Excerpt

## Contents

1. Introduction

2. Literature Review

2.1. Validity oftheCAPM-ß

2.2. Size

2.3. Book-to-Market

3. Data

3.1. DataSources

3.2. Data Preparation

3.3. Data Problems

4. Methodology and Empirical Results

4.1. CAPM-ß

4.2. Size

4.3. Book-to-Market

4.4. Cross-Sectional Regressions

5. Conclusions

Appendix

References

## Preface

This thesis began on April 28, 2010 and was handed in on November 29, 2010 after the editing period of six months was prolonged by one month.

Roman Brückner distibuted an overview of stocks that were relevant for my thesis.

I collected data with Datastream and “Hoppenstedt Aktienführer,” which were provided by the Institute of Banking and Asset Markets, and on the websites of companies. Stefanie Otte and Roman Brückner assisted me with the collection of data.

I would like to thank Prof. Stehle, Ph.D. for entrusting me with an interesting topic and for his helpful suggestions; Roman Brückner for supporting me in a very kind way throughout the process of writing the thesis by giving advice and making himself available for discussion; and Patrick Lehmann for participating in the discussion of the pre-stage of my thesis. For proofreading I would like to thank Colin Adams.

I wish to particularly thank my parents Waltraud and Gerard for supporting me throughout my life and thus giving distinction to my education; my brother Jeroen and sister Saskia for experiencing enjoyable moments together. Last but not least I want to thank my girlfriend Anikó for being in my life.

MLA style for parenthetical references was partially used to quote a sentence period of a citation.

## List of Abbreviations

illustration not visible in this excerpt

## List of figures

Figure 1 Comparison of CDAX performance index, equal-weighted full-sample performance “index”, and value-weighted full-sample performance “index” (calculated with monthly returns):2000-2009

Figure A1 Market Value of “Mineralbrunnen AG” published on the website in the financial statement, 2005-2007

Figure A2 Common Shareholders’ Equity of “Mineralbrunnen AG” published on the website in the financial statement, 2006 and 2007

Figure A3 Market Value of “WMF AG” published on the website in the financial statement, 2002-2008

Figure A4 Common Shareholders’ Equity of “WMF AG” published on the website in the financial statement, 2008 and 2009

## List of tables

Table 1 Correlation Coefficients ofRetums between CDAX, VW index and EW index

Table 2 Summary Statistics for Equally-Weighted Portfolios Sorted by ß: 2005-2009

Table 3 Summary Statistics for Equally-Weighted Portfolios Sorted by ß: 2005-2009 (with year 2008 excluded)

Table 4 Summary Statistics for Equally-Weigthed Portfolios Sorted by Size: 2005-

Table 5 Differences in Portfolio Returns Between Various Equally-Weighted Portfolio Combinations (Size)

Table 6 Summary Statistics for Value-Weighted Portfolios Sorted by Size: 2005-2009

Table 7 Differences in Portfolio Returns Between Various Value-Weighted Portfolio Combinations (Size)

Table 8 Average Monthly Returns on Equally-Weigthed Portfolios Sorted by Size and Month: 2005-2009

Table 9 Average Monthly Returns on Equally-Weighted Portfolios Sorted by Size and Month: 2005-2009 (largest part of financial crisis excluded)

Table 10 Average Monthly Returns on Equally-Weighted Portfolios Sorted by B/M

Table 11 Average Monthly Returns on Value-Weighted Portfolios Sorted by B/M

Table 12 Average Coefficients of the Cross-Sectional Regressions for Individual Stocks: 2005-2009

Table 13 Average Coefficients of the Cross-Sectional Regressions for Individual Stocks: 2005-2009, up-market

Table 13 Average Coefficients of the Cross-Sectional Regressions for Individual Stocks: 2005-2009, down-market

Table Al Market Value of“Mineralbrannen AG” by Datastream, 2006 and 2007

Table A2 Common Shareholders4 Equity of “Mineralbrunnen AG” by Datastream,

2006 and 2007

Table A3 Market Value of “WMF AG” by Datastream, 2005-2009

Table A4 Common Shareholders4 Equity of “WMF AG” by Datastream, 2008 and 2009

Table A5 Summary Statistics of Double-Sorted Portfolios, first by Size then by Beta: 2005-2009

Table A6 Summary Statistics of Double-Sorted Portfolios, first by Size then by Beta: 2005-2009 (year 2008 excluded)

Table A7 Average Monthly Returns ofEqually-Weighted and Equal-Size Portfolios Sorted by Size: 2005-2009

Table A8 Differences in Portfolio Returns between Various Equally-Weighted Portfolio Combinations

Table A9 Average Monthly Returns ofValue-Weighted and Equal-Size Portoflios Sorted by Size: 2005-2009

Table Al0 Differences in Portfolio Returns between Various Value-Weighted Portfolio Combinations 6l

Table All Average Monthly Returns on Value-Weighted Portfolios Sorted by Size and Month: 2005-2009 6l

Table Al2 Average Monthly Returns on Value-Weighted Portfolios Sorted by Size and Month: 2005-2009 (largest part of financial crisis excluded)

## 1 Introduction

The renowned and still popular Capital Asset Pricing Model (CAPM) developed by Sharpe (1964), Lintner (1965), and Black (1972) is the standard Asset Pricing Model for theory and practice. While the first empirical studies which tested for the validity of the CAPM-S and were conducted by, among others, Black, Jensen, and Scholes (1972) and Fama and MacBeth (1973), confirmed the validity of the CAPM-S as significantly related to average stock returns, later studies, which used more recent data could not confirm its validity.

Thus, other models were developed to improve the standard model in order to show a more significant relation between beta and average stock returns. Many different approaches to improve the standard model introduced by Sharpe-Lintner-Black were developed that attempted to uphold the hypothesis of“efficient capital markets”.

But other models were developed that were not closely related to the CAPM. In some cases fundamental or macroeconomic data were used to make up for the CAPM’s lack of explanatory power. Although some of these factors were able to describe security returns, the widely-tested alternative to the standard CAPM version is the extension of the model by both size and book-to-market (B/M) ratio; the Three Factor Model introduced by Fama and French (FF, 1992). For its high explanatory power in many studies of different markets, it was often mentioned as a reasonable extension of the standard CAPM.^{[1]} ^{[2]} ^{[3]}

Most of the studies concentrated onUS equity capital markets, but in Germany the same tendency was also detected in most empirical studies.^{[4]}

Beta alone has rarely shown a significant relation to average stock returns. Adding the characteristics size and B/M to the CAPM or testing them alone, usually leads to better results for relating these characteristics to average stock returns than with beta alone.

One important goal of this study is to find out, whether the most recent data also shows the same tendency as earlier studies of the German market:

- A very low relation between beta and average stock returns

- A higher relationship between size and average stock returns

- An even higher relation between B/M ratio and average stock returns.

In many studies the methodology used to test for the relationship between beta, size, B/M ratio, and stock returns are cross-sectional regressions and two-sorted portfolios. In this study, more weight is put on the ability to predict stock returns by testing these characteristics alone. Usually researchers are interested in the statistical relationship between the characteristics and stock returns. In contrast to this approach, which is especially reasonable for long-term series, this study will focus on the problems with the data and methodology of “anomaly”^{[5]} studies, and will discuss the different economic reasons respective to beta, size, and B/M effects in stock returns. Most of the published studies use long-term series of longer than 30 years, where the stock market returns are quite stable and only small shocks are included.

Having a short-term period of only five years and a very turbulent stock market during this time, a direct result of the worst global financial crisis since the Great Depression of the 1930s, one can expect a different structure of the relationship between average stock returns and the three characteristics.

Therefore, testing different procedures may help provide a better view on how the results of the different procedures are affected when the observed time period is very short and turbulent.

One motivation for concentrating on different methodologies is the tremendous amount of discussion in the academic literature about approaches, results, and interpretations of beta, size, and В/M. While some researchers encourage the academic world to reconsider the CAPM, others criticize the procedures used in many studies, which have shown the superiority of models where size and/or В/M are added. To conclude that the CAPM-S should be adjusted by the two characteristics or even be ignored by another model consisting of these characteristics is premature, since not all studies have shown the same tendency when different markets and methodologies were tested.

Another motivation for the topic in general is whether it would be possible, assuming that the characteristics size and В/M have some predictive power and are related to average stock returns, to generate an advantageous position when trading in equity capital markets. Instead of time-consuming information gathering for stock-picking and very cost-intensive trading when market matching, one could earn higher returns using a passive strategy by just buying stocks which show the value of the characteristics that seem to predict the highest returns,^{[6]} if it can be proven in general that size and B/M have predictive power for stock returns.

This thesis is organized as follows: In section 2, findings and economic interpretations in the literature about beta, size and В/M, are discussed. The first findings, especially about size and В/M, are briefly reconsidered and recent developments are presented and further discussed. Section 3 describes the data used for the empirical study and discusses the specialties of the data preparation used, when testing for size and B/M effects. The methodologies and results are then presented in section 4. Concluding remarks are found in section 5.

## 2 Literature Review

### 2.1 Validity of the CAPM-j3

In the earlier studies of Black, Jensen, and Scholes (1972) and Fama and MacBeth (1973), a positive relation between average stock returns and beta was found. While these results hold especially for the pre-1969 period, more recent empirical studies testing the validity of the CAPM-S did not satisfy the conditions. For a large data set including stocks from the three largest stock exchanges^{[7]} in the United States from 1963 to 1990, FF (1992) cannot confirm the validity of the CAPM any longer. For the longer time period of 1941 to 1990, they only find a weak relation between beta and average stock returns.^{[8]}

For Germany, empirical evidence cannot support the theory of the CAPM. Testing for beta alone, Wallmeier (2000) even finds a slightly negative, but insignificant relationship between beta and average stock returns (45). For his study he uses a wider data set than other studies for German markets, which usually only include the “Amtlicher Handel” for its higher liquidity and superior transparency compared to minor market segments. But the results of empirical studies concentrated only on the “Amtlicher Handel” also do not have an outcome that suggests for the validity of the CAPM.

Sattler (1994)^{[9]} and Bunke, Sommerfeld, and Stehle (1998)^{[10]} test the explanatory power of beta together with size and B/M cross-sectionally, and do not have significant values for beta.

One approach used to adjust the model to the empirical evidence that the standard form of the CAPM does not show a significant relation to average stock returns, is to loosen the strict assumptions of the model or to implement some practical features which are not accounted for in the standard theoretical model. Merton’s Intertemporal CAPM, which accounts for hedges of investors against shortfalls in consumption or savings; the post-tax CAPM, which factors taxes paid for dividends and capital gains into the model; or the Consumption-based CAPM, where instead the returns on a market index the growth rates of consumption are used as the independent variable, are a few examples. Several other extensions were developed, while most of them are impractical to test or did not show significant results.

But more interesting in the context of this study are the different methodologies, used to test the standard CAPM. Three of them will be discussed now. First, the model itself does not give any details about the time interval over which it should be tested. Meanwhile, the time interval of stock returns and market returns seem to be an important aspect of testing the validity of the CAPM. Handa, Kothari, and Wasley (1989) find that betas of high risk (low capitalized) securities increase when the time interval is lengthened from one day to one year, while betas of low risk (high capitalized) securities decrease (84-99).^{[11]} Their finding is that “an asset’s covariance with the market and the market’s variance do not change proportionally as the return interval is changed.” Brailsford and Josev (1997) come to the same result for the Australian equity market (366-375). Additionally Handa et al. (1989) find that the standard error of calculating beta increases with the return interval (84-86).

Roll (1981) also compares betas of different time intervals, and underlines the problem of autocorrelation and infrequent trading in average stock returns especially for small firms (881-887). Stocks of firms with a low market capitalization have a lower average trading volume than those of big firms. Therefore the probability that stocks of small firms are not traded for one or more days can be quite high. This problem of infrequent trading induces autocorrelation in stock returns. If one uses these stock returns to calculate beta, the model suffers from a statistical bias.

So, while monthly stock returns may also suffer from infrequent trading, it offers a good compromise between the high autocorrelation caused by short return intervals and the high standard error caused by longer return intervals. Since most of the recent studies use monthly return data because of its statistical quality, monthly returns are also used in this study.

Secondly, an important empirical problem ignored by researchers in most cases is the previously mentioned problem of “infrequent trading.” Dimson (1979) argues that if betas are calculated of shares that are not traded frequently, the beta can suffer from an underestimated covariance with the market and will be downward biased. In contrast, betas calculated with frequently-traded shares are upward biased. Sorting the stocks of the London Stock Exchange from 1955 to 1974 into portfolios of different trading frequencies and calculating betas of different time intervals from monthly to half- yearly, he finds that the downward bias in betas of infrequently-traded shares and the upward bias in betas of frequently-traded shares diminishes when the time interval is increased. Concluding that this phenomenon is caused mainly by the different trading frequencies, Dimson suggests a model where the beta does not only consist of a regression from stock returns with the market returns of the same point in time. He includes lagged and leading market returns which should take into account the problem of infrequent trading, and aggregates those together with the beta of the matching returns in order to get a less biased beta. The aggregation of the lagged, matched, and leading beta coefficients truly results in a less biased and more significant estimator of beta compared to the beta without lags and leads in his study. As Dimson shows, the first lag contains most of the explanatory power when compared with the matched marketreturns. (218)

**[...]**

^{[1]} Intertemporal CAPM, post-tax CAPM, Consumption-based CAPM and other forms of the standard Capital Asset Pricing Model were developed and tested.

^{[2]} An important assumption of the CAPM is that the share price reflects the information available (three different levels for the extent of information-availability exist: weak, semi-strong and strong form)

^{[3]} For example Chan, Karceski, and Lakonishok (1998) use ratios of cash-flows, earnings and dividends to the market value of equity as fundamental factors and growth rate of monthly industrial production, default premium, and real interest rates as macroeconomic factors for US data. Wallmeier (2000) uses factors including cash-flow and financial leverage for German data. In both papers, many other factors are tested and numerous other papers exist where different factors are tested for many different markets.

^{[4]} See Schulz and Stehle (2002) for an overview of seven empirical studies in Germany. Four of them test the relationship between beta and average stock returns and do not show a significant relationship. In contrast, size and book-to-market show a significant relation to average stock returns. While the procedures and data are different between the studies, similar tendencies can be observed.

^{[5]} Whether the explanatory power of size and B/M ratio should be called anomaly can be debated. According to standard Asset Pricing Models such as CAPM, other factors or characteristics should not have any explanatory power. Therefore the size and B/M effect is often called anomaly. Berk (1995) mentions that merely empirical evidence of a size effect is not definite evidence that it can be seen as an additional risk factor, but instead as an anomaly, which is not taken into account by standard Asset Pricing Models. Serving a theoretical corroboration of the size effect, he advises that it should not be called an anomaly anymore.

^{[6]} Daniel, Grinblatt, Titman, and Wermers (1997) find that, for a large data set of mutual funds over the time period from 1975 to 1994, actively managed mutual funds have the ability to outperform a mechanical strategy which would be based on investing in stocks where size and В/M values seem to predict high returns, but conclude that the management fees paid for investing in these funds do not take into account the level of outperformance of these mutual funds over an mechanical strategy based on size and B/M.

^{[7]} New York Stock Exchange (NYSE), American Stock Exchange (AMEX) and NASDAQ were the three largest stock exchanges in the United States by trading volume during this time period.

^{[8]} See FF(1992), p.428.

^{[9]} See Schulz and Stehle (2002), p. 12.

^{[10]} See Bunkeetal. (1998),p. 9-11.

^{[11]} In the paper of Handa et al. (1989), betas calculated with different return intervals are compared using size-sorted portfolios. The whole sample of stock returns and their betas are sorted into portfolios separated by the level of their of market capitalization. So reasoning that high risk betas are in the portfolio with the low market capitalization implies that Handa et al. consider size to be a risk factor in this context.

- Quote paper
- David Bosch (Author), 2010, Size and Book-to-Market Effects in the German Stock Market, 2005-2009, Munich, GRIN Verlag, https://www.grin.com/document/364783

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