Master Thesis
Finance
Investigating the size effect: evidence from Southern Europe
By Lodewijk Bramer
ABSTRACT
This paper investigates whether firms from the Southern European region with a small market capitalization tend to offer a better return than those with a large capitalization. The data analyzed covers the companies listed on the stock exchanges of Spain, Italy, Portugal, Greece and France for the period 2002-2013. The evidence contradicts the expectation based on the literature about the American market: it appears that Southern European large-caps perform better than small-caps. Moreover, examining the size effect exposed a negative relation between a stock’s return and beta. Contradictory to the CAPM theory, firms with the lowest betas yield the highest returns.
Article info:
Keywords: Size effect, CAPM, stock return JEL classification: G12, G15
Date and location: 16 January 2015, Groningen
Author: Lodewijk Bramer
Mail: lodewijkb89@gmail.com
Student number: 1719106
2
1. Introduction
In recent decades, academics have attempted to explain cross-sectional returns of stocks listed on exchanges by using and testing a wide variety of variables and models, hoping that their variables explaining past returns will retain their prognostic value in the future. Attempts to create models that have more predictive value over future stock prices than simply tossing a coin is not surprising. A model that proves to be valuable in making accurate predictions about future stock returns is the aspiration of every market participant. A reliable model that accurately predicts stock movement over time would benefit money managers and analysts to a great extent. Significant key performance indicators of stock returns with predictive value for longer periods of time are therefore of high value and the search for these indicators is a never ending pursuit for investment engineers.
At present, there does not appear to be a single asset pricing model that is accepted as the unambiguous supreme asset pricing model in the academic and business world. However, there seems to be a general consensus surrounding one aspect. It is generally accepted that, under the assumptions of the efficient-market hypothesis, there is a positive relation between risk and return. Intuitively, this makes sense. To convince investors to invest in company X with a higher risk level relative to company Y, higher returns are needed to persuade them. Consequently, based on the relation between risk and return, efficient asset pricing models should incorporate factors that account for the risk factors of stocks to make accurate predictions of future returns.
The most well-known and widely taught asset pricing model is the capital asset pricing model (CAPM) developed by Sharpe (1964), Lintner (1965) and Black (1972). To estimate required rates of return for stocks, the CAPM uses a single parameter, beta (β). Following their theory, the beta of a company represents all non-diversifiable risk, or market risk, of a security. The CAPM has dominated portfolio theory for several decades. However, despite its popularity, research has shown that other variables aside from beta add to the explanatory powers of the return model.
Banz (1981) finds that firm size, measured by market equity (number of shares multiplied by share price), adds to the predictive power of the CAPM. His findings are based on United States data for the period 1936-1975. Banz (1981) discovered that stocks of small firms generate higher returns than stocks of large firms. The observed phenomenon of small firms yielding higher returns compared to large firms is referred to in the literature as the size effect, or the small-cap premium.
3 three-factor-model, together with a firm’s book-to-market ratio, as a means of improving the predictive value of the CAPM. Primarily based on the findings of Banz (1981) and Fama and French 1992), size is recognized as an anomaly of the CAPM model.
1.1. Problem statement
Although the size effect is still widely used as a factor in asset pricing models, academics are not unanimous about the merits of the size effect. After the publications of Banz (1981) and Fama and French (1992), researchers found no evidence for the size effect in the same markets after the year 1980 (Dichev, 1998; Horowitz et al., 2000a). Moreover, Horowitz et al. (2000a) found that the size effect only occurred in the month of January, and was absent in the remaining months. Furthermore, no unambiguous evidence of the size effect has been reported in leading financial papers within the last two decades. One can thus question whether the current usage of size as a factor in asset pricing models can be justified based on academic findings of previous decades. Especially since research on the size effect was carried out predominantly in the United States, whereas circumstances are likely to vary across markets in the world.
The majority of the size effect literature is based on stocks listed on United States stock indexes, and to a smaller extent on European markets. Since firm size is also used as a factor by investors when making inquiries about future returns of stocks listed on stock indexes outside of the United States, additional research is needed to test for the validity of the size factor. Although the size effect is investigated outside the markets of the United States, the results are inconclusive and contradictory. Furthermore, investigations outside of the United States focus predominantly on Northern Europe and Japan. In this paper, an attempt is made to lessen the gap within the literature of the size effect by studying the size effect in the Southern European region.
1.2. Research objective
The objective of this paper is to investigate whether firms from the Southern European region with a small market capitalization tend to offer a better return than those with a large market capitalization. To reach this objective, data from major stock exchanges in Spain, Italy, Portugal, Greece and France are grouped together and examined on a monthly basis to see whether the size effect is observable in the period 2002-2013. The stocks of these markets are compiled to represent the Southern European region.
4 model of Fama and Macbeth (1973). Both the portfolio formation method and a regression analysis are applied in this paper.
Based on traditional size effect literature, it is expected that returns of smaller firms in terms of market equity are higher than the returns of larger firms. However, the anomaly of the size effect and its occurrence throughout time and space is not obvious. Therefore, the results might differ from the expectations.
The timespan of this study includes the subprime mortgage crisis that affected the entirety of the global economy, which began towards the end of 2007. As a result of this crisis, the market value of stock indexes around the world suffered a decrease in market value. The effect of the economic crisis among companies with varying market equity is examined. This provides a good scenario to investigate whether the trends hold regardless of the economic cycle.
The discussion whether or not size should still be used as a factor in asset pricing models has increased because the evidence for the small-cap premium in recent years is meager. Although the size effect has been previously observed, the small-cap premium is not beyond dispute, and additional evidence of its merits within a recent timespan might help to settle the debate. This study adds to the current literature by investigating a specific region within Europe that has not been investigated in this capacity. The results in this paper can be used as an indication as to whether or not the usage of the small-cap premium by financial practitioners is still justifiable in present markets.
This paper is organized as follows: Section 2 provides an overview of the literature of the CAPM and the size effect. Section 3 discusses the methodology, followed by Section 4, which describes the data collection method. The results constitute Section 5. Section 6 presents the conclusions based on the results. Lastly, Section 7 describes the limitations and suggestions for further research.
2. Literature review
5 2.1. Capital asset pricing model
Sharpe (1964), Lintner (1965) and Black (1972) developed the capital asset pricing model to estimate the required rate of return for any given stock using a single parameter, beta, which represents the market risk of a stock. The CAPM claims a positive linear relation between the return of an asset and its market risk. Based on an asset’s beta, the level of risk of the asset is determined, and hence, the required rate of return. Based on the CAPM, theory investors are able to create mean-variance efficient portfolios consisting of a ‘risk free’ asset in combination with risky assets constructed in line with the personal risk preference of investors.
The CAPM has dominated portfolio theory for several decades, and Sharpe won the Nobel Prize in Economic Science in 1990 for his contribution to the development of the model. Despite its popularity and the fact that the CAPM is still the centerpiece model for the most prestigious investment courses, the empirical record of the CAPM is far from indisputable (Fama and French, 2004). Academics investigating the validity of the CAPM found only a weak relation between beta and average return after 1963 (Reinganum, 1981; Lakonishok and Shapiro, 1986; Fama and French, 1992). These findings challenge the predictive value of the CAPM regarding the rate of return based on the beta of stocks. Moreover, Fama and French (1992) found that the correlation between rate of return and a company’s beta is also insubstantial during the period of 1941-1990. Consequently, they see no evidence pleading for the validity of the CAPM.
Researchers found that other firm specific characteristics aside from beta add to the predictive power of the CAPM model. Variables other than beta holding predictive power for stock returns are known as anomalies. Following Brennan and Xia (2001), an anomaly is the significant difference between the actual return associated with certain characteristics of a security and the predicted returns by the asset pricing model. Some of the anomalies are briefly described below.
6 2.2. The size effect
Banz (1981) was the first to discover the size effect while testing the CAPM based on American data. He finds that firm size adds to the explanatory power of the CAPM, and shows that during 1936-1975, small firms measured by market equity, outperform large firms in terms of average risk-adjusted return. More specifically, after ranking stocks in portfolios based on their market equity, he finds that firms in the quintile with the lowest market equity generated an additional 0.4% risk-adjusted monthly return. Based on his findings, investors could benefit from the size effect anomaly by investing in small firms if the size effect was to prevail.
Fama and French (1992) add to the size effect literature by finding evidence that small firms also outperform large firms for the 1962-1989 time period based on all nonfinancial firms listed on the NYSE, AMEX and NASDAQ in the United States. Aside from the size effect, they also reveal a relation between stock return and the book-to-market ratio. Based on their findings, Fama and French (1993) presented their three-factor-model, which includes market risk, size and book-to-market equity in their asses pricing model However, despite the inclusion of size within the three-factor-model other academics question the relation between size and return.
2.3. Criticism of the size effect
Despite an extensive amount of articles investigating the size effect, the academic world is not unanimous regarding its merits. There is an absence of consensus within the literature regarding the occurrence of the size effect and the correct methodologies to identify the small-cap premium. Furthermore, different theoretical explanations of the small-cap premium emerged.
Lo and MacKinlay (1990) challenge the methodology of ranking stocks based on size or beta, as is done by Fama and French (1992), Banz (1981), and other academics, and accuse their methodology of data-snooping. Consequently, they question the results of the size effect studies. Moreover, they argue that no real theoretical framework underwrites the findings of Banz (1981), namely that size and expected rate of return are related, and therefore dismiss the results.
7 2.4. Possible size effect explanations
After Banz (1981) revealed the size effect, academics attempted to theoretically explain the small-cap premium. Whether size is in itself the reason for the higher return or whether it functions as a substitution for other risk factors related to firm size is debatable. Some of the theories attempting to explain the observed size effect are presented below.
2.4.1. Liquidity
Amihud and Mendelson (1986) argue that the small-cap premium functions as a proxy for liquidity risk. They emphasize that the risk of not being able to sell a stock at low costs or without influencing the price is compensated with a premium in the form of higher returns. Because stocks with lower market equity are deemed to be less liquid, smaller companies are faced with a higher cost of capital. Liu (2006) also advocates that the size effect partially functions as a proxy for liquidity issues. In his article, he concludes that the size effect and other anomalies of the CAPM are absorbed when he controls for liquidity factors. Amihud (2002) shows that expected market illiquidity positively influences stock market returns, and defines this as the illiquidity premium. According to his findings, illiquidity risks have a stronger effect on small stocks because stocks of small firms are generally less liquid, explaining the observation of the size effect. Depending on market conditions, the liquidity preference of investors in their portfolios can fluctuate. Following Amihud (2002), this explains the observed variety in the occurrence of the size effect. Furthermore, he states that the diminishing small-cap premium can be explained by the increased liquidity of small firms over time.
2.4.2. Information asymmetry
Zhang (2006) argues that firm size is a proxy for information asymmetry. He argues that information asymmetry is larger for smaller firms because large firms receive more attention from institutional investors and generally provide investors with more firm-specific information. Additionally, attention from the press and from analysts seems to be positively related to firm size, adding to the arguments of Zhang (2006). Banz (1981) also argues that investors are more reluctant to buy stocks of smaller market equity values since less information is generally available. Ergo, a higher return is needed to convince investors to include stocks of small firms into their portfolios. Following their line of reasoning, the size effect could thus be explained to some extent by a relatively limited amount of available information for (potential) investors.
2.4.4. Delisting bias in CRSP
8 bankruptcy of a firm, are often not included in the database. Therefore, the actual returns of small firms are, in fact, lower than is reported by the CRSP. Banz (1981), who first described the size effect, relied on CRSP data, similarly to the majority of academics studying the size effect. When Shumway and Wartner (1999) control for the delisting bias, they conclude that the size effect is no longer significant during the period of 1972-1995.
2.4.5. The January effect
In general, and particularly within the United States, markets are known to experience what is known as the January effect. The January effect is a phenomenon wherein, on average, stock returns in January are higher than in the remainder of the year (Haugen and Jorion, 1996). In relation to the January effect, Keim (1983) identifies that the size effect primarily occurs in the first month of the year, even more specifically, in the first week of January. Roll (1983) also finds that the size effect seems to be linked to the January effect, and states that the January effect is stronger for smaller firms than for their larger counterparts. Horowitz et al. (2000a) confirm this finding when they conclude that the size effect varies throughout the year. They show that when the January effect is taken into consideration, the size effect does not occur during the rest of the year. Likewise, they indicate that the size effect seems to reverse in the remaining months, indicating that smaller firms are actually yielding lower returns relative to larger firms.
Keim (1983) attempts to explain the findings of the January effect with the theory that taxpayers sell stocks that generated a loss before the end of the year to profit from tax advantages. Hence, selling loser stocks in December provides a tax shield for investors in the US. Roll (1983) criticizes this theory by stating that the laws of arbitrage will ensure that the January effect does not systematically occur. Similar to the seasonality reason of selling loser stocks in December, Poterba and Weisbenner (2001) argue that institutional investors rebalance their portfolios towards more volatile stocks at the beginning of January. This could potentially contribute to the intensified January effect for the size effect, when following their line of reasoning that small market equity stocks tend to be more volatile.
9 2.5. The occurrence of the size effect
Adding to the puzzle of the size effect is that the occurrence and extent of the small firm premium seems to vary in time and between regions. After its discovery by Banz (1981) the premium was no longer measured in the United States markets after 1980 (Dichev, 1998). Horowitz et al. (2000a) found that the size effect only occurs in firms with market equity below 5 million dollars and is not present in the period of 1984-1998 in United States markets. Also, Barry et al. (2002) found no size effect in 35 emerging countries during the years of 1985-2000.
Fama and French (2008) find that the size effect has the most impact on the bottom 20th percentile of NYSE stocks, and merely a minor effect on the rest of the market during 1963-2005, indicating that the size effect only affects the smallest companies and is not observable throughout the entire market.
The inconsistent findings of the size effect across regions and time periods are puzzling academics in their attempt to explain the small firm premium. If the size effect is indeed a proxy for risks such as information asymmetry, it is hard to explain why this risk is not more consistent. Moreover, it does not explain why it differs between nations and across time periods. Evidently, the most important size effect studies examined the size effect within the American markets. Investigating the Southern European market within a more recent timeframe therefore fills a gap in literature. The next section describes the methodology used in this paper.
3. Methodology
As described in the previous two sections, an effort is made to minimize a gap in the size effect literature by investigating whether the size effect occurs in the Southern region of Europe. This paper examines securities listed on the major stock exchanges of Spain, Italy, Portugal, Greece and France. The objective is to find whether the size effect is present for the time period 2002-2013. To test for the occurrence of the size effect, two methods of testing are applied. The two methods are based on previous size effect research.
To test for the size effect in Southern Europe as a region, the included stocks of the five countries are combined into one data set. The argument for the creation of one data set is that this paper investigates the size effect in a specific region, not for specific countries. Combining the database increases the number of available stocks for data analysis and thus increases the power of the results. If each country would be dealt with individually, the statistical power would vary among the nations.
10 countries, they argue, is that there should be sufficient reason to believe that there is strong market integration within the region. In their article, they claim that market integration is a reasonable assumption in the case of European countries since most nations are part of the European Union and all operate within the open market provisions of the European Union. In this paper, all countries are European Union member states and all use the Euro as their currency. Moreover, the countries of interest are in close geographic proximity, making the assumption of market integration increasingly plausible.
3.1. Research model
Fig. 1. displays the research model of this paper. The emphasis of this paper is on the relation between stock returns and firm size. However, following conventional size effect literature, the relation between beta and stock returns is included in the analysis. Banz (1981) uses a generalized version of the CAPM in his analysis. The model used in this paper is derived from his approach. Fama and French (1992) also include beta in their analysis. Their reason for including beta is that Chan and Chen (1988) among others have shown that beta and size exhibit a strong correlation in previous research. Beta is incorporated in the analysis to control for possible correlation between beta and size and to disentangle their influence on average returns.
The literature review section of this paper shows that other factors than size and beta influence stock returns, e.g. the book-to-market ratio. Although other variables are beyond the scope of this paper it is important to note that firm size and beta are only two factors of influence. Variables other than firm size and beta are represented by ‘other variables’ in Figure 3.2.
Fig. 1. Research model
11 3.3. Research techniques
Two different methods investigating the occurrence of the size effect can be identified within the traditional size effect literature. The first method is to compare average returns between different size-based portfolios. The second method is a regression analysis on the full data set. Both approaches are applied in this paper. The reason for the inclusion of both methods is because they provide different insights into the relation between the variables, as well as the fact that the usage of multiple methods increases the validity of results.
The first two steps of the analysis encompass the technique of portfolio formation, which is a conventional method in the size effect literature (see, e.g., Fama and French, 1992; Horowitz et al., 2000a). This method is used to analyze and compare average returns among different size-based and beta-based portfolios, as well as to form a graphical overview of the data set.
The second step is to perform regression analysis based on separate securities in all 144 months. The following section contains an elaborate description of the methods.
3.4. Portfolio formation
The formation of portfolios is based on the methodology of Fama and French (1992). The method of comparing returns among portfolios ranked on size or beta provides a simple and clear image of the existing relations between the variables (Fama and French, 2008).
Following Fama and French (1992), the securities are placed into two types of portfolios. One portfolio is based on the size of securities, while the other portfolio is based on the beta of securities. To construct the portfolios, stocks are ranked each month based on their beta or size. After the stocks are ranked from low values to high values, breakpoints are determined as such that ten portfolios are formed, each consisting of an equal number of securities. This process is repeated monthly. Thus, every month securities are ranked based on their new size or beta values and consequently, new portfolios are created. By constructing portfolios and updating them monthly, the average value of size and beta in each portfolio is held relatively constant through time. Thus, despite the fact that the composition of securities within each portfolio might change, the beta or size value of the portfolios remains relatively stable through time. The average returns of the portfolios are based on equally weighted returns.
After the portfolios are constructed, the average monthly return among portfolios is compared to see whether different returns are observable among different size and beta portfolios. The average equally weighted returns of the portfolios are presented in Section 5.
12 is investigated based on previous studies described in Section 2. The two tests are discussed in more detail below.
3.4.1. The subprime mortgage crisis
The timespan of this paper of 2002-2013 includes the subprime mortgage crisis (Purnanandam, 2011). In an attempt to examine whether the crisis has had an impact on the occurrence of the size effect, the data set is divided in half. The period of 2002 until 2007 (referred to as period I) is considered as the timespan before the financial crisis, while the period of 2008-2013 (referred to as period II) represents the post-crisis period. The portfolios of period I and period II are composed in a similar fashion as the portfolios that cover the entire timespan. The approach of dividing the data set into different timespans is similar to the approach applied by Horowitz et al. (2000a).
3.4.2. The January effect
Based on the findings of Keim (1983), Roll (1983) and Horowitz et al. (2000a), the small-cap premium is primarily observable in the first month of the year. Therefore, the difference between average portfolios returns in January is compared with the average return of the rest of the year. Then, similar to Horowitz et al. (2000a), the difference between the return in January and the return in the remaining months of each portfolio is examined to see whether or not the difference in return is larger for the smallest companies. When the relation between size and return differs among portfolios in January, investors can exploit this anomaly, regardless of the occurrence of the size effect in the remaining months. Thus, irrespectively of the average relation between return and size over the entire year, the January effect is examined to test whether the relation between size and return in January differentiates strongly from the other months, as suggested by previous studies.
3.5. Graphical outlay
To portray the cumulative returns of the ten size-sorted and beta-sorted portfolios, a €1 investment is made in each portfolio in the beginning of 2002. Hereafter, the cumulative return of each portfolio over the 144 months is portrayed in a graph. The returns can then be observed in terms of movements throughout the years within the different portfolios. Graphical representations of time series data in the form of graphs provide a meaningful way of visualizing information and conducting comparative analyses (Phillips, 2001).
3.6. Regression analysis
13 beta make up the two independent variables in the model. The regression is performed based on the data set as a whole for separate stocks. No portfolios are formed prior to this analysis. The regression is performed using the ordinary least squared (OLS) method to estimate the relation of the variables.
The following linear regression is performed in all 144 months:
Rit= γ0t+ γ1tβit+ γ2tln(S)it+ εit (1)
Where,
Rit = the monthly return of security i in month t γ0t = the intercept
γ1t = coefficient ofβit βit = β of stock i in month t γ2t = coefficient of ln(S)it
ln(S)it = the natural logarithm of the market value of security i in month t εit = residual with zero expectation
The εit’s are residuals with zero expectation. They may not be homoscedastic, and they may not be independent (autocorrelated). Since OLS assumes that the residuals are independent and identically distributed, the standard errors provided by OLS might be biased when analyzing the time-series data in this paper due to the time effect (Petersen, 2009). The time effect describes the possible correlation between the residuals of firms in one specific moment in time (Petersen, 2009). To allow for possible heteroskedasticity and autocorrelation, the Newey-West approach is applied in this paper (Newey and West, 1987). The Newey-West procedure modified for panel data corrects potential biases in the error terms provided by the standard OLS method. Following the conventional size effect literature, the means of the resulting slopes of the time series are interpreted as the estimators of the parameters. The outcomes of the regressions are presented in Section 5.
3.6.1. Beta estimation method
14 To estimate the beta of a stock, a substitution for the market index is needed. According to the CAPM theory, the true -market index- should be used to make correct inferences about the outcomes of the model (Roll, 1977). However, since from a practical perspective a true market index encompassing all available assets is not obtainable, a substitution for the market index is chosen as the second best estimate. Since this paper investigates the size effect in a European setting, the MSCI European index is used as a market proxy. The MSCI European market index is designed to measure the performance of the developed markets in Europe. According to the definition of the MSCI (2014), the MSCI European market index is a free float-adjusted market capitalization weighted index which measures the equity performance of 15 developed countries in Europe. Countries that are included in the index are: Austria, Belgium, Denmark, Finland, France, Germany, Ireland, Italy, the Netherlands, Norway, Portugal, Spain, Sweden, Switzerland, and the United Kingdom (MSCI, 2014). The usage of the MSCI European index as a proxy for the market index provides an average beta of 1.09 for the data set.
4. Data Collection
The definitions of variables are explained to fully comprehend what data is collected to form the data set. First, the dependent variable stock return is defined. Then, the two independent variables are delineated.
4.1.1. Return
Average monthly return figures are based on the total return index of each stock on a monthly basis. The adjusted stock price is used to determine the return figures to account for capital arrangements such as dividend payouts and stock splits.
4.1.2. Firm size
Firm size is measured by the market equity of a firm, which is determined by multiplying share price with the number of issued ordinary shares outstanding. Market equity is expressed in millions of Euros. In line with previous size effect literature, this paper uses the natural logarithm of the market equity of firm size in the analysis (see, e.g., Fama and French, 1992; Horowitz et al. 2000b).
4.1.3. Beta
15 4.2. Collection of data
The data set of this paper consists of stocks that are listed on stock exchanges from Italy, Spain, Greece, Portugal and France during 2002-2013. The data set includes Italian stocks listed on the Borsa Italiana, Spanish stocks listed on the Bolsa de Madrid, Greek Stocks listed on the Athens Stock Exchange and Portuguese stocks listed on the Euronext Lisbon. Finally, stocks listed on the CAC All-Tradable account for the French input. The countries with the corresponding stock exchanges are displayed in Table 1.
Table 1. Stock indexes encompassed in the data set.
Country: Included stock index:
Italy Borsa Italiana
Spain Bolsa de Madrid
Greece Athens Stock Exchange
Portugal Euronext Lisbon
France CAC All-Tradable
The adjusted price, adjusted return, and market equity data for each stock are extracted on a monthly basis from Thomson Reuters’ Datastream (2014). In this paper, the return of the MSCI European Market Index is used as a proxy for the market return. The returns of the MSCI European Market Index are extracted from the MSCI website (2014). Stocks that are listed for less than three years are excluded from the analysis since monthly return data of 36 months is used to make beta-estimations for the included stocks. Only securities meeting the data requirements are included in the analysis, resulting in a data set consisting of 846 stocks distributed over five countries.
5. Results
The results of the methods described in Section 3 are presented in this section. Firstly, the results of the portfolio formation and the corresponding graphical outlay are presented and described, followed by the results of the regression analysis.
5.1. Portfolio Formation
16 5.1.1. Size portfolios: 2002-2013
The characteristics of the ten size-based portfolios for the period 2002-2013 are displayed in Table 2. Portfolio 1 consists of the smallest firms and portfolio 10 consists of the largest firms. The table shows that portfolio 1 generates the lowest average monthly returns and is the only portfolio that yields a negative monthly return of -1.02% over the 144 months. Moreover, the average return of the size based portfolios increases from portfolio 1 to portfolio 7. The highest average returns are generated by portfolio 7 and portfolio 8. Although portfolios 9 and portfolio 10 yield returns that are higher than the 0.43% average, 0.63% and 0.62% respectively, they do not constitute the portfolios with the highest average yields. The difference between portfolio 1 with the lowest average return and portfolio 7 with the highest average return is 1.93% per month.
When interpreting the average return distribution among the portfolios, the relation between size and return appears to be positive rather than negative, contradicting expectations based on size effect theory. Additionally, the fact that the smallest size portfolio generates the lowest average return is also not in line with the expectations of the small-cap premium.
The average betasof the size-based portfolios do not differ substantially between the portfolios. The beta of size portfolio 1 and portfolio 10 are 1.10 and 1.09 respectively, indicating that no strong relation between beta and size exist in portfolios ranked by size.
5.1.2. Beta portfolios: 2002-2013
The results of the ten beta-based portfolios for the period 2002-2013 are displayed in Table 2. The return distribution of the beta-ranked portfolios is not in line with the prediction of the CAPM, which suggests a positive linear relation between beta and return. Based on the returns of the beta portfolios, there is no evidence for such a positive linear relation within the scope of this paper. Portfolio 1 with the smallest betas yields the highest average monthly return of 0.73%, while portfolio 9 and portfolio 10 with the highest beta values produce below average returns of 0.19% and 0.35%, respectively. These findings seem to suggest that the CAPM is misspecified, or at least appears to fail in elucidating average monthly returns for this particular data set.
The observation that low beta stocks yield higher returns than high beta stocks is defined in the literature as the ‘small beta anomaly’ and is in line with the findings of recent academic articles. The small beta anomaly is discussed in greater detail in the next section of this paper.
17 Table 2. Characteristics of the portfolios listed on size or beta: 2002-2013
The return of each portfolio is the average equally weighted monthly return in percentages of the time-series. The values of beta and size are the monthly average values of the variables. Size is measured as the natural logarithm of market equity in millions of Euros.
Results for portfolios based on size, 2002-2013
Size
portfolios Average size Average beta
Average monthly return (%) (1) (2) (3) (4) Small 1,85 1,10 -1,02 2 2,87 1,19 0,15 3 3,56 1,08 0,33 4 4,17 0,98 0,49 5 4,76 1,02 0,62 6 5,35 1,06 0,72 7 6,03 1,13 0,91 8 6,95 1,11 0,90 9 8,07 1,11 0,63 Large 9,68 1,12 0,62
Results for portfolios based on beta, 2002-2013
Beta
portfolios Average size Average beta
18 5.1.3. Size-based portfolios: 2002-2007 and 2008-2013.
The characteristics of the size-based portfolios of both periods are shown in Table 3. Considerable differences regarding the general relation between size and monthly return are not found for portfolios ranked on size between 2002-2007 and 2008-2013. Within both periods, stocks in the smallest portfolio underperform relative to other portfolios. The difference between return of stocks in the smallest portfolio and stocks in the portfolio with the highest return is almost identical between time period I and time period II, 2.01% and 1.99% respectively. Based on the observed pattern, the relation between size and return seems to be consistent throughout the sampling period, and the financial crisis did not change the general tendency observed within period I. This finding is of importance, as firms varying in their size could in theory react differently on a period of economic downturn. The returns in period II are lower than in period I, which is explained by the subprime mortgage crisis that occurred in the time span of this study. The financial crisis has negatively affected stock prices of stocks listed on the exchanges under investigation. The financial crisis is therefore a good scenario to see whether the observed pattern is consistent. A clearer picture of the financial crisis is visible in the graphical outlay.
5.1.4 Beta-based portfolios: 2002-2007 and 2008-2013.
19 Table 3. Characteristics of the portfolios listed on size: 2002-2007 and 2008-2013.
The return of each portfolio is the average equally weighted monthly return in percentages of the time-series. The values of beta and size are the monthly average values of the variables. Size is measured as the natural logarithm of market equity in millions of Euros.
Results for portfolios based on size 2002-2007
Size
portfolios Average size Average beta
Average monthly return (%) (1) (2) (3) (4) Small 2,29 1,22 -0,35 2 3,24 1,18 0,89 3 3,86 1,04 0,89 4 4,40 0,95 1,26 5 4,94 0,98 1,54 6 5,50 0,99 1,56 7 6,16 1,08 1,66 8 7,08 1,00 1,50 9 8,18 1,03 0,99 Large 9,75 1,11 0,96
Results for portfolios based on size 2008-2013
Size
portfolios Average size Average beta
20 Table 4. Characteristics of the portfolios listed on beta: 2002-2007 and 2008-2013.
The return of each portfolio is the average equally weighted monthly return in percentages of the time-series. The values of beta and size are the monthly average values of the variables. Size is measured as the natural logarithm of market equity in millions of Euros.
Results for portfolios based on beta 2002-2007
Beta
portfolios Average size Average beta
Average monthly return (%) (1) (2) (3) (4) Small 4,86 -0,04 1,30 2 5,60 0,33 0,99 3 5,81 0,52 1,04 4 5,86 0,69 1,11 5 5,77 0,86 1,30 6 5,83 1,04 1,29 7 5,80 1,23 0,97 8 5,56 1,47 1,25 9 5,18 1,81 0,75 Large 5,15 2,67 0,86
Results for portfolios based on beta 2008-2013
Beta
portfolios Average size Average beta
21 5.1.5. The January effect
To examine the January effect, the average monthly returns of January are compared to the returns in the remaining eleven months. Then, differences between size portfolios considering returns in January and the rest of the year are compared in a similar fashion as performed by Horowitz et al. (2000a). The portfolio characteristics are shown in Table 5. The average monthly return in January is 2.41%, while the average return in the remaining months is 0.26%. Average returns in January are on average 2.15% higher. This finding supports the results of Kohers and Kohli (1991) who find that on average, stocks in January generate higher returns.
According to Keim (1983), the January effect is more profound for small firms. The difference between the January returns for the smallest size portfolio is 2.80% per month, indicating that the January effect for the smallest size portfolio is above average. However, the differences in return of portfolio 2, portfolio 4 and portfolio 8 are even larger, 3.01%, 3.17% and 3.13% respectively. Remarkably, portfolio 10 is the only portfolio that generates lower returns in January compared to the rest of the year.
Thus, based on the observations in this paper, the exceptional strong size effect in January as suggested by Keim (1983) and Horowitz et al. (2000a) is not found. Although the January effect is observable, the relation between firm size and the January effect is not as strong as was concluded in previous studies.
The difference in returns between January and the rest of the year may seem generous. However, the difference in return in this paper is smaller than that reported by Horowitz et al. (2000a). In their study, the difference between average return from January and the rest of the year is 2.90% for the period of 1963-1997 based on stock listed on the NYSE, AMEX and NASDAQ.
22 Table 5. Characteristics of the portfolios listed on size: January returns and February - December returns for 2002 to 2013.
The return of each portfolio is the average equally weighted monthly return in percentages of the time-series. The values of beta and size are the monthly average values of the variables. Size is measured as the natural logarithm of market equity in millions of Euros.
Results for portfolios based on size: January, 2002-2013
Size portfolio Average size Average beta
Average monthly return (%) (1) (2) (3) (4) Small 1,94 1,11 1,55 2 2,94 1,16 2,91 3 3,61 1,08 2,31 4 4,21 0,99 3,39 5 4,77 1,02 3,12 6 5,36 1,04 2,81 7 6,05 1,13 2,67 8 6,96 1,12 3,78 9 8,08 1,09 1,61 Large 9,68 1,12 -0,04
Results for portfolios based on size: Feb to Dec, 2002-2013
Size portfolio Average size Average beta
23 5.2. Graphical outlay
The result of the graphical outlay of the size portfolios and the beta portfolios are described and shown below.
5.2.1 Cumulative returns of size portfolios
Fig. 2 displays the cumulative returns of a €1 investment in the ten different size portfolios between 2002 and 2013. The results are based on the portfolios composed in the previous section. Portfolio 1 thus consists of the smallest firms and portfolio 10 contains the largest firms. The graph shows cumulative returns instead of only the average return in the previous section. The additional value of the graphical outlay is that it provides an image of the variations in return over time. The downward trend of stock returns in the countries within the data set as a result of the mortgage crisis, which started in the year 2007, is clearly observable in the graph.
Generally, portfolios with larger firms generate higher returns, and their above average performance is consistent throughout time. When looking at portfolio 1, the value of the invested €1 is consistently the lowest throughout time and never rises above its original investment value. After 144 months, the average value of the investment of all ten portfolios is €1.75. Portfolio 1 is worth €0.17 and portfolio 10 is worth €1.95. Portfolios 7 and 8 yield the highest returns, which is consistent with the averages returns described in the previous section, since the figure is based on those returns.
5.2.2 Cumulative returns of beta portfolios
Fig. 2. Cumulative return of a €1 investment in each of the 10 size-based portfolios. 0 0,5 1 1,5 2 2,5 3 3,5 4 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014
Value of €1 invested in 2002
Portfolio 1 portfolio 2 portfolio 3 portfolio 4 portfolio 5 portfolio 6 portfolio 7 portfolio 8 portfolio 9 portfolio 10
0 0,5 1 1,5 2 2,5 3 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014
Value of €1 invested in 2002
Portfolio 1 Portfolio 2 Portfolio 3 Portfolio 4 Portfolio 5 Portfolio 6 Portfolio 7 Portfolio 8 Portfolio 9 Portfolio 10
5.3. Regression analysis
As described in the methodology, regression (1) is performed on a monthly basis in this analysis.
Rit= γ0t+ γ1tβit+ γ2tln(S)it+ εit (1)
Since the regression analysis is performed in each month, a total of 144 regression outputs are generated. In line with previous size effect studies (see, e.g., Banz, 1981; Fama and French, 1992; Horowitz et al., 2000b), the average parameters of the variables are interpreted as final estimators. Contrary to most size effect literature, which only reports the average slope and corresponding t-statistic, an overview of all slopes and t-statistics of the regression is provided. In line with previous studies, a 95% confidence interval is used in this paper.
The values and direction of the slopes of size and beta differ between months and are not always significant. The parameter of the natural logarithm of size is significant in 85 of the 144 months, or 59.03% of the time. The parameter of beta is significant in 84 months, which is equivalent to 58.33%.
The results of the regression analysis are shown in Table 6, Fig. 4 and Fig. 5. The average size coefficient is 0.16, indicating a positive relation between average monthly returns and firm size. The average coefficient of beta is negative with a slope of -0.11. The average value of the intercept is -0.25. However, it should be noted that these averages of the parameters include coefficients that are not significant at the 5% confidence level. The average t-statistic for size including all months is 2.72. Beta has an average t-statistic of -2.82 over all 144 months. As some outlying t-statistics have high values the average is tilted upwards.
When the insignificant slopes are excluded and only significant parameters are taken into consideration, the average slope of size is 0.31. Without the insignificant coefficients, the average parameter of beta has a negative slope of -0.10. The slopes and t-statistics of the significant parameters are shown in Table 7, Fig. 6 and Fig. 7.
Fig. 6. shows that the parameter values of beta are more dispersed than the parameters of size. The average slope of beta is influenced by some extreme values. Particularly the significant beta coefficient of 16.05, measured in April 2009, has a significant impact on the average parameter. If this month is excluded from the analysis, the average coefficient of beta decreases to a value -0.22. However, as the outlier is positive, exclusion of the slope of April 2009 would leave the direction of the relation unchanged.
27 The average R² of the analysis over all 144 months is 0.04. Which means that on average, only 4% of the variation in return is explained by beta and size. An R² of this magnitude might be troubling for investment engineers attempting to create a first-class return predictive model. However, this paper is not attempting to create an asset pricing model, which predicts returns as accurately as possible. The intent of this paper is to test for the occurrence of the trends described by previous literature. The low R² does indicate however, that the model used does not suffice in explaining the return to a large extent.
The results of the regression analysis do not provide evidence that the size effect exist within the scope of this paper. When interpreting the positive average parameter of size the relation appears to be positive, indicating that firms with more market equity outperform firms with less market equity. The average negative slope of beta suggests a negative relation between beta and average returns.
. Table 6.
Average coefficients and t-statistics of the monthly regressions of percentage stock returns on beta and the logarithm of market equity: 2002-2013.
Rit= γ0t+ γ1tβit+ γ2tln(S)it+ εit
Beta Size
Average coefficient -0,11 0.16
Averate t-statistic 2,84 2,72
Fig. 4.
Boxplots showing the coefficients of the monthly regressions of percentage stock returns on beta and the logarithm of market equity: 2002-2013.
28 Fig. 5.
Boxplot showing the t-statistics of the coefficients of the monthly regressions of percentage stock returns on beta and the logarithm of market equity: 2002-2013.
Rit= γ0t+ γ1tβit+ γ2tln(S)it+ εit
Table 7.
Average coefficients and t-statistics of the monthly regressions of percentage stock returns on beta and the logarithm of market equity, excluding insignificant coefficients: 2002-2013.
29 Fig. 6.
Boxplots showing the coefficients of the monthly regressions of percentage stock returns on beta and the logarithm of market equity, excluding insignificant coefficients: 2002-2013.
Rit= γ0t+ γ1tβit+ γ2tln(S)it+ εit
Fig. 7.
Boxplot showing the t-statistics of the coefficients of the monthly regressions of percentage stock returns on beta and the logarithm of market equity, excluding insignificant coefficients: 2002-2013. Rit= γ0t+ γ1tβit+ γ2tln(S)it+ εit 0 2 4 6 8 10 12
Significant beta
Significant size
30 6. Conclusion
As stated in the introduction, the objective of this paper is to determine whether or not small firms listed on the stock exchanges of Southern Europe generate higher average returns than stocks of large firms during 2002-2013. The countries in the data set that form this region are Spain, Portugal, Italy, Greece and France. The occurrence of the size effect in Southern European countries is examined because the small-cap premium is relatively under-investigated in this region. Following conventional methodologies, practiced and described in the size effect literature, no evidence is found for small firms yielding higher average returns compared to larger firms, within the scope of this study. In fact, the relation seems to be reversed. Firms with small market capitalization yield lower returns relative to firms with large market capitalization.
While examining the size effect, a negative relation between average return and beta becomes apparent. Based on the results of this paper, there is no evidence for the positive relation between return and beta as proposed by the CAPM. The model does not suffice in explaining the cross-section of returns found in the scope of this paper. Firms with the lowest betas generate the highest returns, while high beta securities produce below average yields.
Conclusions about the size effect and the relation between a firm’s return and beta are discussed separately below.
6.1 Size effect
The results of the two methods that are used to examine the relation between return and size are consistent with one another. The size-ranked portfolios that are created following the method of Fama and French (1992) suggest a positive relation between size and return. Portfolio 1 consisting of the smallest firms generates the lowest average returns and average returns increase along the first seven portfolios. The lower average returns for small firm portfolios are consistent throughout the data set and no different pattern is observed after the start of the subprime mortgage crisis. A graphical representation of the cumulative returns of a €1 investment in each of the ten portfolios over the 144 months confirms the positive relation between return and size.
31 Finally, the regression analysis provides no evidence pleading for the existence of the size effect within the timespan of this data set. In line with previous size effect studies (see, e.g., Banz, 1981; Fama and French, 1992; Horowitz et al., 2000b), the average slope of the size parameter is interpreted as the final predictor. The relation between size and return is significant in 85 of the 144 months. The average significant size coefficient is 0.31, which suggests that the relation between size and return is positive rather than negative. Thus, based on the results of this paper the negative relation between size and return suggested by size effect theory is not only nonexistent, the relation between return and size seems to be positive.
Although the evidence that no size effect exists within this paper contradicts the findings of Banz (1981) and Fama and French (1992), this is not the first paper challenging the merits of the small-cap premium. In recent years, doubt has been casted over the size effect theory. The absence of the size effect is in line with the studies of Dichev (1998), Horowitz et al. (2000a) and Barry et al. (2002), who also do not find evidence for the size effect in their studies. Likewise, Horowitz et al. (2000a) find the relation between size and returns to appear positive in the months of February-December.
The results of this study challenge the inclusion of the small-cap premium in asset pricing models. Based on this study, the inclusion of the small-cap premium is unfounded for Southern European countries. Moreover, the results challenge the theories described in Section 2 that attempt to theoretically explain the observed size effect revealed by Banz (1981). The theories explaining the size effect are either inaccurate or the underlying risk factors are not robust in other regions and/or time periods. The risk factors associated with smaller stocks does not suffice in explaining the observed return distribution among the size portfolios in this data set. Possibly, the risks associated with small firms described in Section 2 are only valid for firms with smaller market equity values than those included within this analysis.
Following the efficient-market hypothesis, which states that there is a positive relation between risk and return (Fama, 1970), it could be argued that the seemingly positive relation between size and return can be explained by risk factors associated with stocks of larger firms. Similar to line of reasoning of academics explaining the small-cap premium, larger market capitalization might be a proxy for risks that are associated with large market equity value. However, this is only a hypothesis, potential underlying risk factors of large-cap firms are beyond the scope of this paper.
6.2 Small beta anomaly
32 while the two highest beta portfolios, portfolio 9 and portfolio 10, yield below average returns. The graph showing cumulative returns of a €1 investment in each of the beta portfolios portrays this finding. The relation between return and beta is significant in 84 of the 144 months and the average of the significant beta coefficients is -0.11. The negative slope seems consistent with the image of the portfolio formation method and provides further indication that the relation between return and beta is negative within this data set.
Although the negative relation between beta and return might be unforeseen if one follows the theories of the CAPM, the results are in line with recent findings that low beta stocks generate better returns compared to high beta stock. This anomaly proposes that higher returns are obtainable with lower risk exposure, contradicting the basics of the efficient-market hypothesis.
The phenomenon that low beta stocks outperform high beta stocks is known in literature as the low beta anomaly and is initially reported by Haugen and Heins (1975). More recently, Baker et al. (2013) find that low beta portfolios outperform high beta portfolios in United States equity markets during 1968-2012. Withal, they find that the low beta anomaly holds for the markets of 31 developed countries for the period 1989-2012, including all countries studied in this paper. Their finding that small beta stocks outperform high beta stocks is in line with the results from this paper. Furthermore, Frazzini and Pedersen (2014) show that the creation of portfolios with long positions in low beta stocks and short positions in high beta stocks yield significantly higher risk adjusted returns. They argue that due to leverage constraints investors with limited leverage possibilities overweight risky assets in their portfolios. The tendency of investors to tilt towards assets with high betas suggests that high-beta stocks require relatively lower returns than low-beta stocks (Frazzini and Pedersen, 2014).
Based on the above, low beta stocks seem to outperform high beta stocks. The evidence found in this paper combined with other small beta anomaly literature is at odds with the assumptive relation between risk and return.
Although the focus of this paper is on the size effect, an attempt to further explore the low beta anomaly is made. To examine whether the low beta firms share basic characteristics such as the industry in which they operate or the country of origin, the firms with the smallest betas are examined. However, after analyzing the firms within beta portfolio 1, no pattern is observable among the low beta stocks. Further research is needed to explore the small beta anomaly observed in this data set.
6.3 Final remarks
33 still be used as a factor when predicting stock returns, at least within the region investigated in this paper. Similarly, the predictive value of the CAPM should not be regarded a matter of fact, as the CAPM seems to be misspecified in the context of this analysis. Although it might be tempting to rely on models and anomalies that once seemed to have predictive power, practitioners and academics should never turn a blind eye to evidence conflicting with notions of the past.
7. Limitations and suggestions for further research
This paper is bound to certain restrictions that need to be taken into consideration when interpreting the results. First of all, data availability constraints must be addressed. The data set only includes stocks that are currently listed on stock exchanges of the included countries, resulting in a survivorship bias within the data. Because stocks that have been delisted within the time span of this study are not included, it is likely that the actual average monthly return is lower than the average return portrayed in this paper. Furthermore, not all stocks listed of the selected stock indexes are incorporated, because the market equity values of some of the smaller stocks could not be obtained. A data set including all listed and delisted stock of the countries under investigation would increase the validity of the results. When more small stocks are included in future analyses, it might be valuable to include value-weighted size groups, in addition to equally-weighted size groups, because small firms generally outnumber large firms (Fama and French, 2008). An increase of the timespan of the study would add to the power of the results as well. However, because the number of stocks would become limited in the first years of the analysis, the timespan of this study is restricted to 144 months.
In this paper, stocks from different countries are compiled into one cluster. Although grouping countries together has a positive effect on the power of the test, it also has potential drawbacks. For example, the results might be influenced by stocks from one specific country. Monitoring the country of origin of firms in future analyses counters this potential bias.
Some evidence has been found for the small beta anomaly within this paper. Further research exploring this anomaly in depth is needed to gain a deeper understanding of its merits and its occurrence within Southern Europe. Exploring whether low beta stocks within the analysis share certain characteristics that are not exposed in this paper could be valuable for future asset pricing models.
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