Citation |

- Permanent Link:
- http://ufdc.ufl.edu/UF00102818/00001
## Material Information- Title:
- A study of the systematic component of risk in common stocks
- Creator:
- Goldenberg, David Harold, 1949-
- Copyright Date:
- 1981
- Language:
- English
## Subjects- Subjects / Keywords:
- Assets ( jstor )
Cash flow ( jstor ) Market prices ( jstor ) Monopoly power ( jstor ) Net income ( jstor ) Prices ( jstor ) Regression coefficients ( jstor ) Statistical discrepancies ( jstor ) Subject terms ( jstor ) Systematic risks ( jstor )
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- All applicable rights reserved by the source institution and holding location.
- Resource Identifier:
- 07883262 ( oclc )
ABS1742 ( ltuf ) 0028130688 ( ALEPH )
## UFDC Membership |

Downloads |

## This item has the following downloads: |

Full Text |

A STUDY OF THE SYSTEMATIC COMPONENT OF RISK IN COMMON STOCKS BY David Harold Goldenberg A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1981 ACKNOWLEDGEMENTS I wish to thank Professor Fred D. Arditti for intro- ducing me to the subtleties in the content and methodology of finance through his invaluable lectures and personal instruction, for providing characteristically acute insights into the nature of systematic risk, and for his constant encouragement. I am grateful to Professor G. S. Maddala for providing valuable insights into the econometric methodology appropriate to the modelling and estimation of systematic risk and for his very helpful assistance during the execu- tion of the study. Helpful discussions with Professor Raymond Chiang and Professor Richard Cohn are greatly appreciated. TABLE OF CONTENTS Page ACKNOWLEDGEMENTS........................................... ii LIST OF TABLES...... ..... .. ............................ vi ABSTRACT ............................................... x INTRODUCTION............... .....................* ....... 1 CHAPTER I REVIEW OF THE LITERATURE................ 5 Introduction...................... ...... 5 1. Ad Hoc Studies Using Accounting Numbers ............................. 6 Ball and Brown..................... 6 Beaver, Kettler and Scholes......... 11 Gonedes........... .. ........ .... 22 2. Theoretical Basis Study.............. 25 3. Barr-Rosenberg and Associates' Work. 28 Barr-Rosenberg and McKibben......... 28 Barr-Rosenberg and Marathe.......... 32 Barr-Rosenberg and Marathe.......... 40 4. Study of the Effect of Financial Leverage............... ..... ........ 41 5. Studies of the Effect of Market Power................................ 48 Sullivan........................... 48 Thomadakis......................... 51 Sullivan............................ 54 6. Studies of the Effect of Operating Leverage............................. 56 Lev................................. 56 Rubinstein......................... 62 Subrahmanyam and Thomadakis......... 68 iii Page CHAPTER II MICROECONOMIC FACTORS AFFECTING EQUITY BETAS.............. ..................... 70 Introduction............................ 70 1. Financial Leverage.................. 71 2. Volatility of Operating Earnings.... 74 3. Growth............................... 78 Assumptions....................... 82 Single Period Betas in a Multi- period setting................. 92 Lemma........................ 96 Proof ......................... 96 Duration and Asset Betas............ 100 4. Monopoly Power and the Labor-Capital Ratio ............................... 101 Sources of Uncertainty............. 102 Uncertainty in the price of output.. ................. 102 Uncertainty in the wage rate.. 103 Relationship between Demand Uncer- tainty and Uncertainty in the Wage Rate..................... 104 Competitive Equilibrium Risk Deter- mination ...................... 106 Derivation of the Systematic Risk of the Purely Competitive Firm 108 Systematic Risk and Monopoly Power. 112 Optimal Valuation and Beta for the Monopolistic Firm............. 115 5. The Model of Systematic Risk and Hypotheses to be Tested.............. 116 CHAPTER III ESTIMATION TECHNIQUES AND EMPIRICAL RESULTS... ............................. 119 1. Estimation Techniques ............... 119 Fixed Effects....................... 119 Random Effects..................... 121 Prior Likelihood Estimation........ 121 2. Empirical Results with Five Descrip- tors ................................ 125 Fixed Effects Estimation........... 125 Page Definitions and data sources for descriptors.......... 125 Parameter Estimates........... 127 Beta Estimates................ 129 GLS Estimation..................... 133 Prior Likelihood Estimation........ 134 Prediction......................... 135 Out of sample prediction...... 135 Within sample prediction...... 138 3. Empirical Results with Twenty-nine Descriptor Data Set................. 141 Fixed Effects Estimation............ 141 Names of descriptors.......... 141 Classification of descriptors. 143 Descriptor definitions........ 144 Parameter estimates (a assumed constant) ................ 149 Parameter estimates (a variable) ............... 149 Beta estimates................ 149 GLS Estimation..................... 155 Prior Likelihood Estimation......... 156 Prediction......................... 156 4. Comparison of the Predictive Perfor- mance of the LSDV29 and the LSDV Betas ............................... 159 SUMMARY AND CONCLUSIONS OF THEORETICAL AND EMPIRICAL RESULTS................................................ 160 APPENDIX 1 RELATIVE MAGNITUDES OF THE FIRM AND LABOR BETAS ............................. 164 APPENDIX 2 DERIVATION OF THE MONOPOLY BETA ......... 167 REFERENCES...... ........ ............................... 168 BIOGRAPHICAL SKETCH.................................... 171 LIST OF TABLES Table Page 1.1 Coefficients of Correlation between Various Measures of the Proportion of Variability in a Firm's Income That Is Due to Market Effects Variables Not Standardized........... 8 1.2 Coefficients of Correlation between Various Measures of the Covariance between a Firm's Index and a Market Index of Income Variables Standardized................................ .10 1.3 Association between Market Determined Risk Measure in Period One (1947-56) Versus Period Two (1957-65) ......................... 15 1.4 Contemporaneous Association between Market Determined Measure of Risk and Seven Account- ing Risk Measures............................ 20 1.5 Analysis of Forecast Errors.................. 23 1.6 Correlation Coefficients, R, between Estimates from Market Model (M) and Four Accounting Number Models (A)................. 26 1.7 Correlation Analysis......................... 29 1.8 Prediction Rules for Systematic Risk Based on Fundamental Descriptors and Industry Groups.. 35 1.9 Prediction Rules for Systematic Risk Based on Fundamental Descriptors Including Market Variability Descriptors...................... 37 1.10 Unbiased Estimates of the Performance of Alternative Prediction Rules in the Histor- ical Period (Predicted Variance as a Multiple of the Variance Predicted by a Widely Util- ized Prediction Rule) ........................ 39 1.11 Summary Results over 304 Firms for Levered and Unlevered Alphas and Betas............... 45 Table Page 1.12 Market Adjustment Factor Regressions over Alternative Periods.......................... 47 1.13 Mean and Standard Deviation of Industry Betas 49 1.14 Industry Concentration and Future Monopoly Power......................................... 53 1.15 Monthly Betas, Leveraged and Unleveraged, Regressed on Market Power and Control Variables..................................... 55 1.16 Estimates of Average Variable Cost Per Unit.. 61 1.17 Regression Estimates for Systematic Risk on Average Variable Costs Per Unit.............. 63 3.1 Estimated Descriptor Coefficients Used in Generating Betas for the Five Descriptor Data Set.......................................... 128 3.2 Yearly Means of Fixed Effects, Historical, and Varicek Betas for the Five Descriptor Data Set..................................... 130 3.3 Variances of Fixed Effects, Historical, and Vasicek Betas Based on the Five Descriptor Data Set..................................... 131 3.4 Estimated Descriptor Coefficients for Constant Alpha Model for the Five Descriptor Data Set............ ......................... 132 3.5 Estimated Descriptor Coefficients Used in Generating Alphas and Betas for Out of Sample Prediction............................ 136 3.6 Correlation Matrix of Descriptors and Descriptors Times the Market for the Five Descriptor Data Set.......................... 137 3.7 Out of Sample Prediction Results for the Five Descriptor Data Set.......................... 139 3.8 Within Sample Prediction Results for the Five Descriptor Data Set......................... 140 3.9 Estimated Descriptor Coefficients Used in Generating Betas for the Twenty-nine Descrip- tor Data Set (Alpha Constrained to Be Con- stant) ....................................... 150 vii 3.9 Estimated Descriptor Coefficients Used in Generating Betas for the Twenty-nine Descriptor Data Set (Alpha Constrained to Be Constant) .......................... ...... 150 3.10 Estimated Descriptor Coefficients Used in Generating Betas for the Twenty-nine Descrip- tor Data Set (Alpha Allowed to Vary According to Equation 3.2).............................. 151 3.11 Estimator Descriptor Coefficients Used in Generating Alphas for the Twenty-nine Descriptor Data Set.......................... 152 3.12 Yearly Means of Fixed Effects Five and Twenty-nine Descriptors and Historical Betas. 153 3.13 Variances of Fixed Effects Five and Twenty- nine Descriptors and Historical Betas........ 154 3.14 Out of Sample Prediction Results for the Twenty-nine Descriptor Data Set.............. 157 3.15 Within Sample Prediction Results for the Twenty-nine Descriptor Data Set.............. 158 viii Page Table Authors Table Page Thomadakis............................... 1.14 .......... 53 Sullivan............................... 1.15 .......... 55 Lev .................................... 1.16 .......... 61 1.17 .......... 63 Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy A STUDY OF THE SYSTEMATIC COMPONENT OF RISK IN COMMON STOCKS By David Harold Goldenberg June 1981 Chairman: Dr. G. S. Maddala Cochairman: Dr. F. D. Arditti Major Department: Finance, Insurance, and Real Estate The theoretical basis for the inclusion of various microeconomic factors as determinants of systematic risk is examined. This set includes financial leverage, variability of operating earnings, and growth in several senses. A recent model incorporating the firm's behavior in its input and output markets is generalized by including risky human capital in the market portfolio. The condition on the covariability between the sources of uncertainty in the model under which beta will be positively related to the labor capital ratio and negatively related to monopoly power is explicated. Much work has been done on this problem by Barr-Rosenberg and associates, but little or no theoretical justification was offered for the choice of variables included in their regressions. The study also corrects some of the deficiencies in the econometric methods employed in those studies. The empirical results on non-stationarity of the coef- ficients of the market model are taken into account by allowing them to vary by firm and time period as a linear function of the set of descriptors plus a firm specific error term that does not vary with time. The fixed effects or least-squares with dummy variables estimation technique is applied. The coefficients of the growth rate in assets and the Lerner Index of monopoly power are negative. The labor-capital ratio has a positive sign. These results conform to the theory. Financial leverage and variability of earnings have negative signs. The former result may be due to the use of book values for debt rather than the appropriate market values. The GLS or random effects estimates do not differ significantly from the fixed effects estimates. The prior likelihood estimates designed to provide best linear unbiased estimates of the firm specific effects also do not differ significantly from the fixed effects estimates. The criterion of mean-square error was employed in comparing classical betas to the descriptor betas as predictors of returns. Predicted returns, conditional upon the market return in the prediction year, were generated from information from the previous set of years for both types of betas. The mean-square error of the naive predic- tor obtained by setting alpha equal to zero and beta equal to unity was also computed. The order of performance was naive, classical, then descriptors. When a within sample prediction was performed by using all the available data to estimate the coefficients used in generating betas, the classical betas obtained by including the prediction year's data in the usual regression never outperformed the naive predictor while the five descriptor based betas did so in two years out of four--1974 and 1975. This suggests that the descriptor based betas may be useful as predictors over periods in which structural changes in the coefficients of the market model take place. The entire set of procedures was repeated on a set of descriptors chosen in the Barr-Rosenberg manner. The signs of the estimated coefficients made no apparent sense. As predictors of returns for the out of sample prediction, they did not outperform the naive. They were outperformed by the five descriptor betas in two years and outperformed them in the other two years. For the within sample predic- tion they were always better than the naive, and they were better than the five descriptor set in three years out of four--1973, 1975 and 1976. Only in 1974 did the five descriptor set provide better predictors. INTRODUCTION The total risk of a firm's equity securities, identi- fied as the variance of the rate of return of those securi- ties over a given period, can be decomposed into two compo- nents. One component is termed systematic: it is the sensitivity of the rate of return on the firm's common stock to the rate of return on the market portfolio. This compo- nent is the market related risk that cannot be diversified away through the process of portfolio formation. Tradition- ally a measure of the non-diversifiable risk component is given by the slope coefficient, assumed constant over the estimating period, in the market model's historical regres- sion of the stock's rate of return on the rate of return of a proxy for the market portfolio of all risky securities. This simple model of the stochastic process generating security returns indicates the second component of the risk of a firm's equity securities. The market model indicates that a portion of the variance of the rate of return cannot be accounted for by movements in the rate of return of the market portfolio. This component of the total risk of the security is specific to the firm in question. However, it is possible to diversify away this risk component by com- bining the security with others in portfolios. In fact, the Capital Asset Pricing Model tells us that equilibrium prices and rates of return on a firm's securities are linear functions solely of the first component of total risk: the systematic component measured by the beta coefficient. It is clearly of great interest to estimate and predict the systematic risk component of a firm's common stock securi- ties. This study will concern itself only with the estima- tion and consequent prediction of this systematic component. While the beta coefficient as defined by the CAPM identifies the systematic component of risk as that component relevant for equilibrium pricing of a firm's risky securi- ties, it provides little information concerning the sources of such systematic risk. The goal is to relate the firm's unobservable beta to the firm's characteristics and to micro-economic variables. Traditionally, financial leverage and volatility of earnings have been considered as micro-economic determinants of systematic risk. Attention has recently focused upon "growth" and the firm's operations in its input and output markets as factors bearing upon the beta coefficient. The present study includes the traditional variables as well as the labor-capital ratio and monopoly power in an attempt to incorporate these factors. The arguments for all of the above variables will be considered in detail as the basis for the inclusion of a set of factors that can be theoret- ically justified as determinants of systematic risk. In the course of this analysis the recent arguments for the labor- capital ratio and monopoly power are generalized. 3 The second major purpose of the study is to correctly apply the available econometric techniques that appear in the literature on pooling cross-sectional and time series data to the problem of estimating systematic risk as a linear function of the set of descriptors developed above plus a firm specific effect. This model has appeared in the early literature on accounting numbers in the form of Beaver, Kettler, and Scholes's instrumental variables estimation procedure for estimating systematic risk. That study did not incorporate firm specific effects. Such effects were added to the model as the residual component of beta that could not be explained by the included set of descriptors. This was done by Barr-Rosenberg and associates. The appro- priate estimation techniques were not applied. Accordingly, the fixed effects and the random effects specification of this model are considered. The model is estimated under the fixed effects specification. Then, the potential gain from applying the generalized least-squares estimation as warranted by the random effects specification is measured in an a priori manner. In principle, with known variances of the firm specific effects, the prior likelihood estimation procedure leads to best linear unbiased predictors of the firm specific effects. The additional explanatory power provided by these estimates is computed by considering the magnitudes of the adjustment factors to be used to derive the prior likelihood estimates. The criterion for usefulness of the betas generated by the procedures outlined above is the mean-square error in the prediction of returns generated by those estimates. The mean-square-error of the fixed effects betas is compared to that of the usual historical beta and that of the naive predictor with alpha set equal to zero and beta set equal to unity. This is done both for out of sample and within sample prediction. Finally, the procedures described are carried out using a set of 29 Barr-Rosenberg type descriptors. The results are compared to those for the five theoretically justified descriptors. CHAPTER I REVIEW OF THE LITERATURE 1. Introduction Those studies that bear directly on the present one are considered here. The review is divided into several sections. (1) Ad Hoc Studies Using Accounting Numbers: Ball and Brown [19691; Beaver, Kettler and Scholes [1970]; and Gonedes [1973]. (2) Study Attempting to Provide a Theoretical Basis for the Use of Accounting Numbers as Proxies for Systematic Risk: Pettit and Westerfield [1972]. (3) Barr-Rosenberg and Associates' Work: Barr-Rosenberg and McKibben [1973], Barr-Rosenberg and Marathe [1975], and Barr-Rosenberg and Marathe [1979]. (4) Study of the Effect of Financial Leverage on Beta: Hamada [1972]. (5) Studies of the Effect of Market Power on Beta: Sullivan [1977], Thomadakis [1977], Sullivan [1978], and Subrahmanyam and Thomadakis [1980]. (6) Studies of the effect of Operating Leverage on Beta: Lev [1974], Rubinstein [1973], and Subrahmanyam and Thomadakis [1980]. The studies concerning the growth variable are consid- ered in Chapter II, where the relationship is explained in the appropriate multiperiod framework. 6 1. Ad Hoc Studies Using Accounting Numbers Ball and Brown The Ball and Brown [1969] study has as its goal the use of "income numbers" as predictors of systematic risk. Their "income numbers" are simple regression coefficients in regressions of accounting income of a given firm in a given year on a market index of accounting income. The income variable was alternatively taken to be operating income, net income, and earnings per share. Initially, two regression models were postulated relating income in levels between the firm and the market and then income in first differences between the firm and the market. Three regressions were run for each of the three income definitions. The simple R2 were computed for these time series regressions for each firm i and then these were correlated with the R2 from the regression of the firm's rate of return on the rate of return on the market. Product correlation and rank-order correlation coefficients were calculated. The regression models employed were as follows: it = a + a2iM Uit income, in levels (1.1) AI = a' + a'.M + U' income, in first (1.2) differences I income/market (1.3) it a + a''M + U' value of equity in it it t it levels I income/market (1.4) A = a''' + a"'''M + U''' value of equity in it it first differences PR. = b + b L +V. (1.5) m bli 2i m + Vim (1.5) where PR. is the price relative for the common stock of im firm i in month m and L is a proxy for the rate of return on the market portfolio in month m. The coefficient b2i is presumably an estimate of the firm's equity beta as this regression is essentially the market model. The other relevant variables are defined as follows: it: accounting income of firm i in year t M. : a market index of accounting income. It Ball and Brown take this market index as an average that, for a given firm i, excludes that firm; being an average over the remaining N-i firms. Why they do this is not obvious as a market index, according to Capital Asset Pricing Theory, contains all firms in the market. The object of the correlation analysis is to determine to what extent estimates of systematic risk from the market model are correlated with the accounting income response coefficients. The regressions were run for 261 firms over the period 1946-1966. Selected results appear in Table 1.1. Ball and Brown conclude from these results that "better" results are obtained when the regressions are run in first differences [see Table 1.1, columns (3), (5) and (7)]. The next set of regressions attempted to control for the size of firm and differences in their accounting prac- tices. To do so, all variables were standardized by dividing Table 1.1 Coefficients of Correlation between Various Measures of the Proportion of Variability in a Firm's Income That Is Due to Market Effects Variables Not Standardized (2) (3) (4) (5) (6) (7) Product Moment Correlation (1) .00 .47 .03 .39 -.05 .42 Spearman's Rank-Order Cor. (1) -.02 .46 .02 .39 -.05 .41 (1) Stock return regression (1.5) (2) Operating income in levels (3) Operating income in first differences (4) Net income in levels (5) Net income in first differences (6) E. P. S. in levels (7) E. P. S. in first differences Source: Ball and Brown [1969, page 319] through by the market value of equity. Using the above income definition Iit, the results appear in Table 1.2. Ball and Brown draw the following conclusions from these results: (1) Comovement's in accounting earnings of firms predict moderately well the firm's systematic risk. From Table 1 the highest product moment correlation coefficient is .47. From Table 2 it is .59. Ball and Brown conclude that comovements in accounting incomes explain approximately 20-25% of the cross-sectional variability in estimated degrees of association with the market (from Table 1) and 35-40% in systematic risk (from Table 2). [p. 319] It is not clear what the difference is between en estimates of comovement of the firm's rate of return with the rate of return on the market and estimates of systematic risk. The extent of the explanatory power is "explained" by the fact that all variables are measured with error. They conclude that accounting numbers may be even better predictors of systematic risk than these results indicate. No evidence is given to support this claim. (2) Better predictions are obtained when the variables are measured in first differences and the results are sensitive to the income definition utilized. The study, being an early attempt to relate accounting variables to systematic risk, suffers from the lack of any theoretical argument showing a relationship between account- ing betas and systematic risk as defined in Capital Market Theory. As such it is not clear whether to expect a consis- tent correlation between such accounting betas and system- atic risk. Table 1.2 Coefficients of Correlation between Various Measures of the Covariance between a Firm's Index and a Market Index of Income Variables Standardized Product Moment Correlation (1) Spearman's Rank-Order Cor. (1) (2) (3) .45 .59 .45 .64 (4) (5) (6) (7) .39 .53 .41 .53 .42 .58 .43 .59 (1) Stock return regression (2) Operating income in levels (3) Operating income in first differences (4) Net income in levels (5) Net income in first differences (6) Available for common in levels (7) Available for common in first differences (Table 2, p. 320) Beaver, Kettler and Scholes Beaver, Kettler and Scholes [1970] attempt to give a rationale for the use of accounting based risk measures as proxies for the systematic risk of the firm's equity securi- ties. They note that such measures highlight several aspects of the uncertainty associated with the earnings (or return) stream of the firm. Further, the accounting risk measures are surrogates for total risk. What is the link between total and systematic risk? If the systematic and individualistic components of risk are positively correlated (at the extreme, per- fectly correlated), then it is reasonable to view the accounting measures as surrogates for systematic risk as well. The evidence does suggest that positive correlation does exist (e.g. securities with a larger than average B tend to have a larger than average variance of the individualistic component. [p. 659, last paragraph] The statement is dubious. There is no a priori reason to believe that factors positively related to systematic risk ought to be positively related to the non-systematic component of risk as well. High beta firms may have small individualistic or firm-specific risk components. They may be affected primarily by market events with little left over in the way of residual risk. The empirical fact referred to may be based upon faulty beta estimation procedures. In any case, if systematic risk and total risk are highly correlated as would be the case if the non-systematic component were highly correlated with the systematic component, Capital Market Theory loses much of its relevance in specifying the systematic component of risk as that component which is relevant for equilibrium pricing. Given this argument, the Beaver, Kettler and Scholes results can be viewed as an attempt to specify the compo- nents of total risk. The basic concern of the study is to answer the ques- tion: To what extent is a strategy of selecting portfolios according to the traditional accounting risk measures equivalent to a strategy that uses the market determined risk measures? The list of factors presented in their study is: (1) dividend payout (2) growth (3) financial leverage (4) liquidity (5) asset size (6) variability of earnings (7) covariability of earnings with the earnings of the market. The arguments given for these factors are: (1) Firms follow a policy of dividend stabilization: once a particular dividend level is established they will be reluctant to cut back. Also, firms are reluctant to pay out more than 100% of earnings in any single fiscal period. Given these tenden- cies on the part of firms, those firms with more volatile earnings' streams will adopt a lower payout ratio. This argument hints at a relationship between the payout ratio and the total risk of the firm's equity securities. (2) Defining growth as the existence of "excessive" earnings opportunities for the firm, Beaver, Kettler and Scholes argue that there is no reason to assume that the growth assets need be more risky than assets already in place. Rather, asset expansion would occur in areas where the prospective earnings stream generated by these newly acquired assets would be more volatile than that generated by the firm's current assets. Presumably a more volatile earnings stream implies a higher beta, although the connection is hard to see. (3) The usual argument is given for financial leverage. (4) Beaver, Kettler and Scholes do not suspect that liquidity in the sense of the fraction of assets that are current assets held to be related to beta. Rather, they suspect that the differences in systematic risk among firms come about as a result of the differences in the riskiness of their non-current assets. However, they use the current ratio as a measure of liquidity. (5) Asset size is included on the grounds that: (a) asset size is highly correlated with the risk of default on bonds outstanding, (b) If individual asset returns are not perfectly correlated then larger firms will have a lower total risk of equity securities than smaller firms. This results because larger firms, by diversifying their asset holdings, reduce their total risk. The connection with systematic risk remains to be demonstrated. (6) Variability of earnings is given an intuitive argument to establish its effect on total risk. (7) Covariability of earnings is introduced as the slope coefficient of the regression of the earnings price ratio of the individual firm on an economy wide average of earnings price ratios as the market variable. No theoretical argument is given for this variable. In the Beaver, Kettler and Scholes regressions, the dependent variable is an estimate of Bi obtained by running a time-series regression of the security's rate of return on a proxy for the rate of return on the market. This was done for two periods: January 1947 January 1957 and December 1956 December 1965. Monthly rates of return were used. The implicit assumption is that beta was stationary over each of these subperiods. Beaver, Kettler and Scholes test their hypothesis by computing correlation coefficients between corresponding beta estimates from the two subperiods at the individual and at the portfolio level. They find a correlation coefficient of .594 at the individual level and .965 for portfolios of size 20. They conclude that the data indicate that stationarity is not violated. One has to ask, however, whether the process of aggregation into portfolios automatically results in higher correlation coefficients (after adjusting as Beaver, Kettler and Scholes have done for the loss of degrees of freedom). Results of this stationarity test are given in Table 1.3. We present here the definitions employed by Beaver, Kettler and Scholes of their accounting measures. T Cash dividends (1) Average Payout = t Income available to common t=l Total assets Total assets0 (2) Average Asset Growth = Total T n T Total senior securities t (t= Total assets (3) Average Leverage = Table 1.3 Association between Market Determined Risk Measure in Period One (1947-56) Versus Period Two (1957-65) Number of Securities in Portfolio Rank Correlation .625 .876 .989 Product-Moment Correlation .594 .876 .965 Table 2, p. 665 (4) Average Asset Size = (5) Average Liquidity = T STotal assets t=l T Current assets. SCurrent liabilities t=l t (6) Earnings Variability = E Income where Ma t P Market v t-i E 2 t E t-1 T available to commont alue of common stock t-1 T Et E t=l t-1 and - T Cov ,Mt t-1 (7) Accounting Beta =t- var(Mt) where Mt t T E. P t=1 i,t-1 N T = the number of years in the subperiod. N = the number of firms in the market. One observes that (6) is a measure of the standard deviation of the rate of return to stockholders and as such is a measure of a.. Variable (7) is an estimate of i , I where rates of return are defined using accounting earnings, and the regression is a time-series one with T = 9 or 10 (depending upon the subperiod). The market index used is equally weighted. Beaver, Kettler and Scholes call this estimate B.. The idea here is that everything is defined here in terms of accounting earnings rather than in terms of prices and dividends. They compare these estimates of the true beta to those from their historical regressions of ordinary rates of return on ordinary rates of return on the market. They find that the dispersion of their accounting betas is almost four times as large as that of the usual historical betas. This loss of efficiency is attributed to relative sample sizes; nine for accounting betas, 120 and 108 for historical betas. Further, the accounting based market measure exhibits first-order serial correlation unlike the market determined market index (used to compute historical betas). Also, 9% and 12% of the accounting B.'s were negative in each subperiod. Beaver, Kettler and Scholes suggest increasing the time period to 19 years for computa- tion of accounting betas. When they do this, they find that the standard deviation drops from 1.164 (1.280 in sub- period 2) to .791 over the entire 19 year period. This is still substantially larger than .336 (.342) the standard deviation of historical betas in subperiod 1 and subperiod 2. Beaver, Kettler and Scholes further find evidence of non- stationarity in accounting betas. While the other accounting variables are relatively stable as measured by rank correla- tion and product moment correlation coefficients, these correlation coefficients come out to be .034 (-.060) for the accounting beta. Recall that their data yielded correlation coefficients of .625 (.594) for the historical betas between the two subperiods. Beaver, Kettler and Scholes conclude that the accounting beta is subject to large errors in measurement and they virtually suggest searching for other accounting measures of risk. One wonders whether the methodology of correlating two variables both measured with a great deal of error, the historical beta and the accounting beta, is fruitful. Any discovered correlation could be the result of correlation between the errors in the measured variables. On the other hand, while the two variables may in fact be highly correlated, the random error terms in their measurements could obscure such correlation. This type of consideration is at a level secondary to that of questioning the adequacy of the historical betas, a variable measured with error and assumed to be stationary, as an estimate of the true beta. On the other hand, it is hard to see what else one could do at the level of correlation analysis. Beaver, Kettler and Scholes go on to suggest the earnings variability measure as an alternative risk measure to the accounting beta. They note in support of this suggestion that it has, over the samples, approximately the same degree of stationarity as the historical beta. This argument is questionable. While a necessary condition for two variables to be highly correlated would appear to be that they have the same degree of stationarity, it is by no means a sufficient condition. To determine the degree of correlation between the accounting based risk measures and the historical beta estimate, Beaver, Kettler and Scholes calculate cross- sectional correlation coefficients between the given account- ing variables and the historical beta at the individual and portfolio levels. They find at the individual level the following ranking by degree of correlation: (1) earnings variability (2) payout variable (3) accounting beta (4) liquidity. At the portfolio level the following ranking is obtained: (1) earnings variability (2) payout variable (3) accounting beta. These results are contained in Table 1.4. In interpret- ing these results, note that a rank correlation in absolute value greater than .10 is significant at the .05 level. The rank correlation coefficients are given in the table. Portfolio correlations were obtained first by forming port- folios of five securities each where the securities were ranked by the magnitude of the given accounting variable. Their betas were calculated and correlated with the usual beta of each portfolio obtained as the arithmetic mean of the betas of the five securities in each portfolio. In interpreting the portfolio results one notes that one would expect the portfolio correlations to be higher Table 1.4 Contemporaneous Association between Market Determined Measure of Risk and Seven Accounting Risk Measures Variable PERIOD Individual Level ONE Portfolio Level PERIOD TWO Individual Portfolio Level Level Payout Growth Leverage Liquidity Size Earnings Variability Accounting Beta Table 5, p. 669 -.49 .27 .23 -.13 -.06 .66 -.79 .56 .41 -.35 -.09 .90 -.29 .01 .22 .05 -.16 .45 -.50 .02 .48 .04 -.30 .82 .44 .23 because aggregation reduces the variance of the unexplained error term. But, Beaver, Kettler and Scholes argue, aggrega- tion could also result in a reduction in the variance of the dependent variable, the portfolio beta estimate. This reduction could offset the increase in correlation described above. Beaver, Kettler and Scholes do not develop the statistics of this argument nor test it on their sample. Rather, they argue that portfolios, not single securities, are the relevant investment instruments held by individuals. The final part of the Beaver, Kettler and Scholes study is concerned with the forecasting ability of the accounting risk measures. Beaver, Kettler and Scholes use an instru- mental variables method to remove the error in the variables. They postulate the following model 1 = +0 + 11 + ... + nZn, (1.6) where 1 is the true unobservable beta, and Z,...,Zn are n accounting risk measures. Note that this model assumes that beta is fully determined as a linear function of the n accounting risk measures without any random firm specific error term. To obtain estimates of ,0'l1",.n the follow- ing regression of the usual beta estimate on the accounting descriptors is run: = C0 + C Z + ... + C Z + W. (1.7) 1 0 1 n n from errors. The instrumental variables used were payout,free from errors. The instrumental variables used were payout, growth, and earnings variability. The resulting beta estimates were used to predict second period betas. These estimates were compared to the naive estimate which assumed that the second period beta would be equal to the first period beta. The results in Table 1.5 show a decrease in mean square error over the naive model. Gonedes Gonedes [1973] sets out to test for an association between accounting based and market based measures of systematic risk. In so doing he rejects the results of the Beaver, Kettler and Scholes study which, he claims are based upon a spurious correlation induced by scaling income numbers by market prices, such prices being implicit in beta. The same basic criticism is applied to Ball and Brown's results. When income numbers are scaled by assets, he does not find the significant association that Beaver, Kettler and Scholes find. Gonedes explains the improvement in results when first differences in income numbers or scaled first differ- ences in income numbers are run as follows: "Presumably, the transformations induce 'better' specifications of the underlying stochastic processes" (p. 433). Why this is so is not made clear. Gonedes runs the following model, I. it AY t t S4i i E + i + 4i,t' (1.8) t A 1 t Table 1.5 Analysis of Forecast Errors NAIVE BETA ESTIMATES Individual Portfolios Securities A(a) B(b) .093 .030 .027 INSTRUMENTAL VARIABLE BETA ESTIMATES Individual Portfolios Securities A(a) B(b) .089 .016 .016 (a) Portfolios ranked according to the historical estimate. (b) Portfolios ranked according to the instrumental variable beta estimate. (Table 7, p. 677) MSE where A represents first differencing, Y = the income number of firm i in period t it Y = the economy-wide income number for period t I. Y = the industry income number for the industry to which the i-th firm belongs for period t A the total assets number of the i-th firm at the beginning of period t A = the economy-wide total asset number at the beginning of period t I A = the total asset number for the industry grouping of the i-th firm at the beginning of period t. The sample consisted of 99 firms randomly drawn from the population for which all the necessary data were available. To obtain beta estimates the market model was run using logarithmic rates of return. Monthly observations were employed. The parameters of the market model were estimated for three year, five year, seven year, ten year, and twenty- one year intervals. The first six monthly observations from 1960 and the first six from 1968 were reserved for prediction tests. Gonedes's results provide evidence of nonstationarity in beta estimates obtained from the market model. He suggests that the proper criterion to be used in deciding upon an appropriate interval over which to estimate the market model is predictive efficiency; the market model estimates are used to derive predicted returns and the mean- square errors of these predicted returns as predictors of the reserved 1960 (or 1968) returns for different time intervals are calculated. His results suggest that a seven year interval provides estimates with the greatest predic- tive efficiency and this interval was used in computing beta estimates for correlation tests with 6 in the above equation. The procedure is to correlate the coefficients of determination for each firm from the market model with the coefficients of determination from the accounting income model for each firm. Again, using annual observations, various intervals were used to calculate the accounting income numbers. Gonedes finds as a result of performing prediction tests that the twenty-one year period provided better estimates than the seven year estimates. The prediction test results suggest that the accounting-number models do not reflect the structural changes reflected by the market model. The results for the model of scaled first differences for various subperiods are given in Table 1.6. 3. Theoretical Basis Study Pettit and Westerfield [1972] attempt to fill a lacuna in the beta literature by providing a rationale for deciding which factors affect systematic risk. They show, using a perpetuity cash flow valuation model for stock prices, that the usual beta of an asset's return can be written as a weighted average of a "capitalization rate beta" and a Table 1.6 Correlation Coefficients, R, between Estimates from Market Model (M) and Four Accounting Number Models (A)(a) Accounting Beta Measured Over 1946-68 1946-52 1953-59 1961-68 Stock Beta Measured Over M2 M3 M4 1946-52 1953-59 1961-68 .18 (a) Ninety-nine Cross-sectional Observations. b/ Significant at a = .05. c/ Significant at a = .01. (Table 5, p. 434) Period A1 A2 A3 A4 "cash-flow beta" each defined from regressions of the firm's capitalization rate and its cash flow on the capitalization rate of the market and the cash flow of the market. They conclude that anything that affects the expected cash flow of the firm or the capitalization rate should affect beta through the cash flow beta and the capitalization rate beta, respectively. The analysis involves a circularity, however. Capital- ization rates are equilibrium expected rates of return. As such, according to the CAPM, they are determined, given RF and EM, solely by beta. Thus a knowledge of which factors affect the expected rate of return on the firm's equity securities would require a knowledge of the factors affect- ing beta. There is no independent knowledge of which fac- tors affect capitalization rates, at least according to the CAPM. Thus, the Pettit and Westerfield decomposition is uninformative. The authors run up against this paradox when they attempt to explain the choice of variables as determi- nants of beta. Because of space limitations we have not given any justification of why these particular variables should or should not be related to asset risk. In some cases we think a relationship is expected, in other cases we feel that any relationship is a spurious one. Never- theless, each variable was included in the analysis because at some time someone proposed that the variable suggested something about the risk associated with an asset (!) [pp. 1661-1662] Their list of variables is: (1) Dividend payout (2) Leverage (3) Firm Size (4) Liquidity (5) Growth. They perform a correlation analysis on the market beta, BM; the capitalization rate beta, /p ; the cash flow beta, SEPS; the operating income beta, 0I ; the payout ratio, PAY; the debt equity ratio, D/E; size, SIZE; liquidity, LIQ.; and growth in earnings per share, GEPS. The correlation analysis was carried out for two periods: Period I, 1947-57, and Period II, 1957-68. Their results appear in Table 1.7. On the individual level the results are not striking. At the portfolio level, the largest correlations are between the market beta and the payout ratio and the market beta and the capitalization rate beta. This may be a spurious correlation though, because grouping of observations into portfolios will, in general, increase the correlation coefficients. 4. Barr-Rosenberg and Associates' Work Barr-Rosenberg and McKibben Barr-Rosenberg and McKibben [1973] attempt to decompose beta into a component that depends upon a set of descriptors based upon accounting data plus a firm specific effect that cannot be accounted for by these descriptors. Their decom- position is similar to the Beaver, Kettler and Scholes Table 1.7 Correlation Analysis INDIVIDUAL FIRMS e/p 'EPS 0I Period I S .329 .259 .197 N=338 Period II .292 .184 .147 N=543 PORTFOLIOS OF Period I 8 .630 .455 .307 Period II .621 .389 .261 Table 3, p. 1663 PAY D/E SIZE LIQ. GEPS -.481 .049 -.074 -.068 .215 -.394 .069 -.182 .013 .249 FIVE FIRMS -.766 .092 -.161 -.204 .409 -.719 .154 -.400 .035 .481 instrumental variable procedure except that firm specific effects are allowed. Their list of descriptors consisted of accounting based variables such as an accounting beta, various financial leverage and growth rate measures and of market based descriptors such as the historical beta, a measure of residual risk, a, in the market model, and market valuation descriptors such as the earnings price ratio. "The 32 descriptors were selected, without any prior fitting to the data, on the basis of studies in the liter- ature and the authors' intuition" (p. 325). They found that the pattern of signs obtained was not as predicted. Their regressions constituted a 2% increase in explanatory power of predicting returns over the naive hypothesis that B = 1. There are several problems with the estimation tech- niques employed in this early study. (1) They estimated betas from 5nt = b'Wnt + (1 by substituting this expression into the market model with constant intercept Rnt = a + ntMt + n (1. This yields Rnt = a + b'WntMt + (nt + Tnt) (1 .9) LO) Ll) Letting Unt = nMt + nt' (1.12) the error term in this regression, applying ordinary least-squares to equation (1.11), yields inconsistent estimates because Unt will be corre- lated with the independent variables WntMt via the market return Mt. (2) The intercept a was taken to be constant in their regressions. If one accepts the CAPM, though, a = R (1- nt) will vary with n and t even if RE is constant because of its dependence on Bnt (3) In estimating the variance of the firm specific effects, n' the authors find that: var(n ) = W < 0. (1.13) This can occur in variance components models if the usual variance formulae are applied. To overcome this problem, alternative estimates of w have been derived in the literature on pooling time-series and cross-sectional data. However, Barr-Rosenberg and McKibben substitute W = 0 whenever W < 0. This leads to problems in pre- dicting ,n In evaluating the predictive power of their betas, Rosenberg and McKibben consider the mean-square error in the prediction of returns generated by those betas. They compare their predicted betas to those generated under alternative assumptions. These include the void predictor a = n = 0, ns ns the naive predictor using historical betas and alphas generated from the stationary market model, the naive predic- tor with a = 0, a Bayesian adjustment of the naive, and the unit beta a = 0, 0 = 1. ns Their results indicated that only their predicted betas did better in a mean-square-error sense than the unit beta predictor. Barr-Rosenberg and Marathe In later work, Barr-Rosenberg and Marathe [1975] classified their descriptors into the following categories: (1) Market Variability e.g., historical beta, sigma, current price (2) Earnings Variability e.g., accounting beta, variability of cash flow (3) Unsuccess and Low Valuation e.g., growth in E.P.S., average proportional cut in dividends over the last five fiscal years (4) Immaturity and Smallness e.g., log (Total Assets), Net Plant/Gross Plant (5) Growth Orientation e.g., Dividend yield, E/P (6) Financial Risk e.g., Total Debt/Assets, Liquidity (7) Indicator of Firm Characteristics e.g., dummy variable for N.Y.S.E. listing, whole- salers, etc. Barr-Rosenberg and Marathe take the market model with a constant intercept nt nt t + nt' (114) where Bnt = b + b Xnt + bX2nt + ... + b nt (1.15) + bJ+1dlnt + ... + bGdGnt. Note that there are no firm specific effects in this speci- fication of nt. That is, the firm beta is a linear combi- nation of J descriptors and G industry dummy variables. The coefficients of the prediction rule for systematic risk are obtained as follows. Equation (1.15) is substituted into equation (1.14) to obtain Rnt = a + b0Mt + bl(X ntMt) + ... + b (XJntM ) (1.16) + b+l(dntMt) + ... + bK(dGntMt) + nt The data are pooled and ordinary least-squares is run on equation (1.16) to obtain estimates 08,6 ,...,b' ,J J ',...,b . 0 J J+1 K Having obtained these OLS estimates the residuals in the market model Ent are obtained nt = nt nt t (1.17) 2 Let o denote the variance of E. In this specifica- nt nt' tion it is assumed that 2 is explained as follows: nt ant = St(S0 + S IXnt + ... + S JXnt + Sj+1d1nt (1.18) + SKdGnt where St is the average cross-sectional standard deviation in month t, and S ,...,SK are the coefficients of the predic- tion rule for residual risk. Let 6nt = E(e ntl) the mean absolute residual return for security n in month t, and a nt c = t-, the coefficient of variation of le The model nt for the residual risk can be rewritten as nt = 6t(S0 + SlX nt + ... + S XJnt + (1.19) S +1dnt + .. + SdGnt), J+1 mnt K Gnt where t is the capitalization weighted cross-sectional average of absolute residual returns. Estimates of residual risk are obtained by running the model ljnt = S t + S1 (Xntt) + ... + S .(X nt ) (1.20) + SJ+1(dlnt t) + ... + SK(dGnt t) Presumably, the assumption here is that E(j nti) does not differ very much from int From this regression one A A obtains OLS estimates S ,...,SK. The next step is to generate estimates of nt using these OLS estimates n8 = c (S, + SX lnt+ .. + SjX t + S d (1.21) nt t 0 1 Int J Jnt J+1 Int + ... + SdGnt) K Gnt The model in equation (1.16) is divided through by 8nt and new estimates of a,b0,...,b ,bj.,...,bK are obtained. Presumably, this provides a GLS estimation procedure for the model. Looking at the results for the generalized least- squares estimates one finds that earnings variability is positively related to beta, growth in E.P.S. is negatively related while growth in total assets is positively related. The results in Table 1.8 do not include market variability variables. Table 1.8 Prediction Rules for Systematic Risk Based on Fundamental Descriptors and Industry Groups Earnings Variability Variance of Earnings Variance of Cash Flow .02266* .02180*** Unsuccess and Low Valuation Growth in E.P.S. -.00416* Immaturity and Smallness Log (Total Assets) .02416*** Growth Orientation Growth in Total Assets .03666 Financial Risk Leverage at Market Debt/Assets -.09150*** .04126*** * Significant at 95% level. *** Significant at 99.9% level. (Table 4, p. 114) One notes that different signs result from alternative specifications of the growth and leverage terms. One wonders what gain is effected by including several different measures of the same effect. Perhaps such a procedure obscures the true relationships underlying the model. When market variability variables such as the histori- cal beta estimates, historical a estimates, and price and share turnover variables were included in the regressions, the following partial results in Table 1.9 were obtained. Barr-Rosenberg is disturbed by the negative adjustment to leverage and finds it inexplicable. The two measures of financial leverage employed are defined in the following manner: (1) Leverage at = Book Value (Long-Term Debt Market + Preferred Stock) + Market Value (Common Stock) Market Value (Common Stock) (2) Debt/Total = Long-Term Debt + Current Liabili- Assets ties Total Assets While theory tells us that the debt/equity ratio at market value is positively related to beta, it turns out to be negatively related in the regressions. The book value measure turns out to have a positive sign. Further, the sign remains negative in the simple regression on market leverage. In conclusion, Barr-Rosenberg asserts: The negative relationship appears to be an empirical fact, but one that we do not now understand. Since the relationship is not comprehensible, we have set the coefficient to zero in actual practice. [p. 122] Table 1.9 Prediction Rules for Systematic Risk Based on Fundamental Descriptors Including Market Variability Descriptors Market Variability Historical Beta Estimate Historical a Estimate Current Price -.03124 -.04546 .05550*** Earnings Variability Variance of Earnings Variance of Cash Flow .00594 .01541 Immaturity and Smallness Growth in E. P. S. -.00453*** Growth Orientation Growth in Total Assets .02290*** Financial Risk Leverage at Market Debt/Assets -.08739*** .02596*** *** Significant at the 99.9% level of significance. (Table 5, p. 124) This represents a highly dubious procedure which would be difficult to justify. The next part of the Rosenberg and Marathe study is concerned with the formulation of measures of predictive accuracy. Two basic historical performance measures are computed as follows: (1) Assuming that all assets have identical risk one runs the regression nt a + b -- (1.22) nt nt One calculates the mean-square error of the predicted returns generated by this naive model. Call the mean square error from this prediction rule MSE . (2) Run a second regression of the form: Rnt 1 RMt HntRMt = ~ a + b0[ n -7 + bl (1.23) nt nt nt nt HBnt is the historical beta and this represents a Bayesian adjustment to the historically generated beta. Call the mean-square-error from this prediction rule MSEB. (3) Any other prediction rule for beta, e.g. a predic- tion rule based upon fundamental descriptors, can be evaluated via the performance index MSE -MSE MSE -MSE 0 (1.24) B 0 where MSE1 is the mean-square error from the prediction rule to be evaluated. That is, one computes the improvement one obtains by using the given prediction rule for beta relative to the improvement over the naive hypothesis achieved by the benchmark procedure. The results appear in Table 1.10. Table 1.10 Unbiased Estimates of the Performance of Alternative Prediction Rules in the Historical Period (Predicted Variance as a Multiple of the Variance Predicted by a Widely Utilized Prediction Rulea) INFORMATION USED IN PREDICTION MEASURE Market Variability Information Only Benchmark 1.00 All market variability descriptors 1.57 Fundamental Information Only Industry adjustments and fundamental 1.45 descriptors Market Variability and Fundamental Information All information except the historical 1.79 estimator All Information 1.86 a The reported figure is the adjusted R2 in the appropriate GLS regression for residual returns, r rMt divided by the adjusted R for the benchmark procedure. (Table 6, p. 134) Barr-Rosenberg's thesis that the use of both market information and fundamental descriptors leads to better beta predictors than the use of each set of information sepa- rately is supported by these results. He also notes that if in predicting beta all information concerning historical betas is discarded, then less than 4% of predictive power is lost. He concludes from this result that "we are able to obtain virtually all of our predictive power for these aspects of risk without relying on historical measures of them" (p. 135). To test for the predictive accuracy of the betas generated into periods other than those in which the predic- tion rules were fitted, Barr-Rosenberg tests for stability of the prediction rule. He finds it to be quite stable over time--the prediction rule estimated for a full history of 230 months is closely similar to that for a recent 101 month subperiod. He also computes the adjusted R2 for the fitted regression lines using various prediction rules. He finds that the betas based upon market and fundamental descriptors were superior to the benchmark betas in five cases out of six with an average R2 of .0768 versus .0502. Barr-Rosenberg and Marathe In their work on testing the Capital Asset Pricing Model, the authors, Rosenberg and Marathe [1979] generate betas where the market model is employed with both alpha and beta varying as linear functions of a set of descriptors plus error terms. This relationship they write as a = a'X + ea (1.25) t = 1,...,T t = b t t n = l,...,N(t) (1.26) where the terms ent' nt are "model errors in prediction with expected value zero" (p. 140). The desire in this study is to test the CAPM. Only the historical beta was used as a fundamental descriptor for the actual beta. The authors note that the prediction rule could be improved by using Bayesian adjustments to the historical beta and fundamental accounting and market based descriptors. This is not done in this study. However, the econometric techniques of their previous studies are improved upon in this one. Generalized least-squares procedures are used in this study. The results are hard to compare to previous results, however, because of the use of a single fundamental descriptor, the historical beta. Part of the purpose of the present study is to apply correct econometric techniques to the model with a set of theoretically justi- fied descriptors. 4. Study of the Effect of Financial Leverage Hamada [1972] tests the effect of the leverage relation he derives using the M-M theory. The relation he derives is S 6u = L- (1.27) u 42 The M-M theory states that V = V + TD so that L u L S L Bu V-TD L L L S L S +D (1-T) L L L This yields SL+DL (1-T) L S u L DL(IT)] - ^^ ---/u- (1.29) To test this relationship, Hamada calculates the following rate of return R ut t Xt(1-T)+AGt S Ut-1 (1.30) the rate of return to stockholders in a firm which has no debt in its capital structure. The change in capitalized growth over the period is represented by AGt. Since S t-1 is unobservable as firms generally employ debt financing, the M-M theory is used to evaluate the denominator Ut-1 = (V D)t- U L t-1 (1.31) The numerator is evaluated using the following identi- ties: Xt(l-T) + AGt = [(X-I)t(l-T) PDt + AGt] + PDt + It(1-T), where PDt denotes preferred dividends at time t, and It is (1.28) (1.32) the interest expense at time t. The corporate tax rate is designated T. Next, the following identity is used: (X-I)t(1-T) PDt + AGt = dt + cgt (1.33) where dt represents dividends paid at the end of the period and cgt represents capital gains. Consequently, dR + cgt + PDt + I (1-T) R = (1.34) t (V-TD) t-1 Next, the observed rate of return to common stockholders is (X-I) t (1-T)-PDt+G d +cgt RL S S (1.35) t L L t-l t-1 Hamada obtains available data to construct R and R the L u t t rate of return to a firm exactly identical to the levered firm except that it has no debt or preferred stock in its capital structure. Using these rates of return he then runs the market models: RJ = a + j R + Ej (1.36) U u U M u,t t t and R = aJ + 8 R + j (1.37) L L L M L,t' t t where R. is the N.Y.S.E. arithmetic stock market rate of t return. The betas obtained from these regressions are estimates of the unlevered and levered betas. Theory implies that j > j . L u Using data on 304 firms, Hamada runs 304 time series regres- sions and calculates mean alphas and mean betas, and the statistics presented in Table 1.11. These results indicate that since .9190 > .7030 lever- age increases systematic risk. Similar results are obtained when continuously compounded rates of return are used in the market model. Hamada concludes, that if the M-M theory is correct, then leverage explains 21-24% of the value of the mean beta. He then goes on to test for which market value rates ought to be used to adjust observed betas to obtain unlevered betas as his formula suggests. To do so, he runs the following regressions: S I S= 1 + b[ SU + u1 j = ,...,102 (1.38) S= a2 + b2 + j = 1,...,102 (1.39) for the 102 firms in his sample that did not have preferred stock in any of the years used. Using average values over the twenty year period for SL and S and the 1947 (beginning u Mea & .02 u u .70 R2 .37 u UL .03 L .91 2 R .38 L * xi-R-X 304 (Table 1, 45 Table 1.11 Summary Results over 304 Firms for Levered and Unlevered Alphas and Betas Mean Absolute Standard Me n Deviation* Deviation Erro: 21 .0431 .0537 30 .2660 .3485 99 .1577 .1896 14 .0571 .0714 90 .3550 .4478 46 .1578 .1905 p. 218) an Standard r of Estimate .0558 .2130 .0720 .2746 period) value for S /SL and then the end of period 1966 value for S /SL, the results were obtained in Table 1.12. Hamada concludes that, if long-run averages are used then the adjustment factor method is appropriate using the derived relationship. An alternative indirect test of the financial leverage effect is carried out on the basis of the following set of considerations. Within a given industry ordinary common stock betas show a certain degree of cross-sectional varia- tion. Some of this cross-sectional variation is presumably due to the differing degrees of financial leverage employed by firms within the given industry. The reason for looking at a given industry in examining cross-sectional beta varia- tion is to confine attention to a given risk class in the M-M sense, that is, to one with the same cost of capital as that of a firm with no leverage but otherwise exactly alike and consequently with the same unlevered beta since P = R + (EM RF)BJu (1.40) In practice it is not possible to specify exactly a risk class so that in practice there will be some variation in unlevered betas of the firms within the industry. The idea of the Hamada test is then the following: If indeed some of the cross-sectional variation in stock betas within a given industry (risk-class) is attributable to the differing debt/equity ratios employed by the firms, then unlevering the betas should result in a set of unlevered 47 0m am * > I 0 a ca o SrH 0H Cdl 00 ON SD 0 o C c 4l co z c o a co ' U) O 0 H O in H o .-- o oo N C n H r-I c a0 C D 0 a UN -) ) l 0 00 2 .N N m *- (Cd ,N 0 C| 0 >* H H o c ( o cP E 48 betas which exhibit less cross-sectional variation than the original levered betas. Thus, by computing the dispersion of the levered betas, unlevering them and computing and comparing that dispersion to that of the resulting set of unlevered betas one should be able to discover a financial leverage effect. The results follow in Table 1.13. These results indicate a positive effect for financial leverage, and Hamada concludes, some support for the M-M theory. 6. Studies of Effect of Market Power Sullivan Various studies have examined the effect of monopoly power on the rate of return to stockholder's and upon a firm's profitability. Firm profitability is usually measured by the ratio of net income to the book value of stockholder's equity. One such study examined the effect of market power on equity valuation. Sullivan [1977] takes the ratio of the market value to the book value of stockholder's equity for a given firm in a given year. Then the arithmetic mean over the years 1961-70 is taken to evaluate relative equity prices. This variable is then regressed on C.: the weighted average four firm concentration ratio 3 for firm j SZ.: the natural log of 1961 sales revenue for firm j MSG.: 1968 market share for firm j divided by 1961 Market share for firm j (representing growth in market share) Table 1.13 Mean and Standard Deviation of Industry Betas Industry Number of Firms Food Chemicals Petroleum Mean 8 o(B) Mean 8 a(B) Mean 8 o(a) Primary Metals Machinery except Electrical Electrical Machinery Transportation Equipment Utilities Department Stores .515 .815 .232 .448 .747 .928 .237 .391 .633 .747 .144 .188 Mean 8 1.036 1.399 o(B) .233 .272 Mean 8 a(B) Mean 8 o(B) Mean 8 a(8) Mean 8 o(8) Mean 8 ao() .878 1.037 .262 .240 .940 1.234 .320 .505 .860 1.062 .225 .313 .160 .255 .086 .133 .652 .901 .187 .282 Table 4, p. 225 ISG.: 1968 estimated sales in firm j's industry Divided by 1961 estimated sales in firm j's industry. A second measure of market power is: S.: the weighted average market share for firm j. Two regressions were run: one with C., the second with 3 S.. In both regressions a positive statistically signifi- cant sign is obtained for these market share variables. Sullivan concludes: These premiums seem to suggest that firms with market power have the ability to set and hold output prices above costs and as a result earn monopoly profits. This ability to control output prices makes the equity shares of powerful firms attractive to investors who bid-up the prices of the equity shares so that all the expected future monopoly profits are capitalized into the existing market prices of the shares. [p. 111] The study purports to control for risk by introducing the standard error of a trend line fitted to book profit- ability over a ten year period. However, this variable is not used in the regression discussed and is a questionable measure of risk in any case. It is possible that the earnings of the monopolistic firm may be capitalized at a lower discount rate because the existence of monopoly power lowers the systematic risk of a firm's equity. Such a decrease would increase stock prices. Sullivan argues that these increased stock prices are the result of the capital- ization of monopoly profits. Thomadakis Thomadakis [1977] carries out a value-based test of profitability and market structure. He attempts to examine the relationship between the capitalized value of monopoly rents and (1) F: a risk measure of the risk of future returns (2) U : the firm's power of oligopolistic restriction with respect to output of currently held assets (3) Uf: the firm's power of oligopolistic restriction in future investment (4) g: the firm's expected rate of growth from exo- geneous demand (5) C: a scale parameter. He runs M= a + alF + a2Uc + a3(gUf) (1.41) where V-A M= (1.42) the difference between total firm value and the book value of assets divided by sales is used as a measure of monopoly power. There may arise problems in using this variable as a measure of monopoly power because of differing accounting methods used to evaluate A and due to the effect of infla- tion on asset values. Thomadakis assumes that the biases introduced as a result of these problems can be disregarded. The results bear some significance to the question of the effect of monopoly power on systematic risk because Thomadakis takes beta from the usual historical regression as his risk measure. Hypothesizing that Uf and U are f c functions of industry concentration IC, the results follow in Table 1.14. Thomadakis finds the sign of the risk factor puzzling. It indicates that the higher the systematic risk, the higher will be the degree of monopoly power. He explains this result by stating that 8 should represent the volatility of excess earnings whereas here it represents the systematic volatility of total earnings. "The only possible interpreta- tion of current results is that 8 is a negative proxy for F, but this appears quite far-fetched and should be viewed with reservation" (p. 183). He notes further that a correct risk measure for the purpose of the study would separate out the risk of the competitive component of return from the supercompetitive component. The use of beta in this context assumes that both components of earnings have the same systematic risk. Presumably, the results give some first indication of the effect of monopoly power on systematic risk. If that relationship is negative then the coefficient of beta in the Thomadakis regression should be negative as well. Of course, the inclusion of other variables measuring market share and growth in the regressions could obscure the possible relationship and its sign. Table 1.14 Industry Concentration and Future Monopoly Power DEPENDENT VARIABLE INDEPENDENT VARIABLES Constant -1.57 I IC gIC 2 0.64a 1.89a 5.12a a Significant at the 5% significance level. (Table 1, p. 182) .199 Sullivan Sullivan [1978] seeks to determine whether the market power of firms, as measured by size and seller concentra- tion, seems to reduce the riskiness of firms. By riskiness is meant systematic risk. To do so, Sullivan regresses monthly betas on (1) SZ. = natural log of sales for firm j (2) C. = four-firm concentration ratio (3) DN. = industry dummy variables (4) SG. = the annual compound growth rate in sales from 1963-1972 for firm j. He runs levered and unlevered betas on these variables. Typical results appear in Table 1.15. The coefficients of C. and SZ. are consistently negative and statistically significant in Sullivan's results. These results do indicate a negative relationship between monopoly power and systematic risk. Sullivan seeks to determine the cause of this relationship. To answer the question he decomposes beta, based on the definition of earnings to equity holders, into three betas: (1) a beta relating the covariability of firm sales with the market (2) a beta relating the covariability of firm expenses with the market (3) a beta relating covariability of the revaluation of the firm's securities in the secondary capital market [a capital gain (loss) less retained earn- ings component]with the market. Table 1.15 Monthly Betas, Leveraged and Unleveraged, Regressed on Market Power and Control Variables Constant 1.831 1.756 SZ. -.0708 -.0739 C. 3 -.2938 -.2846 DN. 3 .2237 .2120 SG. ] 1.3863 .9246 .2696 .3458 25% sample: every firm in which the firm's largest market accounted for at least 25% of firm sales. (Table 1, p. 213) 8j* Then the unlevered beta = Sales beta Total expense beta + Capital gains beta. Each component is measured, and it is found that the sales beta and the total expense beta approximately cancel each other out, leaving most of the effect in the capital gains beta which is virtually identical to the unlevered beta. He concludes that the systematic risk resulting from firm operations seems to be small. 7. Studies of the Effect of Operating Leverage Lev Turning now to theoretical models describing the effect of the labor-capital ratio on beta, we first discuss the work of Lev [1974]. The firm's operating leverage is defined as the ratio of fixed to variable costs. Lev claims that beta is a positive function of operating leverage. Attempting to support his case that higher operating lever- age implies a higher total and systematic risk, Lev appeals to the "better known leverage effect" within a given risk class (a homogeneous industry in our case) the higher the financial leverage, i.e. the relative share of fixed interest charges (fixed costs in our case), the larger the volatility of the earnings residual accruing to common stockholders, and hence, the higher the financial risk associated with the common stocks. [p. 630] This argument is suspect for several reasons: (1) The financial leverage argument does not proceed through the earnings volatility argument. Rather, it follows from the consideration of the precise effect of financial leverage on the rate of return to equity holders. (2) The financial leverage argument is not based upon the existence of fixed debt charges because it holds, as Galais and Masulis [1976] have shown, for risky debt as well. (3) Higher total volatility does not necessarily imply higher systematic risk. Lev goes on, however, to make the following arguments: In an uncertain environment future demand Qjt is a random variable. Then the earnings stream of the firm Xj = (p )jt (v )j F. (1.43) it jt jt jt where p = average price per unit of the product v = average variable costs per unit of the product. Lev asks the question: How does an increase in the uncer- tainty in demand (at any given price) affect the earnings stream of the firm? Lev takes the partial derivative of the earnings stream with respect to Qjt (a random variable) to obtain: = p vj (1.44) aQ j t jt jt Of course, such a partial derivative does not make sense. His result is that the derivative of earnings with respect to demand equals the difference between the product's average price and average variable cost per unit, the contribution margin. In a homogeneous and competitive industry the average product price is the same for all firms. Thus the fluctua- tions in the earnings stream of the firm depends only upon the average variable cost per unit. A firm with a higher average variable cost will have a more volatile earnings stream. A firm with a greater operating leverage, Lev claims, will have a lower variable cost per unit and hence a more volatile earnings stream in accordance with his partial derivative. This higher volatility is transmitted to increase the total volatility of returns to equity holders. This analysis and introduction of uncertainty leaves the question of the firm's response via its factor mix to the uncertainty in demand. It is possible that a firm would in fact respond in such a way as to reduce the variability of its earnings stream. What is missing here is an analysis of the firm's optimal decision behavior under a situation of increased uncertainty in demand. Lev then goes on to attempt to demonstrate the effect of operating leverage on the systematic risk of the firm's equity B.. He does this by writing the rate of return as (Sales. V F )(1-T) + Ag R. t jt jt jt (1.45) SSt- J ,t-1 where Sales. = (pQ) = total revenues jt jt V = total variable costs 3t F = total fixed costs jt Agj = the future growth potential of the firm S = market value of equity at time t-l 3,t-i T = the corporate tax rate. Thus we obtain Cov((PQ) j (1-T),RMt S. 8, = jt Mt (1.46) Var(R t) Cov(V (1-T),Rt) Cov(Agt ,Rt ) Var(RMt) Var(RMt) Consider two firms, exactly alike, including output, stock value, and capitalized growth. They differ only in their use of variable factors of production. The first and last beta will be identical. The firm with the higher operating leverage will have relatively fewer variable costs (a random variable), hence a lower expected value of variable costs. From this, Lev concludes that it will have a lower covariance with the market return. But this state- ment is not correct. Covariance measures comovement between random variables. For three random variables it is possible that: E(x) < E(Y), but that Cov(,2Z) > Cov(Y,Z). Nonetheless, Lev concludes that the firm with the higher operating leverage will have a lower second beta and hence that Sl,t-11 > S2,t-_12* (1.47) He then assumes that the two firms have the same stock value and consequently that operating leverage increases system- atic risk. Turning to the empirical tests of the proposition, the procedure is as follows: using data from three homogeneous industries, electric utility, steel, and oil to ensure cross-sectional equality of sales patterns across states of nature, Lev runs a time series regression of total operating costs of the j-th firm in year t on total physical output. He obtains from this regression the estimated coeffi- cient V., a measure of average variable costs per unit of output (assuming that MC=AVC and that average variable cost is constant over the estimation period). These regressions were run over two different time periods, a 20 year period (1949-1968) and a twelve year period (1957-1968) to test the "stability" of the relation- ship. The results follow in Table 1.16. The intercept is not reported here. When this study was replicated, the intercept, representing fixed costs turned out to be negative. This indicates a serious problem in this model specification to determine average variable costs and/or in the estimation procedure. The next step was to run the market model to obtain estimated beta coefficients. The regression was run using 61 Table 1.16 Estimates of Average Variable Cost Per Unit 20 years V Electric Utilities .00815 (26.85) Steel Manufacturers .892922 (31.98) Oil Producers .74806 (68.12) Table 1, p. 635; t statistics in parentheses. 12 years .00762 (18.67) .81503 (20.97) .71677 (43.27) monthly rates of return over a ten year period. Of course, this assumes that beta was stationary over that period. Lev then runs the estimated B. on the estimated V.: J J j = a2 + b2 + j (1.48) A negative sign for b2 indicates that beta is a decreasing function of operating leverage (as proxied by .). The results follow in Table 1.17. Lev concludes that the hypothesized relationship holds except in the case of an insignificant coefficient for the oil producing industry. He also concludes that operating leverage does not go very far in explaining systematic risk. Rubinstein Rubinstein [1973] lets R.* be the rate of return to a firm without debt, X. the earnings stream of the firm, and V.* its market value. For a similar firm with debt in its capital structure the rate of return to equity holders X.-R D. R. = (1.49) SS. where D. and S. represent the market values of debt and J J equity, respectively. Rubinstein writes the CAPM equilibrium relationship in the following manner: E(R ) = Rp + X*p(Rj,R ) Var(Rj) (1.50) [equation (2), p. 49], where 63 Table 1.17 Regression Estimates for Systematic Risk on Average Variable Costs Per Unit No. of Firms R2 Electric Utilities .08 Steel Manufacturers a2 .5149 (14.790) 2.2833 -6.912 (-2.060) -1.3401 (5.014) (-2.4097) Oil Producers .05 .8101 -.2748 (4.673) (-1.157) Table 3, p. 637; + statistics in parentheses. [E(RM) -RF] X* = (1.51) Var(R ) Now, since X. X. R.* = 3 = (1.52) SV.* V. J ] where V.* = V. S. + D., (1.53) 3 3 it follows that Var(R.) = Var(R.*) 1+ (1.54) Also, p(Rj,R ) = p(Rj*,R ). (1.55) Rubinstein concludes from these two expressions that the full impact of financial risk is absorbed by the standard deviation Var(R.) since the correlation coefficient is invariant to changes in financial leverage. Note that this result holds only under the assumption that debt is not risky. Substitution of these two expressions into the CAPM relationship yields E(R ) = Rf + X*p(R *,R ) Var(R.*) (1.56) + X*p(R *,R ) Var(R.*) D . 3 65 The first component on the right-hand side of this expres- sion is called operating risk by Rubinstein, and the second component is called financial risk. The goal now is to develop the components of operating risk X*p(R.*,R ) Var(R.*) 3 M ] It suffices to drop the X* in this consideration. Let m denote product m for firm j. Rubinstein assumes that the output Qm of product m is a random variable and that the sales price per unit pm is fixed. One can view this as a model in which price is set ex-ante with quantity demanded being uncertain to be determined at the end of the single period. Note that this assumption is contrary to the Subrahmanyam and Thomadakis model to be reviewed below in which the firm is a quantity setter with uncertainty resolved in the product's price. As Leland [1972] has shown, for the monopolistic firm, these are not equivalent behavioral assumptions as far as optimal behavior is concerned. Let v denote variable costs per unit, F total fixed m m costs, and a the proportion of assets (represented by firm value V.) devoted to the production of product m. Then, assuming that all fixed costs of the firm can be allocated, the earnings stream of the firm X = (p p v F ) (1.57) mm mm m m Rubinstein claims that since a X. R. mV (1.58) SaV m j then p(Rj*,Rm) Var(R *) = [a (pm-v )p(QRM ) Var m 1. (1.59) m (m 9 Rubinstein interprets this as follows: a represents m the relative influence of each product line, p -v reflects operating leverage through the contribution margin, p( m,R ) represents the pure influence of economy-wide events on output, and Var(Qm/amVj) the uncertainty of output per dollar of assets, a measure of "operating efficiency." For the case of a firm producing a single product this formula is easily derived. fp-Qv-F p(Rj*,RM) = p v. j m = p( ,Rm) (1.60) Var(R*) = arp -F (p-) 2 Var [-]. (1.61) Hence, p(Rj*,RM) Var(Rj*) = Ip-vp(Q,RM) Var(/V.) (1.62) which is the relationship to be derived. This formula shows that operating risk and consequently systematic risk [the sum of operating and financial risk equation (8), p.57] is a positive function of operating leverage in the following sense. Given the product's price, decreasing variable costs per unit increases the contribu- tion margin jp-vl and with it operating risk which is proportional to the contribution margin. This of course assumes that p(Q,RM) Var(Q/V ) is constant. Another way to view this result is in the following manner: Given the contribution margin operating risk is determined by (1) the pure influence of economy-wide events on output represented by p(Q,RM) (2) Var(Q/V.) the uncertainty of output per dollar of assets. Presumably firms are out to maximize their market values in the presence of uncertain demand. Plausibly, an increase in the use of variable factors would afford a firm with some measure of flexibility in its production process. However, the question becomes: What is the effect of an increase in the use of variable factors on optimal firm value, presumably represented by V.? If, indeed, systematic risk is lower with the use of variable factors, then for a given random income stream, firm value would increase. Without this knowledge, which clearly has to be determined as part of the analysis, firm value could decrease with the use of variable factors and operating (and systematic risk) could increase. This is all for a given output stream j.. As Subrahmanyam and Thomadakis note, the analysis is not very enlightening because it fails to relate the optimal choice of margin (i.e. of factor mix and price) to the uncertainty of output. Subrahmanyam and Thomadakis The model developed by these two authors will be developed and generalized in the next chapter. A brief summary will be given here. The goal of the analysis is to provide a specification of the relationship between monopoly power as represented by the reciprocal of the price elasti- city of demand and the labor capital ratio and the system- atic risk of the firm's equity securities. (There is no debt in the model.) The innovative facet of this work is the introduction of sources of uncertainty into the demand function and into the wage rate. The effects of the above variables on beta then depend upon the co-relationship between the random error terms representing uncertainty in this model. Their basic result is that if wage rate uncer- tainty is relatively less volatile than price uncertainty then beta will be positively related to the labor capital ratio and negatively related to the degree of monopoly power. If wage rate uncertainty is relatively more volatile than price uncertainty, then the reverse will hold: beta will be negatively related to the labor capital ratio and positively related to the degree of monopoly power. The economics underlying these results can be explained with reference to the market portfolio. Since human capital is a risky asset which we can assume to be tradeable (the results generalize to non-tradeable human capital) the risk of the market portfolio is the sum of the risk of firms plus the risk of the labor component of firms' production processes. The risk of labor is essentially determined by the "beta" of wage uncertainty relative to price uncertainty. If that beta is less than unity then in the weighted average repre- senting the beta of the market portfolio relatively less risk will be shared by labor than by firms. This forces the firm betas to be greater than unity. As labor is increased relative to capital, if the basic condition holds, more risk will have to be borne by the relatively smaller amount of capital. Thus beta is a positive function of the labor- capital ratio. This stands in direct opposition to the Lev and Rubinstein results. The reverse will obtain if the beta of wage uncertainty relative to price uncertainty is greater than unity. The model developed here hints at the fruitful- ness of an approach to determinants of systematic risk that takes into account sources of uncertainty, their correlation, and risk sharing among the bearers of risk. CHAPTER II MICROECONOMIC FACTORS AFFECTING EQUITY BETAS 1. Introduction The first purpose of the study is to examine the argu- ments given in support of various microeconomic factors said to affect the systematic risk of a firm's common stock as measured by its beta coefficient. Arguments based upon theoretical models in the literature and upon "economic intuition" will in turn be considered. A large number of factors can be hypothesized to bear a relationship to beta but, of these, only those that appear to be of clear economic interest will be considered. Those that affect beta for purely technical reasons are not of immediate interest here. An example would be non-synchronous trading in that it makes a common differencing interval for return data impossible to calculate and introduces various biases into beta estimation. In principle, one can correct this potential source of bias--in doing so the effect on beta of the phenomenon disappears. An example of a variable of economic interest is financial leverage: through its effect on the rate of return to equity holders it unambiguously affects systematic risk. While the study of the technical factors affecting beta is of great importance, this study concerns itself with the formulation and inclusion of the economic type of variable. Recently, there have appeared theoretical models in the literature supporting the labor-capital ratio, monopoly power, and "growth opportunities" as determinants of betas. More traditional determinants are financial leverage and volatility of operating earnings. In this chapter, the theoretical arguments supporting the above variables will be presented and critically examined. In the absence of theoretical support, the intuitive arguments will be presented and their limitations will be stated. We turn now to a separate consideration of each of the variables pur- ported to affect beta. To motivate the discussion of the effect on beta of monopoly power, it is instructive to compare the two different arguments for the effect on beta of (1) financial leverage and (2) volatility of operating earnings. While the effect of (1) has been theoretically established, the argument for (2) proceeds on an intuitive basis. 2. Financial Leverage To present the argument in its simplest form, taxes will be ignored and it will be assumed that there is no risk of default on the firm's debt securities. This latter assumption guarantees that i, the interest rate on the firm's bonds, is non-stochastic. The rates of return on the levered firm's and unlevered firm's equity are: X-iDL R -S and R - L S u V L u respectively. Evaluating the corresponding betas yields cov(R ,Rm) cov(X-iDL/SL,Rm) cov(X/SL,Rm L 2 2 2 02 (Am) 2 (R) 2a (rm) m m cov(X/SL Vu/Vu,Rm) Vu cov(X/Vu,RM) 2 S 2 S (R ) L 2(R ) V u S U 6U. 8"U L The effect of financial leverage on the firm's equity beta has been established and, following Modigliani and Miller [1958] in a world with no taxes, since Vu = VL we have that leverage has an unambiguous effect on systematic risk: V L S Lu u. This argument for the effect of financial leverage on beta is based upon a precise knowledge of how such leverage affects the rate of return on the firm's equity. In the following generalization of the above argument, the effect of financial leverage is again evident from similar knowl- edge concerning its effect on the rate of return to equity holders. If we are willing to assume that both the assumptions of the Options Pricing Model and the continuous time version of the CAPM hold, then Hamada's result which holds for riskless debt can be generalized to allow for risky debt where all betas are to be interpreted as instantaneous quantities. It is also assumed that dV/V follows a Gauss-Wiener process and hence that the Modigliani and Miller result on the irrelevance of financial structure (for firm value) holds by hypothesis. Viewing the equity of the firm as an option on the underlying firm value with exercise price equal to the face value of the firm's outstanding debt securities, an expres- sion for the dollar return on equity value AS can be derived using Ito's Lemma. AS 1 2S 2 V2 S AS = AV + V At + At 3V 2 + 2 at where V is firm value, a2 is the instantaneous variance of percentage returns on V, and t represents time. Consequently, AS as/av + 1 a2S/av2 G2V2 At as/at At S S 2 S + S and taking the limit of this expression as At 0, we have the expression for the instantaneous rate of return on the firm's equity securities: 74 as AV s av S Multiplying by V/V allows us to write this rate of return in terms of the instantaneous rate of return on total firm value r . as/vV r as/=v s S V S/V v ' where ns represents the elasticity of equity value with respect to firm value. Note that 3S/aV = 1 if 3D/8V = 0. That is, if debt is riskless this formula implies Hamada's result (recognizing that V = V = VL in a world of no taxes). Deriving the corresponding betas, we obtain cov(r ) a/av cov(r ,r ) S = = S/3 V v m V f3 V VBW () S 02 ) S v m m Thus, 8s = sBV. 2. Volatility of Operating Earnings In the absence of precise information concerning the effect of earnings' volatility on the equity's rate of return it is not easy to establish the effect of that vari- able on the systematic risk of the equity securities. The argument that follows will be referred to as the "earnings' volatility argument." It serves to establish the relation- ship between variability in earnings and the total risk of equity securities. The steps in the argument are as follows. (1) Increased volatility of earnings implies increased volatility of end-of-period stock price (a random variable). (2) This induces a more variable rate of return since i,t+l 1,t P plt and var(R. ) Var(P. )t i,t p2 t+1" i,t Thus, earnings volatility affects the total variability of the rate of return of the equity securities. No refer- ence is made in the earnings volatility argument to volatil- ity with respect to the earnings volatility of the market and none is established. To do so, one would have to posit a relationship between the earnings volatility of the market and the earnings volatility of the specific security. Such an assumed relationship would be transmitted to the corre- sponding rates of return. But such a relationship would be very close or identical to a specification of the effect of earnings' volatility on the security's beta; a relationship it is desired to discover. Under a highly restrictive assumption discussed below, it can be shown that the stock beta is proportional to the total volatility of the rate of return. We merely write cov(iR ) Pim 2 = a= i, 1 (R ) m i m where p. is the correlation coefficient between the market im rate of return and the specific security's rate of return, and o. is the standard deviation of the stock's rate of 1 return. If we assume that pim/am is constant, the result is obtained. A sufficient set of conditions for this to hold is that both p. and a are assumed to be constant. Given a constant irn m correlation with the market and a constant market variance, this result is not surprising, as the only factor left to occasion changes in beta is the own variance of the securi- ty's rate of return. Of course, it is important to then know for each different security the proportionality constant Pim/om as it determines the (absolute) magnitude of B., but given that information, the stock beta is then determined solely by all factors influencing the variability of the stock's rate of return, and earnings' volatility is one of those factors. Given the assumed constancy of pim/am and the implied result that systematic risk is then proportional to total risk, what are we to make of the tenets of portfolio theory and the CAPM implying that, in general, it is only the systematic component of risk that matters? The answer is contained in Capital Market Theory and is analogous to the distinction between the Capital Market Line and the Security Market Line. The Capital Market Line is precisely that set of portfolios for which p. assumes the value +1 and for these efficient fully diversified and perfectly-correlated- with-the-market portfolios (the most diversified portfolio), it is indeed a. that becomes the relevant risk measure. The assumption that pim/am is constant for a given security's rate of return is analogous (though not coincident) with the assumption that the security be on the Capital Market Line, that is, that it be an efficient security. The assumed constancy of pim/m/ then is clearly seen to be a highly restrictive assumption and it is this assumption that must be added to be earnings' volatility argument to derive the effect of this factor on the stock beta. As indicated, this assumption is very close to a specification of how the stock's rate of return responds to the market's--a relation- ship it is sought to uncover. After all, the only real difference between cov(R.,R ) and p. is that one is a i m im dimensional measure and the other is not. Either of these quantities is a measure of comovement with the market rate of return. Clearly the former measure determines the stock's beta [with a(Rm)] as the stock beta is a dimensional measure of comovement of the security's return relative to the market. This is the reason that p. and a only deter- Im n mine the proportionality factor. It is left for a. to 1 determine the absolute magnitude of the stock beta. The conclusion of this discussion is straightforward: the assumption that p. /a is constant is an artifice de- signed to guarantee that the earnings volatility argument 78 (combined with it) "works." Given the inherent untestability of that assumption (a defect shared by cost of capital studies) it is safe to assert that by itself the assumption has no real justification. 4. Growth A first distinction is made between growth in earnings, sales, or assets and growth opportunities as opportunities to invest in projects with expected rates of return greater than their costs of capital. Modigliani and Miller [1961] have argued that the latter type of growth is the relevant concept for firm valuation. The essence of 'growth', in short, is not expansion, but the existence of opportunities to invest signifi- cant quantities of funds at higher than normal rates of return. [p. 417] One consequence of this distinction is that, as the firm's stock beta is a valuation concept for measuring that portion of the riskiness of the security which the market rewards in equilibrium pricing, insofar as mere asset expan- sion is not necessarily relevant for valuation, one would expect no necessary relationship to hold between it and the firm's stock beta. Perhaps this accounts for the erratic performance of growth measures in the empirical literature. For in some cases, asset expansion would be indicative of growth in the Modigliani and Miller sense while in other cases not. Of course it is assumed that growth opportunities do affect the stock's systematic risk, a question considered below. It is interesting to examine any technical effects on single period betas of growth in a sense to be defined below. Before doing this, we turn to the question of how growth opportunities in the Modigliani-Miller sense affect beta. The following argument has been presented by Myers [1977]. At any instant in time, a firm consists of tangible assets (in place) and intangible assets or opportu- nities for growth. These growth opportunities can be con- sidered discretionary in the sense that the firm can choose to exercise them or not. In this loose sense, such growth opportunities are "options." Since, according to the Modigliani and Miller valuation model with growth, firm value (in equilibrium) consists of the value of current assets in place plus the present value of future growth opportunities and since options written on stock securities are "riskier" than the underlying security, and, except for special cases, the same is true for options written on real assets, it follows that the greater the proportion of equity value accounted for by growth opportunities the greater will be the "riskiness" of the stock securities. While this argument is suggestive, it contains several difficulties awaiting resolution. The first is to be specific enough about the sense in which growth opportunities are options to allow the application of one of the forms of the Options Pricing Model. The second difficulty is that systematic risk measures (like expected returns) are not to be found in those models. Consequently, by riskiness of the "growth options" is meant total risk and presumably a predominance of such options implies, according to the Myers argument, a higher total risk of stock securities. A central difficulty in testing the hypothesis advocated by the argument is a specification of growth opportunities, a difficulty recognized by Modigliani and Miller and shared by cost of capital studies. As mentioned previously, in some cases asset expansion (used as a proxy for growth opportunities) will be indicative of the growth opportunities and in other cases not. Without knowing a priori the projects' costs of capital there is no way to appropriately define a sample of firms with growth opportunities to be tested. We turn now to a consideration of the effects on the firm's single period stock beta of growth where we define growth to be a predominance of later (positive) cash flows over earlier ones. Presumably, this definition embodies both asset expansion and Modigliani and Miller growth oppor- tunities. Growth clearly takes place in a multiperiod setting while the CAPM equity beta is a measure of the systematic risk of those securities borne over a single period. Thinking of growth as a predominance of later over earlier cash flows, the question is: how does the fact that more of the cash flows from the project are to be received in the future affect the single period CAPM beta? In its essence an answer can be formulated as follows. Later flows will have lower betas relative to a particular 81 initial period than earlier flows. The beta of the entire project (all the flows, early and later) will simply be a weighted average of the betas (relative to the initial period) of the individual flows. If there are relatively more later lower beta flows, then the beta of the entire series of flows will be lower than that for a series with relatively fewer later flows. Thus, growth in the sense of a predominance of later over earlier flows affects any single period beta in a negative direction. The argument rests upon the degree of correlation between the market's cash flow at t = 1 and the cash flow from the project at t = 1,2,.... As a statistical hypothesis, one would expect later cash flows at t = 1 to have a lower statistical depen- dence on the aggregate of all cash flows than earlier flows. Thus, the degree of risk resolution over the given initial period, captured by this correlation, would be lower than for the earlier more highly correlated with the t = 1 market cash flow. One can view the cash flow dependence on the t = 1 market flow as a decay process as we move forward in time. Clearly, this is a purely statistical argument reflecting the effect of technical factors on single period betas, but it must be remembered that the CAPM single period beta is a statistical concept. In a multiperiod setting in which growth takes place, one is looking at single period betas calculated over a given period. To make the model described here more precise we con- sider in detail the multiperiod valuation model proposed by Stapleton and Subrahmanyam [1979] as an alternative to the Myers and Turnbull [1977] model. The advantage of this model is that it avoids the assumption of a particular dependence structure of cash flows. Such an assumption was made by Myers and Turnbull in that expectations of project cash flows were assumed to be generated by a single-index model. Their statement of this are equations MT-2 and MT-3 (p. 322), Xt = E(Xl t_1) (1+t) MT-2 where t is a zero-mean noise term expressing the forecast error as a proportion of the expectation based upon past information. The behavior of 6t is postulated by t = bt + t' MT-3 where It represents unanticipated changes in some general economic index and b represents the sensitivity of 6 to changes in I . Further, the Stapleton and Subrahmanyam model presented here does not make arbitrary assumptions concerning the market prices of risk but derives them endogeneously. Assumptions (1) Investors are expected utility of terminal wealth maximizers where their utility functions are of the constant absolute risk aversion class. (2) Firms generate cash flows Xt which are jointly normally distributed. These two assumptions are sufficient to guarantee that the future market prices of risk are non-stochastic. Further assumptions are: (3) No debt financing is employed. (4) Limited liability does not apply. (5) Future one period interest rates are known with certainty at t = 0. Given these further assumptions, it follows that derived future stock prices are normally distributed. Given the non-quadratic assumption on utility functions, normality is required to apply the single period CAPM. Considering the simple case of a two period model (which can be generalized to any number of periods) the development proceeds as follows. The investor's multiperiod optimal portfolio problem is solved recursively using dynamic programming. For the two period model considered here this problem can be formulated in the following manner. At time t = 1, the individual investor wishes to maxi- mize his expected utility of final wealth W2 by choosing a portfolio of holdings of the cash flows at t = 2 of all firms in the market. This portfolio can be described by a vector {Z k2 of holdings of the cash flows {X }. Given the investors utility function of the CARA class: U(W2) = -a exp(-aW2), the problem is a constrained maximization problem, 84 u(W IXI) = max E[u(W2)] = max E[-ai exp(-a W2)], (2.1) Zi 7i 12 12 subject to W = 2 P + M (2.2) 1 12 12 1 W2 = M2 + 2 X2 (2 where M1 is the amount of risk-free lending undertaken at t = 1, r2 is the period 2 riskless interest rate, and P12 is the vector of t = 1 prices of the X2 cash flows. The solution to the maximization problem in equations (2.1), (2.2) and (2.3) is the vector defined by 2 = 21[E(X2 Xl) r2P12 X], (2.4) where S= [cov(Xxk Xj kj (2.5) is the conditional (on the cash flow at t = 1 vector) variance-covariance matrix. It is assumed here that the characteristics of the state of the world 1 relevant for expectations of the t = 2 cash flows are summarized in the t = 1 cash flows of all firms in the economy X1. This allows us to replace the state of the world information set Sat t = 1 by the cash flow vector of outcomes. The solution in (2.4) and (2.5) is then used to gener- ate equilibrium prices P12 X1 conditional on X1 at time t = 1 of the cash flow vector X2. This is simply a matter of equating supply to demand. Since the Z must aggregate 12 to the unit vector of supplies of the total proportional holdings of the {X k, we obtain from equation (2.4) that i = 2 L a 21[E(X2X1) r2P12Xl]. (2.6) 1 Solving this yields L- (1E( (2.7) 21X r [E(X21X1) - where 1 X (2.8) 2 (i/ai) is the market price of risk. We have, then, equilibrium pricing at t = 1 of the cash flow vector X2 to be received at t = 2 in terms of the relevant parameters of the model. This yields the optimal value of the utility function u(WllX1) = -a exp(-a{Wlr2 + A2}), (2.9) where ,2 A 2= -2 1 (2.10) 2 2a 2 These equations describe the derived utility of wealth function. If r2 and A2 are constants, then u(W IX1) is a 86 non-state dependent exponential function of W1 alone. But we have assumed r2 to be non-stochastic, and 12 will be non- stochastic since it depends only on the coefficients of absolute risk aversion of all individuals: a known con- stant. The assumption of CARA utility functions then guar- antees the non-stochastic character of the market price of risk X2. Further, 02 is non-stochastic, i.e. independent of Xl, as a consequence of the assumed joint normality of the {Xk} [Anderson, 1958]. Thus, u(W1|X1) = u(W1) is a non- state-dependent utility function as is required for the period-by-period application of the CAPM [Fama, 1970]. Essentially, since r2 and X2 are non-stochastic by assump- tion and a is independent of wealth, the only source of state dependence is through Q2' the conditional covariance matrix which represents risky investment opportunities. The assumption of joint normality of all the cash flows in the economy is designed to rule out the state dependence of the (risky) investment opportunity set. One should note that exponential utility and joint normality are sufficient conditions to avoid state dependence of the derived utility function. Given the derived utility function -a exp(-a{W1r2+A2}) = u(W1), equilibrium prices at t = 0 of the cash flows X1 and X2 are derived by solving backwards from the solution derived for t = 1. That is, the individual's maximization problem at t = 0 is as follows: max E[-a exp(-a{W r2 + A2})] (2.11) [Z 0 [202 subject to 0 01 01 02 02 0 W1 = Mr + 1X + 2 (2.13) where Zol are the portfolio weights of proportionate hold- ings of the cash flows X l} and Z02 are those in the vector of cash flows {X2 At t = 1, these assets yield proportions of the cash flows {X1} and values of the {X2} denoted by 12' The solution to equations (2.11), (2.12) and (2.13) are the vectors -1 Z0 = (1/ar2) Q [E(X1) rlP01] (2.14) -1 where Q1 is the variance-covariance matrix of cash flows X1 and prices P 12 S = 1 (2.16) F = [cov(X Xk)] G = [cov(X P )] 1 12 H = [cov(P Pk P 12 12 Equilibrium prices (at t=O) are P01 = /r [E(X) X 1] 011 1 1 1 (2.17) P02 = 1/rl [E(P12) X1i] (2.18) where X = (2.19) 1I/(ai.r2) i The relationship that plays a central role, between X and 12 is simply that X1 = X2r2. (2.20) That is, the current market price of risk equals the future price compounded at the future risk-free interest rate. These results are used now to derive the t = 0 equilib- rium price of the t = 2 cash flow of firm j, X2 as a certain- ty equivalent in terms of the period 2 market price of risk X2 [a simplification allowed by equation (2.20)], the period 1 and period 2 risk-free rates, rl and r2, and covariances between the cash flow at t = 2, X and the 2 market at t = 2, X2, and the compounded market cash flow r X This equilibrium price is derived as follows. Take the expected value (at t=0) of equation (2.7) and substitute the result in equation (2.18). This yields 1 2 S= [2 {E(X2) Q 222t} ti] S1- [E(X2) X t(^l + rl ] (2.21) rl1r 22 2 2" |

Full Text |

PAGE 1 A STUDY OF THE SYSTEMATIC COMPONENT OF RISK IN COMMON STOCKS BY David Harold Goldenberg A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1981 PAGE 2 ACKNOWLEDGEMENTS I wish to thank Professor Fred D. Arditti for intro ducing me to the subtleties in the content and methodology of finance through his invaluable lectures and personal instruction, for providing characteristically acute insights into the nature of systematic risk, and for his constant encouragement. I am grateful to Professor G. S. Maddala for providing valuable insights into the econometric methodology appropriate to the modelling and estimation of systematic risk and for his very helpful assistance during the execu tion of the study. Helpful discussions with Professor Raymond Chiang and Professor Richard Cohn are greatl y appreciated PAGE 3 TABLE OF CONTENTS Page ACKNOWLEDGEMENTS. . . . . . . . . . . . . . . ii LIST OF TABLES. . . . . . . . . . . . . . . . . . . vi ABSTRACT. . . . . . . . . . . . . . . . . . . . . X INTRODUCTION. . . . . . . . . . . . . . . . 1 CHAPTER I REVIEW OF THE LITERATURE .............. 5 Introduction. . . . . . . . . . . . . . 5 1. Ad Hoc Studies Using Accounting Numbers. . . . . . . . . . . . . . 6 Ball and Brown. . . . . . . . . . . 6 Beaver, Kettler and Scholes ....... 11 Gonedes. . . . . . . . . . . . . 22 2. Theoretical Basis Study ........... 25 3. Barr-Rosenberg and Associates' Work. 28 Barr-Rosenberg and McKibben ........ 28 Barr-Rosenberg and Marathe ........ 32 Barr-Rosenberg and Marathe ....... 40 4. Study of the Effect of Financial Leverage............................ 41 5. Studies of the Effect of Market Power . . . . . . . 4 8 Sullivan........................... 48 Thomadaki s . . . . . . . . . . . . 51 Sullivan........................... 54 6. Studies of the Effect of Operating Leverage............................ 56 Lev . . . . . . . . 5 6 Rubinstein......................... 62 Subrahmanyam and Thomadakis ........ 68 iii PAGE 4 CHAPTER II CHAPTER III Page MICROECONOMIC FACTORS AFFECTING EQUITY BETAS 7 0 Introduction ........................... 70 1. F inane ial Lever age. . . . . . . . . 71 2. Volatility of Operating Earnings ... 74 3 Growth. . . . . . . . . . . . . . . 7 8 Assumptions . . . . . . . . . . . . 8 2 Single Period Betas in a Multiperiod setting................ 92 LelllIIla. . . . . . . . . . 9 6 Proof . . . . . . . . . . . . 9 6 Duration and Asset Betas ......... 100 4. Monopoly Power and the Labor-Capital Ratio. . . . . . . . . . . . . . . . 101 Sources of Uncertainty .......... 102 Uncertainty in the price of output ................... 102 Uncertainty in the wage rate .. 103 Relationship between Demand Uncer tainty and Uncertainty in the Wage Rate. . . . . . . . . . 104 Competitive Equilibrium Risk Determination .................... 106 Derivation of the Systematic Risk of the Purely Competitive Firm 108 Systematic Risk and Monopoly Power. 112 Optimal Valuation and Beta for the Monopolistic Firm ............ 115 5. The Model of Systematic Risk and Hypotheses to be Tested .......... 116 ESTIMATION TECHNIQUES AND EMPIRICAL RESULTS.. . . . . . . . . . . . . . . . 119 1. Estimation Techniques .............. 119 Fixed Effects ..................... 119 Random Effects .................... 121 Prior Likelihood Estimation ....... 121 2. Empirical Results with Five Descriptors. . . . . . . . . . . . . . . . 125 Fixed Effects Estimation ........... 125 iv PAGE 5 Page Definitions and data sources for descriptors ......... 125 Parameter Estimates .......... 127 Beta Estimates .............. 129 GLS Estimation ................ 133 Prior Likelihood Estimation ...... 134 Prediction ..................... 135 Out of sample prediction ...... 135 Within sample prediction ...... 138 3. Empirical Results with Twenty-nine Descriptor Data Set .............. 141 Fixed Effects Estimation .......... 141 Names of descriptors ........ 141 Classification of descriptors. 143 Descriptor definitions ...... 144 Parameter estimates (a assumed constant) ............... 149 Parameter estimates (a variable) ............... 149 Beta estimates ................ 149 GLS Estimation ................... 155 Prior Likelihood Estimation ....... 156 Prediction ...................... 156 4. Comparison of the Predictive Perfor mance of the LSDV 29 and the LSDV 5 Betas ............................... 159 SUMMARY AND CONCLUSIONS OF THEORETICAL AND EMPIRICAL RESULTS ............................................... 160 APPENDIX 1 RELATIVE MAGNITUDES OF THE FIRM AND LABOR BETAS .......................... 164 APPENDIX 2 DERIVATION OF THE MONOPOLY BETA ......... 167 REFERENCES ............................................ 168 BIOGRAPHICAL SKETCH 171 V PAGE 6 Table 1.1 1. 2 1. 3 1. 4 1.5 1. 6 1. 7 1.8 1. 9 1.11 LIST OF TABLES Coefficients of Correlation between Various Measures of the Proportion of Variability in a Firm's Income That Is Due to Market Effects Variables Not Standardized ......... Coefficients of Correlation between Various Measures of the Covariance between a Firm's Index and a Market Index of Income Variables Page 8 Standardized.. . . . . . . . . . . . . . . . 10 Association between Market Determined Risk Measure in Period One (1947-56) Versus Period Two (1957-65) ......................... 15 Contemporaneous Association between Market Determined Measure of Risk and Seven Accounting Risk Measures . . . . . . . . . . . . . . 2 O Analysis of Forecast Errors. . . . . . . . . 23 Correlation Coefficients, R, between Estimates from Market Model (M) and Four Accounting Number Models (A) ................ 26 Correlation Analysis......................... 29 Prediction Rules for Systematic Risk Based on Fundamental Descriptors and Industry Groups .. 35 Prediction Rules for Systematic Risk Based on Fundamental Descriptors Including Market Variability Descriptors...................... 37 Unbiased Estimates of the Performance of Alternative Prediction Rules in the Historical Period (Predicted Variance as a Multiple of the Variance Predicted by a Widely Utilized Prediction Rule) ....................... 39 Summary Results over 304 Firms for Levered and Unlevered Alphas and Betas ............ .. 45 vi PAGE 7 Table 1.12 1.13 1.14 1.15 1.16 1.17 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 Page Market Adjustment Factor Regressions over Alternative Periods ......................... 47 Mean and Standard Deviation of Industry Betas 49 Industry Concentration and Future Monopoly Power........................................ 53 Monthly Betas, Leveraged and Unleveraged, Regressed on Market Power and Control Variables.. . . . . . . . . . . . . . . . . . 55 Estimates of Average Variable Cost Per Unit .. 61 Regression Estimates for Systematic Risk on Average variable Costs Per Unit ............. 63 Estimated Descriptor Coefficients Used in Generating Betas for the Five Descriptor Data Set .......................................... 128 Yearly Means of Fixed Effects, Historical, and vaiicek Betas for the Five Descriptor Data Set ..................................... 130 Variances of Fixed Effects, Historical, and Vasicek Betas Based on the Five Descriptor Data Set ..................................... 131 Estimated Descriptor Coefficients for Constant Alpha Model for the Five Descriptor Data Set ..................................... 132 Estimated Descriptor Coefficients Used in Generating Alphas and Betas for Out of Sample Prediction ........................... 136 Correlation Matrix of Descriptors and Descriptors Times the Market for the Five Descriptor Data Set .......................... 137 Out of Sample Prediction Results for the Five Descriptor Data Set 139 Within Sample Prediction Results for the Five Descriptor Data Set .......................... 140 Estimated Descriptor Coefficients Used in Generating Betas for the Twenty-nine Descriptor Data Set (Alpha Constrained to Be Constant) . . . . . . . . . . . . . . . . . . . 15 0 vii PAGE 8 Table 3.9 3.10 3.11 3.12 3.13 3.14 3.15 Page Estimated Descriptor Coefficients Used in Generating Betas for the Twenty-nine Descriptor Data Set (Alpha Constrained to Be Constant) ................................. 150 Estimated Descriptor Coefficients Used in Generating Betas for the Twenty-nine Descrip tor Data Set (Alpha Allowed to Vary According to Equation 3.2) ............................. 151 Estimator Descriptor Coefficients Used in Generating Alphas for the Twenty-nine Descriptor Data Set ..................... 152 Yearly Means of Fixed Effects Five and Twenty-nine Descriptors and Historical Betas. 153 Variances of Fixed Effects Five and Twentynine Descriptors and Historical Betas ........ 154 Out of Sample Prediction Results for the Twenty-nine Descriptor Data Set ............ 157 Within Sample Prediction Results for the Twenty-nine Descriptor Data Set .............. 158 viii PAGE 9 Authors Table Page Thomadaki s ............................ 1.14 . . . . . 53 Sullivan .............................. 1.15 . . . . . 55 Lev .................................. 1.16 . . . . . 61 1.17 . . . . . 63 ix PAGE 10 Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy A STUDY OF THE SYSTEMATIC COMPONENT OF RISK IN COMMON STOCKS By David Harold Goldenberg June 1981 Chairman: Dr. G. S. Maddala Cochairman: Dr. F. D. Arditti Major Department: Finance, Insurance, and Real Estate The theoretical basis for the inclusion of various microeconomic factors as determinants of systematic risk is examined. This set includes financial leverage, variability of operating earnings, and growth in several senses. A recent model incorporating the firm's behavior in its input and output markets is generalized by including risky human capital in the market portfolio. The condition on the covariability between the sources of uncertainty in the model under which beta will be positively related to the labor capital ratio and negatively related to monopoly power is explicated. ix PAGE 11 Much work has been done on this problem by Barr-Rosenberg and associates, but little or no theoretical justification was offered for the choice of variables included in their regressions. The study also corrects some of the deficiencies in the econometric methods employed in those studies. The empirical results on non-stationarity of the coef ficients of the market model are taken into account by allowing them to vary by firm and time period as a linear function of the set of descriptors plus a firm specific error term that does not vary with time. The fixed effects or least-squares with dummy variables estimation technique is applied. The coefficients of the growth rate in assets and the Lerner Index of monopoly power are negative. The labor-capital ratio has a positive sign. These results conform to the theory. Financial leverage and variability of earnings have negative signs. The former result may be due to the use of book values for debt rather than the appropriate market values. The GLS or random effects estimates do not differ significantly from the fixed effects estimates. The prior likelihood estimates designed to provide best linear unbiased estimates of the firm specific effects also do not differ significantly from the fixed effects estimates. The criterion of mean-square error was employed in comparing classical betas to the descriptor betas as predictors of returns. Predicted returns, conditional upon X PAGE 12 the market return in the prediction year, were generated from information from the previous set of years for both types of betas. The mean-square error of the naive predic tor obtained by setting alpha equal to zero and beta equal to unity was also computed. The order of performance was naive, classical, then descriptors. When a within sample prediction was performed by using all the available data to estimate the coefficients used in generating betas, the classical betas obtained by including the prediction year's data in the usual regression never outperformed the naive predictor while the five descriptor based betas did so in two years out of four--1974 and 1975. This suggests that the descriptor based betas may be useful as predictors over periods in which structural changes in the coefficients of the market model take place. The entire set of procedures was repeated on a set of descriptors chosen in the Barr-Rosenberg manner. The signs of the estimated coefficients made no apparent sense. As predictors of returns for the out of sample prediction, they did not outperform the naive. They were outperformed by the five descriptor betas in two years and outperformed them in the other two years. For the within sample predic tion they were always better than the naive, and they were better than the five descriptor set in three years out of four--1973, 1975 and 1976. Only in 1974 did the five descriptor set provide better predictors. xi PAGE 13 INTRODUCTION The total risk of a firm's equity securities, identi fied as the variance of the rate of return of those securi ties over a given period, can be decomposed into two compo nents. One component is termed systematic: it is the sensitivity of the rate of return on the firm's common stock to the rate of return on the market portfolio. This compo nent is the market related risk that cannot be diversified away through the process of portfolio formation. Tradition ally a measure of the non-diversifiable risk component is given by the slope coefficient, assumed constant over the estimating period, in the market model's historical regres sion of the stock's rate of return on the rate of return of a proxy for the market portfolio of all risky securities. This simple model of the stochastic process generating security returns indicates the second component of the risk of a firm's equity securities. The market model indicates that a portion of the variance of the rate of return cannot be accounted for by movements in the rate of return of the market portfolio. This component of the total risk of the security is specific to the firm in question. However, it is possible to diversify away this risk component by com bining the security with others in portfolios. In fact, the 1 PAGE 14 2 Capital Asset Pricing Model tells us that equilibrium prices and rates of return on a firm's securities are linear functions solely of the first component of total risk: the systematic component measured by the beta coefficient. It is clearly of great interest to estimate and predict the systematic risk component of a firm's common stock securi ties. This study will concern itself only with the estima tion and consequent prediction of this systematic component. While the beta coefficient as defined by the CAPM identifies the systematic component of risk as that component relevant for equilibrium pricing of a firm's risky securi ties, it provides little information concerning the sources of such systematic risk. The goal is to relate the firm's unobservable beta to the firm's characteristics and to micro-economic variables. Traditionally, financial leverage and volatility of earnings have been considered as micro-economic determinants of systematic risk. Attention has recently focused upon "growth" and the firm's operations in its input and output markets as factors bearing upon the beta coefficient. The present study includes the traditional variables as well as the labor-capital ratio and monopoly power in an attempt to incorporate these factors. The arguments for all of the above variables will be considered in detail as the basis for the inclusion of a set of factors that can be theoret ically justified as determinants of systematic risk. In the course of this analysis the recent arguments for the labor capital ratio and monopoly power are generalized. PAGE 15 3 The second major purpose of the study is to correctly apply the available econometric techniques that appear in the literature on pooling cross-sectional and time series data to the problem of estimating s y stematic risk as a linear function of the set of descriptors developed above plus a firm specific effect. This model has appeared in the early literature on accounting numbers in the form of Beaver, Kettler, and Scholes's instrumental variables estimation procedure for estimating systematic risk. That study did not incorporate firm specific effects. Such effects were added to the model as the residual component of beta that could not be explained by the included set of descriptors. This was done by Barr-Rosenberg and associates. The appro priate estimation techniques were not applied. Accordingly, the fixed effects and the random effects specification of this model are considered. The model is estimated under the fixed effects specification. Then, the potential gain from applying the generalized least-squares estimation as warranted by the random effects specification is measured in an a priori manner. In principle, with known variances of the firm specific effects, the prior likelihood estimation procedure leads to best linear unbiased predictors of the firm specific effects. The additional explanatory power provided by these estimates is computed by considering the magnitudes of the adjustment factors to be used to derive the prior likelihood estimates. The criterion for usefulness of the betas generated by the procedures outlined above is the mean-square error PAGE 16 4 in the prediction of returns generated by those estimates. The mean-square-error of the fixed effects betas is compared to that of the usual historical beta and that of the naive predictor with alpha set equal to zero and beta set equal to unity. This is done both for out of sample and within sample prediction. Finally, the procedures described are carried out using a set of 29 Barr-Rosenberg type descriptors. The results are compared to those for the five theoretically justified descriptors. PAGE 17 CHAPTER I REVIEW OF THE LITERATURE 1. Introduction Those studies that bear directly on the present one are considered here. The review is divided into several sections. (1) Ad Hoc Studies Using Accounting Numbers: Ball and Brown [1969]; Beaver, Kettler and Scholes [1970]; and Gonedes [1973]. (2) Study Attempting to Provide a Theoretical Basis for the Use of Accounting Numbers as Proxies for Systematic Risk: Pettit and Westerfield [1972]. (3) Barr-Rosenberg and Associates' Work: Barr-Rosenberg and McKibben [1973], Barr-Rosenberg and Marathe [1975], and Barr-Rosenberg and Marathe [1979]. (4) Study of the Effect of Financial Leverage on Beta: Hamada [1972). (5) Studies of the Effect of Market Power on Beta: Sullivan [1977], Thomadakis [1977), Sullivan [1978], and Subrahmanyam and Thomadakis [1980). (6) Studies of the effect of Operating Leverage on Beta: Lev [1974), Rubinstein [1973), and Subrahmanyam and Thomadakis [1980]. The studies concerning the growth variable are consid ered in Chapter II, where the relationship is explained in the appropriate multiperiod framework. 5 PAGE 18 6 1. Ad Hoc Studies Using Accounting Numbers Ball and Brown The Ball and Brown [1969] study has as its goal the use of "income numbers" as predictors of systematic risk. Their "income numbers" are simple regression coefficients in regressions of accounting income of a given firm in a given year on a market index of accounting income. The income variable was alternatively taken to be operating income, net income, and earnings per share. Initially, two regression models were postulated relating income in levels between the firm and the market and then income in first differences between the firm and the market. Three regressions were run for each of the three income definitions. The simple R~ l. were computed for these time series regressions for each firm i and then these were correlated with the R 2 from the regression of the firm's rate of return on the rate of return on the market. Product correlation and rank-order correlation coefficients were calculated. The regression models employed were as follows: I. it I. l. t I = = 6. it n P. it it al, + a2.M + U, l. l. t l. t a1' 1 + a' M + U' 2i t it = a" + a' '.M + U ' li 21 t it a'''+ a'''M + U''' li 2i t it income, in levels income, in first differences income/market value of equity in levels income/market value of equity in first differences ( 1.1) ( 1. 2) (1. 3) (1.4) PAGE 19 7 PR. = b 1 + b 2 .L + V im i i m im ( 1. 5) where PR. is the price relative for the common stock of im firm i in month m and L is a proxy for the rate of return m on the market portfolio in month m. The coefficient b 2 i is presumably an estimate of the firm's equity beta as this regression is essentially the market model. The other relevant variables are defined as follows: I : accounting income of firm i in year t it M. : a market index of accounting income. it Ball and Brown take this market index as an average that, for a given firm i, excludes that firm; being an average over the remaining N-1 firms. Why they do this is not obvious as a market index, according to Capital Asset Pricing Theory, contains all firms in the market. The object of the correlation analysis is to determine to what extent estimates of systematic risk from the market model are correlated with the accounting income response coefficients. The regressions were run for 261 firms over the period 1946-1966. Selected results appear in Table 1.1. Ball and Brown conclude from these results that "better" results are obtained when the regressions are run in first differences [see Table 1.1, columns (3), (5) and (7)]. The next set of regressions attempted to control for the size of firm and differences in their accounting prac tices. To do so, all variables were standardized by dividing PAGE 20 8 Table 1.1 Coefficients of Correlation between various Measures of the Proportion of Variability in a Firm's Income That Is Due to Market Effects Variables Not Standardized ( 2) ( 3) ( 4) ( 5) (6) ( 7) Product Moment Correlation (1) .00 .47 .03 .39 -.05 .42 Spearman's Rank-Order Cor. (1) -.02 .46 .02 .39 -.05 .41 (1) Stock return regression (1.5) (2) Operating income in levels (3) Operating income in first differences (4) Net income in levels (5) Net income in first differences (6) E. P. s. in levels (7) E. P. S. in first differences Source: Ball and Brown [1969, page 319] PAGE 21 9 through by the market value of equity. Using the above income definition Iit' the results appear in Table 1.2. Ball and Brown draw the following conclusions from these results: (1) comovement's in accounting earnings of firms predict moderately well the firm's systematic risk. From Table l the highest product moment correlation coefficient is .47. From Table 2 it is .59. Ball and Brown conclude that comovements in accounting incomes explain approximately 20-25% of the cross-sectional variability in estimated degrees of association with the market (from Table 1) and 35-40% in systematic risk (from Table 2). [p. 319] It is not clear what the difference is between en estimates of comovement of the firm's rate of return with the rate of return on the market and estimates of systematic risk. The extent of the explanatory power is "explained" by the fact that all variables are measured with error. They conclude that accounting numbers may be even better predictors of systematic risk than these results indicate. No evidence is given to support this claim. (2) Better predictions are obtained when the variables are measured in first differences and the results are sensitive to the income definition utilized. The study, being an early attempt to relate accounting variables to systematic risk, suffers from the lack of any theoretical argument showing a relationship between account ing betas and systematic risk as defined in Capital Market Theory. As such it is not clear whether to expect a consis tent correlation between such accounting betas and system atic risk. PAGE 22 ~------------10 Table 1.2 Coefficients of Correlation between Various Measures of the Covariance between a Firm's Index and a Market Index of Income Variables Standardized ( 2) ( 3) ( 4) ( 5) ( 6) ( 7) Product Moment Correlation (1) Spearman's Rank-Order Cor. (1) .45 .59 .39 .53 .41 .53 .45 .64 .42 .58 .43 .59 ( 1) Stock return regression ( 2) Operating income in levels ( 3) Operating income in first differences ( 4) Net income in levels (5) Net income in first differences (6) Available for common in levels ( 7) Available for common in first differences (Table 2, p. 320) PAGE 23 11 Beaver, Kettler and Scholes Beaver, Kettler and Scholes (1970) attempt to give a rationale for the use of accounting based risk measures as proxies for the systematic risk of the firm's equity securi ties. They note that such measures highlight several aspects of the uncertainty associated with the earnings (or return) stream of the firm. Further, the accounting risk measures are surrogates for total risk. What is the link between total and systematic risk? If the systematic and individualistic components of risk are positively correlated (at the extreme, per fectly correlated), then it is reasonable to view the accounting measures as surrogates for systematic risk as well. The evidence does suggest that positive correlation does exist (e.g. securities with a larger than average 6 tend to have a larger than average variance of the individualistic component. [p. 659, last paragraph] The statement is dubious. There is no a priori reason to believe that factors positively related to systematic risk ought to be positively related to the non-systematic component of risk as well. High beta firms may have small individualistic or firm-specific risk components. They may be affected primarily by market events with little left over in the way of residual risk. The empirical fact referred to may be based upon faulty beta estimation procedures. In any case, if systematic risk and total risk are highly correlated as would be the case if the non-systematic component were highly correlated with the systematic component, Capital Market Theory loses much of its relevance in specifying the PAGE 24 12 systematic component of risk as that component which is relevant for equilibrium pricing. Given this argument, the Beaver, Kettler and Scholes results can be viewed as an attempt to specify the compo nents of total risk. The basic concern of the study is to answer the ques tion: To what extent is a strategy of selecting portfolios according to the traditional accounting risk measures equivalent to a strategy that uses the market determined risk measures? The list of factors presented in their study is: (1) dividend payout (2) growth (3) financial leverage (4) liquidity (5) asset size (6) variability of earnings (7) covariability of earnings with the earnings of the market. The arguments given for these factors are: (1) Firms follow a policy of dividend stabilization: once a particular dividend level is established they will be reluctant to cut back. Also, firms are reluctant to pay out more than 100% of earnings in any single fiscal period. Given these tenden cies on the part of firms, those firms with more volatile earnings' streams will adopt a lower payout ratio. This argument hints at a relationship between the payout ratio and the total risk of the firm's equity securities. PAGE 25 13 (2) Defining growth as the existence of "excessive" earnings opportunities for the firm, Beaver, Kettler and Scholes argue that there is no reason to assume that the growth assets need be more risky than assets already in place. Rather, asset expansion would occur in areas where the prospective earnings stream generated by these newly acquired assets would be more volatile than that generated by the firm's current assets. Presumably a more volatile earnings stream implies a higher beta, although the connection is hard to see. (3) The usual argument is given for financial leverage. (4) Beaver, Kettler and Scholes do not suspect that liquidity in the sense of the fraction of assets that are current assets held to be related to beta. Rather, they suspect that the differences in systematic risk among firms come about as a result of the differences in the riskiness of their non-current assets. However, they use the current ratio as a measure of liquidity. (5) Asset size is included on the grounds that: (a) asset size is highly correlated with the risk of default on bonds outstanding, (b) If individual asset returns are not perfectly correlated then larger firms will have a lower total risk of equity securities than smaller firms. This results because larger firms, by diversifying their asset holdings, reduce their total risk. The connection with systematic risk remains to be demonstrated. (6) Variability of earnings is given an intuitive argument to establish its effect on total risk. (7) Covariability of earnings is introduced as the slope coefficient of the regression of the earnings price ratio of the individual firm on an economy wide average of earnings price ratios as the market variable. No theoretical argument is given for this variable. In the Beaver, Kettler and Scholes regressions, the dependent variable is an estimate of B. obtained by running l a time-series regression of the security's rate of return on a proxy for the rate of return on the market. This was done PAGE 26 14 for two periods: January 1947 January 1957 and December 1956 December 1965. Monthly rates of return were used. The implicit assumption is that beta was stationary over each of these subperiods. Beaver, Kettler and Scholes test their hypothesis by computing correlation coefficients between corresponding beta estimates from the two subperiods at the individual and at the portfolio level. They find a correlation coefficient of .594 at the individual level and .965 for portfolios of size 20. They conclude that the data indicate that stationarity is not violated. One has to ask, however, whether the process of aggregation into portfolios automatically results in higher correlation coefficients (after adjusting as Beaver, Kettler and Scholes have done for the loss of degrees of freedom). Results of this stationarity test are given in Table 1.3. We present here the definitions employed by Beaver, Kettler and Scholes of their accounting measures. (1) Average Payout = T T I Cash dividends t=l I Income available to common t=l Total assetsT (2) Average Asset Growth = Total assets 0 T I ( 3) Average Leverage = t=l T n Total senior securities t Total assets t T PAGE 27 15 Table 1.3 Association between Market Determined Risk Measure in Period One (1947-56) Versus Period Two (1957-65) Number of Securities in Portfolio 1 5 20 Table 2, p. 665 Rank Correlation .625 .876 .989 Product-Moment Correlation .594 .876 .965 PAGE 28 16 T (4) Average Asset Size = l Total assets t=l T n T Current assetst (5) Average Liquidity = l Current liabilitiest t=l ( 6) T n Earnings Variability = Et where P t-1 and E p = T Income available to commont = Market value of common stockt-l T E I~ t=l t-1 T E t Cov -p--,Mt (7) Accounting Beta = t-1 T where Mt = I 1 p. 1 t= l.,tN T = the number of years in the subperiod. N = the number of firms in the market. One observes that (6) is a measure of the standard deviation of the rate of return to stockholders and as such is a measure of cri. Variable (7) is an estimate of Si, where rates of return are defined using accounting earnings, and the regression is a time-series one with T = 9 or 10 PAGE 29 17 (depending upon the subperiod). The market index used is equally weighted. Beaver, Kettler and Scholes call this estimate B . The idea here is that everything is defined i here in terms of accounting earnings rather than in terms of prices and dividends. They compare these estimates of the true beta to those from their historical regressions of ordinary rates of return on ordinary rates of return on the market. They find that the dispersion of their accounting betas is almost four times as large as that of the usual historical betas. This loss of efficiency is attributed to relative sample sizes; nine for accounting betas, 120 and 108 for historical betas. Further, the accounting based market measure exhibits first-order serial correlation unlike the market determined market index (used to compute historical betas). Also, 9% and 12% of the accounting B. 's i were negative in each subperiod. Beaver, Kettler and Scholes suggest increasing the time period to 19 years for computa tion of accounting betas. When they do this, they find that the standard deviation drops from 1.164 (1.280 in subperiod 2) to .791 over the entire 19 year period. This is still substantially larger than .336 (.342) the standard deviation of historical betas in subperiod 1 and subperiod 2. Beaver, Kettler and Scholes further find evidence of non stationarity in accounting betas. While the other accounting variables are relatively stable as measured by rank correla tion and product moment correlation coefficients, these correlation coefficients come out to be .034 (-.060) for the PAGE 30 18 accounting beta. Recall that their data yielded correlation coefficients of .625 (.594) for the historical betas between the two subperiods. Beaver, Kettler and Scholes conclude that the accounting beta is subject to large errors in measurement and they virtually suggest searching for other accounting measures of risk. One wonders whether the methodology of correlating two variables both measured with a great deal of error, the historical beta and the accounting beta, is fruitful. Any discovered correlation could be the result of correlation between the errors in the measured variables. On the other hand, while the two variables may in fact be highly correlated, the random error terms in their measurements could obscure such correlation. This type of consideration is at a level secondary to that of questioning the adequacy of the historical betas, a variable measured with error and assumed to be stationary, as an estimate of the true beta. On the other hand, it is hard to see what else one could do at the level of correlation analysis. Beaver, Kettler and Scholes go on to suggest the earnings variability measure as an alternative risk measure to the accounting beta. They note in support of this suggestion that it has, over the samples, approximately the same degree of stationarity as the historical beta. This argument is questionable. While a necessary condition for two variables to be highly correlated would appear to be that they have the same degree of stationarity, it is by no means a sufficient condition. PAGE 31 19 To determine the degree of correlation between the accounting based risk measures and the historical beta estimate, Beaver, Kettler and Scholes calculate cross sectional correlation coefficients between the given account ing variables and the historical beta at the individual and portfolio levels. They find at the individual level the following ranking by degree of correlation: ( 1) earnings variability (2) payout variable ( 3) accounting beta ( 4) liquidity. At the portfolio level the following ranking is obtained: (1) earnings variability (2) payout variable (3) accounting beta. These results are contained in Table 1.4. In interpret ing these results, note that a rank correlation in absolute value greater than .10 is significant at the .05 level. The rank correlation coefficients are given in the table. Portfolio correlations were obtained first by forming port folios of five securities each where the securities were ranked by the magnitude of the given accounting variable. Their betas were calculated and correlated with the usual beta of each portfolio obtained as the arithmetic mean of the betas of the five securities in each portfolio. In interpreting the portfolio results one notes that one would expect the portfolio correlations to be higher PAGE 32 20 Table 1.4 Contemporaneous Association between Market Determined Measure of Risk and Seven Accounting Risk Measures PERIOD ONE PERIOD Individual Portfolio Individual Variable Level Level Level Payout -.49 -.79 -.29 Growth .27 .56 .01 Leverage .23 .41 .22 Liquidity -.13 -.35 .05 Size -.06 -.09 -.16 Earnings 66 .90 .45 Variability Accounting .44 .68 .23 Beta Table 5, p. 669 TWO Portfolio Level -.so .02 .48 .04 -.30 .82 .46 PAGE 33 21 because aggregation reduces the variance of the unexplained error term. But, Beaver, Kettler and Scholes argue, aggrega tion could also result in a reduction in the variance of the dependent variable, the portfolio beta estimate. This reduction could offset the increase in correlation described above. Beaver, Kettler and Scholes do not develop the statistics of this argument nor test it on their sample. Rather, they argue that portfolios, not single securities, are the relevant investment instruments held by individuals. The final part of the Beaver, Kettler and Scholes study is concerned with the forecasting ability of the accounting risk measures. Beaver, Kettler and Scholes use an instru mental variables method to remove the error in the variables. They postulate the following model = (1.6) where S 1 is the true unobservable beta, and z 1 ... ,Zn are n accounting risk measures. Note that this model assumes that beta is fully determined as a linear function of then accounting risk measures without any random firm specific error term. To obtain estimates of ~ 0 ~ 1 ... ~ n the follow ing regression of the usual beta estimate on the accounting descriptors is run: = (1.7) A Then S 1 W = C 0 + c 1 z 1 + ... + CnZn is a measure of S 1 free from errors. The instrumental variables used were payout, PAGE 34 22 growth, and earnings variability. The resulting beta estimates were used to predict second period betas. These estimates were compared to the naive estimate which assumed that the second period beta would be equal to the first period beta. The results in Table 1.5 show a decrease in mean square error over the naive model. Gonedes Gonedes [1973] sets out to test for an association between accounting based and market based measures of systematic risk. In so doing he rejects the results of the Beaver, Kettler and Scholes study which, he claims are based upon a spurious correlation induced by scaling income numbers by market prices, such prices being implicit in beta. The same basic criticism is applied to Ball and Brown's results. When income numbers are scaled by assets, he does not find the significant association that Beaver, Kettler and Scholes find. Gonedes explains the improvement in results when first differences in income numbers or scaled first differ ences in income numbers are run as follows: "Presumably, the transformations induce 'better' specifications of the underlying stochastic processes" (p. 433). Why this is so is not made clear. Gonedes runs the following model, = (1.8) PAGE 35 23 Table 1.5 Analysis of Forecast Errors MSE NAIVE BETA ESTIMATES Individual Portfolios Securities A(a) B(b) .093 .030 .027 INSTRUMENTAL VARIABLE BETA ESTIMATES Individual Portfolios Securities A(a) B(b) .089 .016 .016 (a) Portfolios ranked according to the historical estimate. (b) Portfolios ranked according to the instrumental variable beta estimate. (Table 7, p. 677) PAGE 36 24 where 6 represents first differencing, Y. = the income number of firm i in period t it A it = the economy-wide income number for period t = the industry income number for the industry to which the i-th firm belongs for period t = the total assets number of the i-th firm at the beginning of period t = the economy-wide total asset number at the beginning of period t = the total asset number for the industry grouping of the i-th firm at the beginning of period t. The sample consisted of 99 firms randomly drawn from the population for which all the necessary data,:,..,ere available. To obtain beta estimates the market model was run using logarithmic rates of return. Monthly observations were employed. The parameters of the market model were estimated for three year, five year, seven year, ten year, and twenty one year intervals. The first six monthly observations from 1960 and the first six from 1968 were reserved for prediction tests. Gonedes's results provide evidence of nonstationarity in beta estimates obtained from the market model. He suggests that the proper criterion to be used in deciding upon an appropriate interval over which to estimate the market model is predictive efficiency; the market model estimates are used to derive predicted returns and the mean square errors of these predicted returns as predictors of PAGE 37 25 the reserved 1960 (or 1968) returns for different time intervals are calculated. His results suggest that a seven year interval provides estimates with the greatest predic tive efficiency and this interval was used in computing beta estimates for correlation tests with c 4 i in the above equation. The procedure is to correlate the coefficients of determination for each firm from the market model with the coefficients of determination from the accounting income model for each firm. Again, using annual observations, various intervals were used to calculate the accounting income numbers. Gonedes finds as a result of performing prediction tests that the twenty-one year period provided better estimates than the seven year estimates. The prediction test results suggest that the accounting-number models do not reflect the structural changes reflected by the market model. The results for the model of scaled first differences for various subperiods are given in Table 1.6. 3. Theoretical Basis Study Pettit and Westerfield [1972] attempt to fill a lacuna in the beta literature by providing a rationale for deciding which factors affect systematic risk. They show, using a perpetuity cash flow valuation model for stock prices, that the usual beta of an asset's return can be written as a weighted average of a "capitalization rate beta'' and a PAGE 38 26 Table 1.6 Correlation Coefficients, R, between Estimates from Market Model (M) and Four Accounting Number Models (A) (a) Stock Beta Measured Over Accounting Beta M2 M3 M4 Period Measured Over 1946-52 1953-59 1961-68 Al 1946-68 .32c/ .42c/ .22b/ A2 1946-52 .18 A3 1953-59 .27c/ A4 1961-68 0 ( a) Ninety-nine Cross-sectional Observations. b/ Significant at a = .05. c/ Significant at a = .01 (Table 5, p. 434) PAGE 39 27 "cash-flow beta" each defined from regressions of the firm's capitalization rate and its cash flow on the capitalization rate of the market and the cash flow of the market. They conclude that anything that affects the expected cash flow of the firm or the capitalization rate should affect beta through the cash flow beta and the capitalization rate beta, respectively. The analysis involves a circularity, however. Capital ization rates are equilibrium expected rates of return. As such, according to the CAPM, they are determined, given RF and EM' solely by beta. Thus a knowledge of which factors affect the expected rate of return on the firm's equity securities would require a knowledge of the factors affect ing beta. There is no independent knowledge of which fac tors affect capitalization rates, at least according to t~e CAPM. Thus, the Pettit and Westerfield decomposition is uninformative. The authors run up against this paradox when they attempt to explain the choice of variables as determi nants of beta. Because of space limitations we have not given any justification of why these particular variables should or should not be related to asset risk. In some cases we think a relationship is expected, in other cases we feel that any relationship is a spurious one. Never theless, each variable was included in the analysis because at some time someone proposed that the variable suggested something about the risk associated with an asset ( ) [pp. 1661-1662] Their list of variables is: PAGE 40 (1) Dividend payout (2) Leverage (3) Firm Size (4) Liquidity (5) Growth. 28 They perform a correlation analysis on the market beta, B ; the capitalization rate beta, B / ; the cash flow beta, M e p B ; the operating income beta, 6 01 ; the payout ratio, PAY; EPS the debt equity ratio, D/E; size, SIZE; liquidity, LIQ.; and growth in earnings per share, GEPs The correlation analysis was carried out for two periods: Period I, 1947-57, and Period II, 1957-68. Their results appear in Table 1.7. On the individual level the results are not striking. At the portfolio level, the largest correlations are between the market beta and the payout ratio and the market beta and the capitalization rate beta. This may be a spurious correlation though, because grouping of observations into portfolios will, in general, increase the correlation coefficients. 4. Barr-Rosenberg and Associates' Work Barr-Rosenberg and McKibben Barr-Rosenberg and McKibben [1973] attempt to decompose beta into a component that depends upon a set of descriptors based upon accounting data plus a firm specific effect that cannot be accounted for by these descriptors. Their decom position is similar to the Beaver, Kettler and Scholes PAGE 41 29 Table 1.7 Correlation Analysis INDIVIDUAL FIRMS se/p SEPS SOI PAY D/E SIZE LIQ. GEPS Period I SM .329 .259 .197 -.481 .049 -.074 -.068 .215 N=338 Period II .292 .184 .147 -.394 .069 -.182 .013 .249 N=543 PORTFOLIOS OF FIVE FIRMS Period I SM 6 3 0 4 5 5 3 0 7 7 6 6 0 9 2 161 2 0 4 4 0 9 Period II .621 .389 .261 -.719 .154 -.400 .035 .481 Table 3, p. 1663 PAGE 42 30 instrumental variable procedure except that firm specific effects are allowed. Their list of descriptors consisted of accounting based variables such as an accounting beta, various financial leverage and growth rate measures and of market based descriptors such as the historical beta, a measure of residual risk, cr, in the market model, and market valuation descriptors such as the earnings price ratio. "The 32 descriptors were selected, without any prior fitting to the data, on the basis of studies in the liter ature and the authors' intuition" (p. 325). They found that the pattern of signs obtained was not as predicted. Their regressions constituted a 2% increase in explanatory power of predicting returns over the naive hypothesis that S = l. There are several problems with the estimation tech niques employed in this early study. (1) They estimated betas from = b'W + nt n (1.9) by substituting this expression into the market model with constant intercept R nt = This yields R nt = Letting u = nt a + b'W tMt + ( ~ M + n ) n n t nt (1.10) (1.11) (1.12) the error term in this regression, applying ordinary least-squares to equation (1.11), y ields PAGE 43 31 inconsistent estimates because U will be corre nt lated with the independent variables WntMt via the market return Mt. (2) The intercept a was taken to be constant in their ~egressions. If one accepts the CAPM, though, a= R (1-S ) will vary with n and t even if RF is F nt constant because of its dependence on Snt" (3) In estimating the variance of the firm specific effects, the authors find that: n var( ; ) = w < 0. n (1.13) This can occur in variance components models if the usual variance formulae are applied. To overcome this problem, alternative estimates of w have been derived in the literature on pooling time-series and cross-sectional data. However, Barr-Rosenberg and McKibben substitute w = 0 whenever w < 0. This leads to problems in pre dicting ; n In evaluating the predictive power of their betas, Rosenberg and McKibben consider the mean-square error in the prediction of returns generated by those betas. They compare their predicted betas to those generated under alternative assumptions. These include the void predictor a = S = O, ns ns the naive predictor using historical betas and alphas generated from the stationary market model, the naive predic tor with a = O, a Bayesian adjustment of the naive, and the unit beta a = 0, S = 1. ns Their results indicated that only their predicted betas did better in a mean-square-error sense than the unit beta predictor. PAGE 44 32 Barr-Rosenberg and Marathe In later work, Barr-Rosenberg and Marathe [1975] classified their descriptors into the following categories: (1) Market Variability e.g., historical beta, sigma, current price (2) Earnings Variability e.g., accounting beta, variability of cash flow (3) Unsuccess and Low Valuation e.g., growth in E.P.S., average proportional cut in dividends over the last five fiscal years (4) Immaturity and Smallness e.g., log (Total Assets), Net Plant/Gross Plant (5) Growth Orientation e.g., Dividend yield, E/P (6) Financial Risk e.g., Total Debt/Assets, Liquidity (7) Indicator of Firm Characteristics e.g., dummy variable for N.Y.S.E. listing, whole salers, etc. Barr-Rosenberg and Marathe take the market model with a constant intercept where R nt = = a + 6 M + nt t (1.14) (1.15) Note that there are no firm specific effects in this speci fication of 6 nt" That is, the firm beta is a linear combi nation of J descriptors and G industry dummy variables. The PAGE 45 33 coefficients of the prediction rule for systematic risk are obtained as follows. Equation (1.15) is substituted into equation (1.14) to obtain R nt = The data are pooled and ordinary least-squares is run on equation (1.16) to obtain estimates a, 0 ... ,bJ,bJ+l'"""'bK. Having obtained these OLS estimates the residuals in the market model E are obtained nt E = R S M nt nt nt t (1.17) Let o 2 denote the variance of In this specificant nt tion it is assumed that o 2 is explained as follows: nt o nt = where St is the average cross-sectional standard deviation in month t, and s 0 ,SK are the coefficients of the predic tion rule for residual risk. Let ont = E(lentl) the mean absolute residual return for security n in month t, and 0 nt c = ~ -, the coefficient of variation of le I u nt nt for the residual risk can be rewritten as 0 nt = S 1 d 1 + . + S d ) J+ nt K Gnt The model ( 1.19) PAGE 46 , -34 where 6t is the capitalization weighted cross-sectional average of absolute residual returns. Estimates of residual risk are obtained by running the model (1.20) Presumably, the assumption here is that E(ientl) does not differ very much from Jent!. From this regression one A A obtains OLS estimates s 0 ,sK. The next step is to generate estimates of cr using nt these OLS estimates + + SKd ) Gnt The model in equation (1.16) is divided through by ant and new estimates of a ,b 0 ... ,bJ,bJ+l',bK are obtained. Presumably, this provides a GLS estimation procedure for the model. Looking at the results for the generalized least squares estimates one finds that earnings variability is positively related to beta, growth in E.P.S. is negatively related while growth in total assets is positively related. The results in Table 1.8 do not include market variability variables. PAGE 47 35 Table 1.8 Prediction Rules for Systematic Risk Based on Fundamental Descriptors and Industry Groups ~arnings Variability Variance of Earnings Variance of Cash Flow .02266* .02180*** Unsuccess and Low Valuation Growth in E.P.S. -.00416* Immaturity and Smallness Log (Total Assets) Growth Orientation Growth in Total Assets Financial Risk Leverage at Market Debt/Assets Significant at 95% level. .02416*** .03666 -.09150*** .04126*** *** Significant at 99.9% level. (Table 4, p. 114) PAGE 48 36 Cne notes that different signs result from alternative specifications of the growth and leverage terms. One wonders what gain is effected by including several different measures of the same effect. Perhaps such a procedure obscures the true relationships underlying the model. When market variability variables such as the histori cal beta estimates, historical cr estimates, and price and share turnover variables were included in the regressions, the following partial results in Table 1.9 were obtained. Barr-Rosenberg is disturbed by the negative adjustment to leverage and finds it inexplicable. The two measures 0 financial leverage employed are defined in the following manner: ( 1) ( 2) Leverage at Market Debt/Total Assets = Book Value (Long-Term Debt + Preferred Stock) + Market Value (Common Stock) 7 Market Value (Common Stock) = Long-Term Debt+ Current Liabili ties 7 Total Assets While theory tells us that the debt/equity ratio at market value is positively related to beta, it turns out to be negatively related in the regressions. The book value measure turns out to have a positive sign. Further, the sign remains negative in the simple regression on market leverage. In conclusion, Barr-Rosenberg asserts: The negative relationship appears to be an empirical fact, but one that we do not now understand. Since the relationship is not comprehensible, we have set the coefficient to zero in actual practice. [p. 122] PAGE 49 37 Table 1.9 Prediction Rules for Systematic Risk Based on Fundamental Descriptors Including Market Variability Descriptors Market Variability Historical Beta Estimate Historical a Estimate Current Price Earnings Variability Variance of Earnings variance of Cash Flow -.03124 -.04546 .05550*** .00594 .01541 Immaturity and Smallness Growth in E. P. s. Growth Orientation Growth in Total Assets Financial Risk Leverage at Market Debt/Assets -.00453*** .02290*** -.08739*** .02596*** *** Significant at the 99.9% level of significance. (Table 5, p. 124) PAGE 50 38 This represents a highly dubious procedure which would be difficult to justify. The next part of the Rosenberg and .Marathe study is concerned with the formulation of measures of predictive accura .::y. Two basic historical performance measures are computed as follows: (1) Assuming that all assets have identical risk one runs the regression R nt nt = (1.22) One calculates the mean-square error of the predicted returns generated by this naive model. Call the mean square error from this prediction rule MSE 0 (2) Run a second regression of the form: = (1.23) H S nt is the historical beta and this represents a Bayesian adjustment to the historically generated beta. Call the mean-square-error from this prediction rule MSE 8 (3) Any other prediction rule for beta, e. ~ a predic tion rule based upon fundamental descriptors, can be evaluated via the performance index MSE 1 -MSE 0 MSE 8 -MSE 0 ( 1. 24) where MSE 1 is the mean-square error from the prediction rule to be evaluated. That is, one computes the improvement one obtains by using the given prediction rule for beta relative to the improvement over the naive hypothesis achieved by the benchmark procedure. The results appear in Table 1.10. PAGE 51 39 Table 1.10 Unbiased Estimates of the Performance of Alternative Prediction Rules in the Historical Period (Predicted variance as a Multiple of the Variance Predicted by a Widely Utilized Prediction Rulea) INFORMATION USED IN PREDICTION Market Variability Information Only Benchmark All market variability descriptors Fundamental Information Only Industry adjustments and fundamental descriptors Market Variability and Fundamental Information All information except the historical estimator All Information MEASURE 1.00 1.57 1.45 1. 79 1.86 a The GLS reported figure is the adjusted R 2 in the appropriate regression for residual returns, r rMt' divided by 2 nt the adjusted R for the benchmark procedure. (Table 6, p. 134) PAGE 52 40 Barr-Rosenberg's thesis that the use of both market information and fundamental descriptors leads to better beta predictors than the use of each set of information sepa rately is supported by these results. He also notes that if in predicting beta all information concerning historical betas is discarded, then less than 4% of predictive power is lost. He concludes from this result that "we are able to obtain virtually all of our predictive power for these aspects of risk without relying on historical measures of them" (p. 135). To test for the predictive accuracy of the betas generated into periods other than those in which the predic tion rules were fitted,Barr-Rosenberg tests for stability of the prediction rule. He finds it to be quite stable over time--the prediction rule estimated for a full history of 230 months is closely similar to that for a recent 101 month subperiod. He also computes the adjusted R 2 for the fitted regression lines using various prediction rules. He finds that the betas based upon market and fundamental descriptors were superior to the benchmark betas in five cases out of . h 2 f six wit an average R o .0768 versus .0502. Barr-Rosenberg and Marathe In their work on testing the Capital Asset Pricing Model, the authors, Rosenberg and Marathe [1979] generate betas where the market model is employed with both alpha and beta varying as linear functions of a set of descriptors plus error terms. This relationship they write as PAGE 53 41 a = a'X + e a } (1.25) nt nt nt t 1, ... T = s = b'X + es n = l, ... ,N{t) (1.26) nt nt nt where the terms e a es are "model errors in prediction nt nt with expected value zero" (p. 140). The desire in this study is to test the CAPM. Only the historical beta was used as a fundamental descriptor for the actual beta. The authors note that the prediction rule could be improved by using Bayesian adjustments to the historical beta and fundamental accounting and market based descriptors. This is not done in this study. However, the econometric techniques of their previous studies are improved upon in this one. Generalized least-squares procedures are used in this study. The results are hard to compare to previous results, however, because of the use of a single fundamental descriptor, the historical beta. Part of the purpose of the present study is to apply correct econometric techniques to the model with a set of theoretically justi fied descriptors. 4. Study of the Effect of Financial Leverage Hamada [1972] tests the effect of the leverage relation he derives using the M-M theory. The relation he derives is = (1.27) PAGE 54 42 The M-M theory states that VL = = = This yields = = V u + 1"DL so that To test this relationship, Hamada calculates the following rate of return Xt(l-1)+6.Gt = s ut-1 (1.28) (1.29) (1.30) the rate of return to stockholders in a firm which has no debt in its capital structure. The change in capitalized growth over the period is represented by 6Gt. Since S ut-1 is unobservable as firms generally employ debt financing, the M-M theory is used to evaluate the denominator (1.31) The numerator is evaluated using the following identities: = (1.32) where PDt denotes preferred dividends at time t, and It is PAGE 55 43 the interest expense at time t. The corporate tax rate is designated T. Next, the following identity is used: (X-I) (l-1) PD + 6 G t t t = (1.33) where dt represents dividends paid at the end of the period and cgt represents capital gains. Consequently, = dt + cgt +Pot+ It(l-T) (V-1D) l t(1.34) Next, the observed rate of return to common stockholders is = (X-I)t(l-r )-PDt+ 6 G s Lt-1 (1.35) Hamada obtains available data to construct R and R the Lt ut rate of return to a firm exactly identical to the levered firm except that it has no debt or preferred stock in its capital structure. Using these rates of return he then runs the market models: Rj = a) + B j R + e ) u u u M u,t t t (1. 36) and Rj = a ) + B j RM + j L L L SL, t t t (1.37) PAGE 56 where RMt return. 44 is the N.Y.S.E. arithmetic stock market rate of The betas obtained from these regressions are estimates of the unlevered and levered betas. Theory implies that Using data on 304 firms, Hamada runs 304 time series regres sions and calculates mean alphas and mean betas, and the statistics presented in Table 1.11. These results indicate that since .9190 > .7030 lever age increases systematic risk. Similar results are obtained when continuously compounded rates of return are used in the market model. Hamada concludes, that if the M-M theory is correct, then leverage explains 21-24% of the value of the mean beta. He then goes on to test for which market value rates ought to be used to adjust observed betas to obtain unlevered betas as his formula suggests. To do so, he runs the following regressions: j al+ b [su ~j l + uj B L = 1 SL u 1 j = 1, ... ,102 (1.38) j b [sLsj] + uj 6 = a 2 u 2 S L 3 u j = 1, ... ,102 (1.39) for the 102 firms in his sample that did not have preferred stock in any of the years used. Using average values over the twenty year period for S ands and the 1947 (beginning L u PAGE 57 A a. u s u R2 u A a. L A S L R2 L 45 Table 1.11 Summary Results over 304 Firms for Levered and Unlevered Alphas and Betas Mean Absolute Mean Deviation* .0221 0431 .7030 .2660 .3799 .1577 .0314 .0571 .9190 .3550 .3846 .1578 Standard Deviation .0537 .3485 .1896 .0714 .4478 .1905 Mean Standard Error of Estimate .0558 .2130 .0720 .2746 IJx.-xJ l. -3~ (Table 1, p. 218) PAGE 58 46 period) value for Su/SL and then the end of period 1966 value for Su/SL, the results were obtained in Table 1.12. Hamada concludes that, if long-run averages are used then the adjustment factor method is appropriate using the derived relationship. An alternative indirect test of the financial leverage effect is carried out on the basis of the following set of considerations. Within a given industry ordinary common stock betas show a certain degree of cross-sectional varia tion. Some of this cross-sectional variation is presumably due to the differing degrees of financial leverage employed by firms within the given industry. The reason for looking at a given industry in examining cross-sectional beta varia tion is to confine attention to a given risk class in the M-M sense, that is, to one with the same cost of capital as that of a firm with no leverage but otherwise exactly alike and consequently with the same unlevered beta since = R F + (E R ) S j. M F u (1.40) In practice it is not possible to specify exactly a risk class so that in practice there will be some variation in unlevered betas of the firms within the industry. The idea of the Hamada test is then the following: If indeed some of the cross-sectional variation in stock betas within a given industry (risk-class) is attributable to the differing debt / equity ratios employed by the firms, then unlevering the betas should result in a set of unlevered PAGE 59 Table 1.12 Market Adjustment Factor Regressions over Alternative Periods TWENTY YEAR AVERAGE 1947 VALUE 1966 VALUE Equation a b R2 a b R2 a b R2 1. 38 -.022 1.062 .962 .150 .842 781 .085 .905 .859 (.021) (.021) (.048) (.045) (.041) (.038) 1. 39 .030 .931 .969 .112 .843 .888 .080 .898 .902 (.016) (.017) (.028) (.030) ( 2 7) (.030) -..J Table 2, p. 220 PAGE 60 48 betas which exhibit less cross-sectional variation than the original levered betas. Thus, by computing the dispersion of the levered betas, unlevering them and computing and comparing that dispersion to that of the resulting set of unlevered betas one should be able to discover a financial leverage effect. The results follow in Table 1.13. These results indicate a positive effect for financial leverage, and Hamada concludes, some support for th~ M-M theory. 6. Studies of Effect of Market Power Sullivan Various studies have examined the effect of monopoly power on the rate of return to stockholder's and upon a firm's profitability. Firm profitability is usually measured by the ratio of net income to the book value of stockholder's equity. One such study examined the effect of market power on equity valuation. Sullivan [1977] takes the ratio of the market value to the book value of stockholder's equity for a given firm in a given year. Then the arithmetic mean over the years 1961-70 is taken to evaluate relative equity prices. This variable is then regressed on C : J sz. : J MSG.; J the weighted average four firm concentration ratio for firm j the natural log of 1961 sales revenue for firm j 1968 market share for firm j divided by 1961 market share for firm j (representing growth in market share) PAGE 61 49 Table 1.13 Mean and Standard Deviation of Industry Betas Industry Number of Firms Bu SL Food 30 Mean S .515 .815 (J ( s) .232 .448 Chemicals 30 Mean S .747 .928 (J ( s) .237 .391 Petroleum 18 Mean B .633 .747 cr ( B) .144 .188 Primary Metals 21 Mean B 1.036 1. 399 (J (8) .233 .272 Machinery 28 Mean B .878 1.037 except Electrical (J ( s) .262 .240 Electrical 13 Mean B .940 1.234 Machinery (J ( s) .320 .505 Transportation 24 Mean B .860 1.062 Equipment (J ( s) .225 .313 Utilities 27 Mean B .160 .255 (J ( s) .086 .133 Department Stores 17 Mean B .652 .901 (J ( s) .187 .282 Table 4, p. 225 PAGE 62 ISG : J 50 1968 estimated sales in firm j's industry divided by 1961 estimated sales in firm j's industry. A second measure of market power is: S : the weighted average market share for firm j. J Two regressions were run: one with c the second with J s . In both regressions a positive statistically signifi J cant sign is obtained for these market share variables. Sullivan concludes: These premiums seem to suggest that firms with market power have the ability to set and hold output prices above costs and as a result earn monopoly profits. This ability to control output prices makes the equity shares of powerful firms attractive to investors who bid-up the prices of the equity shares so that all the expected future monopoly profits are capitalized into the existing market prices of the shares. [p. 111] The study purports to control for risk by introducing the stand~rd error of a trend line fitted to book profit ability over a ten year period. However, this variable is not used in the regression discussed and is a questionable measure of risk in any case. It is possible that the earnings of the monopolistic firm may be capitalized at a lower discount rate because the existence of monopoly power lowers the systematic risk of a firm's equity. Such a decrease would increase stock prices. Sullivan argues that these increased stock prices are the result of the capital ization of monopoly profits. PAGE 63 51 Thomadakis Thomadakis (1977] carries out a value-based test of profitability and market structure. He attempts to examine the relationship between the capitalized value of monopoly rents and (1) F: a risk measure of the risk of future returns ( 2) U: the firm's power of oligopolistic restriction C with respect to output of currently held assets (3) Uf: the firm's power of oligopolistic restriction in future investment (4) g: the firm's expected rate of growth from exo geneous demand (5) C: a scale parameter. He runs A = where A = V-A s (1.41) (1.42) the difference between total firm value and the book value of assets divided by sales is used as a measure of monopoly power. There may arise problems in using this variable as a measure of monopoly power because of differing accounting methods used to evaluate A and due to the effect of infla tion on asset values. Thomadakis assumes that the biases introduced as a result of these problems can be disregarded. PAGE 64 52 The results bear some significance to the question of the effect of monopoly power on systematic risk because Thomadakis takes beta from the usual historical regression as his risk measure. Hypothesizing that Uf and Uc are functions of industry concentration IC, the results follow in Table 1.14. Thomadakis finds the sign of the risk factor puzzling. It indicates that the higher the systematic risk, the higher will be the degree of monopoly power. He explains this result by stating that S should represent the volatility of excess earnings whereas here it represents the systematic volatility of total earnings. "The only possible interpreta tion of current results is that Sis a negative proxy for F, but this appears quite far-fetched and should be viewed with reservation" (p. 183). He notes further that a correct risk measure for the purpose of the study would separate out the risk of the competitive component of return from the supercompetitive component. The use of beta in this context assumes that both components of earnings have the same systematic risk. Presumably, the results give some first indication of the effect of monopoly power on systematic risk. If that relationship is negative then the coefficient of beta in the Thomadakis regression should be negative as well. Of course, the inclusion of other variables measuring market share and growth in the regressions could obscure the possible relationship and its sign. PAGE 65 53 Table 1.14 Industry Concentration and Future Monopoly Power DEPENDENT VARIABLE M INDEPENDENT VARIABLES Constant s IC -1.57 a Significant at the 5% significance level. (Table 1, p. 182) .199 PAGE 66 54 Sullivan Sullivan [1978] seeks to determine whether the market power of firms, as measured by size and seller concentra tion, seems to reduce the riskiness of firms. By riskiness is meant systematic risk. To do so, Sullivan regresses monthly betas on ( 1) ( 2) ( 3) (4) sz = natural log of sales for firm j J C. = four-firm concentration ratio J DN = industry dummy variables J SG. = the annual compound growth rate in sales J from 1963-1972 for firm j. He runs levered and unlevered betas on these variables. Typical results appear in Table 1.15. The coefficients of C and SZ are consistently J J negative and statistically significant in Sullivan's results. These results do indicate a negative relationship between monopoly power and systematic risk. Sullivan seeks to determine the cause of this relationship. To answer the question he decomposes beta, based on the definition of earnings to equity holders, into three betas: (1) a beta relating the covariability of firm sales with the market (2) a beta relating the covariability of firm expenses with the market (3) a beta relating covariability of the revaluation of the firm's securities in the secondary capital market [a capital gain (loss) less retained earn ings component]with the market . PAGE 67 s. J s J 55 Table 1.15 Monthly Betas, Leveraged and Unleveraged, Regressed on Market Power and Control Variables Constant SZ. C. DN. SG __ J _J __ ] __ J 1. 831 -.0708 -.2938 .2237 1. 3863 1. 756 -.0739 -.2846 .2120 .9246 R2 .2696 .3458 25% sample: every firm in which the firm's largest market accounted for at least 25% of firm sales. (Table 1, p. 213) PAGE 68 56 Then the unlevered beta = Sales beta Total expense beta + Capital gains beta. Each component is measured, and it is found that the sales beta and the total expense beta approximately cancel each other out, leaving most of the effect in the capital gains beta which is virtually identical to the unlevered beta. He concludes that the systematic risk resulting from firm operations seems to be small. 7. Studies of the Effect of Operating Leverage Lev Turning now to theoretical models describing the effect of the labor-capital ratio on beta, we first discuss the work of Lev [1974]. The firm's operating leverage is defined as the ratio of fixed to variable costs. Lev claims that beta is a positive function of operating leverage. Attempting to support his case that higher operating lever age implies a higher total and systematic risk, Lev appeals to the "better known leverage effect" within a given risk class (a homogeneous industry in our case) the higher the financial leverage, i.e. the relative share of fixed interest charges (fixed costs in our case), the larger the volatility of the earning s residual accruing to com m on stockholde rs, and hence, the higher the financial risk associated with the common stocks [p. 630] This argument is suspect for several reasons: PAGE 69 57 (1) The financial leverage argument does not proceed through the earnings volatility argument. Rather, it follows from the consideration of the precise effect of financial leverage on the rate of return to equity holders. (2) The financial leverage argument is not based upon the existence of fixed debt charges because it holds, as Galais and Masulis [1976] have shown, for risky debt as well. (3) Higher total volatility does not necessarily imply higher systematic risk. Lev goes on, however, to make the following arguments: In an uncertain environment future demand Qjt is a random variable. Then the earnings stream of the firm = (1. 43) where p = average price per unit of the product v = average variable costs per unit of the product. Lev asks the question: How does an increase in the uncer tainty in demand (at any given price) affect the earnings stream of the firm? Lev takes the partial derivative of the earnings stream with respect to Q. (a random variable) to Jt obtain: a~jt PAGE 70 58 to demand equals the difference between the product's average price and average variable cost per unit, the contribution margin. In a homogeneous and competitive industry the average product price is the same for all firms. Thus the fluctua tions in the earnings stream of the firm depends only upon the average variable cost per unit. A firm with a higher average variable cost will have a more volatile earnings stream. A firm with a greater operating leverage, Lev claims, will have a lower variable cost per unit and hence a more volatile earnings stream in accordance with his partial derivative. This higher volatility is transmitted to increase the total volatility of returns to equity holders. This analysis and introduction of uncertainty leaves the question of the firm's response via its factor mix to the uncertainty in demand. It is possible that a firm would in fact respond in such a way as to reduce the variability of its earnings stream. What is missing here is an analysis of the firm's optimal decision behavior under a situation of increased uncertainty in demand. Lev then goes on to attempt to demonstrate the effect of operating leverage on the systematic risk of the firm's equity S He does this by writing the rate of return as J R. = J where (Sales. VJ t F t) (1-T) + 6 g. Jt J Jt s 1 J, t(1.45) PAGE 71 59 Sales j t vjt = = (pQ)jt = total revenues total variable costs = total fixed costs = the future growth potential of the firm s = market value of equity at time t-1 j,t-1 T = the corporate tax rate. Thus we obtain S t l S J J = Cov( (PQ) jt (1T ) ,RMt) Var(RMt) Cov(v (1-T) ,R ) Jt M t Cov( .6. gjt'RMt) + Var (RMt) (1.46) Consider two firms, exactly alike, including output, stock value, and capitalized growth. They differ only in their use of variable factors of production. The first and last beta will be identical. The firm with the higher operating leverage will have relatively fewer variable costs (a random variable), hence a lower expected value of variable costs. From this, Lev concludes that it will have a lower covariance with the market return. But this state ment is not correct. Covariance measures comovement between random variables. For three random variables it is possible that: E ( x ) < E ( Y ) but that Cov ( x z ) > Cov ( y z ) Nonetheless, Lev concludes that the firm with the higher operating leverage will have a lower second beta and PAGE 72 60 hence that (1.47) He then assumes that the two firms have the same stock value and consequently that operating leverage increases system atic risk. Turning to the empirical tests of the proposition, the procedure is as follows: using data from three homogeneous industries, electric utility, steel, and oil to ensure cross-sectional equality of sales patterns across states of nature, Lev runs a time series regression of total operating costs of the j-th firm in year ton total physical output. He obtains from this regression the estimated coeffi cient V., a measure of average variable costs per unit of J output (assuming that MC=AVC and that average variable cost is constant over the estimation period). These regressions were run over two different time periods, a 20 year period (1949-1968) and a twelve year period (1957-1968) to test the stability'' of the relation ship. The results follow in Table 1.16. The intercept is not reported here. When this study was replicated, the intercept, representing fixed costs turned out to be negative. This indicates a serious problem in this model specification to determine average variable costs and/or in the estimation procedure. The next step was to run the market model to obtain estimated beta coefficients. The regression was run using PAGE 73 61 Table 1.16 Estimates of Average Variable Cost Per Unit 20 years V 12 years Electric Utilities .00815 .00762 (26.85) (18.67) Steel Manufacturers .892922 .81503 (31.98) (20. 97) Oil Producers .74806 .71677 (68.12) (43.27) Table 1, p. 635; t statistics in parentheses. PAGE 74 62 monthly rates of return over a ten year period. Of course, this assumes that beta was stationary over that period. A A Lev then runs the estimated S. on the estimated V : J J s. = (1.48) J A negative sign for b 2 indicates that beta is a decreasing function of operating leverage (as proxied The J results follow in Table 1.17. Lev concludes that the hypothesized relationship holds except in the case of an insignificant coefficient for the oil producing industry. He also concludes that operating leverage does not go very far in explaining systematic risk. Rubinstein Rubinstein [1973] lets R.* be the rate of return to a J firm without debt, X the earnings stream of the firm, and J V its market value. For a similar firm with debt in its J capital structure the rate of return to equity holders R. J = X -R D J F J S (1.49) J where D and S represent the market values of debt and J J equit y respectively. Rubinstein writes the CAPM equilibrium relationship in the following manner: E (R ) J = R F + >.. p ( R R ) Var ( R ) J M J [equation (2), p. 49], where (1.50) PAGE 75 63 Table 1.17 Regression Estimates for Systematic Risk on Average Variable Costs Per Unit No. of Firms R 2 Electric Utilities 75 .08 .5149 (14. 790) Steel Manufacturers 21 .23 2.2833 (5.014) Oil Producers 26 .05 .8101 (4. 673) Table 3, p. 637; + statistics in parentheses. -6.912 (-2.060) -1.3401 (-2.4097) -.2748 (-1.157) PAGE 76 \ = N ow, since R. = J where V = J [E (R M ) -RF] Var (RM) X X J = _J V V J J I V S + D. I J J J 64 it follows that Var (R ) = var ( R i J [ 1 1 ~; ]J Also, p (R j ,R M ) = p (R *,R ). J M (1.51) (1.52) ( 1. 53) (1.54) (1.55) Rubinstein concludes from these two expressions that the full impact of financial risk is absorbed by the standard deviation Var(R ) since the correlation coefficient is J invariant to changes in financial leverage. Note that this result holds onl y under the assumption that debt is not risk y Substitution of these two expressions into the CAPM relationship yields E (R ) J = R f + \ p (R *,R) Var(R *) J M J (1.56) + 11. p (R *,R) Var(R *) [ 0 j] J M J S J PAGE 77 65 The first component on the right-hand side of this expres sion is called operating risk by Rubinstein, and the second component is called financial risk. The goal now is to develop the components of operating risk It suffices to drop the \ in this consideration. Let m denote product m for firm j. Rubinstein assumes that the output Q of product mis a random variable and that the m sales price per unit pm is fixed. One can view this as a model in which price is set ex-ante with quantity demanded being uncertain to be determined at the end of the single period. Note that this assumption is contrary to the Subrahmanyam and Thomadakis model to be reviewed below in which the firm is a quantity setter with uncertainty resolved in the product's price. As Leland [1972] has shown, for the monopolistic firm, these are not equivalent behavioral assumptions as far as optimal behavior is concerned. Let v denote variable costs per unit, F total fixed m m costs, and a the proportion of assets (represented by firm m value V ) devoted to the production of product m. J Then, assuming that all fixed costs of the firm can be allocated, the earnings stream of the firm X. J = \ ( Q p Q V l m m m m m F ) m (1.57) Rubinstein claims that since PAGE 78 then R J = a X m J av.' m J 66 Rubinstein interprets this as follows: a represents m (1.58) the relative influence of each product line, p -v reflects m m operating leverage through the contribution margin, p(Qm,RM) represents the pure influence of economy-wide events on output, and Var(Q /a V ) the uncertainty of output per m m J dollar of assets, a measure of "operating efficiency." For the case of a firm producing a single product this formula is easily derived. p (R *,R) J M Var (R *) J Hence, = = [ Q p-Qv-F R l p V m J = p ( Q ,R) m [ Q p-Qv-Fl Var V j = = \p-v \p ( Q ,R) Var( Q /V ) M J which is the relationship to be derived. (1.60) (1.61) (1.62) This formula shows that operating risk and consequently systematic risk [the sum of operating and financial risk equation (8), p.57) is a positive function o f operating PAGE 79 67 leverage in the following sense. Given the product's price, decreasing variable costs per unit increases the contribu tion margin Ip-vi and with it operating risk which is proportional to the contribution margin. This of course assumes that is constant. Another way to view this result is in the following manner: Given the contribution margin operating risk is determined by ( 1) the pure influence of economy-wide events on output represented by p(Q,RM) ( 2) Var(Q/V ) the uncertainty of output per dollar of assets.J Presumably firms are out to maximize their market values in the presence of uncertain demand. Plausibly, an increase in the use of variable factors would afford a firm with some measure of flexibility in its production process. However, the question becomes: What is the effect of an increase in the use of variable factors on optimal firm value, presumably represented by V ? If, indeed, systematic J risk is lower with the use of variable factors, then for a given random income stream, firm value would increase. Without this knowledge, which clearly has to be determined as part of the analysis, firm value could decrease with the use of variable factors and operating (and systematic risk) could increase. This is all for a given output stream Q .. J PAGE 80 68 As Subrahmanyam and Thomadakis note, the analysis is not very enlightening because it fails to relate the optimal choice of margin (i.e. of factor mix and price) to the uncertainty of output. Subrahmanyam and Thomadakis The model developed by these two authors will be developed and generalized in the next chapter. A brief summary will be given here. The goal of the analysis is to provide a specification of the relationship between monopoly power as represented by the reciprocal of the price elasti city of demand and the labor capital ratio and the system atic risk of the firm's equity securities. (There is no debt in the model.) The innovative facet of this work is the introduction of sources of uncertainty into the demand function and into the wage rate. The effects of the above variables on beta then depend upon the co-relationship between the random error terms representing uncertainty in this model. Their basic result is that if wage rate uncer tainty is relatively less volatile than price uncertainty then beta will be positively related to the labor capital ratio and negatively related to the degree of monopoly power. If wage rate uncertainty is relatively more volatile than price uncertainty, then the reverse will hold: beta will be negatively related to the labor capital ratio and positively related to the degree of monopoly power. The economics underlying these results can be explained with reference to the market portfolio. Since human capital is a PAGE 81 69 risky asset which we can assume to be tradeable (the results generalize to non-tradeable human capital) the risk of the market portfolio is the sum of the risk of firms plus the risk of the labor component of firms' production processes. The risk of labor is essentially determined by the "beta" of wage uncertainty relative to price uncertainty. If that beta is less than unity then in the weighted average repre senting the beta of the market portfolio relatively less risk will be shared by labor than by firms. This forces the firm betas to be greater than unity. As labor is increased relative to capital, if the basic condition holds, more risk will have to be borne by the relatively smaller amount of capital. Thus beta is a positive function of the labor capital ratio. This stands in direct opposition to the Lev and Rubinstein results. The reverse will obtain if the beta of wage uncertainty relative to price uncertainty is greater than unity. The model developed here hints at the fruitful ness of an approach to determinants of systematic risk that takes into account sources of uncertainty, their correlation, and risk sharing among the bearers of risk. PAGE 82 CHAPTER II MICROECONOMIC FACTORS AFFECTING EQUITY BETAS 1. Introduction The first purpose of the study is to examine the argu ments given in support of various microeconomic factors said to affect the systematic risk of a firm's common stock as measured by its beta coefficient. Arguments based upon theoretical models in the literature and upon "economic intuition" will in turn be considered. A large number of factors can be hypothesized to bear a relationship to beta but, of these, only those that appear to be of clear economic interest will be considered. Those that affect beta for purely technical reasons are not of immediate interest here. An example would be non-synchronous trading in that it makes a common differencing interval for return data impossible to calculate and introduces various biases into beta estimation. In principle, one can correct this potential source of bias--in doing so the effect on beta of the phenomenon disappears. An example of a variable of economic interest is financial leverage: through its effect on the rate of return to equity holders it unambiguously affects systematic risk. While the study of the technical factors affecting 70 PAGE 83 71 beta is of great importance, this study concerns itself with the formulation and inclusion of the economic type of variable. Recently, there have appeared theoretical models in the literature supporting the labor-capital ratio, monopoly power, and "growth opportunities" as determinants of betas. More traditional determinants are financial leverage and volatility of operating earnings. In this chapter, the theoretical arguments supporting the above variables will be presented and critically examined. In the absence of theoretical support, the intuitive arguments will be presented and their limitations will be stated. We turn now to a separate consideration of each of the variables pur ported to affect beta. To motivate the discussion of the effect on beta of monopoly power, it is instructive to compare the two different arguments for the effect on beta of (1) financial leverage and (2) volatility of operating earnings. While the effect of (1) has been theoretically established, the argument for (2) proceeds on an intuitive basis. 2. Financial Leverage To present the argument in its simplest form, taxes will be ignored and it will be assumed that there is no risk of default on the firm's debt securities. This latter assumption guarantees that i, the interest rate on the firm's bonds, is non-stochastic. The rates of return on the PAGE 84 72 levered firm's and unlevered firm's equity are: X-iDL = ---,,--and R SL u = X V u respectively. Evaluating the corresponding betas yields cov (RL, Rm) o2(Rm) cov(X-iDL/S ,R) L m cov(X/S ,R ) L m = = cov(X/SL V /V ,R) u u m = o 2 (R ) m = = o 2 (R l m ~ ~ V cov ( X/V R ) u u m = SL o 2 (R ) m The effect of financial leverage on the firm's equity beta has been established and, following Modigliani and Miller [1958] in a world with no taxes, since Vu= VL we have that leverage has an unambiguous effect on systematic risk: = This argument for the effect of financial leverage on beta is based upon a precise knowledge of how such leverage affects the rate of return on the firm's equity. In the following generalization of the above argument, the effect of financial leverage is again evident from similar knowl edge concerning its effect on the rate of return to equity holders. PAGE 85 73 If we are willing to assume that both the assumptions of the Options Pricing Model and the continous time version of the CAPM hold, then Hamada's result which holds for riskless debt can be generalized to allow for risky debt where all betas are to be interpreted as instantaneous quantities. It is also assumed that dV/V follows a Gauss-Wiener process and hence that the Modigliani and Miller result on the irrelevance of financial structure (for firm value) holds by hypothesis. Viewing the equity of the firm as an option on the underlying firm value with exercise price equal to the face value of the firm's outstanding debt securities, an expres sion for the dollar return on equity value 6S can be derived using Ito's Lemma. = as 1 a 2 s 0 2 v2 At+ aast av 6 V + 2 av2 u 6t 2 where Vis firm value, a is the instantaneous variance of percentage returns on V, and t represents time. Consequently, 6 S s = and taking the limit of this expression as ~t + 0, we have the expression for the instantaneous rate of return on the firm's equity securities: PAGE 86 r s = as tN av s 74 Multiplying by V/V allows us to write this rate of return in terms of the instantaneous rate of return on total firm value rv. r s = as;av v tN S V = as;av S/V rv = where n represents the elasticity of equity value with s respect to firm value. Note that as/av= l if an;av = o. That is, if debt is riskless this formula implies Hamada's result (recognizing that V =Vu= VL in a world of no taxes) Deriving the corresponding betas, we obtain = cov(r ,r) s m 0 2 (r ) m = as;av v s cov(rv,rm) a 2 (r) m 2. Volatility of Operating Earnings = In the absence of precise information concerning the effect of earnings' volatility on the equity's rate of return it is not easy to establish the effect of that vari able on the systematic risk of the equity securities. The argument that follows will be referred to as the "earnings' volatility argument." It serves to establish the relation ship between variability in earnings and the total risk of PAGE 87 75 equity securities. The steps in the argument are as follows. (1) Increased volatility of earnings implies increased volatility of end-of-period stock price (a random variable). (2) This induces a more variable rate of return since R t 1. = and var(R t) 1. I P. 1 1.,t+ 1 P. 1. It = 1 p~ 1. t Var (P. t 1 ) 1.' + Thus, earnings volatility affects the total variability of the rate of return of the equity securities. No refer ence is made in the earnings volatility argument to volatil ity with respect to the earnings volatility of the market and none is established. To do so, one would have to posit a relationship between the earnings volatility of the market and the earnings volatility of the specific security. Such an assumed relationship would be transmitted to the corre sponding rates of return. But such a relationship would be very close or identical to a specification of the effect of earnings' volatility on the security's beta; a relationship it is desired to discover. Under a highly restrictive assumption discussed below, it can be shown that the stock beta is proportional to the total volatility of the rate of return. We merely write PAGE 88 6 1. = cov(R.R) l. m o 2 ( R l m 76 = a l. where p. is the correlation coefficient between the market l.ffi rate of return and the specific security's rate of return, and a is the standard deviation of the stock's rate of l. return. If we assume that p. /a is constant, the result im m is obtained. A sufficient set of conditions for this to hold is that both p and a are assumed to be constant. Given a constant im m correlation with the market and a constant market variance, this result is not surprising, as the only factor left to occasion changes in beta is the own variance of the securi ty's rate of return. Of course, it is important to then know for each different security the proportionality constant P. /a as it determines the (absolute) magnitude im m of B. but given that information, the stock beta is then 1. determined solely by all factors influencing the variability of the stock's rate of return, and earnings' volatility is one of those factors. Given the assumed constancy of p. /a and the implied im m result that systematic risk is then proportional to total risk, what are we to make of the tenets of portfolio theory and the CAPM implying that, in general, it is only the systematic component of risk that matters? The answer is contained in Capital Market Theory and is analogous to the distinction between the Capital Market Line and the Security Market Line. The Capital Market Line is precisely that set PAGE 89 77 of portfolios for which p. assumes the value +land for im these efficient fully diversified and perfectly-correlatedwith-the-market portfolios (the most diversified portfolio), it is indeed o. that becomes the relevant risk measure. The l. assumption that p. /cr is constant for a given security's l.ID m rate of return is analogous (though not coincident) with the assumption that the security be on the Capital Market Line, that is, that it be an efficient security. The assumed constancy of p. /cr then is clearly seen to be a highly im m restrictive assumption and it is this assumption that must be added to be earnings' volatility argument to derive the effect of this factor on the stock beta. As indicated, this assumption is very close to a specification of how the stock's rate of return responds to the market's--a relation ship it is sought to uncover. After all, the only real difference between cov(R. ,R) and p is that one is a l. m im dimensional measure and the other is not. Either of these quantities is a measure of comovement with the market rate of return. Clearly the former measure determines the stock's beta [with cr(R )] as the stock beta is a dimensional m measure of comovement of the security's return relative to the market. This is the reason that p and cr only deteri m m mine the proportionality factor. It is left for cr to l. determine the absolute magnitude of the stock beta. The conclusion of this discussion is straightforward: the assumption that p. /o is constant is an artifice de im m signed to guarantee that the earnings volatility argument PAGE 90 78 (combined with it) "works." Given the inherent untestability of that assumption (a defect shared by cost of capital studies) it is safe to assert that by itself the assumption has no real justification. 4. Growth A first distinction is made between growth in earnings, sales, or assets and growth opportunities as opportunities to invest in projects with expected rates of return greater than their costs of capital. Modigliani and Miller [1961] have argued that the latter type of growth is the relevant concept for firm valuation. The essence of 'growth', in short, is not expansion, but the existence of opportunities to invest signifi cant quantities of funds at higher than normal rates of return. [p. 417] One consequence of this distinction is that, as the firm's stock beta is a valuation concept for measuring that portion of the riskiness of the security which the market rewards in equilibrium pricing, insofar as mere asset expan sion is not necessarily relevant for valuation, one would expect no necessary relationship to hold between it and the firm's stock beta. Perhaps this accounts for the erratic performance of growth measures in the empirical literature. For in some cases, asset expansion would be indicative of growth in the Modigliani and Miller sense while in other cases not. Of course it is assumed that growth opportunities do affect the stock's systematic risk, a question considered PAGE 91 79 below. It is interesting to examine any technical effects on single period betas of growth in a sense to be defined below. Before doing this, we turn to the question of how growth opportunities in the Modigliani-Miller sense affect beta. The following argument has been presented by Myers [1977]. At any instant in time, a firm consists of tangible assets (in place) and intangible assets or opportu nities for growth. These growth opportunities can be con sidered discretionary in the sense that the firm can choose to exercise them or not. In this loose sense, such growth opportunities are "options." Since, according to the Modigliani and Miller valuation model with growth, firm value (in equilibrium) consists of the value of current assets in place plus the present value of future growth opportunities and since options written on stock securities are "riskier" than the underlying security, and, except for special cases, the same is true for options written on real assets, it follows that the greater the proportion of equity value accounted for by growth opportunities the greater will be the "riskiness" of the stock securities. While this argument is suggestive, it contains several difficulties awaiting resolution. The first is to be specific enough about the sense in which growth opportunities are options to allow the application of one of the forms of the Options Pricing Model. The second difficulty is that systematic risk measures (like expected returns) are not to be found in PAGE 92 80 those models. Consequently, by riskiness of the "growth options" is meant total risk and presumably a predominance of such options implies, according to the Myers argument, a higher total risk of stock securities. A central difficulty in testing the hypothesis advocated by the argument is a specification of growth opportunities, a difficulty recognized by Modigliani and Miller and shared by cost of capital studies. As mentioned previously, in some cases asset expansion (used as a proxy for growth opportunities) will be indicative of the growth opportunities and in other cases not. Without knowing a priori the projects' costs of capital there is no way to appropriately define a sample of firms with growth opportunities to be tested. We turn now to a consideration of the effects on the firm's single period stock beta of growth where we define growth to be a predominance of later (positive) cash flows over earlier ones. Presumably, this definition embodies both asset expansion and Modigliani and Miller growth oppor tunities. Growth clearly takes place in a multiperiod setting while the CAPM equity beta is a measure of the systematic risk of those securities borne over a single period. Thinking of growth as a predominance of later over earlier cash flows, the question is: how does the fact that more of the cash flows from the project are to be received in the future affect the single period CAPM beta? In its essence an answer can be formulated as follows. Later flows will have lower betas relative to a particular PAGE 93 81 initial period than earlier flows. The beta of the entire project (all the flows, early and later) will simply be a weighted average of the betas (relative to the initial period) of the individual flows. If there are relatively more later lower beta flows, then the beta of the entire series of flows will be lower than that for a series with relatively fewer later flows. Thus, growth in the sense of a predominance of later over earlier flows affects any single period beta in a negative direction. The argument rests upon the degree of correlation between the market's cash flow at t = 1 and the cash flow from the project at t = 1,2, .... As a statistical hypothesis, one would expect later cash flows at t = 1 to have a lower statistical depen dence on the aggregate of all cash flows than earlier flows. Thus, the degree of risk resolution over the given initial period, captured by this correlation, would be lower than for the earlier more highly correlated with the t = 1 market cash flow. One can view the cash flow dependence on the t = 1 market flow as a decay process as we move forward in time. Clearly, this is a purely statistical argument reflecting the effect of technical factors on single period betas, but it must be remembered that the CAPM single period beta is a statistical concept. In a multiperiod setting in which growth takes place, one is looking at single period betas calculated over a given period. To make the model described here more precise we con sider in detail the multiperiod valuation model proposed by Stapleton and Subrahmanyam [19791 as an alternative to the PAGE 94 82 Myers and Turnbull [1977] model. The advantage of this model is that it avoids the assumption of a particular dependence structure of cash flows. Such an assumption was made by Myers and Turnbull in that expectations of project cash flows were assumed to be generated by a single-index model. Their statement of this are equations MT-2 and MT-3 (p. 322), MT-2 where o t is a zero-mean noise term expressing the forecast error as a proportion of the expectation based upon past information. The behavior of 8t is postulated by = MT-3 ~ where It represents unanticipated changes in some general ~ economic index and b represents the sensitivity of o t to changes in It. Further, the Stapleton and Subrahmanyam model presented here does not make arbitrary assumptions concerning the market prices of risk but derives them endogeneously. Assumptions (1) Investors are expected utility of terminal wealth maximizers where their utility functions are of the constant absolute risk aversion class. (2) Firms generate cash flows X~ which are jointly normally distributed. PAGE 95 83 These two assumptions are sufficient to guarantee that the future market prices of risk are non-stochastic. Further assumptions are: (3) No debt financing is employed. (4) Limited liability does not apply. (5) Future one period interest rates are known with certainty at t = 0. Given these further assumptions, it follows that derived future stock prices are normally distributed. Given the non-quadratic assumption on utility functions, normality is required to apply the single period CAPM. Considering the simple case of a two period model (which can be generalized to any number of periods) the development proceeds as follows. The investor's multiperiod optimal portfolio problem is solved recursively using dynamic programming. For the two period model considered here this problem can be formulated in the following manner. At time t = 1, the individual investor wishes to maxi mize his expected utility of final wealth w 2 by choosing a portfolio of holdings of the cash flows at t = 2 of all firms in the market. This portfolio can be described by a k k vector {z 12 } of holdings of the cash flows {X 2 }. Given the investors utility function of the CARA class: U (W 2 ) = the problem is a constrained maximization problem, PAGE 96 = subject to = = max i zl 2 = 84 max E[-a. exp(-a.W 2 )], l. l. i zl2 ( 2 .1) ( 2 2) ( 2 3) where M 1 is the amount of risk-free lending undertaken at t = 1, r 2 is the period 2 riskless interest rate, and P 12 is the vector oft= 1 prices of the x 2 cash flows. The solution to the maximization problem in equations (2.1), (2.2) and (2.3) is the vector defined by = ( 2 4) where = ( 2. 5) is the conditional (on the cash flow at t = 1 vector) variance-covariance matrix. It is assumed here that the characteristics of the state of the world ~ 1 relevant for expectations of the t = 2 cash flows are summarized in the t = 1 cash flows of all firms in the economy x 1 This allows us to replace the state of the world information set ~ 1 at t = 1 by the cash flow vector of outcomes. The solution in (2.4) and (2.5) is then used to gener ate equilibrium prices P 12 ix 1 conditional on x 1 at time PAGE 97 85 t = 1 of the cash flow vector x 2 This is simply a matter of equating supply to demand. i Since the z 12 must aggregate to the unit vector of supplies of the total proportional k holdings of the {X 2 }, we obtain from equation (2.4) that 1 = = ( 2 6) Solving this yields = ( 2 7) where = 1 ( 2 8) I (1/a.) 1 i is the market price of risk. We have, then, equilibrium pricing at t = 1 of the cash flow vector x 2 to be received at t = 2 in terms of the relevant parameters of the model. This yields the optimal value of the utility function = ( 2. 9) where (2.10) These equations describe the derived utility of wealth function. If r 2 and A 2 are constants, then u(W 1 1x 1 ) is a PAGE 98 86 non-state dependent exponential function of w 1 alone. But we have assumed r 2 to be non-stochastic, and \ 2 will be non stochastic since it depends only on the coefficients of absolute risk aversion of all individuals: a known con stant. The assumption of CARA utility functions then guar antees the non-stochastic character of the market price of risk \ 2 Further, n 2 is non-stochastic, i.e. independent of x 1 as a consequence of the assumed joint normality of the {x:} [Anderson, 1958]. Thus, u(w 1 jx 1 ) = u(W 1 ) is a non state-dependent utility function as is required for the period-by-period application of the CAPM [Fama, 1970]. Essentially, since r 2 and \ 2 are non-stochastic by assump tion and a is independent of wealth, the only source of state dependence is through n 2 the conditional covariance matrix which represents risky investment opportunities. The assumption of joint normality of all the cash flows in the economy is designed to rule out the state dependence of the (risky) investment opportunity set. One should note that exponential utility and joint normality are sufficient conditions to avoid state dependence of the derived utility function. Given the derived utility function -a exp(-a{w 1 r 2 +A 2 }) = u(W 1 ), equilibriwn prices at t = 0 of the cash flows x 1 and x 2 are derived by solving backwards from the solution derived fort= 1. That is, the individual's maximization problem at t = 0 is as follows: max E[-a exp(-a{W 1 r 2 + A 2 })] [zo1] [Z02] (2.11) PAGE 99 subject to = = 87 (2.12) (2.13) where z 0 1 are the portfolio weights of proportionate hold ings of the cash flows { X 1 } and z 02 are those in the vector of cash flows {X 2 }. At t = 1, these assets yield proportions of the cash flows { x 1 } and values of the { x 2 } denoted by p1 2 The solution to equations (2.11), (2.12) and (2.13) are the vectors = (2.14) = (2.15) where ~ 1 is the variance-covariance matrix of cash flows x 1 and prices P 12 ~ l = [ :. : l (2.16) k F = [cov( X i X 1 )] k G = [cov( X iP 1 2 )] k H = [cov(Pi 2 P1 2 )] Equilibrium prices (at t=O) are = ( 2 .17) PAGE 100 88 where 1 = The relationship that plays a central role, between A 1 and A 2 is simply that = {2.18) ( 2 .19) (2.20) That is, the current market price of risk equals the future price compounded at the future risk-free interest rate. These results are used now to derive the t = 0 equilib rium price of the t = 2 cash flow of firm j, X~ as a certain ty equivalent in terms of the period 2 market price of risk A 2 [a simplification allowed by equation (2.20)], the period 1 and period 2 risk-free rates, r 1 and r 2 and covariances between the cash flow at t = 2, X~, and the m market at t = 2, x 2 and the compounded market cash flow m r 2 x 1 This equilibrium price is derived as follows. Take the expected value (at t=O) of equation (2.7) and substitute the result in equation (2.18). This yields P02 1 [ .!_{E (X 2 ) A 2Q2i} A lQli] = rl r2 1 [E(X 2 ) A 2 (Q 2 i + r~Ql i]. (2.21) = -rlr2 PAGE 101 89 For any particular cash flow equation (2.21) yields, noting that Q 1 1 adds up elements of n 1 the following equi librium prices = 1 rlr2 2 j m m } + r2cov(P12'P12 + Xl) ] (2.22) Using the assumption that n 2 is non-stochastic, i.e., independent of x 1 this expression can be simplified by noting that and j m cov(P12'P12) j m cov (P 12 ,x 1 ) = = 1 cov (r 2 {E (X~ I x 1 ) (2.23) A 2 cov(X~X~jx 1 ) }, (2.24) The assumption of joint normality of the cash flows implies that PAGE 102 90 = (2.25) and (2.26) Substituting equations (2.23), (2.24), (2.25) and (2.26) into equation (2.22), we obtain = + 2 ( P j P rn + Xrn ) } ] r2cov 12' 12 l = = (2.27) An alternative valuation formula can be stated where we first discount cash flows tot= 1 and then, using the period 1 market price of risk A 1 and the period 1 interest rate r 1 we derive the following expression for P~ 2 : PAGE 103 91 = (2.28) We have used the result in equation (2.20) to derive this from equation (2.27). Since the period 1 beta of the single cash flow at t = 1, Xi, is of interest in the analysis of the effect of growth on one-period betas, we derive the valuation formula j for P 01 now. From equation (2.17) we have j 1 [E (Xi) j m p~2) ] POl = \ 1 cov(X 1 ,x 1 + rl 1 [E (Xi) { j m j m } = ,\1 cov(Xl,Xl) + cov(X 1 ,P 12 ) ] r2 1 [E (Xi) { j m [ E ,x;lx 1 )] = ,\ 1 cov(Xl,Xl) + COV X~, } r2 r2 since = (2.29) = using the assumption that ~ 2 is non-stochastic. Using the assumption of joint normality of the cash flows we obtain that PAGE 104 92 = (2.30) and the resulting valuation is = (2.31) We see, then, that in this multiperiod setting, equilibrium valuation of the end of period l flow involves covariances between that flow and both the market flows at t = 1, X~ and the discounted tot= l market flow at t = 2. A similar interpretation can be given to the result in equation (2.27). Present valuation of the t = 2 cash flow x; involves covariances between that cash flow and both the market flow at t = 2 and the compounded tot= 2 flow of the market at t = 1. These results are the key to showing the effect on single period betas of (multiperiod) growth opportunities. Single Period Betas in a Multiperiod Setting Assuming that X~ and X~ are paid out as dividends and letting P~ refer to the value at t of all future flows X~+ 1 ,xt+ 1 .. using rate of return form, the t-th period beta is defined by: = cov(xtj + Pj ,Xmt + PmJ X 1 ) t t tvar (xm + Pm Ix 1 ) t t t(2.32) PAGE 105 93 To examine how growth affects single period betas we will consider equation (2.32) where we will calculate the Si and S j of the single cash flows to be received at t = 2, xJ 2 . 2 These are defined as follows: = j ml cov(x 2 ,x 2 x 1 ) var (X~ I x 1 ) (2.33) since Xj/Pj represents one plus the rate of return on the 2 1 asset that costs Piatt= 2 and yields an end of period dollar value equal to X~, and since, in this two-period model, there are no flows after t=2 and hence the value of those non-flows is Pm= O. 2 (p j Xm + Pm) cov l' 1 1 = m m var (X 1 + P 1 ) (2.34) j/ j since P 1 P 0 1 2 j / j P 1 2 P 012 represents the rate of return over period 1 of the cash flow to be received at t = 2. To discuss the behavior of S i we use our previous results to evaluate the numerator of equation (2.34): (p j X m + Pm) cov l' 1 1 = = by equations (2.23) and (2.24). By equations (2.25) and PAGE 106 94 (2.26) this reduces to (p j Xm + Pm) cov l' 1 1 = From this formula it is seen that the first period beta of the cash flow at t = 2 depends upon the degree of risk of x 2 which is resolved over period 1. Degree of risk resolved over period 1 is measured by j m cov(x 2 ,x 1 ). One can view the risk of the future cash flow xj 2 as spread out over the two periods at the end of which it is received. Any single period beta is a measure of the amount of the risk spread out or resolved over that given period. To make this idea clear in terms of the previous development we will consider two polar cases. ( 1) None of the risk of the Xj cash flow is 2 over the first period. This means that resolved xj is independent of all end of m 2 period 1 flows xk 1 and hence of x 1 Hence, (2.36) and = (2.37) Substituting equations (2.36) and (2.37) into (2.33), we see that PAGE 107 95 and consequently, = o. (2) The case polar to (1) is that in which all of the uncertainty of the t = 2 flow is resolved over period 1. Thus, given x 1 there is no residual correlation between Xj and xm to be resolved. 2 2 = o. (2.38) We see, then, by substituting equation (2.38) into (2.33) that Also, in this case, we have that = cov(X~/r 2 ,x~) + cov(X~/r 2 ,x;/r 2 ) var(X~ + P~) X (2.39) This is a reasonable result as the total amount of risk of xj to be resolved is the risk arising from 2 covariance with X~, the first term in the numerator of equation (2.39) and the residual risk of x; (discounted tot= 1) with the (discounted to t = 1) cash flow of the market at t = 2, the second term in the numerator of equation (2.39). In between these two polar cases, the behavior of the two betas depends upon the proportion of the total risk (relative to the market at t = 1 and t = 2) resolved over PAGE 108 96 each of the two periods. The more highly correlated is the end-of-period 2 flow with the market at t = 1 [the higher is j m cov(X 2 ,x 1 )] the lower is the residual systematic risk left to be resolved over period 2 [the lower is cov(X~,x;!x 1 )]. A glance at equations (2.33), (2.34) and (2.35) indicate that in this case S{ will decrease as the residual covari ance decreases and si will increase as cov(X~,X~) increases. One consequence of this argument for the question of the stability of the beta series over different periods of a series of cash flows is that equality between 6{ and 6~ necessitates that an equal amount of uncertainty be resolved over each period. There is no a priori reason to expect this to be the case and observed empirical evidence does indicate unstable betas of particular cash flows. The interpretation presented here gives a rationale for that instability in the appropriate multiperiod framework. We will first prove a lemma which will allow us to determine the effect of growth, defined as a predominance of later cash flows over earlier cash flows, on a single period beta. Lemma. The single period beta of a series of cash flows over a particular period is a weighted average of the betas of the individual flows over that period. Proof. We consider a series involving two cash flows, one at t = 1, x 1 and one at t = 2, x 2 The general case is derived from this particular case. PAGE 109 97 Standing at t = 0, we have prices P~ of x 1 and P~ of x 2 At t = 1 the price of x 1 is just x 1 and the price 1 of x 2 is P 2 The rate of return over the initial period t = 0 tot= 1 of the two cash flows is just their terminal value 1 0 0 x 1 + P 2 divided by their initial value P 1 + P 2 We can proceed to determine the relevant covariances. [Xl + pl l cov [p~ :\~,Rm] cov O p~'Rm = pl + (2.40) pl p~, Rm l + cov[ 0 2 pl + Po x 1 l 1 = cov[ 0 p~'Rm pl + Po 2 At t = 0, Po Po 1 and 2 Po + Po P o + Po 1 2 1 2 are non-stochastic so that we can factor them out of the covariance terms. cov = (2.41) PAGE 110 Dividing by 0 2 we obtain m = 98 (2.42) since P~/P~ is the rate of return over the initial period of 0 the cash flow x 2 and x 1 /P 1 is the rate of return over the initial period of the cash flow x 1 Now let us consider the effects of the maturity of the asset and the affect of growth on a single period beta. The argument given will show that they both have a negative effect. To see this note that if 8 2 = 8X the beta of the 2 cash flow x 2 with respect to the initial period is less than 8 1 Bx, the beta with respect to the initial period of x 1 1 we have that Po Po Bx 1 Bx + 2 S x (2.43) = and X 2 Po + Po Po + Po 1 1 2 1 1 2 2 Po Po < 1 sx + 2 sx Po + Po 1 Po + Po 1 1 2 1 2 = Bx. 1 Thus, increasing the maturity of the asset by adding future cash flows decreases beta. The argument rests on the assumed lower beta with respect to the initial period of the later flow. The magnitude of the beta of the later flow with respect to the initial period depends upon the degree of correlation of that flow with the market at t = 1. PAGE 111 99 One would expect x 2 to have a lower degree of correlation with the market at t = 1 than x 1 and consequently its beta should be lower than the beta of x 1 Now consider the effect of growth defined as a predom inance of later cash flows over earlier ones. We have shown in the lemma that the single period beta of an entire project (all cash flows, earlier ones and later ones) is a weighted average of the single period betas of each of the cash flows. Hypothesizing, as we have done, that the statis tical dependence of the later flows with the market at t = 1 ought to be lower than that of the earlier (closer to t = 1) flows, the effect of growth will be to shift weight in the weighted average to the later lower beta (with respect to the initial period) flows. This has the effect of lowering the overall beta with respect to the initial period of the asset. Thus growth has a negative effect on the single period beta. How does this result square with the intuition that a growth asset, by definition, has more total uncertainty to be resolved over its life? While this seems plausible, any single period beta measures the amount of that uncertainty resolved over a given period and a decrease in that amount is consistent with an increase in the sum over all periods of the uncertainty to be resolved. What determines whether an increase or a decrease in systematic risk over a given period takes place will depend upon the timing of the cash flows generated by the growth asset. In turn, this timing PAGE 112 100 can be hypothesized to determine the correlation of those flows with the market at t = 1, and this correlation deter mines the beta of the asset's flows relative to the given initial period. Intuitively, while growth introduces more uncertainty to be resolved over the life of the asset, that increased uncertainty will be resolved later when those cash flows are received. Our argument has focused on the beta of the cash flows of the growth asset over the initial period. The theoretical discussion shows how one would consider the betas relative to each period of the asset's cash flows. Duration and Asset Betas We consider here duration as an alternative to the term to maturity concept and examine the argument given to support its effect on beta. A growth asset is analogous to a longer term bond. Such bonds are more price sensitive to interest rate changes. These interest rate changes, effected through monetary policy, consequently change the value of growth assets more than they would change the value of shorter-term non-growth assets. Presumably the price sensitivity referred to in this argument is price sensitivity relative to the market. An increase in such price sensitivity comes about through an increase in the beta of the asset. This argument fails to consider single period betas in a multiperiod setting. It establishes that the beta of the asset over its life increases which is consistent with the hypothesis that growth introduces more uncertainty to be resolved over the entire life of the asset. PAGE 113 101 4. Monopoly Power and the Labor-Capital Ratio We will consider the following argument for monopoly power as a factor related to the firm's stock beta. It can be thought of as a "strong earnings' variability argument". The existence of monopoly power means that the firm has some control over the variability of its earnings stream relative to that of the market. Suppose that the total earnings stream of the market falls. A firm with monopoly power has some control over the price of its output, hence over total revenues. One would expect that the given monop olistic firm's earnings stream would consequently not fall as much as that of a firm without control over its output price. On the other hand, suppose that the earnings stream of the market were to rise. The monopolistic firm, maximi zing profits as it does by raising price and restricting output (in the absence of regulation) would be already at its optimal profit level. One would expect its earnings not to rise very much relative to the earnings of the market. The hypothesis presented here is that the comovement between the earnings stream of the market and that of the monopo listic firm will be lower than for a non-monopolistic firm. The consequent price fluctuation of the monopolistic firm's securities will be less than the price fluctuation of the market portfolio. (The underlying argument being that earnings variability transmits itself to price through the discounted earnings approach to valuation). A less variable price variable relative to that of the market means a less PAGE 114 102 variable rate of return on the security relative to that of the market. We now consider a generalization of the theoretical model derived by Thomadakis and Subrahmanyam [1980] that provides a specification of the relationship between the stock beta, monopoly power, and the labor-capital ratio. Depending upon the relative uncertainties introduced into the model and discussed below one can demonstrate when the stock beta will be positively related to the labor-capital ratio and negatively related to the degree of monopoly power. The model will be presented in its generality with a correct specification of the market portfolio and no restric tion on the relationship between input and output uncer tainties. One considers a one-period framework at the end of which the firm's assets and productive potential have depreciated completely. The analysis proceeds through a separate consideration of the purely competitive case in which the equilibrium value of the firm equals the repro duction cost of its assets and then of the case in which monopoly power is introduced into the demand function for the firm's single product. We proceed to a detailed speci fication of the model and to the derivation of the relation ship to beta of the labor-capital ratio and monopoly power. sources of Uncertainty Uncertainty in the price of output. This source of uncertainty is introduced through the following random PAGE 115 103 specification of the inverse demand function: p (q.) = p (q.) (1 + e) J J J J (2.44) for the j-th firm, where p. (q.) represents the expected J J value of the output's price and e is a random error term with zero mean and constant variance s 2 representing price e fluctuations about the mean during the period. Note that e is not firm specific. It thus represents an economy-wide disturbance that affects the output prices of all firms in the same way. Uncertainty in the wage rate. The firm's production function q = f(k ,L ) is assumed to be homogenous of J J J ctegree 1. This implies that q /k = f (1,L /k.) and that J J J J J the rate of technical substitution between capital and labor depends only upon the input proportions: q. L _J_,_ q. k J = k J L," J This specific type of production function underlies the analysis that follows. Capital is acquired at the beginning of the period at the certain cost of one per unit. The firm is assumed to be a quantity setter, setting its optimal output at the begin ning of the period. Wages are paid at the end of the period after production has been completed. PAGE 116 104 Wages are subject to a random disturbance w = w ( 1 + v) (2.45) where vis a zero mean error term with constant variance s 2 V representing fluctuations of w about its mean w Note that vis not firm specific so that uncertainty in the wage rate is an economy-wide phenomenon affecting the wage rate of all firms in the same manner. Also note that the wage rate is assumed to be independent of the firm. Since wage rates clearly vary across industries this assumption necessitates that empirical testing be confined to industry samples. Relationship between Demand Uncertainty and Uncertainty in the Wage Rate No assumption is made about the correlation between e and V. We can now proceed to the derivation of an expression for S The cash flow of firm j is random and is defined by J F = p q ( 1 + e) WL ( 1 + v) J J J J (2.46) The cash flow of the market is the aggregate cash flow of all risky assets including human capital F = l p q ( 1 + e) wL ( 1 + v) + l WL ( 1 + v) ( 2 4 7 ) m J J J J = I p q <1 + e). J J ~ ~ The covariance between F. and F is given by J m cov (F F ) J m 2 = p q S o J J e 2 wL S a J e v (2.48) PAGE 117 where s = I p q J J and cr 2 ev 105 = cov (e,v). The variance of the market is given by var(F) m = The general expression for S is J s. = J cov(F ,F) J m V J var(F) m V m According to the CAPM, V. = [F. h cov (F F ) ) J J m J (l+i) (2.49) (2.50) (2.51) (2.52) where V is the market value of firm j, F. is the expected J J value of the random cash flow F F is the aggregate cash J m flow of all risky assets, his the market price of risk, and i is the riskless rate of interest. Using equation (2.57) to evaluate V yields m [S h cov(F ,F )] V m m (2.53) = m (l+i) S(l hScr 2 ) e = (l+i) PAGE 118 106 Substituting equations (2.48), (2.50) and (2.53) into (2.51) yields = 2 2 p.q.a wL.o J J e J ev 2 a e 1 hSo 2 e V. (l+i) J (2.54) This expression for S. is the basis upon which the theory to J be developed is presented. Competitive Equilibrium Risk Determination We examine the systematic risk of firms in perfectly competitive industries. With free entry, the certainty equivalent price must be equal to the certainty equivalent average cost in equilibrium. Adjusting for risk, the cer tainty equivalents are CEQ(p.) = E(p.) h cov(p.,f) J J J m (2.55) = E(p.) h E(p.) So 2 J J e = 2 p. [1 h So ], J e where p = E(p ). J J CEQ(AC) = CEQ[wL + (l+i)k.] J J q. J (2.56) = [ ( 1 + i) k + wL ( 1hS a 2 ) ] J ev q J since PAGE 119 107 CEQ[wL.] = E(wL.) h cov(wL. ,F) J J J rn = WL. h cov[w(l+v)L I p q. (l+e)] J J J J = 2 wL. hwL.Scr J J ev = wL. ( 1 hSo 2 ) J ev Equating the two certainty equivalents above yields or 2 p. (1-hScr ) J e = p q. (1-hSa 2 ) = J J e [ ( 1 + i) k + wL ( 1hS cr 2 ) ] J J ev q J [ ( 1 + i ) k + wL ( 1hS a 2 ) J J ev (2.57) (2.58) ( 2. 58) Using the equation formula (2.52) for v., (2.48) and (2.46) J yields 2 2 [p q wL. h[p.q Sa wL.Scr ]] V. = J J J J J e J ev J (l+i) = [ p q ( 1hS a 2 ) WL ( 1hS a 2 ) ] J J e J ev (l+i) = [(l+i)k. + wL. (1-hSa 2 ) wL. (1-hSa 2 )] J J ev ev (l+i) = k J (2.59) The market value of the competitive firm equals the repro duction cost of its assets regardless of the correlation between the input and output uncertainties. PAGE 120 108 Derivation of the Systematic Risk of the Purely Competitive Firm By substituting the conditions (2.58) and (2.59) into (2.54), we obtain 2 2 2 ( p q cr wL cr ) ( 1 hS 0 ) B = J J e J ev e J 0 2 k.(l+i) e J (2.60) = p q (l-hS0 2 )0 2 wL 0 2 (l-hS0 2 ) J J e e J ev e cr 2 k. (l+i) e J [(l+i)k + wL. (1-hScr 2 )]0 2 2 2 wL 0 ( 1hS 0 ) J ev e = J J ev e L. 0 2 o 2 = w J e ev l + 1+1. -k 2 J o e Equation (2.60) states that 8. is a linear function of firm J j's labor-capital ratio and does not depend on the laborcapital ratio of all firms as it does in the Subrahmanyam and Thomadakis equation (10) [p. 444]. Note that competi tive firms with no labor have betas equal to unity. Furthermore, the competitive firm's beta will be a positive function of the labor-capital ratio if and only if o 2 o 2 > o. e ev (2.61) 2 2 In the particular case when 0 = 0 the beta of the e ev competitive firm will equal unity regardless of the laborcapital ratio. PAGE 121 109 Condition (2.61) is equivalent to the following condition: 1 > cov(e,v) = S-2 ev a e (2.62) This says that wage uncertainty has a "beta" less than unity relative to price uncertainty. It is equivalent to the condition derived by Subrahmanyam and Thomadakis that Z Z > 0 l 2 in their notation, where = cov(e,F) m In our model, = Scr 2 e and and and the conditions becomes = 2 2 a > a or S~~ < 1. e ev ev = 2 Sa ev (2.63) cov(v,F). m The random error terms e and v are unobservable so that we cannot directly test whether condition (2.61) [equiva lently (2.62)] holds. However, an indirect test is possible. If the coefficient of the labor-capital ratio in the linear (later non-linear) regression turns out to be significantly positive (negative) this can be interpreted in terms of the condition that wage uncertainty has a co-movement relative to price uncertainty less (greater) than unity. PAGE 122 110 If e and v are independent, then equation (2.60) becomes WL L J B 1 + w -1. 1 + l+i = l+i = J k k. J J (2.64) The condition on the volatilities of price and wage uncertainty given in equation (2.62) can be explained by referring to the market portfolio of all risky assets including human capital in this model. The beta of the market portfolio will be a weighted average of the betas of firms plus the betas of labor, 1 = = I x1..B1. + I y .B L i J j We have that the weights add up to unity, Each firm is affected in the same way by price and wage uncertainty so that if one firm has a firm beta greater than unity, as would be the case if B~ ~ < 1 [Appendix 1], then ev all firms would have firm betas greater than unity. The cross-sectional variation in their betas would arise as a result of their different labor capital ratios and degrees of monopoly power. Since all firm betas above are greater than unity, all labor betas must be less than unity. Suppose now that capital is decreased relative to labor. Then firms would have to support more risk than labor PAGE 123 111 but with a relatively smaller amount of capital. This implies that the firm betas are increasing functions of the labor capital ratio. The reverse argument would hold if S~~ > 1. ev How do these results square with traditional wisdom? The usual arguments given for the effect of operating leverage, the ratio of fixed to variable costs, have viewed the capital element as representing the firm's fixed costs while the labor element is linked with the firm's variable costs. By an appeal to the earnings variability argument it is then demonstrated that the stock beta will be an in creasing function of the firm's capital-labor ratio, equiva lently a decreasing function of its labor-capital ratio. This relationship always holds under this interpretation. We have seen that generalizing the earnings volatility argument to new variables expected to affect beta will lead to arguments that are as suspect as the original argument. On the other hand, the financial leverage argument may lead to new insights regarding the determinants of systematic risk. One such argument is the following. We can view labor as analogous to risky debt in our model. We know that the financial leverage argument is not based upon the analogy between debt payments and fixed costs, because it holds for risky debt as well. Then systematic risk might be expected to be a positive function of the labor-capital ratio as it is for the debt-equity ratio. The model presented here gives a specification of the condition on the uncertainty in PAGE 124 112 the labor element relative to the uncertainty in output (through prices) under which this relationship will hold. Systematic Risk and Monopoly Power The firm is assumed to act as a price taker except in its output market. Furthermore, any given firm's output decision has no effect on aggregate output for the industry s = I j p.q and upon aggregate labor for the industry J J D = w I L : J j as aq J = ao aq J = 0. To determine its optimal output, the firm maximizes its market value net of total physical capital costs k .. Firm J value equals the certainty equivalent of total revenues minus the certainty equivalent cf total labor costs minus total capital costs (assumed non-random). V k. = CEQ(p q.) CEQ(wL.) k .. J J J J J J The certainty equivalents of p~. and wL are L J and CEQ (p ) J CEQ (wL.) J = wL [ 1hS cr 2 ] J ev (2.65) (2.66) (2.67) Substituting these certainty equivalents into the formula for V k yields: J J PAGE 125 V. k J J where a = J 113 q [ p ( 1-hS er 2 ) J = l+i J e k. _J ( l+i) ] q. J q. = _]_ l+i L. _J and qj 2 [ p ( 1hS er ) J e C. = J k. _J q J 2 L. _J (2.68) w(l-hSer ) ev q J 2 wa. ( 1-hSer ) c. ( l+i) ] J ev J (2.69) The optimal output is obtained by maximizing V. k. with J J respect to q .. Taking the partial derivative of the above J expression with respect to q. yields: J l 2 2 l+ [p (1-hSer) wa. (1-hSer ) c. (l+i)] l. J e J ev J (2.70) q -~ a p 2 a a 2 a c l + ...1. __ J ( 1-hSer ) w __ J ( 1-hSer ) -2 ( l+i) l +~ '"lqJ e '"I '"I a aq. ev aq. J J 1 2 ap. aa. 2 = i+ [(1-hSer) (p. + q. ~) w(a + q.~) (1-hSer ) J. e J J oq. J J oq. ev J J ac (c + q ~) (l+i}]. J J q J Noting that p + q. ap /aq = p (1-u ), where u. is the J J J J J J J reciprocal of the price elasticity of demand, u. J = q. ap _J __ J P. aq J J 1 = pj ap q. J J PAGE 126 114 and evaluating aa./aq. and ac. /aq. J J J J aa oL. L q. J = J _J = 0 (2.71) J aq. aq. q. J J J ac. ak. k. q J = __ J _i = 0 (2.72) J aq. aq. q. J J J we obtain the first-order condition ( 1-hS a 2 ) p ( 1-u ) 2 c (l+i) 0. (2.73) wa. (1-hScr ) = e J J J ev J Rewriting this expression ( 1-u ) p~Xl = c (l+i) + wajx 2 (2. 74) J J J where Xl 1 hScr 2 and x2 1 hSa 2 (2.75) = = e ev We have used the assumption of constant optimal factor proportions oL /aq = L /q and ak./aq. = k./q .. J J J J J J J J To interpret the first-order condition we must first describe how monopoly power is introduced into this model. This is accomplished by positing a marginal revenue function of the following form: m (q ) = (1-u.)p. (q ) (l+e). J J J J J (2.76) The firm's demand function now depends upon output. Marginal revenue is allowed to diverge from average revenue with u. J equal to the reciprocal of the price of elasticity of PAGE 127 115 demand. Note that a value of zero for u. indicates an J infinitely elastic demand curve or a perfectly competitive firm. Higher values of u. indicate a lower price elasticity J of demand, or equivalently, a greater degree of monopoly power. Now, to interpret the first-order condition, note that 2 (1-hScr )p (1-u ) is simply the certainty equivalent of e J J marginal revenue while wa. (l-hscr 2 ) + c (l+i) is the cerJ ev J tainty equivalent of marginal cost. The first-order condition equates the two. Optimal Valuation and Beta for the Monopolistic Firm where The firm valuation equation is v~ J = l+i = 1 hscr 2 e and 2 x 2 = 1 hScr ev We may substitute the first order condition into this formula to obtain: v~ J = = l+i The general expression for S. is equation (2.79) J = 2 2 p.q cr wL.cr J J e J ev (l+i)cr 2 e (2.78) (2. 79) PAGE 128 116 Using the optimal expected price and optimal firm value in equation (2.79) as derived in Appendix 2, we obtain: f3 J = 1 + (1-u.)wy. (0 2 0 2 ) J J e ev o 2 [(1+i) +X 2 u.wy.] 1 e J J (2.80) where y = L /k .. This expression conforms with the result J J J for the competitive firm. The sign of the difference 2 2 between 0 and 0 plays an important role here. When it is e ev positive (negative) beta is a decreasing (increasing) func. -1 tion of monopoly power as proxied by (1-u.) The relationJ ship between beta and the labor-capital ratio is seen to be non-linear because of the introduction of monopoly power. To determine the sign of the effect of the labor-capital ratio on f3., we take the partial derivative with respect to J y to obtain J where as __ ] ay. J = 0 2 ( 1-u ) w ( 1 + i) [ 0 2 e J e 2 0 ] ev 2 [(l+i) + x 2 u.wy.]0 J J e Again, sign as./ay = sign[0 2 J J e 02 ] ev 5. The Model of Systematic Risk and Hypotheses to be Tested (2.81) Using the microfacotrs discussed to estimate beta the market model will be employed: PAGE 129 117 R = a. + 8 n,tMt + n t' n,t n,t n, (2.82) where B = b'W + s n,t n,t (2.83) and a. n,t = a'W + E n, t' (2.84) where W = {W } the matrix of microfactors for n in year t. n,t Substituting equations (2.83) and (2.84) into (2.82), we obtain (2.85) Using the appropriate techniques discussed in the next chapter, from the econometrics literature on pooling cross sectional and time-series data estimates of band s will n be obtained, and these will be used to generate B by n,t substituting these estimates into the generating formula for B (2.83) and using the set of microfactors { W }. The n,t n,t sign of the coefficients will be tested relative to what the theory we have discussed predicts. The set of hypotheses that arises out of the theory discussed here is the following: as (1) J > 0 a[financial leverage] PAGE 130 118 as ? ( 2) J > 0 < a[volatility of operating earnings] = as. ( 3) J < 0 a [growth] ( 4) as J > 0 a[labor/capital] as. (5) J < 0. a[monopoly power] These hypotheses will be examined within the context of the generating process and estimation procedure to be applied. PAGE 131 CHAPTER III ESTIMATION TECHNIQUES AND EMPIRICAL RESULTS 1. Estimation Techniques Fixed Effects The model to be estimated is: = a + B M + n nt nt t nt ( 3 .1) where a = a'W + s nt nt n ( 3 2) and = b'W + nt n ( 3 3) Substituting (3.2) and (3.3) into (3.1), we obtain: = a'W + + b' (W M) + M + n nt n nt t n t nt ( 3 4) In the fixed effects specification of the model it is assumed thats and~ are fixed constants rather than n n random variables. If this is the case, then we can apply ordinary least-squares to (3.4) and obtain unbiased, consis tent, and efficient estimates of the slope coefficients a' and b' and the constant effects ~n and 119 n PAGE 132 120 The fixed estimation technique is also called the least-squares with dummy variable technique because it amounts to the following. Suppose that there are N firms in the sample over which the pooled cross-sectional and time series model in (3.4) is to be estimated. Introduce N firm dummy variables D 1 ,DN. Then Rnt is run on D 1 ,DN to obtain the fixed effects estimates f 1 ... ,EN on WntMt to obtain the slope coefficients b', and on D 1 Mt, ,DNMt to obtain fixed effects estimates t 1 ... ,tN. Note that there is no intercept in this model specification. All the variables listed above are included on the right-hand side of the regression just described. These estimates are sample statistics and as such we can compute their sampling variances. To do so, we use the following formulae devel oped by Nerlove and designed to avoid the problem of nega tive variance estimates that arises in computing variance estimates in pooled cross-sectional time-series models. These formulae are: 82 1 I 2 ( 3. 5) = E: n, E: N n 82 1 I A 2 ( 3 6) = N E: n, n and T 82 l. [Rnt a'W b'W M A A 2 ( 3 7) = E: ~nMt] n t=l nt nt t n where 2 is the variance of for the n-th firm. CJ nnt n PAGE 133 121 This fixed effects estimator is a within group estima tor in the analysis of covariance sense in that it subtracts out the means over time for each firm of the regressors. In so doing, the between-group (represented by firms) variation in those variables is ignored. If the effects in this model are indeed random effects then such a procedure results in a loss of efficiency. Whether the gain in efficiency by performing a GLS procedure on this model is substantial, how to determine this gain a priori, and whether the GLS estima tors are substantially different from the fixed effects estimators will be considered in the next section. Random Effects Under this specification the firm specific effects ~ n and E: n are treated as random variables with zero means and 2 d 2 variances o ~ an o E: The GLS estimation procedure involves estimating equation (3.4) where the error term is E: + ~ Mt+n t" The variance-covariance matrix is consistently n n n estimated using the estimates in equations (3.5), (3.6) and (3.7). However, it can be shown [Maddala, 1978] that if the 2/ 2 2 l . ratio T o~ To ~ + o n is c ose to unity then the GLS estimator will be almost identical to the fixed effects estimator. Prior Likelihood Estimation The approach concerns itself with best linear unbiased prediction of the firm effects. It is thus in the spirit of the Theil and Goldberger [1960] mixed estimation procedure. PAGE 134 122 For the model in (3.4), we have weak stochastic prior infor mation requiring the prior likelihood generalization of the Theil and Goldberger procedure. An advantage of this approach is that it does not require a Bayesian argument as support. For expository purposes only, assume that the means of the firm effects s and e: are known to be zero. n n The appropriate likelihood function for all the models parameters is: L = 1 ext-1, N + (27T)N/20~ I 2crs n=l 1 ext1 N + I (2n)N/20N 2cr 2 n=l e: e: where xnt = (Wnt ,WntMt) and N 1 2. 2 n=l cr n s~ l l y' = L (R -y'X -s M -e:) T 2] t=l nt nt n t n ( 3 8) (a',b'). ( 3 9) Maximization of the likelihood function is equivalent to minimization of the following quadratic expression: Q = N l T I 2 I n=l a t=l n (R y'X s M e: ) 2 nt nt n t n 2 e: n (3.10) This is to be minimized with respect toy, s and e: The n n PAGE 135 123 first order conditions are: ao N 1 T ay = I I (Rnt y'X c; nMt E: )(-X t) 2 nt n n n=l a t=l (3.11) n = 0' ao 1 T = I 2(R y'X c;n Mt c:: n) (-Mt) 2 nt nt n a t=l (3.12) n + ~(2.;n) a.; = 0, aQ 1 T = I 2(R y'X c;nMt ) (-1) 2 nt nt n n a t=l (3.13) n + .!_ 2 2 n a = 0. Solving these equations one obtains the random effects GLS estimator yGLs The solutions of equations (3.11), (3.12) and (3.13) are the following corrected estimators: where c; = n l [ (R ,\ R ) y I (X -,\X ) ] M t nt n GLS nt n t 2 a \ M2 ,\ T(M)2 + n l t 2 a.; (3.14) PAGE 136 124 2 a A E = a2+a2 ( 3 .15) E n and E* = (R y~LSXn s*M) L n n n (3.16) These estimators, by utilizing the prior stochastic A information, correct the sample based estimates s and f n n by shifting them toward their common means. It can be shown 2 [Lee and Griffiths, 1978], under the assumption that as' 2 2 a, and a are known, thats* and E* are the best linear E n n n unbiased predictors of the random variables s and E n n With yGL 5 ,given Xnt and Mt, these provide the best unbiased linear predictor of Rnt" When the variances are unknown, one approximates the results with consistent estimates of the variances. These are given by formulae (3.7), (3.5) and (3.6) repeated below, respectively. 82 1 T A 2 I [Rn,t A I A = T YGLSXnt E snMt] n t=l n 82 1 I ,...2 = E n' E N n and 82 1 I A 2 = s s N n PAGE 137 125 These consistent estimates of the variance components were suggested by Nerlove (1971] to avoid the problem of negative variance estimates that arises in variance components models. As pointed out by Maddala [1978], little is known about the small sample properties of the prior likelihood estimators when these consistent estimates of the variance components are substituted for the true unknown variances. 2. Empirical Results with Five Descriptors Fixed Effects Estimation Definitions and data sources for descriptors. The set of descriptors employed in the regression results reported here consisted of the following variables: (1) Financial Leverage Wl nt ( 2) Variability of Earninqs w2 nt ( 3) Growth Rate in Assets w3 nt ( 4) Labor Capital Ratio w4 nt ( 5) Monopoly Power ws nt Financial leverage is measured as the ratio of the sum of the book value of long-term debt plus current liabilities to the market value of equity. Variability of earnings is defined as the standard deviation of earnings over the preceeding five years divided by the absolute value of average earnings over that period. PAGE 138 126 Growth is defined as the growth rate in assets measured as an average rate of growth in assets over the last five years. This is the measure which Modigliani and Miller [1966) use in their study of the cost of capital to the electric utility industry. The labor capital ratio is proxied for by the number of employees divided by total generating capability [Atkinson and Halvorsen, 1976). Monopoly power is proxied for by P-MC/P where it is assumed that marginal cost is equal to average cost and hence, multiplying through by output Q, one obtains (Sales Operating Expenses)/Sales as the measure of monopoly power. The descriptors are represented by the set { Wnt} over the time period 1968-1977 for a sample of 74 firms in the electric utilit y industry. The data needed to construct these descriptors is taken from the Compustat tape for this period. The dependent variable in the market model regression is the annualized rate of return for firm n in year t fully adjusted for dividends, stock splits, and other distribu tions. Monthly return data is drawn from the CRSP tapes. A proxy for the market portfolio is the Sand P 500 stock index. Monthly returns are taken from the Sand P PAGE 139 127 stock index, adjusted for dividends available from the Sand P dividend record and are annualized. Parameter estimates. A pooled cross-section time series regression was run of the equation (3.4) over the years 1968-1977. A total of 740 observations are available in principle for the 74 companies over the ten-year period. However, certain descriptors could not always be constructed for every company for every year. In estimating this model, the method of classical least-squares was employed. That is, if any descriptor could not be constructed for any given company in any given year, then all observations for that company for that year were discarded. When this was done, 661 observations were left to estimate the model in equation (3.4). The parameter estimates for the slope coefficients used in generating betas according to equation (3.3) are pre sented in Table 3.1. All the ~ 's were significant with a l mean of 35.3. The E. 's were also significant with a mean of l The signs of growth in assets and monopoly power were negative as expected from the theory presented. These are represented by B 3 and b 5 The sign of the labor-capital ratio was positive, again as suggested by the model presented in Chapter II. In that model it was shown that monopoly power and the labor-capital ratio should have opposite signs if S ~~ < 1. Volatility of earnings was negatively related ve to beta. While this may seem surprising, it can be explained PAGE 140 128 Table 3.1 Estimated Descriptor Coefficients Used in Generating Betas for the Five Descriptor Data Set Estimates t-statistic Financial Leverage bl = 1.013283 (-0.5652) A Variability of Earnings bl = 0.079451 (-0.1078) A Growth in Assets b3 = 1.212741 (-1. 0591) A Labor Capital Ratio b4 = 49.091366 ( 0.4025) A Monopoly Power bs = -33.931048 (-3.6557) PAGE 141 129 as follows: It is possible that earnings volatility may be negatively related to beta in that an increase in total risk results from an increase in the individualistic component of total risk that is greater than the decrease in systematic risk. That is, electric utilities are less affected by market movements than they are by firm-specific non-market related factors. Proxied for by earnings' variability are factors that serve to decrease the systematic component of risk. At the same time the individualistic component of risk is the dominant risk component contained in total risk. Its increase outweighs the decrease in beta resulting in an overall increase in total risk. Beta estimates. Using the parameter estimates obtained, betas were generated. Their mean values are presented in Table 3.2 by year. In addition, mean values of the beta estimates using the historical market model with constant alpha and beta and of the Vasicek betas are reported in Table 3.2. The variance of these alternative betas are reported in Table 3.3. The model was also estimated where beta was allowed to vary but alpha was not as performed by Barr-Rosenberg and McKibben. The coefficients of the descriptors are reported in Table 3.4. These results show a reversal in the signs of the labor-capital ratio 6 4 and of the earnings variability A measure b 2 This contradicts the model presented in Chapter II. Volatility of earnings assumes a positive sign PAGE 142 130 Table 3.2 Yearly Means of Fixed Effects, Historical, and Varicek Betas for the Five Descriptor Data Set A A (a) A ( b) Year B LSDV S H S VASICEK 1968 .95926286 .64044324 .68331437 1969 .92605679 .66711028 .70878930 1970 .91133564 .77261070 .82696451 1971 .92450968 .67966606 .74097535 1972 .93290894 .71908183 .79335239 1973 .94092318 75797930 .84554324 1974 1.07680295 .66917239 .74667592 1975 1.18744822 .77668930 .78576704 1976 1. 26306786 .80088535 .80670761 1977 1.41859204 .78955831 .80319169 (a) Obtained from running the market model R = a + B M + n with constant a and B over the n,t t nt preceding 60 months using monthly returns. (b) A prior beta was obtained by running the sand P 50 stock inde x on the Sand P 500 stock index for each year over the preceding 60 months using monthly returns. The variance of this sample statistic S ~ was B PRIOR calculated in the usual manner. Then each beta obtained in (a) was adjusted by weighting it inversely to its variances ~ to obtain the Vasicek beta defined as: B A A S PRIOR L+ s ~ s ~ B B A PRIOR B VASICEK = 1 + 1 s 2 s ~ s i3 PRIOR PAGE 143 131 Table 3.3 Variances of Fixed Effects, Historical, and Vasicek Betas Based on the Five Descriptor Data Set ~LSDV A ~ VASICEK Year $H 1968 .15199838 .03156987 .00240233 1969 .14694203 .02896324 .00250584 1970 .14647613 .02897895 .00304759 1971 .14045917 .02583410 .00297448 1972 .13666891 .03527560 .00396288 1973 .13577623 .04008501 .00434283 1974 .14952887 .04272373 .00462506 1975 .14831943 .03880162 .00338537 1976 .15372869 .03552042 .00282300 1977 .21096820 .03301176 .00289746 PAGE 144 132 Table 3.4 Estimated Descriptor Coefficients for Constant Alpha Model for the Five Descriptor Data Set Estimates t-statistic bl = 1.392955 (-0.8942) b 2 = 0.412666 ( 0.6809) b 3 = 2.081388 (-2.1333) b 4 = 4.926343 (-0.0465) b s = -48.841175 (-6.0145) PAGE 145 133 in contrast to the previous regression with alpha allowed to vary. In addition, 1.36 % of the betas turn out to be nega tive. This result indicates a misspecification of the model and perhaps some evidence of either a changing risk-free rate or some evidence that the CAPM relationship a = RF(l-S) is correct. GLS Estimation To decide whether to proceed with the GLS estimation 2 2 procedure, 8~ and 8 were estimated using equation (3.6) and equation (3.17): = 1 NT n,t [R a 'W nt nt b'W M nt t These estimates are given below: 8 2 = 1232.71 and f (3.17) n = .01773330. It is clear that the T8 2 is large relative to 8 2 so that the ratio is close to unity. This indicates that the GLS estimates will be virtually identical to the fixed effects estimates. Thus, it was not necessary to construct such estimates independently. Further, 8 2 was calculated using equation (3.5). E: PAGE 146 134 8 2 = 51. 94. E: Prior Likelihood Estimation The prior likelihood estimates are obtained by taking the GLS estimates of the slope coefficients for alpha and for beta, a' and b' and adjusting the firm specific effects t and E toward their means in accordance with the stochasn n tic prior information available on them. Under our specification of the model we have assumed these random effects to be distributed with zero means and with variances 0~ and 2 o respectively. E: To determine whether this adjustment process leads to substantially different estimates for the firm specific effects one has to calculate 1 82 A n ;\ = and 2 82 as 1 + n 82 E: Assuming that 8 2 = 8 2 is independent of the firm n, we n obtain A = .999658704 and = .000014385. For this specification with zero means for s ands n n the prior likelihood estimates are essentially equal to the fixed effects estimators. Thus, independent calculations of such estimators is uninformative. PAGE 147 135 Prediction The procedure in predicting betas is to run the model over a sequence of years and then use the generated coeffi cients from this regression to estimate alphas and betas in the succeeding year by using the values of the descriptors in that year. This procedure was carried out for five time periods: Period I, 1968-1972; Period II, 1968-1973; Period III, 1968-1974; Period IV, 1968-1975; and Period V, 1968-1976 with predictions made in 1973, 1974, 1975 and 1976. (Data were frequently missing in 1977.) A byproduct of these regressions is that they allow us to look at the sensitivit y of the model with respect to addition of obser vations. The estimated a's and b's are presented in Table 3.5. Both the signs of the earnings variability variable and monopol y power change, and growth and financial leverage are consistentl y negative. These coefficients can be compared to the 1968-1977 regression results from Table 3.1. Finan cial leverage is again negative as is the earnings volatil ity variable. An e x planation of these changes in sign is provided by the correlation matrix which indicates that { WntMt } has a high degree of multi-collinearity (see Table 3. 6) Out of sample prediction. Using the market model, returns are generated for the prediction year according to = a + 6 M n,t n,t t PAGE 148 136 Table 3.5 Estimated Descriptor Coefficients used in Generating Alphas and Betas for Out of Sample Prediction I II III IV V 1968-72 1968-73 1968-74 1968-75 1968-76 ai 1.306252 .399553 .232062 .420608 .734357 (1.2918) (. 9739) (.6946) (1.1651) (2.0633) -0.450087 -.222820 -.164285 .010626 -.010349 a 2 (-.6604) (-1.3397) (-1.1590) (.0739) (-.0727) 1. 463421 -.362983 -.259716 -.428413 -.203732 a3 (1.6991) (-1.3967) (-1.1440) (-2.0854) (-1.0014) -59.868058 1.838235 4.861276 -7.372474 -6.721782 a4 (-1. 5610) (.0858) (. 2551) (-.3560) (-.3241) -33.572091 8.177407 13.236058 -6.400697 -1.858883 as (-1.8128) (1.4764) (2. 8830) (-.3560) (-.9199) bl -14.590407 -5.484835 -2.681838 -5.915279 -4.873919 (-2.1605) (-1. 6187) (-1.1261) (-2.8028) (-2.4914) B 2 4.126057 -.105003 -.554261 .751740 .416234 (1.0385) (-.0762) (-.5862) (.9802) (.5565) -3.173927 b3 -14.979603 -.995999 -3.874972 -2.315866 (-2.6893) (-.4400) (-2.5967) (1. 0983) (-1.9459) 1\ 821. 560913 439.359132 371.446468 134.973286 117.49214 (3.5671) (2.9613) (3. 0063) (1.0983) (.9533) 23.926482 -60.843943 -54.828518 b5 323.094999 109.614978 (2.8766 ) (2.2695) (1.1389) (-6.2195) (-5.5001) PAGE 149 Table 3.6 Correlation Matrix of Descriptors and Descriptors Times the Market for the Five Descriptor Data Set Y76 Y77 Y78 Y79 Y80 X76 X77 X78 X79 X80 Wl w2 w3 w4 ws W 1 M W 2 M W 3 M W 4 M W 5 M Y76 Wl 1.0 -.14490 .12346 .27148 .12572 -.10285 -.12885 -.10848 -.07592 -.1219 Y77 w2 1.0 .46341 -.17832 -.34717 .08729 .25093 .11536 .05961 .0922 Y78 w3 1.0 -.27343 -.12597 .02566 .06706 .10716 -.01852 .02739 w4 ..... Y79 1.0 -.19122 -.06197 -.08005 -.09048 .0709 -.06962 w -....J Y80 ws 1.0 -.15306 -.20982 -.14169 -.16190 -.15714 X76 W 1 M 1.0 .88687 .92451 .91285 .99763 X77 W 2 M 1.0 .88691 .81021 .88850 X78 W 3 M 1.0 .80018 .92271 X79 W 4 M 1.0 .90868 X80 W 5 M 1.0 PAGE 150 138 where a t and S t were predicted using the coefficients n, n, from the estimation period (not including the prediction year) and Mt is the rate of return of the market portfolio for the prediction year. This was done for all firms in the sample. Predicted returns were than compared to actual returns in that year and the squared differences A 2 (R R ) were added up over all firms. The results for nt nt the LSDV betas, the historical betas, and the naive predictor of returns ft = Mt are given in Table 3.7. n,t Within sample prediction. A second procedure was employed in predicting returns. The procedure was to test the predictors within the sample in the following manner. The basic regression generating the coefficients of beta was run, as before, over the years 1968-1977. Betas were gener ated for a given year as reported, and the residual sum of squares was calculated for the given year's residuals. This was compared to the naive predictor, using the market return in the given year as a predictor for returns, and to the historical betas predictor generated for the given year from a time-series regression over the preceding 60 months including the prediction year. The results are reported in Table 3.8. It is seen that in two years out of four, the LSDV betas out perform the historical betas and the naive predic tor. The historical beta never out performs the naive predictor. The years of superior predictive performance were 1974 and 1975. An explanation of this is that the LSDV PAGE 151 139 Table 3.7 Out of Sample Prediction Results for the Five Descriptor Data Set Prediction Year (R R ) 2 L nt nt 1973 18.391503 .940242 .713844 1974 4.085162 1.483542 .997645 1975 35.743736 14.298451 8.632252 1976 3.636915 2.026604 1.179439 Beta LSDV 5 B a H a. =0, B=l LSDV 5 BH a. =0, S =l LSDV 5 S H a. =0, S=l LSDV 5 S H a. =0, S=l a Historical betas obtained from the previous year's historical regression over the preceding 60 months were used to predict returns in the next year according to ft = a + s Mt. n,t n n PAGE 152 140 Table 3.8 Within Sample Prediction Results for the Five Descriptor Data Set Prediction Year I (Rnt R )2 nt 1973 .806173 .784918 .713844 1974 .295366 1.696016 .997755 1975 2.557492 10.662360 8.632702 1976 1.693992 1.824591 1.179484 Beta LSDV 5 S a H a =0, S=l LSDV 5 SH a =0, S=l LSDV 5 SH a =0, S=l LSDV 5 SH a =0, S=l a Historical betas obtained from the given year's historical regression over the preceding 60 months including the predictjon year, were used to predict returns in the same year according to Rnt = a nt+ s ntMt. PAGE 153 141 betas are generated from a model that takes into account structural changes that occur over time by estimating alphas and betas that vary with time. The historical betas and the naive beta assume constancy of the model over time by definition in the latter case and by the constant parameter time series market model estimation procedure in the latter case. The years of superior predictive performance were years of structural changes and severe market swings. This points to the predictive power of this model over periods of such changes. 3. Empirical Results with Twenty-nine Descriptor Data Set Fixed Effects Estimation Names of descriptors. The set of descriptors employed in the regression results reported here consisted of the following set of variables: (1) Current Price (2) Leverage at Market (3) Total Debt/Assets (4) Book Value/Price (5) Market Value of Common Equity (6) Earnings/Price (7) Return on Equity (8) Log (Total Assets) (9) Flow (10) Normal Payout (Last 5 years) Wl nt w2 nt w3 nt w4 nt ws nt w6 nt w7 nt WB nt w9 nt WlO nt PAGE 154 ( 11) ( 12) (13) (14) ( 15) ( 16) (17) (18) ( 19) ( 2 0) ( 21) ( 22) (23) ( 24) ( 25) ( 2 6) (27) ( 2 8) ( 2 9) ( 3 0) ( 31) 142 Measure of Proportional Changes in Adjusted E.P.S. in Last 2 Fiscal Years Average Proportional Cut in Dividends (Last 5 years) Variance of Cash Flow (Last 5 years) Variability of Annual Earnings (Last 5 years) Earnings Growth Rate Asset Growth Rate Share Turnover Rate (Last 12 months) Dividend Yield Typical Earnings/Price Ratio (Last 5 years) Indicator of Negligible Earnings Regression Coefficient for Normalized Earnings/Price Ratio of Firm on Normalized Earnings/Price Ratio of Economy Average Absolute Monthly Range of Price, in Logarithms (Last 12 months) Logarithm of (Trading Volume/Variation in Price) in Previous Year Potential Dilution Applicable Federal Tax Rate Historical a 2 Historical Beta x Historical Sigma= Historical Beta Squared= s! Vasicek Beta Historical Beta 0 H Historical a B xa H Wll nt Wl2 nt Wl3 nt Wl4 nt WlS nt Wl6 nt Wl7 nt Wl8 nt Wl9 nt w20 nt w21 nt w22 nt w23 nt w24 nt w2s nt w26 nt w27 nt w2s nt w29 nt w30 nt w31 nt PAGE 155 143 Classification of descriptors. Market Variability ( 1) (17) ( 2 2) (23) ( 2 6) (27) ( 2 8) ( 2 9) ( 3 0) ( 31) Current Price Share Turnover Rate Average Absolute Monthly Range of Price Log (Trading Volume/Variation in Price) Historical Beta: S Historical cr H S~cr s2 H Vasicek Beta cr2 Financial Risk (2) Leverage at Market (3) Total Debt/Assets (9) Flow (24) Potential Dilution Unsuccess and Low Valuation (4) Book Value to Price (12) Average Proportional Cut in Dividends (15) Earnings Growth Rate (20) Indicator of Negligible Earnings (25) Applicable Federal Tax Rate Growth (6) Earnings/Price (10) Normal Payout (16) Asset Growth Rate (18) Dividend Yield (19) Typical Earnings/Price Ratio Immaturity and Smallness (5) Market Value of Common Equity (7) Return on Equity (8) Log (Total Assets) PAGE 156 144 Earnings Variability (13) Variance of Cash Flow (14) Variability of Annual Earnings (21) Regression Coefficient for Normalized Earnings/Price Ratio of Firm on Normalized Earnings/Price Ratio of Economy; "Beaver Beta". Descriptor definitions. Each descriptor is defined for firm n in year t as follows: the full set is represented by {W~t}, j = 1, .. ,31. ( 1) Current Price ( 2) Leverage at Market ( 3) Total Debt/Assets ( 4) Book Value/Price the natural logarithm of the price of the firm's common stock adjusted for stock splits and dividends. long-term debt+ preferred stock+ market value of common equity 7 market value of common equity. total long-term debt+ total current and accrued liabili ties 7 total assets. book value of common equity/market value of common equity. (5) Market Value of Common Equity (as of the end of the previous year). ( 6) Earnings to Price primary E.P.S. (excluding extraordinary items) closing price of the firm's stock. PAGE 157 (7) ( 8) ( 9) ( 10) ( 11) ( 12) (13) 145 Return on Equity I (earnings available for common after adjustments for common stock equivalents) + I (book value of common equity as reported). (The sums are taken over the last 5 years.) Logarithm of Total Assets the logarithm of the aver age of total assets over the last 5 years. Flow the average over the last 5 years of the ratio of (available for common+ total deferred income taxes+ total depreciation) (total current liabilities) Normal Payout (last 5 years) I common dividends + I earnings available for common. (The sums are taken over the last 5 years.) Measure of Proportional Changes in Adjusted E.P.S. in the Last (E.P.S.t E.P.S.t-l) -. (E.P.S.t 2 Fiscal Years Average Proportional Cut in Dividends ( last 5 years) + E.P.S.t_ 1 ) 2 The maximum of O and (D.P.S.t D.P.S.t-l) (D.P.S.t + D.P.S.t_ 1 ) 2 (The sums are taken over the last 5 years.) Variance of Cash Flow (last 5 years) The sample standard deviation of the firm's cash flow over the last 5 years+ the aver age cash flow over the last 5 years. (Cash flow is defined as the sum of available for common, total depreciation, and deferred total income taxes.) PAGE 158 (14) ( 15) ( 16) (17) (18) ( 19) ( 2 0) ( 21) 146 Variability of Annual Earnings (last the sarre definition as for variance of cash flow except in terms of earnings. 5 years) Earnings Growth Rate Asset Growth Rate earnings per share is regressed on time over the last 5 years. This slope coefficient is divided by the sample mean of E.P.S. over the last 5 years. the same definition as in earnings growth rate except total assets is used instead of earnings. Share Turnover I shares traded in month t Rate (The sum Dividend Yield is shares outstanding at t=O taken over the last 12 months.) common dividends paid in last year 7 market value of common equity. Typical Earnings/Price Ratio (last 5 years) (The sums are taken over the last 5 years.) Indicator of Negligible Earnings 0 or 1 depending upon whether the earnings to price ratio is greater than or equal to .005 or less than .005. Regression Coeffi cient for Normal ized Earnings/Price Ratio of Firm on Normalized Earn ings/Price Ratio of Economy E.P.S. is regressed on time over the last 5 years. This slope coefficient is normalized by dividing the previous year's closing stock price. Next the E.P.S. of the S+P 500 Index is regressed on time over the last 5 years. The slope coefficient PAGE 159 ( 2 2) ( 2 3) (24) ( 2 5) ( 2 6) 147 obtained from this regres sion is normalized by dividing by the previous year's closing price of the S+P 500 Index. (This procedure is repeated for each firm for each of ten years to obtain normalized earnings/price ratio trend coefficients for the given firm and for the S+P 500 Index for each of ten years. Finally, the variable is constructed by regressing the given firm's trend coefficient on the S+P 500 Index trend coefficient over the ten year period.) Average Absolute Monthly Range of Price, in Loga rithms (last The minimum of 1 and t (high price in month t) n low price in month t is calculated. This minimum value is averaged over the last 12 months. 12 months) Logarithm (Trading Volume/Variation in Price) in Previous Year t n(shares traded in pre vious year x price at the end of the previous year) (average absolute monthly range of price in loga rithms). Potential Dilution Applicable Federal Tax Rate earnings per share fully diluted earnings per share I (current federal income taxes + total state income taxes) + I (operating income after depre ciation and after federal income tax+ current federal income tax). (The sums are taken over the last 5 years.) Sigma Squared The usual estimate of the variance of the error term in the usual constant coefficient market model. This is calculated for firm n in year t from the regression over the previous 60 months. PAGE 160 ( 2 7) (28) ( 2 9) 8 x cr H Vasicek Beta 148 a is the historical beta estimate. H 82 82 [ _l + 1 l SH ~PRIOR 6 is the beta estimate obtained by regressing PRIOR the S+P 50 Utility Index on the S+P 500 Stock Index over the preceding 60 months. (31) a. The market model first with constant intercept was estimated for 74 firms in the utility industry for which the necessary data, including monthly returns data from the CRSP Tapes, was available. Of the 740 theoretically available observations in the pooled time-series cross-sectional regression 32 could not be constructed due to missing data. The method of classical least squares was employed. That is, if any descriptor could not be constructed for any given company in any given year, then all observations on all the variables for that company for that year were eliminated. Descriptor (9), flow was dropped from the list of descriptors because its calculation required total deferred income taxes, an item frequently missing from the compustat tapes. Indicator of negligible earnings descriptor (20) was always zero and was therefore dropped as well. PAGE 161 149 The final regression of the annualized rate of return for company n in year t had an intercept, 73 dummy variables (one firm was dropped because it always had some missing data) and 29 descriptors. Parameter estimates ( a assumed constant). The param eter estimates for the slope coefficients used in generating betas according to equation (3.3) are presented in Table 3.9. The coefficients of the firm dummy variables times the market ~ were all significant with a mean value of 21.8. i The signs of these slope coefficients do not make any apparent sense. This is due to the high degree of multi collinearity in the set { W~tMt}, j=l, ... ,31 as was verified by examining the correlation matrix of these descriptors times the market's rate of return. Apparently, multiplying the descriptors by Mt creates a high degree of multicol linearity. Parameter estimates ( a variable). The parameter estimates for the model with varying intercept are presented in Table 3.10. These are the slope coefficients for beta in 8 = b'W + The coefficients of the firm dummy nt nt n variables times the market were all significant with a mean value of 13.5. The slope coefficients used in estimating a according to a = a'W + s are presented in Table 3.11. nt nt n Beta estimates. Using the parameters obtained, betas were generated. Their mean values are presented in Table 3.12. In addition, mean values of the previously PAGE 162 150 Table 3.9 Estimated Descriptor Coefficients Used in Generating Betas for the Twenty-nine Descriptor Data Set (Alpha Constrained to Be Constant) Descriptor Estimates t-statistic Wl -2.39647 -3.64 w2 0.19002 1.18 w3 -3.24633 -2.01 w4 -0.75574 -0.46 ws 0.000132 -0.45 w6 -21.61145 -4.48 w7 2.50634 0.31 WB -0.42686 -0.75 wg WlO -3.76163 -2.54 Wll 1.59130 5.25 Wl2 3.01944 1.71 Wl3 0.86416 0.80 Wl4 l. 352 90 1.49 WlS 1.99495 1.14 Wl6 -1. 19352 -0.80 Wl7 0.57222 0.83 Wl8 -3.63630 -0.83 Wl9 24.12272 4.00 w20 w21 -2.04049 -5.62 w22 16.63080 7.59 w23 -0.47134 -4.49 w24 0.66283 0.12 w2s 1.23633 1.14 w26 125.01397 0.88 w21 105.53141 4.57 w2s -0.17025 -0.20 w2 9 9.79795 6.45 w30 -7.57501 -4.94 w31 -106.51715 -5.32 PAGE 163 151 Table 3.10 Estimated Descriptor Coefficients Used in Generating Betas for the Twenty-nine Descriptor Data Set (Alpha Allowed to Vary According to Equation 3.2) Descriptor Estimates t-statistic Wl -1. 983 687 -3.0734 w2 -0.00624001 -0.0399 w3 -2.223361 -1.4780 w4 -0.409485 -0.7384 ws 0.0001833523 0.6428 w6 -16.418656 -3.4246 w7 -1. 646820 -0.2266 wa -0.221570 -0.4150 wg WlO -2.424872 -1.7168 Wll 0.886471 2.9089 Wl2 2.489164 1.3246 Wl3 0.728905 0.7388 Wl4 0.634253 0.7255 WlS 0.835164 0.4790 Wl6 0.547144 0.3914 Wl7 0.484587 0.6841 Wl8 -2.574850 -0.4978 Wl9 12.231158 1.9649 w20 w21 -0.154022 -0.3634 w22 12.862553 6.2561 w23 -0.254991 -2.3215 w24 2.618955 0.5351 w2s 0.957698 0.9636 w26 127.624089 0.9990 w27 51.264587 2.3187 w2a 0.312029 0.3933 w29 2.271084 1. 417 4 w3o -3.848149 -2.6193 w31 -54.525473 -2.7543 PAGE 164 152 Table 3.11 Estimated Descriptor Coefficients Used in Generating Alphas for the Twenty-nine Descriptor Data Set Descriptor Estimates t-statistic Wl .565561 5.3398 w2 .060812 2.2456 w3 .097224 0.4072 w4 -.062789 -0.5748 ws .0000704 1.3947 w6 -0.137958 -0.1428 w7 -3.021723 -2.8668 ws -0~130949 -1. 5091 w9 WlO 0.510178 2.3123 Wll 0.202949 3.1131 Wl2 -0.385754 -1.1710 Wl3 -0.024418 -0.1480 Wl4 0.187891 1.2060 WlS -0.326205 -1.0478 Wl6 -0.421820 -1.8209 Wl7 0.153170 1.1160 Wl8 -0.031255 -0.0318 Wl 9 5.490475 5.6149 w2 o w21 0.032631 0.4498 w22 2.289162 6.6661 w23 -0.093327 -4.6893 w24 1.067035 1.6698 w2s -0.147569 -0.9496 w26 -38.708746 -.9094 w27 6.535861 1. 5867 w2s -0.065932 -0.4966 w29 0.119420 .6732 w3o -0.280607 -1.3053 w31 -2.189074 -0.5249 PAGE 165 153 Table 3.12 Yearly Means of Fixed Effects Five and Twenty-nine Descriptors and Historical Betas A A 6 S LSDV 5 A Year LSDV 29 6H 1968 .57519292 .92037382 .63747671 1969 .63845545 .88851395 .66338814 1970 .92697202 .87388349 .76988186 1971 .52080215 .88702955 .67769500 1972 .38021266 .89508831 .71691386 1973 .54871992 .90277765 .76230000 1974 .69610302 1.01801774 .67662677 1975 1. 08990447 1.11201488 .77998186 1976 .62913812 1.17152057 .80463286 1977 .29020169 1.12784092 .73927570 PAGE 166 154 Table 3.13 Variances of Fixed Effects Five and Twenty-nine Descriptors and Historical Betas A ~LSDVS A 13 SH Year LSDv 29 1967 .14557702 .18203462 .03139353 1969 .12846203 .17471760 .02838508 1970 .15076171 .17355634 .02886257 1971 .10341460 .16838828 .02592867 1972 .06596851 .16536892 .03544830 1973 .06596851 .16510506 .04024436 1974 .07083942 .17666109 .04789180 1975 .24680855 .17516814 .03858309 1976 .10347342 .17789295 .03502365 1977 .18681656 .27382150 .03211513 PAGE 167 155 calculated beta estimates based upon five descriptors and of the historical beta estimates are reported. The variances of the alternative betas are reported in Table 3.13. GLS Estimation To decide whether to proceed with the GLS estimation procedure, 8 2 and 8 2 were estimated using = l l t2 N n and = 1 \ [R Afw bA'W M ~nCnMt]2. NT l nt a nt nt t n,t These estimates were 8 2 = 191.257909 and 8 2 It is clear that the ratio is close to unity. = .007. This indicates that the GLS estimates will be virtually identical to the fixed effects estimates. Thus, it was not necessary to construct such estimates independently. Further, 8 2 was calculated: E: PAGE 168 156 e 2 = 1.057671. E Prior Likelihood Estimation To determine whether this adjustment process leads to substantially different estimates for the firm specific effects, one has to calculate 1 A = 1 + Assuming that 8 2 n obtain 8 2 n e2 E = 1 = 993088243 82 n and 2 e ; 8 2 is independent of the firm n we and = .0000366 For this specification with zero means for ; ands the n n prior likel i hood estimates are essentially equal to the fi x ed effects estimators. Thus, independent calculations of such estimators is uninformative. Prediction The procedure for the out of sample predictors is exactly the same as performed earlier. In testing the usefulness of the alternative betas generated, the same procedure was used as previously described. The results are in Table 3.14. The results of the within sample prediction are listed in Table 3.15. PAGE 169 157 Table 3.14 Out of Sample Prediction Results for the Twenty-nine Descriptor Data Set Prediction Year I (Rnt R )2 nt 1973 16.603964 19.052466 .693174 .506972 1974 3.571238 1.951280 1.303402 .887823 1975 35.701380 10.621703 14.263077 8.535469 1976 3.345887 9.401477 1.778851 1. 015350 Beta LSDV 5 LSDV 29 S a H a =0, S=l LSDV 5 LSDV 29 SH a=0, S=l LSDV 5 LSDV 29 SH a =0, S=l LSDV 5 LSDV 29 SH a =0, S=l a Historical betas obtained from the previous year's histor ical regression over the preceding 60 months were used to to predict returns in the next year according to A A A R = a + S M n,t n n t PAGE 170 158 Table 3.15 Within Sample Prediction Results for the Twenty-nine Descriptor Data Set Prediction Year I (Rnt R )2 nt 1973 .803880 .372810 .762600 .694125 1974 .277284 .284514 1.445686 .887823 1975 2.488151 .738619 10.641859 8.535469 1976 1.627614 .428248 1.706328 1.121995 Beta LSDV 5 LSDV 29 1\ a a =0, S=l LSDV 5 LSDV 29 SH a=0, S=l LSDV 5 LSDV 29 SH a=0, 8=1 LSDV 5 LSDV 29 SH a =0, B=l a Historical betas obtained from the given year's historical regression over the preceding 60 months including the prediction year were used to predict returns in the same year according to A A A R = a + 8 M nt n n t PAGE 171 159 4. Comparison of the Predictive Performance of the LSDV 29 and the LSDV 5 Betas For the out of sample prediction, the same results are obtained as earlier (see Table 3.14). Both the 29 descriptor betas and the five descriptor betas did less well than the naive predictor. The historical beta did not outperform the naive predictor as well. Different results are obtained for the within sample procedure (see Table 3.15). In all years, the 29 descriptor betas outperformed the naive predictor. In only two years, 1974 and 1975, the five descriptor betas outperformed the naive. In addition, in one of these years, 1974, it out performed the 29 descriptor betas. An explanation of these results awaits further explication. PAGE 172 SUMMARY AND CONCLUSIONS OF THEORETICAL AND EMPIRICAL RESULTS This study h~s attempted to clarify the nature of the arguments employed to support various traditionally and recently considered microeconomic facotrs as determinants of systematic risk. The recent arguments given for the labor-capital ratio and monopoly power have been generalized by examination of the sharing of risk between firms and labor in the market portfolio of all risky assets. The condition on the relative volatilities of wage rate uncer tainty and price uncertainty that determines the sign of the effects has been examined. On the empirical front, correct econometric techniques from the literature on pooling cross-sectional and time series data have been applied to the estimation of betas based upon a set of five descriptors for which theory provides some justification. While the multicollinearity introduced into the market model as a result of multiplying the descriptors by the market rate of return may obscure the relationships between these descriptors and beta, the labor-capital ratio, monopoly power, and the asset growth rate emerge with the signs suggested by theory. Financial leverage has a negative sign which may be the result of using book values for debt as opposed to the market values 160 PAGE 173 161 implied as relevant by theory. Volatility of earnings has a negative sign as well. The random effects and prior likelihood refinements of the basic fixed effects beta estimates are considered and are shown not to provide estimates that are numerically substantially different. The betas generated are compared to those arising from the classical market model and those arising from the Vasicek adjustment procedure. They are found to exhibit a higher degree of cross-sectional variability than these classical betas. This cross-sectional variability points to the sensitivity of these betas to changes in fundamental firm information as embodied in the set of descriptors. To test for the comparative usefulness of these betas two tests were performed. Coefficients to be used in generating alphas and betas in the successive year were generated from information from the preceding set of years. Conditional upon the market rate of return in the succeeding year predicted returns were constructed. The mean-square error of these predicted returns for the given year was computed and compared to that of returns generated using the previous year's classical alphas and betas and also to that of the naive predictor obtained by setting alpha equal to zero and beta equal to unity in the market model. This naive predictor outperformed the descriptor generated and the historical constant coefficient market model betas in their roles as predictors of returns. PAGE 174 162 A second test was performed using the coefficients generated over the full estimation period 1968-1977 to generate betas in the given prediction year 1973, 1974, 1975 or 1976. The mean-square error of the returns conditional upon the market rate of return as predictors of the given year's returns was calculated for each of the above predic tion years. In addition, to put historical betas on a similar within sample predictive position, classical betas generated for the given year by performing a regression over the 60 months preceding, including the given year, were used to predict returns in that year. Their mean-square error was computed and compared to that of the descriptor betas as predictors of returns. It was found for this within sample case that the descriptor based betas outperformed both the classical betas and the naive predictor in two years out of four--1974 and 1975. The classical betas never outperformed the naive predictor. The descriptor based betas for this within sample prediction test incorporate fundamental firm information and are thereby more sensitive to periods of sharp market movements than the classical betas or the naive predictor. This increased sensitivity allows them to out perform betas generated by procedures forcing coefficients in the market model to be constant. The entire estimation procedure and consequent predic tion tests were performed for betas generated from a set of twenty-nine descriptors chosen in the Barr-Rosenberg manner to determine whether these betas outperformed the previous PAGE 175 163 set as predictors of returns. There was no apparent explana tion for the signs of coefficients of the descriptors generated. The prediction results indicate that for out of sample prediction these betas did not succeed in outperform ing the naive predictor. In two years they outperformed the five descriptor betas and in two years they were outper formed by them. For the within sample prediction they were better than the naive in all years. They were better than the five descriptor betas in three years out of four--1973, 1975 and 1976. Only in 1974 did the five descriptor betas provide better predictors. PAGE 176 APPENDIX l RELATIVE MAGNITUDES OF THE FIRM AND LABOR BETAS Using the notation in our model, let D = L and i = w( l+v) J D represents units of labor employed, and 1 represents the random wage payment to labor. Further consider the cash flows of the firm and of the market: X J = X ( l+e) J and = By definition, = = 1 ~ \ ~[cov(X L X (l+e)) Ow cov(l+v,X (l+e))] j M J J M = The variance of the market portfolio is given by v ar(R M ) var[::] 1 2 2 = = X M cr e. s 2 M 164 PAGE 177 165 Thus, s 2 2 Dw a 2 ] s. M = XM [ X a X 2 a 2 S S J e ev J = = = Note that and t h at E (R.) J M e J M s M [ X Dw XM S j J CJ X Dw S M XM X J = = J S J CJ Dw CJ S J XM SM X wD J S J CJ 2 e v 2 e 02 ev 02 e 2 ev 2 e T he e x pression for S is thus J s J = E(R J ] 2 2 depending upon t h e sign of a /a ev e PAGE 178 166 Suppose that 8~~ = 1, then the numerator equals E(R ). ve J If 8~ ~ < 1, then the numerator is greater than E(R.). J ve 8~~ ve > 1, then the numerator is less than E(R.). the CAPM expression E (R ) J = we see that 8. must equal unity in order for J 8 J = E (R ) J J But, If using when 8~~ = 1. Thus when 8~~ = 1, then 8. = 1. Also, when ve ve J 8~~ < 1 then 8 > 1 and when 8~~ > 1, then 8 < 1. ve J ve J PAGE 179 APPENDIX 2 DERIVATION OF THE MONOPOLY BETA Note that from equation (2.79), 6 J = 2 2 p~q~o wL o J J e J ev (l+i)o 2 e From equation (2.74) we obtain or x 1 ( 1-u ) p* J J = = (l+i)c + x 1 u p~. J J J Equation (2.78) states that = P*q*X q~a.wx 2 j j 1 J J l+i q~ = ..:L 1 [(l+i)c + x 1 u p~1. +i J J J The denominator of 6 J = o 2 (l+i)V~ e J = = 2 a q~ 1 e J [c (l+i) + wa X 2 u ]. -u J J J J 167 PAGE 180 168 Th t f S = X 1 q* [p*.o 2 wa.o 2 ] e numera or o J J J e J ev = X 2 l q o J e [(l+i)c. + wa x 2 ] J J X 2 1 q wa a J J ev = q~ 1 1 [o 2 (l+i)c + wa.x 2 a 2 x 1 wa.o 2 (1-u.)] J -u. e J J e J ev J J = q~ 1 1 [o 2 (l+i)c + (1-u.)wa x 2 o 2 + u wa x 2 o 2 J -u e J J J e J J e = = J 2 (1-u.)wa.x 1 o ] J J ev q~[o 2 (l+i)c. + u wa X 2 o 2 ] + q~wa [X 2 o 2 J e J J J e J J e q~ J 0 2 [(1+1.')c. + X] + [ 2 1 -u wa 2 q.wa. a -u e J J J J J e J 2 a x 1 ] ev 2 a ] ev Dividing the numerator by the denominator and simplifying, we obtain equation (2.80). PAGE 181 REFERENCES Anderson, T. w. (1958), An Introduction to Multivariate Statistical Analysis. New York: J. Wiley & Sons. Atkinson, s., and Halvorsen, R. (1976), "Interfuel Substi tution in Steam Electric Power Generation," Journal of Political Economy, Vol. 84, No. 5. Ball, R., and Brown, P. (1969), "Portfolio Theory and Accounting," Journal of Accounting Research, Vol. 7. Barr-Rosenberg, B., and Marathe, V. (1975), "The Prediction of Investment Risk: Systematic and Residual Risk," Institute of Business and Economic Research, University of California, Berkeley, Repring, No. 25. Barr-Rosenberg, B., and Marathe, v. (1979), "Tests of Captial Asset Pricing Hypotheses," in Research in Finance, Vol. 1, edited by H. Levy. Greenwich, Connecticut: JAI Press, Inc. Barr-Rosenberg, B., and McKibben, W. (1973), "The Prediction of Systematic and Specific Risk in Common Stocks," Journal of Financial and Quantitative Analysis, Vol. 8. Beaver, w. H., Kettler, P., and Scholes, M. (1970), "The Association between Market Determined and Accounting Determined Risk Measures," Accounting Review, Vol. 45. Fama, E. (1970), "Multiperiod Consumption-Investment Decisions," American Economic Reviews, Vol. 60. Galais, D. and Masulis, R. (1976), "The Option Pricing Model and the Risk Factor of Stock," Journal of Financial Economics, Vol. 3. Gonedes, N. J. (1973), "Evidence on the Information Content of Accounting Nwnbers: Accounting-Based and Market Based Estimates of Systematic Risk," Journal of Financial and Quantitative Analysis, Vol. 8. Hamada, R. (1972), "The Effect of the Firm's Capital Struc ture on the Systematic Risk of Common Stocks," Journal of Finance, Vol. 27. 169 PAGE 182 170 Lee, L. F., and Griffiths, W. (1978), "The Prior Likelihood and Best Linear Unbiased Prediction in Stochastic Coefficient Linear Models," Discussion Paper, University of Minnesota. Leland, H. (1972), "Theory of the Firm Facing Uncertain Demand," American Economic Review, Vol. 42. Lev, B. (1974), "On the Association between Operating Leverage and Risk," Journal of Financial and Quantita tive Analysis, Vol. 9. Maddala, G. s. (1978), "On the Prediction of Systematic and Specific Risk in Common Stocks: A Discussion of Barr-Rosenberg's Model," Manuscript. Modigliani, F., and Miller, M. (1958), "The Cost of Capital, Corporation Finance, and the Theory of Investment," American Economic Review, Vol. 48. Modigliani, F., and Miller, M. (1961), "Dividend Policy, Growth, and the Valuation of Shares," Journal of Business, Vol. 34, No. 4. Modigliani, F., and Miller, M. (1966), "Some Estimates of the Cost of Capital to the Electric Utility Industry, 1954-57," American Economic Review, Vol. 56. Myers, s. (1977), "Determinants of Corporate Borrowing," Journal of Financial Economics, Vol. 5. Myers, s., and Turnbull, S. (1977), "Capital Budgeting and the CAPM: Good News and Bad News," Journal of Finance, Vo 1. 3 3, No. 2. Nerlove, M. (1971), "Further Evidence on the Estimation of Dynamic Relations from a Time-Series of cross-Section Data," Econometrica, Vol. 39, No. 2. Pettit, R., and Westerfield, R. (1972), "A Model of Capital Asset Risk," Journal of Financial and Quantitative Analysis, Vol. 7. Rubinstein, M. (1973), "A Mean-Variance Synthesis of Corporate Financial Theory," Journal of Finance, Vol. 28. Stapleton, R., and Subrahrnanyam (1978), "A Multiperiod Equilib rium Asset Pricing Model," Econometrica, Vol. 46, No. 5. Stapleton, R. and Subrahmanyam (1979), "Multiperiod Equilib rium: Some Implications for Capital Budgeting," in Portfolio Theory: Twenty-Five Years After, TIMS Studies in Management Sciences, Vol. 11, edited by E. Elton and M. Gruber. New York: Oxford University Press. PAGE 183 171 Subrahrnanyam, M., and Thomadakis, S. ( 198 0) "Systematic Risk and the Theory of the Finn," Quarte:cly Journal of Economics, Vol. XCIV, No. 3. Sullivan, T. (1977), "A Note on Market Power and Returns to Stockholders," The Review of Economics and Statistics, Vol. LIX. Sullivan, T. (1978), "The Cost of Capital and the Market Power of Firms," The Review of Economics and Statistics, Vol. LX. Theil, H., and Goldberger, A. (1960) "On Pure and Mixed Statistical Estimation in Economics," International Economic Review, No. 2. Thomadakis, S. (1977), "A Value Based Test of Profitability and Market Structure," The Review of Economics and Statistics, Vol. LIX. PAGE 184 BIOGRAPHICAL SKETCH David H. Gold~nberg was born on May 16, 1949,in Toronto, Canada. He attended the University of Toronto where he studied mathematics and philosophy. Continuing his studies in mathematics and applied mathematics, he received M.Sc. degrees in those disciplines from Brown University. He taught courses in mathematics, finance and econometrics at the University of Florida while completing his studies for the doctorate in finance. 172 PAGE 185 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a disserta tion for the degree of Doctor of Philosophy. G. S. Maddala, Chairman Graduate Research Professor of Economics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a disserta tion for the degree of Doctor of Philosophy. Fred D. Arditti Walter J. Matherly Professor of Finance and Economics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a disserta tion for the degree of Doctor of Philosophy. :,;., I J' .~ Arnold A. Professor of Finance I PAGE 186 This dissertation was submitted to the Graduate Faculty of the Department of Finance, Insurance, and Real Estate in the College of Business Administration and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the Degree of Doctor of Philosophy. June 1981 Dean, Graduate School PAGE 187 UNIVERSIT Y OF FLORIDA 111 1/ IIIIII Ill Ill lllll lllll II IIIII I III I II IIIIII II IIIIII II IIIII I I 3 1262 08553 5853 PAGE 188 UNIVERSIT Y OF FLORIDA 111 1/ IIIIII Ill Ill lllll lllll II IIIII I III I II IIIIII II IIIIII II IIIII I I 3 1262 08553 5853 |