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Full-information industry betas.

A cost of capital estimate is a critical input into the capital-budgeting process of corporations, the valuation of privately held companies, the assessment of property value for tax purposes, and the pricing of services in regulated industries. The cost of equity capital is typically estimated using the Sharpe (1964), Lintner (1965), and Black (1972) capital-asset-pricing model (CAPM). The CAPM defines a linear relationship between the firm's cost of equity (expected return) and the slope coefficient (beta) in a regression of the firm's equity returns on a market index.

The widespread use of the CAPM in practice is no doubt due to its strong theoretical foundation and simplicity. The usefulness of the CAPM and beta has been questioned, most notably, by Fama and French (1992, 1993, and 1997). The evidence against the model has left many practitioners helpless since no theoretically justified alternative model is easily implemented. Largely because of this, researchers are turning their attention to issues involving implementation of the CAPM.(1) This articles focuses on one such issue: estimating industry beta.

It is well known that a beta estimate for an individual firm contains a great deal of statistical noise. To increase precision, analysts estimate the beta of a portfolio of firms who operate solely in the same line of business as the firm, division, or project being valued. The beta of the pure-play portfolio is then used to estimate the cost of capital for the investment. Since the precision of the beta estimate increases with the number of pure plays, analysts prefer to have as many pure plays as possible. However, finding a large sample of firms specializing in a single line of business is often extremely difficult.

The "textbook" pure-play approach described above excludes conglomerates from the set of potential pure plays. Conglomerates can be large firms that account for a significant market share in a particular business segment. For example, Philip Morris is a major player in both the tobacco industry and the food and kindred products industry. Therefore, the observable beta for Philip Morris is a weighted average of the unobservable betas of the individual business segments. As such, Philip Morris would not be included as a pure play in either the tobacco industry or the food and kindred products industry. If the divisions of conglomerate firms are large relative to the pure plays, then excluding them introduces a potential upward bias into the industry beta calculation. This is due to a negative correlation between market capitalization and beta. (We discuss this in more detail in the following section.)

Boquist and Moore (1983) and Ehrhardt and Bhagwar (1991), EB hereafter, develop empirical methodologies for estimating industry segment betas using information contained in the betas of conglomerates. These studies assume that the systematic risk of an industry segment can be applied to all firms and divisions that operate within that segment. We argue that this assumption is too stringent for many practical applications. In practice, it is common to use the industry cost of capital as a benchmark for determining a project's cost of capital. Analysts then make adjustments for more or less risky projects.(2) Our point is that betas do differ across firms within an industry. The methodology we propose for estimating industry beta permits size-related variation in the betas of firms that operate within an industry.

We employ a full-information approach to estimate market-capitalization-weighted industry betas. The approach is similar to EB. The estimation technique utilizes the industry-specific information contained in the betas of conglomerates. The methodology is straightforward. First, firm-specific betas are calculated for all firms. In addition, the percentage of firm sales that is attributable to a particular industry is computed for each industry in the economy. A cross-sectional regression of beta against the industry percentages is then performed using the instrumental variables (IV) regression technique. The instruments are chosen such that the regression coefficients can be interpreted as market-capitalization-weighted industry betas.

Market-capitalization-weighted pure-play and full-information industry betas are estimated for those firms listed in Ibbotson Associates' BetaBook publication. Our results indicate that full-information industry betas are significantly smaller than pure-play betas. This finding implies that traditional estimates of pure-play industry betas are biased upwards, as compared to full-information industry betas.

Section I contains a more detailed discussion of the relationship between market capitalization and beta. Section II describes the data and methodology. Section III presents the main results of our analysis, and Section IV presents our conclusions.

I. The Correlation Between Market Capitalization and Beta

The negative correlation between market capitalization and beta is well established empirically in the finance literature. (See Chan and Chen, 1988; Ibbotson, Kaplan, and Peterson, 1997, IKP hereafter; Kothari, Shanken, and Sloan, 1995, KSS hereafter; and Roll, 1981.) However, the economic factors that underlie this correlation are not well understood.

Berk (1995) provides a theoretical explanation for the negative correlation between market capitalization and systematic risk. The point of his analysis that is relevant here is that high-risk firms have smaller market capitalizations than less risky firms, ceteris paribus, simply because of the additional risk premium embedded in the discount rate of riskier firms. Therefore, if systematic risk is measured by beta, small-market-capitalization firms will tend to have higher betas than large-market-capitalization firms.

Berk's arguments suggest that the negative correlation between market capitalization and beta found in many empirical studies is expected. If market capitalization varies across firms within an industry, then betas may also vary. In the next section, we develop a methodology that accommodates size-related variation in the betas of firms within an industry.

II. Data and Methodology

Our approach closely follows the approach developed in EB. Full-information industry betas are estimated using a two-step process. The first step is to calculate an equity beta coefficient, [Beta], for each firm in the sample. We estimate an equity beta for those firms listed in the 1996 second edition of Ibbotson Associates' BetaBook publication. (This publication uses data through June 1996.) The sample consists of 4,509 firms.(3)

IKP and KSS show that traditional estimates of small-firm betas are too low. We follow Fama and French (1992, 1996) and IKP and estimate [Beta] as the sum of the slope coefficients from the regression of individual security returns on the current and prior month's market returns.(4) These "sum" betas correct for a bias in small-firm betas that most likely results from nonsynchronous trading, trading frictions, or institutional features that induce autocorrelations and cross-autocorrelations in security returns. The CRSP market-capitalization-weighted index of NYSE firms serves as our market proxy. Equity betas are estimated using 36 to 60 months of historical return data. Also, we follow Ibbotson Associates and exclude those firms whose equity betas are greater than five in absolute value.

The full-information approach is based on the premise that a firm can be thought of as a portfolio of assets. If a firm has business units that differ in systematic risk, then the company-wide beta is a weighted average of the betas of the business units. The weight on each beta is determined by dividing the market value of the equity of the business unit by the market value of the firm's equity. Individual business units typically are not traded, so their betas cannot be estimated using historical return data.

If betas differ across industries, then cross-sectional variation in beta should be related to industry participation. Industry participation is measured using industry level sales data. For each firm, total sales and sales by industry are obtained for the most recent fiscal year from Compustat PC Plus. Compustat defines industries using SIC codes. In addition Compustat assigns a primary SIC code to each firm. SIC codes are generally four-digit numbers. However, Compustat often changes the last two digits to zero for those firms with operations throughout an industry group. (An industry group is composed of all firms with the same first two digits.) For each firm, industry weights are computed as the ratio of industry sales to total sales for each two-digit SIC code. Therefore, a firm's industry weights sum to one.(5) The industry classification technique used here was chosen based on data availability. However, the methodology can be applied with other industry classification schemes.

The second step is to impute the full-information betas from the cross-sectional regression

[[Beta].sub.j] = [summation of] [Beta][full.sub.i][[Omega].sub.ji] where i = 1 to n + [[Upsilon].sub.j] (1)

where n is the number of industries, [[Omega].sub.ji] is the weight of firm j in industry i, the [Beta][full.sub.i]s are the regression parameters to be estimated, and [[Upsilon].sub.j] is the regression error term. If Equation (1) is estimated using ordinary least squares (OLS), as in EB, then [Beta][full.sub.i] is interpreted as the equal-weighted average industry-specific beta. We do not equally weight because we want [Beta][full.sub.i] to represent the market-capitalization-weighted industry beta.

To obtain market-capitalization-weighted industry betas, each firm is weighted in proportion to its market capitalization. More precisely, the market-value weight [z.sub.ji] is defined as follows

[z.sub.ji] = [equivalent] [p.sub.j][[Omega].sub.ji] (2)

where [p.sub.j] [equivalent to] [S.sub.j]/[summation of] [S.sub.k] where k = 1 to m, [S.sub.j] is the market capitalization of firm j, and m is the number of firms in the sample. Note that [z.sub.ji] is a function of both the market capitalization of the firm and the industry weight. Thus a market-value weight proxies for the ratio of divisional market capitalization to total stock market value.

The desired point estimators of the [Beta][full.sub.i]s could be obtained by estimating Equation (1) using weighted least squares (WLS) with the [p.sub.j]s as weights. However, the standard errors from a WLS regression would be inappropriate. This is because WLS assumes that the error term is heteroscedastic. Since we assume that the error term is homoscedastic, we do not use the standard errors that result from a WLS analysis. Instead, we estimate Equation (1) using the IV regression technique, where the [z.sub.ji]s represent our choice of instruments. The IV approach yields the same point estimators as WLS, but it produces standard errors that are consistent with the assumption that the error term in Equation (1) is homoscedastic.(6)

The motivation for our full-information approach is that the pure-play technique forces the user to drop conglomerates from the sample, which introduces a bias into the pure-play industry beta estimates. The source of this bias is the negative correlation between beta and market capitalization described earlier. Since conglomerates are typically large players in their industries, omitting them should cause an upward bias in pure-play industry betas. In effect, we are hypothesizing that within an industry beta varies with market capitalization. Although we do not explicitly model this relationship, our IV regression model does allow for correlation between beta and market capitalization through the error term [[Upsilon].sub.j].

To see this, recognize that in the IV regression Equation (1) and with the instruments defined in Equation (2), the sum of the fitted [Upsilon]s is not zero. Rather, [Mathematical Expression Omitted], where [Mathematical Expression Omitted] denotes the fitted value of [[Upsilon].sub.j].(7) Hence, the simple average of the [Mathematical Expression Omitted] may not be zero. The covariance between market capitalization and the error term is of the opposite sign as the simple average of the [Mathematical Expression Omitted].(8) Typically, there will be more small firms than large firms when estimating Equation (1). In this case, the average of the [Mathematical Expression Omitted] will likely be positive. This suggests that the full-information industry beta estimates obtained from estimating Equation (1) using the IV technique will be less than their OLS counterparts.(9)

A final point regarding our methodology merits discussion. We do not control for capital-structure differences across firms within an industry. More precisely, our methodology relies on the assumption that there is an optimal debt-to-equity (D/E) ratio for an industry and that all firms within the industry are at the optimum. An alternative approach would be to unlever the equity beta prior to estimating Equation (1). However, this approach requires choosing an unlevering technique. Moreover, it assumes that all industry segments within a business have the same D/E ratio.

To check the robustness of our results, we replicated the empirical analysis presented in the following section using asset betas. Equity betas are unlevered following the procedure of Hamada (1972). Specifically, [Mathematical Expression Omitted] where [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are the asset beta and equity beta of firm j, respectively. Equation (1) is then estimated using the asset betas as the independent variables. The primary conclusion of our analysis remains unchanged: full-information asset betas are significantly smaller than their pure-play counterparts.

III. Empirical Results

The full-information industry results are compared to those obtained from a pure-play analysis in this section. Only those industries with at least five firms with all their sales in that industry are included in the analysis. The appendix contains a list of the 66 industry groups included in the full-information analysis along with abbreviated names and the two-digit SIC codes that correspond to these groups.

The full-information analysis involves estimating Equation (1) using the IV regression technique where the instruments are defined in Equation (2).The adjusted [R.sup.2] from this regression is 0.14.(10)

The pure-play analysis excludes conglomerates. Any firm with less than 100% of its sales in an industry is removed from the sample. (This restriction eliminates 829 firms.) Firms are then sorted into two-digit industries. Of the 66 industries represented in the sample, seven industries have fewer than five pure plays. Statistics are not computed for these industries. For each industry, the pure-play industry beta is calculated by market-capitalization weighting the betas of the firms in that industry.

A summary of the results obtained from each methodology is presented in Table 1. The average market capitalization of those firms included in the full-information analysis is almost double the average market capitalization of those firms included in the pure-play analysis ($2.13 billion versus $1.11 billion). This evidence supports the contention that pure plays are small firms. Since large firms generally have smaller betas than small firms, our full-information methodology should yield beta estimates that are smaller, on average, than pure-play industry betas. The results in Table 1 confirm this hypothesis. The simple average pure-play beta across all industries is 1.29, whereas the average full-information beta is 1.16. To perform a formal statistical test, [Beta]full-[Beta] is calculated for the 59 industries where both estimates are available. The mean of this difference (-0.13) is significantly less than zero at all conventional significance levels (t-statistic = -3.68). In addition, the proportion of industries where [Beta]full is smaller than [Beta] is 0.68. This proportion is significantly greater than 0.5 (z-statistic = 2.73).

We also computed the standard error for each industry beta estimate in order to assess the degree to which the precision of an industry beta estimate is improved by incorporating the information contained in conglomerate firms.(11) We computed the change in standard errors ([S.sub.[Beta]]full/[S.sub.[Beta]])-1 for each industry that has both estimates of industry beta. The proportion of industries where this change is less than zero is 0.78 (46 out of 59). This proportion of industries is significantly greater than 0.5 (z-statistic = 4.30). The average percentage change in standard errors is -8%, and the median is -14%. Therefore, we find, as do EB, that including conglomerates in the estimation increases the precision of an industry beta estimate.

The statistics for each industry are presented in Table 2. Inspection of these data reveals that many firms participate in multiple industries. The real estate [TABULAR DATA FOR TABLE 1 OMITTED] industry is a good example of an industry that has many conglomerate firms. There are 97 firms that have sales in the real estate industry. However, only 32 firms have 100% of their sales in this industry. The pure-play industry beta is 1.37, whereas the full-information industry beta is 1.10. In addition, the standard error of the full-information beta is 8% lower than its pure-play counterpart.

Some industries are primarily comprised of pure plays. Examples of such industries include clothing, merchandise stores, furniture stores, and banks. As expected, the full-information betas for these industries are very similar to the pure-play betas. For example, the estimates of [Beta] and [Beta]full for the banking industry are 1.61 and 1.60, respectively.

To summarize the results of this section, full-information industry beta estimates have smaller standard errors than pure-play industry betas. Most importantly, we show that traditional pure-play industry beta estimates are biased upwards. EB find very little difference between full-information and pure-play betas. (In their sample, the average full-information beta is 1.08, and the average pure-play beta is 1.10, but the difference is not statistically significant.) Our results differ from theirs for several reasons. First, they use daily data to estimate betas, whereas we use monthly data. Second, we correct for a bias in small-firm betas by estimating sum betas. Third, they equally weight the full-information and pure-play betas, whereas we value weight. Finally, their analysis is for 1986, whereas our analysis is for 1996.

IV. Conclusions

The process of estimating an industry cost of capital is complicated by the fact that many firms operate in multiple industries. The observable beta coefficient for a conglomerate is a weighted average of the unobservable beta coefficients of the individual operations. Therefore, conglomerates are typically excluded from a pure-play industry analysis. However, these firms are often major players in a particular industry. This study reveals that textbook calculations of pure-play industry betas are upwardly biased. Conglomerates can be incorporated into the industry beta estimate via a cross-sectional regression methodology. The resulting full-information industry beta estimates are free from the pure-play bias. Another important advantage of this methodology is that full-information industry beta estimates have smaller standard errors than their pure-play counterparts.

This study has important implications for corporate managers. Managers who use a pure-play analysis in project-selection decisions may reject some positive-net-present-value projects because they are using a discount rate that is too high. Also, managers who use a pure-play analysis to value privately held firms or divisions of conglomerates will tend to understate the value of the firm or division in question. They can avoid these errors by applying the methodology described in this paper.


The authors wish to thank Robert Battalio, Randy Heron, Roger Ibbotson, Bill McDonald, Wayne Mikkelson, and two anonymous referees for helpful comments and suggestions. This paper was completed while Peterson was a Visiting Scholar at Ibbotson Associates and Assistant Professor at University of Notre Dame.

1 For example, Ferson and Locke (1998) argue that better cost of equity capital estimates can be obtained by improving procedures for estimating market risk premiums. Kothari, Shaken, and Sloan (1995) and Ibbotson, Kaplan, and Peterson (1997) show that monthly estimates of small firm betas are biased downward. They also show that stock returns are positively related to beta when betas are adjusted to eliminate this bias.

2 See Brealey and Myers (1996} to further support this procedure.

3 Ibbotson Associates calculates a beta for those NYSE, AMEX and Nasdaq firms listed in the Compustat database. Firms with sales of less than $100,000 in the most recent year or a market capitalization of less than $10,000 in the most recent month are excluded from the analysis.

4 KSS correct for the bias in small-firm betas by estimating betas using annual return data. To get a reasonable number of observations for the annual return regressions, many years of data must be employed, e.g., 20 to 70 years. They implicitly assume in their regression model that beta remains constant over the entire sample period. However, to assume that beta remains constant over several decades is too restrictive for most practical applications.

5 In some cases, sales by industry do not add up to total sales. Compustat collects sales from 10-K reports. However, they often modify total sales to account for differences in reporting methodologies across firms. When industry sales do not add up to total sales, the difference is allocated to the primary SIC code.

6 In a technical appendix, which is available upon request from the authors, we establish this result in detail. In addition, we show that our IV approach collapses to OLS when [p.sub.j] = 1/m i.e., when the observations are equally weighted.

7 This follows from [Mathematical Expression Omitted] together with [summation of] [[Omega].sub.ji] where i = 1 to n = 1.

8 To see this, note that an estimator of Cov([p.sub.j], [[Upsilon].sub.j]) is proportional to [Mathematical Expression Omitted]. Since [Mathematical Expression Omitted] and [summation of] [p.sub.j] where j = 1 to m = 1, this is simply [Mathematical Expression Omitted].

9 The analysis of the error term suggests that a better predictor of a company's beta can be formed from market capitalization and industry participation together rather than from industry participation alone. However, achieving this requires a more explicit specification of the relationship between beta, industry participation, and market capitalization than contained in the model presented here. The specification and testing of such a model is a topic for further research.

10 For comparison purposes, we also estimated Equation (1) using OLS. The adjusted [R.sup.2] value from this regression is 0.07.

11 To compare standard errors, we estimated Equation (1) again using conglomerate firms only. The standard errors from this regression can then be directly compared to those obtained from the regression that incorporates all the data.


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Appendix. Industry Abbreviations

This appendix contains the abbreviated names of the major industry groups, the full names, and the two-digit SIC code for each group.
AgCrop Agricultural Production - Crops 01
AgStock Agricultural Production - Livestock and 02
 Animal Specialties
AgServ Agricultural Services 07
Metal Metal Mining 10
Coal Coal Mining 12
OilGas Oil and Gas Extraction 13
Mines Mining and Quarrying of Nonmetallic Minerals, 14
 Except Fuels
Bldg Building Construction - General Contractors and 15
 Operative Builders
Heavy Heavy Construction other than Building 16
 Construction - Contractors
Trades Construction - Special Trade Contractors 17
Food Food and Kindred Products 20
Smoke Tobacco Products 21
Textls Textiles 22
Clths Apparel and Other Finished Products Made 23
 from Fabrics and Similar Materials
Wood Lumber and Wood Products, Except Furniture 24
Furn Furniture and Fixtures 25
Paper Paper and Allied Products 26
Books Printing, Publishing, and Allied Industries 27
Chems Chemicals and Allied Products 28
Oil Petroleum Refining and Related Industries 29
Rubbr Rubber and Miscellaneous Plastics products 30
Leathr Leather and Leather Products 31
Stone Stone, Clay, Glass, and Concrete Products 32
PrMetl Primary Metal 33
FabMetl Fabricated Metal Products, Except Machinery 34
 and Transportation Equipment
Mach Industrial and Commercial Machining and Computer 35
ElcEq Electronic and other Electrical Equipment and 36
 Components, Except Computer Equipment
TransEq Transportation Equipment 37
LabEq Measuring, Analyzing, and Controlling Instruments 38
MiscMfg Miscellaneous Manufacturing Industries 39
Rail Railroad Transportation 40
Freight Motor Freight Transportation and Warehousing 42
Water Water Transportation 44
Jets Transportation by Air 45
Pipe Pipelines Except Natural Gas 46
TransSv Transportation Services 47
Telcm Communications 48
Util Electric, Gas, and Sanitary Service 49
Dur Wholesale Trade - Durable Goods 50
NonDur Wholesale Trade - Nondurable Goods 51
BldgMat Building Materials, Hardware, Garden Supply, 52
 and Mobile Home Dealers
MerStor General Merchandise Stores 53
GrocStor Food Stores 54
CarSales Automotive Dealers and Gasoline Service Stations 55
ClthStor Apparel and Accessory Stores 56
FurnStor Home Furniture, Furnishings, and Equipment Stores 57
RestBar Eating and Drinking Places 58
MiscRtl Miscellaneous Retail 59
Banks Depository Institutions 60
Credit Nondepository Credit Institutions 61
Fin Security and Commodity Brokers, Dealers, 62
 Exchanges and Services
Ins Insurance Carriers 63
InsAg Insurance Agents, Brokers, and Service 64
RlEst Real Estate 65
HldInv Holding and Other Investment Offices 67
Sleep Hotels, Rooming Houses, Camps, and Other 70
 Lodging Places
PerSv Personnel Services 72
BusSv Business Services 73
AutoSv Automotive Repair, Services, and Parking 75
MiscSr Miscellaneous Repair Services 76
Movie Motion Pictures 78
Amuse Amusement and Recreation Services 79
Health Health Services 80
Educ Educational Services 82
SocSv Social Services 83
EngSv Engineering, Accounting, Research, Management, 87
 and Related Services

Paul D. Kaplan is Vice President and Chief Economist at Ibbotson Associates. James D. Peterson is Vice President at Charles Schwab and Company.
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Title Annotation:includes appendix
Author:Kaplan, Paul D.; Peterson, James D.
Publication:Financial Management
Date:Jun 22, 1998
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