# The Market Anomaly "Size Effect". Literature Review, Key Theories and Empirical Methods

## Leseprobe

## 1. Introduction

The size effect is a market anomaly in asset pricing according to the market efficiency theory. According to the current body of research, market anomalies arise either because of inefficiencies in the market or the underlying pricing model must be flawed. Anomalies in the financial markets are typically discovered form empirical tests. These tests usually rely jointly on one null hypothesis H0= markets are efficient AND they perform according to a specified equilibrium model (usually CAPM). Thus, if the empirical study rejects the H0, the reason could either be due to market inefficiency or due to the incorrect model. Market efficiency theory says that the price of an asset fully reflects all current information and is not predictable (Fama 1970). Fama (1997) states that market anomalies, even long-term anomalies, are not an indicator for market inefficiencies due to the reason that they randomly split between “underreaction and overreaction, (so) they are consistent with market efficiency” (p. 284), they happen by chance and it is always possible to beat the market by chance. This essay will give an overview of the literature of the size effect and will stress the key theories, empirical methods and findings, as well as the existing body of research about this particular anomaly.

## 2. Empirical Methods

The two major methods of testing the size effect are the cross-sectional linear regression or to categorize size-groups and analyse the monthly returns of each group and compare them (Fama and French 2008). Some studies use a both methods but others only use the regression method. The method of sorting companies is a very straightforward method, which presents a “simple picture” (Fama and French 2008, p. 1654). By using this method, researcher just calculate the mean returns of each group over a specific time period and compare them which each other. However, “ *A* *potential problem is that the returns on (...) portfolios that use all stocks can be dominated by stocks* *that are tiny* “ (Fama and French 2008, p.1654).

The cross sectional regression method computes a regression on particular stocks or portfolios. One advantage of the cross-section regression is that it can estimate which “anomaly variable” (Fama and French 2008, p.1654) has what kind of influence on the returns. It is possible to compute minimal effects of each variable. Additionally, according to a diagnostics of the residuals of the regression model it is possible “ *to judge whether the relations between anomaly variables and * *average returns implied by the regression slopes show up across the full ranges of the variables* ” (Fama and French 2008, p.1654). In other words, you can conclude by the different slopes of the regression among the stocks/portfolios if certain anomalies like the size-effect are significant or not.

## 2. Theories and Concepts

The size effect was first discovered by Banz (1981). He firstly describes that the size of a firm is a representation for risk. Banz tested the Capital Asset Pricing Model (CAPM) developed by Sharp (1964) & Lintner (1965). Sharpe (1964) and Lintner (1965) advanced Markowitz’s (1952) Portfolio Theory. They presented a theory in which an investor could combine risk-free investments, like government bonds with risky assets form the market portfolio according to the risk preference of the investor. In addition, the CAPM offered a model to compute the rate of return of an asset with only three different variables, the equity risk premium, the risk-free interest rate and the and the relationship of the investment with the market portfolio (Reilly 2009). The assumption of the CAPM is that the riskiness of a security can be exclusively explained by the systematic market risk. The ratio of the covariance of the return of a particular investment and the return of the market portfolio is estimated by the factor beta (β), which is estimated with an Operation Least Squares (OLS) regression.

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The OLS regression measures the returns of a particular investment on the market portfolio returns. The CAPM is expressed by the following equation (Reilly 2009):

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The size effect is described as a market anomaly according to the CAPM. Since the CAPM has been described, researchers have found other factors, known as CAPM anomalies, which explain asset returns (Berk 1995). The size effect is one of these significant anomalies, which has been discovered by empirical tests of the CAPM (Berk 1995). Banz (1981) describes a negative relationship of size to returns, in other words he discovered that returns of small firms are significantly larger than returns of larger firms. Fama and French (1992) strongly criticised the CAPM model by finding that in the long term stock returns differ form the CAPM prediction.

Fama and French (1992) analysed numerous of previous empirical work. They combined size, , Book / Market Value and β (CAPM) in their study (Brooks 2008). They introduced “ *set of cross-sectional* *regressions* ” (p.653):

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They show that the positive relation between β and the average return is due to the negative correlation between a company’s size and beta (Fama and French 1992) “(…) *average return* *increases with* β *and decreases with size* “ (p. 452). The relationship of β and returns dissolve when you consider this correlation. The positive relationship between return and beta is linear, as predicted by the CAPM. According to this indication, it seems that the CAPM explains the higher returns of small firms. However, when is allowed to differ unconnected to size, the positive, linear -return relationship disappears. This result controverts the prediction of the one-period CAPM. They also indicated that the book / market value and the frim size are those variables, which have the highest explanatory power to returns (Fama and French 1992).

In the following study Fama and French (1993) are describing three factors, which are significant in describing asset returns: the market risk from the CAPM, the firm size and book to market value of an asset. This model is now a factor-based model, which works “ *in the context of a time series* *regression which is now run separately on each portfolio I* ” (Brooks 2008, p. 653).

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To sum up, Fama and French were able to prove “ *variables that have no special standing in asset* *pricing theory show reliable power to explain the cross-section of average returns* ” (Fama and French 1992, p. 3) *.*

## 3. Empirical Evidence

Van Dijk (2011) shows in his paper that various studies, which examined stocks in the U.S. market in the period between 1936 and 1989, showed a size premium.

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Source: Van Dijk (2001)

However, since the 1980’s the size effect seems have disappeared in the U.S. Various studies could not determine a significant impact of company size to their stock returns. Horowitz et al. (2000) showed that small companies outperformed large companies in the period before the discovery of the size-effect (Banz 1981), however underperformed large companies in the period form 1982 - 1997. The empirical test results are shown in the table below (Horowitz et al. 2000):

Avg. Monthly returns of small and large companies:

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