An Empirical Test of the "Capital Asset Pricing Modell" (CAPM) on Current Stock Data

Bachelor Thesis, 2020

57 Pages, Grade: 1,3

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Table of Contents

Table of Contents

Table of Equations

Table of Figures

Table of Tables

Table of Abbreviations

1. Introduction

2. Capital Asset Pricing Model: Theory and Development
2.1 The conception of the model
2.2 Modern Portfolio Theory
2.3 Assumptions of the CAPM
2.4 Uses of the CAPM

3. Components of the CAPM
3.1 The Formula of the CAPM
3.2 The risk-free rate
3.3 The return on market
3.4 Market Risk Premium
3.5 The beta factor
3.6 The Security Market Line

4. Critique of the CAPM
4.1 Early testing of the CAPM
4.2 The Critical Assumptions of the CAPM
4.3 Problems in use of the CAPM
4.4 Anomalies within Capital Asset Models

5. Alternatives to the CAPM
5.1 The Fama-French three-factor model
5.2 The Carhart four-factor model
5.3 Substitution potential of the three-factor model and four-factor model for the German market
5.4 The Arbitrage Pricing Theory

6. Empirical Validation of the CAPM and alternative models
6.1 Datasets used
6.2 The 2019 Dataset
6.3 Stehle Dataset
6.4 French Dataset

7. Methodology

8. Results and interpretation
8.1 Results - CAPM
8.2 Results - Fama three-factor model
8.3 Results - Carhart four-factor model
8.5 Negative risk-free rate
8.6 Difference of Runtime
8.7 Impact of the risk-free rate

9. Interpretation

10. Limitations

11. Conclusion


Table of Appendices:

Table of Equations

Equation 1: The CAPM

Equation 2: The beta factor

Equation 3: The 3FM

Equation 4: The SMB and HML factors

Equation 5: The WML factor

Equation 6: The 4FM

Equation 7: The APT

Table of Figures

Figure 1: The efficient frontier

Figure 2: The effect of Beta on Stock movement

Figure 3: The Security Market Line

Table of Tables

Table 1: Comparison of Datasets used on the CAPM

Table 2: Performance of the 3FM compared to the CAPM

Table 3: Performance of the 4FM compared to the CAPM

Table 4: The 2019 set with and without a negative risk-free rate

Table 5: Change in results due to a shortened runtime

Table 6: Change in results due to a different risk-free rate

Table of Abbreviations

3FM - Fama-French three factor model

4FM - Carhart Four factor model

Adj. - adjusted

ADS - Adidas

ALV - Allianz

APT - Arbitrage Pricing Theory

ARCH Model - Autoregressive conditional heteorskedasticy model

B/M - Book value to Market value


BAYN - Bayer

BEI - Beiersdorf


CAPM - Capital Asset Pricing Model

CEO - Chief Executive Officer

CFO - Chief Financial Officer

CON - Continental

DAI - Daimler

DAX - Deutscher Aktienindex (German Stock Index)

DB1 - Deutsche Börse

DBK - Deutsche Bank

DPW - Deutsche Post

DTE - Deutsche Telekom

E/P effect - earnings to price effect


EURIBOR - Euro Interbank offered rate

FRE - Fresenius

FRME - Fresenius Medical Care

FSE - Frankfurt Stock Exchange

HEI - Heidelberg Cement

HEN3 - Henkel

HML - High minus low

IDW - Insitute der Wirtschaftsprüfer in Deutschland (Institute of Auditors in Germany)

IFX - Infineon

LHA - Lufthansa

LIN - Linde

MBA - Master of Business Administration

MPT - Modern Portfolio Theory

MRK - Merk

MRP - Market Risk Premium

MUV2 - Münchner Rückversicherungs Gesellschaft

OLS regression - Ordinary least square regression

r - return on investment

rf - risk-free rate

rm - return on market



SIE - Siemens

SMB - Small minus big

SML - security market line

U.S. - United States of America VOW3 - VW

WML - Winners minus losers

1. Introduction

The Capital Asset Pricing Model (CAPM) is an important and well researched model in the finance sector.1 It provides an easy formula with only one factor to estimate future returns on stocks or portfolios based on the risk level of securities. The model works by explaining the rate of return on a security as a function of the market portfolio, which is the sum of all securities within a specified market.2

Despite being developed in 1964, the Capital Asset Pricing Model was still used by 73% of CFOs in the U.S. and Canada in 2001 to determine their cost of equity.3 However, the model has been enhanced multiple times over the years to include more than one factor. The most commonly used ones are the Fama and French three-factor model and the Carhart four-factor model. These models take additional risk factors into account for their calculation, thus greatly reducing the simplicity of the formula of the CAPM. This can be remedied by using publicly available data that includes the factors necessary to calculate the future return on investment instead of calculating them individually as is usually done with the CAPM.

An important aspect of the models is that while they can theoretically be applied globally, the factors used are country specific as they depend on each country's system and market portfolio. This paper aims to study the German market, the viability of the models there and which conditions and applications lead to the best results. It will thus use data from German companies and focus on the factors that prevail in German markets. The goal of this study is thus to determine the best available asset pricing model in Germany and whether the use of pre-existing datasets, with the factors already calculated, brings results as accurate as a custom dataset. This is relevant in Germany as the CAPM is still the most commonly used way to compute the cost of equity with 34% of companies using it. Another 16% of companies are using asset pricing models with additional risk factors.4 To determine the answer to this, this study will look into the aforementioned three most commonly used models: the CAPM, the Fama and French three-factor model and the Carhart four-factor model. After explaining the background and functioning of the CAPM, this study will show the flaws within the model and how these flaws led to extensions of the CAPM. Each model will then be statistically analyzed with three distinct sets of data. Two of these are publicly available, while the last has been calculated for this study. Lastly, to understand how the difference in data used can influence the results from asset pricing models, the runtime and underlying factor of datasets will be modified, re-analyzed and compared to the initial results.

2. Capital Asset Pricing Model: Theory and Development

2.1 The conception of the model

The Capital Asset Pricing Model was first conceived by William F. Sharpe in 1964 to determine the return of portfolios under the condition of risk, under the assumption of a riskless asset class.5 However, he was not the only academic coming up with that: J Lintner and J Mossin developed the model independently from each other as well and collaborated to further the theory.6 The base for the CAPM is the modern portfolio theory (MPT) of Markowitz, here Markowitz describes a method to find the maximum return for a given risk or the minimal risk for a given return through optimal diversification of the portfolio. 7 This is done by analyzing the standard deviation of the stocks and bonds to find the expected return for the next period.

2.2 Modern Portfolio Theory

The basis of Markowitz's model is the mean-variance of a portfolio as he derived that the variance of return is a meaningful measure of risk for a given portfolio. He assumes an investor would choose a portfolio while minimizing variance and maximizing return. These investors in the model are risk-averse and only care about the mean return of their investment for a one-year period. Further, the risk of a given portfolio can be reduced by effectively diversifying the securities. 8

The MPT is based on 5 assumptions for the investor’s behavior:

1. Investments are considered by the alternatives of probability distributions of the expected return over the period t-1
2. The investors have a concave utility function and minimize one-period expected utility, which means as mentioned, that they are risk-averse.
3. Risk is estimated by the possible variability of expected returns.
4. Investors are rational and make decisions based only on variance and expected return of securities
5. For a given risk the investors will take the decision with the highest expected return.

Under these assumptions, the portfolio is called efficient if there is no other portfolio that has a higher expected return with a lower or identical risk. 9 If the sum of minimum variance frontiers for each given risk is graphed, they return a line, which is a parabola (figure 1). While the minimum variance frontier is a parabola, only the portfolios lying between point a and point b are efficient and therefore called the efficient frontier as the portfolios between b and c (excluding point b) have a corresponding portfolio with the same risk but a higher expected return.

Figure 1: The efficient frontier

Abbildung in dieser Leseprobe nicht enthalten

Source: Fama, E. F.; French, K. R. (2004) P. 27.

2.3 Assumptions of the CAPM

As with any model the CAPM has underlying assumptions these are partially the same as in the MPT, but it also introduces new ones. It allows evaluations of a trade-off between risk and return for securities and portfolios alike. This is possible through the beta factor, which is the measure of risk in the CAPM.10 The beta factor changes the focus from the total volatility of the securities to only the non-diversifiable risk (or systematic risk) compared to the risk of the market portfolio, which is the market in its entirety. Through this, it is possible to split the expected return for a security into the risk-free rate and the risk premium. 11

Similar to the MPT, investors are also risk-averse and rational in the CAPM. Therefore, all decisions, whether it concerns a single stock or portfolios are made by analyzing statistical data. Relevant data are for example the standard deviation of the return, the rate of return of the asset, and the covariance of the rate of return and other stocks and portfolios.12 This means that every chosen portfolio should lie on the efficient frontier, which is according to the MPT, with the highest possible rate of return for a given risk, and therefore minimizes risk while maximizing the rate of return.

Additionally, to the rational behavior of the investors, the assumption is made that all investors share the same set of expectations. These homogenous expectations mean that all participants share the same knowledge of probability distributions of expected returns. As a consequence of this, all market participants hold the same set of risky assets and vary the proportion of the risk-free asset to reach their accepted level of risk for their portfolio. 13

The arguably most important assumption of the CAPM is the existence of a risk-free rate (rf).14 The investors can lend and borrow using this rate with no restrictions on leverage. This allows the investors to borrow money at a lower rate and invest it into a riskier portfolio with a higher potential rate of return.

Further assumptions for the CAPM are:15

1. All investors have efficient portfolios, i.e. the portfolios are on the efficient frontiers.
2. The investors operate in a perfect market, i.e. there are no transaction costs, no taxes, and lastly the divisibility of factors16.
3. There is no inflation or change of interest.
4. All investors operate within the same period.

2.4 Uses of the CAPM

As mentioned, the CAPM is one of the most influential models in the financial world, this is because of its ease of use and widespread applications. After explaining the basic principles behind the CAPM it is time to present how this model is used. Walter and

Brown list several of these practical applications:17

1. Use in Capital Budgeting:

One of the primary uses of the CAPM is to determine the cost of equity capital for a company. Graham and Harvey found in a 2001 study that determined that 73.5% out of 392 U.S. companies use the CAPM to determine the cost of equity capital, making it a tool in capital budgeting. This is more the case in larger companies, while smaller companies are more likely to use a payback method. There is another increase in likelihood of using the CAPM and multi-beta CAPMs in CEOs with MBAs compared to non-MBA CEOs.18

2. Price setting in regulatory agencies:

The CAPM is now an industry standard when it comes to decisions on the cost of capital for regulatory decisions, as well as the price determination. This exists to regulate squeeze-outs, mergers, control agreements, and delistings in Germany.19 This can be observed for the UK20, the U.S.21

3. Tests on market efficiency:

Tests on market efficiency require a joint hypothesis test.22 Fama states this is necessary because an equilibrium model can only give an answer to a proper price when an asset pricing model defines the proper price. He further states that the Capital Asset Pricing Model passed the test for usefulness in a scientific context and changed the understanding of the behavior of returns for different returns and different time scales.23

4. Further use of Market Risk Premium (MRP):

The Market Risk Premium is an essential part of the CAPM that defines the base risk of investing within a market. In a 2011 study, Fernandez et al. found that 74% out of 6014 participants in 56 countries, predominantly analysts and companies but also professors, used the MRP in their work.24

3. Components of the CAPM

3.1 The Formula of the CAPM

The data to determine these components can originate from different sources. These sources or datasets can either be made by the company itself with its own information or be taken from publicly available data. The generally used option is to determine the variables for the CAPM to calculate it yourself, as they are explained in this chapter. This data can then be introduced in the formula directly or be put into a regression analysis. Through this the quality and fit of the data can be determined, which is relevant when the results need to be within a margin of error.

Now that the assumptions and uses have been established, to understand how the Capital Asset Pricing Model works the formula (equation 1) is explained and each individual component will be described. After it has been broken down, each of them will be examined more thoroughly.

Equation 1: The CAPM

Abbildung in dieser Leseprobe nicht enthalten

Source: adapted from Pankoke, T.; Petersmeier, K. (2009); P.113.

The model's main components are the risk-free asset (rf), the beta ) and the return on market (rm). The subtraction of the risk-free rate from the return on market is called the Market Risk Premium (rm — Tf) .25

3.2 The risk-free rate

As previously discussed, the risk-free asset is only an assumption, because there is no risk-free asset class to be found in the real market. However, to approximate the risk-free interest rate there are several methods; one of those possibilities is the utilization of government bonds. These bonds need to originate from a country with a stable and strong economy and a stable political structure, to approximate the absence of risk.26 The 10 year bonds are preferred, as they are less affected by inflation and have higher liquidity then 30-year bonds. However, using the 10-year bond has the flaw of insufficient runtimes for companies with an indefinite life cycle. To combat this, Svensson formulated the Svensson-method to calculate the interest curves for future interest.27 Another approximation is the use of a one-month rate like the Euribor.

In perfect markets, it is theoretically possible to diversify the portfolio to eliminate all risks specific to an individual stock. This is called the non-systematic risk. However, risks concerning the market are non-diversifiable, because all stocks are within the market and change with it; this non-diversifiable risk is called systematic risk.28 To compensate the investors for partaking in an investment with risk, they receive a premium, this is called the Market Risk Premium. This Market Risk Premium or MRP is determined by the spread of risk-free rate and return on market29, shown in the formula as (rm — ry).

3.3 The return on market

Since a way to estimate the risk-free rate has been determined, it is now necessary to define the return on market. The most common approach is the ex-post approach, which uses the historical data of stock returns to estimate the future performance of the market.30 The return on market is determined via a proxy for the market, as there is no way to determine the actual change in worth of all existing assets within a specified market. These would include the prices on goods and services, as well as prices for land and financial securities. These proxies are usually portfolios with a large number of stocks or indices of stocks, for example the DAX or CDAX in Germany, the EURO STOXX50 for Europe, or the S&P500 in the United States. 31 The opposite of the ex-post approach is the ex-ante approach, which tries to estimate the future performance of the market by extrapolating the data from the latest known values. However, this approach is seldom used as it lacks objectivity and explanation because it is determined by a single analyst and the beliefs he holds.32

3.4 Market Risk Premium

As explained at the beginning of this chapter, the Market Risk Premium in the CAPM is a function of both the risk-free rate and the return on market. The MRP is the measure of systematic risk in a market and is subject to change, hence it needs to be calculated for each observation within a study. Usually, this is done by using rf and rm as determined in the steps prior, in countries with reliable financial information on the return on market and risk-free assets, the MRP can be determined over the past 50 years.33

The difficulty of determining the correct MRP lies within the problems of these approaches, as future returns can be modeled by past information, but past performance is no guarantee for future results. One main reason for the shortcoming of the ex-post approach, as Mayfield describes it, is that the market is continually going through phases of high volatility and lower volatility. The average rate of return during times of low volatility is greater than the observed rate of return, while during phases of high volatility the rate of return is lower.34 Due to this fact and because of the falling transaction costs and improved diversification in the future, the IDW recommends reducing the MRP by 1 to 2 percent.35

3.5 The beta factor

Now that the risk-free rate and market-risk premium have been explained, the centerpiece of the CAPM is next: the beta factor.

The beta factor is determined by the deviation of the portfolio or stock from the market; therefore, it is an indicator of the volatility for that stock and through that it displays the systemic risk.36 The beta factor is defined as the covariance of the stock (or portfolio) and the market divided by the variance of the market (equation 2).

Equation 2: The beta factor

Abbildung in dieser Leseprobe nicht enthalten

Source: adapted from Ziemer, F (2018); P. 218.

Abbildung in dieser Leseprobe nicht enthalten

The beta factor indicates how much a stock changes in comparison to the market, therefore stocks with a beta larger than 1 change more than the market. A higher beta means more risk as a change in the market is multiplied by beta to represent the change of the stock. Vice versa a stock with a beta smaller than 1 but greater than 0 changes less than the market (figure 2), therefore it can be used by investors to decrease the beta of the portfolio at the cost of a lower return. Negative betas are atypical but do exist, as logic dictates stocks with negative betas move contrary to the market and can be used to reduce the risk of a portfolio even more.37 The beta factor of stocks and portfolios has no technical range limit, but extremes like negative betas or high values are rare.

Figure 2: The effect of Beta on Stock movement

Abbildung in dieser Leseprobe nicht enthalten

Source: adapted from Ziemer, F. (2018); P. 142.

The beta commonly used is the so-called levered beta, as the financial structure of the company plays a role in how the company reacts compared to the market. This beta is only available for publicly listed companies - to estimate the beta for unlisted companies a peer group is formed, to account for a difference in financial structure the levered betas need to be unlevered. After they have been unlevered the median of the peer group is formed and re-levered to find the beta for the unlisted company.38

3.6 The Security Market Line

All the factors mentioned before can be represented in a graphical form. This graph is called the Security Market Line or SML (figure 3). In the graph, the x-axis represents the risk, which is represented by beta, while the y-axis represents the expected return of CAPM. In the CAPM, this Security Market Line applies to any security or portfolio in equilibrium.39

Figure 3: The Security Market Line

Abbildung in dieser Leseprobe nicht enthalten

Source: adapted from Ziemer, F. (2018); P. 129.

As explained previously, the expected return on a security changes directly in correlation to the risk, as the investor has to be compensated for taking the risk of that investment. This yields a graph intersecting the y-axis at a beta of 0 i.e. a risk-free asset, with the return of the risk-free rate and increasing linearly with risk, determined by the market risk premium. The other noteworthy point on this graph lies at a beta of 1, where a portfolio exists that mirrors the market perfectly or that is the market portfolio.

When each security and portfolio is plotted, the SML is the line that has the best fit considering return and risk. However, in the real market, not all securities lie among the SML as they can be either undervalued (lying over the SML), when they perform better than expected, or overvalued (lying below the SML) because they offer a smaller return than predicted. These stocks are in disequilibrium - this should not be possible in an efficient market in equilibrium. Research shows that as soon as stocks are undervalued investors will buy them more, increasing the demand and therefore the price, the opposite is also true as overvalued stock will be sold, increasing supply and lowering the price. 40 Now that an understanding of the components of the CAPM has been established as well as its assumptions. The critical nature of the CAPM is explored, as it has been a point of controversy among scholars for almost as long as it has been used.

4. Critique of the CAPM

4.1 Early testing of the CAPM

As with any scientific theory, the CAPM also had to be subjected to testing to verify its predictions. Among the first to do this were Black et al., who used monthly return data on beta specific portfolios to confirm the linear relationship between the return on portfolios and the beta factor. 41 They used portfolios to diversify specific company return and arrive at a more precise beta for the portfolio. Fama and MacBeth also reported a positive linear relationship between beta and the return on portfolios.42 These early tests led many to believe that the CAPM was a promising model due to these positive results. However, further testing has uncovered shortcomings and anomalies in the model, nuancing these results.

4.2 The Critical Assumptions of the CAPM

A predictive model is only as good as the veracity of its predictions, but after further testing the CAPM's results began to fall short. To find an explanation for this, it was necessary to analyze each assumption and find out how the result changes upon modifying variables - this is possible as the CAPM is a rather simplistic model.

Beginning with the assumption of leverage restraint, as with every model that uses the same interest rate for lending and investing this is a strong point of contempt. As there is only one efficient portfolio in the market according to the MPT on which the CAPM is based, the risk aversion is being determined by the levering or de-levering of this portfolio. However, as a single interest rate for lending and borrowing does not exist, just levering the portfolio is not possible to increase the return for most investors. Therefore, investors invest in stocks with a higher beta in accordance with their risk aversion; in turn, stocks with a higher beta are overvalued while low beta stocks are undervalued. This in turn lowers the slope of the Security Market Line.43

Another reason for the insufficiency of the beta is the assumption of utility, the sole focus on the mean and variance of the assets. Fama and French conclude that investors also care about the other factors like the covariance between portfolio return and labor income and opportunities for future investment, therefore the beta cannot represent the risk in its entirety. 44 Thus, changes in expected returns are not explained through the differences in the beta factor on its own.

Furthermore, the assumptions regarding the perfect market with perfect knowledge and rational investors are faulty as well. Each investor has a different set of knowledge influenced by experience, research, and other investors or the market. Therefore, the expectations differ widely, which changes the entire behavior of the market and introduces more reasons for the model to be incorrect, as well as enabling the market to be in disequilibrium for some time before it returns to an equilibrium.45

4.3 Problems in use of the CAPM

However, there are other general problems within the CAPM additionally to the assumptions being unrealistic.

The beta factor requires the return on market in order to be calculated correctly. As explained previously, it is common to use an index as a proxy - however, this does not suffice as the stock market is only a part of the market and does not represent the market in its entirety as not all companies nor all resources are traded on the stock market. This leads to a significant bias of results in CAPM and especially in allocating resources within a company. This bias thus leads to a misallocation of funds46.

Another point of contempt is the behavior of betas over time and the size of the portfolio. As the CAPM is an ex-post model, the beta can only be estimated with past information and is therefore not a perfect representation of the future. While the predictability of betas increases with the size of the portfolio, the betas for single assets have the tendency to be unpredictable. Another problem is the length of the forecast period, while forecasts of 13 to 26 weeks show an increase in predictability only forecasts with more than 52 weeks have usable and repeatable outcomes. Furthermore, the overall tendency of the betas is to regress toward the mean, especially in high-risk portfolios where this is even more pronounced than in low-risk portfolios.47

4.4 Anomalies within Capital Asset Models

Another point of criticism against the CAPM are the anomalies that appear within the market which it does not take into account. All of those anomalies should not exist in an efficient market, which is a requisite for the CAPM.

The size-effect describes how on average smaller firms have a significantly larger risk- adjusted return than bigger firms. One explanation is that most investors buy stock in companies they have information on and information on companies is related to their size, therefore stocks of small companies are less desired. Thus, this leads to a limited diversification and a higher risk-adjusted return.48 Another point that increases the size effect is the systematic bias of the CAPM to undervalue the equity value of small companies while expecting high growth. This in effect means small companies have a higher beta and increased volatility49. The CAPM, however, has no means of measuring this effect, this leads to abnormal results.

The E/P-effect (Earnings-to-Price-effect) is an anomaly that exists based on the E/P ratio, this ratio is determined by the relationship between a company's earnings and its share price. The E/P effect describes that companies with a high E/P earn on average a higher risk-adjusted return compared to their low E/P ratio counterparts.50 This effect is also not considered in the CAPM.

A study by Cook and Rozeff suggests that the Earnings-to-Price-effect and the size-effect are interlinked and are possibly separate aspects of an underlying effect. When a model is adjusted to one of these effects, the other one becomes only marginal.51

Another well established and tested anomaly is the B/M-effect (Book-Value-to-Market- Value-effect) which describes a positive abnormal return on companies with a high B/M ratio - the ratio between the book value of a company to the market value of a company - compared to companies with a low B/M ratio.52 The B/M-effect is widely accepted as a significant risk factor and has been shown by Fama and French to be statistically relevant ; however, Ang and Chen showed in a 2007 paper that they found that using a regression analysis without time-varying betas generates biases. They also found that in the timeframe from 1926 to 1963 when tested for the B/M-Effect, the intercept that explains abnormal returns in regression analysis, was statistically insignificant.53

The last anomaly explored here is the momentum-effect, which is the tendency of falling stocks to continue falling, while well-performing stocks continue to do so in the short­term. Possible explanations for this abnormal behavior of return is for one the positive feedback provided by investors buying or recommending the stock, which has a positive impact on the price. Another explanation is the underreaction to short term information and overreaction to long term information. As short-term information like earnings forecasts are different in nature than the long term goals a company may hold.54 As the CAPM is a one-factor model it does not take this into account, which therefore reduces the predictability.

5. Alternatives to the CAPM

5.1 The Fama-French three-factor model

As we have discussed a lot of shortcomings and faults in the CAPM, it stands to reason that the model is obsolete, due to this the CAPM has evolved over the years. Fama and French, both involved in testing and disproving parts of the CAPM have formulated an evolution of capital asset models in 1993. This multifactor model uses additional risk factors to minimize the distortion of company size, using the size-effect and also the B/M effect, which were both explained in the previous chapter.55

The 3FM takes the CAPM formula and adds two factors to it. This renders the formula more complex, and more time consuming to use. Contrary to the rather simple CAPM, the use of own datasets for this model might not be viable. As the time spent on calculating each factor can be too time consuming - therefore, it can seem reasonable to use publicly available datasets for this, gaining time and simplicity.

The Fama-French three-factor model (3FM) separates the risk for any stock into three factors:56

1. Beta - the measure used in the CAPM that describes the volatility of the stock in comparison to the market. Therefore, companies with a beta larger than one react stronger to the market and have a larger potential for a high increase or decrease in value.
2. Size - the added risk to smaller companies, which in the long run outperform companies with a higher market cap, however this increase in return is also mirrored in the increase of risk.
3. Value - the additional risk of owning companies with a low book to market ratio. These companies perform better than companies with a high book to market ratio, but also have an increased risk

The extended formula of the model is written as follows:

Equation 3: The 3FM

Abbildung in dieser Leseprobe nicht enthalten

Source: adapted from Fama, E. F.; French, K. R. (2004) P. 38.

The first risk factor of the formula is the beta and is calculated in the same was as in the CAPM. The second factor which is the SMB factor (Small Minus Big) is accounting for the size-effect. This can be done by dividing the companies within the market by market capitalization. Through this the difference in return is shown and can be accounted for. The last factor the HML factor (High Minus Low) removes the bias of the B/M effect from the CAPM, by splitting companies in portfolios determined by their B/M ratio. The respective betas of all factors are estimated with a linear regression, similar to the beta in the first part of the equation. 57 Both additional factors are determined by creating so- called factor-mimicking portfolios, these are split into six types:

1. S-H/Small-High: Small market capitalization, with a high B/M ratio
2. S-M/Small-Medium: Small market capitalization, with a medium B/M ratio
3. S-L/Small-Low: Small market capitalization, with a low B/M ratio
4. B-H/Big-High: Large market capitalization, with a high B/M ratio
5. B-M/ Big-Medium: Large market capitalization, with a medium B/M ratio
6. B-L/ Big-Low: Large market capitalization, with a low B/M ratio

Fama and French evenly divide the companies into “large” and “small” companies, by their market capitalization. For the HML-factor they divide companies into three groups: the companies within the 40th percentile are in the “low” group, the companies above the 40th percentile but under the 70th percentile are in the “medium” group and the companies above the 70th percentile are in the “high” group.58

The SMB for the factor mimicking portfolios is being determined by subtracting the arithmetic mean of the return on the Big Portfolios from the Small Portfolios as shown in equation 4. Analog to this the HML is determined by subtracting the Low Portfolios from the High Portfolios. After every year these portfolios are analyzed again to make sure the portfolios have not changed, and the factors remain correct.

Equation 4: The SMB and HML factors

Abbildung in dieser Leseprobe nicht enthalten

Source: adapted from Vogler, O. (2009); P. 384.

5.2 The Carhart four-factor model

The four-factor model (4FM) was developed in 1997 by Mark M. Carhart just shortly after the 3FM to explain another standing anomaly of the CAPM: the momentum-effect, the tendency of well-performing stocks to continue to do so while badly performing stocks continue to perform worse comparably.59 The method of finding the WML (Winner Minus Losers) is similar to the 3FM, as it is determined by generating six factor­mimicking portfolios. The relevant factors for these portfolios are market capitalization and the previous year's return. The companies are again evenly divided by market capitalization into “large” and “small”. For the return they are split into three groups: “winners”, “neutral”, and “losers”.60

The “winners” are the 30% of companies who have performed the best within the last year. The “losers” are the 30% of companies who have performed the worst in the previous year. The “neutral” are the remaining 40% of Stocks. However, only the “winners” and “losers” are of relevance for the calculation of the WML factor as seen in. The WML is the mean of the winning portfolios minus the mean of the losing portfolios. This factor is recalculated each year alongside the factors in the 3FM.61

Equation 5: The WML factor

Abbildung in dieser Leseprobe nicht enthalten

Source: adapted from Hanauer, M., et al (2012); P. 11.

In the formula of the 4FM, the WML factor is added to the other factors of the 3FM to mitigate the momentum effect and increase the accuracy and veracity of the 4FM compared to the 3FM. However, as mentioned for the 3FM this additional factor also increases the time spent on the model and using pre-existing data could be more favorable.

Equation 6: The 4FM

Abbildung in dieser Leseprobe nicht enthalten

Source: adapted from Hanauer, M., et al (2012); P. 5.

5.3 Substitution potential of the three-factor model and four-factor model for the German market

As explained before, the 3FM and 4FM are evolutions of the CAPM and made specifically to improve known anomalies, which the CAPM does not take into account. Therefore, it is only logical to expect a higher validity of the results from the 3FM and 4FM. This higher empirical quality can be measured by the comparatively higher coefficient of determination R2.62 As the models have a different number of factors and runtime, the adjusted R2 (adj. R2) will be used, to increase the comparability of the results.

As the 3FM was developed by observing the U.S. market and has no underlying explanation as to why these factors exist or how they work. It is important to question its validity in the German market. And studies have shown that the factors indeed do not seem to be universal, as the SMB factor is negative in Germany.63 Hanauer proposes that the size effect was not as prevalent in Germany, to begin with, and has stopped existing or even having the opposite effect since 1995.64 This of course also plays a role in the 4FM.

While the SMB factor is negative, the HML factor is still relevant, as well as the entire model as the coefficient of determination is greater than the one for the CAPM. In a study done by Hanauer, the adj. R2 for the 3FM is 72.5% while it is 48.5% for the CAPM. The 4FM has a further increase of 0.6% compared to the 3FM.65 This increase is due to momentum strategies having a similar result in Germany as in the United States.66

A further common critique of the 3FM and 4FM is the increased complexity and the work necessary to get results, which might be a reason why only roughly a tenth of CFOs use multi-risk factor models, while around three quarters still use the CAPM.67 As mentioned for the 3FM and 4FM public datasets exist for both models. These can remedy the increase in time spent on determining the future returns. While this the case, the quality of these sets is not guaranteed. Thus, making a theoretically more accurate model possibly less desirable than the CAPM.

The last standing points of criticism are the single period nature of these models which are the same as in the CAPM, and the absence of other influences on return, especially the momentum effect in the 3FM.

5.4 The Arbitrage Pricing Theory

The first major evolution of the CAPM was the Arbitrage Pricing Theory (APT) by Stephen A. Ross in 1976, changing one of the main underlying assumptions. As the name implies, the model assumes that arbitrages cannot exist in the market i.e. securities cannot be mispriced.68 The APT is a multifactor model, which is unlimited in kind and number of factors it uses, but the factors are generally grouped into macroeconomic and company­specific factors. Factors in the APT are represented as [p(rk) - i].

Equation 7: The APT

Abbildung in dieser Leseprobe nicht enthalten

Source: adapted from Pankoke, T.; Petersmeier, K. (2009); P.129.

Common factors used in the APT are:69

1. Inflation
2. Treasury-Bill rate
3. Oil Prices
4. Stock market index
5. Change in risk premium (as derived from differences in yield of Aaa and Baa rated entities)

While the APT has a greater degree of explanation of returns than the CAPM, due to its undefined nature it will not be used to contrast the CAPM. The lack of intersubjectivity and possible individual risk factors make it impossible to give a general answer as to the nature of that model in comparison to the CAPM.

6. Empirical Validation of the CAPM and alternative models

Knowing the limitations and uses of the CAPM, this paper now intends to look into the viability and pertinence for the use in Germany or whether it should be replaced with multi-factor models like the 3FM or 4FM.

Also, what dataset should be used, does it make sense to create a set for your own or are publicly available factors an alternative to this. In order to answer this, each model will be analyzed with three different sets of data, where two are publicly available and one is made for this study.

6.1 Datasets used

For the main part of the paper the CAPM, 3FM, and 4FM are being examined empirically within Germany. The stocks under examination are those with a continuous presence in the DAX from 01.01.2015 to 31.12.19. These companies are listed in table 1.

The analysis is done on a monthly basis in three different sets of relevant factors, consisting of each the risk-free rate, the return on market, the HML factor and SMB factor for the 3FM, as well as the WML factor for the 4FM.

One set is the European data as published by Kenneth R. French70, one is published by Richard Stehle71 and the last one has been calculated for this paper. This is done to contrast different ways of using the models in a scenario outside of an academic study. To look at the differences in results and if it is necessary to calculate the factors for your own set as Creating a data set is time-consuming, using an already existing set might be a viable option.

Due to the availability of data the time frame initially is not identical for these sets, it will however be adjusted later to determine how this impacts the results. The data set calculated for this study, which will be named the 2019 set, has a range of five years starting with the beginning of 2015 and ending on 31.12.2019. The Stehle set begins on 01.01.2000 and ends in June 2016, while the set by French ranges from 01.01.2000 to 31.12. 2019. All of them are within the standards of the IDW, which requires at least five years of monthly data. 72

6.2 The 2019 Dataset

For the 2019 set the data used will consist of the following:

The monthly stock data is the adjusted Xetra closing price of the last trading day of each month as taken from yahoo finance. The adjustment of prices takes into account the accumulation of dividends, subscription rights, and other gains from stocks as well as corrections and splits. Therefore, it displays the actual growth of stocks analog to the DAX and thus the data is consistent. The DAX is a performance index that is comprised of the 30 German stocks with the largest market capitalization and order book volume in the prime standard of the Frankfurt stock exchange (FSE).73

The DAX, even though it is only comprised of 30 companies, holds around 80% of the entire value and therefore is still a valid approximation especially as all companies used for this study are present within the DAX, which is the reason it was chosen as the market proxy. Analogically to the stock closing prices, the DAX closing was also taken from yahoo finance. To determine the continuous rate of return for the stocks and the index, the natural logarithm was used, as it is possible to approximate the rate of return with a normal distribution.

Initially, the yield of the 10-year German government bond is used as the risk-free rate, but as the other two sets use a more short-term risk-free rate, the one-month Euribor is also used as it is the one used for the Stehle set.

For the dataset of this study, the factors have been recalculated for each year according to the Fama/French and Carhart model and can be found in the appendix. The beta factor has been calculated in two different ways. The first way has been done adhering to the mathematical formula, which means allowing a negative risk-free rate. However, as this does not happen in a real setting another calculation was necessary. This set has a minimum risk-free rate of 0, therefore it will be called the 0+ set from now on to reduce confusion. This has been done as the set by Stehle allows negative risk-free rates, but since the set by French does not, it is necessary to find out how this does affect the outcome.

6.3 Stehle Dataset

The dataset taken from the Website of Stehle gives a monthly overview from 1958 to 2016 of all necessary factors for the CAPM, 3FM, and 4FM.74 The data used for determining the factors is further broken up into "TOP" and "ALL" segments. The “ALL” segment contains all firms with at least one share in either the top segment, the middle segment or the “Neuer Markt” until 2007, and from 2007 onwards all shares within the regulated market.

The “TOP” segment, which is being used as a comparison to the other datasets, is comprised of all firms traded in the top segment of the Frankfurt stock exchange until 2007, from then on all stocks in the regulated market that were formerly in the "Amtliche Markt” are used, with the exclusion of firms traded in the “Neuer Markt” and the Middle Segment of the FSE.

For each segment they have one set including and excluding tax credit, this however is only of significance for a timeframe before the year 2000, but still, the data set without the tax credit has been chosen, for a better comparison.

The data of the market portfolio is each stock in the segment weighed by its market value, as compared to entire value of the market, for each month.

The Fama/French factors were calculated in accordance with the original papers done by Fama and French as illustrated previously. The factor mimicking portfolios for the WML factor, however, have been calculated on a monthly basis instead of a yearly one as proposed by Carhart. The Stehle dataset utilized in this comparison uses the averaged one-month money market rates as reported by Frankfurt banks until 2012, and from then on continued using the one-month Euribor.75

6.4 French Dataset

The data obtained on the website of French is the updated version from Fama and French's paper “Size, Value, and Momentum in International Stock Returns”.76 The sample data goes back to November of 1989, with the newest data dating from April of 2020. They defined their risk-free rate as the one-month U.S. Treasury bill rate. For the European data sample, they used stocks from Austria, Belgium, Denmark, Finland, France, Germany, Greece, Ireland, Italy, the Netherlands, Norway, Portugal, Spain, Sweden, Switzerland, and the United Kingdom. To determine the return on market, French used all stocks with available market equity data in Europe. The calculation for all factors was done by using companies with available data for market equity and book equity.77

To calculate each factor, they used method described earlier; the exception is the size factor, as they defined the biggest companies that encompass 90% of the Market cap as “Big” with the companies that make up the remaining 10% as “Small”. The factor­mimicking portfolios were rearranged yearly in June and calculated monthly.

7. Methodology

For all stocks a regression analysis has been done for each model and set, to see how the difference in parameters changes the outcome. Through the regression, the adjusted R2 is being determined as well as the p-values to find the veracity and significance of the models. As previously discussed, the adjusted R2 is the coefficient of determination and concludes how many of the measurements are explained by the model.

To determine whether to reject or accept the hypothesis of the model, the significance level will be 5%. P-values between 1% and 0.1% will be mentioned as they are very significant as well as values below the 0.1% threshold as they are highly significant. This will be done for the models overall as well as for the factors used in the model.

An additional factor used to understand how this model behaves is the intercept of the regression analysis, this factor is called Jensen's alpha. The Jensen's alpha explains any deviation of actual result to the expected result of the models - a positive alpha shows that a positive abnormal return exists, while a negative intercept means the stock or portfolio performed worse than predicted.78

To increase comparability between each dataset, one analysis has been done for the entire time and one for a runtime of the last five whole years to reduce seasonal factors. This is necessary for the set done by Stehle as it ends in June of 2016. Subsequently, the 5-year period will last from the beginning of 2011 to the end of 2015.

8. Results and interpretation

In this chapter, the results will be presented and analyzed beginning with each model comparing the sets with their maximum runtime. Beginning with the Capital Asset Pricing Model, then adding the factors to compensate for size and the book-to-market effect and lastly adding the factor for the momentum effect. This will show how much the veracity of the model is improving. Afterward, to minimize the bias, each set will be reviewed with the same runtime, to find how the component of time changes the findings. Additionally, to the factor of time the risk-free rate will be changed to reflect the differences between the sets. This includes whether negative risk-free rates are permissible in the data, as well as changing the risk-free rate.

8.1 Results - CAPM

The Following table (table 1) shows the results of the regression analysis done for the three different sets tested on CAPM:

Table 1: Comparison of Datasets used on the CAPM

Abbildung in dieser Leseprobe nicht enthalten

Source: Own calculation (Data taken from Yahoo finance)

As seen in table 1 the results vary widely for each set of data. For the 2019 set the t-test, which shows the confidence level of the hypothesis to our model, reveals that the median of our p-values for each stock lies below the 0.001 threshold required for a highly significant result at 1.1 X 10_7 : in terms of numbers, that is 20 out of the total 26 stocks with 4 values classified as very significant as they lie between 0.001 and 0.01. The last 2 values of the companies RWE and Beiersdorf are still significant with a value of 0.015 and 0.037 respectively.

The results for the 2019 set of data show that the average adjusted R2 lies at 0.369, with the median at 0.378. The range of values goes from 0.081 to 0.684 with a standard deviation of 0.187. This shows that the CAPM alone explains around 37% of the variance in the data.

In contrast to this, the other German by Stehle set performs worse in both significance and determination. The median of the p-value lies at 0.028 with the average lying at 0.113, relying on this set the model would be rejected 10 out of 26 times. Of the remaining 16, 6 stocks show a significant result, 1 a very significant and 9 a highly significant result. The decrease in adjusted R2 is severe as well with an average of 0.051 and a median of 0.019, the standard deviation for adj. R2 is 0.069. The range of adjusted R2 is from -0.004 to 0.258, therefore even the highest value is lower than the median of the set of 2019.

The last set provided by French falls in between those two sets for determination but is the most significant out of the three datasets, with a median p-value of 1.7 x 10_17. Henkel is one outlier, with a value of 0.14. The coefficient of determination lies below the 2019 set with an average of 0.255 and a median of 0.267 and has a standard deviation of 0.127.

Regarding the Jensen's alpha all fall within a narrow margin, the 2019 set averages at 0.0013 with the median at 0.0008, therefore the model predicted less of a return than the factual returns show. The set by French behaves similarly, but with a lower average of 0.0003 and a higher median of 0.0013. The Stehle set averages at an alpha of 0 with the median being slightly negative at -0.0003.

8.2 Results - Fama three-factor model

In the next step, the CAPM is being compared to the 3FM model as seen in table 2 in view of the significance and the coefficient of determination, as well as an into the factors to determine the how they change the results. For all sets, the added factors also increase the adjusted R2 in average as well as the median. While the average of the p-value for the sets increases, i.e. reduce significance, for the French and Stehle sets the median decreases. While the average of the 2019 set as well as all medians decrease.

Table 2: Performance of the 3FM compared to the CAPM

Abbildung in dieser Leseprobe nicht enthalten

Source: Own calculation (Data taken from Yahoo finance)

For the 2019 set the average of the p-value is 0.0025, which is a decrease of —9.8 x 10_5, while the median decreases by —5.2 x 10_9 to 1.1 x 10_7, compared to the CAPM. While there is an overall measurable decrease, 13 stocks have an increase in p-value. Most of them are rather slight with the exception of E.ON, where it increases by 0.06 which would lead to a rejection of the 3FM for E.ON. In terms of significance for the 3FM, one stock is rejected, one value is very significant and 24 are highly significant. The coefficient of determination increases on average by 0.076 to 0.446 and the median increases by 0.063 to 0.441. Similar to the p-values, 8 stocks behave negatively compared to the CAPM with the biggest decrease being -0.44 by the Deutsche Telekom AG, while the biggest increase is SAP with a plus of 0.55. Judging only by the overall results of the model this seems like an improvement, however, when examining the p-value of factors in detail they reveal a slightly different picture. These can be found in appendix 1. The HML-factor for the 2019 set is rejected 16 out of 26 times while 2 stocks are significant, 3 very significant, and 5 highly significant. The p-value for the SMB-factor is even higher and average at 0.42 with the median at 0.39. For this factor all but 6 of them are be rejected with 3 being significant, 2 being very significant, and 1 highly significant.

Similar to the 2019 set the Stehle set has a decrease in the p-value at least in the median, which decreased by -0.02, while the average increased by 0.01 resulting in a p-value of 0.008 and 0.122 respectively. Also, similarly, this change is not positive for all stocks as 12 increase in p-value, this change does result in fewer stocks being rejected, only 9 stocks are rejected compared to the 10 in the CAPM. The level of significance for the remainder of stocks increases as now 3 show a significant result, 2 a very significant result, and 11 a highly significant result. The positive change in adjusted R2 is weaker than observed in the 2019 set with an increase of 0.023 in average and 0.029 of the median. Among the stocks, 8 have a decreased value, while 18 increase. The new range of results is spanning from -0.012 to 0.252. As expected, the rejection of the other factors is also higher than the 2019 set, due to the overall lower significance. This leads to 21 rejections for the HML factors, 1 significant value, and two values each highly and very significant, with an average of 0.44 and a median of 0.48. While this set performs worse overall compared to the 2019 set, the SMB factor is more accurate as the median of p- values lies at 0.14, leading to 18 rejections, 3 significant values, 2 very significant results, and 3 highly significant values.

Lastly, the results of the French dataset. Here as well the average of the p-value increased while the median decreased from 0.0055 to 0.01 and 1.7 x 10_17 to 1.07 x 10_17 respectively. Both average and median are well within significant results. In contrast to the other sets, the distribution of stocks in different categories of significance has remained the same. The increase of determination is similar to the set by Stehle. While the average rose by 0.025 to 0.292, the median rose to 0.292 by 0.03. The HML-factor presents the best results so far, with a median of 0.058, however still more than 53% of stocks would be rejected, 3 are significant, 4 are very significant and 5 are highly significant. The SMB-factor has 18 rejections set with a median p-value of 0.18, one of the values is significant, 3 are very significant and 4 are highly significant. With these results, it is easy to see how the 3FM has improved our primary model, even if the factors themselves have to be rejected.

For the 3FM, the alphas for all sets reduced. This means the average and median of alpha got closer to 0 for the 2019 set and the French set. For the 2019 set the average and median reduced to 0.001, and 0.0003 respectively. While for the French set both average and median reduced to 0.0008. That also means the average and median of alpha for the Stehle set now are both negative, -0.0008 and -0.0002, i.e. the set predicts higher returns than the achieved returns.

8.3 Results - Carhart four-factor model

Now that the 3FM has been examined and the improvement overall has been shown, it is time to see how the further increase in factors changes the results with the last model the 4FM in comparison to the CAPM. As before we will begin the analysis with the 2019 set done for this paper and work our way to the set done by French. As we have discussed the previous factors already, only the additional factor will be illuminated as seen in table 3.

Table 3: Performance of the 4FM compared to the CAPM

Abbildung in dieser Leseprobe nicht enthalten

Source: Own calculation (Data taken from Yahoo finance)

With the addition of the WML factor all sets improved their adjusted R2 and the median of the p-values. The Stehle set was the only set with an improvement of average p-value, while the other two sets fare slightly worse. The p-values of the 2019 set average at 0.005 with the median being 2.67 x 10_8, this is a change by 0.002 and —8.5 x 10_8 respectively compared to the CAPM. Here as well as in the 3FM there is one outlier the Deutsche Bahn AG that has to be rejected with a p-value of 0.012, but with the exception of this all other stocks have a highly significant value. The same cannot be said for the WML-factor: it is the most significant factor after the beta; however, the median of its p- value is 0.096 and a total of 17 stocks have to be rejected. The coefficient of determination however increased on average by 0.11 to 0.446, this increase is almost 50% higher than the increase of the factors of the 3FM alone. The change in adj. R2 ranges from -0.43 to 0.57 with only three decreases in adj. R2.

The Stehle set also improved results compared to the CAPM. The median and average of the overall p-value decreased by -0.026 and -0.022 respectively. There are still 9 rejections as with the 3FM, three significant and very significant results, as well as 11 highly significant results. The adjusted R2 increased on average by 0.033 to 0.083, while the median increased to 0.054 by 0.035. In this set the WML-factor performs similarly like the other factors and has a rejection rate of 73%, while the median lies at 0.21.

Now to the final results of this section, the French set has a minute decrease of —6 x 10_18 of p-value in the median, with a slight increase of 0.004 of the average compared to the CAPM. Both average and median are still significant or better with an average of 0.01 and a median of 1.06 x 10_17. The coefficient of determination is an average of 0.298, which is an improvement of 0.042 with the median being 0.306, an improvement of 0.039. As with the other sets, the WML-factor is not very significant as its median is 0.08 and an even higher average of 0.24, 14 out of 26 stocks need to reject the factor, which is the best result so far.

And for the alphas of the 4FM: while the alphas of the 2019 set continued to drop to an average of 0.0007 and a median of -0.0005, both other sets had an increase in alpha. The French set increased to 0.003 in both median and average, and the Stehle set turned positive again with an average of 0.001 and a median of 0.0006.

Now that the results have been shown for all sets and models, it is necessary to understand why they have such different results. The overall results so far show that the 2019 set performs best in determining results, while the French set places second in determination but has the highest significance. The set done by Stehle is so far the worst in both categories. Possible reasons for these results are the amount of data that is both significantly longer than in the 2019 set, the chosen market proxy, or the risk-free rate. To determine the answer to this, the sets will be changed to facilitate these factors and then compare the results.

8.5 Negative risk-free rate

First of all, as previously mentioned, is the minute difference between the previously existing sets by French and Stehle, which is the existence of a negative risk-free rate. Stehle allows a negative rate, while French does not. To find the difference in results, the 2019 set was changed to find the difference and as mentioned earlier is called the 0+ set as no negative risk-free rates are permitted. The 2019 set as it has been used so far is designated as 0- in table 4.

Table 4: The 2019 set with and without a negative risk-free rate

Abbildung in dieser Leseprobe nicht enthalten

Source: Own calculation (Data taken from Yahoo finance)

In table 4, it is clear that even if the change is small, the 0+ set performs worse in both significances and determination. This is true for all 3 models; however, the beta does not change uniformly, as the beta for the CAPM decreases, while it increases in the other two models.

8.6 Difference of Runtime

A major difference between the three sets is the length of the runtime - thus, to mitigate a false conclusion, all three sets have been reduced to five years as mentioned at the beginning of this chapter. In table 5 a strong improvement can be observed for the Stehle set due to the shortening of its runtime. The same cannot be said about the set by French as the CAPM performs worse in both categories, while the median of the p-value increases for all sets, the average decreases for the 3FM and 4FM. The adjusted R2 increases slightly as well for the multifactor models, but not as drastic as the Stehle set.

Table 5: Change in results due to a shortened runtime

Abbildung in dieser Leseprobe nicht enthalten

Source: Own calculation (Data taken from Yahoo finance)

8.7 Impact of the risk-free rate

Now to the last-mentioned aspect, which is the choice of the risk-free rate. To determine how strong the impact of the risk-free rate is, the one-month Euribor has been used as the risk-free rate for the 2019 set instead of the 10-year German government bond. While the 10-year German government bond has been negative for short periods, the mean and average of the yield was positive. The Euribor however was negative for all but two months and decreased over time to a minimum of -0.45%.

Table 6: Change in results due to a different risk-free rate

Abbildung in dieser Leseprobe nicht enthalten

Source: Own calculation (Data taken from Yahoo finance)

With this change, as seen in Table 6, the coefficient of determination has increased by at least 82% and up to 128% compared to the 10-year government bond. While the CAPM has increased most, the model with the highest adjusted R2 is still the 4FM. However not only has the adjusted R2 improved, but the total p-value also improved, making the 2019 set the most significant of sets. Even though the overall significance improved, the significance of the additional factors to the beta factor has not changed to explain this strong improvement.

9. Interpretation

After examining all results, it is time to understand what this means for the model and in effect, the users.

Beginning with the most basic differences, the runtime. As shown, a shorter runtime did improve the quality of the model if the market is affected by crises. This is especially evident if a large part of the runtime lies within these periods, as with the Stehle set. When contrasted with the French set, which has a longer runtime overall these changes normalize and shortening the runtime can reduce the validity of the set. The positive change in the Stehle set is led by excluding the 2007-2008 subprime financial crisis, and partially the Euro crisis from 2010-2012, as these highly volatile timeframes i.e. high beta,79 influence the beta over the entire runtime of the model. This is the case as the beta is constant in the basic form of the CAPM and of the other models used. Due to the inclusion of these high volatility periods and trying to fit the lower returns that come with them80 into a single model would mean a reduction in expected returns for times with less volatility, and therefore decrease the overall adjusted R2.

Next is the difference in results as observed after changing the risk-free rate for the 2019 set. First with the change from the 10-year German government bond, to the one-month Euribor. The positive change has to be correlated with the reduction of the risk-free rate - in the case of the 2019 set, a reduction of the risk-free rate decreases expected return for companies with a beta smaller than 1, has no effect on companies with a beta of 1 and increases expected returns for companies with a beta higher than 1. This however only is true as long as the absolute value of the risk-free rate is larger than the return on the market, while being negative at the same time. This is the case for 55 out of 59 months in the last 5 years when using the Euribor.

This means the average expected returns of the sample has decreased as the risk-free rate for this set is negative except for two months, while the average and median beta for this set are also smaller than 1. However, Jensen's alpha increased in all models, meaning the stocks had higher returns than expected using the models. This means the reduction of expected return due to the negative risk-free rate was too high.

The other change done to the risk-free rate was whether negative rates should be used or not. In our results, a slight negative change was observed, when negative rates were not used. The change is only marginal as the bond yields were negative only partially negative in comparison to the Euribor. One probable cause for this behavior is that even if a company would not pay to have a risk-free investment, other market participants like banks do and therefore are influenced by the negative risk-free rates. If we remove the negative rates, then this influence is not represented in the model anymore and as a result, the results reflect this move away from reality.

Another key finding in this study is that the factors used in all models are location sensitive, this means that when using proxies for the risk-free rate as well as the market portfolio it is necessary to use the factors found in the targeted country or market. This is in line with previously done studies where multifactor models based on US indices have a lower R2 than the CAPM with German an index.81 Thus any dataset used in Germany has to have the necessary factors from the German market. As the only public dataset utilizes U.S. factors it is rendered useless for important applications.

The last of our findings concerns the use of multifactor models in general. As discussed in our results, the p-values in many of the examined stocks were too high to be of significance. While there is an increase in adjusted R2 in all of the samples in this study, these findings cannot necessarily be compared to findings of the U.S. market. Through the findings in the chapter 5.3 the validity of the size effect was already questioned. The sample of the 2019 set reinforces this, as the SMB-factor performs worst. At the same time, we cannot confirm that either the HML or WML-factors are statistically relevant in any model. This is most prominent in the 2019 set using the one-month Euribor as the risk-free rate, which seems to be a better fit for the tested models. Here the difference in adjusted R2 between the CAPM and the 4FM shrunk from 0.107 to 0.028.

This means that multifactor models in Germany are less relevant than in the U.S. and are not a better option compared to the CAPM. This is especially the case, as the time spent on calculating the additional factors could be used to find a better risk-free rate.

10. Limitations

It is important to talk about the limitations in the analysis that has been done for this study. Beginning with the amount of data that has been collected. While the length of the samples is sufficient for conclusive findings, having a longer timeframe for the 2019 set would have been preferable. This would enable a comparison with the other two sets on more timeframes. However due to limitations in acquiring information this was prevented.

Another point is the number of stocks used in the 2019 set: as the market value of companies was only readily available for companies within the DAX, it was only possible to take these companies into the set, as the market value is needed for HML factor. Due to this constraint, only 26 companies were used in this study. Without this constraint, a larger number of stocks sorted in portfolios by market value or other factors could have improved the validity of this study.

Also due to all stocks being DAX companies while using the DAX as the proxy for the market might have biased the result to be better than they should be. As the DAX performance is representing how the 30 companies within the DAX are performing, this means there is a strong influence of the 26 chosen companies on how the index they are measured by is behaving.

A different point to mention is the performance of the sample by Stehle. This sample is performing much worse than it should be, even though the methodology described in their paper is correct - the results do not mirror that. As shown earlier, the risk-free rate they use was superior to the one initially used for the 2019 set, but it still performed worse. The return on market was different from the return on the DAX but still well within the possible returns i.e. no large increases or decreases. One possible cause for the wide gap between what was expected, and the actual results is that there was a non-comprehensible error where the attributed date and the actual date of the acquired data diverged. This would mean that how the stocks behave is divorced to some degree from the market proxy and delivers inferior results.

And lastly, regarding the method used to obtain the factors, there is also a possible way to increase the veracity of the models. The type of regression used, the Ordinary Least Square (or OLS) regression can only be reliably used in the absence of heteroscedasticity. Heteroscedasticity is the increasing of variance for data points over time. This change in variance can overestimate beta and the other factors when obtained via an OLS regression.82 This is a proven behavior for stock data and an autoregressive conditional heteroscedasticity model (or ARCH) model can correct for this behavior in data. However, this model was not used in this study as the effect usually is minor with changes of less than 0.1 in beta.83 Additionally, the other two sources also used OLS regressions in their factors, and therefore the results would be less comparable if an ARCH regression were used.

11. Conclusion

To conclude the CAPM still holds up and under the right conditions the difference between the CAPM and multifactor models such as the 3FM and the 4FM are minute. This is the case because there seems to be an extremely diminished role of the Size-, Value-, and Momentum anomalies in Germany when compared to the United States. This goes so far that for most companies the factors associated with the anomalies are statistically not significant. There is an improvement in adjusted R2 for the multifactor models when using the 10-year German bond as the risk-free rate. However, this improvement is barely present when the 1-month EURIBOR is used as the risk-free rate. Thus, the factors of the 3FM and 4FM for the set using the 10-year German bond probably reflect how the risk-free rate changes and increase the adjusted R2 in that way.

Another question answered is whether using pre-existing sets make sense for use in a company or a study. Here the answer is negative when the results are of importance. For Germany, the only current potential data sample is from French and is looking at Europe as a whole, while using factors relevant to the United States. Therefore, these results are less reliable than preparing a model for oneself, with the relevant data. This is even the case when using a theoretically more accurate model such as the 3FM and 4FM and comparing it to the CAPM. Additionally, as seen with the Stehle sample, even a theoretically sound set of data can develop less than ideal results, when tested. Thus, calculating a set of data with the right factors is necessary when highly accurate results are needed.

As a closing remark, as this study has a limited extend further research is necessary; this could include the use of an ARCH model to improve the meaningfulness of the results. Another possible improvement would be to increase the database in length and width of data, i.e. using more companies with an extend period of data. This could give a more general answer to the question of usefulness of asset pricing models in Germany. And lastly, with the possible broader scope of a longer thesis, testing more than two different risk-free rates could reveal the optimal rate, as this study has shown that the choice of risk-free rate is more important to the model than an increase in factors.


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Table of Appendices:

A: Results of the OLS Regressions

I. Extended Results of the 3FM
II. Extended Results of the 4FM - 2019 Set
III. Extended Results of the 4FM - French Set
IV. Extended Results of the 4FM - Stehle Set

B: Erklärung der Bachelorarbeit

Abbildung in dieser Leseprobe nicht enthalten


1 Cf, Mabrouk, H. B.; Bouri, A. (2011).

2 Cf. Sharpe, W. F. (1964).

3 Cf. Graham, J. R.; Harvey, C. R. (2001); P. 201.

4 Cf. Brounen, D.; De Jong, A.; Koedijk, K. (2004).

5 Cf. Sharpe, W. F. (1964).

6 Cf. Lintner, J. (1965), Cf. Mossin, J. (1966).

7 Cf. Markowitz, H. (1952).

8 Cf. Markowitz, H. (1952).

9 Cf. Elton, E. J.; Gruber, M. J. (1997); P. 1746.

10 Cf. Sharpe, W. F. (1964); P. 440.

11 Cf. Mossin, J. (1966).

12 Cf. Fama, E. F.; French, K. R. (2004); P. 37.

13 Cf. Lintner, J. (1975); P. 133-136.

14 Cf. Lintner, J. (1965); P. 587.

15 Cf Blitz, D.; Falkenstein, E.; Vliet, P. van (2014); P. 2.

16 Cf. Knight, F. H. (1921); P. 76-80.

17 Cf. Brown, P.; Walter, T. S. (2013); P. 4-6.

18 Cf. Graham, J. R.; Harvey, C. R. (2001); P. 201-203.

19 Cf. Breuer, W.; Jonas, M.; Mark, K. (2007).

20 Cf. Grayburn, J.; Hern, R.; Lay, H. (2002).

21 Cf. Romano, R. (2005).

22 Cf. Brown, P.; Walter, T. S. (2013).

23 Cf. Fama, E. F. (1991); P. 1575-1576.

24 Fernandez, P.; Aguirreamalloa, J.; Avendano, L. C. (2011)

25 Cf. Pankoke, T.; Petersmeier, K. (2009); P. 111.

26 Cf. Damodaran, A. (1999).

27 Cf. Svensson, L. E. O. (1994); P. 6.

28 Cf. Beja, A. (1972).

29 Cf. Ernst, D.; Schneider, S.; Thielen, B. (2017); P. 58.

30 Cf. Scott Mayfield, E. (2004); P. 466.

31 Cf. Ziemer, F. (2018); P. 208.

32 Cf. Jonas, M. (2009); P. 544.

33 Cf. Fazzini, M. (2018); P. 100.

34 Cf. Scott Mayfield, E. (2004); P. 466-467.

35 Cf. Gelhausen, H. F.; Institut der Wirtschaftsprüfer in Deutschland (2008).

36 Cf. Kuhner, C.; Maltry, H. (2006); P. 163.

37 Cf. Ziemer, F. (2018); P. 141-144.

38 Cf. Häcker, J.; Ernst, D. (2017); P. 563-564.

39 Cf. Dybvig, P. H.; Ross, S. A. (1985).

40 Cf. Gomez-Bezares, F.; Ferruz, L.; Vargas, M. (2015).

41 Cf. Black, F.; Jensen, M. C.; Scholes, M. (1972).

42 Cf. Fama, E. F.; MacBeth, J. D. (1973).

43 Cf. Blitz, D.; Falkenstein, E.; Vliet, P. van (2014); P. 62.

44 Cf. Fama, E. F.; French, K. R. (2004); P. 37.

45 Cf. Rosenberg, B. (1981).

46 Cf. Bartholdy, J.; Peare, P. (2003).

47 Cf. Levy, R. A. (1971); P. 9.

48 Cf. Banz, R. W. (1981).

49 Cf. Ibbotson, R. G.; Kaplan, P. D.; Peterson, J. D. (1997).

50 Cf. Basu, S. (1983).

51 Cf. Cook, T. J.; Rozeff, M. S. (1984)

52 Cf. Fama, E. F.; French, K. R. (2006).

53 Cf. Ang, A.; Chen, J. (2007); P. 28-29.

54 Cf. Jegadeesh, N.; Titman, S. (1993).

55 Cf. Fama, E. F.; French, K. R. (1993).

56 Cf. Fama, E. F.; French, K. R. (1993).

57 Cf. Vogler, O. (2009).

58 Cf. Fama, E. F.; French, K. R. (1993); P. 8.

59 Cf. Chan, L. K.; Jegadeesh, N.; Lakonishok, J. (1996).

60 Cf. Carhart, M. M. (1997).

61 Cf. Hanauer, M. X.; Kaserer, C.; Rapp, M. S. (2012); P. 11.

62 Cf. Bello, Z. Y. (2008).

63 Cf. Schrimpf, A.; Schröder, M.; Stehle, R. (2007).

64 Cf. Hanauer, M. X.; Kaserer, C.; Rapp, M. S. (2012); P. 14.

65 Cf. Hanauer, M. X.; Kaserer, C.; Rapp, M. S. (2012); P. 27.

66 Cf. Schiereck, D.; De Bondt, W.; Weber, M. (1999); P. 114.

67 Cf. Graham, J. R.; Harvey, C. R. (2000); P. 7.

68 Cf. Ross, S. A. (1976).

69 Cf. Chen, S.-J.; Jordan, B. D. (1993).

70 Cf.

71 Cf. french-factors-for-germany.

72 Cf. Gelhausen, H. F.; Institut der Wirtschaftsprüfer in Deutschland (2008); P. 195.

73 Cf. Deutsche Börse (2020).

74 Cf. Stehle, R (2020).

75 Cf. Brückner, R.; Lehmann, P.; Schmidt, M. H.; et al. (2015).

76 Cf. Fama, E. F.; French, K. R. (2012); P. 459-460.

77 Cf. French, K.R. (2020b).

78 Cf. Jensen, M. C. (1968).

79 Cf. Liau, Y.-S. (2016).

80 Cf. Scott Mayfield, E. (2004); P. 466-467.

81 Cf. Brückner, R.; Lehmann, P.; Schmidt, M. H.; et al. (2015); P. 20.

82 Cf. Armitage, S.; Brzeszczynski, J. (2011); P. 1535-1537.

83 Cf. Armitage, S.; Brzeszczynski, J. (2011); P. 1525.

57 of 57 pages


An Empirical Test of the "Capital Asset Pricing Modell" (CAPM) on Current Stock Data
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