Conditional CAPM and the effect of leverage on time-variation in Beta
– An analysis of the Dutch Stock Market
Bachelor Thesis in Economics and Business
Specialisation in Economics and Finance
submitted by
Jeff Bausch
11044993
June 2018
This thesis has been supervised by Pascal Golec at the Faculty of Economics and Business of the University of Amsterdam
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Disclaimer
This essay is written by Jeff Bausch who declares to take full responsibility for the contents of this essay.
I declare that the text and the work presented in this essay are original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business at the University of Amsterdam is solely responsible for the supervision of completion of the work, not for the contents.
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Abstract
This paper investigates the effect of firm-leverage on the changing systematic risk responsiveness of Dutch stocks from 1984 until 2017. The analysis is two-fold and begins with testing for time-variation in the beta of the CAPM. Hypothesised by Blume (1971) that beta should be considered as changing from period to period, a short-window rolling regression method is applied, based on the Fama-MacBeth approach, to test for the presence of a varying beta. The research is able to confirm the presence of a time-varying behaviour of market risk sensitivity for all 45 tested stocks. In addition to the econometric implications which arise from these results, factors influencing the beta estimates need to be determined to accurately condition the CAPM. Particularly the effect of leverage, which in theory and practice affects the risk class of a firm, is analysed. First correlations between capital structure and market risk responsiveness was established by Hamada (1972). With a simple linear regression model, a significant predicting effect of the leverage ratio on beta estimates is found. The level however remains low due to probable OVB in the model.
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Contents
1. Introduction 2. Literature Review
2.1 The origins of the CAPM and beta 2.2 Time-variation in beta
2.3 The effect of leverage on the time-variation in beta 3. Data
4. Modelling and Methodology
4.1 Model for time-variation in beta
4.2 Model to test for the influence of leverage in beta 5. Results and Analysis
5.1 Time-variation in the beta of Dutch stocks
5.2 The effect of leverage on the beta time-variation in Dutch stocks 6. Conclusion
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1. Introduction
The measurement of risk and return is one of the most fundamental elements in financial theory and practice (Blume, 1971). Establishing an appropriate relationship between both factors has been the objective of a large body of financial literature (Fama & French, 1997). Out of all the models derived, the Capital Asset Pricing Model, or CAPM for short, by Sharpe (1964) and Lintner (1965) has gained the highest level of popularity amongst practitioners and academics (Fama & French, 2004). The CAPM’s simplicity has made the model and its beta the centre of numerous empirical analyses, which however often result in the conclusion that the model is an inaccurate measure for asset pricing. A potential reason for its shortcomings stems from the aforementioned simplicity (Fama & French, 2004). Namely, amongst the model’s many restrictive assumptions is the time-invariant nature of the beta coefficient which, according to Blume (1971), should be considered just like any other economic variable as varying over time. In other words, the standard CAPM claims that the reaction of stock returns to the market, as measured by the beta, does not vary across different time periods. This is of course, especially for stocks, a crucial assumption since they are known for their volatile behaviour (Berglund & Knif, 1999).
The presence of a time-varying beta has hence been investigated by many authors (Bos & Newbold, 1984; Harvey, 1989; Lewellen & Nagel, 2006). Resulting in unanimous conclusions to reject the hypothesis of beta constancy (Moonis & Shah, 2003). These findings suggest two main considerations when computing a stock’s, or other asset’s, market risk premium (González-Rivera, 1997). First, the econometric implication of using a conditional model which allows for a time-varying risk-return relationship. And second, the importance of determining the factors which cause the varying behaviour of beta. Although these determining factors can range from micro- to macroeconomic factors (González-Rivera, 1997), this research limits itself to firm leverage and tests its ability as a predictor for market risk responsiveness of Dutch stocks.
Reason to analyse the relationship between capital structure and market risk sensitivity does not only allow to link the two major parts of financial theory: Corporate Finance and Investments (Hamada, 1972), it also permits to contribute to the already existing literature on the topic of systematic risk responsiveness. After Hamada (1972) first found that leverage is a significant explaining factor of the standard OLS CAPM beta, many other authors have followed suit. Despite a connection being established between firm debt and the CAPM, it has yet to be analysed whether the amount of a firm’s leverage also helps to predict future market risk sensitivity when beta is considered as changing over time.
This paper will thus address the question whether the amount of leverage in a firm can serve as a predictor of time-varying betas in Dutch stocks, measured over the period of 1984
6 | P a g e until 2017. The contribution of this essay is twofold. First, considering that most studies are conducted in the US and analyse diversified portfolios instead of individual stocks, this paper contributes by shifting the focus to Dutch stocks. Since only the paper by DeJong, Kemna, and Kloek (1992) has analysed for time-variation of betas in Dutch stocks so far, this research first serves as a confirmation of their findings by using additional data and a rolling regression method instead of a GARCH model. Second, it tests the ability of the leverage ratio as a predictor for future beta values. This has so far only been hypothesised but not thoroughly tested when the betas are time-varying. In general, this research has the objective of improving asset pricing accuracy by determining whether the leverage level needs to be included in an adjusted asset pricing model of a conditional nature (Fabozzi & Francis, 1978). The significance of leverage can be assumed following findings of authors such as Mandelker and Rhee (1984) or DeJong and Collins (1985) who all confirmed the existence of a relationship between capital structure and the CAPM beta.
First, to confirm the hypothesis of a time-varying beta in the Dutch stock market, a short-window rolling regression analysis is applied, based on the Fama-MacBeth approach (1973). This methodology is frequently used in other prior literature (Braun, Nelson, & Sunier, 1995; Lewellen & Nagel, 2006). Then, to test for the hypothesis of a predictive effect of leverage on systematic risk sensitivity, a simple linear regression model and a polynomial regression model are used (Hamada, 1972).
The structure of the essay is as follows. The next section discusses the literature written on this topic while giving more insight on the Capital Asset Pricing Model and its beta, time-variation and more details what it implies, and the link between leverage and systematic risk measurement. The third section explains the data used for this research. The fourth section explains the models applied in this paper to test the hypotheses. Section 5 lists and analyses the results of the conducted research. And the last section concludes while giving remarks on the research.
2. Literature Review
2.1 The origins of the CAPM and beta
The CAPM and its beta are indisputably key measures to assess risk and make investment decisions (Rosenberg & Guy, 1976). It is therefore crucial to apply sufficient focus on the understandings of the CAPM and its beta. The model as demonstrated in textbooks can first be observed in the paper by Sharpe (1964), who plotted the expected returns of assets against the returns of a diversified portfolio. While deriving the model, Sharpe (1964) had the intention to measure the risk-return relationship between an asset and solely the systematic risk since he argued that there is no consistent relationship between asset return and total risk. The technique applied in practice today is similar to the CAPM’s first version (Rosenberg & Guy,
7 | P a g e 1976). A financial agent plots a time series of the returns of an asset and the returns of the corresponding market index. Subsequently, the excess returns of the asset are regressed using OLS on the excess returns of the market index, thus obtaining a proportional linear relationship between the measured asset and the systematic risk (Black, Fraser, & Power, 1992). The variation of the asset’s returns with respect to the returns of the diversified portfolio, usually the market portfolio, can then be interpreted as the asset’s systematic risk premium (Sharpe, 1964). The magnitude of the responses by the asset returns towards market risk is reflected by the value of beta, which comprises all the historical data used to chart the linear relationship (Rosenberg & Guy, 1976). Adding to Sharpe’s findings, Lintner (1965) however underlined the importance of accounting for positive residual variances and thus argued that non-systematic risk should be included in the asset pricing model. The significance in accounting for positive residual variances is reflected by a smaller relative effect on excess returns following a change in the market beta. Furthermore, Lintner (1965) mentioned the use of excess returns, in comparison with regular returns as done by Sharpe (1964), meaning that a riskless return is deducted from the asset and market return when measuring the linear regression. Black (1972) later introduced a modified version of the CAPM using a second factor with a zero-beta instead of a risk-free rate. This paper however uses the standard Capital Asset Pricing Model by Sharpe (1964) and Lintner (1965) as it is most common in theory and practice. The analysis described in the remainder of the paper could however be applied to Black’s (1972) version of the CAPM as well.
2.2 Time-variation in beta
Blume (1971) was the first author to hypothesise the time-variation of the beta coefficient within the CAPM model. His reasoning was that no economic variable, including the one measuring systematic risk responsiveness, can be constant from period to period. A big part of the literature was consequently based on Blume’s (1971) assumption (Galai & Masulis, 1975; Klemkosky & Martin, 1975; Bos & Newbold, 1984). The rest of the literature based their research on testing whether an alteration of Sharpe’s assumption, measuring systematic risk through a simple coefficient, would lead towards better outcomes (González-Rivera, 1997; Jacob, 1971; Lewellen & Nagel, 2006). Following Blume’s (1971) and Jacob’s (1971) reasonings, a lot of empirical analyses were conducted on the presence of a time-varying beta in the market. In these analyses, three models in particular have managed repetitive appearances over the last decades. The first model is a short-window rolling regression, influenced by the paper of Fama and MacBeth (1973). This model can furthermore be observed in the papers by Braun et al. (1995) or Fama and French (1997). The second method is the use of a bivariate (G)ARCH model, as seen in Bollerslev, Engle, and Wooldridge (1988)
8 | P a g e or in Harvey (1989). The third model observed in literature is a state-space model, estimated with the Kalman filter techniques (González-Rivera, 1997; Moonis & Shah, 2003).
In addition to a variation in the models applied by the different authors, research about the assumption of a time-varying beta coefficient deviated as well by applying the assumptions to various markets (Berglund & Knif, 1999), using individual stocks or diversified portfolios (Fama & French, 1997), and analysing the differences between the lengths of the periods (Ang & Kristensen, 2012). Focussing briefly on the data sets, Blume’s (1971) hypothesis for a time-varying beta had been confirmed multiple times for the Australian market (Brooks, Faff & Lee, 1992), the Swedish market (Wells, 1994), or the Finnish market (Berglund & Knif, 1999), amongst others. The Dutch market had only been subject in the paper by DeJong, Kemna, and Kloek (1992), who were able to confirm the presence of a moving beta. The authors however applied a GARCH model and used a smaller data set than this paper. Finally, the importance of using stocks instead of portfolios and the length for the windows are both mentioned in Berglund and Knif (1999) and will be further discussed in section 3.
After the findings of beta instability over time, emphasis is put on the practical issues when measuring market risk sensitivity (Fabozzi & Francis, 1978). The authors pointed out two main implications associated with a time-varying beta. First, some econometric implications arise when attempting to improve the Capital Asset Pricing Model. Black et al. (1992) claimed that a stochastic beta would make OLS an unfitting estimation method for predicting returns. The problem with the OLS estimation method is primarily reflected by the damaged linear relationship between asset and market returns (González-Rivera, 1997), which consequently leads to a misspecification of systematic risk. Fabozzi and Francis (1978) had already argued that the OLS model results in an inefficient beta value due to heteroscedastic behaviour in the error term of the regression. As a consequence of the heteroscedastic behaviour by the residuals, systematic and firm-specific risk are summed together and the beta value is no longer a valid representative of systematic risk alone. The authors therefore argued for the importance of the second implication when beta is time-varying. Fabozzi and Francis (1978) claimed that the fluctuations in beta could be explained by factors which influence the riskiness in the relevant investment. This claim has furthermore been supported by other authors such as González-Rivera (1997) or Brooks et al. (1992). Determining the factors which influence the level of beta is motivated by the objective of eventually including these in the econometric asset pricing models and thus leading to more accurate outcomes.
2.3 The effect of leverage on the time-variation in beta
Putting the focus now on the second implication related with a time-varying beta, Fabozzi and Francis (1978) initially listed four major categories of risk factors which could influence the level of beta. These range from micro- and macro-economic factors, to political factors,
9 | P a g e and lastly to market factors. Most of the literature however only considered the effects of micro- and macro-economic variables on systematic risk responsiveness (Brooks et al., 1992). Some of the macro-economic factors analysed in prior literature are inflation or unemployment (Bos & Newbold, 1984). Micro-economic factors which have been researched include merger activity (Dielman & Nantell, 1982) or growth (Myers & Turnbull, 1977). The effect of firm-leverage on beta estimation has frequently been subject to research as well, being analysed for the first time by Hamada in 1972. The author concluded in his research that between 21% and 24% of beta variation can be explained by the amount of leverage a firm has. Mandelker and Rhee (1984) as well found similar results in their analysis. These findings suggest that taking a firm’s debt level into account leads to more accurate estimations of systematic risk responsiveness. Hamada (1972) however measured the beta with a standard OLS regression and did not consider the presence of a time-varying coefficient.
Nevertheless, following the findings of Hamada (1972), the analysis whether the capital structure of a firm plays a significant role in the stock returns with respect to systematic risk has been a compelling topic in literature (DeJong & Collins, 1985). The importance of including leverage in the risk measurement of a stock is mentioned in Galai and Masulis (1976). According to the authors, ignoring the leverage level in a firm would lead to a downward bias of stock returns, as a significant risk factor is not included. Despite Lintner’s (1965) addition of idiosyncratic risk to Sharpe’s model, the capital structure in a firm has been proved to have a strong correlation with systematic risk responsiveness and should thus not be excluded in the analysis of market risk (Galai & Masulis, 1976). This correlation is likely due to the fact that the debt level affects the risk class of the firm which consequently results in higher expected returns. DeJong and Collins (1985) further established a correlation between both variables through their research, concluding that a higher leverage level is connected to a greater magnitude in beta variation. Although correlation has already been confirmed in the literature, causation still needs to be investigated further.
So, in order to test for a predictive function of leverage on systematic risk responsiveness, a simple linear regression method as well as a polynomial regression model are applied, as mentioned in Hamada (1972). These models are further explained in section 4.2.
3. Data
The presence of a time-varying beta in the market model is tested on the Amsterdam Stock Exchange. The Dutch market is used as it represents a developed financial sector (Sonnemans, 2006). The data is composed of monthly stock returns ranging from January 1984 earliest, until December 2017 latest, which provides a maximum of 34 beta values for a company. A minimum of 10 years of stock return data is demanded to be able to observe
10 | P a g e several fluctuations including behaviour during recessions and booms (Jagannathan & Wang, 1996). The shortest time periods observed within the sample amounted to 17 years. All of the regressions conducted range from January until December, no regression was done starting or ending at a random date in the year. Monthly returns are used instead of daily or weekly with the reasoning that beta can be assumed to be stable over such a short period (Lewellen & Nagel, 2006). The dependent variables will be the stock returns of major Dutch companies from various industries. The returns are the monthly Total Index Returns obtained from the Thomson Reuters DataStream.
𝑅𝐼𝑡 = 𝑅𝐼𝑡−1∗𝑃𝑡+ 𝐷𝑡 𝑃𝑡−1
To obtain the monthly return as a percentage value, RIt is divided by RIt-1. As a measure for
the risk-free rate, the EURIBOR monthly rate is used for the period between 1999 and 2017. For earlier observations, the rate of German one-month government bond is used since Dutch rates were pegged against German rates preceding the existence of the Eurozone (Obstfeld & Rogoff, 1995). For the market return, the AEX index will be used.
For this research, 45 companies from different industries were used. The importance to use individual stocks when analysing smaller markets, instead of portfolios, is mentioned in Berglund and Knif (1999). The authors argued that the truthfulness of beta forecasts for individual stocks is more vital than for diversified portfolios as the former are supposed to be more volatile than the latter.
4. Modelling and Methodology
The model used for the research is the market model by Sharpe (1964) and Lintner (1965):
𝑟𝑖𝑡− 𝑟𝑓 = 𝛼 + 𝛽𝑖𝑡(𝑟𝑚𝑡 − 𝑟𝑓) + 𝜀𝑖𝑡
Where βit represents the beta of stock i at time t, which is the covariance of the stock return
with the market return over the variance of the market return. Furthermore, rit represents the
return of stock i at time t, rf is the risk-free return, rmt is the market return at time t, α is a
constant, and εit is the error term (Sharpe, 1964) and (Lintner, 1965). To simplify the regression
process, excess returns are used.
𝑅𝑖𝑡 = 𝛼 + 𝛽𝑖𝑡∗ 𝑅𝑚𝑡+ 𝜀𝑖𝑡
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4.1 Model for time-variation in beta
Most authors vouch for a time-series approach instead of cross-sectional method to test for time-variation in beta as the historical data should theoretically include all possible scenarios and sources of risk (González-Rivera, 1997). Henceforth, to test for time-variation in the beta-coefficient, a short-window rolling OLS regression analysis is applied. This method of measurement is similar to the approach seen in Lewellen and Nagel (2006), and in the paper by Groenewold and Fraser (1999). The various papers in literature differ from each other by changing the regression windows, additionally applying recursive or overlapping regressions, or altering the data sets. Braun et al. (1995) argued in their paper that the rolling regression method is useful mainly due to its simplicity. Although other authors use different models in their research, as briefly stated in section 2.2, there are arguments for applying the short-window rolling regression analysis instead. The bivariate (G)ARCH model is disregarded on account of it providing an econometric analysis opposed to an economic analysis, which is used throughout this paper (Braun et al., 1995). And arguments to not use a state-space model with the Kalman filter technique are mentioned, amongst other things, by Lewellen and Nagel (2006). Although the Kalman filter is known to be more sophisticated than the rolling regression method, as it provides more accurate outcomes. The authors however disregard this method since the Kalman Filter poses econometric restrictions through the unobservability of the correct state variables.
The short-window rolling regression method applied here is based on the Fama-MacBeth approach (1973), however adjusted for a smaller market than the US market (Berglund & Knif, 1999), such as the Dutch market. Fama and MacBeth (1973) used a rolling window of 5 years to measure the betas of portfolios. However, for this study a smaller window of one year is used, as suggested by Berglund and Knif (1999). The authors argued that for the measurement of individual stocks (instead of portfolios), shorter time windows are crucial as beta is assumed to be more volatile. Through shorter time periods, new information is used in a more consistent way to adjust the beta estimations (Berglund & Knif, 1999). Another argument for a shorter time window is the issue of measurement problems caused by the presence of abnormal observations, which would affect the beta over the entire window length (Berglund & Knif, 1999). To measure the betas on a year by year basis, monthly observations for the returns are taken of each specific year. Following, the 12 observations are taken to regress the CAPM for that year. In addition, a rolling-regression with overlapping windows will be applied. This second method is done to receive smoother figures and it serves as a comparison to the firstly mentioned method. For the overlapping method, windows of 24 months with a step of 12 months will be used. The outcomes of the rolling regression methods can be found in Table 1.
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Beta results from the different regression methods
Regression method Rolling Regression
Overlapping
Regression OLS Regression
Mean Beta 0.7827 0.7980 0.8479
Standard Deviation 0.7003 0.5634 0.2951
Note. 12-month window used with a step of 12 months for the rolling regression; 24-month window used with a step of 12 months for the overlapping (rolling) regressions
A preliminary investigation of a time-varying beta coefficient is completed by analysing the values of the standard deviation of the beta estimates, as done in Lewellen and Nagel (2006). Other studies, such as DeJong and Collins (1985), focussed on the variances in order to determine beta variation and the magnitude of variation. Next, the standard deviation of the total beta estimates will be compared to the standard deviation of the regular OLS betas of all companies. Finally, as a more formal test for time-variation, the method as seen in Fama and French (1997) is used. The authors measured the standard deviation of the true beta values. If those ended up being different from zero and positive, time-variation can be assumed. To calculate the standard deviation of the true beta values, first their variance is computed which is the variance of the time-series beta estimates minus the variance of the estimation errors. In other words, the variance of the time-series estimates is the sum of the variance of the true betas and the variance of the estimation errors.
𝜎2(𝑇𝑖𝑚𝑒 − 𝑠𝑒𝑟𝑖𝑒𝑠) = 𝜎2(𝑇𝑟𝑢𝑒 𝐵𝑒𝑡𝑎𝑠) + 𝜎2(𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑖𝑜𝑛 𝐸𝑟𝑟𝑜𝑟𝑠)
The variance of the true betas is then converted to the standard deviation. The results can be seen in Table 2.
Table 2
Averages of True Beta Variances and Standard Deviations
Regression method Variance Time-series Beta Variance Standard Error Variance True Betas Standard Deviation True Betas Rolling regression 0.4940 0.0834 0.4076 0.5563 Overlapping regression 0.2750 0.0251 0.2499 0.4334
Note. Time-series Beta = Beta obtained from rolling regression models
If for instance a standard deviation of 0.15 is obtained and the average for the true beta is equal to 1.0, the actual value can be anywhere between 0.7 and 1.3.
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4.2 Model to test for the influence of leverage in beta
The model applied to test the level of influence leverage has on determining the beta value is one of the four methods suggested by Hamada (1972). Next to the MM valuation model approach, the measurement of systematic risk before and after debt issue, and testing the MM theory, the author mentioned the use of a regression method to analyse the effect of leverage on beta. He however stressed that it is unclear which variables are most useful and which are redundant in the analysis. For this reason, this paper will be the first stepping stone for further research by using a simple linear regression with the leverage ratio as an independent variable. The model used in this research is as follows:
𝛽𝑡 = 𝑎 + 𝑏𝛾𝑡−1+ 𝜀
Where βt is the beta value of at time t, a is a constant, b is the coefficient of the leverage ratio,
and ε is the error term. The leverage ratio of a firm is represented by γ, which is the level of debt divided by the sum of total equity and debt of a firm. The leverage ratio is lagged by one year in order to serve as a predictor. A simple regression is used as it provides the marginal effect of the level of leverage on the value of beta. In the paper by Lewellen and Nagel (2006), the authors also relied on a regression model to test the influence of several state variables on the estimates of beta, however not on the level of leverage within a firm. Although other papers used different models to analyse the relation between independent factors and the beta, firstly applying a rolling regression limits the possibility of using more accurate models. Therefore, as a comparison, leverage will also be regressed on the beta estimates on a higher order, as demonstrated by the following equation:
𝛽𝑡 = 𝑎 + 𝑏1𝛾𝑡−1+ 𝑏2(𝑦𝑡−1)2+ 𝜀
This polynomial regression model is used to analyse whether the relationship between the dependent and the independent variable is non-linear. Statistical significance is tested through the p-value, while the R2 shows the degree of explanation of the leverage ratio on beta
estimates.
5. Results and Analysis
5.1 Time-variation in the beta of Dutch stocks
The preliminary results from the rolling regression analysis may already indicate first signs of time-variation in the beta coefficient of the CAPM model. When taking a look at the graphs in Figure 1 below, which plot the beta values of two randomly selected companies from the sample over the total time span, the behaviour of the coefficient points towards non-constancy.
14 | P a g e Figure 1. Beta estimates from the rolling regressions
Beta estimates for Aegon between 1984 and 2017 Beta estimates for Wolters Kluwer between 1984 and 2017
Figure 1. The beta estimates from the short-window rolling regression method with s window of 12 observations and a step of 12 observations. 34 estimates were obtained through this method for both of the displayed companies.
Then, although the total sample values included some outliers in comparison to ordinary beta values, which can be assumed to range from 0 to 4 (Blume, 1971), the beta estimates overall reflected standard expectations. Next, when taking a first glance at some of the obtained statistics, the outcomes point towards a confirmation of the observation from the graphs. Focussing first on the range of the beta estimates, the average range observed on an individual company basis equalled 2.6555. Analysing the level of volatility, the average standard deviation for the estimated beta values of all 45 tested companies equalled 66.33%. Looking at the total set of beta estimates, the standard deviation equalled 70.02%. In comparison, the standard deviation from the total outcomes of the OLS regressions was only 29.51%. On a per company basis, the smallest degree of volatility for the rolling regression method was found to be 28.14% (Royal Dutch Shell), while the largest was 242.84% (Pharming), which by far exceeds the findings of Lewellen and Nagel (2006). Of course, these values do not provide more than another informal indication. Thus, to obtain further proof for a varying behaviour of the systematic risk measurement, the standard deviations of the true beta values were computed, as done in Fama and French (1997). Through this method, all the tested companies had a positive standard deviation for all the true beta values suggesting a time-varying behaviour of beta for all 45 stocks. The average of the true beta standard deviations summed to 55.63% which in other words means that if the average of the true betas were 1.0, the actual value could range from -0.1127 to 2.1127. On a company by company analysis, the smallest standard deviation was 23.09% and the largest 223.27%.
Therefore, these results provide strong evidence for the confirmation of the hypothesis of a time-varying beta coefficient in Dutch stocks. This is consequently further empirical evidence for Blume’s (1971) initial hypothesis that no economic variable is constant from period to
15 | P a g e period. It is hence of interest to consider the implications which come with a changing beta value. Firstly, from an econometric perspective, the Capital Asset Pricing Model’s shortcomings could be caused by applying standard OLS regressions instead of allowing for time-variations in beta. When measuring and predicting returns of stocks, academics and practitioners should use more advanced models which include the fact that stocks react differently towards market risk for every period. Although there is still an empirical lack in literature proving that a conditional CAPM completely eliminates the misspecifications in measurement of the standard model. It has however been confirmed that the beta value is a function of other economic variables. Which leads to the second implication of time-variation in beta. So, in order to possibly eliminate the misspecification of the model entirely, relevant factors influencing the betas value should be determined.
5.2 The effect of leverage on the beta time-variation in Dutch stocks
The linear regression analysis testing for the ability of the leverage ratio as a predictor of beta estimates can be seen in Table 3 below.
Table 3
Leverage analysis on beta determination
Variable Model 1 Model 2
Constant 0.6847*** (0.0365) 0.7388*** (0.0494)
Leverage ratio 0.2415*** (0.0838) -0.1658 (2639)
Leverage ratio2 0.5031 (0.3092)
R2 0.0098 0.0129
Adj. R2 0.0106
Note. N = 842. Model 1 refers to the simple regression. Model 2 refers to the polynomial regression. *** denotes statistical significance at p<0.01.
The coefficient of the leverage variable in the simple linear regression amounted to 0.2415, with a p-value of 0.004, making it a statistical significant indicator for beta estimates. However, the degree explanation from firm-leverage is fairly low (R2 = 0.0098). When comparing the
results to the outcomes of the polynomial regression method, the degree of explanation improves slightly (R2 = 0.0106), the coefficients however become statistically insignificant, with
p-values of 0.530 (for γ) and 0.104 (for γ2). The findings of the simple linear regression
nevertheless confirm the hypothesis that firm leverage can be used as a predictor for beta estimates in Dutch stocks. The leverage ratio of firms should therefore be included in the derivation of a conditional asset pricing model as a consequence.
Then, for the low level of explanation are of course logical reasons. As already claimed by Fabozzi and Francis (1978), factors influencing systematic risk responsiveness can range
16 | P a g e from four different categories. And while leverage might play a significant role in the determination of the risk-class of a firm, it is very likely that various other factors influence the value in beta as well. The issue of the low degree of explanation could thus be a result of omitted variable bias (OVB). This issue was also mentioned by other authors (Brooks et al., 1992; González-Rivera, 1997), who argued that OVB is likely due to the fact of the wide range of possible explaining factors.
Another reason for the low degree of explanation is that, although the leverage ratio changed for most observations from year to year, it was perceived as less varying in comparison to the beta estimates. Indeed, the behaviour of stocks and their systematic risk responsiveness was claimed by Berglund and Knif (1999) as highly volatile. Overall, the results show that leverage serves as a significant influence of beta behaviour, however the level of explanation is fairly low.
6. Conclusion
In this research it was tested whether firm leverage can serve as a predictor of systematic risk responsiveness if this one is varying from period to period. The analysed data consisted of Dutch stocks from the period between 1984 and 2017. In order to answer the research question, a two-folded analysis had to be done by first testing the presence of time-variation in the beta values of Dutch stocks. And secondly, by testing whether the yearly leverage ratio is a statistically significant predictor of beta estimates. Time-variation was tested through a short-window rolling regression method which resulted in finding a changing beta for all 45 stocks tested. These findings thus confirmed the hypothesis of a time-varying beta in the Dutch stock market which was first stated by DeJong, Kemna, and Kloek in 1992. Then, to test the predictive ability of firm leverage of systematic risk responsiveness, a simple linear regression model was applied. This method resulted in significant outcomes, thus also confirming the second hypothesis posed, despite the low degree of explanation. Although a second regression analysis with a higher order for leverage was able to slightly increase the degree of explanation, the results were statistically insignificant. There are however logical reasons which can explain the low degree of explanation. First, the great number of possible factors make the impact of leverage on its own very small. Second, betas were perceived as more volatile than the yearly changes in the leverage ratio which could as a consequence hinder a clear relationship between both variables.
In the conduction of this research, some limitations were encountered when analysing whether leverage has a predictive ability for systematic risk responsiveness in Dutch stocks. First, the unobservability in the correct state variables in the Kalman filter technique forced to use a rolling regression method instead. Since a state-space model is claimed to provide more accurate results, including the determination of the exact behaviour of the beta estimates,
17 | P a g e more in depth analysis of systematic risk responsiveness was not possible. Second, in the analysis of the explanatory function of leverage for betas, the issue of Omitted Variable Bias was encountered. A higher value for R2 could have been obtained with the use of the correct
control variables in the regression model. It is however not clear which the correct control variables are and further research needs to be done.
Then, to address the issue of a lower level of changes in leverage in comparison to the variation in the beta estimates, it could be of interest to analyse the influence of leverage when measuring betas of diversified portfolios. As diversified portfolios of stocks have been thoroughly analysed in literature, and it has been confirmed that their betas appear a lot more stable over time, the influence of leverage could result in a higher degree of explanation. Lastly, although the theories can be applied to different asset pricing models using linear regressions to measure returns, further empirical analysis with a direct comparison of different models is recommended to see whether outcomes differ significantly from each other.
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