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. BarrRosenberg and Associates' Work. 28
BarrRosenberg and McKibben......... 28
BarrRosenberg and Marathe.......... 32
BarrRosenberg 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 LaborCapital
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 Twentynine
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 (194756) Versus
Period Two (195765) ......................... 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 Twentynine Descrip
tor Data Set (Alpha Constrained to Be Con
stant) ....................................... 150
vii
3.9 Estimated Descriptor Coefficients Used in
Generating Betas for the Twentynine
Descriptor Data Set (Alpha Constrained to
Be Constant) .......................... ...... 150
3.10 Estimated Descriptor Coefficients Used in
Generating Betas for the Twentynine 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 Twentynine
Descriptor Data Set.......................... 152
3.12 Yearly Means of Fixed Effects Five and
Twentynine 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
Twentynine Descriptor Data Set.............. 157
3.15 Within Sample Prediction Results for the
Twentynine 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
BarrRosenberg 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 nonstationarity 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 leastsquares 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
laborcapital 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 meansquare 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 meansquare 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 four1974 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 BarrRosenberg 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
four1973, 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 nondiversifiable 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
microeconomic variables.
Traditionally, financial leverage and volatility of
earnings have been considered as microeconomic 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 laborcapital 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 crosssectional 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 BarrRosenberg 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 leastsquares 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 meansquare error
in the prediction of returns generated by those estimates.
The meansquareerror 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 BarrRosenberg 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) BarrRosenberg and Associates' Work:
BarrRosenberg and McKibben [1973], BarrRosenberg
and Marathe [1975], and BarrRosenberg 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 rankorder
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 Ni 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 19461966. 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 RankOrder 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 2025%
of the crosssectional variability in estimated
degrees of association with the market (from
Table 1) and 3540% 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 RankOrder 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 nonsystematic
component of risk as well. High beta firms may have small
individualistic or firmspecific 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 nonsystematic 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 noncurrent 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 timeseries 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 (194756) Versus Period Two (195765)
Number of Securities
in Portfolio
Rank Correlation
.625
.876
.989
ProductMoment
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
ti
E 2
t E
t1
T
available to commont
alue of common stock
t1
T Et
E t=l t1
and 
T
Cov ,Mt
t1
(7) Accounting Beta =t
var(Mt)
where Mt
t
T E.
P
t=1 i,t1
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 timeseries 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 firstorder 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 economywide income number for period t
I.
Y = the industry income number for the industry to
which the ith firm belongs for period t
A the total assets number of the ith firm at the
beginning of period t
A = the economywide total asset number at the
beginning of period t
I
A = the total asset number for the industry
grouping of the ith 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 twentyone year period provided better estimates
than the seven year estimates. The prediction test results
suggest that the accountingnumber 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
194668
194652
195359
196168
Stock Beta Measured Over
M2 M3 M4
194652 195359 196168
.18
(a) Ninetynine Crosssectional Observations.
b/ Significant at a = .05.
c/ Significant at a = .01.
(Table 5, p. 434)
Period
A1
A2
A3
A4
"cashflow 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. 16611662]
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, 194757, and
Period II, 195768. 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. BarrRosenberg and Associates' Work
BarrRosenberg and McKibben
BarrRosenberg 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 leastsquares 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
timeseries and crosssectional data. However,
BarrRosenberg 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 meansquare 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 meansquareerror sense than the unit beta
predictor.
BarrRosenberg and Marathe
In later work, BarrRosenberg 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.
BarrRosenberg 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 leastsquares 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 crosssectional 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 crosssectional
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.
BarrRosenberg 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 (LongTerm Debt
Market + Preferred Stock) + Market Value
(Common Stock) Market Value
(Common Stock)
(2) Debt/Total = LongTerm 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, BarrRosenberg 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 meansquare 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 meansquareerror 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 meansquare 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)
BarrRosenberg'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, BarrRosenberg tests for stability of
the prediction rule. He finds it to be quite stable over
timethe 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.
BarrRosenberg 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 leastsquares 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 MM theory. The relation he derives is
S
6u = L (1.27)
u
42
The MM theory states that V = V + TD so that
L u L
S
L
Bu VTD L
L L
S
L
S +D (1T) L
L L
This yields
SL+DL (1T)
L S u
L
DL(IT)]
 ^^ /u
(1.29)
To test this relationship, Hamada calculates the
following rate of return
R
ut
t
Xt(1T)+AGt
S
Ut1
(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
t1
is unobservable as firms generally employ debt financing,
the MM theory is used to evaluate the denominator
Ut1
= (V D)t
U L t1
(1.31)
The numerator is evaluated using the following identi
ties:
Xt(lT) + AGt
= [(XI)t(lT) PDt + AGt]
+ PDt + It(1T),
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:
(XI)t(1T) 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 (1T)
R = (1.34)
t (VTD)
t1
Next, the observed rate of return to common stockholders is
(XI) t (1T)PDt+G d +cgt
RL S S (1.35)
t L L
tl t1
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 MM theory is
correct, then leverage explains 2124% 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
* xiRX
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 longrun 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 crosssectional varia
tion. Some of this crosssectional 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 crosssectional beta varia
tion is to confine attention to a given risk class in the
MM 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 crosssectional variation in stock betas
within a given industry (riskclass) is attributable to the
differing debt/equity ratios employed by the firms, then
unlevering the betas should result in a set of unlevered
47
(N in o
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 rI 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 crosssectional 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 MM 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 196170
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
bidup 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 valuebased 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
VA
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 farfetched 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. = fourfirm concentration ratio
(3) DN. = industry dummy variables
(4) SG. = the annual compound growth rate in sales
from 19631972 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 laborcapital 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 )(1T) + Ag
R. t jt jt jt (1.45)
SSt
J ,t1
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 tl
3,ti
T = the corporate tax rate.
Thus we obtain
Cov((PQ) j (1T),RMt
S. 8, = jt Mt (1.46)
Var(R t)
Cov(V (1T),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,t11 > 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
crosssectional equality of sales patterns across states of
nature, Lev runs a time series regression of total operating
costs of the jth 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 (19491968) and a twelve year
period (19571968) 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 righthand 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 exante 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 (pmv )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 economywide 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.
fpQvF
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*) = Ipvp(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 jpvl 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 economywide 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 corelationship
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 nontradeable 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 nonsynchronous 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
biasin 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 laborcapital 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 nonstochastic. The rates of return on the
levered firm's and unlevered firm's equity are:
XiDL
R S and R 
L S u V
L u
respectively. Evaluating the corresponding betas yields
cov(R ,Rm) cov(XiDL/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
GaussWiener 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 endofperiod 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 perfectlycorrelated
withthemarket 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'sa 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 ModiglianiMiller 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 singleindex model. Their statement of this
are equations MT2 and MT3 (p. 322),
Xt = E(Xl t_1) (1+t) MT2
where t is a zeromean 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' MT3
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 nonstochastic. 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
nonquadratic 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 riskfree 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)
variancecovariance 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
nonstate dependent exponential function of W1 alone. But
we have assumed r2 to be nonstochastic, 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 nonstochastic character of the market price of
risk X2. Further, 02 is nonstochastic, i.e. independent of
Xl, as a consequence of the assumed joint normality of the
{Xk} [Anderson, 1958]. Thus, u(W1X1) = u(W1) is a non
statedependent utility function as is required for the
periodbyperiod application of the CAPM [Fama, 1970].
Essentially, since r2 and X2 are nonstochastic 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 variancecovariance 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 riskfree 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 riskfree 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"
