Excerpt

## Contents

1 Introduction

2 Theoretical Background

2.1 Capital Asset Pricing Model

2.2 Fama-French Three Factor Model

3 Empirical Testing

3.1 Data Collection

3.2 Model Assumptions

3.3 Methodology of estimating the CAPM

3.4 Methodology of estimating the Fama-French Three Factor Model

4 Empirical Data Analysis of the CAPM

4.1 First-pass regression test

4.2 Two-pass regression test

4.3 Non-linearity test

4.4 Non-systematic risk test

5 Empirical Analysis of the Fama-French Three Factor Model

5.1 Portfolio Overview

5.2 Regression Analyses

6 Conclusion

References

Appendices

## List of Tables

1 Two-pass regression results

2 Non-linearity test results

3 Non-systematic risk test results

4 Portfolio Returns

5 Portfolio Classificatiom

6 Fama-French regression results

7 Fama-French regression results using MDAX

## List of Figures

1 Investment opportunities

2 SML estimation using DAX

## 1 Introduction

When it comes to the world of business administration and especially to the world of finance, many different models for making financial decisions have been established during the last decades. There are, however, a few models that have gained popularity and recognition in both, the theoretical world as well as in the practical. One of those models is the Capital Asset Pricing Model (CAPM), developed by different authors such as Lintner (1965), Sharpe (1964) and Treynor (1961), resulting in the Nobel price for Sharpe in 1990.

Even though the CAPM was introduced nearly 50 years ago, it is still widely used for different purposes due to its simplicity (Fama and French, 2004). Companies use that technique to calculate the cost of capital for valuing the project or even to determine the systematic risk of a company. Nevertheless, the CAPM has been a critical part of asset pricing and portfolio theory. It can be applied to compute the appropriate return of an individual asset given its systematic risk and investment manager use it to evaluate the performance of portfolios. Over the last years, many studies have examined the CAPM and tried to explain expected returns in order to deduce investment strategies. Examining current research, one can easily find clear evidence for a partial applicability of the CAPM (Hansson et al., 1998, Lau et al., 1974). However, due to major empirical failings, the CAPM is highly controversial among experts. One reason for this are the assumptions of the model. For instance, Roll(1977) claims the CAPM deals with a market portfolio that contains all types of assets that exist, even though the proxy will always contain fewer assets. As long as the true market portfolio is unknown, tests are not fully reliable.

Moreover, empirical tests show the significance of market anomalies. Factors such as size and market to book ratio seem to play an important role in asset pricing (Roll, 1977 and Banz, 1980). Fama and French (1993) took this as an occasion and examined these market anomalies and later developed the Fama-French-Three-Factor Model, which is a modification of the CAPM. They try to explain the return not only through the systematic risk but also through size and value effects.

The aim of this thesis is to apply the CAPM and the Fama-French model on the German stock market and to see whether the models hold or not. The research methodology in this thesis is mostly an empirical analysis and adopts the approach of Pamane et. al (2014) and Fama and French (1993). However, I will use a different data set and run the test for the CAPM on single stocks rather than on portfolios in order to avoid covariance problems. Firstly, we will calculate the security market line in a two-step regression and then evaluate the influence of non-linear factors and non-systematic risk factors. In addition, the effects of the financial crisis have to be taken into consideration which is why, dummy variables will be used. However, before we interpret the regression results, we make sure that the data are reliable in the first place and correct them if necessary.

For the purpose of assessing the Fama-French model, however, we use a quite different approach and follow the original procedure that was used by Fama and French (1993) themselves. This involves classifying the stocks according to size and value and then building a total of four portfolios. Afterwards, returns are computed and regressed against size and value factors. Even though it is quite common to use, for instance, the DAX or the NASDAQ as proxies, I see the chance of facing endogeneity issues when explaining returns of stocks that are listed in the DAX, which is why I will run all tests for a second time but this time using the MDAX instead of DAX as the market portfolio in order to avoid endogeneity problems.

This thesis finds, regarding the Capital Asset Pricing Model, unsupportive results. I do find evidence that non-linear factors and other risk factors than systematic risk do not play a significant role in asset pricing when using the DAX. Even though the estimated security market line looks indeed familiar, the beta coefficient is not significant, and thus the whole model itself has to be questioned. In the course of using the MDAX, we come to the conclusion that non-linear factors influence the expected return of stocks while other risk factors do not.

For the Fama-French model we obtain results that are consistent with the work of Fama and French (1993). I can provide evidence that investors who hold stocks of small companies are more likely to generate a positive return than investors who hold shares of bigger companies. However, we are not able to prove the existence of a connection between higher stock returns and a higher BE/ME ratio. In fact, when making use of the MDAX we find a reverse relationship between a higher BE/ME ratio and the stocks, in a way that portfolios with a higher BE/ME ratio realize smaller returns.

The thesis is structured as follows. Section two deals with the theory of both models through explaining what the implications of the concepts are, what assumptions have to be made, and what the key assertions are. Consequently, I will talk about the limitations one needs to beware of when evaluating the reliability of the models. Furthermore, it discusses different findings and the predictability of both models and introduces modification of the Fama-French model. Section three then addresses the methodology of how the CAPM and the Fama-French model are tested. It considers the data collection, the validity of the data and the tests we perform. Afterwards section four reviews the results, interprets them and finally contextualizes them. Lastly, section five will conclude what has been discussed and give explanations for possible sources of errors.

## 2 Theoretical Background

### 2.1 Capital Asset Pricing Model

#### 2.1.1 Background and Foundation of the CAPM

The Capital Asset Pricing Model is built on the work of Harry Markowitz (1959). Markowitz introduced in his paper ’Portfolio Selection’ his mean-variance model. This model suggests that investors, assuming they are acting risk-averse, make their decision of selecting a portfolio only by two variables, namely return and variance of their return. This means that investors seek to a) minimize the variance of the portfolio return, given the expected return or to b) maximize expected returns, given variance.

After Markowitz laid the foundation for modern capital asset pricing, J. Tobin (1959) made further improvements by introducing his separation theorem. According to this theorem, portfolio formation may be divided into two stages. The first stage starts with choosing the optimal combination of risky assets. The second stage then deals with the allocation between the portfolio of risky assets and the risk-free assets. The allocation depends on the investor’s attitude towards risk and thus can vary as shown in Figure 1.

illustration not visible in this excerpt

The horizontal axis shows the standard deviation whereas the vertical axis shows the expected return. The line abc portrays every combination of risk and expected return while minimizing the variance (note that these combinations do not include risk-free assets). As a result, it can be inferred that every portfolio that lies below point b does not represent an optimal solution since there will always be a portfolio that offers a higher return, given the same variance. Subsequently adding the possibility of risk-free borrowing and lending results in a straight line. Consider a portfolio invests its money x only in risk-free assets. This is illustrated in point Rf where the variance equals zero and the investors obtain a risk-free return. As an alternative, a portfolio can invest a part of its funds x in risk-free assets and the remaining money 1-x in some portfolio g. These combinations lie on the straight line from Rf through g. This line can be expressed as:

illustration not visible in this excerpt

To obtain an efficient portfolio disposable with a risk-free asset, a line is drawn from Rf to the minimum variance frontier. The point in which the line is tangent to efficient frontier is called tagency portfolio T. J. Tobin (1959) shows that a combination of risk-free assets and a risky portfolio T is necessary to get efficient portfolios.

Subsequently Lintner (1965) and Sharpe (1964) added further key assumptions. The first one is complete agreement. Consequently, investors agree on the joint distribution of asset returns and hence they all combine the same risky tangency portfolio T. Since all investors hold the portfolio T, the tangency portfolio has to be the market portfolio. The only difference among all investors is the proportion each invests in either the market portfolio or in risk-free assets. This difference is caused by different risk profiles. The second assumption, introduced by Sharpe (1964), states that all investors are able to lend or to borrow money at a risk-free rate, which is independent of the amount of money borrowed or lent. One of the main results of the work done by Lintner (1965) and Sharpe (1964) was the security market line (SMB). This line, as stated in equation (4), describes the relationship between the expected return of an asset and its systematic risk.

#### 2.1.2 Assumptions and Limitations of the CAPM

One of the significant weaknesses of the CAPM are the assumptions that have to be made for the model in order to work properly. The most important assumptions are the following (Fama and French, 2004):

1. All investors share the same expectations regarding expected returns and standard de viations of individual assets and correlations among several assets.

2. There is no limitation for risk-free borrowing and lending^{[1]}.

3. As mentioned above, it is assumed that all investors are risk averse and act rationally.

Furthermore, they will evaluate portfolios only based on expected returns and standard deviations.

4. Capital markets are perfect in all senses: Expected returns follow a normal distribution; information is freely disposable to everyone; no information or transactions costs; assets are infinitely divisible.

#### 2.1.3 Theory of the CAPM

Lintner (1965) and Sharpe (1964) introduced their version of the Capital Asset Pricing Model called the Sharpe-Lintner-CAPM^{[2]}.

illustration not visible in this excerpt

In order to obtain a better undestanding we can substract Rf from both sides, resulting in:

illustration not visible in this excerpt

where:

1. Rf: Risk-free rate; The risk-free return on an asset with no risk. This return is uncor related with the market return (M. Reinganum, 1981). In Germany, for instance, federal bonds are often referred as risk-free.

2. E(Ri): Expected return on security i.

3. E(Rm) — Rf: Risk premium; The difference between the market portfolio return Rm and the risk-free rate Rf.

4. j5iM: The beta coefficient of asset i. Beta measures the systematic risk of a security, which means it shows the responsiveness of the return of an asset to a movement in the market return. If fiiM=1, the expected return on security i equals the market return.

Since E(Ri) represents the expected return on any security i and Rf the risk-free rate, I can now compute the excess return on security i. This formula gives me a crucial implication, namely, that two different assets with different betas differ in their expected return. Consequently, two assets with different betas cannot experience the same return. The major concern of this thesis is to test this hypothesis and to see whether expected returns are caused only by a difference in beta coefficients. Therefore, it is crucial to compute the asset’s beta:

illustration not visible in this excerpt

There is, however, another interpretation of beta. The risk of the market portfolio Rm, measured by the variance of its returns o[2](RM), is the weighted average of the covariance of the assets cov(Ri,RM), that are contained in the portfolio. Consequently, piM is the covariance of asset i relativ to the average covariance of assets, which equals the variance of the market portfolio^{[3]}.

#### 2.1.4 Debates surrounding the CAPM

As earlier mentioned, the CAPM and especially its assumptions give rise to severe criticism. There is a debate about the presence of a linear correlation between the systematic risk of an asset and its expected return. Several studies on different stock markets indicate a reasonable explanatory power of the CAPM (Fama and MacBeth, 1973 or Lau and Quay 1974). One of the first evidence was given by Black et al. (1972). Using a cross-sectional regression, they provide evidence for the existence of a linear relationship between the return of an asset and its P as implied by the CAPM.

Furthermore, Fama and MacBeth (1973) endorse the findings by Black et al. (1972). They do not reject the hypothesis that an investor should assume a linear relationship between the expected return and a security’s systematic risk. Only one year later the CAPM was tested on the Japanese stock market by Lau and Quay (1974). The intention is to find out whether the CAPM is applicable on the Stock Exchange in Tokyo (TSE). They use data from 1964 to 1969 and achieve results that are consistent with the predictions made by the CAPM. Investors receive compensation in exchange for bearing a higher risk. After evidence was provided in the 1970s, Hansson and Hordahl (1998) came up with further proofs that support the implications of the CAPM. Examing data from 1977 to 1990 and using a GARCH model, they test the Sharpe-Lintner-Mossin CAPM against several hypotheses on the Swedish stock market. After conducting their tests they do not reject the null hypotheses and hence accept the predictability of the CAPM.

However, since many tests on the CAPM have been conducted, test results turned out to be inconclusive. This caused scientists to believe that additional factors may influence asset pricing. Litzenberger and Ramaswamy (1979) find a strong positive relationship between the expected return of stocks and dividend yields. Moreover, Basu (1977) examine the relationship between returns and price/earnings (P/E) ratios. His results indicate the validity of such a relationship. Portfolios with a low P/E ratio had superior returns than portfolios with a high P/E Ratio, and so being consistent with the claims of proponents of this price-ratio hypothesis. Another anomaly was found by Ball (1978) who suggests to determine expected returns through E/P, BE/ME and debt to equity ratios. The most prominent contradiction, however, was given by Banz (1980). He finds that stocks of small firms, and thus a low market equity (ME^{[4]} ) have returns that are too high and stocks of big firms, and thus a high ME, have returns that are too low.

A further criticism was made by Roll (1977). He cast doubt on the fact that the CAPM implies a perfectly well diversifed market portfolio. According to his opinion, however, such an portfolio does not exist. He critices that studies use a proxy (e.g. S& P 500) for the market portfolio. Since a proxy contains less assets than exist, the proxy might turn out to be inefficient, even though the true market portfolio is mean-variance efficient.

### 2.2 Fama-French Three Factor Model

#### 2.2.1 Background and Theory of the Three Factor Model

With these contradictions in mind, Fama and French (1992) evaluated the influence of size, book-to-market equity, E/P ratios and leverage on expected returns. Their key results are: (a) ( does not help in order to explain the cross-sectional returns and (b) the effect of size and book-to-market equity seems to neutralize the effect of leverage and E/P ratios. Moreover, they find that stock returns are multidimensional. The first dimension is proxied by size, the second one is captured by the ratio of book equity to market equity (BE/ME). Consequently, they developed a model that can be seen as an expansion of the CAPM. This new model, the Fama-French Three Factor Model, includes two additional factors, namely the size of a stock (small minus big, SMB) and the book-to-market equity ratio (high minus low, HML) since solely ( is not sufficient enough to explain crosssectional returns. Fama and French hence suggest the following regression to explain the expected return of stocks:

illustration not visible in this excerpt

where Rit is the return of asset i at time t and Rf the risk-free rate. RMt — Rft shows the excess return of the market and SMB and HML denote the size factor and the value factor, respectively. The intercept of the regression is represented by a and £t is the error term at time i. The coefficients Si and hi measure the sensitivity to each factor. The idea behind this is that SMB is positive in case smaller firms outperform larger firms. Since smaller companies are riskier, due to a weaker diversification, investors require a higher risk premium for small companies’ stocks. This is in accordance with the theory of Hu- berman and Kandel (1987). A high value of HML implies that value stocks outperform growth stocks^{[5]}. Companies with value stocks face a higher risk since the market value has decreased due to numerous negative reasons. Fama and French (1992) show that companies with a high BE/ME ratio tend to have lower earnings ratio and are thus persistently distressed. This is in accordance with the theory of Chan and Chen (1991) who call this phenomenon the "relative distress effect”.

#### 2.2.2 The Momentum Effect

Jegadeesh and Titman (1993) published a paper, in which they introduced an additional anomaly, known as the momentum effect. They find that trading strategies that buy stocks, which turned out to be outperforming in the past and sell shares, which performed poorly in the past, are able to achieve abnormal returns. Using the results of Jegadeesh and Titman (1993), Carhart (1997), later, created a factor proxying the momentum effect and created the Four-Factor model to explain the expected return for a security.

illustration not visible in this excerpt

where MOMt measures the excess return of the stocks that perfomed well minus the value lost of the poorly performed shares.

#### 2.2.3 Debates sorrounding the Fama-French Three Factor Model

The criticism of the Three-Factor model behaves in a similar way to the criticism that was drawn regarding the CAPM. One reason for this was given by Fama and French themselves. In their paper, they state that market equity works as a proxy for size and BE/ME as a proxy for risk. However, they do not give an explanation why it works. Several scientists took this as an occasion to conduct their own studies. We have seen that there are still some anomalies, that cannot be explained the Fama-French Model, such as the momentum effect (Carhart 1997).

Secondly, Kothari et al. (1995) insinuate that Fama French did not consider a survivorship bias when conducting their study. Many companies that had a low market equity and a high book equity to market equity ratio may not have survived and are thus not included in the database. Consequently, Kothari et al. (1995) use a different database and find that BE/ME is weakly related to average stock returns.

Thirdly, F. Black (1993), A. MacKinlay (1995) and Lo and MacKinlay (1988) accuse Fama and French of data-snooping. Since they build their portfolios ex-post it is more likely to find deviations. These deviations, however, are due to chance rather than to any model-inherent characteristics. They claim, that if testing the model on other markets than the US market, ME and BE/ME will no longer hold as a proxy for size and risk. There are, however, studies that support the findings of Fama and French. Chen and Fang (2009) test the multifactor models on the Basin Pacific markets and find that additional factors such as size and value indeed outperform the CAPM. Additionally, they cannot find any evidence to support Carhart’s Four-Factor model which might be due to the use of stocks instead of portfolios as Carhart does.

Further evidence is given by Bello (2008). He compares the prediction quality of the CAPM, the Three- and the Four-Factor model. Not finding any evidence for multi- collinearity, he reports that the Three-Factor model clearly outperforms the CAPM whereas, at the same time, the model of Carhart represents a significant improvement compared with the Fama-French model. Liew and Vassalou (2000) support the findings of Fama and French by showing that the performance of SMB and HML is correlated with future economic growth.

## 3 Empirical Testing

The aim of this thesis and especially of the following section is to examine the explanatory power of the Capital Asset Pricing Model and the Fama-French Three Factor Model. Using empirical estimations I gather evidence for the relationship between stocks’ expected returns and their risk. By applying these two models on the German stock market, I test their accuracy and credibility.

### 3.1 Data Collection

**[...]**

^{[1]} Fischer Black (1972), however, developed a CAPM without the scenario of risk-free borrowing since the assumption is not realistic.

^{[2]} Besides the Sharpe-Lintner-CAPM version, many other scientists have suggested amendments. For instance, Merton, R. (1973) introduced the an intertemporal capital asset pricing model.

^{[3]} Technically, the variance of the market portfolio is

illustration not visible in this excerpt

where xm denotes the weight of asset i.

^{[4]} Market equity equals the number of outstanding shares times the price of a share

^{[5]} Stocks with a high book-to-market ratio are called value stocks and stock with a low book-to-market value are called growth stocks.

- Quote paper
- Fabio Martin (Author), 2018, On the Explanatory Power of the CAPM and Multifactor Models on the German Stock Market, Munich, GRIN Verlag, https://www.grin.com/document/424409

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