The Size Effect in Accounting Returns

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The Size Effect in Accounting Returns

Master Thesis Manon Wolters

September 2012

Abstract

This paper investigates the behavior of the book return on equity, with regard to the cost of equity and the size effect for 2367 U.S. companies form the Russell3000 index over the period 1982 to 2011. The analysis includes; (i) whether the return on equity has exceeded the cost and (ii) whether the size effect was present. The result are surprising; the book return on equity was consistently lower than the cost of equity. Furthermore, a statistically significant negative size effect was found. After adjusting for risk, large firms remain more profitable than small firms. The results are robust, however it is difficult to draw solid conclusions due to biased cost and return estimates.

JEL Classification: G12, G14, G30

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The Size Effect in Accounting Returns

University of Groningen Faculty of Economics and Business

MSc Business Administration Specialization Finance Author: H.M. (Manon) Wolters Student number: 1566458 Supervisor:

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Table of content

1. Introduction 4

2. Literature review 6

2.1 The size effect 6

2.1.1 Evidence on the size effect 7

2.1.2. Explanations for the size effect 8

2.2 The book return on equity 9

2.3 The cost of equity 11

2.3.1 Capital Asset Pricing Model 11

2.3.2 Fama-French Three factor model 12

2.3.3 Arbitrage Pricing Theory 12

2.3.4 Historic average market return 13

2.4 Hypotheses 13

3. Data and Methodology 15

3.1 Data 15

3.1.1 Variable Definition 15

3.2 Methodology 18

3.2.1 Portfolio formation 18

3.2.2 ANCOVA on book return versus market cost 19

3.2.3 OLS regression on size effect 20

3.3.4 Industry size effect 21

3.3 Descriptive statistics 21

4. Results 23

4.1 Results size and dependent variables 23

4.2 Results ANCOVA, book return versus market cost 25

4.3 Results OLS regression size effect 28

4.2.1 Results industry size effect 31

4.4 Robustness checks 32

5. Conclusion 33

5.1 Summary and conclusion 33

5.2 Limitations and further research suggestions 34

Appendices

A Overview of number of firms available per year with complete data B Descriptive statistics

B1. Descriptive statistics, per size-based portfolio B2. Descriptive statistics, per industry

C Descriptive statistics, robustness checks

C1. Descriptive statistics, per size-based portfolio C2. Descriptive statistics, per industry

D Results ANOVA and Kruskal-Wallis, robustness checks E ANCOVA- analysis, robustness check

F Results Student’s t-test and Wilcoxon Signed-Ranks test, robustness checks

F1. Per sub-periods and over the whole sample period F2. Per ten size-based portfolio

F3. Per industry

G Results OLS regression, robustness check

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1. Introduction

The average returns for small stocks have been higher compared to large stocks, over the past 80 years (Damodaran, 2002, Brealey et al., 2004, Berk et al., 2012). Figure 1 depicts the cumulative return of a $1, - investment in1925 through 2006, for different portfolios. It illustrates that small stocks undoubtedly outperformed all other portfolios. However, small stocks also experienced greater volatility in returns, for instance when comparing small stocks to the S&P500 or to Treasury bills. The average standard deviation of small stocks was 42.46%, compared to average standard deviations of 20.52% and 3.42% for respectively S&P500 and Treasury bills (Berk et al., 2012). Investors are considered to be risk averse, and therefore dislike greater volatility in returns. The higher returns can be seen as a compensation of the higher risk investors face (Brealey et al., 2004, Berk et al., 2012).

Figure 1. Cumulative market return on different funds for 1925 to 2006

* Data from Brealey, Marcus and Myers, 2004

In an efficient market, returns are influenced by two types of risk: firm specific risk, or unsystematic risk, and risk related to the market, systematic risk. Unsystematic risk can be circumvented by holding a diversified portfolio with a variety of assets. In contrast, systematic risk cannot be avoided. Assuming rational investors, diversification can eliminate unsystematic risk. Hence, the return of a stock is only influenced by the exposure to systematic risk (Fama and French, 1992, Berk et al., 2012). Nevertheless, empirical evidence has shown that after adjustments for market specific risk, small stocks still experience higher average returns than large stocks (Banz, 1981, Reinganum, 1981, Keim, 1983, Fama and French, 1992).

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al., 2012). Still, Banz (1981) found, using a U.S. sample for the period 1936-1975, that a portfolio containing only the smallest firms, yielded a 0.4% higher average market return per month than a portfolio with the largest firms. He constructed the portfolios based on market capitalization. Moreover, Reinganum (1981) presented a similar study using a sample of U.S. firms listed on the NYSE and Amex. In his research the small firm portfolio outperformed the large firm portfolio by a overwhelming 1.77% per month. The persistence of the small firm size premium indicates that the CAPM underestimates the true risk exposure of small firms, the estimates of the cost of capital of small firms (Damodaran, 2002).

Researchers have recognized several variables that are better able to capture the true risk exposure for firms. Amongst these variables are: firm size, the price-earnings ratio and the book-to-market ratio (Banz, 1981, Reinganum, 1981, Roll, 1981, Chan, Chen and Hsieh, 1985, Chan and Chen, 1988, Fama and French, 1992). Additionally, differences in transaction costs, trading infrequencies and momentum, have been named as potential explanations for the size effect (Jegadeesh, 1990, Dimson and Marsh, 1999, Stoll and Whaley, 1983, Chan and Chen, 1991). However, no consensus exists in literature on either of these theories being the explanation. In corporate finance practice some analysts adjust the CAPM, by adding a 2% size premium, in order to correct the cost of capital of small firms (Damodaran, 2002). Overall, there is an apparent difficulty in determining the true cost of capital.

This paper will take a closer look at the relationship of return and cost of equity over the past 29 years. Most academic papers use market return as the measure for firm performance, which is surprising for at the least two reasons. First, market returns are noisy since they include expectations about growth assets and are marked up for the potential future earnings power of current assets (Brealey et al., 2004, Damodaran 2007). Second, corporate finance and valuation use information from financial statement analysis, and are therefore derived from accounting numbers, which do not include market returns Damodaran, 2007). Considering the above, this paper will use a different approach. The book return on equity will be used as performance meassure, instead of the market return. The book return on equity is an accounting number, telling investors how much they have earned with their investment in a firm and identified as the main driver of profitability (Penman and Nissim, 2001), which in turn is the main driver of firm value (Koller et al., 2010). Moreover, the book return on equity effectively measures the return on capital (Damodaran, 2007).

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The two-parted research question of the paper is:

Does the book return on equity exceed the market cost of equity, where the cost of equity is estimated by (i) the average historical market return on equity, and (ii) as the expected rate of return on equity using the CAPM.

If this is true it leads to the second question:

Is the book return on equity of small firms higher than the book return on equity of large firms.

The cost of equity will be estimated in two ways. First, as the average historical market return on equity, and second, by the expected rate of return on equity using the CAPM. The results presented in this paper are based on information from data of 2367 U.S. firms over the period 1982-2011.

The rest of the paper is structured as follows: Section 2 will provide an overview of the literature leading to the hypotheses. Section 3 describes the data and formulate the methodology applied in this paper. Section 4 presents the results, where section 5 concludes.

2. Literature Review

First the empirical evidence found on the ‘size effect’ in market returns is presented. Second, the implications of the size effect on the book return on equity will be described. Third, general corporate finance theory is discussed and the relation between the cost of equity and the return on equity are explained. The chapter will finish with a discussion about the different models available to estimate the cost of equity.

2.1The size effect

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discussed in literature, are the market effect and the leverage-effect. Firms with high book-to-market ratios and higher leverage ratios outperform firms with lower book-to-book-to-market and leverage ratios (Fama and French, 1992) 1. However, the focus of this paper will be on the size effect, which still remains a puzzle to academics. As early as 1973, a relation between return and market value of a firm (measured by market capitalization) was documented by Fama-MacBeth (1973). The next section will provide an in-depth analysis of the firm size anomaly.

2.1.1. Evidence on the size effect

One of the first to explore the effect of market capitalization of stocks returns was Banz (1981). He that the group of smallest stocks yielded a 0.4% higher average market return per month than the largest stocks, over the period 1926 to1975 using stocks listed on the NYSE. However, the lack of theoretical foundation prevents him from concluding that market capitalization is the driving force behind the higher average market returns of smaller firms. He argues that the result could also be caused by another, unidentified, variable which is highly correlated to firm size. Consequently, he argues that the CAPM is not able to properly capture the systematic risk exposure of firms. Reinganum (1981) found a 1.77% higher market return per month for a portfolio consisting of the smallest firms compared to a portfolio containing the largest firms. His study is based on a sample including 566 firms listed at the NYSE and AMEX, over the period 1963 through 1977. Another influential paper pointing out the misspecification of the CAPM was presented by Fama and French in 1992. The paper investigated stocks listed on NYSE, AMEX and NASDAQ over the period 1963 to 1990, and looked at market returns, the risk exposure related to firm size, and the book-to-market ratio. They built portfolios based on market capitalization of the firms. The results show that the portfolios with the smallest firms outperform the portfolio with the largest firms by 0.63% per month. Thereafter, they constructed size portfolios, but subdivided them based on firm betas. They found no evidence of beta having significant explanatory power on market returns. On the contrary, they did found book-to-market ratio to have an effect on market returns. Keim (1983) found relatively high betas for small firms. However, these differences are not able to account for the 2.5% excess market returns he found for the small firm portfolio compared to the largest. Table 1 summarizes the monthly risk premiums found by different papers for several geographical areas as well as over varying time periods. The table is far from exhaustive, but does reflect the scope to which the size effect has been found.

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8 Table 1. Evidence on the size effect

Literature Period Country

Monthly Average Excess Return

Banz (1981) 1926 - 1975 U.S. 0,40%

Reinganum (1981) 1963 - 1977 U.S. 1,77%

Keim (1983) 1963 - 1979 U.S. 2,52%

Fama and French (1992) 1962 - 1989 U.S. 0,63%

Annaert et al. (2002) 1974 - 2000 Europe 1,45%

Ibbotson (2008) 1926 - 2007 U.S. 0,39%

Chan et al. (1991) 1971 - 1988 Japan 0,97%

Bagella et al. (2000) 1971-1997 UK 1,18%

2.3.2. Explanations for the size effect

Although literature has tried, no consensus has been reached so far on the explanation of size effect. In the attempt to find an answer to what causes the size effect, two general strings of researchers can be identified. The first group present theories explaining its existence, mainly related to systematic risk, market inefficiencies and irrational behavior of investors. The second group disputes that the size effect actually exists, and claims the empirical findings are caused by methodological issues. The different theories will be elaborated briefly in this chapter.

Starting with Banz (1981), who concludes in his paper that the CAPM is misspecified. In his opinion the persistence of the size effect indicates that beta is not able to measure the systematic risk. Either size itself, or some other variable correlated to size, should be added to the CAPM to capture the true systematic risk exposure of firms. Fama and French (1993) identify size and market-to-book ratio as good proxies for risk exposure of firms. Chan et al., 1985, state that distress, measured by the default spread, and other macroeconomic variables that capture changes in economic conditions, influence small and large firms differently, causing differences in returns. Reinganum (1981) uses the APT model and includes up to five different macroeconomic factors in his analysis. The results show that firm size has a significant impact on market returns.

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Yet another set of researches tried to explain the size effect by turning to behavioral finance. The reasoning behind this is that the size effect is caused by less than fully rational investors. Chan and Chen (1991) find that small firms are often “fallen angels”, firms which have performed bad in the past. This is related to the arguments of Lakonishok at al. (1994), who stress that the value effect might also explain the size effect. The value effect is caused by investors’ overreaction to bad performance of firms in the past, driving the current value of a stock down too far. When the market in time corrects the mispricing, high returns can be earned by investors. Others have claimed that investors simply do not want to hold smaller stocks. Especially institutional investors are said to prefer holding large and liquid stocks (Gompers and Metrick, 2001). Also, less information is often available about smaller stocks, increasing the risk perception of investors (Banz, 1981, Merton, 1987). An additional feature of investors is that they can be overconfident about their own judgment (Daniel et al., 2001). Investors will tend to overreact to available information, for example information on the market value of a firm, temporarily causing mispricing, for it takes time to be corrected by the market.

Lastly, a concept explaining the size effect is the January effect, that returns are much higher in January than during the rest of the year. Correcting for the January effect in average stock returns, reduces the size premium found for small firms over large firms by half, over the period 1963 to 1979 (Keim, 1983). Although no consensus exists about the causes of the January effect, it does seem to influence the statistical validity of the size effect (Ibbotson, 2008).

2.2 The book return on equity

As noted, most academic papers addressing the size effect are based on an investigation of market returns, which is surprising. Intuitively, it makes more sense to use an accounting return as performance measure of the firm, since practitioners in corporate finance base their valuation on accounting numbers (Brealey et al., 2004, Damodaran, 2010, Koller et al., 2010). Most models are built around assumptions of future cash flow and residual earnings, which critically depend on the growth and profitability of a firm (Koller et al., 2010). In this study the focus is on the book return on equity (net income divided by the book value of equity) as performance measure. The book return on equity is identified as the main driver of profitability and therefore firm value (Penman and Nissim, 2001, Koller et al., 2010). The book return on equity effectively measures the overall profitability of the firm, where all financial holdings are included (Penman and Nissim, 2001, Damodaran, 2007). An extra pro of the book return on equity is that it tells investors how much they have earned with their investment, after all costs have been deducted.

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Moreover, Basu (1997) states that accounting numbers respond differently to good and bad news. Bad news is incorporated in the numbers faster than good news, biasing the book value downwards. However, Zhang (2000) stresses that not only book value is decreased by conservative accounting but net income as well. Furthermore, potential measurement errors are consistent over time, leading Zhang (2000) to conclude that on average the book return on equity will be equal to the true economic income of a firm.

Economic theory states that an investment is only profitable when the return on an investment exceeds the costs (Berk et al., 2012). Investors are not so much interested in earnings growth, since earnings do not take into account the cost of capital. Any project with a positive rate of return creates earnings, but true value creation only occurs when the return is higher than the costs. Assuming that Zhang is right and the book return does equal the true economic income of a firm, the return will be measured by the book return on equity. If managers act in the best interest of investors, they will aim to maximize the book return on equity (Brealey et al., 2004). In other words, in an efficient market, the book return on equity over the past 29 years should have exceeded the cost of equity. Penman and Nissim (2001) also expected the book return on equity to be higher than the cost of equity. Even more so because conservative accounting applied by firms, depresses the book value, raising the book return on equity. However, the results contradict the theory. Between 1963 and 1999, the median book return on equity was lower than the cost of equity for most years. Penman and Nissim (2001) blamed the results on the estimate used for cost of equity, the risk-free rate plus 6%, which apparently was too high. The first part of the research question is concerned with the relationship between the return on equity and the cost of equity.

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Fama and French (1993) also establish that there are variations in the book return on equity based on size and the book-to-market ratio. However the results suggest that, if the market is efficient and investors are rational, conditional on the book-to-market ratio, small firms were significantly less profitable compared to large firms after the 1980s (Fama and French, 1993). This suggests a negative relationship between size and book return. Horowitz et al. (2000) claim that the results of Fama and French (1993) are biased since data after the 1980s are used. They find in their paper that the crisis in 1980 has left small firms in an “earnings depression”, causing a reversal of the size effect. Furthermore, Fama and French (1993) themselves argue that the book return might be biased since accounting numbers are only available on an annual basis, compared to daily numbers on the market return. The second part of the research question focuses on whether investors in small firms have indeed earned higher returns over investors in large firms.

2.3. The cost of equity

An important factor in the investment decisions of a firm is the cost of equity. The cost of equity represents the expected rate of return on the firms stocks and can be calculated in different ways. The most commonly used asset pricing model is the CAPM. Other multi-factor pricing models have gained popularity in literature, especially the Fama-French three-factor model and the Arbitrage Pricing Theory model (Koller et al., 2010). A different method to calculate the cost of equity is to use historic market return as an estimate. The asset pricing models and the historic market return are discussed one-by-one.

2.3.1. Capital Asset Pricing Model

Practitioners mostly use the Capital Asset Pricing Model to calculate the cost of equity (Graham and Harvey, 2001, Koller et al., 2010). The theory behind the CAPM is that investors want to be compensated for taking on more risk. Therefore, the required rate of return on equity will be determined by an evaluation of the relationship between expected return and risk. The expected return depends on two types of risk, market risk and firm-specific risk. The market risk is the same for all firms and cannot be avoided, while the firm-specific risk can be diversified away by holding a broad portfolio of assets. The formula of the CAPM looks as follows;

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estimate the true risk exposure of a firm (Banz, 1981, Fama and French, Reinganum, 1981). The fact that, for instance size, the book-to-market ratio, and other macroeconomic variables are found to have explanatory power on returns has led to the development of other models claiming to better predict the cost of equity. The two most influential ones are the Fama-Frech three-factor model and the arbitrage pricing theory model (henceforth APT) which will be discussed hereafter.

2.3.2. Fama-French Three factor model

Fama and French (1992) questioned the functioning of the CAPM. They found no relation between beta and average stock returns, looking at stocks listed on the NYSE, AMEX and NASDAQ between 1963 to 1990. Going back even further to the 1940s, the expected positive relation between beta and average stock market returns was only weak. Fama and French did identify two other factors that help explain cross-sectional variations in average stock returns, size (measured as market capitalization) and the book-to-market ratio. According to theory, both variables are proxies for risk. Size ought to correct for the fact that smaller firms are found to have higher average market returns, implying higher risk exposure than larger firms. The book-to-market ratio is used as an indicator of the level of financial distress a firm is in, a higher book-to-market ratio is related to a higher level of financial distress. The basis of the Fama-French three-factor model (1993) is the CAPM extended with these two factors. The three-factor model regresses excess stock returns on three different factors; (1) excess market returns, (2) excess returns from small over big firms and (3) excess returns from high book-to-market firms over low book-to-book-to-market firms. Empirics show that the model captures most of the average market return differences between small and large firms, which are not explained by the CAPM (Fama and French, 1993). However, these results are based solely on empirics, and lack theoretical foundation.

2.3.3. Arbitrage Pricing Theory

Another model put forward in finance to calculate the expected return on an asset is the Arbitrage Pricing Theory model (Roll and Ross,1980). The APT model regresses the excess returns over certain risk factors as well. Besides, incorporating the two risk factors familiar from the CAPM, the risk-free rate the market risk premium (Ibbotson, 2008).However, the APT model allows for an infinite number of other factors that potentially influence the return of an asset. The APT model looks as follows:

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model specifies neither the nature nor the number of other risk factors that should be include in the calculations Therefore it is hard to implement, and open to much debate (Koller et al., 2010).

2.3.4. Historic average market return

Having discussed all the doubts about the different asset pricing models, Pastor and Stambaugh (1998) claim that the historic average market return is a good estimate of the past cost of equity. The market return of one year is not a good proxy for the cost of equity, since it is highly variable. However, assuming that the risk exposure of a firm is time-invariant and the market is efficient (Koller et al., 2010) calculating the average market return over the past 29 years should provide a valid estimate of the cost of equity of the firms.

Although scholars disagree on the appropriateness of the CAPM (Banz, 1981, Reinganum, 1981, Fama and French 1992), fact is that the CAPM is the only model built upon a sound theoretical base. In academia as well as corporate finance practices, the CAPM still is the mostly used model in cost of equity computations (Graham and Harvey, 2001, Damadoran, 2007, Koller et al., 2010). One reason to use the CAPM, is definitely its simplicity (Koller et al., 2010). Furthermore, the CAPM is based on numerous assumptions. However, both the three-factor model and the APT model have similar implementation issues. For instance, the question on how many years of data should be used to determine the risk factor loadings, and whether monthly or annual numbers are best to use (Koller et al., 2010). This paper will use two different methods to estimate the cost of equity to satisfy the skeptic crowd; (1) the historic average market return and (2) the CAPM.

2.4 Hypotheses

The previous section has clearly outlined the theoretical background on the general functioning of the economy with regard to the cost and return of an investment, and the evidence on the size effect. Based on the literature the following two-part research question emerges.

Does the book return on equity exceed the market cost of equity, where the cost of equity is estimated by (i) the average historical market return on equity, and (ii) as the expected rate of return on equity using the CAPM.

Therefore, the second part of the question the paper tries to answer is:

Is the book return on equity of small firms higher than the book return on equity of large firms.

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14 Hypothesis 1

Ho:: The book return on equity exceeds the market cost of equity. H1: The book return on equity does not exceed the market cost of equity.

The hypothesis will be tested first, with the cost of equity estimated by the average historical market return on equity and second by the expected return on equity based on the CAPM estimation.

The second subject explained above is the empirical anomaly of the size effect. Given an efficient market, the return of a firm should only depend on its exposure to risk, market risk as well as firm specific risk. However, size is found to have explanatory power on a firms’ market return after adjusting for risk (Banz, 1981, Reinganum, 1981, Fama and French, 1992). The second hypothesis checks whether the data confirms the size effect described by academics:

Hypothesis 2:

Ho: The cost of equity of smaller firms exceeds the cost of equity of larger firms.

H1: The cost of equity of smaller firms does not exceed the cost of equity of larger firms.

Again the cost of equity are estimated using the historic market return on equity, and the expected return on equity estimates of the CAPM, over the period 1982 to 2011. If hypothesis 2 holds and small firms experience higher cost of equity than large firms, this implies that the returns of small firms should exceed the returns of larger firms. The third hypothesis focuses on the presence of a size effect in the book return on equity; the driver behind profitability (Penman and Nissim, 2001).

Hypothesis 3:

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3. Data and Methodology

3.1 Data

The paper’s initial sample included all U.S. listed firms of the Russell3000 Index as of June 2011. The Russell3000 index is a broad index, covering almost 98% of the total value of all equity traded on the U.S. stock markets. Koller et al. (2005) argue that the market risk premium in the U.S. is more or less stable over time; which leads them to propose the use of data from the longest time period possible, in order to reduce estimation errors. Thomson Reuters DataStream presents the source of the data, and the period studied is from 1982 to 2011, since accounting data are generally not available before 1982. The sample also includes firms that have been delisted during the period of 1982 and 2011, since the data until the point of delisting is still useful (Brouwer et al., 1996). All firms must be listed for a minimum of 5 years (60 month) between 1982 and 2011 in order to be included. To calculate the book return on equity and the market return, data on net income, common shareholders’ equity and the market value is needed. The absence of any of these numbers led to the exclusion of the firm from the sample. Table 2 presents an overview of the construction of the final sample, including 2.367 firms Appendix A provides a list with the number of firms with complete data available, per year. In the end 633 firms had to be excluded. The sample covers ten different industries based on the NYSE criteria: Basic Materials, Consumer Goods, Consumer Services, Financials, Health Care, Industrials, Oil & Gas, Technology, Telecommunication and Utilities.

Table 2. Data construction of final sample, 2.367 firms

Russell3000 Index Number of firms Total number of firms in Russell3000 Index 3.000 Firms not present in DataStream 59 No accounting data for min. 5 years 353

No market return data 221

Final Sample 2.367

3.1.1 Variable definition

This section provides an overview of both the calculations and the origin of the different variables (table 3). Starting with the average book return on equity and proceeding with the two different estimates for the market cost of equity, following Koller et al. (2010) and Damodaran (2006). Years in which firms experienced negative common shareholders’ return on equity were excluded in the calculations of book return on equity (Penman and Zhu, 2011).

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cost of equity is estimated by the Capital Asset Pricing Model, and will be denoted by CROE. As mentioned earlier the CAPM is based on three factors; the risk-free rate, the market risk premium and the beta. The choice of these three factors has a large influence on the final estimate, therefore the exact definition of each factor will discussed one by one. The formulas are shown in table 2.

Risk-free rate

The risk-free rate should represent the return on a riskless asset. However, such an asset does not exist in reality and only the construction of a complex portfolio with no covariance to the market portfolio would suffice. The next best alternative mostly used in practice is a default-free government bond, where there is almost no risk of default (Koller et al., 2010). Since this paper only contains U.S. based firms, the 10-year U.S. treasury bill is chosen as an estimate for the risk-free rate (Damodaran, 2010).

Market risk premium

The market risk premium is calculated by subtracting the risk-free rate from the expected market return. Since the expected market return is unobservable, an estimate has to be employed once more. Koller et al. (2010) find the market risk premium over the past 100 years to range between 4.5% and 5.5%. Damodaran (2010) advocates the use of a 5% as estimate for the market risk premium, based on historic numbers. The use of a historic risk premium of 5% is therefore selected. As a robustness check, calculations are repeated using a risk premium of 3%.

Beta

The third and last factor needed for the estimation of the return on equity by the CAPM, is the firm specific beta. The CAPM beta measures the sensitivity of the return of a firm to changes in the return of the market portfolio (Brealey et al., 2004).The market portfolio ideally includes all assets, traded and untraded (Koller et al., 2010), therefore the use of a broad benchmark is appropriate.

All firms included in the sample are from the Russell3000 index, which is a broad benchmark in itself. Therefore the average annual return of the total Russell3000 index is used as the market portfolio index here. Total return data is used to accurately include dividends.

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Table 3. Variable Description

The different formulas used to arrive at the dataset are described and their sources are presented.

Variable Name Description

Nit Net Income before preferred dividends

BVt Common shareholders' equity

BROEi Book return on equity

Vt Market Value on common equity

MROEi Market return on equity

Rf Risk-free interest rate

Beta, index Russell3000

Rm – rf Market risk premium

CROEi

Expected rate on return on equity estimated by the CAPM

Βadj i Adjusted beta

ACROEi Average market return on equity

NOSH Number of shares outstanding

Pi,t Stock price

The annual historic market cost of equity and the CAPM estimates of the cost of equity, are converted into geometric averages (formula 3). The conversion normalizes the averages over the 29 year period, which enables a comparison between the book retur

the cost of equity for the different measures.

Another important variable in this study is firm size (MV). The firm size is estimat capitalization at the end of each of year

firm (Fama and French, 1992). Since the focus is on the equity capital market, market capitalization is used instead of total assets, following Fama and French (1992).

The different formulas used to arrive at the dataset are described and their sources are presented.

Computation

Net Income before preferred dividends DataStream Worldscope item WC01651 Common shareholders' equity DataStream Worldscope item WC03501

2 Book return on equity

Market Value on common equity DataStream Worldscope item WC07210

Market return on equity DataStream RI

est rate _{10-year U.S. treasury bill}

, index Russell3000 ,

Market risk premium 5.00 % Expected rate on return on equity

estimated by the CAPM

Average market return on equity

Number of shares outstanding DataStream key NOSH Stock price DataStream key P

The annual historic market cost of equity and the CAPM estimates of the cost of equity, are converted into geometric averages (formula 3). The conversion normalizes the averages over the 29 year period, which enables a comparison between the book return on equity per year and the central tendency of the cost of equity for the different measures.

, ,… . ,

Another important variable in this study is firm size (MV). The firm size is estimat

capitalization at the end of each of year t (number of shares outstanding times the stock price) of the firm (Fama and French, 1992). Since the focus is on the equity capital market, market capitalization is

ollowing Fama and French (1992).

17 The different formulas used to arrive at the dataset are described and their sources are presented.

DataStream Worldscope item WC01651 taStream Worldscope item WC03501

DataStream Worldscope item WC07210

The annual historic market cost of equity and the CAPM estimates of the cost of equity, are converted into geometric averages (formula 3). The conversion normalizes the averages over the 29 year period, n on equity per year and the central tendency of

(3)

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3.2 Methodology

In this section the procedure for testing the hypothesis is discussed. First the size-based portfolio formation will briefly be explained, followed by the statistical methods used to answer the two-parted research question. Hypothesis 1, whether book returns have exceeded market costs, will be tested by using an ANCOVA analysis. The hypothesis on the size effect in book return data will be tested using the Kruskal-Wallis test and an simple Ordinairy Least Square regression, following Lerman and Parliament (1991).

3.2.1 Portfolio formation

To obtain the different size-based portfolios, the firms are ranked based on their market capitalization at the end of each year, and subsequently divided into ten equally-sized groups (Fama and French, 1992). The portfolios are updated annually, where portfolio 1 contains the smallest 10% of the U.S. firms, and portfolio 10 the largest 10% of the U.S. firms. Since the number of firms with complete datasets over the entire period of 1982 to 2011 is limited, the portfolios contain a different number of observations per year. In 1982, each portfolio merely contains 49 firms, while in 2011 the number has grown to 231 firms per portfolio. Appendix B shows an overview of data descriptives of the different size-based portfolios. For further calculations the median values are used since they are better able to correct for extreme outliers.

The ten size-based portfolios represent the basis of the analysis for testing whether size has an effect on both the cost of and the return on equity. However, first it is important to know whether size has an impact on the return on equity and the cost of equity at all. This is tested by performing a one-way analysis of variance (hereafter ANOVA) on the ten size-based portfolios.

The ANOVA tests the null hypothesis that all means of the groups within the sample are the same. If firm size has no influence on the dependent variables under consideration, the means and variance of the ten portfolios should not differ significantly. Moreover, the ANOVA is applied to look for differences between the ten industries and differences per year, in order to see whether one of the variables has an impact on the book return or market cost of equity. Fairfield et al. (2009) found significant influence of industry membership on forecasting growth and profitability.

The ANOVA is based on the assumption of a normal distribution of the residuals. However, the skewness and kurtosis (appendix B) indicate that the data is not normally distributed, therefore the non-parametric Kruskal-Wallis test is applied to make sure non-normality does not bias the results.2 The Kruskal-Wallis tests for significant differences in the medians between the portfolios. After ranking the pooled medians of the different portfolios, and summing the ranks for the pooled samples,

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it tests whether the differences between the summed ranks are significantly different from one another (Lerman and Parliament, 1991).

3.2.2 ANCOVA on book return versus market cost

The first part of the research question is whether the book return on equity has exceeded the market cost of equity, over the past 29 years. The cost of equity is estimated (1) as the average historical market return on equity, and (2) as the expected rate of return on equity using the CAPM. The paired two-sample Student’s t-test, which is a parametric test that checks whether the means of two samples are significantly different from one another, is used to test this theory (Brooks, 2008).

The results of the Student’s t-test tell us which group has the higher mean and whether the difference between the groups is statistically significant. When using the Students’ t-test, a critical assumption again is the normal distribution of the population underlying the sample data. Preferably, also the variances of both samples should be equal. However, Markowski and Markowski (1990) find that if the sample sizes are close to equal, which is the case here, the t-test is highly robust and variance heterogeneity does not affect the strength of the results. Therefore, despite the variances, the two-sample Student’s t-test should deliver reliable results. Since the data is not normally distributed, as noted, the non-parametric Wilcoxon Signed-Ranks test is performed, which is basically a the Kruskal-Wallis test for only two categories. It simply counts the number of times the book return on equity is higher, lower or the same as the cost of equity and determines whether the differences are statistically significant.

First a simple test on the aggregate level of book return on equity versus costs of equity will be performed over the entire 29 year period. The results provide a direction about whether the book return on equity on average is higher than the cost of equity.

To get a better insight into possible industry, size and sub-period specific differences, an analysis of covariance (hereafter ANCOVA) is performed. The ANCOVA test is similar to the ANOVA, comparing the medians of the ten portfolios but it allows for the inclusion of control variables. Three different control variables are used to construct the following four models:

(1) Testing the difference between the means of the ten portfolios, controlling for industry (2) Testing the difference between the means of the ten portfolios, controlling for year

(3) Testing the difference between the means of the ten portfolios, controlling for size-based portfolio

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20 3.2.3 OLS regression on size effect

The second part of the research question was whether the investors in small firms have earned higher book returns on equity and experienced higher cost of equity, than investors in large firms. Since the ANOVA and the Kruskal-Wallis test only show whether there is a difference between respectively the mean and the median of the ten size-based portfolios, but provides no information on the nature of the relationship, further testing is needed. For this purpose a simple ordinary least square regression (hereafter OLS) will be prepared. Thereafter, a more industry specific analysis is carried out.

In the OLS, three different dependent variables are defined; the book return on equity, the historic market return and the CAPM estimated cost of equity. These dependent variables will be regressed over the independent variable size, measured by the ten portfolios based on the market capitalization. The OLS regression is the following:

_,_{= ∝}_!_+∝ _"#$%_{+ ∈}_, ₍₃₎

Where,

',( = the equally weighted return on the size-based portfolio i

∝₀ = the intercept

∝₁ = the coefficient

"#$%' = the number of the size-based portfolio i

∈',(= error term

As noted before, the book return and market cost of equity, are also influenced by the industry they operate in (Koller et al., 2010) and certain industries contain significantly larger firms, measured by the median market capitalization. Furthermore, different time-periods might have an effect on the dependent variables. Therefore several different regressions are performed. The first model (1) includes only the dependent variable and the portfolio number as independent variable. The second model (2) adds beta as a control variable, since beta is supposed to capture a firm’s risk exposure (Koller et al., 2010). The third model (3) will, besides size and beta, also include dummy variables for three sub-periods in the data, 1982 to 1991, 1992 to 2001 and 2002 to 2011. In the last model (4) the sub-period dummies are exchanged for industry dummies.

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variable is excluded from the regression in model 3 and 4. In model 3, the dummy for the period 1982 to 1991 is excluded, and in the 4th model the dummy variable for the Utilities industry is excluded. The exclusion of one of the dummies means, that the intercept of the regression can be interpreted as the average value of the dependent variable for the excluded dummy variable. The coefficients of the other dummy variables are then the difference between the average value of the dependent variable in their period or industry, and the coefficient for the omitted dummy. In other words, in model 3, the intercept is the average value of the dependent variable over the period 1982 to 1991, while the coefficients of the over two sub-periods represent the difference to the reference value in 1982 and 1991.

3.2.4 Industry size effect

In order to get more insight in the presence of the size effect in the different industries, the data is reorganized per industry and ranked per firm size. Thereafter, the firms are split into two portfolios; portfolio 1 containing the 50% of the smallest firms and portfolio 2 the 50% largest firms. The reason for building just two portfolios instead of dividing each industry into 10 size-based portfolios is that some industries only contain a small number of firms, for instance the Telecommunication industry containing only 29 firms. in addition, splitting each industry up into 10 portfolios would result in 100 new portfolios. A simple two-sample Student’s t-test is applied on the two portfolios per industry, to identify the industry specific size-effect.

3.3 Descriptive statistics

This chapter will render an overview of the characteristics of the relevant data set. First, the ten size-based portfolios are discussed, followed by the industry specific differences. The summary statistics can be found in appendix B, where B1 contains the descriptive statistics per size-based portfolio and B2 the data organized per industry.

Starting with the size-based portfolios the table reveals that the expected relationship of higher book returns for smaller firms compared to larger firms is not confirmed. Instead largest 10% of the stocks seem to have outperformed the smallest 10% over the past 29 years. The portfolio containing the smallest 10% of the stocks has yielded a median return of 4.33% on equity, compared to 16,55% for the portfolio containing the largest 10% of the stocks. A similar pattern is found for the historic market cost of equity.

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the market return of the smallest firms 12.33. In contrast, the portfolios with the largest firms only have a standard deviation of respectively 18.15 and 6.30. The higher volatility is combined with slightly higher median betas for portfolio 1 when compared to portfolio 10. However, portfolios 1 to 7 all have betas in the range of 1.19 to 1.23, so they are very close. Only portfolios 8 to 10 are somewhat lower, with betas of respectively 1.12, 1.12 and 1.08. The higher betas for portfolios containing smaller firms is in line with findings in literature (Banz, 1981, Keim 1983). The differences however, are only found when comparing the portfolio containing the smallest firms to the portfolio holding the largest firms. The central portfolios barely differ from each other with regard to their betas.

Another noteworthy finding from the summary statistics is that the dependent variables are not perfectly normally distributed, as can be seen from the skewness and kurtosis variables. This underlines the importance of the use of non-parametric testing procedures as robustness checks.

Figure 2. Median and standard deviation per size-based portfolio

The book return on equity and the historic market cost of equity, and their standard deviations are graphed below per size-based portfolio. Portfolio 1 holds the 10% the smallest firms and portfolio 10 holding the 10% largest firms, as measured by their market capitalization. The returns are the medians of the portfolio.

Moving on to the summary statistics per industry; the Financials, followed by the Industrials industry are most presented in the sample, respectively covering 21,34% and 18,76% of the total of 2367 firms over the 29 year period. The least number of firms are available for the Telecom (29 firms) and Utilities industry (79 firms), for the detection of the size effect within those industries the sample sizes might be too small to arrive at a valid conclusions. Brooks (2008) argues for example that for a Student’s t-test the number of observations per sample should be no less than 20. The Telecom and Utilities, besides being low on firm count, do represent the two industries holding the largest firms, measured by median market capitalization. The industry containing the smallest firms is Healthcare, followed by Financials and Industrials. The fact that Financials and Industrials contain mainly small

0,00 5,00 10,00 15,00 20,00 25,00 30,00 35,00 1 2 3 4 5 6 7 8 9 10 Portfolios

Book return on equity

Median St.Deviation 0,00 2,00 4,00 6,00 8,00 10,00 12,00 14,00 1 2 3 4 5 6 7 8 9 10 Portfolios

Historic market cost of equity

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firms and together cover 40% of the sample implies that portfolio 1, containing the smallest firms, includes a large number of firms from these two industries. This needs to be taken into account when interpreting the results.

The data also shows that industry variations in betas exist. The median industry betas range from 0.41 to 1.77. The Utilities industry, holding comparably large firms and having the lowest standard deviation of all industries of only 2.64%, has the lowest beta of 0.41. Considering these characteristics of the industry, it is consistent with the expectations that Utilities have a relatively low beta compared to the other industries. Relatively low median betas are furthermore found for Financials (0.92), Healthcare (1.10) and the Telecom industry (1.07). The highest beta is found for the Technology industry, which is to be expected, since the standard deviation is at the high end and it is a fast moving industry.

4. Results

In this chapter the results will be presented. First, the results on of the ANOVA and Kruskal-Wallis test are given. Followed by the results on whether book return has exceeded market costs, tested by the ANCOVA. The third part focuses on the regressions results regarding the size effect.

4.1 Results size and dependent variables

The first step is identifying whether size has an influence on the dependent variables. Table 4 presents the results of the ANOVA and Kruskal-Wallis test, which test for the influence of size on the three different dependent variables; the book return on equity, the historic market cost of equity and the CAPM estimated cost of equity. As noted, it is hypothesized that significant differences between the ten portfolios are present, which would convey to us that size has indeed an impact on the dependent variables. The findings are all statistically significant at the level of α_{= 0.01 and both tests are in}

agreement with each other. The implication is that the means and medians of the dependent variables are significantly different per size portfolios. Also included are the results for the ANOVA and Kruskal-Wallis test when grouping the data per industry and per year. Again, the differences between the groups are statistically significant, suggesting that not only size but also the industry membership and the year has an influence on the dependent variables. Model 4 furthermore shows, that the returns also differ per sub-period investigated.3

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24 Table 4. Analysis of differences between respectively portfolios, industry and years.

The first part of the table describes the results for the ten size-based portfolios. The second part is based on the ten different industries. Part three compares all the 29 years to each other, where the 4th part looks at differences per sub-period; 1982 to 1991, 1992 to 2001 and 2002 to 2011.

ANOVA Kruskal-Wallis

F-test Chi-square

1. Portfolio

Book return on equity 322,36*** 4366,91*** Historic market cost of equity 383,78*** 3900,99*** CAPM cost of equity 11,34*** 68,93***

2. Industry

Book return on equity 169,72*** 1300,87*** Historic cost of equity 110,12*** 1143,44*** CAPM cost of equity 1141,81*** 1095,02***

3. Year

4. Sub-periods

*** significant at α = 0.01, ** significant at α = 0.05, * significant at α = 0.10.

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25 4.2 Results ANCOVA, book return versus market cost

This part will investigate the first hypothesis;

Hypothesis 1:

H0: The book return on equity exceeds the market cost of equity.

H1: The book return on equity does not exceed the market cost of equity.

The book return on equity is compared to (i) the historic market cost of equity and (ii) to the cost of equity as estimated by the CAPM. Table 5 presents the results for the paired Student’s t-test and its non-parametric equivalent the Wilcoxon signed-ranks test. The tests are performed on three sub-periods; 1982 to 1991, 1992 to 2001 and 2002 to 2011, and on the whole 29 year period.

Looking at the results for the whole sample period in column 5, the sign of the t-static is completely opposite to the expectations. A highly significant negative value is obtained, indicating that the book return on equity did not exceed the historic market cost of equity, nor the CAPM estimated cost of equity. The Wilcoxon singed-ranks test confirms the results. Focusing on the sub-periods it can been seen that the book return on equity was higher than the market cost of equity during period 1 (1982 to 1991) and significantly lower during period 3 (2002 to 2011). For the second period the results are less clear cut. The Student’s t-test suggests the book return was significantly lower for both market cost measures, while the Wilcoxon signed-ranks test states the opposite relation with regards to the CAPM estimated cost of equity.4

Table 5. Results Student’s t-test and Wilcoxon Signed-Ranks, sub-periods

The tests are performed on the book return on equity against the variables in column 1, respectively the historic market cost of equity and the CAPM estimated cost of equity. The test results and their significance level are presented below, for three different periods. The last column shows the findings for the whole period. The row named “Conclusion” contains the general result for that particular period, where > indicates that the book return on equity was higher than the cost of equity and < if the opposite is true.

4

The results for the cost of equity are also confirmed when using different underlying assumptions for the CAPM (i) Bloomberg adjusted betas and (ii) a market risk premium of 3% instead of 5%, appendix E.

Historic market cost of equity 1982 to 1991 1992 to 2001 2002 to 2011 Whole period

Student's T-test 1,82**** -9,85*** -20,27*** -9,46*** Wilcoxon Singed-Ranks test -5,80*** -7,40*** -0,08 -9,64***

Conclusion > < < <

CAPM cost of equity

Student's T-test 10,61*** -8,10*** -26,65*** -14,48*** Wilcoxon Singed-Ranks test -25,02*** -7,13*** -14,38*** -12,96***

Conclusion > > < <

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Based on the findings on this data set as presented above, one of the general assumptions in finance, that the return on investment must exceed the cost in order to be profitable (Koller et al., 2010, Damodaran, 2010, Brealey et al, 2007, Berk et al., 2012), cannot be confirmed with these data.

Next, the same test is performed on the ten different size-based portfolios. Results can be found in table 6. The first part looks at the historic market cost of equity. It reveals that the book return on equity has only significantly exceeded the of equity for portfolio 10, which contains the 10% largest U.S. stocks. The other negative t-statistics indicate that historical cost of equity was significantly higher than the book return in portfolios 1 to 8, and negative but insignificant for portfolio 9. The smaller the firms, the larger the difference between the return and cost of equity.

For the cost of equity as estimated by the CAPM; the t-statistics show that portfolios 1 to 6, have had significantly lower returns compared to the cost of equity and portfolios 7 to 10 the other way around. Where the results for portfolio 7 and 9 are positive but insignificant. The Wilcoxon signed-rank tests slightly differs from the results found obtained by the Student’s t-test. Here, only portfolios 1 to 4 had negative returns compared to the costs, while from portfolio 5 onward the returns have exceeded the cost of equity. Overall the findings indicate that size influences the return-cost relationship of equity, where investments in smaller firms turn out unprofitable more often than investments in larger firms. It is striking that for a large part of the portfolios, the returns have been lower than the costs, contradicting our null hypothesis.

Table 6. Results Student’s t-test and Wilcoxon Signed-Ranks, per size-based portfolio

The tests are performed on the book return on equity against the variables in column 1, respectively the historic market return on equity and the CAPM estimated cost of equity. The test results and their significance level are presented below, for the 10 different portfolios. The last column shows the findings for the portfolio 10, containing the 10% largest firms. The row named “Conclusion” contains the general result for that particular portfolio, where > indicates that the book return on equity was higher than the cost of equity and < if the opposite is true.

Portfolios Small 1 2 3 4 5 6 7 8 9 Large 10

Student's T-test -21,61*** -13,19*** -8,92*** -6,22*** -4,58*** -3,97*** 3,12*** -2,70*** -0,723 8,42*** Wilcoxon Singed-Ranks test -22,64*** -10,66*** -7,30*** -4,92*** -4,31*** -4,53*** -3,89*** -4,50*** -2,03** -10,17***

Conclusion < < < < < < < < < >

Student's T-test -30,01*** -22,08*** -15,51*** -11,78*** -7,34*** -3,93*** 0,75 4,63*** 0,2 22,46*** Wilcoxon Singed-Ranks test -31,52*** -21,61*** -13,54*** -8,98*** -4,36*** -0,96 -7,69*** -13,16*** -20,29*** -31,13***

Conclusion < < < < > > > > > >

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At last we take a look at the profitability per industry level. Do certain industries more often result in positive returns by exceeding the costs? The answer is yes. Industry characteristics are linked to the cost and return measures of firms, which is confirmed by the results in table 7.

Table 7 . Results Student’s t-test and Wilcoxon Signed-Ranks, per industry

The tests are performed on the book return on equity against the variables in column 1, respectively the historic market return on equity and the CAPM estimated cost of equity. The test results and their significance level are presented below, for the ten different industries.The row named “Conclusion” contains the general result for that particular industry, where > indicates that the book return on equity was higher than the cost of equity and < if the opposite is true.

The results are quite different for the two measure of the cost of equity. While the CAPM estimates per sub-period and portfolio more often confirmed the expectations of returns exceeding the costs, for the industries this is not found. Only the Consumer Goods industry has positive returns for both cost of equity measures, but these results are not statistically significant. For most of the industries the results are significantly negative. The Basic Materials, Healthcare, Oil and Gas, Technology and Telecom industry show that the cost has significantly exceeded the return. Where the numbers for the Telecom industry are based on a rather small sample size and should therefore be interpreted with caution. Also, since innovative industries are considered more risky, it is no surprise that the Technology industry has large, significant, negative t-values for both the CAPM cost of equity and the historic market cost of equity.

A conclusion that can be drawn is that industry differences exist between the book return and market cost of equity relationship. Furthermore, the findings indicate again that for a large part of this sample, the book return on equity did not exceed the market cost of equity.

In order to arrive at a solid conclusion about whether the book return on equity has exceeded the market cost of equity over the past 29 years an ANCOVA analysis is performed. The ANCOVA test, similar to the ANOVA, tests whether differences between groups are statistically significant. However, the ANCOVA is a little more sophisticated as it allows for the inclusion of control variables. The results above have already demonstrated that size, industry and year are important influences on

Basic Consumer Consumer Oil

Industry Materials Goods Services Financials Healthcare Industrials and Gas Technology Telecom Utilities

Student's T-test -2,24** 0,55 -0,28 -4,55*** -9,37*** -0,86 -4,21*** -4,29*** -2,26** -5,69*** Wilcoxon Singed-Ranks test -0,88 -0,45 -5,48*** -5,33*** -8,54*** -1,6 -4,60*** -3,27*** -2,22** -6,01***

Conclusion > > > < < < < < < <

Student's T-test -3,85*** 1,71* -1,18 -2,68*** -11,69*** -3,96*** -6,85*** -12,57*** -2,88*** 10,80*** Wilcoxon Singed-Ranks test -2,97*** -1,92* -2,13** -1,68* -9,41*** -4,03*** -6,63*** -10,99*** -2,39** -6,75***

Conclusion < > < > < < < < < >

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the relationship between the book return and the market cost of equity. Below, in table 8, the results of the ANCOVA are revealed.

Table 8. Results ANCOVA- analysis

The ANCOVA analysis describes the differences between (i) the book return on equity minus the historic market cost of equity and (ii) the book return on equity minus the CAPM estimated cost of equity, while controlling for three different variables. In model 1 the size-based portfolios are included as control variable. In the second model the test is controlled for industries. The third accounts for differences per year and the fourth model includes all 3 control variables. The sign indicates the nature of the relationship found, while the row below represent the F-value and the significance level of the differences found.

1 2 3 4

Model Portfolio Industry Year All

Book return on equity minus - - - -

Historic market cost of equity 410,73*** 391,69*** 392,47*** 415,05***

Book return on equity minus - - - -

CAPM cost of equity 595,74*** 580,36*** 580,65*** 598,90***

As can be seen the relationships found are all negative and statistically significant, while theory would suggest a positive outcome. Controlling for the different variables that potentially influence the book return-market cost relationship has only strengthened the results that the book return did not exceed the market cost of equity over the past 29 years.

Therefore, hypothesis 1, surprisingly, needs to be rejected on the basis of these results.

4.3 Results OLS regression size effect

The following section will discuss the results of the OLS regression used for testing the second and third hypothesis. Previous results have already indicated that industry membership and year are variables influencing the book return and cost of equity. Therefore, both variables are included in the regression. The hypothesis are as followed;

Hypothesis 2:

Ho: The cost of equity of smaller firms exceeds the cost of equity of larger firms.

H1 : The cost of equity of smaller firms does not exceed the cost of equity of larger firms. Hypothesis 3:

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The regression results can be found in table 9. The main coefficient of interest is from the portfolio, which represents the number of size-based portfolio a firm belongs to. Four OLS regression have been performed in order to control for risk, sub-period and industry differences.5

The first four columns show the size effect on the book return on equity. The coefficient of the portfolio is positive and significant for all four models, while initially a negative relationship has been hypothesized. A higher portfolio number therefore leads to an unexpected increase in the book return on equity. The second model includes beta, which has a significant negative coefficient, indicating that a higher beta leads to a lower book return on equity. Again, this results contradicts theory. The idea that investors are compensated for higher risk exposure, as signaled by the higher beta, with higher returns does not seem to hold here. Model three confirms the results from the ANOVA and Kruskal-Wallis test that the three sub-periods in the data have significantly different book returns. During the third sub-period, 2002 to 2011, the book returns were lower than in the second period, 1992 to 2001, and the book returns were again lower in the second than in the first period, 1982 to 1991.

Looking at the industry dummies, Healthcare, Technology and Telecommunications perform the worst measured by the book return, which is in line with the data descriptives in appendix B2.

The historic market cost of equity displays similar results for the portfolio coefficient. Again the overall results are positive and significant indicating that larger firms are related to higher market cost of equity. The beta is still negatively related to the market cost of equity. The descripitives (appendix B2) show that the Utilities industry has the highest median historic market cost of equity compared to the other industries. This is in line with the overall negative coefficients found in the regression for all industries. Again, Healthcare and Technology have the lowest coefficients. The R-squared for both the book return and the cost of equity is quite low, but improves with the inclusion of either beta or the industry dummies.

At last, the CAPM estimated cost of equity. This is the only one with a significant negative coefficient for the size variable, but still only very small with -0.034. When beta is included in the regression, the size coefficient turns positive.

The regression results confirm the pervious results found by the ANOVA and Krukal-Wallis tests, and are also in line with basic descriptives. Size is positively related to the book return on equity and the historic market cost of equity. These findings force me to also reject the second and the third hypothesis.

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Table 9. Results OLS regression

The table shows the results of the OLS regression performed on the models 1 to 4, for the book return on equity, the historic market cost of equity and the CAPM cost of equity. *** significant at α = 0.01, ** significant at α = 0.05, * significant at α = 0.10.

Book return on equity Historic market cost of equity CAPM cost of equity

Model 1 2 3 4 1 2 3 4 1 2 3 4 Constant 1,939*** 6,342*** 2,208*** -1,676*** 6,439*** 9,302*** 7,735*** 8,452*** 11,853*** 5,528*** 11,279*** 8,176*** Portfolio 1,976*** 1,855*** 1,978*** 1,932*** 0,907*** 0,865*** 0,908*** 0,886*** -0,034*** 0,059*** -0,034*** -0,019*** Beta -5,853*** -2,023*** 4,471*** Sub-period dummies 1992-2001 -3,171*** -1,060*** 0,250*** 2002-2011 -5,986*** -1,826*** 0,952*** Industry dummies Basic Materials 0,124 -2,418*** 4,467*** Consumer Goods 5,135*** -0,582* 3,286*** Consumer Services 2,271*** -1,420*** 4,023*** Financials 2,055*** -0,21 2,408*** Healthcare -9,751*** -4,681*** 3,003*** Industrials 1,918*** -1,597*** 3,860***

Oil and Gas -3,189*** -1,523*** 4,230***

Technology -3,859*** -5,499*** 6,426***

Telecommunication -7,653*** -2,405*** 3,486***

R-squared 0,059 0,089 0,067 0,089 0,048 0,062 0,051 0,071 0,001 0,952 0,017 0,195