Common risk factors in the German stock market

Index

List of Tables

Abbreviations

1 Introduction

2 The Efficient Market Theory and Asset Pricing Models
2.1 The Efficient Market Hypothesis
2.2 The Joint Hypothesis Problem
2.3 Asset Pricing Models - Literature Review
2.4 Arbitrage Pricing Theory
2.5 The Intertemporal CAPM
2.6 Finding the Factors
2.7 The Fama and French Three-Factor Model

3 Behavioral Explanations of Market Anomalies
3.1 The Limits to Arbitrage
3.2 Investor’s Psychology
3.3 Empirical Evidence on Behavioral Phenomena
3.4 Behavioral Models

4 Empirics - The Fama-French Three-Factor Model for the German Market
4.1 Purpose of the Empirical Study
4.2 Testing Approach
4.3 Data and Methodology
4.4 The 16 Dependent Portfolios
4.5 The Common Risk Factors: RM-RF, SMB and HML
4.6 Results for the Time Series Regression
4.7 Analysis of the Alpha Constant
4.8 Testing for E/P and C/P
4.9 Risk in Value Strategies
4.10 Testing for Seasonality in Returns

5 Risk or Irrationality?
5.1 Rational Interpretations of the Three-Factor model
5.2 Behavioral Interpretations of the Three-Factor Model

6 Conclusion

References

A. Appendix

List of Tables

Table 1 - Descriptive Statistics for the 16 Dependent Portfolios

Table 2 - Monthly Excess Returns and Traditional Risk Measures for 16 portfolios. July 1990 – June 2007

Table 3 - Summary Statistics for Monthly Explanatory Returns, July 1990 to June 2007

Table 4 - Regression Results for One-Factor Regression

Table 5 - Regression Results for Two-Factor Regression

Table 6 - Regression Results for Three-Factor Regression

Table 7 - Analysis of the Alpha Constants

Table 8 - Summary Statistics for Portfolios based on E/P and C/P, July 1990 to June 2007

Table 9 - Regression Results for Portfolios based on E/P and C/P

Table 10 - Performance of Value Stocks Relative to Growth Stocks over Different Time Horizons

Table 11 - Performance of Value vs. Growth in Different States of the Market

Table 12 - January and Non-January Returns for Risk Factors and the 16 Dependent Portfolios, July 1990 to June 2007

Table 13 - Regression Results Using a January Dummy Variable

Abbreviations

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1 Introduction

Modern academic thinking and theory about asset pricing can be divided into two broad categories. The first one bases asset pricing on rational factor models such as the traditional Capital Asset Pricing Model (CAPM) or expanded versions of it such as the Arbitrage Pricing Theory (APT). Basic assumptions of these models are that markets are efficient and that investors are rational. One of the most influential models has been the Fama-French (1993) three factor model, which extends the CAPM and explains the cross-sectional variation in asset prices by their exposure to three common risk factors proxying for some underlying economic variables. The risk factors are the market premium, size measured by a firm’s market capitalization and the ratio of a company’s book value to its market value.

The second category of modern finance theory focuses on investor’s psychology and non-rational explanations for asset pricing and differences in stock returns. This new field, called behavioral finance, came up in the 1990s as an alternative to the traditional, risk-based asset pricing models. Behavioral finance models allow for investor irrationality and limits to arbitrage, two important assumptions in that field to explain market anomalies.

The possibility of return predictability is a crucial point for investors and thus for portfolio advice in academic finance. Supporters of the risk-based models in the first category do not rule out that it is possible for an investor to predict stock returns in the long-run.^[1] However, since superior returns only appear with additional risk involved, this predictability is not a sign for market inefficiency and thus completely rational. For supporters of behavioral models, return predictability is a market anomaly caused by irrational investors. Some investors constantly act in a non- or bounded rational way in the stock market and drive stock prices away from their fundamental value. Hence, it is possible to exploit these mispricings by smart investors who can earn superior returns without additional risk involved. Clearly, this return predictability arises from market inefficiencies.

Most of the research in both categories of asset pricing so far has been done for the US market; studies for the German market are rare and are limited in various ways. Thus, this paper redresses these limitations, extends prior work for the German market and provides out-of-sample (non-US) evidence for asset pricing models.

The study focuses on the German stock market and deals with the question if it is possible to explain the cross-section in average returns by a multifactor asset pricing model. The only paper using the Fama-French approach has been recently done by Stehle et al. (2007). In that study, the Fama and French model is empirically verified for the time period between July 1968 and June 1995 in the German market. However, the paper does not test the validity of the model during the very interesting periods of the stock market boom of the late 1990s, the sharp downturn afterwards and the following large recovery which is still ongoing. Therefore, this paper focuses on the more recent developments. It is tested if the Fama-French three-factor model is able to explain return differences for portfolios in the German stock market for the period July 1990 to June 2007. The results show that the model works very well during the considered period and for longer-term horizons. It also extends the Stehle et al. analysis by testing the model for additional market anomalies, for example the return predictability caused by high earnings-to-price (E/P) or cash flow-to-price (C/P) ratios of stocks.

In a second step, the paper examines if the validity of the three-factor model can be explained by rational explanations, or if the field of behavioral finance is superior in explaining the return predictability caused by the market anomalies found. This is an approach which has not been done so far in studies for the German market. Previous studies for that market only focus on rational explanations for stock market anomalies. The conclusion is that many objections against a risk-based explanation can be found empirically. A non-parametric analysis of portfolio returns shows that superior returns come up without additional risk involved; a different test highlights the existence of a January effect and seasonality in returns. Both effects cannot be explained by a rational multifactor model. Thus, this paper argues that behavioral models are superior in describing market anomalies compared to traditional, risk-based models.

The paper is structured as follows: Section 2 gives an overview about the efficient market theory, the most important risk-based asset pricing models and the empirical results for those models. Aspects of investor psychology, behavioral models of asset pricing and their empirical foundations and implications are presented in section 3. Section 4 contains the main focus of the paper, the empirical analysis of the Fama-French three factor model for the German market. It also includes tests for possible non-rational based asset-pricing. Section 5 summarizes the empirical results and gives possible interpretations of the findings, section 6 concludes.

2 The Efficient Market Theory and Asset Pricing Models

2.1 The Efficient Market Hypothesis

Since the late 1960s, the central proposition for financial markets has been the Efficient Market Hypothesis (EMH). Under investor rationality and frictionless markets, a stock’s price should reflect its underlying fundamental value, given by the available public information. One common measure of fundamental value is the Discounted Cash Flow model, which states that the stock price today is just the sum of its future expected dividends, discounted by expected returns:

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, where Pt is the stock price today, Dt+j is the expected dividend in period t+j and rt+k is the expected return for the stock in period t+k. Instead of expected dividends, it is also common to use expected cash-flows.

If this relationship and the EMH hold, it should not be possible to have free lunch in the market, meaning that no investment strategy can earn a superior risk-adjusted return. If there is new information available, prices should immediately adjust in a correct way according to (I). ‘Correct’ is the crucial term here since it depends on how investors form their expectations in response to new information.

The main research on market efficiency can be divided into three major categories.^[2] The first category, the weak-form tests, analyzes if it is possible to predict future stock returns using past return data. The second alternative is the semi-strong form, which tests if all publicly available information is integrated into today’s market price. Fama (1990) suggests the title event studies for that kind of test. The third form of market efficiency, the strong one, includes private information into the test.

2.2 The Joint Hypothesis Problem

This study mainly focuses on the first category of market efficiency, meaning to ask if there is return predictability from using past data. However, the analysis in this paper does not directly test if market returns are indeed predictable by using past returns, instead it tests if the asset-pricing model considered by Fama and French (1993) is valid or not. This relates to the joint-hypothesis problem, which states that on the one hand, a successful test on market efficiency is conditional on a valid asset-pricing model, and on the other hand, one can only test an asset-pricing model conditional on market efficiency. This important problem of testing market efficiency was pointed out by Fama (1970). Stale information should be of no value to investors for making money, if prices react quickly and correctly to new available information. But the definition of ‘making money’ in academic literature means that an investor can earn superior returns only by bearing additional risk.

Thus, a strategy that has positive superior (or abnormal) returns over time is not by itself a sign for market inefficiency but has to be tested with an underlying equilibrium model describing what risk means from an investment perspective. As a result, every indication of market inefficiency is depending on the asset-pricing-model used. Campbell et al. describe the phenomenon as follows: ‘any test of efficiency must assume an equilibrium model that defines normal security returns. If efficiency is rejected, this could be because the market is truly inefficient or because an incorrect equilibrium model has been assumed.’^[3]

2.3 Asset Pricing Models - Literature Review

The first important asset pricing model was developed independently by Sharpe (1964), Lintner (1965) and Black (1972), usually referred to as the Capital Asset Pricing Model (CAPM), and has since then influenced the way how investors and academics should think about average return and risk. The CAPM assumes a single-period, utility maximizing investor who bases his investment choice solely on risk and return, trying to achieve the highest possible Sharpe Ratio (measured by the excess return on a portfolio divided by its standard deviation). No transaction costs are considered and all investors have homogeneous views on variances and covariances of securities’ returns. Furthermore, all investors can borrow and lend at the risk free rate.^[4] The model’s most important assumption is that there is a simple linear relationship between the expected return of a stock and its market risk, covered by the beta of a stock. Beta, also called systematic risk, is defined as the covariance of a stock’s return with the return on the market portfolio divided by the market portfolio’s variance:

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Moreover, the market portfolio rm in equilibrium pricing should be ex ante mean-variance efficient according to Markowitz (1959) which implies that it must include all possible investment opportunities for an investor. As a result, a stock’s expected return should be described by the simple equation:

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so that all returns should lie on the so-called securities market line. Furthermore, beta should be the only measure of risk needed to explain the cross-section of returns. The implication of the CAPM is as follows: A strategy which earns high (low) returns must have by definition a higher (lower) systematic risk. Another basic insight of the linear regression estimation conducted in that study is that if alpha is defined as:

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this constant should be zero for every single asset. A non zero-alpha implies that it is possible for an investor to form a portfolio with the same risk measured by beta, but a higher return. Thus, the market portfolio would not be mean-variant efficient if the alpha constant differs from zero. That can be interpreted as a zero-beta, zero-investment strategy.

Early studies such as Black et al. (1972) or Fama and MacBeth (1973) suggested that equations (II) and (III) are empirically verifiable. Both studies give evidence to the fact that the beta factor has a significant influence in explaining security returns. However, the problem with the model is that not only the beta of a stock influences the stock returns, but also other characteristics or factors such as the Price-to-Earnings Ratio (PE), size (captured by market capitalization of a stock) or the ratio between book value and market value (BM) of a stock seem to play an important role for variation in stock returns. Without considering these factors in an asset-pricing model, average returns for several strategies are either higher or lower than predicted by beta as shown by the studies presented in the following.

The first criticism comes from Roll (1977), where it is argued that the proxy used for the market portfolio in the early studies did not capture the overall investment opportunities which are available for an investor. Stambaugh (1982) shows that tests for the single-period asset pricing model are not sensitive to these choices, but the problem to find the appropriate market portfolio still cannot be ruled out. In a fully-fledged CAPM, the market portfolio must contain all available investment opportunities for an investor. However, indices such as the Standard & Poor 500 or the Deutsche Aktien Index (DAX) 100, which do not fulfill these requirements, are commonly used as proxy for the market portfolio.

A strong and prominent argument against the validity of the single-factor CAPM is reported by Banz (1981). Banz shows for the 1936-1976 time period that small size stocks had on average higher-risk adjusted returns than big size ones in the US market. Banz leaves the question open if size is another factor itself or if it is just a proxy for a true but unknown factor correlated with size. The effect has been known as the size effect since then.

An empirical confirmation for the size effect in the German market was first shown by Domke (1987) and Schnittke (1989). Beiker (1993) extends these two studies and shows that the time period chosen is very important for the relevant results, which is also validated by Stehle (1997). Stehle shows that stocks of small size firms have a risk-adjusted return of 6% more than big size stocks for the period from 1954 to 1990 and reports that this size-effect is mostly due to returns in January. Another important fact is that small stocks had a beta of less than one, big stocks a beta greater than one. This is opposite to the findings for the US market, where small stocks tend to have a beta greater than one, big stocks a beta smaller than one.

A study of Rosenberg et al. (1985) for the first time highlights a relationship between BM and the return of a stock. It is shown that stocks with a high ratio in that category do on average have higher returns than the corresponding stocks with low BM ratios. Depending on the time period analyzed, the excess return lies between four and five percent per year. This is called the value-growth anomaly in returns. Fama and French (1991) and Chan et al. (1991) confirm this result and report a positive relationship between returns and BM with explanatory power when controlling for beta. Studies for the German market exist from Capaul et al. (1993), Beiker (1993), Gehrke (1994) and Sattler (1994). They all come to similar results as in the US studies, stating that the anomaly is existent in the German market as well.

Another anomaly which was discovered in empirical research was based on the earnings-to-price ratio (E/P). Basu (1983) shows that expected returns are positively related to E/P when controlling for beta for the time period 1963-1980. Additionally, he finds that after controlling for E/P and systematic risk, the size effect in the market virtually disappears.

Fama and French (1992) discover a very important result from analyzing the presented anomalies through multivariate cross-sectional regressions. The bottom-line is that the beta factor is not helpful in explaining the cross-section of average-returns and that a combination of BM and size ‘seems to absorb the roles of leverage and E/P in average stock returns’^[5].

All these reported discrepancies from the one-factor CAPM suggest that the model should be extended. A possible model which is suitable for capturing many factors is the Arbitrage Pricing Theory developed by Ross (1976) and the Intertemporal CAPM by Merton (1973).

2.4 Arbitrage Pricing Theory

The Arbitrage Pricing Theory (APT)^[6] is a very appropriate and helpful alternative to the single-period CAPM described above. There are two main and important differences to the CAPM: First, the APT allows for more than one factor, therefore being a multi-factor model. Second, the APT bases its foundation on the condition of no arbitrage, implying that the market portfolio is not necessarily mean-variance efficient. The condition of ruling out riskless arbitrage profits leads directly to the APT and to an equilibrium which is characterized by a linear relationship between the expected return of a stock (or portfolio) and its loading on common factors. Another feature of the APT is that it allows for multiple periods.

The APT theory assumes, much like the CAPM, perfectly competitive and frictionless asset markets. The return generating process from an individual’s point of view is (homogenously) believed to be of the form:

illustration not visible in this excerpt, (V)

where E(ri) is the expected return on asset i, Fk is the mean-zero factor common to the returns of all assets and bi is the coefficient which quantifies the sensitivity of an asset’s returns to the movement in the common factor. In the APT, the common factors are the systematic risk components for the return generating process. The random noise term ε captures the unsystematic or idiosyncratic risk of the asset, which is due to specific information unrelated to other assets, and is assumed to be mean-zero. The assumption of mean-zero factors highlights that it is only the deviation from the mean which is important for the random return of the asset. The APT does not tell which ones are the common or systematic factors; it just assumes that there are some. That means that the model can be tested for any possible combination of possible common risk factors.

The APT is derived from the return-generating process in the following way: An individual with wealth invested in the market is searching for an investment alternative. By changing his portfolio, the sum of the investment proportions xi changes must equal zero, since overall wealth must be unchanged. These new portfolios are called arbitrage or zero-cost portfolios. These portfolios are assumed to be well-diversified, thereby having no idiosyncratic risk for large n, and additionally, they will have no systematic risk.^[7] For such portfolios, the random return for the arbitrage portfolio will equal its expected return and it is not possible to earn higher returns without being exposed to greater risk. It is always possible for an investor to mimic his original portfolio with an investment in the market without using additional money. By the no arbitrage condition, it is not possible that a combination of zero-cost portfolios with the same exposure to common factors earns a higher return than the original portfolio. The resulting expected return for a portfolio j then is:

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where λj is the risk premium on factor Fj.^[8] If lambda is non-negative, it is a priced factor. The coefficients bj are often called the factor loadings.

Hence, the APT states that it is only systematic risk which matters for security pricing in an arbitrage-free world. As stated before, the beneficial feature of the APT is that it can easily accommodate any realistic factor structure of returns. But it does not tell us what the factors and the risk premia are. This is left to empirical research.

2.5 The Intertemporal CAPM

Merton (1973) developes the Intertemporal CAPM (ICAPM), an extension of the CAPM, using a different approach than the APT. Merton assumes an infinite investment horizon for a single individual and a stochastic investment opportunity set. This means that it is possible for the interest rate and the market risk premium to fluctuate randomly. Due to that, investor’s want to hedge future investment opportunities to ensure an appropriate wealth level in different states of the world. The expected return for the stock i in the ICAPM is:

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where λk describes the premium for bearing the exposure of changing investment opportunities k. The ICAPM differs from the APT in the way that it only assumes priced factors to influence the expected return, hence the factors affecting marginal utility or changes in the investment opportunity set. The model’s important implication is that for an empirical test, it is not necessary to find a high R2 close to 100 percent, but zero alphas.

The CAPM, APT and ICAPM share one important assumption: it is the marginal value of wealth which is affecting investor decisions. Investors would like to hold assets which give high payoffs when the marginal value of wealth is high and vice versa. This is captured by the covariance of a stock or a portfolio with the various factors (the betas) in the different models. Thus, investors will drive up prices of these assets, thus implying low expected returns and low risk premia for those assets. That means that stocks with high expected returns should pay off badly in bad times of the stock market.^[9]

2.6 Finding the Factors

Research from the 1980s onwards concentrated on finding possible common factors for the cross-sectional variation in stock returns by using the multi-factor models presented in the previous chapter, either using stock specific factors as book-to-market ratio, or macroeconomic factors as economic growth. Ball (1978) argues that the ratio of companies’ earnings to its stock price is a general proxy for risk and expected return. Comparing two stocks with the same earnings but different risk pattern, the stock with a higher risk should have a higher expected return, leading to a lower price and thus higher E/P. Hence, E/P would be of high explanatory power when asset-pricing follows a multi-factor model since it proxies for the factors not used in the model. Another approach was made by Chen et al. (1986). They look at macroeconomic variables as industrial production or inflation among other factors and estimate their factor betas from a time-series regression. As a result, average returns and macroeconomic indicators are related but they can not explain the value or size anomaly in stock returns.

2.7 The Fama and French Three-Factor Model

For this paper, the most important analysis so far has been done by Fama and French (1993). They use an APT approach with zero-cost, factor-mimicking portfolios trying to explain the cross-section of average returns. Following their previous work, Fama and French, FF henceforth, define book-to-market and size as proxies for common risk factors in the stock market. If an APT model is capable of describing the cross-sectional variation in average returns, the factor sensitivities to BM and size should have explanatory power for this variation. The first zero cost portfolio, RM-RF, is the excess return on the broad market portfolio, the second one, called SMB (small minus big) uses the difference in returns between small and big stocks as proxy for risk, and the third one, HML (high minus low), covers the difference in returns between value and growth stocks.^[10] The expected excess return on portfolio i then is:

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where ß, s, and h are the slopes in the time-series regression:

illustration not visible in this excerpt. (IX).

The basic finding of the study is that the presented three-factor model does a very good job in explaining the cross-section of average returns in the US market in the period between July 1963 and December 1991. Not only the size and value anomaly can be described by the model, but also patterns in returns due to E/P, cash flow-to-price (C/P) or past sales growth.^[11] SMB and HML seem to mimic two underlying risk factors which are of special hedging concern to investors, which is in line with Merton’s (1973) ICAPM or Ross’s (1976) APT.^[12] The question remains which economic factors or variables are covered by SMB and HML. FF (1993) interpret HML as a factor proxying for a risk premium on relative distress, since low BM stocks are usually associated with persistently low earnings. However, the evidence is still weak and doubtful. Section 5 will continue to raise that question when interpreting the empirical results for the German market.

Similar studies explaining differences in the cross-section of stock returns for the German market are rare. Wallmeier (2000) uses the Fama-MacBeth approach to test for determinants of expected returns in the German stock market for the time period 1967-1994. He finds that there is a significant relationship between average returns and the ratios BM and C/P.

The only study for the German market using the presented three-factor model and a time-series regression was conducted very recently by Stehle et al. (2007) for the time period July 1968 to June 1995. The paper shows that a multi-factor model is indeed superior to the CAPM in describing excess returns on portfolios formed on BM and size as well. However, the study does not address the question if the three-factor model does capture cross-sectional variation in stock returns for portfolios formed on E/P or C/P. Section 3 presents the empirical work on the more recent period, July 1990 to June 2007, and extends the previous work for the German market by testing the three-factor model for differently formed portfolios.

But before dealing with this question, the next section describes a relatively new theory of academic thinking about capital markets, named Behavioral Finance, and its definition and explanation of stock market anomalies.

3 Behavioral Explanations of Market Anomalies

The efficient market hypothesis and the asset pricing models described in section 2 combine one important and basic assumption: the existence of rational investors. Rationality of investors means that they update their beliefs correctly according to Bayes’ law, and that they maximize their (subjective) expected utility.^[13] A basic problem which arises when looking at market anomalies, for example in the cross-section of average returns, is that rational theories can only describe or explain these market imperfections in a very limited sense, at least in the view of the academics favoring the behavioral approach. Thus, there is the need for modeling investor’s behavior more realistically which is done by behavioral theory.

3.1 The Limits to Arbitrage

Theories of efficient markets usually assume frictionless markets, meaning that transactions are costless and it is possible to borrow and lend at a risk free rate. Furthermore, it is commonly supposed that arbitrage is possible and involves no risk. One example is the APT, whose basic principle is an arbitrage-free market. If an arbitrage opportunity arises, speculative traders will exploit and correct it immediately. The problem of that assumption is that in the real investment world, arbitrage can indeed be risky and additional costs can be involved. Thus, prices can persist which deviate far away from their fundamental value. Two important types of these limits to arbitrage are noise trader risk and fundamental risk. To highlight this issue, the distinction between rational and so-called noise traders is introduced. The term noise trader was introduced by Black (1986). Black contrasts noise and information in financial markets. Whereas information trading is based on fundamental facts, ‘noise trading is trading on noise as if it were information.’^[14] This non-rational investment approach of some stock market participants can lead to prices which do not reflect the true fundamental value since it reflects opinions of both information and noise traders. Concerning arbitrage opportunities, this means that even if the smart stock market participants have better and true information about a stock’s fundamental value, it will not be possible for them to take the opposite position eliminating the mispricing without facing risk, at least in the short-run. Thus, arbitrage in financial markets can be risky and stocks prices can deviate from their fundamental value. As Black puts it, ‘noise creates the opportunity to trade profitably, but at the same time makes it difficult [risky] to trade profitably.’^[15]

Other important forms of risky arbitrage are implementation costs such as commissions, bid-ask spreads, price impacts or short-sale constraints. Arbitrageurs are faced with costs to eliminate mispricing. If these costs are too high, the speculative traders may not take a position if the cost of the strategy is higher than its return. Costs for searching and learning about a mispricing also fall into this category (Merton 1987). However, these latter reasons for mispricings do not include investor behavior and are thus of minor importance for this study.

The implication of limits to arbitrage on the validity of asset pricing models is shown in section 3.3. Before, additional important aspects of investor’s psychology are presented.

3.2 Investor’s Psychology

To understand the term ‘noise’ in a more detailed way, some evidence from psychological research is presented to have a background on what drives noise traders and individual behavior and beliefs in financial markets. Of the several findings in psychology only the most important ones for the analysis in this paper are presented here.^[16] These are Overconfidence, Representativeness, Conservatism and Positive Feedback Trading.

Overconfidence means two things: first, people assign their confidence intervals too narrow, for example about the future level of the DAX. Second, people are poor in estimating probabilities. Fischhoff et al. (1972) show that events which are certain in people’s belief occur only 80% of the time, and events which they think are impossible occur with a probability of about 20%.

The representativeness bias was first shown by Kahnemann and Tversky (1974). In forming their beliefs, people do not tend to follow statistical theories of prediction. Instead, they ‘rely on limited number of heuristics […] which sometimes lead to severe and systematic errors.’^[17] Individuals do not apply Bayes’s law in a correct way, putting to much weight on conditional information. One of the outcomes of this heuristic representativeness is that people deduce from very small samples on the overall population. For example, if a soccer player scored three games in a row, the likelihood of scoring in the next game is overestimated.^[18] This is called the “law of small numbers”.

Conservatism is very similar to the latter bias, but with a different outcome. People seem to underweight new information or data if the new one is “too” different from any model in which people believed in before. Thus, individuals cling too much to their prior beliefs.

Positive feedback trading can be seen as a follow-up of representativeness and the “law of small numbers”. This behavior is due to noise traders who buy stocks after prices rise and sell them after prices fall. The trend-chasing behavior gives rise to stock bubbles like the tech-bubble in the late 1990s.^[19]

3.3 Empirical Evidence on Behavioral Phenomena

One of the first and most influential papers in the field of behavioral finance was published in 1985 by De Bondt and Thaler. The paper tests the overreaction hypothesis by building portfolios of stocks with prior high three-year cumulative return (winner portfolios) and prior low three-year performance (loser portfolios) and then analyzes their post-formation returns for different periods, every three years from 1926 to 1982. As a result, the loser portfolios outperform the winner portfolios three years after portfolio formation by about 25%.^[20] Furthermore, most of the superior returns of the winner portfolios are earned in January, an indication for a seasonality in returns. The overall results can be seen as evidence that investor’s do overreact to unexpected information or news and that their probability estimation violates Bayes’s law as presented in section 3.2.

A related, but different anomaly is the momentum effect first shown by Jegadeesh and Titman (1993). For the time period of January 1963 to December 1989, they use the same methodology as De Bondt and Thaler (1985), but sort the stocks according to their prior 6-month return and analyze the returns for pre-formation periods of two to twelve months. For these short-term periods, a trend-continuation effect can be seen. The biggest winner portfolios outperform the loser portfolios by an average of 10 percent per year. The conclusion drawn from the empirical evidence is that for short-term periods, investors seem to underreact to new information.

Lakonishok et al. (1994) reconsider the value effect by Fama and French (1992) and give alternative explanations for it. While FF argue (see section 2.7) that these strategies are fundamentally riskier due to a common factor proxying for financial distress, Lakonishok et al. base their analysis on the existence of ‘naive’ strategies, referring to the model of noise traders by Black (1986). Noise traders do extrapolate past information about earnings too far into the future, thereby pushing today’s stock price above its fundamental value and creating an overpriced glamour (growth) stock. At the same time, they oversell stocks with a bad history in earnings so that these value stocks become underpriced.^[21] Contrarian strategies which exploit these facts by selling glamour and buying value stocks give an outperformance of more than 10 percent per year in the period from April 1968 - April 1990 in the US market for an investment horizon of more than three years. On the other hand, several tests show that these value strategies are not riskier in terms of traditional measures of risk (standard deviation) or by controlling for systematic risk (beta).

3.4 Behavioral Models

The recent findings of predictability in returns due to investors who do not act strictly rational offer a hard challenge to the presented asset-pricing models in Section 2. They try to trace back differences in average returns on different loadings on economic risk factors, whereas behavioral finance tries to accept mispricing in the market due to investor’s misbehavior. New theories of asset pricing are only beginning to emerge. Two of them are presented here to provide explanation alternatives against the rational, risk-based asset pricing-models.

The first one is from Barberis et al. (1998) and belongs to the category of belief-based models. The authors combine the psychological findings of conservatism and representativeness, thereby showing that investors are systematically wrong in forming their expectations and in reacting to new available public information. Investors do not believe that earnings follow a random walk, but belief that the investment world has two possible regimes. Under the first one, companies' earnings are mean-reverting, under the second one, earnings trend. This means that the possibility of lower earnings in the future after a positive shock is higher under regime one than under regime two. Investors who belief in a process according to the first regime are subject to a conservatism bias; they react too little to new earnings announcements. Under the second regime, investors extrapolate current earnings too far into the future, thereby having a representativeness bias. The decision in which regime investors believe at any point in time is modeled by an underlying regime-switching process which leads investors to belief in this or that regime. The simulation outcomes of the model of investor sentiment are highly representative to the empirical findings of the value effect and short-term momentum. For a short-term horizon, stocks with high past returns do outperform loser stocks, and on the other hand, high E/P stocks do outperform their low counterparts over a long-term horizon without additional risk involved.^[22]

A second type of models is based on beliefs as well, but also incorporates institutional frictions. These models mainly explain differences in cross-sectional returns by investor's overconfidence and short-sale constraints. Overconfidence implies difference in opinion about a stock's future prospect. Without it, private investors would listen to information by other investors (for example institutional ones) about a stock's expected future earnings growth and opinions would converge rapidly. With overconfidence, the arising difference in opinion can lead to short-sale constraints, meaning that some stock is more expensive to short than another. As presented in section 3.2, higher costs of shorting can lead to mispricings, since arbitrage becomes costly.^[23] Empirical findings by Cheng et al. (2002), who use 'breadth of ownership' (the amount of institutional investors holding a stock) as a measure for difference in opinion about a stock validate this theory. When breadth is low, prices are high because no or only few pessimistic opinions are valuated in the stock price. Only investors who are optimistic about the stock's future prospects are invested and short-selling is costly. Thus, the current stock price is too high and future expected returns are low. The study shows that 'breadth of ownership' is indeed a valuation indicator and that reductions (increases) in breadth do forecast lower (higher) future returns.

The presented facts from theory and empirical research in the field of behavioral finance are an interesting challenge to the standard asset-pricing models. To verify the validity of these models in the German market, the next section tests the Fama-French three-factor model for that market. Furthermore, it addresses the question of risk-based superior returns to test what underlies the market anomalies, additional risk or investor irrationality.

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^[1] Return predictability through variables such as the dividend-to-price ratio over long horizons and business cycles has been shown in various studies; see for example Fama and French (1988).

^[2] See Fama (1990), pp.1576-1577.

^[3] Campbell et al. (1997), p. 24.

^[4] Black et al. (1972), pp. 1-2.

^[5] Fama and French (1992a), p. 428.

^[6] The presented version of the APT follows the empirical work of Roll and Ross (1980).

^[7] See Roll and Ross (1980), pp. 1078-1079.

^[8] For a complete derivation of the APT see Roll and Ross (1980).

^[9] For a detailed description and overview see Cochrane (2006), pp. 2-5.

^[10] For a detailed description on the portfolios see empirical work in Chapter 4.

^[11] See Fama and French (1996).

^[12] See Fama and French (1996), p. 57.

^[13] See Barberis and Thaler (2003), Chapter I.

^[14] Black (1986), p. 531.

^[15] Black (1986), p. 534.

^[16] For an introductory survey, see Barberis and Thaler (2003), chapter 3.

^[17] Kahnemann and Tversky (1974), p. 237.

^[18] See Barberis and Thaler (2003), p.1164.

^[19] See Shleifer (2000), chapter 6: Positive Feedback strategies.

^[20] See De Bondt and Thaler (1985), p. 799.

^[21] See Lakonishok et al. (1994), p. 2.

^[22] Investors are assumed to be risk-neutral in the Barberis et al. model.

^[23] For specific models, see Miller (1977) or Scheinkman and Xiong (2003).