The ISE Equity Market where publicly-held companies from various sectors are traded
offers a liquid, transparent and safe investment floor for all investors. The securities of
companies that do not satisfy the listing requirements of the Exchange, but promise
development and growth potential can be traded on the Emerging Companies Market.
The Debt Securities Market is the only organized market for both fixed income
securities trading and repo-reverse repo transactions. Foreign debt securities which have
been issued by the Treasury of Turkey and listed on the Exchange (“Eurobonds”) are
traded on the Foreign Securities Market. Equity Market transactions are conducted on
the four main markets e.g. the national market, collective products market, second
national market, and the watch-list companies market. In addition, issuers and savers
meet directly on the Primary Market, equity trades above a certain amount with or
without pre-determined buyers are executed in an organized market called Wholesale
Market which is a reliable and transparent environment, and finally Free Trade Platform
(FTP) has been established for trading of the equities which are to be determined by the
Capital Markets Board of Turkey (CMB) (ISE, 2012).
13
On the ISE markets where trust and transparency are the fundamental principles, it is
essential to provide investors with access to company and market data. Investors can
reach any information disclosed by the companies traded on the ISE via ISE web page
(www.ise.org) as well as the Public Disclosure Platform (www.kap.gov.tr). As an
emerging market, the number of listed companies on the ISE has been changing over
time due to de-listing (e.g. because of bankruptcy, failing to satisfy the
listing/registration criteria, etc.) or new IPOs. The total number of companies listed on
the ISE equity market has increased for the last 10 years from 288 to 389. Table 1
shows the number of companies traded on ISE markets between 2000 and 2012.
Table 1 - Number of companies traded on the ISE equity markets
Number of companies traded on the ISE equity markets
National
Market
Second
National
Market*
Watch List
Companies
Market
CPM**
General Total
2000
287
13
15
-
315
2001
279
13
18
-
310
2002
2003
262
264
14
16
12
5
-
288
285
2004
2005
274
282
18
18
5
4
-
297
306
2006
290
18
8
-
322
2007
2008
292
284
17
21
10
12
-
327
326
2009
2010
233
241
22
33
12
11
58
65
325
350
2011
2012/6
237
238
61
71
11
12
64
68
373
389
Year
(*):The New Economy Market merged with the Second National Market on January 03.2011.
Data before January 03, 2011 belonging to the New Economy Market was consolidated with
the Second National Market data.
(**) :The Collective Products Market became operational as of November 13, 2009. The Exchange
Traded Funds, Real Estate Investment Trusts Venture Capital Investment Trusts and Investment Trusts
which are previously classified under National Market classified under Collective Products Market
since November 13, 2009.
14
In accordance to the increase in the total number of firms traded on the ISE equity
markets, the daily average trading volume (both, number of shares / number of shares
times price) has also increased remarkably and reached a daily volume of one billion
USD daily volume during the last five years. Table 2 shows the daily average trading
volume on a yearly basis in ISE equity markets between 2000 and 2012 July.
When compared, it can be easily seen that the daily average trading volume (the number
of traded shares) in 2012 is more than ten times the daily average trading volume in
2000 at Istanbul Stock Exchange equity markets.
Table 2 - Daily average of traded volume (USD) of the ISE equity markets
Year
2000
Daily Average of Traded Volume (USD) on the ISE Equity Markets
Second National
Watch List
National Market
CPM**
Market*
Companies Market
732,207,677
6,464,772
894,913
2001
322,357,140
1,515,944
-
320,257
2002
277,737,525
2,177,915
-
863,822
2003
404,087,461
2,675,677
-
413,104
2004
588,398,908
4,639,648
-
355,396
2005
775,882,348
11,497,118
-
369,559
2006
889,594,024
15,671,558
-
319,941
2007
1,156,386,578
15,271,311
-
520,337
2008
1,002,839,084
8,234,567
-
661,370
2009
1,188,299,176
25,558,007
4,236,618
1,551,978
2010
1,548,013,512
48,455,627
65,399,784
1,870,022
2011
1,451,656,567
64,747,934
125,552,993
387,859
2012/7
1,184,139,451
58,868,203
82,922,504
249,918
* Including New Economy Market
** Excluding warrant and ETFs
Daily average volume in national market decreased to the level of less than USD
300,000,000 in 2002 but increased sharply and reached a level of USD 1.5 billion in
2010. Due to the increase in the number of companies traded on the second national
market, from 13 in 2000 to 71 in 2012, the daily average volume has increased and
reached to the level of approximately USD 60 million.
15
The daily average trading volume between 2000 and 2012 July has been shown on a
yearly basis in table 3.
Table 3 - Daily average of traded volume of the ISE equity markets
Year
2000
Daily Average of Traded Volume (Number of shares) on the ISE equity markets
Second National
Watch List
National Market
CPM**
Market*
Companies Market
44,669,928
280,449
72,731
2001
95,589,835
567,568
-
367,390
2002
132,586,653
1,278,282
-
790,822
2003
236,980,340
2,775,707
-
486,962
2004
274,228,526
4,575,600
-
772,787
2005
311,031,079
6,169,904
-
908,034
2006
356,951,436
6,784,947
-
1,055,533
2007
453,236,623
6,780,252
-
1,668,216
2008
446,855,451
6,799,112
-
1,769,377
2009
780,425,874
25,125,364
4,611,340
4,534,915
2010
713,042,249
38,446,479
57,597,098
3,285,799
2011
618,867,948
50,085,500
95,232,059
848,731
2012/7
549,370,654
45,668,435
80,307,869
554,625
* Including New Economy Market
** Excluding warrant and ETFs
Although it is an emerging market, the ISE is already the member of the world’s leading
organizations such as the World Federation of Exchanges - WFE, the Federation of
Euro-Asian Stock Exchanges - FEAS (ISE is also the founder and President of FEAS
which was established in 1995), the Federation of European Securities Exchanges FESE, the International Securities Services Association - ISSA, the International
Capital Market Association - ICMA, the European Capital Markets Institute - ECMI,
and finally the International Organizations of Securities Commissions, IOSCO.
16
4. DATA
The daily (adjusted) price data of shares to calculate daily returns on portfolios were
taken from DATASTREAM (Datastream 5.1 was developed by Thomson Reuters) and
BE/ME ratios, size value, and all other related data were taken from the Istanbul Stock
Exchange official website (ISE Data, 2012)
4.1. The Sample Stocks and Returns
Fama and French (1992) use only non-financial firms because besides size and BE/ME
ratio they also take leverage as a risk factor, and they postulate that high leverage which
is normal for financial firms is more likely to indicate distress for non-financial firms.
Since the leverage ratio is not tested as a risk factor in this paper, the distinction of
financial and non-financial firms is not made.
First, since the ISE national market represents more than 99% of all ISE equity markets,
only firms traded on the ISE national market were involved in the investigation.
Secondly, to ensure data continuity we excluded firms which came onto the ISE
markets or de-listed from the ISE markets on a date later than December 2003, and we
ended up with 231 stocks. And finally, at the end of each year we eliminated firms
which have negative BE/ME ratios, as Fama French 1993 did. The chosen stocks
represent a considerable portion of the ISE equity markets in terms of capitalisation and
number. Table 5 shows the number of chosen companies and of firms traded ISE Equity
Markets, their capitalisations.
17
Table 4 - The capitalisation of chosen firms and the ISE equity markets
Year
2004
Chosen Firms Traded on National
Market**
Market Capitalisation*
Number
(TRL) (a)
225
85,811,383,604
All Firms Traded on ISE Equity
Markets***
Market Capitalisation*
Number
(TRL) (b)
297
93,397,479,405
Ratio
(%)
(a/b)
91.9
2005
227
111,274,863,995
306
122,669,558,085
90.7
2006
227
208,229,886,736
322
243,894,564,062
85.4
2007
228
220,216,741,476
327
257,167,961,184
85.6
2008
229
218,653,114,056
326
269,223,489,407
81.2
2009
229
163,471,658,229
325
218,957,879,598
74.7
2010
229
297,247,989,418
350
403,222,974,590
73.7
(*) Market Capitalisation is calculated by multiplying the total number of shares and the closing price
at the end of April each year.
(**): The Collective Products Market became operational as of November 13, 2009. The Exchange
Traded Funds, Real Estate Investment Trusts Venture Capital Investment Trusts and Investment
Trusts which are previously classified under National Market, have been classified under Collective
Products Market since November 13, 2009.
(***) : ETFs are excluded
The shares not included in the sample either are very small in terms of market
capitalization or do not have accounting/price information on a continuous basis for
investigation period.
Our return data are based on the daily percentage return from May 2004 to April 2011
consisting of 1762 observations. The daily returns have been calculated by using the
price data obtained from DATASTREAM. The daily return of a stock has been
calculated according to the following formula,
Gi = [ ln(Pi/ Pi-1) ] *100
Where
(7)
Gi : Return for the day i
Pi : The closing price of a stock of the day i
Pi-1 : The closing price of a stock of the day i-1
18
4.2. The Market Proxy
Fama and French (1993) and some other authors use the return on the value-weighted
portfolio of the stocks in the portfolios, however some authors use an market index
representing the big part of the stocks in the portfolios. The latter method is followed in
this paper.
Various indices have been calculated by ISE such as IMKB All index, IMKB 100
index, IMKB 50 index, IMKB 30 index, and City indices. IMKB All index consists of
the stocks of companies traded on all IMKB markets except those in Investment Trusts.
The IMKB 100 index is used as the basic index for IMKB Stock Market, and it consists
of 100 stocks which are selected among the stocks of companies listed on the National
Market (excluding list C companies) and the stocks of real estate investment trusts and
venture capital investment trusts listed on the Corporate Products Market. As the
correlation between IMKB All index and IMKB 100 is 0.9989, it does not make much
difference which index to be used, the IMKB All or IMKB 100.
Since we covered more than 200 companies representing 74%-92% of all ISE markets
and the IMKB All index, which is a value-weighted index, represents ISE markets as a
whole, we used IMKB All index as the market proxy in the model.
4.3. The Risk Free Interest Rate Proxy
The return on Government Debt Securities (GDS) is assumed to be the best proxy for a
risk-free interest rate. 18 indices are calculated in three groups on the ISE Government
Debt Securities (DBS) markets where treasury bills and government bonds are traded.
These are GDS price indices, GDS performance indices, and GDS portfolio
performance indices.
GDS performance indices measure the yield earned by the investor with reference to
both the variation in interest rate and the reduction in the number of days to maturity,
and they are calculated on a daily basis taking into account the reduction in the number
of days to maturity. In this regard, we use the daily return of 3-month GDS performance
index as the risk-free interest rate proxy for investigation period.
19
5. METHODOLOGY
In this paper, we use the time-series regression approach. Daily returns on 12 portfolios
are simply regressed on the returns to a proxy of market portfolio of stocks and
mimicking portfolios, SMB and HML, for size and BE/ME ratio. The structure of the
three-factor model suggested by Fama French is as follows,
E(Ri) - Rf = βi [ E(Rm) - Rf ] + βis (SMB) + βih E(HML)
(8)
To test their three-factor model, we used excess returns on 12 portfolios as dependent
variables, and returns on mimicking portfolios, SMB and HML and excess returns on
the market proxy as explanatory variables in our regressions. The approach used to
create 12 portfolios, SMB and HML is explained in section 5.2. and 5.3.
5.1. The Return Period
Previous work postulates that accounting data are made public within three months of
fiscal yearend (e.g. Basu (1983)) and uses March commencement date. Fama and
French say that 6-month gap between the end of the fiscal year and the portfolio
formation date can be considered as convenient and conservative and they use 6-month
gap.
In Turkey, most of firms listed on the ISE are required to announce their financial
reports within ten weeks of their fiscal yearend, and this period of time is twelve weeks
(at the end of March) for the firms preparing consolidated financial reports. A few firms
don’t comply with this rule, and ask for additional time from the regulatory body
(CMB) in Turkey. Because of this reason and being conservative, the four-month gap
between the end of fiscal year and the return calculation commencement (portfolio
formation) date is considered convenient for Turkey. Hence, to ensure that all
accounting variables are known by investors, we assume that all accounting information
is made public by the end of April, and we use daily returns from the beginning May to
the end of following year April. And each year at the end of April, we reform the
portfolios.
20
5.2. SMB and HML Portfolios
To test the three-factor model by running multiple regressions for ISE, SMB (small
minus big) and HML (high minus low) portfolios were created by using a similar
portfolio constructing approach to that of Fama and French (1993, 1996) as follows. Six
portfolios (S/L, S/M, S/H, B/L, B/M, B/H) were formed on the basis of size and BE/ME
as the intersections of the two ME and the three BE/ME groups as follows. At the end
of April of each year t, selected stocks were separated into two groups (small or big, S
or B) based on whether their size value (ME) is below or above the median size of
selected stocks. Then the stocks were sorted in ascending order into three groups (low,
medium, or high; L, M or H) based on the breakpoints for the bottom 30%, middle 40%,
and top 30% of the values of BE/ME ratio. Each year t, the BE/ME ratios for each stock
were calculated by using year t-1 December book value and market equity value. At the
end of April of each year t, six portfolios (S/L, S/M, S/H, B/L, B/M, B/H) were
established as the intersections of the two size and the three BE/ME groups.
Despite of the fact that Fama and French (1993) used value-weighted returns for six
size-BE/ME portfolios, they admitted (Fama and French 1996) that equal-weighted
returns do better job than value-weighted returns in explaining returns by three-factor
model. Lakonishkok, Shliefer and Vishny (1994), and Munesh and Segal (2001) also
suggest to use equally-weighted portfolios to investigate the relationship between risk
factors and stock returns. Hence, the equal-weighted daily returns on each portfolio
were calculated starting in May of each year to be sure that book value for December of
each year t-1 is known. For example, at the end of April 2004, we first ranked stocks by
using BE/ME ratios of December 2003 in ascending order then separated them into
three groups, Low (30% top), Medium (40% middle) and High (30% bottom). Then, by
using 2004 April-end ME values, we independently sorted stocks in ascending order
then separated them into two groups, small and high, based on whether their ME value
is below or above the median. And finally, we constructed 6 portfolios (S/L, S/M, S/H,
B/L, B/M, B/H) as the intersection of the two size and the three BE/ME groups at each
year, and calculated the equal-weighted daily return on each portfolio between May
2004 and April 2005. We repeat this procedure at every April year during the period
under investigation.
21
SMB is calculated for each day as the difference between the simple average of the
returns on the three small portfolios (S/L, S/M, and S/H) and the average of the returns
on the three big portfolios (B/L, B/M, and B/H). Hence, we ensure that SMB is simply
the difference between the returns on small-stock and big-stock portfolios with identical
BE/ME, and so this difference is mostly free of the effect of BE/ME, focusing on the
different return patterns of small and big stocks. (Fama and French, 1993)
HML is calculated for each day as the difference between the simple average of the
returns on the two high BE/ME portfolios (S/H and B/H) and the simple average of the
returns on the two low BE/ME portfolios (S/L and B/L). HML is just the difference
between the returns on high-BE/ME and low-BE/ME stock portfolios with the identical
size, and so this difference is mostly free of the effect of size, focusing on the different
return patterns of high-BE/ME and low-BE/ME stocks (Fama and French, 1993). For
instance, SMB and HML were calculated as follows for day i,
SMBi = [Ri(S/L)+Ri(S/M)+Ri(S/H)]/3 - [Ri(B/L)+Ri(B/M)+Ri(B/H)]/3,
HMLi = [Ri(B/H)+Ri(S/H)]/2 - [Ri(B/L)+Ri(S/L)]/2,
where Ri( ) is the equal-weighted daily return.
The correlation between the monthly mimicking returns for the size (SML) and BE/ME
(HML) factors was only -0.28, which justified this simple method of forming
mimicking factors.
5.3. Returns on Twelve Portfolios as Independent Variables
We tested the three-factor model by using 12 portfolios’ excess returns as dependent
variables in the regression, instead of using each individual stock’s return. The reason to
use size and BE/ME basis is to determine whether or not the mimicking portfolios HML
and SMB capture the risk factors in returns related to size and book-to-market equity.
We constructed 12 portfolios on the basis of size and book-to-market equity in a similar
way that we used to form the six size-BE/ME portfolios explained earlier. At the end of
April each year, we first sorted stocks by size and then by BE/ME ratio. Size was
22
calculated at the end of April by multiplying the number of shares and closing price of
April. To measure BE/ME ratio, we used market equity and book equity of December
year t-1. We allocated stocks into three groups (small sized, medium sized and big sized
firms) based on the breakpoints for the bottom 30%, middle 40%, and top 30% of the
values of ME. And we splited stocks in to 4 equal groups on basis of the BE/ME ratio.
Twelve portfolios were constructed as the intersection of the three size and four BE/ME
based groups and the equal-weighted daily returns on these portfolios were calculated
from May of year t to April year t+1. The excess returns on these 12 portfolios from
May 2004 to April 2011 (1762 observation for each portfolio) are used as dependent
variables in the time-series regression.
Table 5 shows some statistics for 12 stock portfolios on size and BE/ME ratio.
Comparing portfolios in terms of size and the number of firm in the portfolio, it is
remarkable that the big/high portfolio’s size is bigger than other two high BE/ME
portfolios (Small/High and Medium/High) even though Big/Medium portfolio has just
seven stocks.
Table 5 - Descriptive statistics for 12 stock portfolios
Size
Group
Book-to-market equity (BE/ME) Group
Low
2
3
High
Average of annual averages of firm size
Small
Medium
Big
22,413,396
24,540,711
23,125,461
22,200,605
147,383,086
150,273,947
128,757,839
125,457,867
3,580,787,172
2,262,961,390
1,466,432,973
1,276,216,379
Average of annual percent of market value in portfolio
Small
0.11
0.21
0.24
0.30
Medium
1.57
1.69
1.68
1.80
53.19
23.65
10.92
4.63
Big
Average of annual BE/ME ratios for portfolio
Small
0.38
0.75
1.11
1.89
Medium
0.36
0.73
1.11
2.02
Big
0.39
0.73
1.09
1.67
Average of annual number of firms in portfolio
Small
9
16
19
24
Medium
20
21
24
27
Big
28
20
14
7
23
Table 6 gives the main statistics for the dependent (12 size-BE/ME based stock
portfolios) and explanatory (market proxy, size based factor, SMB and BE/ME based
factor, HML) returns in the time-series regressions. The average returns on the
explanatory portfolios refer to the average risk premiums per unit of risk for the
common risk factors in returns (Fama French 1993).
The 12 stock portfolios yield a range of daily average excess returns, from -0.023% to
0.053%. The Fama-French (1993) evidence that there is an positive relationship
between BE/ME and average return is confirmed by the portfolios. This positive
relationship can be seen in the relationship between low-BE/ME groups and highBE/ME groups portfolios. For example, the average return on small sized-low BE/ME
portfolio, -0.023%, increases to 0.027% for small sized/high BE/ME portfolio; the
average return on medium sized/low BE/ME portfolio, -0.018%, increases to 0.053%
for medium sized/high BE/ME portfolio; and the average return on big sized/low
BE/ME portfolio, -0.007%, increases to 0.038% for big sized/high BE/ME portfolio.
But the other Fama French (1993) evidence that there is a negative relationship between
size and average return is not apparent for all BE/ME group portfolios. But in middle
BE/ME goups, average returns decrease from the small to big-size portfolios. It seems
that the ISE markets display a strong BE/ME effect but not an apparent size effect.
24
Table 6 - Summary statistics of dependent and explanatory variables in the regressions
Excess return on 12 ME-BE/ME based stock portfolios as dependent variables
Size
groups
Book-to-market equity groups
Low
2
3
High
Low
2
3
High
Small
-0.023
0.018
Means
0.032
0.027
1.92
Medium
-0.018
0.014
0.012
0.053
1.78
1.67
1.66
1.60
Big
-0.007
-0.006
0.008
0.038
1.55
1.74
1.73
1.86
Small
0.51
t-statistics for means
0.41
0.80
0.69
Medium
0.43
0.34
0.30
1.39
Big
0.20
0.14
0.18
0.87
Standard deviations
1.80
1.68
1.63
Explanatory variables
Auto corr. for lag
Correlation
Name
RM-Rf
Mean
0.033
Std.
1.784
t (means)
0.785
1
0.063
2
-0.006
12
-0.015
SMB
-0.008
0.563
0.560
0.118
0.005
0.029
HML
0.055
0.569
4.042
0.088
-0.010
0.006
RM-Rf
1
SMB
0.21
HML
-0.02
1
-0.28
1
In the time-series regression, the average risk premiums for the explanatory factors in
returns are just the mean values of the explanatory variables (Fama and French 1993).
The daily average return of RM (the average premium per unit of market B) is 0.033%,
the daily average return of size factor SMB (the average premium for the size-related
factor in returns) is minus 0.008%, and the daily average return of book-to-market
factor HML is 0.055%.
25
6. EMPIRICAL ANALYSIS
We use the standard multivariate (OLS) time-series regression approach to test FamaFrench model. Regression analysis is concerned with describing and evaluating the
relationship between a given variable (dependent) and one or more other variables
(independent). Let (Rit-Rf) denote the excess return on size-BE/ME portfolio i in time t
as dependent variable, and (Rmt-Rf) excess return on the market proxy, SMBt the return
on the size factor portfolio, and HMLt the return on the book-to-market equity factor
portfolio as independent (explanatory) variables. Given these variables, we estimate the
regression as follows:
(Rit-Rf) = ai+ bi (Rmt-Rf)+ si SMBt + hi HMt +εt ; i=1,…, N; and t=1, …, T
(9)
Where bi, si, and hi are respectively the market, size and book-to-market value factor
sensitivities of portfolio i, ai is the abnormal return of portfolio i, and εt is the mean-zero
random return of portfolio j.
In the time-series regression, the coefficients of the explanatory factors (slopes) and R2
values are the most informative evidence as to whether risk factor in the model (RM,
SMB, and HML) explains the common variation in size and BE/ME portfolios’ return.
Before starting the empirical analysis, we should test whether our explanatory variables
are stationary. Therefore, we use Augmented Dickey-Fuller (ADF) test statistics. Since
Dickey-Fuller (ADF) test statistics are less than the test critical values, we reject the null
hypotheses that our explanatory variables (have a unit root) are not stationary at any
level of significance. ADF test statistics are given in table 8.
Table 7 – The root test statistics of explanatory variables
Augmented Dickey-Fuller
Test critical values
Market Proxy
SML
-28,98
-28.08
1% level
5% level
10% level
HML
-18.84
-3,43
-2,86
-2,57
In the following section (6.1.), we first examine the explanatory power of excess return
on market proxy to explain excess portfolio returns; in section 6.2., we test explanatory
power of size and BE/ME factors, respectively SMB and HML, together excluding
26
market factor in the model, and finally in section 6.3., we use all three factors, excess
return of market proxy and mimicking portfolios (SMB and HML) to examine whether
they capture common variation in stock returns.
6.1. Regressions of excess stock portfolios returns on the excess-market proxy
return
Table 7 shows the result of the regression that uses the excess market factor return as
the explanatory variable. In term of R2, our results are consistent with those of Fama
and French 1993. For all portfolios, excess return on market proxy has the explanatory
power on portfolio returns. The values of R2 are less than 0.65 for small-sized firms,
which are the portfolios for which SMB and HML factors will have their greatest
explanatory power. The biggest R2 , 0.87, belongs to big-sized/low BE/ME portfolio.
Table 8 - Regressions of excess returns of 12 portfolios on market factor
(Rt-Rf) = a + b (Rmt-Rf) + εt
Dependent variable: Excess returns on 12 stock portfolios formed on size and BE/ME
Size
groups
Book-to-market equity groups
Low
2
3
High
Low
b
2
3
High
t(b)
Small
0.73
0.75
0.73
0.73
38.45
46.24
51.88
55.31
Medium
0.80
0.79
0.78
0.76
55.61
66.03
65.72
65.26
Big
0.81
0.90
0.88
0.87
109.77
99.79
90.79
64.58
R2
s(e)
Small
0.46
0.55
0.60
0.63
1.41
1.21
1.05
0.99
Medium
0.64
0.71
0.71
0.71
1.07
0.90
0.89
0.87
Big
0.87
0.85
0.82
0.70
0.55
0.67
0.73
1.01
The values of t-statistics show that all slopes, b, in the regressions are statistically
significant at all reasonable significance levels, which means that the market proxy
explains the stock portfolio returns. But we will find that even more variation in stock
returns will be explained when other two factors, SMB and HML, are included in the
model.
27
6.2. Regressions of excess stock portfolios returns on the returns of the SMB and
HML factors
Table 8 shows the result of regressions that use the mimicking returns for size and
BE/ME factors, SMB and HML as explanatory variables. Although R2 is not high,
when we test the coefficients (t-values of s and h) we suggest that SMB and HML
capture the variation in portfolio returns without the factor of market. But they leave a
common variation in stock returns to market proxy as seen in table 7.
Table 9 - Regressions of excess returns of 12 portfolios on SMB and HML factors
(Rt-Rf) = a+ s SMBt + h HMt +εt
Dependent variable: Excess returns on 12 stock portfolios formed on size and BE/ME
Book-to-market equity groups
Size
groups
Low
2
3
High
Low
2
3
s
t(s)
0.82
0.77
0.66
0.68
10.41
10.00
9.08
Small
0.09
0.03
0.14
0.22
1.19
0.42
1.94
Medium
Big
-0.49
-0.58
-0.53
-0.50
7.35
h
7.74
High
9.71
3.10
7.03
6.26
t(h)
Small
-0.70
-0.15
0.08
0.45
9.00
1.96
1.08
6.49
Medium
-0.72
-0.12
0.04
0.38
9.51
1.64
0.53
5.47
Big
-0.47
-0.29
-0.02
0.30
7.06
0.32
3.83
1.64
1.59
Small
0.13
0.07
0.05
0.06
1.79
3.90
s(e)
1.74
Medium
0.06
0.00
0.00
0.02
1.73
1.67
1.66
1.59
Big
0.04
0.04
0.03
0.04
1.51
1.71
1.71
1.82
2
R
The F-statistic in a regression is a test of the hypothesis that all of the coefficients (s and
h) in the regression (excluding the intercept) are zero. If the p-value of F-statistic is less
than the significance level that we are testing, we reject the null hypothesis that all
coefficients jointly are equal to zero. Even if each coefficient in the regression is not
statistically significant, all coefficients can be highly significant. We use the F-statistic
to test the significance of our regression coefficients. In our model with two factors,
SMB and HML, the joint null hypotheses for F-test is H0: s=0 and h=0 for each
portfolio. Except medium/2 and medium/3 (respectively size and BE/ME) portfolios, for
all other portfolios the p-values confirm that we reject the null hypothesis at all
significance level, which means that the coefficients s and h are statistically significant
and can jointly explain the returns on the portfolios.
28
6.3. Regressions of excess stock portfolios returns on the excess return on the
market proxy and returns on SMB and HML factors.
Table 8 shows the result of regression that uses the mimicking returns for size and
BE/ME factors, SMB and HML as explanatory variables. When used together in the
model, the three factors (market factor, size factor and BE/ME factor) capture the
variation in portfolios returns more strongly than when they are used alone.
Comparing the R2-values of table 7 with those of table 9, it is clear that the explanatory
power of the model has increased when we use the market, SMB and HML factor all
together in the regression. This result is consistent with that of Fama French 1993, they
also find that the three factors (market proxy, SMB, and HML) capture strong common
variation in portfolio returns.
Table 10 - Regressions of excess returns of 12 portfolios on market, SMB and HML factors
(Rt-Rf) = a+ b RMt + s SMBt + h HMt +εt
Dependent variable: Excess returns on 12 stock portfolios formed on size and BE/ME
Size
groups
Book-to-market equity groups
Low
2
3
High
Low
b
2
3
High
t(b)
Small
0.82
0.84
0.82
0.82
54.71
64.57
72.59
85.64
Medium
0.84
0.83
0.84
0.81
66.16
73.44
76.39
82.26
Big
0.81
0.90
0.89
0.89
112.75
98.09
90.83
66.93
s
t(s)
Small
1.43
1.39
1.26
1.29
28.92
32.35
33.94
40.70
Medium
0.71
0.65
0.76
0.82
16.98
17.28
21.04
25.08
Big
0.11
0.08
0.13
0.15
4.69
2.78
3.99
3.50
h
t(h)
Small
-0.47
0.09
0.31
0.68
-9.94
2.08
8.54
22.22
Medium
-0.48
0.11
0.27
0.61
-11.88
3.14
7.80
19.21
Big
-0.24
-0.04
0.23
0.55
-10.26
-1.28
7.20
13.05
R
2
s(e)
Small
0.68
0.72
0.76
0.82
1.09
0.95
0.82
0.70
Medium
0.73
0.75
0.77
0.80
0.93
0.83
0.80
0.72
Big
0.88
0.85
0.83
0.73
0.53
0.67
0.71
0.97
29
The t-statistics for the all slopes of market proxy and SMB factor are more than 2.78,
which means that these two explanatory variables are statistically significant at all
significance levels and differ from zero.
And with one exception, the t-statistic for the slope of HML factor is more than 2.08,
which means that HML factor as an explanatory variable in the regression is significant
at least at the %5 significance level. The t-statistic of h slope for big/3 (respectively size
and BE/ME) portfolio is 1.28, which is not statistically significant.
We find that the slopes on SMB factor decrease from smaller-sized portfolios to biggersized portfolios for every BE/ME group, this confirms that the coefficients of SMB are
related to size. Similarly, the slopes on HML factor are related to BE/ME, and in every
size group the slopes on HML factor increase from low-BE/ME portfolios towards
high-BE/ME portfolios.
6.4. Intercepts from regressions of excess returns on stock portfolios
The slopes and R2 values in tables 7 to 9 confirm that the market index returns, SMB
and HML proxy for risk factors in portfolio returns. In time series regression, the
estimate of the risk premium factor (explanatory variables) is just the sample mean of
the explanatory variable, and we should test how well the sample mean of the proxy risk
factors (market index, SMB, HML) explain the cross-section of average returns on the
size-BE/ME portfolios, and whether the regression intercepts are indistinguishable from
zero.
30
Table 11- Intercepts from regressions of the excess return on the 12 stock portfolios
Book-to-market equity groups
Size
groups
a
t(a)
Low
2
3
High
Low
(Rit-Rf) = a + b (Rmt-Rf)+ εt
2
3
High
Small
-0.048
-0.007
0.008
0.003
-1.415
-0.252
0.308
0.108
Medium
-0.045
-0.013
-0.014
0.028
-1.759
-0.603
-0.663
1.356
Big
-0.034
-0.036
-0.022
0.009
-2.612
-2.216
(Rit-Rf) = a+ s SMBt + h HMt +εt
-1.263
0.382
Small
0.021
0.032
0.033
0.007
0.497
0.760
0.836
0.194
Medium
0.022
0.020
0.011
0.034
0.524
0.508
0.278
0.896
Big
0.015
0.006
0.005
0.018
0.401
0.146
0.120
0.410
(Rit-Rf) = a+ b (Rmt-Rf) + s SMBt + h HMt +εt
Small
-0.014
-0.005
-0.002
-0.028
-0.537
-0.204
-0.125
-1.678
Medium
-0.015
-0.016
-0.025
-0.001
-0.658
-0.793
-1.313
-0.064
Big
-0.021
-0.033
-0.034
-0.020
-1.639
-2.055
-1.962
-0.883
The intercepts of regressions refer to the pricing errors of each model. If the model
prices assets correctly, all the regression intercepts should be zero. We can use t-tests to
find whether each intercept is zero for each individual regression (Cochrane, 2001,
p.231).
When we use the t-statistics value for each intercept in the model (i), except 2 out of 12
intercepts, we cannot reject the null hypothesis that intercepts are statistically
indistinguishable from zero at the significance level of 5%. In the big-size group, the
two lowest portfolios’ intercepts
(-0.034 and -0.036) are not
statistically
indistinguishable from zero. We rejected the null hypotheses at any level of
significance, that each intercept is zero for these portfolio groups.
For other alternative model (ii), the t-statistics are less than critical values at any level of
significance, and we cannot reject the null hypotheses that intercept is zero for each
regression result. Therefore, all each intercept is statistically indistinguishable from
zero.
31
In three-factor model of Fama and French, except in the case of four out of the 12
intercepts, we find that other intercepts are statistically indistinguishable from zero at
any reasonable level of significance. But we cannot reject the null hypotheses at the
significance level of 10% for the three lowest BE-ME portfolios in the big-size group
and small-high portfolio.
Besides testing all intercepts individually, we also should test whether all the pricing
errors are jointly equal to zero (Cochrane, 2001, p.231). Gibbons, Ross and Shanken
(1989) suggested GRS test statistic (J) to test whether all pricing errors are zero in
CAPM and the three factor model.
Table 12 – The GRS statistics for all alternative models
Model
Rt-Rf = a + b (Rmt-Rf) + εt
GRS statistic
p - value
2.3672
0.0051
Rt-Rf = a+ s SMBt + h HMt +εt
0.6784
0.7736
Rt-Rf = a+ b (Rmt-Rf) + s SMBt + h HMt +εt
1.3376
0.1902
We test the intercepts according to the following hypothesis:
H0: a1 = a2 = … = a12 = 0
H0: a1, a2, … , a12 ≠ 0
The critical value for J is F distributed with F(N, T-N-k), where N is the number of
assets/portfolio, T is the number of observation, and k is the number of factor.
If Fcritical > J, we can’t reject the null that all intercepts are jointly zero, and therefore
there are pricing errors (abnormal returns) in the model.
For our sample period, the F (12, 1758) is found as 2.1847. When we compare our
GRS-statistics with F critical value, we result in that the null can’t be rejected for threefactor model but for the CAPM. It simply mean that three-factor model eliminate
pricing error in the CAPM for our sample period.
32
7. CONCLUSION
In this paper we examined the three-factor model of Fama and French for Istanbul Stock
Exchange. First, we investigated the size and BE/ME effects in ISE markets. We found
that the Fama-French (1993) evidence, there is a positive relation between BE/ME and
average return, is confirmed by our empirical results. But the evidence that there is a
negative relation between size and average return is not apparent for all BE/ME group
portfolios. But in the middle BE/ME goups, average returns decrease from the small to
big-size portfolios. It seems that ISE markets exhibit a strong BE/ME effect but not an
apparent size effect.
Second, we investigated the relationship between the excess returns on 12 portfolios
based on size and book-to-market value and the return on market proxy, and mimicking
portfolios for size and book-to-market value factors, and finally tested whether the
market, size and book-to-market value were the risk factors which explain the returns in
Turkey Markets. Fama and French (1993) suggest that their three-factor model explain
common variation in returns better than CAPM. Our empirical results confirm that the
SMB and HML portfolios capture the risk factor related to size and BE/ME ratio, and
when these two factors are added to the CAPM, the significance of the model increases.
Therefore, the three- factor model captures common variation in returns better than
CAPM.
The intercepts of regressions refer to the pricing errors of each model. If the model
prices assets correctly, the regression intercept should be zero. When we used t-tests to
find whether each intercept is zero for each individual regression, except three of twelve
portfolios, we could not reject the null hypothesis that the intercept is zero at all
reasonable significance level for CAPM and the three factor model. Therefore, we
found that except three of twelve portfolios, all intercepts of CAPM and the three factor
model are significantly indistinguishable from zero
33
Besides testing all intercepts individually, we also tested whether all the pricing errors
were jointly equal to zero. Gibbons, Ross and Shanken (1989) suggested GRS test
statistic to test whether all pricing errors are zero. Our GRS test results confirm that all
intercepts of the three-factor model are jointly indistinguishable from zero but we found
that CAPM has pricing errors. Therefore, we found that the three factor model
eliminates the pricing error in CAPM for our sample period.
Consequently, our empirical results are reasonably consistent with Fama and French
(1993) findings, and the three-factor model was found viable and superior to CAPM for
ISE markets since the pricing error values (intercept) of the regression Three Factor
Model were found zero.
34
8. REFERENCES
Ball, Ray, 1978, Anomalies in relationships between securities’ yields and yieldsurrogates, Journal of Financial Economics 6:2, pp. 103–126.
Banz, Rolf W., 1981, The relationship between return and market value of common
stocks, Journal of Financial Economics 9, pp. 3-18.
Basu, Sanjoy, 1983, The relationship between earnings yield, market value, and return
for NYSE common stocks: Further evidence, Journal of Financial Economics 12, pp.
129-156.
Beaver, William H., Paul Kettler and Myron Scholes, 1970, The Association between
market determined and accounting determined risk measures, Accounting Review 45,
pp. 654-682.
Bhandari, Laxmi Chand, 1988, Debt/Equity ratio and expected common stock returns:
empirical evidence, Journal of Finance 43:2, pp. 507–28.
Black, Fisher, 1972, Capital market equilibrium with restricted borrowing, journal of
business 45, pp. 444–455.
Black, Fischer, Michael C. Jensen, and Myron Scholes, 1972, The capital asset pricing
model: some empirical tests, Studies in Theory of Capital Markets (Praeger, New York,
NY) 79-121.
Breeden, Douglas, Michael Gibbons, and Robert Litzenberger, 1989, Empirical test of
the consumption-oriented CAPM, Journal of Finance 44, pp.231-262.
Chan, Louis K.C., Yasushi Hamao and Josef Lakonishok, 1991, Fundamentals and
stock returns in Japan, Journal of Finance 46:5, pp. 1739-1789
Cochrane, John H., 2001, Asset Pricing, Princeton University Press, Princeton New
Jersey.
Connor, Gregory, and Sehgal, Sanjay, 2001, Tests of the Fama and French Model in
India, working paper.
De Bondt, W.F.M., and R.H. Thaler, 1985, Does the stock market overreact?, Journal of
Finance 40:3, pp. 793-808.
Douglas, George W., 1968, Risk in the equity markets: an empirical appraisal of market
efficiency, Ann Arbor, Michigan, University Microfilms Inc.
Elton, Edwin J, and Martin J. Gruber, 1997, Modern portfolio theory: 1950 to date,
Journal of Banking and Finance 21, pp.1743-1759.
Elton, Edwin J, Marin J. Gruber, Stephen J. Brown, William N. Goetzman, 2011,
Modern portfolio theory and investment analysis, 8th edition, John Wiley and Sons Inc.
Fama, Eugene F. and Kenneth R. French, 1992, The cross-section of expected stock
returns, Journal of Finance 47:2, pp. 427–465.
Fama, Eugene F. and Kenneth R. French, 1993, Common risk factors in the returns on
stocks and bonds, Journal of Financial Economics 33, pp. 3-56.
Fama, Eugene F. and Kenneth R. French, 1996, Multifactor explanations of asset
pricing anomalies, Journal of Finance 51:1, pp. 55–84
35
Fama, Eugene F., and James D. MacBeth, 1973, Risk, return, and equilibrium:
empirical tests, The Journal of Political Economy 81:3, pp. 607-636.
Gibbons, Michael R., Stephen A. Ross and Jay Shanken, 1989, A test of the efficiency
of a given portfolio, Econometrica 57, 1121-1152.
Gokgoz, Fazil, 2007, Testing the asset pricing models in Turkish Stock Markets: CAPM
vs three factor model, International Journal of Economic Perspectives 1:2, pp. 103-117.
Istanbul
Stock
Exchange
Data
and
information
http://www.ise.org/Data/StocksData.aspx (accessed 12/06/2012)
available
at:
Lakonishok, Josef, Andrei Schleifer and Robert W. Vishny, 1994, Contrarian
investment, extrapolation and risk, Journal of Finance 49, pp. 1541-1578.
Lintner, John, 1965, The valuation of risk assets and the selection of risky investments
in stock portfolios and capital budgets, Review of Economics and Statistics 47:1, pp.
13–37
Markowitz, Harry, 1952, Portfolio selection, Journal of Finance 7:1, pp. 77–99.
Mossin, J., 1966, Equilibrium in a capital asset market, Econometrica 34, pp. 768-783.
Reinganum, Marc R., 1981, Misspecification of capital asset pricing: empirical
anomalies based on earnings’ yield and market value, Journal of Financial Economics
9, pp. 19-46
Rosenberg, Barr, Kenneth Reid, and Ronald Lanstein., 1985, Persuasive evidence of
market inefficiency, Journal of Portfolio Management 11, pp. 9-17.
Ross, Stephen A., 1976, The arbitrage theory of capital asset pricing, Journal of
Economic Theory 13:3, pp. 341–360.
Sharpe, William F. 1964, Capital asset prices: a theory of market equilibrium under
conditions of risk, Journal of Finance 19:3, pp. 425–442.
Sharpe, W.F. and Cooper, G.M., 1972, Risk-return classes of New York Stock
Exchange common stocks, Financial Analysts Journal March-April.
Stattman, Dennis. 1980, Book values and stock returns, The Chicago MBA: A Journal
of Selected Papers 4, pp. 25–45.
Vos, Ed and Pepper, Byron, 1997, The size and book to market effects in New Zealand,
The New Zealand Investment Analyst 18, pp. 35-45
36