The unconditional CAPM holds for unlevered returns
By
Gosse Rienks
*January 2015
Abstract
During economic downturns, a decline in equity value increases both financial leverage and the risk premium, which makes financial leverage a priced risk factor. Equity returns are moreover unable to price all default risk, as the change in debt value is not observable. This thesis shows that unlevered returns capture the cross-sectional variation in average stock returns associated with financial leverage and default risk. To do so, I develop a general methodology to compute maximum likelihood estimates of unlevered returns using the theoretical value of a derivative contract. Moreover, I conduct a two-stage regression procedure with Dutch data, and find that the unconditional CAPM using unlevered returns performs better than alternative models.
Keywords: Asset Pricing, CAPM, Unlevered Returns, Derivative Contract, Maximum Likelihood. JEL: C14, G10, G12, G13
* University of Groningen: Faculty of Economics and Business, The Netherlands. I thank my supervisor
1. Introduction
The unconditional CAPM is a profound model in mathematical finance which, however, is subject to empirical failure for levered returns. Therefore, several academics propose to use alternatives such as the conditional CAPM and multi-factor models. The conditional CAPM however still shows large pricing errors and multi-factor models seem to focus on optimizing empirical performance. I show that the unconditional CAPM still holds for unlevered returns and, therefore, renders alternative models obsolete in many scenarios.
The empirical research of Fama and French (1992) examines the independent explanatory power of risk factors for levered returns associated with market beta, size, leverage, book-to-market equity and earnings-price ratio. They find that the size and
value factors capture all variance explained by other risk factors. Chan, Chen and
Hsieh (1985), and Chan and Chen (1991) suggest that the risk factors found by Fama and French (1992) are implicit default factors. Another explanation for pricing errors in the CAPM is financial leverage. Financial leverage is the correlation between the market premium and levered betas, as they both increase during an economic downturn (see: Choi, 2013 and Dam, and Qiao, 2016).
Unlevered returns capture the risk factors associated with both default and
leverage, as they incorporate debt returns. The idea of using unlevered returns is not
new, but unlevered returns are not directly observable in the market. Several construct them using either book values of debt (Dam and Qiao, 2016) or market values of debt from a restricted data source (Choi, 2013 and Korteweg, 2010). The disadvantages of these methods are that they take the risk free rate as proxy for the return on debt or are not transferable to all companies. I introduce a widely applicable and realistic method to construct unlevered returns using the theoretical value of a derivative contract (Black and Scholes, 1973, and Merton, 1973b). The method I develop incorporates changes in the market value of debt and generates unlevered returns that capture
default and leverage factors.
Having generated unlevered returns, I adopt the Fama and Macbeth (1973) two-stage procedure to test the performance of the CAPM and the Fama and French (1992) three-factor model using unlevered vis-à-vis levered returns. First, I run time-series regressions to construct vectors of relevant betas. Next, I perform Fama and Macbeth (1973) cross-sectional regressions on a constant and the vectors of estimated betas from stage one. For the time-series regressions (stage one), I use an informal test to determine how much betas and intersects are respectively different from zero, and zero. The formal significance of the cross-sectional regression coefficients (stage two) determines whether the betas from the time-series regressions (stage one) are priced risk factors and the 𝑅2 indicates how much cross-sectional variation models
explain.
I carry out unlevered return estimations and the Fama and Macbeth (1973) two-stage procedure for 108 listed firms from the Dutch AEX, AMX and AScX during the period January 2002 until December 2014. The dataset is manually adjusted for survivorship bias, stock splits and acquisitions. As robustness check I perform the same regressions using a dataset without financials and real estate companies.
The informal test for time-series regressions (stage one) shows that models using unlevered instead of levered returns display less significant intersects and more significant market premiums, which implicates that unlevered betas are better predictors. Models moreover perform better if the Fama and French (1992) factors are excluded. The results of the cross-sectional regressions (stage two) show that unlevered returns explain more variance, though risk premiums are negative while these should be positive. The results indicate that unlevered returns capture risk factors associated with default and leverage and show that the unconditional CAPM is still the best asset pricing model.
2. Background
Markowitz (1952) introduces the concept of diversification in portfolio selection and recommends investors to price assets relative to other assets and the market portfolio, which is assumed to be the most efficient portfolio. With Tobin (1958) adding the notion of a risk free instrument the market portfolio becomes super-efficient, as investors can select an efficient portfolio with a differing risk profile than the market portfolio. Sharpe (1964), Lintner (1965a,b) and Mossin (1966) formalize these findings into the (unconditional) CAPM. The CAPM relates the price of an asset to its systematic risk, since idiosyncratic risk can be diversified away and, hence, the latter should not be rewarded.
Several papers challenge the assumptions of the CAPM. Black (1972) develops a less restricting CAPM by suggesting the assumption of a non-zero return asset uncorrelated with the market portfolio instead of assuming that investors can lend unlimited money at a risk free rate. Mayers (1972) suggests the same but with an unmarketable asset. Jensen (1972) empirically tests the CAPM under the assumptions of a risk free rate, an uncorrelated asset, and an unmarketable asset, stating that if the expected return on an asset is directly proportional to its sensitivity to market fluctuations, intersect 𝛼𝑖 has to be zero. The cross-sectional regressions of Jensen
(1972) reject that asset returns are directly, and only proportional to systematic risk, which results in the rejection of the unconditional CAPM and the conception of various versions of the conditional CAPM such as the inter-temporal CAPM (Merton (1973a), and several unconditional multi-factor models, such as the Arbitrage Pricing Theory (Ross, 1976). The conditional CAPM still shows large pricing errors (Lewellen and Nagel, 2006), but multi-factor models are an empirical success (see e.g.: Roll and Ross, 1980).
Multi-factor models suggest a linearity function of the systemic risk factor. In accordance, asset returns are explained by multiple risk factors. Empirical research on priced risk factors document larger companies to give lower returns than smaller companies (the size effect) (Banz, 1981), and companies with higher market leverage1 to obtain higher returns (Bhandari, 1988). Rosenberg, Reid and Lanstein (1985) find
1 Fama and French (1992) define market leverage as market value of equity divided by book value of debt and
that value stocks (stocks with a relatively high book value of equity compared to its market value of equity) give higher returns. Basu (1983) mentions a company’s earnings-price ratio as a potential risk factor, though Ball (1978) argues that an earnings-price ratio simply catches all unnamed risk factors. Fama and French (1992) examine the independent explanatory power of risk factors associated with market beta, size, leverage, value and earnings-price ratio, and find that the size and value factors capture all variance explained by other risk factors. Other studies discuss the effects of among which momentum (Carhart, 1997), volatility (Adrian and Rosenberg, 2008) and liquidity (Acharya and Pedersen, 2005). These other factors are however less obviously present than the Fama and French (1992) size, and value factors and are not covered in this thesis.
The Fama and French (1992) size and value factors both only exist in segments of the market with high default risk (Vassalou and Xing, 2004). Chan and Chen (1991) see the value effect as a relative distress factor, since a high book-to-market equity implies more default risk. Chan, Chen and Hsieh (1985) relate the size effect to the difference between the yield on low-, and high-grade corporate bonds, which in principle also captures the size effect as a priced default factor.
Roll (1977) fairly expresses critique on the validity of empirical tests of researchers such as Fama and French (1992, 1993), as empirical researchers regress levered returns on a levered market portfolio, while the CAPM suggests that risk-averse investors invest in an unlevered market portfolio (including among others real estate, commodities and debt). Equity value alone moreover gives a skewed image of the market value of assets, since equity has a subordinated claim on a firm and therefore only captures limited downside movements. Subordination explains why the
size and value factors are often found for companies close to default: the extra factors
simply offset the non-observable changes in the value of debt.
Another explanation for pricing errors in the CAPM is financial leverage. Financial leverage is the correlation between the market premium and levered betas, as they both increase during an economic downturn (Choi, 2013 and Dam, and Qiao, 2016).
the risks associated with financial leverage and default factors, as unlevered returns represent all asset fluctuations instead of linking them to equity returns. Equity returns do not capture all asset fluctuations due to the limited liability attribute of equity. The assumption that asset values are directly related to equity values and the leverage ratio is therefore too simple. Also because not all debt yields the risk free rate. Unlevered returns do not have the limited downside attribute and reflect a risky return on debt.
To generate unlevered returns, the problem arises that not all assets have exchange-quoted prices. Previous researchers generate unlevered returns by using corporate bond data from the Reuters Fixed Income Database and the Loan Pricing Corporation database (Choi, 2013), using the NAIC database in which US insurance companies have to file all their trades (Korteweg, 2010) or by using book values of debt as a proxy for market values of debt (Dam and Qiao, 2016). The disadvantage of the methods of Choi (2013) and Korteweg (2010) is that they are not transferable to companies not covered by their respective datasets and the method of (Dam and Qiao, 2016) takes the risk free rate as proxy for the return on debt.
I introduce an unique method, which is transferable and shows a realistic return on debt. I calculate the market value of assets as done by Duan (1994), who develops a general methodology that uses the theoretical price of a derivative contract to compute maximum likelihood estimates for an asset value process. With the estimated asset values I generate unlevered returns. I am to my knowledge the first to employ the method of Duan (1994) to test asset pricing models.
3. Methodology
Section 3.1. shows how I generate an asset value process using a maximum
likelihood function for transformed data to exploit the BSM (Black and Scholes, 1973, and Merton, 1973b) option pricing formula. Section 3.2. describes how I construct unlevered returns. This thesis follows the methodology of Fama and French (1993), although the returns on size, and value portfolios2 are from external sources. Fama and French (1993) use the Black, Jensen and Scholes (1972) time-series regressions to perform Fama and Macbeth (1973) cross-sectional regressions. This is a two-stage procedure. Section 3.3. elaborates on how I use time-series regressions (stage one) to construct vectors of relevant betas. Section 3.4. explains how I perform Fama and Macbeth (1973) cross-sectional regressions (stage two) with the betas from the time-series regressions.
3.1. A maximum likelihood function for transformed data
Assume that a firm’s asset value 𝑉𝑡 follows a geometric Brownian motion with
drift 𝜇 and asset volatility 𝜎:
(1) 𝑑𝑉𝑡 = 𝜇𝑉𝑡𝑑𝑡 + 𝜎𝑉𝑡𝑑𝑊𝑡
in which 𝑊𝑡 is a Wiener process. Then, following Black and Scholes (1973), equity 𝐸𝑡 can be seen as a call option on a firm’s asset value, with the strike price
being the market value of debt 𝐷𝑡:
(2) 𝐸𝑡= 𝑉𝑡𝑁(𝑑𝑡) − 𝐷𝑡𝑁(𝑑𝑡− 𝜎√𝑀) where (3) 𝑑𝑡 = ln (𝑉𝑡 𝐷𝑡 ⁄ ) + (𝜎2⁄ ) 𝑀2 𝜎√𝑇
2 To construct factor portfolios representing the size and value effect, Fama and French (1993) split a stock
in which M represents the maturity of debt. The function in (2) is dependent on the two unobserved parameters 𝜎 and 𝑉𝑡. Since the function is a nonlinear two-equation system, the unknown parameters cannot be derived. Duan (1994) uses the Ronn and Verma (1986) numerical procedure to solve the nonlinear two-equation system, utilizing the functional relationship specified in (2) for linking the unobserved parameters and observed random variables. I replicate (and improve) the method developed by Duan (1994) in MATLAB. The script is a likelihood function to estimate 𝜎 and 𝑉𝑡, that iterates different values for 𝜎 until equation (2) holds:
(4) 𝐿(𝑬, 𝜇, 𝜎) = −𝑚 − 1 2 ln(2𝜋) − 𝑚 − 1 2 ln 𝜎2 − ∑ ln (𝑁( 𝑚 𝑡=2 𝑑̂𝑡)) − 1 2𝜎2∑ [ln ( 𝑉̂𝑡(𝜎) 𝑉̂𝑡−1(𝜎)) − 𝜇] 2 𝑚 𝑡=2
in which 𝑬 is a sequence of observed equity values, t ∈{1,…, 𝑚}. If I maximize the equation in (4), 𝑉̂𝑡(𝜎) is the unique solution for 𝑉𝑡 in equation (2) and (3). With 𝑉̂𝑡(𝜎) in place of 𝑉𝑡 in equation (2) and (3), 𝑑̂𝑡 corresponds to 𝑑𝑡. This makes it
possible to compute maximum likelihood estimate(s) of 𝜎̂ (and 𝜇̂).
I use the sum of the market value of equity and the discounted book value of debt as starting 𝑉̂𝑡, and I take -log(0.3) as starting 𝜎̂ (following Lo, 1986). Since the data
transformation is on an element-by-element basis (for each firm separately), the script is able to compute estimates of 𝜎̂ from 𝑬 only (the discounted book value of debt is used as rough starting point for estimations). For each year and for each firm, I calculate 𝜎̂ using 𝑬 for around 252 trading days. Alike Duan (1994), I set the maturity of debt 𝑀 at a 1 year, since no mathematical method to calculate debt maturity exists (only methods with ordinal data from annual reports). Although Duan (1994) assumes a constant risk free rate of 1%, I use time varying Euribor rates with maturity 𝑀 (this rate is however only used for the rough estimation of starting asset value).
3.2. Unlevered returns
With the asset value process of each firm known, I am able to generate unlevered returns. I compose unlevered returns by combining the weighted returns on equity, and debt, since a firm distributes wealth inter-temporarily in the form of dividends, and interest. The unlevered return is therefore not simply the change in asset value:
(5) 𝑅𝑈,𝑡+1= 𝐸𝑡 𝐸𝑡+𝐷𝑡𝑅𝐸,𝑡+1+ 𝐷𝑡 𝐸𝑡+𝐷𝑡𝑅𝐷,𝑡+1 with (6) 𝑅𝐷,𝑡+1= 𝑅𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡,𝑡+ 𝑅∆𝐷,𝑡+1
in which 𝑅𝑈,𝑡, 𝑅𝐸,𝑡, and 𝑅𝐷,𝑡 are the unlevered return, equity return, and debt return
respectively. Equity value 𝐸𝑡 is observable in the market and the market value of debt naturally follows 𝐷𝑡 = 𝑉𝑡− 𝐸𝑡. The return on equity 𝑅𝐸 comes from the return on an individual stock index including the re-investment of dividends. Return on debt 𝑅𝐷
consists of the capital gains on the market value of debt 𝐷𝑡, and an implied interest rate, which comes from the difference between market value of debt 𝐷𝑡 and the notional value of debt with maturity 𝑀. This implied interest rate follows from the maximum likelihood estimations specified in (4), which gives surprisingly realistic interest rates that become higher when a company comes closer to default.
Having generated unlevered returns, I adopt the Fama and Macbeth (1973) two-stage procedure to determine whether the unconditional CAPM holds for unlevered returns.
3.3. First stage: time-series regressions
The first stage in the two-stage procedure is performing multivariate time-series regressions with each asset on the market portfolio and if relevant on size, and value portfolios, to estimate the 𝑁 beta coefficients, 𝛽𝑖,𝑛, of the relative risk factors and to
determine whether intersects 𝛼𝑖 are zero. I perform regressions including, and excluding the risk factors proposed by Fama and French (1992):
(7) 𝑅𝑖,𝑡 = 𝛼𝑖 + 𝑅𝑀𝑃,𝑡′𝛽𝑀𝑃,𝑖+ 𝜀𝑖,𝑡, 𝑡 = 1, … , 𝑇
in which 𝑅𝑖, and 𝑅𝑀𝑃 are respectively the excess return, and the market premium. And:
(8) 𝑅𝑖,𝑡 = 𝛼𝑖 + 𝑅𝑀𝑃,𝑡′𝛽𝑀𝑃,𝑖+ 𝑅𝑆𝑀𝐵,𝑡′𝛽𝑆𝑀𝐵,𝑖+ 𝑅𝐻𝑀𝐿,𝑡′𝛽𝐻𝑀𝐿,𝑖+ 𝜀𝑖,𝑡, 𝑡 = 1, … , 𝑇
in which 𝑅𝑆𝑀𝐵 is the difference in return of a portfolio containing small companies and a portfolio consisting of big companies, and 𝑅𝐻𝑀𝐿 is the difference in return of
portfolios with high- and low book-to-market equity companies. Fama and MacBeth (1973) use rolling 5-year regressions, but I use the technique with full-sample betas (Lettau and Ludvigson, 2001) which is more appropriate with a small dataset. Beta estimates come from an Ordinary Least Squares regression framework.
I perform time-series regressions with functions (7) and (8) using levered returns, unlevered returns, and using individual levered returns and an unlevered market portfolio. The 3 different time-series, and tests with the single systemic risk factor and including the Fama and French (1992) risk factors result in 6 different models.
In all time-series regressions, significant beta values 𝛽𝑖 should leave pricing errors
𝛼𝑖 zero. Looking at the significance of 𝛼𝑖’s is a better test than for example looking at the amount of variance a model explains, while individual firms have idiosyncratic risk that is not meant to be capture by asset pricing models. I informally test for significant 𝛼𝑖’s (these should be insignificant) by looking at the amount of significant 𝛼𝑖’s instead of performing the Gibbons-Ross-Shanken (GRS) test, since the GRS test
3.4. Second stage: cross-sectional regressions
In the second stage of the Fama and Macbeth (1973) two-stage procedure I perform cross-sectional regressions of excess returns on a constant, and the vectors of estimated betas, 𝛽̂𝑖, retrieved from the time-series regressions in stage one. In the case of respectively the CAPM, and the Fama and French (1992) 3-factor model the specifications are:
(9) 𝐸[𝑅𝑖,𝑡] = 𝛼 + 𝛽̂𝑀𝑃,𝑖′𝜆𝑀𝑃, 𝑖 = 1, … , 𝑁
and
(10) 𝐸[𝑅𝑖,𝑡] = 𝛼 + 𝛽̂𝑀𝑃,𝑖′𝜆𝑀𝑃+ 𝛽̂𝑆𝑀𝐵,𝑖′𝜆𝑆𝑀𝐵+ 𝛽̂𝐻𝑀𝐿,𝑖′𝜆𝐻𝑀𝐿, 𝑖 = 1, … , 𝑁
in which 𝛼 is the intercept, 𝛽̂𝑖 are vectors of beta values from stage one, and 𝜆𝑖 are the cross-sectional regression coefficients of risk factors. To calculate the average estimated 𝛼, and 𝜆𝑖 (and associated standard errors) for each month, I run 𝑇
cross-sectional regressions following the Fama and Macbeth (1973) procedure. The essential test is whether the risk factors are positively priced (reflected by the significance, and values of 𝜆𝑖’s) and whether intercept 𝛼 is zero. As standard errors
provided by the Fama and Macbeth (1973) regressions are biased when applied on a time-series (Peterson, 2009), I use Newey-West standard errors to adjust for possible autocorrelation, or heteroskedasticity. In the second stage of the Fama and Macbeth (1973) procedure the 𝑅2-statistic becomes relevant, since risk factors are market estimates and idiosyncratic risk is therefore not present. Fama and Macbeth (1973) cross-sectional regressions limit me from using the adjusted-𝑅2, so I use the regular
𝑅2-statistic to compare the explanatory power of the 6 tested models.
4. Data
The analyses in this thesis are done with a dataset consisting of 108 companies from the Dutch AEX, AMX, and AScX during the period January 2002 until December 2014. The simple capitalization-weighted composition of Dutch indices supports their usage, as well as the fact that the three indices represent all company sizes. For each year, I only include active companies by looking at the active constituent list from Thomson Reuters Datastream to avoid a survivorship bias. The list does not cover periods prior 2002. The three Dutch indices each year contain 75 active companies. As the maximum likelihood estimates in MATLAB and manual index calculations require a significant amount of time, this small dataset is ideal. Also because the development of the new methodology forces me to re-do all analyses for every new insight.
I take company data for daily stock price, outstanding amount of shares, book value of liabilities, and total return index from Thomson Reuters Datastream with respective codes P~E, NOSH, WC03351~E, and RI~E. The equity value equals the share price times the amount of shares outstanding and the last known book value of liabilities is a proxy for the book value of debt. I omit companies with only 1 year of observations and leave out a year of data when the stock returns of a company are influenced by a successful takeover bid3, or when a year of data is incomplete due to a company’s recent IPO. The maximum likelihood calculations request full year samples. I moreover adjust the stock price of a company when Datastream has not properly treated stock splits, and reverse stock splits (see Appendix A).
The maximum likelihood calculations require assumptions for debt maturity and the risk free rate. Since literature has not yet provided a reliable method to calculate debt maturity, I assume all debt to have a maturity of 1 year consistent with Ronn and Verma (1986) and Duan (1994). Instead of assuming a fixed risk free rate as done by Duan (1994), I use the 1-year Euribor rate from Datastream. With the observed parameters, I use MATLAB to exploit the BSM (Black and Scholes, 1973, and
3 2003: KLM, Volker Wessels, CMG, Vodafone Libertel. 2004: Gucci, KLM, Maxeda. 2005: P&O Nedlloyd,
Merton, 1973b) formula, resulting in different daily asset volatilities for each firm. I then annualize daily asset volatilities with a standard of 250 trading days to retain comparability. The extra parameter asset volatility in the BSM formula yields in a relatively simple calculation of daily asset values.
Instead of taking the change in asset value, I compose unlevered returns with weighted returns of equity, and debt. The equity value is observable and the market value of debt follows 𝐷𝑡 = 𝑉𝑡− 𝐸𝑡. Equity return is the natural logarithm of the total
return index (re-investing dividends). I compose debt returns by adding the capital gains on the market value of debt to an implied interest rate, which comes from the difference between the last known book value of debt with maturity 1, and the market value of debt. This implied interest rate follows from the maximum likelihood estimations specified in (4), which gives surprisingly realistic interest rates that become higher when a company comes closer to default. Using this unique method to calculate debt returns differentiates this thesis from other studies. You can find a representation of market returns on debt in figure 1.
Figure 1 – Monthly Return on Debt of the Market vis-à-vis the Risk Free Rate, Netherlands, Jan. 2002 – Dec. 2014
This figure shows the development of the implied debt yield of the market index during the period January 2002 to December 2014. The y-axis represents the monthly return, whereas the x-axis shows the years. The highlighted areas represent recessions. The thick grey line is the market’s return on debt per month and the striped line represents the monthly risk free rate. The big dots reflect decisions of the Federal Open Market Committee bigger than 50 basis points. The return on debt is constructed with an implied annual bond yield, and the change in market value of debt. Source: Datastream.
I subtract the risk-free rate from all calculated returns to obtain excess returns. I use the 3-month Euribor rate from Datastream as risk free rate, since this instrument is widely acknowledged as liquid and as having low risk.
Daily returns are initially essential for the maximum likelihood function in (4) to estimate asset volatility, since 12 monthly data points are not enough for robust estimates. Fama and Macbeth (1973) cross-sectional regressions however perform
-1% 0% 1% 2% 3% 2002 2004 2006 2008 2010 2012 2014 R et ur n Year
best using monthly data4, so I convert the excess returns from daily to monthly geometric returns. Excess levered, and unlevered returns of the market portfolio come from a manually value-weighted index.
To determine the added value of the Fama and French (1992) factors in asset pricing models, I take returns of the European Small-Minus-Big, and High-Minus-Low portfolios from Kenneth French’s website5. Note that the Small-Minus-Big, and High-Minus-Low factor portfolios do not come from the intersections of 6 self-formed size, and value portfolios2 as done by Fama and French (1993), since the amount of active companies is too small to form 25 portfolios. Table 1 presents a summary of monthly returns of the Small-Minus-Big, and High-Minus-Low factor portfolios along with a summary of excess levered, and unlevered returns. The data forms an unbalanced panel of 108 firms from the AEX, AMX, and AScX for the period January 2002 to December 2014.
Table 1 – Means and Standard Deviations for Key Variables, Netherlands, Jan. 2002 – Dec. 2014
Variable Observations Mean S.D.
Aggregate Market
Market Excess Levered Return (%) 156 0.19 5.68
Market Leverage (%) 156 84.73 3.05
Market Excess Unlevered Return (%) 156 0.08 0.94
Small-Minus-Big (%) 156 0.19 1.96
High-Minus-Low (%) 156 0.30 2.18
Individual Firms
Excess Levered Return (%) 9900 -0.33 12.26
Leverage (%) 9900 49.90 23.74
Excess Unlevered Return (%) 9900 0.16 6.06
Total Equity, Market Value (Billion €) 9900 6.21 14.08 Total Assets, Market Value (Billion €) 9900 41.27 158.08
Note: Subtracting the daily Euribor-3M rate from the natural logarithm of the total return index forms
daily excess levered returns. Daily asset values come from the maximum likelihood function specified in (4) with daily market value of equity (shares outstanding multiplied by the closing price of stocks), book value of liabilities and the Euribor-12M rate from Datastream. Daily debt returns follow from the addition of capital gains on market value of debt to an interest rate implied by the difference between the book value of debt with maturity 1, and the market value of debt. I compose excess unlevered returns with weighted equity and debt returns (both minus the Euribor-3M rate). Subsequently, I convert daily returns to monthly returns to perform Fama and Macbeth (1973) regressions. Aggregate market data comes from the sum and weighted sum of company data. The Fama and French (1992) size and value factors are from Kenneth French’s website. I define leverage as the market value of debt divided by the market value of assets. The data reflects an unbalanced panel of 108 firms with data varying from 24 to 156 months.
4 Monthly returns are used most often, which makes the results of regressions better comparable with equivalent
studies. Another support for using monthly instead of daily returns is that the assumption of normality of returns seems more reasonable for monthly than for daily returns. Aggregating further from monthly to yearly, returns results in too much loss of information.
Table 1 shows that the equity class is more risky than combined assets, since the standard deviation of excess levered returns is higher than that of excess unlevered returns for both the aggregate market, and individual firms. The big difference in excess levered return of the market, and individual firms (0.19% versus -0.33%) is explained by the fact that bigger companies in the dataset outperform smaller companies (this is also reflected by the Small-Minus-Big coefficient, as shown in a later stage).
Other statistics in Table 1 that attract attention are the big difference in leverage between the market, and individual firms (84.73% against 49.90%) and the high stand-alone market leverage. These statistics show that the inclusion of financials and
real estate6 companies gives a skewed view, as their balance sheets consist of huge amounts of financial claims on other firms/assets, which makes their liabilities account artificially high. As a robustness check, I therefore also perform the Fama and Macbeth (1973) regressions using 88 companies that are not active in the financial, or real estate sector. Appendix C shows their characteristics. In Appendix D you find a representation of returns on debt of the market without financials, and real estate companies.
6 I consider ABN AMRO, AEGON, BinckBank, Delta Lloyd, Fortis, ING, KAS Bank, Phoenix, Vandermoolen,
5. Results
To determine whether unlevered returns capture the risks associated with default, and leverage and whether the unconditional CAPM holds for unlevered returns, I follow the Fama and Macbeth (1973) two-stage procedure and perform 1,176 time-series regressions, and 336 cross-sectional regressions using a dataset of 108 Dutch firms (and the dataset without financials, and real estate companies).
The 3 tested CAPM models are the Traditional CAPM using only levered returns, the Unlevered CAPM using unlevered returns, and a model that uses individual levered returns and unlevered market returns, which I call the ‘Hybrid CAPM’.
I test 3 additional models by adding the returns of Fama and French (1993) size, and value portfolios to regressions, resulting in the Fama and French (1992)
three-factor model, Unlevered three-three-factor model, and Hybrid three-three-factor model.
Table 2 shows some descriptive statistics of 648 time-series regressions for the 6 different models. In Appendix E you find an overview of the distribution of the intersect and beta values in Table 2.
Column 3 in Panel A of Table 2 shows an average sensitivity to the market premium of 1.05, 5.92, and 2.74 for the Traditional CAPM, Hybrid CAPM, and Unlevered CAPM respectively. The high average 𝛽𝑀𝑃 for the Unlevered CAPM can be attributed to the large weight of financials, and real estate companies in the sample (financials and real estate companies push down unlevered market returns significantly) and the highest value for the Hybrid CAPM is explained by the fact that levered returns are higher than unlevered returns. The amount of significant 𝛽𝑀𝑃’s (Column 7, at a 5% level) of the Traditional CAPM is with 92 slightly higher than the 91 of other models, implicating that the sensitivity to the market premium of the Traditional CAPM explains the returns of more companies. The amount of significant 𝛼𝑖’s, which is the most important statistic in the two-stage procedure, however
Panel B in Table 2 shows that including the Fama and French (1992) risk factors does not change the fact that the Unlevered three-factor model with 12 significant 𝛼𝑖’s performs better than its equivalents. Panel B shows 𝛽𝑀𝑃 values of 1.01, 5.45, and 2.65, which are nearer to the theoretical sensitivity of one than the market betas in Panel A. Adding the Fama and French (1992) factors thought does not improve the models, as it makes the amount of significant 𝛼𝑖’s increase and the amount of significant 𝛽𝑀𝑃’s decrease for each model.
Table 2 – Time Series Regressions of each asset on Risk Factors, Netherlands, Jan. 2002 – Dec. 2014 Panel A: Time Series Regressions without Fama and French (1992) factors
Model Mean Min Max S.D. #Sign. (%)
Traditional CAPM 𝛼 -0.01 -0.10 0.03 0.02 13 (12%) 𝛽𝑀𝑃 1.05 -0.14 2.69 0.57 92 (85%) Hybrid CAPM 𝛼 -0.01 -0.10 0.03 0.02 14 (13%) 𝛽𝑀𝑃 5.92 -0.47 15.24 3.29 91 (84%) Unlevered CAPM 𝛼 0.00 -0.07 0.02 0.01 11 (10%) 𝛽𝑀𝑃 2.74 -1.10 11.10 1.94 91 (84%)
Panel B: Time Series Regressions with Fama and French (1992) factors
Model Mean Min Max S.D. #Sign. (%)
Fama and French (1992) three-factor model
𝛼 -0.01 -0.10 0.02 0.02 17 (16%)
𝛽𝑀𝑃 1.01 -0.07 3.03 0.61 88 (81%)
𝛽𝑆𝑀𝐵 0.69 -2.41 2.97 0.93 35 (32%)
𝛽𝐻𝑀𝐿 0.37 -6.68 4.59 1.21 36 (33%)
Hybrid three-factor model 𝛼 -0.01 -0.10 0.03 0.02 16 (15%)
𝛽𝑀𝑃 5.45 -2.03 16.03 3.26 87 (81%) 𝛽𝑆𝑀𝐵 0.54 -2.49 2.42 0.89 31 (29%) 𝛽𝐻𝑀𝐿 0.56 -6.35 4.71 1.22 35 (32%) Unlevered three-factor model 𝛼 0.00 -0.07 0.02 0.01 12 (11%) 𝛽𝑀𝑃 2.65 -1.29 12.33 2.02 85 (79%) 𝛽𝑆𝑀𝐵 0.30 -1.96 1.82 0.54 36 (33%) 𝛽𝐻𝑀𝐿 0.16 -1.24 2.16 0.44 21 (19%)
Note: This table reports the estimation results of time-series regressions for 108 companies with the
form:
𝑅𝑖,𝑡= 𝛼̂ + 𝑓𝑡′𝛽̂ 𝑖
On the basis of the results of 648 time-series regressions reflected in Table 2, I state that adding Fama and French (1992) factors does not improve asset pricing models, as it makes the amount of significant intersects 𝛼𝑖’s increase and the amount of significant 𝛽𝑀𝑃’s decrease for each model. The Unlevered CAPM shows the lowest amount of significant 𝛼𝑖’s and an average intersect of 0.00, giving the Unlevered
CAPM the best performance in the first stage of the two-stage procedure.
Table 3 shows the result of 168 (one for every month) cross-sectional regressions from the second stage of the Fama and Macbeth (1973) two-stage procedure. All 𝜆𝑀𝑎𝑟𝑘𝑒𝑡 and 𝜆𝐻𝑀𝐿coefficients in Column 3 and 5 show negative signs (the market
premium of the Traditional CAPM even shows a -8.3% return annually), whereas asset pricing theories suggest a positive relationship between asset returns, and risk factors. All coefficients, except 𝛼 and 𝜆𝑆𝑀𝐵 (Column 2 and 4) of the Unlevered three-factor model, are moreover insignificant at a 10% level. Since asset pricing theories require intercept 𝛼 to be zero, the significant 𝛼 of the Unlevered three-factor model suggests the use of other models.
The 𝑅2-statistic becomes relevant for cross-sectional regressions and the values of
8%, and 20% for the Unlevered models exceed those of other models. This higher 𝑅2
is especially important for models including Fama and French (1992) factors while arbitrage opportunities should be non-existent in a multi-factor model. As such, the higher explanatory power of the Unlevered models advocates the use of unlevered returns.
The results from 168 cross-sectional regressions in Table 3 show the highest 𝑅2 for
Table 3 – Cross-sectional Regressions for Individual Dutch firms, Jan. 2002 – Dec. 2014 Model 𝛼 (%) 𝜆𝑀𝑎𝑟𝑘𝑒𝑡 (%) 𝜆𝑆𝑀𝐵 (%) 𝜆𝐻𝑀𝐿 (%) Avg. 𝑅2 Traditional CAPM 0.30 -0.69 0.07 (0.89) (-1.09) Hybrid CAPM 0.12 -0.09 0.07 (0.37) (-0.85) Unlevered CAPM 0.16 -0.01 0.08 (1.08) (-0.14) Fama and French (1992)
three-factor model
0.23 -0.40 0.09 -0.36 0.17
(0.71) (-0.68) (0.38) (-1.28) Hybrid CAPM with Fama and
French factors
0.11 -0.03 0.09 -0.43 0.16
(0.35) (-0.27) (0.38) (-1.51) Unlevered CAPM with Fama
and French factors
0.23* -0.00 0.16* -0.53 0.20
(1.75) (-0.03) (0.54) (-1.80)
Note: This table reports the estimation results of Fama and Macbeth (1973) cross-sectional regressions
of asset pricing models with the relation:
𝐸[𝑅𝑖,𝑡] = 𝛼̂ + 𝛽𝑖′𝜆̂
I estimate betas by performing time-series regressions of individual excess levered, and unlevered returns on their relative factors. The returns are from 108 companies in the Dutch AEX, AMX, and AScX for the period January 2002 to December 2014. The Traditional CAPM only uses excess levered returns, the Hybrid CAPM uses individual excess levered returns, and excess unlevered returns for the market factor and the Unlevered CAPM uses excess unlevered returns for individual firms as well as for the market factor. The models adopting Fama and French (1992) factors include estimates of sensitivity to Small-Minus-Big, and High-Minus-Low portfolios to the analysis. Unlevered returns follow from a maximum likelihood function that exploits the theoretical price of a derivative. This table only reports the cross-sectional averages of the second-stage coefficients from the Fama and Macbeth (1973) procedure. The table reports t-statistics, adjusted for autocorrelation using the Newey-West procedure with 1 lag, of the estimated coefficients in parentheses and the last column of the table reflects the average R2 of the cross-sectional regressions. *, ** and *** denote a significance level at 10%, 5%, and 1% respectively. Sources: Datastream, and Kenneth French’s website.
Lower amounts of significant 𝛼𝑖’s and average intersects of 0.00 in the time-series regressions, and the highest 𝑅2 in cross-sectional regressions show that asset pricing
Robustness check
As the results of the time-series, and cross-sectional regressions indicate that the unconditional CAPM using unlevered returns is the best model, I perform a robustness check with a dataset of 88 companies in which financials, and real estate6 companies are omitted. Excluding financials, and real estate companies provides us with a dataset in which individual firm leverage, and market leverage are more alike. This primarily yields in less biased unlevered market returns, since these are highly influenced by great amounts of debt (the liabilities account of financials, and real estate companies is artificially high).
Again, I adopt the Fama and Macbeth (1973) two-stage procedure and first perform 528 time-series regressions for 6 different models, of which you find a summary in Table 4. In Appendix F you find an overview of the distribution of the intersect, and beta values in Table 4. In Column 2 of Table 4 the average 𝛽𝑀𝑃 of the
Hybrid CAPM, and the Unlevered CAPM stand out, as the values of 2.10, and 1.08 are closer to one than their equivalents in Table 2, indicating a superior sample.
Although the Fama and French (1992) three-factor model notably improves in the robustness check, the Unlevered CAPM again shows the best results, as its 9 significant 𝛼𝑖’s are the lowest amount (Column 7, at a 5% level), it’s 74 significant
Table 4 – Time Series Regressions of each asset on Risk Factors, Netherlands without Financials and Real Estate companies, Jan. 2002 – Dec. 2014
Panel A: Time Series Regressions without Fama and French (1992) factors
Model Mean Min Max S.D. #Sign. (%)
Traditional CAPM 𝛼 -0.01 -0.10 0.03 0.02 11 (13%) 𝛽𝑀𝑃 1.09 -0.44 3.14 0.62 71 (81%) Hybrid CAPM 𝛼 -0.01 -0.10 0.03 0.02 9 (10%) 𝛽𝑀𝑃 2.10 -0.72 5.82 1.19 72 (82%) Unlevered CAPM 𝛼 0.00 -0.07 0.02 0.01 9 (10%) 𝛽𝑀𝑃 1.08 -0.44 3.83 0.64 74 (84%)
Panel B: Time Series Regressions with Fama and French (1992) factors
Model Mean Min Max S.D. #Sign. (%)
Fama and French (1992) three-factor model
𝛼 -0.01 -0.10 0.03 0.02 10 (11%)
𝛽𝑀𝑃 1.03 -0.51 3.05 0.63 72 (82%)
𝛽𝑆𝑀𝐵 0.64 -2.01 2.37 0.84 25 (28%)
𝛽𝐻𝑀𝐿 0.56 -5.44 4.38 1.11 26 (30%)
Hybrid three-factor model 𝛼 -0.01 -0.10 0.03 0.02 12 (14%)
𝛽𝑀𝑃 1.97 -0.81 5.56 1.20 73 (83%) 𝛽𝑆𝑀𝐵 0.61 -1.94 2.41 0.83 25 (28%) 𝛽𝐻𝑀𝐿 0.61 -5.40 4.31 1.10 24 (27%) Unlevered three-factor model 𝛼 0.00 -0.07 0.02 0.01 11 (13%) 𝛽𝑀𝑃 1.04 -0.51 4.12 0.66 73 (83%) 𝛽𝑆𝑀𝐵 0.34 -2.00 1.80 0.57 28 (32%) 𝛽𝐻𝑀𝐿 0.27 -1.07 1.95 0.43 23 (26%)
Note: This table reports the estimation results of time-series regressions for 88 companies with the
form:
𝑅𝑖,𝑡= 𝛼̂ + 𝑓𝑡′𝛽̂ 𝑖
Betas and intersect estimates come from regressions of individual asset returns on relevant risk factors. The table shows the average, maximum, and minimum value of the individual regressions, as well as the standard deviation of the individual outcomes, and the amount of significant outcomes at a 5% level. The asset returns are from 88 companies from the Dutch AEX, AMX, and AScX for the period January 2002 to December 2014. I omit companies active in the financial, or real estate sector. The Traditional CAPM only uses excess levered returns, the Hybrid CAPM uses individual excess levered returns, and excess unlevered returns for the market factor and the Unlevered CAPM uses excess unlevered returns for individual firms as well as for the market factor. The models including the Fama and French (1992) factors add the returns of Small-Minus-Big, and High-Minus-Low portfolios. Unlevered returns follow from a maximum likelihood function that exploits the theoretical price of a derivative contract. Sources: Datastream, and Kenneth French’s website.
Table 5 shows the result of 168 Fama and Macbeth (1973) cross-sectional regressions performed with the beta estimates of the dataset excluding companies active in the financial, and real estate sector. The intersect values (Column 2) of both Unlevered models show significant signs, implicating a weak relative performance to other models. The higher 𝑅2 value (Column 6) of 20% versus the 18% of equivalent
The negative and significant values for the 𝜆𝐻𝑀𝐿 coefficients (Column 5) however discourage the inclusion of Fama and French (1992) factors. The Traditional CAPM, the Hybrid CAPM, and the Unlevered CAPM all show the same 8% 𝑅2-statistic. As
such, all models show to perform equally well (or equally poor) in the cross-sectional regressions.
Table 5 – Cross-sectional Regressions for Individual Dutch firms excluding the Financial, and Real Estate sectors, Jan. 2002 – Dec. 2014
Model 𝛼 (%) 𝜆𝑀𝑎𝑟𝑘𝑒𝑡 (%) 𝜆𝑆𝑀𝐵 (%) 𝜆𝐻𝑀𝐿 (%) Avg. 𝑅2 Traditional CAPM 0.10 -0.40 0.08 (0.25) (-0.64) Hybrid CAPM 0.05 -0.18 0.08 (0.13) (-0.56) Unlevered CAPM 0.41** -0.26 0.08 (2.24) (-0.88) Fama and French (1992)
three-factor model
0.07 0.11 0.06 -0.70** 0.18
(0.18) (0.18) (0.25) (-2.30)
Hybrid three-factor model 0.05 0.09 0.05 -0.71** 0.18 (0.14) (0.31) (0.21) (-2.33)
Unlevered three-factor model 0.48*** -0.20 0.21 -0.61* 0.20 (2.63) (-0.71) (0.69) (-1.96)
Note: This table reports the estimation results of Fama and Macbeth (1973) cross-sectional regressions
of asset pricing models with the relation:
𝐸[𝑅𝑖,𝑡] = 𝛼̂ + 𝛽𝑖′𝜆̂
Betas are estimated by performing time-series regressions of individual excess levered, and unlevered returns on their relative factors. The returns are from 88 companies in the Dutch AEX, AMX, and AScX for the period January 2002 to December 2014. I omit companies active in the financial, or real estate sector. The Traditional CAPM only uses excess levered returns, the Hybrid CAPM uses individual excess levered returns, and excess unlevered returns for the market factor and the Unlevered CAPM uses excess unlevered returns for individual firms as well as for the market factor. The models adopting Fama and French (1992) factors include estimates of sensitivity to Small-Minus-Big, and High-Minus-Low portfolios to the analysis. Unlevered returns follow from a maximum likelihood function that exploits the theoretical price of a derivative. This table only reports the cross-sectional averages of the second-stage coefficients from the Fama and Macbeth (1973) procedure. The table reports t-statistics, adjusted for autocorrelation using the Newey-West procedure with 1 lag, of the estimated coefficients in parentheses and the last column of the table reflects the average R2 of the cross-sectional regressions. *, ** and *** denote a significance level at 10%, 5%, and 1% respectively.
The average 𝛽𝑀𝑃 values of the robustness check in Table 4 are closer to one than their equivalent in Table 2, which leads to the conclusion that the dataset without financials, and real estate companies is a better sample. The more consistent leverage ratio (see Table 1 and Table 8) supports this statement.
The cross-sectional regression results in Table 5 (of the robustness check) do not provide sufficient evidence to state which of the models performs best. Table 4, showing the time-series regressions of the robustness check, indicates that the Unlevered CAPM has the lowest amount of significant 𝛼𝑖’s, the highest amount of significant 𝛽𝑀𝑃’s and an intersect of 0.00. This, along with the cross-sectional
regression results from Table 2 and the high 𝑅2 in Table 3 show that the Unlevered
CAPM performs best.
6. Conclusion
Using levered returns in tests of the unconditional CAPM results in the addition of
default, and leverage factors in a multi-factor model. As unlevered returns capture
these risk factors, I determine whether using unlevered returns improves the performance of the unconditional CAPM. The idea of using unlevered returns is not new, but unlevered returns are not directly observable in the market. I introduce a widely applicable and realistic method to construct unlevered returns, using a maximum likelihood function to exploit the theoretical value of a derivative contract. Having generated unlevered returns, I adopt the Fama and Macbeth (1973) two-stage procedure to perform regressions using both levered, and unlevered returns. I carry out the regressions on asset returns of 108 listed firms from the Dutch AEX, AMX, and AScX during the period January 2002 until December 2014. As robustness check I perform the same regressions using a dataset without financials, and real estate companies.
The informal test for the time-series regressions (first stage of the two-stage procedure) shows that models using unlevered returns display fewer significant intersects and more significant market premiums than models using levered returns. Another finding is that asset pricing models work better if the Fama and French (1992) factors are excluded. The results indicate that the Fama and French (1992) three-factor model is no better alternative than the unconditional CAPM, and the unconditional CAPM still holds. Models using unlevered returns moreover show better performance than models using levered returns. This confirms the findings of Choi (2013) and Dam, and Qiao (2016), who state that unlevered returns capture
financial leverage risk.
The conclusive results in this thesis are subject to some limitations. I have for example not back-tested the suggested models as alternative method of performance. Another shortcoming is that the best performance of the unconditional CAPM using unlevered returns may be the result of including debt returns, as debt returns in principle show less variance than equity returns and have high inter-company correlation. To check this, I suggest to compare the performance of the models in this thesis with models using the risk free rate as debt return instead of the estimated return on debt from the maximum likelihood calculations (as done by for example Dam and Qiao, 2016). This replication also shows whether the robust performance of the unconditional CAPM using unlevered returns is due to the fact that leverage factors are abolished, or also because it captures default factors. Another addition to check the performance of the unconditional CAPM is the inclusion of other types of models such as the conditional CAPM with time-varying betas (Merton, 1973a).
Although the developed maximum likelihood methodology in this thesis works surprisingly well, some improvements can be made. A future theory for the calculation of debt maturity makes it possible to implement different debt maturities (for now studies only look at ordinal maturity classes), as I now assume a fixed maturity of 1 year. Different debt maturities however result in the necessity to calculate individual risk free rates via the term structure of Euribor rates, because the maturity of the risk free rate then also varies.
The disadvantage of the developed methodology is that it is very labour intensive and takes a lot of computation time. Maximum likelihood estimates in MATLAB for the small Dutch dataset already take 14 hours, and I manually assemble all unlevered returns and index returns. A more diversified sample for a longer time period sees to more significant results and allows the construction of Fama and French (1993) portfolio’s to generate applicable factor betas instead of generating them with returns of Kenneth French’s website. Generating size, and value portfolios using unlevered returns can moreover capture the potential influence of leverage on the Fama and French (1992) factors, as the size and value factors from Kenneth French’s website come from levered returns. Being able to build 25 portfolios of stocks with joint characteristics moreover results in more stable coefficients, and variances, since the bundling of data filters random movements. A bigger dataset also makes it possible to calculate betas using rolling 5-year regressions (Fama and Macbeth, 1973) instead of full-sample betas (Lettau and Ludvigson, 2001) used in this thesis.
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Appendix A
Table 6 - Overview of stock splits during the period Jan. 2002 – Dec. 2014
Panel A: Forward Stock Splits Execution date Split factor
Aalberts 18 May 2007 1:4 Accell 1 June 2011 1:2 Arcadis 16 May 2008 1:3 BAM 11 May 2006 1:5 Boskalis 21 May 2007 1:3 Brunel 3 June 2014 1:2 DSM 5 September 2005 1:2 Fugro 20 June 2005 1:4 Grontmij 4 June 2007 1:4 Mediq 17 April 2007 1:4 Nutreco 2 May 2013 1:2
Royal Imtech 4 October 2007 1:2
SBM Offshore 2 June 2006 1:4
Smit 26 October 2006 1:2
Ten Cate 18 April 2006 1:4
Unilever 22 May 2006 1:3
USG 13 October 2006 1:2
Value8 27 December 2013 1:4
Vopak 18 May 2010 1:2
Panel B: Reverse Stock Splits Execution date Split factor
Ahold 22 August 2007 5:4 Antonov 21 February 2005 4:1 Antonov 11 May 2009 10:1 Corus 16 June 2006 5:1 Getronics 29 June 2005 7:1 Heijmans 23 September 2009 10:1 Pharming 5 March 2013 10:1
Royal Imtech 28 October 2014 500:1
Super de Boer 4 November 2004 10:1
Wavin 3 May 2010 8:1
Note: This table provides an overview of stock splits in the AEX, AMX, and AScX during the period
Appendix B
Table 7 - Monthly return on debt (%) of individual companies during the period Jan. 2008 – Dec. 2008
Company Jan Feb Mar Apr May Jun Jul Aug Sep Okt Nov Dec
Ordina 0.69 0.34 0.04 0.27 0.40 0.28 0.39 0.39 -0.17 0.51 -0.89 2.39 Pharming 0.60 0.47 -0.52 0.30 -0.23 1.06 0.50 0.57 -0.09 0.21 1.98 1.04 Philips 0.70 0.34 0.03 0.27 0.40 0.31 0.41 0.40 0.10 0.94 1.18 1.28 Post NL 0.68 0.30 -0.02 0.32 0.43 0.21 0.46 0.49 -0.10 0.72 1.29 0.73 Randstad 0.65 0.36 0.33 0.19 0.76 0.15 0.24 0.63 -0.04 0.64 1.19 1.30 Reed Elsevier 0.70 0.34 0.03 0.27 0.40 0.31 0.41 0.40 0.11 0.97 1.23 1.25 SBM 0.67 0.38 0.02 0.32 0.41 0.30 0.13 0.58 0.04 0.90 1.06 0.59 Shell 0.70 0.34 0.02 0.28 0.40 0.31 0.41 0.41 0.10 0.97 1.23 1.23 Sligro 0.70 0.34 0.03 0.27 0.40 0.31 0.41 0.40 0.11 0.96 1.23 1.25 Smit 0.68 0.37 0.03 0.28 0.40 0.31 0.41 0.39 0.13 0.95 1.18 1.12 SNS Reaal 0.69 0.35 0.02 0.28 0.41 0.30 0.40 0.41 0.05 0.88 1.18 1.18 Super de Boer 0.71 0.34 0.03 0.27 0.40 0.31 0.42 0.40 0.11 0.96 1.23 1.25 Telegraaf 0.70 0.34 0.03 0.27 0.40 0.31 0.41 0.40 0.11 0.96 1.19 1.29 Ten Cate 0.67 0.40 0.04 0.27 0.40 0.31 0.43 0.41 0.09 0.83 1.07 1.41 TKH 0.69 0.36 0.04 0.28 0.41 0.29 0.42 0.41 0.08 0.96 0.83 1.35 TomTom 0.69 0.31 -0.04 0.16 0.48 -0.03 -0.06 0.91 -0.64 -3.49 -0.69 3.26 Unibail Rodamco 0.71 0.34 0.03 0.27 0.40 0.31 0.41 0.40 0.11 0.96 1.22 1.25 Unilever 0.70 0.34 0.03 0.27 0.40 0.31 0.41 0.40 0.11 0.97 1.23 1.25 Unit 4 0.68 0.37 0.02 0.27 0.40 0.30 0.37 0.44 0.05 0.94 0.73 1.23 USG People 0.66 0.36 0.03 0.27 0.41 0.23 0.30 0.59 -0.06 0.77 1.34 1.36
Van der Moolen 0.72 0.29 0.15 0.24 0.77 0.29 0.66 0.48 -0.24 0.78 1.46 0.78
Van Lanschot 0.70 0.34 0.03 0.27 0.40 0.31 0.41 0.40 0.11 0.97 1.23 1.25 Vastned Offices 0.70 0.34 0.03 0.27 0.40 0.30 0.41 0.40 0.10 0.77 -0.36 2.30 Vastned Retail 0.70 0.34 0.03 0.27 0.40 0.31 0.42 0.40 0.11 0.96 1.14 1.34 Vopak 0.70 0.34 0.03 0.27 0.40 0.31 0.41 0.40 0.10 0.94 1.21 1.29 Wavin 0.68 0.34 0.04 0.28 0.40 0.12 0.44 0.47 -0.21 0.32 1.28 0.99 Wereldhave 0.70 0.34 0.03 0.27 0.40 0.31 0.41 0.40 0.11 0.97 1.23 1.25 Wessanen 0.70 0.34 0.03 0.27 0.40 0.31 0.41 0.41 0.11 0.93 1.20 1.27 Wolters Kluwer 0.70 0.34 0.03 0.27 0.40 0.31 0.41 0.41 0.10 0.97 1.23 1.25
Note: This table provides an overview of the monthly debt returns of companies in the AEX, AMX, and
Appendix C
Table 8 – Means and Standard Deviations for Key Variables, Netherlands without Financials, and Real Estate companies, Jan. 2002 – Dec. 2014
Variable Observations Mean S.D.
Aggregate Market
Market Excess Levered Return (%) 156 0.26 5.12
Market Leverage (%) 156 47.96 4.57
Market Excess Unlevered Return (%) 156 0.22 2.62
Small-Minus-Big (%) 156 0.19 1.96
High-Minus-Low (%) 156 0.30 2.18
Individual Firms
Excess Levered Return (%) 7968 (0.22) 12.09
Leverage (%) 7968 43.14 19.44
Excess Unlevered Return (%) 7968 0.16 6.60
Total Equity, Market Value (Billion €) 7968 6.12 14.66 Total Assets, Market Value (Billion €) 7968 11.75 30.00
Note: Subtracting the daily Euribor-3M rate from the natural logarithm of the total return index forms
Appendix D
Figure 2 – Monthly Return on Debt of the Market vis-à-vis the Risk Free Rate, Netherlands without Financials, and Real Estate companies, Jan. 2002 – Dec. 2014
This figure shows the development of the implied debt yield of the market index excluding companies active in the financial, or real estate sector during the period January 2002 to December 2014. The y-axis represents the monthly return, whereas the x-y-axis shows the years. The highlighted areas represent recessions. The thick grey line is the market’s return on debt per month and the striped line represents the monthly risk free rate. The big dots reflect decisions of the Federal Open Market Committee bigger than 50 basis points. The return on debt is constructed with an implied annual bond yield, and the change in market value of debt. Source: Datastream.
-1% 0% 1% 2% 3% 2002 2004 2006 2008 2010 2012 2014 Ret ur n Year
Figure 3 – Histograms indicating the distribution of intersect values, and beta values, Netherlands, Jan. 2002 – Dec. 2014
Model 𝛼 𝛽𝑀𝑃 𝛽𝑆𝑀𝐵 𝛽𝐻𝑀𝐿
Traditional CAPM
Hybrid CAPM
Unlevered CAPM
Fama and French (1992) three-factor model
Hybrid three-factor model
Unlevered three-factor model
Note: This figure presents graphical representations of the distribution of the intersect values, and beta values, which are reflected on the x-axes. Betas, and intersect
estimates come from time-series regressions of individual asset returns on the market premium, a Small-Minus-Big factor portfolio, and a High-Minus-Low factor portfolio. The regressions are performed with asset returns of 108 companies from the Dutch AEX, AMX, and AScX for the period January 2002 to December 2014. The Traditional CAPM only uses excess levered returns, the Hybrid CAPM uses individual excess levered returns, and excess unlevered returns for the market factor and the Unlevered CAPM uses excess unlevered returns for individual firms as well as for the market factor. The models including the Fama and French (1992) factors add the returns of Small-Minus-Big, and High-Minus-Low portfolios. Unlevered returns follow from a maximum likelihood function that exploits the theoretical price of a derivative contract.
Sources: Datastream and Kenneth French’s website.
Figure 4 – Histograms indicating the distribution of intersect values, and beta values, Netherlands without Financials, and Real Estate companies, Jan. 2002 – Dec. 2014
Model 𝛼 𝛽𝑀𝑃 𝛽𝑆𝑀𝐵 𝛽𝐻𝑀𝐿
Traditional CAPM
Hybrid CAPM
Unlevered CAPM
Fama and French (1992) three-factor model
Hybrid three-factor model
Unlevered three-factor model
Note: This figure presents graphical representations of the distribution of the intersect values, and beta values, which are reflected on the x-axes. Betas, and intersect
estimates come from time-series regressions of individual asset returns on the market premium, a Small-Minus-Big factor portfolio, and a High-Minus-Low factor portfolio. The regressions are performed with asset returns of 88 companies from the Dutch AEX, AMX, and AScX for the period January 2002 to December 2014. Companies in the financial, and real estate sector are omitted from the sample. The Traditional CAPM only uses excess levered returns, the Hybrid CAPM uses individual excess levered returns, and excess unlevered returns for the market factor and the Unlevered CAPM uses excess unlevered returns for individual firms as well as for the market factor. The models including the Fama and French (1992) factors add the returns of Small-Minus-Big, and High-Minus-Low portfolios. Unlevered returns follow from a maximum likelihood function that exploits the theoretical price of a derivative contract. Sources: Datastream and Kenneth French’s website.
