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The Unlevered CAPM: A New Alternative Asset-Pricing Model
for Predicting European Stock Returns?
Jeroen Workel
(s2349736)
University of Groningen
Faculty of Economics and Business
MSc. Finance
Supervisor: Dr. L. Dam
June 2016
ABSTRACT
One of the known downsides of the risk-return relationship of the traditional CAPM is heavily influenced by the leverage effect. To overcome this pitfall, Dam & Qiao (forthcoming) suggest an unlevered CAPM based on unlevered returns and unlevered betas. The unlevered CAPM will eliminate the leverage effect since the unlevered CAPM focusses on (unlevered) asset returns and betas, rather than the firm (levered) equity returns and betas. To assess whether the unlevered CAPM outperforms the traditional asset-pricing models, i.e. the traditional CAPM and the Fama & French (1993) three-factor model, 2,561 listed firms are collected from twelve different countries in the Eurozone. The sample period ranges from January 1991 to December 2015. The results are inconclusive as none of the asset-pricing models clearly outperform the other models. Also, the results do not select a model to be regarded to as most robust asset-pricing model.
Key words: Fama & French (1993) three-factor model, leverage effect, traditional CAPM, unlevered CAPM, unlevered betas, unlevered returns
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I. INTRODUCTION
Over the past decades many asset-pricing models are introduced and criticized by a great deal of scientific studies. The capital asset-pricing model (CAPM) and the Fama & French (1993) three-factor model are often used in finance practise for analysing stock returns. Though, both models are heavily criticised for both their empirical and theoretical power. These pitfalls cause that researchers develop extensions and try to eliminate these downsides for both these asset-pricing models. For instance, Dam & Qiao (forthcoming) present their improvement of the traditional CAPM: the unlevered CAPM. The unlevered CAPM is developed to analyse the leverage affect by disregarding the firm’s financing structure. Dam & Qiao (forthcoming) analyse 7,563 firms located in the United States. The unlevered CAPM tests unlevered returns and unlevered betas, whereas the traditional CAPM focusses on levered returns and levered betas. Unlevered returns could be considered as the return on assets and unlevered betas could be seen as asset betas. As the unlevered CAPM focusses on asset returns, it ignores the financing structure firms have adopted. The traditional CAPM does not disregard the capital structure of the firm, as it uses levered returns to calculated levered betas. The only exception is for cases where firms are financed only by equity rather than by debt. In this full-equity financing case, the unlevered returns and unlevered betas are equal to levered returns and levered betas. Dam and Qiao (2016) compare the performance of their developed asset-pricing model to benchmark asset-asset-pricing models, i.e. the traditional CAPM and the Fama & French (1993) three-factor model and conclude that their unlevered CAPM significantly outperforms both the two well-known asset-pricing models for the United States. This thesis focuses on the European stock market to assess whether the unlevered CAPM also outperforms the benchmark models outside the U.S. Using the methodology of Dam and Qiao (2016) I test whether the unlevered CAPM also outperforms the two benchmark asset-pricing models for European data.
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errors are reduced, the unlevered CAPM theoretically should outperform the traditional asset-pricing models.
This thesis will use the same approach as Dam and Qiao (2016) have introduced in their papers to examine whether the unlevered CAPM holds for the Euro-zone. In their paper they define unlevered returns as the firm’s levered returns multiplied by the lagged leverage ratio of the firm. This lagged leverage ratio is defined as the market value of total equity divided by the market value of total assets. I will perform two regressions using monthly data to analyse and compare the performance of the unlevered CAPM with the traditional CAPM and the Fama and French (1993) three-factor model. To do so, several time-regressions are performed, which will be combined with the Gibbons, Ross, and Shanken (1989) (GRS) test. After the GRS-test, the Fama-MacBeth (1973) two-step procedure is performed where several cross-sectional regressions are used to examine the performance of the models.
The European stock market is assessed to compare the different asset-pricing models. West-European countries are chosen as the majority of these countries have the same currency (euro), which eliminates potential exchange rate difficulties in the data. Moreover, the Eurozone is chosen as this area is not assessed extensively in the existing asset-pricing literature. The literature rather focusses on the U.S. stock market as the data is more available compared to European data. The final dataset contains 2,561 listed European companies, originated in twelve different European countries. The sample period starts in January 1991 to December 2015, which covers a total of 300 months. As there are 3 different asset-pricing models to be assessed, I created three individual portfolios for each of these asset-pricing models. Each of these individual portfolios consists of 25 individual portfolios. An additional portfolio sorted on industry is created to test the robustness of the analysis. This portfolio sort contains 10 portfolios as there are 10 different industry classes used in this thesis.
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unlevered CAPM compared to the traditional CAPM and the Fama & French (1993) three-factor model. As the results are inconclusive for both the performance as the robustness of the asset-pricing models, it is hard to choose one asset-pricing models that outperforms the other models. When following the GRS-test, the Fama & French (1993) three-factor model could be chosen as the best performing asset-pricing model as it passes for two portfolio sorts. However, the results of the Fama & MacBeth (1973) two-step procedure may favour the unlevered CAPM and the traditional CAPM, as these perform better in this test than the Fama & French (1993) three-factor model.
In several ways, this thesis contributes to the existing literature on asset-pricing models. First, it adds a performance assessment of traditional asset-pricing models, i.e. the traditional CAPM and the Fama & French (1993) three-factor model, but also analyses the empirical strength and robustness of these models. Second, the performance regarding robustness and empirical power of the unlevered CAPM is assessed for the European stock market and compared to the benchmark asset-pricing models. Third, the European stock market is analysed which creates new insights on the stock market in the Eurozone. As the literature on performance assessment of asset-pricing models in Europe is not extensive due to limited data availability, this thesis will contribute to the literature by fulfilling this literature gap.
This thesis is structured as follows. Section II provides an overview of relevant background literature on the several asset-pricing models. Section III will introduce the used methodology and states the tested hypothesis. Section IV discusses the data and provides an overview of the descriptive statistics. Section V lists the results of the different asset-pricing models assessment for the different portfolio sorts. These results will be discussed in discussed in section VI. Finally, section VII will provide a conclusion of this thesis.
II. LITERATURE REVIEW
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market. The CAPM shows the risk-return relationship for individual securities or for portfolios. The great advantage of the traditional CAPM is that the model is very simple to apply. However, the model comes with many assumptions which do not hold in the real-world, for example the assumption that trade occurs without taxes.
After the traditional CAPM become a widely accepted asset-pricing model, many researchers were interested in testing the empirical power of the model. Black, Jensen & Scholes (1972) test the empirical justification of the CAPM by performing several empirical tests. For these empirical tests, they collected monthly prices, dividends and adjusted prices for all listed NYSE securities in the period of January 1926 until March 1966. To improve the precision of the estimated betas, they create portfolios based on the security’s betas. The 10% largest betas were assigned to the first portfolio, and so on. They conclude that the beta factor seems to be an important determinant of security returns, resulting in that the CAPM is empirically justified. Another paper, written by Fama & MacBeth (1973) confirms the empirical strength of the CAPM. They test two implications of the two-parameter asset-pricing model (CAPM). First implication is that the expected returns are implied by the fact that investors all hold efficient portfolios in the world of the CAPM. The second implication states that return behaviour through time is implied by the assumption that markets are perfect and frictionless, i.e. no transaction costs. Fama & MacBeth (1973) conclude that their results support both the testable implications of the CAPM. Consequently, they conclude that the two-factor model is functioning well.
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perfectly short-selling mechanism of the CAPM potentially causes the failure of the CAPM-theory. Also Banz (1981) states that the CAPM is misspecified, he finds that the total market equity add to the cross-sectional explanation of stock returns, i.e. small stocks show average returns which are too high given their beta estimates. Large stocks show the opposite relationship. Therefore, Banz (1981) states that the CAPM should include a size-factor to capture this misspecification caused by this size-effect.
All these scepticism on the CAPM resulted in new research on explaining these shortfalls and finding possible solutions to overcome these shortfalls. Fama & French (2004) try to explain why the CAPM lacks to have empirical strength and to find possible causes of this weak empirical strength. They confirm they statement made by Blume & Friend (1973) that one of the main causes why the CAPM is lacking performance is that the model is based on unrealistic simplifying assumptions, i.e. unrestricted risk-free borrowing or unrestricted short selling of risky assets. Though, they admit that all interesting models have unrealistic simplifications. Moreover, they conclude that the CAPM is rather difficult to test empirically. Though, besides all the critic notes, Fama & French (2004) emphasize that the model remains a fundamental concept in both asset pricing as in portfolio theory. The CAPM should function as a fundament for future more sophisticated asset pricing models.
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defined as the difference between small and big stocks in a portfolio with a comparable weighted average book-to-market equity. Third, a new factor called High-minus-Low (HmL) is introduced and captures the risk factor returns related to value. After performing several time-series regressions using the average returns calculated using their three factor model shows that the three factor model does better explain cross-section variation in average stock returns than the traditional CAPM.
Also the development of the three-factor model resulted in new literature on the models’ performance. Rossi (2012) confirms the empirical strength of the three-factor model. He concludes that adding a size-factor (SmB) creates that the asset-pricing model has a greater explanatory power than an asset-pricing model without a size-factor. Hence, the three-factor model of Fama & French (1993) shows an improvement in explanatory power comparing to the traditional CAPM. Also Xie & Qu (2016) use the three-factor model to test the Chinese stock market. They created twenty-five portfolios based on size and value. They conclude that the obtained size and value premiums are significant and fir for the Chinese stock market. Consequently, the three-factor model also shows empirical strength in case for the Chinese markets.
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country-specific factors to the three-factor model drastically reduce the pricing errors compared to global factors.
One of the reactions on the criticism of the Fama & French (1993) three-factor models is the four-factor model developed by Cahart (1997). He extends the Fama & French (1993) three-factor model with a new added momentum three-factor (MOM). This three-factor captures the momentum effect, which is defined as the tendency of an increasing stock to increase also in the future. Also, when stock have a decreasing tendency whether they maintain this decreasing trend in the future. To test his four-factor model, he forms portfolios sorted on one-year lagged returns on mutual funds and estimate the performance of the created portfolios. He concludes that for the first year a momentum effect is observable. However, after the first year this momentum effect of stocks seems to disappear. Recently, Fama & French (2015) introduced an improved version of their own three-factor model. The extended their initial three-factor model with 2 additional factors resulting in a new five-factor asset-pricing model. The model still includes the original three-factor models factors, i.e. an market factor (𝑅𝑀− 𝑅𝐹), an size factor (SmB) and the value factors (HmL), now also factors for
profitability (RmW) and investment (CmA) are included. The RmW factor looks at the differences between portfolio returns of firms with high and low profitability. The CmA factor is defined as the differences between returns of firms with high and low investment levels. They conclude that the five-factor model fails the GRS-test for the U.S. market, though it explains between 71% and 94% of returns in the cross-section for the different portfolios sorted based on the five individual factors.
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This trading over time makes the ICAPM a two-period model as investors create portfolios now, but also take the future developments into account.
Nowadays, the turbulent economic times have created new thoughts for analysing stock returns. For instance, Choi (2013) analyses the relationship of both asset risk and leverage on the equity risk dynamics of values and growth stocks. The rationale of analysing asset betas and leverage is the different reactions for both the value and growth firms. For value firms, the equity betas will increase as both the asset risk and leverage ratios rise, whereas for growth firms the equity betas remain constant as a result of low leverage and low asset betas. Choi (2013) concludes that leverage and the unlevered asset betas alone do not drive the value premium. Moreover, research by Choi & Richardson (forthcoming) confirms that the leverage effect should not be neglected when analysing stock returns. They analyse equity volatility using the asset volatility and assess the performance in the cross-section. One of the conclusions of Choi & Richardson (forthcoming) is that financial leverage heavily influences the equity volatility and that a fundamental difference between assets and equity of firms is observed.
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Dam & Qiao (forthcoming) state two critical assumptions for their unlevered CAPM to hold. First, the market portfolio of assets is mean-variance efficient. This is in line with the modern portfolio theory of Markowitz (1952). Second, the unlevered betas and the risk premium are uncorrelated. When both of these assumptions hold the leverage effect is removed, hence the unlevered CAPM will perform better than the traditional CAPM.
In order to test the Unlevered CAPM, Dam & Qiao (forthcoming) test their model on individual firms and for several portfolio sorts using U.S. firms. Moreover, they compare their results with the traditional asset-pricing models, i.e. the traditional CAPM and the Fama & French (1993) three-factor model for the same sample. Dam & Qiao (forthcoming) conclude that the unlevered CAPM is significantly outperforming the traditional asset-pricing models for the United States stock market. Consequently, the unlevered asset betas explain cross-sectional variation in the unlevered average stock returns for the United States. Also the robustness of the unlevered CAPM is tested using different portfolios sorts, i.e. portfolios sorted on industry sectors.
III. METHODOLOGY
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III.a The unlevered CAPM
The methodology of this thesis is based on the unlevered CAPM developed by Dam & Qiao (forthcoming). The main difference of the unlevered CAPM compared to the traditional CAPM is that unlevered returns and betas are used rather than levered returns and betas. The unlevered excess returns are calculated as levered excess returns multiplied by the lagged firm leverage ratio:
𝑅𝑖,𝑡𝑢 − 𝑟
𝑡= (𝑅𝑖,𝑡− 𝑟𝑡)𝐸𝐴𝑖,𝑡−1
𝑖,𝑡−1, (1)
where 𝑅𝑖,𝑡𝑢 − 𝑟𝑡 is defined as the unlevered excess return for firm 𝑖, at time 𝑡, (𝑅𝑖,𝑡− 𝑟𝑡) as the levered excess return for firm 𝑖, at time 𝑡, 𝐸𝑖,𝑡−1 as the lagged market value of total equity for firm 𝑖, at time 𝑡, and 𝐴𝑖,𝑡−1 as the lagged market value of total assets for firm 𝑖, at time 𝑡. To create a difference between levered and unlevered returns, the subscript 𝑢 is added to indicate unlevered returns. Equation (1) is used to calculate the unlevered returns, which are used to calculate the unlevered betas for each individual firm. To calculate the unlevered betas, equation (2) is used:
𝑅𝑖,𝑡𝑢 − 𝑟
𝑡= 𝛼𝑖+ 𝛽𝑖𝑢(𝑅𝑀,𝑡𝑢 − 𝑟𝑡) + 𝜀𝑖,𝑡, (2)
where 𝛼𝑖 and 𝛽𝑖𝑢 are the coefficients of firm 𝑖 which need to be estimated, 𝑅𝑀,𝑡𝑢 − 𝑟𝑡 is defined
as the unlevered market excess returns, and 𝜀𝑖,𝑡is equal to the asset-specific residual of firm 𝑖, at time t.
III.b Traditional asset-pricing models
For the traditional asset-pricing models the betas are calculated using the equations mentioned in the papers of Black, Jensen & Scholes (1972) and Fama & French (1993):
𝑅𝑖,𝑡 − 𝑟𝑡= 𝛼𝑖+ 𝛽𝑖(𝑅𝑀,𝑡− 𝑟𝑡) + 𝜀𝑖,𝑡; (3)
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III.c Time-series regressions
To obtain for all asset-pricing model’s alphas and betas, three time-series regressions will be performed on each individual asset. The times-series regressions will be based on equation (2) for the unlevered CAPM, equation (3) for the traditional CAPM and equation (4) for the Fama & French (1993) three-factor model. Hence, for the unlevered CAPM, the unlevered excess returns are regressed on the market excess returns; for the traditional CAPM, the levered excess returns are regressed on the market excess returns. Also, for the Fama & French (1993) three factor model, the levered excess returns are regressed on the market, SmB and HmL factors.
III.d GRS-test
The GRS test is created by Gibbons, Ross & Shanken (1989) and is used to test whether the estimated alphas, the intercept coefficients from the multiple regression models, are jointly zero. To explain the relationship between stock returns and factor returns, I will use a multivariate linear panel regression. According to the methodology of Gibbons et al. (1989), the main point of the GRS-test is to test whether the obtained alphas are jointly zero and significant. In cases when the intercepts are jointly zero and significant the GRS-test indicates whether the asset-pricing model does perform well. Transforming the GRS-test into a hypothesis, the null hypothesis of the GRS test states that the intercepts of the asset-pricing models are jointly and significantly equal to zero:
𝐻0: 𝛼 = 0, ∀𝑖 = 1, … , 𝑁 (5)
where 𝛼 is the alpha of firm 𝑖 or portfolio 𝑝.
The calculated alphas for each of the individual asset-pricing model can be seen as the pricing error of these asset-pricing models. When a model is observing any pricing-error, the null hypothesis should be rejected. Hence, as the GRS test states that the alphas are jointly and statistically equal to zero, a good asset-pricing model do not have the problem of pricing errors. By performing time-series regressions using equations (2), (3) and (4), it is possible to examine whether the asset-pricing models do show any pricing error.
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𝐹𝐺𝑅𝑆= 𝑇(𝑇−𝑁−1)𝑁(𝑇−2) [𝛼̂
′∑̂−1𝛼̂
1+𝜃2 ], (6)
𝜃̂2 = 𝑟/𝑠, (7)
where 𝐹𝐺𝑅𝑆 is the F-statistic of the GRS test, T is the number of observations in the time-series on returns, N is the number of firms, 𝛼̂ are the alphas of the portfolios, ∑̂−1 is the unbiased residual covariance matrix, 𝑟 is the mean of the excess returns of each individual portfolios and 𝑠 is the sample standard deviation of the excess returns on each individual portfolio.
The GRS test, the null hypothesis of no pricing errors is rejected in case the F-statistic is large and insignificant. When the null hypothesis is rejected, the asset-pricing model does not pass the GRS test, which indicates that the model shows pricing errors which are significantly different form zero. Consequently, the asset-pricing model is not able to price assets. On the contrary, when the F-statistic is small and insignificant, the null hypothesis cannot be rejected. In this case, the pricing errors are jointly equal to zero, resulting in that the asset-pricing model passes the GRS-test. This indicates that the asset-pricing model is performing well as it is able to price assets. One of the assumptions for performing the GRS test is that a balanced dataset is needed. As the panel dataset for individual firms is unbalanced, whereas the panel dataset for the constructed portfolios is balanced, the test is only applicable for the constructed portfolios.
III.e Cross-sectional regressions
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cross-sectional variations in average unlevered stock returns. The cross-sectional performance of the asset-pricing model is also reflected when the intercept, or alpha, is equal to zero.
𝐸(𝑅𝑖,𝑡− 𝑟𝑡) = 𝛼𝑖+ 𝛽𝑖′𝜆 (8)
where 𝛼𝑖 is the alpha (intercept) of asset 𝑖, 𝛽𝑖′ is the estimated beta of asset 𝑖 and 𝜆 represents
the factor prices. The test is also performed for the individual firm as for the portfolios on a monthly basis.
To provide a clear overview of all the performed regressions, the cross-sectional fit of each regression will be visualized. The actual average returns are plotted against the average returns predicted by the asset-pricing models. In the situation that all the observations are evenly centred around the 45-degree, the model is performing optimal as the pricing errors are low. Therefore, the 𝑅2 of each of the performed regressions is included in the plotted graphs.
III.f Portfolios
As this thesis examines three different asset-pricing models, three different portfolio sets will be created using the calculated unlevered betas. Each of these sorts contains 25 portfolios, where portfolio 1 consists of firms with low betas and portfolio 25 includes firms with high betas. To increase the creditability and robustness check of this analysis, another portfolio set based on the firm’s industry is added. In this portfolio sorted on industry sectors 10 different industry classes are used, as the industry consists of 10 different sectors. An overview of these 10 different sectors is provided in Appendix C2.
IV. DATA
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To test the unlevered CAPM, I need monthly data on stock returns, stock prices and the European stock market index. I selected the Euro Stoxx 50 as a proxy for this market index as this market index includes the 50 largest listed European companies. Also, this market index is known to be one of the most liquid market indices for the Eurozone, which makes it an ideal benchmark for the aggregate stock market. To retrieve all these data, I consult the Thomson Reuters’ DataStream database. The needed factors of the Fama & French (1993) three-factor model, SmB and HmL, are found on the personal website of Kenneth R. French. This website also provides an appropriate risk-free interest rate for the Eurozone on a monthly basis. Though, this risk-free interest rate is denominated in US Dollars, which results in that this risk-free rates needs to be transformed into Euros using the Euro/Dollar exchange rate. The monthly Euro/Dollar exchange rate is obtained via the Thomson Reuters’ DataStream database. The DataStream database is also used to collect individual firm data on the book value of total assets, the book value of total equity and the number of total outstanding shares.
One of the limitations of the DataStream database is that the book values do not change on a yearly basis, but rather change on a monthly basis. To transform this yearly data into monthly data, I use the transformation method introduced by Dam & Qiao (forthcoming). They state that variation in the leverage ratio is mainly caused by variation in the value for equity rather than variation in the value of debt. Consequently, the assumption of a constant debt level will be valid for this thesis and the transformation of yearly into monthly data is possible.
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rates for the firm’s individual monthly returns. To obtain the unlevered excess returns, the levered excess returns are multiplied by the firm’s lagged leverage ratio.
For the aggregate market, the market return is calculated by summing all the firm’s individual returns. The market excess return is the market return reduced by the risk-free interest rate. To calculate the value weighted market leverage ratio, the sum of the market value of total equity is divided by the sum of the total market value of total assets. Finally, the unlevered market excess returns are obtained by multiplying the market excess returns with the lagged market leverage ratio.
Several data is omitted from the dataset in order to create a cleaned and useable dataset. First, any observation that misses a value for any of the variables are removed, i.e. missing (unlevered) returns, missing book values for equity or assets. Second, as DataStream repeats the last observation for dead or delisted firms, these repetitive observations are removed as these create misleading data. Moreover, firms which do not report data on returns for two months in a row are regarded as a delisting or dead and hence are removed from the dataset. Also, firms which are active for less than 60 consecutive months are eliminated from the data set, as they create strange firm’s betas and hence are not representative for this analysis. Third, to eliminate potential outliers in excess returns, all returns which are lower than 1% or higher than 99% are dropped. After the dataset is cleaned, the unbalanced panel dataset consists of 2,561 listed European firms, from twelve different European countries. Since the personal website of Kenneth R. French only reports Euro-specific factors from June 1990, the sample period is a total of 300 months, ranging from January 1991 until December 2015.
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Table 1. An overview of means and standard deviations of the key variables, Eurozone for the sample period January 1991 – December 2015
Variable Observations Mean S.D.
Aggregate market # Observations Mean S.D
Market excess returns (%) 300 0.41 5.50
Market leverage (%) 300 49.66 21.69
Market unlevered excess returns (%) 300 0.12 3.01
Risk-free interest rate 300 0.19 0.16
SmB (%) 300 0.01 2.01
HmL (%) 300 0.35 2.26
Individual firms
Levered excess returns (%) 458,774 0.38 11.14
Leverage ratio (%) 458,774 45.33 28.89
Unlevered excess returns (%) 458,774 0.02 6.24
Book value of total equity (million €) 458,774 1.06 4.48
Book value of total assets (million €) 458,774 8.98 69.30
Market value of total equity (million €) 458,774 7.28 291.00
Market value of total assets (million €) 458,774 15.20 300.00
This table shows the descriptive statistics of monthly data collected from January 1991 to December 2015 for 2,561 European companies. Source: Thomson Reuter’s DataStream database.
Also, both the market values of total equity and total assets show quite high standard deviation. This is created by the fact that the market value of equity is calculated using the stock price times the number of outstanding shares for each individual firm. As the market value of total assets is calculated as the sum of the market value of total equity and the book value of debt, also explains the standard deviation of market value of total assets. These high standard deviations area caused as there are quite some differences in the market values for equity and assets across firms.
V. RESULTS
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To test the robustness of these results and to account for criticism that the performance of asset-pricing models highly depend on how portfolios are sorted, three additional portfolio sorts are introduced. The results of these additional portfolio sorts are introduced after the results unlevered CAPM. Consequently, the results for the traditional CAPM and the Fama & French (1993) three-factor model and the industry portfolio sorts are mentioned after the unlevered CAPM.
V.a Unlevered CAPM portfolio sort
The unlevered portfolio sort is sorted based on the unlevered betas and contains 25 equally sized individual portfolios. Since this portfolio sort uses unlevered betas, it examines whether unlevered betas are able to explain the average unlevered returns. The descriptive statistics for this portfolio sort are mentioned in Appendix A1. Appendix A1 shows that the means and the standard deviations for the unlevered excess returns are lower compared to the excess returns. The low standard deviation for the unlevered excess returns indicates that the unlevered excess returns do vary less than the levered excess returns. Moreover, the mean leverage ratios range from 2% to 95%, with standard deviations ranging from 1% to 32%, which indicates that the leverage ratios also contributes to the variations. All these facts are in line with the hypothesis of this thesis.
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Table 2. Alphas, betas and GRS-test for 25 portfolios sorted on Unlevered CAPM betas, Eurozone January 1991 – December 2015
Pf# Unlevered CAPM Traditional CAPM Fama & French (1993)
GRS: F=1.33 GRS: F=0.88 GRS: F=0.81 𝛼(%) 𝛽𝑚𝑎𝑟𝑘𝑒𝑡 𝑅2 𝛼(%) 𝛽𝑚𝑎𝑟𝑘𝑒𝑡 𝑅2 𝛼(%) 𝛽𝑚𝑎𝑟𝑘𝑒𝑡 𝛽𝑆𝑚𝐵 𝛽𝐻𝑚𝐿 𝑅2 1 -0.06 0.05** 0.02 -0.18 0.08* 0.01 -0.16 0.03 0.42*** -0.4 0.04 2 0.00 0.02*** 0.30 -0.02 0.49*** 0.37 -0.09 0.47*** 0.14 0.21** 0.38 3 0.00 0.06*** 0.64 0.08 0.99*** 0.71 -0.02 0.97*** 0.26** 0.30*** 0.72 4 0.00 0.09*** 0.65 0.17 1.04*** 0.74 0.08 1.02*** 0.23** 0.26*** 0.75 5 -0.02 0.12*** 0.55 -0.06 1.00*** 0.65 -0.20 1.01*** -0.08 0.37*** 0.67 6 0.00 0.15*** 0.55 0.04 0.98*** 0.73 0.00 0.99*** -0.10 0.11 0.73 7 0.05 0.22*** 0.40 -0.03 0.56*** 0.42 -0.15 0.54*** 0.17 0.34*** 0.45 8 0.01 0.21*** 0.30 0.01 0.72*** 0.47 -0.08 0.71*** 0.06 0.26** 0.48 9 -0.03 0.25*** 0.45 -0.06 0.87*** 0.58 -0.12 0.86*** 0.10 0.16 0.58 10 0.00 0.28*** 0.37 -0.04 0.64*** 0.51 -0.14 0.65*** -0.01 0.28*** 0.53 11 0.15** 0.35*** 0.50 0.38* 0.65*** 0.52 0.30 0.63*** 0.22** 0.25*** 0.53 12 0.09 0.33*** 0.54 0.12 0.68*** 0.66 0.04 0.67*** 0.11 0.22*** 0.67 13 0.07 0.41*** 0.52 0.23 0.88*** 0.66 0.09 0.88*** -0.03 0.40*** 0.69 14 0.06 0.48*** 0.69 0.11 0.79*** 0.73 0.08 0.78*** 0.14* 0.09 0.74 15 -0.25 0.55*** 0.08 -0.36 0.69*** 0.20 -0.47 0.65*** 0.41* 0.32* 0.22 16 -0.08 0.50*** 0.59 -0.13 1.01*** 0.74 -0.21 1.00*** 0.14 0.23*** 0.74 17 0.24 0.67*** 0.23 0.49 0.88*** 0.35 0.61 0.85*** 0.20 -0.33** 0.36 18 -0.03 0.70*** 0.45 -0.21 0.84*** 0.51 -0.30 0.80*** 0.41*** 0.28** 0.53 19 0.13 0.76*** 0.52 0.28 0.79*** 0.56 0.35 0.77*** 0.12 -0.17* 0.56 20 0.19 0.81*** 0.38 0.19 0.75*** 0.43 0.09 0.76*** -0.08 0.26** 0.44 21 0.05 0.98*** 0.68 0.07 1.00*** 0.69 0.06 0.97*** 0.22* 0.07 0.69 22 -0.13 1.08*** 0.21 -0.13 0.83*** 0.27 -0.06 0.72*** 1.00*** -0.12 0.31 23 0.35 1.30*** 0.21 0.18 0.91*** 0.31 0.03 0.92*** -0.09 0.41** 0.32 24 -0.03 1.39*** 0.28 -0.20 0.86*** 0.27 -0.05 0.81*** 0.45* -0.38* 0.29 25 -0.52 1.89*** 0.49 -0.66 1.22*** 0.43 -0.48 1.20*** 0.10 -0.48** 0.44 This table reports the pricing errors (𝜶), betas and 𝑹𝟐s for the three asset-pricing models. 25 portfolios are sorted using the
unlevered beta. Portfolio 1 includes stocks with the lowest unlevered beta, portfolio 25 consists of stocks with the highest unlevered beta. Monthly data is used from January 1991 to December 2015. The GRS-test reports an F-statistic, which is reported at the top of the table. Significance is reported using * p<0.10, ** p<0.05, *** p<0.01. Pf# represents the portfolio number. Sources: Thomson Reuter’s DataStream database and K. French personal website.
Besides the insignificant alphas, all the market premiums are also insignificant, which indicates poor performance. Also, the SmB and HmL factors of the Fama & French (1993) three-factor model are insignificant, meaning that they are not priced. Moreover, the SmB factor is negative, which indicates the poor performance of the three-factor model on the unlevered CAPM beta sort.
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Table 3. Cross-sectional regressions and GRS-test for 25 portfolios sorted on Unlevered CAPM betas
Panel A: Portfolios sorted on Unlevered CAPM betas
Model 𝛼(%) 𝜆𝑚𝑎𝑟𝑘𝑒𝑡 (%) λSmB (%) λHmL (%) Avg. 𝑅2 GRS
Unlevered CAPM 0.05 0.05 0.24 1.33
(1.12) (0.20)
Traditional CAPM 0.06 0.36 0.10 0.88
(0.28) (0.86)
Fama & French (1993) 0.13 0.33 -0.24 0.12 0.26 0.81
three-factor model (0.53) (0.80) (-0.64) (0.30)
This table reports the results of the Fama & MacBeth (1973) cross-sectional regressions. The t-statistics are mentioned in parentheses. Avg 𝑹𝟐 reports the average 𝑹𝟐 of the performed cross-sectional regressions. The last column reports the F-statistic of the GRS test. Significance is reported using: * p<0.10, ** p<0.05, *** p<0.01. Source: Thomson Reuter’s DataStream database.
Figure 1. Cross-sectional fit of actual and predicted unlevered returns using the unlevered CAPM portfolio sort
This figure shows a scatterplot of the actual and predicted unlevered returns for 25 portfolios sorted on unlevered betas. The reported 𝐑𝟐 represents the cross-sectional fit of the OLS regression of actual on predicted average returns. GRS represents the F-statistic of the GRS test. The significance of the GRS F-statistic is reported using: * p<0.10, ** p<0.05, *** p<0.01.
Figure 1 shows that both the traditional CAPM (𝑅2 = 6%) and the Fama & French (1993) three-factor model (𝑅2 = 15%) observe a better fit compared to the unlevered CAPM; however still the cross-sectional fit of these models is still rather weak.
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the hypothesis, which states that the unlevered CAPM outperforms the traditional asset-pricing models.
V.b Traditional CAPM portfolio sort
The traditional CAPM portfolio sort includes 25 individual 25 portfolios of equal size. Table 5 displays the results of time-series regressions of these 25 portfolios, all sorted on levered betas. The majority of the pricing errors are insignificant and close to zero. Though, the unlevered CAPM shows two alphas significant at the 1%-level. Besides, almost all betas are significant and show a nice pattern for both the traditional CAPM and Fama & French (1993) three-factor model from low betas for portfolio 1 to high betas for portfolio 25. For the Fama & French three-factor model, close to half of the SmB-factor betas are significant at the 1%-level, which also applies for the HmL-factor betas for the similar three-factor model. Looking at the GRS-test, both the unlevered CAPM and the traditional CAPM show a significant F-statistic (both at the 10%-level).
Table 4. Cross-sectional regressions and GRS-test for 25 portfolios based on three alternative portfolios sorts
Panel A: 25 portfolios sorted on traditional CAPM betas
Model 𝛼(%) 𝜆𝑚𝑎𝑟𝑘𝑒𝑡 (%) λSmB (%) λHmL (%) Avg. 𝑅2 GRS
Unlevered CAPM -0.05 0.33 0.23 1.92*
(-0.75) (1.10)
Traditional CAPM 0.12 0.26 0.15 1.43*
(0.80) (0.71) Fama & French (1993)
three-factor model 0.25 (1.63) 0.20 (0.54) -0.44 (-1.38) 0.21 (0.50) 0.27 1.35
Panel B: 25 portfolios based on Fama & French (1993) sorts
Model 𝛼(%) 𝜆𝑚𝑎𝑟𝑘𝑒𝑡 (%) λSmB (%) λHmL (%) Avg. 𝑅2 GRS
Unlevered CAPM 0.08 -0.07 0.36 3.22***
(1.62) (-0.23)
Traditional CAPM 0.17 0.60 0.20 5.10***
(0.66) (1.36) Fama & French (1993)
three-factor model -0.36 (-1.36) 0.53 (1.23) 1.76*** (4.06) 1.13** (2.39) 0.37 4.92*** Panel C: 25 portfolios sorted on Industry
Model 𝛼(%) 𝜆𝑚𝑎𝑟𝑘𝑒𝑡 (%) λSmB (%) λHmL (%) Avg. 𝑅2 GRS
Unlevered CAPM 0.32* -0.37 0.18 2.22**
(1.84) (-1.14)
Traditional CAPM 1.88** -1.82* 0.14 2.36**
(2.33) (-1.78) Fama & French (1993)
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Table 5. Alphas, betas and GRS-test for 25 portfolios sorted on Traditional CAPM betas, Eurozone January 1991 – December 2015
Pf# Unlevered CAPM Traditional CAPM Fama & French (1993)
GRS: F=1.92* GRS: F=1.43* GRS: F=1.35 𝛼(%) 𝛽𝑚𝑎𝑟𝑘𝑒𝑡 𝑅2 𝛼(%) 𝛽𝑚𝑎𝑟𝑘𝑒𝑡 𝑅2 𝛼(%) 𝛽𝑚𝑎𝑟𝑘𝑒𝑡 𝛽𝑆𝑚𝐵 𝛽𝐻𝑚𝐿 𝑅2 1 -0.16* -0.02 0.00 -0.33 0.00 0.00 -0.22 -0.03 0.22 -0.27 0.02 2 0.04 0.04** 0.02 0.01 0.09*** 0.04 0.03 0.07** 0.19** -0.05** 0.06 3 0.12* 0.08*** 0.03 0.38 0.18*** 0.06 0.40* 0.14*** 0.42*** -0.01*** 0.09 4 0.05 0.10*** 0.11 0.17 0.25*** 0.16 0.10 0.22*** 0.28*** 0.21*** 0.21 5 -0.47 0.22 0.01 -0.80* 0.33*** 0.05 -0.86* 0.28*** 0.52** 0.21** 0.06 6 0.01 0.16*** 0.18 0.06 0.31*** 0.23 0.05 0.27*** 0.34*** 0.06*** 0.27 7 -0.01 0.11*** 0.14 0.02 0.32*** 0.22 -0.09 0.32*** 0.02 0.30 0.25 8 -0.09 0.22*** 0.11 -0.12 0.42*** 0.22 -0.09 0.37*** 0.45*** -0.01*** 0.25 9 0.40* 0.27*** 0.03 0.59 0.47*** 0.10 0.66 0.45*** 0.22 -0.16 0.11 10 0.17 0.31*** 0.19 0.50** 0.45*** 0.30 0.44* 0.45*** 0.02 0.18 0.31 11 -0.11 0.26*** 0.19 -0.30 0.48*** 0.27 -0.28 0.47*** 0.14 -0.05 0.27 12 0.15* 0.33*** 0.29 0.44** 0.52*** 0.37 0.46** 0.55*** -0.31*** -0.10*** 0.39 13 0.05 0.55*** 0.15 0.09 0.65*** 0.26 -0.01 0.61*** 0.32* 0.27* 0.27 14 0.15* 0.34*** 0.28 0.43* 0.58*** 0.38 0.41* 0.56*** 0.10 0.06 0.38 15 0.01 0.24*** 0.27 -0.05 0.66*** 0.48 -0.14 0.65*** 0.14 0.23 0.49 16 0.28* 0.66*** 0.35 0.42 0.67*** 0.41 0.37 0.64*** 0.33** 0.17** 0.43 17 -0.16 0.69*** 0.13 -0.29 0.77*** 0.24 -0.26 0.67*** 0.93*** 0.01*** 0.28 18 -0.10 0.45*** 0.36 -0.30 0.70*** 0.40 -0.32 0.67*** 0.28** 0.07** 0.41 19 0.16 0.82*** 0.34 0.33 0.80*** 0.40 0.24 0.77*** 0.28* 0.26* 0.41 20 0.28 1.31*** 0.18 0.14 0.95*** 0.28 -0.05 0.98*** -0.20 0.46 0.29 21 -0.13*** 0.35*** 0.60 -0.26 0.94*** 0.67 -0.17 0.94*** 0.05 -0.21 0.67 22 -0.01 0.42*** 0.74 -0.11 1.03*** 0.72 -0.06 1.01*** 0.15 -0.09 0.72 23 -0.06 0.65*** 0.63 -0.20 1.07*** 0.67 -0.22 1.03*** 0.37*** 0.10*** 0.68 24 -0.03 0.27*** 0.80 -0.05 1.18*** 0.82 -0.06 1.15*** 0.26*** 0.04*** 0.82 25 -0.15*** 0.34*** 0.53 -0.51** 1.30*** 0.76 -0.57** 1.30*** -0.02 0.14 0.77 This table reports the pricing errors (𝜶), betas and 𝑹𝟐s for the three asset-pricing models. 25 portfolios are sorted using the
traditional CAPM beta. Portfolio 1 includes stocks with the lowest beta, portfolio 25 consists of stocks with the highest bet a. Monthly data is used from January 1991 to December 2015. The GRS-test reports an F-statistic, which is reported at the top of the table. Significance is reported using * p<0.10, ** p<0.05, *** p<0.01. Pf# represents the portfolio number. Sources: Thomson Reuter’s DataStream database and K. French personal website.
This indicates that for both the unlevered and the traditional CAPM the pricing errors are not jointly equal to zero at the 10%-significance level. The only model that passes the GRS-test is the Fama & French (1993) three-factor model.
For the Fama & MacBeth (1973) two-step procedure, Panel A in Table 4 shows that all models have an insignificant alpha. Again, the unlevered CAPM shows the lowest pricing error compared to the other asset-pricing models. Also all the market premiums are insignificant, but show a reasonable market premium. Moreover, the Fama & French (1993) SmB and HmL factors both are insignificant again, indicating that both factors are not priced. The negative sign for the SmB-factor indicates poor performance of the three-factor model.
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traditional CAPM is the lowest of all three models as 𝑅2 = 6%, and The Fama & French (1993) three-factor model shows an 𝑅2 of 9%. In case for the traditional CAPM portfolio sort, there is no asset-pricing model which clearly outperforms the other models, therefore the hypothesis could not be accepted nor rejected.
V.c Fama & French (1993) three-factor model
The Fama & French (1993) three-factor model portfolio sort is based on levered betas for the factors SmB and HmL. The portfolio sort contains 25 individual portfolios of approximately the same size. The results of the time-series regressions are reported in Table 6. For each asset-pricing model half of the pricing errors are significant. For the unlevered CAPM the pricing errors are nearly to zero. On the contrary, for the other asset-pricing models, some of the pricing errors are closer to one than to zero. Though, for all asset-pricing models the betas are highly significant at the 1%-level. Also, for the Fama & French (1993) three-factor model, half of the SmB factor-betas are highly significant at the 1%-level, indicating the reliability of the predictive power of the Fama & French (1993) three-factor model. For the HmL factors, the factor-betas are all significant at the 10%-level, except for 8 observations. Looking at the GRS-test, all models fail the test as all F-statistics are highly significant at the 1%-level. Hence, for all asset-pricing models the alphas are not jointly equal to zero, which indicates that pricing errors do exist for all the asset-pricing models. Table 4, Panel B reports the results for the Fama & MacBeth (1973) cross-sectional test.
Table 4 Panel B shows that all the pricing errors are significant again, and that the unlevered CAPM reports an alphas closest to zero (0.08%). The observed market premiums are all insignificant and show risk premiums which are reasonable. However, the observed market premium for the unlevered CAPM is quite low (-0.07%). The SmB and HmL factor-betas of the Fama & French (1993) three-factor model both show positive values and are significant at 1%-level (SmB) and 5%-level (HmL) respectively. This implies that both the SmB- and HmL factors are priced factors. Appendix B3 plots the predicted average (un)levered returns on the actual average (un)levered returns. The unlevered CAPM shows the poorest cross-sectional fit (𝑅2 = 2%) compared to traditional CAPM (𝑅2 = 9%). Although the Fama & French (1993)
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Table 6. Alphas, betas and GRS-test for 25 portfolios sorted on Fama & French (1993) betas, Eurozone January 1991 – December 2015
Pf# Unlevered CAPM Traditional CAPM Fama & French (1993)
GRS: F=3.22*** GRS: F=5.10*** GRS: F=4.92*** 𝛼(%) 𝛽𝑚𝑎𝑟𝑘𝑒𝑡 𝑅2 𝛼(%) 𝛽𝑚𝑎𝑟𝑘𝑒𝑡 𝑅2 𝛼(%) 𝛽𝑚𝑎𝑟𝑘𝑒𝑡 𝛽𝑆𝑚𝐵 𝛽𝐻𝑚𝐿 𝑅2 1 -0.09 0.18*** 0.06 -0.28 0.21*** 0.05 -0.33 0.21*** 0.03 0.13 0.05 2 -0.02 0.34*** 0.28 0.07 0.40*** 0.21 0.11 0.39*** 0.10 -0.10 0.22 3 0.01 0.24*** 0.23 0.16 0.34*** 0.19 0.08 0.32*** 0.20* 0.19* 0.21 4 0.10* 0.20*** 0.36 0.48*** 0.41*** 0.37 0.43** 0.38*** 0.26*** 0.14* 0.40 5 0.05** 0.05*** 0.16 0.91*** 0.44*** 0.34 0.79*** 0.40*** 0.38*** 0.31*** 0.39 6 -0.23 0.50*** 0.29 -0.36 0.52*** 0.33 -0.39 0.50*** 0.12 0.08 0.34 7 -0.01 0.39*** 0.41 -0.03 0.50*** 0.43 -0.09 0.47*** 0.32*** 0.14* 0.45 8 0.09 0.29*** 0.38 0.29 0.48*** 0.44 0.23 0.45*** 0.32*** 0.14* 0.47 9 0.17** 0.25*** 0.31 0.72*** 0.51*** 0.47 0.65*** 0.49*** 0.15 0.18** 0.48 10 0.06*** 0.05*** 0.26 0.82*** 0.49*** 0.42 0.72*** 0.45*** 0.40*** 0.25*** 0.46 11 -0.36*** 0.60*** 0.42 -0.55*** 0.54*** 0.44 -0.55*** 0.53*** 0.04 -0.01 0.44 12 0.05 0.42*** 0.49 0.20 0.58*** 0.49 0.19 0.55*** 0.23** 0.02 0.50 13 0.11* 0.32*** 0.49 0.33* 0.55*** 0.52 0.27 0.52*** 0.32*** 0.15* 0.54 14 0.14*** 0.18*** 0.36 0.57*** 0.52*** 0.51 0.51*** 0.49*** 0.27*** 0.14** 0.53 15 0.09*** 0.08*** 0.27 0.76*** 0.59*** 0.51 0.62*** 0.55*** 0.39*** 0.35*** 0.57 16 -0.20* 0.70*** 0.55 -0.29 0.65*** 0.57 -0.30 0.63*** 0.24** 0.03 0.58 17 0.11 0.45*** 0.58 0.27 0.67*** 0.61 0.25 0.63*** 0.31*** 0.06 0.62 18 0.10** 0.27*** 0.48 0.57*** 0.66*** 0.58 0.47** 0.63*** 0.28*** 0.26*** 0.61 19 0.07** 0.16*** 0.50 0.64*** 0.62*** 0.61 0.55*** 0.59*** 0.32*** 0.21*** 0.63 20 0.01 0.07*** 0.37 0.51** 0.73*** 0.59 0.36* 0.71*** 0.28*** 0.37*** 0.62 21 0.07 1.11*** 0.27 -0.07 0.90*** 0.39 -0.24 0.92*** -0.06 0.42** 0.40 22 0.05 0.40*** 0.73 0.21* 0.88*** 0.86 0.19 0.85*** 0.26*** 0.05 0.87 23 0.04* 0.22*** 0.74 0.37*** 0.95*** 0.91 0.33*** 0.93*** 0.14*** 0.09** 0.91 24 0.05** 0.16*** 0.67 0.39*** 0.95*** 0.85 0.33** 0.95*** 0.05 0.17*** 0.85 25 0.01 0.13*** 0.34 0.49** 0.93*** 0.64 0.36 0.89*** 0.35*** 0.32*** 0.66 This table reports the pricing errors (𝜶), betas and 𝑹𝟐s for the three asset-pricing models. 25 portfolios are sorted using the
Fama & French (1993) three-factor model SmB and HmL factors. The portfolios are sorted on size and book-to-market values. Monthly data is used from January 1991 to December 2015. The GRS-test reports an F-statistic, which is reported at the top of the table. Significance is reported using * p<0.10, ** p<0.05, *** p<0.01. Pf# represents the portfolio number. Sources: Thomson Reuter’s DataStream database and K. French personal website.
Overall, all the asset-pricing models do empirically fail the GRS-test. Also the Fama & MacBeth (1973) procedure does not favour one particular asset-pricing model as the unlevered CAPM has the lowest pricing error, but Fama & French (1993) three-factor model observes the highest cross-sectional fit. Consequently, the results are inconclusive for the Fama & French (1993) three-factor portfolio sort. Therefore, I cannot accept nor reject the hypothesis of this thesis
V.d Industry
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Table 7. Alphas, betas and GRS-test for 25 portfolios sorted on Industries, Eurozone January 1991 – December 2015
Pf# Unlevered CAPM Traditional CAPM Fama & French (1993)
GRS: F=2.22** GRS: F=2.36** GRS: F=2.26** 𝛼(%) 𝛽𝑚𝑎𝑟𝑘𝑒𝑡 𝑅2 𝛼(%) 𝛽𝑚𝑎𝑟𝑘𝑒𝑡 𝑅2 𝛼(%) 𝛽𝑚𝑎𝑟𝑘𝑒𝑡 𝛽𝑆𝑚𝐵 𝛽𝐻𝑚𝐿 𝑅2 1 0.33 0.93*** 0.32 0.48 0.83*** 0.39 0.45 0.76*** 0.62*** 0.13 0.42 2 0.11* 0.49*** 0.69 0.27* 0.75*** 0.70 0.23 0.73*** 0.21** 0.15** 0.71 3 -0.70*** 1.19*** 0.39 -1.03** 1.00*** 0.39 -0.93* 0.98*** 0.15 -0.26 0.40 4 -0.03 0.48*** 0.18 0.01 0.98*** 0.34 -0.17 1.00*** -0.18 0.47** 0.35 5 0.19* 0.79*** 0.59 0.41** 0.74*** 0.60 0.47* 0.74*** 0.01 -0.15* 0.61 6 0.02 0.63*** 0.75 -0.06 0.93*** 0.76 -0.04 0.90*** 0.31*** -0.01 0.77 7 -0.07 0.63*** 0.14 0.01 0.67*** 0.26 -0.09 0.62*** 0.46** 0.31** 0.29 8 -0.30 1.21*** 0.35 -0.45 1.01*** 0.43 -0.29 0.94*** 0.54*** -0.40** 0.46 9 0.05 0.79*** 0.37 0.11 0.88*** 0.37 0.27 0.84*** 0.33* -0.40** 0.39 10 -0.08 0.39*** 0.32 -0.30 0.80*** 0.51 -0.36 0.82*** -0.14 0.16 0.51 This table reports the pricing errors (𝜶), betas and 𝑹𝟐s for the three asset-pricing models. 25 portfolios are sorted using the
10 industry classes. For industry class specification, see Appendix C1. Portfolio 1 includes stocks with the lowest beta, portfolio 25 consists of stocks with the highest beta. Monthly data is used from January 1991 to December 2015. The GRS -test reports an F-statistic, which is reported at the top of the table. Significance is reported using * p<0.10, ** p<0.05, *** p<0.01. Pf# represents the portfolio number. Sources: Thomson Reuter’s DataStream database and K. French personal website.
For the traditional CAPM and the Fama & French (1993) three-factor model, the majority of the alphas are insignificant; however the pricing errors which are significant are closer to one. Besides, all the betas are large and highly significant at the 1%-level. Also the majority of the SmB and HmL factor-betas are significant, which implies the explanatory power of the model. However, some of these SmB and HmL factors have negative signs, which is not in line with the Fama & French (1993) three-factor model. The GRS-test results show that all the asset-pricing models fail the GRS-test at the 5% significance level.
Table 4, Panel C shows the results of the cross-sectional regressions of the Fama & MacBeth (1973) methodology. Panel C displays significant alphas for all the asset-pricing models. Comparing the values of these alphas, for the unlevered CAPM the alpha is close to zero, whereas for the traditional CAPM and the Fama & French (1993) three-factor model the alphas are rather high, 1.88% and 1.54% respectively. For the market premiums, in case of the traditional CAPM, the market premium is significant at the 10%-level, but the market premium has a negative sign. Also for the traditional CAPM and the Fama & French (1993) three-factor model the market premiums are negative and insignificant. When focussing on the Fama & French SmB and HmL factors, both are insignificant which indicates that they are not priced factors. However, both are positive, which is in line with the model.
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cross-sectional regressions of the Fama & MacBeth (1973) procedure. The traditional CAPM has the best cross-sectional fit (𝑅2 = 27%), followed by the unlevered CAPM (𝑅2 = 15%). The Fama & French (1993) three-factor model shows even a non-existing fit between the actual average excess returns and the actual predicted excess returns as 𝑅2 is equal to 0%.
Overall, all the asset-pricing models do perform poor on the industry portfolio sort. Since, all the models show significant alphas and large negative insignificant market premiums all the models fail the GRS-test. Consequently, there is no evidence that the unlevered CAPM does outperform the other benchmark models. Hence, the hypothesis cannot be accepted nor rejected.
V.e Robustness of the asset-pricing models
In general, the results make it hard to conclude which asset-pricing model is the most robust among the three models. For the unlevered CAPM, the pricing errors are always close to zero and are insignificant for three portfolios cases. Looking at market premiums, for the Fama & French (1993) sort and the industry portfolio sort these are negative and insignificant, which indicates poor performance. However, for the unlevered CAPM and the traditional CAPM portfolio sorts, the market premiums both are positive and also insignificant. Looking at the GRS-test, the unlevered CAPM only passes the test for the unlevered portfolio sort. For all other portfolio sorts, i.e. traditional CAPM, Fama & French (1993) and the industry portfolio sorts, the unlevered CAPM fails the test. However, this pattern is also observed by the two benchmark models. The cross-sectional fit for the unlevered CAPM ranges between 2% and 24%.
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pricing error observed for the industry class sort is significant and is rather high. For the other portfolios the alphas are all insignificant and close to zero. The market premiums for the industry class sort are significant but rather low, for all other portfolios the market premium is reasonable but insignificant. With the GRS-test, the Fama & French (1993) three-factor model does perform better compared to the other asset-pricing models as the model passes the unlevered and the traditional CAPM sorts. However, when the cross-sectional fit is considered, the Fama & French (1993) three-factor model shows weak results. The cross-sectional fits ranges from 0% for the industry class sort to 24% for the portfolio sorted on SmB and HmL factors.
All in all, when solely considering pricing errors, the unlevered CAPM does perform best for all four portfolio sorts. When considering solely the market premiums, the traditional CAPM and the Fama & French (1993) three-factor model do perform better regardless how the portfolio is sorted. Also for the GRS-test, the Fama & French three-factor model outperforms the two other asset-pricing models when we disregard how the portfolios are sorted. The cross-sectional fit favours the unlevered CAPM and the traditional CAPM, as the range of the fit is smallest for these portfolios. Consequently, there is not a clear asset-pricing model that performs more robust compared to the other asset pricing models.
VI. DISCUSSION
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unlevered CAPM regarding the GRS-test than Dam & Qiao (forthcoming). The results for the market premium of the unlevered CAPM show all insignificant premiums ranging from -0.37% to +0.33%. The other two asset pricing models do slightly perform better compared to the unlevered CAPM. They both show insignificant premiums; however for both models all premiums are positive except for the industry class sort. There the premiums are rather low and significant for both models.
There are several explanations why this thesis yields these unexpected outcomes. First, the used sample period might be too short. The more extensive the time span is, the less noisy the average returns are as the dataset will include more observations. Dam & Qiao (forthcoming) construct a dataset using a sample period ranging from January 1952 to December 2013, which includes 744 months. Due to the fact that European data is only available for the last few decades, it is not possible to replicate such a large sample period for the Eurozone. Moreover, the Eurozone specific Fama & French (1993) factors, which are provided on the personal website of Kenneth R. French, start from June 1990 onwards. All these data limitations resulted in that the maximum sample period obtainable for this thesis ranges from January 1991 to December 2015. To obtain a representative conclusion from the analysis, it seems necessary to have a longer time span to analyse the true relationship between the unlevered returns and the unlevered betas. Moreover, to a similar sample period of Dam & Qiao (forthcoming) is necessary to create a more representative comparison of the results of the research by Dam & Qiao (forthcoming).
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Despite the fact that the empirical results of this thesis do not support the earlier states hypothesis and the listed limitations of this thesis, this thesis still contributes to the existing literature. The introduction of the potential impact of the leverage effect on average stock returns is confirmed by this thesis. The descriptive statistics of all the individual time-series regressions for each of the 4 portfolio sorts display that the means and the standard deviations for the unlevered excess returns are always lower compared to the levered excess returns. Moreover, the rather high reported leverage ratios in these tables also stress the potential impact of the leverage effect. Further research could build on these results.
VI. CONCLUSION
This thesis analysed the performance of the unlevered CAPM. The unlevered CAPM uses unlevered returns (asset returns) and unlevered betas (asset betas). The model has an interesting feature as it disregards the impact of the firm’s capital structure. Consequently, the leverage effect on equity returns is eliminated as the equity betas and the risk premiums are not correlated. This eliminates potential pricing errors caused by this phenomenon.
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When considering the descriptive statistics of all the time-series regressions, it becomes clear that the leverage effect plays an important role in comparing the unlevered with the levered excess returns. Though, this does not imply that the unlevered CAPM clearly outperforms the other asset-pricing models which are based on levered returns and levered betas. Solely facing on the results of the GRS-test, there is no clear outperformance of the model. The Fama & French (1993) three-factor model does pass the GRS-test twice, for the unlevered CAPM and the traditional CAPM portfolio sorts. The other asset-pricing models fail the GRS-test for the three different portfolio sorts. Moreover, based on the Fama & MacBeth (1973) cross-sectional regressions, all models perform rather poorly. One could argue that the unlevered CAPM only outperforms the two asset-pricing models for the unlevered CAPM sorts. For the unlevered CAPM in this portfolio sort, the GRS-test is passed and both the pricing error is insignificant and close to zero. Though, the market premium is low and insignificant, which indicates poor performance of the asset-pricing model. The other portfolio sorts do not yield a clear outperformance of one asset-pricing, resulting in that the results are inconclusive. Consequently, the stated hypothesis that the unlevered CAPM is outperforming the other two benchmark asset-pricing models, i.e. the traditional CAPM and the Fama & French (1993) three-factor model, is not supported.
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APPENDIX
A1. Descriptive statistics for the 25 portfolios based on the unlevered CAPM betas sorted
𝜷𝒖 Unlevered excess return (%)
Excess return (%) Leverage
Mean S.D. Mean S.D. Mean S.D.
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A2. Descriptive statistics for the 25 portfolios based on the traditional CAPM betas sorted
𝜷𝑳 Unlevered excess return (%)
Excess return (%) Leverage
Mean S.D. Mean S.D. Mean S.D.
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A3. Descriptive statistics for the 25 portfolios based on Fama & French (1993) sorted on size and book-to-market ratios
Small/Big High/Low Unlevered excess return (%)
Excess return (%) Leverage
Mean S.D. Mean S.D. Mean S.D.
Big Low -0.07 2.36 -0.19 5.42 0.46 0.14 4 Low 0.02 1.98 0.23 4.90 0.40 0.13 3 Low 0.04 1.56 0.30 4.34 0.34 0.14 2 Low 0.12 1.05 0.64 3.77 0.24 0.13 Small Low 0.06 0.40 1.09 4.25 0.06 0.05 Big 2 -0.17 2.87 -0.15 5.02 0.56 0.11 4 2 0.04 1.86 0.17 4.28 0.43 0.12 3 2 0.13 1.44 0.48 4.06 0.33 0.14 2 2 0.20 1.36 0.92 4.15 0.26 0.15 Small 2 0.06 0.30 1.02 4.27 0.06 0.05 Big 3 -0.29 2.85 -0.34 4.56 0.59 0.11 4 3 0.10 1.88 0.43 4.62 0.38 0.14 3 3 0.15 1.42 0.55 4.30 0.31 0.14 2 3 0.16 0.95 0.78 4.08 0.20 0.12 Small 3 0.10 0.50 1.00 4.60 0.08 0.06 Big 4 -0.11 2.96 -0.03 4.86 0.57 0.14 4 4 0.17 1.85 0.54 4.81 0.36 0.11 3 4 0.14 1.23 0.84 4.81 0.23 0.12 2 4 0.09 0.69 0.89 4.50 0.14 0.08 Small 4 0.02 0.36 0.80 5.34 0.06 0.04 Big High 0.21 6.66 0.29 8.16 0.81 0.07 4 High 0.10 1.45 0.56 5.31 0.27 0.10 3 High 0.07 0.79 0.75 5.58 0.13 0.07 2 High 0.07 0.59 0.78 5.81 0.10 0.06 Small High 0.03 0.71 0.86 6.50 0.08 0.06
A4. Descriptive statistics for the 10 portfolios based on industry sectors
Pf# Unlevered excess return (%)
Excess return (%) Leverage
Mean S.D. Mean S.D. Mean S.D.
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B1. Cross-sectional fit of actual and predicted levered returns sorted using the unlevered CAPM portfolio sort
B2. Cross-sectional fit of actual and predicted (un)levered returns sorted using the traditional CAPM portfolio sort
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B4. Cross-sectional fit of actual and predicted (un)levered returns sorted using the industry class portfolio sort
40 D1. Specification of the 10 industry portfolios Pf# Industry Group 1 Basic Materials 2 Consumer Goods 3 Consumer Services 4 Financials 5 Health Care 6 Industrials 7 Oil & Gas
8 Technology
9 Telecommunications
