The Estimation of the Unlevered Beta Revised
Friso Willem Heikens 1
University of Groningen Faculty of Economics and Business
MSc Finance 2018/2019 Supervisor: Dr. Ing. N. Brunia
Date: 12-06-2019
ABSTRACT
This study estimates and evaluates three sets of unlevered betas following the best practices of Koller et al. (2015). Monthly stock returns and accounting data is used on a sample set of 1511 US companies over the sample period January 1970 until January 2019. The three sets of unlevered betas are estimated using rolling monthly regressions following the CAPM. The findings in this study show that all three unlevered betas on a sector level vary over time and that significant correlations are observable between all three sets of unlevered betas and the relative market capitalization of the corresponding sector. Furthermore, the variance of the unlevered smoothed beta is significantly lower than the variance of the unlevered beta and the median unlevered beta of companies in the same sector. Finally, the unlevered smoothed beta suffers from less cross-sectional standard deviation than the unlevered beta. Based on the findings of this study it is advised to use the median of the unlevered smoothed beta when one wishes to estimate the future beta of an arbitrary company. When one wishes to estimate the future beta of a specific company it is advised to use the median of the unlevered betas of companies in the same sector.
Keywords: Sector beta; Capital Asset Pricing Model; Rolling Regression JEL-Code: G12; G32
Word Count: 8,086 (excl. Appendix)
I. Introduction
Estimation errors are the key problem in determining unlevered betas. Betas of individual companies can at any moment in time be heavily influenced by one-time events. When valuing a company, it is not the object to estimate a company’s historical beta but rather to estimate a company’s future beta. So, these estimation errors can give difficulties in estimating a company’s future beta. Koller et al. (2015) propose two methods to deal with these estimation errors. The first step, in both methods, is to estimate the levered beta. Following this, both methods unlever and smooth the levered beta. The first method, as proposed by Koller et al. (2015) is determining sector betas by unlevering the levered betas and then smooth these betas in full to the median of the industry. Koller et al’s (2015) company specific beta, as opposed to industry betas, first smooths the levered beta and then unlevers the smoothed beta in the same way as the first method. Damodaran (2012) deals in a similar way with solving the estimation problem in determining unlevered betas. The only difference compared to Koller et al. (2015) is that Damodaran’s (2012) industry beta is a value weighted average of the unlevered betas in the same industry.
Koller et al. (2015) provide the following steps for calculating an industry beta. First, the levered equity beta is estimated for every company in the sector. Following this, the levered beta is unlevered at each company specific debt-to-equity ratio. The unlevered beta of a company is then estimated as the median of the unlevered betas of the companies in the same sector. As was shown by Fama and French (1997) estimates for the cost of equity for industries are imprecise for both the CAPM and the three-factor model. Koller et al’s (2015) industry beta might solve this problem by setting a company’s unlevered beta equal to that of its sector. However, this method assumes that the unlevered beta of a sector is stable over time. The estimation problems are company specific and not industry specific.
This study will include three estimates of the unlevered beta of a company. The unlevered beta of a company equals (1.) the unlevered beta on a company-specific level, (2.) the unlevered smoothed beta on a company-specific level and (3.) the median of the unlevered betas of the companies in the same sector. (1.) and (2.) are based on the particulars of individual companies only. (3.) is based on the particulars of comparable companies, so companies within the same sector. Furthermore, (1.) does not make an adjustment for estimation errors, where (2.) and (3.) do make such an adjustment by either smoothing the levered betas or by taking the median of companies in the same industry. Koller et al. (2015), and the first results from this study, show that unlevered beta on a sector level shows relatively much variation over time. Following this, the research question for this study will be as follows;
The monthly data used in this study will run from January 1970 until January 2019. For the estimation of the rolling levered equity betas 60 monthly returns are used as is recommended by Koller et al. (2015), and in line with Fama and French (1997) and Gregory and Michou (2009). The levered equity betas will be unlevered with a 5-year median debt-to-equity ratio. This implies that 530 monthly rolling unlevered betas, unlevered smoothed betas and the median of the unlevered betas of the companies in the same sector can be estimated from December 1974 until January 2019.
II. Literature
This section will discuss the different methods that can be used to estimate levered betas. These two methods are the Capital Asset Pricing Model and the Fama-French Five Factor Model. These two models measure company risk by estimating the relationship, or the co-movement, between the stock price and changes in the market, which is known as beta.
II.1. The Capital Asset Pricing Model
According to Fama and French (1997) and Koller et al. (2015), the Capital Asset Pricing Model (CAPM) is the common asset pricing model choice. In the CAPM, the expected return on stock i at time t, is:
𝐸(𝑅𝑖𝑡) = 𝑅𝑓𝑡 + 𝛽𝑖[𝐸(𝑅𝑚𝑡) − 𝑅𝑓𝑡] (1)
where,
𝑅𝑓𝑡 = the risk-free interest rate.
𝐸(𝑅𝑚𝑡) = the expected return on the value-weighted market portfolio. 𝛽𝑖 = the CAPM risk of stock i.
𝐸(𝑅𝑚𝑡) − 𝑅𝑓𝑡 = the return on the value-weight market portfolio m in excess of the risk-free return f for period t.
In the CAPM, the risk-free interest rate and the market risk premium are equal for all companies in the market. Only the beta, the CAPM risk of a stock, varies across companies. So, the CAPM assumes that the expected returns on securities are a positive linear relationship of the market. However, Fama and French (1992) argue that the market beta does not suffice to explain expected stock returns. In this paper by Fama and French, the conducted tests do not support the prediction of the CAPM that average stock returns are positively related to market betas. II.2. Fama-French Five Factor Model
The Fama-French Three Factor Model captures many of the average-return anomalies of the capital asset pricing model according to Fama and French (1996). However, in 2015, Fama and French introduced the five-factor model that is directed at capturing the size, value, profitability, and investment patterns in average stock returns. This five-factor performs better than the three-factor model according to Fama and French (2015). The formula for the Fama-French Five Factor Model is the following;
𝐸(𝑅𝑖𝑡) − 𝑅𝑓𝑡 = 𝛼𝑖 + 𝛽𝑖(𝐸(𝑅𝑚𝑡) − 𝑅𝑓𝑡) + 𝑠𝑖(𝑆𝑀𝐵) + ℎ𝑖(𝐻𝑀𝐿) + 𝑟𝑖(𝑅𝑀𝑊) + 𝑐𝑖(𝐶𝑀𝐴) (2)
Where,
𝐸(𝑅𝑖𝑡) − 𝑅𝑓𝑡 = the return on security i in excess of the risk-free return f for period t.
𝐸(𝑅𝑚𝑡) − 𝑅𝑓𝑡 = the return on the value-weight market portfolio m in excess of the risk-free return f for period t.
𝑆𝑀𝐵 = excess return small stocks over big stocks.
𝐻𝑀𝐿 = excess return high-book-to-market stocks over low-book-to-market stocks. RMW = the difference between the returns of stocks with robust and weak profitability. CMA = the difference between the returns of stocks with of low and high investment firms.
The Fama-French five factor model states that the excess return of stock i is explained by the sensitivity to five factors. The first one is identical to the CAPM and is the excess return of a well-diversified market portfolio. The second factor (SMB) describes the relation between market capitalization and the average return. The third factor (HML) links the average return to the book-to-market ratio. The fourth factor (RMW) tries to link profits to average stock returns. Finally, the fifth factor (CMA), captures the relationship between the growth of total assets and average stock returns.
III. Estimation Method and Data
This section will deal with the estimation of the key ingredients that are necessary for estimating the three sets of unlevered betas. First, the estimation method of the company specific levered betas will be discussed. Following this, the method used for unlevering these company specific levered betas will be analyzed. The study will then continue by discussing and motivating how the companies in the data set will be separated across different sectors. Finally, this section will provide the data and the descriptive statistics for this study.
III.1. Levered Beta
To estimate an unlevered industry beta, the levered beta, 𝛽𝑖𝑡𝐿, has to be estimated first on a company-specific level. The levered equity beta is estimated through the CAPM. The following formula is used to estimate a company specific levered beta:
𝛽𝑖𝑡𝐿 = cov(𝑅𝑖𝑡−𝑟𝑓𝑡,𝑅𝑚𝑡−𝑟𝑓𝑡)
var(𝑅𝑚𝑡−𝑟𝑓𝑡) (3)
where,
𝛽𝑖𝑡𝐿 = company specific levered beta i at time t. 𝑅𝑖𝑡 = total monthly return company i at time t. 𝑟𝑓𝑡 = risk-free rate at time t.
𝑅𝑚𝑡 = total return market at time t.
This formula is also used by Koller et al. (2015) and Damodaran (2012) for estimating company specific levered betas. The numerator is the covariance between the firm specific return on asset i and the return of the market portfolio m. Whereas the denominator is the variance of the returns of the market portfolio. The returns that are used in this study are total returns. Dividends and stock-splits are taken into account by using total returns.
For estimating the levered betas, Koller et al. (2015) provide the following two recommendations. First, the estimation period should have at least 60 return periods, which is five-years of monthly returns. Secondly, a well-diversified portfolio has to be used as the market portfolio. In this study the returns of the S&P 500 will be used as the returns of the market portfolio.
III.1.A Levered Smoothed Beta
According to Koller et al. (2015) many academics and practitioners adjust a company’s levered beta by bringing this beta closer to its mean. This can be done through a so-called “beta-smoothing” formula. The formula Koller et al. (2015) provide is the beta smoothing process used by Bloomberg:
𝛽𝑖𝑡𝐴𝐿 = 13+23𝛽𝑖𝑡𝐿 (4)
where 𝛽𝑖𝑡𝐴𝐿 stands for the adjusted levered equity beta. The theoretical underpinning of this smoothing technique is that all estimated levered beta coefficients tend to regress to the grand mean of all levered betas over time, as was shown by Blume (1975). So, the above smoothing formula brings the levered beta closer to one, which is in theory the mean of all betas.
III.2. Unlevered Beta
After the rolling, company specific, levered betas are estimated, over the sample period December 1974 until January 2019, the unlevered betas can be calculated. Koller et al. (2015) use the following formula for unlevering the company-specific levered beta and levered smoothed beta:
𝛽𝑖𝑡𝑈 =𝛽𝑖𝑡𝐿+(𝐷𝐸)𝑖𝑡∗𝛽𝑖𝑡𝐷
1+(𝐷𝐸)𝑖𝑡 (5)
where,
𝛽𝑖𝑡𝐿 = company specific levered beta i at time t. 𝛽𝑖𝑡𝑈 = company specific unlevered beta i at time t. (𝐷𝐸)𝑖𝑡 = company specific debt-to-equity ratio i at time t. 𝛽𝑡𝐷 = the cost of debt at time t.
The term (𝐷𝐸)𝑖𝑡 is, as defined earlier, the five-year median debt-to-equity ratio of a company in a single point. This implies that from December 1974, just as with the levered betas, rolling monthly five-year median debt-to-equity ratios can be estimated. The company specific debt to equity ratio will be calculated using the following formula:
(𝐷𝐸)𝑖𝑡 = 𝐵𝑉𝐷𝑖𝑡
where,
𝐵𝑉𝐷𝑖𝑡 = book value of debt of company i at time t.
𝑀𝑉𝐶𝐸𝑖𝑡 = market value common equity of company i at time t.
Table 1: The Estimation Method for Three Sets of Unlevered Betas
Table 1 shows the estimation method for the three sets of unlevered betas. The first column shows the estimation method for a company specific unlevered beta. The second column shows the estimation method for a company specific unlevered smoothed beta. The third shows the estimation of the median unlevered beta of companies in the same sector.
1. Unlevered Beta 2. Unlevered Smoothed Beta 3. The Median Unlevered
Beta of Companies in the Same Sector
𝛽𝑖𝑡𝐿 = cov(𝑅𝑖𝑡− 𝑟𝑓𝑡, 𝑅𝑚𝑡− 𝑟𝑓𝑡) var(𝑅𝑚𝑡− 𝑟𝑓𝑡) 𝛽𝑖𝑡𝐿 = cov(𝑅𝑖𝑡− 𝑟𝑓𝑡, 𝑅𝑚𝑡− 𝑟𝑓𝑡) var(𝑅𝑚𝑡− 𝑟𝑓𝑡) 𝛽𝑖𝑡𝐿 = cov(𝑅𝑖𝑡− 𝑟𝑓𝑡, 𝑅𝑚𝑡− 𝑟𝑓𝑡) var(𝑅𝑚𝑡− 𝑟𝑓𝑡) 𝛽𝑖𝑡𝑈 = 𝛽𝑖𝑡 𝐿 + (𝐷 𝐸)𝑖𝑡∗ 𝛽𝑖𝑡𝐷 1 + (𝐷𝐸)𝑖𝑡 𝛽𝑖𝑡𝐴𝐿 = 1 3+ 2 3𝛽𝑖𝑡𝐿 𝛽 𝑖𝑡𝑈 = 𝛽𝑖𝑡𝐿 + (𝐷 𝐸)𝑖𝑡∗ 𝛽𝑖𝑡𝐷 1 + (𝐷𝐸)𝑖𝑡 𝛽𝑖𝑡𝑈𝑆 = 𝛽𝑖𝑡 𝐴𝐿+ (𝐷 𝐸)𝑖𝑡∗ 𝛽𝑖𝑡𝐷 1 + (𝐷𝐸)𝑖𝑡 𝛽𝑖𝑡𝑀𝑒𝑑. = 1 ∗ 𝛽 𝐼𝑀𝑒𝑑.+ 0 ∗ 𝛽𝑖𝑡𝑈
III.3. Industry Classification
After estimating the company specific rolling monthly unlevered equity betas, over the sample period December 1974 until January 2019, median industry betas can be estimated. The 1511 companies in the sample set will be distinguished into industries according to the Global Industry Classification Standard (GICS). There are many industry classification schemes available, however Hrazdil, Trottier and Zhang (2013) show that the Global Industry Classification Standard is superior compared to other industry classifications in grouping stocks with similar operating characteristics. The choice of using the Global Industry Classification Standard is further supported by Kile and Phillips (2009) who show that GICS codes provide improvements over other industry classifications in targeting technology firms, which is however only one specific industry. Finally, Bhojrai, Lee and Oler (2003) find that GICS classifications are significantly better in explaining stock return co-movements compared to the Fama and French algorithm, SIC- and NAICS codes. Koller et al. (2015) recommend using Standard Industrial Classification (SIC) codes or the GICS system.
The Global Industry Classification Standard is widely used, since 2006 34,000 publicly traded companies were classified by this standard according to Standard & Poor’s (2006). There have recently been changes within the GIC groups and industries in 20182. These changes in classifications is mainly due to technological progress, and the way people communicate, in the Technology, Media and Telecommunications (TMT) Sector. For the other industries the classification standards have been stable since the introduction in 2006. Appendix A shows the 11 different sectors combined with their industry groups.
III.4. The Data
This study uses 1511 distinctive companies that were present in the S&P 500 during the sample period January 1970 until January 2019. These companies are equal in terms of size, data availability and the number of companies that were/are included in this index. In this list of 1511 companies, there are companies included that have been delisted, gone bankrupt and were merged or acquired over the sample period. The required data is obtained from the Compustat database3.
This study will make use of monthly total returns to estimate monthly rolling levered betas from December 1974 onwards. For the calculation of the book value of debt and the market value of common equity quarterly fiscal end accounting data is used. These two different data sets contained a different number of companies. Therefore, the data was trimmed to match the companies in both data sets. So, companies that are not present in one data set but are present
2 https://www.msci.com/gics
in the other are not included in this study. Furthermore, the monthly and quarterly data of these 1511 companies is available before they entered the index, when they were present in the index and when they possibly left the index.
Table 2 Panel A provides an overview of the variables that are obtained from the Compustat database. All the required data for this study could be acquired for Compustat except from the monthly risk-free rate and the monthly market returns. In both the data set for the monthly total return and the quarterly debt-to-equity ratio, Compustat does not provide industry classifications for 171 companies. These companies were at last active until May 1991. The companies without an industry classification will remain in the data set. Since, if these companies were removed from the sample set, there would be decrease in monthly observations in the early period of the sample set. The distribution of the number of companies over the entire sample period December 1974 until January 2019 can be seen in Figure 1.
Figure 1: The distribution of companies over the entire sample period December 1974 – January 2019
Table 2: Compustat Variable Description
Table 2 presents the Compustat variables needed for this study. Panel A displays the monthly total return- and quarterly accounting data; Panel B displays the GICS Codes; Panel C shows the factor risk premiums for the CAPM for which the number of monthly observations is 589.
Panel A: Compustat Variable Description Accounting Data Accounting Data Abbreviation
Compustat
Variable Calculation
Monthly Total Return Rit TRT1M
Monthly Risk-free Rate4 rft
Monthly Market Return5 Rmt
Monthly Levered Beta 𝛽𝑖𝑡𝐿
cov(𝑅𝑖𝑡− 𝑟𝑓𝑡, 𝑅𝑚𝑡− 𝑟𝑓𝑡) var(𝑅𝑚𝑡− 𝑟𝑓𝑡)
Quarterly Total
Short-Term Debt DD1Q
Quarterly Total
Long-Term Debt DLTTQ
Quarterly Book Value
Debt BVDit DD1Q+DLTTQ
Quarterly Common Shares
Outstanding CSHOQ
Quarterly Price Close PRCCQ
Quarterly Market Value
Common Equity MVCEit CSHOQ*PRCCQ
Quarterly Debt-to-Equity DEit (DD1Q+DLTTQ)/(CSHOQ*PRCCQ) Panel B: Company Sector Codes
Code Name Compustat Variable Number of Categories
Company Name Company Name
GIC Sector GSECTOR 11
GIC Groups GGROUP 24
GIC Industries GIND 67
GIC Sub-Industries GSUBIND 147
Panel C: Factor risk premiums for the CAPM Rmt - rft
Monthly Average Premium 0.29 Standard Deviation (SD) 4.38 Standard Error (SD/5891/2) 0.18
4 The monthly risk-free rate is acquired from the public online library of Kenneth R. French:
https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html
5 The monthly market returns are calculated from the monthly returns on the S&P 500. These returns
Table 3: Descriptive Statistics Median Levered Beta and Median Relative Industry Market Capitalization
Table 3 presents the descriptive statistics for the median levered beta and the median sector relative market capitalization. The total number of monthly levered betas is 461,090. The total number of monthly sector medians is 530 for all sectors.
Panel A: Descriptive Statistics Median Levered Beta December 1974 – January 2019 Sector
GIC Code
# of
Companies MAX MIN MEAN MEDIAN ST. DEV MAX - MIN Energy 10 92 1.63 0.56 1.02 0.93 0.27 1.07 Materials 15 106 1.49 0.72 1.11 1.15 0.18 0.76 Industrials 20 192 1.34 0.71 1.09 1.13 0.14 0.63 Consumer Discretionary 25 230 1.46 0.80 1.13 1.15 0.17 0.66 Consumer Staples 30 108 1.13 0.27 0.74 0.75 0.21 0.86 Health Care 35 121 1.32 0.38 0.96 1.02 0.24 0.94 Financials 40 172 1.49 0.47 1.04 1.06 0.22 1.02 Information Technology 45 157 2.51 1.07 1.48 1.42 0.32 1.43 Communication Services 50 49 1.51 0.60 0.97 0.96 0.18 0.91 Utilities 55 75 0.86 0.06 0.46 0.47 0.17 0.80 Real Estate 60 38 1.82 0.09 0.74 0.57 0.43 1.74 All Sectors 1511 1.23 0.69 1.02 1.05 0.12 0.54
Panel B: 1-year Relative Market Capitalization per Sector (in %) January 1971 – January 2019 Sector GIC Code Median Market Cap. Maximum Market Cap. Minimum Market Cap. Energy 10 0.11 0.24 0.06 Materials 15 0.06 0.09 0.02 Industrials 20 0.12 0.15 0.08 Consumer Discretionary 25 0.10 0.17 0.07 Consumer Staples 30 0.11 0.16 0.07 Health Care 35 0.10 0.15 0.06 Financials 40 0.12 0.20 0.06 Information Technology 45 0.12 0.29 0.07 Communication Services 50 0.07 0.10 0.01 Utilities 55 0.05 0.08 0.00 Real Estate 60 0.01 0.03 0.00
What stands out from Table 3 Panel A is that there are substantial differences observable between the largest- and the smallest levered beta across all sectors. The Industrials sector, for example, exhibits the smallest difference (0.63) between the largest and lowest beta. For the Real Estate sector, the difference between the largest and lowest beta even reaches 1.74. Even the total market, where all the monthly observations are included, exhibits a large difference of 0.54 over the entire sample period. This implies that there are relatively large swings in the levered beta on a sector level. Figure 2 shows how the total median levered beta behaves over time together with the first- and third quartile. What stands out from this graph is that total median levered beta decreases from around 1996 to 0.69 in 2003. As can be observed, after 2003 the total median levered median increases back to its old level before 1996.
Figure 2: Median, Q1 and Q3 of the monthly levered betas.
Figure 3 shows the difference between the first- and third quartile of the monthly levered betas. The first quartile states that around 75% of the levered betas lies above this number and 25% of the betas lies below this number. The third quartile can be defined as the upper part of the dataset. This implies that 25% of the levered betas lies above this number and 75% lies below this number. So, what stands out from figure 2 and 3 is that around the dot-com bubble the difference between the first- and third quartile sharply increases. This means that the highest 25% of levered betas increase and the 25% lowest betas decrease over this period. From around 2005 until January 2019 the first- and third quartile are closing the gap, which is observable in Figure 2 and 3, where the median of the total monthly levered beta is again moving around one.
Figure 4: Median levered beta Information Technology sector; median levered beta other sectors excluding the Information Technology Sector; relative market capitalization Information Technology Sector.
Figure 4 and 5 provide an explanation for what is observed in Figure 2 and 3. In Figure 4 it is observable that the median levered beta of the Information Technology sector is increasing from around 1.5 in 1999 to 2.51 in 2005, where the median levered of all other industries is decreasing to around 0.55 over the same period. The explanation to this is that the Information Technology sector became an important part of the market portfolio, with a relative market capitalization of almost 30%. This is also in line with the findings of Koller et al. (2015). Furthermore, the sharp increase in market share together with the increase of the levered beta for the Information Technology sector explains the increase in difference between the first- and third quartile as depicted in Figure 3. Since the beta of the Information Technology sector is relatively higher those of the other sectors. What can be concluded as well from Figure 4 is that company or sector betas can be highly influenced by one-time events that for example change the relative market capitalization of a specific company or sector.
Table 4: Descriptive Statistics 5-year Median Debt-to-Equity Ratio December 1974 – January 2019
Table 4 shows the descriptive statistics for the monthly 5-year median debt-to-equity ratio over the sample period December 1974 until January 2019.
Sector
GIC Code
# of
Companies MAX MIN MEAN MEDIAN
ST. DEV MAX - MIN Energy 10 92 0.45 0.13 0.29 0.28 0.08 0.32 Materials 15 106 0.51 0.18 0.31 0.28 0.10 0.33 Industrials 20 192 0.41 0.16 0.24 0.22 0.06 0.26 Consumer Discretionary 25 230 0.49 0.13 0.22 0.19 0.09 0.36 Consumer Staples 30 108 0.29 0.10 0.17 0.16 0.05 0.19 Health Care 35 121 0.18 0.03 0.08 0.07 0.04 0.16 Financials 40 172 0.50 0.00 0.29 0.29 0.09 0.50 Information Technology 45 157 0.13 0.00 0.04 0.03 0.03 0.13 Communication Services 50 49 1.14 0.15 0.42 0.36 0.21 0.99 Utilities 55 75 1.55 0.55 0.88 0.74 0.33 1.00 Real Estate 60 38 0.82 0.00 0.57 0.56 0.13 0.82 All Sectors 1511 0.43 0.14 0.23 0.21 0.08 0.29
regarding the largest and smallest to-equity ratio. For the Utilities sector the largest debt-to-equity ratio is 1.55 and the lowest is 0.55, which results in a difference of 1.00.
Table 5: Descriptive Statistics Unlevered Beta and Unlevered Smoothed Beta Table 5 shows the descriptive statistics for the monthly median unlevered beta, the unlevered smoothed beta, and the median unlevered beta of companies in the same sector over the sample period December 1974 – January 2019; Panel A shows the descriptive statistics for the unlevered betas; Panel B shows the descriptive statistics for the unlevered smoothed betas; Panel C shows the descriptive statistics of the median unlevered beta for companies in the same sector.
Panel A: Descriptive Statistics Unlevered Beta (𝛃𝐢𝐭𝐔) December 1974 – January 2019
Sector
GIC Code
# of
Companies MAX MIN MEAN MEDIAN ST. DEV MAX- MIN Energy 10 92 1.29 0.46 0.81 0.72 0.22 0.83 Materials 15 106 1.20 0.59 0.86 0.87 0.15 0.61 Industrials 20 192 1.07 0.60 0.88 0.93 0.11 0.48 Consumer Discretionary 25 230 1.25 0.65 0.90 0.89 0.13 0.60 Consumer Staples 30 108 0.91 0.27 0.63 0.64 0.16 0.65 Health Care 35 121 1.16 0.37 0.85 0.91 0.19 0.79 Financials 40 172 1.02 0.43 0.76 0.78 0.12 0.59 Information Technology 45 157 2.40 0.92 1.38 1.31 0.31 1.48 Communication Services 50 49 1.07 0.40 0.71 0.69 0.13 0.67 Utilities 55 75 0.61 0.14 0.35 0.36 0.08 0.47 Real Estate 60 38 1.23 0.17 0.57 0.47 0.28 1.07 All Sectors 1511 0.96 0.57 0.81 0.83 0.08 0.39
Panel B: Descriptive Statistics Unlevered Smoothed Beta (𝜷𝒊𝒕𝑼𝑺)December 1974 – January 2019
Sector
GIC Code
# of
Table 5 Continued
Panel C: Descriptive Statistics Median Unlevered Beta of Companies in the Same Sector (𝛃𝐢𝐭𝐌𝐞𝐝.)
December 1974 – January 2019 Sector
GIC Code
# of
Companies MAX MIN MEAN MEDIAN ST. DEV MAX- MIN Energy 10 92 1.29 0.46 0.81 0.72 0.22 0.83 Materials 15 106 1.20 0.59 0.86 0.87 0.15 0.61 Industrials 20 192 1.07 0.60 0.88 0.93 0.11 0.48 Consumer Discretionary 25 230 1.25 0.65 0.90 0.89 0.13 0.60 Consumer Staples 30 108 0.91 0.27 0.63 0.64 0.16 0.65 Health Care 35 121 1.16 0.37 0.85 0.91 0.19 0.79 Financials 40 172 1.02 0.43 0.76 0.78 0.12 0.59 Information Technology 45 157 2.40 0.92 1.38 1.31 0.31 1.48 Communication Services 50 49 1.07 0.40 0.71 0.69 0.13 0.67 Utilities 55 75 0.61 0.14 0.35 0.36 0.08 0.47 Real Estate 60 38 1.23 0.17 0.57 0.47 0.28 1.07 All Sectors 1511 0.96 0.57 0.81 0.83 0.08 0.39
Table 5 Panel A shows the descriptive statistics for the median unlevered beta over the sample period December 1974 until January 2019. Observable from this table is that, after unlevering the company specific levered beta with the corresponding 5-year median debt-to-equity ratio, there is hardly a reduction observable in the sector specific standard deviations. Furthermore, for some industries the differences between the largest and smallest beta have decreased but for others this difference has increased compared to the sector specific levered beta.
If Table 5 Panel A is compared to Panel B, it is observable that there is a slight reduction in standard deviations if the levered betas are smoothed first. This implies that over the entire sample period, the sector specific monthly unlevered betas fluctuate relatively more than the sector specific monthly unlevered smoothed betas. Furthermore, the differences between the largest- and smallest beta is also relatively lower for the unlevered smoothed betas compared to the unlevered betas and the median unlevered beta of companies in the same sector. This is not surprising, since potential outliers are smoothed towards their theoretical mean of 1.00 before they are unlevered with a company specific debt-to-equity ratio. Table 5 Panel C is similar to Panel A. This is because these are the descriptive statistics based on the monthly median of the unlevered betas. In Panel C, each company’s monthly unlevered beta is equal to the median of that industry in that particular month.
IV. Methodology
To test whether it may be assumed that a company specific unlevered beta is equal to the long-term industry median, as proposed by Koller et al. (2015), first the largest and smallest value of the three monthly median unlevered betas will be reviewed. Following this sector specific correlation coefficients will be estimated.
This correlation coefficient will be estimated between the relative market capitalization and the sector specific unlevered beta, unlevered smoothed beta and the median unlevered beta of companies in the same sector. As was shown in the previous section of this study, sector specific levered betas and the total levered beta can react relatively strong to changes in relative market capitalization of a certain sector. This correlation coefficient shows the strength and direction of the potential linear relationship between these two populations. A hypothesis test has to be conducted to test for the significance of the correlation coefficient to observe whether the relationship is strong enough. The test for significance of the correlation coefficient consists of the following null- and alternative hypothesis;
𝐻0: 𝜌 = 0 𝐻𝑎: 𝜌 ≠ 0
The null hypothesis states that the correlation coefficient is not significantly different from zero. This implies that there is not a linear relationship between the relative market capitalization and the industry specific unlevered beta. The alternative hypothesis states that the correlation coefficient is significantly different from zero, and that there is linear relationship.
After the estimation of sector specific correlation coefficients for all three sets of unlevered betas, a variance F-test will be conducted on a sector level. Due to the rolling estimation in levered betas there is a large overlap in observations which results in highly autocorrelated unlevered betas. This implies that the variance in these betas is underestimated which will result in biased standard deviations according to Brooks (2014). So to tackle this problem of autocorrelation the (unconditional) variances will be calculated using an autoregressive model of order, which is also known as an AR(1) model. The model is derived from the book “Introductory Econometrics for Finance” by C. Brooks (2014).
𝐻0: 𝑠12= 𝑠22 𝐻𝑎: 𝑠12 > 𝑠
22
The null hypothesis states that the variance of one of the unlevered betas is equal to the variance of one of the other two unlevered betas. This implies that the two populations will have the same variance over time. Whereas the alternative hypothesis states that the variance of one of the unlevered beta is significantly larger than the variance of one of the other two unlevered betas. If the null-hypothesis can be rejected, it is safe to say that the variance of one of the unlevered betas is significantly larger than the one of the other two unlevered betas.
Table 6 provides the results from the described methodology. The first column shows the difference between the largest and smallest value for the median unlevered beta within each industry over the entire sample period. The second column gives the correlation coefficient for each sector between one of the unlevered betas and the corresponding one-year median relative market capitalization. The third column shows the F-statistic for the variances test between two sets of populations. The fourth and final column provides the median cross-sectional standard deviation within each industry over the entire sample period (in %).
As can be seen in the first column the difference between the largest and smallest value of the sector-specific median of the three sets of betas is the smallest for the unlevered smoothed beta. This implies that the unlevered smoothed betas, on a sector level, have relatively the least variation over the entire sample period compared to the other two unlevered betas. For the unlevered beta and the median unlevered beta of companies in the same industry the numbers are similar. This is because these two unlevered betas are based upon the same fundamentals as was mentioned earlier in this study.
As can be observed in Section III.4. Table 3 there are also relatively large differences in relative market capitalization over the entire sample period. The differences in relative market capitalization can be up to 0.22. The correlation coefficient in the second column shows the strength of the linear relationship of the three sets of unlevered betas with the sector-specific one-year median relative market capitalization. For the unlevered beta and the median unlevered beta of companies in the same industry, the null-hypothesis of no linear can be rejected at the one percent significance level for eight sectors. This implies that for eight sectors there is a significant linear relationship between the unlevered beta and the relative market capitalization. Furthermore, the correlation coefficients of the unlevered beta and the median of the unlevered for companies in the same sector are again similar. For the unlevered smoothed beta nine sectors have a significant correlation coefficient. Here, the utilities sector also shows a significant correlation coefficient. If the correlation coefficients within sectors are compared to each other it is observable that some linear relationships become stronger and some become weaker for the unlevered smoothed beta. Therefore, no substantiated conclusion can be drawn from the correlation coefficients in comparing the three sets of unlevered betas.
significance level. Meaning that for each sector the variance of the unlevered smoothed beta is significantly lower compared to the variance of the unlevered beta. The second column shows the F-statistic between the unlevered beta and the median unlevered beta of companies in the same sector. Since both variances are equal to each other over the entire sample period the null-hypothesis cannot be rejected for every sector. The third column provides the F-statistic between the unlevered smoothed beta and the median unlevered beta of companies in the same sector. These statistics are equal to the statistics in the first column, since the variance of the median unlevered beta of companies in the same sector is equal to the variance of the unlevered beta. So, in this case the null-hypothesis of equal variances can be rejected again for every sector at the one percent significance level.
VI. Conclusion
The central question is whether there are significant differences present between the unlevered beta, the unlevered smoothed beta and the median of the unlevered betas of the companies in the same sector. When valuing a company, it is not the objective to estimate a company’s historical beta but rather to estimate a company’s future beta.
First, all three unlevered betas show much variation over the entire sample period. Secondly, except from the correlation coefficient there are significant differences observable between these three unlevered betas. The difference between the largest and smallest value of the unlevered beta over the sample period December 1974 until January 2019 is the smallest for the unlevered smoothed beta for all sectors. Furthermore, the unlevered smoothed beta suffers from significantly less variance compared to the other two sets of unlevered betas for all sectors. Finally, the unlevered smoothed beta shows less cross-sectional standard deviation compared to the unlevered beta. The median of unlevered betas of companies in the same sector, the method proposed by Koller et al. (2015), has no cross-sectional standard deviation over the entire sample period. These results indicate that either smoothing the unlevered beta or taking the median of unlevered betas of companies in the same sector reduces estimation errors.
Reference List
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Internet Sources
Public online library Kenneth R. French for the risk-free interest rate. Retrieved from:
https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html
Yahoo Finance for the monthly market returns S&P-500. Retrieved from:
https://finance.yahoo.com/quote/^GSPC?p=^GSPC&.tsrc=fin-srch
Wharton Compustat for the company specific monthly total returns and accounting data:
https://wrds-web.wharton.upenn.edu/wrds/index.cfm
MSCI for stability in industry classification codes:
https://www.msci.com/gics
Appendix
Appendix A: Industry Classification
Global Industry Classification Standards (GICS) Codes
Sector Code Industry Group Code
Energy 10 Energy 1010
Materials 15 Materials 1510
Industrials 20 Capital Goods 2010
Commercial & Professional Services 2020
Transportation 2030
Consumer Discretionary 25 Automobiles & Components 2510 Consumer Durables & Apparel 2520
Consumer Services 2530
Retailing 2550
Consumer Staples 30 Food & Staples Retailing 3010 Food, Beverage & Tobacco 3020
Household & Personal Products 3030
Health Care 35 Health Care Equipment & Services 3510
Pharmaceuticals, Biotechnology & Life
Sciences 3520
Financials 40 Banks 4010
Diversified Financials 4020
Insurance 4030
Information Technology 45 Software & Services 4510 Technology Hardware & Equipment 4520
Semiconductors & Semiconductor Equipment 4530
Communication Services 50 Telecommunication Services 5010
Media & Entertainment 5020
Utilities 55 Utilities 5510
