Size premium in Europe

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Size premium in Europe

An explanatory study on the size effect in the post 2000

North-Western Eurozone

Abstract

This thesis examines the existence of the size effect in the North-Western Eurozone in the period 2001-2013. After the publication of the first empirical evidence on the size effect in 1980's, different strands of theory and empirical work have given contradicting results. Some empirical studies have even asserted that the size effect has disappeared after the early 1980's. This thesis uses both Fama-Macbeth (1973) style cross-sectional regressions on beta-based portfolios of stocks and time-series regressions based on size-based portfolios to test for the existence of the size effect in the North-Western Eurozone in the years 2001-2013. It is found that for the period of the analysis a significant size effect can be observed, though not as outspoken as hypothesised. The review of the existing literature leads to the conclusion that up to this point a holistic theory, explaining the size effect, has not been found yet. The implications of this thesis for practitioners in the field of corporate finance are that they confirm the validity of the current common practice of adding a size factor to e.g. WACC calculations.

Key words: Size effect, Europe, CAPM, Anomalies JEL classification: C33, G12, G15, G30

Author: R.A. Bos

Student ID number: 1765035

Date: 12/01/2014

Supervisor: prof. dr. T.K. Dijkstra Co-assessor: dr. P.P.M. Smid

MSc thesis Finance

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Size premium in Europe 2

TABLE OF CONTENT

LIST OF TABLES AND FIGURES ... 3

LIST OF ABBREVIATIONS ... 3 1 | INTRODUCTION ... 4 2 | LITERATURE REVIEW ... 6 2.1 Origination ... 6 2.2 Risk ... 7 2.2 Liquidity ... 8 2.3 Dividend yield ... 9 2.4 Seasonality ... 10

2.5 Stochastic analysis of the size effect ... 12

2.6 Empirical evidence since 1990 ... 13

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LIST OF TABLES AND FIGURES

Table I Descriptive statistics of the data set ... 16

Table II Descriptive statistics of beta-based portfolios for cross-sectional regressions ... 24

Table III Regression output for the cross-sectional regressions ... 26

Table IV Descriptive statistics of size-based portfolios for time-series regressions ... 29

Table V Regression output for the time-series regressions ... 30

Figure I Spread of the estimated coefficients in model 2 ... 27

Figure II Spread of the estimated coefficients in model 1 ... 40

Figure III Spread of the estimated coefficients in model 3 ... 41

LIST OF ABBREVIATIONS

AMEX American Stock Exchange

ASX Australian Securities Exchange

CAPM Capital Asset Pricing Model

HML High Minus Low

NYSE New York Stock Exchange

SMB Small Minus Big

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1 | INTRODUCTION

Banz was in 1981 the first to empirically prove the relationship between a company's size and its stock return. For the period 1936-1975 he found that small stocks have outperformed large stocks based on risk-adjusted returns. Since then a vast amount of financial academic literature focused on providing empirical evidence and theoretical explanations for the relationship between size and the risk-adjusted return of stocks. The most recognised of these papers is the Fama and French paper of 1993, in which they introduced their three-factor model which explains stock returns through three factors: beta, size and the book-to-market ratio. This specific explanation of the size factor is based on an underlying risk factor other than beta. Besides the article by Fama and French, a vast body of literature came into existence on the size effect since the 1980’s. Scholars have sought to empirically confirm the results and theoretically explain the anomaly. But did they succeed? After the first evidence, contradictory results were found in the late 1990’s indicating that for some time frames and for some geographical markets, the size effect either didn’t exist or affected returns in the opposite direction, with large-cap firms performing better than small-cap firms. Other evidence from that same period and later did however show the existence of a significant and negative size effect. This raises the question whether the size effect can in fact be found in a recent time period. This will be tested for the North-Western Eurozone as in most studies on the size effect a sample consisting of US data is used, this would be the first time that these countries within the monetary union are tested. Beyond the empirical proof, theoretical explanations offered for the size effect have been far from conclusive. This means that the size effect is still an anomaly in the sense that academic literature has yet to explain the cause of the observed effect. Different strands of literature have tried to explain the effect based on either firm characteristics, underlying risk factors or time varying risk loadings. Up this point none of the tested explanations of the size effect have sufficient empirical support to truly explain the size effect. However, models in which the size effect rises endogenously as a firm characteristic seem to have a more significant impact than risk-based explanations as the latter lack theoretical backing.

Another issue that exists around the size effect is the apparent gap that separates the academic world from practitioners in the field of corporate finance. Whereas many scholars and academic text books focus on Fama’s and French’s three factor asset pricing model when discussing the size effect, practitioners still use an extended version of the CAPM which adds a size factor to the standard CAPM.

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Size premium in Europe 5 This set of countries is chosen for a specific reason: it is the first time that the size effect is analysed for this specific set of markets. Furthermore, it is an interesting sample since the markets closely resemble each other in culture and share the same currency (the Euro). The evidence that this thesis seeks to find can fill the gap that emerged after the late 1990’s when the empirical evidence on the size effect gave mixed results and the attention academic research gave to the subject, vastly decreased. Furthermore, most evidence on the subject is based on either USA data or broad international samples covering a wide range of geographical areas. This thesis will provide empirical evidence on the existence of the size effect in recent years for a selected group of countries in the North-western Eurozone. The first main research question in this thesis is:

Can the existence of a significant size effect be empirically proven in the North-Western Eurozone for the years 2001-2013?

The second main research question is:

How can the size effect be explained?

In order to answer the first question, a combination of cross-sectional and time series-regressions is run on data from the 7 countries in North-Western Europe that have adopted the Euro as their currency. Two separate statistical tests are done: first, monthly cross-sectional Fama-Macbeth (1973) style regressions are run on monthly rebalanced beta-based portfolios; and second, a set of time-series regressions is run on size-based portfolios. The results indicate that a significant size effect cannot be rejected over the course of the years in the sample, however the sign of the regression coefficients is not as consistent as hypothesised. The second question aims to find a holistic answer to the possible cause behind of the size effect based on a literature review. This is an important question as it could explain the discrepancy between academic literature and common practice which was mentioned earlier.

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2 | LITERATURE REVIEW

2.1 Origination

The first empirical evidence that a size premium existed was presented in the article by Banz (1981) on the relationship between stock return and the market value of stocks. The study used a similar portfolio approach as Black and Scholes (1974) in which the securities are divided into five portfolios based on market capitalisation and then subdivided into portfolios based on the asset's beta. The results are significant and show that for the 1936-1975 period, small stocks have an on average higher risk-adjusted return than large firms. The relationship between size and return is not linear and is more pronounced for the smallest stocks in the sample. At that moment no theoretical equilibrium model existed to explain this anomaly in the CAPM model, so Banz concluded: "It is not possible to determine conclusively whether market value per se matters or whether it is only a proxy for unknown true additional factors correlated with market value" (Banz, 1981, p.4). The study empirically showed that the CAPM model as developed by Sharpe (1964) and Lintner (1965) had its flaws in explaining the return on capital assets which quickly after being introduced gained serious theoretical and empirical support (Black, 1972; Black, Jensen and Scholes, 1972; Fama and Macbeth, 1973) and is still today the most commonly used model to explain asset returns. However, since Fama and MacBeth, a discussion has been going on about the explanatory capabilities of the CAPM model and its completeness in using the beta measure of risk as the only variable in explaining return (Blume and Friend, 1973; Roll, 1977). Several academics have shown the CAPM to be oversimplified and have introduced a wide range of anomalies which should be taken into consideration when predicting returns (Basu, 1977; Litzenberger and Ramaswamy, 1979; Bhandari, 1988;). One of the more influential articles of such kind is the Fama and French 1993 article in which the well known Fama-French three-factor model is developed (which follows the line of their previous work from 1992). This model shows evidence for a statistical significant relation between both size and book-to-market factors and returns, furthermore an interesting finding of this analysis is that when sorting stocks in equally weighted beta-sorted portfolios no evidence is found for a relation between beta and return.

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Size premium in Europe 7 factors. What becomes clear is that academic literature is far from finding a sound theoretical framework that can explain the size anomaly.

2.2 Risk

As Fama and French (1993) describe in their article: "One of our central themes is that if assets are priced rationally, variables that are related to average returns, such as size and book-to-market equity, must proxy for sensitivity to common (shared and thus non-diversifiable) risk factors in returns" (Fama and French, 1993, p.4). The portfolios mimicking the risk factors related to size (SMB), Book-to-market (HML) and Market (Beta) absorb time-series variation and sufficiently explain cross-section of average stock returns. This leads Fama and French to conclude that size and book-to-market are justifiable proxies for common risk factors. This is in conjunction with Chan, Chen and Hsieh (1985) who find that their multifactor pricing model essentially captures the size effect. The variables in the regression are macro-economic variables such as inflation, industry production and the risk premium. These variables are used to proxy for changes in the economy. The higher return of smaller firms is justified by the additional risks borne in an efficient market. Fama and French (1995; 1996) suggest later, partially based on the conclusions from Chan and Chen (1991), that one of the state variables is financial distress. Vassalou and Xing (2004) examine default risk as measured through Merton's (1974) model. The cross-sectional regression captures the size effect for all created portfolios except for the one with most default risk. Asset-pricing tests are conducted to successfully test whether default risk is a measure for systematic risk. In the cross-sectional analysis of Petkova (2006), empirical evidence is provided for correlation between the SMB and HML factors and variables that describe investment opportunities such as dividend yield, term spread , default spread and the one-month Treasury-bill yield. When these variables are added to the model, factor loadings on SMB and HML lose their explanatory power for the cross-sectional regression of returns. It is shown that SMB proxies for the default spread factor while HML proxies for the term spread factor, which confirms previous research by Hahn and Lee (2006)1. These examples show that there is some evidence for the statement that size is a proxy for systematic risk with financial distress as the state variable, however there are scholars who argue on the contrary. Dichev (1998) and Campbell, Hilscher and Szilagyi (2008) empirically show that the size effect cannot be explained by distress factors. Daniel and Titman (1997) find that the Fama-French factor loadings cannot explain cross-sectional variance as well as firm characteristics can. Results show that after controlling for size, there is no association

1

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Size premium in Europe 8 between average returns and SMB. This argument gained support and was built upon in later papers (Teoh, Welch, and Wong, 1998; Heston, Rouwenhorst, and Wessels, 1999; Baker and Wurgler, 2006). This is in essence an argument to justify the use of the size premium as it is currently used in the field of corporate finance, where the size effect is treated as a firm characteristic rather than a factor loading.

2.2 Liquidity

In literature, stock liquidity is generally described as the ability to trade large quantities quickly at low cost with little price impact (De Moor and Sercu, 2013). Since the early 1980's empirical research has tried to explain the relation between the size effect and liquidity. Stoll and Whaley (1983) find that after adjusting for increased transaction costs it is not possible to earn risk-adjusted returns in excess of CAPM in their sample of firms listed on the NYSE. The sample is extended to AMEX stocks by Schultz (1983) who concludes that the size effect cannot be solely attributed to difference in transaction costs between small and large firms. Amihud and Mendelson (1986) construct a model which takes into consideration the compensation required by investors for future trading costs. Empirical evidence is found for the hypothesis that investors with longer holding periods tend to hold assets with larger bid-ask spreads since large trading costs are amortized over a longer period. When the spread is larger, a smaller compensation is required for any increase in this spread, therefore the relation between expected return and bid-ask spread should be concave. Their regression which includes a measure for risk, spread and size, gives significant return for the first two factors however, the size effect is rendered insignificant. When the analysis is repeated (Eleswarapu and Reinganum, 1993) with a broader data set and without excluding very small stocks from the sample, the cross-sectional variation in the bid-ask spread does not render the size effect insignificant anymore.

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Size premium in Europe 9 trading speed. Liu (2006) develops a two-factor model with liquidity and market which, after adjusting for these factors, no longer finds a significant size effect as well as other well-known CAPM anomalies. However, this result is not robust over time and various studies using a wide range of other measures for liquidity have been tested in relation to stock returns (Brennan and Subrahmanyam, 1996; Datar, Naik, and Radcliffe, 1998; Pástor and Stambaugh, 2003; Acharya and Pedersen, 2005; Lee, 2011). The models developed in these studies have different measures for liquidity and vary in scope and data analysed. What is notable is that these studies independently conclude that liquidity and return are negatively related and find that portfolios of small firms have the highest loadings on the liquidity factor. However, regarding the relation between liquidity and size, the liquidity factor is not capable of fully subsuming the size effect.

2.3 Dividend yield

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Size premium in Europe 10 loadings is tested. Cross-sectional and time-series regressions are run and Sercu and De Moor (2013) conclude that smaller firms appear to be affected more by time-varying risk loadings related to dividend yield than large firms. Finally, dividend yield is linked to a missing risk factor and again dividend yield shows to be a significant factor in explaining cross-sectional variance in size-based portfolio returns. However, none of the above models can fully explain the size effect and subsume the size-based alpha in the regressions regardless of the model being based on firm characteristics, a missing risk factor or time- varying risk loading. Results do show that the underlying factor that explains the size effect does show some correlation with dividend yield, however further research is required to investigate whether this implication can fully explain the size anomaly and which specific factor underlies these results.

2.4 Seasonality

It is often claimed that much of the size effect originates from the high relative performance of small cap securities in January. This has lead to the idea that the size effect is fully contributable to turn-of-the-year-effects and therefore no more than a behavioural fluke. Keim (1983) found that small stocks outperformed large stocks by 15% in the first month of the year. This explains a large part of the full year size effect. Furthermore, most of this effect is found in the first five trading days of January. Brown et al. (1983), Lamoureux and Sanger (1989), and Daniel and Titman (1997) all find strong size effects in the first month of the year. The sample covers multiple periods in between 1963-1993 for data from both the NYSE and the ASX. More recently, Moller and Zilca (2008) show that the effect did not disappear over the last years. In fact it is shown to be even more concentrated during the earlier days in the month. The effect is seen to be more pronounced for small-stock portfolios whereas the large-stock portfolios show little seasonality. There are two possible explanations for this finding. First, small stocks tend to be more illiquid; and since illiquid stocks tend to be more sensitive to changes in demand, the January effect is more pronounced for small stocks. Second, a portfolio of small stocks has a bias towards stocks that have experienced a large price decline, which would lead to these portfolios being more sensitive to the-year effects; small stock portfolios could instead be defined as turn-of-the-year portfolios.

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Size premium in Europe 11 December. Roll (1983) finds evidence of stock returns in January being significantly negatively related to the returns in the previous years, which is consistent with the tax-loss selling hypothesis. However, international evidence shows that the tax benefits cannot capture the full size effect. Brown et. al (1983) find that the size effect for Australian markets is constant throughout the year. This is an important finding since the Australian tax years ends in June, so no turn-of-the-(fiscal)-year effect is present in this market. Berges, McConnell, and Schlarbaum (1984) performed their analysis on Canadian data before and after the introduction of the capital gains tax in Canada in 1973. The results are significant, both before and after the introduction. Kato and Schallheim (1985) find a strong January effect with heavier results for small firms in Japan despite the fact that capital-losses are not tax deductible in Japan, providing the incentives of the turn-of-the-year hypothesis.

Another explanation for the January effect which is used to explain the size effect as a turn-of-the-year effect is the window-dressing hypothesis. In order to present sound portfolio holdings, institutional investors have an incentive towards the end of the year to buy stocks that have performed well in the last year (or stocks with a low risk) and sell securities that have underperformed during the year. In January, the portfolios have to be rebalanced with more risky securities and the turn-of-the-year effect is caused by the same absence of selling pressure as in the tax-loss selling hypothesis. This explanation is considered in several articles (Ritter and Chopra, 1989; Sias and Starks, 1997; Poterba and Weisbenner, 2001). Despite results confirming this hypothesis, the latter two articles also show that the effect of the tax-loss selling hypothesis is far larger than the window-dressing hypothesis.

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2.5 Stochastic analysis of the size effect

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Size premium in Europe 13 the contribution has to be attributed to an implied liquidity premium. The analysis leads to the conclusion that the asymptotic rate of return of a stock is equal to the stock’s liquidity premium. This mathematical explanation of the size effect and the substantiation of its persistence in history provides yet another string of evidence that a. the size effect is a persistent phenomenon over time and b. it is based on firm characteristics rather than an underlying risk factor. However these theories have yet to find empirical support.

Both STP and its subsequent arguments regarding the size effect have not gained much academic attention (e.g. Fernholz (1998) has only 23 citations in the Google scholar database). This is largely due to the nature of his compelling mathematical arguments; these are not easy to read as they are based on mathematics of a far more advanced level than used in most financial economic articles in academic literature. In practice, however, stochastic portfolio theory has had its impact. Intech, which is part of the Janus Capital Group, offers different mutual funds with strategies based on this mathematical approach for constructing the portfolios. The existence of mutual funds based on this mathematical approach further strengthens the validity of the arguments which confirm the existence of the size effect independent of relative risk.

2.6 Empirical evidence since 1990

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Size premium in Europe 14 large-cap passive index funds, which left less funding available for small-cap funds. Finally, as investors sought to capture the size effect as identified by academic scholars, the market corrected a mispricing which under-priced small-cap stocks in the decades before 1980. Horowitz et al. (2000) again try to empirically show the existence of a size effect in the years after 1980. In this paper three methods are tested to find the effect. Annual compounded returns, monthly cross-sectional regression and linear spline regressions fail to find a significant size effect in the 1980-1996 period. Barber and Lyon (1997) and Dimson and Marsh (1999) find similar results. Despite this evidence, other scholars have found evidence on the contrary (Rouwenhorst, 1999; Heston et al., 1999). Heston et al. (1999) find a significant size premium in a dataset consisting of 12 European companies in the years 1978-1995. At the same time Heston et al. (1999) find that when size is added as a variable, the Fama-French SMB factor loses its significance. De Moor and Sercu (2013) analyse the size effect based on data from the 1980-2009 period from an international database and come to the same conclusion; the size effect is still observable. Their analysis is extended by testing whether some variables described extensively in the literature such as liquidity, Fama-French SMB and dividend yield subsume the size effect. Their conclusion is that none of the factors considered in the current literature can fully explain the size effect by creating alpha’s close to zero. However, models based on firm-characteristics are more viable than models based on underlying risk factors. The lack of evidence on the size effect in the late 1990´s combined with the success of the Fama-French three factor model, has caused the diminish of attention for the size effect since then.

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3 | DATA

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Size premium in Europe 16 Table I Descriptive statistics of the data set

Table I presents the mean, median, minimum and maximum of three variables that will be used in the analyses. Panel A gives the beta descriptives per country. This beta is constructed as the slope of a securities monthly returns over the monthly return of the market index which is set as the MSCI World Index. Panel B shows the monthly return data and Panel C the market value in ‘000 $. The dollar is chosen since during the sample period 2001-2013, the local currency was changed by all countries in the sample to Euro.

Mean Median Minimum Maximum

Panel A: Beta Netherlands 0.90 0.79 -5.60 11.85 Germany 0.97 0.87 -23.73 29.52 France 0.93 0.81 -81.14 112.39 Belgium 0.67 0.59 -34.08 16.80 Luxembourg 0.53 0.48 -0.72 2.10 Austria 0.71 0.62 -10.18 50.12 Finland 1.01 0.95 -7.28 31.72 Total 0.90 0.79 -81.14 112.39 Panel B: Return Netherlands 0.83 % 0.17 % -90.00 % 342.78 % Germany 1.38 % 0.72 % -78.23 % 418.69 % France 1.08 % 0.70 % -67.40 % 230.86 % Belgium 0.69 % 0.25 % -68.24 % 112.58 % Luxembourg 0.97 % 0.00 % -51.02 % 45.80 % Austria 1.10 % 0.55 % -55.56 % 85.46 % Finland 1.21 % 1.05 % -51.52 % 107.84 % Total 1.11 % 0.59 % -90.00 % 418.69 %

Panel C: Market Value ($) x1000 Netherlands 4,151,448 321,192 739 113,871,478 Germany 5,705,442 667,978 1,062 134,808,943 France 6,811,645 1,006,998 6,929 217,617,581 Belgium 2,660,022 468,082 13,251 169,525,993 Luxembourg 3,437,792 318,186 2,261 19,762,729 Austria 1,817,321 587,788 17,762 24,267,019 Finland 4,684,113 1,320,444 25,892 226,285,993 Total 5,268,448 736,693 739 226,285,993

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4 | METHODOLOGY

Banz (1981) used Fama-Macbeth style regressions (1973) to come to his conclusion regarding the size effect. Today it is still in use and known for its robustness and reliability in measuring return premiums in panel data (Petersen, 2009). Therefore, in order to obtain the results for this analysis, this study will use a methodology similar to that of Banz (1981), by using Fama-Macbeth (1973) style regressions. The analysis consists of two parts; first, the significance of size in explaining expected returns will be tested using Fama-Macbeth style cross sectional regressions (1973) for the following formula:

(1)

The second part focuses on testing the zero intercept hypothesis for the formula:

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4.1 Analysis 1

Starting with the first equation, the first problem that needs to be addressed is the estimation of beta. In equation (1), the relative risk measure is the true value of a securities beta. However in order to run a regression, an estimate of this measure needs to be constructed; this causes an errors-in-the-variables problem well known through previous analyses of beta (Fama and Macbeth, 1973; Banz, 1981; Bhandari, 1988). This study estimates a securities' beta as the regression slope of the securities' returns over the value-weighted MSCI World Index. The MSCI World index is a standard proxy for the market return. This slope uses three years of monthly data which precede the specific month for which the regression is run. This gives the following formula:

(3)

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Size premium in Europe 19 portfolios instead of individual securities, the analysis will group portfolios based on ranked beta in order to create a wide range of portfolio betas. However, this causes a serious regression problem. In a cross section of estimated betas, high observed betas tend to be above their corresponding true value and low betas tend to be below their true value. Thus, when constructing portfolios the large-beta portfolio is expected to have an overstated beta where the small-beta portfolio will have an understated beta, thus clustering positive and negative sampling errors. This can be dealt with in several ways: Banz (1981) uses a method where a long position in small firms and a short position in large firms is used. Heston et al. (1999), in line with Fama and French (1992) first construct 10 portfolio's based on size, which are subsequently subdivided into 100 size-beta portfolios. In the cross-sectional regressions the corresponding post ranking portfolio beta is used as independent variable. The method employed in this thesis will be as proposed in the original Fama-Macbeth analysis: first, portfolios will formed based on ranked beta computed from data in one time period. Subsequently, the beta from the preceding month will be used as the independent variable. This new data will, to a large extent redistribute errors in individual betas across the portfolio, thereby solving the regression phenomenon.

The specifics of the analysis are as follows: to measure the return premiums associated with beta and size, first the stocks are divided in 10 beta-based equally-weighted portfolios, let N be the total number of securities in the sample then each portfolio contains N/10 securities. To construct the portfolios, securities are ranked according to their beta and placed in descending order, this leads to portfolio 1 containing the largest-beta securities and portfolio 10 the smallest. Then betas of the preceding period are used to form the portfolio's beta. Each portfolio's beta is computed as the simple average of the individual stocks in the portfolio:

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The component beta for securities is updated monthly, this allows the portfolios to be monthly adjusted. By using monthly rebalanced portfolios, the analysis allows for both the shift to another portfolio on a monthly basis and a possible delisting of the security without consequences for the regression, thus solving any survival bias problem encountered in the sample. The monthly returns for the sample period are computed. Now for each month a cross-sectional regression, the empirical analogue of equation (1) is run:

(5)

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Size premium in Europe 20 where is the return on the beta-based portfolio in month ; is the estimated portfolio

beta for the lagged period as discussed in length above; is the logarithm of

market value of equity, measured in dollars in the previous month and is an error term. The tests of the model are predictive in the sense that independent variables and are inputs from periods prior to the period in which the regression is run. This is in line with the CAPM, being a normative theory meaning it allows users to predict future results based on current information. In other words, there should be a relation between future returns and a measure of risk that is currently observable. Additionally, it has the benefit of making the regressions suitable for use in the field of corporate finance. Besides equation (5) two other models will be used for testing the hypothesis using the same parameters (6) and (7):

(6)

this model, equation (6) tests the introduction of the theoretical value of as the intercept

in the model. Whereas the theoretical value of is estimated in the first model of equation (5), this model fixes the intercept at the observed , this is suggested in the

original size effect article by Banz (1981) as a test to increase the stability of the model when the observed value for in equation (5) differs from its theoretical value The last

model is model (7):

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in the third model, equation (7), the intercept is taken out of the formula and the regression is run on the difference of the independent variables and their mean. By doing so, the results are expected to be more stable. Since the model is a cross-sectional regression, the assumption of constant variance around the mean can be assumed as opposed to time-series date where this is often violated. This model is a simpler version of the two previous ones which implies that its goodness of fit measure, adjusted (see below) will be higher than for the previous models.

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p-Size premium in Europe 21 values are reviewed under the assumption of normality, the hypotheses are on firmer ground considering the thicker tails. The hypothesis to be tested is:

This test allows for conclusions to be drawn on the significant influence of size on expected returns for the securities in the sample.

4.1.1 Model selection

In order to select the most true model from the given set of models, this thesis will use the adjusted R squared ( ) given by the formula:

(8)

where is the number of degrees of freedom in the model; is the sample size and is the non adjusted sample given by the formula:

(9)

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Size premium in Europe 22 Regarding the matter of consistency, the has some unfavourable options. Kloek (1975) shows that the method is strongly consistent when the sample size increases. However, Schmidt (1974) shows that it is a weakly consistent method if there is an issue of autocorrelation in the model. And Giles and Smith (1977) add that this remains an issue when the model's residuals are autocorrelated. For the analysis in this thesis however, there is no evidence of such problems. Therefore, it can be assumed that the model selected based on in this thesis, is the correct one on average.

4.2 Analysis 2

Since many practitioners in different fields of finance use CAPM plus a size factor as an estimator for a company's cost of equity (Koller et al., 2010), it is relevant to test this implication. This size factor is the empirical alpha that is found when comparing size portfolios in time-series regressions. In the second part of the analysis the assumption of a zero intercept ( ) for the following time series regression for ten size-based portfolios, is tested:

(10)

where is the return on a size based portfolio p in month ; is the return on a

one-month German T-bill in the preceding one-month. The analysis uses the German T-bill because this the most frequently traded T-Bill in the Eurozone which most closely resembles the true risk-free rate (Koller et al., 2010); is the portfolios estimated beta constructed in a

similar manner to the first analysis; is the value-weighted average monthly return of

the MSCI World Index in the preceding month; is an error term and is the constant for which the following hypothesis is tested:

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5 | RESULTS

5.1 Analysis 1

As elaborated in the previous section, the data is clustered in portfolios based on the securities' beta in month t, after which data from the preceding month is used for the explanatory variables. Table II provides the portfolio descriptive statistics for the 10 portfolios over 156 months in the analysis. Means on return, portfolio betas in t-1 and market values in t-1 are given.

Table II Descriptive statistics of beta-based portfolios for cross-sectional regressions

Table II presents the mean of the input variables in the monthly cross-sectional regressions, over the 156 regression months per beta based portfolio. At the end of each month, the beta of each security is calculated based on the monthly returns three years prior to the specific month and the MSCI World Index. The stocks are then placed in ten beta based portfolios. The portfolio variables: return, beta and market value are equally weighted averages. The market capitalization is shown in '000 $.

Portfolio Return BetaT-1 MVT-1 ($) x1000

1 1.7908% 2.3301 4,748,856 2 1.3191% 1.6097 6,357,957 3 1.3910% 1.2914 6,226,920 4 1.3984% 1.0680 6,642,028 5 1.1404% 0.8986 5,840,463 6 1.3756% 0.7428 5,348,730 7 1.2029% 0.5911 5,241,835 8 1.1925% 0.4550 5,225,153 9 1.1994% 0.2775 4,201,368 10 0.9014% -0.1073 3,042,245 Total 1.2911% 0.9157 5,287,555

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Size premium in Europe 25 beta and market value is run. These 156 regression each produce a slope for the two independent variables and the intercept (if present in the particular model). Table III shows the regression output for the three models of the 156 independent regressions.

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Size premium in Europe 26 Table III Regression output for the cross-sectional regressions

Table III shows the regression output for the three different versions of the model:

where panel A consist of the results of

the original model. Panel B shows the model were the intercept is fixed with the introduction of . Panel C presents the model in which the intercept is taken out of the consideration by deducting from the dependent variable and the regression is run on the difference of the independent variables and their mean. The results of the 156 individual monthly cross-sectional regressions are confined by giving the mean and median of the monthly regression coefficients. The and are calculated for each month and their mean and median values are given. For each individual month and explanatory variable, a two-sided t-test is performed and the results are displayed in the table by giving the values for their mean, median and max. Furthermore, a count is given of the number of times the test confirmed the significance of the coefficient at a 10-, 5-, and 1% level.

Panel A: Original model

Mean 0.0367 0.0048 -0.0019 0.5485 0.4195 Median 0.0075 0.0017 -0.0002 0.595 0.4793 P-values (t-test) Mean <0.0001* 0.0075 <0.0001 Median <0.0001 0.0031 <0.0001 Max <0.0001 0.1705 <0.0001 0.1>p>0.05 0 2 0 0.05>p>0.01 0 27 0 0.01>p 156 126 156

Panel B: fixed intercept

Mean 2.0842 0.0048 -0.0019 0.5515 0.4233 Median 1.6298 0.0013 -4.8E-06 0.5976 0.4827 P-values (T-test) Mean 0.0692 0.0075 <0.0001 Median 0.0003 0.0031 <0.0001 Max 0.9425 0.1705 <0.0001 0.1>p>0.05 10 2 0 0.05>p>0.01 18 27 0 0.01>p 109 126 156

Panel C: demeaned factors

Mean 0.0048 -0.0019 0.1917 0.0906 Median 0.0017 -0.0002 0.1166 0.0061 P-values (T-test) Mean 0.8473 0.7871 Median 0.8821 0.8259 Max 0.9989 0.9991 0.1>p>0.05 0 0 0.05>p>0.01 0 0 0.01>p 0 0

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Size premium in Europe 27 Figure I Spread of the estimated coefficients in model 2

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5.2 Analysis 2

In the previous section the significant influence of both beta and size was established for the 156 month sample of equity markets in the North-Western Eurozone. In the next part of the analysis, these results are sought to be confirmed using equation (10). The time-series regressions, test for a zero intercept in size-based portfolios over the sample period of 156 months. Stocks are placed in size-based portfolios and rebalanced monthly, the use of portfolio returns instead of individual stock returns increases the power of the test. The descriptive statistics of the portfolios can be found in Table IV.

Table IV Descriptive statistics of size-based portfolios for time-series regressions

Table IV exhibits the mean and median for the three variables used in the second part of the analysis. The mean and median market value, monthly return and beta are given of the 156 data points in the time-series regression per portfolio. The portfolios are size-based, each month the portfolio is rebalanced based on the market value of the individual securities in that specific month. The market value is given in `000’s of dollars.

Portfolio Market Value ($) x1000 Return Beta

Mean Median Mean Median Mean Median

1 36,900,092 37,418,904 1.13% 1.76% 0.99 0.93 2 8,073,962 8,100,950 1.26% 1.50% 1.06 1.02 3 3,374,707 3,673,790 1.53% 2.00% 1.03 0.97 4 1,720,642 1,875,826 1.57% 2.02% 0.87 0.88 5 1,010,987 1,098,329 1.39% 1.91% 0.93 0.90 6 627,021 668,598 1.70% 2.10% 0.94 0.85 7 401,859 445,501 1.40% 2.12% 0.91 0.82 8 254,673 274,348 1.31% 1.94% 0.79 0.77 9 154,100 159,269 1.07% 1.83% 0.80 0.76 10 56,637 56,618 0.54% 0.46% 0.85 0.83

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Size premium in Europe 30 Table V Regression output for the time-series regressions

Table V presents the output of the time-series regressions. For each size-based portfolio the following regression is run: where is the

return on each portfolio in month ; is the return on the 3 month German T-bill in the month preceding month ; is the portfolios beta in the preceding month and is the return on the

market index MSCI World in the month preceding month . Stock are divided in size-based portfolios. Portfolio 1 contains the largest-cap stocks and portfolio 10 the smallest-cap stocks. Regression estimations for the values for the intercept and independent variable are

given together with the p-value of a two-sided t-test, testing their significance. Column six gives the value and p-value for the Jarque-Bera test which tests for autocorrelation an heteroscedasticity in the residuals. The last two columns give the value of the regressions and .

Portfolio p-value p-value

Jarque-Bera p-value 1 0.009 0.021 0.131 0.116 2.084 0.353 0.014 0.008 2 0.010 0.013 0.228 0.003 4.017 0.134 0.048 0.041 3 0.013 0.003 0.248 0.003 0.355 0.837 0.057 0.051 4 0.013 0.003 0.294 <0.001* 3.182 0.204 0.074 0.068 5 0.011 0.015 0.326 <0.001 3.828 0.147 0.085 0.079 6 0.014 0.003 0.411 <0.001 1.716 0.424 0.116 0.110 7 0.012 0.007 0.412 <0.001 5.547 0.062 0.130 0.124 8 0.010 0.009 0.465 <0.001 5.557 0.062 0.170 0.164 9 0.008 0.092 0.468 <0.001 2.436 0.296 0.121 0.116 10 0.003 0.449 0.458 <0.001 1.774 0.412 0.141 0.135

* "<0.001" means: a value smaller than 0.001.

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6 | CONCLUSION

This thesis re-examined the size effect (i.e. the tendency of small-cap securities to have higher risk adjusted returns than large-cap securities) as first introduced by Banz (1981) for the North-Western Eurozone.

The empirical analysis of this thesis focussed on answering the question: "Can the

existence of a significant size effect be empirically proven in the North-Western Eurozone for the years 2001-2013?". In order to do so, a sample was taken from stock exchanges from 7

countries in the North-Western Eurozone (The Netherlands, Belgium, Luxembourg, France, Germany, Austria and Finland). This sample was used to perform 156 individual Fama-Macbeth (1973) style cross-sectional regressions on beta-based portfolios and 10 time-series regressions run on size-based portfolios. The results show that the existence of a negative and significant size effect cannot be rejected. However, the results are not as solid as predicted in the hypothesis and existing literature; the sign of the coefficient is not as consistent throughout the analysis as expected. In the second part, a positive and significant intercept is found when regressing size-based portfolios over time. However, the results do not show the monotonically, increasing pattern which was expected based on previous literature; and the two smallest-stock portfolios have the lowest intercept out of the ten portfolios. This means that the size related premium is the lowest for the smallest-cap securities, however this result is not statistically significant. The results of this particular analysis are in line with a trend that has emerged since the late 1990's, where theories have been developed in which the size effect rises endogenously from firm- and market-characteristics. Inconsistent with this is the variance in the empirical evidence found by this thesis. Results are not consistent enough in the sample to exactly pin-point the size of the size effect in the time period examined. Despite this, the presence of a size effect cannot be rejected, for a sample that has not been tested before. So, to answer to the second research question: yes, based on this research the existence of a significant size effect in the North-Western Eurozone during the years 2001-2013 cannot be rejected.

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Size premium in Europe 33 bubble and the financial crisis of 2008. This is likely to have affected the stability of the size effect in the tests, since in times of deep recession the volatility of small-cap securities is affected more than that of large-cap securities.

Practitioners in the field of corporate finance, and valuations in particular, to this date still use the CAPM model plus a size factor to construct the cost of equity of a business. This method is widely spread and advocated in text books on this matter, of which "Valuation: Measuring and Managing the Value of Companies" (Koller, Goedhart and Wessels, 2010) is most used and regarded as the standard work in the field. The question that was proposed in the beginning of this thesis was: " How can the size effect be explained?". First of all, the academic literature has shown that the validity of the results by Fama and French (1993) regarding their risk factors are not as solid as often appears in university text books. Work by Dichev (1998), Baker and Wurgler (2006) and Petkova (2006) shows that the assumption of the SMB factor being based on an underlying risk factor is not robust. This argument is confirmed, although from a different angle, by Fernholz who also concludes that the presence of a size premium is not based on any underlying risk factors. During the course of the years since the first empirical evidence on the size effect, several different explanatory variables have been tested; e.g. liquidity, dividend yield and seasonality, none of which have successfully subsumed the size effect in empirical tests. This issue is more directly addressed in work by e.g. Heston et al. (1999) and De Moor and Sercu (2013). Heston et al. (1999) find that after controlling for the SMB factor, size is still a significant variable in explaining variance in stock returns, De Moor and Sercu (2013) conclude that after controlling for different independent variables, the size effect is not subsumed by any of them. It is clear that a satisfactory explanation for the size effect that can be empirically tested and is robust enough to be valid for practical implementation, is not yet found. An important implication of this result, is that as long as research cannot come up with a model that that is robust over time and over samples and shows no ambiguity about the cause of the size effects incorporated in the model, practitioners will use the empirical alpha plus CAPM, a method that has been consistently robust over time. This thesis suggests that this method is more robust than an asset pricing model, as it allows for exceptions or outliers in the hypothesised monotonically increasing pattern of the size effect for increasingly small-cap companies.

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Size premium in Europe 34 when evaluating European companies or stocks, the size premium should be a factor, and subsequently data from this market can be used instead of the usual US data. Furthermore, the approach which is used (i.e. the use of a size premium in access of a CAPM calculated cost of equity) is confirmed as preferable over the use of an empirically inspired asset pricing model, based on underlying risk-factors such as the Fama-French model. Applying such a model is a precarious matter when there is ambiguity about the robustness and underlying risk-factors, which is the case for the models considered in this thesis.

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8 | APPENDIX

Figure II Spread of the estimated coefficients in model 1

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Size premium in Europe 41 Figure III Spread of the estimated coefficients in model 3