Assessing the Performance of the Unlevered Capital
Asset Pricing Model for the United Kingdom
EMMA J. KRUIJSWIJK s2200317
University of Groningen Faculty of Economics and Business
MSc. Finance Supervisor: Dr. L. Dam
January 2016
Abstract
The leverage effect on equity returns obscures the return-beta relationship derived from the CAPM. Dam and Qiao (2015) propose an unlevered CAPM based on unlevered betas and unlevered returns, which is not affected by the firm’s capital structure and therefore removes the leverage effect. This thesis compares the performance of the unlevered CAPM with the traditional CAPM and the Fama and French (1993) three-factor model for 2,127 U.K. firms over a period from January 1988 to December 2014. The results are inconclusive and do not identify a best performing asset-pricing model. The unlevered CAPM outperforms the two benchmark models only for the portfolio sorted based on unlevered betas. Nonetheless, the unlevered CAPM turns out to be the most robust model among the different sets of portfolios.
1. Introduction
There is no consensus yet on which asset-pricing model is best in describing the relationship between risk and expected return. The most well-known asset-pricing models, the CAPM and the Fama and French (1993) three-factor model, are questioned for their empirical and theoretical justification, respectively. Dam and Qiao (2015) suggest an unlevered CAPM and test the model using a sample of 7,563 firms originated in the United States. The unlevered CAPM adopts unlevered returns and unlevered betas, thereby focusing on asset returns and ignoring the financial structure of the firm. Asset returns are the equivalent of unlevered returns. Similarly, asset betas are also called unlevered betas. In case a firm is fully financed with equity, unlevered returns and unlevered betas are similar to regular (levered) returns and betas. Levered returns are also known as equity returns and levered betas are the same as equity betas. Dam and Qiao (2015) compare the performance of their model with that of the traditional CAPM and Fama and French (1993) three-factor model and find that the unlevered CAPM significantly outperforms the benchmark models in the U.S. In this thesis the methodology of Dam and Qiao (2015) is applied to the United Kingdom in order to test whether their results hold outside the U.S. Specifically, this thesis examines whether the unlevered CAPM is a superior asset-pricing model compared to the traditional CAPM and the Fama and French (1993) three-factor model for U.K. stocks.
the financial structure of the firm. Consequently, the unlevered CAPM eliminates the leverage effect on returns and thereby the potential bias in the pricing errors.
In order to test the asset-pricing ability of the unlevered CAPM for the U.K. market, this thesis applies the same approach as in Dam and Qiao (2015). They develop an unlevered CAPM in which the unlevered excess returns are calculated by multiplying the levered excess returns with the lagged firm leverage ratio. The firm’s leverage ratio is defined as the market value of total equity divided by the market value of total assets. The leverage ratios as well as the levered excess returns are calculated on a monthly basis. The performance of the asset-pricing models is then tested through two regressions. First, time-series regressions are performed and these are combined with the Gibbons, Ross, and Shanken (1989) (GRS) test. Second, cross-sectional regressions are performed using the Fama-MacBeth (1973) two-step procedure.
The comparison of the three asset-pricing models is performed for firms operating in the U.K. market. The U.K. market is chosen because it is located in the western part of Europe, which countries are relatively similar to the U.S., and because it has many data available for a reasonable time span. The final dataset includes 2,127 U.K. firms that are examined over a period from January 1988 to December 2014, covering 324 months. The performance assessment is executed for the complete sample of individual firms and four different portfolio sorts, consisting of either 25 or 10 U.K. portfolios each.
asset-pricing model. Nevertheless, the unlevered CAPM is the most robust model among the different sets of portfolios.
By testing the performance of the unlevered CAPM in the U.K. market and comparing it to two other asset-pricing models, this thesis contributes to existing literature in three ways. First, it tests the performance, empirical validity and robustness of the two most well-known asset-pricing models, the traditional CAPM and the Fama and French (1993) three-factor model. Second, it compares the models’ performances with the performance of the unlevered CAPM. Third, it tests the applicability of the unlevered CAPM outside the U.S., specifically, in the U.K. To avoid confusion the term ‘traditional CAPM’ is used for the original CAPM and ‘unlevered CAPM’ is used for the leverage adjusted CAPM throughout the rest of this thesis.
This thesis is structured as follows. Section two reviews the existing literature on the main asset-pricing models together with their empirical results. Section three discusses the methodology and clarifies the consequential hypothesis. In Section four describes the data and presents the relevant descriptive statistics. Section five provides the results, which are discussed concisely in Section six. Lastly, a final conclusion of the thesis is provided in Section seven.
2. Literature review
model is its simplicity. However, the numerous assumptions that it makes results in the model being an unrealistic asset-pricing model that is not applicable to real-world investing.
Black et al. (1972) perform several empirical tests for the traditional CAPM and find that the model is scientifically justified and that the beta factor is an important determinant of stock returns. However, a more recent study by Dempsey (2013), involving a re-examination of this research, reveals that the data does not provide an empirical justification of the traditional CAPM, as claimed before by Black et al. (1972).
On the other hand, Brown and Walter (2013) criticise Dempsey (2013) by his assumption that the scientific evidence is valid. Brown and Walter mention that this assumption is inconsistent with Roll’s critique on the traditional CAPM. Roll (1997) argues that is impossible to test the traditional CAPM since one is unable to prove that the model’s underlying assumption, that portfolios are mean-variance efficient, is correct. The market portfolio should include the returns of all possible investment opportunities, but not all of them are observable. Consequently, it is impossible to test whether the market portfolio is mean-variance efficient and the empirical results of the traditional CAPM are invalid. Black and Walter (2013) further state that efficient benchmarks are required to ensure the validity of the traditional CAPM tests and these are yet proven indefinable. Consequently, they claim that Dempsey’s work is invalid and the traditional CAPM might still be an appropriate asset-pricing model.
Fama and French (2004) also state that even though the CAPM might not be empirically justified, it remains a fundamental concept in portfolio theory and asset pricing and must be used to build on more complicated models. However, they also warn that ‘whether the model’s problems reflect weaknesses in the theory or in its empirical implementation, the failure of the traditional CAPM in empirical tests implies that most applications of the model are invalid’. This quote suggests that the traditional CAPM is an imperfect asset-pricing model, but currently is the best foundation for other asset-pricing models as there is a lack of a superior model.
In response to the criticism on the traditional CAPM, various enhancements of the model are proposed. The most common one is the Fama and French (1993) three-factor model. Fama and French (1992) claim that their model explains more than 90% of the time variation in portfolios’ realized returns, while the traditional CAPM clarifies on average 70%. The Fama and French (1993) three-factor model adds size and value three-factors in addition to the market risk three-factor in the traditional CAPM. The size factor is called Small-minus-Big (SmB) and is related to a firm’s market capitalization. It measures the historic excess returns of small caps over large caps. The value factor is called High-minus-Low (HmL) and measures the historic excess returns of value stocks over growth stocks. The model thereby accounts for the tendency of small caps stocks to outperform large caps stocks and value stocks to outperform growth stocks. The origin of this outperformance tendency is not quite clear and there exist two opposed opinions. On the one hand, in the situation of market efficiency, small cap and value firms face excess risk resulting from additional business risk and a higher cost of capital and therefore obtain higher returns. On the other hand, in case of market inefficiency, market participants misprice the value of small cap and value firms and this provides investors of small cap and value stocks with excess returns in the long run, when their true value is discovered.
Blanco, 2012; Rossi, 2012) compare the traditional CAPM and the Fama and French (1993) three-factor model for various markets worldwide and conclude that the Fama and French (1993) three-factor model is superior to the traditional CAPM. However, several other researchers (see, e.g., Eraslan, 2013; Odera, 2013) whom apply the same model and research approach find that the model is limited in its potential to explain variations in portfolio returns.
The differences in results are yet unaccounted for, but two factors that potentially influence the model’s results are examined. First, studies such as Kandel and Stambaugh (1995), Lewellen, Nagel and Shanken (2010), and Blanco (2012) find that the performance of an asset-pricing model varies significantly depending on how test portfolios are constructed. Second, Griffin (2002) discovers that the Fama and French factors are country specific. From his research covering Canada, Japan, the U.S. and the U.K. he concludes that local factors rather than global ones should be included when applying the Fama and French (1993) three-factor model. Nonetheless, despite taking these two factors into account, there is still no consensus on whether the Fama and French (1993) three-factor model is an accurate asset-pricing model.
Another well-known extension of the traditional CAPM is the Intertemporal CAPM (ICAPM) developed by Merton (1973). This model differs from the traditional CAPM in the sense that it is a consumption based asset-pricing model that takes into account how investors participate in the market over time. The rationalization of the model is that investors usually participate in the market for longer time periods and therefore wish to construct portfolios that hedge against uncertain changes in the future investment opportunity set. The traditional CAPM includes only a single factor that does not account for these uncertain changes and therefore is an incomplete measure of the risks investors face. The ICAPM is criticised for the assumption that consumer expectations are homogenous, implicating that it does not consider individual risk preferences. Furthermore, the empirical justification of the model is doubtful. Several researchers (see, e.g., Lutzenberger, 2014; Maio, and Santa-Clara, 2012) find that numerous multifactor models do not satisfy the ICAPM restraints and also cannot be justified by the ICAPM.
textbook leverage effect, which states that over time there is a positive correlation between financial leverage and both equity betas and equity return volatility (see e.g., Hillier, Grinblatt, and Titman, 2011; Schwert, 1989; Nelson, 1991). Second, the interaction between financial leverage ratios and the risk premium (Jangannathan and Wang, 1996). The intuition behind this interaction is that when the market is in a downward trend, the debt/equity ratio will increase while investors will simultaneously require a higher risk premium. The opposite occurs when the market is in an upward trend. This source of leverage effect is more present within value firms than growth firms (Choi, 2013). In case of an economic downturn both leverage and asset betas increase in value, leading to an increase in equity betas. Growth firms are less sensitive to economic conditions than value firms and therefore their leverage increases less resulting in relatively more stable equity betas.
This leverage effect on equity returns obscures the return-beta relationship in the traditional CAPM in the following manner. Changes in financial leverage cause equity betas to vary over time and financial leverage is correlated with the risk premium, resulting in a correlation between equity betas and the risk premium over time as well. This correlation potentially yields substantial pricing errors in the traditional CAPM (see e.g., Jagannathan and Wang, 1996; Lewellen and Nagel, 2006).
bias in the alphas. Consequently, the unlevered CAPM is theoretically a better model than the traditional CAPM.
Dam and Qiao (2015) examine the scientific validity of the unlevered CAPM for individual firms and various test portfolios in the U.S. Furthermore, they compare the performance of the unlevered CAPM with the performance of the traditional CAPM and the Fama and French (1993) three-factor model for this market. The study presents promising results: the unlevered CAPM outperforms the two benchmark models in the U.S. This result indicates that the unlevered betas justify the cross-sectional variation in average unlevered stock returns of U.S. firms. Moreover, the results are robust among various portfolio sorts. Since the theoretical underpinnings of the unlevered CAPM are independent of the geographical location of firms, the same results are expected for the U.K. sample. Dam and Qiao (2015) further investigate the relationship between unlevered betas and levered returns by implementing a two-beta model that accounts for both the leverage-risk premium interaction and portfolio selection. However, this aspect will not be addressed in this thesis as it is outside its scope.
3. Methodology
asset-pricing models are compared based on their performance for a large set of individual firms and four different portfolio sorts.
3.1. The unlevered CAPM
This thesis uses the unlevered CAPM model that is developed by Dam and Qiao (2015). This model implies that the unlevered excess return is equal to the levered excess return times the lagged firm leverage ratio:
(1) 𝑅!
!,!− 𝑟! = 𝑅!,! − 𝑟! !!!,!!! !,!!! ,
where 𝑅!
!,! − 𝑟! is defined as the unlevered excess return for asset i at time t,
𝑅!,! − 𝑟! as the levered excess returns for asset i at time t, 𝐸!,!!! as the lagged market value of total equity for asset i at time t, and 𝐾!,!!! as the lagged market value of total assets for asset i at time t. Similarly, the unlevered market excess return is equal to the levered market excess return times the lagged market leverage ratio. The superscript u added to the returns in this equation highlights that the returns are unlevered returns. The unlevered returns that are calculated using equation (1) are used to measure the unlevered betas. For this equation (2) is used, which is an adjustment to the traditional CAPM equation through replacing the levered returns and levered betas by the unlevered returns and unlevered returns, respectively:
(2) 𝑅!
!,!− 𝑟! = 𝛼!,+ 𝛽!! 𝑅!!,!− 𝑟! + 𝜀!,!,
where 𝛼!, and 𝛽!! are the coefficients of asset i to be estimated, 𝑅!!,!− 𝑟! is the
unlevered market excess return, and 𝜀!,! is the equity-specific residual of asset i at time t. The unlevered beta measures the volatility of a security that is financed solely by equity, with respect to the market.
3.2. Alternative models
the levered betas are obtained using the equations proposed by Black et al. (1972) and Fama and French (1993), respectively:
(3) 𝑅!,!− 𝑟! = 𝛼!,+ 𝛽! 𝑅!,!− 𝑟! + 𝜀!,!,
(4) 𝑅!,!− 𝑟! = 𝛼!,+ 𝛽! 𝑅!,!− 𝑟! + ℎ!𝐻𝑚𝐿 𝑡 + 𝑠!𝑆𝑚𝐵 𝑡 + 𝜀!,!,
where 𝐻𝑚𝐿 𝑡 and 𝑆𝑚𝐵 𝑡 are defined as the value (High-minus-Low) and size (Small-minus-Big) factors, respectively, and ℎ! and 𝑠! are their coefficients. 3.3. Time-series regressions
For every asset three time-series regressions will be performed on a monthly basis, one for each asset-pricing model. For the unlevered CAPM equation (2) is used and for the traditional CAPM and the Fama and French (1993) three-factor model equation (3) and (4) are used, respectively. In case of the unlevered CAPM the unlevered excess returns are regressed on the unlevered market excess returns to obtain the alphas and unlevered betas. The same applies for the traditional CAPM and the Fama and French (1993) three-factor model using levered excess returns and levered market excess returns to obtain the alphas and levered betas for the market factor and the HmL and SmB factors.
3.4. GRS test
The GRS test is developed by Gibbons et al. (1989) and is used to examine whether the estimated intercepts from a multiple regression model are jointly zero. This thesis performs a multivariate linear panel regression in which I try to explain stock returns with respect to its exposure to factor return series. Theoretically, a good factor model will have an intercept statistically indistinguishable from zero. In the GRS equation the intercepts are displayed as the alphas. Hence, in statistical terms the null hypothesis of the GRS test states that the alphas of the asset-pricing models are jointly and significantly zero:
where α is the alpha of asset i or portfolio p.
In the asset-pricing models the alphas represent the pricing errors. Consequently, in verbal terms the null hypothesis states that the asset-pricing model is able to price all assets correctly. The pricing errors are obtained by performing the time-series regressions using equation (2), (3) and (4).
It is outside the scope of this thesis to go into depth about the origination of the test and therefore the GRS equation is assumed as given:
(6) 𝐹!"#= !(!!!!!)!(!!!) [!(!!!!∑!!!!)],
(7) 𝜃! = 𝑟/𝑠,
where 𝐹!"# is the F-statistic of the GRS test, T is the number of time-series observations on returns, N is the number of assets, 𝛼 are the alphas, ∑!! is the
unbiased residual covariance matrix, 𝑟 is the sample mean of the excess returns on the portfolios and 𝑠! is the sample variance of the excess returns on the
portfolios.
3.5. Cross-sectional regressions
In this thesis the Fama-MacBeth (1973) two-step procedure is used for the cross-sectional regressions. The first step involves the previously described time-series regressions to determine the assets’ betas for every risk factor. For this equation (2) is used for the unlevered excess returns, and equation (3) and (4) are used for the levered excess returns of the traditional CAPM and the Fama and French (1993) three-factor model, respectively. In the second step, the risk premium for each factor is determined by regressing the time average of all assets’ returns on the estimated betas. These cross-sectional regressions are performed using equation (8). It determines whether the factors are priced risk factors, which is reflected by the significance of lambda, and whether the asset-pricing model is able to explain the cross-sectional variation in average unlevered returns, reflected by the intercept (alpha) being zero.
(8) 𝐸 𝑅!,!− 𝑟! = 𝛼! + 𝛽!!𝜆,
where 𝛼! is the intercept (alpha) of asset i, 𝛽!! is the estimated beta of asset i and
𝜆 reflects the factor price(s). This test is performed on a monthly basis for both the set of individual firms and the portfolio sets. The test output is corrected for standard errors and provides the average of the alpha, lambda, and R2.
Lastly, to visualize the cross-sectional fit of the models the actual average returns are plotted against the models’ predicted ones. The related R2 of the
simple OLS cross-sectional regression is used as a natural measure of fit. In case of an optimal model the observations should be evenly centred around the 45-degree line, resulting in low pricing errors.
3.6. Portfolios
obtained from the time-series regressions. The main portfolio is sorted based on unlevered betas, the others are sorted based on the traditional CAPM betas, on the betas of the Fama and French (1993) three-factor model, and on industry classes. The first three portfolios consist of 25 portfolios each, Portfolio 1 containing the stocks with the lowest betas and Portfolio 25 covering the stocks with the highest betas. The industry portfolio comprises 10 portfolios that represent different industry classes. The sub portfolios (either 25 or 10) are of roughly equal size in terms of number of assets.
4. Data
Since only yearly data are available for the book value of total assets and equity, a constant debt level is assumed for each year. This enables the transformation of annual data to monthly data. This transformation method is similar to the one used by Dam and Qaio (2015), whom stated that although this not a completely accurate method, the leverage ratio is the variable of interest. Variation in the leverage ratio is mainly caused by variation in equity rather than in debt and therefore the assumption of a constant debt level does not have a substantial affect on the area of interest of this thesis, namely the impact of a firm’s capital structure on asset-pricing models.
In order to acquire all data required for the time-series and cross-sectional regressions several computations are performed with the data obtained. First, the total amount of debt is derived from reducing the book value of total assets by the book value of total equity. Second, the market value of total equity is calculated by multiplying the number of shares with the firm’s share price. The market value of total assets is then computed by adding the total amount of debt to the market value of total equity. Third, the firm leverage ratio is derived by dividing the market value of total equity by the market value of total assets. Fourth, the firm’s excess return is calculated by subtracting the risk-free rate from the firm’s return. According to Dam and Qiao (2015), the unlevered excess return is then obtained by multiplying the firm excess return with the lagged firm leverage ratio.
Table I – Means and Standard Deviations for Key Variables, U.K. January 1988 – December 2014
firm return. Lastly, the unlevered market excess return is calculated by multiplying the market excess return by the lagged market leverage ratio.
Several observations are omitted from the dataset. Firstly, all observations including missing values for any of the variables are removed. Second, Datastream does not always represent delisted firms as missing data, but rather repeats the last market price in the dataset. Firms that delisted during the time period are identified by a return of exactly zero for two months in a row, and are consequently dropped. Third, excess returns that fall below the 1% or above the 99% quantile are considered outliers and removed. Fourth, highly levered firms with a leverage ratio below zero or above one and shortly lived firms with a threshold of 60 months are eliminated from the dataset. This results in an unbalanced panel data set of 2,127 U.K. firms for the period January 1988 to December 2014, which covers 324 months.
Table II – Cross-Sectional Regressions for Individual U.K. Firms, January 1988 – December 2014
leverage effect that levered returns are on average higher and more volatile (see e.g. Hillier, Grinblatt, and Titman, 2011; Schwert, 1989; Nelson, 1991). Furthermore, for both samples the leverage ratio is approximately 21% on average and the standard deviation is substantial, nearly 5% for the market and more than 21% for the individual firms. Accordingly, the leverage ratio varies considerably, corroborating the expectation that leverage might have an important impact in explaining differences in average returns.
5. Results 5.1. Individual firm tests
The comparison of the models’ performance for the set of individual firms is solely based on the Fama-MacBeth cross-sectional regressions, as the GRS test is not suitable for this unbalanced sample. Besides, the Fama and French (1993) three-factor model is not tested for the individual firms sample due to data constraints. The results of the test are presented in Table II. Note that the unlevered CAPM uses unlevered excess returns and presents unlevered betas while the traditional CAPM uses levered excess returns and displays levered betas.
-0,02% and 0,12%, respectively. This result implies that, in case of the unlevered CAPM, assets will generate the risk-free rate when the firm’s unlevered return is uncorrelated with the unlevered market return. The same applies for the traditional CAPM considering levered returns and levered market returns. On the other hand, both models show an insignificant market risk premium equal to 0,22%, indicating poor performance of the models. The market risk premium represents the excess return rate for (un)levered assets and is remarkably small, at about 2,66% annually. Comparing this estimated risk premium with the historical annual average of 4,75% over the time period, I conclude that for both models the security market line is too flat.
These results do not support the hypothesis that the unlevered CAPM is superior to the traditional CAPM on the individual firm level. However, the traditional CAPM is not convincing either in terms of its performance. Besides, the performance of the Fama and French (1993) three-factor model is not tested and compared to the other models yet. For this reason, the same test and an additional GRS test are applied to the three models in order to determine whether the unlevered CAPM is an accurate asset-pricing model in comparison to the traditional CAPM and the Fama and French (1993) three-factor model. The tests are performed for three sets of 25 portfolios and one set of 10 portfolios. 5.2. Unlevered CAPM
Table III – Betas, Pricing Errors, and GRS tests for 25 Portfolios Based on Unlevered Beta Sorts, U.K. January 1988 – December 2014
10% to 81% with standard deviations varying from 5% to 18%. The fact that the leverage ratios include considerable variation corresponds with the hypothesis. Table III shows the individual pricing errors (𝛼), betas, and R2s for each portfolio
Table IV – Cross-Sectional Regressions and GRS Tests for 25 Portfolios Based on Unlevered Beta Sorts
Fama and French (1993) three-factor model fails the test at a 5% significance level.
Table IV presents the results of the Fama-MacBeth (1973) two-step procedure for the unlevered CAPM beta sort. The results do not show an obvious best asset-pricing model. First, all models show an insignificant alpha, the unlevered CAPM presenting the one closest to zero. However, the unlevered CAPM also displays a low and insignificant market premium, as well as the traditional CAPM. Although the Fama and French (1993) three-factor model has a significant market risk premium at a 1% level, the sign is negative. Besides, the SmB and HmL factor betas of the Fama and French (1993) three-factor model are insignificant, indicating that they are not priced. Furthermore, the SmB factor beta has a negative sign, suggesting poor performance of the model.
Lastly, Figure 1 visualizes the relationship between the actual and predicted values of the average unlevered returns. The portfolios are reasonably well centred around the 45-degree line, hence on average the pricing errors are low. Furthermore, the model shows a high cross-sectional fit with an R2 of 60%.
Figure 1 – Actual and Predicted Unlevered Returns of 25 Portfolios Sorted on Unlevered CAPM Betas
average there is more variation in the pricing errors and the cross-sectional fit is also poorer (R2=38%). The Fama and French (1993) three-factor model exhibits
the poorest fit as the average returns hardly fit and the R2 is fairly low (22%).
Overall, for portfolios sorted based on unlevered betas the unlevered CAPM and the traditional CAPM perform roughly equal, the unlevered CAPM showing a slightly better cross-sectional fit. The Fama and French (1993) three-factor model clearly performs the poorest. These results support the hypothesis, stating that the unlevered CAPM is superior to its benchmark asset-pricing models.
5.3. Robustness to alternative test portfolios
Table V – Cross-Sectional Regressions and GRS Tests for Three Alternative Portfolio Sorts, U.K. January 1988 – December 2014
Table V presents the results of the second stage cross-sectional regressions of the Fama-MacBeth (1973) two-step procedure and the GRS-test statistics for the traditional CAPM beta sort, the Fama and French (1993) portfolio sorts, and industry sorts, respectively. These models include levered betas and therefore indirectly test the relation between unlevered betas and the average unlevered returns. The descriptive statistics of the three portfolio sorts are shown in Table A2-A4 in the Appendix.
5.3.1. Traditional CAPM
Table VI – Betas, Pricing Errors, and GRS test for 25 Portfolios Based on Traditional Beta Sorts, U.K. January 1988 – December 2014
majority of the SmB factor betas are significant, but most of the HmL factor betas are not. This implies that the size effect has greater explanatory power than the value effect. However, both factors show a few significant betas that have a negative sign, which is not in line with the model. Regarding the GRS test, all models exhibit an insignificant F-value for the GRS statistic and thereby all pass the test.
and traditional CAPM are insignificant and rather low. The betas of the SmB and HmL factors in the Fama and French (1993) three-factor model are insignificant, implying both are not priced factors. The HmL factor beta even shows a negative sign and a value close to zero, suggesting poor performance of the Fama and French (1993) three-factor model.
Lastly, for all models the relation between the actual and predicted values of the average (un)levered returns is pictured in Figure B2 in the Appendix. All models exhibit a moderate to high average of the pricing errors. Remarkable is that the traditional CAPM shows the poorest cross-sectional fit with an R2 of 12%
compared to the unlevered CAPM with an R2 of 17% and the Fama and French
(1993) three-factor model with an R2 of 25%. I would expect the traditional
CAPM to display the best fit as the portfolios are sorted based on the traditional CAPM betas. Overall, the results do not point out one best model for the portfolios based on traditional CAPM betas sorts, and the hypothesis cannot be accepted nor rejected.
5.3.2. Fama and French (1993) three-factor model
Table VII – Betas, Pricing Errors, and GRS test for 25 Fama and French (1993) Portfolios, U.K. January 1988 – December 2014
Lastly, Figure B3 in the Appendix plots the value of the average actual (un)levered returns against the average predicted ones. The unlevered CAPM shows on average the relatively smallest pricing errors while at the same time presenting a non-existing fit between the average returns (R2=0%). The
traditional CAPM and Fama and French (1993) three-factor model display on average moderate to high pricing errors and a cross-sectional fit of 20% and 56%, respectively.
Overall, all models fail empirically for both the GRS test and the Fama-MacBeth (1973) test and results are again inconclusive. The outcomes for the Fama and French (1993) portfolios sorted on the betas of the SmL and HmL factors do not point out an unequivocal best performing asset-pricing model. As a consequence, no strong evidence is provided supporting the hypothesis that the unlevered CAPM outperforms the benchmark models.
5.3.3. Industry
The industry portfolio incorporates 10 portfolios, each portfolio representing a different industry class. The industry classification is presented in Table C1 in the Appendix. Table VIII shows the results of the time-series regression. All individual pricing errors are insignificant and close to zero and all market betas are significant at a 1% level and have a high value. The majority of the betas for the SmB and HmL factors are significant at 1% level, implying reliable explanatory power. Conversely, a few SmB and HmL factor betas show a negative sign, which does not correspond with the model. Furthermore, all models fail the GRS test at a 10% or 5% significance level.
Table VIII – Betas, Pricing Errors, and GRS test for 25 Portfolios Based on Industry Sorts, U.K. January 1988 – December 2014
traditional CAPM even shows a negative market premium. The Fama and French (1993) three-factor model performs the ‘least bad’ at this aspect, with a market premium of 0,45%. The SmB factor beta is positive but insignificant, suggesting its size effect might be diminishing. The HmL factor beta is negative and insignificant, which questions the accurateness of the model.
The graphs showing the relation between the values of the average actual and predicted (un)levered returns are presented in Figure B4 in the Appendix. All models show on average high pricing errors and the cross-sectional fit is the lowest for the unlevered CAPM (R2=9%), followed by the traditional CAPM
(R2=28%) and the Fama and French (1993) three-factor model (R2=37%). In
general, all three models perform poorly for portfolios sorted based on industry classes. Subsequently, there is again no evidence that supports the hypothesis that the unlevered CAPM outperforms the other models.
5.4. Robustness of models
traditional CAPM betas, the unlevered CAPM passes the test. However, for the portfolios sorted based on the levered betas of the SmB and HmL factors and the ones sorted based on industry classes, the model fails the test. Although this diversity is undesirable, the two benchmark models show the same variety regarding this test. Relative to the other models, the cross-sectional fit is the only measure that causes variety among the results of the unlevered CAPM, R2
ranging from 0% to 60% among the different portfolios, which is substantial. The traditional CAPM shows more variation in its results. The alpha differs from being insignificant and close to zero to being significant and different from zero depending on the manner in which a portfolio is sorted. Furthermore, the market premium is negative for the portfolio sorted based on industry classes whereas it is positive for the other portfolios. Besides, it also differs per portfolio sort whether the market premium is significant or not. The Fama and French (1993) three-factor model also exhibits weak results in terms of robustness. The value of the alpha varies from being close to zero to being relatively high at 1,28 for the portfolio sorted on industry classes. The alpha is also significant for the portfolio sorted on the levered betas of the Fama and French (1993) three-factor factors, while being insignificant for all other portfolio sorts. The market premium and the SmB and HmL factor betas change in terms of its sign and significance among the different portfolios. Overall, the manner in which portfolios are sorted has a greater impact on the performance of the traditional CAPM and the Fama and French (1993) three-factor model than on the unlevered CAPM. Consequently, although the results do not present a best asset-pricing model, the unlevered CAPM is superior to the other models in terms of its robustness.
6. Discussion
performs fairly well. The model fails the test at a 5% significance level only for the portfolio based on Fama and French (1993) sorts. The traditional CAPM fails the GRS test for the portfolio based on Fama and French (1993) sorts at a 1% significance level and the Fama and French (1993) three-factor fails the test for all portfolios except the ones sorted based on traditional CAPM betas. Hence, the unlevered CAPM outperforms the benchmark models based on the GRS test. Furthermore, in Dam and Qiao (2015) the unlevered CAPM fails the GRS test at a 1% significance level for the portfolio based on Fama and French (1993) sorts and at a 5% significance level for the portfolio sorted based on traditional CAPM betas. Consequently, in terms of the GRS test this thesis provides better results for the unlevered CAPM than Dam and Qiao (2015). However, based on the Fama-MaBeth (1973) cross-sectional regressions all asset-pricing models perform poorly. The market premium of the unlevered CAPM is for every portfolio set insignificant and nearly zero. The traditional CAPM and Fama and French (1993) three-factor model perform evenly poor or slightly better.
limited research factor contributing to the unexpected result that the unlevered CAPM does not outperform the two benchmark models in the U.K.
Furthermore, consideration must be given to two other aspects of this thesis that could potentially improve its quality. First, one could argue that the companies operating in the financial sector should be eliminated from the dataset. Despite the fact that financial companies have their own, publicly known leverage ratio, their balance sheets mainly consist of financial claims on other firms or assets. For these firms or assets the underlying leverage ratio is unobservable and therefore the ‘true’ leverage ratio of financial firms is unknown. This caveat is recognized and further research is encouraged to exclude financial companies, thereby uncovering the effect of financials on unlevered betas and unlevered returns. Second, one could criticize the methodology of calculating the market leverage. Taking the mean of all firms’ leverage ratios to derive the market leverage, as performed in this thesis, implies that firm sizes are disregarded. A solution could be to appoint a weight to every firm based on its assets value. In this manner, the leverage ratio of large firms contributes more to the market leverage ratio than that of smaller firms. Further research should examine whether this change in approach would have a significant impact on the results. Despite these caveats and the fact that the hypothesis is not confirmed in this thesis, the presented results still have a valuable contribution to the world of finance. The theoretical foundations regarding the firm’s capital structure and the impact of its related leverage effect remain valid, as proven by the lower means and standard deviations of unlevered excess returns compared to levered excess returns and the high leverage ratios presented descriptive statistics. These results provide a basis for further research.
7. Conclusion
is an interesting adjustment to the traditional CAPM since it is not affected by a firm’s capital structure. The model does not involve a leverage effect on equity returns as there is no correlation between equity betas and the risk premium over time. This avoids potential pricing errors.
The performance of the unlevered CAPM is tested and compared with the performance of two benchmark models, the traditional CAPM and the Fama and French (1993) three-factor model. The methodology of Dam and Qiao (2015) is applied to measure the unlevered excess returns, which are derived by multiplying the levered excess returns with the lagged firms’ leverage ratios. The firm’s leverage ratio is defined as the market value of total equity divided by the market value of total assets. The unlevered betas are calculated using a transformation of the standard formula for the traditional CAPM. The levered excess returns are replaced by the unlevered excess returns and the levered betas are substituted by the unlevered betas. The comparison of the three models is executed for a set of individual firms and four portfolios and is based on time-series regressions, the GRS test, the cross-sectional regressions based on Fama-MacBeth (1973), the goodness of fit of the models and the robustness of their results among the different portfolios.
Fama and French (1993) three-factor model. Nonetheless, the unlevered CAPM is the most robust model among the different portfolios.
As discussed, the results might not support the hypothesis due to the short time period. The same research applied to a longer time span will be valuable and might confirm the hypothesis. Reanalysing the data set of this thesis by excluding financial companies from the sample and including a market leverage ratio that accounts for firm’s size would also be interesting.
References
Abbas. N., Khan, J., Aziz, R., Sumrani, Z., 2015. A study to check the applicability of Fama and French, three-factor model on KSE 100-Index from 2004-2014. International Journal of Financial Research 6, 90-100.
Black, F., Jensen, M., Scholes, M., 1972. The capital asset pricing model: some empirical tests. In: M.C. Jensen (E.d), Studies in the Theory of Capital Markets. New York, pp. 79-124.
Blanco, B., 2012. The use of CAPM and Fama and French three factor model: portfolios selection. Public and Municipal Finance 1, 61-70.
Brown, P., Walter, T., 2013. The CAPM: theoretical validity, empirical intractability and practical applications. A Journal of Accounting, Finance, and Business Studies 49, 44-50.
Campbell, H., Yan, L., Heqing, Z., 2014. … and the cross-section of expected returns. Unpublished working paper. Duke University, Durham.
Carhart, M., 1997. On persistence in mutual fund performance. The Journal of Finance 52, 57-82.
Choi, J., 2013. What drives the value premium?: The role of asset risk and leverage. Review of Financial Studies 26, 2845-2875.
Czapkiewicza, A., Wójtowicza, T., 2014. The four-factor asset pricing model on the Polish stock market. Economic Research 27, 771-783.
Dempsey, M., 2013. The capital asset pricing model (CAPM): the history of a failed revolutionary idea in finance? A Journal of Accounting, Finance, and Business Studies 49, 7-23.
Eraslan, V., 2013. Fama and French three-factor model: evidence from Istanbul stock exchange. Business and Economics Research Journal 4, 11-22.
Fama, E., French, K., 1992. The cross-section of expected stock returns. The Journal of Finance 47, 427-465.
Fama, E., French, K., 1993. Common risk factors in the returns on stocks and bonds. Journal of Financial economics 33, 3-56.
Fama, E., French, K., 2004. The capital asset pricing model: theory and evidence. Journal of Economic Perspectives 18, 25–46.
Fama, E., MacBeth, J., 1973. Risk, return, and equilibrium: empirical tests. Journal of Political Economy 81, 607-636.
Gibbons, M., Ross, S., Shanken, J., 1989. A test of the efficiency of a given portfolio. Econometrica 57, 1121-1152.
Griffin, J., 2002. Are the Fama and French factors global or country specific? Review of Financial Studies 15, 783–803.
Hillier, D., Grinblatt, M. Titman, S., 2011. Financial markets and corporate strategy.
Kandel, S., Stambaugh, R., 1995. Portfolio inefficiency and the cross-section of expected returns. The journal of Finance 50, 157-184.
Lewellen, J., Nagel, S., 2006. The conditional CAPM does not explain asset-pricing anomalies. Journal of Financial Economics 82, 289-314.
Lewellen, J., Nagel, S., Shanken, J., 2010. A skeptical appraisal of asset-pricing tests. Journal of Financial Economics 96, 175-194.
Lintner, J., 1965. The valuation of risk assets and the selection of risky
investments in stock portfolios and capital budgets. Review of Economics and Statistics 47, 13-37.
Lutzenberger, F., 2014. Multifactor models and their consistency with the ICAPM: evidence from the European stock market. European Financial Management 21, 1014-1052.
Maio, P., Santa-Clara, P., 2012. Multifactor models and their consistency with the ICAPM. Unpublished working paper. Hanken School of Economics and Nova School of Business and Economics.
Markowitz, H., 1952. Portfolio selection. Journal of Finance 7, 77-91.
Mossin, J., 1996. Equilibrium in a capital asset market. Econometrica, 34, 768-783.
Nelson, D., 1991. Conditional hetroskedasticity in asset returns: A new approach. Econometrica 59, 347-370.
Odera, J., 2013. The validity of Fama and French three factor model: evidence from the Nairobi securities exchange. Unpublished working paper. University of Nairobi, Nairobi.
Roll, R. (1977). "A Critique of the Asset Pricing Theory’s Tests". Journal of Financial Economics 4: 129–176.
Rossi, F., 2012. The three-factor model: evidence from the Italian stock market. Research Journal of Finance and Accounting 3, 151-160.
Schwert, W., 1989. Why does stock market volatility change over time? The Journal of Finance 44, 1115-1153.
Sharpe, W., 1964. Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance, 19 425-442.
Treynor, J., 1961. Market value, time, and risk. Unpublished manuscript.
Treynor, J., 1999. Toward a theory of market value of risky assets. In: Korajczyk, R. (Ed.), Asset Pricing and Portfolio Performance: Models, Strategy and
Performance Metrics. London, United Kingdom, pp. 15-22.
University of Exeter, The Business school – Xfi centre for Finance and Investment: http://business-school.exeter.ac.uk/research/areas/centres/xfi/
Appendix
B.1. – Actual and Predicted Unlevered Returns of 25 Portfolios Sorted on
Unlevered CAPM Betas
Traditional CAPM Fama and French (1993)
B.2. – Actual and Predicted Unlevered Returns of 25 Portfolios Sorted on Traditional CAPM Betas
Unlevered CAPM Traditional CAPM Fama and French (1993)
B.3. – Actual and Predicted Unlevered Returns for 25 Fama and French (1993) Portfolios Sorted on Size and Book-to-Market Ratio
Unlevered CAPM Traditional CAPM Fama and French (1993)
B.4. – Actual and Predicted Unlevered Returns Statistics for 25 Portfolios Sorted on Industry Classification
Unlevered CAPM Traditional CAPM Fama and French (1993)
C.1. – Classification of the 10 Industry Portfolios
