Master of Science in
Business Economics: Finance
Master Thesis
Asset Pricing Controversy
Name: Lukas Ripper Student ID: 10828346
Address: Schuchardstraße 11, 64354 Reinheim, Germany Submitted to: Dr. Liang Zou
Statement of Originality
This document is written by Student Lukas Ripper who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
In the financial landscape, there exist single-factor and multi-factor asset-pricing models that try to capture and explain the riskiness of investments and its implications on the expected return of an asset or portfolio. Based on the Modern Portfolio Theory, the Capital Asset Pricing Model (CAPM) is an equilibrium single-factor model that enables investors to calculate asset or portfolio’s expected return by pricing the market risk and neglecting the idiosyncratic risk. In contrast, the multi-factor Fama-French Three-Factor Model (FF3) is based on the Arbitrage Pricing Theory and explains the variation in stock returns by asset or portfolio’s sensitivity to the size and value factor in addition to the market risk factor. However, there are debates in the related literature about the model with the higher explanatory power. The research topic of this thesis is thus to compare the explanatory power of the CAPM and FF3. By constructing randomly uninformed equally and value-weighted portfolios out of empirical data from stocks of the S&P 500 Index for the sample period 1987 to 2014, and by further using two sub-samples, this thesis compares the explanatory power for a recent time frame. The methodology is based on time-series regression. Jensen’s Alpha (1967) and the Model Performance Ratio (2014) are used as the thesis’ measures to rank the models based on their explanatory power and thus come up with a conclusion to the research topic. The evidence presented in the analysis suggests that the CAPM is the superior model for pricing uninformed value-weighted portfolios, whereas the FF3 is the superior model for pricing uninformed equally weighted portfolios; given the respective settings of this research. The analysis contributes to the empirical literature aimed at ranking the explanatory power of the CAPM and FF3 by using US American data for a recent time frame.
List of Abbreviations ... I List of Figures ... II List of Tables ... III
1 Introduction ... 1
1.1 Problem Definition and Objectives ... 1
1.2 Course of Investigation ... 2
2 Literature Review ... 3
2.1 Foundations of Portfolio Theory ... 3
2.2 From CAPM to Multi-‐Factor Models ... 9
2.3 Fama-‐French Three-‐Factor-‐Model ... 11
2.4 CAPM vs. FF3 Related Literature ... 15
3 Data and Measures ... 16
3.1 Data ... 16
3.2 Measures ... 19
3.2.1 Jensen’s Alpha ... 19
3.2.2 Model Performance Ratio ... 21
4 Methodology ... 23
4.1 CAPM Assumption Testing ... 27
4.1.1 CAPM Assumption Testing for Equally Weighted Portfolio ... 27
4.1.2 CAPM Assumption Testing for Value-‐Weighted Portfolio ... 31
4.2 FF3 Assumption Testing ... 35
4.2.1 FF3 Assumption Testing for Equally Weighted Portfolio ... 36
4.2.2 FF3 Assumption Testing for Value-‐Weighted Portfolio ... 41
5 Results ... 47
5.1 CAPM and FF3 With Equal Weightings ... 47
5.1.1 Equally Weighted Portfolios with 15 Stocks ... 47
5.1.2 Equally Weighted Portfolios with 30 Stocks ... 48
5.2 CAPM and FF3 With Value Weightings ... 50
5.2.1 Value-‐Weighted Portfolios with 15 Stocks ... 50
5.2.2 Value-‐Weighted Portfolios with 30 Stocks ... 51
6 Conclusion ... 54 Sources ... 57 Appendix ... 62
List of Abbreviations
AMEX American Stock Exchange
APT Arbitrage Pricing Theory
BCAPM Best-Beta CAPM
CAL Capital Allocation Line
CAPM Capital Asset Pricing Model
CF4 Carhart Four-Factor-Model
CL Confidence Level
CLT Central Limit Theorem
EMH Efficient Market Hypothesis
EW Equally weighted
FF3 Fama-French Three-Factor-Model
HML High Minus Low
MOM Momentum
MPR Model Performance Ratio
MPT Modern Portfolio Theory
NASDAQ National Association of Securities Dealers Automated Quotations
NYSE New York Stock Exchange
RESET Regression Equation Specification Error Test S&P 500 Standard & Poor’s 500
SMB Small Minus Big
SML Security Market Line
Q-Q Plot Quantile-quantile plot
VIF Variance inflation factor
List of Figures
Figure 1: Efficient E, V Combinations Figure 2: The Security Market Line
Figure 3: Linear Relationship MKTRF EW15_1 (CAPM) Figure 4: Residual plot EW15_1 (CAPM)
Figure 5: Distribution of Standardized Residuals EW15_1 (CAPM) Figure 6: Q-Q Plot EW15_1 (CAPM)
Figure 7: Linear Relationship MKTRF VW30_85 (CAPM) Figure 8: Residual plot VW30_85 (CAPM)
Figure 9: Distribution of Standardized Residuals VW30_85 (CAPM) Figure 10: Q-Q Plot VW30_85 (CAPM)
Figure 11: Linear Relationship SMB EW15_1 (FF3) Figure 12: Linear Relationship HML EW15_1 (FF3) Figure 13: Residual plot EW15_1 (FF3)
Figure 14: Distribution of Standardized Residuals EW15_1 (FF3) Figure 15: Q-Q Plot EW15_1 (FF3)
Figure 16: Linear Relationship SMB VW30_85 (FF3) Figure 17: Linear Relationship HML VW30_85 (FF3) Figure 18: Residual plot VW30_85 (FF3)
Figure 19: Distribution of Standardized Residuals VW30_85 (FF3) Figure 20: Q-Q Plot VW30_85 (FF3)
List of Tables
Table 1: Regression results for EW portfolios with 15 stocks at the 95% CL Table 2: Regression results for EW portfolios with 30 stocks at the 95% CL Table 3: Regression results for VW portfolios with 15 stocks at the 95% CL Table 4: Regression results for VW portfolios with 30 stocks at the 95% CL Table 5: Accumulated regression results at the 95% CL
1 Introduction
1.1 Problem Definition and Objectives
As stated by Brealey, Myers, and Allen (2011), “the stock market is risky because there is a spread of possible outcomes” (Brealey, Myers, Allen, 2011, p. 185). Researchers have developed models that try to capture and explain this riskiness of the stock market by factors and their implications on expected returns. The question “what the factors are that affect expected returns on assets” (Elton, 1999, p. 1199) has been tried to be answered by many scholars. This thesis will elaborate on two asset-pricing models and their respective risk factors’ explanatory power. Based on the modern portfolio theory (MPT) by Harry Markowitz (1952), the Capital Asset Pricing Model (CAPM) “marks the birth of asset pricing theory” (Fama & French, 2004, p. 25). Developed by William Sharpe (1964), John Lintner (1965), and Jan Mossin (1966), the CAPM is an equilibrium model that states that the asset’s sensitivity to the market risk factor affects asset’s expected returns. Its equation predicts that the expected excess return of an asset over the risk-free asset is explained by the asset’s sensitivity to the excess return on the market over the risk-free asset, assuming the asset’s idiosyncratic risk can be eliminated by diversification. Herein, the CAPM makes it possible to calculate the expected excess return of an asset or portfolio by pricing the asset or portfolio’s sensitivity to the market risk factor.
According to empirical tests by scholars such as Black, Jensen, and Scholes (1972), Ross (1976), and Banz (1981), there are other factors that aid in the explanation of the variation in stock returns. Based on Ross’ (1976) Arbitrage Pricing Theory (APT), Eugene Fama and Kenneth French (1993) developed the Fama-French Three-Factor-Model (FF3). Their empirical tests showed that the single-factor CAPM model is unable to fully explain this variation. According to their FF3, two more factors, namely the size factor and value factor in addition to the market risk factor, are required to explain this variation. Nevertheless, scholars started to criticize the FF3. Black (1993) has claimed that Fama and French’s (1993) proof of the FF3 are a result of data mining, for example.
In the financial landscape, even more scholars have come up with different asset-pricing models, such as Mark Carhart’s (1997) Carhart Four-Factor-Model (CF4), which includes an additional momentum (MOM) factor, as well as Liang Zou who developed the Best Beta CAPM (BCAPM) “that maintains the CAPM’s theoretical appeal and analytical simplicity yet unambiguously improves its pricing accuracy” (Zou, 2006, p. 131).
The research topic of this thesis is thus to compare the explanatory power of the CAPM and FF3. In order to come up with a quantitatively measurable conclusion on the research topic, this thesis is going to use empirical data from the S&P 500 index for the years 1987 – 2014, which will further be split into two sub-samples. Out of the data, uninformed random portfolios are constructed, which differ in the amount of stocks and weighting technique and thus allow for a comparison of its implications on the explanatory power. The thesis methodology is based on time-series regression. Hence, the time-series regression tests if the monthly excess returns of the uninformed randomly constructed portfolios are explained by the respective risk factors of both asset-pricing models. For the purpose of analysing the results of the time-series regression this thesis uses two measures, which will be explained in detail in the later part of this paper. First, Jensen’s Alpha (1967), which is the amount of statistically significant Alphas, will be used to report on the explanatory power of the CAPM and FF3. Any statistically significant Alpha indicates a mispricing of the asset-pricing model. Moreover, the Model Performance Ratio (MPR) by Rasa Karapandza and Jose M. Marin (2014) will be used as a second measure. The MPR is the ratio of the observed percentage of statistically nonsignificant Alphas to the theoretical percentage of statistically nonsignificant Alphas at a certain confidence level.
1.2 Course of Investigation
By providing a literature review on the theoretical aspects of the research question, chapter 2 is the starting point of this thesis. Thus, chapter 2.1 introduces Markowitz’ (1952) MPT and the development of the CAPM. Furthermore, model controversies, which have led to the development of multi-factor models, are mentioned in chapter 2.2. A detailed explanation and model-related literature on the FF3 can be
found in chapter 2.3. Finally, chapter 2.4 provides empirical evidence for different researchers’ findings regarding both models explanatory power and functions as the summary of the literature review.
After having expounded on the theoretical and practical applications of both models, chapter 3.1 presents the data used for the analysis and the construction of portfolios. In what follows, proper measures, which are needed to quantify the results of the empirical analysis and the resulting null hypothesis, are defined, before further going into the methodology and analysis. Thus, chapter 3.2 provides an explanation of Jensen’s Alpha measure and the MPR.
Following the explanations on data and measures, chapter 4 describes the methodology that is relevant for the research question at hand. Since linear regression and respectively the method of ordinary least squares (OLS) are subject to certain assumptions, this chapter tests the associated assumptions and describes the results in detail.
Chapter 5 is dedicated to the results of the regressions, which are further divided into two weighting techniques as well as two portfolio sizes. In chapter 4.3, the accumulated results of the regressions are then presented.
Finally, chapter 6 will serve as the conclusion to this thesis and will sum up the key findings of the analysis.
2 Literature Review
This chapter provides an explanation of the development of the above-mentioned asset-pricing models, their practical applications, as well as the criticism that these models face. The goal is to explain the theories of the existing asset-pricing literature, the associated predictions about risk and expected return, and their debates by providing some already existing empirical evidence.
2.1 Foundations of Portfolio Theory
Harry Markowitz’ (1952) MPT is an investment theory based on the idea that rational investors diversify their assets within a portfolio in order to decrease risk. His
theory allows for the measurement of the riskiness of investments and thus the constructing of portfolios to maximize the expected return based on a given level of risk. Markowitz (1952) emphasized that being exposed to a higher risk is rewarded by a higher return. According to his theory, investors are able to construct an efficient frontier of optimal portfolios offering the highest return possible for a given level of risk.
Markowitz (1952) assumed that investors “consider expected return a desirable thing and variance of return an undesirable thing” (Markowitz, 1952, p. 77). Hence, rational investors try to diversify their investments by constructing portfolios in order to reduce their risk exposure given an expected portfolio return or increase expected return given the portfolio risk. By allocating capital across multiple assets, firm-specific risk or so-called idiosyncratic risk is reduced and portfolio risk thus decreases. Standard deviation is used as a proxy of risk and can be calculated by taking the square root of the variance. The expected return of a portfolio and the variance can be calculated by equations 1 and 2, which are taken from Markowitz (1952, p. 81), and can further be applied to any amount of stocks of a given portfolio.
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑅𝑒𝑡𝑢𝑟𝑛 = ! 𝑋!µμ! !!! [1]
𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = !!!! !!!!𝜎!"𝑋!𝑋!; where 𝜎!" = ρ!"σ!σ! [2]
𝑋! = Weight of stock i in portfolio
µμ! = Expected return of stock i
𝜎!" = covariance between stock i and stock j; which is equal to the correlation coefficient 𝜎!"= ρ!"σ!σ!
According to Markowitz’ MPT (1952), portfolio diversification per se is not a simple concept, as “the returns from securities are too intercorrelated. Diversification cannot eliminate all variance. The portfolio with maximum expected return is not necessarily the one with minimum variance” (Markowitz, 1952, p. 79). Hence, the correlation between the assets strongly influences the portfolio risk. Equation 2 demonstrates that the correlation coefficient ρ!" influences the variance of a portfolio and thus the risk. This coefficient can vary from +1 to -1. In case of a coefficient of +1, if one asset moves by 1%, the other asset moves by 1% in the same direction. In case of a correlation coefficient of 0, the asset’s movements are completely independent of each other and asset A’s movements do not provide any information about asset B’s
movements. Thus, the correlation between the assets held in a portfolio has an impact on the riskiness of the portfolio, but not on the return. Markowitz (1952) concluded that a rational investor would choose a portfolio, which would offer a minimum variance for a given expected return and maximizes return given variance. With respect to Figure 1, the best expected return and variance (E, V) combinations are presented on the thick black line and Markowitz (1952) therefore called them efficient portfolios.
.
Figure 1: Efficient E,V combinations, from: Markowitz, 1952, p. 82
Hence, according to the MPT by Markowitz (1952), investors should diversify across industries in order to achieve a successful diversification strategy, for example: “A portfolio with sixty different railway securities […] would not be as well diversified as the same size portfolio with some railroad, some public utility, mining, various sort of manufacturing, etc. […] Similarly in trying to make variance small it is not enough to invest in many securities. It is necessary to avoid investing in securities with high covariances among themselves” (Markowitz, 1952, p. 89).
But, diversification of stocks within a portfolio cannot reduce risk that affects all stocks. This risk is called market risk or systematic risk. All assets are subject to economic influences and the investor cannot eliminate this market risk by means of diversification. From a statistical point of view, the market risk is the standard deviation
of a value-weighted market portfolio and the market portfolio is a value-weighted portfolio consisting of all risk-bearing assets traded. According to Roll’s critique (1977), the market portfolio is unobservable and hence has to be proxied by certain indices. For example, Kenneth French uses the “value-weight return of all CRSP firms incorporated in the US and listed on the NYSE, AMEX, or NASDAQ” (French, 2015) to proxy the market portfolio.
Given the findings of Fama (1970) who derived the efficient market hypothesis (EMH) and which expounds that prices “fully reflect” (Fama, 1970, p. 383) all information of the market, and given the insights of the MPT by Markowitz (1952), every investor will consequentially invest in the same portfolio and no investor has superior information. As a result, every investor will invest in the market portfolio. By investing in the market portfolio, every investor bears the market risk and gets the market return. According to Fama (1970), it is thus impossible for an investor to outperform the market portfolio. The only way to obtain a higher return than the market is by investing in riskier investments.
By definition, the market portfolio includes all assets available. Thus, it is possible to diversify every asset’s idiosyncratic risk and focus on the asset’s market risk. Now, the CAPM explains that a stock should not be priced according to its idiosyncratic risk as this risk is reduced by the means of diversification, but focus on the stock’s sensitivity to the market risk. Every asset in a portfolio has a different sensitivity to the market risk. The Greek letter β (Beta) was assigned to measure this sensitivity. Beta can be calculated by taking the covariance of a stock with the market portfolio and by dividing it by the variance of the market portfolio. The Beta factor therefore allows for the quantifying of the co-movements of a stock with the market and hence is a measure of the non-diversifiable market risk.
𝐵𝑒𝑡𝑎 = β!,! =!!,!
!!! [3]
β!,! = Beta of market portfolio m and stock i
σ!,! = Covariance of stock i with the market portfolio m
The left hand side of the equation shows the sensitivity of stock i returns to market movement, and the right hand side is the covariance of stock i with the market portfolio divided by the statistical variance of the market portfolio. (Fama & French, 2004)
We further assume that there is a risk-free rate 𝑟!at which an investor can invest in and borrow at. In reality, there is no such thing as a risk-free asset, since any investment involves risk. But there are investments, such as treasury bills, for which the risk is marginal. Consequentially, investors can invest one portion of their money in a risk-free asset and the remainder in a portfolio, according to their level of risk aversion. The risk-free rate 𝑟! has a Beta of 0, since it is independent of market risks. Thus, it offers a risk premium of 0. The market correlates perfectly with itself and hence has a Beta of 1, offering the market risk premium 𝑟!− 𝑟!. Stocks with a Beta larger than one have a higher sensitivity to the market and offer higher returns to the investor, while being exposed to a higher risk. Stocks with a Beta lower than one offer lower returns than the market, but are also less risky. The graphical combination of the risk-free asset with the market Beta is the Security Market Line (SML), which is shown in figure 2. It clearly demonstrates that a Beta > 1 requires a higher expected return on an investment than the market return due to the higher riskiness. The slope of the SML equals the market risk premium 𝑟!− 𝑟!.
As such, the SML illustrates that investors should not be concerned about idiosyncratic risk, but should rather focus on the sensitivity of the asset’s return to the market risk, hence the Beta. The Beta has a direct impact on the risk and return of an asset, whereas the asset’s return is independent of the asset’s idiosyncratic risk. In case of a Beta of 0 or 1, the expected risk premium is easily quantifiable. But how can the expected risk premium be calculated if Beta ≠ 0 or 1? William Sharpe (1964), John Lintner (1965), and Jan Mossin (1966) have come up with an answer to this question by providing an equation for the graphical representation of the SML.
The authors have summarized the theoretical concepts in one framework in order to calculate the expected or required return of an asset after a single holding period. To come up with the expected risk premium, an investor needs to know the risk-free rate, the market return, and the Beta.
Figure 2: The Security Market Line, from: Brealey, Myers, and Allen, 2011, p. 192
These findings were used to form the CAPM, a model that explains that an asset should be priced according to its sensitivity to the market risk, neglecting the asset’s idiosyncratic risk.
E(𝑟!) = 𝑟!+ β!,! ∗ (𝑟!− 𝑟!) [4]
𝐸(𝑟!)= Expected return of asset i
𝑟! = Risk free rate
β!,! = Sensitivity of asset i to market movements
𝑟!− 𝑟!= Risk premium on the market
In this equation, E(𝑟!)is the expected return of an individual asset, 𝑟! is the risk-free rate, β!,! measures the sensitivity of the individual asset with the market, and 𝑟!− 𝑟! is the risk premium on the market. The CAPM thus demonstrates that the asset’s expected risk premium (E(𝑟!) − 𝑟!) is priced by an asset’s individual sensitivity (β!,!) with the excess return of the market, which is the market risk premium (𝑟!− 𝑟!). Therefore, the formula can be rearranged as follows:
E(𝑟!) − 𝑟! = β!,!∗ (𝑟!− 𝑟!) [5]
Sharpe (1964), Lintner (1965), and Mossin (1966) have independently developed the CAPM by using the underlying assumptions that there exist no transaction costs on the capital markets and that investors are risk averse and seek for the highest return linked to the lowest risk. Moreover, all investors can borrow and lend at the risk-free rate and everyone holds the market portfolio. (Black et al., 1972) Given these assumptions, “the CAPM says that all variation in Beta across stocks is compensated in the same way in expected returns” (Fama & French, 2006, p. 2164), hence investors demand a risk premium proportional to its Beta.
In summary, investors are able to link risk to return to all asset classes for investment decisions by using the equilibrium single-factor CAPM. This model enables to derive how much return an investment should yield pricing the systematic risk and neglecting the idiosyncratic risk. By employing this model for investment decisions, investors can calculate asset or portfolio’s expected excess return over the risk-free rate based on the excess return of the market over the risk-free asset and the Beta.
2.2 From CAPM to Multi-Factor Models
Scholars have criticized the CAPM, its explanatory power and underlying assumptions. Some have tested this model empirically and have stated that “evidence suggests the existence of additional factors which are relevant for asset-pricing” (Banz, 1981, p. 3). Black, Jensen, and Scholes (1972) have empirically tested the CAPM using data of all stocks from the NYSE from 1931 to 1965. In turn, this led them to the conclusion that “the expected excess return on an asset is not strictly proportional to its Beta” (Black et al., 1972, p.1).
Due to the existence of empirical studies about the explanatory power of the CAPM, other scholars discovered correlations between different firm-specific characteristics and average returns. These firm specific characteristics are not included in the CAPM and hence contradict the model. Basu (1977) used a sample of price-earnings (P/E) ratios from stocks by industrial firms listed on the NYSE from September 1956 to August 1971, which amounted to over 1400 firms, and analysed
their rates of return. He stated that “low P/E portfolios seem to have, on average, earned higher absolute and risk-adjusted rates of return than the high P/E securities” (Basu, 1977, p. 680). According to the CAPM, the expected excess return of an asset should be explained by pricing the systematic risk and not by the P/E ratio. Levy and Lerman (1985) have supported Basu’s (1977) empirical study by conducting similar tests.
Banz (1981) sorted NYSE stocks according to their market capitalization and showed that “in the 1936 - 1975 period, the common stock of small firms had, on average, higher risk adjusted returns than the common stock of large firms” (Banz, 1981, p. 3-4). He concluded that this “size effect has been in existence for at least forty years and is evidence that the capital asset pricing model is misspecified” (Banz, 1981, p. 3).
Another contradiction to the CAPM is the work by Rosenberg, Reid and Lanstein (1985) and Stattman (1980), who pointed out that the book-to-market ratio is a factor influencing the return of a stock. The book-to-market ratio can be calculated by dividing the book value of a firm by the market value. The scholars analysed stock returns and came up with the conclusion of a positive relation between high returns and a high book-to-market ratio. This effect is known as the value effect.
Chan, Hamao and Lakonishok (1991) provided further proof for the findings of Rosenberg et al. (1985) and Stattman (1980). They used a sample of stocks from manufacturing and nonmanufacturing firms, delisted securities and companies from the Tokyo Stock Exchange. Their empirical study provides evidence that the “book to market ratio and cash flow yield have the most significant positive impact on expected returns” (Chan et al., 1991, p. 1739).
Bhandari (1988) provided evidence that the “expected common stock returns are positively related to the ratio of debt […] to equity, controlling for the beta and firm size” (Bhandari, 1988, p. 507). His results, which he derived using his chosen sample size from the NYSE between 1948 and 1979, contradict the CAPM yet again, since Bhandari (1988) showed that the Beta is inadequate in explaining expected stock returns, but rather leverage has a positive effect on stock returns. Jegadeesh and Titman (1993) demonstrated that strategies to buy well performing stocks and sell poorly
performing stocks produced positive returns for the sample period between 1927 and 1989 for NYSE and AMEX stocks with different sub-samples. This effect on stock returns is known as the momentum (MOM) factor and was further used by Carhart (1997) for his CF4. Fama and French (2004) criticized that the “CAPM is based on many unrealistic assumptions [… such as] the assumption that investors care only about the mean and variance of one-period portfolio returns is extreme” (Fama and French, 2004, p. 37).
Due to the empirical tests of the CAPM, other scholars have come up with different asset-pricing models. The next chapter is dedicated to the development of one of these models, the FF3.
2.3 Fama-French Three-Factor-Model
Contradictory to the CAPM, the FF3 is not based on Markowitz (1952) MPT, but on the arbitrage pricing theory (APT) by Ross (1976). This chapter provides an explanation of Ross’ APT and further describes the development of the FF3.
Ross’ (1976) APT was proposed as an alternative asset-pricing model besides the CAPM which "has become the major analytical tool for explaining phenomena observed in capital markets for risky assets” (Ross, 1976, p. 341). The APT is not a modification of the CAPM, but rather a multi-factor model. In contrast to the CAPM, the APT assumes that an asset’s expected return depends on the asset’s sensitivity to macroeconomic factors and not on the asset’s sensitivity to the market risk premium. However, Ross’ did not specify these factors due to the uniqueness of every portfolio. For example, a spike in oil prices will have a higher impact on a portfolio consisting mainly of stocks issued by energy companies than another portfolio consisting mainly of stocks issued by telecommunication companies. Ross (1976) assumed that there could not be only one single factor that took every exposure to risk into account.
𝐸(𝑟!) = 𝑟!+ β! * (𝑟!− 𝑟!) + β!∗ (𝑟!− 𝑟!) + ⋯ + β!∗ (𝑟!− 𝑟!) [6]
𝑟! = Risk free rate
β! = Sensitivity of asset n to macroeconomic factor n
𝑟!− 𝑟!= Risk premium associated on the respective factor
In this equation, E(𝑟!)is the expected return of an individual asset, 𝑟! is the risk-free rate, β! measures the sensitivity of the individual asset to the respective risk factor, and 𝑟! − 𝑟! is the risk premium associated with the respective factor. In comparison to the CAPM, the APT replaces the market risk factor by unspecified risk factors and Betas relative to these factors replace the Beta relative to the market. If no asset of a portfolio is sensitive to any factor, the portfolio yields the risk-free rate.
One modification of the APT is the Three-Factor Model developed by Fama and French (1993). In addition to the asset’s sensitivity to the market risk factor, the FF3 also includes the asset’s sensitivity to the size and value factors, since “size and book-to-market equity do a good job explaining the cross section of average returns on NYSE, Amex, and NASDAQ stocks for the 1963-1990 period” (Fama & French, 1993, p.4). While the explanatory power of both factors has already been proven by Banz (1981) as well as by Rosenberg et al. (1985) and Stattman (1980), Fama and French (1993) have used the combination of both the size and value effect and added these factors to the traditional CAPM by creating variables that proxy these effects. The variable equaling the size effect is called SMB (small minus big) and is the difference between the return on small firm stocks and large firm stocks, hence the performance of small stocks relative to big stocks. The value effect is mirrored by the variable HML (high minus low) and is the difference between the return on stocks with high book-to-market ratios and the return on stocks with low book-to-book-to-market ratios, hence the performance of value stocks relative to growth stocks. The expected return of asset i according to the FF3 is therefore based on the exposure of the asset to these three factors.
𝐸(𝑟!) = 𝑟!+ β!,! ∗ (𝑟!− 𝑟!) + β!"#,! ∗ 𝑆𝑀𝐵 + β!"#,!∗ 𝐻𝑀𝐿 [7] or
𝐸(𝑟!) − 𝑟!= β!,!∗ (𝑟!− 𝑟!) + β!"#,!∗ 𝑆𝑀𝐵 + β!"#,!∗ 𝐻𝑀𝐿 [8]
𝑟! = Risk free rate
β!,! = Sensitivity of asset i to market movements
𝑟!− 𝑟!= Risk premium on the market
β!"#,! = Sensitivity of asset i to size factor
SMB = Risk premium on size risk
β!"#,! = Sensitivity of asset i to value factor HML = Risk premium on value risk
In the equation, E(𝑟!)is the expected return, β!,! represents the sensitivity to market risk, 𝑟!− 𝑟! is the risk premium on the market, β!"#,! is the sensitivity to size risk, SMB is the risk premium on size risk, β!"#,! is the sensitivity to value risk, and HML is the risk premium on value risk. The FF3 formula thus demonstrates that the asset’s expected risk premium (E(𝑟!) − 𝑟!) is a function of an asset’s individual sensitivity (β!,!) with the market risk premium plus the asset’s individual sensitivity (β!"#,!) with the risk premium on size risk and the asset’s individual sensitivity (β!"#,!) with the risk premium on value risk.
Even though the FF3 includes more explanatory factors than the CAPM, scholars have questioned both models’ explanatory power and have come up with opposing empirical evidence. In the next step, the literature concerning the explanatory power of the FF3 is presented, whereas chapter 2.4 will present some empirical tests concerning both asset-pricing models.
Black (1993) has blamed Fama and French (1993) for using data mining to achieve their results by arguing that Fama and French (1993) used only the results that were supportive of their research, but that these results are not repeatable for periods of time other than the one chosen for the initial research. Black (1993) has further argued that Fama and French (1992) were unable to provide evidence for Banz’ (1981) findings on the size effect after 1981.
Kothari, Shanken, and Sloan (1995) have criticized Fama and French’s interpretation of the statistical results. They have accused Fama and French (1993) of a survivorship bias in the Compustat database within their research, which is to say that their database contained an error due to the exclusion of failed companies. This
exclusion has a significant effect on book-to-market ratios, since firms with earning anomalies are more likely to have high or low book-to-market ratios.
Breen and Korajczyk (1993) examined the value effect on stock returns and came up with a much weaker explanatory power than Fama and French (1992) for their 1974 – 1995 sample size for NYSE, AMEX, and NASDAQ firms. Daniel and Titman (1997) further criticized the FF3 results and proved evidence that, in the period from 1973 – 1993 for the NYSE, “the return premia on small capitalization and high book-to-market stocks does not arise because of the comovements of these stocks with pervasive factors. It is the characteristics rather than the covariance structure of returns that appear to explain the cross-sectional variation in stock returns” (Daniel & Titman, 1997, p. 1). However, Davis, Fama, and French (2000) challenged the arguments made by Daniel and Titman (1997) by using a larger sample size of July 1929 to June 1997 of US stocks and thus showed that there is “a positive relation between average return and book-to-market” (Davis et al., 2000, p. 389).
Davis (1994) was able to confirm the explanatory power of the book-to-market ratio and came up with similar results for the 1940 – 1963 period for NYSE and AMEX stock returns. He was further able to provide counterevidence that the findings of Fama and French (1993) were in fact not based on data mining and survivorship bias, as claimed by Black (1993) and Kothari et al. (1995). Moreover, the work of Chan, Jegadeesh, and Lakonishok (1995), who concluded that the survivorship bias in Compustat data is small, also supports the findings of Davis (1994). Additionally, their work proved that “the superior performance of value stocks is confirmed for the top quintile of NYSE-Amex stocks” (Chan et al., 1995, p. 269). Furthermore, Barber and Lyon (1997) were able to show the positive relation between book-to-market ratio and securities by using monthly returns files for the period from July 1973 to December 1994 for the NYSE, Amex, and NASDAQ.
Due to the empirical evidence of scholars, the FF3 has been widely accepted and the explanatory of the SMB and HML have been confirmed. (Alves, 2013) Moreover, Fama and French (1996) have continued to defend the FF3. Even though there are proofs for the explanatory power of the FF3, Fama and French (2012) had to admit that returns could not be fully explained by their model.
2.4 CAPM vs. FF3 Related Literature
In comparison to two firm characteristics used by Fama and French (1993), Karapandza and Marin (2014) used more than 44 million firm characteristics from “Compustat data for fiscal years 1962 – 2009 and CRSP data for the period July 1963 through June 2010” (p. 11). They were thus able to give evidence for the FF3 and C4 showing more unexplainable economic abnormal returns than the CAPM. Hence, they concluded that the CAPM has a higher explanatory power and that the “good performance documented elsewhere for multifactor models is either sample specific or driven by tautological considerations” (Karapandza & Marin, 2014, p. 20-21).
Simpson and Ramchander (2008) used a sample of 12 years from January 1991 to December 2002 of all non-financial stocks listed on the NYSE, AMEX, and NASDAQ, and created six intersecting portfolios. The study compared which model captured news related to underlying economic state variables to a greater degree. They came up with “strong evidence to suggest that the FF3 model is relatively better at controlling for macroeconomic surprises than the CAPM […] thus, there is overwhelming evidence to support the superiority of the FF3 model to the CAPM” (Simpson & Ramchander, 2008, p. 807) This thesis analysis will further give evidence about the explanatory power of both models for another setting.
To summarize the findings so far, “each of these different models of risk and return has its fan club. However, all financial economists agree on two basic ideas: (1) Investors require extra expected return for taking on risk, and (2) they appear to be concerned predominantly with the risk that they cannot eliminate by diversification” (Brealey et al., 2011, p. 204) Aside from the evidence for and against the CAPM and FF3, Bartholdy and Peare (2004) state that “neither model is useful” (2004, p. 17). After having elaborated on the development of asset-pricing models in general, having introduced the CAPM and FF3 and taking the empirical evidence into consideration, the question of which asset-pricing model offers the higher explanatory power has not yet been answered. The next chapter thus deals with the empirical part of the research.
Therefore, the data, measures and methodology utilized in answering the research question are going to be explained.
3 Data and Measures
This chapter is dedicated providing specific details about the data, the construction of the random portfolios and the corresponding null hypotheses. Moreover, the measures used for this thesis will be presented in chapter 3.2.
3.1 Data
This thesis draws on monthly stock returns, which were retrieved from different sources. Stock specific data, such as stock prices and market capitalization, were retrieved from Thomson Reuters Datastream Service, whereas the three factors market risk premium, HML and SMB were retrieved from the French’s data library of the Tuck Business School of Dartmouth College1. The focus of the analysis is on the time period January 1987 to March 2014 in order to ensure a coherent set of data and to avoid potential skews. Moreover, the goal is to use a recent time frame. The sample was further split into two sub-samples, one from January 1987 to December 2000 and one from January 2001 to March 2014. These sub-samples include almost the same amount of observations and allow for comparison of the changes in the explanatory power of the CAPM and FF3. Since this thesis focuses on the US market, the portfolios include stocks from the S&P 500 Index.
The setup was completed as follows. To prepare the data for the regression analysis, monthly stock prices were used to first of all calculate monthly stock returns. The dividend yield was not included for calculating the monthly stock returns. By using the following formula, the returns have thus been calculated.
𝑟!,! = !!,!
!!,!!!− 1 [9]
𝑟!,! = Return of stock i at time t
𝑝!,!= Price of stock i at time t
𝑝!,!!! = Price of stock i at time t – 1
In the next step, market capitalization and returns were merged in one Excel spreadsheet. Out of the 500 stocks, those with missing market capitalizations had to be excluded. The market capitalization is essential to create value-weighted portfolios. 221 stocks were excluded due to missing market capitalizations, leaving 279 stocks for the time series regression analysis.
With the help of the software program Excel, portfolios of two sizes and types were randomly constructed out of the 279 stocks by applying the command ‘RANDBETWEEN(1;279)’. The portfolios constructed were either made up of 15 or 30 stocks per portfolio and the types were either equally weighted or value-weighted. The software program Excel checked that any stock does not occur more than one time per randomly constructed portfolio by applying the command ‘Conditional Formatting – Highlight Cells Rules – Duplicate Values’. For both types and both sizes, 100 random portfolios were created. In total, 400 portfolios were created. The goal of this analysis is further not to test the performance of the asset-pricing models in a small set of suitably designed portfolios, such as Fama and French (1993) did by using 25 informed portfolios sorted by size and value and tested by the FF3 that includes the size and value factor. Therefore, the portfolios were uninformed, meaning that the stocks of the portfolios had not been ranked according to any figure, such as size, but were randomly chosen out of the 279 stocks.
According to asset-pricing theory, the CAPM and the FF3 are models that price securities and assets; hence they can also price portfolios. Similar to the work by Black, Jensen, and Scholes (1972) this analysis reports on the explanatory power of pricing portfolios.
Fama and French (2004) state that “since expected returns and market betas combine in the same way in portfolios, if the CAPM explains security returns it also explains portfolio returns” (Fama & French, 2004, p. 31). A benefit of using randomly constructed portfolios is that the resulting estimates of the time-series regression analysis for the randomly diversified portfolios used “are more precise than estimates for individual securities” (Fama & French, 2004, p. 31). Since the volatility of a portfolio’s return decreases with an increase in stocks in the respective portfolio, the
usage of randomly diversified portfolios makes it possible to draw more accurate conclusions about the explanatory power of the asset-pricing models. Moreover, the differentiation within portfolio size and type allows for comparison of the effect of the size of portfolios as well as type of portfolios on the explanatory power of the respective models.
The portfolio return for an equally weighted portfolio was calculated by dividing the sum of stock returns in the portfolio by the number of stocks included in the respective portfolio, which can be represented using the following formula:
(𝑟!,!!") = !
!∗ 𝑟!,! [10]
𝑟!,! = Return of asset i at time t
N = Number of assets of the equally weighted portfolio
By contrast, the stocks returns of the portfolio had to be weighted according to their market capitalization in t – 1 in order to calculate the portfolio return for a value-weighted portfolio. The respective formula is:
𝑟!,!!" = !!!!(!!,!!!∗ !!,!) (!!,!!!) ! !!! [11] (𝑥!,!!!) !
!!! = Entire value of portfolio at t – 1
𝑥!,!!! = Market capitalization of asset i at t – 1
N = Number of assets of the equally weighted portfolio
In an Excel spreadsheet, the data was merged with the market risk premium, HML and SMB factor, and this data sheet was then imported to STATA. STATA is the statistical software used for the regression analysis performed as part of this paper. As a last step, the time-series regressions were run in STATA. In total, this analysis ran 2400 regressions. For more information on the Do File, please see page 67 in the appendix.
For the purpose of analysing the performance of the respective models, it is necessary to use proper measures to arrive at meaningful results. After having regressed monthly excess returns of the randomly constructed portfolios on the explanatory
variables of the two asset-pricing models, two measures for the purpose of ranking the model’s performances are used within this thesis. These measures are Jensen’s Alpha (1967) and the Model Performance Ratio by Karapandza & Marin (2014). Furthermore, it has to be mentioned that formal statistical tests, such as the GRS test statistic by Gibbons, Ross, and Shanken (1989)2, are ignored within the analysis of the performance of both asset-pricing models. In their seminal paper, Fama and French (1993) have created 25 portfolios sorted on size and book-to-market ratio and then claim to show that the FF3 is better in explaining Alphas of these 25 portfolios than the CAPM by plotting the number of statistically significant Alphas in a table, which is lower in the case of the FF3. Moreover, they do a GRS (1989) test to check if the Alphas of these 25 portfolios are jointly significantly different than zero. Their tests yield the result that both the CAPM and FF3 have F-statistics such that Alphas are jointly different than zero, hence none of these models seems to be really good. Reverse, they focus back on the Alphas and argue that they are smaller in the case of the FF3 without any statistical test and this is how they came up with the FF3 as the model with the higher explanatory power. Hence, this thesis will also elaborate on the amount of statistically significant Alphas.
3.2 Measures
3.2.1 Jensen’s Alpha
Jensen (1967) derived this measure to measure the performance of mutual funds. Alpha is a statistically significant estimate for a y-axis intercept. Jensen (1967) concluded that any outperformance of a manager of his or her respective mutual fund could be observed via the Jensen’s Alpha measure.
Basically, for both the CAPM and the FF3, any occurring of such an Alpha indicates a mispricing of the respective model. After having performed the time-series regression analysis, any occurring of an Alpha indicates a mispricing at a certain confidence level. The confidence levels used in this thesis are 90%, 95%, and 99% respectively. For the purpose of using Jensen’s Alpha, it is necessary to add an α to the
2 The GRS test enables to test whether multiple regression estimated intercepts, such as Alphas, are jointly zero or jointly different than zero.
models empirical formulas. The α actually indicates the difference in performance of the managed portfolio. Moreover, the error term 𝜀!,! was added to the empirical formulas (see formulas 12 and 13). According to Fama and French (1993), the usage of Jensen’s Alpha provides “a simple return metric and a formal test of how well different combinations of the common factors capture the cross-section of average returns. Moreover, judging asset-pricing models on the basis of the intercepts in excess-return regressions imposes a stringent standard” (Fama & French, 1993, p. 5).
To further elaborate on this measure, let us thus return to the case of the CAPM and the SML (see figure 2). Jensen’s Alpha is the distance between the managed portfolio and the SML. A positive Alpha, which is to say that the fund performed better than indicated by the CAPM, is located above the SML. By contrast, a negative Alpha, which is to say that an investor would be better off investing in the market, lies below the SML. If the fund’s performance were equal to the market’s performance, there would be no statistically significant Alpha. Since the CAPM explains that, in the case of this analysis, the equally or value-weighted portfolio’s excess return is priced by the portfolio’s sensitivity to the market risk factor, the equation of the CAPM for the time-series regression analysis thus reads as follows:
𝑟!,! − 𝑟!,! = α!+ β!,!∗ (𝑟!,!− 𝑟!,!) + 𝜀!,! [12]
𝑟!,!= Return of portfolio p at time t
𝑟!,! = Risk free rate at time t; by definition return of a US American one month Treasury Bill
β!,! = Sensitivity of portfolio p to market risk premium
𝑟!,!− 𝑟!,! = Risk premium on the market at time t
α! = Jensen’s alpha; if α! ≠ 0: indicates mispricing of asset-pricing model
𝜀!,!= E(𝜀!,!) = 0; difference between observed and proposed return value
The FF3 explains that the equally or value-weighted portfolio’s excess return is priced by the portfolio’s sensitivity to two further risk factors in addition to the market risk factor. The corresponding time-series regression equation thus reads as follows:
𝑟!,! − 𝑟!,! = α!+ β!,! ∗ (𝑟!,!− 𝑟!,!) + β!"#,!∗ 𝑆𝑀𝐵!+ β!"#,!∗ 𝐻𝑀𝐿!+ 𝜀!,! [13]
𝑟!,! = Risk free rate at time t; by definition return of a US American one month Treasury Bill
β!,! = Sensitivity of portfolio p to market risk premium
𝑟!,!− 𝑟!,! = Risk premium on the market at time t
β!"#,! = Sensitivity of portfolio p to size factor
𝑆𝑀𝐵! = Risk premium on size risk at time t
β!"#,! = Sensitivity of portfolio p to value factor
𝐻𝑀𝐿! = Risk premium on value risk at time t
α! = Jensen’s alpha; if α! ≠ 0: indicates mispricing of asset-pricing model
𝜀!,!= E(𝜀!,!) = 0; difference between observed and proposed return value
As already mentioned, the market risk premium is taken from French’s data library of the Tuck Business School of Dartmouth College. Furthermore, the risk-free rate was defined as the return of a US American “one month Treasury Bill” (Fama & French, 1993, p. 5), which French (2015) gathered from Ibbotson and Associates, Inc.
If the underlying assumptions of both models hold, we should not end up with any statistically significant Alpha, meaning we theoretically expect α! = 0. Hence, the goal of the analysis of this thesis is to compare the amount of statistically significant Alphas at the different confidence intervals. The corresponding null hypothesis thus reads:
H0: There are no statistically significant Alphas.
This hypothesis will be tested at the three different confidence levels. Jensen’s Alpha provides the foundation for the second measure, the MPR.
3.2.2 Model Performance Ratio
The MPR is a measure that was developed by Karapandza and Marin (2014), which can be used to calculate the “ratio of observed to theoretical […] percentage of statistically nonsignificant alphas” (2014, p. 40). The MPR has been developed in their working paper “The Rate of Market Efficiency” (2014), which is currently being revised and has to be resubmitted for a publication in The Journal of Finance.
The purpose of the MPR is to rank asset-pricing models based on their ability to price uninformed randomly constructed portfolios relative to what would be expected
by theory. Moreover, it is explained that “differences in the MPR for different asset-pricing model for a fixed set of portfolios allows assessment of these factor-induced alphas” (Karapandza & Marin, 2014, p. 8). The MPR thus fits to the setting of this thesis, since the purpose of the analysis is to rank both asset-pricing models according to their explanatory power for a fixed set of uninformed portfolios. Karapandza and Marin (2014) state “To the extent that this ratio is less than 1, the chosen asset pricing model has failed to price (portfolios of) securities accurately” (2014, p. 19).
A MPR of one indicates that the model perfectly priced stock returns, since the observed percentage of statistically nonsignificant Alphas equals the theoretical percentage of statistically nonsignificant Alphas at the certain confidence levels. In order to illustrate this point assume a calculation of the MPR at the 95% CL. For the purpose of calculating the MPR, it is necessary to divide the percentage of observed nonsignificant Alphas, assume 90%, by the theoretical percentage of 95% due to the 95% CL. Hence, the MPR would be 90% divided by 95% = 94.7%. The respective formula is:
MPR = !!!"#!$%&'( % !" !"#!"#!"$%&&' !"!#$%!$&$'(!) !"#!!" !" !" ! !" !!! !"#$%"&'"(!"#$%&$' % !" !"#"$!"$%#&&' !"!#$%!$&$'(!) !"#!!" !" !" ! !" !!! !"#$%"&'"( [14]
CL: Statistical confidence levels: 90%, 95%, 99%
With regard to the equation, Karapandza and Marin (2014) explain that “Deviations from a MPRkj = 1 may be due to model misspecification and/or market inefficiency, but the latter should have a minimal effect in the case of randomly constructed portfolios. For such Uninformed portfolios, then, differences between the empirical and the theoretical distribution of alphas are driven mainly (if not entirely) by the accuracy of the pricing model” (2014, p. 19).
Hence, the benefit of using the MPR as another measure for this thesis analysis is that it measures, by definition, the accuracy of the pricing model. Moreover, the MPR was developed as a tool for evaluating the explanatory power of asset-pricing models for randomly constructed uninformed portfolios. It characteristic formula comes up with
percentage values as a result, thus allowing for comparison of the explanatory power in another way than Jensen’s Alpha. Whereas a lower number equals a higher explanatory power in the case of Jensen’s Alpha, the MPR reports with a higher percentage value for a higher explanatory power. This contrariness is beneficial for the presentation of the results of the author’s analysis in chapter 5. Moreover, the possibility of comparing the MPR at different confidence levels is another benefit.
After having run the time series regression analysis, the amount of Alphas, representing a mispricing of the model, have to be checked for in the STATA output. Hence, every Alpha’s p-value was checked if it is statistically significant at the respective confidence level. If this is the case, the aforementioned null hypothesis is rejected because there is a statistically significant Alpha. These Alphas should be appearing as infrequently as possible and are the basis for both measures. By having these two measures at hand, this author will be able to evaluate the explanatory power of the respective models.
Further, it has to be mentioned that other measures are ignored within this thesis. For example, the measure of R2, which is a widely used measure of fit, is ignored within this thesis, since this value increases with an increase in regressors, independent of the explanatory power of the respective model. (Stock & Watson, 2010) Furthermore, the adjusted R2, which is a default output of STATA, is not used as a measure of asset-pricing explanatory power, as this thesis just concentrates on Jensen’s Alpha and the MPR. Moreover, the values for the factor loadings β!,!, β!"#,!, and β!"#,! are ignored within this thesis, as they explain the sensitivity of the randomly generated portfolios to the respective risk factor and thus do not provide information about the explanatory power.
Since this thesis offers a comparison of the explanatory power of the CAPM and FF3, it is necessary to report on the relevant statistical assumptions of the regression analysis. The respective STATA Do File can be found on page 69 in the appendix.
In the following, this chapter deals with multiple linear regression as described in Backhaus, Erichson, Wulff, and Weiber (2003) and Stock and Watson (2010). During this statistical procedure, a dependent variable 𝑌!, 𝑖 = 1, … , 𝑛 is explained by independent variables 𝑋!!, … , 𝑋!". The index i indicates the sample size. For multiple linear regression, we assume a linear additive relationship, which can be described by the following mathematical model:
𝑌! = 𝛽! + 𝛽!𝑋!!+ 𝛽!𝑋!!+ ⋯ + 𝛽!𝑋!"+ 𝑢! [15]
Here, 𝛽!, 𝛽!, … , 𝛽! are the parameters of the mathematical model, which are estimated using the ordinary least squares method (OLS). 𝑢! are the associated error terms of the model. The linear regression and respectively the method of OLS are subject to certain assumptions, which will be tested to be able to draw valid conclusions about the reliability of the coefficients resulting from the regression analysis and the model in general. The assumptions are:
1. The model is correctly specified, which implies that a. It is linear within ins parameters 𝛽!, … , 𝛽!, b. It contains all relevant explanatory variables
2. The error terms have an expected value of zero: 𝐸 𝑢! = 0;
3. There exists no correlation between the other factors contained in 𝑢! and the explanatory variables, hence: 𝐶𝑜𝑟 𝑢!, 𝑋!" = 0, where 𝑗 = 1, … , 𝑘.
4. Homoscedasticity, which is explained by constant variances 𝜎! for the error terms: 𝑉𝑎𝑟 𝑢! = 𝜎!.
5.
The error terms are not correlated with each other: 𝐶𝑜𝑟 𝑢𝑖, 𝑢𝑖+𝑟 = 0 with 𝑟 ≠ 0. Thus, there is no autocorrelation.6. There is no linear relationship between the explanatory variables 𝑋!, hence there is no multicollinearity.
If assumptions 1 – 6 are met, the OLS method provides appropriate estimators of the regression parameters. Hereby, the estimators are unbiased and efficient in a statistical context. Assumption 7 is relevant for the performance of significance tests of the parameters of the model.
Linearity, see assumption 1a, is tested via the correlation between the dependent and independent variables. Hereby, the Pearson product moment correlation coefficient is used as a measure of the linear relationship between metrically scaled variables. Furthermore, scatterplots showing the relationship between dependent and independent variables can be used to test for linearity. In addition, this paper employs a Regression Equation Specification Error Test (RESET) by Ramsey (1969). The RESET enables to test if the dependent and independent variables have a linear relationship or if the model might have, for example, a squared relationship. If the test result is not significant, the dependent and independent variables are linearly related.
If assumption 2 is violated, the constant parameter 𝛽! is biased. This assumption will be tested via scatterplots of estimated values of the regression models and the standardized residuals. Here, the plots should evenly spread around the value zero.
The violation of assumption 1b due to missing independent variables might lead to biased estimators, if the missing independent variable correlates with one or more independent variables included in the regression model. The effect of missing variables diminishes in the error term, which results in a violation of assumption 3, since there now might be a correlation between the error term and the independent variables of the regression model. Nevertheless, it can be counterproductive to include more independent variables in the regression model, as the OLS method cannot provide efficient estimators in this case, since the variance of the estimators is no longer minimal. Given these circumstances, it is advisable for practical applications to use efficient models. In this paper, the CAPM and FF3 include all relevant explanatory variables and hence assumptions 1c and 3 are fulfilled.
The assumption of homoscedasticity, explaining that the error terms have constant variances, will be tested via residual plots. Heteroscedasticity, as well as autocorrelation, might influence the standard error of the model parameters. In case of a violation of assumption 4 and 5, the OLS method generally provides to small standard errors. Standard errors are used for calculating confidence levels and conducting significance tests, hence biased standard errors may lead to incorrect conclusions. In
this paper, assumption 4 will be tested via residual plots. The plots should be evenly distributed. For example, a funnel-shaped arrangement of the residuals would suggest a violation of the assumption of homoscedasticity. Moreover, this assumption will be tested via the Breusch-Pagan-Test (Breusch & Pagan, 1979). A significant test result shows that the assumption of homoscedasticity is violated. Furthermore, the autocorrelation will be tested via residual plots. There should not be any recognizable structure within the plots, i.e. a curve.
In the case of multicollinearity, it is not possible to compute the OLS estimator, as one explanatory variable would be a perfect linear combination of the other regressors. To rule out multicollinearity, the correlations between the different independent variables will be analyzed. Moreover, this paper tests for multicollinearity by employing a variance inflation factor (VIF) test, similar to Fahrmeir et al. (2008, p. 171). In case of a VIF test result of VIF > 10, there is multicollinearity. Since the CAPM includes one independent variable, assumption 6 only needs to be tested for the FF3.
Assumption 7 is essential for the tests of the regression parameters. For large sample sizes, this assumption can be considered as fulfilled due to the central limit theorem3 (CLT). Nevertheless, this paper will report on the distribution of the error terms by doing a residual analysis via a quantile-quantile-plot (Q-Q plot), similar to Hartung, Elpelt, and Klösener (2005, p. 845-847). Hereby, the quantiles of the residuals and the quantiles of a normal distribution will be plotted in a coordinate system. If the points lie almost on the same straight line, the assumption of a normal distribution will be met. The residuals are the difference between the estimated values of the dependent variables by regression and the observed values of the dependent variables. Thus, the equation can be described as
𝑢! = 𝑦! − 𝑦! [16]
Here, 𝑢! is the residual and 𝑦! is the estimated value of the dependent variable by the OLS method. Furthermore, this paper considers modified residuals, which are called standardized residuals. As already mentioned, assumptions 4 and 5 will also be tested
3 The CLT is a statistical theory explaining that the mean of all samples from the same population will almost be equal to the mean of the population, given a sufficiently large sample size from a population.
via residuals. Normal residuals are generally autocorrelated and heteroscedastic (Fahrmeir Kneib & Lang, 2009, p. 108), which standardized residuals are not. Hence, this thesis will draw conclusions on standardized residuals. STATA is calculating these standardized residuals.
4.1 CAPM Assumption Testing
In the next step, the aforementioned assumptions will be tested for the CAPM. The analysis is further divided into equally and value-weighted portfolios. The results are exemplified for a randomly chosen uninformed portfolio consisting of 15 equally weighted stocks, called EW15_1, as well as for a randomly chosen portfolio consisting of 30 value-weighted stocks, called VW30_85, which are 2 out of 400 randomly created uninformed portfolios for the regression analysis.
4.1.1 CAPM Assumption Testing for Equally Weighted Portfolio
Figure 3: Linear relationship for EW15_1
-3 0 -2 0 -1 0 0 10 20 Exce ss R et urn o f R a nd om Po rt fo lio -20 -10 0 10 20
Excess Return of the Market
| EW15_1 MKTRF ---+--- EW15_1 | 1.0000 | | MKTRF | 0.5429 1.0000 | 0.0000 |
Figure 3 as well as the STATA output above show us the result of the investigation of an equally weighted portfolio and prove that there is a linear relationship between the independent variable and dependent variable of the CAPM, the excess return of the market portfolio and the excess return of random portfolio respectively. The scatterplot indicates that the values can be described by a straight line. Moreover, the estimated regression line is shown in the graph. To verify the strength of the linear regression, the correlation between the two variables was further determined. The correlation is r = 0.5429 and is significant at the 5% level: p = 0.000 < 0.05. The correlation is medium to strong and further significant. Moreover, the Ramsey (1969) RESET, which results are shown below, yields a non-significant result. F(3, 320) = 2.51 and p = 0.0586. Thus, the RESET provides proof that there is no violation of the linearity assumption. Hence, assumption 1a has been fulfilled.
Ramsey RESET test using powers of the fitted values of EW15_1
Ho: model has no omitted variables F(3, 320) = 2.51 Prob > F = 0.0586
Figure 4: Residual plot for EW15_1
Breusch-Pagan / Cook-Weisberg test for heteroskedasticity Ho: Constant variance
Variables: fitted values of EW15_1
chi2(1) = 2.84 Prob > chi2 = 0.0921
Figure 4 demonstrates that the plots distribute around the value zero, hence the error terms have an expected value of zero 𝐸 𝑢! = 0. Thus, condition 2 has been fulfilled. Moreover, the Breusch-Pagan-Test for heteroscedasticity does not yield a significant test result, as 𝜒! 1 = 2.84 and p = 0.0921. Hence, assumption 4 of homoscedastic error terms has been fulfilled. Moreover, figure 4 shows no recognizable structure in the distribution of the residuals. Thus, there is no autocorrelation and assumption 5 has been met.
-6 -4 -2 0 2 4 st an da rd ize d re si d ua ls -15 -10 -5 0 5 10 fitted values
Figure 5: Distribution of standardized residuals for EW15_1
Figure 6 demonstrates that the assumption of normally distributed standardized residuals is not violated. The distribution of standardized residuals is almost symmetrically. On the left margin, there are indeed outliers. They differ, however, apparently not too far away from the rest of the distribution, so it can be assumed that they do not have a too prominent impact on the tests of the parameters. Moreover, figure 7 is a Q-Q plot and further provides proof that the standardized residuals are normally distributed by plotting the empirical quantiles of standardized residuals on the y-axis and the quantiles of a normal distribution on the x-axis. It is demonstrated that the points lie almost on the same straight line and thus both figures show that assumption 7 has been fulfilled.
0 .1 .2 .3 .4 .5 D en si ty -6 -4 -2 0 2 4 Standardized residuals
