# A full-information approach for estimating divisional betas.

* Modern finance theory suggests that the identification of systematic risk is a necessary prerequisite to solving many types of financial problems. One example is capital budgeting, in which an estimate of systematic risk is required for computing the cost of capital. Systematic risk is often measured by beta, within the context of the capital asset pricing model (CAMP). Beta is typically estimated with time-series regression, in which the dependent variable is the stock return and t he independent variable is the market return.If the security in question is not publicly traded or if the objective is to find the systematic risk of a division or project, then the necessary inputs for a time-series regression are not available. At least two major approaches to resolving this problem are currently advocated in the finance literature: (i) accounting-based approaches (Hill and Stone [9], Rosenberg and McKibben [11]) and (ii) the analogous firm approach, which is often called the pure-play approach (Fuller and Kerr [7]). As shown in this study, these approaches suffer from both theoretical and empirical difficulties.

A new approach is proposed and applied in this study. This approach is based on the solid theoretical premise that a firm is simply a portfolio of projects; therefore, the beta of a firm is the weighted average of its project betas. Unlike the pure-play approach, which is limited to the subset of firms competing in one and only one line of business, the proposed full-information approach uses all available firms; the increased sample size due to using all the available information increases the power of the approach. Despite this increase in sample size, the data requirements are modest, comprising only the estimated market beta for publicly traded firms and the percentage of sales (or net revenue) in each line of business for these firms; these data are readily available through CRSP and COMPUSTAT. The methodology is simple, requiring only a multiple linear regression. In contrast to goal programming, there is no ambiguity or need for interpretation of the results; i.e., the estimates are unique. Finally, empirical tests indicate that the estimates of beta are more accurate than those obtained using previous approaches.

Section I of this paper is a brief review of related research. Section II describes the proposed methodology. The methodology is applied and results are presented in Section III. Section IV is a brief summary.

I. Review of Prior Research

Estimation of beta is an important step in determinating the cost of capital. When a corporation's common stock is publicly traded, systematic risk can be estimated with the following time-series regression: [Mathematical Expression Omitted] where [R.sub.it] is the observe return on security i in period t, [R.sub.ft] is the observe risk-free rate in period t, [R.sub.mt] is the observed market return in period t, [e.sub.it] is a security specific error of firm i in period t, [Mathematical Expression Omitted] is an estimated intercept for firm i, and [Mathematical Expression Omitted] is the estimate of systematic risk for firm i.

Although the approach is straightforward, typical empirical results for individual firms are not precise. For example, the average [R.sup.2] values for estimating the beta of individual stocks range from 0.20 to 0.45; the typical 95% confidence interval for the estimated beta is approximately 0.6 to 1.4.(1)

Even if the lack of precision inherent in this approach is acceptable, it cannot be used when the objective is to determine the systematic risk of a privately held corporation, a division, or a single project. Following is a very brief review of two major approaches advocated in the finance literature.

A. Accounting-Based Approaches

At least two major subapproaches to estimating beta with accounting information exist. The first accounting-ased subapproach relies on the concept of accounting betas. Hill and Stone [9] examine the relationship between accounting-based measures of systematic risk and market-based measures of systematic risk. The first step in their approach is to compute an accounting-based measure of return, such as operating earnings as a percentage of total assets. The market return on earnings is computed as the weighted average of individual company earnings returns. These values are computed using a time-series of accounting data.

Hill and Stone define four measures of accounting-based systematic risk and two measures of market-based systematic risk. They find statistically significant but low correlations, ranging from 0.156 to 0.351, between the accounting-based measures of beta and the market-based measures of beta. Given the noise in estimating the accounting beta, it is no surprise that it is not highly correlated with the market beta.

Kulkarni, Powers and Shannon [10] recently proposed a method for estimating the systematic risk for different division of a diversified firm. This method utilizes public information on business divisions as required by the Financial Accounting Standards Board (FASB) Statement No.14. The approach is similar to that of Hill and Stone, except that Kulkarni, Powers, and Shannon find accounting betas for divisions. In addition to sharing the same problems of Hill and Stone, the proposed approach has not been empirically validated, since it is not possible to compare the estimated divisional accounting betas with the unobservable divisional market betas.

The second subapproach is that of Rosenberg and McKibbin [11], who use both market-based returns and accounting-based information to predict systematic risk. They first estimate the ex-post beta for a publicly traded firm by using a time-series regression. their objective is to explain this ex-post beta by using a set of independent variables termed "descriptors." Some of these descriptors are means from a time-series of accounting data, some are variances from a time-series of accounting data, some are contemporaneous accounting data. Other descriptors are derived from market data. One of these market-data descriptors is the historical beta, which is simply the beta that is estimated in the prior period. The initial set contains 30 descriptors; the final model includes 18. Despite using so many independent variables, one of which is actually the historical beta, the [R.sup.2] of the various models range only from 0.364 to 0.382. Even if the method were more accurate, it cannot be used to estimate beta for a division, since some of the descriptors depend on market data for the firm as a whole.

B. The Analogous Firm Approach

Fuller and Kerr [7] employ the analogous firm approach to find a beta for a nontraded firm or for a division; this is often known as the pure-play method. Consider a division for which a measure of systematic risk is desired. The first step is to identify another firm with publicly traded stock that is engaged solely in the same line of business as the division. The systematic risk of such a firm can be estimated and used as a proxy for the systematic risk of the division. Fuller and Kerr empirically test this approach; their test involves 60 multidivision firms and 142 analogous pure-play identified. In some cases, several proxy firms may be identified. In these situations, Fuller and Kerr suggest the use of a median beta value. However, Bower and Jenks [3] observe that firms with publicly traded securities and in identical lines of business are not easily identifiable.

C. Goal Programming

The general applicability of the pure-play technique is questioned by Boquist and Moore [2]. Like Bower and Jenks, they note that it may me impossible to find a publicly traded company that operates solely in the desired line of business.

They suggest a mathematical programming method to resolve this dilemma. Their approach is based on the premise that a firm is simply a portfolio of projects. Thus, the systematic risk of a firm is the weighted average of the betas in the lines of business that constitute the portfolio. They first estimate the beta of several publicly traded firms. For each firm they measure the weight of each business segment. Since they have more segments than firms, in their sample they use a type of linear programming called goal programming. The objective is to find segment betas that minimize the sum of absolute deviations from the firm betas.

The use of goal programming is based on the assumption that full information is not available; i.e., there are more business segment than there are firms. If this is the case, then Crum and Bi [5] note that a unique optimal solution does not exist. Hence, the estimated division betas are ambiguous.

II. The Full-Information Methodology

The full-information approach proposed in this study relies on the same theoretical foundation as the goal programming approach of Boquist and Moore. A corporation having separate divisions of lines of business is a portfolio of assets, in which each asset is defined as a business segment. It is assumed that the beta for a company is simply the weighted average of the betas of its business segments: [Mathematical Expression Omitted] where [BETA.sub.i] is the overall beta for firm i, [[Beta].sub.j] is the beta for segment j, N is the number of segments, and [w.sub.ij] is the market value of segment j for firm i divided by the total market value of firm i.

There are several assumptions underlying this approach. The systematic risk of a particular business segment is assumed to be identical for all firms. In other words, all companies engaging in a particular line of business face the same systematic market risk due to competition in that line of business. This does not require that the nonsystematic risk be identical for all companies. For example, firms operating in different geographical areas may have different nonsystematic risk, even if they compete in the same line of business. The systematic risk, however, is assumed to be identical for the firms. Different multidivision firms may also have unique risks. The systematic risk of a division, however, is assumed to be independent of any other firm-specific characteristics.

It is also possible that the very act of creating a multidivision firm alters the firm's risk-return characteristic. If some assets of the division are fungible with respect to other corporate uses, then there exists an option effect. This will certainly affect nonsystematic risk; possible effects on systematic risk are ambiguous.

Even if segment betas are constant, differences in capital structure between firms may cause differences in systematic risk. This study assumes that there is a unique optimal level of debt for each line of business and that all firms are operating at that optimum.(2) This implies that the resulting estimates of segment betas will reflect the optimum leverage.

As an alternative to this assumption, it is possible to unlever all of the estimated levered company betas with a procedure such as that of Hamada [8]. The resulting business segment betas, [Mathematical Expression Omitted], would be unlevered betas. The reliability of this approach depends on the validity of the chosen procedure for unlevering. It also assumes that all divisions within a firm are supported by the same degree of leverage. If all divisions are not supported by the same degree of leverage, then the resulting estimates of division betas are incorrect. Consistent with this are the tests of Fuller and Kerr [7], who found that unlevering betas did not improve their results. Therefore, no attempt is made in this study to unlever firm betas.

Consider a sample of M publicly traded firms for which [BETA.sub.i] and [w.sub.ij] are measurable. If there are more companies than segments, then the following regression provides an estimate of the individual betas of the N business segments: [Mathematical Expression Omitted] where [e.sub.i] is an error term. This approach is much simpler than goal programming, but it offers several advantages: (i) it uses all available information, not just a subset of companies; and (ii) the resulting estimates of division betas can be interpreted without ambiguity.

III. Application and Results

A. Company-Specific Systematic Risk Data

Data for stock returns are obtained from the daily stock returns file on the NYSE, Amex, and OTC exchanges; this data is compiled by the Center for Research in Security Prices (CRSP) of the University of Chicago. The sample includes all firms with at least 200 nonmissing daily returns during 1986. Data for 2,170 firms from the NYSE/Amex and 3,520 firms from the OTC are available.

B. Industry-Segment Data

In the industry Segment file, COMPUSTAT reports selected financial data for up to ten business segments within a company. Each industry segment is assigned a four-digit code based upon the Standard Industrial Classification (SIC) Manual's definition of industries.(3) To reduce the number of segments to a manageable level, the four-digit industry segment code is truncated in this study to two digits; e.g., industry segments with codes ranging from 0100 to 0199 are merged to create a single line of business with a two-digit code of 01. This requires the further assumption that all segments with the same two-digit SIC code have the same beta. To the extent that betas of four-digit segments differ significantly within a two-digit classification, the likelihood of finding significant regression results is decreased, causing the resulting empirical tests to be conservative. Additional tests are conducted later in this section using three-digit codes.

Unfortunately, the market values of assets for business segments are not available. A proxy for the percentage of market value for a segment is defined as the ratio of segment sales to total sales of the company; this is similar to the procedure of Fuller and Kerr [7]. Business segment data with valid sales data for 1986 are available for 6,728 companies.

The ratio of segment net revenue (defined as segment sales less the allocated share of cost of goods sold, selling, general, and administrative expenses, and depreciation, depletion, and amortization) to total net revenue of the company is also used as a proxy for percentage of market value. Firms with negative net revenue in any segment are rejected. Business segment data with valid net revenue data for 1986 are available for 4,069 firms.

An alternative is to proxy [w.sub.ij] with the book value of assets in segment j for firm divided by the book value of total assets for firm i. Book values of assets are based on historical costs after adjustment for depreciation and may not reflect the current market value of the assets. Sales and net revenue are used instead because they reflect the current earning ability of the segment.

C. Estimation of Firm Betas

An estimate of systematic risk for each firm, [BETA.sub.i] is obtained via the following time-series regression: [Mathematical Expression Omitted] where [R.sub.it] is the daily return (including dividends) at time t for firm i, [R.sub.Mt] is the equal weighted market return (provided by CRSP) at time t, and [u.sub.t] is an error term.(4)

The average beta is 0.99 for the NYSE/Amex sample and is 1.01 for the OTC sample.

D. Estimation of Segments Betas

The first part of this section tests the proposed methodology using two-digit SIC codes with weights defined by segment sales; a comparison is made with the pure-play approach. The second part tests the methodology using selected three-digit codes. Additional tests are also conducted using net revenue to define the weights.

Tests of Two-Digit SIC Codes. After merging the samples containing betas with the sample containing industry segment data based on sales, the resulting sample contains 4,287 firms. Exhibit 1 provides descriptive information for each segment. There are 70 two-digit segments with valid data represented in the sample. For each segment, a substantial number of companies have at least a portion of their sales in that particular line of business; on average, 99.8 companies are represented in each segment.

The pure-play approach is limited to firms in a single line of business. Of the 4,287 firms in the sample, 3,073 have sales in one and only one segment. This is an average of 43.9 pure-play companies per segment; although significant, this is much smaller than the 99.8 companies per segment from the full-information approach. In the pure-play approach many segments have only a few companies; 20 have fewer than 10 companies, and three have none. Restricting the sample to companies that trade in one and only one segment, which is the pure-play approach, arbitrarily eliminates a large source of information.

The model specified in Equation (3) is estimated.(5) The regression is highly significant, with an adjusted [R.sup.2] of over 0.69. This is extremely high in comparison with the explanatory power of previous cross-sectional approaches. For example, Rosenberg and McKibben [11] have an adjusted [R.sup.2] of approximately 0.37.

The parameter estimates are shown in Exhibit 1. Only four of the 70 estimates are not significant at a level of 5%. The standard errors are relatively small, implying a tight confidence interval around the estimated beta. In fact, most confidence intervals around the segment betas actually are smaller than those in a typical timeseries regression used to estimate beta for a firm.

The results of the full-information approach are compared with the analogous firm approach. For those segments having companies that trade in that and only that segment, a mean value of beta is calculated. This is also reported in Exhibit 1, as well as the standard error of the mean. Of the 64 segments for which the standard error can be computed, 41 have larger standard errors than those based on the full-information approach. In other words, the full-information approach yields an estimated divisional beta with a tighter confidence interval for over 64% of the business segments than does the pure-play approach. Using a two-tailed binomial test, this is significant at an alpha of less than 5%. The average standard error of 0.156 using the full-information approach is 18% smaller than the standard error of 0.191 when using the pure-play approach; this average difference is -0.035, which is not significant in a t-test. [TABULAR DATA OMITTED]

The full-information approach produces an estimated divisional beta that has lower variance vis-a-vis the pure-play approach. Unfortunately, the inability to actually observe [W.sub.ij] may cause an errors-in-variables problem and produce biased estimators. This is in contrast to the pure-play approach, which should produce unbiased estimators.

It is impossible to measure the degree of any possible attenuation bias. However, the mean divisional beta estimated from the pure-play approach is 1.10 and the mean divisional beta estimated with the full-information approach is 1.08. This difference is statistically insignificant using a conventional a conventional t-test. A nonparametric comparison yields the same conclusion. Of the 64 segments, 32 have larger mean betas when estimated using the pure-play approach. This evidence suggests that any potential attenuation bias is small enough to be neglected.

The full-information and pure-play approach are also tested using a hold-out sample. The original sample of 4,287 firms is randomly divided into two samples, one with 3,675 firms and the other with 612. Business segment betas are estimated from the larger sample using both the full-information and pure-play approaches. These estimates are used to predict the company beta for the hold-out sample of 612. The full-information approach produces a mean error (estimated beta-actual beta) of -0.036; the mean error for the pure-play approach is -0.042. The maen-squared errors are also slightly smaller for the full-information approach. Although these differences are not statistically different with either a t-test or nonparametric binomial test, they suggest that the full-information approach is at least as good as the pure-play approach in prediciting out-of-sample betas.

Other Tests. Betas may not be constant within segments based on two-digit SIC codes. Selected two-digit segments are divided into segments based on the three-digit SIC code. The selected segments are Food (20), Chemicals (28), Industrial Machinery (35), Communication (48), Wholesale Trade - Durables (50), Wholesale Trade - Nondurables (51), Holding and Investment (67) and Business Services (73). The total number of segments with at least one firm having sales in that segment increases from 70 to 110. This reduces the average number of firms per segment from 99.8 to 65.0.

Equation (3) is estimated using the three-digit sample; the adjusted [R.sup.2] is 0.66. As in the case of the two-digit sample, the full-information approach provides better estimates of beta than does the pure-play approach. The average standard error is 0.221 for the full-information approach and is 0.228 for the pure-play method; it is smaller in 60 of 105 segments for which a comparison is possible.

The additional independent variables in the regression are almost orthogonal to the original independent variables. Therefore, the regression estimates for the original variables in Exhibit 1 are virtually unchanged. The results for the 40 additional three-digit segments are reported in Exhibit 2. [TABULAR DATA OMMITTED]

The same tests are replicated for the data using net revenues to define segment weights. Due to the requirement that the weights be nonnegative, the resulting sample has only 2,666 firms. The adjusted [R.sup.2] using Equation (3) is 0.69. The full-information betas are better than the corresponding pure-play betas; the average standard error is 0.214 for the full-information approach and is 0.218 for the pure-play method; it is smaller in 55 out of 98 cases.

The average standard error of the full-information betas based on sales data is slightly lower than the average standard error of full-information betas based on net revenue data. To determine whether the estimates of beta significantly differ, the betas based on sales are regeressed against the betas based on revenues. The intercept is 0.05 with a standard error of 0.17 and the slope coefficient is 0.95 with a standard error of 0.06; the adjusted [R.sup.2] is 0.68. This test indicates that the revenue beta is an unbiased estimator of the sales beta. Tables for the revenue betas corresponding to Exhibits 1 and 2 for the sales beta are not reported here but are availabe from the authors.

IV. Summary

The full-information approach is a new method for estimating the systematic risk of nontraded corporation, division, or project. The approach has a sound theoretical foundation, relying on the concept that a firm is a portfolio of projects. The data requirements are modest and are easily obtained. The approach uses a simple statistical tool, linear regression. In contrast to the goal programming procedure, the new approach allows an ambiguous interpretation of the estimated betas. Unlike the pure-play approach, the full-information approach is not restricted to a subset of the available data.

The empirical application shows that the full-information approach explains a large portion of the cross-sectional variation in firm betas. Despite numerous simplifying assumptions, the accuracy of the estimated business segment betas exceeds that of previous methods, including the pure-play approach.

References

[1.] F. Black, M. C. Jensen, and M. Scholes, " The Capital Asset Pricing Model: Some Empirical Tests," in Studies in the Theories of Capital Markets, Michael Jensen (ed.), Praeger Publishing, 1972, pp. 79-121. [2.] J. Boquist and W. Moore, "Estimating the Systematic Risk of an Industry Segment: A Mathematical Programming Approach," Financial Management ( Winter 1983), pp. 11-18. [3.] R. Bower and J. Jenks, " Divisional Screening Rates, " Financial Management ( Autumn 1975), pp. 42-49. [4.] M. Bradley, G. A. Jarrell, and E. H. Kim, " On the Existence of an Optimal Capital Structure: Theory and Evidence, " Journal of Finance ( July 1984), pp. 857-880. [5.] R. Crum and K. Bi, " An Observation on Estimating the Systematic Risk of an Industry Segment, " Financial Management (Spring 1988), pp. 60-62. [6.] M. Ehrhardt, " Diversification and Interest Rate Risk, " Journal of Business Finance and Accounting ( January 1991), pp. 43-59. [7.] R. Fuller and H. Kerr, " Estimating the Divisional Cost of Capital: An Analysis of the Pure-Play Technique, " Journal of Finance (December 1981), pp. 997-1009. [8.] R. Hamada, " The Effect of the Firm's Capital Structure on the Systematic Risk of Common Stocks, " Journal of Finance (May 1972), pp. 435-452. [9.] N. Hill and B. Stone, " Accounting Betas, Systematic Operating Risk, and Financial Leverage: A Risk-Composition Approach to the Determinants of the Systematic Risk," Journal of Financial and Quantitative Analysis ( September 1980), pp. 595-637. [10.] M. Kulkarni, M. Powers, and D. Shannon, " The Estimation of Product Line Betas as Surrogates of Divisional Risk Measures," Financial Management (Spring 1989), pp. 6, 7. [11.] B. Rosenberg and W. McKibben, " The Estimation of Systematic and Specific Risk in Common Stocks," Journal of Financial and Quantitative Analysis ( March 1973), pp. 317-333.

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Author: | Ehrhardt, Michael C.; Bhagwat, Yatin N. |
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Publication: | Financial Management |

Date: | Jun 22, 1991 |

Words: | 4106 |

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