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J I, 1; fits, a 1: .H. .Nmr ‘=. ' 9‘ S{m-;=;:)m‘w=m£§m.Jam» ris- .1}; L11 r. ..d ‘5 ' I! .. ”my“ m’ ' “m - at; .J n '- 3‘ “MARY Michigan State University ” _ ,l - u. 3"." " Tconrv- .. ..., - - THE!!! This is to certify that the thesis entitled Rig paw W 7) a)": Afléféifi. W 147%?" 13—5“ c: i him/rum presented by has been accepted towards fulfillment of the requirements for I Ph D figgree in P111...;c n 79 117%‘4 /:/ Mil/ig/Q/ . H (1.1)") Ii)"; 2; “/JJL/ 0 ii Mdflw Major professor Date foil]. 021/757 0-7639 OVERDUE FINES: 25¢ per (in per item RETURNING LIBRARY MATERIALS: Place in book return to rmve charge from c1 rculat1on records REPLACEMENT COST DATA AND CAPITAL MARKET EQUILIBRIUM By Jack L. Freeman A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Accounting and Financial Administration 1981 © Copyright by Jack L. Freeman 1981 All rights reserved. ii ABSTRACT REPLACEMENT COST DATA AND CAPITAL MARKET EQUILIBRIUM By Jack L. Freeman This research investigates the relationship between security returns and replacement cost disclosures mandated by the Securities and Exchange Commission's Accounting Series Release 190. In this regard, other recent replacement cost studies have focused solely on the mar- ket's initial reaction to these disclosures. Accordingly, they were limited to the use of historical cost data and Value Line estimates of replacement cost in deriving expectations. Furthermore, these investi- gations were done in isolation, i.e., their information hypotheses pertain only to replacement cost variables. As‘a result, their re— search designs could not provide evidence regarding the Commission's contention that replacement cost data provide user information not otherwise available. To correct for the first limitation, a firm's 1976 replacement cost data are used in formulating the market's 1977 expectation of cur- rent cost income. Deviations from each firm's actual 1977 replacement cost income then form realizations of that information variable. To overcome the second limitation, the historical cost income forecast error variable is included so that portfolio returns are conditioned on various realizations of both variables. This inclusion provides a mechanism for determining whether the current cost income numbers reflect information beyond that reflected by the historical cost income numbers. Jack L. Freeman One hundred and eight firms are used to construct six infor- mation portfolios. Each are conditioned on the various realizations of the two income forecast error variables. Exploiting the properties of the capital asset pricing model, pre—experimental equivalence is assumed to be attained. Thus, detection of significant return differ- ences implies that the forecast error realizations (signals) reflect information. The use of two conditioning variables necessitates an examination of the relationship between them since it is crucial to appropriate design selection and interpretation of test results. The inferred variable relationship resulted in a one-factor design being selected and reparameterization of its underlying model resulted in the 3 351951 contrasts of interest. A_prigri contrasts are employed since they eliminate the need for control portfolios (unconditional portfolio returns) and increase the power of the test. Included are specific contrasts which test the primary research hypo- thesis that current cost income signals do not reflect information beyond that of the historical cost income signals. The test period consists of the fifty work weeks subsequent to the March 31, 1977 portfolio formation date. The results are consistent with the hypo- thesis stated above and, therefore, provide evidence that required replacement cost disclosures provide no information to the market. DEDICATION To the Professor, and all like him, may their efforts be appreciated on earth as I am sure they are in heaven. A C K N O W L E D G M E N T S I wish to thank my dissertation committee of Professor Rick Simonds (Chairman), Professor Randall Hayes, and Professor Maureen Smith for their assistance and constructive criticism. Their guidance throughout this research effort was extremely beneficial. I also wish to express my deep felt appreciation to Carole Boomer, Tom McEvoy, Maxine Quinlan, Spud Wolfe, and Elaine Womack for their unselfish and invaluable assistance in completing this work. iv TABLE OF CONTENTS Chapter INTRODUCTION AND OVERVIEW ............... THEORETICAL FOUNDATIONS ................ Overview ...................... Expectation Models ................. Distribution Properties, Assumptions, and Implications of Security Returns ......... COMPARISON OF RETURN METRICS AND MODEL EQUIVALENCE The Information Hypothesis ............. The Market Model .................. Residual Return Metric ............... Difference in Total Return Metric .......... Comparison of Metric Approaches in the Two Realization Case . . . . . . . . . . ....... The General Linear Model .............. Two-Factor (Zero-Beta) and General Linear Return Models ................... Empirical Considerations .............. Concluding Remarks ................. LITERATURE REVIEU OF REPLACEMENT (CURRENT) COST ACCOUNTING . . . . . . . , .............. Current Cost Concepts ................ Object of Prediction ................ The Theoretical Ability of Current Cost Accounting to Predict Future Distributable Operating Flow .................. Empirical Research ................. DATA AND PORTFOLIO CONSTRUCTION ............ Firm Selection ................... External Validity .................. Derivation of Income Forecast Errors ........ Weekly Returns ................... Portfolio Construction ............... Results of Portfolio Construction .......... Internal Validity .................. Page 1 8 8 11 36 Chapter VI. HYPOTHESES AND EXPERIMENTAL DESIGN ......... Statement and Test of the Omnibus Hypothesis . . . . Experimental Design Approach Multivariate Analysis of Variance (MANOVA) .. ........ Main-Class Model ................. Interaction Effects ............... Variable Relationships and the One-Factor Design ..................... Comparison of Design Implementation ....... VII. EMPIRICAL RESULTS ................. Relationship Between Conditioning Variables Mixed-Model Assumption . . . . . . . . . . . . I : . Test Results of the Information Hypothesis . . . . VIII. CONCLUSION ..................... Summary ..................... Recommendations for Future Research .’ ...... APPENDICES Appendix A - Comparative Analysis, "Classical" Control Groups and the Reparameterization Process .................... vi Page 60 60 64 7O 73 76 86 90 90 92 94 .102 LIST OF TABLES Table Page 5-1 Security and Portfolio Betas ............ 56 7-1 Contingency Table for Sample Firms ......... 93 7-2 Estimated Error Correlation Matrix for Transformed Portfolio Returns .......... 95 7-3 Portfolio Weights and Resultant Contrasts ..... 96 7-4 Summary Statistics for Tests on the Transformed Mean Return Vector for the Fifty Week Test Period ...................... 98 7-5 Summary Statistics for Subperiod Tests ....... 100 vii LIST OF FIGURES Figure Page 5-1 Beta Estimation and Test Period .......... 51 5-2 Review of Construction Process and Summary of Forecast Error Realizations ........... 53 6-1 Two-Factor Design ................. 65 6-2 Some Possible Outcomes of a 3 x 2 Factorial Design ...................... 74 6-3 Variable Relationships ............... 77 6-4 One-Factor Design ................. 79 A-l The K, (T')'1, and P Matrices for the One- Factor Case ................... 113 viii ANOVA ASR CAPM COP COPFE CRSP DOF ECCI EEI FAS GLM GLRM HCI HCIFE MANOVA PCCI PEI RCS R'edCS SEC SIC UCOP UEI ABBREVIATIONS Analysis of variance Accounting Series Release Capital asset pricing model Current operating profit Current operating profit forecast error Center for Research in Security Prices Distributable Operating flow Expected current cost income Expected economic income Financial Accounting Standard General linear model General linear return model Historical cost income Historical cost income forecast error Multivariate analysis of variance Past current cost income Past economic income Realizable cost savings Realized cost savings Securities and Exchange Commission Standard Industrial Classification Unexpected current operating profit Unexpected economic income ix CHAPTER I INTRODUCTION AND OVERVIEW The Securities and Exchange Commission's No. 190 (ASR 190) re- quiring particular firms to disclose certain replacement cost (current cost) data was issued March 23, 1976. This requirement is effective for fiscal years ending on/or after December 25, 1976. The purpose of this study is to examine the information content of these required replacement cost data. Specifically, the purpose is to examine the relationship between current cost data and security returns, i.e., to investigate the extent to which these disclosures reflect information pertinent to assess- ing firms' equilibrium prices (hereafter, called expected returns). ASR 190 required that certain large registrants disclose the estimated current replacement cost Of inventories and productive capacity and the approximate cost of sales and depreciation based on replacement cost. Having imposed this requirement, the Commission recognized that the firms might incur non-trivial costs in obtaining, storing, and report- ing this data. Research by O'Connor and Chandra [1977, pp. l66-167] supports this possibility. Although the initial reaction to the require- ment by the business community was overwhelmingly negative (see O'Connor [l977, pp. 37-42]), the commission was adamant in its conviction that the benefits to be derived from such disclosures would exceed the costs. In arriving at its decision the Commission stated its belief ". . . that such data are important and useful to investors and are not otherwise obtainable." The Commission further states: . .that under current eCOnomic conditions, data about the impact of changes in the prices of specific goods and services on business firms is of great significance to investors in developing an under- standing of any firm. While the current general rate of inflation has been reduced from l974 levels, it is still at a level such that 1 unsupplemented historical cost based data do not adequately reflect current business economics. If the Securities and Exchange Commission (SEC) is to continue requiring disclosure, it is reasonable to expect evidence suggesting that investors use this data. Now that a limited amount of data have been made available, it would be appropriate to empirically investigate whether investors behave as if their assessed distribution of securities' future values are formed conditionally on this data. With the Financial Accounting Standards Board's adoption of the Statement Of Financial Accounting Standards No. 33 (FAS No. 33), ASR l90 was rescinded in favor of FAS No. 33. FAS No. 33 requires particular firms to disclose certain current cost data in their published annual reports. Thus, while ASR l90 is technically revoked the replacement cost requirement is not. There are, however, reporting requirement differences between FAS NO. 33 and ASR 190. The fundamental difference between the two is their definition of current cost. FAS No. 33 defines current cost as the cost of acquiring the existing processes of produc- tion at todays prices, whereas ASR l90 defines current cost as the cost currently required to replace existing processes with currently avail- able technology. It follows that changes in technology could cause a difference in the two information sets. If, however, one assumes that there exists no material differ- ence between the two information sets, then the results of this study could be used in evaluating the information content Of FAS No. 33's current cost disclosure requirements. 0n the other hand, if one assumes there is a difference, then this research would still be valuable. Evidence on the information content of both requirements would be needed to make comparative evaluations. Research results could imply that the ASR 190 required data reflect information, whereas FAS No. 33 do not. It is reasonable to expect that evaluation of all evidence could have an impact on future current cost reporting requirements. Previous research studies regarding replacement cost disclosures have shown no significant evidence of an information effect. However, all of these studies focused on the market's initial reaction to these disclosures. Accordingly derivation of their replacement cost expecta- tion models were limited to the use of historical cost data and Value Line estimates of replacement costs which were announced prior to the lO-K filings. This study differs from these studies in two important ways. First, it uses prior reported replacement cost data in forming the mar- ket's expectation of current cost income. Secondly, it incorporates the historical cost income variable as well as the current cost income vari- able to analyze the security return behavior of firms. This, in turn, provides a mechanism for determining whether the current cost income numbers reflect information beyond that reflected by the historical cost income numbers. Regarding these two variables, Gonedes [l978, p. 27] incorpor- ating the concepts espoused by Spence [l974] states that: Taken by itself, a signal is effective if agents behave as if their assessed distributions of securities' future values are formed conditionally on the signal (or a perfect substitute for it). Reported income numbers and, in particular, current cost income numbers may be effective signals. This would occur if these numbers reflect information about attributes of firms' production, investment, and financing decisions (e.g., distribution functions Of cash flows) deemed important by investors. Prior to ASR l90 reported income numbers were derived primarily from historical cost data. For disclosing firms, however, required dis- closures Of current cost data now enable investors to derive income numbers based on current cost data (i.e., cost of sales and depreciation). Furthermore, this additional data allows investors to disaggregate his- torical cost income (HCI) into (using the terminology employed by Edwards and Bell [l96l]) current operating profit (COP) and realized cost savings (R'edCS). This disaggregation of the HCI would ideally correspond to the operating and holding activities Of the firm. The COP concept, defined as the difference between revenue and current cost of assets used to create that revenue, attempts to measure the firm's current period oper- ating efficiency. The R'edCS concept, defined as the difference between current cost and historical cost of assets used during a period, attempts to measure the contribution to total income from the firm's holding activities realized during the period. Since the HCI concept does not provide measures which distinguish between these two income producing activities, current cost advocates have contended that the HCI concept does not provide unambiguous signals about attributes of firms' decisions (Edwards and Bell [196l, pp. lO-ll, 223-2271). Given certain assumptions, Revsine [l970] contends that COP is a surrogate for expected economic income. Expected economic income is the difference between the beginning and end of period values of discounted expected future cash flows as envisioned at the beginning of the period. An equivalent formulation defines expected economic income as the product of the discount rate and the initial discounted value. .Most valuation models that are derived from partial equilibrium theories of asset valu- ation under both certainty and uncertainty are variants of the expected economic income model.1 Within the context of these models, accounting measures which provide information about changes in expected cash flows from operations would be useful. Thus, the COP potential to approximate expected economic income supports the contention that replacement cost data may provide information useful to investors that is not already reflected in the HCI number. This study consists of eight chapters. Chapter II introduces the theoretical foundation underlying the study's experimental design and hypo- theses. Specifically, information content is defined, implications of market efficiency are explained, properties of the two-parameter, two- factor capital asset pricing model are exploited, an income expectation model is introduced and the assumed distribution properties of security returns and their resultant implications are explored. ‘s-guu‘... -mm‘—-l—-n,‘-_ip_ , A comparative analysis of the econometric properties of the resi- dual return and the difference in total returns metrics is presented in Chapter III. In addition, the general linear design model from extant statistical theory is introduced and its equivalence with the two-factor (zero-beta) model is derived. This design model is subsequently used to facilitate metric comparison in the general case and to formulate the con- trasts to be tested. Chapter IV reviews the theoretical rationale set forth in the literature which has explained why replacement cost data should provide useful information to market agents in making their investment decisions. The primary methodological distinction between this study and other recent replacement cost studies, along with a brief summary of their findings, is also presented. Chapter V presents criteria used in selecting firms, describes the processes employed to derive income forecast errors and weekly returns, and explains procedures in constructing information portfolios. The beta estimation and test periods are set forth. In addition, issues of external and internal validity are addressed. Statement of the omnibus hypothesis is presented in Chapter VI. Moreover, the specific hypotheses comprising the omnibus hypothesis are set forth and interpreted for both the two-factor and one-factor (fixed effects) design cases. The relationship between the two income forecast error variables regarding appropriate design choice is analyzed. Empirical findings are presented in Chapter VII and Chapter VIII concludes by summarizing the key aspects of the study and by making I recommendations for future research. FOOTNOTE TO CHAPTER I 1Hayes [1978] relates the economic income model with a represen- tative model from valuation theory. rind-.2 4.- 1 man—C. CHAPTER II THEORETICAL FOUNDATIONS Overview The primary purpose of this study is to examine the relationship between reported replacement cost data and security returns. An equiva- lent formulation of the purpose is to determine if the current replace- ment cost data has information content within the context of the capital markets. It is this latter statement of the purpose that will be used as a framework for testing the hypotheses addressed in this study. 1 Let 6 be another random variable Let R be a random variable. (or vector of random variables). Information theory defines 5 as having information content, if for some realization, e of 5, the conditional distribution F(R/e) is not identical to the unconditional distribution F(R) (see Spence [1974, pp. 7-10] and Demski [1971, p. 14]). If, on the other hand, F(R/e) = F(R) for all realizations, e of 5, then 6 does not have information content. Accordingly, agents do not behave as if values of 5 affect their probabilistic assessment of R. Reported accounting numbers are random variables 5, whereas their realizations represent accounting events 6. Inferences can be made about the information content of these accounting numbers by observing the be- havior of some (dependent) random variable R during a period in which it is reasonable to expect that the event may be related to this random variable. Therefore, it is necessary to determine three criteria: the event, the random variable, and the time period. The event to be consi- dered in this study is the value of the income forecast error. The random variable to be observed is the return on a security (or a portfolio of securities). Previous research has provided evidence that indicates that an association exists between reported accounting numbers and security re- turns (e.g., Ball and Brown [1968], Beaver [1968] and Beaver and Dukes [1972]). The implication is that accounting numbers will have informa- tion content if either or both of the following two conditions exists: (1) the market uses these numbers in setting prices or (2) these numbers are associated with other sources of information used by the market in setting prices. Under the first condition there exists both correlation r and causality between accounting numbers and stock returns, whereas under I the second condition, only correlation exists. These two conditions along with the assumption that the market is efficient with respect to ‘- publicly available information will be used to identify a time period that one would expect to capture any potential information effects. Market efficiency implies that the market adjusts prices fully and in- stantaneously when new information becomes available (see Fama [1970]). It is assumed in this study that the market is efficient with reSpect to publicly available information. With respect to market efficiency, and condition (1), one should expect a market reaction immediately following public disclosure Of the accounting event. If, on the other hand, condition (2) exists, one might expect a market reaction prior to public disclosure of the accounting event. This would occur whenever the other source of information used by the market becomes publicly available prior to disclosure of the ac- counting event. The findings of previous research noted above, suggests that other sources of information that are reflected in accounting numbers impound in stock returns several months before public disclosure of these numbers. In summary, it appears that the appropriate time period should include both periods immediately following and preceding the disclosure 10 of the accounting event. These three criteria have been discussed within the context of information theory and the semi-strong form of market efficiency. As- suming that the distribution function F is normal, then one condition necessary for inferring information content is that E(R/e) : E(R). How- ever, to analyze conditional and unconditional expected return behavior Of securities (or portfolios), it is necessary to make an assumption as to how equilibrium prices (or expected returns) are established by the market. The two-parameter, two-factor capital asset pricing model (CAPM) provides an equilibrium expectation of the return from holding an asset. The model is a two-parameter model in the sense that the joint distribution of returns is assumed to be multivariate normal. This distribution is de- fined by two parameters, the vector Of means and the variance-covariance matrix. The model is a two-factor model in the sense that the dependent variable, expected return, is a function of two independent variables. The model implies that there exists a linear relationship between an as- set's return and its systematic risk. Jensen [1972] provides a review of theory and evidence supporting the various forms of the model. His as- sumptions of the model are adopted here. The Sharpe-Lintner version of this model is given by the expression (2-1) E(Rit) = th + Bit[E(Rmt) - th], where Rit ; the return on a security (or portfolio) i in period t, th = the risk free rate Of return in period t, Bit = the measure of systematic risk for security (or portfolio) i in period t, and 11 ~ Rmt = the return on the market in period t. Within an efficient market, values of the model's respective para- meters are established, conditioned on the information available at time t. Since the only parameter unique to asset i is beta Bit’ the expected returns of any two assets, i and j, are to be treated as equal, given that their betas are equal. In other words, if Bit = Bjt’ then E(Rit) = E(Rjt) because th and E(Rmt) are constant for all assets during a given . I t. Gonedes [1975, p. 224, 1978, pp. 48-49] contends it is this property of the Sharpe-Litner model that enables one to control "other things" so that an assessment can be made of the information content of a random W‘s—“Thzc .- variable O. The following discussion of its main facets will provide the rationale for its use in detecting information content. Omitting the subscript t for convenience, let portfolio p be a portfolio consisting of only firms that report the realization, e of 5, after establishing equilibrium at time t. In contrast, let portfolio r consist of randomly selected securities. Consequently, portfolio r is not formed conditionally on any realization of 5. Furthermore, suppose that both portfolios are constructed such that SD = Br‘ If 6 does not have information content beyond that available when equilibrium was estab- lished at time t, then the two-parameter model implies that E(Rp/e) = E(Rr). On the other hand, if these expected returns were unequal then 5 must have information content. This is because all "other things" which might effect the expected returns are held constant by setting SD = Br' Expectation Models Examining the information content of replacement cost data, the particular attribute considered is the income forecast error. The meth- odology employed will analyze the security return behavior of firms 12 conditioned on different realizations of both their historical cost in- come (HCI) and current operating profit (COP) forecast errors. These two accounting random variables are denoted by 51 and 52, respectively, and are the components of the 2 x 1 "information" vector of random vari- ables 5 = (51, 52). In calculating realized values of these two forecast error vari- ables, it is necessary to specify an investor expectation model. All conclusions are conditioned on the prOpriety of the model(s) assumed. Other researchers have attempted to garner stronger support for their conclusions by incorporating in their studies a number of expectation models (e.g., Beaver and Dukes [1972, pp. 322-324]). However, in choosing ‘TI'T-T-T’T‘T - Ann- "‘7‘_ an expectation model for calculating the COP forecast error values, the available data limit the selection to the martingale model. Fortunately, there is some theoretical support (to be discussed below) for using this model to calculate the COP forecast error realizations. The martingale model is given by where E(ut+1/ut, ut_1,...) = 0. The drift factor 6 may equal zero. This model is less restrictive than the random walk model since the ut residual series does not have to be independently and identically distributed. The results of previous empirical research has suggested that given only the past values of the HCI sequence, the martingale model is a descriptively valid expectation model of future expected HCI (Ball and Watts [1972]). This evidence is consistent with the statement that the 13 HCI forecast error reflects information about a firm's value when the martingale model is used as a surrogate for investor's income expectation (e.g., Ball and Brown [1968] and Gonedes [1978]). Within the framework of the martingale model, Gonedes [1978, p.37] has argued that some other signal (e.g., the COP forecast error) which becomes available when the HCI number is realized may reflect information if that signal alters the expected values of the HCI predictive distributions for future periods. His contention supports the use of the martingale model to investigate the COP forecast error. The martingale model will be used in this study to derive both COP and HCI forecast errors. Applying this model to the COP variable, data limitations prevent the statistical calculation of the drift factor 6, and will therefore be assumed to be zero. There is, however, under both the opposing assumptions of stability and instability some theoretical support for using this period COP as the best estimate of the following period's COP. In the case where stability is assumed, Edwards and Bell [1961, p. 99] state: The significance of current operating profit may extend to periods other than the current period if certain assumpstions are valid. Current operating profit can be used for predictive purposes if the existing production process and the existing conditions under which that process is carried out are expected to continue into the future; current operating profit then indicates the amount that the firm can expect to make in each period over the long run. Under the instability assumption Revsine [1973, p. 127] states: In an environment in which relative prices, risk, the technological processes are constantly changing, one can seldom make very accurate estimates of future current operating profits. Furthermore, when changes in Operating variables have no discernable pattern, detailed trend analyses are of limited benefit. For lack of a better method, a reasonable basis for estimating future flows is to extrapolate the most recent periods' results under the assumption that no further 14 changes will occur. If indeed, no further changes occur (and if volume is constant), then the following period's current operating profit will be equal to the present period's current operating profit. ‘ Ideally, one attempts to use the expectation model that reflects aggregate market behavior. When constructed and tested, most (if not all) expectation models become public information. Assuming the semi-strong form of the efficient market hypothesis, a researcher might further assume that the expectation model that "best" predicts is the one used by market agents. However, at this point in time, there exists a limited amount of data for specifying COP expectation models because . of the recency of Securities and Exchange Commission (SEC) reporting standards requiring this information. Therefore, in addition to the above theoretical arguments, it seems reasonable to expect that market agents act as if they use the martingale model. The Distribution Properties, Assumptions, and Implications of Security Returns The weak-form of the efficient market hypothesis implies that a security's returns (i.e., price changes over uniform intervals) are independent. If, in addition, one assumes that a security's returns are drawings from the same population, then returns are independently and identically distributed. The simple return for a period (interval) involves the product of simple returns for each of the intermediate sub-intervals. By taking the logarithmic function of one plus the simple return, this expression is transformed into the logarithmic return and is referred to as the return with continuous compounding. This return is also independently and identically distributed. However, unlike the simple return for a period, this return is equal to the sum of logarithmic returns for each 15 Of the intermediate periods. Consequently, some financial theorists have argued through the use of the central limit theorem that this distribu- tion approximates normality if the variance is finite and the number of subperiods is large (Fama [1976, pp. 17-20]). Furthermore, the continu- ously compounded return expressed as In (1 + Rt) is approximately equal to the simple return Rt where Rt is less than 15% in absolute value (Fama [1976, p. 20]). This reasoning has provided the theoretical rationale for hypothesizing a normal distribution for a security's return. The empirical results provided by Fama [1976, pp. 21-38] imply that both daily and monthly returns are leptokurtic relative to normal distributions. These distributions are members of the symmetric stable family of distributions of which the normal distribution is a member. However, unlike the normal distribution they have infinite variances. The degree of leptokurtosis detected in monthly returns is less pro- nounced than that of daily returns. In fact, their departures from normality are not sufficient enough to completely invalidate the normal assumption. A sufficient, but not necessary condition for the theoretical construction of the CAPM is that the joint distribution Of security returns is multivariate normal.2 Therefore, at this level, departures from normality are not an impediment for using the model. Accordingly, this study will employ the equilibrium property derived from those versions of the model which assume that the market is a minimum variance portfolio (Fama [1976, pp. 391-302 and pp. 320-370]). Specifically, if 8i = Bj, then E(Ri) = E(Rj). Hereafter, this property will be referred to as the equilibrium assumption. Although one could choose not to 16 employ this assumption and/or the CAPM, both provide a theoretical rationale for constructing sample portfolios with pre-experimental equivalence.3 With respect to a variant of the CAPM for conditional returns, this equivalence provides the means for eliminating from total returns that systematic portion attributable to economy-wide events (see (3-25)). The remaining variation can then be dichotomized into two parts. The first part, the remaining systematic variation in the data, is assumed to be accounted for by fixed effects in the model. The second part, the remaining random variation, is assumed to arise from small independ- ent influences which produce normally distributed residuals. Consequently, it is the joint distribution properties of these residuals or equivalently the associated conditional returns which are of primary concern to the researcher. This joint distribution is assumed to be multivariate normal. It differs from the assumption that the unconditional returns are jointly distributed multivariate normal. In other words, the normality (or lack of it) of one distribution does not imply the normality Of the other. Furthermore, neither assumption is less restrictive than the assumption that the joint distribution, which includes the returns and the conditioning variable(s), is multi- variate normal. An important point not to be overlooked is that probability statements in tests of significance refer to the sampling distribution of the statistic and not to the distribution of Observed conditional returns. Of course, if the joint distribution of the conditional returns is multivariate normal, then the statistic's distribution will be multivariate normal. However, if the joint distribution is not multi- variate normal, the central limit theorem (assuming finite variance) 17 implies that the distribution of the statistic is essentially multi- variate normal for large sample sizes (see Rao [1965, p. 108]). Currently, there is no practical method available for deter- mining whether a sample is drawn from a multivariate normal population (Bock [1975, p. 155]). However, one necessary condition is that the marginal distributions are univariate normal. The evidence cited above regarding monthly returns is not sufficient to reject the hypothesis that they are normally distributed and thus the multivariate assumption. Therefore, both joint conditional and unconditional distributions are assumed to be multivariate normal. 18 FOOTNOTES TO CHAPTER II 1 2Fama [1971] shows that any probalistic distribution which is a member of the symmetric stable class with characteristic exponent a > 1 is sufficient for the theoretical construction of the CAPM. 3Without invoking this assumption, one could use the market model to derive a residual return metric. Employing this metric, construction of sample portfolios with pre-experimental equivalence is also possible. The market model and the residual metric are dis- cussed in Chapter III. Tilde (~) denotes a random variable. CHAPTER III COMPARISON OF RETURN METRICS AND MODEL EQUIVALENCE The Information Hypothesis Financial accounting research inVolVed with studying relation- ships between accounting data and security returns (hereafter referred to as market based research) has employed two different return metrics. They will be referred to as (1) the residual return metric and (2) the difference in total returns metric. Metric terminology was introduced by Beaver [1980] to differentiate between types of return measures and their underlying distribution functions. Thus, comparison of the econo- metric properties Of different return metrics is equivalent to comparing each of the functions' corresponding parameters. To facilitate compari- son, all conditional and unconditional returns will assumed to be dis- tributed multivariate normal. Both Of these metrics are derivations of the most fundamental Of return metrics, i.e., the total return. This metric is defined as the percentage Change in price for a period after adjusting for dividends. In symbols, it is expressed as (3_1) fi 3 Pzt T th ' Pzt-1 zt Pzt-I where th = the total return for security 2 during period t, zt the price Of security 2 at the end of period t, zt dividends paid during period t, and 19 20 Pzt-l = the price of security 2 at the end of period t-l. The comparison will be made in the context Of expected returns,1 [which has been the primary focus of most previous studies. In this con- text, a researcher interested in testing the information content of an accounting random variable 5 would investigate the expected conditional return, giVen some realization (signal) 6 of 5. This expected condi- tional return is then compared to the expected unconditional return. The complete statement of both null and alternative hypothesis is (3-2) H0: E(th/ezt) - E(th) = O for all realizations Of e , and zt (3-2 ) H1: E(th/ezt) - E(th) x O for at least one realization of ezt' An equivalent formulation of the null and alternative hypotheses discussed by Gonedes [1975, p. 222] is given by (3-3) H0: E(th/eizt) - E(th/ejzt) = 0 for all i and j, i a j, and Both sets require a test of the equivalence of means, The number of realizations of 5 determines the number of means. Thus, subsequent reference to the n realization case will imply a test of the equality of n means. Beaver [1980] explored the characteristics of each metric's mean and variance parameters. His comparative analysis was made in the con- text Of analyzing a single realization of the conditioning variable 5. The conclusion he reached in this setting is that both metrics have the same expected values, however, the form of their variances differ. . . 2 ~ 2 ~ 2 ~ - Spec1f1cally, they are q (apt) and o (apt) + 0 (sq - 2Cov(e t) pt’eqt) 21 where p and q represent two different securities (portfolios). There- fore, the size of one in relation to the other depends on the value of the correlation coefficient associated with the latter. For example, if the disturbance terms (residuals) E and E are P q uncorrelated, then Cov(E ,E ) = O and the variance term 02(Ep) would be q the smaller of the two. pHoweVer, Beaver acknowledges that one can not exclude the non-existence of interdependence (cross sectional correlation) between disturbance terms. Assuming that the security return process is generated by a single factor (the market return) and no omitted variables exist, Fama [1976, p. 74] shows mathematically the interdependence of security residuals and provides empirical eVidence consistent with this phenomenon ([pp. 351-3551). Gonedes [1978, pp. 54-57] proVides eVidence that portfolio residuals are also correlated. Existence of large positive correlation could make the latter variance term the smaller of the two. Thus, Beaver considers this as- pect of his comparative analysis important since along with Type I error rate and sample size, the magnitude of the variance affects the power of a test (Slakter [1972, p. 273]). Consequently, the choice of a metric could have an impact on actual test results. Although a researcher may choose to investigate the effect of only one realization, a random variable must have at least two realiza- tions. It will be shown that by employing the appropriate statistical testing procedure in cases involving all realizations of 5, the residual return metric approach is transformed into the difference in residual returns metric approach. Furthermore, the distribution parameters of both difference in returns metric approaches are identical. In cases including all realizations Of 5, three alternative 22 procedures are available for testing the omnibus null hypothesis: (1) make n independent tests comparing each conditional return with the un- conditional return; (2) use a joint test of the equality of means incor- porating, if necessary, post hog contrasts as an adjunct procedure; (3) use a joint test of the equality of means with 3 251251 contrasts.2 The primary disadVantage of alternatiVe (1) is that multiple independent tests "inflate" the total probability of making a Type I I error. In general, the probability of accepting all the null hypotheses f using n independent tests when in fact they are all true is n .(3-4) H (1 ‘ 3k). k=1 . ‘31- rv Thus, the probability of rejecting at least one of the null hypotheses when in fact they are all true is (3-5) a* n l'gU-ak): k 1 where 3* is the total probability of making a Type I error for the col- lective hypothesis set (Bock [1975, p. 190]). For example, in the case with two tests, if a1 = a2 = .05, then a* = .0975. On the other hand, if the researcher selects a total Type I error rate Of .05, then a1 = a2 a .025 for each test. Reducing the a level reduces the power of a test (Slakter [1972, p. 273]) and the dif- ference between ak and a* is the price a researcher would pay for follow- ing this approach. Joint testing procedures called for in alternatives (2) and (3) control the overall Type I error rate. Under both Of these alternatives, if the joint test of the omnibus null hypothesis is rejected, then sub- sequent testing to detect the cause of this rejection is usually desired. 23 The advantage of using a prior: contrasts for detecting causes is that they provide more powerful tests than pgst Egg contrasts (Bock [1975, pp. 266-2673). From the viewpoint of maximizing the power of all tests, the third alternative is preferable. It is this procedure that will be used as a standard to provide a testing framework in the comparison of return metrics. The Market Model The earliest studies in market based research employed the resi- dual return metric. Researchers originally used the market model in deriving this metric and it will be introduced here. The discussion will be presented in a conceptual setting invol- ving only one asset and time period. Therefore, asset and time subscripts will be omitted. The market model is expressed as (3-6) R=a+sfim+;~. where 30: II the return of a market index for period t, the intercept and slope parameters, and 393 the disturbance term. ('0 ll With the assumption that returns are distributed multivariate normal, then E(E) = O, and E and Rm are independent. The expected value and variance are, respectively, (3-7) E(R) = a + BE(Rm), and 24 (3-8) Var(R) = 8202(Rm) + 02(E). The conditional return given some realization 9i of 5 is expres- sed as (3-9) (R/ei) a + sRm + (2x91) + ~ + . + ". “ BRm YT eT ’ where Rm and 5 are assumed to be independent. Thus, given an actual realization Oi of 5 implies the fixed effect parameter Yi' This Version of the model for conditional returns differs from the usual presentation only in that it decomposes the residual random variable into two compon- ents. They are the expected Value (i.e., E(E/ei) = yi) and a disturbance random variable éi’ where E(Ei) = O. A The conditioning Variable 5 is generally assumed to be an ordinal scale variable (see e.g., Gonedes [1978] and BeaVer [1980]). therefore, an equivalent formulation could be developed employing dummy variables. Assuming n possible realizations of 5, the market model could be expan- ded as (3-10) R = a + 35m + ylil + 7222 +-... + y X + e, where each variable ii assumes the value one when Oi is realized and zero otherwise. Moreover, their respectiVe coefficients are the fixed effect parameters. Under both the unconditional and conditional versions of the model, the term a + sRm is assumed to reflect economy-wide eyents, where- as, terms E and (Yi + éi) reflect firm-specific events. Both the residual return and difference in total returns metrics will be described using the 25 version introduced here. Residual Return Metric A researcher concerned with the information content of 5 might employ a residual return metric and reformulate the information hypo- thesis given by (3-2) and (3-3) in terms of residual returns. The market model can be used to derive this metric. However, unlike the market model, the total return is conditioned on a realization of Rm. In addi- ; tion, if a realization 6i of 5 is given, then the metric is expressed as ~ (3-11) (E/Rm’ei) = (R/Rm,ei) - a - sRm = y, + e1. i In words, the metric is the difference between the total return L and that portion attributable to economy-wide events. The expected value and variance are, respectively, (3-12) E(E/Rm,ei) = Y] + E(éi) = 71, and 2(a). (3-13) Var(E/Rm,ei) = o 1 The reason for its construction is to eliminate from the total return variability attributable to Rm (see (3-8)). Difference in Total Returns Metric In contrast, a researcher concerned with eliminating variability attributable to Rm might employ a metric as the difference in total re- turns. This metric can be expressed as ~ (R/ei) - R 0.: ll (3-14) (0 + BRm + Yi + ei) - (a + BRm + a) ll ..< + (D: I ('l 26 The expected value and variance are, respectively, ~ (3-15) E(di) = Yi + E(Ei) - E(E) = Yi’ and (3-15) Var(di) = 02(51) + 02(5) - 2Cov(Ei,E). In comparing the two metrics the results are the same as Beaver's. That is, the expected returns are the same but the variances differ. However, Beaver was analyzing only one realization of 5, therefore, he used the assets unconditional return in constructing the difference in total returns. Alternatively, if one takes the difference between two conditional returns, then the properties of the difference in total re- turns metric change. For example, let 6i and 5‘j be two realizations of 5, then (3-17) di,j (R/GT) ' (R/ej) (a + BRm + Y, + e1) - (a + BRm + Yj + ej) The expected value and variance are, respectively, (3‘18) E(dT,j) = Y1 ' Yj + E(eT) ' E(ej) = YT ‘ Yj’ and (3-19) Var(di’j) = 02(ei) = 52(ej) - 2Cov(éi,éj). The form of the variance terms in (3-16) and (3-19) are the same. How- ever, in comparing (3-15) and (3-18) the former is an expected conditional return, whereas, the latter is a difference in two expected conditional returns. Furthermore, the residual and difference in total returns metric approaches are no longer appropriately comparable in this context. This is true since the residual return metric involves only one realization of 27 5, while the difference in total returns metric involves two realizations of 5. Comparison of Metric Approaches in the Two Realization Case With respect to the information hypothesis given by (3-2) and (3-3), comparing two different realizations 6i and ej of 5 involves a test of the equality of two means. Recall that the standard testing pro- cedure is a joint test incorporating ghppippi contrasts. The simplest testing procedure available in this case is a Z test (assuming known variance) of the difference in two means. This test is given by (T -1) -0 (3-20) 2 = Y' YJ [(oz(51) + °2(éj) - 2Cov(5i,5j))/n]5 The variance term in the denominator of the test is identical to the variance of the difference in returns metric (see (3-17) - (3-19)). Furthermore, the numerator is an estimate of this metric's expected value. The residual return metric presented in (3-11) involves only one realization of 5. However, a new metric could be constructed by taking the difference between two individual residual return metrics. This metric is given by (3-21) (E/Rm.e,) - (E/Rm.ej) E(R/Rm.e,) - a - BRmJ ’ [(R/RM’ej) ‘ a ‘ BRm] (Y1 + e1) ‘ (Yj + ej) ll 4 I .4 + m I (D 28 The expected value and variance are, respectively, (3-22) E[(E/Rm,ei) - (E/Rm.ej)1 vi - ,j + E(éi) - E(é.) J Y1 ’ Yjs and (3-23) Var[(E/Rm,ei) - (E/Rm,ej)] = 02(éi) + 02(éj) - 2Cov(e,,ej). Thus, the difference in the residual returns metric distribution parameters are the same as those of the difference in total returns metric. Moreover, this result is generalized for cases involving more than two realizations in Appendix A. Therefore, in these cases, the im- pact that each metric's variance has on the power of a test is identical. Generalizing the results of Beaver's analysis to the two (or more) realization case implies one of two things. First, it might imply that 61 can have information content, whereas, its compliment ej does not. If this holds, construction of the difference in residual returns metric would not be necessary in testing both realizations. This is true since the compliment's expected conditional return is assumed to be zero. Beaver [1980, p. 22], in fact, assumes this possibility in his analysis. In this regard, Gonedes [1974, p. 28] argues that the expected value of all conditional expected returns is zero in an efficient market. If this was not true, then E(R) : a + eE(Rm). For example, if 61 and ej each occur fifty percent of the time, E(R/ei) = E(R) + C, and E(R/ej) = E(R), then E(R) = (.5)E(R/ei) + (.5)E(R/ej) = (.5)[E(R) + C] + (.5)E(R) = E(R) + (.5)C, This, of course, is impossible. The expected unconditional return cannot be equal to itself plus the additional term (.5)C. The second possible implication is that the difference in total returns metric requires an unconditional return in its construction. Gonedes [1978], for example, uses unconditional returns in constructing 29 the difference in total returns metric. If this requirement is necessary, then taking the difference of this difference would result in a variance which is not identical in form to that of the difference in two condi- tional residual (or total) returns (see (A-1) - (A-3)). However, employ- ing 2 priori contrasts eliminates the need for this requirement. The General Linear Model Implemention of the standard testing procedure in cases involving more than two realizations of 5 requires the use of analysis of variances (ANOVA) or in cases involving more than one dependent variable, multi- variate analysis of variance (MANOVA). Both of these methods involve the formulation of a general linear model. More Specifically, this is a model in which the dependent variable(s) is expressed as a linear function of the independent variable(s). In market based research, the dependent variable(s) will be to- tal return(s) or residual return(s). 'Accordingly, the model proposed later in this context will be referred to as a general linear return model (GLRM). Currently, the discipline of finance also expresses re- turns as a linear function using either the one-factor (market) model or the two-factor (zero-beta) model. The following discussion will explain: (1) the relationship between the GLRM and the two-factor model, and (2) .the difference in total returns metric, incorporating both the two-factor model and GLRM. Two-Factor (Zero-Beta) and General Linear Return Models In general, the two-factor model is expressed as ' (3-24) R = wRo + sRm + u, ~ where Ro is the return on a minimum variance zero-beta portfolio. 3O Imposition of the requirement that the market portfolio of positive variance securities be a minimum variance (and usually an efficient) portfolio implies that p = (1 - 8) (see Fama [1976, p. 301]). This, in turn, implies for any two securities (portfolios) z and q that if 82 = Bq, then E(Rz) = E(Rq). The requirement is referred to as the equili- brium assumption and will be assumed throughout the following discussion. The two-factor model can be expanded for conditional returns by including in the firm-specific component a fixed effect parameter. The model is given by (3-25) (R/ei) = (1 - B)Ro + BRm + vi + n1 . This expansion is the same as introduced earlier regarding the one- factor model (see (3-9)). The proposed general linear return model (GLRM) for conditional returns is given by (3-26) (ii/e1.) u + Y, + (i + F11.) + . + “. Ll Y] 81 9 where the grand mean, 11: Yi = the ith level effect of Bi, and 5. = (V + 5.) = the error term. 1 T The components 9 and Bi are'independent and each have an expected value of zero. This model is hypothesized for the one-factor (fixed effects) design case and is presented in detail in Chapter VI (see pages 80-81). 31 In this context, the term "factor(s)" refers to the fixed effect para- meter(s) associated with the conditioning variable(s). For the two- . factor (fixed effects) design case see Chapter VI pages 64-67. The following derivation will demonstrate the equivalence of the two-factor model and the general linear return model for conditional returns. (3-27) (R/ei) (1 - B)Ro + eRm + Y, + n, [(1 ‘ B)RO + BRm] + YT T “I [(1 - e)E(Ro) + eE(Rm) + 91 + y, + a, + 7 + . + ”. (u V) Y1 R1 = + . + ” + ”. u v, (V n1) + . + e. . u Y1 1 In discussing the two models, their respective components will be dicho- tomized into economy-wide and firm-specific effects. The GLRM component (u + V) is a random variable with expected value v. The two-factor model component [(1 - B)Ro + sRm] is a random variable with expected value (1 - B)E(Ro) + 8E(Rm). Both of these components are assumed to reflect economy-wide events, where u equals (1 - B)E(Ro) + eE(Rm). The other component in each model is identical, i.e., vi + 5 This compon- 1'. ent has as its expected value the term vi and is assumed to reflect firm-specific events. Given this equivalence, either model can be used to construct a difference in total returns metric. (3-28) di,j = (R/ei) - (R/ej) 32 [(1 ‘ B)RO + BRm + Y1 + n1] [(1 - B)RO + BRm + vj + nj] nu-emdp+su%)+O+yi+a1 IUI-mq%)+emp+c)+fi+59 [(U + V) + Yi + 51] [(U + g) + Yj + fij] [u + Yi + (V + 51)] - [u + vj + (V + fij)] [U + YT + ET] ' [U + Yj + éj] (YT + éi) ' (Yj + éj) ll .< The expected value is the same as when the one-factor (market) model was employed. Furthermore, the variance has the same form, however, it will be smaller assuming the second factor R0 explains some of the total variation in R. Empirical Considerations The discussion thus far has been in a conceptual setting. In more realistic environments, it is impossible using only the SE p233 re- turn of an asset for a single period t to empirically estimate more than one value of 7. Even for one value, it would be impossible to estimate and statistically test for significance. When obtaining more observa- tions by using an asset‘s ex pest returns from various periods, though, concern must be given to possible changes in parameter values over time. 33 Alternatively, additional observations could be obtained by employing different asset returns for a given period. Using either alternative, the effect of e is assumed to be homogeneous over time and/or among firms. In the construction of "treatment" groups (portfolios), a re- searCher must control confounding variables. A confounding variable may be described as any variable which offers an alternative explanation regarding differences in the dependent variable(s) other than the "treat- ment“. Total returns can differ among assets regardless of “treatment" effects since there may exist structural differences in their economy- wide components. This is attributable to differences in systematic risk measured by beta. The two methods control this component differently. Residual return metric may employ properties of either the market model or the zero-beta model which would eliminate from the Observed total return this component. Employing the market model for asset 2 results in (3-29) ez = R2 - oz - Bsz. Portfolios of the estimated residuals can then be constructed. With respect to the difference in total returns metric, portfo- lios are constructed so that the estimated parameters of the economy-wide components of each portfolio are equal. Under either model, imposition of the equilibrium assumption requires estimation of just one economy-. wide parameter, i.e., beta. Hence, for two portfolios p and q their respective economy-wide components excluding estimation error would be e l 'f A = A . qua 1 8p Bq 34 Concluding Remarks Both the residual return and difference in total return methods eliminate the "undesired" variability attributable to Rm contained in the total return metric. However, interdependence among residuals might exist. If interdependence is detected, then statistical tests which assume independence are inappropriate. Comparison of the two metrics employing a standard testing pro- cedure is made in a conceptual setting. In cases involving two reali- zations of an "information" variable, it is shown that their econometric properties are identical. To provide the framework for cases involving more than two realizations, the general linear model from extant statis- tical theory is shown to be equivalent to the two-factor model. Although the same results can be reached without its use, this model is employed (see Appendix A) in the general case to demonstrate metric equivalence. This demonstration involves reparameterization Of the model. This, in turn, results in g pgiggi contrasts which are differences in total returns. 35 FOOTNOTES TO CHAPTER 111 1An expected return represents only one parameter of the distri- bution. For an explanation of the general case see page 8. 2A contrast is defined as a weighted average of two or more population parameters such that the weights (coefficients) sum to zero. CHAPTER IV LITERATURE REVIEW OF REPLACEMENT (CURRENT) COST ACCOUNTING Current Cost Concepts Hicks [l946, p. l76] offers the following concept of income: . .a person's income is what he can consume during the week and still expect to be as well Off at the end of the week as he was at the beginning. However, to operationalize this concept, one has to decide how wealth ("well-offness") is measured. Current cost advocates propose to define wealth as the current cost market value of assets. Within the Hicksian framework, a firm's expected current cost income is the amount of divi- dend a firm could distribute at the end of the period without impairing the current cost market value of its assets. One of the major purported advantages to the current cost method of valuation is that it allows for dichotomization of the total income. Edwards and Bell refer to the two resultant components as current oper- ating profit and realizable cost savings (RCS). Recall that current operating profit is defined as the difference between revenue and current cost of assets used to create that revenue. Realizable cost savings are defined as the difference between the current cost of assets at the end of a period or at time of sale and their current cost at the beginning of the period or at time or purchase if the assets are acquired in that interval. Purportedly, they measure changes in value attributable to firm operating and holding activities, respectively.1 Specifically, current operating profit recognizes changes in value due to operating activities at time Of sale, whereas, realizable cost savings recognizes Changes in value due to holding activities when they occur. Edwards and Bell [l96l, p. 73] explain the importance of distinguishing between the two different 36 37 activities, stating: These two kinds of gains are often the result of quite different sets of decisions. The business firm usually has considerable free- dom in deciding what quantity of assets to hold over time at any or all stages of the production process and what quantity of assets to commit to the production process itself. The opportunity to make profit through holding activities, that is, by holding assets while their prices rise, is probably not such an important alternative for most business firms as is the opportunity to make profits through operating activities, that is, by using asset services and other inputs in the production and sale of a product or service. The diff- erence between the forces motivating the business firm to make profit by one means rather than by another and the difference between the events on which the two methods of making profit depend require that the two kinds of gain be carefully separated if the two types of decision involved are to be meaningfully evaluated. Historical cost income (accounting profit) based solely on histor- ical cost data cannot differentiate between these two activities. This is true since changes in value attributable to both activities are recognized at only one point in time, i.e., at time of sale. Therefore, any change in value through holding activities is not recognized when earned. How- ever, by incorporating the current cost Of those assets used in creating revenue, one could disaggregate accounting profit into current Operating profit and realized cost savings. Edwards and Bell [l96l, pp. ll7-ll8] referring to this dichotomized accounting profit as realized profit state: A logical first step toward improving the accounting concept of pro- fit is the reclassification of gains realized through use as cost savings rather than as operating profit. . .Such a measurement would have the advantage of drawing a sharp distinction in the records between current operating profit and realized cost savings. Manage- ment could better appraise its operating decisions because the results of current operations would no longer be confused with holding acti- vities. National income statisticians would be able to accumulate aggregate profit figures for the economy with less difficulty and more accuracy. Similar benefits would accrue to financial analysts, creditors, and the public. ObjeCt of Prediction It has been argued that financial reports should provide in- formation useful in predicting future values of relevant variables to 38 decision makers. Evaluation of this reporting function is commonly referred to as the predictive ability criterion (Beaver, Kenelly, and Voss [l968]). Implementation problems arise, however, when one attempts to discover relevant variables or equivalently the real object of pre- diction since they may vary among users. Present knowledge of users' decision models is limited, therefore, theorists have attempted to isolate certain variables that would be of interest to a large user group. For example, one often suggested variable of interest to investors is future income. Income, by definition, is an artifact and as such has no neces- sary external relevance. If, however, an income concept represents something of interest to users, then prediction of its own future value would provide through surrogation a future value of the real object of prediction. In this regard Revsine [l97l] states: . . .the crucial issue in predictive ability is not the relative ability Of an income concept to predict itself, but rather the ability of a concept to predict whatever object should be of con- cern to users. Some theorists have viewed current cost income as a surrogate for economic income or equivalently distributable operating flow. Since the determinates of economic income are future cash flows (explained in more detail below), then distributable operating flow is the amount of cash that a firm can distribute, within a period, without impairing the value of its assets. Normative valuation models imply that investors desire information about their future cash flows. Therefore, income concepts that enhance predictive ability Of future distributable oper- ating flows should be benefiCial. 39 The Theoretical Ability Of Current Cost Accbunting to Predict FUture Distributable Operating Flow In this section, it will be demonstrated in a theoretical setting that current Operating profit is equal to distributable operating flow. In addition, two approaches for predicting future values of current Operating profit and therefore distributable Operating flow will be set forth. Both the interrelation and relative merits of each approach will be explored. The subscript notation i/j will be employed in the following dis- cussion. Thus, the symbol V represents the value at time i of the i/j discounted future net cash flows expected to be generated from a firm's assets measured as of time j. More formally, n (4 1) v - kiiRHC_C ee_toa “woe Amxmu gee: oomv mxmmz >Hx_m ee_toa ee_oee_omu eoom muo_cma amok ace cowuee_umm mumm H-m oc=m_o 52 will be formed conditioned on different HCI and COP forecast error reali- zations, which are assumed to become available any time within the twelve month period subsequent to the formation date. This twelve month period is equivalent to the fifty week test period described above. There has been concern over the imprecise knowledge of the event date in market based research. Watts and Zimmerman [1980, p. 104] sum- marize this problem in stating: Does the market revise expectations on the day the information is realised or prior to and through the release date due to alternative sources of information?. . .This problem forces the researcher to make a trade-Off. The fewer the number of days being examined, the greater the signal to noise ratio and the more powerful the test. But at the same time, the fewer the number of days, the greater the likelihood that some or all of the total price change has al- ready occurred due to alternative sources Of information. Since all one hundred and eight firms have December 31 fiscal years, additional tests will be made for each of the following three subperiods: (1) twelve weeks before and after December 31; (2) twelve weeks before December 31; and (3) twelve weeks after December 31. In constructing information portfolios, firms in the information sample will be ranked according to their standardized HCI forecast errors and then divided into three non-overlapping groups: high (H), middle (M), and low (L). These three designating group values will represent the possible realizations of 51. Then, firms within each of these groups will be ranked according to their standardized COP forecast errors and divided into two groups: high (H) and low (L). These two designating group values will represent the realizations of 52. The result of this procedure will be six (3 x 2) different information groups, each con- taining eighteen firms. FigUre 5-2 pictorially reviews the construction process and summarizes the six forecast error realizations Of 5 = (51,52). By construction these realizations will form the various conditioning 53 Figure 5-2 Review of Construction Process and Summary of Forecast Error Realizations HCI Ranking a COP Rankings b “T ~ / H (e1. e2) = (H.H) H 61 = H \ _ L (91, 52) = (H,L) ~ 1.. H (é,é)=(M,H) M 61 = M \ 1 2 "F' L (61, 92) = (H,L) .1- fi— '7' - ‘- H (6.5)=(L.H) L 91 = L \ 1 2 L (51, 62) = (L,L) Ji— _— a One group of one hundred and eight firms ranked according to their HCI forecast errors b Three groups of thirty-six firms each ranked according to their COP forecast errors 54 values of the information portfolios. To proceed further with the construction of portfolios, the motivatiOn for such construction must be explained. Within the experimental design context, one attempts to construct samples (portfolios) so that pre-experimental sample equivalence exists; i.e., one attempts to hold all things other than the "treatment(s)" equal. Recall that according to the version of the CAPM given by (2-1) the only parameter which differentiates between portfolio equilibrium expected returns at time t, the formation date, is beta. All information portfolios would be constructed with equal betas. Therefore, the only difference between each information portfolio would be the "treatments". These "treatments" are by construction the different realizations of 5 which become available to the market after the formation date. Portfolios will be constructed from the six groups as follows: (1) Upon aggregation of daily returns, into weekly returns, beta for each form will be estimated by applying sixty weeks of return data from the beta estimation period to the market model; (2) for each group, firms will be ranked according to their estimated betas and then divided at the median into equally weighted high and low risk portfolios; (3) for each group, the high and low risk portfolios will be combined with weights which add to one so that the resulting portfolio has an esti- mated beta equal to one. More formally, (5-6) 1= er +(1- x)3,_, where the estimated beta of an equally weighted high risk portfolio, = the estimated beta of an equally weighted low risk portfolio, and 55 x,(1-x) = the weights. Results of Portfolio Construction The actual estimated betas used in constructing each set of equally weighted high and low risk portfolios are presented in Table 5-1. For each set, the table also shows betas for each high and low risk portfolio along with their respective weights. .As mentioned above, these weights are used to combine each of the portfolio sets to form the six equal beta information portfolios. Recall, in this study equality is attained by setting beta equal to one. Reviewing the results, there is only one case (M,H) that the prespecified value of unity did not fall within the range of values defined by the equally weighted high and low risk portfolios. In this case the low risk portfolio weight has a value less than zero. Negative weight implies that this portfolio is sold short. This construction process then determines how each portfolio return is established from the observed returns of securities comprising it. The weekly returns for each information portfolio were then cal- culated in two steps. In step one, the weekly return for each high and low risk portfolio was calculated by 9 (5-7) R3 = .2 R1/9. H(L) 1=1 Then in step two, the return for each information portfolio was calcula- ted by (5-8) R = xR- + (1 - x)R~ . I 8H BL Each resulting set of six weekly returns represent the observed measures of the design. At the portfolio formation date, their expected returns 56 Table 5-1 Security and Portfolio Betas Information Portfolios (L,L) (L.H) (M.L) (M,H) (H,L) (H.H) Individual Security Betas for the Equally Weighted High Risk Portfolios 1.667029 1.775445 2.649679 1.172304 1.752493 1.787479 1.434967 1.562208 1.820371 1.134310 1.520941 1.500210 1.242371 1.474098 1.347095 1.043278 1.211616 1.404480 1.057531 1.274420 1.170336 1.006278 1.173266 1.217990 1.045827 1.174276 1.146689 .947367 1.129478 1.135438 .903220 1.161185 1.005060 .896346 1.048634 1.005791 .899631 1.120045 .917951 .872150 1.029532 .955707 .882591 1.036045 .884583 .828662 1.026163 .916336 .870760 .859770 .853825 .797978 .740036 .913178 Total 10.003927 11.437492 11.795589 8.698664 10.632159 10.836609 Average 1.111547 1.270832 1.310621 .966518 1.181351 1.204068 High Risk Portfolio ' Weight (x) .848927 .692525 .625532 1.083675 .791051 .781932 Individual Security Betas for the Equally Weighted Low Risk Portfolios .733547 .811465 727968 .797423 .583013 .801533 .653992 .692757 667873 .797316 .398901 .593414 .614145 .690822 666884 .637696 .394231 .560295 .540192 .661564 661152 .597071 .323071 .514206 .312438 .583049 653871 .594562 .272701 .244482 .266101 .355764 319238 .496981 .251648 .229146 .141359 .303181 287887 .441027 .245837 .201222 .121035 .210854 265266 .424493 .188272 .144283 (.024178) (.799404) 079953 .310813 .163219 (.874167) Total 3.358631 3.510052 4.330092 5.097382 2.820893 2.414414 Average .373181 .390005 481121 .566376 .313432 .268268 Low Risk Portfolio Weight (1-x) .151073 .307475 374468 (.083675) .208979 .218068 57 are assumed to be equal. Therefore, any significant differences over the fifty week test period would imply information content. IntErnal Validity Ideally, in the experimental setting one attempts to assure pre- experimental equivalence by randomly assigning subjects to treatments. Randomization, of course, is not possible in this study. In lieu of this, imposition of the equilibrium assumption, if Bi = ej, then E(Ri) = E(Rj) provides the rationale for obtaining pre-experimental equiva- lence. However, in attempting to set all portfolio betas equal, errors might arise. This is because true beta values are not available, and must be estimated from ex.ppst_returns (Fama [1976, p. 344]. Departures from equality arising due to estimation errors could therefore affect the obtainment of pre-experimental equivalence. Unfortunately, the absence of this equivalence provides an alternative explanation for any systematic difference(s) (or lack thereof) discovered. For example, if in comparing two portfolios‘returns, one portfolio had an actual beta greater than the other, then according to the CAPM their expected re- turns should differ regardless Of any "treatment" effect. The researcher should attempt to minimize this threat to inter- nal validity by obtaining the most precise beta estimates possible. This problem of estimating beta with precision is one reason for group- ing securities into portfolios. Fama and McBeth have shown the effect- tiveness of the portfolio approach. Their test results revealed that S(iglei) = S(ep) is one-third to one-seventh the magnitude of 1.EIS(e1.)/n, where S(e) is the estimated standard derivation of the error. Further- more, this result implies that the standard error of a portfolio beta estimate S(5p) is also one-third to one-seventh the magnitude of the 58 standard error Of an individual security beta estimation 5(51) (see Fama [1976, pp. 351-3563). S9 FOOTNOTES TO CHAPTER V 1There was one exception. Kaiser Steel Corporation had in- complete data. Missing return data was obtained from the Standard and Poor's Daily Stock Price Record. 2ASR 190 excludes those firms whose total inventory and gross property, plant, and equipment are either less than one hundred million dollars or ten percent of the total asset value. 3Although one might take exception to the term operating, this figure can be considered as the historical counterpart to current oper- ating profit as defined by Edwards and Bell [1961, pp. 111-121]. 4Ball et al. [1976] presented evidence that the longer the period employed to compute 6, the lower the mean absolute forecast error. 5Ben and Brown [1953, pp. 166-167] found that seventy-five percent of the 1957 firms in their sample of December 31 year-end firms had made a preliminary year-end report by March 10 of the following year. The length of time between the year-end and the release of these reports gradually shortened for the years included in their sample so that by the final year, 1965, seventy-five percent of the firms re- leased a preliminary report by February 21 of the following year. The entire annual report must be made available to the public by at least the ninetieth day after fiscal year-end, which is the last day for filing Form 10-K with the SEC (see Rappaport [1972, pp. 14.5-14.7]). 60 CHAPTER VI HYPOTHESES AND EXPERIMENTAL DESIGN Statement and Test of the Omnibus Hypothesis Omitting the subscript t for convenience and using underlined notations to refer to vectors, let-fl1 and 32 denote two 6x1 vectors of returns for the information and control samples, respectively. By con- struction, the components of each vector are formed to have a beta equal to one. The estimate, then, of E(BI) will differ from the estimate Of E(Ez) only because each component of the former will be an estimated mean return conditioned on a realization (e.g., H,L) Of the "information" vector 5, By following this construction, the properties of the two- parameter model will be exploited so that any difference in the estimates will be attributed to realizations of §_rather than difference in the specific assets represented ing1 and 32' To emphasize this point nota- tional changes will be made by substituting E(E/g) and E(E) for E(Bi). and E(Bz), repectively. Recall, the following two assumptions were made: (1) capital markets are efficient and (2) all joint conditional and unconditional security return distribution functions are multivariate normal. The second assumption regarding the nature of these distribution functions as disucssed in Chapter II is sufficient but not necessary for testing the hypotheses. The omnibus null hypothesis is E(E/g) = E(R).. Setting p = -d E(E7p) - E(E) this hypothesis is equivalent to: (6-1) Ho: [THE/p) - E(E)] = g'pd = pa = 0 for all values Of the 6x1 vector _w_. 61 The multivariate hypothesis is stated in terms of mean vectors so that joint simultaneous tests will be performed on all realizations of the "information" variable(s). Each component Of the difference vector pd represents a proposition that the mean return difference between the in- fOrmation portfolio and a control portfolio is zero. The weight vector ! generalizes the hypothesis to include all linear combinations of the six mean return differences. Acceptance of this hypothesis would imply that §_reflects no information. 1 In the alternative hypothesis, there exists at least one reali- zation, §_ of 5, so that the equality in (6-1) does not hold. The alternative hypothesis is given by (6-1') H1: [Ed r. 30 = 9_ for some _w_. Acceptance of this hypothesis would imply that §_reflects information. .The following procedure is used in estimating the mean return difference parameter Ed. The year consists of fifty five work day periods each Of which is considered a week. Weekly returns Of each portfolio are computed throughout the fifty week period subsequent to the March 31, 1977 portfolio formation date. Each control sample portfolio is arbi- trarily paired with one information sample portfolio forming six pairs. For each week in the test period the difference between the weekly return of an information sample portfolio and its corresponding control sample portfolio is calculated. Each resulting set of fifty calculations corres- ponding to portfolio pairs is used to estimate each of the six components Of the expected mean difference vector parameter Ed. The statistical test that is used in testing the omnibus null 2 hypothesis is Hotelling's T . This test, a generalization of the uni- variate t test, is an appropriate test of the hypothesis given the above 62 mentioned assumption that security returns are multivariate normal.1 The T2 statistic to be used in this test has the form 2 (6-2) I = max tzhd = ma - [and - Harm/(wow), where g is previously defined in (6-1); 51 is the estimate of the expec- ted mean difference parameter pd; N is the sample size; and S is the sample covariance matrix of (IR/g) - E = i. The hypothesized value of [[0 is the null vector, and may be dropped from the expression. In constructing a single test statistic for the mean vector hypothesis, (6-2) employs a weight vector wahich forms a linear combi- nation with the components of the mean difference vector 8, Although any 2 'set of values can be assigned to the weight vector !,.the T statistic uses that set which maximizes the value of T2. The statistic t2(g) is unaffected by a change in scale of the components of g, This property creates a problem of indeterminancy with respect to the set of values of g which maximizes T2. This problem can be resolved by imposing the constraint 1'5! = 1. The imposition of the constraint leads to an equivalent form of (6-2). 2 T I ‘1 '- (5-3) T = N(Q.‘ Ho) 5 (g,- no). This invariance of t2(g) to scalar multiples of g 8150 allows the set of values of !_associated with any value of t2(w) to be normalized so that the set sums to unity. The normalized set of values of that g which maximizes t2(!) can be calculated by multiplying the vector y.= S'1(§ - 20) p by the scalar I/ZyE The resulting normalized set of values form a P=. ‘ legitimate set of portfolio weights. 63 When the null hypothesis is true, the T2 statistic can be trans- formed to the F statistic: (6-4) F = (N - P) T2 FTN‘FTTT ‘ This F distribution has P and N - P degrees of freedom where P equals the number of portfolio pairs, and N is the number of weekly observations. Departures of pd from 30 increase the expected value of T2, and therefore the value of the calculated F statistic. The decision rule for a given level of significance a will be to accept the null hypothesis if the calculated F statistic is less than the critical F value. Otherwise, reject the null hypothesis and infer that the income forecast errors have information content. However, under the null hypothesis given by (6-1) each component of the vector of conditional expected returns, E(R/pj), i = 1,...,6, is equal to the unconditional expected return, E(R).. Therefore, an equiVa- lent statement Of this hypothesis is that all conditional expected re- turns are equal to each other: (6-5) E(R/gjl - E(ngd) = 0. i.i = 1,...,5, i = 1. Viewing the hypothesis in this form lends itself more clearly than the hypothesis given by (6-1) to the experimental-design model approach to hypothesis formulation and testing. Furthermore, control portfolios are unnecessary and thus the distribution assumption regarding unconditional returns can be dropped. 64 Experimental-Design Approach Multivariate Analysis of Variance (MANOVA) Under this approach a general linear model is hypothesized and the parameters are estimated. Hypothesis testing will be introduced as an adjunct to this estimation procedure.2 In addressing the model issue recall that the method to be used in constructing portfolios will yield six information portfolios “identical" except for the treatments, i.e., the six different combinations of the three realizations of 51 and the two realizations Of 52. This can be viewed as a design with two factors over measures, where the measures are simple weekly returns on portfolios. The first factor 51 has three levels: high (H), middle (M) and low (L). The second factor 52 has two levels: high (H) and low (L). This design will be referred to as a 3x2 factorial design over measures (3x2 D/M). See Figure 6-1 for a pictorial representation. The following linear model is hypothesized for this 3x2 D/M: (6-6) (Ii/9.)ijt u + v]. + SJ. + (iii)ij + e.. i = 1.2.3. I 1.2. t = 1,2,...,50 where (R/g)ijt = the observed portfolio return in the (ij)th treatment or week t, U = the grand mean; pi. = the mean for the ith treatment; ".j = the mean fer the jth treatment; ”1' = u + 7i + Bj + (78)ij = the mean for the (ij)th treatment; 65 Figure 6-l Two-Factor Design c3 2 high (H) low (L) marginal é means 1 "‘9“ (H) yll. ’12. y1.. midd‘e (M) y21. yzz. Y2.. 1°“ (L) y31. - y32. Y3.. marginal y 1 y 2 9 means ° Key: - denotes average is used to replace the t, i, or j and indicates that the t, i, and/or j have been "summed over". - is the observed average portfolio return in yij. the (ij)th treatment for the 50 week period. - is the observed average portfolio return in yi. the ith treatment for the 50 week period. - is the observed average portfolio return in y.j. the jth treatment for the 50 week period. - is the observed average portfolio return for y.. the 50 week period. 66 vi = (pi. - p) = the ith level HCI forecast error effect; Bj = (u.j - u) = the jth level COP forecast error effect; (YBTij = "ij - (u + v1 + Bj) = the (ij)th interaction effect; éijt = the error term. Moreover, Yi class j in B, and (v8), j is an effect specific to subclass i,j. This is the main effect of class i in 1,8j is the main effect of model is the two-factor version of the general linear return model given by (3-26). In this model, the numbers one and Zero are implied coefficients of the u. Yi’ Bi and (YB)ij terms. Thus, the first equation may be written as (Vi/911,; = 1P + 171+ Ovz + 0Y3 + 181 + 052 + 1(y8)11 + 0(yB)12 + 0(44121 + 0(43122 + 0(ys)31 + 0(v8)32 + allt This equation states that the return in week t of a portfolio in treatment 1, 1 is the sum of an effect u general to all treatment combi- nations, plus an effect v. due to treatment 1 of the first experimental factor, plus an effect 81 due to treatment 1 of the second factor, plus an interaction effect (v8)11 due to the treatment combination of both factors, plus a random component allt' For clarity and ease of discussion, the model will be presented in matrix form before proceeding with estima- tion and hypothesis testing. Let 2, denote the 6x1 vector of means §ij. frOm a hypothetical random sample, then the matrix representation of the model in (6-6) is given by the expression (6-7) .i. ='A§ +'§, 67 where 111 0 0 1 010 o O o 0. 110 0 01010 0 o O _1 0 1 O 1 o 0 010 0 0 A.1010 O 1 o O o 1 0 o 10 011 o 0 0 o 010 __1'0 0 1 O 10 0 0 0 O 1] Fu" *1 I2 I3 '31 82 Tape)“ (48),, (48),, (“M22 (M31 (M32. b .J and 2: denotes the 6x1 random error vector and is assumed to be distri- buted N(Q,z), where Q_is a 6x1 vector of zeros and z is the variance- covariance matrix- For convenience, the tildes on 2, and E, are dropped. In the matrix representation of the model, rows of matrix A contain ones and zeros that were implied implicitly in (6-6). Since there are twelve unknown parameters and only six equations in the model, 68 a unique solution does not exist (that is, the matrix A is singular and therefore, AA.1 does not exist). This problem can be overcome by repara- meterizing the model so that the number of unknowns is reduced to six. The reparameterization process will yield a new set of parameters which are a linear combination of the old set. Furthermore, by selecting the appropriate reparameterization matrix, the new parameters will represent the g_p§iggi_contrasts of interest to the researcher. Mechanics of this process are presented below. Reparameterization is achieved by factoring the A matrix into the product of two matrices K and L. (6-8) A = KL. Matrix L is the row basis of A and matrix K is the column basis of A. (5-9) L = (k'k)‘1kA. (6-10) K a AL'(LL')‘1. Substituting KL into the original model and premultiplying §_by L results in a 6x1 vector p, The components of 9_represent a new set of parameters which are a linear combination of the components of g, The last five combinations form the g_priori contrasts Of interest. The new model is (6-11) L = KL§_+ e = K(L§) + g: = Kp_+ 5, In this study the parameterization matrix L is 69 1/3 1/3 1/3 1/2 1/2 1/6 1/6 I O -1 O O 1/2 1/2 1 -1 O O O O O 1 -1 1/3 -1/3 0 O 1 -1 '1 O o 0 o .9 0000 000 O 0 0 0 0 0 Premultiplying §_by L results in u + (Y1 + v2 + v3)/3 + (31 + 82)/2 + (Y1 ' Y3) + [(YB)11 + (YB)12 ' (YB)31 (72 ‘ Y3) + [(YB)21 + (YB)22 ' (YB)31 be u (YBT11 ‘ (TB)12 ' (T3)13 T (T3)23 (TB)21 ‘ (7“)22 ' (T8)13 T (75’23 1/6 1/6 1/6 0 O -1/2 1/2 1/2 -1/2 1/3 -1/3 1/3 0 O -1 [(vslu + ‘ (YB)32]/2 ‘ (Y37321/2 1/5' -1/2 -1/2 -1/3 1 + (Y87321/6 (81 ' 52) + [(YB)11 ' (YB)12 + (YB)21 ‘ (YB)22 + (Y8731 ' (YB)32]/3 Addressing the issue of parameter estimation and thereby attemp- ting to find the most parsimonious form of the model to fit the data, the following set of hypotheses will be tested: (6-12a) Ho: yl = 72 = Y3 = o; (6-12b) Ho: 81 82 - o; and (6-12c) Ho: (78)ij = O, i f 1,2,3, j = 1,2. This set of hypotheses is consistent with the omnibus hypotheses given by (6-1) and (6-5). Acceptance of all three hypotheses would imply that 70 all conditional expected returns are equal to each other. The follow- ing discussion will relate these hypotheses with the five contrasts of vector 2, Furthermore, assuming no interaction effects an equivalent form of contrasts one through three will be derived, possible results of the testing will be interpreted, and estimation of parameters will be explained. Main-Class Model It is evident from the vector g.that main-class and interactive effects are inextricably confounded and cannot be estimated separately. In the preliminary discussion of the contrasts, it will be assumed for east of interpretation that no interaction effects exist, i.e., (6-12c) is not rejected statistically. Under this assumption the interaction parameters are set equal to zero and the model is referred to as a main- class model. The vector 1_reduce to u + (v1 + v2 + Y3)/3 + (81 + 82)/2 71-73 Yz‘Y3 1: 81‘32 0 [_ O ‘__. The contrast yl - v3 equals (Pl. - u) - (W3. - u) 71 T (“1. T “3.) = (U11 + U12)/2 ' (U31 + U32)/2 H,5 L)]/2 - ll H m A In \ (D H II = H) + E(R/51 II I U (D II 2 [E(R/e1 = 1.52 = H) + E(R/a1 = 1.52 = L)1/2 = E(R/é1 = H,5 = A) - E(R/é2 = 1,5 = A) 2 2 H) ‘ E(filél = L). II on A In \ CD I II This difference holds for all levels of the COP forecast error (reali- zations of 52). The letter "A" indicates the values of 52 averaged out. One could have gone directly from (p1. - u3.) to E(R/51 = H) - E(R/51 = L) in this progression. The additional steps were inserted to help clarify the contrasting of this design with an alternative to be discussed later. Similarly 72 - y3 equals (“2. - u) - (W3. - u) T (“2. ‘ "3.) (H21 + U22)/2 ‘ (U31 + U32)/2 = [E(R/é1 = M, 52 = H) + E(R/51 = M, 52 = L]/2 - [E(R/é1 = 1,6 = H) + E(R/é1 = L, 52 = L)]/2 = E(R/e1 = M, 62 = A) - E(R/O1 = L, 92‘: A) = E(R/é1 = M) - E(R/é = L). 1 These two contrasts will be used to test the effects that dif- ferent levels of the HCI forecast error have on expected returns. If 72 their differences are not statistically different from zero, then the null hypothesis given by (6-12a) would be accepted. This result would be consistent with E(R/51 = H) = E(R/51 = M) = E(R/51 = L) = E(R) and implies that the HCI forecast error signal reflects no information beyond that available at time t, the portfolio formation date. 0n the other hand, rejection Of this hypothesis would imply that the HCI fore- cast error signal does reflect information. Furthermore, this main- class model assumption enables one to estimate these effect contrasts by directly employing the estimated marginal means of the HCI forecast A - error factor. These estimates are: §1 - 73 = y1 - 93 and 72 - p3 T 92.. ' Y3..' The contrast 81 - 82 equals (v.1 - U) - (v.2 - u) (u.1 - v.2) (“11 T "21 T "31”3 ‘ (“12 T “22 T "32”3 [E(R/e1 = H,e2 = H) + E(R/e1 = M e = H) + E(R/51 = 1,52 = H)]/3 - [E(R/é1 = H,52 = L) + E(R/51 = M,52 = L) + E(R/5l = 1,52 = L)]/3 = E(R/51 = A,‘é2 = H) - E(R/é1 = A,52 = L) 5(R/52 H) - C(R/e2 = L). This contrast will be used to test the effects that two dif- ferent levels of the COP forecast error haVe on expected returns. If 73 the difference is not statistically different from zero, then the null hypothesis given by (6-12b) would be accepted. This result would be consistent with E(R/52 = H) = E(R/52,= L) = E(R) and implies that the COP forecast error signal reflects no information beyond that available at time t. On the other hand, rejection of this hypothesis would imply that the COP forecast error signal does reflect information. Given the main-class assumption, the effect contrast can be estimated directly by employing the estimated marginal means of the COP forecast error. The estimate is 31" §24= 9.1. - y 2.. Interaction Effects If the interaction terms in the multivariate analysis of variance are significant ((6-12c) is rejected), then the main-class model does not hold. Main class effects and interaction effect are inextricably con- founded and cannot be estimated separately. In this case, the marginal means of the two-way design are not informative and analysis of cell means is necessary to interpret the interactive effects. Figure 6-2 reviews pictorially some of the possible outcomes which will be dis- cussed next. Figure 6-2a depicts the case where no interaction effect exists. Recall, this was the preliminary assumption in the discussion above. Returns associated with the high COP forecast error level are greater than the returns associated with the low COP forecast error. Since the lines are parallel, the magnitude of the difference is uniform through- out the various HCI forecast error levels. If there was no COP forecast error effect, the lines would be coincident. Figure 6-2b also depicts a case where the high COP forecast error level is associated with greater returns than the low COP forecast error leVel, but the difference 74 Figure 6-2 Some Possible Outcomes of a 3 x 2 Factorial Design COP(H) Average Return HCI(L) HCI(M) HCI(H) Solution a COP(H) Average . Return /C0P(L) HCI(L) HCI(M) HCI)H)A Solution b COP(H) Average Return COP(L) HCI(L) HCI(M) HCI(H) Solution c 75 increases throughout the HCI forecast error levels. In this case the COP forecast error effect is greater at the high HCI forecast error level than it is at the low HCI forecast error level. Therefore, the magnitude of the COP forecast error effect is dependent on the HCI fore- cast error level. Lines cross in Figure 6-2c depicting the case where the return associated with low COP forecast error level is greater than the return associated with high COP forecast error level at the low HCI forecast error level. The Opposite condition holds at the high HCI forecast error level. In the design discussed above, it was assumed that the two factors are crossed, i.e., every level of one of the factors appears with every level of the other factor. In a design over measures, the design should be balanced, i.e., an equal number of observations per cell (Cox [1958, p. 30]. In this type of research setting, the researcher does not have control over which firms receive the various "treatments". Therefore, if correlation exists between the two factors, the crossed design would not be balanced with respect to firms. However, the ultimate experimental units are portfolios each of which is comprised of those firms that are conditioned on a particular realization of the conditioning variables. Therefore, if it is possible to construct portfolios for each realiza- tion, then technically the crossed design would be balanced. Even if this construction is possible, it will be argued below that the design should not be treated as crossed given that a "high" degree of correla- tion exists. In those cases where the two factors should not be treated as crossed, modifications resulting in a one-factor design might be appro- priate for analyzing the data. Introduction of this design along with the discussion of design choice will first be presented where relational 76 extremes exist between the two factors. Variable Relationships and the One-Factor Design For ease in discussing design modifications that could arise due to the degree of correlation existing between the factors, the COP forecast error will also be divided into three levels. The degree of correlation will be examined by ranking the firms twice. One ranking will be based on the HCI forecast error, while the other will be based on the COP forecast error. Each ranking will be divided into three levels (ranges): high (H), middle (M), and low (L). Given these sub- divisions Of each ranking, Figure 6-3 illustrates the two extreme cases that could result from assigning firms to the nine difference combina- tions. Figure 6-3a-1 and 6-3a-2 depict the case where there exists an equal number of firms at each level of the COP forecast error factor for every level of the HCI forecast error factor. This implies, for example, that firms with high HCI forecast error realization might have corres- ponding COP forecast error realizations of either high, middle, or low. There is no relationship between the factors. Hence, the factors are both crossed and balanced, and the data analysis would proceed as dis- cussed above. Figure 6-3b-1 and 6-3b-2 depict the case where only the "H,H", "M,M", and "L,L" level combinations (cells) have firms. This implies that firms with high HCI forecast error realizations have corresponding high COP forecast error realizations. Therefore, at this level of grouping, there exists a relationship between the two factors. Further examination of factor relationships within each cell is then necessary and leads to two additional extreme possibilities within this case. 77 (I «A II I. o 4 am 4 z 2 em 2 I I em : .mmm .mmm o z I _u= mmewxcmz aou .1.. N-a H-a o 3 NS NS N“ s z ‘ z .2 2 S 2 I . = NH NH NH 2 ago Ho: 3 z 1, H8: mmcwxcmm sou Amscme meg :o commav mawgmco_ucpmm mpnmwce> m-e otsm_L 78 These possibilities will be discussed next. First, if firms within each of the three cells are perfectly correlated with respect to the two factors, then these factors are perfect substitutes for each other. One factor would not reflect any information not already reflected by the other. This finding, in itself, would have important implications since it implies that the COP forecast error does not reflect information not already reflected in the HCI forecast error. Finally, if the factors are uncorrelated within these cells, further analysis might reveal that the COP forecast error signal reflects information beyond that Of the HCI forecast error signal. These are the extreme cases, however, and the analysis would be made as long as the correlations within each diagonal cell are not high. Furthermore, the analysis will be made by employing a one-factor design, i.e., the two factors will be considered as one. Given the original two-level classi- fication of the COP forecast error (i.e., H and L), the factor will have six levels with no interaction effects possible. See Figure 6-4 for a pictorial representation of this design. In this case, factor order in the construction of portfolios becomes important in interpreting the variables of the study. Given the factor order employed in the construction above, the variables, to be analyzed are 51 and 52/61, rather than 51 and 52. The variable 51 is defined as before. Whereas, 52/61 is the COP forecast error given the HCI forecast error. For convenience, let 5i = 52/61. There are six different realizations of 6] (i.e., H/H, H/M, H/L, L/H, L/M, L/L). As an example, H/H would be interpreted as the COP forecast error realiza- tion being Hgiven that the HCI forecast error realization is H. For brevity, one might refer to the first three as H realizations Key: 79 Figure 6-4 One-Factor Design (61,5é) (H. H/H) 91. (H, L/H) 92. (M, H/M) 93. (M, L/M) 94. (L, H/L) 95. (L, L/L) i6. 9.. denotes average is the observed average portfolio return ' in the ith treatment for the 50 week period. is the observed average portfolio return ° for the 50 week period. 80 and the second three as L realizations. The reader, however, should not improperly interpret these H and L realizations as being the respective values assigned to the upper and lower ranges of the COP forecast error ranking of all firms. Instead the H and L realizations are the respec- tive values assigned to each of three upper and lower ranges. These ranges correspond to three COP forecast error rankings, which are derived after first grouping firms according to their HCI forecast error in the portfolio construction process. In summary, it should be noted that range ordering of the realizations Of 5é according to the COP fore- cast error ranking of all firms is H/H > L/H > H/M > L/M > H/L > L/L. The model hypothesized for the one factor design is (6-13) (Ii/p)” = u + v,- + eit’ i = 1,2,...,6, t =-1,2,...,50, where the observed portfolio return in the ith treatment (R/g) for week t, u = the grand mean, Yi = (pi - p) = the ith level HCI-COP forecast effect, and eit the error term. The matrix representation of the model is given by the expression (54.“ l. = A5; + 5., where In N b - 00000 0000 and 5, is defined as in (6-7). OOO 81 The reparameterization matrix 'O O O O O 1/6 1/2 0 l O O 1/6' 1/2 0 -1 Premultiplying g by L results in 1/6 r1/4 1/2 COO 0000 selected for this study is 1/6 -1/4 1/2 0 -1 0 000°C) 1/6 -1/4 -1/2 0 O 1 1/5' -1/4 -1/2 82 P 7 u + (Y1 + Y2 + V3 + Y4 + Y5 + Y6)/6 (Y1 + Y2)/2 ‘ 2L(Y3 + Y47/2 + (Y5 + Y6)/2] (Y3 + Y4)/2 ‘ (Y5 T Y6)/2 $3 Y1 T Y2 Y3 T Y4 Y5 ' Y6 b d o The omnibus hypothesis related to the five contrasts of vector .9 is given by (5-15) H0: 71 = 72 = Y3 = Y4 3 Y5 = Y5 = 0. If these five contrasted differences are not statistically different from zero then the omnibus hypothesis given by (6-5) will be accepted. This would imply that neither forecast error variables reflect infor- mation. If there are, however, statistically significant differences, the resulting implications will depend on the contrasts involved. The third, fourth, and fifth contrasts and their equivalent form are v1 - v2 - E(R/e1 = H,5é = H/H) - E(R/E1 - H, Eé = L/H). v3 - Y4 - E(R/é1 = M,5§ = H/M) - E(R/é1 = M, 55 = L/M). Y5 - Y5 = E(R/51 = L;5§ = H/L) - E(R/é1 = L, éé = L/L). The equivalent forms result from substitution, e.g., E(R/e1 .< H II 83 H, 5é = H/H) - p. If any of their differences are significant, then the implication is that the COP forecast error signal reflects additional information to that of the HCI forecast error signal. The first and second contrasts are comprised of three distinct components. These components and their equivalent forms are (41 + 42)/2 = E(R/e‘1 = H.6é = H/H)/2 + E(R/é1 = H.9é = L/H)/2 - a. (Y3 + 741/2 = E(R/é1 = M,5é = H/M)/2 + E(R/51 = M,5é = L/M)/2 - u, (Y5 + Y6)/2 = E(R/é1 = 1,6] = H/L)/2 + E(R/é1 = 1,6] = L/L)/2 - u. That segment of each component, other than n, represents expected returns of portfolios. These portfolios are a simple average of two from the original six portfolios. Specifically the average of two H, two M, and two L HCI forecast error portfolios, respectively. Interpretation of any significant differences is not as straightforward as above. This is because the COP forecast error realizations for each segment do not average out and, therefore, this variable cannot be dropped as in the crossed design case. They do not average out since each pair (e.g., H/H, L/H) does not represent H and L values from the full range of COP fore- cast error ranked values. Recall, however, that at this level Of group- ing (i.e., the resulting three portfolios), firms with, for example, high (H) HCI forecast error realizations also have high (H) COP forecast error realizations. Therefore, at this level, it is possible for both forecast error variables to reflect the same information. In symbols, this is explained by E(R/é1 = H,eé = H/H)/2 + E(R/e1 = H,e2 = L/H)/2 _ 84 A/H) E(R/é1 = H,5 E(R/e1 = H,e2 = H) E(R/é1 = H) E(R/e2 II I W (Note that 5é is replaced by 52 in the third step and the H realiza- tion is based on the full range of COP forecast error ranked values.) Thus, if there are any significant differences for these two contrasts, then the implication is that either forecast error signal reflects in- formation. Other firm configuarations between the two extremes depicted in Figure 6-3 are possible. For example, one or more firms might exist in each off-diagonal cell without the total configuration being balanced. Thus, the crossed design would not be balanced with respect to firms. Recall, however, that the ultimate experimental unit is a portfolio. Conceptually, measures of the dependent variable for each observation (week) are assumed to be returns on a "single" portfolio where each return corresponds to a different realization of 5. In reality, of course, it is impossible for a single portfolio to receive six different "treatments" each week. Rather, six individual portfolios, viewed as having identical properties except for the "treatments" are employed. Since the number of weekly returns observed for each of these portfolios is equal, it is technically possible to treat the design as crossed. Pre-experimental portfolio equivalence is essential to internal validity. Therefore, in constructing portfolios, it is important to approximate relative risk equivalence. However, this equivalence is 85 impaired, as Gonedes explains, where the number of firms comprising each portfolio varies. In a study examining the information content of special items he states [1975, p. 244]: Both the portfolio constructed for each type of special item and its matching portfolio (based upon firms reporting no special items) should have about the same number of components for each fiscal year. This is to avoid inducing heteroscedasticity; recall that the variance of a portfolio return is directly related to the number of securities in the portfolio. In addition, the number of components in the port- folio for a given type and fiscal year should have roughly the same number of components as its matchin portfolio for that year. The reason for this is given in remark R.6) in the Appendix. It is clear from table 1 that neither Of these size requirements can be met with- out eliminating some firms from the analysis. So, firms were randomly excluded so as to satisfy those requirements. In the appendix he states [p. 254]: This is important for. . .comparisons. . .Of the estimated relative risks of portfolios. If the groups' sizes were not balanced, our results would be affected by the mechanical effects of different sample sizes. Therefore, given the additional concern of portfolio equivalence, con- figurations that are not approximately balanced with respect to firms will not be treated as crossed. These cases also have an impact on how the contrasts are inter- preted in a one-factor désign. In this regard, the following relations between COP forecast error realizations hold: H/H > L/H; H/M > L/M; and H/L > L/L. However, some realizations would contain firms from both the upper and lower ranges of the COP forecast error ranking of all firms. For example, L/H might contain firms in the lower range, whereas H/L might contain firms in the upper range. Therefore, one cannot interpret the effect of averaging over-two conditional COP forecast error realiza- tions (e.g., H/H and L/H), in the same manner as when all firms fall on the diagonal. Thus, the variables are not perfect substitutes at this 86 level of grouping. Accordingly, if there are any significant differ- ences for the two HCIFE contrasts, interpretable implications are restricted to HCIFE variable.- Comparison of Design Implementation The two designs discussed above are adaptations of a design developed by Gonedes [1978]. He examined the information contents of earnings, dividends, and extraordinary items. The primary variable of interest in this study was not included in his set. In addition, there; are three major design implementation differences. Each of these differ- ences will be discussed within the context of the variables examined here. Following Gonedes' method of implementation, one would first test the omnibus hypothesis given by (6-1) to determine whether there exists any significant difference in expected portfolio returns. If the null hypothesis is rejected, further exploration would be made into which forecast error realizations probably contributed to this rejection. This, of course, is the primary Objective of the research effort. The explora- tion would be made by formulating appropriate pg§t_hpg_contrasts of interest. Mechanically, these contrasts are constructed by assigning specific values to the components of the vector g'in (6-1). These con- trasted differences would then be tested for significance. The hypotheses that are introduced as ppgt hpg contrasts after rejecting the omnibus hypothesis could, however, be introduced as gwprigri contrasts and tested initially. In this study, 2.221251 contrasts are in- troduced through the reparameterization process. The advantage to this procedure is that it provides more powerful tests (and shorter conficence intervals) for these contrasts than the classical omnibus approach. This 87 is desirable because increasing power of the tests, increases the proba- bility of detecting differences when in fact they exist. Gonedes' approach would also involve construction of control portfolios to be arbitrarily paired with the information porthlios. The portofolio pairs would be used to estimate each of the six components of the expected means difference vector‘pd in (6-1). However, the effect of these control portfolios are cancelled out when the post hoc contrasts are formed. For example, [E(R/é1 = H,52 = H) - E(R)] - [E(R/é1 = H,52 =U-Emn=EmN1=m%_ 2 For clarity, an equivalent statement of the hypothesis given by = H) - E(R/51 = H,5 = L). (6-1) was introduced in this study. This statement given by (6-5) hypothesizes that all conditional expected returns are equal to each other. Since unconditional expected returns are not included in this formulation, control groups are not needed to estimate their values. Even if (6-1) had been solely relied upon in formulating the original model, control groups still would not have been necessary. As with pgst hpg contrasts, their effects are cancelled out when the g'pripri_con- trasts are formed during the reparameterization process. Elimination of this need for control portfolios avoids problems that might arise in their construction and increases the power of the test. For a thorough discussion of this issue see Appendix A. The most important difference between approaches is that consi- eration is given here to relationships between "information" variables. This is essential since appropriate design choice and interpretation of results is dependent on these relationships. In his study, Gonedes does not present the correlations between his “information" variables. His interpretation of test results on the post hpg contrasted differences 88 would lead one to assume that the variables are crossed and balanced. This is because he averages out variables upon forming contrasts. If, in fact, variable relationships justify this assumption, then a two (three in his design) factor crossed design would have been warranted. This design would have allowed testing of the interaction effects. Furthermore, if the results are consistent with a main-class model, one could then estimate the contrasted effects each "information" variable has on expected returns. His overall approach, however, is consistent with a one-factor design. This design would be appropriate when variable realtionships prohibit one from assuming variables are crossed. This in turn, requires a different interpretation Of the test results as discussed above. 89 FOOTNOTES TO CHAPTER VI 1Morrison [1967] provides a detailed discussion of this test statistic. 2See Bock [1975] for a thorough develOpment Of the models and many of the procedures discussed in this chapter. CHAPTER VII EMPIRICAL RESULTS Relationship Between Conditioning Variables The sequential portfolio construction procedure results in portfolios comprised of an equal number of firms; i.e., the portfolios are balanced. It is this balanced construction that enhances pre- experimental equivalence among the information portfolios. Recall, that in this construction, all firms are initially ranked according to their HCI forecast errors and then divided into three non-overlapping groups: high (H), middle (M) and low (L). The firms in each of these groups are ranked according to their COP fore- cast errors and then divided into two groups: high (H) and low (L). This procedure results in six different information groups, each con- taining eighteen firms. An inherent facet of this procedure is that the COP forecast error values are conditioned on HCI forecast error values. For example, 5; = 62/61 = L/H, where 52 is the COPFE realiza- tion and 01 is the HCIFE realization. However, this aspect of the process does not imply that an actual relationship exists between the two variables. Determination of this relationship between the two variables is necessary for the selection of an appropriate design. Recall that the following two relational extremes are possible: (1) the variables are independent or (2) the variables are perfectly correlated. If the first extreme exists, then a two-factor crossed design would be employed. The portfolio construction process does not deter implementation of this design, since independence negates the conditioning aspect of the pro- cedure. For example, 52 = 62/61 = L/H is equivalent to 52 = 92 = L. 90 91 If minor departures from equal cell size exist, modifications at the pre- portfolio construction stage can be made to accommodate implementation of the two-factor design (i.e., random elimination of firms from the over-populated cells). If the second extreme exists, then analysis of a single variable's realizations is sufficient for investigating the information effect of the other._ This is true, since the two variables are perfect substitutes for each other. Intermediate relationships would require the use of a one-factor design. The relationship that exists between the two variables is tested first by using each variable's raw scores (i.e., its calculated quanti- tative forecast errors) obtained from the one hundred and eight sample firms. The calculated Pearson product-moment correlation is .6698 and the probability of making a Type I error in rejecting the hypothesis of no association is zero. From this result it is inferred that correla- tion exists between the variables, although not a perfect one. Recall, however, that conditioning variables are not treated quantitatively. Rather, the raw scores are grouped and 51 and 52 are treated as ordinal scale variables. With respect to analyzing the re- lationship that exists between 51 and 52, the full range of COP forecast errors are trichotomized for clarity. Therefore, at this level of grouping both variables can assume anyone of three values, namely, high (H), middle (M) and low (L). Given this partitioning on an ordinal scale, one must examine the joint frequency distribution of cases (firms) according to the two conditioning variables in order to determine the nature of the relation- ship. The three levels for each variable result in a (3 x 3) contingency table. The test of association employed is Kendall's Tau b. This sta- tistic takes on the value of + 1 when all firms fall on the major 92 diagonal and - 1 when all firms fall on the minor diagonal. If each cell frequency is equal (implying no association), then the Tau b value is zero. The observed sample frequencies (percentages) are shown in Table 7-1. The calculated Tau b is .32587 and the probability of re- jecting the null hypothesis of no association is .0001. Not unexpectedly, the statistical result (at this level of grouping) also reveals a posi- tive relationship, although not a perfect one. Furthermore, visual inspection of the contingency table reveals a pattern of cell frequencies consistent with the statement that neither relational extreme exists. However, the existence of a positive relationship is supported by the largest firm frequencies occurring on the major diagonal. These diagonal frequencies (percentages) are HH = 23 (21.3), MM = 19 (17.4), and LL = 23 (21.3). The smallest cell frequency of 3 (2.8) occurs in HL (column value given first). Overall, the degree of association revealed does not appear to be exceedingly strong. However, departures from equal cell size are deemed excessive for employment of the two-factor crossed design. Accordingly, the one-factor design is employed to test the information hypothesis. Mixed-Model Assumption Traditionally, the univeriate mixed model was used to analyze information portfolio returns. The advantage of this model is that it provides a more powerful test than the multivariate model. However, the assumption of independence of error terms (i.e., zero covariances after transformationl) is essential for prOper implementation of the mixed-model. See Bock [1975, pp. 449-460] for a complete model des- cription. 93 Table 7-1 Contingency Table For Sample Firms HCIFE COPFE .Row H M L Total H 23 (21)* 7 (6) 6 (6) 36 M 10 (9) 19 (18) 7 (6) 36 L 3 (3) 10 (9) 23 (21) 36 Column Total 36 36 36 108 Kendall's Tau b = .32587: .Significance = .0001 * Percentage of total firms 94 The determinant of the correlation matrix given by (7-1) A = lCorr(N-1)P £P| provides the likelihood-ratio criterion for testing the hypothesis that P XP is diagonal against a general alternative. Percentage points of the null distribution of (7-1) are well approximated by equating (7-2) -[(N-1) - (2p + 5)/6]lnA to the central x2 distribution on [p(p-1)]/2 degrees of freedom, where p is the number of dependent variables and N is the number of Observa- tions. Significance of this x2 rejects the hypothesis that P'XP is a diagonal matrix. The calculated value of (7-2) based on the sample error correla- tion matrix in Table 7-2 is 1.03. This result is consistent with the assumption of independence of error terms for the transformed Observa- tions. Therefore, the mixed-model approach is deemed to be apprOpriate for testing the data. Thus, the univariate F-tests for each contrast are statistically independent, and the overall error rate may be calcu- lated by (3-5). Except for the computational format, multivariate analysis of variance can be adapted to provide the same analysis as the univariate mixed-model. However, as discussed below the resultant increase in power does not appear to noticeably affect the results. Test Results of the Information Hypothesis The 2.221221 contrasts to be tested jointly are summarized in Table 7-3. Each set of weights used in their construction correspond to the last five rows of the one-factor design reparameterization matrix L. In parentheses are related normalized sets of weights. These sets 95 Table 7-2 Estimated Error Correlation Matrix For Transformed Portfolio Returns Transformations Constant 1.000 Contrast 1 -.091 1.000 Contrast 2 .076 -.259 1.000 Contrast 3 -.425 -.021 .187 1.000 Contrast 4 .197 -.265 .381' .099 1.000 Contrast 5 .170 .378 -.064 -.121 -.222 1.000 Calculated x2 (15) = 1.03 Alpha Points Value of x2 (15) .100 22.31 .050 25.00 .025 27.49 .010 30.58 €96 .muom “gm.oz uo~._e3co=-oca o:.>o_auo woucomoca a. umacucou gun“ « Aaxa.dvn I Aa\=.4._ m A:\J.:.— I Ax\:.xv. e A:\4.=_~ . A=\:.:.~ m A4\4.4V«\~ - .u\:.4.~\_ - A:\u.:.~\_ + Az\=.:.~\_ ~ .4\4.4.e\~ I .4\:.4.o\_ - Ax\4.zvvx. . Ax\:.:vc\_ I A=\4.:v~\~ e .:\=.=.~\~ . mumucucou . coaezz «auscucou acaupamoz .. .an-~ .Nxmx-._- - .~\_rvm\~- .e\m\-.e\_ o\o.o e Auxmxv— .~\_-.~\_- .c\m\-.o\_- o\:.4 m .N\N\-..- A~\_.~\_ .o\m\-ve\.- x\4.x q .Nxmxv. .~\_.~\. Acxmx-.e\_- :x:.: n .Nxmx-._- , An\m\.~\_ :x4.= N .~\N\.~ .n\m\.~\_ =\=.= _ E S E E 3 mass: ease... ( gowuan—aua Aeo~._ascez. woea.o= to seem me.eo.o_eeou I mumacueou seem ca mo.—Ouuccm ca no._aa< maga.oz «a moon t- .amIN nanosecou acau—awoa use mago.oz o._ouucom mum «peep 97 correspond to the last five columns of the transformation matrix P presented in Figure A-1. It is the normalized set which is used in the omnibus test. Result of the omnibus test regarding information content of both HCI and COP forecast errors are consistent with the no information hypothesis. The omnibus test statistic for the overall test period along with other summary data is shown in Table 7-4. Furthermore, the joint test result obtained from employing the mixed-model does not alter this inference. For this model, the calculated level of signifi- cance is .128. This value is insufficient to reject the omnibus null hypothesis given a critical value of .10. Notwithstanding this insig- nificance, the signs of the observed contrast mean values will be reviewed. It was anticipated, constructing contrasts, that the observed mean differences would be positive, given that on a relative basis, positive weight(s) were assigned to the more favorable portfolio(s). Only three of the five,mean differences reported in Table 7-4 are posi- tive. Included in this group are contrasts (1) and (2) which are em- ployed to test the information content of the HCI forecast errors. These two positive value outcomes are consistent with the results of previous research (e.g., Ball and Brown [1969] and Gonedes [1978]). With respect to contrasts (3), (4), and (5), which are formu- lated to test mean return differences between portfolios conditioned on different COP forecast error realizations, only the reported value of the fifth contrast is positive. Although the mean differences of the other two contrasts are negative, both values are close to zero. How- ever, since the sample data failed to reject the omnibus null hypothesis, further discussion of possible implications seems to be highly speculative 98 Table 7-4 Summary Statistics For Tests On The Transformed Mean Return Vector For The Fifty Week Test Period 4 t l j - L 102 x Estimated 103 x Estimated Standard Contrast Mean Deviation Univeriate F P Less Than 1 2.20 .88 3.12 .08 2 1.93 1.31 1.09 .30 3 -.93 .97 .45 .50 4 -.18 1.40 .01 .92 5 2.73 .85 5.21 .03 F (5,45) Statistic for Multivariate Test of Equality of Means = 1.64 with P Less Than .17 Value of Value of Value Of Value of Alpha Points F (1,40) F (1,60) F (5,40) F (5,60) .100 2.84 2.79 2.00 1.95 .050 4.08 4.00 2.45 2.37 .025 5.42 5.29 2.90 2.79 .001 7.31 7.08 3.70 3.51 99' at best. Contrasted mean differences and multivariate F-statistics for each of the other three test periods are summarized in Table 7-5. The test results are highly insignificant, and yet the most interesting result pertains to the twelve week test period after December 31. The sign of each contrast is as expected for both this period and the twenty- four week test period. However, the resultant level of significance is smaller for this twelve week period than for each of the other two sub- periods. Furthermore, the value of each mean difference(s) for this twelve week subperiod is greater than both the corresponding actual and absolute values of all the other test periods. One possible interpre- tation of this result is that the appropriate test period is twelve weeks after the fiscal year end. If this is true, two further comments are in order. First, additional observations obtained from using a larger test period would tend to dilute the effect. This would therefore explain the lack Of significance resulting from the use of the larger test periods. Secondly, the lack of significance for the twelve week subperiod may be attributable to the small number of observations since the power of the test is affected by sample size. 100 Table 7-5 Summary Statistics For Subperiod Tests 3 Multivariate P Period 10 x Estimated Contrast Mean F Less Than (1) (2) (3) (4) (5) 24 Weeks Centered 1.08 1.85 1.78 .05 2.35 F(5,19)= .66 .66 On December 31 12 Weeks Before -.96 1.67 -.30 -2.41 1.78 F(5,7)= .55 .73 December 31 12 Weeks After 3.13 2.04 3.86 2.50 2.92 F(5,7)=1.05 .46 December 31 VEHJ‘o no . :- 101 FOOTNOTE TO CHAPTER VII 1Transformation of the observed means and the transformation matrix p are discussed in Appendix A. “'9" CHAPTER VIII CONCLUSION Summary This study is designed to investigate whether both the historical cost income and current operating profit forecast error values reflect information pertinent to assessing firms' equilibrium expected returns. Specifically, the study tests the hypothesis that the expected return differences between portfolios conditioned on various realizations of these two forecast error variables are zero. Since pre-experimental equivalence is assumed to be attained by exploiting the properties of the capital asset pricing model, detection of significant differences would imply that the forecast error realizations reflect information. This study differs from previous replacement cost research studies in its formulation of expectations. Since these other studies focused on the initial disclosure year, prior replacement cost data were not available to formulate expectations. As a consequence, they were limited to the use of historical cost data and Value Line estimates of replacement cost in deriving expectations. In contrast, forecast error values for both variables are calculated using the martingale model to obtain proxies for investors' expectations. Since only one pre-1977 COP value exists, the drift factor is assumed to be zero in calculating the 1977 COP expectation. However, theorists have argued that the current period's COP is the best estimate of the following period's COP. Conditioning portfolio returns on various realizations of both the historical cost income and current operating profit forecast error 102 1 L 103 variables necessitates an examination of the realationship between them. since it is crucial to appropriate design selection and interpretation of test results. In this regard, two extreme outcomes are possible: (1) the variables are independent or (2) the variables are perfectly correlated and thus perfect substitutes for each other. Neither extreme is inferred from the sample data. However, a positive systematic re- lationship is detected. Accordingly, a one-factor design is deemed appropriate. Regarding the one-factor design, the inferred variable relation- ship is consistent with the formulation of'g_pripri_contrasts that test the following two hypotheses: (1) the COP signals do not reflect infor- mation beyond that of the HCI signals and (2) the HCI signals do not reflect information. Test results Of the first hypothesis are considered important with respect to the Securities and Exchange Commission's con- tention that replacement cost disclosures provide information useful to investors which is not otherwise obtainable. Testing of the secohd hypothesis attempts to replicate the results of previous research which has shown that the HCI forecast error variable reflects information beyond that available at time t. The test period consists of the fifty work weeks subsequent to the March 31, 1977 portfolio formation date. Tests are conducted on weekly returns pertaining to the full fifty week period as well as three subperiods. The three subperiods comprise: (1) twenty-four weeks centered at December 31, 1977, (2) twelve weeks before December 31, and (3) twelve weeks after December 31, respectively. This segmen- tation of the overall period is made, since all sample firms have December 31 fiscal year ends and the existence Of uncertainty with respect to the event date. 104 Empirical results of this study constitute evidence consistent with the hypothesis that the Securities and Exchange Commission's man- dated replacement cost disclosures provide no information to the market. In this regard, one might question this study's conclusion since mispe- cification of the expectation models, choice of the time period and sample size, together with the use of ex_ppst_data to calculate ex_gpte' expectations may have prevented detection of an effect. It is suggested, however, that the results in conjunction with those of other studies, each of which have utilized diverse procedures, contribute to the body of evidence necessary to the formulation of a compelling case. Methodological procedures employed are largely adaptations of Gonedes' work. Several extensions of his methodology are introduced and believed to be of value. In testing the information content Of an accounting random variable, the difference in total returns metric intro- duced by Gonedes has been compared to the residual return metric. This comparison has been made within the framework of maximizing the power of all tests. .Maximization is accomplished by employing the statistical procedure that tests jointly the contrasts of interest to the researcher, g_prigri. To provide a setting for formulating these contrasts when investigating the information content of more than one realization of an accounting variable, the general linear model from extant statistical theory has been introduced. This model has been shown to be equivalent to the two-factor, zero-beta model and its reparameterization results in the 3_prigri contrasts of interest. The simplest contrast is the difference between two returns. As a consequence, it has been shown within a one-asset, one-period concep- tual setting that the econometric properties of both metric approaches are identical. Thus, this choice between the two within an empirical 105 setting should only be contingent upon the assumption that the expected returns of two or more securities (portfolios) are equal if their syste- matic risk parameters are equal. In the market based research context, there is yet a further consequence of using g_pripri contrasts. Forming the difference in total returns metric, this procedure has been shown to eliminate the necessity of employing an unconditional return to ex- tinguish variability attributable to the economy-wide factor. Corres- pondingly, it has been shown that this elimination increases the power of the test. The most important extension is the explicit consideration given to the relationships between accounting variables. This is essential in studies employing more than one accounting variable, since appropriate design choice and interpretation of results is dependent on their relationships. If, in fact, variables are independent, then an n-factor crossed design is warranted. This design allows explicit testing of the interaction (joint) effects.( Furthermore, if the results are consistent with a main-class model (i.e., interactions effects are inferred to be zero), one could then estimate the contrasted effects each accounting variable has on expected returns. 0n the other hand, if systematic relationships exist, although not perfect ones, then a one-factor design is employed. This in turn, requires a different interpretation of test results. Recommendations For Future Research The empirical phase of this study is subject to one primary line of criticism. The test periods include data from only one year and hence provide a small number of observations. This is particularly true regarding the subperiods tested. Furthermore, demands for an 106 ‘ increased sample size requirement are intensified by the relatively large number of parameters estimated and hypotheses tested. The reason for using just one year in examining the information content of the two forecasts error variables is attributable to the availability of data at the time the study was initiated. Of course, one possible remedy that can be incorporated in future studies is to increase the number of years included in the test periods. However, another possibility is the use of daily rather than weekly returns. The methodological refinements which have been presented are recommended for all studies investigating the information content of a random accounting variable or a vector of such variables. ‘. E “T" —'—'——p——_ "o _ APPENDICES d: m. —‘ . with. -l ‘- . APPENDIX A COMPARATIVE ANALYSIS, "CLASSICAL" CONTROL GROUPS AND THE REPARAMATERIZATION PROCESS Comparative analysis is used to measure difference(s) in the effects of two or more treatments. If the researcher is concerned with the effect of only one treatment, then a control group (i.e., a group receiving no treatment) would be formed. The control would be considered a “treatment" enabling the researcher to make the comparative analysis. In general, however, the researcher is concerned with analyzing more than one treatment. . Analyzing more than one treatment, the presentation will be made in the context of market based research. In this context, treatments are considered to be different realizations of the information random vari- able 5. The metric to be employed is the difference in total returns discussed in Chapter III. Recall, that this metric was designed to elimi- nate from the total return metric any variability attributable to Rm. In the two treatment case, let 61 and pi be the two realizations of 5. Testing the null hypothesis given by (3-3) that the two expected conditional returns are equal, it was demonstrated in Chapter III that the appropriate test, (3-20), employed the difference in total returns metric. Specifically, the metric's standard deviation and an estimate of its expected value form the test denominator and numerator, respec- tively. The use of controls, which are in this context unconditional returns on portfolios, will be reviewed next. Employment of these controls in the two realization case would result in forming differences by matching,against each conditional return 107 108 an unconditional return and then taking the difference of the differences. This is redundant! In symbols, (A-1) [(Rz/ei) - RC1] - [(Rq/ej) - R621 = [((1 - BZ)RO + 825m + y, + £2) - ((1 - ecl)Ro + Selim + ic1)1 - 1((1 - sq)Ro + equ + vj + iq) - ((1 - scleo + scsz + £6211 = [((1 - leRo + Bsz + v, + 52) - ((1 - sq)Ro + equ + vj + fiq)] - [((1 - 6C1)Ro + Satin + ficl) - ((1 - sc2)Ro + chRm + ic2)1 ~ [(4, - Yj) + (a, - fiq)1 + (ac1 - 162). As a consequence, this usage results in the additional error term (Eel - ficz) in contrast to the case where controls are not employed. The expected value of the difference in total returns metric is y, - yj with or without the employment of control portfolios. However, the researcher pays a price for this redundancy because the variance with controls is greater than the variance without controls. In presen- ting the proof, it will be assumed that all error variances are equal and depicted by 05' Furthermore, all error covariances are assumed equal and will be-expressed as puci, where pH is the correlation coef— ficient. In the non-control approach the variance is given by ~ ~ - 2 - (A 2) Var(vi - Yj + “z - uq) - Zou(1 9“). Whereas, under the control approach the variance is given by - - ~ ~ _ 2 (A-3) var(Yi ' Yj + “Z ’ uq + “C1 ' “C2) - 40H(1 ' Du). 109 The variance in (A-3) is twice as large as the variance in (A-2) for all values of pH except, Of course, the value of one. In that case both variances are zero. Examining cases with more than two realizations, for example three realizations, involves an additional complication. Letting 9k be the third realization, the null and alternative hypotheses would then be (A-4) H0: E(Rz/ei) = E(Rq/ej) = E(Rp/ek), and (A-4 ) H1: H0 is false. To test for differences in these expected values jointly and thereby control Type I error rates, procedures such as ANOVA or MANOVA would be employed. However, as presently formulated, the metric employed would be the total return metric. This metric contains "Undesired" variability attributable to the factor Rm and therefore reduces the power of the test. Gonedes [1978] solVes this problem by matching against each conditional portfolio return a control portfolio. By matching, he creates the difference in total returns metric. In summary, his pro- cedure accomplishes both of the following objectives: (1) eliminates variability attributable to Rm by employing the difference in total returns metric and (2) controls the error rate by comparing differences using a joint test. Both of these objectives increases the power of the omnibus testing procedure. This paper proposes another approach to accomplish both of these goals. This approach is presented within the framework of the GLRM associated with the ANOVA and MANOVA testing techniques. Using this model, one formulates a priori contrasts through the reparameterization process. (See Chapter VI pages 69 and 82). The entire process involves 110 three steps. Although step 1 and part of step 2 are presented in Chapter V1, for continuity of thought all steps will be presented below: (A-5) (A-6) Step 1 - Formulation of the GLRM Model 1 is the GLRM in matrix form and is given by r=fi+s where y. = the vector of observed total return means, A = the design matrix, .5 = the vector Of parameters, and E: = the random error vector which is assumed to be distributed N(0.2). Step 2 - Reparameterization of the GLRM The purpose of the reparameterization process is to reduce the number of unknown parameters so that the new set ofparameters equals the number Of equations in the model. This new set of parameters are a linear combination of the old set and they allow for a unique solution. Furthermore, by selecting the appropriate reparameterization matrix L, the new parameters will represent the e_priori contrast of interest to the researcher. Reparameterization is achieved by factoring the design matrix into the product Of two matrices (i.e., A = KL), where L is the row basis of A and K is the column basis. The derivation of model 2 is given by “ 3=A€+e (KL); + g: 111 = K(L§) +Ig = K3 + 2:, where p.= the vector of new parameters, L = (K'K)'1KA, and K = ALWLL')"1 Let L .3 g. = (x. - (gm. - Kg). 0 III then differentiating yields 30/3o = -2Ky. + 2101(3). Equating 30/36 to the null vector yields the normal equations used for the least squares estimation. They are given as K'Kp_= rp., or §= (K'K)"Ky_.. where E(§) = p_, and Var(§) = (K'K)-1£ . Step 3 - Orthonormalization of the Reparameterized GLRM Let p_= T'p_, where KiK = T'T and T is called the ChOlesky factor. It results from the triangular decomposition of K'K. . Furthermore, if K is column-wise orthogonalL and the design is 112 balanced (equal cell size), then the Cholesky is a diagonal matrix (i.e. T = T'). The derivation of model 3 is given by K¢+e. (A-7) y. KUE'TEE' K[(T')TIT']g+g. [K(T')'11[T'gj +‘e, P11: + e. where K is orthonormalized by the inverse of the transpose of the Cholesky factor. The resulting matrix P is orthonormal, i.e. PP' = I (see figure A-I). Furthermore, y, can be transformed as (A-8) ' P'y. - P'Pp + P'E. * = V + e. . where_ E°* =N(O,P'2P). Therefore, where up=g=Ig. Thus, ‘§=Ou?[ The transformation matrix P transforms the Observed total return means. Moreover, this transformation process will yield a new set of observed means which are a linear combination of the old set. For each combination, except the first, the coefficients will add to zero. This implies that each of the components Of the new set which are associated 113 Figure A-l The K, (T')'1, and P Matrices For The One-Factor Case K (T')‘1 P = m)1 HHO—‘I—‘HH 1776 00000 1/76' 1//6' 1//6' 1//6' 1/76' 1/76' 2/3 2/3 -1/3 -1/3 -1/3 -1/3 7372 0000 7373 7573 -/376 -/§76 -/376 -/376 0 1/2 1/2 -1/2 -1/2 OOOHOO 1/2 1/2 -1/2 -1/2‘ 1/2 -1/2 0 O 1/2 -1/2 0000 1/2 -1/2 ooooo ..5 0000 7272 -/272 114 with the g_prjpri contrasts parameters of direct interest to the resear- cher have been transformed into differences in total mean returns. Whereas, the combination forming the first component has equal coeffi- cients and its associated contrast parameter can be viewed as the expected market return. Estimating and testing this parameter is not a part of the researcher's omnibus hypothesis. Before concluding, a formal consolidated analysis of the issues will be presented. The statistical test employed is Roy's largest-root criterion, which in this case equals Hotelling's T2 (Bock [1975, pp. 150- 'k 152]). This test criterion can be expressed as the largest root A of L (A-9) |sh - Ise | = 0, where Sh = the sum of squares and cross products matrix (SSCP) for the hypotheSTs based on N independent observations, Se = the SSCP matrix for the error, and 1 = the Lagrangian multiplier. These two matrices can be viewed as the multivariate counterparts to the univariate ANOVA sum of squares between and sum of squares within parti- tions of the sum of squares total. Their expected values are (2 + N 3_3f) and (N - 1):, respectively (Bock [1975, p. 458]). In addition, the vector 'Of independent variables y, will be expressed as residual returns. There- fore, I_is the vectOr of expected conditional residual returns and 2 is the SSCP error matrix. More formally, (A-10) 1,- II p.) + ('1 do to w 3 Q. (A-11) 1/N( 1 II M 2 ,__, . 1 V II CF ll lo—3 + [In 115 where ‘yj = the (r x 1) vector of dependent variables for the ith subject, i = 1,2,...,n, where each dependent variable (measure) is (e/Oj), j = 1,2,...,r, .3 = the (r x 1) vector of expected values, where each expected value is yj, j = 1,...,r, .51 = the (r x 1) vector of sampling errors distributed N(O,Z). Substituting the expected SSCP matrices results in ((1-12) H: + N 33') + 1[(N -1)21|= 0 The largest root 1* is equal to T2. Thus, statistically, this is equi- valent to the approaCh given by (6-3) which was employed by Gonedes [1978]. Using the GLRM approach, recall that the reparameterization process results in the following transformations (A-13) P'_]_I_. = P': + P'E. = p + P'e. , where P'I_equals p_. Furthermore, * (A-14) Sh P'(z + N 13'” and (N - 1)P'2P , * (A-IS) Se where P'P = I. The impact that this transformation has on the test criterion is explored below. (A-16) O IIP'(2 + N 31%] - 11[(N - 1)P'£P1| |P||[P'(z + N 11')P] - 7(1[(N -1)P'2P]||P'| (ppm; + N 33')PP' - 11[(N -1)PP'EPP']| |[I(2 + N 3.1')I] - 11[(N - 1)IzI]l 116 = II): + 1111‘] - 111(1) - 11211. * * Therefore, T1 = A , since the transformation does not change the deter- minant. The discussion to this point has assumed the use of the residual return metric. For comparison let x, be a vector of observed total con- ditional return means and 9_be the vector of expected conditional total returns. More formally, each of the respective vector components is 1 given by _ (A-17) v.3. = (1 - BZ)RO + 6sz + vj + u-j i ) ( . + :L + . + .. .xJ n J) (YJ u 3 (xj + Yj) + ("'j + u.j) o + o. . = ,0.., OJ H J for j 1 r and in vector notation (A-18) V. =g+g. , where g, T N(0,Zu). After transformation, both matrices P'gflng and P'ZuP differ from matrices P'IHIfP and P‘zP, respectively, in only the first row and column. By partioning each matrix into upper (1 x 1) and lower [(r - 1) x (r - 1)] matrices, then (P E-E-PTr-l is identical to (P I_2_P)r_1 and (P zuP)r_1 is identical to (P'ZP)r_1. This is true because the transformation matrix P transforms the last (r-l) compOnents in both g_and I_and their correspon- ding error terms 3, and 5. into return differences. TO clarify, recall that the only difference between the residual and total return metrics is 117 the economy-wide component contained in the latter. The transformation matrix, however, removes this component in the (r - 1) partitioned matrices since the coefficients of each of the (r - 1) transformations equal zero. Furthermore, either of the partitioned matrices (P'9_pr)r_1 or (P'g 3'P)r_1 can be tested directly and both represent the omnibus hypothesis of interest. Therefore, in cases involving more than two realizations the residual return metric approach is equivalent to the difference in total returns metric approach. The use of control portfolios is thought to have a specific function in market based research. This function is to reduce varia- bility by creating a difference in total returns metric. It has been shown in cases involving more than one realization of 5 that this is un- necessary. Therefore, in market based research, as in general, control groups enable the researcher to determine the absolute effects of each treatment, but are entirely unnecessary for determining difference(s) in effects between treatments. “I; - .a.n.-siu-_Fr 118 FOOTNOTE T0 APPENDIX A 1Two vectors are orthogonal if their inner product equals zero. SIC Code 1000 1520 1600 2020 2046 2065 2085 2086 2200 2270 3570 3600 3610 3622 3630 3662 3699 119 APPENDIX B INDUSTRIES OF SAMPLE FIRMS Industry Metal Mining General Building Contractors Construction - Not Building Construction Dairy Products 1 Wet Corn Milling Candy and Other Confectionery Distilled Rectif Blend Beverage Bottled - Canned Soft Drinks Textile Mill Products Floor Covering Mills Apparel and Other Finished Products Lumber and Wood Products Paper and Allied Products Convert Paper-Paperbd Pd. Nec. Paperboard Containers - Boxes Books - Publishing and Printing Commercial Printing Chemicals and Allied Products Plastic Matr. and Synthetic Resin Drugs Perfumes Cosmetics Toil Prep. Paints - Varnishes - Lacquers Industrial Organic Chemicals Misc. Chemical Products Petroleum Refining Rubber and Misc. Plastics Products Cement Hydraulic Concrete Gypsum and Plaster Abrasive Asbestos and Misc. Mis Blast Furnaces and Steel Works Second Smelt - Refin Nonfer Mt. Rolling and Draw Nonfer Metal Engines and Turbines Construction Machinery and Equipment Metal Working Machinery and Equipment Special Industry Machinery General Industrial Machinery and Equipment Office Computing and Accounting Machinery Elec and Electr Machinery Equipment and Supplies Elec. Transmission and Distr. Equipment Industrial Controls Household Appliances Radio - T.V. Transmitting Equipment - AP Electrical Machinery and Equipment NEC Firm Frequency Number r—II—u—iI—ANI—nI—u—tNo—IHHi—IHHI—IHwNwNHHHHHmHHHNwHHHwNHHHHHHH SIC Code 3714 3721 4927 4931 5661 6199 6790 7011 7500 7810 9997 120 Industry Motor Vehicle Parts - Accessories Aircraft Aircraft Parts and Aux. Equip. Railroad Equipment Photographic Equipment and Supplies Railroads - Line Haul Operating Air - Transportation - Certified Telephone Communication . Radio - T.V..Broadcasters Electric Services Natural Gas Tramsmis. - Distr. Natural Gas Distribution Electric and Other Serv. Comb. Wholesale - Groceries and Related Products Wholesale - Nondurable Goods NEC Retail - Shoe Stores Finance - Services Miscellaneous Investing Hotels - Motels _ Service - Auto Repair and Service Service - Motion Picture Products 1Conglomerates Total Firm Frequency Number NNHNHHHHHmHAHmNNNo-JHHHHH 108 '“fi . .'- ".- I I LIST OF REFERENCES LIST OF REFERENCES Ball, Ray. "Changes in Accounting Techniques and Stock Prices," Empirical Research in Accounting: Selected Studies, 1972, SUppTementTto Journal of Accounting Research 10”T1972): 1.-38. Ball, Ray, Baruch, Lev, and Watts, Ross. "Income Variation and Balance Sheet Composition," Journal of Accounting Research (Spring 1976 : 1-9. Ball, Ray and Brown, Phillip. "An Empirical Evaluation of Accounting Income Numbers.“ Journal of Accounting Research 6 (Autumn 1968): 159-177. Ball, Ray and Watts, Ross. "Some Time-Series Properties of Accounting Income." Journal of Finance 27 (June 1972): 300-323. Beaver, William H. 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