¹ Fama, Jensen, and Beja among others have shown that, under conditions well approximated by empirical behavior of security prices, the beta variable used here is approximately equal to the beta factor from the market model: Fama, E. F., “Risk, Return and Equilibrium: Some Clarifying Comments,” Journal of Finance (March 1968), pp. 29–40CrossRefGoogle Scholar; Jensen, M. C., “Risk, the Pricing of Capital Assets, and the Evaluation of Investment Portfolios,” Journal of Finance (April 1969), pp. 167–247Google Scholar; Beja, A., “On Systematic and Unsystematic Components of Financial Risk,” Journal of Finance (March 1972), pp. 37–46CrossRefGoogle Scholar.

² Hamada, R. S., “Portfolio Analysis, Market Equilibrium and Corporation Finance,” Journal of Finance, vol. 24, no. 1 (March 1969), pp. 13–32CrossRefGoogle Scholar.

³ Beaver, William, Kettler, Paul, and Scholes, Myron, “The Association between Market Determined and Accounting Determined Risk Measures,” Accounting Review (October 1970), pp. 654–682Google Scholar.

⁴ Accounting returns are defined as accounting earnings deflated by some measure of the investment base, such as market value of common equity or net worth (book value of common equity). Market price returns are defined as dividends plus change in market price divided by initial market price.

⁵ For example, see Lintner, J., “The Aggregation of Investor's Diverse Judgments and Preferences in Purely Competitive Security Markets,” Journal of Financial and Quantitative Analysis (December 1969), pp. 347–400CrossRefGoogle Scholar.

⁶ It is important to note that empirically the security price change is taken as overt manifestation of changes occurring at the individual investor level. Unless information can induce an alteration in the behavior of one or more of the individual investors, there is no information value regardless of the price changes that occur as a result of the information arrival.

⁷ Sharpe, William F., “Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk,” Journal of Finance (September 1964), pp. 425–442Google Scholar. Lintner, John, “The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets,” Review of Economics and Statistics (February 1965), pp. 13–37CrossRefGoogle Scholar. Lintner, John, “Security Prices, Risk and Maximal Gains from Diversification,” Journal of Finance (December 1965), pp. 587–616Google Scholar. Mossin, Jan, “Equilibrium in a Capital Asset Market,” Econometrica (October 1966), pp. 768–782CrossRefGoogle Scholar.

⁸ Black, F., “Capital Market Equilibrium with Restricted Borrowing,” Journal of Business, vol. 45, no. 3 (July 1972), pp. 444–454CrossRefGoogle Scholar. Black, F., Jensen, M. C., and Scholes, M., “The Capital Asset Pricing Model: Some Empirical Tests,” in Jensen, M. C., ed., Studies in the Theory of Capital Markets (Praeger, 1972)Google Scholar.

⁹ Kraus, A. and Litzenberger, R., “Skewness Preference and the Valuation of Risk Assets,” Journal of Finance (forthcoming)Google Scholar. Also Working Paper No. 130, Graduate School of Business, Stanford University, December 1972.

¹⁰ Black, “Capital Market Equilibrium,” pp. 444–54. Black, Jensen, and Scholes, “The Capital Asset Pricing Model: Some Empirical Tests.” Friend, I. and Blume, M., “Measurement of Portfolio Performance under Uncertainty,” The American Economic Review, vol. 60, no. 4 (September 1970), pp. 561–575Google Scholar.

¹¹ Lintner's work is unpublished but is summarized in the Douglas article. Douglas, G., “Risk in the Equity Markets: Empirical Appraisal of Market Efficiency,” Yale Economic Essays, vol. 9 (Spring 1969), pp. 3–45Google Scholar.

¹² Miller, M. H. and Scholes, M., “Rates of Return in Relation to Risk: A Reexamination of Some Recent Finds,” in Studies in the Theory of Capital Markets, edited by Jensen, M. C. (Praeger, 1972), p. 47–58Google Scholar. Fama, E. and Mac-Beth, J., “Risk, Return, and Equilibrium: Empirical Tests,” Journal of Political Economy (May–June 1973), pp. 607–636CrossRefGoogle Scholar.

¹³ With respect to the cross-sectional valuation studies, among others, see Miller, M. and Modigliani, F., “Some Estimates of the Cost of Capital to the Electric Utility Industry, 1954–1957,” American Economic Review (June 1966), pp. 333–391Google Scholar; McDonald, J., “Required Return on Public Utility Equities: A National and Regional Analysis, 1958–1969,” Bell Journal of Economics and Management Science (Autumn 1971), pp. 503–514CrossRefGoogle Scholar; and Litzenberger, R. and Rao, C., “Estimates of the Marginal Rate of Time Preference and Average Risk Aversion of Investors in Electric Utility Shares, 1960–66,” Bell Journal of Economics and Management Science (Spring 1971), pp. 265–277CrossRefGoogle Scholar. With respect to the second class of studies, see Ball, R. and Brown, P., “An Empirical Evaluation of Accounting Income Numbers,” Journal of Accounting Research (Autumn 1968), pp. 159–178CrossRefGoogle Scholar; Beaver, W., and Dukes, R., “Interperiod Tax Allocation, Earnings Expectations, and the Behavior of Security Prices,” Accounting Review (April 1972), pp. 320–332Google Scholar; Beaver, W. and Dukes, R., “Delta Depreciation Methods: Some Empirical Evidence,” Accounting Review (July 1973), pp. 549–559Google Scholar.

¹⁴ Pettit, R. and Westerfield, R., “A Model of Market Risk,” Journal of Financial and Quantitative Analysis (March 1972), pp. 1649–1658CrossRefGoogle Scholar.

¹⁵ Rosenberg, B. and McKibben, W., “The Prediction of Systematic and Specific Risk in Common Stocks,” Journal of Financial and Quantitative Analysis (March 1973), pp. 317–334CrossRefGoogle Scholar. Their approach differs from that of BKS in two major respects: (1) The variable forecasted was future conditional return (i.e., conditional upon the beta forecast and perfect knowledge of future return on the market portfolio, and (2) their beta forecast assumed perfect knowledge of the accounting variables in the forecast period.

¹⁶ Ball, Ray and Brown, Philip, “Portfolio Theory and Accounting Theory,” Journal of Accounting Research (Autumn 1969), pp. 300–323CrossRefGoogle Scholar. The rationale for the first-differences transformation will be explained later.

¹⁷ Gonedes, N., “Evidence on the Information Content of Accounting Messages: Accounting-Based and Market-Based Estimates of Systematic Risk,” Journal of Financial and Quantitative Analysis (June 1973), pp. 407–444CrossRefGoogle Scholar.

¹⁸ The reasons for this contention are not immediately apparent. The statistical literature cited by Gonedes deals with a substantially different problem and lends little or no support to his contention. The statistical issue as originally set forth by Kuh and Meyer refers to a single-equation, two-variable structure involving undeflated variables. The issue concerns spurious correlation induced because of the deflation of the two variables (i.e., the dependent and the independent variables) by a single common variable. The problem is not applicable here for several reasons. (1) As Kuh and Meyer point out, the problem does not arise when the theory states the original relationship in ratio (i.e., deflated) form, which is the case for the market betas. (2) This is a system of several equations with a common deflator used in the dependent variables of two equations. It is not clear how this structure induces correlated error in the estimation of the slope coefficients of the two equations. See Kuh, E. and Meyer, J., “Correlation and Regression Estimates When Data Are Ratios,” Econometrica (October 1965), pp. 400–416Google Scholar.

¹⁹ Lintner, “Security Prices, Risk and Maximal Gains from Diversification,” pp. 587–616.

²⁰ Ball and Brown, “An Empirical Evaluation”; Beaver and Dukes, “Interperiod Tax Allocation”; and Brown, P. and Kennelly, J., “The Informational Content of Quarterly Earnings: An Extension and Some Further Evidence,” Journal of Business (July 1972), pp. 403–415CrossRefGoogle Scholar.

²¹ Blume, M., “On the Assessment of Risk,” Journal of Finance (March 1971), pp. 1–10CrossRefGoogle Scholar.

²² Vasichek, O., “A Note on Using Cross-Sectional Information in Bayesian Estimation of Security's Beta,” Journal of Finance (December 1973), pp. 1233–1239CrossRefGoogle Scholar.

²³ Bogue, M., “The Behavior and Estimation of Systematic Risk.” (Graduate School of Business, Stanford, 1972, unpublished.)Google Scholar

²⁴ Johnston, J., Econometric Methods (New York: McGraw-Hill, 1972), pp. 228–238Google Scholar. The estimates of the regression coefficients are still unbiased but have greater sampling error.

²⁵ Malinvaud, E., Statistical Methods of Econometrics (Rand-McNally, 1966), pp. 359–362Google Scholar. This problem has been also confronted in Black, Jensen, and Scholes, “The Capital Asset Pricing Model,” and Fama and MacBeth, “Risk, Return and Equilibrium.”

²⁶ One virtue of this solution is simplicity. The variable is readily at hand, and it does not have to be altered as the independent variable (i.e., accounting beta) is varied. However, it is by no means optimal. A more elaborate procedure would be to construct instrumental variable equations for each of the accounting betas. Since there will be 54 different forms of accounting betas examined, this would be a costly procedure, and in our view not cost-benefit effective at the present time.

²⁷ Johnson, Econometric Methods, pp. 232–238. This problem is present in Fama and MacBeth, “Risk, Return and Equilibrium,” and Kraus and Litzenberger, “Skewness Preference and the Valuation of Risk Assets.”

²⁸ Manegold, J., “A Comparison of Risk Measures: Accounting Betas versus Market Betas,” (Stanford University, 1972, unpublished), p. 20Google Scholar.

²⁹ Ball and Brown, “Accounting Income Numbers,” pp. 159–178. Ball and Brown, “Portfolio Theory and Accounting Theory,” pp. 300–323. Beaver and Dukes, “Interperiod Tax Allocation, Earnings Expectations, and the Behavior of Security Prices,” pp. 320–332. Beaver and Dukes, “Delta Depreciation Methods,” and Gonedes, “Accounting-Based and Market-Based Estimates of Systematic Risk.”

³⁰ Johnston, Econometric Methods, pp. 261–265.

³¹ Durbin, J., “Estimation of Parameters in Time Series Regression Models,” Journal of the Royal Statistical Society, Series B, vol. 22 (1960), pp. 139–153Google Scholar.

³² The first-stage estimating equation is of the form:

Y_t = b₀ + b₁ Y_t−1 + b₂ x_t + b₃X_t−1 + e_t, where b₁ is the estimate of the auto-regressive coefficient (ρ). In this context, Y_t is the individual security's return in period t and X_t is the economy-wide index of returns in period t. In all cases, ρ was estimated using all data available, which was 1951–1969.

³³ Rao, P. and Griliches, Z., “Small Sample Properties of Several Two-Stage Regression Methods in the Context of Autocorrelated Errors,” Journal of the American Statistical Association, vol. 64 (1969), pp. 253–272CrossRefGoogle Scholar.

³⁴ Because of a nonstationarity in the underlying betas, the use of observations from a noncontemporaneous time period involves a trade-off between a potential reduction in sampling error versus drawing from a different population. This issue was addressed in Gonedes, “Accounting-Based and Market-Based Estimates of Systematic Risk.”

³⁵ For ten of the firms, the last year's data were missing. The accounting betas were estimated from the data available.

³⁶ Austin Nichols was dropped because of its merger with Ligget and Myers, which caused it to be deleted from our version of the Compustat tape. Four firms, General Cable, Sterling Drug, Ward Foods, and Bendix Corporation were on Compustat but did not appear on our current version of the CRSP tape. The prime sample of 254 firms was reduced from 257 because of the absence of three of the above four firms on the CRSP tape.

³⁷ Security i's returns were deleted from R_m before beta was estimated. This procedure was applied to both the market and accounting betas.

³⁸ The numbers appearing below refer to the numbers assigned by Compustat to the variables. For a complete description of these variables, see compustat Manual (Investors Management Sciences, 1972)Google Scholar. Note that all of the return series are defined before extraordinary items. However, the betas from an after extraordinary items series are highly correlated (i.e., approximately .95) with the betas reported here.

³⁹ See Jensen, “Risk, the Pricing of Capital Assets, and the Evaluation of Investment Portfolios,” pp. 167–247 for a discussion of some aspects of the issue of the logarithmic transformation.

⁴⁰ Manegold, “Accounting Betas versus Market Betas.”

⁴¹ Vasichek, “A Note on Using Cross-Sectional Information.” In general, the procedure is the same as that found in most Bayesian texts, where the prior and sample means are weighted by the precision of each. For example, see Raiffa, H. and Schlaiffer, R., Applied Statistical Decision Theory (Boston: Division of Research, Graduate School of Business Administration, Harvard University, 1961), p. 295Google Scholar.

⁴² Bogue, “The Behavior and Estimation of Systematic Risk.”

⁴³ One reason is the use of fewer observations in calculating the accounting betas (annual versus monthly). On this basis alone, greater error is expected. Moreover, accounting betas possess all the estimation problems of market beta plus some additional ones of their own, as indicated in an earlier section.

⁴⁴ The Spearman correlation coefficient (r_s) is distributed approximately as a t distribution with a standard deviation of with N–2 degrees of freedom. The product moment correlation coefficient, assuming the null of ρ=0 and normality of one of the variables, is distributed as a t distribution with an analogous standard deviation with N–2 degrees of freedom. The critical r_s (and r_p−m) at the .05 level of significance is .11, .23, and .32, for one security, five security and ten security portfolios, respectively. The critical coefficients at the .01 level are .15, .32, and .45, respectively. The test of r_p−m against a null of ρ=0 is equivalent to testing the null of β=0 on the slope coefficient of the implied linear regression between the two variables being correlated. See Snedecor, G. and Cochran, W., Statistical Methods (Iowa State University Press, 1967), p. 184Google Scholar

⁴⁵ Blume, “On the Assessment of Risk,” pp. 1–10, and Bogue, “The Behavior and Estimation of Systematic Risk.”

⁴⁶ For all of the portfolio results, the portfolios were constructed by arraying the securities according to the market beta for the total period, and then picking the first N securities for portfolio one, etc. (where N equals the number of securities on each portfolio). Research is currently underway to test the sensitivity of the results to the variable on which the portfolios are selected.

⁴⁷ Manegold, “Accounting Betas versus Market Betas.”

⁴⁸ See footnote 44 for computation of t-value. It is important to note that the various specifications of accounting betas are not independent of one another. Hence, if one form exhibits positive correlation with the market beta, positive correlation would also be expected for the other forms.

⁴⁹ Because of the large number of correlations computed, there has been ample opportunity to overfit the data. As a result, the “best” specifications observed here suffer from this potential bias. Their apparent superiority will have to be verified on a testing sample. To a limited extent, an examination of the persistency of superiority across subperiods and forms of specification provide a check on the magnitude of such bias. However, as indicated at the outset, our primary purpose is to determine if a relationship (as indicated by some goodness of fit metrica) could be detected across a wide variety of specifications. The number of correlations computed prevents a straightforward interpretation of the levels of significance between any pair of betas. However, it seemed important to fully report all of the relationships examined rather than giving a misleading impression by reporting only a hand-selected subset.

⁵⁰ This is lower than that observed by Ball and Brown (35 to 40 percent).

⁵¹ Plots of the observations revealed no obvious violations of the assumptions of OLS regression, such as linearity and homoscedasticity. A representative plot is shown in Figure 5.

⁵² However, care must be taken in drawing such an inference. To the extent there are omitted variables, their correlation with the accounting beta is probably increasing as aggregation takes place. As a result, the slope coefficient is reflecting the contribution of the accounting beta plus some portion of the contribution of omitted, correlated variables. The magnitude of this problem can only be assessed by a more complete specification of the set of other variables that affect market betas. However, there is no obvious reason a priori why this would affect one class of accounting betas more than any other.

⁵³ See comment made in footnote 48. Unless otherwise indicated, reference will be made to the Spearman rank, rather than the product moment, correlation coefficient.

⁵⁴ In all these comparisons of market versus accounting betas, Bayesian was compared with Bayesian and non-Bayesian with non-Bayesian. The observed improvement in correlation is due to the joint effect of the Bayesian adjustment to both market and accounting betas. Results not reported here indicate that most of the improvement is due to the adjustment of the accounting betas. While there is clearly merit to adjusting the market betas, there appears to be even more to be gained from adjusting the accounting betas. This is consistent with our beliefs, expressed earlier, that the error of estimation in the accounting data is probably greater.

⁵⁵ See footnote 49.

⁵⁶ It is important to note that this section is a replication of our analysis on Gonedes sample, as opposed to a replication of Gonedes procedures on Gonedes sample. For example, in a recent note, Gonedes lists four differences. Our research indicates, however, that none of these factors accounts for the differences to be reported later. See Gonedes, N., “A Note on Accounting-Based and Market-Based Estimates of Systematic Risk” (unpublished, 1973), footnote 8Google Scholar.

⁵⁷ Unless otherwise indicated, the anlayses will refer to the Spearman rank correlations. The critical coefficients at the .05 significance level are .17, .39, and .54 at the one, five and ten security portfolio levels. The critical values at the .01 level are .24, .52, and .75, respectively.

⁵⁸ Because of the superiority of the Bayesian betas, the rest of the analysis will restrict itself to the Bayesian betas.

⁵⁹ With the possible exception of the hypothesis concerning the superiority of total period accounting betas.

⁶⁰ See our comments in footnote 56.

⁶¹ Manegold, “Accounting Betas versus Market Betas.”