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Empirical Researches of the Capital Asset Pricing Model and the Fama-French Three-factor Model on the U.S. Stock Market

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Mälardalens University Västerås, 2013-06-04

The School of Business, Society and Engineering (EST) Division of Economics

Bachelor Thesis in Economics Supervisor: Clas Eriksson

Empirical Researches of the Capital Asset Pricing Model and the

Fama-French Three-factor Model on the U.S. Stock Market

Dingquan Miao

Xin Yi

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Abstract

The aim of this paper is to use the US stock market index to construct different portfolios and test the possible differences in the validity between the capital asset pricing model (CAPM) and the Fama and French three-factor model for the US market. We perform a comprehensive analysis of the two models, and form risk factors that are applied with advanced methods from recent literatures. By using the tool of MS EXCEL 2007, we estimate regression equations and test which factor model can better explain the return of stock. We use a time-series regression approach and different hypotheses tests to check the statistical significance of key parameters (intercepts, market beta, SMB beta, HML beta). We also examine whether the Ordinary Least Squares (OLS) assumptions are fulfilled. Furthermore, we compare the estimated parameters from the different models and check which model has a better explanation on the relationship between risk factors and stock returns. The paper concludes that our testing results show that the Fama and French three-factor model has more explanatory power than the single-factor CAPM, in explaining the variation of the stock returns. We also find that the market beta is the key factor, no matter if we look at the capital asset pricing model or the FF three-factor model.

Key words: expected return and risk, single-factor CAPM, Fama-French three-factor model,

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Acknowledgements

We would like to thank our supervisor Clas Eriksson for his invaluable experience when we were writing this paper. We would also like to express thanks to our lecturers who educated us the knowledge on economics, their lectures have also been a great help in writing this thesis.

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Contents 1.  Introduction ... 6 1.1.  Problem  Formulation ... 6 1.2.  Literature  Review ... 6 1.3.  Aim ... 7 1.4.  Limitations ... 7 1.5.  Methodology ... 8 2.  Model  Analysis ... 9

2.1.  Capital  Asset  Pricing  Model ... 9

2.1.1.  Background  of  the  CAPM ... 9

2.1.2.  Assumptions  and  limitations  of  the  CAPM ... 10

2.1.3.  Theory  of  the  CAPM ... 10

2.1.4.  Methodology  of  estimating  the  CAPM ... 12

2.1.5.  The  CAPM  Testing ... 13

2.2.  Fama  and  French  three-­‐factor  model ... 15

2.2.1.  Background  of  the  FF  three-­‐factor  model ... 15

2.2.2.  Limitations  of  the  FF  three-­‐factor  model ... 17

2.2.3.  Methodology  of  the  FF  three-­‐factor  model ... 17

2.2.4.  The  FF  three-­‐factor  model  Testing ... 21

3.  The  results  (or  estimates) ... 24

4.  Summary  and  conclusions ... 25

5.  References ... 26

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List of Tables

Table 1: Monthly return statistics for the 18 beta portfolios with betas estimated

using monthly returns, a value-weighted estimator ... 28

Table 2: Regressions of excess stock returns (in percent) on the excess stock-market return,  ! !!" − !!": August 2010 to March 2013, 30 months. ... 29

Table 3: Regressions of excess stock returns (in percent) on the mimicking returns for the size (SMB) and book-to-market equity (HML) factors: August 2010 to March 2013, 30 months. ... 29

Table 4: Regressions of excess stock returns (in percent) on the excess market return ! !!" − !!" and the mimicking returns for the size (SMB) and book-to-market equity (HML) factors: August 2010 to March 2013, 30 months. ... 30

Table 5: Intercepts from excess stock return regressions for 16 stock portfolios formed on size and book-to-market equity: August 2010 to March 2013, 30 months. ... 31

Table 6: The Durbin-Watson d Statistic of Fama-French Model ... 31

Table 7: Variance Inflation Factor (VIF) of Fama-French Model ... 33

Table 8: White Test (NR^2 chi-square distribution) of Fama-French Model ... 33

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1. Introduction

1.1. Problem Formulation

As economies have been growing for centuries, people now have more money (real purchasing power) than before. Some people start consuming more leisure, but there are some people who want to invest their money in financial markets. Most people are risk averse, and how to maximize returns and minimize risks becomes an important problem. In response to this, some financial models were developed in the academic world, and these models were aiming to estimate the stocks’ expected returns and evaluate their performances with respect to their risks.

This thesis uses two typical models to analyze data from Standard and Poor’s during 30

months. The data of this paper are chosen from S&P 5001, and all selected stocks are traded in

NYSE (New York Stock Exchange). We made this choice since many investors consider the S&P 500 the best representation of the market as well as the U.S. economy. Moreover, the NYSE is the world’s largest stock exchange by market capitalization.

The problem is that even though some models can explain the expected return with risk in some degree under specific limitations, there is no model that can explain the expected return completely. So our choice of working with two models is not just reinventing the wheel, we are aimed to get different testing results based on our own empirical studies. On the other hand, we can get a better understanding of the basic concepts of the models, and get to know in what circumstances the models can be successfully applied.

1.2. Literature Review

The fundamental usefulness of the capital asset pricing model and its extension model lies in its ability to determine the expected return and risk. The development of the model has a long history. In 1959, Harry Markowitz developed the “mean-variance model”, and this portfolio selection approach had been considered the early form of the CAPM. Markowitz’s model assumes that people are risk averse, and that the efficient portfolios for investors satisfy two conditions: “(1) minimize the variance of portfolio return, given expected return, and (2) maximize expected return, given variance.” Based on the mean-variance model, William Sharpe (1964) and John Lintner (1965) added two key assumptions (Complete Agreement and Borrowing and lending at a risk-free rate) to Markowitz’s model, and in 1972 Fischer Black adjusted one of the assumptions. (In the Black version the beta of market risk premium is                                                                                                                

1 “The Standard & Poor's 500 Composite Stock Price Index is a capitalization-weighted index of 500

stocks intended to be a representative sample of leading companies in leading industries within the U.S. economy. Stocks in the Index are chosen for market size, liquidity, and industry group representation”, U.S. Securities and Exchange Commission,

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positive.) After Sharpe and Lintner create the CAPM model, the model has been extended, and researchers not only focus on individual securities, but also try to work with portfolios in order to improve the precision of estimated betas. For instance, Friend and Blume (1970) and Black, Jensen and Scholes (1972) worked with diversified portfolios to get more precise estimations. Now their method of building portfolios is the common standard in empirical tests, and we have decided to use this method in our empirical studies.

In 1993, Fama and French publish a paper2 on the empirical tests with the CAPM model,

where the problems are mainly related to the security beta. In Fama and French’s research they use empirical evidences to show that the security beta does not suffice to explain expected returns. In their empirical works, they choose stocks from the New York Stock Exchange, and add two factors (market size and book to market equity ratio) in order to have a better explanatory power for the expected return. Based on previous empirical studies, it is meaningful and interesting to test the validity of the CAPM model and the FF three-factor model, and also the impact of three factors on average expect return in the real world stock market during the past recent years.

1.3. Aim

The aim of this thesis is to use empirical estimations to represent the relationship between stocks’ expected returns and their risks. We are in particular interested in the differences in performance between the Capital Asset Pricing model and the FF three-factor model. This comparison will be a way to test the validity of two models.

1.4. Limitations

From Fama and French (1992) we know that the pre-1962 data are tilted toward big historically successful firms, which caused selection bias. And from 1963 to 1990, Beta does not seem to help explain the cross-section of average stock return.

In order to test the practicality of the models, we therefore try to analyze the models in a different time period. Our sample data (72 stocks) are selected from New York Stock Exchange during 2010/10-2013/03 (30 months). Unfortunately, after a first inspection of these raw data we found that two limitations appear in the two models. First, in the CAPM model, we use S&P 500 index as market proxy to estimate the market returns, but in some months we get negative market returns. In other words, we sometimes get a negative risk premium. To tackle this potential problem, we decide to use the Sharpe-Lintner version of CAPM rather than the Sharpe-Lintner-Black version of the CAPM model, which means that a negative risk

premium is not forbidden3.

                                                                                                               

2 Fama, Eugene F.; French, Kenneth R. (1993), ‘Common Risk Factors in the Returns on Stocks and

Bonds’, Journal of Financial Economics 33:1, pp. 3-56.

3 Fama, Eugene F.; French, Kenneth R. (2004), ‘The Capital Asset Pricing Model: Theory and

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Second, in the Fama and French three-factor model, data needs to be divided into different groups, and the breakpoints for size and for BE/ME (book to market equity ratio) are difficult to define. Even though the University of Chicago’s Center for Research in Security Prices (CRSP) database has organized the data based on different standards, it’s not applicable in our empirical study since we choose different samples. We decide to adjust the breakpoints based on our sample data, and the solution for this problem is to rank them in order and divide the data equally.

The methodologies of constructing the portfolios in the two methods are different. In the CAPM model the standard testing method is two-pass regression, and we construct the portfolio based on the stocks’ betas. In the Fama and French three-factor model, on the other hand, we divide the securities into different portfolios, based on the stocks’ market sizes and book value to market value ratios. Thus, we use a different method to construct the portfolio when we study the two models. Since they are two different models, it is reasonable to use two different methods to construct portfolios.

1.5. Methodology

In this thesis, our research investigates the stock price movement on the NYSE, and includes the time period from October 2010 to March 2013, which covers 30 months. We select 72 stocks from different companies as our sample database, and all of them are chosen from 4 different industries listed on the S&P 500. For choosing those stocks we require that they have been listed on the NYSE for at least two years. Furthermore, those stocks whose book-to-market equity ratios are negative will be excluded and replaced by other stocks. As a replacement for using a regular price index, we propose to use the return index. Since the close price has already been adjusted for dividends and splits, we do not need to take dividends and splits into consideration when calculating the returns. We use the following formula to calculating the return index:

!! = !!

− ! !!!

!!!!

Where R denotes the return at time t, !! denotes the adjusted close price at time t and  !

!!!

is the stock price of time t-1.

In the Capital Asset Pricing Model, we use the S&P 500 index as the market proxy to calculate the beta with the risk premium in order to get the expected return on individual securities. In the estimation process, we will use the Ordinary Least Squares method and standard statistical and econometrical tests to test the reliability of two models. The methodology for testing the CAPM is to run two-pass regression. The purpose of the first-pass regression is to estimate each security’s beta by using Ordinary Least Squares Methods, and then we can sort securities based on their betas to form the portfolios. The

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second-pass regression is a cross-sectional regression, and we choose to use hypothesis tests and other econometrical tests in the second-pass regression.

After the study of the CAPM model, we use the same data as in our CAPM model to replicate

the Fama and French three-factor model empirical tests4. First, we will analyze the two added

factors compared with the one-single factor model. The two additional factors are the security market size factor and the BE/ME factor. Second, we add the stock’s beta factor in the regression equation, which gives the FF three-factor model. We then run the linear regression to make comparison with the one-single factor regression and two-added factors regression. To be more specific, we divide the Fama and French three-factor model in three levels, which are the one-factor (market factor) level; the two-factor (size-BE/ME) level; and the three-factor (size-BE/ME with Market factor) level.

Finally, we can, (1) test the validity of the CAPM model and the FF three-factor model in the real stock market; (2) in the Fama and French three-factor model, we compare the explanatory power among the one-factor and two-factor and the three-factor model; (3) based on the test results we judge whether the conclusions are reliable.

In order to analyse the validity of the FF three-factor model, we use the time-series regression approach that the same as the researches of Black, Jensen, and Scholes in 1972. Monthly excess returns on stocks or portfolios are regressed on the excess stock-market return and the mimicking returns for the size (SMB) and book-to-market equity (HML) factors.

2. Model Analysis

In this section, we will explain the theory of the Sharp-Linter CAPM model, to show the whole picture of the model. We will also analyze the Fama and French three-factor model thoroughly, and compare the one-factor (market factor), two-factor (size and BE/ME) and three-factor versions of the Fama and French three-factor model.

2.1. Capital Asset Pricing Model 2.1.1. Background of the CAPM

Sharpe and Lintner first developed the Capital Asset Pricing Model from Harry Markowitz’s mean-variance model. The fundamental use of the mean-variance model is to help investors to select efficient portfolios (maximize expected return and minimize the risk). A few years later, two key assumptions (complete agreement and borrowing and lending at a risk-free rate) were added to the mean-variance model, and this was the original CAPM model. However in                                                                                                                

4 Fama, Eugene F.; French, Kenneth R. (1993), ‘Common Risk Factors in the Returns on Stocks and

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todays academic world, the most frequently used CAPM is the SLB (Sharpe-Lintner-Black CAPM) CAPM model, which was developed by Fischer Black in 1972.

In our thesis, we decide to use the Sharpe and Lintner version of the CAPM model. It is worth to notice that the only difference between this model and the SLB-CAPM model is that Black’s version requires that the risk premium for beta is positive.

2.1.2. Assumptions and limitations of the CAPM

The capital asset pricing model has some important assumptions5, and these assumptions are:

1. All investors are risk averse and they have homogenous expectations regarding risk and return (investors are rational). Investors will evaluate their investment portfolios solely in terms of expected return and standard deviation of return measured over the same single holding period.

2. All investors have a similar time horizon and select according to similar risk-return criteria; in other words, all investors have access to the same investment opportunities. 3. Capital markets are perfect in several senses: all assets are infinitely divisible and the

expected return is normally distributed; there are no transactions costs, short selling restrictions or taxes; information is costless and available to everyone; and all investors can borrow and lend at the risk-free rate; no one can influence the interest rate or the share prices.

4. All investors make the same estimates of individual assets expected returns, standard deviations of return and the correlations among asset returns.

We use S&P 500 index as our market portfolio, but it is not the true market portfolio, and it is practically impossible to include every single security. Our sample data consists only of 72 companies, observed in a period of 30 months. Even though we try to avoid survivorship bias, by using companies from different industries and with different market sizes, there may still be some errors.

2.1.3. Theory of the CAPM

In this section we will present the CAPM model in more detail. The formula of the Sharpe-Lintner (-Black) CAPM model is:

! !! = !!+ !!" !(!!)  − !! , ! = 1, … , !. (1)                                                                                                                

5 Merton, Robert C. (1973), ‘An Intertemporal Capital Asset Pricing Model’, Econometrica 41(5): pp.

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In order to have a better understanding of the model, we can modify equation (1) to its

equivalent equation by moving !!  to the left side of the equation (use excess return as

dependent variable).6 The variables are explained as follows:

 !!:Risk-free rate of return that means zero risk, or return on an asset is uncorrelated

with the market return.7 It hardly exists in the real world, but the U.S Treasure bill is

often used as the risk-free rate. In our paper, we collect daily bank discount rates for 30

days, and then weight them equally to get the monthly treasure bill rate. 8

 !(!!)  − !!:Risk premium, which is the difference between the expected return of

the market and risk-free interest rate. Since CAPM is an equilibrium model, !(!!)  −

!! is the slope of the Security Market Line, and this line is an upward-sloping line (as

long as the expected return on the market is higher than the risk-free rate) with beta of a security as the horizontal axis; SML (Security Market Line) depicts the relationship between the expected return of a security (or portfolio) with its beta. For higher risk, investors require higher returns, and the risk premium is the compensation of the difference between the market return and the risk-free return.

 ! !! : Expected rate of return on security i. The formula explains the relationship

between the stock excess return (! !! − !!) and the security beta (!!"), and in order to

figure out the relation between them we need to calculate the explanatory variable of the equation. Hence it is important to compute the beta:

!!" = !"# !!,!! !! ! ! (2) !!" = ! !!!!! !(!!)  !!! (3) ! !! − !!= !"# !!! !!,!!! !(!!)  − !! , ! = 1, … , ! (4)

 !!": The beta coefficient, a measure of systematic risk (which cannot be eliminated by

diversification); this coefficient also shows the linear relationship between !!and !!9.

“The beta measures the responsiveness of a security to movements in the market

portfolio”.10 For instance, when !!" = 0, ! !! = !!, in other words, if the security’s

beta is zero, the security has no relevant risk with the market portfolio. When !!" = 1,

                                                                                                               

6 ! !

! − !! = !!" !(!!)   − !! , ! = 1, … , !

7 Reinganum, Marc R. (1981), ‘A New Empirical Perspective on the CAPM’, Journal of Financial

and Quantitative Analysis, Vol. 16, no. 4, pp. 440.

8 Data from the

“http://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/TextView.aspx?data= billrates” 9 β !"=!"# !!!!!,!!! = !"# !!,!! !!!! = r!"∙ !!! !!!

10 Hillier, David.; Ross, Stephen A.; Westerfield, Randolph W.; Jaffe Jeffrey F.; Jordan, Bradford D.

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we get ! !! = !(!!), the expected return of a security then equals the expected return of the market portfolio (neutral share).

We can see the implications of the CAPM equation [derived from equation (1)]: ! !! − !! = !(!!)  –!!

!! ∗ !!"!!     (5)

!! = !(!!!)  !!!

! (6)

Here we have defined:

 !!"!!: Systematic risk, which is the risk that cannot be diversified away. It is the risk intrinsic to the entire market or to the entire market segment.

 !!:market price of risk. The investors need a compensation for the excess part of the

risk-free rate, and thus the market price of risk is the excess return for investors as compensation for taking excess risk.

2.1.4. Methodology of estimating the CAPM

In this section, we will introduce econometrical estimations as our main method to test the performance of the CAPM model. By using our sample data we will examine, for instance, whether the econometric results are consistent with the economic theory.

The most common way of testing the CAPM model is to run two-pass regression:

1. First-pass regression: We use equation (2) to calculate the stock’s beta (!!") for the 72

companies and then rank them in order (from low to high) and form the portfolios (4 stocks in a group from up to down) based the value of the beta. At the same time we calculate the expected return of each stock corresponding to the stock beta in order to formulate the regression equation. The expected return of each stock is computed by the average of the individual stock’s 30 months return.

2. Second-pass regression: As we have ranked the beta and formed the portfolio in the previous section, the first thing we need to do in this second-pass regression is to work out the portfolios’ beta and the expected return. We compute the beta of each portfolio by computing the average beta of the four stocks in the portfolio, and we equally weight the expected returns of the four stocks to get the portfolio’s expected return. It is worth to mention that the risk-free rate is computed as the average of the 30 monthly (2010/10-2013/03) U.S Treasury bill. Finally we can use equation (1) to run cross-sectional regression, and observe and analyze its statistical properties.

To conclude, the methodology of the CAPM model is to use the Ordinary Least Squares method to estimate betas and then use cross-sectional (second pass) regression to test the

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hypothesis derived from the CAPM model. The following cross-sectional regression is

estimated11:

!!− !! = !!+ !!!" + !! (7)

Based on the theory of the CAPM model from the equation (1) we know that:

The expected return on a security or portfolio is the sum of the risk-free rate plus the risk inherent in the market portfolio. For a higher risk, investors need a higher return to compensate for the risk. The following should be found:

1. The relationship between excess return (difference between the expected return of a

portfolio and risk-free rate) and the explanatory variables (!!") should be linear; any

nonlinear factors (ex. factor with power or root) should not be significantly different from zero.

2. !!", the security beta, should be the only relevant variable in the CAPM model, and has significant explanatory power in this model.

3. !! should not be significantly different from zero, in other words no intercept should

exist in the CAPM model.

4. !  should be equal to the risk premium !(!!)  − !!, and this can be seen by comparing

the estimated equation (6) with equation (1).

2.1.5. The CAPM Testing

This section reports the results of our empirical tests of the Capital Asset Pricing Model, and studies whether the portfolios with different beta have consistency with their expected returns. After we have collected and organized the data of 72 companies from S&P 500 and formed

18 portfolios, we run the linear regression with the dependent variable !!− !!, and one

independent variable !!, which we refer to as the beta of the portfolio. The beta factor is the

essential factor in the CAPM model and its extension model, and we use the beta of the portfolio to evaluate the responsiveness of the dependent variable to the movement of the market portfolio. The estimated regression equation is:

!!− !!= −0.07104 + 0.015185!!"

     Estimated  standard  errors       0.007783       0.004401      

     ! =       −9.12719        3.45040      !! = 0.390797

! = 18, !" = ! − 1 − 1 = 16, !"#$%&'  ! − !"#$% = 2.120                                                                                                                

11  The stochastic error term is included to take care of all sources of variation in excess return on

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In order to analyze our linear regression equation, we need some statistical measurements.

The t-test is often used to test hypotheses about individual regression slope coefficients12

(and we will also use a t-value to judge the significance of the intercept term). The t-test is logical to use since it considers the standard deviation of the estimated coefficients, and a critical t-value is the value that distinguishes the non-rejection region and rejection region.

The t-value of the !th coefficient can be computed by the formula:

!! = !!

!" !!    (! = 1,2,3, … , !)

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Here we define

!!: the estimated regression coefficient of the !th variable, in our hypothesis test this is the

market risk premium, the estimated coefficient of the market factor (security beta).

SE !! :the estimated standard error (the square root of the estimated variance of the

distribution) of !!. In our hypothesis test, SE !! refers to the estimated standard error of

the estimated coefficient of the market factor.

To use the above measurements, we need to state the null and alternative hypotheses of the estimated intercept and slope coefficient:

Null hypothesis !!: ! = 0 (! can be intercept and slope coefficient)

Alternative hypothesis !!:  ! ≠ 0

In our empirical tests, we find that:

1. The critical t-value for 16 degrees of freedom at a two-tailed 5-percent significance level

is 2.120. We can observe that the t-value of !! is -9.12719, which is significantly

different from zero, and thus we reject the null hypothesis. Even though we can see from

the estimated equation that the estimated coefficient of !! is rather small (-0.07104), this

cannot prove that the !! does not exist in the observed linear regression equation.

2. The critical t-value of the estimated coefficient b is also 2.120. As can be seen from the estimated equation, the t-value of the estimated b (3.45040) is also significantly different from zero. Based on the theory of the model, the estimated coefficient is the estimated value of the market risk premium (0.015185), which is fairly small, compared with historical empirical evidence; and the average return per year on the market portfolio over

the past 80 years was 8.5% above the risk-free rate.14

                                                                                                               

12 Studenmund, A. H. (2010), Using Econometrics: A Practical Guide, International edn of 6th

revised edn, Pearson Education (US), Upper Saddle River: p 128.

13 !

! =!!" !!!!!!

!  (! = 1,2,3, … , !) the border value in our hypothesis is zero, in other words, !!! = 0.

14 Hillier, David.; Ross, Stephen A.; Westerfield, Randolph W.; Jaffe Jeffrey F.; Jordan, Bradford D.

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3. How fit is the regression? !! can be computed as the ratio of the explained sum of

squares to the total sum of squares.(Given a linear model Ordinary Least Squares provides

the largest possible !!). Statistically, !! measures the ‘goodness of fit’15, in other words,

it measures how much of the variation in Y around ! that can be explained by the

regression equation16. The alternative statistic !! is used to adjust for degrees of freedom,

and it is a better measurement of the quality of the fit of an equation. In our regression

equation, the value of !! is 0.390797, which is an acceptable value in our cross-sectional

regression. However, there is an outlier in our observations17, and this outlier makes the

regression line less reliable, even with a sufficient !! value.

To sum up, the estimated intercept coefficient is significantly different from zero in our regression equation (because of the extremely small standard errors), which is not as we expected, and we consider the value of the estimated intercept not small enough to be omitted. The regression tends to be linear and upward sloping, and the functional form of the CAPM

model is supported. However, the !! value is not showing a high level of fit and the outlier

appears in the linear regression. In other words, the ‘goodness of fit’ is not strongly proving the trend of the security market line. It thus seems that beta is not the only factor priced by the market. Also the estimated coefficient for beta has limited impact on the expected return, and the security market line is rather ‘flat’. Because of the above reasons, our empirical tests of the CAPM cannot be strongly proved, and the market factor is lacking explanatory power. Thus, other explanatory variables or functional form adjustment may be needed.

2.2. Fama and French three-factor model

2.2.1. Background of the FF three-factor model

The traditional CAPM is based on some unrealistic assumptions. In order to explain the contradictions of empirical tests of the CAPM, a number of economists extended it to some more complicated asset pricing models. For example, Merton’s (1973) intertemporal capital asset pricing model (ICAPM) is an important extension of the CAPM model. Based on previous contradictions, that the security beta does not provide a complete description of an asset’s risk, we should not be astonished that the differences in expected return are not completely explained by differences in the beta.

Fama (1996) explains that the ICAPM generalizes the logic of the CAPM.18 It implies that if

                                                                                                                15 !!=!"" !""= !""!!"" !"" = 1 − !"" !""

16 Studenmund, A. H. (2010), Using Econometrics: A Practical Guide, International edn of 6th

revised edn, Pearson Education (US), Upper Saddle River: p 49.

17 Outlier is the ‘High Beta’ group (see table 1), without outlier the regression line tends to be

downward sloping with an extremely insignificant value of !!(0.05739).

18 Fama, Eugene F.; French, Kenneth R. (2004), ‘The Capital Asset pricing Model: Theory and

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the CAPM is allowed to borrow and lend at the riskless interest rate and sell risky assets, market clearing prices mean that the market portfolio is multifactor with efficiency. Furthermore, even there is empirical evidence shows that the security beta has explanatory power on expected returns, it is still necessary to have additional betas to help explain the expected returns that include the security beta, which is implied by multifactor efficiency. Subsequently, Fama and French (1993) specify the state variables (size, BE/ME) combine with the security beta together has influence on expected returns. They use a more indirect method than the ICAPM, which could mostly in the idea of Ross’s (1976) arbitrage pricing

theory. They also state that though the size and book-to-market equity are state variables if

they are combined with the security beta, they are not so by themselves.

There is a list of variables which are proposed and tested by empirical researches [by Banz (1981), Bhandari (1988), Basu (1983), Rosenberg, Reid, and Lanstein (1985).]: size (ME, stock price multiplied by the number of shares outstanding), leverage, earnings/price (E/P), and book-to-market equity (ratio of the book value of a firm’s common stock, BE, to its market value, ME). There are several empirical evidences of contradictions of the CAPM model. For example, Banz (1981) finds that market equity, ME, has explanatory power to the cross-section of average returns combined with market beta. Given their security beta estimates, average returns on small stocks (low ME) are too high, and average returns on big stocks (high ME) are too low. Bhandari (1988) states that there is a positive relation between leverage and average return, which is another contradiction of the CAPM model. Moreover, leverage helps the size (ME) together with the market beta to explain the cross-section of average stock returns. Rosenberg, Reid, and Lanstein (1985) find that there is a positive relation between book-to-market equity, BE/ME and the average return on U.S. stocks. Finally, Basu (1983) shows that earnings-price ratios (E/P) also help the size together with the market beta to explain the cross-section of average returns on U.S. stocks.

What make size and book-to-market ratios together become two factors in FF three-factor model? Fama and French (1992a) do research on the joint roles of security beta, size, leverage, book-to-market equity and earnings-price ratio in the cross-section of average stock returns. They find that security beta has little explanation on average returns when it is used alone or in combination with other variables, and when the other four risk factors are used alone all of them have explanatory power. However, when each one of them is combined with others, size (ME) and book-to-market equity (BE/ME) appear to absorb the other two apparent roles of leverage and earnings-price ratio in explaining the average returns. Finally, the conclusion is that two empirically determined variables (ME and BE/ME) do a good performance in explaining the cross-section of average returns on NYSE, AMEX, and NASDAQ stocks for the 1963-1990 period. Based on this evidence, Fama and French (1993, 1996) formed the FF three-factor model for expected return, which can capture much more of the cross-sectional variation in average stock returns and can describe a cross-section of average stock returns better than the CAPM model.

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2.2.2. Limitations of the FF three-factor model

This thesis is aiming to compare the explanatory power between the CAPM model and the FF three-factor model to measure the captured common variation in stock returns in two different models. In order to make the two models more comparative, we have to select a different database than Fama and French (1993) did, and there might be some uncertainty and unexpected results might be obtained. On the one hand, the criterion of choosing the evaluation time period is limited for our data; thus, some results might not be significant. And the breakpoints that we use to divide stocks into different groups for market size and book-to-market equity are not the same as Fama and French (1993) used in their empirical tests. On the other hand, the approach of computing the book-to-market equity ratio in our thesis is different from what Fama and French (1993) did. First, we search the different Price/Book value for every company during time period 2010/10/01 to 2013/03/31 from the

financial website19, i.e. the Price/Book value for Apple is 3.264 (March 28, 2013). Second,

since we can only obtain the daily Price/Book value from the financial website, we calculate the average value for each month and this is regarded as the monthly Price/Book value. The definition of book-to-market equity is the reciprocal of the Price/Book value. As a result, we are able to obtain the book-to-market equities for 72 firms from 2010/10/01 to 2013/03/31.

2.2.3. Methodology of the FF three-factor model

In the Fama and French three-factor model, we analyze and decompose the model in three

levels20: the first level is the FF one-factor regression, which we based on the Fama and

French portfolio constructing method to analyze the excess returns and market beta. It is worth to note that the FF one-factor model is different from the CAPM model in the portfolio construction process. The second level is the FF two-factor regression, and the two factors are the ME and BE/ME ratios. The third level is the FF three-factor model in which we put all the above factors together. For all three levels we use the same method to form the portfolio, in order to compare the explanatory power of three different regressions.

To help understand our testing method, we need to introduce the Fama and French ‘six

portfolios’ method21. Fama and French divide the NYSE, AMEX, and NASDAQ stocks into

three book-to-equity groups, based on the breakpoints for the bottom 30% (Low), middle 40% (Medium), and top 30% (High) of the ranked values of the BE/ME ratios. At the same time, they split the NYSE, AMEX and NASDAQ stocks into two groups (small and big) by using the median of ranked NYSE stocks on the CRSP (Center for Research in Security Prices) database. Based on the above data division, they form six portfolios (S/L, S/M, S/H, B/L, B/M, B/H) from the intersections of the two ME (market size) groups and the three BE/ME                                                                                                                

19 http://ycharts.com/companies/stock_center

20 Fama, Eugene F.; French, Kenneth R. (1992), ‘Common risk factors in the returns on stocks and

bonds’, Journal of Financial Economics 33, p. 19.

21 Fama, Eugene F.; French, Kenneth R. (1992), ‘Common risk factors in the returns on stocks and

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(book to market equity) groups. For instance, the S/H portfolio contains the stocks in the Small-ME group that are also in the High-BE/ME group.

We construct portfolios that are much like the six size-ME/BE portfolios in Fama and French (1993), but with more groups, in order to make a comparison between the CAPM model and the FF three-factor model. We divide all stocks that we have chosen from the NYSE into four book-to-market equity (BE/ME) groups based on the breakpoints for the bottom 25% (Low), the second 25% (2), the third 25% (3), and the top 25% (High) of our 72 companies’ stocks. At the same time, we rank the stocks based on their market equity (ME), and divide them into four groups based on the breakpoints for the bottom 25% (Small), the second 25% (2), the third 25% (3), and the top 25% (Big). Then we can construct sixteen portfolios from the intersections of the four BE/ME groups and the four ME groups (S/L, S/2, S/3, S/H, 2/L, 2/2, 2/3, 2/H, 3/L, 3/2, 3/3, 3/H, B/L, B/2, B/3, B/H). There is a way to increases the power of our tests by adding more observations. We take this strategy by using overlapping portfolios, where new portfolios are constructed each month. We calculate the monthly return for each portfolio by weighting the average monthly return of the stocks from the same portfolio. There is however a risk that this strategy may induce autocorrelation. After the portfolio construction process, we will estimate the time-series regression equations on the one-single factor (security beta), two additional risk common factors (size and book to market ratio) and the FF three-factor model respectively, based on these sixteen portfolios.

In order to run the time-series regressions, we require as variables the risk factor of risk

premium, SMB and HML. The risk premium (!!− !!) is the excess market return in stock

returns. Nevertheless, our proxy for the monthly return on the market portfolio is the monthly return on the S&P 500 index. This is not the same as what Fama and French (1993) did for their market portfolio. The variable SMB (small minus big) mimics the risk factor in returns related to size. In the first place, we need to calculate the simple monthly average returns on the four small-stock portfolios (S/L, S/2, S/3 and S/H) and the simple monthly average returns on the four big-stock portfolios (B/L, B/2, B/3 and B/H) respectively. With these two sets of values we calculate, in the first step, SMB as the difference between the returns on the small-stock portfolio and the big-stock portfolio with the value weighted book-to-market equity. The construction of the HML variable follows the same approach that we used for SMB. These two variables can thus be acquired by using the following formulas:

!"# = ! ! + ! 2 + ! 3 + ! ! − (! ! + ! 2 + ! 3 + ! !)

4

!"# = ! ! + 2 ! + 3 ! + ! ! − (! ! + 2 ! + 3 ! + ! !)

4

When we analyze the results obtained from the time-series regression on the FF three-factor model, by aid of MS Excel 2007, we have sixteen different regression results. Through these results we can observe the estimated coefficients, the t-values of these estimated coefficients

and the overall fit of this regression equation (R!). We can operate different hypotheses tests

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among the stock returns that the regression can explain.

Our hypothesis tests are testing whether an estimated beta is statistically different from zero (two-sided t-tests). In this case, we have the following hypotheses:

Null hypothesis: !!: !! = 0 (!! can be !,  ! and ℎ)

Alternative hypothesis: !!: !! ≠ 0

By aid of MS Excel 2007, we run the regressions and obtain some estimated t-values. The default of the level of significance in MS Excel is 5 percent, which is also recommend in

research projects22, so we decide to keep the level of significance at 5 percent. By applying

the standard decision rule, if the estimated t-value is greater than the critical t-value, then we can conclude that the estimated coefficient is significant, which means this risk factor can explain some of the variation in the stock returns.

We estimate several regressions for different factor models by using MS Excel and collect

different values of !!. We are looking at the overall fit of an estimated factor model, not only

for evaluating the quality of the regression, but also for comparing different factor regressions that have different combinations of explanatory variables. In order to perform a comparison in performance between the FF three-factor regression and different factor regressions (Market factor regression, size and BE/ME factors regression), we apply the F-test that is used in regression analysis to deal with the hypothesis test:

Null hypothesis: !!:  ! = ! = ℎ = 0

Alternative hypothesis: !!:  !!  !" not true

We apply the F-test on each one of the 16 regression results and compare each F-value with a critical F-value (at the 5-percent significance level). If the F-value is greater than the critical F-value that would lead us to reject the null hypothesis and declare that the regression equation is statistically significant at the 5-percent level.

Finally, before we undertake any further tests for FF three-factor model, we should check whether our time-series regressions fulfill the necessary Ordinary Least Squares (OLS) assumptions. In particular, we will test whether the error terms show any signs of heteroskedasticity and if there is a serial correlation among the error terms. We will also check whether the independent variables are multicollinear. If some assumptions of OLS are violated, our OLS estimates (coefficients or standard errors) will be biased. Therefore, it is important that we test these assumptions. In the following, we will introduce the methods that we use to detect these three possible characteristics:

                                                                                                               

22 Lind, Douglas A.; Marchal, William G.; Wathen, Samuel A. (2010), Statistical Techniques in

Business and Economics, International edn of 14th edn, McGraw-Hill/Irwin, 1221 Avenue of the

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(1) The Variance Inflation Factor (VIF)

One way to detect severe multicollinearity is to investigate the variance inflation factors between the independent variables. This is done by looking at the extent to which a given independent variable can be explained by all the other independent variables in the regression equation. The VIF aims to judge whether multicollinearity is so serious that it increases the estimated variance of an estimated coefficient (and the standard error). The particular high VIFs for estimated coefficients indicate that the probability of existing high-level multicollinearity among these explanatory variables will be increased. Calculating the VIF for

a given !!, involves two steps: 1. Constructing the regression equation where !! is the

dependent variable as a function of all the other explanatory variables in the equation. 2.

Given this secondary regression, we can obtain the !!! for this equation, and calculate the

variance inflation factor for an estimated !!.

!"# !! = (!!!!

!!) (9)

(2) The Durbin-Watson d Test

The Durbin-Watson d statistic is aimed to test for first-order serial correlation in the residuals, which we get from the regression result. The hypothesis test that we will use is a one-sided test:

Null hypothesis: !!:  ! ≤ 0 (no positive serial correlation)

Alternative hypothesis: !!:  ! > 0 (positive serial correlation)

The decision rule that we apply is:

If ! < !! Reject !!

If ! > !! Do not reject !!

If !! ≤ ! ≤ !! Inconclusive

Since Excel cannot calculate the d-value in the regression result, we must use Excel to calculate the residuals as a part of output. Having the residuals, we can apply the following formula to compute the d-value:

! = !!(!!!!!!!)!

!!! !

! (10)

(3) The White Test

In our paper, we will utilize the White test to detect possible. The OLS assumption states that the error terms are homoscedastic (have the same variance). If the error terms are heteroskedastic, then the OLS estimated coefficients are unbiased but their standard errors are biased, which would lead to unreliable hypothesis tests. There are three steps to run a Whites

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test23: 1. By aid of Excel, we obtain the residuals of the estimated regression equation. 2. By using the squared residuals as the dependent variable in a secondary regression equation, that includes each X from the original equation, the square of each X and the product of each X with other X’s as independent variables. 3. Testing the overall significance of the secondary regression equation with the chi-square test. In order to apply the chi-square test we have to

formulate the appropriate test statistic, !"!, which is the product of the sample size (N)

times the coefficient of determination (!!) of the secondary regression equation, and NR!

has a chi-square distribution. If !"! is greater than the critical chi-square value, we can

reject the null hypothesis (homoscedasticity). If !"! is less than the critical value of

chi-square distribution, we cannot reject the null hypothesis.

2.2.4. The FF three-factor model Testing

The Fama and French three-factor model is:

! !!" − !!! = !!" !(!!") − !!" + !!"! !"!! + !!!!(!"#!) (11) where:

! !!"  is the expected return of asset i during 30 months from October 2010 to March 2013. !!" is the weighted daily U.S Treasure bill rate from October 2010 to March 2013. !(!!") is the S&P 500 index return during 30 months from October 2010 to March 2013. !(!!") − !!" is the excess return between and market return and risk-free rate during 30

months.

!!!! is the difference between the returns on diversified portfolios of small and big (ME)

stocks (small minus big).

!"#! is the difference between the returns on diversified portfolios of high and low (BE/ME)

stocks (high minus low).

The risk-factor sensitivities !!", !!", and !!! are the slopes at time-series regression at time

t period, and our estimated multivariate regression is:

! !!" − !!" = !! + !!" !(!!") − !!" + !!"! !"#! + !!!! !"#! + !!" (12) The expected return equation of the three-factor model has one implication in the light of

theory, namely that the intercept !! in the time-series regression is zero for all assets i. Fama

and French (1993, 1996) found that the FF three-factor model explains much more of the variation in average return for different portfolios than the CAPM. The variables formed on size, book-to-market equity ratio thus have some explanatory power.

In this section, we will analyze the FF three-factor model in three steps. First, we run

regressions that only use the excess market return, !(!!)  − !!, to explain excess stock

                                                                                                               

23 Studenmund, A. H. (2010), Using Econometrics: A Practical Guide, International edn of 6th

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returns. Second, we use only SMB (mimicking returns for the size factor) and HML (mimicking returns for the book-to-market factors) as explanatory variables. Third, we run the

regression equation in (12). That is, we use all the variables !(!!)  − !!, SMB and HML to

show the impact on the excess stock returns.

We will first explain the market-factor regression. Table 2 shows that the excess return on the

market portfolio of stocks, !(!!)  − !!, the entire 16 portfolio have an ‘aggressive’ !!"

value higher than 1, and the highest estimated portfolio beta explains the stock return up to 1.68, which means the firm is affected by 0.68 times more than the market movement. As we can observe from the table, the average estimated coefficients from low to high BE/ME (book to market) ratio are 1.11375, 1.21966, 1.19387, and 1.49877 respectively; we can find that companies with high BE/ME ratio have more impact (more explanatory power) on companies

with low market BE/ME ratio. Moreover, from the !!value we can conclude that most

observations can be well explained by the CAPM model regression; only one-forth of the

!!values is below the 0.6, and the average of the rest of the !!value is over 0.7.

In our regression model, the observations are 30 months, and therefore, at the 5% significance

level, the two-sided test critical t-value is 2.045. All of the estimated ! coefficients are

significantly different from zero; and most of them are highly significant, except for the small-stock low-book-to-market portfolio and the big-stock high-book-to-market portfolio. However, are these significant t-values caused by biased standard errors of the estimated

coefficients !, due to serial correlation? We choose to use the Durbin-Watson d Statistic to

test the first-order serial correlation. As we can see from Table 6, thirteen out of sixteen are proved to be ‘Do not reject’, and the other three are ‘Inconclusive’. The ‘Do not reject’ result indicates that there is no evidence of positive serial correlation in our one factor Fama-French regression, thus the hypothesis test is reliable. From the above evidence, we can conclude that during our study period, the market factor has some degree of explanatory power to stock excess returns.

Table 3 depicts the empirical results of the FF model that uses the SMB and HML factors. According to the table, the size and book to market factor individually has less power to

explain the portfolios’ returns, compared with the market factor, !(!!)  − !!: from table 2;

the mean of estimated coefficients for SMB and HML are 0.68 and 0.798 respectively. However, the size and book to market factor together have a significant impact on portfolios’ returns. The big stock portfolio has a relatively low explanatory power (compared to the other size factors) to the stock excess return, and the low book-to-market portfolio explains less

variation (among the book to market factors) to the stock excess return. The !! ranges from

0.1 to 0.799, and 9 of 16 portfolios are below 0.3, while 3 are above 0.5. In comparison with the market portfolio, SMB and HML capture less of time-series variation in stock returns. As can be seen from the table, the t-values in the two-factor hypothesis tests are generally lower than the t-values in the market factor tests from table 2. There are some t-values that

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even indicate that the coefficients are not significantly different from zero, and these values are mainly found in big-stock portfolios (there are 8 t-value smaller than the critical t-value). To make the results more trustable, we need to test the result from three perspectives: multicollinearity, serial correlation and heteroskedasticity.

It can be seen from table 7 that the variance inflation factor for both the size factor and the book to market factor is 1.8653 (according to a common of thumb, it should be lower than 5). There is thus no sign of severe multicollinearity between two explanatory variables. From table 6 we observe that only one of the two-factor regressions results in a Durbin-Watson d test that implies that the null hypothesis should be rejected, which implies that positive serial correlation exists only in 1 out of 16 portfolios. This is an acceptable result, and we can conclude that for the two-factor model there is no strong evidence of positive serial correlation. Finally, we use the White test to test for hetereoskedasticity from table 8. It is obvious that all the results are ‘Do not reject’, and thus we cannot reject the null hypothesis of homoscedasticity. To conclude, size and book to market factor leave common variation in stock returns that is captured by the market portfolio in table 2; and the size-BE/ME two-factor show less explanatory power than the market factor regression.

It is interesting to analyze three-factor regression after comparing the results between the market factor and the size-BE/ME regressions. As expected, table 4 shows that the three-factor model captures a robust common variation in stock returns. All the market betas (!) are significantly different from zero; and even the lowest t-value is as high as 2.176 (the largest is 10.921). However, the t-values of the SMB slope coefficients for portfolios are not as significant as the results from table 3, and also the t-values of the HML slope coefficients for portfolios are less significant compared to the table 3 t-values of estimated HML coefficient. Even though the t-values are generally lower in the three-factor regression, it is still interesting to examine the magnitudes of the slope coefficients. The slopes coefficients of the size factor (SMB) for portfolios are related to size. For each book to market quartile, the slopes decrease monotonically from small-size to big-size quartiles. The book-to-market factor (HML) slope coefficients increase monotonically from the lowest book-to-market quartile to the highest book-to-market quartile. We can observe, from a comparison of table 2

to table 4, that the three-factor model has increased !! largely, and that the market factor

alone only produces five values greater than 0.7 (in table 2); in the two-factor regression only one value is greater than 0.7 (in table 3); in the three-factor model there are twelve portfolios

with an !! higher than 0.7.

Since we have added more factor(s) in the three-factor model, and the t-values of SMB and HML tend to be smaller than when we just run them alone, it is important to test for possible mulitcollinearity among the three variables. As can be seen from table 7, the variance inflation factor for all three variables are less than 5, indicating that there is no severe multicollinearity among three factors.

In order to make our hypothesis testing more reliable, we need to test for serial correlation and heteroskedasticity. The results of the test for serial correlation can be seen in table 6. Only

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three results are ‘Inconclusive’ and the remaining thirteen results are ‘Do not reject’. Pure serial correlation is thus not supported by our tests. Table 8 shows the testing results of heteroskedasticity. The results imply that we cannot reject the hypothesis that the error term has a constant variance.

There is something we cannot omit from the regression equation, which is the intercept. As can be seen from the table 5, most of the t-values of estimated intercepts for two-factor regression are significantly different from zero (14 out of 16 intercepts are significantly different from zero), which is not consistent with our expectations from the theory. However,

in three-factor regression hypothesis test, only one t-value24 of the intercept is higher than the

critical t-value, overall we cannot reject the null hypothesis based on the critical t-value in three-factor regression. Thus our three-factor regression seems to have a good explanation on the average stock returns of the time series regression.

To sum up, our FF three-factor hypothesis testing proves the model to be reliable, and all the three factors have some degree of the explanatory power to the portfolios’ excess returns. When we compare the overall significance and fitness between the two-factor regression and the three-factor regression from table 9, we can see that six out of sixteen portfolios’ F test results are ‘Do not reject’ in the two-factor regression, whereas all the F-test results in the three-factor regression are ‘Reject’. ‘Do not reject’ means we cannot reject the null hypothesis, which estimated coefficients are overall indistinguishable from 0. This sign of improvement permits the interpretation that some of our estimated two-factor regression cannot prove the two estimated slope coefficients to be significantly different from zero simultaneously, while the three-factor regressions indeed have a significant overall fit.

3. The results (or estimates)

Consistently with the assumptions of the CAPM model, our results show that the trend of the security market line is a linear upward sloping line. However, the slope is relatively ‘flat’ and the hypothesis test shows that the intercept is significantly different from zero. The CAPM model can thus not be strongly proved. On the other hand, the regression line has captured some degree of the variations, and it is still meaningful to study this model with future data sets.

In the FF-three factor studies, it has been found that there is no evidence of positive serial correlation, no severe multicollineartiy and no reason to reject homoscedasticity among all three regressions. This means that the hypothesis tests prove to be reliable. From our empirical tests, we found that the market factor captures a big share of variation among all three variables, and that the Size-BE/ME factors significantly helped to explain the portfolios’                                                                                                                

24 See table 5, under the (iii) equation 3/H portfolio shows the t-value is 1.8 higher than critical t-value

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excess returns. However, the use of the Size-BE/ME factors alone captures less variation of returns.

To sum up, the results have shown that the FF three-factor model has the strongest explanatory power. At the same time, we found that the market factor plays an indispensable role in the FF model. This puts emphasis on the importance of the development of the CAPM model along the lines suggested by Fama and French (1993).

4. Summary and conclusions

Our main results for this paper are obvious to conclude. Within the scope of our research, stock portfolios were constructed to mimic risk factors related to size book-to-market ratio that capture more common variations in returns, when we run several different types of time-series regressions. This evidence shows that size and book-to-market equity as proxies seem to have a quite good explanation power for sensitivity to common risk factors in stock returns. Furthermore, for the sixteen stock portfolios we examined in order to run the time-series regressions for the FF three-factor model, the intercepts we obtained from these regressions are close to zero. Therefore, the market factor and our two additional risk factors related to size and book-to-market equity seem to do a good job in explaining the cross-section of average stock returns. When we run time-series regressions for the one-single factor (market beta) and the two additional risk factors (ME, BE/ME) respectively, the slopes

and the R! values show that the FF three-factor model has more explanatory power than the

single-factor CAPM model when mimicking portfolios for risk factors related to size and book-to-market equity, which capture shared variation in stock returns.

Through the time-series regression, we found that through the !! values for different

regressions we can conclude that the size and book-to-market factors combined with the market factor can explain the differences in average returns across stocks. However, these two factors alone cannot explain the differences in average stock returns. Thus, the market factor plays an important role no matter whether it is in the CAPM model or the FF three-factor model.

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5. References

Citing sources from Books:

Greene,William H. (2008), Econometric Analysis, Sixth edn, Pearson Education, Upper Saddle River, New Jersey 07458: p. 1020-1022.

Hillier, David; Ross, Stephen A.; Westerfield, Randolph W.; Jaffe Jeffrey F.; Jordan, Bradford D. (2010), Corporate finance, European edn, McGraw-Hill Education (UK) Limited, New York, p. 280.

Rachev, Svetlozar T.; Stefan Mittnik, Frank J.; Fabozzi, Sergio M.; Focardi, Teo Jašić (2007), Financial Econometrics From Basics to Advanced Modeling Techniques, Wiley & Sons, Inc., Hoboken, New Jersey, Regression Applications in Finance: p. 169–191.

Studenmund, A. H. (2010), Using Econometrics: A Practical Guide, International edn of 6th revised edn, Pearson Education (US), Upper Saddle River: p. 35-66, 248-361.

Lind, Douglas A.; Marchal, William G.; Wathen, Samuel A. (2010), Statistical Techniques in Business and Economics, International edn of 14th edn, McGraw-Hill/Irwin, 1221 Avenue of the Americas, New York: p. 330-373.

Citing sources from Journal Articles:

Ross, Stephen A. (1976), ‘The Arbitrage Theory of Capital Asset Pricing’, Journal of Economic Theory 13(3): pp. 341-60.

Black, Fischer. S; Jensen, Michael C; and Scholes, Myron S. (1972), ‘The Capital Asset Pricing Model: Some Empirical Tests’, Michael C. Jensen, edn, Praeger Publishers Inc., Studies in the Theory of Capital Markets.

Merton, Robert C. (1973), ‘An Intertemporal Capital Asset Pricing Model’, Econometrica 41(5): pp. 867-887.

Banz, Rolf W. (1981), ‘The relationship between return and market value of common stocks’, Journal of Financial Economics 9: pp. 3-18.

Reinganum, Marc R. (1981), ‘A New Empirical Perspective on the CAPM’, Journal of Financial and Quantitative Analysis, Vol. 16, no. 4, pp. 439-462.

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. Bhandari, Laxmi C. (1988), ‘Debt/equity ratio and expected common stock returns: Empirical

Figure

Table 1: Monthly return statistics for the 18 beta portfolios with betas estimated using  monthly returns, a value-weighted estimator
Table 3: Regressions of excess stock returns (in percent) on the mimicking returns for the size  (SMB) and book-to-market equity (HML) factors: August 2010 to March 2013, 30 months
Table 4: Regressions of excess stock returns (in percent) on the excess market return
Table 5: Intercepts from excess stock return regressions for 16 stock portfolios  formed on size and book-to-market equity: August 2010 to March 2013, 30
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References

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The portfolios are then rebalanced at the beginning of each month following the procedure explained above and the resulting factor is given by a long position on the winner portfolio

Vårt Diebold Mariano test visar att CAPM och FF3 är statistiskt signifikant olika för alla portföljer, förutom för portföljen B/M där den inte kan säga om det är någon