Group Title: study of the systematic component of risk in common stocks
Title: A study of the systematic component of risk in common stocks
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Title: A study of the systematic component of risk in common stocks
Physical Description: Book
Language: English
Creator: Goldenberg, David Harold, 1949-
Copyright Date: 1981
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Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
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David Harold Goldenberg





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.



ACKNOWLEDGEMENTS........................................... ii

LIST OF TABLES...... ..... .. ............................ vi

ABSTRACT ............................................... x

INTRODUCTION............... .....................* ....... 1


Introduction...................... ...... 5
1. Ad Hoc Studies Using Accounting
Numbers ............................. 6
Ball and Brown..................... 6
Beaver, Kettler and Scholes......... 11
Gonedes........... .. ........ .... 22
2. Theoretical Basis Study.............. 25
3. Barr-Rosenberg and Associates' Work. 28
Barr-Rosenberg and McKibben......... 28
Barr-Rosenberg and Marathe.......... 32
Barr-Rosenberg and Marathe.......... 40
4. Study of the Effect of Financial
Leverage............... ..... ........ 41
5. Studies of the Effect of Market
Power................................ 48
Sullivan........................... 48
Thomadakis......................... 51
Sullivan............................ 54
6. Studies of the Effect of Operating
Leverage............................. 56
Lev................................. 56
Rubinstein......................... 62
Subrahmanyam and Thomadakis......... 68



BETAS.............. ..................... 70

Introduction............................ 70
1. Financial Leverage.................. 71
2. Volatility of Operating Earnings.... 74
3. Growth............................... 78
Assumptions....................... 82
Single Period Betas in a Multi-
period setting................. 92
Lemma........................ 96
Proof ......................... 96
Duration and Asset Betas............ 100
4. Monopoly Power and the Labor-Capital
Ratio ............................... 101
Sources of Uncertainty............. 102
Uncertainty in the price of
output.. ................. 102
Uncertainty in the wage rate.. 103
Relationship between Demand Uncer-
tainty and Uncertainty in the
Wage Rate..................... 104
Competitive Equilibrium Risk Deter-
mination ...................... 106
Derivation of the Systematic Risk
of the Purely Competitive Firm 108
Systematic Risk and Monopoly Power. 112
Optimal Valuation and Beta for the
Monopolistic Firm............. 115
5. The Model of Systematic Risk and
Hypotheses to be Tested.............. 116

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


Definitions and data sources
for descriptors.......... 125
Parameter Estimates........... 127
Beta Estimates................ 129
GLS Estimation..................... 133
Prior Likelihood Estimation........ 134
Prediction......................... 135
Out of sample prediction...... 135
Within sample prediction...... 138
3. Empirical Results with Twenty-nine
Descriptor Data Set................. 141
Fixed Effects Estimation............ 141
Names of descriptors.......... 141
Classification of descriptors. 143
Descriptor definitions........ 144
Parameter estimates (a assumed
constant) ................ 149
Parameter estimates (a
variable) ............... 149
Beta estimates................ 149
GLS Estimation..................... 155
Prior Likelihood Estimation......... 156
Prediction......................... 156
4. Comparison of the Predictive Perfor-
mance of the LSDV29 and the LSDV
Betas ............................... 159

RESULTS................................................ 160

LABOR BETAS ............................. 164


REFERENCES...... ........ ............................... 168

BIOGRAPHICAL SKETCH.................................... 171


Table Page

1.1 Coefficients of Correlation between Various
Measures of the Proportion of Variability in
a Firm's Income That Is Due to Market
Effects Variables Not Standardized........... 8

1.2 Coefficients of Correlation between Various
Measures of the Covariance between a Firm's
Index and a Market Index of Income Variables
Standardized................................ .10

1.3 Association between Market Determined Risk
Measure in Period One (1947-56) Versus
Period Two (1957-65) ......................... 15

1.4 Contemporaneous Association between Market
Determined Measure of Risk and Seven Account-
ing Risk Measures............................ 20

1.5 Analysis of Forecast Errors.................. 23

1.6 Correlation Coefficients, R, between
Estimates from Market Model (M) and Four
Accounting Number Models (A)................. 26

1.7 Correlation Analysis......................... 29

1.8 Prediction Rules for Systematic Risk Based on
Fundamental Descriptors and Industry Groups.. 35

1.9 Prediction Rules for Systematic Risk Based on
Fundamental Descriptors Including Market
Variability Descriptors...................... 37

1.10 Unbiased Estimates of the Performance of
Alternative Prediction Rules in the Histor-
ical Period (Predicted Variance as a Multiple
of the Variance Predicted by a Widely Util-
ized Prediction Rule) ........................ 39

1.11 Summary Results over 304 Firms for Levered
and Unlevered Alphas and Betas............... 45

Table Page

1.12 Market Adjustment Factor Regressions over
Alternative Periods.......................... 47

1.13 Mean and Standard Deviation of Industry Betas 49

1.14 Industry Concentration and Future Monopoly
Power......................................... 53

1.15 Monthly Betas, Leveraged and Unleveraged,
Regressed on Market Power and Control
Variables..................................... 55

1.16 Estimates of Average Variable Cost Per Unit.. 61

1.17 Regression Estimates for Systematic Risk on
Average Variable Costs Per Unit.............. 63

3.1 Estimated Descriptor Coefficients Used in
Generating Betas for the Five Descriptor Data
Set.......................................... 128

3.2 Yearly Means of Fixed Effects, Historical,
and Varicek Betas for the Five Descriptor
Data Set..................................... 130

3.3 Variances of Fixed Effects, Historical, and
Vasicek Betas Based on the Five Descriptor
Data Set..................................... 131

3.4 Estimated Descriptor Coefficients for
Constant Alpha Model for the Five Descriptor
Data Set............ ......................... 132

3.5 Estimated Descriptor Coefficients Used in
Generating Alphas and Betas for Out of
Sample Prediction............................ 136

3.6 Correlation Matrix of Descriptors and
Descriptors Times the Market for the Five
Descriptor Data Set.......................... 137

3.7 Out of Sample Prediction Results for the Five
Descriptor Data Set.......................... 139

3.8 Within Sample Prediction Results for the Five
Descriptor Data Set......................... 140

3.9 Estimated Descriptor Coefficients Used in
Generating Betas for the Twenty-nine Descrip-
tor Data Set (Alpha Constrained to Be Con-
stant) ....................................... 150


3.9 Estimated Descriptor Coefficients Used in
Generating Betas for the Twenty-nine
Descriptor Data Set (Alpha Constrained to
Be Constant) .......................... ...... 150

3.10 Estimated Descriptor Coefficients Used in
Generating Betas for the Twenty-nine Descrip-
tor Data Set (Alpha Allowed to Vary According
to Equation 3.2).............................. 151

3.11 Estimator Descriptor Coefficients Used in
Generating Alphas for the Twenty-nine
Descriptor Data Set.......................... 152

3.12 Yearly Means of Fixed Effects Five and
Twenty-nine Descriptors and Historical Betas. 153

3.13 Variances of Fixed Effects Five and Twenty-
nine Descriptors and Historical Betas........ 154

3.14 Out of Sample Prediction Results for the
Twenty-nine Descriptor Data Set.............. 157

3.15 Within Sample Prediction Results for the
Twenty-nine Descriptor Data Set.............. 158




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



David Harold Goldenberg

June 1981

Chairman: Dr. G. S. Maddala
Cochairman: Dr. F. D. Arditti
Major Department: Finance, Insurance, and Real Estate

The theoretical basis for the inclusion of various

microeconomic factors as determinants of systematic risk is

examined. This set includes financial leverage, variability

of operating earnings, and growth in several senses. A

recent model incorporating the firm's behavior in its input

and output markets is generalized by including risky human

capital in the market portfolio. The condition on the

covariability between the sources of uncertainty in the

model under which beta will be positively related to the

labor capital ratio and negatively related to monopoly power

is explicated.

Much work has been done on this problem by

Barr-Rosenberg and associates, but little or no theoretical

justification was offered for the choice of variables

included in their regressions. The study also corrects some

of the deficiencies in the econometric methods employed in

those studies.

The empirical results on non-stationarity of the coef-

ficients of the market model are taken into account by

allowing them to vary by firm and time period as a linear

function of the set of descriptors plus a firm specific

error term that does not vary with time. The fixed effects

or least-squares with dummy variables estimation technique

is applied. The coefficients of the growth rate in assets

and the Lerner Index of monopoly power are negative. The

labor-capital ratio has a positive sign. These results

conform to the theory. Financial leverage and variability

of earnings have negative signs. The former result may be

due to the use of book values for debt rather than the

appropriate market values.

The GLS or random effects estimates do not differ

significantly from the fixed effects estimates. The prior

likelihood estimates designed to provide best linear unbiased

estimates of the firm specific effects also do not differ

significantly from the fixed effects estimates.

The criterion of mean-square error was employed in

comparing classical betas to the descriptor betas as

predictors of returns. Predicted returns, conditional upon

the market return in the prediction year, were generated

from information from the previous set of years for both

types of betas. The mean-square error of the naive predic-

tor obtained by setting alpha equal to zero and beta equal

to unity was also computed. The order of performance was

naive, classical, then descriptors. When a within sample

prediction was performed by using all the available data

to estimate the coefficients used in generating betas,

the classical betas obtained by including the prediction

year's data in the usual regression never outperformed the

naive predictor while the five descriptor based betas did

so in two years out of four--1974 and 1975. This suggests

that the descriptor based betas may be useful as predictors

over periods in which structural changes in the coefficients

of the market model take place.

The entire set of procedures was repeated on a set of

descriptors chosen in the Barr-Rosenberg manner. The signs

of the estimated coefficients made no apparent sense. As

predictors of returns for the out of sample prediction,

they did not outperform the naive. They were outperformed

by the five descriptor betas in two years and outperformed

them in the other two years. For the within sample predic-

tion they were always better than the naive, and they were

better than the five descriptor set in three years out of

four--1973, 1975 and 1976. Only in 1974 did the five

descriptor set provide better predictors.


The total risk of a firm's equity securities, identi-

fied as the variance of the rate of return of those securi-

ties over a given period, can be decomposed into two compo-

nents. One component is termed systematic: it is the

sensitivity of the rate of return on the firm's common stock

to the rate of return on the market portfolio. This compo-

nent is the market related risk that cannot be diversified

away through the process of portfolio formation. Tradition-

ally a measure of the non-diversifiable risk component is

given by the slope coefficient, assumed constant over the

estimating period, in the market model's historical regres-

sion of the stock's rate of return on the rate of return of

a proxy for the market portfolio of all risky securities.

This simple model of the stochastic process generating

security returns indicates the second component of the risk

of a firm's equity securities. The market model indicates

that a portion of the variance of the rate of return cannot

be accounted for by movements in the rate of return of the

market portfolio. This component of the total risk of the

security is specific to the firm in question. However, it

is possible to diversify away this risk component by com-

bining the security with others in portfolios. In fact, the

Capital Asset Pricing Model tells us that equilibrium prices

and rates of return on a firm's securities are linear

functions solely of the first component of total risk: the

systematic component measured by the beta coefficient. It

is clearly of great interest to estimate and predict the

systematic risk component of a firm's common stock securi-

ties. This study will concern itself only with the estima-

tion and consequent prediction of this systematic component.

While the beta coefficient as defined by the CAPM

identifies the systematic component of risk as that component

relevant for equilibrium pricing of a firm's risky securi-

ties, it provides little information concerning the sources

of such systematic risk. The goal is to relate the firm's

unobservable beta to the firm's characteristics and to

micro-economic variables.

Traditionally, financial leverage and volatility of

earnings have been considered as micro-economic determinants

of systematic risk. Attention has recently focused upon

"growth" and the firm's operations in its input and output

markets as factors bearing upon the beta coefficient. The

present study includes the traditional variables as well as

the labor-capital ratio and monopoly power in an attempt to

incorporate these factors. The arguments for all of the

above variables will be considered in detail as the basis

for the inclusion of a set of factors that can be theoret-

ically justified as determinants of systematic risk. In the

course of this analysis the recent arguments for the labor-

capital ratio and monopoly power are generalized.


The second major purpose of the study is to correctly

apply the available econometric techniques that appear in

the literature on pooling cross-sectional and time series

data to the problem of estimating systematic risk as a

linear function of the set of descriptors developed above

plus a firm specific effect. This model has appeared in the

early literature on accounting numbers in the form of Beaver,

Kettler, and Scholes's instrumental variables estimation

procedure for estimating systematic risk. That study did

not incorporate firm specific effects. Such effects were

added to the model as the residual component of beta that

could not be explained by the included set of descriptors.

This was done by Barr-Rosenberg and associates. The appro-

priate estimation techniques were not applied. Accordingly,

the fixed effects and the random effects specification of

this model are considered. The model is estimated under the

fixed effects specification. Then, the potential gain from

applying the generalized least-squares estimation as

warranted by the random effects specification is measured in

an a priori manner. In principle, with known variances of

the firm specific effects, the prior likelihood estimation

procedure leads to best linear unbiased predictors of the

firm specific effects. The additional explanatory power

provided by these estimates is computed by considering the

magnitudes of the adjustment factors to be used to derive

the prior likelihood estimates.

The criterion for usefulness of the betas generated by

the procedures outlined above is the mean-square error

in the prediction of returns generated by those estimates.

The mean-square-error of the fixed effects betas is compared

to that of the usual historical beta and that of the naive

predictor with alpha set equal to zero and beta set equal to

unity. This is done both for out of sample and within

sample prediction.

Finally, the procedures described are carried out using

a set of 29 Barr-Rosenberg type descriptors. The results

are compared to those for the five theoretically justified




1. Introduction

Those studies that bear directly on the present one are

considered here. The review is divided into several


(1) Ad Hoc Studies Using Accounting Numbers: Ball and
Brown [19691; Beaver, Kettler and Scholes [1970];
and Gonedes [1973].

(2) Study Attempting to Provide a Theoretical Basis
for the Use of Accounting Numbers as Proxies for
Systematic Risk: Pettit and Westerfield [1972].

(3) Barr-Rosenberg and Associates' Work:
Barr-Rosenberg and McKibben [1973], Barr-Rosenberg
and Marathe [1975], and Barr-Rosenberg and
Marathe [1979].

(4) Study of the Effect of Financial Leverage on
Beta: Hamada [1972].

(5) Studies of the Effect of Market Power on Beta:
Sullivan [1977], Thomadakis [1977], Sullivan [1978],
and Subrahmanyam and Thomadakis [1980].

(6) Studies of the effect of Operating Leverage on
Beta: Lev [1974], Rubinstein [1973], and
Subrahmanyam and Thomadakis [1980].

The studies concerning the growth variable are consid-

ered in Chapter II, where the relationship is explained in

the appropriate multiperiod framework.


1. Ad Hoc Studies Using Accounting Numbers

Ball and Brown

The Ball and Brown [1969] study has as its goal the use

of "income numbers" as predictors of systematic risk. Their

"income numbers" are simple regression coefficients in

regressions of accounting income of a given firm in a given

year on a market index of accounting income. The income

variable was alternatively taken to be operating income, net

income, and earnings per share. Initially, two regression

models were postulated relating income in levels between the

firm and the market and then income in first differences

between the firm and the market. Three regressions were run

for each of the three income definitions. The simple R2

were computed for these time series regressions for each

firm i and then these were correlated with the R2 from the

regression of the firm's rate of return on the rate of

return on the market. Product correlation and rank-order

correlation coefficients were calculated.

The regression models employed were as follows:

it = a + a2iM Uit income, in levels (1.1)

AI = a' + a'.M + U' income, in first (1.2)

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
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.

Ball and Brown take this market index as an average

that, for a given firm i, excludes that firm; being an

average over the remaining N-i firms. Why they do this is

not obvious as a market index, according to Capital Asset

Pricing Theory, contains all firms in the market.

The object of the correlation analysis is to determine

to what extent estimates of systematic risk from the market

model are correlated with the accounting income response

coefficients. The regressions were run for 261 firms over

the period 1946-1966. Selected results appear in Table 1.1.

Ball and Brown conclude from these results that "better"

results are obtained when the regressions are run in first

differences [see Table 1.1, columns (3), (5) and (7)].

The next set of regressions attempted to control for

the size of firm and differences in their accounting prac-

tices. To do so, all variables were standardized by dividing

Table 1.1

Coefficients of Correlation between Various Measures
of the Proportion of Variability in a Firm's Income
That Is Due to Market Effects Variables
Not Standardized

(2) (3) (4) (5) (6) (7)

Product Moment Correlation (1) .00 .47 .03 .39 -.05 .42

Spearman's Rank-Order Cor. (1) -.02 .46 .02 .39 -.05 .41

(1) Stock return regression (1.5)
(2) Operating income in levels
(3) Operating income in first differences
(4) Net income in levels
(5) Net income in first differences
(6) E. P. S. in levels
(7) E. P. S. in first differences

Source: Ball and Brown [1969, page 319]

through by the market value of equity. Using the above

income definition Iit, the results appear in Table 1.2.

Ball and Brown draw the following conclusions from

these results:

(1) Comovement's in accounting earnings of firms
predict moderately well the firm's systematic
risk. From Table 1 the highest product moment
correlation coefficient is .47. From Table 2 it
is .59. Ball and Brown conclude that comovements
in accounting incomes explain approximately 20-25%
of the cross-sectional variability in estimated
degrees of association with the market (from
Table 1) and 35-40% in systematic risk (from
Table 2). [p. 319]

It is not clear what the difference is between en
estimates of comovement of the firm's rate of
return with the rate of return on the market and
estimates of systematic risk. The extent of the
explanatory power is "explained" by the fact that
all variables are measured with error. They
conclude that accounting numbers may
be even better predictors of systematic risk than
these results indicate. No evidence is given to
support this claim.

(2) Better predictions are obtained when the variables
are measured in first differences and the results
are sensitive to the income definition utilized.

The study, being an early attempt to relate accounting

variables to systematic risk, suffers from the lack of any

theoretical argument showing a relationship between account-

ing betas and systematic risk as defined in Capital Market

Theory. As such it is not clear whether to expect a consis-

tent correlation between such accounting betas and system-

atic risk.

Table 1.2

Coefficients of Correlation between Various Measures
of the Covariance between a Firm's Index
and a Market Index of Income
Variables Standardized

Product Moment Correlation (1)

Spearman's Rank-Order Cor. (1)

(2) (3)

.45 .59

.45 .64

(4) (5) (6) (7)

.39 .53 .41 .53

.42 .58 .43 .59

(1) Stock return regression
(2) Operating income in levels
(3) Operating income in first differences
(4) Net income in levels
(5) Net income in first differences
(6) Available for common in levels
(7) Available for common in first differences

(Table 2, p. 320)

Beaver, Kettler and Scholes

Beaver, Kettler and Scholes [1970] attempt to give a

rationale for the use of accounting based risk measures as

proxies for the systematic risk of the firm's equity securi-

ties. They note that such measures highlight several

aspects of the uncertainty associated with the earnings (or

return) stream of the firm. Further, the accounting risk

measures are surrogates for total risk. What is the link

between total and systematic risk?

If the systematic and individualistic components of
risk are positively correlated (at the extreme, per-
fectly correlated), then it is reasonable to view the
accounting measures as surrogates for systematic risk
as well. The evidence does suggest that positive
correlation does exist (e.g. securities with a larger
than average B tend to have a larger than average
variance of the individualistic component. [p. 659,
last paragraph]

The statement is dubious. There is no a priori reason

to believe that factors positively related to systematic

risk ought to be positively related to the non-systematic

component of risk as well. High beta firms may have small

individualistic or firm-specific risk components. They may

be affected primarily by market events with little left over

in the way of residual risk. The empirical fact referred to

may be based upon faulty beta estimation procedures. In any

case, if systematic risk and total risk are highly correlated

as would be the case if the non-systematic component were

highly correlated with the systematic component, Capital

Market Theory loses much of its relevance in specifying the

systematic component of risk as that component which is

relevant for equilibrium pricing.

Given this argument, the Beaver, Kettler and Scholes

results can be viewed as an attempt to specify the compo-

nents of total risk.

The basic concern of the study is to answer the ques-

tion: To what extent is a strategy of selecting portfolios

according to the traditional accounting risk measures

equivalent to a strategy that uses the market determined

risk measures?

The list of factors presented in their study is:

(1) dividend payout
(2) growth
(3) financial leverage
(4) liquidity
(5) asset size
(6) variability of earnings
(7) covariability of earnings with the earnings of the

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

(3) The usual argument is given for financial leverage.

(4) Beaver, Kettler and Scholes do not suspect that
liquidity in the sense of the fraction of assets
that are current assets held to be related to
beta. Rather, they suspect that the differences
in systematic risk among firms come about as a
result of the differences in the riskiness of
their non-current assets. However, they use the
current ratio as a measure of liquidity.

(5) Asset size is included on the grounds that:
(a) asset size is highly correlated with the
risk of default on bonds outstanding,
(b) If individual asset returns are not perfectly
correlated then larger firms will have a
lower total risk of equity securities than
smaller firms. This results because larger
firms, by diversifying their asset holdings,
reduce their total risk. The connection with
systematic risk remains to be demonstrated.

(6) Variability of earnings is given an intuitive
argument to establish its effect on total risk.

(7) Covariability of earnings is introduced as the
slope coefficient of the regression of the earnings
price ratio of the individual firm on an economy
wide average of earnings price ratios as the
market variable. No theoretical argument is
given for this variable.

In the Beaver, Kettler and Scholes regressions, the

dependent variable is an estimate of Bi obtained by running

a time-series regression of the security's rate of return on

a proxy for the rate of return on the market. This was done

for two periods: January 1947 January 1957 and

December 1956 December 1965. Monthly rates of return were

used. The implicit assumption is that beta was stationary

over each of these subperiods. Beaver, Kettler and Scholes

test their hypothesis by computing correlation coefficients

between corresponding beta estimates from the two subperiods

at the individual and at the portfolio level. They find a

correlation coefficient of .594 at the individual level and

.965 for portfolios of size 20. They conclude that the data

indicate that stationarity is not violated. One has to

ask, however, whether the process of aggregation into

portfolios automatically results in higher correlation

coefficients (after adjusting as Beaver, Kettler and Scholes

have done for the loss of degrees of freedom). Results of

this stationarity test are given in Table 1.3.

We present here the definitions employed by Beaver,

Kettler and Scholes of their accounting measures.

Cash dividends
(1) Average Payout = t
Income available to common

Total assets
Total assets0
(2) Average Asset Growth = Total

T Total senior securities
(t= Total assets
(3) Average Leverage =

Table 1.3

Association between Market Determined Risk Measure
in Period One (1947-56) Versus Period Two (1957-65)

Number of Securities
in Portfolio

Rank Correlation








Table 2, p. 665

(4) Average Asset Size =

(5) Average Liquidity =

STotal assets

T Current assets.
SCurrent liabilities
t=l t

(6) Earnings Variability =

E Income
where Ma t
P Market v

E 2
t E

available to commont
alue of common stock

T Et

E t=l t-1
and -

Cov ,Mt
(7) Accounting Beta =t-

where Mt

T E.
t=1 i,t-1

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 ,
where rates of return are defined using accounting earnings,

and the regression is a time-series one with T = 9 or 10

(depending upon the subperiod). The market index used is

equally weighted. Beaver, Kettler and Scholes call this

estimate B.. The idea here is that everything is defined

here in terms of accounting earnings rather than in terms of

prices and dividends. They compare these estimates of the

true beta to those from their historical regressions of

ordinary rates of return on ordinary rates of return on the

market. They find that the dispersion of their accounting

betas is almost four times as large as that of the usual

historical betas. This loss of efficiency is attributed

to relative sample sizes; nine for accounting betas, 120

and 108 for historical betas. Further, the accounting based

market measure exhibits first-order serial correlation

unlike the market determined market index (used to compute

historical betas). Also, 9% and 12% of the accounting B.'s

were negative in each subperiod. Beaver, Kettler and Scholes

suggest increasing the time period to 19 years for computa-

tion of accounting betas. When they do this, they find that

the standard deviation drops from 1.164 (1.280 in sub-

period 2) to .791 over the entire 19 year period. This is

still substantially larger than .336 (.342) the standard

deviation of historical betas in subperiod 1 and subperiod 2.

Beaver, Kettler and Scholes further find evidence of non-

stationarity in accounting betas. While the other accounting

variables are relatively stable as measured by rank correla-

tion and product moment correlation coefficients, these

correlation coefficients come out to be .034 (-.060) for the

accounting beta. Recall that their data yielded correlation

coefficients of .625 (.594) for the historical betas between

the two subperiods. Beaver, Kettler and Scholes conclude

that the accounting beta is subject to large errors in

measurement and they virtually suggest searching for other

accounting measures of risk. One wonders whether the

methodology of correlating two variables both measured with

a great deal of error, the historical beta and the accounting

beta, is fruitful. Any discovered correlation could be the

result of correlation between the errors in the measured

variables. On the other hand, while the two variables may

in fact be highly correlated, the random error terms in

their measurements could obscure such correlation. This

type of consideration is at a level secondary to that of

questioning the adequacy of the historical betas, a variable

measured with error and assumed to be stationary, as an

estimate of the true beta. On the other hand, it is hard to

see what else one could do at the level of correlation


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







Individual Portfolio
Level Level








Table 5, p. 669



























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 [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,

it AY t t
S4i i E + i + 4i,t' (1.8)
t A 1

Table 1.5

Analysis of Forecast Errors


Individual Portfolios
Securities A(a) B(b)


.030 .027


Individual Portfolios
Securities A(a) B(b)


.016 .016

(a) Portfolios ranked according to the historical estimate.
(b) Portfolios ranked according to the instrumental variable
beta estimate.

(Table 7, p. 677)


where A represents first differencing,

Y = the income number of firm i in period t

Y = the economy-wide income number for period t

Y = the industry income number for the industry to
which the i-th firm belongs for period t

A the total assets number of the i-th firm at the
beginning of period t

A = the economy-wide total asset number at the
beginning of period t

A = the total asset number for the industry
grouping of the i-th firm at the beginning of
period t.

The sample consisted of 99 firms randomly drawn from the

population for which all the necessary data were available.

To obtain beta estimates the market model was run using

logarithmic rates of return. Monthly observations were

employed. The parameters of the market model were estimated

for three year, five year, seven year, ten year, and twenty-

one year intervals. The first six monthly observations from

1960 and the first six from 1968 were reserved for prediction

tests. Gonedes's results provide evidence of nonstationarity

in beta estimates obtained from the market model. He

suggests that the proper criterion to be used in deciding

upon an appropriate interval over which to estimate the

market model is predictive efficiency; the market model

estimates are used to derive predicted returns and the mean-

square errors of these predicted returns as predictors of

the reserved 1960 (or 1968) returns for different time

intervals are calculated. His results suggest that a seven

year interval provides estimates with the greatest predic-

tive efficiency and this interval was used in computing beta

estimates for correlation tests with 6 in the above


The procedure is to correlate the coefficients of

determination for each firm from the market model with the

coefficients of determination from the accounting income

model for each firm.

Again, using annual observations, various intervals

were used to calculate the accounting income numbers.

Gonedes finds as a result of performing prediction tests

that the twenty-one year period provided better estimates

than the seven year estimates. The prediction test results

suggest that the accounting-number models do not reflect the

structural changes reflected by the market model. The

results for the model of scaled first differences for various

subperiods are given in Table 1.6.

3. Theoretical Basis Study

Pettit and Westerfield [1972] attempt to fill a lacuna

in the beta literature by providing a rationale for deciding

which factors affect systematic risk. They show, using a

perpetuity cash flow valuation model for stock prices, that

the usual beta of an asset's return can be written as a

weighted average of a "capitalization rate beta" and a

Table 1.6

Correlation Coefficients, R, between Estimates from
Market Model (M) and Four Accounting Number Models (A)(a)

Accounting Beta
Measured Over





Stock Beta Measured Over
M2 M3 M4
1946-52 1953-59 1961-68


(a) Ninety-nine Cross-sectional Observations.
b/ Significant at a = .05.
c/ Significant at a = .01.

(Table 5, p. 434)






"cash-flow beta" each defined from regressions of the firm's

capitalization rate and its cash flow on the capitalization

rate of the market and the cash flow of the market. They

conclude that anything that affects the expected cash flow

of the firm or the capitalization rate should affect beta

through the cash flow beta and the capitalization rate beta,


The analysis involves a circularity, however. Capital-

ization rates are equilibrium expected rates of return. As

such, according to the CAPM, they are determined, given RF

and EM, solely by beta. Thus a knowledge of which factors

affect the expected rate of return on the firm's equity

securities would require a knowledge of the factors affect-

ing beta. There is no independent knowledge of which fac-

tors affect capitalization rates, at least according to the

CAPM. Thus, the Pettit and Westerfield decomposition is

uninformative. The authors run up against this paradox when

they attempt to explain the choice of variables as determi-

nants of beta.

Because of space limitations we have not given any
justification of why these particular variables should
or should not be related to asset risk. In some cases
we think a relationship is expected, in other cases we
feel that any relationship is a spurious one. Never-
theless, each variable was included in the analysis
because at some time someone proposed that the variable
suggested something about the risk associated with an
asset (!) [pp. 1661-1662]

Their list of variables is:

(1) Dividend payout
(2) Leverage
(3) Firm Size
(4) Liquidity
(5) Growth.

They perform a correlation analysis on the market beta,

BM; the capitalization rate beta, /p ; the cash flow beta,

SEPS; the operating income beta, 0I ; the payout ratio, PAY;

the debt equity ratio, D/E; size, SIZE; liquidity, LIQ.; and

growth in earnings per share, GEPS. The correlation analysis

was carried out for two periods: Period I, 1947-57, and

Period II, 1957-68. Their results appear in Table 1.7.

On the individual level the results are not striking.

At the portfolio level, the largest correlations are between

the market beta and the payout ratio and the market beta and

the capitalization rate beta.

This may be a spurious correlation though, because

grouping of observations into portfolios will, in general,

increase the correlation coefficients.

4. Barr-Rosenberg and Associates' Work

Barr-Rosenberg and McKibben

Barr-Rosenberg and McKibben [1973] attempt to decompose

beta into a component that depends upon a set of descriptors

based upon accounting data plus a firm specific effect that

cannot be accounted for by these descriptors. Their decom-

position is similar to the Beaver, Kettler and Scholes

Table 1.7

Correlation Analysis


e/p 'EPS 0I

Period I S .329 .259 .197

Period II .292 .184 .147


Period I 8 .630 .455 .307

Period II .621 .389 .261

Table 3, p. 1663


-.481 .049 -.074 -.068 .215

-.394 .069 -.182 .013 .249


-.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





Unt = nMt + nt' (1.12)

the error term in this regression, applying
ordinary least-squares to equation (1.11), yields

inconsistent estimates because Unt will be corre-
lated with the independent variables WntMt via
the market return Mt.

(2) The intercept a was taken to be constant in their
regressions. If one accepts the CAPM, though,
a = R (1- nt) will vary with n and t even if RE is
constant because of its dependence on Bnt

(3) In estimating the variance of the firm specific
effects, n' the authors find that:

var(n ) = W < 0. (1.13)

This can occur in variance components models if
the usual variance formulae are applied. To
overcome this problem, alternative estimates of w
have been derived in the literature on pooling
time-series and cross-sectional data. However,
Barr-Rosenberg and McKibben substitute W = 0
whenever W < 0. This leads to problems in pre-
dicting ,n

In evaluating the predictive power of their betas,

Rosenberg and McKibben consider the mean-square error in the

prediction of returns generated by those betas. They compare

their predicted betas to those generated under alternative

assumptions. These include the void predictor a = n = 0,
ns ns
the naive predictor using historical betas and alphas

generated from the stationary market model, the naive predic-

tor with a = 0, a Bayesian adjustment of the naive, and the

unit beta a = 0, 0 = 1.
Their results indicated that only their predicted betas

did better in a mean-square-error sense than the unit beta


Barr-Rosenberg and Marathe

In later work, Barr-Rosenberg and Marathe [1975]

classified their descriptors into the following categories:

(1) Market Variability
e.g., historical beta, sigma, current price

(2) Earnings Variability
e.g., accounting beta, variability of cash flow

(3) Unsuccess and Low Valuation
e.g., growth in E.P.S., average proportional cut
in dividends over the last five fiscal years

(4) Immaturity and Smallness
e.g., log (Total Assets), Net Plant/Gross Plant

(5) Growth Orientation
e.g., Dividend yield, E/P

(6) Financial Risk
e.g., Total Debt/Assets, Liquidity

(7) Indicator of Firm Characteristics
e.g., dummy variable for N.Y.S.E. listing, whole-
salers, etc.

Barr-Rosenberg and Marathe take the market model with a

constant intercept

nt nt t + nt' (114)


Bnt = b + b Xnt + bX2nt + ... + b nt (1.15)

+ bJ+1dlnt + ... + bGdGnt.

Note that there are no firm specific effects in this speci-

fication of nt. That is, the firm beta is a linear combi-

nation of J descriptors and G industry dummy variables. The

coefficients of the prediction rule for systematic risk are

obtained as follows. Equation (1.15) is substituted into

equation (1.14) to obtain

Rnt = a + b0Mt + bl(X ntMt) + ... + b (XJntM ) (1.16)

+ b+l(dntMt) + ... + bK(dGntMt) + nt

The data are pooled and ordinary least-squares is run on

equation (1.16) to obtain estimates 08,6 ,...,b' ,J J ',...,b .
0 J J+1 K
Having obtained these OLS estimates the residuals in

the market model Ent are obtained

nt = nt nt t (1.17)

Let o denote the variance of E. In this specifica-
nt nt'
tion it is assumed that 2 is explained as follows:

ant = St(S0 + S IXnt + ... + S JXnt + Sj+1d1nt (1.18)

+ SKdGnt

where St is the average cross-sectional standard deviation

in month t, and S ,...,SK are the coefficients of the predic-

tion rule for residual risk. Let 6nt = E(e ntl) the mean

absolute residual return for security n in month t, and

c = t-, the coefficient of variation of le The model

for the residual risk can be rewritten as

nt = 6t(S0 + SlX nt + ... + S XJnt + (1.19)

S +1dnt + .. + SdGnt),
J+1 mnt K Gnt

where t is the capitalization weighted cross-sectional

average of absolute residual returns.

Estimates of residual risk are obtained by running the


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
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


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


Table 1.8

Prediction Rules for Systematic Risk Based on
Fundamental Descriptors and Industry Groups

Earnings Variability

Variance of Earnings
Variance of Cash Flow


Unsuccess and Low Valuation

Growth in E.P.S.


Immaturity and Smallness

Log (Total Assets)


Growth Orientation

Growth in Total Assets


Financial Risk

Leverage at Market


* Significant at 95% level.

*** Significant at 99.9% level.

(Table 4, p. 114)

One notes that different signs result from alternative

specifications of the growth and leverage terms. One

wonders what gain is effected by including several different

measures of the same effect. Perhaps such a procedure

obscures the true relationships underlying the model.

When market variability variables such as the histori-

cal beta estimates, historical a estimates, and price and

share turnover variables were included in the regressions,

the following partial results in Table 1.9 were obtained.

Barr-Rosenberg is disturbed by the negative adjustment

to leverage and finds it inexplicable. The two measures of

financial leverage employed are defined in the following


(1) Leverage at = Book Value (Long-Term Debt
Market + Preferred Stock) + Market Value
(Common Stock) Market Value
(Common Stock)

(2) Debt/Total = Long-Term Debt + Current Liabili-
Assets ties Total Assets

While theory tells us that the debt/equity ratio at

market value is positively related to beta, it turns out to

be negatively related in the regressions. The book value

measure turns out to have a positive sign. Further, the

sign remains negative in the simple regression on market


In conclusion, Barr-Rosenberg asserts:

The negative relationship appears to be an empirical
fact, but one that we do not now understand. Since the
relationship is not comprehensible, we have set the
coefficient to zero in actual practice. [p. 122]

Table 1.9

Prediction Rules for Systematic Risk Based on
Fundamental Descriptors Including
Market Variability Descriptors

Market Variability

Historical Beta Estimate
Historical a Estimate
Current Price


Earnings Variability

Variance of Earnings
Variance of Cash Flow


Immaturity and Smallness

Growth in E. P. S.


Growth Orientation

Growth in Total Assets


Financial Risk

Leverage at Market


*** Significant at the 99.9% level of

(Table 5, p. 124)

This represents a highly dubious procedure which would be

difficult to justify.

The next part of the Rosenberg and Marathe study is

concerned with the formulation of measures of predictive

accuracy. Two basic historical performance measures are

computed as follows:

(1) Assuming that all assets have identical risk one
runs the regression

nt a + b -- (1.22)
nt nt

One calculates the mean-square error of the
predicted returns generated by this naive model.
Call the mean square error from this prediction
rule MSE .

(2) Run a second regression of the form:

Rnt 1 RMt HntRMt
= ~ a + b0[ n -7 + bl (1.23)
nt nt nt nt

HBnt is the historical beta and this represents a
Bayesian adjustment to the historically generated
beta. Call the mean-square-error from this
prediction rule MSEB.

(3) Any other prediction rule for beta, e.g. a predic-
tion rule based upon fundamental descriptors, can
be evaluated via the performance index

MSE -MSE 0 (1.24)
B 0
where MSE1 is the mean-square error from the
prediction rule to be evaluated. That is, one
computes the improvement one obtains by using the
given prediction rule for beta relative to the
improvement over the naive hypothesis achieved by
the benchmark procedure.

The results appear in Table 1.10.

Table 1.10

Unbiased Estimates of the Performance of Alternative
Prediction Rules in the Historical Period
(Predicted Variance as a Multiple of the Variance
Predicted by a Widely Utilized Prediction Rulea)


Market Variability Information Only

Benchmark 1.00
All market variability descriptors 1.57

Fundamental Information Only

Industry adjustments and fundamental 1.45

Market Variability and Fundamental Information

All information except the historical 1.79

All Information 1.86

a The reported figure is the adjusted R2 in the appropriate
GLS regression for residual returns, r rMt divided by
the adjusted R for the benchmark procedure.

(Table 6, p. 134)

Barr-Rosenberg's thesis that the use of both market

information and fundamental descriptors leads to better beta

predictors than the use of each set of information sepa-

rately is supported by these results. He also notes that if

in predicting beta all information concerning historical

betas is discarded, then less than 4% of predictive power is

lost. He concludes from this result that "we are able to

obtain virtually all of our predictive power for these

aspects of risk without relying on historical measures of

them" (p. 135).

To test for the predictive accuracy of the betas

generated into periods other than those in which the predic-

tion rules were fitted, Barr-Rosenberg tests for stability of

the prediction rule. He finds it to be quite stable over

time--the prediction rule estimated for a full history of

230 months is closely similar to that for a recent 101 month

subperiod. He also computes the adjusted R2 for the fitted

regression lines using various prediction rules. He finds

that the betas based upon market and fundamental descriptors

were superior to the benchmark betas in five cases out of

six with an average R2 of .0768 versus .0502.

Barr-Rosenberg and Marathe

In their work on testing the Capital Asset Pricing

Model, the authors, Rosenberg and Marathe [1979] generate

betas where the market model is employed with both alpha and

beta varying as linear functions of a set of descriptors

plus error terms. This relationship they write as

a = a'X + ea (1.25)
t = 1,...,T
t = b t t n = l,...,N(t) (1.26)

where the terms ent' nt are "model errors in prediction

with expected value zero" (p. 140).

The desire in this study is to test the CAPM. Only the

historical beta was used as a fundamental descriptor for the

actual beta. The authors note that the prediction rule

could be improved by using Bayesian adjustments to the

historical beta and fundamental accounting and market based

descriptors. This is not done in this study. However, the

econometric techniques of their previous studies are improved

upon in this one. Generalized least-squares procedures are

used in this study. The results are hard to compare to

previous results, however, because of the use of a single

fundamental descriptor, the historical beta. Part of the

purpose of the present study is to apply correct econometric

techniques to the model with a set of theoretically justi-

fied descriptors.

4. Study of the Effect of Financial Leverage

Hamada [1972] tests the effect of the leverage relation

he derives using the M-M theory. The relation he derives is

6u = L- (1.27)


The M-M theory states that V = V + TD so that
L u L


S +D (1-T) L

This yields

SL+DL (1-T)
L S u

- ^^ ---/u-


To test this relationship, Hamada calculates the

following rate of return




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
is unobservable as firms generally employ debt financing,

the M-M theory is used to evaluate the denominator


= (V D)t-
U L t-1


The numerator is evaluated using the following identi-


Xt(l-T) + AGt

= [(X-I)t(l-T) PDt + AGt]

+ PDt + It(1-T),

where PDt denotes preferred dividends at time t, and It is



the interest expense at time t. The corporate tax rate is

designated T.

Next, the following identity is used:

(X-I)t(1-T) PDt + AGt = dt + cgt (1.33)

where dt represents dividends paid at the end of the period

and cgt represents capital gains.


dR + cgt + PDt + I (1-T)
R = (1.34)
t (V-TD)

Next, the observed rate of return to common stockholders is

(X-I) t (1-T)-PDt+G d +cgt
RL S S (1.35)
t L L
t-l t-1

Hamada obtains available data to construct R and R the
L u
t t
rate of return to a firm exactly identical to the levered

firm except that it has no debt or preferred stock in its

capital structure. Using these rates of return he then runs

the market models:

RJ = a + j R + Ej (1.36)
U u U M u,t
t t


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

The betas obtained from these regressions are estimates

of the unlevered and levered betas. Theory implies that

j > j .
L u

Using data on 304 firms, Hamada runs 304 time series regres-

sions and calculates mean alphas and mean betas, and the

statistics presented in Table 1.11.

These results indicate that since .9190 > .7030 lever-

age increases systematic risk. Similar results are obtained

when continuously compounded rates of return are used in the

market model. Hamada concludes, that if the M-M theory is

correct, then leverage explains 21-24% of the value of the

mean beta.

He then goes on to test for which market value rates

ought to be used to adjust observed betas to obtain

unlevered betas as his formula suggests. To do so, he runs

the following regressions:

S= 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


& .02

u .70

R2 .37

UL .03

L .91

R .38

* xi-R-X


(Table 1,


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





period) value for S /SL and then the end of period 1966

value for S /SL, the results were obtained in Table 1.12.

Hamada concludes that, if long-run averages are used

then the adjustment factor method is appropriate using the

derived relationship.

An alternative indirect test of the financial leverage

effect is carried out on the basis of the following set of

considerations. Within a given industry ordinary common

stock betas show a certain degree of cross-sectional varia-

tion. Some of this cross-sectional variation is presumably

due to the differing degrees of financial leverage employed

by firms within the given industry. The reason for looking

at a given industry in examining cross-sectional beta varia-

tion is to confine attention to a given risk class in the

M-M sense, that is, to one with the same cost of capital as

that of a firm with no leverage but otherwise exactly alike

and consequently with the same unlevered beta since

P = R + (EM RF)BJu (1.40)

In practice it is not possible to specify exactly a risk

class so that in practice there will be some variation in

unlevered betas of the firms within the industry.

The idea of the Hamada test is then the following: If

indeed some of the cross-sectional variation in stock betas

within a given industry (risk-class) is attributable to the

differing debt/equity ratios employed by the firms, then

unlevering the betas should result in a set of unlevered


(N in o

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SD 0 o C

c 4l co

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o a co '

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betas which exhibit less cross-sectional variation than the

original levered betas. Thus, by computing the dispersion

of the levered betas, unlevering them and computing and

comparing that dispersion to that of the resulting set of

unlevered betas one should be able to discover a financial

leverage effect. The results follow in Table 1.13. These

results indicate a positive effect for financial leverage,

and Hamada concludes, some support for the M-M theory.

6. Studies of Effect of Market Power


Various studies have examined the effect of monopoly

power on the rate of return to stockholder's and upon a

firm's profitability. Firm profitability is usually measured

by the ratio of net income to the book value of stockholder's

equity. One such study examined the effect of market power

on equity valuation.

Sullivan [1977] takes the ratio of the market value to

the book value of stockholder's equity for a given firm in a

given year. Then the arithmetic mean over the years 1961-70

is taken to evaluate relative equity prices. This variable

is then regressed on

C.: the weighted average four firm concentration ratio
3 for firm j

SZ.: the natural log of 1961 sales revenue for firm j

MSG.: 1968 market share for firm j divided by 1961
Market share for firm j (representing growth in
market share)

Table 1.13

Mean and Standard Deviation of Industry Betas


Number of Firms




Mean 8

Mean 8

Mean 8

Primary Metals

except Electrical




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

Mean 8

Mean 8

Mean 8

Mean 8

.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

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
S.. In both regressions a positive statistically signifi-

cant sign is obtained for these market share variables.

Sullivan concludes:

These premiums seem to suggest that firms with market
power have the ability to set and hold output prices
above costs and as a result earn monopoly profits.
This ability to control output prices makes the equity
shares of powerful firms attractive to investors who
bid-up the prices of the equity shares so that all the
expected future monopoly profits are capitalized into
the existing market prices of the shares. [p. 111]

The study purports to control for risk by introducing

the standard error of a trend line fitted to book profit-

ability over a ten year period. However, this variable is

not used in the regression discussed and is a questionable

measure of risk in any case. It is possible that the

earnings of the monopolistic firm may be capitalized at a

lower discount rate because the existence of monopoly power

lowers the systematic risk of a firm's equity. Such a

decrease would increase stock prices. Sullivan argues that

these increased stock prices are the result of the capital-

ization of monopoly profits.


Thomadakis [1977] carries out a value-based test of

profitability and market structure. He attempts to examine

the relationship between the capitalized value of monopoly

rents and

(1) F: a risk measure of the risk of future returns

(2) U : the firm's power of oligopolistic restriction
with respect to output of currently held assets

(3) Uf: the firm's power of oligopolistic restriction
in future investment

(4) g: the firm's expected rate of growth from exo-
geneous demand

(5) C: a scale parameter.

He runs

M= a + alF + a2Uc + a3(gUf) (1.41)


M= (1.42)

the difference between total firm value and the book value

of assets divided by sales is used as a measure of monopoly

power. There may arise problems in using this variable as

a measure of monopoly power because of differing accounting

methods used to evaluate A and due to the effect of infla-

tion on asset values. Thomadakis assumes that the biases

introduced as a result of these problems can be disregarded.

The results bear some significance to the question of

the effect of monopoly power on systematic risk because

Thomadakis takes beta from the usual historical regression

as his risk measure. Hypothesizing that Uf and U are
f c
functions of industry concentration IC, the results follow

in Table 1.14.

Thomadakis finds the sign of the risk factor puzzling.

It indicates that the higher the systematic risk, the higher

will be the degree of monopoly power. He explains this

result by stating that 8 should represent the volatility of

excess earnings whereas here it represents the systematic

volatility of total earnings. "The only possible interpreta-

tion of current results is that 8 is a negative proxy for F,

but this appears quite far-fetched and should be viewed with

reservation" (p. 183).

He notes further that a correct risk measure for the

purpose of the study would separate out the risk of the

competitive component of return from the supercompetitive

component. The use of beta in this context assumes that

both components of earnings have the same systematic risk.

Presumably, the results give some first indication of

the effect of monopoly power on systematic risk. If that

relationship is negative then the coefficient of beta in the

Thomadakis regression should be negative as well. Of

course, the inclusion of other variables measuring market

share and growth in the regressions could obscure the

possible relationship and its sign.

Table 1.14

Industry Concentration and Future Monopoly Power





I IC gIC 2

0.64a 1.89a 5.12a

a Significant at the 5% significance level.

(Table 1, p. 182)



Sullivan [1978] seeks to determine whether the market

power of firms, as measured by size and seller concentra-

tion, seems to reduce the riskiness of firms. By riskiness

is meant systematic risk. To do so, Sullivan regresses

monthly betas on

(1) SZ. = natural log of sales for firm j

(2) C. = four-firm concentration ratio

(3) DN. = industry dummy variables

(4) SG. = the annual compound growth rate in sales
from 1963-1972 for firm j.

He runs levered and unlevered betas on these variables.

Typical results appear in Table 1.15.

The coefficients of C. and SZ. are consistently

negative and statistically significant in Sullivan's

results. These results do indicate a negative relationship

between monopoly power and systematic risk. Sullivan seeks

to determine the cause of this relationship. To answer the

question he decomposes beta, based on the definition of

earnings to equity holders, into three betas:

(1) a beta relating the covariability of firm sales
with the market

(2) a beta relating the covariability of firm expenses
with the market

(3) a beta relating covariability of the revaluation
of the firm's securities in the secondary capital
market [a capital gain (loss) less retained earn-
ings component]with the market.

Table 1.15

Monthly Betas, Leveraged and Unleveraged, Regressed
on Market Power and Control Variables


















25% sample: every firm in which the firm's largest market
accounted for at least 25% of firm sales.

(Table 1, p. 213)


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


He concludes that the systematic risk resulting from

firm operations seems to be small.

7. Studies of the Effect of Operating Leverage


Turning now to theoretical models describing the effect

of the labor-capital ratio on beta, we first discuss the

work of Lev [1974]. The firm's operating leverage is

defined as the ratio of fixed to variable costs. Lev claims

that beta is a positive function of operating leverage.

Attempting to support his case that higher operating lever-

age implies a higher total and systematic risk, Lev appeals

to the "better known leverage effect"

within a given risk class (a homogeneous industry in
our case) the higher the financial leverage, i.e. the
relative share of fixed interest charges (fixed costs
in our case), the larger the volatility of the earnings
residual accruing to common stockholders, and hence,
the higher the financial risk associated with the
common stocks. [p. 630]

This argument is suspect for several reasons:

(1) The financial leverage argument does not proceed
through the earnings volatility argument. Rather,
it follows from the consideration of the precise
effect of financial leverage on the rate of return
to equity holders.

(2) The financial leverage argument is not based upon
the existence of fixed debt charges because it
holds, as Galais and Masulis [1976] have shown,
for risky debt as well.

(3) Higher total volatility does not necessarily imply
higher systematic risk.

Lev goes on, however, to make the following arguments:

In an uncertain environment future demand Qjt is a random

variable. Then the earnings stream of the firm

Xj = (p )jt (v )j F. (1.43)
it jt jt jt


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


= p vj (1.44)
aQ j t 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


In a homogeneous and competitive industry the average

product price is the same for all firms. Thus the fluctua-

tions in the earnings stream of the firm depends only upon

the average variable cost per unit. A firm with a higher

average variable cost will have a more volatile earnings

stream. A firm with a greater operating leverage, Lev

claims, will have a lower variable cost per unit and hence a

more volatile earnings stream in accordance with his partial

derivative. This higher volatility is transmitted to

increase the total volatility of returns to equity holders.

This analysis and introduction of uncertainty leaves

the question of the firm's response via its factor mix to

the uncertainty in demand. It is possible that a firm would

in fact respond in such a way as to reduce the variability

of its earnings stream. What is missing here is an analysis

of the firm's optimal decision behavior under a situation of

increased uncertainty in demand.

Lev then goes on to attempt to demonstrate the effect

of operating leverage on the systematic risk of the firm's

equity B.. He does this by writing the rate of return as

(Sales. V F )(1-T) + Ag
R. t jt jt jt (1.45)
J ,t-1


Sales. = (pQ) = total revenues
jt jt
V = total variable costs
F = total fixed costs
Agj = the future growth potential of the firm

S = market value of equity at time t-l
T = the corporate tax rate.

Thus we obtain

Cov((PQ) j (1-T),RMt
S. 8, = jt Mt (1.46)
Var(R t)

Cov(V (1-T),Rt) Cov(Agt ,Rt )

Var(RMt) Var(RMt)

Consider two firms, exactly alike, including output,

stock value, and capitalized growth. They differ only in

their use of variable factors of production. The first and

last beta will be identical. The firm with the higher

operating leverage will have relatively fewer variable costs

(a random variable), hence a lower expected value of

variable costs. From this, Lev concludes that it will have

a lower covariance with the market return. But this state-

ment is not correct. Covariance measures comovement between

random variables. For three random variables it is possible


E(x) < E(Y), but that Cov(,2Z) > Cov(Y,Z).

Nonetheless, Lev concludes that the firm with the

higher operating leverage will have a lower second beta and

hence that

Sl,t-11 > S2,t-_12* (1.47)

He then assumes that the two firms have the same stock value

and consequently that operating leverage increases system-

atic risk.

Turning to the empirical tests of the proposition, the

procedure is as follows: using data from three homogeneous

industries, electric utility, steel, and oil to ensure

cross-sectional equality of sales patterns across states of

nature, Lev runs a time series regression of total operating

costs of the j-th firm in year t on total physical output.

He obtains from this regression the estimated coeffi-

cient V., a measure of average variable costs per unit of

output (assuming that MC=AVC and that average variable cost

is constant over the estimation period).

These regressions were run over two different time

periods, a 20 year period (1949-1968) and a twelve year

period (1957-1968) to test the "stability" of the relation-

ship. The results follow in Table 1.16.

The intercept is not reported here. When this study

was replicated, the intercept, representing fixed costs

turned out to be negative. This indicates a serious problem

in this model specification to determine average variable

costs and/or in the estimation procedure.

The next step was to run the market model to obtain

estimated beta coefficients. The regression was run using


Table 1.16

Estimates of Average Variable Cost Per Unit

20 years V

Electric Utilities .00815

Steel Manufacturers .892922

Oil Producers .74806

Table 1, p. 635; t statistics in parentheses.

12 years




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 = 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 [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)

where D. and S. represent the market values of debt and
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


Table 1.17

Regression Estimates for Systematic Risk
on Average Variable Costs Per Unit

No. of Firms R2

Electric Utilities


Steel Manufacturers






(5.014) (-2.4097)

Oil Producers




(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)


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


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 .


The first component on the right-hand side of this expres-

sion is called operating risk by Rubinstein, and the second

component is called financial risk. The goal now is to

develop the components of operating risk

X*p(R.*,R ) Var(R.*)
3 M ]

It suffices to drop the X* in this consideration. Let

m denote product m for firm j. Rubinstein assumes that the

output Qm of product m is a random variable and that the

sales price per unit pm is fixed. One can view this as a

model in which price is set ex-ante with quantity demanded

being uncertain to be determined at the end of the single

period. Note that this assumption is contrary to the

Subrahmanyam and Thomadakis model to be reviewed below in

which the firm is a quantity setter with uncertainty

resolved in the product's price. As Leland [1972] has

shown, for the monopolistic firm, these are not equivalent

behavioral assumptions as far as optimal behavior is


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

Rubinstein claims that since

a X.
R. mV (1.58)
m j


p(Rj*,Rm) Var(R *) = [a (pm-v )p(QRM ) Var m 1. (1.59)
m (m 9

Rubinstein interprets this as follows: a represents
the relative influence of each product line, p -v reflects

operating leverage through the contribution margin, p( m,R )

represents the pure influence of economy-wide events on

output, and Var(Qm/amVj) the uncertainty of output per

dollar of assets, a measure of "operating efficiency."

For the case of a firm producing a single product this

formula is easily derived.

p(Rj*,RM) = p v. j m = p( ,Rm) (1.60)

Var(R*) = arp -F (p-) 2 Var [-]. (1.61)


p(Rj*,RM) Var(Rj*) = Ip-vp(Q,RM) Var(/V.) (1.62)

which is the relationship to be derived.

This formula shows that operating risk and consequently

systematic risk [the sum of operating and financial risk

equation (8), p.57] is a positive function of operating

leverage in the following sense. Given the product's price,

decreasing variable costs per unit increases the contribu-

tion margin jp-vl and with it operating risk which is

proportional to the contribution margin. This of course

assumes that

p(Q,RM) Var(Q/V )

is constant.

Another way to view this result is in the following

manner: Given the contribution margin operating risk is

determined by

(1) the pure influence of economy-wide events on
output represented by p(Q,RM)

(2) Var(Q/V.) the uncertainty of output per dollar of

Presumably firms are out to maximize their market

values in the presence of uncertain demand. Plausibly, an

increase in the use of variable factors would afford a firm

with some measure of flexibility in its production process.

However, the question becomes: What is the effect of an

increase in the use of variable factors on optimal firm

value, presumably represented by V.? If, indeed, systematic

risk is lower with the use of variable factors, then for a

given random income stream, firm value would increase.

Without this knowledge, which clearly has to be determined

as part of the analysis, firm value could decrease with the

use of variable factors and operating (and systematic risk)

could increase. This is all for a given output stream j..

As Subrahmanyam and Thomadakis note, the analysis is

not very enlightening because it fails to relate the optimal

choice of margin (i.e. of factor mix and price) to the

uncertainty of output.

Subrahmanyam and Thomadakis

The model developed by these two authors will be

developed and generalized in the next chapter. A brief

summary will be given here. The goal of the analysis is to

provide a specification of the relationship between monopoly

power as represented by the reciprocal of the price elasti-

city of demand and the labor capital ratio and the system-

atic risk of the firm's equity securities. (There is no

debt in the model.) The innovative facet of this work is

the introduction of sources of uncertainty into the demand

function and into the wage rate. The effects of the above

variables on beta then depend upon the co-relationship

between the random error terms representing uncertainty in

this model. Their basic result is that if wage rate uncer-

tainty is relatively less volatile than price uncertainty

then beta will be positively related to the labor capital

ratio and negatively related to the degree of monopoly

power. If wage rate uncertainty is relatively more volatile

than price uncertainty, then the reverse will hold: beta

will be negatively related to the labor capital ratio and

positively related to the degree of monopoly power. The

economics underlying these results can be explained with

reference to the market portfolio. Since human capital is a

risky asset which we can assume to be tradeable (the results

generalize to non-tradeable human capital) the risk of the

market portfolio is the sum of the risk of firms plus the

risk of the labor component of firms' production processes.

The risk of labor is essentially determined by the "beta" of

wage uncertainty relative to price uncertainty. If that

beta is less than unity then in the weighted average repre-

senting the beta of the market portfolio relatively less

risk will be shared by labor than by firms. This forces the

firm betas to be greater than unity. As labor is increased

relative to capital, if the basic condition holds, more risk

will have to be borne by the relatively smaller amount of

capital. Thus beta is a positive function of the labor-

capital ratio. This stands in direct opposition to the Lev

and Rubinstein results. The reverse will obtain if the beta

of wage uncertainty relative to price uncertainty is greater

than unity. The model developed here hints at the fruitful-

ness of an approach to determinants of systematic risk that

takes into account sources of uncertainty, their correlation,

and risk sharing among the bearers of risk.



1. Introduction

The first purpose of the study is to examine the argu-

ments given in support of various microeconomic factors said

to affect the systematic risk of a firm's common stock as

measured by its beta coefficient. Arguments based upon

theoretical models in the literature and upon "economic

intuition" will in turn be considered. A large number of

factors can be hypothesized to bear a relationship to beta

but, of these, only those that appear to be of clear economic

interest will be considered. Those that affect beta for

purely technical reasons are not of immediate interest here.

An example would be non-synchronous trading in that it makes

a common differencing interval for return data impossible to

calculate and introduces various biases into beta estimation.

In principle, one can correct this potential source of

bias--in doing so the effect on beta of the phenomenon

disappears. An example of a variable of economic interest

is financial leverage: through its effect on the rate of

return to equity holders it unambiguously affects systematic

risk. While the study of the technical factors affecting

beta is of great importance, this study concerns itself with

the formulation and inclusion of the economic type of


Recently, there have appeared theoretical models in the

literature supporting the labor-capital ratio, monopoly

power, and "growth opportunities" as determinants of betas.

More traditional determinants are financial leverage and

volatility of operating earnings. In this chapter, the

theoretical arguments supporting the above variables will be

presented and critically examined. In the absence of

theoretical support, the intuitive arguments will be

presented and their limitations will be stated. We turn now

to a separate consideration of each of the variables pur-

ported to affect beta. To motivate the discussion of the

effect on beta of monopoly power, it is instructive to

compare the two different arguments for the effect on beta

of (1) financial leverage and (2) volatility of operating

earnings. While the effect of (1) has been theoretically

established, the argument for (2) proceeds on an intuitive


2. Financial Leverage

To present the argument in its simplest form, taxes

will be ignored and it will be assumed that there is no risk

of default on the firm's debt securities. This latter

assumption guarantees that i, the interest rate on the

firm's bonds, is non-stochastic. The rates of return on the

levered firm's and unlevered firm's equity are:

R -S and R -
L S u V
L u

respectively. Evaluating the corresponding betas yields

cov(R ,Rm) cov(X-iDL/SL,Rm) cov(X/SL,Rm
L 2 2 2
02 (Am) 2 (R) 2a (rm)
m m
cov(X/SL Vu/Vu,Rm) Vu cov(X/Vu,RM)
2 S 2
S (R ) L 2(R )

S U 6U.

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


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


If we are willing to assume that both the assumptions

of the Options Pricing Model and the continuous time version

of the CAPM hold, then Hamada's result which holds for

riskless debt can be generalized to allow for risky debt

where all betas are to be interpreted as instantaneous

quantities. It is also assumed that dV/V follows a

Gauss-Wiener process and hence that the Modigliani and

Miller result on the irrelevance of financial structure (for

firm value) holds by hypothesis.

Viewing the equity of the firm as an option on the

underlying firm value with exercise price equal to the face

value of the firm's outstanding debt securities, an expres-

sion for the dollar return on equity value AS can be derived

using Ito's Lemma.

AS 1 2S 2 V2 S
AS = AV + V At + At
3V 2 + 2 at

where V is firm value, a2 is the instantaneous variance of

percentage returns on V, and t represents time.


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:


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


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


(1) Increased volatility of earnings implies increased
volatility of end-of-period stock price (a random

(2) This induces a more variable rate of return since

1,t P


var(R. ) Var(P. )t
i,t p2 t+1"

Thus, earnings volatility affects the total variability

of the rate of return of the equity securities. No refer-

ence is made in the earnings volatility argument to volatil-

ity with respect to the earnings volatility of the market

and none is established. To do so, one would have to posit

a relationship between the earnings volatility of the market

and the earnings volatility of the specific security. Such

an assumed relationship would be transmitted to the corre-

sponding rates of return. But such a relationship would be

very close or identical to a specification of the effect of

earnings' volatility on the security's beta; a relationship

it is desired to discover.

Under a highly restrictive assumption discussed below,

it can be shown that the stock beta is proportional to the

total volatility of the rate of return. We merely write

cov(iR ) Pim
2 = a= i,
1 (R ) m i

where p. is the correlation coefficient between the market
rate of return and the specific security's rate of return,

and o. is the standard deviation of the stock's rate of
return. If we assume that pim/am is constant, the result

is obtained.

A sufficient set of conditions for this to hold is that

both p. and a are assumed to be constant. Given a constant
irn m
correlation with the market and a constant market variance,

this result is not surprising, as the only factor left to

occasion changes in beta is the own variance of the securi-

ty's rate of return. Of course, it is important to then

know for each different security the proportionality

constant Pim/om as it determines the (absolute) magnitude

of B., but given that information, the stock beta is then

determined solely by all factors influencing the variability

of the stock's rate of return, and earnings' volatility is

one of those factors.

Given the assumed constancy of pim/am and the implied

result that systematic risk is then proportional to total

risk, what are we to make of the tenets of portfolio theory

and the CAPM implying that, in general, it is only the

systematic component of risk that matters? The answer is

contained in Capital Market Theory and is analogous to the

distinction between the Capital Market Line and the Security

Market Line. The Capital Market Line is precisely that set

of portfolios for which p. assumes the value +1 and for

these efficient fully diversified and perfectly-correlated-

with-the-market portfolios (the most diversified portfolio),

it is indeed a. that becomes the relevant risk measure. The

assumption that pim/am is constant for a given security's

rate of return is analogous (though not coincident) with the

assumption that the security be on the Capital Market Line,

that is, that it be an efficient security. The assumed

constancy of pim/m/ then is clearly seen to be a highly

restrictive assumption and it is this assumption that must

be added to be earnings' volatility argument to derive the

effect of this factor on the stock beta. As indicated, this

assumption is very close to a specification of how the

stock's rate of return responds to the market's--a relation-

ship it is sought to uncover. After all, the only real

difference between cov(R.,R ) and p. is that one is a
i m im
dimensional measure and the other is not. Either of these

quantities is a measure of comovement with the market rate

of return. Clearly the former measure determines the

stock's beta [with a(Rm)] as the stock beta is a dimensional

measure of comovement of the security's return relative to

the market. This is the reason that p. and a only deter-
Im n
mine the proportionality factor. It is left for a. to
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


(combined with it) "works." Given the inherent untestability

of that assumption (a defect shared by cost of capital

studies) it is safe to assert that by itself the assumption

has no real justification.

4. Growth

A first distinction is made between growth in earnings,

sales, or assets and growth opportunities as opportunities

to invest in projects with expected rates of return greater

than their costs of capital. Modigliani and Miller [1961]

have argued that the latter type of growth is the relevant

concept for firm valuation.

The essence of 'growth', in short, is not expansion,
but the existence of opportunities to invest signifi-
cant quantities of funds at higher than normal rates of
return. [p. 417]

One consequence of this distinction is that, as the

firm's stock beta is a valuation concept for measuring that

portion of the riskiness of the security which the market

rewards in equilibrium pricing, insofar as mere asset expan-

sion is not necessarily relevant for valuation, one would

expect no necessary relationship to hold between it and the

firm's stock beta. Perhaps this accounts for the erratic

performance of growth measures in the empirical literature.

For in some cases, asset expansion would be indicative of

growth in the Modigliani and Miller sense while in other

cases not. Of course it is assumed that growth opportunities

do affect the stock's systematic risk, a question considered

below. It is interesting to examine any technical effects

on single period betas of growth in a sense to be defined

below. Before doing this, we turn to the question of how

growth opportunities in the Modigliani-Miller sense affect


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


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


initial period than earlier flows. The beta of the entire

project (all the flows, early and later) will simply be a

weighted average of the betas (relative to the initial

period) of the individual flows. If there are relatively

more later lower beta flows, then the beta of the entire

series of flows will be lower than that for a series with

relatively fewer later flows. Thus, growth in the sense of

a predominance of later over earlier flows affects any

single period beta in a negative direction. The argument

rests upon the degree of correlation between the market's

cash flow at t = 1 and the cash flow from the project at

t = 1,2,.... As a statistical hypothesis, one would expect

later cash flows at t = 1 to have a lower statistical depen-

dence on the aggregate of all cash flows than earlier flows.

Thus, the degree of risk resolution over the given initial

period, captured by this correlation, would be lower than

for the earlier more highly correlated with the t = 1

market cash flow. One can view the cash flow dependence on

the t = 1 market flow as a decay process as we move forward

in time. Clearly, this is a purely statistical argument

reflecting the effect of technical factors on single period

betas, but it must be remembered that the CAPM single period

beta is a statistical concept. In a multiperiod setting in

which growth takes place, one is looking at single period

betas calculated over a given period.

To make the model described here more precise we con-

sider in detail the multiperiod valuation model proposed by

Stapleton and Subrahmanyam [1979] as an alternative to the

Myers and Turnbull [1977] model. The advantage of this

model is that it avoids the assumption of a particular

dependence structure of cash flows.

Such an assumption was made by Myers and Turnbull in

that expectations of project cash flows were assumed to be

generated by a single-index model. Their statement of this

are equations MT-2 and MT-3 (p. 322),

Xt = E(Xl t_1) (1+t) MT-2

where t is a zero-mean noise term expressing the forecast

error as a proportion of the expectation based upon past


The behavior of 6t is postulated by

t = bt + t' MT-3

where It represents unanticipated changes in some general

economic index and b represents the sensitivity of 6

to changes in I .

Further, the Stapleton and Subrahmanyam model presented

here does not make arbitrary assumptions concerning the

market prices of risk but derives them endogeneously.


(1) Investors are expected utility of terminal wealth
maximizers where their utility functions are of
the constant absolute risk aversion class.

(2) Firms generate cash flows Xt which are jointly
normally distributed.

These two assumptions are sufficient to guarantee that

the future market prices of risk are non-stochastic. Further

assumptions are:

(3) No debt financing is employed.

(4) Limited liability does not apply.

(5) Future one period interest rates are known with
certainty at t = 0.

Given these further assumptions, it follows that derived

future stock prices are normally distributed. Given the

non-quadratic assumption on utility functions, normality is

required to apply the single period CAPM.

Considering the simple case of a two period model

(which can be generalized to any number of periods) the

development proceeds as follows.

The investor's multiperiod optimal portfolio problem is

solved recursively using dynamic programming. For the two

period model considered here this problem can be formulated

in the following manner.

At time t = 1, the individual investor wishes to maxi-

mize his expected utility of final wealth W2 by choosing a

portfolio of holdings of the cash flows at t = 2 of all

firms in the market. This portfolio can be described by a

vector {Z k2 of holdings of the cash flows {X }. Given the

investors utility function of the CARA class:

U(W2) = -a exp(-aW2),

the problem is a constrained maximization problem,


u(W IXI) = max E[u(W2)] = max E[-ai exp(-a W2)], (2.1)
Zi 7i
12 12

subject to

W = 2 P + M (2.2)
1 12 12 1

W2 = M2 + 2 X2 (2

where M1 is the amount of risk-free lending undertaken at

t = 1, r2 is the period 2 riskless interest rate, and P12 is

the vector of t = 1 prices of the X2 cash flows. The

solution to the maximization problem in equations (2.1),

(2.2) and (2.3) is the vector defined by

2 = 21[E(X2 Xl) r2P12 X], (2.4)


S= [cov(Xxk Xj kj (2.5)

is the conditional (on the cash flow at t = 1 vector)

variance-covariance matrix. It is assumed here that the

characteristics of the state of the world 1 relevant for

expectations of the t = 2 cash flows are summarized in the

t = 1 cash flows of all firms in the economy X1. This

allows us to replace the state of the world information set

Sat t = 1 by the cash flow vector of outcomes.

The solution in (2.4) and (2.5) is then used to gener-

ate equilibrium prices P12 X1 conditional on X1 at time

t = 1 of the cash flow vector X2. This is simply a matter

of equating supply to demand. Since the Z must aggregate
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)

Solving this yields

L- (1E( (2.7)
21X r [E(X21X1) -


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)


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


non-state dependent exponential function of W1 alone. But

we have assumed r2 to be non-stochastic, and 12 will be non-

stochastic since it depends only on the coefficients of

absolute risk aversion of all individuals: a known con-

stant. The assumption of CARA utility functions then guar-

antees the non-stochastic character of the market price of

risk X2. Further, 02 is non-stochastic, i.e. independent of

Xl, as a consequence of the assumed joint normality of the

{Xk} [Anderson, 1958]. Thus, u(W1|X1) = u(W1) is a non-

state-dependent utility function as is required for the

period-by-period application of the CAPM [Fama, 1970].

Essentially, since r2 and X2 are non-stochastic by assump-

tion and a is independent of wealth, the only source of

state dependence is through Q2' the conditional covariance

matrix which represents risky investment opportunities. The

assumption of joint normality of all the cash flows in the

economy is designed to rule out the state dependence of the

(risky) investment opportunity set. One should note that

exponential utility and joint normality are sufficient

conditions to avoid state dependence of the derived utility


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

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

The solution to equations (2.11), (2.12) and (2.13)

are the vectors

Z0 = (1/ar2) Q [E(X1) rlP01] (2.14)


where Q1 is the variance-covariance matrix of cash flows X1

and prices P 12

S = 1 (2.16)

F = [cov(X Xk)]

G = [cov(X P )]
1 12
H = [cov(P Pk P
12 12

Equilibrium prices (at t=O) are

P01 = /r [E(X) X 1]
011 1 1 1


P02 = 1/rl [E(P12) X1i] (2.18)


X = (2.19)

The relationship that plays a central role, between X

and 12 is simply that

X1 = X2r2. (2.20)

That is, the current market price of risk equals the future

price compounded at the future risk-free interest rate.

These results are used now to derive the t = 0 equilib-

rium price of the t = 2 cash flow of firm j, X2 as a certain-

ty equivalent in terms of the period 2 market price of risk

X2 [a simplification allowed by equation (2.20)], the

period 1 and period 2 risk-free rates, rl and r2, and

covariances between the cash flow at t = 2, X and the
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"

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