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The unlevered CAPM – evidence from Germany

VIRGINIA J. WASIKOWSKA*

June 2016

ABSTRACT

Variation in financial leverage is a primary source of cross-sectional differences in average stock returns. The unlevered CAPM of Dam and Qiao (2015) eliminates the influence of leverage and is able to capture variation in average returns left unexplained by the standard CAPM and Fama and French (1993) model. Using standard test procedures, this study investigates the empirical performance of the unlevered CAPM in the German market. Results are highly reliant on the applied portfolio formation method and do not unequivocally identify the best performing model. The unlevered CAPM performs best in the unlevered and levered beta sorts, however these results are not robust.

Keywords: Leverage Effect, Unlevered Equity Return, CAPM, Asset Pricing.

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I. Introduction

Central to the entire asset pricing discipline is the CAPM developed by Sharpe (1964) and Lintner (1965), which played an instrumental role in shaping the way practitioners and academics alike think about risk and return. The model embodies the theory, that differences in average returns across firms, can be explained by variations in market exposure. For nearly half a century now, the CAPM has been an object of numerous empirical tests, and despite a lack of full consensus, most researchers now agree that, empirically, market beta has no power to explain cross-sectional differences in average stock returns (Fama and French, 1992; Frazzini and Pedersen, 2010; Baker, et al., 2011; Asness, et al., 2013). Most of these studies have addressed this issue by proposing various multifactor extensions to the basic model, however, there is a new and growing body of literature that reconsiders the standard CAPM in light of financial leverage and its impact on the covariance of stock returns with the market (Choi, 2013; Dam and Qiao, 2015; Doshi et al., 2015; Rienks, 2015). These studies focus on the idea that market-induced fluctuations in financial leverage cause variation in market betas, leading to the observed cross-sectional dispersion in average returns. They confront this phenomena, by transforming equity returns into unlevered asset returns, which represent return on real capital of the firm under the assumption that it is financed entirely by equity. This research provides new evidence with respect to this approach by comparing the performance of the unlevered CAPM with the standard CAPM and Fama and French (1993) three-factor model.

Throughout this paper I argue that cross-sectional variation in financial leverage is a source of cross-sectional differences in average stock returns, thus leading to the failure of standard CAPM and emergence of various asset pricing anomalies. The unlevered CAPM eliminates the influence of leverage, and therefore, is able to capture cross-sectional variations in average returns left unexplained by the standard CAPM and Fama and French (1993) three-factor model. The key objective of this study is to investigate the empirical performance of the unlevered CAPM in the German market. More specifically, I test the following hypotheses:

Hypothesis 1: standard CAPM holds (pricing errors are zero leaving no unexplained patterns in average returns).

Hypothesis 2: Fama and French (1993) three-factor model holds (pricing errors are zero; SMB and HML factors capture effects of size and value anomalies leaving no unexplained variation in average risk-adjusted returns).

Hypothesis 3: unlevered CAPM holds (pricing errors are zero, leaving no unexplained cross-sectional variation in unlevered average returns).

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non-U.S. sample. Asset pricing literature to date has investigated the impact of leverage mainly by including it as a factor in a multifactor setup (for review and criticism see, e.g. Harvey, et al., 2016). Research centred on unlevering the returns first and then modelling them is still fairly new and empirical evidence regarding this approach is sparse, especially outside of the U.S. It is particularly interesting to evaluate the model in the context of Germany due to differences in dominant corporate financial structures between Germany and the U.S. Indeed, in the past the distinction between bank-based and market-bank-based financial systems has been often exemplified by comparative studies of Germany and the U.S. Traditionally, bank debt constitutes the primary source of financing for German companies, compared to predominance of equity financing in the U.S. (Doupnik and Perera, 2015). In general, German companies tend to be more leveraged than their counterparts from the U.S. and rely more on intermediated bank debt, as opposed to public debt financing (Doupnik and Perera, 2015). Table 1 shows a brief comparison of stock markets and banking sectors in Germany and the U.S. Total market capitalizations and numbers of domestic listed companies indicate that German stock market has historically been less developed relative to the U.S., while at the same time banking sector was comparatively large. With German companies on average carrying more leverage, I anticipate that their leveraged returns are even more affected by the leverage effect, and hence less representative of the true asset returns. Therefore, I expect differences in the performance of the unleveraged and leveraged CAPM to be more pronounced in my German sample, compared to the U.S. sample of Dam and Qiao (2015).

Additionally, this study contributes to the existing debate on empirical performance of the standard CAPM and Fama and French (1993) in the German market, by providing new evidence. The significance of this study and its implications for practitioners should also be recognised, as currently

Table 1. Comparison of stock markets and banking sectors in Germany and the U.S.

Germany United States

1990 2001 2010 2014 1990 2001 2010 2014 Listed domestic companies per

million of population 5.2 9.1 8.4 7.3 25.8 21.7 13.8 13.7 Market capitalization of listed

domestic companies (% of GDP) 20.1 54.9 41.8 44.9 51.7 131.7 115.5 151.2 Domestic credit to private sector by

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the standard CAPM, despite of its empirical failure, is one of the most commonly used methods for estimating cost of equity (Jagannathan and Meier, 2002).

In the current study, I follow the methodology proposed by Dam and Qiao (2015) and obtain unlevered excess returns and betas by multiplying leveraged excess returns by a 1-period lag of the leverage ratio of a given company. Leverage ratio is defined as the market value of equity divided by the market value of total assets. These unlevered returns then serve as inputs in the unlevered version of CAPM. Throughout this research I employ a sample of 545 German companies listed on the Frankfurt Stock Exchange in the period from January 1990 to December 2014. I asses the empirical performance of the unlevered CAPM and the two benchmark models by applying Black-Jensen-Scholes (1972) test on individual time-series regressions, its multivariate extension known as the Gibbons-Ross-Shanken (1989) test, and the cross-sectional test of Fama-MacBeth (1973). I perform the assessments for the main sample of individual firms, the extended sample including 118 companies from the financial and real estate sector, and four various portfolio sorts.

The results that emerge from my study suggest that performance of the models highly depends on the applied portfolio formation method. Overall, findings from all portfolio sorts are not very encouraging and do not unequivocally identify the best performing model. The unlevered CAPM appears to perform best in the unlevered and levered beta sorts, although these results are not robust. In general, Fama and French (1993) model demonstrates inconsistent behaviour of size and value factors, with significance and sign of the premiums varying across different sorts. Standard CAPM exhibits the weakest performance, confirming its inability to explain cross-section of average returns. Lastly, findings from tests on individual company level do not provide additional insight, as both the unlevered CAPM and the standard CAPM achieve similar results.

Throughout the rest of this paper I refer to the standard CAPM of Sharpe (1964) and Lintner (1965) as “standard CAPM” or simply “CAPM”, and to the version developed by Dam and Qiao (2015) using unlevered returns as “unlevered CAPM”.

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II. Background

The standard CAPM is the most widely researched model for modelling stock returns and a benchmark in most empirical studies. Ever since its construction by Sharpe (1964) and Lintner (1965) its application has sparked a vigorous debate. Despite its heroic assumptions, the model provides an important insight into the relationship between average return on an asset and its beta.

A considerable asset pricing literature has documented evidence inconsistent with the model and the current consensus in academic community is that the CAPM is an empirical failure. Most specifically, the CAPM has failed on two main fronts. First, the existence of the so-called beta anomaly - portfolios of low-beta stocks are observed to have higher average returns than predicted by the model, while portfolios of high-beta stocks show lower average returns (Frazzini and Pedersen, 2010; Baker, et al., 2011). Second, the existence of other systematic patters in returns - factors able to predict average returns independently of the systemic risk of the stocks, most notably the value and momentum anomalies (Asness, et al., 2013). The inability of the CAPM to explain these asset pricing anomalies has instigated the development of other approaches and extensions.

Multiple studies resorted to consideration of multi-factor asset pricing models, which attempt to explain cross-sectional variation in average returns by decomposing them into a linear function of a benchmark model and additional risk factors (for review and criticism see, e.g. Harvey, et al., 2016). Financial literature has considered tradable factors based on various firm characteristics, such as Fama and French (1993) SMB and HML factors, as well as momentum (Carhart, 1997), liquidity (Acharya and Pederson, 2005) or geographic location (García and Norli, 2012). Yet another group of studies focused on macroeconomic factors, such as labour income (Jagannathan and Wang, 1996), industrial production, inflation rate, credit risk spread, and disposable income (Chen et al., 1986). Despite their empirical success, the proliferation of different multi-factor models generated controversy regarding their economic justification, which is often unclear. Many factors were challenged on the grounds that they arise from spurious relationships and data mining and that their results are sensitive to different portfolio-weighting schemes (Harvey, et al., 2016).

Another branch of asset pricing research has proposed that inability of the standard CAPM to explain cross-sectional variation in stock returns arises from the false assumption that betas are constant over time. Variation of betas across time has been widely documented (Fama and French, 1997, 2006; Lewellen and Nagel, 2006; Ang and Chen, 2007) giving raise to various versions of conditional CAPM, which attempt to capture this variation. The results from this studies are often contradictory and there is still no consensus on the ability of conditional CAPM to address many asset pricing anomalies.

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to the observed cross-sectional dispersion in average returns. The theoretical underpinnings for the leverage effect link to Modigliani and Miller's (1958) propositions: more debt increases the volatility of company returns in relation to market fluctuations. Although debt level is usually considered relatively stable, changes in market value of equity have an influence on the level of leverage, which in turn affects the covariance between company returns and the return on the market portfolio. When such market-induced changes in firms’ leverage ratios occur during the beta estimation window, significant measurement errors may arise (Drobetz, et al., 2014).

In the study employing U.S. data, Choi (2013) examines the relationship between financial leverage and both market risk premium and market volatility in the context of business cycle risk. He finds that unlevered (asset) betas and leverage increase during economic downturns, which leads to spikes in levered (equity) betas. This effect is primarily observed in value stocks, which face higher disinvestment costs and tend to carry more debt. On the other hand, growth firms are typically less levered due to high proportion of intangibles and higher present value costs of financial distress, thus their asset betas are less sensitive to economic conditions. This relative stability of equity betas of growth companies means that value firms become relatively more risky at times when the risk premium is high, thus giving raise to the value premium. Countercyclical variations in levels of leverage translate into greater variability of levered (equity) betas, which might be a reason behind empirical failure of the CAPM when applied to levered returns. Doshi et al. (2015) design a somewhat similar approach, but focus on capturing the asymmetric relationship between leverage and leveraged returns, which is subject to unlevered return being positive or negative. They find that value premium mostly disappears when unlevered stock returns are used. In a related study, Dam and Qiao (2015) propose the unlevered CAPM with unlevered betas constructed by employing firms’ leverage ratios. They examine the performance of the model against the standard CAPM of Sharpe (1964) and Lintner (1965), as well as the Fama-French (1993) three factor model, both of which use leveraged betas. Based on a large sample of U.S. companies, they find solid support for the leverage effect hypothesis, as evident through good and robust performance of their unlevered CAPM.

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then constructed by value-weighting individual returns of those three components. This method of estimating unlevered returns, although robust, is not easily transferable to other samples. Technique proposed by Dam and Qiao (2015) is analogical to that of Choi (2013), however, it assumes the risk-free rate as a proxy for debt return. Authors argue, that this approach is likely to produce less precise estimates of unlevered betas, but with sufficiently long sample period and adjustment for outlying leverage and book leverage observations, it is sufficient for applications in cross-sectional studies on average stock returns. In general, due to considerably lower data requirements, the method of Dam and Qiao (2015) offers a wider range of applications, particularly for samples where corporate bond returns are not easily available.

Doshi et al. (2015) employ various structural credit risk models, such as that of Merton (1974), to compute unlevered return series using techniques commonly applied in option pricing. This method takes into account the asymmetry in the relationship between leverage and levered returns, by viewing equity as a call option on the firm’s assets. In addition to addressing the leverage effect, this method captures the interaction between leverage and default risk, while in their linear approximation Dam and Qiao (2015) essentially assume no default risk. In his paper on unlevered CAPM with Dutch sample, Rienks (2015) uses a similar methodology, but applies the maximum likelihood method to infer the volatility of unlevered equity returns, while Doshi et al. (2015) use Itô's lemma. The latter approach assumes constant volatility of levered returns, which is argued to produce estimates inconsistent with the applied model (Duan, 1994; Ronn and Verma, 1986). Both approaches are computationally complex and time-consuming as the required calculations are performed individually for each year and for each security.

Performance of the CAPM in Germany

To better position this research in the context of German market, the following is a brief review of findings from frequently cited studies evaluating empirical performance of the standard CAPM and multifactor models using German data.

A paper by Artmann, et al. (2012b) looks at empirical performance of the standard CAPM versus Fama and French (1993) three-factor model and Carhart’s (1997) momentum model in various samples of underlying assets. They use a unique and extensive dataset, which extends from 1960 to 2006 and includes all companies listed on Frankfurt Stock Exchange during that period. Based on Gibbons-Ross-Shanken (1989) test, they conclude that none of the three models tested is able to consistently explain cross-sectional differences in average returns, with Carhart’s (1997) model exhibiting the best results. Adapting a somewhat broader perspective, authors strongly advocate in favour of tests on non-U.S. data. They argue that as most studies employ the same U.S. dataset, all are affected by the same patterns in the underlying data, thus producing little new insight.

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in expected returns. Using Fama and MacBeth (1973) testing procedure, they find that equity book-to-market ratio and earnings-to-price ratio have considerable explanatory power and produce significant factor risk premiums. The ability of both of these ratios to capture value effect is in line with the leverage effect hypothesis discussed earlier in this section (Choi, 2013; Dam and Qiao, 2015). In general, Artmann, et al. (2012a and 2012b) document strong relationship between the empirical performance of the models and the selection of underlying test assets.

Brückner, et al. (2012) conduct an extensive study of the standard CAPM using companies listed in the top segment of Frankfurt Stock Exchange between 1960 and 2007 (sample period very similar to that of Artmann, et al. (2012b)). They perform a robust analysis including various portfolio sorting methods, weighting schemes, frequencies of return data, and lengths of sample periods. They show that results vary substantially, especially with the portfolio grouping and weighting procedures. They find greatest support for the CAPM in the full sample (47 years), and contrary to Artmann, et al. (2012a), argue that evidence against CAPM’s ability to explain cross-sectional differences in average stock returns is weak in case of Germany. In addition, Brückner, et al. (2012) also investigate the existence of size and value anomalies, and document their highly erratic behaviour (for instance, in their paper size effect reverses from significantly positive prior to 1990 to significantly negative afterwards).

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III. Research design

A. Methodology

Throughout this paper I compare empirical performance of the unlevered CAPM of Dam and Qiao (2015) with the standard (leveraged) CAPM, and the Fama-French (1993) three factor model for the German market. I employ three most commonly used testing procedures for evaluating asset pricing models: the test of Black, et al. (1972) on individual time-series regressions; its multivariate extension introduced by Gibbons, et al. (1989); and the cross-sectional test of Fama and MacBeth (1973).

The Models

In their unleveraged CAPM model, Dam and Qiao (2015) construct unleveraged excess returns, as a product of a company’s leveraged excess return and its lagged leverage ratio:

,

− =

,

,, (1)

where , − is the unleveraged excess return for company at time ; , − is the leveraged

excess return; , is the lagged market value of equity for company ; and , is the

lagged market value of assets for company . Analogically, they define the unlevered excess return on a market portfolio as a product of levered excess return on a market and a one-period lag of the market leverage ratio. Unlevered excess return on individual companies and the aggregate market (further identified by a subscript ) are then used as inputs in the standard CAPM model, to produce estimates of unlevered market betas. Thus, the unlevered CAPM derived by Dam and Qiao (2015) takes the form:

,

− =

+

,

+

, (2)

where , − is the unleveraged excess return for company at time ; , − is the unleveraged

excess return of the aggregate market portfolio at time ; the coefficient is the unlevered market beta, which measures the sensitivity of the company's returns to the market returns, irrespective of its capital structure. It quantifies the systematic risk of the company, under the assumption that it is financed fully by equity. The alpha ( ) is an estimated intercept.

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,

− =

+

,

+

,

(3)

,

− =

+

,

+

+ ℎ

+

,

(4)

where , − is the levered excess return of company at time ; , − is the leveraged excess

return of the aggregate market portfolio at time ; coefficients , , and ℎ measure the company’s exposure to the market and Fama and French (1992) size and book-to-market factors, respectively. Black-Jensen-Scholes (1972) test

The first method used for evaluating performance of the models is the Black-Jensen-Scholes (1972) test on individual time-series regressions. The BJS test works by comparing the actual excess return to the theoretical excess return predicted by a given model and examining the resulting pricing errors. For the unlevered CAPM these pricing errors1 are defined as:

=

,

,

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where , − is the actual unleveraged excess return; and , − is the unlevered excess

return predicted by the model. The test is carried out by estimating time-series regressions based on Eq. (2-4) and conducting a standard -test to evaluate statistical significance of the intercepts (pricing errors). Naturally, a good asset pricing model is expected to produce pricing errors statistically indistinguishable from zero.

As pointed out by Black, et al. (1972), the test is inefficient when evaluated on the level of an individual company, because it ignores the large amount of data available from time-series regressions of other firms. To circumvent this problem, the authors advise aggregating stocks into portfolios, which I perform in this study based on multiple portfolio sorts. In addition, I informally evaluate performance of the models in the sample of individual firms, by looking at the number of statistically significant pricing errors across the whole sample and their average absolute magnitude.

Time-series regressions are estimated using OLS method with Newy-West (HAC) heteroscedasticity and autocorrelation consistent standard errors, to ensure consistency with the assumptions of BJS testing procedure (Black, et al., 1972; Newey and West, 1987).

Gibbons-Ross-Shanken (1989) test

A criticism of the BJS test given by Gibbons, et al. (1989) is that in each time period residuals from time-series regressions for individual portfolios exhibit cross-sectional interdependence, and thus, cases when only some portfolios produce significant pricing errors are difficult to interpret. To

1 In studies evaluating performance of mutual fund managers, this measure is also known as Jensen’s alpha, after

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address this issue, in addition to performing the BJS test I employ its multivariate extension known as the Gibbons-Ross-Shanken (1989) test. The GRS test evaluates the hypothesis that alphas from time-series regressions in Eq. (2-4) are jointly zero. Rejection of the null hypothesis means that intercepts are jointly statistically different from zero, which violates the assumption that market portfolio is mean-variance-efficient and indicates poor performance of the model under consideration.

The test requires a balanced sample and that the number of assets or portfolios is smaller than the number of time-series observations. This prohibits me from applying this test to the sample of individual companies, therefore I employ it only in tests on various portfolio sorts.

Fama-MacBeth (1973) test

Furthermore, I use a two-stage testing procedure developed by Fama and MacBeth (1973) to test whether unlevered excess market return is a priced risk factor, and if so, estimate the average risk premium resulting from exposure to this factor. The advantage of this procedure is that it is able to accommodate unbalanced data, which enables me to apply it both on the level of individual companies and portfolios. The first stage involves running time-series regressions in Eq. (2-4) for each of assets. From this stage I extract vectors of estimates of full-sample factor exposures. Fama and MacBeth (1973) apply 5-year rolling regressions, as opposed to full-sample beta estimates, but the latter is more suited for small samples (Lettau and Ludvigson, 2001), and Fama and French (1992) document that the application of full-sample betas generally produces the same inferences as rolling betas.

In the second stage, for each time period I compute cross-sectional regressions of the unlevered excess returns on the factor exposures from the previous stage, where is the number of time periods in the first stage time-series regressions. For the unlevered CAPM, this allows to test whether larger exposure to the (unlevered) market factor translates into higher unlevered excess return on the firm:

,

= +

,

+

, (6)

where , − is a transposed matrix of unlevered excess returns from the previous stage; , are

vectors of risk premium coefficients across time; and is the vector of unlevered betas for assets obtained from time-series regressions. Analogically, the cross-sectional regressions corresponding to the standard CAPM and Fama and French (1993) three-factor model take the form:

,

= +

,

+

, (7)

,

= +

,

+

,

̂ +

,

ℎ +

, (8)

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Cross-sectional regressions corresponding to all three models employed in this study are estimated using the OLS framework with the Newy-West (HAC) heteroscedasticity and autocorrelation consistent standard errors (Newey and West, 1987). This adjustment is necessary when applying the Fama-MacBeth (1973) testing procedure, because any autocorrelation in excess returns (which at the monthly frequency is typically negligible) can lead to considerable autocorrelation in gamma estimates (Peterson, 2009; Goyal, 2011).

From the second-stage regressions, I obtain average risk premiums ( ) of the individual gammas ( , ) from the cross-sectional regressions, as well as average . Significant

indicates that a given risk factor is indeed priced, and contains information on average excess returns, which is not captured by exposures to the remaining factors. Thus, Fama-MacBeth (1973) testing procedure can also be considered a test for factor redundancy. This allows me to address one of the shortcomings of the standard CAPM outlined above, namely the existence of size and value anomaly. Also, an average intercept ( ) not significantly different from zero indicates that the model is able to fully explain the cross-sectional variation in average returns2.

B. Data

My initial dataset consists of all publicly traded German companies, listed on Frankfurt Stock Exchange at any point in time during the sample period. Therefore, my sample excludes any companies that trade exclusively on other exchanges, which introduces bias when it comes to coverage of the whole German market. Nevertheless, this bias is overall small and diminishes towards the end of the sample (Stehle and Schmidt, 2015, note that in 1995, only 65% of German securities traded in Frankfurt, but 89% in 2008).

The main source of company data is Thomson Reuters Datastream and Worldscope Financial Database, from which I extract company price, number of shares outstanding, book value of assets and book value of equity (plus deferred taxes). Codes and definitions of these series are presented in Appendix C. My sample spans the period from January 1990 until December 2014, which is mainly driven by the quality of data. Brückner (2013) analyses the quality of German equity data available via Datastream, and concludes that before 1990 Datastream is not a reliable source due to coverage issues and systematic errors. The sample ends in December 2014, because for many companies Worldscope is missing book values of equity and book values of assets after that date. With monthly frequency, that gives a 300-month window (25 years).

It is also important to note that throughout this paper I use simple company returns calculated from the price series observations. This choice was necessitated by severe data quality issues in total return

2 Note that the intercept ( ) in the Fama-MacBeth (1973) two-pass procedure is sometimes defined as a return

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series available via Datastream for Germany (also observed by Brückner, 2013). Although this choice might deflate observed returns and limit comparability of my findings, I argue that it will not have a big impact on my analysis, which focuses on cross-sectional differences in average returns (Agrrawal and Borgman, 2010).

Treatment of preference shares

There are two types of share classes in Germany: Stammaktien (common stock) and Vorzugsaktien (non-voting stock). Unlike in the U.S., where preference shares exhibit bond-like characteristics, German non-voting stock is more comparable to common equity and has similar risk-return features (see Stehle and Schmidt, 2015, for more detailed discussion on the topic). Brückner (2013) argues that exclusion of preference shares in German datasets can introduce coverage issues and bias market capitalization estimates, which are crucial for my calculation of firms’ leverage ratios. Also, for non-U.S. corporations, Worldscope includes non-voting stock in the book value of equity (Table C1, Appendix C). I include both common equity and preference shares in my sample and treat them as a single unit. Hence, the analysis in this paper focuses on individual firms and not on individual securities. For the market value of equity calculation I consolidate market values of individual share classes. For the return calculation the returns of individual securities are value-weighted by multiplying them by their respective market values, expressed as a percentage of the combined market value of the company. I match security level records with main company records by Worldscope Identifiers (codes: W06035 and W06036). Brückner et al. (2015b), also argues for the inclusion of the unlisted share classes in calculations of firms’ size, however I do not make this adjustment due to data quality concerns.

Data filters

After matching multiple share classes to individual companies, I impose several filters on the dataset. First, I filter out any non-equity securities and listings of foreign companies (German companies are identified by ISIN numbers with “DE” prefix). I also remove any companies with less than 36 monthly observations. Subsequently, I screen the dataset for inferred missing observations by looking at the number of identical consecutive monthly prices, and remove companies with significant proportion of data points missing. I also exclude any penny stocks, as such securities tend to be very illiquid and are often subject to price manipulation, which goes against the CAPM assumption of efficiently priced securities. Most studies define penny stocks as companies whose price falls below 1 Euro, however, Stehle and Schmidt (2015) point out that German stock exchanges do not automatically delist stocks which trade below 1 (unlike other stock exchanges, e.g. NASDAQ). The authors give example of

Infineon Technologies AG, which despite being traded below 1 Euro in

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companies that trade consistently below 1 Euro (companies with more than 10% of their price observations below 1).

I exclude observations for which any one of the records - price, number of shares outstanding, book value of assets, or book value of equity – is unavailable. Furthermore, following Dam and Qiao (2015), I exclude companies with leverage ratio below 0 and above 1, and book leverage of less than 0 (book leverage ratio is defines as the book value of equity divided by the book value of assets). This has a considerable impact on my sample size (14% reduction in the number of companies), however, it allows to eliminate cases when leverage effect is no longer linear, which would be inconsistent with the applied methodology (Dam and Qiao, 2015). In addition, I remove companies with expected return in the top and bottom percentiles to eliminate the impact of outliers. Finally, I exclude all financial and real estate companies from the main sample, as their true leverage ratio is not observable (they are later included in an extended sample as a robustness check). Companies are classified based on Worldscope Industry Group classification (code: WC06011).

A resulting unbalanced sample constitutes of 545 companies, which seems to adequately represent the German stock market, both in terms on industry and exchange segment coverage. The sample is also largely free of survivorship bias, including 322 active and 223 dead companies, as per Datastream equity status. In terms of exchange segment coverage, the dataset contains 313 companies from Regulated Market (Regulierter Markt), and 106 companies from Free Market (Freiverkehr). For the remaining 126 firms exchange segment information was not available. These are primarily dead companies, for which the series discontinues before the introduction of the latest market segment classification in 2007 (Stehle and Schmidt, 2015, provide an overview of the changes in Frankfurt Stock Exchange segmentation).

Calculation of unlevered returns for individual companies

Before running time-series regressions, I conduct a series of preliminary calculations. After calculating simple returns from company share prices I obtain the estimates of unlevered returns as follows: first, I calculate market value of equity, by multiplying the number of shares outstanding by the official closing price for each security. Because most corporate debt is traded infrequently and over the counter, the market value of total debt is not publicly available. Thus, I use book value of debt, calculated as a difference between the book value of assets and the book value of equity. Value of total debt is added to market value of equity, to arrive at the market value of assets. Finally, leverage ratio is calculated as a quotient of market value of equity and market value of assets.

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volatility of leverage ratios in my sample of individual firms is roughly in line with the one observed by Dam and Qiao (2015), as discussed later in this section.

Construction of market variables

I construct the market excess return series as the value-weighted average of the returns of all individual companies in the sample less the risk-free rate. Analogically, the market leverage ratio is the asset-weighted average of leverage ratios of all individual companies. Finally, I calculate the unlevered excess return for the aggregate market by multiplying the (leveraged) market excess return series by the 1-month lag of the market leverage ratio. All of these series exclude financial and real estate companies form computations, as due to the unique nature of their business their true leverage ratio is not observable. I use a monthly risk-free rate proxy and Fama-French (1993) size and value factors for the German market constructed by Brückner et al. (2015a) and available on the Humboldt University of Berlin website3. Until June 2012, risk-free rate series employs the average of one-month

money market rates reported by Frankfurt banks. Afterwards, the one-month EURIBOR rate is used, as in June 2012 the Deutsche Bundesbank stopped collecting data on the former.

Descriptive statistics

Table 2 shows descriptive statistics for the set of individual companies and the aggregate market. The mean leverage ratio for individual firms is 47% - lower than 62% for the U.S. sample of Dam and Qiao (2015). This is consistent with the theory that debt financing is more prevalent in Germany than the U.S. (note that our leverage ratio is defined as market value of equity divided by market value of assets). Somewhat lower mean leverage ratio of the aggregate market in relation to that of individual firms average, can be attributed to the former being asset-weighted.

For both individual firms and the aggregate market, unlevered excess returns have lower means than the (leveraged) excess returns. Also, in both samples unlevered excess return series show lower standard deviations and narrower ranges, indicating that leveraged excess returns exhibit more variation than unlevered excess returns. For the aggregate market this is visualised in Figure A1 in Appendix A, which plots unlevered and levered excess market return index.

3 The time-series used do not account for the corporate income tax credit (Körperschaftsteuergutschrift). See

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Table 2. Summary statistics for the sample of individual companies and key market variables.

Excess return Leverage Unlevered excess return PANEL A: Individual firms

Observations 86441 86986 86441 Mean (%) 0.36 46.90 0.04 Median (%) -0.28 46.40 -0.05 Standard deviation (%) 12.52 28.44 7.62 Maximum (%) 247.27 99.99 219.11 Minimum (%) -73.54 0.07 -72.91

PANEL B: Aggregate market

Observations 299 300 299 Mean (%) 0.74 28.45 0.23 Median (%) 1.17 32.54 0.17 Standard deviation (%) 5.30 11.72 1.75 Maximum (%) 15.54 46.60 7.16 Minimum (%) -15.19 8.34 -5.64

Note: The table presents descriptive statistics for the unbalanced sample of 545 German companies listed on the Frankfurt Stock Exchange between January 1990 and December 2014. Excess return is a simple price return calculated from monthly share prices less the risk-free rate. Leverage ratio is defined as market value of equity divided by market value of assets. Unlevered excess return is calculated as excess return multiplied by the 1-period lag of the leverage ratio. Monthly prices (adjusted for capital events), number of shares outstanding, book value of assets and book value of equity are obtained from Thomson Reuters Datastream and Worldscope Financial Database. Market value of equity is calculated as number of shares outstanding multiplied by their corresponding closing price. Market value of assets is calculated as the market value of equity plus the difference between book value of assets and book value of equity. A proxy for the risk-free rate is constructed by Brückner et al. (2015a) and available on the Humboldt University of Berlin website. Excess returns for the aggregate market are constructed by value-weighting returns of individual companies. Analogically, market leverage ratio is obtained by asset-weighting leverage ratios of individual companies. The sample includes preference shares which are consolidated with the corresponding primary share by value-weighting their returns series and asset-weighting other company-level series. The sample excludes companies operating in financial and real estate sectors.

Construction of portfolio sorts

Following the established practice in empirical studies of cross-sectional patterns in average returns, I group individual firm series into portfolios. This approach is said to diversify away the idiosyncratic return movements, thus reducing the noise and the impact of outliers on estimation outputs. Moreover, portfolio formation allows to circumvent the challenges of working with unbalanced panel dataset, and thus allowing the application of the GRS statistic.

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Within each sorting method, I construct the excess return series for each portfolio by value-weighting returns of individual portfolio constituents. Analogically, I calculate the leverage ratio for each portfolio, as an asset-weighted average of leverage ratios of its individual firms. The portfolio unlevered excess return is then computed by multiplying the portfolio excess return by a lagged portfolio leverage ratio.

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IV. Results

A. Sample of individual firms

I begin the analysis by examining the main sample of 545 individual companies. For each of the three models tested - unlevered CAPM, standard CAPM, and Fama and French (1993) three-factor model - I run 545 time-series regressions involving excess returns of individual companies. I then estimate Fama-MacBeth (1973) cross-sectional regressions and evaluate the performance of the models tested.

Table 3 presents descriptive statistics summarising results from time-series regressions. For indicative purposes, distributions of coefficient estimates are also provided in Figure A2, Appendix A. Disappointingly, the unlevered CAPM indicates the poorest performance based on BJS tests. Most importantly, the model produces the largest number of pricing errors, which are statistically different from zero at 5% level: 45 out of 545 regressions, compared to 29 for the standard CAPM and 24 for the Fama and French (1993) model. On average, the model also yields pricing errors of the largest magnitude (in absolute terms), as indicated by the average intercept equal to -0.27% versus -0.25% for the standard CAPM and -0.17% for the Fama and French (1993) model. These findings are somewhat disheartening, as for a good asset pricing model, we would expect pricing errors (intercepts) to be insignificant and close to zero on average.

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Table 3. Time-series regressions for the sample of individual companies.

Unlevered CAPM Standard CAPM Fama and French (1993)

(%) (%) (%) ℎ Mean -0.27 1.16 0.11 -0.25 0.73 0.13 -0.17 0.72 0.18 0.10 0.15 Median -0.14 0.75 0.08 -0.16 0.63 0.10 -0.11 0.62 0.15 0.12 0.12 Standard deviation 0.84 1.22 0.12 1.12 0.52 0.12 1.20 0.52 0.35 0.43 0.12 Maximum 3.86 7.25 0.60 4.29 2.89 0.63 7.49 2.90 2.83 2.93 0.63 Minimum -6.36 -1.77 0.00 -6.36 -0.87 0.00 -5.98 -1.26 -0.90 -1.63 0.00 No. of significant 45 374 29 413 24 410 59 59 % significant 8% 69% 5% 76% 4% 75% 11% 11%

Note: This table summarises the results of the 545 time-series regressions for each of the three models: unlevered CAPM, standard CAPM, and Fama and French (1993) three-factor model, which take the general form:

, − = + + ,

For standard CAPM and Fama and French (1993) model the dependent variable is monthly excess returns of individual companies, while unlevered CAPM uses monthly unlevered excess returns, which are constructed by multiplying (levered) excess returns by the 1-period lag of the leverage ratio. Leverage ratio is defined as market value of equity divided by market value of total assets. The risk factor used for the unlevered CAPM is the unlevered excess return on a market portfolio. Standard CAPM uses (levered) market excess return, while Fama and French (1993) also adds and factors. The table also gives the amount of coefficient estimates significant at 5% level, expressed as a percentage of all estimated regressions. The regressions employ an unbalanced sample of 545 German companies listed on Frankfurt Stock Exchange in the period from January 1990 to December 2014. Companies operating in financial and real estate sectors are omitted from the sample. The company-level data is obtained from Thomson Reuters Datastream and Worldscope Financial Database. German equivalents of Fama and French (1993) and factors, as well as the risk-free rate proxy are constructed by Brückner et al. (2015a) and available on the Humboldt University of Berlin website.

To assess the performance of the unlevered CAPM and the proposed two benchmark models on individual firm level, I rely on the Fama-MacBeth (1973) cross-sectional model, as the unbalanced nature of the dataset and high number of assets inhibits the application of GRS statistic.

Table 4 summarises the results of the second-stage Fama-MacBeth regression on the sample. The tests for all three models produce statistically insignificant intercept and significant coefficient. For the unlevered CAPM, the annual market risk premium of 2.16%4 is significant at 10% level. The two

benchmark models produce higher market premiums, 6.13% annually for the standard CAPM and 6.31% for the Fama and French (1993) three-factor model (both significant at 5% level). Market risk premium for the unlevered CAPM is considerably lower, because it represents excess return on companys’ assets assuming that the company is financed solely by equity, while for the remaining two models, market risk premium represents excess return on companys’ equity, which is influenced by leverage. Using the average leverage ratio for aggregate market (see: Table 2), it is possible to calculate the implied (average) annual equity risk premium from unlevered CAPM, as:

. %

.

= 7.59%

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4 Average annual market risk premium is calculated by multiplying the estimated monthly average market risk

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This estimate seems more in line with the remaining two benchmark models. It also appears the closest to the documented historical average for Germany: 8.88% per annum, as shown in Table 2. The remaining gamma coefficients of Fama and French (1993) model show that SMB and HML factors are not priced, as both are not significantly different from zero. The average is near zero for each of the models, indicating that they have virtually no ability to explain the variability of data around their respective means. This finding could possibly be driven by the large amount of noise in factor exposures estimated in the time-series regressions. On the basis of these results, one cannot differentiate between the models in question, although Fama and French (1993) model appears the weakest. Therefore, I sort the dataset on unlevered betas and replicate the test for 16 portfolios, but before that, I conduct a robustness check using the extended sample including financial and real estate companies.

B. Sample including financial and real estate companies

As a robustness check, I also repeat the testing procedure for the sample including companies from financial and real estate sectors. Due to the nature of their business, companies operating in these sectors are likely to carry higher amounts of debt, which is unrelated to their financial structure, and hence less stable (Viale, et al., 2009). Because of that, their true leverage ratio is not observable, and

Table 4. Fama-MacBeth (1973) cross-sectional regressions for the sample of individual companies.

(%) (%) (%) (%) Average

Unlevered CAPM 0.00 0.18*** 0.06

(0.56) (1.42)

Standard CAPM 0.08 0.51** 0.05

(1.19) (1.73)

Fama and French (1993) 0.11 0.53** -0.09 -0.22 0.07

(1.18) (1.78) (-0.45) (-0.29)

Note: This table presents results of the cross-sectional regressions, estimated following the Fama-MacBeth (1973) two-stage testing procedure, which take the form:

, − = + + ,

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the reported ratio is likely to deviate more from the market average. Table A1 in Appendix A shows descriptive statistics for the sample of 663 individual companies, including 545 companies from the previous sample and 118 operating in the financial and real estate sectors. As expected, leverage ratio exhibits lower mean and higher variation compared to the previous sample and the aggregate market (Table 2).

Table A2 in Appendix A shows summary statistics of the 663 time-series regressions for each of the three models: unlevered CAPM, standard CAPM, and Fama and French (1993) three-factor model. Histograms presenting distributions of estimated coefficients are also provided in Figure A2 (Appendix A). The results appear to be mostly in line with insights from my previous sample, but with few notable exceptions. For all three models, I report slightly higher amounts of significant intercepts (at 5% level), which on average exhibit higher magnitudes (in absolute terms). Again, the unleveraged CAPM produces the highest number of statistically significant alphas: 46 out of 663 regressions, compared to 33 and 26 for standard CAPM and Fama and French (1993) model. Additionally, all models show less statistically significant market betas, while average sensitivities to the market are lower than those reported in the previous sample. Similarly, SMB and HML factors are also mostly insignificant, and thus do not help in explaining excess returns of higher number of companies. Higher average pricing errors and less significant market betas corroborate with my expectation that the sample including financial and real estate companies is likely to contain more noise, resulting in worse average performance of the models.

Following the procedure applied to the first sample, I also perform Fama-MacBeth (1973) cross-sectional regressions for the expanded sample, results of which are summarised in Table A3, Appendix A. Unsurprisingly, the results appear less convincing than the baseline sample. Again, all models show average intercepts statistically indistinguishable from zero. Market premiums are somewhat higher than in the previous sample, however only those for the standard CAPM and Fama and French (1993) three-factor model are significant (both at 1% level). Also, these two benchmark models both produce market risk premiums of 7.2% per annum – slightly higher than those estimated from my previous sample. Insignificant market beta in the unlevered CAPM, can be explained by less consistent leverage ratio of financial and real estate firms, which is likely to introduce more unexplained variation in unlevered excess returns. Similarly to the previous sample, both SMB and HML factors are not statistically significant, and thus are not priced risk factors.

C. Portfolio tests based on unlevered beta sort

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and provides a robustness test for my findings. These sorts are based on: (a) unlevered betas, (b) traditional (levered) betas, (c) Fama and French (1992) size and book-to-market values, and (d) industry classification. Dam and Qiao (2015) point out that the first three of the above-listed sorts are likely to favour unlevered CAPM, standard CAPM and Fama-French (1993) model respectively, therefore the results need to be interpreted with caution.

Table B1 in Appendix B presents descriptive statistics for the portfolio sort based on unlevered betas, which is aimed specifically at assessing the relationship between unlevered market betas and mean unlevered excess returns of underlying portfolios of stocks. This sort allows to minimise the measurement error in estimated unlevered beta coefficients, while enduring their wide dispersion. Individual portfolios are ranked in decreasing order, with Portfolio 1 consisting of the companies with the highest unlevered beta. Standard deviation of both unlevered and levered excess returns increases together with unlevered beta of the portfolio, which is in line with expectations.

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Table 5. Time-series regressions and GRS test for 16 portfolios sorted based on unlevered beta. Unlevered CAPM Standard CAPM Fama and French (1993)

GRS = 2.26*** = 1.76** = 1.79** P # (%) (%) (%) ℎ 1 0.47 4.00*** 0.39 0.72 1.96*** 0.46 0.85 1.95*** 0.25 0.10 0.47 2 -0.17* 1.52*** 0.67 -0.38** 1.41*** 0.70 -0.31 1.40*** 0.07 -0.04 0.70 3 -0.01 1.25*** 0.73 0.14 1.26*** 0.70 0.12 1.26*** 0.01 0.06 0.70 4 -0.14 1.35*** 0.69 -0.32 1.16*** 0.71 -0.34 1.17*** -0.10 -0.09 0.72 5 0.00 0.73*** 0.56 0.03 1.00*** 0.67 -0.01 0.99*** 0.11 0.23* 0.68 6 0.13 0.90*** 0.44 0.15 0.86*** 0.48 0.23 0.85*** 0.25** 0.19** 0.50 7 0.02 0.93*** 0.49 0.03 0.89*** 0.58 0.03 0.88*** 0.17** 0.24*** 0.60 8 -0.09 0.45*** 0.30 -0.33 0.74*** 0.38 -0.28 0.73*** 0.13 0.09 0.39 9 0.16 0.49*** 0.18 0.37 0.58*** 0.31 0.42 0.57*** 0.17** 0.14 0.32 10 0.19 0.61*** 0.23 0.45 0.55*** 0.27 0.34 0.55*** -0.10 0.07 0.28 11 0.09 0.53*** 0.25 0.12 0.48*** 0.32 0.05 0.48*** 0.05 0.19** 0.33 12 0.35** 0.54*** 0.13 0.72** 0.40*** 0.15 0.89*** 0.38*** 0.32*** 0.13 0.19 13 0.04 0.26*** 0.05 -0.01 0.25*** 0.08 0.00 0.25*** 0.11 0.11 0.09 14 0.20*** 0.27*** 0.09 0.40** 0.24*** 0.13 0.39** 0.24*** 0.02 0.03 0.13 15 0.03 0.13** 0.04 0.08 0.18*** 0.07 0.03 0.18*** 0.00 0.10 0.07 16 0.24** -0.02 0.00 0.74*** 0.03 0.00 0.74*** 0.02 0.08 0.12** 0.01 Note: This table reports the results of time-series regressions for each of the three models: unlevered CAPM, standard CAPM, and Fama and French (1993) three-factor model, which take the general form:

, − = + + ,

For standard CAPM and Fama and French (1993) model the dependent variable is monthly excess return of each portfolio, while unlevered CAPM uses monthly unlevered excess returns, which are constructed by multiplying (levered) excess returns by the 1-period lag of the leverage ratio. Leverage ratio is defined as market value of equity divided by market value of total assets. The risk factor used for the unlevered CAPM is the unlevered excess return on a market portfolio. Standard CAPM uses (levered) market excess return, while Fama and French (1993) also adds and factors. The regressions employ a balanced sample of 16 portfolios formed based on unlevered market betas ( ) estimated in the time-series regressions of unlevered monthly excess returns of 545 German companies listed on Frankfurt Stock Exchange during the period January 1990 and December 2014. P # designates the portfolio number, with Portfolio 1 consisting of stocks with the highest unlevered beta. The company-level data is obtained from Thomson Reuters Datastream and Worldscope Financial Database. German equivalents of Fama and French (1993) and factors, as well as the risk-free rate proxy are constructed by Brückner et al. (2015a) and available on the Humboldt University of Berlin website.

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compered to tests on individual firms, with average of the unlevered CAPM considerably higher than the benchmark models (72% versus 0.43% and 0.52% for the standard CAPM and Fama and French model respectively). In general, results from cross-sectional regressions point at the superior performance of the unlevered CAPM model in this sample.

Table 6. Fama-MacBeth cross-sectional regressions for 16 portfolios sorted based on unlevered betas.

(%) (%) (%) (%) Average

Unlevered CAPM 0.05 0.29*** 0.72

(0.86) (4.55)

Standard CAPM 0.30** 0.59** 0.43

(1.76) (2.11)

Fama and French (1993) 0.20 0.55*** 1.21*** 0.07 0.52

(1.20) (2.34) (2.45) (0.08)

Notes: This table presents the estimation results of Fama-MacBeth (1973) cross-sectional regressions, which takes the general form:

, − = + + ,

Vectors of exposures to factors are obtained from the time-series regressions: unlevered CAPM uses unlevered excess return of the market; standard CAPM uses (levered) excess return on the market; and Fama and French (1993) three-factor model uses (levered) excess return on the market and and factors. Standard CAPM and Fama and French (1993) three-factor model use (levered) excess returns, while Unlevered CAPM uses unlevered excess returns for individual companies, which are constructed by multiplying leveraged excess returns by the 1-period lag of leverage ratio. Leverage ratio is defined as market value of equity divided by market value of total assets. Portfolios are formed from an unbalanced panel of monthly excess returns of 545 German companies listed on Frankfurt Stock Exchange during the period January 1990 and December 2014. The table reports average estimates following from the second stage of the Fama-MacBeth (1973) testing procedure. Reported -statistics were obtained using the Newy-West autocorrelation adjustment with 1-period lag. Statistical significance at 10%, 5%, and 1% is denoted by *, **, and *** respectively. The company-level data is obtained from Thomson Reuters Datastream and Worldscope Financial Database. German equivalents of Fama and French (1993) and factors, as well as the risk-free rate proxy are constructed by Brückner et al. (2015a) and available on the Humboldt University of Berlin website.

D. Tests of additional portfolio sorts

It is interesting to see whether insights from sorting procedure based on unlevered betas are robust across different portfolio sorts. Results from time-series regressions as well as GRS statistics for the remaining three portfolio sorts are presented in Tables B6-B8 in Appendix B. Table 7 below outlines results of Fama-MacBeth (1973) second stage regressions – Panel A displays results for the levered beta sort, Panel B for industry sort, and Panel C for Fama and French (1992) two-way sort. Portfolio sort based on levered betas

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based on the unlevered CAPM, versus 1.91% and 1.90% for the two benchmark models. For all three models, time-series regressions seem to show alphas for individual portfolios mostly insignificant, with unlevered CAPM showing intercepts of the lowest magnitude (in absolute terms). Nevertheless, all models do not pass the GRS test, indicating that their pricing errors are jointly significant (at 1% level) and the tested factors fail to fully explain expected returns of the portfolios. Again, exposures to size and value factors appear significant only for some portfolios, and do not seem to reduce the magnitude of the pricing errors.

Looking at the summary of cross-sectional regressions in Panel A of Table 7, the unlevered CAPM displays the best performance in the levered beta sort. Its intercept is statistically indistinguishable from zero, while annual market risk premium of 3.96% is priced and highly significant at 1% level. As expected, the standard CAPM seems to perform slightly better in this sort, with its intercept now insignificant and market premium of 7.8% per annum (significant at 5%). Performance of the Fama and French (1993) model disappoints: intercept is significant at 10% level, while market risk premium is zero in statistical sense. Both size and value premiums are now highly significant (1% level), however value premium is negative, which is contradictory to the underlying theory that value stocks earn higher risk-adjusted returns than growth stocks. Size premium is higher compared to unlevered beta sort (2.25% versus 1.21%). These results are likely driven by the insignificant market premium for this model. Similarly to the previous sort, the unlevered CAPM shows much better fit than the remaining two models, with average of 82%, compared to 32% and 48% for the standard CAPM and Fama and French (1993) model respectively. Overall, this portfolio sort provides greatest support for the unlevered CAPM.

Portfolio sort based on industry classification

Turning now to evidence from the portfolio sort based on industry classification, estimation results from time-series regressions show that market betas for all portfolios are highly significant at 1% for all models. In addition, for each model tested, betas appear to be less dispersed and more closely around one, compared to the previous sorts. Review of the pricing errors for individual portfolios, points at superior performance of the Fama and French (1993) model, with only one portfolio showing intercept marginally different from zero at 10%. This is further confirmed by the GRS test, which shows that both the standard CAPM and Fama and French (1993) model produce jointly insignificant pricing errors. In contrast, the unlevered CAPM does not pass the GRS test, with alphas jointly different from zero at 1%. This indicates that unlevered market beta is not sufficient to fully explain expected returns of the portfolios. Consistent with findings from previous portfolio sorts, exposures to SMB and HML factors are insignificant for most portfolios.

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underlying financial theory. Average is effectively zero for both the unlevered CAPM and the standard CAPM, thus implying that the models are unable to explain average returns. With regard to the Fama and French (1993) model, average of 19% shows better fit, however this relative outperformance is most likely due to inclusion of additional size and value factors. Findings from this sort should be viewed with caution due to unequal number of companies in individual portfolios (Table B5 in Appendix B illustrates portfolio compositions). In consequence, time-series betas of some portfolios are estimated with lower precision, which is a likely explanation of particularly poor performance of all the models in Fama-MacBeth (1973) cross-sectional test.

Portfolio sort based on size and book-to-market value of equity

Table B8 in Appendix B documents time-series results for the two-way sort based on size and book-to-market of equity. Compared to the previous sorting procedures, here all three models exhibit significant intercepts for considerably more portfolios. This is especially visible in case of Fama and French (1993) three-factor model, which is rather surprising as it was expected to perform best in this sort. Consistently with the above, all three models fail the GRS test producing pricing errors jointly different from zero at 1% level. Again, I document market betas all statistically significant at 1% level. Exposures to size and value factors are significant for roughly half of the portfolios, however without corresponding decrease in the pricing errors. Consistent with the expectation, positive and significant exposure to the SMB factor is observed mostly in portfolios containing smaller stocks, although the pattern is not strong. Similarly, positive and significant exposure to the HML factor emerges mostly in portfolios with higher book-to-market ratio (note specifically Portfolios 1, 5, 9, and 13), which again is in line with the financial theory.

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Table 7. Fama-MacBeth cross-sectional regressions for additional portfolio sorts.

(%) (%) (%) (%) Average

PANEL A: Portfolio sort based on levered betas

Unlevered CAPM 0.00 0.33*** 0.82

(0.00) (4.62)

Standard CAPM 0.30 0.65** 0.32

(1.14) (2.26)

Fama and French (1993) 0.41* 0.41 2.25*** -1.43*** 0.48

(1.48) (1.26) (2.76) (-2.02)

PANEL B: Portfolio sort based on industry classification

Unlevered CAPM 0.22*** -0.02 0.00

(2.89) (-0.21)

Standard CAPM 0.50*** 0.13 0.02

(4.09) (0.78)

Fama and French (1993) 0.53*** 0.13 -0.91* 0.75 0.19

(6.60) (0.90) (-1.43) (1.02) PANEL C: Portfolio sort based on size and book-to-market value of equity

Unlevered CAPM 0.38*** 0.05 0.01

(5.77) (0.49)

Standard CAPM 2.32*** -1.46*** 0.15

(4.48) (-2.38)

Fama and French (1993) 1.04 -0.25 0.84 1.77* 0.32

(0.83) (-0.20) (0.31) (1.50)

Notes: This table reports the estimation results of Fama-MacBeth (1973) cross-sectional regressions, which takes the general form:

, − = + + ,

Vectors of exposures to factors are obtained from the time-series regressions: unlevered CAPM uses unlevered excess return of the market; standard CAPM uses (levered) excess return on the market; and Fama and French (1993) three-factor model uses (levered) excess return on the market and and factors. Standard CAPM and Fama and French (1993) three-factor model use (levered) excess returns, while Unlevered CAPM uses unlevered excess returns for individual companies, which are constructed by multiplying leveraged excess returns by the 1-period lag of leverage ratio. Leverage ratio is defined as market value of equity divided by market value of total assets. Portfolios are formed from an unbalanced panel of monthly excess returns of 545 German companies listed on Frankfurt Stock Exchange during the period January 1990 and December 2014. Each sort involves 16 portfolios. The table reports average estimates following from the second stage of the Fama-MacBeth (1973) testing procedure. Reported -statistics were obtained using the Newy-West autocorrelation adjustment with 1-period lag. Statistical significance at 10%, 5%, and 1% is denoted by *, **, and *** respectively. The company-level data is obtained from Thomson Reuters Datastream and Worldscope Financial Database. German equivalents of Fama and French (1993) and factors, as well as the risk-free rate proxy are constructed by Brückner et al. (2015a) and available on the Humboldt University of Berlin website.

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ones. Overall, the standard CAPM and Fama and French (1993) model exhibit more dispersion in the magnitude of pricing errors, and show greater deviations from the 45° line in sort based on size and book-to-market ratio, as well as leveraged beta sort (for Fama and French model). The observed values corroborate with results obtained from cross-sectional test of Fama-MacBeth (1973), particularly for the unlevered CAPM and the standard CAPM. Large differences between cross-sectional fit across various portfolio sorts are visible for all models, however they are most striking for the unlevered CAPM. Its varies from 82% in the levered beta sort (the highest across all tests) to 0% in the industry classification sort (the lowest across all tests). This contrasts with findings obtained by Dam and Qiao (2015), who document that the unlevered CAPM exhibits the least erratic fit. In general, it seems that there is simply less cross-sectional variation in unleveraged returns to explain in my sample and less dispersion in betas, which might be a possible reason behind this difference. Dam and Qiao (2015) also discuss this effect in their paper and it seems to be more pronounced in my sample.

V. Conclusions

In this study I investigate the empirical performance of unlevered CAPM of Dam and Qiao (2015) in the sample of 545 German companies listed on the Frankfurt Stock Exchange in the period from January 1990 to December 2014. I argue that cross-sectional variation in financial leverage is a reason behind observed cross-sectional differences in average stock returns, which obscures the true relationship between return and beta. By eliminating the influence of leverage, the unlevered CAPM is able to capture cross-sectional variation in average returns left unexplained by the standard CAPM and Fama and French (1993) three-factor model.

Throughout my analysis I employ three commonly used tests to evaluate asset pricing models: Black-Jensen-Scholes (1972) test on individual time-series regressions, its multivariate extension known as the Gibbons-Ross-Shanken (1989) test, and the cross-sectional test of Fama-MacBeth (1973). I perform analysis for the main sample of individual firms, an extended sample including 118 companies from the financial and real estate sector, and four various portfolio sorts.

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of Dam and Qiao (2015). Also, authors conduct the test on 25 portfolios, while the current study uses 16, resulting in differences in the power of the test. Taken together, my results from GRS tests suggest that all models in question produce significant and positive pricing errors, and thus fail to fully explain expected returns of the portfolios.

Looking at Fama-MacBeth (1973) cross-sectional tests, I find the strongest support for the unlevered CAPM in sorts based on unlevered and levered betas. These cases show positive and highly significant market risk premium and insignificant pricing errors, while at the same time showing the highest values across all sorts and models. In the remaining two portfolio sorts all models produce market risk premium indistinguishable from zero or negative, whereas most intercepts are significant at 1% level and tends to be low.

In general, Fama and French (1993) model demonstrates inconsistent behaviour of size and value factors, with significance and sign of the premiums varying across different sorts. This corroborates with findings of Brückner, et al. (2012), who observe similar behaviour in their sample. In my time-series regressions exposures to these factors are statistically zero for roughly half of the portfolios, which is visible across all sorts. Based on cross-sectional tests, Fama and French (1993) model seems to perform best in the unlevered beta sort, however it is still overpassed by the unlevered CAPM. Standard CAPM demonstrates the most erratic behaviour, with market risk premium varying from positive to insignificant or negative, depending on the sorting procedure used. With the exception of levered beta sort, where the model performs best, the standard CAPM shows average intercepts in cross-sectional regressions significantly different from zero. Nevertheless, in this sort the model shows considerably worse fit compared to the unlevered CAPM.

Based on the above, I can conclude that results are highly dependent on the applied portfolio formation method. Overall, findings from all portfolio sorts are not very encouraging and do not unequivocally identify the best performing model. The unlevered CAPM appears to perform best in the unlevered and levered beta sorts, however, in contrast to findings of Dam and Qiao (2015), these results are not robust. My results on the standard CAPM are in line with those obtained by Artmann, et al. (2012a, 2012b), indicating general inability of the model to explain differences in average returns. Lastly, findings from tests on individual company level do not provide additional insight, as the unlevered CAPM and the standard CAPM achieve similar results.

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available via Datastream (Brückner, 2013) and the use of total returns could improve the comparability of my findings. Also, the proxy for the market portfolio might not be efficient, which could be addressed by testing several other proxies and generally including a larger, more diversified sample for its construction.

It is also possible that the unlevered CAPM holds, but the chosen methodology for unlevering the returns is not adequate for German companies, which on average exhibit higher leverage compared to the U.S. As mentioned earlier, the methodology proposed by Dam and Qiao (2015) does not take into account the non-linearity of the relationship between leverage and levered returns and thus, might produce higher pricing errors when applied to companies with higher leverage. Therefore, it would be interesting to compare performance of several approaches to generating unlevered returns in the same sample.

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