The Precision of Asset Beta Estimates

The Precision of Asset Beta Estimates by Vance P. Lesseig Associate Professor of Finance McCoy College of Business Administration Texas State University San Marcos, TX 78666 vlesseig@txstate.edu and Janet D. Payne* Associate Professor of Finance McCoy College of Business Administration Texas State University San Marcos, TX 78666 jpayne@txstate.edu *contact author

The Precision of Asset Beta Estimates Abstract The CAPM has fundamentally changed the way finance is taught and practiced since its development in 1964. However, one problem with the use of the model is estimating the systematic risk of untraded assets. Academics and practitioners have dealt with the problem by using traded assets as proxies for the untraded asset. Some academic research has attempted to measure the validity of this technique using the average difference in the true beta of a traded firm and the proxy beta using a sample of similar firms. However, the use of the average difference across a number of comparisons is not necessarily useful to a practitioner. This paper examines the absolute difference between a firm s unlevered beta and a proxy beta calculated using the formula given in Hamada, 1972, and the pure play method. We find that the estimates are not reliably close to the true value. Using both deciles of relevant variables and a matching method similar to that used by practitioners, we examine a variety of different characteristics to identify similar firms. However, we do not find any matching criteria that improves the estimate to a level we believe would be acceptable to practitioners attempting to measure cost of equity capital for their untraded firm or asset. We conclude that further research should be done in order to identify a better way for managers of untraded firms or assets to proxy their systematic risk.

The Precision of Asset Beta Estimates I. Introduction The Capital Asset Pricing Model (CAPM) has been a topic in most finance courses and widely used in practice since its introduction in 1964. This level of adoption has occurred despite a vast body of theoretical and empirical research that has questioned the estimation and validity of beta as the sole measure of relevant risk. 1 Over the past half century almost every aspect of the CAPM has been researched and it has often been found lacking with respect to its ability to explain returns. However, its importance in practice makes the ability to precisely estimate its parameters still very relevant. Practitioners have long used the CAPM to estimate the cost of equity capital for firms as well as for individual projects. The simplicity of the model is clearly one of its most attractive features. For practitioners the most difficult aspect of using the CAPM is the estimation of the systematic risk (Beta) of an untraded asset. Beta is the only asset-specific characteristic required in CAPM, but the asset must be traded for beta to be correctly estimated. The difficulty in practice is for situations in which betas are needed for non-traded assets, particularly for private firms, specific divisions within a firm, or individual projects. Even if the division or project is part of a traded firm, if it does not have the same systematic risk as the overall firm, its beta must be estimated separately from the firm as a whole. Thus a good deal of work is focused on the estimation of beta using something other than a regression estimation of the asset returns versus the market. The most common technique is to estimate a beta based on the beta of traded assets sharing similar systematic risk, then adjusting this beta to fit the characteristics of the asset whose beta is being estimated. 1 See, for example, Fama and French (2004) for a review of empirical history related to CAPM as well as Shih, et.al (2014) and Harissis (2000) for a review of some of the theoretical issues associated with CAPM. 1

But how precise are these estimates? Most studies that evaluate this technique (Bowman and Bush, 2006, Sarmiento-Sabogal and Sadeghi, 2014) use Hamada s, 1972, formula (or some variant) to estimate the asset betas of traded firms and then compare the estimated (or proxy) beta to the true asset beta of the firm. However, they tend to report only the average difference between the estimate and the actual asset beta. Unfortunately, a small average difference could mean that the estimates are accurate, or only that they are unbiased. For a practitioner, the average difference across many firms isn t nearly as important as just how close a particular estimate is to the true value. Our approach is to mimic as closely as possible the techniques a practitioner would use to estimate beta by matching traded firms on various characteristics. We select a traded firm with a known beta and treat it as an untraded asset. By following the technique of er and Kerr (1981) we unlever the betas of firms matching our test firm then re-lever the average asset beta with the capital structure of the firm being examined to create a proxy beta, much like previous work. However, instead of examining only the average difference between proxy and actual betas across firms, we also examine the absolute different between individual firm proxy betas and their actual betas. We argue that for a practitioner, this absolute difference is a more relevant measure. In examining the difference between the true and proxy asset betas for traded firms we find the mean difference to be effectively zero, consistent with previous studies. However, we show that the average absolute difference is 0.415. Considering that the average asset beta is around 0.7, this estimation error is quite large. As practitioners try to estimate CAPM, an error in beta of that magnitude can lead to a gross over- or underestimation of the cost of equity capital. The error could easily be so large as to make the estimation no better than using a predetermined hurdle rate based on an educated guess, which comes at a much lower cost. Therefore, the purpose of this paper is twofold, first to point out the magnitude of errors in traditional approaches of determining betas for untraded assets, and second to try to improve those 2

estimations. We use a multitude of measures to identify competitors for use in the pure-play method, as well as two different methods for calculating the asset beta of the subject firm. All are woefully lacking in their precision, leading us to believe that further research must be done to identify a better way to estimate this important metric. II. Background A. CAPM and Leverage Hamada, 1972, is the first to decompose the observed (market) beta into its operating (asset beta) and financial risk (financial leverage) components. Hamada s process of removing financial leverage is still the most common adjustment technique used to determine an asset beta. Hamada s process decomposes systematic risk into operating and financial risk by isolating the firm s unlevered beta (asset beta). In this decomposition, as modified by Rubenstein, 1973, the effects of financial leverage are removed from the firm s leveraged equity beta in the following familiar way: (1 ττ)dd ββ UU = ββ LL 1 EE This seminal work has contributed a great deal to the literature, both from an academic and a practical perspective. Academics frequently use a firm s asset beta as a control variable to measure operating risk in various types of analysis. More relevant to this paper, practitioners also use the measure to estimate the systematic risk and thus the cost of equity capital for a firm, project, or division when a true measure of beta cannot be calculated. The typical method often referred to as the pureplay method is to take the observable market beta of another firm, or group of firms, then find their average unlevered beta and re-lever the beta to fit the firm, project, or division being estimated. The term pure-play refers to the idea that the observable firms are in the same line of business as the firm, project or division that is being estimated. 3

Mandelker and Rhee, 1984, test the hypothesized association between market beta and operating and financial risk, and find that both operating risk and financial leverage are positively related to beta. They also find that firms with higher (lower) operating risk tend to have lower (higher) levels of financial leverage, indicating that firms act as though both components impact the firm s overall systematic risk. Gahlon and Gentry, 1982, develop and illustrate a model in which the degrees of operating and financial leverage, as well as the coefficient of variation of revenue and the correlation coefficient of cash flows affect a security s systematic risk. Faff, et.al (2002) demonstrate that using the time series of a firm s leverage gives a better estimate of a firm s asset beta than an annual cross-section. While this appears to be a useful technique, we feel that practitioners are unlikely to use this approach, due to the data requirements as well as the sophistication necessary for the amount of precision gained. B. The Estimation Process er and Kerr, 1981, formalize the pure-play technique used by practitioners to estimate the beta for use in the CAPM to calculate equity cost of capital for assets. They test the method described above for multidivisional firms, using firms that are comparable to their subject firms divisions to proxy for the asset beta of each division. They then take the weighted average of the divisions in the firm and find that, on average, the weighted average is a good approximation for the overall beta of the traded firm. Several empirical papers have tested the efficacy of various common methods used by practitioners to estimate beta as described above. Sarmiento-Sabogal and Sadeghi, 2014, address the issue of which estimation procedure (with or without taxes) results in a better estimate of asset beta. Their procedure is to calculate a proxy levered firm beta by (1) unlevering market betas of observable firms, 2) calculating the exogenous (not including the subject firm) yearly mean asset beta for each industry of observable firms, and 3) re-levering the industry beta for the leverage of each firm. They use 4

four different techniques to create the proxy betas 2 and compare each technique to the actual beta. To account for differences in operating risks between firms within an industry they compute a statistic, λ, as: λ ii = ββuu ββuu ii where ββuu = the industry average unlevered beta and ββuu ii = the unlevered beta for firm i. This statistic is then multiplied by the re-levered beta to create the proxy beta. Finally, they run regressions to identify the best estimate for beta. They find that the beta recommended by Hamada, 1972, and extended by Rubenstein, 1973, using the market value leverage ratio, provides the best estimate of the firm s actual beta. Bowman and Bush, 2006 use a similar approach to test the effectiveness of the practitioner method for estimating beta. They identify comparable companies using Bloomberg s industry groups then unlever the beta according Hamada s equation including taxes 3. They then compare the actual asset beta to the calculated asset beta and report the average difference. They find that while their estimates are close to the actual betas, the relative sizes of the comparable firms versus the firm of interest is an important determinant in the difference between the proxy and actual betas. Ingram and Margetis (2012) use cluster analysis to group peers across various accounting measures. They use regression analysis to test the hypothesis that their proxy estimates are a close approximation of firms estimated betas. They find a fairly good fit, indicating an improvement in the ability to estimate cost of equity capital. However, we again believe that their technique is not likely to be used by most practitioners. 2 For the four proxies, they use market value of equity (with and without adjusting for taxes) and book value of equity (with and without adjusting for taxes). 3 Bowman and Bush, 2006 use the corporate tax rate as of 2003, which was 40%, as their tax rate for all firms. Because this rate is the same for all firms, we assert that this method is effectively the same as using the formula without taxes. 5

As noted above, previous researchers testing proxies for firm betas do not report the absolute differences between actual and proxy betas. Sarmiento-Sabogal and Sadeghi, 2014, report the average proxy beta and the average λ, while Bowman and Bush, 2006, report the average difference between the proxy beta and the actual beta. While this average is the customary method of examining differences for empirical research, from the perspective of the practitioner the absolute difference is more relevant. It is certainly conceivable that even if the average difference across a number of firms is very small, the average absolute difference between the betas for individual firms can be quite large. If an individual beta estimate is far from the actual value it invalidates the process. This is especially true given that in practice, most are trying to estimate a beta for one specific firm or division; in this case, the absolute difference matters a great deal. Ingram and Margetis, 2010, report tests on equity cost of capital rather than beta. In a regression of cost of equity capital on proxy cost of equity capital (using the proxy beta) they find an intercept of 0.1758 and a slope of 0.8426. Given these estimates, a firm with a proxy cost of equity capital of 0.07 would have a predicted cost of equity capital of 0.235. The authors acknowledge that more work should be done to calibrate their procedure. We assert that an estimate that high is likely to be biased upward, resulting in rejection of (most likely) positive NPV projects. This paper addresses the practitioner s problem by examining proxy asset betas more closely. First, we assert that if asset beta is an effective measure of operating risk, firms in the same industry should have the same (statistically) asset beta, after adjusting for differences in operating leverage. If there are statistical differences between operating leverage-adjusted asset betas of firms in the same industry, there are two possible explanations: either asset beta does not measure business risk or asset beta is not measured correctly. We assume that asset beta measures the systematic risk of the firm s assets, as described by Hamada, 1972. If asset beta is not being measured correctly, our goal is to 6

identify the other firm characteristics that may keep unlevered betas from reaching the true asset beta. We begin by documenting the variation of asset betas among firms in the same industry. We test whether this variation is due to differences in operating leverage, which is certainly a legitimate reason for such variation. If this is the case, our asset betas must be adjusted for operating leverage to continue. We use a modification of the approach of Sarmiento-Sabogal and Sadeghi, 2014, to examine the precision of proxy asset beta estimates when estimated using the pure play method under several different assumptions about the correct choice of peers. We then determine whether it is necessary to adjust for operating leverage. Rather than using average difference between asset beta as directly measured by OLS (which we refer to as the actual asset beta) and our proxy asset beta, we examine the absolute value of the difference between proxy and actual asset beta. We begin by using all firms within a subject firm s industry, then further refine the sample by matching firms on other characteristics. 4 III. Methods and Results Our sample consists of 75,304 firm-year observations representing 9,942 firms from 1992 to 2013. Given that practitioners tend to use the S&P 500 (a value-weighted index) as a market proxy, we estimate our betas using the CRSP value-weighted index with monthly returns 5. Following common practice, we compute betas over the previous five years. We also calculate market value of equity at the 4 We include a cluster method, as well, as used by Ingram and Margetis (2010). We find that under every clustering method the results are worse than those using the full industry and matching techniques. Given that industry practitioners are unlikely to use cluster analysis, we do not include those results in this paper. 5 We also estimate three additional betas (CRSP equally weighted daily computed annually, CRSP equally weighted monthly using rolling five years, and CRSP value weighted daily computed annually). Results are substantially the same, so we restrict our reporting to CRSP value weighted monthly for parsimony in reporting. 7

end of each year using CRSP data. We exclude firms with fewer than 30 available returns over the observation period. We use Compustat data for the book value of debt, revenue, EBIT, net income, SIC code, book value of equity, total assets, and cash flow from operations. We remove the effect of leverage from the betas using Hamada s formula as typically applied 6. That is, we calculate debt-to-equity (D/S) ratios using only long-term debt and the current portion of long-term debt in the numerator 7 and market value of equity in the denominator. Using Fama and French s 12 industries 8,9, we take the industry average beta for each firm, not including the firm in question 10. Following previous work, we exclude financial firms from the sample ( 11). To mitigate the effect of outliers, we winsorize the D/S ratio at 1 percent. Given that practitioners seldom use all firms in the industry as comparable companies for the purpose of calculating asset beta, we construct a second control sample comprised of firms that are the closest in terms of various metrics to the subject firm, in keeping with the findings of Bowman and Bush, 2006. Table 1 displays the summary statistics for each of the 11 remaining Fama and French industry designations. The first thing that becomes readily apparent is the large difference in characteristics across industries. Specifically, the average market value of equity (column 3) ranges from $1.8 billion for industry 2 to $7.8 billion for industry 4. The difference in leverage ratios (column 4) shows a low of 0.2 in industry 6 to 1.23 for industry 7. One of the more consistent values across industries is the beta. 6 We report results using Hamada s formula without taxes. However, results are substantively unchanged when the adjustment for taxes is made. 7 While the market value of a firm s debt is preferable it is typically more difficult to obtain and less likely to be used by a practitioner. See Sweeney, et al., 1997 for an analysis. 8 We use several definitions of industry, including Fama and French s 49 industries, the Hoberg-Phillips, 2010, definitions and 2, 3, and 4-digit SIC codes. The results are similar to those using the Fama and French industry classifications, which we report here for brevity. 9 The industry portfolios are available from Kenneth French s Data Library: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html 10 This is the average that Bowman and Bush (2006) call the exogenous industry average. We prefer not to use that term in order to avoid the implication that the average is truly exogenous. 8

Since they represent the industry averages, the lack of variation should not be surprising as they tend to range near one. The most notable exceptions are industry 8 with an average beta of.48 and industry 6 with a beta of 1.45. However, the variations in firm characteristics do lend credence to the strategy of using industry as a key matching factor, a common practice in both academic research and practice. [TABLE 1 ABOUT HERE] Next, we document the large variation in asset beta among firms in the same industry in Table 2. For the overall sample the mean asset beta is 0.801, with a standard deviation of 0.654, but we see a greater variation in asset betas across industries than is readily apparent for the equity betas in Table 1. The consistency of the levered betas reinforces previous findings and capital structure theory that firms with lower levels of business risk tend to take on more leverage. [TABLE 2 ABOUT HERE] Table 3 gets at the heart of the first issue we are investigating: comparing proxy betas to actual betas. The results are striking. Column 1 shows the average unlevered beta of the firm of interest for the total sample and by industry segment while column 2 shows the average proxy asset beta, again for the total sample and by industry. As shown in previous studies, the average difference between the betas is quite small (Column 3), as is the average difference for the sample and by industry (Column 4). The worst estimates occur in industries 6 and 12. For industry 12 the average difference is approximately 13% of the average actual asset beta for the industry. The small average differences are consistent with Bowman and Bush, 2006, and give the illusion that the method of creating proxy asset betas using Hamada s formula and the pure-play method is quite effective. Column 5 shows the lambda parameter as defined by Sarmiento-Sabogal and Sadeghi, 2014 to be the adjustment factor for a firm s asset beta relative to the industry. The values are comparable to those shown by Sarmiento-Sabogal and Sadeghi, 2014, and again imply that the estimate is reasonably precise. 9

However, when the average of the absolute difference between the actual and proxy asset betas is examined in column 6, a different story emerges. For the overall sample, the average absolute difference of 0.415 is over 50% of the full sample unlevered beta. This level is fairly consistent across industries, as the absolute difference is greater than 50% of the industry asset beta in eight of the twelve industry classifications. In no industry is the absolute difference less than 47% of the actual unlevered beta. Clearly this kind of error should be concerning to practitioners. It is hard to imagine that a difference of that magnitude gives anyone any confidence in the estimate. In economic terms, using a market risk premium of 4.5%, an error of that size results in a cost of equity that is off by more than 2.8%. [TABLE 3 ABOUT HERE] To attempt to improve the precision of our proxy betas, as most practitioners are unlikely to use the entire industry to form their estimates, we divide the industries along several dimensions and form decile portfolios based on each dimension. Using characteristics determined by previous literature, including Bowman and Bush (2006), we examine their correlations with asset beta 11. The correlations between asset beta and the firm characteristics we selected are reported in Table 4. [TABLE 4 ABOUT HERE] Small firms tend to have larger asset betas, so firm size is negatively related to asset beta. Contrary to our original assertion, degree of operating leverage is not a significant determinant of asset beta. Given this finding, we do not adjust asset beta for operating leverage in further tests. All other variables, including industry code, are significantly related to asset beta. Given the significance of size in the above regression we create deciles for three different measures of size: total assets, revenues, and the market value of equity. Additionally, we create deciles by earnings-to-price ratio, growth in 11 We use correlations rather than regression analysis as many of the assumptions of OLS are violated in the structure of our data. 10

revenues over the previous year, market-to-book ratio, Tobin s q, capital expenditures, leverage, longterm debt, and volatility of operating cash flows 12. Because the attributes of potential peers are not available until financial statements are produced, we use the lagged values of the firm-specific variables to estimate the decile and to calculate the proxy asset beta. The firms used as proxies are the other firms in the same decile as the firm of interest. The results are shown in Panel A of Table 5. Unfortunately, the average absolute difference is not much closer than when the entire industry is used. Because firms with similar characteristics and risk profiles should be similarly valued no matter their industry, we repeat the analysis using deciles of the full sample (disregarding industry). Again, we find no improvement in the proxy. The results in Panel B of Table 5 show the value of maintaining the industry match. The absolute differences using the full sample are greater than when only the firms in the same industry are used. [TABLE 5 ABOUT HERE] To try to find the best matches possible to form the proxy betas, we move beyond deciles by matching the firm to be estimated with the firms in its industry that are closest to it in terms of the same characteristics as used in previous tests. A maximum of six matching firms are used for each firm of interest. Table 6 shows the results of this more refined matching procedure, and the results are similar to our original results. The average differences are small across the entire sample as well as in each industry, but the average absolute difference is still impractically large. [TABLE 6 ABOUT HERE] 12 Size has been shown to be a significant factor in many corporate finance applications, and is commonly used in the practice of identifying peers. Bowman and Bush, 2006 suggest earnings to price, growth in revenues, and market to book. The use of Tobin s q and capital expenditure is motivated by the work of Kim and Sun, 2016. Volatility of cash flows is suggested by Damodaran, 2008 as a measure of firm comparability. The two leverage ratios are discussed earlier in this section. 11

IV. Conclusions This paper examines the estimation of divisional asset beta and asset beta for privately held firms using Hamada s formula and the pure-play method. We first examine the impact of degree of operating leverage on the intra-industry differences in asset beta, and conclude that there is no significant relationship. We then identify several measures that can be used by firms to choose comparable companies, and test each of these measures for efficacy in calculating an adequate proxy asset beta. We make several contributions to the literature. First, we compare actual to proxy betas by means of absolute differences rather than average differences, because average differences are close to zero by definition. We find absolute difference to be quite large relative to average asset beta; hence we attempt to improve the estimate using several different metrics to identify peers. No metric we examined improves the estimate to a level of precision we deem acceptable to a practitioner trying to estimate equity cost of capital. In addition, we use lagged variables for the decile matching in order to simulate the information that would actually be available to practitioners when they are choosing peers. We conclude that the pure-play technique using common matching metrics does not provide a precise estimate for asset beta when compared to actual asset betas of publicly traded firms. We are not testing the validity of Hamada s formula in estimating systematic risk of the firm s assets; we simply conclude that further research is necessary to address the practitioner s problem of correctly implementing the pure-play method using that tool. 12

References Bowman, R.G. and Bush, S.R. (2006), Using comparable companies to estimate the betas of private companies, Journal of Applied Finance, Fall/Winter, pp. 71-81. Damodaran, A., 2008. Damodaran on valuation. John Wiley & Sons. Fama, E.F. and K.R. French (2004), The Capital Asset Pricing Model: Theory and evidence, Journal of Economic Perspectives, Vol. 18, No. 3, pp. 25-46. er, R.J. and Kerr, H.S. (1981), Estimating the divisional cost of capital: An analysis of the pure-play technique, The Journal of Finance, Vol. 26, No. 5, pp. 997-1009. Gahlon, J.M. and Gentry, J.A. (1982), On the relationship between systematic risk and the degrees of operating and financial leverage, Financial Management, Vol. 11, No. 2, pp. 15-23. Hamada, R.S. (1972), The effect of the firm s capital structure on the systematic risk of common stocks, The Journal of Finance, Vol. 27, No. 2, pp. 435-452. Harissis, H. F. (2000)m The Capital Asset Pricing Model: A review of the issues, European Research Studies, Vol. 3, Nos. 3-4, pp. 111-130. Hoberg, G., and Phillips, G. (2010), Product market synergies and competition in mergers and acquisitions: A text-based analysis, Review of Financial Studies, Vol. 10, pp. 3773-3811. Kim, M.H., and Sun, L. (2016), Dynamic conditional correlations between Chinese sector returns and the S&P 500 index: An interpretation based on investment shocks, SSRN Working Paper 2713602. Mandelker, G.N. and Rhee, S.G. (1984), The impact of the degrees of operating and financial leverage, Journal of Financial and Quantitative Analysis, Vol. 19, No. 1, pp. 45-57. Rubenstein, M.E. (1973), A mean-variance synthesis of corporate financial theory, Journal of Finance, Vol. 28, pp. 167-182. Sarmiento-Sabogal, J., and Sadeghi, M. (2014), Unlevered betas and the cost of equity capital: An empirical approach, North American Journal of Economics and Finance, Vol. 30, pp. 90-105. Shih, Y.C., S.S. Chen, C.F. Lee, and P.J. Chen (2014), The evolution of capital asset pricing models, Review of Quantitative Finance and Accounting, Vol. 42, No. 3, pp. 415-448. Sweeney, R.J., Warga, A.D., and Winters, D. (1997), The market value of debt, market versus book value of debt, and returns to assets, Financial Management, Vol. 26, No. 1, pp. 5-21. 13

Table 1: Mean values for variables under consideration. Number of firm years is in italics. Interestbearing debt PV of Operating Leases Market Value of Equity Debt to Equity Ratio without operatin g leases Debt to Equity Ratio with operatin g leases Total Revenue EBIT NI Total Asset s Beta estima te (Value - weight ed month ly index) Sample 3,380 110 3,608,520 0.699 0.780 3,738 549 262 75,303 75,304 73,643 73,642 73,642 75,304 75,304 75,304 ByFama and French (12 ies) 11,96 5 0.960 75,27 9 63,501 1 1,130 75 3,989,711 0.62 0.70 3,223 452 256 3,603 0.82 4,120 4,120 4,018 4,018 4,018 4,120 4,120 4,120 4,119 3,588 2 3,504 90 1,805,811 0.64 0.71 7,911 476 270 9,766 1.05 1,963 1,963 1,904 1,904 1,904 1,963 1,963 1,963 1,963 1,715 3 890 56 2,263,460 0.59 0.65 2,977 292 147 3,301 1.05 8,151 8,151 7,990 7,990 7,990 8,151 8,151 8,151 8,147 7,123 4 13,78 2,554 263 7,833,334 0.67 0.71 13,462 1,646 1,021 4 0.98 3,054 3,054 2,991 2,991 2,991 3,054 3,054 3,054 3,053 2,568 5 1,430 98 5,435,931 0.59 0.65 4,705 584 351 5,221 0.94 1,903 1,903 1,865 1,865 1,865 1,903 1,903 1,903 1,902 1,677 6 7 8 453 70 4,402,181 0.20 0.26 2,263 309 197 2,756 1.45 10,56 4 8,601 10,566 10,566 10,303 10,303 10,303 10,566 10,566 10,566 17,60 5,846 336 6,955,493 1.23 1.34 8,449 1,446 543 3 1.27 2,497 2,497 2,378 2,378 2,378 2,497 2,497 2,497 2,496 2,014 10,53 4,042 8 3,921,852 1.18 1.18 4,255 657 290 5 0.48 2,521 2,521 2,478 2,478 2,478 2,521 2,521 2,521 2,521 2,304 9 741 277 2,780,651 0.53 0.78 5,114 264 137 2,598 0.99 7,668 7,669 7,447 7,446 7,446 7,669 7,669 7,669 7,666 6,514 10 838 65 5,751,670 0.33 0.39 2,272 513 315 3,710 0.94 14

4,675 4,675 4,587 4,587 4,587 4,675 4,675 4,675 4,671 4,018 11 9,127 46 2,794,850 1.09 1.13 2,614 727 249 18,639 18,639 18,376 18,376 18,376 18,639 18,639 18,639 32,46 6 0.67 18,63 4 15,630 12 1,668 155 2,638,939 0.70 0.82 2,521 371 193 4,813 1.01 9,546 9,546 9,306 9,306 9,306 9,546 9,546 9,546 9,543 7,749 15

Table 2: Variability of Asset Beta show average unlevered beta using Hamada s formula by industry. Mean Median Standard Deviation Sample 0.801 0.670 0.654 1 0.594 0.507 0.496 2 0.741 0.679 0.537 3 0.772 0.658 0.597 4 0.685 0.614 0.488 5 0.679 0.611 0.486 6 1.298 1.186 0.832 7 0.702 0.568 0.545 8 0.242 0.217 0.227 9 0.746 0.660 0.563 10 0.793 0.688 0.609 12 0.693 0.609 0.525 16

Table 3: Average asset beta compared to proxy asset beta, for the full sample and by industry. Proxy is obtained using industry averages that do not include the subject firm. Actual asset beta Proxy asset beta Average difference Average λ as defined by SSS Average absolute difference Error as a percent of asset beta Difference Sample 0.7006 0.7007 0.00 0.0000 1.0497 0.3737 53.3% 1 0.5942 0.5948 0.00-0.0005 1.3306 0.3349 56.4% 2 0.7409 0.7684-0.03-0.0272 1.3981 0.3624 48.9% 3 0.7716 0.7805-0.01-0.0088 0.5828 0.3913 50.7% 4 0.6852 0.6850 0.00 0.0005 1.7042 0.3331 48.6% 5 0.6791 0.7106-0.03-0.0315 1.2311 0.3252 47.9% 6 1.2975 1.2005 0.10 0.0970 3.1626 0.6120 47.2% 7 0.7018 0.7018 0.00 0.0000 0.1097 0.3833 54.6% 8 0.2419 0.2419 0.00 0.0000 0.2344 0.1314 54.3% 9 0.7461 0.7436 0.00 0.0026-0.9277 0.4081 54.7% 10 0.7933 0.7933 0.00 0.0000 1.0168 0.4426 55.8% 11 0.3946 0.3945 0.00 0.0002 1.8967 0.2474 62.7% 12 0.6927 0.7822-0.09-0.0895-0.8894 0.3980 57.5% 17

Table 4: Correlations between asset beta and variables chosen from previous literature and used as comparable companies for further analysis. Asset beta Total Assets Revenues Market Value of Equity Earnings to Price Growth in Revenues Market to book ratio of equity Tobin s q estimate Capital expenditures Leverage ratio Total long-term debt Volatility of cash flows Degree of operating leverage Asset beta 1 Total Assets -0.049 1 Revenues -0.047 0.436 1 Market Value of Equity 0.019 0.312 0.559 1 Earnings to Price -0.020 0.019 0.029-0.004 1 Growth in Revenues 0.039-0.009-0.011-0.005 0.000 1 Market to book ratio of equity -0.004-0.002-0.002 0.002-0.001-0.002 1 Tobin s q estimate 0.235-0.042-0.043 0.100-0.043 0.040 0.182 1 Capital expenditures -0.055 0.209 0.758 0.394 0.034-0.007-0.001-0.048 1 Leverage ratio -0.379 0.198 0.119-0.050 0.069-0.014-0.006-0.264 0.128 1 Total longterm debt -0.051 0.772 0.311 0.303 0.010-0.003-0.007-0.039 0.141 0.191 1 Volatility of cash flows -0.036 0.767 0.528 0.452 0.015-0.012 0.000-0.030 0.390 0.120 0.544 1 Degree of operating leverage -0.001 0.000-0.001-0.002 0.000 0.000 0.000-0.001 0.000-0.001-0.001-0.001 1 * All correlations with asset beta are statistically significant, except for market to book ratio (p value = 0.30) and degree of operating leverage (p value = 0.87). We kept market to book ratio because of its use in previous papers. 18

Table 5: Average asset beta compared to proxy asset beta by decile. Proxy is obtained using average asset beta within the decile for each of the candidate criteria from the previous year (not including the subject firm). Panel A: Average asset beta compared to proxy asset beta, by decile of firm characteristic. Proxy is obtained using average asset beta within the decile for each of the candidate criteria from the previous year (not including the subject firm) by industry. Actual asset beta Proxy asset beta Average absolute difference Median absolute difference Deciles by total 0.801 0.795 0.400 0.304 assets Deciles by 0.801 0.796 0.400 0.303 revenue Deciles by MVE 0.801 0.800 0.395 0.298 Deciles by EP 0.801 0.797 0.393 0.295 Deciles by growth Deciles by market to book Deciles by Tobin's Q Deciles by capital expenditures Deciles by leverage ratio Deciles by longterm debt Deciles by cash flow volatility 0.801 0.796 0.397 0.302 0.801 0.802 0.375 0.275 0.801 0.801 0.370 0.271 0.801 0.801 0.382 0.283 0.801 0.798 0.373 0.269 0.801 0.800 0.377 0.277 0.801 0.803 0.384 0.285 Panel B: Same as Panel A, but disregarding industry in decile match Deciles by total 0.801 0.798 0.447 0.360 assets Deciles by 0.801 0.798 0.450 0.450 revenue Deciles by MVE 0.801 0.804 0.452 0.452 Deciles by EP 0.801 0.800 0.435 0.435 Deciles by growth Deciles by market to book Deciles by Tobin's Q Deciles by capital expenditures Deciles by leverage ratio Deciles by longterm debt Deciles by cash flow volatility 0.801 0.800 0.444 0.358 0.801 0.805 0.435 0.337 0.801 0.803 0.424 0.326 0.801 0.804 0.450 0.361 0.801 0.800 0.397 0.294 0.801 0.801 0.434 0.328 0.801 0.806 0.451 0.358 19

Table 5: Average asset beta compared to proxy asset beta, for the full sample and by industry. Proxy is obtained using closest matches (up to 6). Matching is done by industry first, then five different matching methods. Actual asset beta Proxy asset beta 20 Difference Average differen ce Average absolute differenc e Averag e λ as defined by SSS Matched by Total Assets Sample 0.7007 0.7040-0.0033-0.0004 0.4084 1.3066 1 0.5942 0.5952-0.0011 0.0011 0.3780 1.1976 2 0.7409 0.7451-0.0042 0.0010 0.3723 1.4126 3 0.7716 0.7697 0.0019-0.0012 0.4271 0.9642 4 0.6852 0.6858-0.0006-0.0011 0.3504 1.7428 5 0.6791 0.6801-0.0010 0.0024 0.3468 1.2615 6 1.2975 1.2868 0.0108 0.0012 0.6543 2.7501 7 0.7018 0.7060-0.0042-0.0009 0.4075 0.2369 8 0.2419 0.2426-0.0007 0.0003 0.1446 0.2645 9 0.7461 0.7436 0.0026-0.0009 0.4499-0.6200 10 0.7933 0.8022-0.0089-0.0006 0.4564 0.9403 y 11 0.3946 0.3932 0.0014-0.0007 0.2990 1.8224 y 12 0.6927 0.6916 0.0011-0.0015 0.4136 1.2575 Matched by revenue Matched by MVE Sample 0.7007 0.7028-0.0022 0.0002 0.4060 0.9145 y 1 0.5942 0.5946-0.0005-0.0013 0.3755 1.1340 y 2 0.7409 0.7489-0.0080 0.0017 0.3738 1.2640 y 3 0.7716 0.7701 0.0015 0.0000 0.4255 0.6921 y 4 0.6852 0.6861-0.0009-0.0019 0.3473 1.6446 y 5 0.6791 0.6801-0.0010 0.0000 0.3434 1.3272 y 6 1.2975 1.2830 0.0145 0.0005 0.6593 3.0083 y 7 0.7018 0.7087-0.0069-0.0007 0.4163 0.0028 y 8 0.2419 0.2422-0.0003-0.0005 0.1403 0.7949 y 9 0.7461 0.7445 0.0016 0.0022 0.4523-1.8888 y 10 0.7933 0.8052-0.0119-0.0013 0.4555 0.8616 y 11 0.3946 0.3887 0.0059 0.0000 0.2852 1.7841 y 12 0.6927 0.6915 0.0012 0.0011 0.4178-0.8321 Sample 0.7007 0.7006 0.0001 0.0003 0.3978 1.3430

Matched by EP Matched by growth y 1 0.5942 0.5896 0.0045-0.0005 0.3771 1.2914 y 2 0.7409 0.7398 0.0011 0.0035 0.3712 1.5090 y 3 0.7716 0.7664 0.0051 0.0007 0.4158 0.5693 y 4 0.6852 0.6882-0.0029 0.0024 0.3548 1.5961 y 5 0.6791 0.6747 0.0044 0.0020 0.3440 1.2833 y 6 1.2975 1.2912 0.0064 0.0001 0.6366 2.7977 y 7 0.7018 0.6980 0.0038 0.0008 0.4152 0.1029 y 8 0.2419 0.2414 0.0006-0.0004 0.1433 0.8685 y 9 0.7461 0.7401 0.0060-0.0009 0.4297 0.2037 y 10 0.7933 0.7979-0.0046 0.0021 0.4589 0.7809 y 11 0.3946 0.3884 0.0063 0.0000 0.2829 1.8598 y 12 0.6927 0.6898 0.0029-0.0001 0.4030 1.0145 Sample 0.7007 0.7050-0.0044 0.0012 0.4013 1.2379 y 1 0.5942 0.5939 0.0002 0.0027 0.3751 1.1371 y 2 0.7409 0.7488-0.0079 0.0075 0.3534 1.3818 y 3 0.7716 0.7665 0.0051 0.0010 0.4152 0.5609 y 4 0.6852 0.6833 0.0019 0.0036 0.3322 1.4754 y 5 0.6791 0.6802-0.0011 0.0073 0.3363 1.2990 y 6 1.2975 1.2960 0.0015-0.0003 0.6403 3.1594 y 7 0.7018 0.7051-0.0033 0.0036 0.3975-1.5669 y 8 0.2419 0.2451-0.0031 0.0011 0.1420 0.6423 y 9 0.7461 0.7452 0.0009 0.0014 0.4337-0.6767 y 10 0.7933 0.8018-0.0085 0.0031 0.4649 0.7876 y 11 0.3946 0.3952-0.0006-0.0002 0.3004 2.5246 y 12 0.6927 0.6916 0.0011 0.0003 0.4048-0.2237 Sample 0.7007 0.7112-0.0106-0.0009 0.4087 1.3954 21

y 1 0.5942 0.5968-0.0026-0.0031 0.3811 1.1622 y 2 0.7409 0.7475-0.0066 0.0008 0.3729 1.3476 y 3 0.7716 0.7749-0.0033-0.0007 0.4219 0.6650 y 4 0.6852 0.6836 0.0016-0.0019 0.3598 1.6491 y 5 0.6791 0.6847-0.0056-0.0039 0.3516 1.1651 y 6 1.2975 1.3085-0.0109-0.0032 0.6502 3.3239 y 7 0.7018 0.7096-0.0078-0.0016 0.3909-0.7202 y 8 0.2419 0.2424-0.0004-0.0010 0.1462-0.1401 y 9 0.7461 0.7520-0.0059 0.0001 0.4440 0.9520 y 10 0.7933 0.8091-0.0158-0.0047 0.4741 0.9241 y 11 0.3946 0.3987-0.0041 0.0002 0.3025 1.9857 y 12 0.6927 0.7000-0.0073 0.0021 0.4135 0.4341 Table 8: Non-leasing firms only, matched with non-leasing firms only. Average asset beta compared to proxy asset beta, for the full sample and by industry. Proxy is obtained using closest matches (up to 6). Matching is done by industry first, then five different matching methods. Matched by Total Assets Actual asset beta Proxy asset beta Average difference Average absolute difference Average λ as defined by SSS Difference Sample 0.4193 0.4266-0.0073 0.0002 0.2796 2.2432 1 0.5942 0.4528 0.1414 0.0018 0.3164 1.2344 2 0.7409 0.6354 0.1055 0.0070 0.3720 2.3480 3 0.7716 0.6617 0.1098-0.0011 0.4224 1.6273 4 0.6852 0.6085 0.0767 0.0014 0.3791 1.9328 5 0.6791 0.4950 0.1841 0.0009 0.2675 0.7293 6 1.2975 1.1554 0.1422 0.0043 0.6991 12.5285 7 0.7018 0.6055 0.0963 0.0055 0.4250 2.0671 22

Matched by revenue Matched by MVE 8 0.2419 0.2351 0.0068 0.0004 0.1272-0.0150 9 0.7461 0.5407 0.2055-0.0028 0.4029 0.6963 10 0.7933 0.8129-0.0197-0.0026 0.4431 2.2025 11 0.3946 0.2762 0.1185-0.0002 0.2027 1.9953 12 0.6927 0.5569 0.1358-0.0018 0.3885 2.3072 Sample 0.4193 0.4250-0.0057 0.0004 0.2797 1.9316 1 0.5942 0.4503 0.1439-0.0042 0.3129 1.4129 2 0.7409 0.6362 0.1047 0.0089 0.3518 2.1150 3 0.7716 0.6579 0.1137-0.0002 0.4320 1.7863 4 0.6852 0.6112 0.0741-0.0086 0.3923 1.6142 5 0.6791 0.4965 0.1826-0.0024 0.2649 0.8749 6 1.2975 1.1448 0.1528 0.0002 0.6922 9.3945 7 0.7018 0.6032 0.0986 0.0083 0.4388 2.1135 8 0.2419 0.2341 0.0079-0.0006 0.1232 1.0197 9 0.7461 0.5485 0.1977 0.0050 0.4094 0.7693 10 0.7933 0.8131-0.0199 0.0037 0.4404 2.7671 11 0.3946 0.2753 0.1193 0.0004 0.1994 1.4303 12 0.6927 0.5522 0.1405 0.0021 0.4088 2.4179 Sample 0.4193 0.4227-0.0034-0.0003 0.2707 1.8554 1 0.5942 0.4474 0.1467 0.0014 0.3028 0.9560 2 0.7409 0.6112 0.1297-0.0115 0.3701 2.1136 3 0.7716 0.6526 0.1190-0.0035 0.4085 1.0374 4 0.6852 0.6040 0.0813-0.0046 0.3918 1.8209 5 0.6791 0.4940 0.1852 0.0026 0.2648 0.9621 6 1.2975 1.1363 0.1612 0.0042 0.6724 7.4337 7 0.7018 0.6030 0.0988-0.0053 0.4128 1.9990 23

Matched by EP Matched by growth 8 0.2419 0.2345 0.0074 0.0004 0.1256 0.9195 9 0.7461 0.5250 0.2212-0.0065 0.3976 1.8921 10 0.7933 0.8161-0.0229 0.0065 0.4432 2.9909 11 0.3946 0.2745 0.1201 0.0002 0.1946 1.5902 12 0.6927 0.5517 0.1410-0.0004 0.3661 2.1295 Sample 0.4193 0.4290-0.0097 0.0029 0.2746 2.1898 1 0.5942 0.4565 0.1376 0.0101 0.2799 1.0441 2 0.7409 0.6421 0.0988 0.0094 0.3172 2.0221 3 0.7716 0.6594 0.1121 0.0025 0.3948 1.3850 4 0.6852 0.6230 0.0622 0.0138 0.3770 1.4159 5 0.6791 0.4919 0.1872 0.0123 0.2580 0.6634 6 1.2975 1.1601 0.1374 0.0063 0.6775 11.2867 7 0.7018 0.6050 0.0968 0.0093 0.4095 2.0377 8 0.2419 0.2364 0.0055 0.0009 0.1252 0.8215 9 0.7461 0.5428 0.2033 0.0122 0.3746 1.3663 10 0.7933 0.8210-0.0278 0.0114 0.4512 2.7425 11 0.3946 0.2785 0.1161-0.0002 0.2090 1.8713 12 0.6927 0.5547 0.1380 0.0058 0.3614 2.3496 Sample 0.4193 0.4305-0.0112 0.0000 0.2852 1.7954 1 0.5942 0.4464 0.1478 0.0001 0.3077 0.9699 2 0.7409 0.6417 0.0992 0.0088 0.3592 2.2874 3 0.7716 0.6626 0.1089-0.0006 0.4326 1.3450 4 0.6852 0.6040 0.0812-0.0058 0.4122 1.7478 5 0.6791 0.5028 0.1763 0.0011 0.2824 0.8563 6 1.2975 1.1598 0.1377-0.0079 0.7000 3.3469 7 0.7018 0.6026 0.0992-0.0043 0.4107 1.9007 24

8 0.2419 0.2352 0.0067-0.0004 0.1292-0.1075 9 0.7461 0.5429 0.2032 0.0092 0.4017 1.2573 10 0.7933 0.8127-0.0194 0.0025 0.4518 2.4407 11 0.3946 0.2812 0.1134 0.0002 0.2089 2.0512 12 0.6927 0.5637 0.1290 0.0032 0.3997 2.3430 Table 8: Non-leasing firms only, matched with non-leasing firms only. Average asset beta compared to proxy asset beta, for the full sample and by industry. Proxy is obtained using closest matches (up to 6). Matching is done by industry first, then five different matching methods. Matched by Total Assets Matched by revenue Actual asset beta Proxy asset beta Average difference Average absolute difference Average λ as defined by SSS Difference Sample 0.4193 0.4266-0.0073 0.0002 0.2796 2.2432 1 0.5942 0.4528 0.1414 0.0018 0.3164 1.2344 2 0.7409 0.6354 0.1055 0.0070 0.3720 2.3480 3 0.7716 0.6617 0.1098-0.0011 0.4224 1.6273 4 0.6852 0.6085 0.0767 0.0014 0.3791 1.9328 5 0.6791 0.4950 0.1841 0.0009 0.2675 0.7293 6 1.2975 1.1554 0.1422 0.0043 0.6991 12.5285 7 0.7018 0.6055 0.0963 0.0055 0.4250 2.0671 8 0.2419 0.2351 0.0068 0.0004 0.1272-0.0150 9 0.7461 0.5407 0.2055-0.0028 0.4029 0.6963 10 0.7933 0.8129-0.0197-0.0026 0.4431 2.2025 11 0.3946 0.2762 0.1185-0.0002 0.2027 1.9953 12 0.6927 0.5569 0.1358-0.0018 0.3885 2.3072 Sample 0.4193 0.4250-0.0057 0.0004 0.2797 1.9316 25

Matched by MVE Matched by EP 1 0.5942 0.4503 0.1439-0.0042 0.3129 1.4129 2 0.7409 0.6362 0.1047 0.0089 0.3518 2.1150 3 0.7716 0.6579 0.1137-0.0002 0.4320 1.7863 4 0.6852 0.6112 0.0741-0.0086 0.3923 1.6142 5 0.6791 0.4965 0.1826-0.0024 0.2649 0.8749 6 1.2975 1.1448 0.1528 0.0002 0.6922 9.3945 7 0.7018 0.6032 0.0986 0.0083 0.4388 2.1135 8 0.2419 0.2341 0.0079-0.0006 0.1232 1.0197 9 0.7461 0.5485 0.1977 0.0050 0.4094 0.7693 10 0.7933 0.8131-0.0199 0.0037 0.4404 2.7671 11 0.3946 0.2753 0.1193 0.0004 0.1994 1.4303 12 0.6927 0.5522 0.1405 0.0021 0.4088 2.4179 Sample 0.4193 0.4227-0.0034-0.0003 0.2707 1.8554 1 0.5942 0.4474 0.1467 0.0014 0.3028 0.9560 2 0.7409 0.6112 0.1297-0.0115 0.3701 2.1136 3 0.7716 0.6526 0.1190-0.0035 0.4085 1.0374 4 0.6852 0.6040 0.0813-0.0046 0.3918 1.8209 5 0.6791 0.4940 0.1852 0.0026 0.2648 0.9621 6 1.2975 1.1363 0.1612 0.0042 0.6724 7.4337 7 0.7018 0.6030 0.0988-0.0053 0.4128 1.9990 8 0.2419 0.2345 0.0074 0.0004 0.1256 0.9195 9 0.7461 0.5250 0.2212-0.0065 0.3976 1.8921 10 0.7933 0.8161-0.0229 0.0065 0.4432 2.9909 11 0.3946 0.2745 0.1201 0.0002 0.1946 1.5902 12 0.6927 0.5517 0.1410-0.0004 0.3661 2.1295 Sample 0.4193 0.4290-0.0097 0.0029 0.2746 2.1898 26

Matched by growth 1 0.5942 0.4565 0.1376 0.0101 0.2799 1.0441 2 0.7409 0.6421 0.0988 0.0094 0.3172 2.0221 3 0.7716 0.6594 0.1121 0.0025 0.3948 1.3850 4 0.6852 0.6230 0.0622 0.0138 0.3770 1.4159 5 0.6791 0.4919 0.1872 0.0123 0.2580 0.6634 6 1.2975 1.1601 0.1374 0.0063 0.6775 11.2867 7 0.7018 0.6050 0.0968 0.0093 0.4095 2.0377 8 0.2419 0.2364 0.0055 0.0009 0.1252 0.8215 9 0.7461 0.5428 0.2033 0.0122 0.3746 1.3663 10 0.7933 0.8210-0.0278 0.0114 0.4512 2.7425 11 0.3946 0.2785 0.1161-0.0002 0.2090 1.8713 12 0.6927 0.5547 0.1380 0.0058 0.3614 2.3496 Sample 0.4193 0.4305-0.0112 0.0000 0.2852 1.7954 1 0.5942 0.4464 0.1478 0.0001 0.3077 0.9699 2 0.7409 0.6417 0.0992 0.0088 0.3592 2.2874 3 0.7716 0.6626 0.1089-0.0006 0.4326 1.3450 4 0.6852 0.6040 0.0812-0.0058 0.4122 1.7478 5 0.6791 0.5028 0.1763 0.0011 0.2824 0.8563 6 1.2975 1.1598 0.1377-0.0079 0.7000 3.3469 7 0.7018 0.6026 0.0992-0.0043 0.4107 1.9007 8 0.2419 0.2352 0.0067-0.0004 0.1292-0.1075 9 0.7461 0.5429 0.2032 0.0092 0.4017 1.2573 10 0.7933 0.8127-0.0194 0.0025 0.4518 2.4407 11 0.3946 0.2812 0.1134 0.0002 0.2089 2.0512 12 0.6927 0.5637 0.1290 0.0032 0.3997 2.3430 27

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