Value Stocks beat Growth Stocks: An empirical Analysis for the German Stock Market

Table of contents

List of tables

List of figures

List of abbreviations

List of symbols

1. Introduction
1.1. Motivation
1.2. Structure and objective

2. Conceptual definitions
2.1. Value investing
2.2. Growth investing
2.3. Links between Value and Growth investing

3. Asset pricing theories
3.1. Capital Asset Pricing Model
3.1.1. Risk-free interest rate
3.1.2. Market risk premium
3.1.3. Beta factor
3.1.4. Criticism and extensions
3.2. Fama and French three factor model
3.3. Explanation approaches for the value premium
3.4 Carhart four factor model

4. Determinants of expected stock returns
4.1. Price-to-book
4.2. Price-to-earnings
4.3. Dividend yield
4.4. Size
4.5. Momentum
4.6. Further determinants

5. Empirical studies for the German market

6. Own empirical analysis
6.1. Data and methodology
6.2. Descriptive statistics
6.3. Seasonality
6.4. Univariate and multivariate regressions

7. Conclusion

List of references

List of tables

Table 1: Characteristics of value and growth stocks

Table 2: Historical market risk-premiums for the German market

Table 3: Market risk premiums – Analysts’ and professors’ forecasts for 2011

Table 4: Determination of the Deutsche Bank beta factor using three approaches

Table 5: Fama and French (1996) beta factor summary statistics

Table 6: Fama and French (1993) portfolio excess returns

Table 7: Fama and French (1993): Explanatory power of the market model

Table 8: Fama and French (1993): Explanatory power of the three factor model

Table 9: Components of stock returns

Table 10: Average number of firms

Table 11: Summary statistics 1992 - 2011

Table 12: Average Pearson Correlation Coefficients

Table 13: Average monthly returns

Table 14: Hedge portfolio (5 - 1) average monthly returns

Table 15: Returns related to seasonality

Table 16: Univariate regressions

Table 17: Explanatory power of different sets of models

List of figures

Figure 1: Development of German government bond yields with different maturities

Figure 2: Yield curves as at 07/06/2012

Figure 3: Momentum of the Adidas stock

Figure 4: German Interbank one-month offered rate

Figure 5: Development of firm characteristics.

Figure 6: Monthly average hedge portfolio returns

Figure 7: Seasonal hedge portfolio returns

Figure 8: Seasonal hedge portfolio returns in excess of the equity risk-premium

List of abbreviations

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List of symbols

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1. Introduction

1.1 Motivation

There has been an ongoing discussion for many years of whether value investing-, or growth investing strategies achieve higher stock market returns. Value firms can be defined as those that have a poor past performance and are expected to perform below average in the future, whereas growth firms as those that performed strongly in the past and are expected to have a high performance in the future. In literature value (growth) firms are characterized by low (high) price-to-earnings ratios [P/E], low (high) price-to-book ratios [P/B], low (high) price-to-cash-flow ratios [P/CF], low (high) price-to-sales ratios [P/S], and high (low) dividend yields [DY = D/P]. The rising academic interest in value and growth investment strategies can be traced back to the two seminal papers of Fama and French in 1992 and 1993. Their three-factor model containing additional risk premiums for “low minus high P/B stocks” and “small- minus big- sized stocks” meant a blow to the explanatory power of the Capital Asset Pricing Model (CAPM) of Sharpe (1964) and Lintner (1965) and rose the question of the “death” of the beta factor.^[1] Based on the evidence from numerous empirical investigations on the P/B effect and related anomalies, the consensus has been evolved that value investing strategies, on average, outperform growth investing strategies and thus generate a value premium. Chan et al. (1991), for example, find significant value premiums for the Japanese market with the P/B ratio being the best proxy for expected stock returns.^[2] Fama and French (1998) document that value stocks outperform growth stocks in 12 out of 13 major markets. They further test whether a global asset pricing model can explain stock returns and find that an international two-factor model does a good job.^[3] However, Griffin (2002), amongst others, states that domestic factor models had a higher power to explain time-series variation in stock returns than a world factor model.^[4] Beside the value premium proxies, a bulk of empirical research has been done on the size premium, i.e. that small companies, on average, perform better than big companies.^[5] In addition, recent literature has shown a high interest in a technical firm characteristic – the Momentum factor.^[6] Findings on these factors arouse the motivation to examine which trading strategies perform best for German stocks.

1.2. Structure and objective

Based on a “free of survivorship-bias” sample of German stocks listed at the Frankfurt stock exchange, the study investigates the ability of hedge portfolio formation structures, built of three value premium proxies (P/B, P/E, and DY), the size factor, and the technical momentum factor, to generate excess returns in the period 1992 to 2011. The study differs in three main aspects from common methodologies: First, financial firms are not excluded from the sample. Second, risk-factors are built from one-dimensional sorts, and not from double-sorts. And most important, third, all portfolios are rebalanced on a monthly basis, whereas common studies rearrange their portfolios on firm characteristics only once a year. These deviations should, in best, yield higher hedge returns in comparison to relevant studies.

The remainder is structured as follows: Section 2 gives an overview of characteristics and definitions that are associated with value and growth investing. Section 3 discusses the theoretical framework of asset pricing with the CAPM, the Fama and French three-factor model, and the Carhart extension. The most important determinants of expected stock returns are described in section 4. Related studies for the German stock market are presented in section 5. Before section 7 concludes, the empirical analysis is described in detail in section 6. Three main questions should be addressed:

1) Is there a value premium in the German market between 1992 and 2011, is there a reversed size premium like recent empirical findings^[7] suggest, and do high momentum stocks perform better than low momentum stocks?
2) Is there a significant seasonal pattern in hedge portfolio returns?
3) The combination of which factors best explains expected stock returns?

2. Conceptual definition

2.1. Value investing

The principles of value investing can be traced back to the two fundamental books of Benjamin Graham – “Security Analysis”, which he wrote in 1934 coauthored by David Dodd, and the “Intelligent Investor”, which he wrote in 1949. Although Graham has not used the term to describe his approach, he is known as the father of value investing.^[8] Graham’s core principle remains the “margin of safety”, i.e., a security should only be purchased, if its price is substantially below its intrinsic value.^[9] The intrinsic value is the “actual value of a company or an asset based on an underlying perception ofits true value including all aspects of the business, in termsof both tangible and intangiblefactors”^[10]. Thus, value investors believe that the true value of a security is not reflected in its price. Value stocks tend to be cheap relative to their current intrinsic value. Market participants may not be willing to pay more for companies which are out of favor. Thus, the task of value investors is to indentify companies that are likely to manage a turnaround, leading to higher earnings and higher stock prices. To identify undervalued stocks, a comprehensive fundamental analysis has to be done.

John Burr Williams extended the concept of value investing by determining a company’s intrinsic value through discounting its probable future cash flows.^[11] Estimating future cash flows and the appropriate discount rate is difficult, takes some time and depends on realistic assumptions. Martin J Whitman (1999) lists some further properties of value investing:

- Value investing treats every accounting number or financial ratio equally and businesses are valued as wholes.
- The primary goal is to value a business independent of potential interests from passive minority investors.
- Macroeconomic factors like the Gross domestic product (GDP) or employment forecasts are not considered. The analysis aims only on the idiosyncratic factors which exert a long term influence on a company.
- The company analysis is strictly separated from market analysis.
- Value investing uses only financial analysis, no technical and chartist analyses.^[12]

The measures of what defines a value company can usually be found in the balance sheet. Value companies have typically high assets in the form of tangibles like machines, real estate and inventory. They often belong to industries like auto manufacturing, timber and metal. These companies usually operate with very low profit margins. Thus, value companies are usually characterized by a strong balance sheet.

An easy way to determine the intrinsic value of a company and to make stocks comparable is the application of multiples. This concept is used in finance literature as well as in practice. The intrinsic or fair value of a company is determined by means of financial ratios. If the current market value is below the intrinsic value of a company, it seems to be worth to buy the stock. The intrinsic value is often used synonymous with the book value of a firm, or more exactly the net asset value (NAV). The NAV corresponds to the equity of the firm. The market value (price times number of outstanding shares) of the company divided by the book value (equity in the balance sheet) results in the financial ratio “price-to-book” (P/B). The lower this ratio, the cheaper is the stock in relation to its peer group. Further ratios of the value approach are a low price-to-earnings ratio (P/E), a low price-to-sales ratio (P/S), and a low price-to-cash-flow ratio (P/CF).^[13] Thus, value managers invest mainly in companies with low valuations and stable earnings- and growth forecasts. Because of these attributes value investing is regarded as the appropriate investment style for defensive investors. At the moment, for example Volkswagen can be seen as a typical value company, trading at a P/B of 0.84 and a P/E of 2.9 in December 2011.^[14]

2.2. Growth investing

Growth companies or glamour firms have a stronger past performance than the average company and are expected to perform very strong in the future.^[15] According to Benjamin Graham “[t]he term “growth stock” is applied to one which has increased its per-share earnings in the past at well above the rate for common stocks generally and is expected to continue to do so in the future.”^[16] Growth investors believe that these high growth companies would enable them to outperform the stock-market in the long run. The market is willing to pay more for growth stocks, since these are leading companies with the potential for fast earnings growth, and may thus be worth considerably more in the future. Philip A. Fisher is regarded the forerunner in the field of growth investing. Beside the fundamental analysis, he also focused on qualitative aspects of a company like management ability, risk management, innovation and business strategy. Intangible assets like the brand-name and the goodwill of a firm became important.^[17] Fisher extended the number of value investing candidates by emphasizing that also cheap stocks can be unprofitable investments, in fact, if the stock trades below its intrinsic value because of bad management.^[18]

Simultaneously, the high potential of growth stocks bears the danger that expectations will be disappointed. The demand for growth stocks arised, above all, in the late 1990s with the internet boom. There were numerous initial public offerings (IPO’s) of companies with very high earnings growth expectations. In the German market, a well-known case is the case of EMTV. The German media company was issued on the “Neuer Markt” platform for a price of 35.50 Deutsche Mark (DM) (adjusted for various share splits: 0.35 EUR) in October 1997. In the beginning of 2000 the stock peaked at a price of 120 Euro (EUR). This meant a gain of over 30,000 percent since the IPO. In 1999, the company generated sales of DM 317 million (m) with 220 employees, whereas the market value exceeded DM 15 billion (bn).^[19] At this point of time, EMTV was valued as high as the Deutsche Aktienindex (DAX)^[20] company Lufthansa and had a very high P/S ratio of approximately 22. In the mid of 2000 the decline of the stock started and in April 2003 the Chief Executive Officers (CEO’s) were convicted to pay high fines due to wrong presentations of the condition of the company. EMTV was not the only company that failed to achieve the high expectations. After the burst of the “dot-com bubble”, a lot of companies of the so-called “New Economy” went bankrupt.

A similar phenomenon, happening at the moment, is the IPO of the social network Facebook. Since the first listing on May 18, the stock declined from USD 38 to USD 26.81^[21], a minus of approximately 30 per cent. This means a decline in the market value of USD 34.5 bn in a few weeks.^[22] On 7 June 2012, the company trades at a 2012 P/E of 49.27 and a 2013 P/E of 41.20.^[23] These high ratios are reminiscent of the “Neue Markt” valuations.

A growth-oriented investor sets the future in the foreground. Not the current sales or earnings are important, but the expectation of strong sales- and earnings growth. Hence, growth stocks are characterized by high P/B ratios, high P/E ratios and low dividend yields. Growth stocks trade with a growth premium. An extreme example for growth investing would be the investment in a start-up company. These companies have typically a low amount of assets and often negative earnings, but have at the same time a high potential for growth. Positive examples of recent growth companies are Apple and Google. The future will show, if Facebook becomes a “second Google” or a “second EMTV”.

2.3. Links between Value and Growth investing

Value stocks are considered to have a poor past performance and are expected to perform poorly in the future, whereas growth stocks performed strongly in the past and are expected to have a strong performance in the future. However, to draw this borderline and categorize stocks in either value or growth remains challenging in practice. The discrepancy between the price of a stock and its intrinsic value can make a value company into a growth stock and vice versa.^[24] This phenomenon arises, for example, if a well established and saturated company discloses low or no earnings. If valued only by means of the P/E ratio, the company will be classified as a growth stock. Furthermore, classifications can change over time. A company that is now classified as a growth company can be classified as a value company within a few years. The other way round – the development from a value company to a growth company – is also possible, even if this is more difficult. A prominent representative can be seen in Apple. After starting as a growth company in 1981, the stock developed more and more to a value company in the 1990s, by means of a decline in the P/E ratio. However, since 2004 the P/E has risen again. Today analysts and financial experts are at odds with Apple to be labeled as growth stock or value stock.^[25] Both, value and growth portfolio managers invest in Apple. This example shows again the difficulty of categorizing stocks into value and growth. Also Warren Buffet, the most famous student of Benjamin Graham and successful value investor said that the value and growth investment style are “joined at the hip”^[26], since growth is always part of value investing. However, table 1 tries to summarize different characteristics of value and growth investing:

Table 1: Characteristics of value and growth stocks

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(Source: Own representation based on Naumer et al. (2009), p. 5)

3. Asset pricing theories

3.1. Capital Asset Pricing Model

The Capital Asset Pricing Model (CAPM) is still the most important tool in practice to determine the expected stock return or cost of equity of a specific company. The CAPM is a one-period equilibrium model which is based on the portfolio theory of Markowitz^[27]. The model is based on several assumptions:^[28]

- Investors are risk-averse and thus choose the portfolio with the lowest risk out of all portfolios with the same return, i.e. minimizing the volatility at a given expected value (all investors use Markowitz optimization).
- Any fraction of securities can be traded at any point of time and there are no short-selling restrictions.
- Investors are price-taker, i.e. investments have no influence on the market price.
- There are no transaction costs and no taxes.
- Investors can borrow and lend money at the risk-free rate.
- Information is costless and available to everyone.
- All forms of securities can be traded.
- All market participants have homogenous expectations regarding returns, variances and covariances.
- Investors control risk through diversification.

According to the CAPM the expected return of a security i can be determined as a linear function of the firm’s idiosyncratic beta factor (β):

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Thus, according to the CAPM only three factors are necessary to calculate the expected stock return: the risk-free interest rate, the market risk-premium and the beta factor. Methods, how these components can be derived, are discussed in the following.

3.1.1. Risk-free interest rate

The basic interest rate is the return of a risk-free investment with identical terms at the valuation point.^[29] Since there is no risk-free debtor in a theoretical context, the risk-free interest rate is approximated by the returns^[30] of high-rated^[31] government bonds. In this context, in particular, two issues are discussed in literature: First, the question of which market data should be used – historical average returns of government bonds, the return that applies on the date of valuation, or the interest of zero-coupon bonds (spot rates). And second that there is no risk-free comparison security with an endless maturity.^[32] Most valuation approaches assume an infinite maturity, whereas government bonds are only available up to a maturity of 30 years.

Figure 1 illustrates the development of German government bond yields with different maturities. It is obvious that long-term bonds should have higher returns than short-term bonds, since they face a higher interest rate risk. In rare cases short-term bonds pay a higher interest than long-term bonds. This situation is called inverse yield curve and occurs, if the market expects decreasing interests in the future. Whereas average yields accounted for about seven per cent in 1994, a ten-year German government bond pays only 1.45 percent on 7 June 2012, and the one-year German government bond is close to zero with 0.03 percent.^[33] Figure 2 shows the yield curves of German government bonds, Eurobonds (red line) and Corporate bonds with differing ratings. Due to these historical low yields of German government bonds, in particular, in comparison to Eurobonds, it can be questioned, whether the derivation of the risk-free interest rate only out of German government bonds can be maintained.

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(Source: Own representation, Data from Thomson Reuters Datastream)

Figure 2: Yield curves as at 07/06/2012

illustration not visible in this excerpt(Source: Boerse Stuttgart https://www.boerse-stuttgart.de/en/toolsandservices/yieldcurves/yieldcurves.html, as of 20.06.2012)

Ballwieser (2007) proposes the Svensson approximation as the most favorable method to determine the risk-free interest rate.^[34] The “Institut der Wirtschaftsprüfer” (IDW)^[35] also follows this approach.^[36] The Svensson method is a continuous term structure approach that enables the estimation of spot rates for any maturity. The resulting equation for the zero-coupon yield curve is modeled using six parameters:^[37]

With Risk-free spot rate term structure

Vector of parameters describing the yield curve^[38]

Maturity at which the spot rate is calculated

Yield curves calculated with the Svensson method have a high degree of economic interpretation and are more consistent with the interest rate expectation theory.^[39] Thus, the Svensson method is also applied by most central banks.^[40] However, even if this approach gained a lot of approval in the recent past, a bulk of international analysts still use ten-year government bonds for approximation.

3.1.2. Market risk-premium

The market risk-premium (MRP) is the difference between the expected return on a market portfolio^[41] and the risk-free interest rate. Technically, it corresponds to the slope of the Security market line (SML) in the CAPM.^[42] There are two main approaches to derive the MRP: First by using historical data and second by using forward-looking approaches.^[43] Table 2 summarizes some studies that measure historical market risk premiums for the German market. The most influencing one is the paper of Stehle (2004). The IDW adapted the results and suggests a range of 4.5 to 5.5 per cent for business valuations.^[44] Due to the currently low basic interest rate and high capital market risks, the IDW proposes a market risk premium on the upper range.^[45] Stehle determines the market risk premium by subtracting the yearly return of the “Renten Performance Index” (REXP)^[46] from the “Composite-DAX” (CDAX) and the DAX respectively. The REXP serves as proxy for the risk free interest rate. Depending on what method of mean (arithmetic vs. geometric) is used, the historical market risk premium differs between 5.46 and 2.66 percent (CDAX) and 6.02 and 2.76 per cent (DAX). Stehle recommends the arithmetic mean for valuations. Furthermore, he prefers the CDAX due to its higher market breath, in particular for small and medium- sized companies (SME’s).^[47] The questions – what method of calculating the mean should be applied, what is the Benchmark index, what is the length of the observation period and are the numbers pre- or after- tax – lead to significant differences among the studies. The application of historical data has the advantage of an intersubjective confirmability. On the other side, it is not compatible with the CAPM as “ex-ante”- model and the future-oriented DCF-valuation. Furthermore, the method to build an average over historical data remains questionable, since market risk-premiums are not constant, but change over time. Stehle suggests a deduction of 1 to 1.5 per cent from the historical mean to take the contemporary higher possibilities of diversification into account.^[48]

illustration not visible in this excerpt^[49]

Table 2: Historical market risk-premiums for the German market

(Source: Own representation with regard to Stehle (1999), p. 19.)

The problem with a historical approach is that it is backward-looking. One possibility of a forward-looking approach is a simple dividend discount model (DDM). Within this model, the equity value can be calculated by discounting the expected future dividend payments. A second approach to achieve future-oriented risk-premiums is by linking the market risk premiums of equities to the default spread^[50] of corporate bonds.^[51] This approach bases on the assumption that risky securities of different asset classes should be priced consistently. Damodaran (2008) finds that the average ratio of the market risk premium to a Baa rated default spread from 1960 to 2007 is 2.41 and the median 2.02. He finds a high variation in the ratio (MRP – Baa^[52] default spread) which would oppose the advantage of the approach. However, this disadvantage is compensated by a reverting median.^[53] A third forward-looking approach is to ask analysts, investors, financial managers, or professors about their expectations regarding MRPs for single countries. Table 3 illustrates a survey with forecasts of financial experts for different countries in 2011. The analysts estimate significantly lower MRPs for countries like Germany, the USA, or the United Kingdom (median of 5.0 percent) than for countries like Argentina (9.0 percent) and China (7.8 percent). Surprisingly, regarding the European debt crisis, Spain (5.5 percent) and Italy (5.0 percent) are still estimated very low.

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Table 3: Market risk premiums – Analysts’ and professors’ forecasts for 2011

(Source: See Fernandez et al. (2011), p. 6.)

The question remains, which of the discussed approaches performs best? Since valuation is future-oriented, a forward-looking approach should be superior. Damodaran (2008) finds that an implied market risk premium at the end of the prior period reaches the highest correlation with the implied premium next year (0.758) and the actual risk premium the next ten years (0.376), whereas the historical premium approach performs worst.^[54] Summing up, it is difficult to determine the right MRP, an implied one seems to be preferable and the MRP changes over time. These facts should be considered by analysts.

3.1.3. Beta factor

The beta factor^[55] is a measure of the systematic risk that cannot be diversified. It expresses the correlation of a stock with the market portfolio. If a company is listed on a stock exchange, the beta factor can be determined using a linear regression of historical data. To obtain a valid result, historical data over a period of five years should be available.^[56] In practice, beta factors are often regressed on a daily basis over the past one year (for example Deutsche Börse). The financial services provider Bloomberg, amongst others, delivers two year-betas on the basis of weekly returns.^[57] Dörschell et al. (2008) compare the range of the beta factor of the Deutsche Bank to the DAX, applying the three calculation methods that are common in practice.^[58] They find a difference of 0.224 between the “one-year /daily”- beta and the “two-year /weekly”- beta. Since the beta factor influences the cost of equity of a company, the denominator in the valuation model will be influenced.^[59] As a consequence, the risk premium rises up to 1.12 per cent for the Deutsche Bank.^[60] It is evident that this would have a significant impact on the valuation of the company. For valuation purposes, Dörschell et al. (2008) suggest the application of a five-year beta on a monthly basis. The two remaining approaches should be used for plausibility checks.^[61]

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Table 4: Determination of the Deutsche Bank beta factor using three approaches

(Source: Own representation based on Dörschell et al. (2008), p. 1155.)

Beside the observation period and the return interval, the beta factor can also be influenced by the so-called “day-of-the-week effect”^[62], and tax-, currency-, or financing aspects.

If companies are not listed on a stock exchange, the beta factor cannot be determined using regressions of historical data. The beta factor can be calculated via a benchmark approach, instead. For this purpose, listed companies have to be identified that face a similar business risk than the target company.^[63] According to Geginat et al. (2006), this should include the following five steps:^[64]

1. Longlist: On the basis of wider filter criteria, a long list of comparable companies will be worked out.
2. Shortlist: Idiosyncratic filter criteria are determined, like the balance sheet, sales, financing conditions etc.
3. Peer-group: The companies of the shortlist are analyzed in detail and the best matches form the peer-group.
4. Un-levering: All peer-group beta factors are adjusted for the capital structure, i.e. hypothetical un-levered beta factors are calculated. This can be done using the Modigliani-Miller^[65] formula:
(3.4) With = unlevered beta factor, = levered beta factor, t = tax rate, D = debt and E = equity. It is important to use market figures. Equity is the market capitalization. Since it is difficult to obtain a market debt value, debt can be proxied by the book debt. After calculating the unlevered betas of the peer-group, it is reasonable to use an average value (either equally-weighted, or according to subjective estimations).
5. Re-levering: The average value of the un-levered peer-group betas is plugged in the following formula to obtain the beta factor for a company which is not listed on an exchange.

3.1.4. Criticism and extensions

The previous pages discussed various approaches to determine the single components (risk-free interest rate, MRP and beta factor) of the CAPM. It was shown that even small deviations in the methods of calculation, lead to significant different values. But not only the determination of the components, but also the CAPM as a comprehensive theoretical concept, faces more and more criticism in the recent past. Although, the CAPM is still the main concept in use,^[66] Fama and French already wrote an article in the Journal of Finance in the year 1996, with the header: “The CAPM is Wanted, Dead or Alive”.^[67] The problems of the CAPM mainly base on its restrictive assumptions:

- Since homogenous expectations of investors cannot be identified in practice, historical data have to be used to determine the MRP and the beta-factor. Thus, the CAPM becomes an “ex-post”-model.
- The assumptions that there are no transaction costs and no taxes, are not maintainable in practice. In addition, the idiosyncratic risk cannot be diversified completely.
- In particular, the CAPM includes no components that consider financial difficulties or insolvency risks.^[68] Portfolio managers do not think in terms of variance as a measure of risk. They rather use tracking error measures.^[69]
- Even Markowitz (2005) argues for more realistic assumptions. He, in particular, doubts the assumption that all investors can borrow and lend money without any limit. If this assumption would be substituted by a real-world version, the market portfolio would no longer be an efficient portfolio. Furthermore, he argues that in this situation the assumption of a representative investor could not be maintained and that the expected returns would no longer be linearly related to betas.^[70]

Since the implementation of the CAPM, several empirical contradictions have been detected.^[71] Banz (1981) finds that size (stock’s price times shares outstanding) increases the explanatory power of the cross-section of average returns. Using only the beta factor as a proxy for risk, would overestimate the average returns for small stocks (low size) and underestimate the average returns for high stocks (high size).^[72] Bhandari (1988) identifies a positive relationship between leverage and the average return.^[73] This contradicts the CAPM, since the beta factor should already include leverage risk. Rosenberg et al. (1985) figure out that average stock returns are positively correlated to the ratio of a company’s book value to its market value (B/P).^[74] Chan et al. (1991) confirm the results of Rosenberg et al. for Japanese stocks.^[75] Basu (1983) demonstrates that the ratio of earnings-to-price (E/P) helps to explain the cross-section of average returns on U.S. stocks in calculations that also include size and beta.^[76] Fama and French (1996) calculate the returns for 1928 – 1993 on β-deciles of NYSE^[77] stocks.^[78] They form ten portfolios in June of each year using betas on the NYSE value-weight market portfolio, estimated of two to five years of past monthly returns. They find that the relation between beta and the average return is rather flat,^[79] that is, that high beta portfolios have no significant higher return than low beta portfolios. They come to the conclusion that beta alone does not suffice to explain expected stock returns.^[80]

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Table 5: Fama and French (1996) beta factor summary statistics

(Source: See Fama / French (1996), p. 1951.)

The CAPM has been extended in a variety of forms. Some of the extensions include allowing heterogeneous beliefs; restricting the possibility of risk-free borrowing and lending; allowing a multi- time period; extensions to an international context; tax considerations;^[81] and the paradigm shifting Fama-French three factor model.

3.2. Fama and French three factor model

Fama and French (1992) find that the variables market equity (price of a stock) and the ratio of book equity to market equity (B/P), as well as size (stock price times shares outstanding) explain much of the cross-section of average stock returns.^[82] They argue that under the assumption of rational pricing, these variables must represent the sensitivity to common risk factors in stock returns. Fama and French (1993) use the time-series regression approach of Black, Jensen, and Scholes (1972). Monthly returns on stocks are regressed on the returns to a market portfolio and hedging portfolios for size and book-to-price ratios (B/P).^[83] They find that a three-factor pricing model that includes the market factor beta and risk factors pertained to size and B/P seems to have a significant explanatory power for the cross-section of average stock returns:^[84] The model states that the expected return of a portfolio in excess of the risk free rate is explained by the sensitivity of its return to three factors: (i) the excess return on a broad market portfolio , (ii) the difference between the returns on diversified portfolios of small stocks and big stocks (SMB) and (iii) the difference between the returns on diversified portfolios of high book-to-market (value) stocks and low book-to-market (growth) stocks (HML). The model is as follows:

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To do a historical regression analysis, a new variable is introduced and the terms in the above formula are rearranged the following:

is the intercept of the regression and should be statistically ^[85] indistinguishable from zero. If is not equal to zero, the factors in the model cannot fully explain the portfolio returns. In the context of portfolio management, indicates the relative performance of a fund manager to the market. If an investment fund generated a positive alpha, the manager outperformed the market, vice versa. Therefore, can serve as a backward-looking performance measure.

The SMB and HML factors are constructed as follows: At the end of June of each year all stocks are ranked according to the median into two size portfolios, small and big (S and B). Both size portfolios are then split into three BE/ME portfolios based on the quantiles 30% (low), 30% to 70% (medium), and the top 30% (high). From the intersections six portfolios (S/L, S/M, S/H, B/L, B/M, B/H) are constructed. For example B/L represents the portfolio of big sized stocks and low book-to-market stocks. The portfolios are rebalanced once a year in the beginning of July. Value-weighted returns are calculated monthly for each portfolio. The factor mimicking portfolios SMB and HML can be interpreted as zero-investment portfolio, since the simultaneous long and short position requires no investment costs. SMB and HML are calculated as:^[86]

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The factors are constructed this way in order to minimize the correlations among each other. Thus HML can be interpreted as the return of a portfolio that is long in high book-to-market stocks and short in low book-to-market stocks, and one that is widely independent from the market factor and the size effect.^[87]

Fama and French (1993) investigate U.S. stock returns between 1963 and 1991. As dependent variables, they form 25 value-weighted portfolios according to size (market equity ME) and book-to-market equity (BE/ME). The portfolios generate a range of excess monthly returns from 0.32% to 1.05%.^[88] The results indicate a positive relation between the BE/ME ratio and average returns and a negative relation between size and average returns. However, most of the portfolios of the two lower quintiles of the BE/ME portfolios show excess returns that are less than two standard errors from zero (t-statistics (t) < 2). This is due to the high standard deviations of the stocks that range between 4.27 and 7.76 among the different portfolios.^[89]

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Table 6: Fama and French (1993) portfolio excess returns

(Source: See Fama / French (1993), p. 15.)

Fama and French (1993) further calculate an average risk premium of the market beta (corresponds to the average value of ) of 0.43% per month (t = 1.76), an average SMB return of 0.27% per month (t = 1.73), and an average HML premium of 0.40 % per month (t = 2.91).^[90] They finally find that adding the two factors SMB and HML leads to strong increases in R² ^[91]. As can be seen in table 7, only two out of 25 portfolios have a R² that equals to 0.90 or above. In particular, low size and high BE/ME portfolios face a low explanatory power. The three factor model creates 21 portfolios that are greater than 0.90, instead (table 7). Thus, adding the two factors SMB and HML, increases the explanatory power on average about 15.2 percent points (from 0.7792 to 0.9312).

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Table 7: Fama and French (1993): Explanatory power of the market model

(Source: See Fama and French (1993), p. 20.)

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Table 8: Fama and French (1993): Explanatory power of the three factor model

(Source: See Fama and French (1993), p. 25)

Fama and French (1998) extend the three-factor model to a global context. They point out that value stocks, sorted on B/M, perform better than growth stocks in twelve out of thirteen markets between 1975 and 1995. In their calculations, global portfolios on value stocks outperform portfolios on growth stocks on average by 7.68 per cent per year (t = 3.45). They further confirm that a two-factor model, including the world market and the world B/M ratio, has a better explanatory power than an international CAPM.^[92]

3.3. Explanation approaches for the value premium

The academic consensus agrees that value investing strategies, on average, outperform growth investing strategies, i.e. that there is a value premium. Challenging is the question, how this premium can be explained. Basically, there are two points of view. The first is represented by Fama and French, the second by the so called behavioralists. Fama and French take the position of the efficient market hypothesis and assign the higher returns of value investing to its increased risk.^[93] The efficient market hypothesis implies that markets are very efficient in reflecting information about individual securities and the whole market. The theory further states that when new information arises, it will be immediately priced in any security. Hence, neither technical, nor fundamental analyses would help an investor to achieve excess returns in comparison to a randomly selected portfolio of individual stocks. Zhang (2005) suggests an explanation for the view that value stocks are more risky than growth stocks. He argues that because of costly reversibility and countercyclical price of risk, value firms are less flexible than growth firms to mitigate the impact of negative shocks. As a consequence, value firms face a higher risk than growth firms in recessions when the price of risk is high.^[94] Gulen et al. (2008) confirm these countercyclical variations of expected stock returns by applying a Markov switching framework.^[95]

The behavioralists do not believe that the value premium can be explained by the efficient market hypothesis. De Bondt and Thaler (1985) rank portfolios according to three- to five-year past returns and find that prior losers tend to outperform prior winners in the future, and vice versa. They explain their results with the overreaction hypothesis, i.e. investors consider too much the past and recent performance of stocks in their decision making process, whereas they underweight base rate data, like the fact that stock returns tend to mean-revert.^[96] Lakonishok et al. (1994) suggest that the value strategies perform better than growth strategies, since they exploit the irrational behavior of investors.^[97] Consistent with the empirical evidence, they find an annually value premium of 10 to 11 percent. They suggest that their results can be best explained by the preference of investors for growth strategies. Since the high expectations, that are reflected in these stocks, are often not fulfilled, value strategies perform better than growth strategies.^[98] Beside the existence of the value premium, its persistence over a long time period has been discussed extensively. Basically, every anomaly is expected to be arbitraged after its detection. Behavioral aspects may explain several impediments in arbitrage.^[99]

An additional explanation for the value premium is proposed by Kothari et al. (1995). They find that the higher returns of value strategies would be due to data selection biases.^[100] However, Chan et al. (1995) oppose this explanation. They point out that there is no severe selection bias on the database “Compustat”.^[101]

3.4. Carhart four factor model

Carhart (1997) extends the three-factor model of Fama and French by adding a momentum factor. The model can then be written as:

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captures Jegadeesh and Titman’s (1993)^[102] one-year momentum anomaly and is long in last year’s short-term winners and short in last year’s short-term losers. is the corresponding factor loading. The momentum is the cumulative return from month to . The most recent month is not considered, to avoid the short-term reversal effect documented by Jegadeesh and Titman (1993).^[103] The UMD factor is calculated analogous to the HML factor of Fama and French (1993). According to its past year performance stocks are sorted on the 30% and 70% quantile breakpoints into portfolios of losers (D), portfolios of neutrals (N), and portfolios of winners (U). These three portfolios are again intersected with the two portfolios ranked according to size, which results in six portfolios (S/U, S/N, S/D, B/U, B/N, and B/D). UMD is then the equal-weighted return of the winner portfolios minus the loser portfolios:

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Carhart explains short-term persistence in equity mutual fund returns with common components in stock returns and investment costs. According to his study, a long position in last year’s top-decile mutual funds and a short position in last year’s worst-decile mutual funds generate an average yearly return of eight per cent. 4.6 per cent of this spread could be explained by differences in the market value and momentum, 0.7 per cent by differences in expense ratios, and one per cent by transaction costs.^[104] He finds that the four-factor model does a better job in avoiding average pricing errors in comparison to the CAPM and the three-factor model of Fama and French.^[105] He concludes with three rules-of-thumb for mutual fund investors. First, funds with durable poor performance should be avoided. Second, funds that performed well in the previous year, have higher returns in the next year, but not in later years. And third, the total expense ratio and transaction costs directly influence the performance.^[106] Since the study of Carhart, the four-factor model has become one of the most prominent asset pricing models and its explanatory power has been confirmed in several empirical papers.^[107]

4. Determinants of expected stock returns

After discussing the most important models to value stock returns, this chapter describes the potential determinants in detail. As has been demonstrated, the CAPM does a poor job in explaining stock market returns. Ever since its publication, a bulk of empirical studies has attempted to identify additional components. Assuming a rational investor, capital market anomalies would be expected to disappear after their detection due to arbitrage. However, there seem to be some anomalies with a continuous persistence. Table 9 indicates the expected direction of the different firm characteristics with the stock returns based on empirical evidence. Literature proposes that the higher the P/B and the P/E ratio, the lower is the expected stock return. Thus, a negative relationship is expected. In contrast, the dividend yield should have a positive relation with expected stock returns. The direction of the size characteristic is set in brackets due to recent findings of a reversal of the size effect.

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Table 9: Components of stock returns

4.1. Price-to-book

The price-to-book ratio (P/B) is used to compare a company’s current market price to its book price. The ratio can be useful for identifying value stocks, since it indicates whether a firm’s asset value is comparable to its share price. The validity differs among industries.

It is above all useful when valuing companies that are composed of mostly liquid assets, like insurance companies or banks. The price-to-book ratio is not as useful for companies with high research- and development expenditures (R&D) or firms with high levels of property or other fixed assets. Since long-term assets are held on the balance sheet at the original cost, book values can differ strongly from market values, if market prices change.

The P/B ratio is one of the most discussed determinants in finance literature. Already Stattman (1980) finds a negative relationship between the P/B figure and the cross-section of expected stock returns,^[108] i.e., higher P/B measures indicate a lower expected return than lower P/B ratios (value premium or HML). Fama and French (1992) report that the average monthly returns rise from 0.30 per cent for the highest P/B portfolio to 1.83 per cent for the lowest portfolio. This difference of 1.53 per cent is twice as high as the difference between portfolios that are ranked relating to size.^[109] They further argue that the combination of P/B and size absorbs the explanatory power of the P/E ratio and leverage.^[110] Fama and French (1998) find value premiums in almost all countries around the world^[111], which they confirm in later studies^[112]. There are few studies which neglect the influence of the P/B measure. For example, Kothari et al. (1995) find that P/B is only weakly related to average stock returns.^[113]

4.2. Price-to-earnings

The P/E ratio is a method to compare a firm’s current share price to its earnings per share. It can be interpreted as the unit^[114] of net income that investors are paying for a share. Thus, it can be seen as the number of years that a company had to cumulate earnings, in order to justify the purchase price. Similar to the P/B ratio, P/E ranges that are considered to be priced fairly, differ among industries. Hence, the P/E ratio can be used to evaluate certain sectors or to compare the prices of companies within the same sector. In order to account for differences in growth, the P/E ratio can be adjusted by the expected growth in EPS and becomes to the price- earnings-growth ratio (PEG) (with g = growth rate).

Empirical research starts with Ball (1978). He points out that the P/E ratio can be seen as a direct proxy for expected stock returns.^[115] Basu (1983) argues that stocks of low P/E firms generate significant higher risk-adjusted returns than stocks of high P/E firms.^[116] Fama and French (1992) find that the P/E determinant loses its explanatory power , when the regression consists of the beta factor, size and the P/B.^[117] In contrast, Artmann et al. (2011) figure out that the value effect is not only captured by the P/B, but also by the P/E ratio. They find that a four factor model that consists of beta, P/B, P/E, and momentum does the best job in explaining German stock returns.^[118]

Studies on the PEG ratio are limited. Peters (1991) finds that within the period 1982 till1989, USD 1 invested in the lowest PEG portfolio would have turned to USD 15.36, whereas the highest PEG portfolio would have turned only to USD 1.38.^[119] Easton (2004) develops a model of earnings and earnings growth and compares the results with the estimates implied by the P/E and PEG ratio. He finds that the results implied by the PEG ratio have a higher correlation to his model and a lower downward-sloping bias than the results implied by the P/E ratio.^[120]

4.3. Dividend yield

The dividend yield can be interpreted as the return in per cent that an investor receives via dividends in relation to the current share price. There is an inverse relationship between the yield and the stock price. If the share price doubles, the dividend yield will be the half, and vice versa. It is not mandatory for companies to pay dividends. In particular, growth firms often only pay low or no dividends, since they retain earnings to invest in future growth. Stock markets react very fast on companies’ announcements to change their dividend policy. Historically, many investors preferred companies that paid high dividend yields, since these imply an underpriced stock and a higher stability. The discussion of the influence of the dividend yield on stock returns has a long history in scientific papers.

Whereas Black and Scholes (1973) find no difference in expected returns of high yield and low yield stocks,^[121] Fama and French (1988) point out that regressions of returns related to dividend yields have an explanatory power, in particular, when considering longer return horizons.^[122] The results of Naranjo et al. (1998) suggest a strong positive relation between the dividend yield and the expected stock returns, even after applying the Fama/French three-factor model. They further argue that the effect would be larger for small size companies.^[123] Kothari and Shanken (1997) also support evidence that the dividend yield tracks variation in expected stock returns between 1926 and 1991. In the sub-period 1941 – 1991, they find that the D/P effect is stronger than the P/B effect.^[124]

4.4. Size

Market capitalization or size represents the total value of a company’s shares and thus, the public consensus on its equity value. Traditionally, companies are classified in large-, mid-, and small- caps. Large-caps face a market capitalization greater than EUR 2 bn, mid-caps are between EUR 500 m and EUR 2 bn, and small-caps below EUR 500 m. However, there is no uniform definition about the cutoffs. Beside the turnover of traded shares, the market capitalization is a crucial criterion regarding the admittance of the stock in a certain index.

Whereas, earlier studies usually find evidence for a size premium^[125], current works often negate any significant impact. Banz (1981) points out that smaller firms had higher risk-adjusted returns than larger firms.^[126] Fama and French (1992) also figure out a significant size premium and use size in combination with beta and the P/B ratio for their extension of the CAPM. There are many other studies that identify a size effect.^[127] In contrast, Dimson and Marsh (1999) find evidence that the size premium reversed in the United Kingdom (U.K.) in the 1990s.^[128] Horowitz et al. (2000) investigate the relationship between expected returns and the size effect between 1980 and 1996 and find no size premium.^[129] Artmann et al. (2011) detect a negative, but statistically insignificant size premium for the German market.^[130]

4.5. Momentum

Momentum = Cumulative past return of a stock over a certain time period

The implementation of momentum is an investment strategy that aims to generate profit on the continuance of trends in the market. Momentum investors believe that stocks that had a large price increase will continue to have additional gains, and vice versa. To execute this strategy, a trader will take a long position in stocks that show an upward trending price and short sell stocks with a downward trend.

Momentum is no fundamental, but a technical firm characteristic that is based on the technical chart analysis. Figure 3 shows the share price development of the Adidas stock within the past one year and the related momentum indicator. The momentum is measured within a 12-days period and can be interpreted in percent.^[131] A momentum above the 100 per cent line indicates an upward-sloping trend, and vice versa. Some investors even consider the breakthrough of the line to be a buy- or sell- signal.

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(Source: www.onvista.de)

[...]

^[1] See for example Fama and French (1996), p. 1947.

^[2] See Chan et al. (1991), p. 1761.

^[3] See Fama and French (1998), p. 1997.

^[4] See Griffin (2002), p. 783.

^[5] See for example Banz (1981).

^[6] See for example Carhart (1997).

^[7] See for example van Dijk (2011).

^[8] See Lowe (1996), p. 1.

^[9] See Greenwald et al. (2001), p. 3.

^[10] www.investopia.com, (http://www.investopedia.com/terms/i/intrinsicvalue.asp#axzz1x1cuQOjQ), as of 15.06.2012.

^[11] See Cunningham (2004), p. 7.

^[12] See Whitman (1999), pp. 3-5.

^[13] See Fama and French (1998), p. 1975.

^[14] Data are from Datastream request.

^[15] See Ghargori et al. (2011), p. 2.

^[16] Graham and Zweig (2003), p. 115.

^[17] See Cunningham (2004), p. 10.

^[18] See Cunningham (2004), p. 10.

^[19] See Manager Magazin Online (2003), http://www.managermagazin.de/finanzen/artikel/0,2828,147003,00.html, as of 10.06.2012.

^[20] The „Deutsche Aktienindex“ (DAX) ist the index of the 30 largest German stocks.

^[21] Closing price on 7 June 2012, source: www.onvista.de.

^[22] See Eddy (2012), http://www.geekosystem.com/facebook-value/, as of 05.06.2012.

^[23] These P/E ratios are average values of analyst’s forecasts, see www.onvista.de.

^[24] See Martin et al. (2008), p. 15.

^[25] See Bloch (2012), http://seekingalpha.com/article/325572-is-apple-a-growth-or-value-stock-who-cares, as of 16.06.2012.

^[26] See Buffet, Warren (2003), Chairman’s letter – 1992, http://www.berkshirehathaway.com/letters/1992.html, as of 16.06.2012.

^[27] See Markowitz (1952).

^[28] See Perridon and Steiner (1993), p. 250.

^[29] See Drukarczyk (2003), p. 352.

^[30] The term return corresponds to the yield-to-maturity (YTM) of a bond.

^[31] High-rated government bonds are bonds with an AAA rating like German government bonds.

^[32] See Reese, Wiese (2005), p. 4.

^[33] See figure 2.

^[34] See Ballwieser (2007), p. 84.

^[35] The IDW is a privately run organisation that serves the interests of its members who comprise both individual German Public Auditors and German Public Audit firms.

^[36] See Reese and Wiese (2005), p. 3.

^[37] See Svensson (1994), p. 6.

^[38] These parameters are published on a daily basis by the Deutsche Bundesbank.

^[39] See Shen, Huang (2011), p. 43.

^[40] See Benkert (2004), p. 50.

^[41] A market portfolio can be represented by the S&P 500, the HDAX, the DAX etc., depending on the investment horizon.

^[42] See www.investopia.com, http://www.investopedia.com/terms/m/marketriskpremium.asp#axzz1xCL99zcx, as of 24.06.2012.

^[43] See Damodaran (2008), p. 1.

^[44] See FM-IDW 2009, S. 696f.

^[45] See www.idw.de, http://www.idw.de/idw/portal/d616038/index.jsp, as of 01.06.2012.

^[46] The REXP is the “Deutsche Rentenindex” as performance index. It expresses the value of a representative segment of the German bond market.

^[47] See Stehle (2004), p. 921.

^[48] See Stehle (2004), p. 921.

^[49] See Stehle (2004), p. 921.

^[50] The default spread of a corporate bond is the difference in yield of a risky bond in comparison to a risk-free bond, to compensate the investor for the default risk. The worse the rating, the higher is the yield.

^[51] See Damodaran (2008), p. 63 f.

^[52] Baa is a rating by Moody’s. It stands for an investment grade bond with a relatively low risk of default.

^[53] See Damodaran (2008), p. 64.

^[54] See Damodaran (2008), p. 69 f.

^[55] Stocks with a beta > 1 are riskier than the market portfolio, stocks with a beta < 1 are less risky than the market portfolio and stocks with a beta = 0 are risk-less. The beta factor includes no idiosyncratic risks like management failures or production blackouts.

^[56] See Nguyen (2008), p. 193.

^[57] See Dörschell et al. (2008), p. 1156.

^[58] See table 4.

^[59] Assume a simplified valuation model of: with V = Entity value of the company, CF = Cash Flows, = cost of equity, and t = time. Applying the highest beta factor (1.281) in comparison to the lowest (1.057), leads to an increase in the cost of equity of 1.12 per cent in the calculation of Dörschell et al. (2008).

^[60] See Dörschell et al. (2008), p. 1155.

^[61] See Dörschell et al. (2008), p. 1156.

^[62] The „week-of-the-day“ - effect describes the influence of the day (Monday, Tuesday, Wednesday etc.) (when the beta factor regression starts) on the beta factor. Watrin et al. (2011) find that the selection of the week-day exerts a significant influence on the beta factor and the valuation of a company. See Watrin et al. (2011), p. 176.

^[63] See Nguyen (2008), p. 195.

^[64] See Geginat et al. (2006), p. 23.

^[65] See Modigliani and Miller (1958).

^[66] According to a study by Roland Berger, two third of the surveyed companies still work with the classical (not-adjusted) beta factor. See Geginat et al. (2006), p. 15.

^[67] Fama/ French (2006), p. 1947.

^[68] See Cummins / Trainar (2009), p. 471.

^[69] See Montier (2009), p. 25.

^[70] See Markowitz (2005), p. 27 f.

^[71] A detailed discussion about the determinants that can explain stock returns will be done in section 4.

^[72] See Banz (1981), p. 3.

^[73] See Bhandari (1988), p. 507.

^[74] See Rosenberg et al. (1985), p. 9.

^[75] See Chan et al. (1991), p. 1739.

^[76] See Basu (1983), p. 129.

^[77] NYSE stands for New York Stock exchange.

^[78] See table 5.

^[79] See Fama and French (1996), p. 1952.

^[80] See Fama and French (1996), p. 1947.

^[81] See Brennan (1970); Wiese (2006).

^[82] See Fama and French (1992), p. 427.

^[83] See Fama and French (1993), p. 4.

^[84] See Fama and French (1993), p. 5.

^[85] is analogous to the classical CAPM beta, but not equal, due to the additional factors SMB and HML.

^[86] See Fama and French (1993), pp. 8 – 10.

^[87] See Ammann and Steiner (2008), p. 9.

^[88] See table 6.

^[89] See Fama and French (1993), p. 13.

^[90] See Fama and French (1993), p. 13.

^[91] The coefficient of determination R² gives an indication of the explanatory power (goodness of fit) of a model.

^[92] See Fama and French (1998), p. 1997.

^[93] See for example Fama (1998), p. 287.

^[94] See Zhang (2005), p. 95.

^[95] See Gulen et al. (2008), p. 1.

^[96] See De Bondt and Thaler (1985), p. 793.

^[97] See Lakonishok et al. (1994), p. 1541.

^[98] See Lakonishok et al. (1994), p. 1576 f.

^[99] See Shleifer / Vishny (1997), p. 54.

^[100] See Kothari et al. (1995), p. 221.

^[101] See Chan et al. (1995), p. 292.

^[102] See Jegadeesh and Titman (1993), p. 65 – 91.

^[103] See Jegadeesh and Titman (1993), p. 89.

^[104] See Carhart (1997), p. 79 f.

^[105] See Carhart (1997), p. 62.

^[106] See Carhart (1997), p. 80 f.

^[107] See Daniel et al. (1997), pp.1035 – 1058; Eberhart et al. (2004), pp. 623 – 650.

^[108] See Stattman (1980), p. 25.

^[109] See Fama and French (1992), p. 441.

^[110] See Fama and French (1992), p. 445.

^[111] See Fama and French (1998), p. 1997.

^[112] See for example Fama and French (2010), p. 22 f.

^[113] See Kothari et al. (1995), p. 186.

^[114] Market capitalization/net income corresponds to the P/E ratio, only if there was no change in the number of outstanding shares during the observed periods (market capitalization = price times current number of shares, whereas earnings per share = net income/weighted average number of shares).

^[115] See Ball (1978), p. 103.

^[116] See Basu (1983), p. 129.

^[117] See Fama and French (1992), p. 445.

^[118] See Artmann et al. (2011), p. 20.

^[119] See Peters (1991), p. 49.

^[120] See Easton (2004), p. 73.

^[121] See Black and Scholes (1974), p. 1.

^[122] See Fama and French (1988), p. 4.

^[123] See Naranjo et al. (1998), p. 2031.

^[124] See Kothari and Shanken (1997), p. 169.

^[125] The size premium is a stock market anomaly that states that small companies on average perform better than large companies.

^[126] See Banz (1981), p. 3.

^[127] See, for example, Rouwenhorst (1999); Barber and Lyon (1997).

^[128] See Dimson and Marsh (1999), p. 53.

^[129] Horowitz et al. (2000), p. 143.

^[130] See Artmann et al. (2011), p. 20.

^[131] Momentum=((Closing price today (t))/(Closing price (t-12) ))*100.