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The Performance of the Unlevered Captital Asset Pricing model on an intermediate sample size, the Standard and Poors 500
Edzard van der Ploeg S1889370
University of Groningen Faculty of Economics and Business
MSc. Finance Supervisor: L. Dam
June 2016
Abstract
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1. Introduction
The academic literature is searching for an asset pricing model which is able to price assets effectively. Two models are famous in attempting to price assets, the Capital Asset Pricing Model(traditional CAPM) and the Fama and French thee factor model(FF93). Both the traditional CAPM and FF93 rely on the expected return-beta relationship. The unlevered CAPM likewise relies on the expected return-beta relationship and is an extension of the traditional CAPM proposing an superior asset pricing model. Empirical interest into stock volatility introduces the leverage effect(Schwert, 1989; Nelson, 1991). Choi(2013) and Dam and Qiao(2016) are the first who touch upon the unlevered CAPM, and recognize leverage as important determinant of the return-beta relationship. Leverage intervenes with returns, especially in economic downturns for value portfolios. During economic downturns the risk premium is high, leverage increases and the equity betas rise consequently. Growth portfolios are less levered and less affected by economic downturns. The conditional traditional CAPM betas are correlated with the risk premium creating pricing errors in an unconditional
traditional CAPM (Jagannathan and Wang, 1996; Lewellen and Nagel, 2006). The pricing errors in the conditional traditional CAPM arise from financial leverage meddling in the return-beta relationship. Dam and Qiao(2016) test the unlevered CAPM on the traditional CAPM and FF93 on a sample of 7,563 firms situated in the United States. The unlevered CAPM proves less erratic than traditional CAPM and FF93 when tested in test portfolios sorted on the betas of the unlevered CAPM, the traditional CAPM and FF93. In this thesis the unlevered CAPM is tested on an intermediate size sample, the Standard and Poors
500(S&P500), to provide further insight in the applicability of the unlevered CAPM.
The unlevered CAPM is an interesting extension of the traditional CAPM because the financial structure is ignored by the unlevered CAPM. The traditional CAPM and FF93 use levered excess returns and levered betas to price assets, and are vulnerable to the leverage effect (Choi, 2013; Dam and Qiao, 2016). If the pricing errors created by the traditional CAPM and FF93 do not surface in the unlevered CAPM, a more effective asset pricing model is designed.
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degree of financial leverage are provided every month. Classic unconditional test of the traditional CAPM are performed in the cross section on the individual firms and different test portfolios. The performance of the models is tested through two regressions. First, a time series regression is performed and tested combined with the Gibson, Ross, and Shanken (1989) test. Second, a Fama MacBeth two step procedure is estimated for the cross sectional regressions. Lastly, the cross sectional fit of the models is visualized by making a scatter plot of the predicted and the realized returns of each of the models based on the beta sorts.
The S&P 500 denotes the 500 firms in the United States with rather large market
capitalizations, and could be considered an intermediate size sample compared to the total economy of the United States. The S&P 500 is considered a leading indicator and lagging indicator (Renshaw 1995). Whether the S&P 500 is a leading indicator or lagging indicator is not discussed in this thesis, however the S&P500 is an important indicator of economic performance in the United States, and this is made clear by Renshaw. After omitting firms which are for different reasons not representative, a sample consisting of 2,966 firms remains. The sample is spanning 538 months, from March 1969 and ends in December 2013. 25 portfolios are created for each of the 3 models based on the betas of the models examined. There is no direct evidence for the unlevered CAPM outperforming the other models. The unlevered CAPM, despite not being able to outperform the traditional CAPM and FF93 model consistently, is more robust. When the test portfolio is selected based on the beta of the
unlevered CAPM, the unlevered CAPM definitely outperforms the other models. When the test portfolios are selected based on the betas of the traditional CAPM or FF93 model, the unlevered CAPM is not able to outperform the traditional CAPM nor the FF93 model. The FF93 model stand out in a notable manner, when selecting the test portfolio using the betas of the FF93 model . The model performs worse than when using the betas of the other models to create test portfolios.
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United States sample is used(Choi, 2013; Dam and Qiao, 2016). This counters some of data issues mentioned in the articles of Choi(2013) and Dam and Qiao(2016).
The structure maintained in this thesis is as follows. In section two the literature containing the main findings concerning asset pricing models is reviewed. The third section explains the methodology used in this thesis and the manner in which this methodology is established. The fourth section describes the data collected to perform tests on. Section five presents the results found by the comparison of the models. Section six presents a discussion and a conclusion of the main findings of this thesis.
2. Literature review
The traditional CAPM is the most used and critically appraised model in asset
pricing(Jagannathan and McGrattan,1995). The traditional CAPM is devised by Lintner, Mossin and Sharpe in different articles ( Lintner 1965, Mossin 1966 and Sharpe 1964). The traditional CAPM relates expected returns to the beta and risk premium of an asset. The traditional CAPM relies on the mean variance relationship of modern portfolio theory developed by Markowitz (1952) to explain the expected return-beta relationship of the traditional CAPM. The traditional CAPM is subject to several forms of criticism due to the strict assumptions needed to justify the model.
Lintner(1969) discusses the existence of a risk-free asset and states that a riskless assets is nonexistent. A riskless asset should pay out a certain amount after a defined period.
Governments are suggested as providers of a risk-free asset, ex ante a specified amount is set. Ex post general commodity and consumer goods price levels may have changed creating an uncertain real return. The real costs of borrowing are hereby also made uncertain, and the risk-free asset is no longer truly risk free. The traditional CAPM created by Black (1972) is without the need for a risk-free asset, but with fully or partially restricted risk-free borrowing the results of Black, Jensen and Scholes(1972) remain empirically justifiable.
Black, Jensen and Scholes(1972) performed several tests on the traditional CAPM,
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financial market do not exhibit the risk and return rationality of the traditional CAPM, undermining the foundations of the traditional CAPM. Investors exhibit more a Keynesian type of crowd psychology, investors respond positively to good news and negatively to bad news. Dempsey(2013) in his turn is criticized by Brown and Walter(2013).
Brown and Walter (2013) criticize Dempsey’s(2013) perspective upon two points. Starting with Dempsey(2013) not considering modern portfolio theory in criticizing the traditional CAPM. Dempsey(2013) rejects investor rationality by declaring a Keynesian type of crowd psychology is exhibited, but does not explicitly describes the mean variance relationship. The critique of Dempsey(2013) is related to the expected return-beta relationship. Further not mentioning Roll’s(1977) critique concerning tests of the traditional CAPM is a point of criticism. Roll(1977) alludes it is impossible to test the traditional CAPM, as it is impossible to observe the true market portfolio. To observe the true market portfolio the returns of all possible investments should be known. Not all returns are known, so it cannot be stated that the market portfolio is mean variance efficient. In extension to Brown and Walter(2013), Diacogiannis and Feldman(2013) overcome the problem of the market portfolio being inefficient. They propose Linear Beta Pricing restriction with Inefficient Benchmarks(LBPI). Using LBPI rather than using an efficient alternative for testing the traditional CAPM makes the tests more meaningful.
After all the critiques on the traditional CAPM, Fama and French (2004) provide insight in why the traditional CAPM should be considered as an useful and insightful asset pricing model. The traditional CAPM is an intuitively strong and simple model for asset pricing. Fama and French(2004) do state that there are empirical shortcomings within the traditional CAPM, mainly originating from the simplifying assumptions needed to justify the traditional CAPM . The main issue Fama and French(2004) observe is the implementation and testing of the model. The traditional CAPM is a point of departure to develop more complex and more effective performing models.
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French(2012), however cannot observe the SmB effect, in the 80’s and early 90’s the SmB effect seems to perish. Chan, Hamao and Lakonishok(1991) confirm the HmL factor when examining the Japanese market. Fama and French (2006) confirm the existence of the HmL effect on a longer time horizon than in their previous Fama and French(1993) article. For both effects two possible explanations arise. Ross(1976) developed the arbitrage pricing theory, thereby developing the concept of factor investing. Factors are created and with each factor, undiversifiable factor risk is associated with the factor. An example of such a factor would be the market factor of the traditional CAPM. The SmB and HmL factors have an associated risk. These risk are consistently mispriced by the market and lead to the existence of a premium. The rationalizing of the SmB effect no longer being present, the SmB effect might now be priced in a correct manner. Another explanation, which is tightly linked to the other explantion, is firms sensitive to the SmB and HmL effect face excess risk from the market due to different causes. Excess risk leads to higher cost of capital and thus higher returns.
Black(1993) criticizes the FF93 model being data mining from Fama and French. The
anomalies found by Fama and French (1993) are coincidence according to Black(1993). Black states “Once in a while, just by chance, a strategy will seem to have worked consistently in the past. The researcher who finds it writes it up, and we have a new anomaly. But it generally vanishes as soon as it's discovered”. The results of applying the FF93 model differs among the articles, and counterarguments as Black(1993) exist. Some researchers find the FF93 model outperforming the traditional CAPM for different markets around the world (Ajilli, 2002; Arioglu and Canbas, 2008 ;Blanco, 2012; Rossi, 2012). On the other side of the spectrum, multiple academics oppose the FF93 model as it does not explain all the variation in the returns(Bartholdy and Peare, 2005; Faff, 2004; Griffin, 2002). So far no consensus is achieved regarding the FF93 model.
Another model proposed is the Carhart four -factor model. The Carhart four-factor
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Nwani(2015) assesses the model for U.K. market and encounters a significant MoM factor. The MoM factor is only significant for stock with a large market capitalization, thus a high HmL factor from the FF93 model, both factors appear to be related. As with the FF93 model, the Carhart four-factor model(1997) is not without doubt the best performing or most
effective asset pricing model.
In search of a best performing and effective asset pricing model Fama and French (2015) propose a five factor model to explain the variation in the returns of assets. The FF five factor model has three factors identical to the FF93 model, the market factor, the SmB factor and the HmL factor. Two factor are introduced, the profitability(RMW) and investment(CMA)
factors. The RMW factor is a set of diversified portfolios subdivided on robust minus weak profitability firms. Novy-Marx (2013) uses gross profit to assets as an proxy for the measure of profitability. When assessing the relation between the profitability and average stock returns a relation is discovered, comparable to HmL factor of the FF93model. When controlling for profitability, the value portfolios performance is enhanced. This is a
confounding discovery and is until now not explained. The CMA factor is a set of diversified portfolios subdivided on conservative minus aggressive investment firms. Titman, Wei, and Xie (2004) provide proof for the relation of average stock returns and the capital investment of firms. They find a negative relationship in the capital investment-return relation. Firms able to execute large capital investments are more prone to this negative relationship. The
conclusion is in line with the hypothesis of Titman, Wei, and Xie (2004), investors tend to underreact to the empire building implications of increased investment expenditures. In line with the effects of the added factors Fama and French (2015) find their five factor model explaining variations in the average stock returns for 71% to 94%.
Fama and French(2015) deliver critique on Hou, Xue and Zhang(2012) for not including the HmL factor. Nwani(2015) finds the MoM factor significant for stock with a high HmL factor. Interaction between HmL and MoM occurs, and possibly the factors are intertwined.
Excluding MoM in the FF five factor model could create biased results. The five factor model of Fama and French(2015) is liable to more forms of critique. Volatility as an factor is
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and Stambaugh(2013) establish a relation between average stock returns and the aggregate liquidity of the market. The MoM and liquidity factor are not included in the FF five factor model because the two factors have regression slopes close to zero and change the model performance significantly(Fama and French, 2015). The criticism presented is as well theoretical as empirical. Racicot and Rentz(2016) perform robustness tests for the FF five factor model. The results of the tests are mixed, when using standard OLS estimation of the FF five factor model the results are robust and the model seems quite effective in explaining variation of the average stock returns. When a more sophisticated model is used, the General Method of Moments, the models robustness and explanatory power diminishes. The true value of the FF five factor model remains to be seen as empirical studies still need to be performed to assess the robustness and effectiveness of the model.
To test models for effectiveness and performance often test portfolios are used, another choice is testing on the individual firms themselves. Test portfolios create a potential bias, the
performance of an asset pricing model varies significantly with the choice of construction of the test portfolio(Kandel and Stambaugh, 1995; Lewellen, Nagel and Shanken, 2010; Blanco, 2012). Construction of test portfolios based on the betas of the model are potentially biased and Dam and Qiao (2016) therefore compare the models based on multiple beta sorts associated with each of the models. By sorting for each of the betas and comparing each of the models Dam and Qiao(2016) overcome the bias.
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Hamada(1972) examined the effect of leverage created by corporate debt on the riskiness of firms. Hamada concludes that a significant part of the added systematic risk can be explained by taking on debt claims. Bhandari(1988) finds a positive relationship between leverage and average returns. Fama and French(1992) note that the HmL factor could be associated with financial distress. Leverage aids in explaining the cross section of average returns.
The leverage effect is explained in two ways. First, a positive correlation between leverage and equity betas and equity return volatility emerges(Hillier Grinblatt and Titman 2011; Schwert 1989 and Nelson 1991). This positive correlation implies , when taking leverage into account, the levered betas and return volatilities are higher than should be expected. Second, the risk premium is affected positively by leverage(Jagannathan and Wang 1996). The risk premium increases when leverage increases, investors demand a higher risk premium when their investment has more exposure to leverage. This demand is line with economic sense, investors are junior in case of default compared to debt holders. If equity is junior combined with a higher probability of default in case of an increase of leverage implies a higher risk to the investors. Choi(2013) finds the HmL factor to even further increases the effect leverage. Growth firms are firms less affected by the HmL factor, and therefore are less prone to the further increase of leverage effect. When economic conditions are less favorable, growth firms experience less difficulty from the economic conditions than value firms do, growth firms have less leverage compared to value firms.
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removing leverage they remove the correlation between risk premiums and leverage, thereby removing biases relating to leverage.
Dam and Qiao(2016) test their unlevered CAPM and the results are promising. The unlevered CAPM is evaluated next to the traditional CAPM and the FF93 model. They sort the firms in the sample based on the betas of these firms into portfolios, creating twenty five test
portfolios. The unlevered CAPM provides better results, and is more robust compared to the traditional CAPM and FF93 model. Dam and Qiao(2016) use a two beta model for further testing. The betas consisted out of the unlevered beta and a beta with regard to the portfolio selection, this does not necessarily improve the results compared to the unlevered CAPM. The logic why the unlevered CAPM provides promising results is a result of the wider spread of unlevered excess returns. The wider spread is a consequence of the negative correlation between the unlevered betas and the leverage in the cross section. Keim(1983) finds levered equity betas tend to one over longer periods, and not being able to account for the variation of average stock returns. The unlevering of returns causes a wider spread, thus more variation in the cross section which the model can explain. Now the unlevered CAPM is put to the test on the S&P500.
3. Methodology
The main purpose of this theis is to assess the unlevered CAPM, the traditional CAPM and the FF93 three-factor model on the S&P 500. For every asset pricing model, the firms are sorted and divided in twenty five test portfolios. To make the comparison between the asset pricing models, multiple regressions are estimated, the purpose of these regressions is
obtaining the respective betas of the firms and portfolios. After obtaining these betas, a Fama MacBeth(1973) two step cross sectional regression is estimated. The output of the Fama MacBeth procedure gives insight in whether the estimated risk factors are priced risk factors, this is done by examining the significance of the factors. More specifically the unlevered beta’s are tested for significance to examine if they are priced risk factors. The Fama
MacBeth procedure is performed on each of the asset pricing models. Further a Gibson-Ross-Shanken test(GRS) is executed to observe whether the pricing errors are jointly significantly different from zero. As last analysis, the actual average returns are plotted versus the
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The unlevered CAPM , as proposed by Dam an Qiao(2015), starts out with the unlevered excess return. (1) 𝑅𝑖,𝑡+1𝐴 = 𝐸𝑖,𝑡 𝐸𝑖,𝑡+𝐵𝑖,𝑡𝑅𝑖,𝑡+1 𝐸 + 𝐵𝑖,𝑡 𝐸𝑖,𝑡+𝐵𝑖,𝑡𝑅𝑖,𝑡+1 𝐵
Ei,t is the total equity ,Bi,t is the total debt , and 𝑅𝑖,𝑡𝐸 and 𝑅𝑖,𝑡𝐵 are the returns of equity and debt
at time is t of firm i. This formula proposes the unlevered return of the total assets is driven by the return on equity and return on debt. Debt is assumed constant and approximately risk free, making the return equal to the short term risk free interest rate. In result the process of
creating an unlevered excess return from a levered excess return is linear. The formula for the unlevered excess returns is stated below.
(2) 𝑅𝑖,𝑡𝑢 = 𝑅𝑖,𝑡
𝐿𝑖,𝑡−1
𝑅𝑖,𝑡𝑈 = 𝑅𝑖,𝑡𝐴 − 𝑟𝑡 represents the unlevered excess return, 𝑅𝑖,𝑡𝐴 = 𝑅𝑖,𝑡𝐸 − 𝑟𝑡 is the excess levered return, and 𝐿𝑖,𝑡−1 = 𝐸𝑖,𝑡−1+𝐵𝑖,𝑡−1
𝐸𝑖,𝑡−1 is the leverage ratio. Equation (3) is used to estimate the
regression concerning the unlevered CAPM.
(3) 𝑅𝑖,𝑡𝑢 = 𝛼𝑖 + 𝛽𝑖𝑢(𝑅𝑀,𝑡𝑢 − 𝑟𝑡) + 𝜀𝑖,𝑡
𝛼𝑖and 𝛽𝑖𝑢 are the coefficients of the asset i which are estimated by the formula, 𝑅𝑀,𝑡𝑢 − 𝑟𝑡 is
the unlevered market excess return and 𝜀𝑖,𝑡 is the residual of asset i at time is t. The unlevered
beta estimates the relationship of the asset i to the market. 3.2 Traditional CAPM and the FF93 model
The unlevered CAPM is compared to the traditional CAPM and FF93 model, and equation (4) is used to estimate the traditional CAPM.
(4) 𝑅𝑖,𝑡− 𝑟𝑓,𝑡 = 𝛼𝑖,𝑡+ 𝛽𝑖,𝑡(𝑅𝑀,𝑡− 𝑟𝑓,𝑡) + 𝜀𝑖,𝑡
Where 𝛼𝑖,𝑡 and 𝛽𝑖,𝑡 are the coefficients with respect to the levered excess return, 𝑅𝑀,𝑡− 𝑟𝑡 is
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(5) 𝑅𝑖,𝑡− 𝑟𝑡 = 𝛼𝑖 + 𝛽𝑖(𝑅𝑀,𝑡− 𝑟𝑡) + ℎ𝑖𝐻𝑚𝐿(𝑡) + 𝑠𝑖𝑆𝑚𝐵(𝑡) + 𝜀𝑖,𝑡
Equation (5) is the FF93 three-factor model regression equation where 𝛼𝑖, 𝛽𝑖, ℎ𝑖 and 𝑠𝑖are the
coefficients of the asset i with respect to the factors of the model. 𝑅𝑀,𝑡− 𝑟𝑡 is the levered market excess return, 𝐻𝑚𝐿(𝑡)is the High-minus-Low or value factor, 𝑆𝑚𝐵(𝑡) is the Small-minus-Big or size factor and 𝜀𝑖,𝑡 are the residuals of the asset i at time is t.
3.3 Fama MacBeth cross sectional regression
A further comparison of the models is made by the two step Fama Macbeth cross sectional regression. The first step consists of regressing each of the assets i to the factors determined by each of models, with the rationale of relating the betas of each asset with respect to the factor. The equations (3), (4) and (5) are used for this purpose. The second step involves determining the risk premium for each factor and regressing the returns of the assets for a fixed period to these risk premiums. Formula (6) is used to estimate the regression of returns of the assets and the risk premium.
(6) 𝐸(𝑅𝑖,𝑡 − 𝑟𝑡) = 𝛼 + 𝛽𝑖′𝜆
𝛼 is the intercept, 𝛽𝑖′ is the beta associated with asset i, and reflects the sensitivity of the asset to the market risk premium 𝜆, the associated standard errors are as well calculated. The purpose of this regression is testing every factor for being a priced risk factor. The regression is estimated every month for the portfolios.
3.4 Gibson-Ross-Schanken test
The Gibson-Ross-Schanken test(1989) is a multivariate linear regression test, suitable to deal with panel data. The GRS-test uses alphas representing the intercepts created by the model described in equation (3), (4) and (5). This lead to the following hypothesis.
(7) 𝐻𝑜: 𝛼 = 0, ∀𝑖 = 1, 2, 3, 4, … , 𝑁
Where 𝛼 is the alpha associated with each of the test portfolios. The rationale of the test is provided the alphas of the model are jointly indistinguishable from zero the model performs well in explaining the stock returns, put differently little residual explanatory power remains in the model. Confirming the Ho concludes in the model being able to correctly price the
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any portfolio, not necessarily the market portfolio. The GRS-test is not totally explained in this thesis, as this is outside the scope of this thesis. However, the 𝐹𝐺𝑅𝑆 statistic formula is provided give insight in how the F-statistic is calculated.
(8) 𝐹𝐺𝑅𝑆= 𝑇(𝑇 − 𝑁 − 1) 𝑁(𝑇 − 2) [ 𝛼̂´∑̂−1𝛼̂′ (1 + 𝜃2)], (9) 𝜃̂2 = 𝑟 𝑠
Where 𝑇 is the number of time series observations on returns, 𝑁 is the number of assets, 𝛼̂ are the alphas. ∑̂−1 is the unbiased residual covariance matrix, 𝑟 is the sample mean of the excess returns, levered or unlevered, of the portfolios and 𝑠2 is the sample variance of the levered or unlevered excess returns on the portfolios.
When the 𝐹𝐺𝑅𝑆 statistic is large and significant H0 is rejected, significance is defined by the
degrees of freedom of the model. Rejecting H0 implies that the alphas are significantly
distinguishable from zero, and the model has residual explanatory power outside of the betas. The assets are not correctly priced by the model if H0 is rejected, and significant pricing
errors exists. The GRS test requires a balanced dataset, as a consequence the GRS test cannot be performed on the individual firms and only on the test portfolios.
As last analysis the actual average returns are plotted against the predicted average returns by the model. This visualization provides insight regarding the fit of the model, the 𝑅2 is also calculated. The 𝑅2 is calculated by simple ordinary least squares regression to represent the fit, and forms the slope of a line. When the line is a 45 degree line, the fit is 1 or can be seen as an exact fit.
4. Data
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The returns of the stock, the prices of the stock, the shares outstanding, the book value of total assets, market value of equity and the book value of equity are provided by COMPUSTAT and CRSP. The COMPUSTAT database provides the market value of equity if possible, otherwise the CRSP database fills in the missing values. To generate book leverage ratios total equity is divided by total assets. Total equity is divided by the market value of equity to obtain a market to book ratio. Debt is calculated by subtracting the total book value of equity of the total book value of assets. The market value of assets is calculated by adding debt and the market value of equity. Lastly, the leverage ratio is calculated by dividing the market value of equity and the market value of assets. All firm specific variables are now created. The Fama and French factors, excess return of the market and the risk free rate are provided by the Wharton School of the University of Pennsylvania database, which in their turn receive the data from the University of Darthmouth. The risk free rate is the one-month treasury bill of the United States of America.
The value of total assets and the book value of total assets are available quarterly while the stock returns and total equity are available on a monthly basis, leading to gaps in the variables. To solve for the missing values, debt is assumed constant and missing values are interpolated. This approach is similar to the methodology used by Dam and Qiao(2016). The argument at hand is, although the constant debt level can be assumed, the point of interest is the leverage ratio. Variation in the leverage ratio is mainly the result of variation in equity, so debt is assumed constant.
The Flow-of-Funds of the Federal reserve supplies the data for the total debt of the market and the market value of total equity, this data comes at quarterly intervals. The sum of debt and the market value of equity is assumed equal to the market value of assets, the total debt and total market value of equity are equal to the total market value of assets. The total market value of equity is retrieved by netting the total value of “Nonfinancial corporate business, corporate equities; liability” and “Nonfinancial corporate business, mutual fund shares; asset”. Total debt is collected by taking “Nonfinancial corporate business; credit market instruments; liability” and subtracting “Nonfinancial corporate business; money market fund shares,
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equity is combined with the aggregate monthly stock returns to estimate monthly total equity, the leads to the total market value of assets similarly as above.
Within the dataset several observations are missing and when a variable is missing the observation is removed. As mentioned firms denoted at the S&P500 shorter than 60 months are removed. The rationale behind removing short quoted firms is, when a firm is denoted so shortly, including this firm would distort the model. Further extremely levered firms are dropped. Outliers in the 1% and 99% quantile are omitted. The starting month and ending of the data changes when all non-suitable observations are omitted. The data starts in March 1969 and finishes in December 2013. The total number of firms in the sample is reduced to 2,966 firms. A further point of interest is the dataset being an unbalanced dataset.
Having described the variables at hand, a short description of some of the variables is presented in table 1.
Table 1. Observations, means, standard deviations, minima and maxima for the sample.
Variable Obs. Mean(%) S.D.(%) Min(%) Max(%)
Individual firms
Excess return(%) 697,060 0.68 12.19 -41.07 63.68
Leverage(%) 697,060 54.48 25.00 0.00 100.00
Unlevered excess return(%) 697,059 -0.76 6.79 -25.75 36.63 Market value of total equity(in 000
$) 697,060 4,361,899 2,840,236 0.00 16,110,430 Market value of total assets(in 000
$) 697,060 9,894,719 6,706,874 0.00 31,097,610
Aggregate
Market excess return(%) 538 0.24 2.54 -11.87 6.76
Market leverage(%) 538 55.61 8.27 39.49 72.89
Market unlevered excess return(%) 537 0.47 4.57 -26.57 14.77
SmB(%) 538 0.16 3.15 16.70 22.32
HmL(%) 538 0.36 3.00 -13.11 13.91
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at the market level. The market levered excess return is smaller than the unlevered excess return, this can be caused by the missing months in the market excess returns versus the unlevered excess returns. The last point concerns the leverage, the leverage varies a great deal amongst firms, one standard deviation increases or decreases leverage by half of its original value.
5. Results
5.1 The unlevered CAPM sort
The unlevered CAPM 25 portfolios are created by sorting on the respective unlevered betas, the test portfolios are created and the test portfolios have a near equal size. The unlevered betas are used in creating the test portfolios as a consequence the relationship with unlevered excess return is tested directly. Only the unlevered CAPM is tested directly as the other two models use the levered excess return. The first portfolio consists of firms possessing the lowest unlevered betas and the twenty-fifth portfolio including the firms with the highest unlevered betas. The portfolios almost are buy-and-hold portfolios, only firms leaving and entering the sample modify the portfolio. The factor risk premiums are restricted for this model.
Observing the descriptive statistics in appendix A table 1, the means of the portfolios
unlevered excess returns are lower than the excess return including leverage. Theory predicts lower means for unlevered excess returns, and the means observed are lower, however the standard deviations are equal, the theory predicts the opposite. Leverage varies largely from 10.53 % to 86.88%, the variety in leverage explains the variety in the unlevered excess return. Table 2 shows the pricing errors by α, the β of the market and the R2 for all models and
addition for the FF93 model the SmB and HmL factor. The pricing errors are insignificant for almost all alphas, with some exceptions. The combined GRS test for each of the models is insignificant. The alphas of each model are considered jointly zero, leaving no explanatory power within the alphas.
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Table 2. The betas pricing errors and GRS test of the unlevered beta sort for 25 portfolios.
Pf# Unlevered CAPM Traditional CAPM Fama and French GRS tes F=0.70 GRS test F=0.84 GRS test F=1.39 α(%) Βmarket R2 α(%) βmarket R2 α(%) βmarket βSmB βHmL R2
1 0.01 0.13*** 0.07 0.14 0.77*** 0.12 -0.21 0.94*** -0.11** 0.73*** 0.15 2 0.01 0.25*** 0.43 0.13 0.58*** 0.48 -0.04 0.71*** -0.24*** 0.40*** 0.66 3 0.01 0.28*** 0.39 0.12 0.73*** 0.44 -0.11 0.86*** -0.13*** 0.49*** 0.56 4 0.02 0.34*** 0.50 0.09 0.70*** 0.54 -0.14 0.83*** -0.14*** 0.48*** 0.64 5 0.03 0.43*** 0.56 0.12 0.76*** 0.58 -0.07 0.86*** -0.06 0.40*** 0.70 6 -0.03 0.47*** 0.44 0.06 0.88*** 0.50 -0.10 0.95*** -0.04 0.33*** 0.59 7 0.01 0.52*** 0.65 0.08 0.81*** 0.69 0.01 0.92*** -0.31*** 0.20*** 0.73 8 -0.01 0.61*** 0.60 0.03 0.79*** 0.66 -0.05 0.87*** -0.20*** 0.22*** 0.69 9 0.09 0.80*** 0.55 0.21 0.83*** 0.64 0.11 0.89*** -0.28*** 0.27*** 0.69 10 0.12 0.69*** 0.43 0.22** 0.77*** 0.53 0.17* 0.77*** -0.28*** 0.16*** 0.58 11 0.13 0.80*** 0.60 0.22 0.83*** 0.66 0.16 0.90*** -0.22*** 0.15*** 0.71 12 0.05* 0.94*** 0.43 0.16 0.94*** 0.49 0.13 0.96*** -0.03 0.07 0.52 13 0.04 0.96*** 0.56 0.13 0.97*** 0.63 0.10 0.96*** 0.08* 0.05 0.64 14 -0.06 0.94*** 0.71 -0.04 1.03*** 0.78 -0.07 1.08*** -0.20*** 0.10** 0.80 15 0.07 1.05** 0.70 0.13 0.85*** 0.75 0.19 0.92*** -0.38*** -0.05 0.77 16 0.05 1.07*** 0.68 0.09 0.99*** 0.75 0.04 1.03*** -0.06 0.12*** 0.77 17 -0.02 1.24*** 0.60 0.01 0.96*** 0.67 0.07 1.01*** -0.34*** -0.05 0.71 18 -0.02 1.21*** 0.62 -0.01 0.94*** 0.65 0.02 0.95*** -0.11** -0.05 0.68 19 0.08 1.17*** 0.69 0.13 1.00*** 0.74 0.19 0.99*** -0.11*** -0.11*** 0.75 20 0.13 1.28*** 0.76 0.14 0.99*** 0.79 0.15 1.01*** -0.11*** -0.01 0.80 21 0.00 1.50*** 0.74 0.01 1.16*** 0.77 0.06 1.12*** 0.05 -0.12** 0.77 22 0.08 1.67*** 0.72 0.12 1.26*** 0.76 0.17 1.20*** 0.19*** -0.14*** 0.77 23 -0.08 1.84*** 0.64 -0.11 1.37*** 0.66 0.09 1.21*** 0.33*** -0.48*** 0.72 24 0.28 2.07*** 0.66 0.33* 1.35*** 0.68 0.71*** 1.18*** 0.06 -0.78*** 0.76 25 0.20 2.53*** 0.56 0.22 1.60*** 0.59 0.50* 1.40*** 0.35*** -0.65*** 0.65
* denotes p<.10, ** denotes p<.05 and *** denotes p<.001, Pf# indicates the portfolio number
For the FF93 model the same applies, the lowest beta being portfolio 2 and the highest portfolio 25 and presenting an erratic pattern. All βmarkets still are highly significant, the betas possess explanatory power, the R2 indicates explanatory power. The βSmB and βHml
are significant for majority of the time and the R2 is rather high, the outlier is portfolio 1 with a R2 of 0.15. The βSmB is mostly negative and significant, the FF93 model predicts another
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Table 3. Cross sectional regressions for 25 portfolio sorted on unlevered betas.
Model α(%) λmarket(%) λSmB(%) λHmL(%) avg R2
Unlevered CAPM -0.01 0.32 0.38 (-0.15) (2.35)** Traditional CAPM 0.05 0.57 0.24 (0.20) (1.71)* FF93 -0.06 0.68 -0.07 -0.13 0.37 (-0.25) (2.12)** (-0.28 ) (-0.67)
The alphas are insignificant, which is promising as no explanatory power is contained by the alphas. All alphas are close to zero, thus insignificant. The lambdas of each of the models are significant, the lambdas of the unlevered CAPM and FF93 model at the 5% level and CAPM at the 10% level. All risk premiums are positive as expected. The SmB and HmL gammas are not significant and are considered not priced by the model. The risk premium of the FF93 model is the highest with 0.68 and unlevered CAPM the lowest with 0.32. The explanatory power of the unlevered CAPM and FF93 model is nearly equal.
As last analysis the actual and predicted average returns are plotted for the unlevered CAPM in figure 1. In the appendix B the plots of the traditional CAPM and FF93 model are included.
Figure 1. Realized versus predicted unlevered excess returns, sorted on the unlevered CAPM betas. R² = 0.8799 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 P re dict ed un lev er d ex ce ss re turns ( %)
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By introducing a plot the fit between the realized and predicted unlevered excess returns is examined. On first sight the test portfolios are close to the 45-degree line. The R2 of around
88% confirms the test portfolios centering around the 45-degree line. Two outliers appear on the right side of the scatterplot, however with a high R2 this is not a concern. Compared to the plots in appendix B, the unlevered CAPM has a superiour cross sectional fit. The traditional CAPM achieves a R2 of 63% and the FF93 model a R2 of 5%. The FF93 has a large number of pricing errors, and thus performs the weakest of the three models.
Overall the unlevered CAPM outperforms the other two models for the unlevered beta sort. The unlevered CAPM is outperforming the traditional CAPM and FF93 model, however the criticism of forming the test portfolios on the betas of the unlevered CAPM should be kept in mind. The betas of the unlevered CAPM make sense as portfolio 1 has the lowest and
portfolio 25 the highest, and the betas ascend overall. Regarding the cross sectional analysis, the unlevered CAPM outperforms the traditional CAPM, the unlevered CAPM does not outperform the FF93 model. Also little pricing error occurs in the unlevered CAPM, considering the plot.
5.2 The traditional CAPM sort
As comparison test portfolios are created based on the traditional CAPM betas. Appendix A table 2 shows the unlevered excess returns are lower than the normal excess return, as theory predicts. The standard deviations however are still virtually equal for both the unlevered excess returns and the normal excess returns. The leverage moves irregular, caused by the test portfolios are sorted on the traditional CAPM betas which are not adjusted for leverage. The GRS tests in table 4 show that pricing errors still exist in the unlevered CAPM and the FF93 model if the portfolios are sorted based on the traditional CAPM betas. The returns produced by the model are not only explained by the factors included in the model and other factors not captured by the model influence the average stock returns.
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Table 4. The betas pricing errors and GRS test of the traditional beta sort for 25 portfolios.
Pf# Unlevered CAPM Traditional CAPM Fama and French GRS test F=1,59** GRS test F=1.39 GRS test F=1.77**
α(%) Βmarket R2 α(%) Βmarket R2 α(%) βmarket βSmB βHmL R2
1 0.12* 0.21*** 0.12 0.32** 0.27*** 0.12 0.20 0.36*** -0.17*** 0.28*** 0.19 2 0.01* 0.23*** 0.17 0.31* 0.37*** 0.18 0.11 0.50*** -0.22*** 0.47*** 0.34 3 0.12* 0.34*** 0.29 0.29* 0.50*** 0.33 0.08 0.63*** -0.16*** 0.45*** 0.46 4 0.07 0.31*** 0.32 0.20 0.51*** 0.36 0.00 0.63*** -0.16*** 0.43*** 0.49 5 0.12* 0.56*** 0.44 0.29* 0.61*** 0.49 0.28** 0.69*** -0.31*** 0.09** 0.56 6 0.18** 0.59*** 0.44 0.35*** 0.63*** 0.48 0.29** 0.74*** -0.37*** 0.19*** 0.58 7 0.14 0.64*** 0.40 0.32** 0.68*** 0.51 0.42*** 0.74*** -0.44*** -0.12*** 0.60 8 0.08 0.71*** 0.45 0.18 0.72*** 0.53 0.13 0.81*** -0.31*** 0.16*** 0.59 9 0.11 0.67*** 0.47 0.22 0.78*** 0.57 0.16 0.89*** -0.37*** 0.19*** 0.65 10 0.02 0.91*** 0.61 0.10 0.88*** 0.66 0.14 0.93*** -0.26*** -0.03 0.69 11 -0.03 0.89*** 0.61 0.01 0.85*** 0.67 0.03 0.89*** -0.23*** 0.00 0.69 12 0.07 0.70*** 0.66 0.14 0.95*** 0.71 0.08 1.02*** -0.21*** 0.16*** 0.74 13 -0.04* 1.08*** 0.60 -0.01 1.01*** 0.70 0.05 1.01*** -0.10** -0.10** 0.71 14 -0.09* 0.69*** 0.71 -0.05 1.04*** 0.79 -0.14 1.05*** 0.15*** 0.16*** 0.80 15 -0.02 0.73*** 0.69 0.01 1.02*** 0.77 -0.11 1.10*** -0.11*** 0.27*** 0.80 16 -0.04 0.63*** 0.72 -0.02 1.08*** 0.81 -0.19* 1.16*** -0.08** 0.36*** 0.84 17 0.08 1.14*** 0.79 0.13 1.11*** 0.82 0.03 1.14*** 0.08** 0.19*** 0.83 18 -0.03 0.87*** 0.71 -0.03 1.15*** 0.77 -0.14 1.23*** -0.18*** 0.25*** 0.80 19 -0.05 0.55*** 0.73 -0.09 1.19*** 0.82 -0.18 1.23*** -0.02 0.19*** 0.83 20 0.03 1.40*** 0.69 0.05 1.23*** 0.73 0.20 1.18*** -0.01 -0.29*** 0.74 21 -0.09* 0.61*** 0.63 -0.22 1.24*** 0.74 -0.40*** 1.32*** -0.03 0.38*** 0.77 22 -0.03 1.62*** 0.64 -0.01 1.41*** 0.69 0.14 1.25*** 0.45*** -0.38*** 0.74 23 0.02 1.56*** 0.67 0.10 1.42*** 0.71 0.26 1.27*** 0.38*** -0.39*** 0.76 24 -0.13 1.10*** 0.60 -0.01 1.54*** 0.71 0.11 1.39*** 0.44*** -0.32*** 0.75 25 -0.04 0.87*** 0.54 0.00 1.84*** 0.72 -0.10 1.65*** 0.62*** -0.40*** 0.77
Twostep Fama MacBeth cross sectional regressions are estimated based on the 25 portfolio sort by traditional CAPM betas. The results are presented in table 5. The alphas for the traditional CAPM are insignificantly different from zero the alphas of the other models are significantly different from zero.
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Table 5. Cross sectional regressions for 25 portfolio sorted on traditional CAPM betas.
Model α(%) λmarket(%) λSmB(%) λHmL(%) avg R2
Unlevered CAPM 0.09 0.18 0.28 (1.29) (1.27) Traditional CAPM 0.44 0.17 0.31 (2.22)** (0.61) FF(1993) 0.61 0.03 0.04 -0.27 0.44 (2.68)*** (0.08) (0.15) (-1.39)
The last method of analysis is the plot of the realized excess returns and predicted excess returns of the portfolios. The R2 of the models can be found in appendix B. The unlevered
CAPM R2 is 44%, the R2 of the traditional CAPM is 33% and the R2 of the FF93 model is
8%. The unlevered CAPM produces the fewest pricing errors, and the FF93 model has rather large pricing errors. Sorting based on the traditional CAPM, the traditional CAPM is the most effective asset pricing model. Still the evidence in favor of the traditional CAPM is not overwhelming, and pricing errors still exist.
5.3 The FF93 model size and book-to-market sort
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Table 6. The betas pricing errors and GRS test of the FF93 coefficients sort for 25 portfolios.
Pf# Unlevered CAPM Traditional CAPM Fama and French GRS test F=2.71*** GRS test F=2.58*** GRS test F=1.82*** α(%) βmarket R2 α(%) βmarket R2 α(%) βmarket βSmB βHmL R2
1 -0.35** 0.97*** 0.29 -0.44 0.87*** .29 -0.63** 0.77*** 0.79*** 0.20** .39 2 -0.26** 0.77*** 0.33 -0.50** 0.86*** .38 -0.64*** 0.76*** 0.69*** 0.13* .48 3 0.20* 0.67*** 0.33 0.34 0.80*** .37 0.07 0.71 0.87*** 0.35*** .56 4 0.21** 0.57*** 0.39 0.42** 0.76*** .43 0.18 0.66*** 0.86*** 0.29 .67 5 -0.01 0.14*** 0.17 0.31* 0.75*** .43 0.02 0.69*** 0.76*** 0.40 .64 6 -0.23 1.29*** 0.50 -0.28 1.09*** .53 -0.29 0.90*** 0.86*** -0.14** .68 7 0.22** 1.04*** 0.57 0.34** 1.01*** .61 0.18 0.90*** 0.81*** 0.16*** .77 8 0.23*** 0.77*** 0.57 0.42*** 0.93*** .61 0.15 0.85*** 0.80*** 0.35*** .81 9 0.24*** 0.62*** 0.56 0.57*** 0.85*** .61 0.23** 0.82*** 0.74*** 0.49*** .83 10 0.07** 0.19*** 0.30 0.53*** 0.89*** .57 0.11 0.88*** 0.79*** 0.64*** .81 11 -0.07 1.37*** 0.67 -0.08 1.15*** .71 -0.06 0.98*** 0.70*** -0.18*** .84 12 0.17** 0.99*** 0.67 0.27** 1.05*** .73 0.32 1.00*** 0.61*** 0.33*** .85 13 0.16*** 0.65*** 0.62 0.40*** 0.90*** .69 0.11 0.90*** 0.54*** 0.46*** .84 14 0.15*** 0.51*** 0.65 0.49*** 0.89*** .70 0.15** 0.89*** 0.61*** 0.53*** .89 15 0.05 0.22*** 0.37 0.54*** 0.92*** .57 0.09 0.96*** 0.60*** 0.74*** .77 16 -0.05 1.37*** 0.74 -0.05 1.12*** .81 -0.04 1.07*** 0.24*** -0.06*** .83 17 0.15*** 0.90*** 0.76 0.29*** 1.02*** .83 0.13 1.02*** 0.26*** 0.27*** .87 18 0.10** 0.62*** 0.71 0.28*** 0.90*** .77 0.01 0.96*** 0.23*** 0.47*** .86 19 0.14*** 0.44*** 0.66 0.41*** 0.83*** .72 0.11 0.91*** 0.17*** 0.55*** .84 20 0.05 0.27*** 0.47 0.40*** 0.93*** .63 -0.02 1.02*** 0.31*** 0.73*** .78 21 -0.02 1.21*** 0.79 -0.05 0.93*** .84 0.09 0.94*** -0.30*** -0.22*** .88 22 0.07 0.76*** 0.77 0.09 0.87*** .86 0.07 0.94*** -0.19*** 0.08*** .88 23 0.06 0.55*** 0.70 0.23** 0.86*** .80 0.08 0.94*** -0.12*** 0.30*** .85 24 0.08** 0.38*** 0.60 0.25** 0.82*** .71 -0.05 0.96*** -0.04 0.59*** .85 25 0.03 0.36*** 0.41 0.37** 0.80*** .53 0.08 0.93*** -0.05 0.58*** .64
The betas of the SmB and HmL factors are mostly significant and positive. Compared to the other portfolio sorts the FF(1993) sort improves the coefficients of the SmB and HmL factor as should be expected.
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Table 5. Cross sectional regressions for 25 portfolio sorted on FF93 model coefficients.
Model α(%) λmarket(%) λSmB(%) λHmL(%) avg R2
Unlevered CAPM 0.15 0.13 .28
(4.33)** (0.96)
Traditional CAPM 1.04 -0.37 .11
(3.01)*** (-0.90)
Fama and French -0.31 0.81 0.14 0.68 .37 (-0.76) (1.71)* (0.88 ) (4.16)***
The lambda of the market is significant for the FF93 model exclusively, and only weakly significant at the 10% level. The lambda relating te the SmB factor is not significant, although the lamba of the HmL Factor is highly significant at a 1% level. The lambda of the HmL factor thus proves to have more explanatory power.
In conclusion to visualize the fit the realized excess return is plotted versus the predicted excess return, this plot is shown in appendix D. The R2 of the unlevered CAPM is 9 % , the
R2 of the traditional CAPM is 2% and lastly the R2 of the FF93 model is 44%. The pricing errors are noticeably larger for the unlevered CAPM and traditional CAPM. The R2 of the FF93 model is with 44% the highest R2. The GRS tests show significant for each model, and non-significant risk premiums at the 5% level, I am unable to state which model outperforms the other models based on the size and book-to-market sort.
5.4 The different tests discussed
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6. Conclusion
The unlevered CAPM is not without doubt the best performing and most effective asset pricing model when tested on the S&P 500. The unlevered CAPM and traditional CAPM do not pass all the GRS tests, although at least they pass the GRS test when the test portfolios are selected based on their own betas. In the GRS test, the FF93 model exhibits a large number of pricing errors. The robustness of the FF93 model decreases therefore. The betas are mostly erratic when the test portfolios are sorted on the criteria provided by other models, Dam and Qiao(2016) find the same results with respect to the betas. The unlevered CAPM beta sort results of the cross section are comparable to Dam and Qiao(2016), however in the FF93 model the HmL factor is no longer significant. The traditional CAPM beta sort cross sectional regression does not provide a significant market risk premium, and therefore does not explain variation in the average stock returns and the origin of the average stock returns does not become clear. The traditional CAPM exhibits a negative market risk premium. So investors pay to face risk, which would be contra dictionary as investors as investors need to be
rewarded for risk. When sorting for the FF93 coefficients, the HmL factor is the best predictor in the cross section, and the market is only marginally significant. The HmL factor drives the model mostly in this case. The HmL factor is seen as a financial distress factor and thereby reflecting leverage. The SmB factor drops out as an significant factor in the cross section. The most effective model does not come forth from this thesis, nevertheless the unlevered CAPM does generate less unexpected results, and is more robust when tested.
The omission of financial firms potentially improve the validity of the models. Financial firms are included in the sample, and financial companies present a challenge due to their structure. A financial firm consists out of financial claims on other firms or assets, this blurs their leverage ratios. The true leverage ratios of the financial firms cannot be calculated, as all the claims on both sides of the balance of financial firms are not known. The assumption
25
considered an intermediate sample size. A smaller sample size possess a new challenge, and provide more insight in the behavior of the unlevered CAPM. Forming test portfolios than becomes harder, less assets are available in a smaller sample size. Other tests have to be performed to ensure robustness and validity.
26
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Appendix A
Table 1. The descriptive statistics of the portfolios sorted on the unlevered CAPM betas.
Pf# Sorted by Unlevered CAPM
Unlevered excess
return Excess return Leverage
βu Mean(%) S.D(%) Mean(%) S.D.(%) Mean(%) S.D.(%)
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Table 2. The descriptive statistics of the portfolios sorted on the traditional CAPM betas.
Pf# Sorted by traditonal CAPM
Unlevered excess
return Excess return Leverage
βu Mean (%) S.D(%) Mean(%) S.D.(%) Mean(%) S.D.(%)
31
Table 3. The descriptive statistics of the portfolios sorted on the FF93 betas.
Pf# Sorted by FF(1993) Model
Unlevered excess
return Excess return Leverage
βu Mean (%) S.D(%) Mean(%) S.D.(%) Mean(%) S.D.(%)
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Appendix B. Realized and actual excess returns based on 25 portfolios sorted on the unlevered CAPM betas.
Traditional CAPM FF93 model R² = 0.6304 R² = 0.6304 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 P re dict ed ex ce ss re turn ( %)
Realized excess return (%)
R² = 0.0532 R² = 0.0532 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 P re dict ed ex ce ss re turn ( %)
33
Appendix C. Realized and actual excess returns based on 25 portfolios sorted on the traditional CAPM betas.
Unleverd CAPM Traditional CAPM R² = 0.4414 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 P re dict ed unlev er ed excess re tirn (%)
Realized unlevered excess return(%)
R² = 0.3344 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 P re dicet ed ex ce ss re turn ( %)
34 FF93 model
Appendix D. Realized and actual excess returns based on 25 portfolios sorted on the FF93 models betas. Unlevered CAPM R² = 0.0755 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 P re dict ed ex ce ss re turns ( %)
Realized excess returns (%)
R² = 0.0938 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 P re dict ed un lev er ed ex ce ss re turns ( %)
35 Traditional CAPM FF93 model R² = 0.0201 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 P re dict ed ex ce ss re turns (%)
Realized excess returns (%)
R² = 0.4414 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 P re dict ed ex ce ss re turns ( %)
