Systematic risk and market power : an application of Tobin's q

BEBR

FACULTY WORKING PAPER NO. 1144
College of Commerce and Business Administration
University of Illinois at Urbana- Champaign
April, 1985

Systematic Risk and Market Power
An Application of Tobin's q

K. C. Chen, Assistant Professor
Department of Finance

David C . Cheng
University of Alabama

Gailen L. Hite
Southern Methodist University

Abstract

We investigate the relationship between systematic risk and market
power measured by Tobin' s q, the ratio of market value to replacement
cost. We demonstrate there is a one-to-one relationship between the
Tobin' s q ratio and the S&T measure of market power. Our theoretical
model predicts that as a firm's market power increases the systematic
risk will, ceteris paribus, decrease. Our empirical results ultimately
confirm this negative association as predicted by S&T.

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Systematic Risk and Market Power:
An Application of Tobin's q

I. Introduction

One of the most important advances in the field of financial eco-
nomics has been the development of an equilibrium model of security
price determination under uncertainty. This model, known as the
capital asset pricing model (CAPM) , was developed by Sharpe [1964],
Lintner [1965] and Mossin [1966]. The CAPM holds that securities will
be priced in equilibrium to yield an expected return that is a linear
function of the systematic, or non-diversif iable , risk. As originally
developed, the CAPM does not provide a direct linkage between a firm's
systematic risk and the underlying microeconomic variables of the
firm, e.g., demand uncertainty, input mix, market power, etc.

The lack of a well-defined model of the determinants of systematic
risk has hampered the investigation of industrial organization econo-
mists in their ability to interpret rates of return across industries
with different market structures. It is difficult to determine how
much of the return variation is explained by systematic risk differen-
tial and how much is attributable to the existence of market power.
Aggravating this problem is the fact that systematic risk itself may
be influenced by market power. In fact, the work of Hurdle [1974],
Melicher, et. al. [1976], Sullivan [1977, 1982], and Curley, et. al.
[1982] have produced mixed results as to the direction and magnitude
of the relationship between systematic risk and market power.

One of the difficulties with the empirical work stems from the
lack of a precise measure of market power. Typically, concentration

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ratios have been used as proxies for market power but the difficulty
of inferring the existence of economic rents from mere concentration
ratios is well-known. Thus, the empirical work to date sheds little
light on the relationship between systematic risk and market power.
One of the purposes of this paper is to develop and test an integrated
model of firm decisions that links systematic risk to a well-defined
measure of market power.

Early attempts at integrating systematic risk and firm variables
may be found in Thomadakis [1976], Hite [1977], and, most notably,
Subrahmanyam and Thomadakis [1980]. While these papers vary widely in
approach and emphasis, there is general agreement that for a given
level of cashflow risk, greater market power results in lower systematic
risk. As Subrahmanyam and Thomadakis [1980, p. 447] state, "Thus,
irrespective of the source of uncertainty, monopoly power unambiguously
reduces beta."

In this paper we model the production and output of a quantity-
setting firm facing stochastic demand. We start with the Subrahmanyam
and Thomadakis (henceforth, S&T) model and develop a precise form for
the relationship between systematic risk and market power. We note
that if a firm is earning economic rents, these will be capitalized
into the market prices of the firm's outstanding securities. The
existence of positive rents would result in the market value of the
firm exceeding the replacement cost of its capital stock. Using the
ratio of market value to replacement cost, a ratio commonly referred
to as "Tobin's q," we have a measure of economic rents that can be

mo

-3-

re easily estimated than the S&T measure of market power. We demon-
strate there is a one-to-one relationship between the q-ratio and the
S&T measure of market power. Furthermore, we confirm the negative
relationship between market power and systematic risk hypothesized by
S&T.

Tobin introduced the q-ratio in an attempt to explain aggregate
investment behavior in the economy. Thus, using q to measure economic
rents represents an extension of Tobin's concept to microeconomic
analysis. Recently, Lindenberg and Ross [1981 J have used q-ratios to
measure economic rents and market power. The current paper extends
the Lindenberg and Ross insight to investigate the relationship between
market power and systematic risk. Our theoretical model predicts that
as a firm's market power increases the systematic risk will, ceteris
paribus, decrease. Our empirical results ultimately confirm this
negative association as predicted by S&T.

The rest of the paper is organized as follows. Section II pre-
sents the standard optimization problem of the firm in an economy in
which uncertain return streams are priced according to the single-
period CAPM. We derive an explicit expression for the firm's systematic
risk as a function of its q-ratio. The third section describes the
sample and the estimation of the variables for the empirical tests.
Section IV contains our findings followed by a short summary.

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II. The Model

A. Firm Equilibrium

Following the S&T approach, we consider a firm facing Che follow-
ing stochastic demand function:

P = P(Q)(l+e) (1)

Here and throughout the paper a indicates a random variable. The
function P(Q) is assumed to be a negatively sloped function but e is
a random demand variable which is independent of the quantity of out-
put with E(e) = 0 so that E(P) = P(Q). The marginal revenue function
is assumed to be given by

MR(Q) = (l-u)P (2)

where u is the reciprocal of the price elasticity of demand. Note that
(1-u) is the Lerner index of monopoly power. The firm makes its out-
put decision before the price is known, i.e., before e is revealed.
For simplicity, assume that the firm has a constant proportions
production function calling for labor L and capital K inputs given by

L = aQ (3a)

and

K = bQ. (3b)

It is assumed that capital is exhausted in the production process.
The net cashflow of the firm after paying the wage rate w on L
units of labor is given by

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Y = PQ(l+e) - wL. (4)

Note that we assume that the wage rate w is deterministic and paid at

end of the period when output is sold. According to the cashflow

version of the CAPM the value of the firm V is the present value of

the net cashflow given by

E(Y) - A cov(Y,R )
v= r— = =- (5)

where A is the market price of systematic risk, s is the covariance

em

of e with the market portfolio, R is tne uncertain rate of return on

m

the market portfolio, and r is the risk free rate of interest. If we

define 4 = E(l+e) - As as the certaintv equivalent of (1+e) , then

era

since e has a zero expected value tne certainty equivalent price is

simply E(P(l+e)) - As = 4>P. The value of the firm simplifies to

em

V = W - wL (5')

1 + r K J

= <frP - Wa . Q

1 + r

Note that in the normal case we would expect firms to have positive

systematic risk so with s > 0 the certainty equivalent term 5 would

em

be less than one. Consequently, uncertain revenue is valued at less
than its expected value due to the discount for systematic risk.

The goal of the firm is to maximize its net present value, i.e.,
the difference between its market value and its capital expenditure,

NPV = V - K (6)

-6-

Substituting the production requirenents for L and K in (3) and de-
fining c = wa + (l+r)b as the constant marginal and average cost of
production we have the net present vaiue as

<i>PO - cQ
NPV = I + r ' (7)

The first and second order conditions for maximizing (7) are
3 NPV (i(l-u)P - c

3Q 1 + r

= 0 (8)

and

3 NPV 6(l-u) dP

— = 1 + r ' dQ < ° (9)

3Q

respectively. Downward sloping demand is sufficient to assure that

(9) holds. Re-arranging (3) we have

(fr(i-u)p = c (yJ)

which states that the optimum is where the firm equates the certainty
equivalent marginal revenue and marginal cost. Thus, the firm sets
output such that the certainty equivalent price is

<J>P = ~— (10a)

1 - u

or such that expected price is

P = 777 — r (10b)

Note that 9P is the certainty equivalent price and c is average and
marginal cost. If the firm possesses market power then u > 0 and the
certainty equivalent price will be set above marginal cost c.

-7-

To see the meaning of u under uncertainty, we can solve (10) for
u Co yield u = ($P-c)/<£P. Thus u represents in certainty equivalent
form the spread between price and marginal cost as a proportion of
price. In the competitive case, this spread should be non-existent
and u would be zero. When the firm possesses monopoly power, a posi-
tive spread would exist indicating a positive value of u.

These equilibrium relationships can be used to express the system-
atic risk of the firm as a function of the firm's market: power u.

3. Systematic Risk

According to the CAPM, systematic risk is measured by the relation-
ship between the rate of return on the firm's securities and the rate
of return on the market portfolio. Define the rate of return on the
firm as R where

Then the firm's systematic risk, or g, is given by

2
$ = cov(R,R )/s~

ra m

= cov(Y,R )/(Vs2) (11)

m m

where cov(R,R ) is the covariance between the rates of return on the
m

2

firm and the market portfolio and s** is the variance of the return on

m

the market portfolio. This covariance term may be simplified to

cov(Y,R ) = PQs

m em

cs
= — • Q (12)

<>(l-u)

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from (10b). If we substitute (10a) into (5'), then the value of the

firm can be written as

c

- wa

V - '"x + r • °- (13)

Finally, using (12) and (13) we can express (11) as

i s

1 + r era c ....

d> 2c- wa(l-u)

in

Equation (14) shows that 8 is negatively related to market power u
as S&T showed earlier. The difficulty with this relationship is that
market power as measured by u is extremely difficult to observe from
publicly available data. To be useful, S&T's insight must be trans-
lated into a form that can be estimated from market data.

C. Tobin's q

Recall that Tobin's q is defined as the ratio of the firm's market
value to the replacement cost of its capital stock. A value of q

exceeding unity implies that a firm is earning above a normal rate of

return on its capital. To see this, consider a firm that has no

monopoly power, i.e., u = 0. Then from (10) we see that the optimum

calls for 6p = c, i.e., the certainty equivalent price equals marginal

cost. The value of the firm in (13) is simply

V = (c-wa)Q/(l+r) = bQ

= K

Thus, Tobin's q would be 1.

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In the more general case, we have ^P(l-u) = c at the optimum
yielding

q = V/K

- 1 + nLh • t^— • <15)

(H-r;b 1 - u

Thus, for u > 0 we see that q > 1. Furthermore, q is positively re-
lated to u. Being able to estimate q obviates the difficulties asso-
ciated with trying to estimate u.

If we solve (15) for u as a function of q we get

u - q ' ' (16)

wa

TlTrTb + q
which can be substituted into (14) to give
1 + r • Spti r -, wa 1

_em

s

s-^ 2" !i+TT^Tb-f) <17>

m

Here it can be seen that 0 is negatively related to q. That is, there

is a one-to-one positive relationship between u and q, and since 3 and

u are negatively related, then 0 and q will also be negatively related.

In fact, we can compare this general result to the special case of

the competitive firm for which u = 0 and, therefore, q = 1. In that

case, the systematic risk 3 is given by

c

s
1 + r era , wa

6c=~ r-u + o^- (18)

v s

m

Finally, we can write the systematic risk, of the non-competitive firm
as

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1 + r era • wa , . 1 , . , _ .

&'*c-— r or?)b-[i-7!- (19)

s
m

The departure of (3 from 3 is negative and tlie spread increases in

c

absolute magnitude with the size of q.

The expression for 8 in (17) shows that systematic risk, is depen-
dent on both q and s among other variables. The first of these re-
em

lationships shows that 6 is negatively related to q. Thus, a firm
that has market power as measured by q will, ceteris paribus, have
lower systematic risk. The second relationship snows that 3 is posi-
tively associated with s . Note that s shows the covariance of

em era

the stochastic demand term with the return on the market portfolio.
Thus, if the firm's stochastic demand is highly correlated with the
demand for the output of other firms in the market portfolio, then the
firm's systematic risk will, ceteris paribus, be higher. In other
words, firms with product demands that are highly correlated with
sales of other firms in the economy will have higher 6's and higher
risk preraia in the CAPM context.

Our model indicates that systematic risk as measured by 3 should
be negatively related to market power and positively related to the
covariance of the firm's sales with the rest of the economy. To test
these hypotheses requires that (17) be arranged into a more convenient
form that can be estimated from available public data.

U. Estimable Forms

The difficulty with (19) in its current form is that s , or the

em

covariance of price with the market portfolio, is not directly

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observable. Using aggregate sales revenue for Che firm will overcome
this problem. That is, re-write (12) as

Cov(Y,R ) = Cov(PQ(l+e),R )
rn m

= Cov(S,R ) (20)

m

where S is the uncertain total revenue or dollar volume of total
sales. Then from the definition of c we can express the value of the
firm in (13) as

C • 11

= 'TI^'r^I + lllc (21)

where bQ is the capital requirement in (3b). Then if we substitute

(20) and (21) into (11) we have systematic risk expressed as

Cov(S,R )
8 = a (22)

[1 +-^— • -^-JKs2
(l+r)b 1 - u m

Finally, we can substitute for u in terms of q from (16) to give

0=1-. Cov(|, R ) • -. (23)

2 K 21 q
s
m

Here, the ratio S/K is the capital turnover ratio or the sales revenues
generated per dollar of investment in capital. This is commonly re-
ferred to as the "asset turnover ratio."

The final difficulty in estimating the model is that the system-
atic risk in (23) is a "firm" 3. Unfortunately, most major corporations

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have both debt and equity outstanding so there is no single security
that allows us to estimate the overall firm 3. Instead, we are left
with estimating the 0 for a security instead of the entire firm.
Specifically, we estimate the 3 for the equity of the firm which re-
flects both overall firm risk and the additional effects of financial
leverage. To account for the effect of deot usage on the equity 3 of
the firm we make an adjustment developed by Hamada [1972].

Denote the total value of the firm as the sum of the values of the
outstanding securities

V = V + V,. (24)

3 L

where V and V are the respective values of the bonds and the equity
B E

securities. Then as Hamada shows, the 3 for the equity can be ex-
pressed as

3£ - B("VJL). (25)

£

In other words, the 3 for a levered firm's equity is the overall firm

3 multiplied by the ratio of firm value to equity value.

Finally, we can express the equity 3 as

1 S ~ 1 VR

g^ -A- • Cov(£, R) . - . (1 +-£). (26)

E 2 K m q V_
s n E

m

S
The first firm-specific term, Cov(— , R ), represents the systematic

K. m

"business risk" of the firm as measured by the relationship of the
firm's capital turnover in relationship to the rest of the market
portfolio. The second terra, 1/q, is the inverse of the firm's market

-13-

power as measured by Tobin's q. The final term, V'/V , reflects Che
capital structure or leverage of the firm. This multiplicative rela-
tionship indicates that the firm's systematic risk, is positively re-
lated to business risk and leverage and negatively related to the
firm's market power.

Furthermore, as noted by Bowman [1980], the risk class concept of
Modigliani and Miller [1958] points to the possibility of differential
business risk across industries. That is, a portion of beta could be
explained by the use of intercept dummy variables (D). In the context
of multiple regression analysis, the intercept dummy variables based
upon industries will capture that portion of beta whicn varies system-
atically between industries. Thus, the model at this point can be
stated functionally as:

S - VR

6£ = f(Cov(~, Rm), q, ^, D).

III. Sample Selection and Measurement of Variables

Because of the technical problem involved in the measurement of
Tobin's q, firms comprising the 1978 Standard and Poor's 400 provide the
initial sample for our empirical analysis. Then, data availability
criteria are imposed to ensure continuous data on the COMPUSTAT and
CRSP tapes for the period from 1969 to 1978. The remaining 116 firms
are then classified according to their two-digit Standard Industry
Classification (SIC) Code. Because of the use of dummy variables in
the model, the inclusion of firms from industries with only a small
number of firms is insufficient. In an attempt to preserve high

-14-

degrees of freedom, the seven largest industry groups are chosen. This
gives a sample of 94 firms distributed according to SIC Code as shown
in Table 1.

Insert Table 1

In what follows we address the measurement issues. First, beta
defined by the CAPM is not directly observable. The market model is
commonly used empirically to obtain a surrogate. We employ 120 monthly
excess returns in the market model so chat the nonstationari ty problem
of beta is reduced to a minimum.

The systematic business risK. variable COV is the covariance
between capital turnover ratio and annual return on the mark.ec port-
folio. This variable is very similar to the accounting beCa used in
the literature.

The firm's q ratio, it is recalled, is the ratio of the firm'
market value to the replacement cost of its assets. The estimation of
q is similar to procedures described in Lindenberg and Ross [1981 j and
Chappell and Cheng [1982, 1984]. It includes adjustments for the
baises induced by inflation in the reported values of property, plant
and equipment, and inventories and by interest rate changes in the
reported value of debt. Details of the estimation procedure are avail-
able from the authors. For the firms in our sample, we have averaged
q ratios for the period from 1969 to 1978.

vB

As shown in section II, the leverage variable -rr~ is measured as
the ratio of market value of debt to market value of common equity.
Since the book value measure of leverage has been intensively used in

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the literature, both market value and book value measures (LM and LB)
are tested. In order to obtain a stable measure, we average the debt
to equity ratio over the ten year period.

IV. Hypothesis and Empirical Results

As shown in section II of this paper, market power as measured by
Tobin's q is theoretically negatively correlated with beta. Kence ,
the primary null hypothesis is:

H,-> : Market power as measured by Tobin's q is not
statistically correlated with market beta.

A multiple regression analysis of the full model presented in (26)
constitutes the principal test of the null hypothesis. Since all three
variables, COV , q, and LM (or LB) are in a multiplicative relationship
with 3„, we take natural logarithms on all variables and add industrv
dummy variables in the following multiple regressions:

6

In 0„ = an + a. In COV + a~ In LM + a, In q + Z a.x,D. + e (27)
L 0 1 2 3 l+J l

i=l

where a, and a2 are hypothesized to be positive and a-, negative. The
significance of cc. is a direct test of the null hypothesis.

To investigate the hypothesized relationship further, we first exa-
mine the correlation matrix for the variables used as shown in Table 2.
It is found that there is a problem of raulticollineari ty between market

power and leverage measures, with a correlation coefficient of -0.526.

2
This finding has been documented in industrial organization studies."

Insert Table 2

-16-

Furthermore, examination of the latent vectors and roots of Che
nine independent variables in equation (27), which are presented in
Table 3, indicates more precisely that these variables are highly
correlated and that the market power and leverage variables are the
main sources of multicollinearity . Because the presence or multi-
collinearities in a data base generally inflates ordinary least squares
(OLS) estimator variances, the OLS estimates are very unstable, even
though unbiased.

The principal component regression (PCR) technique is used to
alleviate the problem of multicollinearity. Gunst and Mason [198UJ
show that principal component coefficient estimates eliminate raulti-
collineari ties from the OLS estimators, thereby greatly reducing esti-
mator variance while attempting to introduce only a small amount or
bias. Principal component estimates are biased, but more stable with
smaller variances than the OLS estimate. We first transform the inde-
pendent variables into an equal number of components that are linear
combinations of the independent variables and then eliminate the com-
ponents associated with very small (near zero) latent roots. The
critical value (the small cutting-off value) for the latent roots

should be large enough to eliminate multicollinearity, yet small enough

4
to minimize the bias resulting from the omitted components.

The PCR results of equation (27J are shown in Table 4. The coef-
ficient of the market power variable, a~, is negative and significant
at 5 percent level. Therefore, the null hypothesis is rejected.
Market power as measured by Tobin's q indeed is statistically corre-
lated with market beta. This finding is important in the sense that it

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supports Sullivan [1973, 1982] in that market power reduces a firm's
systematic risk and thus its cost of equity capital. Most importantly,
this result supports the hypothesis testing derived from an integrated
theoretical model, which was absent in Sullivan and Curley, et . al .

Furthermore, our results confirm the existence of other variables
in addition to market power as determinants of systematic risk. First,
the coefficient of the systematic business risk term is positive
although not statistically significant. Second, the leverage variable
is positive and significant at the 1% level, a result consistent with
earlier findings in the finance literature. Finally, the industry
dummy variables play a significant role in explaining inter-industry
differences in betas. In fact, the industry dummy variables may pick
up systematic risk differences between industries more effectively than
the business risk term, COV. Nevertheless, firm-specific variables
explain a significant portion of individual firm betas.

Insert Table 3

Insert Table 4

VI. Summary

In this paper we investigate the relationship between a firm's sys-
tematic risk and it's market power as measured by Tobin's q, the ratio
of market value to replacement cost of the capital stock. Starting
from the earlier theoretical work of Subrahmanyam and Thomadakis
[ 1 980 J , we develop a testable model relating the firm's beta to its
q-ratio. Our model predicts and our empirical results confirms that

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beta is positively related to business risk as measured by the sensi-
tivity of firm sales to aggregate economy sales, positively related Co
financial leverage as measured by debt-equity ratios, and negatively
related to market power as measured by Tobin's q.

Our results would seem to have several implications for the study
of industry structure and performance. First, it is important to under-
stand the underlying cause of the relationship between systematic risk
and market power. Note that by virtue of using the capital asset
pricing model as the description of the underlying security market
equilibrium we are assuming that the firm is a price-tatcer in the capi-
tal market. In other words, the existence of market power in the pro-
duct market in no way produces market power in the securities market.
The lower systematic risk and resultant lower cost of equity capital,
ceteris paribus, for a firm with a q-ratio in excess of unity arises
because it generates higher expected dollar returns per unit of product
price risk, s , and not from any non-competitive access to the capital
market. Thus, it is the higher expected return per unit risk that pro-
duces a lower beta and a lower cost of equity capital.

Second, it should be noted that assuming that there exists a common
cost of capital for an entire industry is not appropriate even as an
approximation. While our empirical results indicate that there is a
strong industry effect, or "business risk" component in beta, it can
also be seen that individual firm variables, in particular leverage and
market power, have significant influences on individual firm betas.
Thus, there is good reason to believe that assigning a single "industry
beta" to individual firms within an industry will obscure important
cross-sectional differences between firms.

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Footnotes
1
"For example, see Curley, Hexter, and Choi [1982].

See Tobin and Brainard [1968, 1977] and Tobin [1969, 1978].
2.

3
The critical value for the latent roots in the PCR is 0.2.

4
For more detailed information on principal component regression

analysis, see Mansfield, Webster, and Gunst [1977], and Gunst and

Mason [1980].

Because the PCR results with book value leverage measure, LB, are
analogous, we do not report them here.

We also include two ad hoc variables in the test, growth (the
geometric mean annual rate of growth in sales from 1969 through 1978)
and size (1978 sales in natural logarithm). The coefficient of the
market power variable is still negative and significant at 10 percent
level.

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-21-

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D/21

Table 1

Firms Distribution by SIC Code

2-Digit

SIC Code Industry Description Firr.s

20 Manufacturing-Food 15

28 Apparel and other Finished Products 11

29 Petroleum Refining 3

35 Manufacturing-Machinery 30

36 Electrical & Electronic Machinery 3

37 Transportation Equipment 14

38 Measuring and Analyzing Instruments

94

Table 2

Correlation Matrix

P£ COV LB LM q

6E 1.000 0.234 0.273 0.282 -0.031

COV 1.000 0.251 0.095 -0.220

LB 1.000 0.594 -0.414

LM 1.000 -0.526

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