V- "
i' '•
EPA-230/3-76-001
November 1975
ESTIMATION OF THE COST OF CAPITAL
FOR MAJOR UNITED STATES INDUSTRIES
WITH APPLICATION TO POLLUTION CONTROL INVESTMENTS
Contract No. 68-01-2848
Environmental Protection Agency
Office of Planning and Evaluation
Economic Analysis Division
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ESTIMATION OF THE COST OF CAPITAL
FOR MAJOR UNITED STATES INDUSTRIES
WITH APPLICATION TO POLLUTION-CONTROL INVESTMENTS
Dr. Gerald A. Pogue
4 Summit Drive
Manhasset, New York 11030
November 1975
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ESTIMATION OF THE COST OF CAPITAL
FOR MAJOR UNITED STATES INDUSTRIES
WITH APPLICATION TO POLLUTION-CONTROL INVESTMENTS
TABLE OF CONTENTS
Section
Paee
PREFACE x
PART 1. THEORY
I INTRODUCTION 1-1
(a) Background and Purpose of Study ...... 1-1
II ORGANIZATION OF THE REPORT - 1-3
PART 2. THEORY: COST OF CAPITAL CONCEPTS
I INTRODUCTION 2-1
II THE COST OF CAPITAL FOR AN "ALL-EQUITY" FIRM . . 2-2
(a) Definition 2-2
(b) Decision Rules 2-5
III SECURITY VALUATION THEORY FOR ESTIMATING
THE COST OF EQUITY CAPITAL 2-8
IV THE WEIGHTED AVERAGE COST OF CAPITAL 2-15
(a) The Effects of Leverage on the Market
Value of the Firm 2 15
(b) The Weighted Average Cost of Capital . . . 2-24
(c) Relationship Between p* and Leverage . . . 2-3(
ii
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TABLE OF CONTENTS (Continued)
Section Page
PART 2 — APPENDIX A: DEVELOPMENT OF
SECURITY VALUATION MODELS !-41
FOOTNOTES FOR PART 2 2-45
PART 3. THEORY: RISK AND RETURN CONCEPTS
1 INTRODUCTION 3-1
II A BRIEF INTRODUCTION TO THE THEORY OF
RISK AND THE REQUIRED RATE OF RETURN 3-3
(a) The Risk of Individual Common Stocks ... i-3
(b) Basic Risk-Return Concepts: The Capital
Asset Pricing Model 3-13
III RISK AND THE REQUIRED RATE OF RETURN 3-18
IV THE RELATIONSHIP BETWEEN REAL AND
FINANCIAL MEASURES OF RISK 3-49
(a) Empirical Studies on the Real Determinants
of Beta (Through Mid-1973) 3-51
Footnotes for S. C. Myers1 Article .... 3-78
References for Excerpt from
S. C. Myers' Article 3-80
(b) Recent Studies 3-83
(c) Summary 3-88
FOOTNOTES FOR PART 3 3-92
PART 4. ESTIMATION
I INTRODUCTION 4-1
II METHODOLOGY 4-2
iii
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TABLE OF CONTENTS (Continued}
Section Page
(a) Combining the Cost of Capital and
Risk-Return Models 4-2
(bj Estimation Equation (1-3) 4-5
(c) Model 4—The Long-Run Risk-Return Line. . 4-9
(d) Cross-Sectional Regression Procedures . . 4-11
(ej Combining the Results of Models
1 through 4 4-14
III DEFINITION OF THE PRIMARY AND SECONDARY
SAMPLES 4-16
IV MEASUREMENT OF EQUITY RISK 4-19
(aj Calculation of Betas 4-19
V COST OF EQUITY CAPITAL: EMPIRICAL RESULTS . . 4-24
(a) Variable Estimation for Cross-Sectional
Regression Models 1, 2, and 3 4-24
(b) Estimation of Weights for Cross-
Sectional Regression Equations 4-27
(c) Weighted Regression Results 4-29
(d) The Cost of Equity Capital: Results. . . 4-35
VI THE WEIGHTED AVERAGE COST OF CAPITAL:
EMPIRICAL RESULTS 4 42
(a) The Cost of Debt and Preferred
Stock Capital 4-42
(b) The Capital Structure Proportions .... 4-44
(c) The Weighted Average Cost of Capital . . 4-46
VII FORECASTING THE WEIGHTED AVERAGE COST
OF CAPITAL 4-52
FOOTNOTES FOR PART 4 4-58
IV
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TABLE OF CONTENTS (ContinuedJ
PART 5. APPLICATIONS
Section Page
I INTRODUCTION 5-1
II CAPITAL BUDGETING 5-2
(a) The Weighted Average Cost of Capital . . 5-2
(c) The All-Equity Cost of Capital 5-7
III MEASURING THE FINANCIAL IMPACT OF INVESTMENTS
IN POLLUTION-CONTROL DEVICES 5-12
(a) A Financial Impact Index 5-12
(t>) The Risk and Financing of Pollution-
Control Investments
(cj The Offset Rate of Return 5-20
(d) Example 5-23
FOOTNOTES FOR PART 5 5-28
REFERENCES R-l
v
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ESTIMATION OF THE COST 01-' CAPITAL
FOR MAJOR UNITED STATliS I NDUSTR] I-S
WITH APPLICATION TO POLLUTION-CONTROL INVESTMENTS
LIST OF TABLES
PART 2. THEORY:' COST OF CAPITAL CONCEPTS
Table No. . Page
2-1 The Relationship Between the Earnings Price
Ratio and Reinvestment Rates of Return .... 2-12
2-2 Financial Leverage—Example 2-19
2-3 Necessary and Sufficient Conditions for
Cost-of-Capital Formulas 2-35
2-4 Calculation of the Weighted Average Cost
of Capital: Example 2 39
PART 3. THEORY: RISK AND RETURN CONCEPTS
3-1 Risk Versus Diversification: Randomly
Selected Portfolios 3-5
PART 4. ESTIMATION
4-1 Definition of Sample Sizes
Number of Companies 4-18
4-2 Stock Risk and Capital Structure Data
Primary Group Averages ... 4-22
4-3 Stock Risk and Capital Structure Data
Secondary Group AVerages 4-23
VI
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LIST OF TABLES (Continued)
Table No. Page
4-4 Weighted Regression Coefficients —
Model 1 4-30
4-5 Weighted Regression Coefficients —
Model 2 4-31
4-6 Weighted Regression Results —
Model 3 4-33
4-7 Model 3 Iteration Summary 4-34
4-8 Parameters for Model 4 4-36
4-9 Relationship Between Cost of Equity Capital
and Beta Resulting from Combining
Models 1 through 4 ' 4-37
4-10 Cost of Equity Capital (% Per Year):
Year End 1971 1974 — Primary Groups 4-38
4-11 Weighted Average Cost of Capital (I Per Year):
Year End 1971 - 1974 — Primary Groups 4-48
4-12 Projected Weighted Average Cost of Capital
(I Per Year): Year End 1975 - 1984 — Primary
Groups 4-55
4-13 Weighted Average Cost of Capital (I Per Year)
— Secondary Groups 4-56
PART 5. APPLICATIONS
5-1 Possible Risk and Capital Structure Consider-
ations for Pollution-Control Investments .... 5 17
5-2 Adjusted Present Value for Example Pollution-
Control Investment 5-21
5-2 Example: The Impact of Pollution-Control
Investments on Firms of Different Size 5-25
VII
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ESTIMATION OF THIi COST OF CAPITAL
FOR MAJOR UNITED STATES INDUSTRIES
WITH APPLICATION TO POLLUTION-CONTROL INVESTMENTS
LIST OF FIGURES AND EXHIBITS
Figure/Exhibit
Number Page
2-1 Impact of leverage on the value of
the firm , 2-25
2-2 Effects of financial leverage on costs
of debt and equity financing and the
weighted average cost of capital 2-37
PART 3, THEORY: RISK AND RETURN CONCEPTS
3-1 Systematic versus unsystematic risk . . . 3-7
3-2 Calculation of a security's market
sensitivity index from past data 3-9
3-3 Results of Black, Jensen and Scholes
Study—Average monthly returns versus
systematic risk for the 35-year period
1931 1965 for ten portfolios and the
market portfolio 3-15
3-4 Scatter diagram: market beta versus
accounting beta 3-86
PART 4. ESTIMATION
4-1 Normalized regression weights
versus beta 4-28
4 2 Pooled estimated cost of equity capital
versus beta 4-40
VI 11
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LIST OF FIGURES AND EXHIBITS (Continued)
Figure/Exhibit
Number
4-
i _
4-
3
4
5
Error
versus
Pooled
versus
Error
estima
range
beta
estir
beta
range
te vei
for
nated
for
rsus
pooled
total
pooled
beta .
equity estimate
cost of capital
total cost
Pa
4-
4-
4-
ge
41
49
51
PART 5. APPLICATIONS
5-1 Plot of impact index versus firm size
for Table 5-3 example 5-27
IX
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PRIiFACH
In this report I have attempted to draw together a
number of elements of capital markets and corporate finance
theories in order to derive new procedures for estimating
the costs of capital for major United States industries. The
report presents no new theory; instead, the major task lias
been to combine and apply existing theories to practical esti
mation and decision problems.
The major part of the report is concerned with the cost
of capital estimation methodology and empirical results. In
addition, I have begun the development of an analytical framework
for analyzing the economic impact on corporations of mandated
investment programs in pollution-control devices. My approach
combines the cost of capital estimates with capital budgeting
procedures to produce Financial Impact Indices of the effect
on corporations of pollution control requirements. However, no
attempt has been made to present a detailed analysis of this
question, but rather to point the direction in which the
analysis might fruitfully proceed.
The report deals with many concepts from finance theory,
some of which are relatively new. I have tried to make the
report reasonably self contained by including two lengthy parts
dealing with the relevant theoretical background; Part 2 deals
with the cost of capital theory, Part 3 with modern portfolio
x
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and capital markets theories. However, even these discussions
assume some prior knowledge of financial management concepts.
It is assumed that the reader has had a course in financial
management of about the level of the ames C. Van Home text
(Financial Management and Policy, Second Edition, Prentice Hall,
1971J or the J. Fred Western and Eugene Brigham text (Managerial
Finance, Fourth Edition, Holt Rinehart § Winston, 1972). For
those without this background, particularly diligent reading of
Parts 2 and 3 will be necessary. In addition, since the cost
of capital estimation procedures use econometric models, some
knowledge of basic econometrics and statistics would be helpful.
A particularly good source is the J. Johnston text (Econometric
Methods, Second Edition, McGraw-Hill, 1972).
In the conduct of the research and the writing of this
report, I have relied heavily on the writings of my former
colleague, Professor Stewart C. Myers of MIT. Professor Myers
has done much pathbreaking research in the corporate-finance
area. I have relied particularly on his writings in the cost
of-capital and capital-budgeting areas. This research has
also benefited from my private conversations with Professor
Robert Litzenberger of Stanford University.
Finally, I would like to express my appreciation to my
research associate Mai Pogue, to my research assistants C. Nguyen
and M. Castellino, and to Anna Beliveau who has diligently typed
the drafts of this report and provided valuable editorial
assistance.
Gerald A. Pogue
Manhasset, New York
November 1975
xi
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ESTIMATION OF THE COST OF CAPITAL
FOR MAJOR UNITED STATES INDUSTRIES
WITH APPLICATION TO POLLUTION-
CONTROL INVESTMENTS
PART 1. INTRODUCTION
Dr. Gerald A. Pogue
4 Summit Drive
Manhasset, New York 11030
November 1975
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I. INTRODUCTION
(a) Background and Purpose of Study
The cost of capital has long be-^n recognized as an
important ingredient in financial decision-making. In capital
budgeting, for example, it permits the condensation of a stream
of future cash flows into a single figure of merit for an
investment project or strategy—its discounted present value.
The cost-of-capital concept is prominent in both the literature
and practice of finance: textbook definitions abound; the term
is common in management conversations.
The application of the concept in practice, however, is
not at all simple or straightforward. While much agreement
exists regarding the general concept, less exists as to how
the cost of capital should be measured and applied in practice.
While part of the difficulty lies in the abstraction of the
usual definitions, the major problem stems from the lack of
simple and reliable estimation procedures.
Most of the past academic studies of cost of capital have
focussed on public utilities. This was due in part to the
regulated nature of the utility industry (and thus the public
interest and availability of data) and in part to the risk
structure of the industry. As most public utilities face
reasonably similar investment risks, the industry could be
approximately considered as made up of companies of similar
1-1
-------
risk, thus sidestepping the difficult question of how 1 he
cost of capital varied with investment risk.
The purpose of this study is to estimate capital costs
for a broad cross-section of American industries. As such,
it is one of the first empirical studies to explicitly
consider differences in investment risk. To achieve this end,
the study combines traditional security valuation approaches
to estimation of capital costs with rrpdern capital markets
theory to develop new estimation procedures which reflect th'
current state of finance theory.
While the procedures used in this study represent improve-
ments over previous studies, they must 'not be considered as
either infallible or the final word on the subject. Finance
theory is evolving rapidly and new methods for estimating
capital costs will undoubtedly appear. Nevertheless, while
the final answer to the estimation probleip is not available
(and will not be for some time), a great deal is known upon
which to base useful estimates of capital costs.
In conjunction with micro-economic analyses being performed
for and by the Environmental Protection Agency, cost-of-capital
estimates are produced for six basic industry groups: pulp and
paper, chemicals, petroleum refining, iron and steel, non-ferrou:
metals, and utilities. Estimates were made as of the year-ends
1971 through 1974 and projections prepared for each year of the
1975-1984 period. The estimates include the cost of equity
capital and the weighted average cost of capital.
1-2
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II. ORGANIZATION OF THE REPORT
The report is divided into five parts. Part 2 presents
the relevant cost-of-capital concepts which underlie the
estimation procedures. Section I contains a definition of the
cost of equity capital. Section II describes three related
discounted cash flow security valuation models which are used
to estimate the cost of equity capital. Section III deals with
the impact of debt financing on the market value of the firm and
procedures for measuring the weighted average cost of capital.
Part 3 summarizes the necessary capital markets and port-
folio theory concepts. Sections II and III discuss how the risk
of common stocks can be measured, and how risk is related to the
cost of capital. Section IV summarizes the empirical literature
on the relationship between stock-market-oriented risk measures
and the accounting characteristics of firms' assets.
Part 4 presents the cost-of-capital estimation methodology
and empirical results. Section II shoivs how the discounted
cash flow cost of capital models and the risk-return concepts
of Parts 2 and 3, respectively, can be combined to produce
cross-sectional estimation equations. Section III describes
the primary and secondary company samples used in the study.
Section IV presents the equity risk measures for sample firms.
Section V describes the procedures used to estimate the cross-
1-3
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sectional regression variables and presents the cost of
equity capital estimates. Section VI contains the methodology
used to estimate weighted average costs of capital and presents
the results. Finally, Section VII shows how the weighted
costs are likely to change during the 1975-34 period given
estimates of future interest rates.
The purpose of Part 5 is to show how the cost-of-capital
estimates developed in Part 4 can be used for making capital
budgeting decisions in practical situations. Section II
reviews and extends the capital budgeting procedures intro-
duced in Part 2, Section III begins the development of
analytical procedures which can be used for measuring the
financial impact on corporations of investments in pollution-
control devices.
1-4
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ESTIMATION OF THE COST OF CAPITAL
FOR MAJOR UNITED STATES INDUSTRIES
WITH APPLICATION TO POLLUTION-
CONTROL INVESTMENTS
PART 2. THEORY: COST OF CAPITAL CONCEPTS
Dr. Gerald A. Pogue
4 Summit Drive
Manhasset, New York 11030
November 1975
-------
I. INTRODUCTION
The purpose of Part 2 is to present the relevant cost-
of-capital concepts underlying the models and estimation
procedures of Part 4.
Section II contains definitions of the cost of capital.
For ease of exposition it is assumed in Section II that the
firms under consideration are all-equity financed. The defini-
tion is broadened later to include debt financing (See Section
IV).
In Section III three related "discounted cash flow"
security valuation models are discussed. The models provide
the basic framework for estimating the firm's cost of equity
capital.
Section IV deals with the impact of debt financing on
the market value of the firm and generalizes the cost-of-
capital definitions and Net Present Value (NPV) decision rules
to include the effects of debt financing.
2-1
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II. THE COST OF CAPITAL FOR AN
"ALL-EQUITY" FIRM
(a) Definition
The cost of capital for an all-equity-financed company
is the rate of return that investors expect to earn on the
company's securities, as well a,s on al} other securities of
equivalent risk. The expected rates of return on securities
of equivalent risk must be equal. If one security offers a
higher rate of return than other equivalent risk securities ,
then there will be excess demand for this security and excess
supplies of the others. Conversely, a security offering a
/
relatively low return will be in excess supply. Prices rise
when there is an excess demand and fall when there is an
excess supply. Thus prices of equivalent risk securities
will tend to adjust so that they will offer the same ex-
pected rate of return.
The cost of capital is the minimuiji acceptable rate of
return or "hurdle rate" for new investments by the firm.
The logic in developing a cost of capital on an all-equity-
financed firm's investment goes as follows:
1. The firm is one of a class with similar risk
characteristics—Call this class "j".
2-2
-------
2. At any point in time there is a unique expected
rate of return prevailing in capital markets for
this degree of risk—call it R..
3. The share price of the fi: m in question will
adjust so that it offers an expected rate of
return R- to investors.
4. This rate, the shareholders' opportunity cost,
should be the minimum acceptable expected rate
of return on new investment, assuming the projects
under consideration have risk characteristics
similar to currently held assets.
(Suppose the firm undertakes investments offering less than R.;
its shareholders will be worse off than if the firm had fore-
gone the investment and paid an extra dividend, since the
shareholders can always invest at R.. On the other hand,
investors can obtain no more than R. by investing directly,
so the firm should never pass up investments offering more
than R..)
For any stock in risk class j, the cost of equity capital
is formally defined by the relation
Dt+l + Pt+l i
?tJ = 1 + R. (2"1J
where
P . = ex-dividend price of the stock at the end of
period t
2-3
-------
2
D dividends expected during period t+1
P . expected ex-dividend price for t+1
Of course, it is not literally true that everyone has the
same expectations of future return, but for purposes of
analysis this is assumed so that it is permissible to speak of
"the market's" expectations.
The above logic must be extended when the project under
consideration has risk characteristics different from currently
held assets. Suppose the project under consideration has
risk characteristics more like firms in risk class k. For
example, assume a paper company is considering expanding into
the chemical business. In this case, the stockholder's oppor-
tunity cost for this investment is R, , the going rate of return
of firms in risk class k. Thus, the rate R, should be used
to discount projects of this type.
For investments in risk class k, the cost of equity
capital can be formally defined by extending Equation (2-1).
ADt + l + APt + l
t + 1
_ _ _
t 1 + R --
where
APt = the change in the price at time t due to the
adoption of the project in risk class k
ADt = the expected incremental cash flow from
adopting the project
APt+l = cnanSe in price at time t+1 due to the
adoption of the project
2-4
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The cost of capital, as defined above, is relatively
straightforward. The underlying proposition is that at any
point in time securities are so priced that all securities
of equivalent risk (i.e., all securities in a risk class)
offer the same expected rate of return. For a given stock,
however, the basic problem is to determine the expected rate
of return for the class in which the stock falls, that is,
estimating the rates prevailing in the market. There is no
simple way of doing this; measurement of expectations is
intrinsically difficult. The security valuation theory which
permits estimation of stockholders' required returns is
presented in Section III.
(b) Decision Rules
The premise in finance theory is that corporate managers
will make decisions in order to increase the wealth of the
owners of the firm. In other words, the corporation is assumed
to be run primarily for the long-run benefit of the common
stockholders. The decision rules proposed by the theory are
designed to help achieve that end. The cost-of-capital concept
is no exception.
Consider first a firm which is all-equity financed. The
cost of capital for such a firm is the rate of return that
investors expect to earn on the company's common stock, as
well as all other securities of equivalent risk. (The expected
2-5
-------
rate of return on all securities of equivalent risk must be
equal.) ThJs rate is the stockholders' opportunity cost,
since they can always expect to earn this rate by investing
in other firms of comparable risk.
Thus, the cost of capital is the minimum acceptable rate
of return, or "hurdle rate", for new investments by the firm.
If the firm were to undertake an investment with rate of return
less than the cost of capital, its shareholders would be worse
off than if the firm had foregone the investment and paid an
extra dividend, since the stockholders can always invest else-
where and earn the cost of capital. On the other hand,
investors can expect to earn no more than the cost of capital
by investing elsewhere (at the same risk level), so the firm
should never pass up investments offering more than the cost
of capital.
Thus, a decision rule to increase stockholder wealth is
to accept all projects which offer an internal rate of return
(IRR) greater than the cost of equity capital (R). That is
Accept Project if IRR > R (2-3)
Another version of this rule is to accept all projects
for which the sum of all future cash flows discounted at the
cost of capital exceeds the investment. The difference
between discounted benefits and costs is called the net
present value (NPV) of the project
2-6
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NPV = —-^ I (2-4)
t=l (1+R)
where
C = the expected project cash flow in year t
In = the investment (assumed to be made at t = 0)
T = the economic lifetime of the project
R = the cost of (equity) capital
Accept Project if NPV > 0 (2-5)-
When the firm has both debt and equity financing, the
goal is the same. Accept only projects which will increase
stockholder wealth. The NPV decision rule will be generalized
in Section IV to include debt financing.
2-7
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III. SECURITY VALUATION THEORY FOR ESTIMATING
THE COST OF EQUITY CAPITAL
The usual approach to estimating the cost of equity
capital is through the use of various security valuation
models. The valuation models relate the cost of equity capital
to the expected future dividends on the firm's outstanding
stock. The rate of return implicit in this stream of
dividends is that discount rate which equates the present
value of the dividends to the current share price.
The general form of the valuation model will be considered
first, followed by the three specific forms -used in this study.
General Form
The basic discounted-cash-f low model for security valua-
tion follows directly from Equation (2-1). The current price
PO (the subscript j is now unnecessary) is given by
D + P
p = -± _ £
0 (1 + R)
It similarly follows from Equation (2-1) that PI can be replace
by (D + P?)/(l + R) . Therefore,
t-t C->
P°
(1
If the substitution is continued for P2, P3, etc., we obtain
Dt
P° = (2-6)
-------
Equation (2-6) states that the current common stock share
price is the present value of the stream of future expected
dividends, where the discount rate (the cost of equity
capital) reflects both the time va]ue of money and the
riskiness of the stream.
The discounted-cash-flow method is useful because it is
often possible to arrive at reasonable estimates of the firm's
dividend stream. Since Pn is known, estimates of D-, D? • • •
U J_ i->
yields an estimate of R. Three simplifications of Equation
(2-6) which permit reasonably straightforward estimation of
future dividends will now be considered.
Perpetual-Growth Model
Suppose the dividend stream D is expected to grow
indefinitely at some rate g which is less than R. Then
Equation (2-6) can be simplified to
D
1 0 R g ^ ' >
Therefore,
D,
R = JT- + g (2-8)
0
That is, the cost of capital will be equal to the dividend
yield plus the growth rate if the stated assumption is correct.
(See Appendix A of Part 2 for the development of Equation (2-7).)
For companies for which a constant long-term trend in
earnings and dividends is realistic (for example, utilities),
2-9
-------
Equation (2-8) can be a reasonable rule of thumb for esti
mating R.
For some companies, however, if Equation (2-8) is applied
in a mechanical way, the estimates of R will be upward biased.
Companies in their early stages typically have a period of
rapid growth of earnings and dividends followed by a much
slower growth rate as they enter maturity (for example, IBM).
During this period of rapid growth, if estimates of future
growth rates are mechanically produced from past trends, the
estimated g will overstate the market's expectations. The
result will be an upward-biased estimate of R.
It is often stated that estimates produced from
Equation (2-8) based on mechanical extrapolations of past
trends tend to overstate the cost of capital. This view is
based on the assumption that future growth rates will tend to
be less than or equal to historical rates; non-mature
companies will mature (i.e., their real growth opportunities
will diminish) and mature companies will continue or expire.
If this view is correct, then estimates of R from Equation (2-8)
can be considered as maximum values for the cost of equity
capital.
No-"Real"-Growth Model
Consider next the case where all future earnings not
paid out as dividends are reinvested in projects which offer
an expected return exactly equal to the cost of equity capital.
9 _
10
-------
In this case, the general-valuation model, Equation (2-6),
reduces to
E
P = —
*0 R
Therefore
El
R = p1 (2-10)
F0
The cost of equity capital is equal to the expected earnings
per share during the next year (E, ) divided by the current
share price (Pn) • (See Appendix A for the development of
Equation (2-9) . )
The expression "no real growth" needs further explanation.
Real growth refers to reinvestment at rates of return different
from the cost of capital. If the firm reinvests future earnings
in projects yielding exactly the cost of capital, nothing will
change. The investors will simply have given up current
dividends at the time of the reinvestment for a future stream
which has exactly .the same present value. Thus, the adoption
of these projects will make the firm no more or less attractive
to investors. The firm's stock price will not change, nor will
its earnings per share (E-,). Therefore, the earnings -price
ratio will still equal the cost of equity capital.
If the project offers a future rate of return in excess
of the cost of equity capital, the stock price will rise, current
current earnings per share will remain constant, and the
earnings-price ratio will understate the cost of equity capital.
2-11
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For example, consider a firm initially earning and paying
out $10 per share per year. This dividend is not expected to
grow or decline over time. R, the cost of equity capital, is
10%, so stock price is $100. Now the fj.rjn announces that H
will invest the first year's earnings in a project expected
to yield x dollars per year in perpetuity, starting in t = I.
The expected rate of return on this project is x/10.
Table (2-1) shows, for different levels of x, the errors in
using the earnings-price ratio as a measure of R. Note that
the measure is correct only in the case in which the additional
investment offers a 10% return.
Table 2-1
THE RELATIONSHIP BETWEEN THE EARNINGS PRICE RATIO AND
REINVESTMENT RATES OF RETURN
X
(Dollar
Return
Per Year)
$ .50
$1.00
$1.50
$2. 00
x/10
(Expected
Yield on
Proj ect)
0.05
0.10
0.15
0. 20
po
(100 + Net
Present Worth
of Project)
I '
95.45
100.
104.55
109.09
EPSI/PQ
(Earnings-
Price
Ratio)
p. 105
0,100
0.096
0.92
R
(Cost of
Equity
Capital)
0.10
0.10
0.10
0. 10
Source: S. C. Myers [18], Table 1, p. 14.
The usual view is that earnings^price ratios tend to
understate the cost of equity capital. This is based on the
assumption that corporate managers will accept only those
projects with expected returns greater than or equal to R.
2-12
-------
To the extent that real growth opportunities exist, then,
the earning-price ratio will understate the cost of equity
capital. In this case, the estimates of R from Equation (2-10')
can be considered as minimum values for the cost of equity
capital.
Finite-Horizon-Growth Model
The third valuation model makes assumptions about
investment behavior which are between the extremes of
the previous two. Suppose the firm can reinvest earnings
at expected rates or return greater than the cost of equity
capital, but only for a finite number of years, T. Beyond T
all real growth opportunities cease; any further reinvestment
earns only the cost of capital. Thus, the firm will have T
years of rapid growth, followed by an indefinite period of
normal growth. Given these assumptions, the share price is
given by
II
R
Rl
T
(2 11)
where
the investment per share in period 1 (i.e.,
retained earnings)
the rate of return on reinvested earnings for
years 1 through T
(r =£ R)
2-13
-------
Therefore ,
R = + i [r R]T (2-12)
As seen from Equation (2-12), the absence of real growth
opportunities (i.e., r = R) , R is simply the earnings-price
ratio. It can also be shown that when the real growth
opportunities are expected to last indefinitely, R is again
given by 3-,/Pn + g- (See Appendix A for the development
of Equation (2-11.) . )
Because of the more flexible and realistic assumptions
of this model, there is no a priori reason to expect that
estimates of R will be either upward or downward biased.
2-14
-------
IV. THE WEIGHTED AVERAGE COST OF CAPITAL3
(a) The Effects of Leverage on the Market Value of the Firm
Modigliani and Miller (MM) [15] were the first to
rigorously consider the question of how the value of a firm
changes with financial leverage. Their main conclusion is
that in perfect capital markets and no taxes, the value of
the firm will be independent of the proportion of debt and
equity in the firm's capital structure.
The adjective "perfect" is used to describe markets in
which no trader is large enough to have any control over
market prices, and in which distorting elements such as
transaction costs and taxes can safely be ignored.
The Modigliani and Miller Argument
Suppose that we observe a group of firms which are
identical except for financing. We observe the relative value
of firms which are all-equity financed (class A) and of the
others (class B) which have $4.00 of debt outstanding. The
MM proposition implies that
V. = DR + EB (2-13)
110.00 = $4.00 + $6.00
2-15
-------
where
V. = the market value of firm A
DB = the market value of the debt of firm B
£„ = the market value of the equity of firm B
JD
There are markets for unleveraged firms, for the equity of
leveraged firms, and for debt.
To "prove" the proposition we consider what happens if
the prices are not in the predicted relationship. What if
(2-14
$10.00 > $4.00 + $£.QQ?
In this case, who would ever buy shares in the unleveraged
firm? Since we assume that all firms under consideration are
exactly the same except for financing, an investor who buys
up the outstanding debt and equity of a leveraged firm (for
$9.00) obtains exactly the same commodity as one who buys up
all the shares on an unleveraged firm (for $10.00).
On the other hand, what if
VA < D + E (2-15)
$10.00 < $4.00 + $7.00?
2-16
-------
Consider the options open to investors who wish to hold the
equity of leveraged firms. Direct purchase costs $7.00,but
they also have the option of buying unleveraged firms for
$10.00, selling $4.00 of debt, and thereby obtaining a
residual claim for a net cost of $6.00. The residual claim
is exactly equivalent to the leveraged equity of class B firms.
In this case we would expect the value of unleveraged firms to
rise and the value of leveraged firms to fall until the MM
proposition is satisfied.
In this case, at least, we see that market processes tend
to the result that the market value of firms is independent
of the proportions of debt and equity financing used. Since
advantageous trading opportunities are created at every point
where the MM proposition is violated, and since equilibrium
in capital markets may be defined as the absence of favorable
trading opportunities, the MM proposition is a necessary condi-
tion for equilibrium in security markets. (That is, at equilib
rium all securities of equivalent risk must have the same
expected rates of return, thus eliminating the possibility of
making "easy money" through combinations of trades.)
Effects on Equilibrium Expected Rates of Return on the
Firm's Stock
Increased financial leverage increases the expected rate
of return on the firm's stock, but it also increases the risk
2-17
-------
borne by stockholders. Table (2-2) presents an example illustr. ting
this. The firm is expected to generate $10.00 per year
operating income, but with $7.00 per year standard deviation.
Assume that MM are right, and that the firm can borrow at 8
percent. Then the expected rate of return to equity goes up.
But so does the risk: as debt increases, the risk of the
operating earnings is concentrated on a smaller and smaller
equity base.
Modigliani and Miller Propositions I and II
The results of these arguments are the famous MM proposi
tions:
Proposition I. The value of the firm is independent of
the proportion of debt and equity in the firm's capital
structure. It will always be equal to its value under all
equity financing (Vn).
V = D + E = VQ (2-16)
Proposition II. The cost of equity capital increases as
a function of the leverage ratio, (D/E). The relationship .s
given by
k = P0 + (P0- *) ' (§) C2-17)
where
k = the cost of equity capital
PQ = the cost of equity capital with all-equity
financing (i.e., D = 0)
i = the interest rate on debt
2-li
-------
Table 2-2
FINANCIAL LEVERAGE—EXAMPLE;
The firm has assets of $100 which are expected to
produce $10 annually in perpetuity before interest.
Assume no taxes. The interest rate is 8 percent.
Total return
Return to debt
Return to equity
Equity investment
Rate of return relative
to original equity
investment
Standard deviation of
equity earnings
Standard deviation of
return to equity
relative to original
investment
I Debt
0 25
Financing
50
75
10 10
0 2
10 8
100 75
.10 .107
7 7
.07 .093
10
4
6
50
. 12
7
. 14
10
6
4
25
.16
7
. 28
Source: S. C. Myers [23], Table 2, p. 9.
2-19
-------
The reader should note that these proportions are true only if
MM's assumptions (perfect capital markets and no taxes) are
true. Also assumed is that bankruptcy is costless, that is
the firm can be reorganized at no cost to the stockholders.
Corporate Income Taxes and Bankruptcy Costs
Thus far we have ignored a number of things. The most
important is the corporate income tax. Since interest payments
are deducted from taxable income, increases in leverage reduce
taxes and increase the aggregate amount the firm can pay out
to investors. Thus, one would expect to increase both the
overall value of the firm and the market price of its stock.
More precisely, borrowing should increase the market value of
the firm by the present value of tax savings due to borrowing.
Tax savings can be extremely valuable. Suppose a firm
issues $10.0 million of 6-percent perpetual debt, and that the
corporate income tax rate is T 0.5. Then the present value
4
of taxes saved is $5.0 million, or one-half the principal
amount of the debt issued. The computations are as follows.
The tax "rebate" due to deducting $600,000 from taxable in-
come is $300,000. The present value of $300,000 in perpetuity is
oo
Y 500,000 300,000 tc nnn ._.
(1.06) t = 0.06 = $5,000,000 (2-18)
2-20
-------
Six percent is the appropriate discount rate because the risk
characteristics of the tax rebates are essentially the same as
those of the interest payments on the firm's debt." If
investors discount the latter at six percent, the same rate
should be applied to the former.
Since the present value of tax savings on perpetual debt
is simply the tax rate times the amount of debt issues, the
MM Proposition I for perpetuities must be modified to:
V=D+E=VQ+ TCD (2-19)
where
T = the corporate income tax rate
VQ = what the firm would be worth if it were
unleveraged
In general, the proposition is:
present value of tax
V=D+E=V^+ savings due to current
and future debt
(2-20)
Example. At the end of 1972, the XYZ Company's book
equity was $565 million. There was no long-term debt. How
would the value of the firm change if it issued debt to retire
$170 million of stock? (This would give a 30-percent debt
ratio, which is not excessive for an established firm.) Accord
ing to MM, the firm's value would increase by $85 million (AV).
2-21
-------
AV = AVQ + TCAD
= 0 + 0.5 (170,000,000)
= 85,000,000
In other words, the tax subsidy to debt financing is a strong
incentive indeed. Taken literally, Equation (2-19) implies that
the optimal capital structure is 99.99 percent debt.
Of course, this is a lopsided result. It indicates that
shareholders always benefit from increased leverage, which is
nonsense. A more sensible conclusion follows when we intro-
duce another consideration. This is the apparent existence
of "costs of financial distress."
Costs of Financial Distress
At no point has the possibility of financial distress
been ignored. We have not assumed that the payment which
the holders of the firm's bond,5 receive will always be the
amount promised in the bond contract. What has been assumed
is that financial distress is costless, in the special sense
that its occurrence in any contingency does not affect the
amount which the firm pays out to investors.
However, if the financial distress will reduce the aggre-
gate amount which current investors will receive, then the
current value of the firm's securities will be reduced. The
amount of the reduction will depend on the probability of
"trouble".
2-22
-------
The costs associated with financial distress are difficult
to pin down precisely, but they can he roughly grouped into
three categories.1
1. Suppose the firm is threatened with the imminent
possibility that available cash flow will be less
than its promised interest and/or principal payments.
One way to avoid bankruptcy is to change the firm's
investment strategy; that is, the firm can pay more
to its creditors by investing less. But if valuable
investments are thus foregone, then the changes will
be costly from the point of view of present investors,
even though the cost may be justified if bankruptcy
is thereby avoided.
2. Another evasive tactic is to obtain additional
financing. In fact, if the firm could always obtain
additional financing or revise its capital structure
when threatened by financial distress , then there
would be no reason for distress actually to occur,
and firms could operate at very high debt ratios.
However, distress often arrives without many early
warning signals, and once the firm is in distress,
negotiations between stockholders and creditors are
vastly complicated. Both parties find themselves
holding high-risk securities, and neither has much
incentive to bail the other out.
3. If the firm has enough debt, there will be some
contingencies in which bankruptcy cannot be avoided.
2-23
-------
Aside from the direct costs> confusion and delay
associated with the legal process of reorganization,
the firm may be forced to forego investment oppor-
tunities which would be feasible and profitable if
the firm had avoided bankrupcy by borrowing less.
These costs will likewise reduce the present value
of the firm.
Practical Implications of the Modigliani-Miller Argument
If the MM argument is accepted in principle, but taxes
and costs of distress are taken into account, the market value
of the firm will depend on leverage in t\\e way shown by the
solid line in Figure 2-1. The optimum is reached when the
present value of tax savings due to additional borrowing is
just offset by increases in the present value of costs of
distress due to additional borrowing. Of course, this assumes
that the firm can actua-lly find takers for the relatively
high-risk debt that would have to be issued to reach the
optimum. If this is not feasible, the MM recommendation
becomes "borrow as much as you can." Unfortunately, there is
no satisfactory procedure for predicting the debt level at rhich
the optimum would be reached for a given firm.
(b) The Weighted Average Cost of Capital
In the case of all-equity financing, the problem of
capital budgeting is reduced to the task of identifying
projects with net present values (NPV) greater than zero.
The NPV is an estimate of the project's market value (net of
2-24
-------
Firm Value
V = D + E
A
Cost of
Financial Distress
Present Value
of Tax Shield
D/V
Financial
Leverage
All Equity
Financing
Theoretical
Opt imum
Figure 2-1.
Impact of
(Source:
leverage on
S. C. Myers
2-25
the value of
[23], Figure
the firm.
2, p. 18,
-------
investment required) if offered to the market as a separately
financed enterprise.
When a project is partially financed with debt, the NPV
assuring all-equity financing must be adjusted to allow for
the positive or negative effects associated with debt
financing. The basis for this adjustment is the previously
discussed MM valuation formula for the firm (see Equation (2-20 )
Assuming a modest amount of debt financing (i.e., the debt
ratio lies to the left of the theoretical optimum in Figure 2 1 ,
the value of the firm is equal to the all-equity value plus
the present value of the tax savings on current and future
debt.
The adjusted present value for a project can similarly
be computed from the all-equity NPV plus the present value
of the tax savings on debt supported by the project. That is,
APV = NPV(p0) + PVTS (2-21)
where
APV = the adjusted present value
NPV(pg) = the net present value assuming the project
is all-equity financed
PVTS = present value of tax savings on debt
supported by the project
The new decision rule is
Accept Project if APV > 0 (2-22)
2-26
-------
For a project to be acceptable under the base case
assumptions of all-equity financing, its internal rate of
return (IRR) had to exceed R, the cost of equity capital.
It is clear, however, from a comparison of Equations (2-5) and
(2-22) that as long as PVTS is greater than zero, the mini
mum acceptable IRR will now be less than R.
Example. Consider a project which requires an invest-
ment of $1000 and produces an expected perpetual return of
$90 per year. The cost of capital for all-equity financing
is 10 percent (pn).
The base case net present value for the project is given
by
NPV(pn) = I 9° f 1000
U t=l (1-1)
100
Thus, assuming all-equity financing, this project should be
rejected. The internal rate of return is 9 percent, while
the hurdle rate is 10 percent.
Now assume that the project will support $400 of perpetual
debt. The interest rate on the debt is 7 percent. The present
value of the tax savings is given by
PVTS = I (.Q7)(4QO)(.5)
t=l (1.07)*
= 200
2-27
-------
The adjusted present value for the project is $100. Hence it
should now be accepted. In the debt financing case, note that
the minimum acceptable rate of return on the project is only
8 percent, at which point the APV is exactly equal to zero.
This minimum acceptable level of project profitability is
called the weighted average cost of capital. It is designated
at p*.
Formal Definition of the Weighted Average Cost of Capital, p*
The weighted average cost of capital, p*, is the minimum
acceptable rate of return for a project so that acceptance of
the project does not reduce the value of the firm's common
stock.
The net present value of a project that is partially
financed by debt is given by
T C,
NPV(p*) = I L_^ I (2-23)
t=l (l+p*)r U
where
Ct = the expected project cash flow in year t
Ig = the initial investment (assumed to be
made at t = 0)
T = the estimated lifetime of the project
p* = the weighted average cost of capital
The weighted average cost of capital p* must reflect the
financing proportions of the project, the riskiness of the
2-28
-------
investment and, the relationship between these factors and
the economic lifetime of the project.
Now comparison of Equations (2-21) and (2-23) makes it
perfectly clear that the adjusted present value and the net
present value computed using p* (NPV(p*)) are measuring
exactly the same thing. A project is acceptable if the APV
is greater than zero, or correspondingly, if the NPV(p*) > 0.
Thus, we can use this relationship between the APV and
the NPV(p*) to define p*.
The weighted average cost of capital p* for a given
project is the value such that
NPV(P*) = APV (2-24)
or
T C. T C.
I T- = I f + PVTS (2 24a)
t=0 (1+P*) t=0 (1+Pn)
(Note that in the above equations the investment term, !„,
has been included as the t = 0 term of the summations.)
The minimum acceptable rate of return on a project is
obtained by solving Equation (2-24) when the APV is equal
to zero.
Example. Consider a project which requires a $1,000
investment and is expected to last for one year only. Assume
Pn = 0.20. In the base case,
2-29
-------
NI'Vl (]) = —j 1,000
where
C, = the expected year-end cash flow
Now, assume the project will add $400 to the firm's debt
capacity for one year, and that the borrowing rate is 8 percent
Then the project's total contribution to the firm's value (see
Equation (2-21)) is
APV = 1000 + •5(-°8
cl
— - 1000 + 14.8
1. 2
Setting APV equal to zero and solving for C, yields the minimum
acceptable C, equal to 1182.2. Therefore, the minimum acceptab e
rate of return on the project, p*, is equal to 0.182 (18.2
percent) .
Rules of Thumb for Computing the Weighted Average Cost
of Capital
Now we need 'a simple formula for computing p* as a function
of R and other factors. Unfortunately, there is no general
purpose fOKjnula. There are some reasonable rules of thumb
however,
2-30
-------
i. The MM Formula
0* = Pn(.l T L) (---M
(.i C '
where
e i = the basic hurdle rate for the firm under
all-equity financing
L = the f i rm's target debt ratio
T = the corporate tax rate.
c r
This formula applies only to projects having the same business
risk as the firm's existing assets and do not lead to a shift
in the firm's target debt ratio.
ii. Generalized MM Formula
P * = p , . (1 T _ L . ) (1 - 2 6 ')
where
.: v = the hurdle rate for project j under all-equity
financing
L. = the debt capacity ratio for proj ect j (i.e. , the
proportion of project j that can be debt financed
without reducing the firm's other debt capacity).
iii. The Textbook Formula
1(1 Tc> r + R v
2-31
-------
where
i = the current interest rate on the firm's bonds
R = the cost of equity capital
D = market value of debt
E = market value of equity
V = D + E
Example. Consider a firm, which is initially all-equi;y
financed, that is expected to earn $100,000 per year in
perpetuity. If PQ = 0.10 then
VQ = = 1,000,000
Now it changes its capital structure to include $500, 000 od
debt at 7 percent. (The proceeds are used to repurchase
$500,000 of equity.) Following MM we assume the value of
the firm, V, increases by the present value of the tax savings
generated by the debt
V = VQ + PVTS
1,000,000 + 17>50°
1,250,000
2-32
-------
The firm now has a (market value) debt ratio of
D = 500,000 „ .n
V 1 ,250,000
The market value of the equity is 1,250,000 500 = 750,000.
The expected rate of return on equity is
100,000 .07(500,000) + 0.35(500,000)
750,000
= 0. 11
We now have the data to compute p* from the textbook formula.
p* = i(l T } ^ + R | (2-27)
= .07(.S) (.4) + .lie.6)
= .08
This is the appropriate hurdle rate for a project which offers
constant expected cash flows for the indefinite future and
which does not change the risk characteristics of the firm's
assets or debt ratio of the firm.
We could also have used the MM formula to compute p*.
2-33
-------
P0(l TcL) (2-25)
.101
.08
Practical Usefulness of the Three Cost-of-Capital Formulas
The various rules for computing the weighted average cost
of capital clearly depend on special assumptions. The complete
list is given in Table 2-3. The table seems to indicate that
the generalized MM rule is superior to the other two, but even
its realism must be questioned.
However,, we are not really concerned with whether the
rules are exactly true, but whether they are useful rules
of thumb in a practical context. It has been shown by
Myers [21] that the generalized MM rule is reasonably accurate
if capital markets are perfect and the rule is properly used.
It is beyond the scope of this survey to deal with the question
of robustness of the formulas under various violation of the
assumptions. This question has been examined in detail by
Myers [21] and the interested reader is referred to this
source.
In Part 5 the APV formula and its application to invest-
ment problems will be further examined.
2-34
-------
Table 2-3
NECESSARY AND SUFFICIENT CONDITIONS FOR
COST-OF-CAPITAL FORMULAS
Equation
Condition
Formula
MM
Generalized
MM
Textbook
Dividend policy
irrelevant
(lla) Leverage irrelevant
except for corporate
income taxes
(lib) Investment projects
are perpetuities
(lie) Project does not
change firm's risk
characteristics
(lid) Project makes a
permanent contri
bution to debt
capacity
(lie)
(llf)
(15a)
Acceptance of project
does not lead to
shift of target debt
ratio
Risk-independence
Firm's assets
expected to generate
a constant and
perpetual earnings
s t r e am
(15b) Firm is already at
target debt ratio
x
n. a.
n. a.
* n.a. = not applicable.
Source: S. C. Myers, [21], Table 1, p. 14
2-35
-------
fc) Relationship Between .P^J
When the all-equity-financed firm revises its capital
structure to include debt, the cost of equity capital (R)
will rise because financial leverage makes the firm's common
stock even riskier than before.
Figure 2-2 shows how R, i, and p* vary as a function of
financial leverage. Because of the tax shield provided by
the debt in the capital structure, the weighted average cost
of capital will decline with increasing leverage. However,
beyond some point (B in Figure 2-2), the possibility of
bankruptcy will begin to have increasingly negative conse-
quences. As leverage is further increased, a point will be
reached (C in Figure 2-2) beyond which the tax benefits of
additional debt are more than offset by the increasing risl-
of bankruptcy. At this point p* will cease to decline and
start to rise.
Generalization of the p* Textbook Formula to Include Other
Types of Financing
The formula assumes there are only two kinds of financing
instruments. But the weighting principal remains the same
even if there are others. Consider the more general case
where the firm is financed with debt, preferred and common
stock, and financial leases. Since financial leases are
simply an alternative form of debt, the first step in computing
2-36
-------
Required Rate
of Return
(Cost of Capital]
Cost of
Equity
(R)
Weighted Average
Cost of Capital
Cost of
Debt
Financial
Leverage
(D/V)
All-Equity
Financing
Probability
of Bankruptcy
Negligible
Probability
of Bankruptcy
Not Negligible
Figure 2- 2.
Effects of financial leverage on costs
of debt and equity financing and the
weighted average cost of capital.
2-37
-------
p*is to determine the capitalized value of financial leases.
This is done by capitalizing the expected lease payments at
the corporate debt rate i.
The overall cost of capital is given by
P* = i(l - T) £ + i(l T)£ + kp • £ + R ' |
(2- 28)
where
L = the capitalized value of financial leases
P = the market value of outstanding preferred stool-
kp = the yield on the preferred stock (i.e., the cost
of capital for preferred stock)
V = D + L + P + E
All other symbols are as defined for Equation (2-27)
Example. Consider the case of the Winco Distribution
Company. Winco has $5.3 million in long-term debt outstanding,
$5.0 million of preferred stock, and $27.1 million of commo i
stock (all figures are market values). Its current cost of
borrowing is 5-1.4%, its preferred stock cost is 6%, and its
current cost of equity capital is 10.6%. In addition, Winco
has financial leases with annual payments of $850,000 per year
(assumed to continue indefinitely). The capitalized value of
the lease payments (at 5-1/4%) is $16.2 million. The weighted
average cost of capital using Equation (2-28) is 7%. The
details of the calculation are given in Table 2-4.
2-38
-------
Table 2-4
CALCULATION OF THE WEIGHTED
AVERAGE COST OF CAPITAL: EXAMPLE
Source
Long-terra debt
Leases
Preferred Stock
Equity
Total
Amount
($ Millions)
5.3
16.2
5. 0
27.1
53.6
Pro-
portion
0.099
0.302
0.093
0. 506
1.000
After-
Tax
Cost
2.625
2.625
6.000
10.600
Weighted
Cost
0. 26
0. 79
0.56
5.3&
6.97
Of the variables used to compute the weighted average
cost of capital, only R, the cost of equity capital, is not
directly observable. (R was computed using Equation (2-8)
D-,
with
0
. 6% and g =
The Effect of Debt Financing on the Estimation Equations for R
The estimation equations developed for R in Section II
assumed the firm was all-equity financed. How will the form
of these equations change when other types of financing are
used as well?
The answer is, not at all, providing the proper assumptions
are made. In Section II the use of a single value of R for
o
all future periods assumed that the business risk of the firm
2-39
-------
will remain stable over time, or else differing R's would
g
be required to reflect changing business risks. When debt
(or other) financing is introduced, the same type of consicer-
ation is involved. The use of debt increases the equity rsk
(and thus increases R), but the same value of R for all ful ure
periods is appropriate as long as the financial risk of th«
firm is assumed to remain stable over time. Under these
assumptions, the same discounted cash flow models developed
in Section II can be used to estimate R for a firm which is
only partially equity financed. Further, the estimating
equations are not affected by the issue of new equity since
we deal with per-share values. The necessary assumption is
that new equity is issued at P , the per-share price of old
equity.
2-40
-------
PART 2 — APPENDIX A
DEVELOPMENT OF SECURITY VALUATION MODELS
The basic valuation is given by
oo D
° t = l (1 + R)1"
Model 1—Perpetual Growth
Suppose that in each period the firm invests a fraction
b of its earnings in new projects (note that b < 1 else the
firm would never pay dividends). These projects produce a
perpetual yield on equity of r, beginning with the next period
By definition, the earnings and dividends in period t are
given by
Et = Et_x + rbEt_x
= Et_1(l + br)
= E0(l + g)t (2A-1)
where g is the growth rate of dividends per share (g = br)
Dt = DQ(1 + g)1 (2A-2)
Substituting Equation (2A-2) into Equation (2-6).
2-41
-------
t=l (1 +
D
i
L- (g < R) C2A-3J
g
Note the condition that the growth rate must be less than the
discount rate. If the product of the reinvestment rate r
times the retention rate b equals R, PQ will be infinite.
Model 2—-No "Real Growth"
Suppose now that all equity reinvestment produces an
expected return equal to the cost of capital R. Then the
growth rate of per-share earnings and dividends will equal
bR. (Note that g by definition will now be less than R, since
b < 1.) Substituting g bR into Equation (2A-3)
D,
P
0 R(l b)
= if C2A-4)
Note that this equation also applies when b = 0, i.e., the
firm pays out all of its earnings and does not grow.
Model 5 — Finite Growth
Consider now a more realistic variant of the perpetual-
growth model. The special investment opportunities (r > R)
are not available in perpetuity but only for some finite
interval of T years.
2-42
-------
The share value at t = 0 is equal to the no- real- growth
value, plus the present value of the T years of real growth
opportunities (PVGO) .
El
P0 = if + PVGO (2A-5)
In year t the firm invests I dollars per share (withheld
earnings) which generates a perpetual return r. The value
in period t of this investment (PVGO ) is given by
oo I • r
PVGO = I — * - - I
1 t=l (1 + R)r r
' ' r
- R -
I (r R)
(2A-6)
The value at t = 0 of these T growth opportunities is given
by
T I.(r - R)
PVGO = I — - - (2A-7)
t=l R(l + R)
Now
It = 1^1 + g)1'1
Therefore ,
PVGO = 1^ - I, I (1 + g^'1
K i t=l (1 + R)1
- R I (1 + gj1'1
--
tl u + R)t-i
.1 . Y Cl + g)T f2A_8
(
2-43
-------
It now remains to sum the terms in i The sum of the geo
metric series with T 1 terms (ST_1) is given by
1
1 + R
'T-l
.
1 + R
1 + R
1
1 +
1 + R
C2A-9)
R G
„ __j
If, as we generally would expect, (1 + g)/(l + R) is close
to one, and T is not too large, then the right-hand term in
Equation (2A-9) admits to a convenient approximation
T
1 +
1 + R
=: 1
TCg
(2A-10)
Substituting the expression for the sum (2A-9) and this approxi
mation (2A-10) into Equation (2A-8) and simplifying, we obtain
PVGO =
R
R
VT
(2A-11)
We can now state the final form of Equation (2A-5)
-
R
R
R
VT
(2A-12)
2-44
-------
FOOTNOTES FOR PART 2
1. This definition of the cost of capital is based on a
definition by S. C. Myers in [19], pp. 63-65.
2. Precisely speaking, Equation (2-1) assumes that dividends
are paid at the end of period T + 1. If dividends were
paid and invested during the period, Equation (2 1) would
have to be adjusted to allow for the additional return
received. (See Part 3, Section III, item 2 for a detailed
discussion of investment returns.) Whether the dividends
are reinvested or consumed at the end of period T + 1 makes
no difference in the return calculation. The effect will
show up in the investment base for the next period (i.e.,
the stockholder will either have Pt + 1 or Pf+i + Dt+l
to invest in period t + 2).
3. Large portions of Section IV, including footnotes 4-6,
have been reproduced verbatim from two unpublished manu-
scripts by Stewart C. Myers. These are, "Optimal Capital
Structure" [23] and "Interaction of Investment and Financing
Decisions" [24]. Both copyright 1975 by S. C. Myers. The
excerpts from these manuscripts are included with the kind
permission of the author.
4. The firm's taxable income is assumed to always exceed
the interest payments. If this is not the case, the
2-45
-------
exact present value of the tax savings becomes extremely
elusive. One reason is the fact that accounting losses
can be carried forward for tax purposes.
5. Actually, there are some problems with this assumption.
What if the firm will issue more debt if earnings rise
more than expected, and will retire debt if earnings are
disappointing? Then the tax benefits are uncertain:
they take on the risk characteristics of the firm's
assets. However, it is difficult to specify precisely
what the discount rate should be under these circumstances.
Whatever the discount rate, the favorable tax treatment
of debt provides a strong incentive for the use of debt
financing.
6. These costs are discussed more fully in Robichek and
Myers, "Problems in the Theory of Optimal Capital structure,
Journal of Finance and Quantitative Analysis, June 1966,
pp. 15-19. The existence of bankruptcy costs has been
documented by Nevins D. Baxter: "Leverage, Risk of Ruin
and the Cost of Capital," Journal of Finance, September
1967, p. 96.
7. This condition is actually more restrictive than
necessary. What is really required is a rule for def ning
2-46
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P* such that NPV(p*) > 0 whenever APV x 0. Howcvei , this
generalized requirement does not provide a practical
procedure for estimating p*, as does Equation (2-24).
8. The business risk of the firm refers to the riskiness
of the firm's assets independently of how they are
financed. For example, one can speak of the (business)
risk of being in the electric utility or machine tool
business. Financial risk refers to the additional risks
borne by the stockholders when debt (or other funds of
senior financing) is used by the firm. Since the stock-
holders have a residual claim to the firm's earnings (and
assets), the use of debt will tend to magnify the
business risk of the firm as it is passed along to the
stockholders. This magnification is referred to as
financial risk. An example of this effect is shown in
Table 2-2.
9. To be precise, the use of a constant discount rate R
assumes that the risk of the firm's dividend stream
increases at a constant rate as a function of time.
That is, risk increases at a constant rate as one looks
further and further into the future. This is not an
unreasonable assumption, and is almost universally
2-47
-------
assumed in studies using discounted cash flow valuation
models. For a fuller discussion of this matter, sec
Robichek and Myers, "Conceptual Problems in the use of
Rick-Adjusted Discount Rates," The Journal of Finance,
Vol. XXI (Dec. 1966), pp. 727-730.
10. This is precisely true only in the absence of equity
issue costs. From the point of view of the existing
stockholders, the required rate of return on new issues
must be slightly higher than R to offset the effect of
transactions costs. This will allow the new stockholders
to earn the rate R on the gross proceeds of the issue
(i.e., their investment).
2-48
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ESTIMATION OF THE COST OE CAPITAL
FOR MAJOR UNITED STATES INDUSTRIES
WITH APPLICATION TO POLLUTION-CONTROL INVESTMENTS
PART 3. THEORY: RISK AND RETURN CONCEPTS
Dr. Gerald A. Pogue
4 Summit Drive
Manhasset, New York 11030
November 1975
-------
I. INTRODUCTION
As discussed in Part 2, the cost of equity capital is
the rate of return (dividends plus capital gains) investors
expect to earn by holding a firm's common stock. Assuming
that investors are by-and-large averse to bearing risk, the
required rate of return will depend on the stock's risk level:
The higher the risk, the higher the expected rate of return
(cost of capital).
But this definition of the cost of capital raises as
many questions as it answers. How is risk to be defined and
measured? What are the current theories about the relationship
between risk and required returns? How are stock market (i.e.,
financial) risk measures related to the characteristics of
firms' (real) assets, such as the variability of corporate
earnings, the degree of financial leverage, and the growth
rate of earnings?
These questions are the subject of Part 3. Part 3 is
organized in four sections. Sections II and III provide a
nontechnical introduction to the current risk and return
theories. Section IV summarizes the existing empirical studies
relating to the real determinants of a stock's financial risk.
Sections II and III perform overlapping functions.
Section II provides a brief introduction to the "high points"
3-1
-------
of the theory. Section III presents a detailed survey for
the reader intent on a greater depth of understanding. It
contains a two-part article written by Professors Franco
Modigliani and Gerald Pogue entitled, "An Introduction to
Risk and Return: Concepts and Evidence," [16]. Readers
familiar with the basic notions of portfolio and capital
markets theories could well skip Sections II and III.
3-2
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II. A BRIEF INTRODUCTION TO THE THEORY OF
RISK AND THE REQUIRED RATE OF RETURN1
(a) The Risk of Individual Common Stocks
The measurement of common stock risk is a complex subject.
No-one has "all the answers". Yet enough is known and
reasonably well agreed upon to identify the main elements of
stock risk and to measure them using past data. The purpose
of this section is to explain this methodology.
The Risk of Holding a Portfolio of Securities
Some investors hold stock in only one company, others
in hundreds of companies. But in any case, they are interested
in the rate of return realized on their portfolios, not the
rate of return on any individual security per se.
Let R stand for the rate of return realized on an
P
investor's portfolio over a given period of time. R is
equal to dividends plus capital gains, divided by the value
of the portfolio at the start of the period:
D +
R _ _P
R - -^y
F P
3-3
-------
where
n = dividends received from the portfolio
P
V = market value of the portfolio at the start
P
of the period
AV = capital gain on the portfolio during the period
The investor is obviously concerned with the predicta-
bility of R —i.e., with the possible difference between the
anticipated and the actual rate of return. The usual
statistical measure of the extent of possible deviation is
a , the standard deviation of R .
P P
Diversification
Intelligent diversification can substantially reduce
the risk of a portfolio return. But the risk cannot be
eliminated entirely.
Wagner and Lau [28] have reported an experiment which
clearly illustrates this. They formed portfolios of varying
size by randomly selecting stocks from a sample of firms
rated A+ in Standard and Poor's 1960 Earnings and Dividend
Rankings. Then they determined what the month-by-month
portfolio returns would have been during the 1960-70 period,
and calculated the standard deviation of this monthly
return. The results are shown in Table 3-1.
The reduction in risk due to diversification is dramatic
However, it is also evident that additional diversification
3-4
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Table 3 1
RISK VERSUS DIVERSIFICATION:
RANDOMLY SELECTED PORTFOLIOS*
Number of
Securities in
Portfolio
1
o
^
5
4
5
10
15
20
Standard Deviation
n~F Rp1"ii~m
Percent/Month
7.0
5. 0
4. 8
4.6
4.6
4. 2
4. 0
3.9
Correlation with
the Return on
a Market Index**
0. 54
0.63
0. 75
0.77
0. 79
0. 85
0. 88
0. 89
* Wagner and Lau examined ten distinct portfolios
for each level of diversification. Thus the 3.9
percent standard deviation at the 20-security
diversification level is the average standard
deviation of 10 randomly selected 20-security
portfolios.
** The index in this case was the average return on
all N. Y. Stock Exchange stocks.
Source: Wagner and Lau [28].
yields a rapidly diminishing reduction in risk. The improve-
ment is slight when the number of securities is increased
beyond, say, ten.
Note that the return on a diversified portfolio "follows
the market" very closely. The 20-security portfolios had a
3-5
-------
correlation of 0.89 with the market (perfect positive corre-
lation results in a coefficient of 1.0). The implication is
that the risk remaining in the 20-security portfolio is
predominantly a reflection of uncertainty about the performance
of the stock market in general.
Thus, one is led to the distinction between a security's
unsystematic risk, which can be eliminated by diversification,
and its systematic risk, which cannot. The relationship
between the two is illustrated in Figure 3-1.
The systematic risk is due to the fact that the returr
on nearly every stock depends to some degree on the overall
performance of the stock market. Investors are thus exposed
to "market uncertainty" no matter how many stocks they hold.
Consequently, the returns on diversified portfolios are
highly correlated with the market.
The Risk of Individual Securities
Since the investor is interested in the risk of his
portfolio, he will judge the risk of any security by
assessing its contribution to the portfolio risk.
However, the nature of this contribution depends on
how diversified the portfolio is. At one extreme is the
investor who holds only one stock. In this instance, the
portfolio risk can be measured by the standard deviation
of the stock's return.
At the other extreme is the investor who holds "the
market", or a portfolio sufficiently diversified that its
3-6
-------
Standard
Deviation
Portfolio
Return
of
Unsystematic or
Diversifiable
Systematic or
Market-Related
Risk
Number of
Stocks in
Portfolio
Figure 3-1. Systematic versus unsystematic risk
3-7
-------
return is highly correlated with the market. In this case,
standard deviation is not a good risk measure since some
fluctuations in the stock's return will tend to "cancel out"
against fluctuations in the returns on other securities. A
stock that seems highly risky to an undiversified investor
may contribute very little to the risk of a diversified
portfolio.
Obviously, the problem is to measure a stock's systematic
risk—that part of its risk that cannot be diversified awa).
The key to the measurement is the fact that the risk cf
a diversified portfolio is predominantly a reflection on
uncertainty about the performance of the market. Thus, in
measuring systematic risk we can concentrate on the extent
to which individual securities' returns depend on market
performance.
This is why a stock's systematic risk is measured by its
beta (B) , sometimes called the "market sensitivity index".
Beta can be thought of as the slope of a line fitted to a
plot of rates of return on the stock versus rates of return
on a "market portfolio" composed of all stocks. This is
shown in Figure 3-2.
The hypothetical security shown in Figure 3-2 has a beta
of 1. The meaning of this is that, if the actual return 01
the market is 101 in excess of expectations, the stock's
return will tend to be 10% above expectations. To put it n
-------
The Security's
Rate of Return
R
x
X
3, the market sensitivity
index, is the slope of the
line. In this case, 3=1.
R*
The Rate of
Return on
the Market
Figure 3-2.
Calculation of a security's market sensitivity
index from past data.
3-9
-------
every-day terms, the price of a stock with 6=1 will tend
to rise 101 when the market rises 10%, and fall 10% when the
market falls 10%. An average stock will have a beta of 1.0.
Consider a well-diversified portfolio made up of
securities having g = 2. The value of such a portfolio will
tend to go up 20% when the market goes up 10%. On the other
hand, if the market falls, the portfolio's value will fall by
double the amount. Such a portfolio will have twice the
standard deviation of the market portfolio, and four times
the standard deviation of a portfolio composed of stocks with
3=0.5.
In other words, the standard deviation of a well-diversified
portfolio is approximately proportional to the portfolio's beta.
Any portfolio's beta is simply a weighted average of the betas
of the securities in the portfolio. Therefore, we can take a
stock's beta as a measure of the stock's contribution to port-
folio risk, assuming the portfolio is well diversified.
The analysis so far can be summarized as follows:
1. The risk of a portfolio can be measured by the
standard deviation of its rate of return.
2. The risk of an individual security is its contri
bution to portfolio risk.
3. The standard deviation of a stock's return is the
relevant measure of risk for the undiversified
investor.
3-10
-------
4. However, a stock's standard deviation partly
reflects unsystematic risk—risk that can be
eliminated by diversification. Only the systematic
component of stock risk is relevant to the well
diversified investor.
5. A stock's systematic risk is measured by its beta,
or "market sensitivity index".
Needless to say, there are investors holding only a
handful of stocks who cannot be described as either undiversi
fied or well diversified. A stock's risk to such an investor
will depend on both its standard deviation and its beta.
Since most investors have extensive opportunities for
diversification (even the proverbial widow or orphan can buy
a mutual fund), beta is usually considered the more relevant
risk measure.
Measuring Betas
Obviously there is no risk in hindsight. The investor
is worried about the unpredictability of the future return
on his portfolio, and in individual stocks' contributions to
this unpredictability.
It is not feasible to obtain direct measurements of how
investors assess the risk of various stocks at this point in
time. But these assessments will certainly depend on past
experience. A stock's past behavior can provide strong
evidence pertaining to its current and future risk.
3-11
-------
The basic data for estimating betas are past rates of
return earned over relatively short intervals—usually weeks
or months. For example, the beta calculations in Part 4
are based on monthly rates of return which occurred during
the period February 1962 to December 1974.
Beta is calculated by fitting a straight line to a plot
of observed stock returns versus observed returns on the
market (see Figure 3-2). The return on the market as a whcLe
is measured by a broadly based market index such as the
Standard and Poor's Composite 500 Stock Index.
The equation of the fitted line is
R. = a. + 8. R,, + e . (3-2)
1 1 J M 3
where
a. = the intercept of the fitted line
e. = the variation around the fitted line
/\
6- = stock j's systematic risk
It is customary to put a hat (") over the estimated values
a., a., and-e.. It is important to remember that these estimated
values may differ from the true values because of statistical
difficulties. However, the extent of possible error can be
measured, providing a range within which the true value is
almost certain to lie.
3-12
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(h) Basic Risk-Return Concepts: The Capital _As_s_et
Pricing Model
In the past several years a great deal of empirical
research has been conducted on the relationship between risk
and rate of return in United States capital markets. Several
dozen articles have been published, representing diverse
points of view. However, there is a common theme in these
studies: over long periods of time, higher risk securities
have, on the average, achieved higher rates of return.
The Capital Asset Pricing Model (CAPM)
The typical starting point for current research in this
area is the "capital asset pricing model". According to the
model, E(R-), "the rate of return expected by shareholders of
the jt firm, is equal to the rate on a risk-free asset (R£)
plus a risk-premium which is proportional to the security's
systematic risk (3-J- That is,
E(R.) = R£ + 3j • [E(RM) - R£] (3-3)
The constant of proportionality, [E (R,,) — R£] , is the
expected risk premium on the market portfolio. The market
portfolio could be represented by the Standard and Poor's
500 Stock Composite Index,for example, and the risk-free rate
by yields on short-term treasury bills.
The predictions of the model are inherently sensible.
For safe investments (3- = 0), the model predicts that inves-
tors would expect to earn the risk-free rate of interest.
3-13
-------
For a risky investment (6- > 0) investors would expect ;i rate
of return proportional to the stock's beta. Thus stocks with
lower-than-average betas (such as most utilities) would offer
expected returns less than the expected market return.
Stocks with above-average values of beta (such as most air-
line securities) would offer expected returns in excess of
the market.
Tests of the Capital Asset Pricing Model
If the capital asset pricing model is right, the em-
pirical tests should show the following:
1. On the average, and over long periods of time, the
securities with high systematic risk should have
high rates of return.
2. On the average, there should be a linear relation-
ship between systematic risk and return.
3. Unsystematic risk should play no significant role
in explaining differences in security returns.
These predictions have been tested in several recent
statistical studies. One such study was by Black, Jensen,
and Scholes [ 3 ]. Their results showed that, over the 35-year
period from 1931 to 1965, average stock returns increased
with increasing betas, but not as much as predicted by the
capital asset pricing model. Also, their results indicated
that there was little reason to question the linearity of the
relationship over the test period. Figure 3-3 shows a plot of
3-14
-------
1931 — 1965
.it
.10
.06
to
z
o
Ul
cc
.06-
.04-
.02-
.00-
-.02
T
INIEfCErT
STD.ERfl.
0.00519
0.00053
SLOPE - 0.01C91
3TD.ERR. . O.COQSO
0.0
0.5 1.0
STSTEHflTIC RISK
1.5
Figure 3-3.
Results of Black, Jensen and Scholes Study-
Average monthly returns versus systematic
risk for the 35-year period 1931 1965 for
ten portfolios and the market portfolio
(Black, Jensen and Scholes [3], Figure 7,
p. 104).
3-15
-------
;iverat;e month returns for the Bl;irk, Jensen ;nu! Scholes
test portfolios versus their sys t OIIKI t i e risk
Briefly,the major result of this and other studies can
he summarized as follows:
1. The evidence shows a significant positive relation-
ship between realized returns and systematic risl .
However, the relationship is not always as strong
as predicted by the capital asset pricing model.
2. The relationship between risk and return appears
to be linear. The studies give no evidence of
significant curvature in the risk-return.relation-
ship .
3. Tests which attempt to discriminate between the
effects of systematic and unsystematic risk do not
yield definitive results. Both kinds of risk
appear to be positively related to security returns.
However, the relationship between return and un-
systematic risk is at least partly spurious—that
is, partly reflecting statistical problems rather
than the true nature of capital markets.
Obviously, one cannot claim that the capital asset pricing
model is absolutely right. On the other hand, the empirical
tests do support the view that beta is a useful risk measure
and that investors in high beta stocks expect correspondingly
high rates of return.
3-16
-------
The Two-Factor Model
The difficulty with the capital asset pricing model is
that the observed tradeoff between return and risk is smaller
than predicted. In the Black, Jensen and Scholes tests, for
example, the average return during the 35-year test period
increased by approximately 1.08 percent per month for a one-
unit increase in beta; this is only about three quarters of
the amount predicted by the capital asset pricing model.
Black [ 2 ] has proposed a variant of the CAPM known as
the two factor model. This model results from a somewhat
different theoretical reason but the result is structurally
similar. In the two-factor model, the risk-free rate of the
CAPM is replaced by the expected return on a "zero-beta"
portfolio, designated ER_. The model is given by
ERj = ERZ + 0. (ERM - ERZ) (3-4)
The R_ factor represents the expected return on a portfolio
whose returns are uncorrelated with the market. This model
conforms to the empirical evidence much more closely than
the CAPM. However, the theoretical nature of the R7 factor
is less well understood.
3-17
-------
III. RISK AND THE REQUIRED RATE OF RETURN*
by Franco Modigliani and Gerald A. Pogue
An Introduction
to
Risk and Return
Concepts and Evidence
1. Introduction
Portfolio theory deals with the selection of optimal
portfolios by rational risk-averse investors: that is,
by investors who attempt to maximize their ex-
pected portfolio returns consistent with individual-
ly acceptable levels of portfolio risk. Capital
markets theory deals with the implications for
security prices of the decisions made by these in-
vestors: that is, what relationship should exist be-
tween security returns and risk if investors behave
in this optimal fashion. Together, portfolio and
capital markets theories provide a framework for
the specification and measurement of investment
risk, for developing relationships between expected
security return and risk, and for measuring the per-
formance of managed portfolios such as mutual
funds and pension funds.
The purpose of this article is to present a non-
technical introduction to portfolio and capital
markets theories. Our hope is to provide a wide
class of readers with an understanding of the foun-
Franco Modigliani is Professor of Finance, Sloan
School of Management, M.I.T. Gerald A. Pogue is
Professor of Finance at Baruch College, City Univer-
sity of New York. This article was originally pre-
pared as Chapter 2 of A. Study of Investment Perform-
ance Fees (Heath-Lexington Books, forthcoming
1974). The research was supported by u grant from
the Investment Company Institute, Washington, D.C.
The article will also appear in the forthcoming
Financial Analysts Handbook (Sumner Levine, editor;
to be published by Dow Jones-Irwin). Because of its
length, part of the article is deferred to the next
dation upon which the modern risk and per-
formance measures are based, by presenting the
main elements of the theory along with the results
of some of the more important empirical tests. We
are attempting to present not an exhaustive survey
of the theoretical and empirical literature, but
rather the main thread of the subject leading the
reader from the most basic concepts to the more
sophisticated but practically useful results of the
theory.
2. Investment Return
Measuring historical rates of return is a relatively
straightforward matter. We will begin by showing
how investment return during a single interval can
be measured, and then present three commonly
used measures of average return over a series ol
such intervals.
The return on an investor's portfolio during ;
given interval is equal to the change in value of tht
portfolio plus any distributions received from the
portfolio expressed as a fraction of the initial port-
folio value. It is important that any capital or in-
come distributions made to the investor be in-
cluded, or else the measure of return will be
deficient. Equivalently, the return can be thought
of as the amount (expressed as a fraction of the
initial portfolio value) that can be withdrawn at the
end of the interval while maintaining the principal
intact. The return on the investor's portfolio,
designated RP, is given by
RP =
V, Vp + PI
(la
where
V, = the portfolio market value at the end of the
interval
V0 — the portfolio market value at the beginning
of the interval
D, = cash distributions to the investor during the
interval.
The calculation assumes that any interest or
dividend income received on the portfolio
securities and not distributed to the investor is
reinvested in the portfolio (and thus reflected in
V,). Furthermore, the calculation assumes that any
* Footnotes and references for this section appear at the end of
the articles. Reproduced with permission of Authors and Publisher
3-18
-------
distributions occur at the end of the interval, or are
held in the form of cash until the end of the in-
terval. If the distributions were reinvested prior to
the end of the interval, the calculation would have
to be modified to consider the gains or losses on
the amount reinvested. The formula also assumes
no capital inflows during the interval. Otherwise,
the calculation would have to be modified to
reflect the increased investment base. Capital in-
flows at the end of the interval, however, can be
treated as just the reverse of distributions in the
return calculation.
Thus given the beginning and ending portfolio
values, plus any contributions from or distributions
to the investor (assumed to occur at the end of the
interval), we can compute the investors return
using Equation (la). For example, if the XYZ pen-
sion fund had a market value of $100,000 at the
end of June, capital contributions of $10,000,
benefit payments of $5,000 (both at the end of
July), and an end-of-July market value of $95,000,
the return for the month is a loss of 10 per cent.
The arithmetic average return is an unweighted
average of the returns achieved during a series of
such measurement intervals. For example, if the
portfolio returns [as measured by Equation (la)]
were -10 per cent, 20 per cent, and 5 per cent in
July, August, and September respectively, the
average monthly return is 5 per cent. The general
formula is
Rpl
^PN
N
where
RA = the arithmetic average return
RpK = the portfolio return in
interval k, k= 1, . ., N
N = the number of intervals in the performance-
evaluation period.
The arithmetic average can be thought of as the
mean value of the withdrawals (expressed as a frac-
tion of the initial portfolio value) that can be made
at the end of each interval while maintaining the
principal intact. In the above example, the investor
must add 10 per cent of the principal at the end of
the first interval and can withdraw 20 per cent and
5 per cent at the end of the second and third, for a
mean withdrawal of 5 per cent of the initial value
per period.
The time-weighted return measures the com-
pound rate of growth of the initial portfolio during
the performance-evaluation period, assuming that
all cash distributions are reinvested in the port-
folio. It is also commonly referred to as the
"geometric" rate of return. It is computed by
taking the geometric average of the portfolio
returns computed from Equation (la). For exam-
ple, let us assume the portfolio returns were -10
per cent, 20 per cent, and 5 per cent in July,
August, and September, as in the example above.
The time-weighted rate of return is 4.3 per cent per
month. Thus one dollar invested in the portfolio at
the end of June would have grown at a rate of 4.3
per cent per month during the three-month period.
The general formula is
RT= [(1
RP2) ...
where
RT = the time-weighted rate of return
RPK — tne portf0''0 return during the
interval k, k= 1, .. ., N
N = the number of intervals in the performance-
evaluation period.
In general, the arithmetic and time-weighted
average returns do not coincide. This is because, in
computing the arithmetic average, the amount in-
vested is assumed to be maintained (through ad-
ditions or withdrawals) at its initial value. The
time-weighted return, on the other hand, is the
return on a portfolio that varies in size because of
the assumption that all proceeds are reinvested.
The failure of the two averages to coincide is
illustrated in the following example: Consider a
portfolio with a $100 market value at the end of
1972, a $200 value at the end of 1973, and a $100
value at the end of 1974. The annual returns are
100 per cent and -50 per cent. The arithmetic and
time-weighted average returns are 25 per cent and
zero per cent respectively. The arithmetic average
return consists of the average of $100 withdrawn at
the end of Period 1, and $50 replaced at the end of
Period 2. The compound rate of return is clearly
zero, the 100 per cent return in the first period
being exactly offset by the 50 per cent loss in the
second period on the larger asset base. In this
example the arithmetic average exceeded the time-
weighted average return. This always proves to be
true, except in the special situation where the
returns in each interval are the same, in which case
the averages are identical.
The dollar-weighted return measures the average
rate of growth of all funds invested in the portfolio
during the performance-evaluation period—that is,
the initial value plus any contributions less any dis-
3-19
-------
tnbutions. As such, the rate is influenced by the
timing and magnitude of the contributions and dis-
tributions to and from the portfolio. The measure
is also commonly referred to as the "internal rate
of return." It is important to corporations, for
example, for comparison with the actuarial rates of
portfolio growth assumed when funding their em-
ployee pension plans.
The dollar-weighted return is computed in ex-
actly the same way that the yield to maturity on a
bond is determined. For example, consider a port-
folio with market value of $100,000 at the end of
1973 (V,,), capital withdrawals of $5,000 at the end
of 1974, 1975, and 1976 (C,, C,, and C:)), and a
market value of $110,000 at the end of 1976 (V:l).
Using compound interest tables, the dollar-
weighted rate of return is found by trial and error
to be 8.1 per cent per year during the three-year
period. Thus each dollar in the fund grew at an
average rate of 8.1 per cent per year. The formula
used is
., C, C,
(I+RD)2
c,
(l+RD)a
(Id)
where
Rp = the dollar-weighted rate of return.
What is the relationship between the dollar-
weighted return (internal rate of return) and the
previously defined time-weighted rate of return? It
is easy to show that under certain special con-
ditions both rates of return are the same. Consider,
for example, a portfolio with initial total value V0.
No further additions or withdrawals occur and all
dividends are reinvested. Under these special cir-
cumstances all of the C's in Equation (Id) are zero
so that
PI'
P2'
P.V
where RP's are the single-period returns. The
numerator of the expression on the right is just the
value of the initial investment at the end of the
three ptriods (V:l). Solving for RD we find
R =[(1 +RP|)(I + Rp2)(l+Rp3)]'/:' -I ,
which is the same as the time-weighted rate of
return RT given by Equation (Ic). However, when
contributions or withdrawals to the portfolio oc-
cur, the two rates of return are not the same.
Because the dollar-weighted return (unlike I e
time-weighted return) is affected by the magnitu c
and timing of portfolio contributions and distrih i-
tions (which are typically beyond the portfo'. o
manager's control), it is not useful for measurii g
the investment performance of the manager. FIT
example, consider two identical portfolios (desig-
nated A and B) with year-end 1973 market values
of $100,000. During 1974 each portfolio has a 20
per cent return. At the end of 1974, portfolio A
has a capital contribution of $50,000 and portfolio
B a withdrawal of $50,000. During 1975, both
portfolios suffer a 10 per cent loss resulting in
year-end market values of $153,000 and $63,000
respectively. Now, both portfolio managers per-
formed equally well, earning 20 per cent in \9~ \
and -10 per cent in 1975, for a time-weighu 1
average return of 3.9 per cent per year. The dollai -
weighted returns are not the same, however, due to
the different asset bases for 1975, equaling 1.2 per
cent and 8.2 per cent for portfolios A and B
respectively. The owners of portfolio B, unlike
those of A, made a fortuitous decision to reduce
their investment prior to the 1975 decline.
In the remainder of this article, when we men-
tion rate of return, we will generally be referring to
the single interval measure given by Equation (la).
However, from time to time we will refer to the
arithmetic and geometric averages of these returns.
3. Portfolio Risk
The definition of investment risk leads us into
much less well explored territory. Not everyone
agrees on how to define risk, let alone how to
measure it. Nevertheless, there are some attribute1
of risk which are reasonably well accepted.
If an investor holds a portfolio of treasun
bonds, he faces no uncertainty about monetan
outcome. The value of the portfolio at maturity of
the notes will be identical with the predicted value.
In this case the investor bears no monetary risk.
However, if he has a portfolio composed of com-
mon stocks, it will be impossible to exactly predict
the value of the portfolio as of any future date. The
best he can do is to make a best guess or most-
likely estimate, qualified by statements about the
range and likelihood of other values. In this case,
the investor does bear risk.
One measure of risk is the extent to which the
future portfolio values are likely to diverge from
the expected or predicted value. More specifically,
risk for most investors is related to the chance that
future portfolio values will be less than expected.
3-20
-------
Thus if the investor's portfolio has a current value
of $ 100,000, and an expected value of $ 110,000 at
the end of the next year, he will be concerned
about the probability of achieving values less than
$110,000.
Before proceeding to the quantification of risk,
it is convenient to shift our attention from the ter-
minal value of the portfolio to the portfolio rate of
return, Rp, since the increase in portfolio value
is directly related to Rp.'
A particularly useful way to quantify the un-
certainty about the portfolio return is to specify the
probability associated with each of the possible
future returns. Assume, for example, that an in-
vestor has identified five possible outcomes for his
portfolio return during the next year. Associated
with each return is a subjectively determined
probability, or relative chance of occurrence. The
five possible outcomes are:
Possible Return
50%
30%
10%
-10%
-30%
Subjective Probability
Ol
0.2
0.4
0.2
0.1
1.00
Note that the probabilities sum to 1.00 so that the
actual portfolio return is confined to take one of
the five possible values. Given this probability dis-
tribution, we can measure the expected return and
risk for the portfolio.
The expected return is simply the weighted
average of possible outcomes, where the weights
are the relative chances of occurrence. The ex-
pected return on the portfolio is 10 per cent, given
by
5
E(Rn
•= L
P, R,
= 0.1 (50.0) + 0.2 (30.0) + 0.4 (10.0)
+ 0.2 (-10.0) + 0.1 (-30.0)
= 10%, (2)
where the R,'s are the possible returns and the P/s
the associated probabilities.
If risk is defined as the chance of achieving
returns less than expected, it would seem to be
logical to measure risk by the dispersion of the
possible returns below the expected value.
However, risk measures based on below-the-mean
variability are difficult to work with and are ac-
ExmeiT 1
POSSIBLE SHAPES FOR
PROBABILITY D03TR8BUTIONS
Symmetric Probability Distribution
Prob
Probability Distribution Skewed to Left
Prob
Probability Distribution Skewed to Right
Prob
I. Footnotes appear at end of article.
tually unnecessary as long as the distribution of
future return is reasonably symmetric about the ex-
pected value.2 Exhibit I shows three probability
distributions: the first symmetric, the second
skewed to the left, and the third skewed to the
right. For a symmetric distribution, the dispersion
of returns on one side of the expected return is the
same as the dispersion on the other side.
Empirical studies of realized rates of return on
diversified portfolios show that skewness is not a
significant problem.1' If future distributions are
shaped like historical distributions, then it makes
little difference whether we measure variability of
returns on one or both sides of the expected return.
If the probability distribution is symmetric,
measures of the total variability of return will be
twice as large as measures of the portfolio's
variability below the expected return. Thus if total
variability is used as a risk surrogate, the risk
rankings for a group of portfolios will be the same
as when variability below the expected return is
used. It is for this reason that total variability ol
3-21
-------
1
2
3
4
7
m
11
1 2
17
18
19
20
RATE OF RETURN
RANGE
-13.6210 -12.2685
-12.2685 -10.9160
-10.9160 -9.5635
-9.5635 -8.2110
.5060 -4. IOOD
-0.0960 1.2565
5O-1 ACl R fififiC;
8.0190 9.3715
9.3715 10.7240
10.7240 12.0765
12.0765 13.4290
SCALING FACTOR = 1
Average Return = 0.91% per
Standard Deviation = 4.45%
Number of Observations = 3
EXHIBIT 2
I DISTRIBUTION FOR A PORTFOLIO OF 100 SECURITIES
(EQUALLY WEIGHTED)
January 1945 - June 1970
FREQ. 1 i i i 5 i i n10i i i |15, , i |20, , , ,25, , , ,30| , , (35| i i |40, , , |45| , , ,50|
1 *
2 **
2 **
3 8**
30 ******************************
4 * ***
2 **
2 8*
3 ***
month
per month
06
returns has been so widely used as a surrogate for
risk.
It now remains to choose a specific measure of
total variability of returns. The measures most
commonly used are the variance and standard
deviation of returns.
The variance of return is a weighted sum of the
squared deviations from the expected return.
Squaring the deviations ensures that deviations
above and below the expected value contribute
equally to the measure of variability, regardless of
sign. The variance, designated )
= 0.1(50.0 -
+ 0.4(10.0- io.o)2 + o.iT-'ro.o
+ O.H-30.0 - IO.O)2
= 480 per cent squared.
10.0)-
(3)
The standard deviation (erp) is defined as the
square root of the variance. It is equal to 22 per
cent. The larger the variance or standard deviation,
the greater the possible dispersion of future
realized values around the expected value, and the
larger the investor's uncertainty. As a rule of
thumb for symmetric distributions, it is often
suggested that roughly two-thirds of the possible
returns will lie within one standard deviation either
side of the expected value, and that 95 per cent
will be within two standard delations.
Exhibit 2 shows the historical return distribu-
tions for a diversified portfolio. The portfolio is
composed of approximately 100 securities, with
each security having equal weight. The month-by-
month returns cover the period ftS'm January 1945
to June 1970. Note that the distribution is approxi-
mately, but not perfectly, symmetric. The arith-
metic average return for the 306-month period is
0.91 per cent per month. The standard deviation
about this average is 4.45 per cent per month.
3-22
-------
EXH6BIT 3
RATE OF RETURN D9STRIBUTION FOR NATIONAL
DEPARTMENT STORES
January 1945 - June 1970
RANGE
1
2
3
4
5
6
7
Q
1 9
1 ^
1 7
1 fl
19
20
-32.3670
-29.4168
-26.4666
-23.5163
-20.5661
-17.6159
-14 Rfi^7
-11 71 £>f\
_Q 7fie;Q
OnftCLQ
o nQRC
5QQCT
20.7366
23.6868
-29.4168
-26.4666
-23.5163
-20.5661
-17.6159
-14.6657
-11 71 ^R
_C Q1 CH
5QQC T
1 7 7AfiR
on 7Tfifi
23.6868
26.6370
FREQ. 1I1I5III 1101 1 1 1151 1 1 1201 1 1 1251 1 1 1301 1 1 1351 1 1 1401 1 ' 145' 1 1 '50
1
0
0
1
1
3
25
5
2
8
8
8
8«*
8 ****
8*
SCALING FACTOR = 1
Average
Standard
Return = 081% per month
Deviation = 9
Number of Observations
02% per
= 306
month
Exhibit 3 gives the same data for a single
security. National Department Stores. Note that
the distribution is highly skewed. The arithmetic
average return is 0 81 per cent per month over the
306-month period. The most interesting aspect,
however, is the standard deviation of month-by-
month returns—9.02 per cent per month, more
than double that lor the diversified portfolio. This
result will be discussed further in the next section.
Thus far our discussion of portfolio risk has
been confined to a single-period investment
horizon such as the next year; that is, the portfolio
is held unchanged and evaluated at the end ot the
year. An obvious question relates to the effect of
holding the portfolio for several periods—say for
the next 20 years: Will the one-year risks tend to
cancel out over time'.' Given the random-walk
nature of security prices, the answer to this
question is no. If the risk level (standard deviation)
is maintained during each year, the portfolio risk
for longer horizons will increase with the horizon
length. The standard deviation of possible terminal
portfolio values after N years is equal toVN times
the standard deviation after one year.' Thus the in-
vestor cannot rely on the "long run" to reduce his
risk of loss.
A final remark should be made before leaving
portfolio risk measures. We have implicitly
assumed that investors are risk averse, i.e., that
they seek to minimize risk for a given level ol
return. This assumption appears to be valid for
most investors in most situations. The entire theory
of portfolio selection and capital asset pricing is
based on the belief that investors on the average
are risk averse.
4. Diversification
When one compares the distribution of historical
returns for the 100-stock portfolio (Exhibit 2) with
the distribution for National Department Stores
(Exhibit 3), he discovers a curious relationship.
While the standard deviation of returns lor the
3-23
-------
security is double that of the portfolio, its average
return is less. Is the market so imperfect that over a
long period of time (25 years) it rewarded substan-
tially higher risk with lower average return?
Not so. As we shall now show, not all of the
security's risk is relevant. Much of the total risk
(standard deviation of return) of National Depart-
ment Stores was diversifiable. That is, if it had
been combined with other securities, a portion of
the variation in its returns could have been
smoothed out or cancelled by complementary
variation in the other securities. The same port-
folio diversification effect accounts for the low
standard deviation of return for the 100-stock
portfolio. In fact, the portfolio standard deviation
was less than that of the typical security in the
portfolio. Much of the total risk of the component
securities had been eliminated by diversification.
Since much of the total risk could be eliminated
simply by holding a stock in a portfolio, there was
no economic requirement for the return earned to
be in line with the total risk. Instead, we should ex-
pect realized returns to be related to that portion
of security risk which cannot be eliminated by
portfolio combination.
Diversification results from combining securities
having less than perfect correlation (dependence)
among their returns in order to reduce portfolio
risk. The portfolio return, being simply a weighted
average of the individual security returns, js not
diminished by diversification. In general, the lower
the correlation among security returns, the greater
the impact of diversification. This is true regardless
of how risky the securities of the portfolio are
when considered in isolation.
Ideally, if we could find sufficient securities with
uncorrelated returns, we cculd completely
eliminate portfolio risk. This situation is un-
fortunately not typical in real securities markets
where returns are positively correlated to a con-
siderable degree. Thus while portfolio risk can be
substantially reduced by diversification, it cannot
be entirely eliminated. This can be demonstrated
very clearly by measuring the standard deviations
of randomly selected portfolios containing various
numbers of securities.
In a study of the impact of portfolio diversifica-
tion on risk, Wagner and Lau [27]* divided a sam-
ple of 200 NYSE stocks into six subgroups based
on the Standard and Poor's Stock Quality Ratings
as of June 1960. The highest quality ratings (A+ )
formed the first group, the second highest rating?
(A) the next group, and so on. Randomly selectei
portfolios were formed from each of the subgroup^
containing from 1 to 20 securities. The month-b}
month portfolio retu.ns for the I0-ye;u perioi
through May 1970 were then computed for eac!
portfolio (portfolio composition re;n-iir'ng un
changed). The exercise was repeated ten times u>
reduce the dependence on single samples, and the
values for the ten trials were then averaged.
Table 1 shows the average return and standard
deviation for portfolios from the first subgroup
(A+ quality stocks). The average return is
unrelated to the number of issues in the portfolio.
On the other hand, the standard deviation of
return declines as the number of holdings in-
creases. On the average, approximately 40 per cent
of the single security risk is eliminated by forming
randomly selected portfolios of 20 stocks.
* References appear at end of article.
TABLE 1. RISK
VERSUS DIVERSIFICATION FOR RANDOMLY *
SELECTED PORTFOLIOS OF
Number of
Portfolio
1
2
3
4 *"-*'^5&
5
• ! JQ
,W'¥s
""' 20
Source: Wagner
Average
(%/month)
0.88
0.69
0.74
wssffZS^'*'- • 0.65
0.71
0.68
0.69
0.67
and Lau [ 27) , Table C,
A+ QUALITY SECURITIES
June 1960— May 1970
Std. Deviation
(%/month)
7.0
5.0
4.8
4.6
4.6
4.2
4.0
3.9
p. 53.
Correlation
R
0.54
0.63
0.75
0.77
0.79
0.85
0.88
0.89
with Market
R2
0.29
0.40
0-56..,
0.59
0.62
0.72
0.77
0.80
3-24
-------
EXHIBIT 4
STANDARD DEVSATION VERSUS
NUMBER OF ISSUES 9N PORTFOLIO
STANDARD
DEVIATION
12
12345 10 15 20
Source: Wagner and Lau [27], Exhibit 1, p.50.
EXHIBIT 8
CORRELATION VERSUS NUMBER OF
ISSUES IN PORTFOLIO
R-SQUARE
12345 10 15 20
Source: Wagner and Lau [27), Exhibit 2, p.50.
However, it is also evident that additional diversi-
fication yields rapidly diminishing reduction in
risk. The improvement is slight when the number
of securities held is increased beyond, say, 10.
Exhibit 4 shows the results for all six quality
groups. The figure shows the rapid decline in total
portfolio risk as the portfolios are expanded from 1
to 10 securities.
Returning to Table I, we note from the next to
last column in the table that the return on a diver-
sified portfolio follows the market very closely.
The degree of association is measured by the
correlation coefficient (R) of each portfolio with
an unweighted index of all NYSE stocks (perfect
positive correlation results in a correlation coeffi-
cient of 1.0).n The 20-security portfolio has a
correlation of 0.89 with the market. The im-
plication is that the risk remaining in the 20-stock
portfolio is predominantly a reflection of un-
certainty about the performance of the stock
market in general. Exhibit 5 shows the results for
the six quality groups.
Correlation in Exhibit 5 is represented by the
correlation coefficient squared, R'2 (possible values
range from 0 to 1.0). The R2 coefficient has a
useful interpretation: it measures the proportion of
variability in portfolio return that is attributable to
variability in market returns. The remaining
variability is risk, which is unique to the portfolio
and, as Exhibit 4 shows, can be eliminated by
proper diversification of the portfolio. Thus, R-
measures the degree of portfolio diversification. A
poorly diversified portfolio will have a small R-
(0.30 - 0.40). A well diversified portfolio will have
a much higher R'2 (0.85 0.95). A perfectly diver-
sified portfolio will have an R2 of 1.0; that is, all
the risk in such a portfolio is a reflection of market
risk. Exhibit 5 shows the rapid gain in diversifica-
tion as the portfolio is expanded from 1 to 2
securities and up to 10 securities. Beyond 10
securities the gains tend to be smaller. Note that
increasing the number of issues tends to be less ef-
ficient at achieving diversification for the highest
quality A + issues. Apparently the companies com-
3-25
-------
prising this group arc more homogeneous than the
companies grouped under the other quality codes.
THfc Ifestflts show that while some risks can he
eliminated via diversification, others cannot. Thus
we are led to distinguish between a security's "un-
systematic" risk, which can be washed away by
mixing the security with other securities in a diver-
sified portfolio, and its "systematic'1 risk, which
cannot be eliminated by diversification. This
proposition is illustrated in Exhibit 6. It shows
total portfolio risk declining as the number of
holdings increases. Increasing diversification
gradually tends to eliminate the unsystematic risk,
leaving only systematic, i.e., market-related risk.
The remaining variability results from the fact that
the return on nearly every security depends to
some degree on the overall performance of the
market. Consequently, the return on a well diver-
sified portfolio is highly correlated with the
market, and its variability or uncertainty is
basically the uncertainty of the market as a whole.
Investors are exposed to market uncertainty no
matter how many stocks they hold.
5. The Risk of Individual Securities
In the previous section we concluded that the
systematic risk of an individual security is that por-
tion of its total risk (standard deviation of return)
which cannot be eliminated by combining it with
other securities in a well diversified portfolio. We
now need a way of quantifying the systematic risk
of a security and relating the systematic risk of a
portfolio to that of its component securities. This
can be accomplished by dividing security return
into two parts: one dependent (i.e., perfectly
correlated), and a second independent (i.e., un-
correlated) of market return. The first component
of return is usually referred to as "systematic", the
second as "unsystematic" return. Thus we have
Security Return = Systematic Return
+ Unsystematic Return. (4)
Since the systematic return is perfectly
correlated with the market return, it can be ex-
pressed as a factor, designated beta (ft), times the
market return, Rm. The beta factor is a market
sensitivity index, indicating how sensitive the
security return is to changes in the market level.
The unsystematic return, which is independent of
market returns, is usually represented by a factor
epsilon (e'). Thus the security return, R, may be
expressed
EXHIiST 8
SYSTEMATIC AND UNSYSTEMATIC RISK
Unsystemat c or
Diversiflabl' Risk
'Systematic or
Market-Related
Risk
For example, if a security had a /? factor of 2.0
(e.g., an airline stock), then a 10 per cent market
return would generate a systematic return for the
stock of 20 per cent. The security return for the
period would be the 20 per cent plus the un-
systematic component. The unsystematic com-
ponent depends on factors unique to the company,
such as labor difficulties, higher than expected
sales, etc.
The security returns model given by Equation
(5) is usually written, in a way such that the average
value of the residual term, e', is zero. This is ac-
complished by adding a factor, alpha (a), to the
model to represent the average value of the un-
systematic returns over time. That is. we set e' =
a + e so that
R = a + /3 Rm
+ e
where the average e over time is equal to zero
The model for security returns given by
Equation (6) is usually referred to as the "market
model". Graphically, the model can be depicted a;
a line fitted to a plot of security returns againsi
rates of return on the market index. This is showr
in Exhibit 7 for a hypothetical security.
The beta factor can be thought of as the slope ol
the line. It gives the expected increase in security
return for a one per cent increase in market return.
In Exhibit 7, the security has a beta of 1.0. Thus, a
ten per cent market return will result, on the
average, in a ten per cent security return. The
3-26
-------
EXHIBIT 7
THE MARKET MODEL FOR SECURITY RETURNS
Security
Return
Market Return R m
Beta (p), the market sensitivity index, is the slope of
the line.
Alpha (a), the average of the residual returns, is the
intercept of the line on the security axis.
Epsilon (e), the residual returns, are the perpendicular
distances of the points from the line.
market-weighted average beta for all stocks is 1.0
by definition.
The alpha factor is represented by the intercept
of the line on the vertical security return axis. It is
equal to the average value over time of the un-
systematic returns (e') on the stock. For most
stocks, the alpha factor tends to be small and un-
stable. (We shall return to alpha later.)
Using the definition of security return given by
the market model, the specification of systematic
and unsystematic risk is straightforward—they arc
simply the standard deviations of the two return
components."
The systematic risk of a security is equal to /3
3-27
-------
times the standard deviation of the market return:
Systematic Risk =/3am (7)
The unsystematic risk equals the standard
deviation of the residual return factor e:
Unsystematic Risk = vf (8)
Given measures of individual security systematic
risk, we can now compute the systematic risk of
portfolio. It is equal to the beta factor for the port-
folio, /?p, times the risk of the market index,
-------
security systematic risk is equal to the security beta
times \
-------
[4] Brealey, Richard A. An Introduction to Risk and
Return from Common Stocks. (Cambridge, Mass.: MIT
Press, 1969.)
[5] Fama, Eugene F. "Components of Investment Per-
formance." The Journal of Finance, Vol. XXVII (June
1972), pp. 551-567.
[6] Fama, Eugene F., and MacBeth, James D. "Risk, Return
and Equilibrium: Empirical Tests." Unpublished Working
Paper No. 7237, University of Chicago, Graduate School
of Business, August 1972.
[7] Francis, Jack C. Investment Analysis and
Management. (New York: McGraw-Hill, 1972.)
[8J Friend, Irwin, and Blume, Marshall E. "Risk and the
Long Run Rate of Return on NYSE Common Stocks."
Working Paper No 18-72, Wharton School of Commerce
and Finance, Rodney L. White Center for Financial
Research.
[9] Jacob, Nancy. "The Measurement of Systematic Risk for
Securities and Portfolios: Some Empirical Results." Jour-
nal of Financial and Quantitative Analysis, Vol VI
(March 1971), pp. 815-834.
[10] Jensen, Michael C. "The Performance of Mutual Funds in
the Period 1945-1964." Journal of Finance, Vol. XXI11
(May 1968), pp. 389-416.
[11] Jensen, Michael C. "Risk, the Pricing ot Capital Assets,
and the Evaluation of Investment Portfolios." Journal of
Business, Vol. 42 (April 1969), pp. 167-247.
[12] Jensen, Michael C "Capital Markets: Theory and
Evidence." The Bell Journal of Economics and
Management Science, Vol. 3 (Autumn 1972), pp. 357-
398.
[13] Levy, Robert A. "On the Short Term Stationarity of Beta
Coefficients." Financial Analysts Journal, Vol. 27
(November-December 1971), pp. 55-62.
[14] Lintner, John. "The Valuation of Risk Assets and the
Selection of Risky Investments in Stock Portfolios and
Capital Budgets" Review of Economics and Statistics,
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[15] Lintner. John. "Security Prices, Risk, and Maximal Gains
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[16] Mains, Norman E. "Are Mutual Fund Beta Coefficients
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[17] Markowitz, Harry M. "Portfolio Selection." Journal i'f
Finance, Vol VII (March 1952), pp. 77-91.
[18] Markowitz, Harry M. Portfolio Selection: Efficient
Diversification of Investments. (New York: John Wile>
and Sons, 1959.)
[19] Miller, Merton H., and Scholes, Myron S. "Rates of
Returns in Relation to Risk: A Reexamination of Recent
Findings." Published in Studies in the Theory of Capital
Markets, edited by Michael Jensen. (New York: Praeger.
1972), pp. 47-78.
[20] Modigliani, Franco, and Pogue, Gerald A. A Study of In-
vestment Performance Fees. (Lexington, Mass.: Heath-
Lexington Books. Forthcoming 1974.)
[21 ] Pogue, Gerald A., and Conway, Walter. "On the Stability
of Mutual Fund Beta Values." Unpublished Working
Paper, MIT, Sloan School of Management, June 1972
[22] Securities and Exchange Commission, Institutional In-
vestor Study Report of the Securities and Exchange
Commission, Chapter 4, "Investment Advisory Com-
plexes", pp. 325-347. (Washington, D.C.: U.S. Govern-
ment Printing Office, 1971.)
[23] Sharpe, William F. "Capital Asset Prices: A Theory of
Market Equilibrium under Conditions of Risk." Journal
of Finance, Vol. XIX (September 1964), pp. 425-442.
[24] Sharpe, William F. Portfolio Theory and Capital
Markets. (New York: McGraw-Hill, 1970.)
[25] Treynor, Jack L. "How to Rate the Management of In-
vestment Funds." Harvard Business Review, Vol. XLII1
(January-February 1965), pp. 63-75.
[26] Treynor, Jack L. "The Performance ol Mutual Funds in
the Period 1945-1964: Discussion." Journal of Finance,
Vol. XXHI (May 1968). pp. 418-419.
[27] Wagner, Wayne H., and Lau, Sheila. "The Effect of
Diversification on Risk." Financial Analysts Journal,
Vol. 26 (November-December 1971), pp. 48-53.
3-30
-------
by Franco Modigliani and Gerald A. Pogue
An Introduction
to
Risk and Return
Concepts and Evidence - II
6. The Relationship Between
Expected Return and Risk:
The Capital Asset Pricing Model*
The first part of this article developed two
measures of risk: one is a measure of total risk
(standard deviation), the other a relative index of
systematic or nondiversifiable risk (beta). Thfe beta
measure would appear to be the more relevant for
the pricing of securities. Returns expected by in-
vestors should logically be related to systematic as
opposed to total risk. Securities with higher sys-
tematic risk should have higher expected returns.'
The question to be considered now is the form of
the relationship between risk and return. In this
section we describe a relationship called the
"Capital Asset Pricing Model" (CAPM), which is
based on elementary logic and simple economic
principles. The basic postulate underlying finance
theory is that assets with the same risk should have
1. Footnotes appear at end of article.
* This article is the second part of a two-part article
which was originally prepared as Chapter 2 of A
Study of Investment Performance Fees (Heath-
Lexington Books, forthcoming 1974). The re-
i search was supported by a grant from the Invest-
ment Company Institute, Washington, D.C, The
article will also appear in the forthcoming Finan-
cial Analysts Handbook (Sumner Levine, editor; to
be published by Dow Jones-Irwin). Franco
Modigliani is Professor of Finance, Sloan School
of Management, M.I.T. Gerald Pogue is Professor
of Finance at Baruch College, City University of
New York.
the same expected rate of return. That is, the prices
of assets in the capital markets should adjust until
equivalent risk assets have identical expected
returns.
To see the implications of this postulate, let us
consider an investor who holds a risky portfolio2
with the same risk as the market portfolio (beta
equal to 1.0). What return should he expect?
Logically, he should expect the same return as that
of the market portfolio.
Let us consider another investor who holds a
riskless portfolio (beta equal to zero). The investor
in this case should expect to earn the rate of return
on riskless assets such as treasury bills. By taking
no risk, he earns the riskless rate of return.
Now let us consider the case of an investor who
holds a mixture of these two portfolios. Assuming
he invests a proportion X of his money in the risky
portfolio and (1 - X) in the riskless portfolio,
wtiat risk does he bear and what return should he
expect? The risk' of the composite portfolio is
easily •computed when we recall that the beta of a
portfolio is simply a weighted average of the com-
ponerit ^security betas, where the weights are the
portfolio proportions. Thus the portfolio beta, j8p,
is a weighted average of the beta of the market
portfolio and the beta of the risk-free rate.
However, the market beta is 1.0, and that of the
risk-free rate is zero. Therefore
0 + X
= X.
(11)
Thus /3p is equal to the fraction of his money in-
vested in the risky portfolio. If 100 per cent or less
of the investor's funds is invested in the risky port-
folio, his portfolio beta will be between zero and
1.0. If he borrows at the risk-free rate and invests
the proceeds in the risky portfolio, his portfolio
beta will be greater than 1.0.
The expected return of the composite portfolio
is also a weighted average of the expected returns
on the two-component portfolios; that is,
RF + X • E(Rm),
(12)
where E(Rp), E(Rm), and Rf are the expected re-
turns on the portfolio, the market index, and the
risk-free rate. Now, from Equation (11) we know
3-31
-------
that X is equal to /3p. Substituting into Equation
(12), we have
= (1-/3P) • Rf + ftp - E(Rm),
E(RP) = RF + ftp . (E(Rm) - RF) (13)
Equation (13) is the Capital Asset Pricing Model
(CAPM), an extremely important theoretical
result. It says that the expected return on a port-
folio should exceed the riskless rate of return by an
amount which is proportional to the portfolio beta.
That is, the relationship between return and risk
should be linear.
The model is often stated in "risk-premium"
form. Risk premiums are obtained by subtracting
the risk-free rate from the rates of return. The ex-
pected portfolio and market risk premiums
(designated E(rp) and E(rm) respectively) are given
by
E(rp) = E(RP) - RF, (14a)
E(rm) = E(RM) - RF. (14b)
Substituting these risk premiums into Equation
(13), we obtain
E(rp) = ftp • E(rm). (15)
In this form, the CAPM states that the expected
risk premium for the investor's portfolio is equal to
its beta value times the expected market risk
premium.
We can illustrate the model by assuming that
short-term (risk-free) interest rate is 6 per cent and
the expected return on the market is 10 per cent.
The expected risk premium for holding the market
portfolio is just the difference between the 10 per
cent and the short-term interest rate of 6 per cent,
or 4 per cent. Investors who hold the market port-
folio expect to earn 10 per cent, which is 4 per
cent greater than they could earn on a short-term
market instrument for certain. In order to satisfy
Equation (13), the expected return on securities or
portfolios with different levels of risk must be:
Expected Return for Different Levels of Portfolio Beta
Beta
0.0
0.5
1.0
1.5
2.0
Expected Return
6%
8%
10%
12%
14%
The predictions of the model are inherently sen-
sible. For safe investments (/3 = 0), the model
predicts that investors would expect to earn the
risk-free rate of interest. For a risky investment (/3
> 0) investors would expect a rate of return pro-
portional to the market sensitivity (/3) of the in-
vestment. Thus, stocks with lower than average
market sensitivities (such as most utilities) would
offer expected returns less than the expected
market return. Stocks with above average values oi
beta (such as most airline securities) would offei
expected returns in excess of the market.
In our development of CAPM we have made a
number of assumptions that are required if the
model is to be established on a rigorous basis.
These assumptions involve investor behavior and
conditions in the capital markets. The following is
a set of assumptions that will allow a simple
derivation of the model.
(a) The market is composed of risk-averse investors
who measure risk in terms of standard deviation of
portfolio return. This assumption provides a basis
for the use of beta-type risk measures.
(b) All investors have a common time horizon for in-
vestment decision making (e.g., one month, one
year, etc.). This assumption allows us to measure in-
vestor expectations over some common interval,
thus making comparisons meaningful.
(c) All investors are assumed to have the same ex-
pectations about future security returns and risks.
Without this assumption, the analysis would become
much more complicated.
(d) Capital markets are perfect in the sense that all
assets are completely divisible, there are no trans-
actions costs or differential taxes, and borrowing
and lending rates are equal to each other and the
same for all investors. Without these conditions,
frictional barriers would exist to the equilibrium
conditions on which the model is based.
While these assumptions are sufficient to derive
the model, it is not clear that all are necessary in
their current form. It may well be that several of
the assumptions can be substantially relaxed
without major change in the form of the model. A
good deal of research is currently being conducted
toward this end.
While the CAPM is indeed simple and elegant,
these qualities do not in themselves guarantee that
it will be useful in explaining observed risk-return
patterns. In Section 8 we will review the empirical
literature on attempts to verify the model.
7. Measurement of Security
and Portfolio Beta Values
The basic data for estimating betas are past rates of
return earned over a series of relatively short in-
tervals— usually days, weeks, or months. For
example, in Tables 3 and 4 we present calculations
3-32
-------
TABLE 3: REGRESSION STATISTICS FOR 30 RANDOMLY SELECTED SECURITIES*
January
SECURITY
1 City Investing Co.
2 Foster Wheeler
3 Pennsylvania Dixie
4 National Gypsum Co.
5 Radio Corp. Of America
6 Fox Film Corp.
7 Intercontinental Rubber
8 National Department
9 Phillips Jones Corp.
10 Chrysler Corp.
11 American Hide & Leather
12 Adams Express
13 Caterpillar Tractor
14 Continental Steel Co.
15 Marland Oil Co.
16 Air Reduction Co.
17 National Aviation
18 NA Tomas Co.
19 NYSE Index
20 American Ship Building
21 James Talcott
22 Jewel Tea Co. Inc.
23 International Carrier
24 Keystone Steel & Wire
25 Swift & Co.
26 Southern California
27 Bayuk Cigars
28 First National Store
29 National Linen Service
30 American Snuff
31 Homestake Mining Co.
32 Commercial Paper
.. .Mean Sec. Values
...Standard Deviations
d)
NOBS
306.00
306.00
306.00
306.00
306.00
306.00
306.00
306.00
306.00
306.00
306.00
306.00
306.00
306.00
306.00
306.00
306.00
306.00
306.00
306.00
306.00
306.00
306.00
306.00
306.00
306.00
306.00
306.00
306;00
306.00
306.00
306.00
306.00
0.0
(2)
ALPH
0.30"
-0.12
-0.20
-0.18
0.02
-0.04
0.69
-0.05
0.36
-0.26
0.55
0.11
0.43
0.21
0.06
-0.59
0.22
0.28
0.0
0.31
0.33
0.21
0.34
0.18
-0.09
0.00
-0.04
-0.08
0.61
0.17
0.16
0.0
0.13
0.28
'Based on monthly data, regression results sorted
0)
SE.A
0.53
0.49
0.47
0.32
0.38
0.53
0.64
0.45
0.44
0.37
0.66
0.23
0.32
0.36
0.29
0.29
0.39
0.63
0.0
0.52
0.42
0.32
0.26
0.30
0.30
0.22
0.39
' 0.31
0.33
0.25
0.38
0.0
0.39
0.12
by beta
1945 - June
(4)
BETA
1.67
1.57
1.40
1.38
1.35
1.31
1.28
1.26
1.25
1.21
1.16
1.16
1.14
1.12
1.11
1.08
1.04
1.01
1.00
0.99
0.98
0.95
0.93
0.84
0.81
0.77
0.71
0.67
0.63
0.54
0.24
0.0
1.05
0.31
(column 4).
1970
(5)
SE.B
0.14
0.13
0.12
0.08
0.10
0.14
0.17
0.12
0.12
0.10
0.17
0.06
0.08
0.10
0.08
0.08
0.10
0.17
0.0
0.14
0.11
0.08
0.07
0.08
0.08
0.06
0.10
0.08
0.09
0.07
0.10
0.0
0.10
0.03
(6)
SE.R v
9.20
8.36
8.15
5.45
6.60
9.15
10.95
7.73
7.54
6.29
11.36
3.93
5.45
6.22
4.99
4.98
6.71
10.88
0.0
9.01
7.23
5.42
4.39
5.19
5.08
3.77
6.76
5.33
5.75
4.33
6.60
0.0
6.76
2.10
(7)
R**2
31.43
32.98
29.33
47.29
37.02
22.35
16.13
27.05
27.89
34.12
12.78
54.87
38.09
31.31
40.69
39.73
25.15
10.72
0.0
14.53
20.43
30.14
38.41
26.90
26.08
36.60
13.49
18.01
14.50
17.74
1.77
0.0
27.25
11.85
(8)
ARPJ
1.45
0.96
0.77
0.77
0:95
0.87
1.58
0.81
1.22
0.58
1.35
0.91
1.22
0.99
0.82
0.16
0.94
0.98
0.69
0.99
1.01
0.87
0.98
0.76
0.47
0.53
0.45
0.38
1.04
0.54
0.33
0.28
0.86
0.33
(9)
SD.R
11.09
10.20
9.67
7.49
8.30
10.36
11.94
9.04
8.86
7.73 '
12.14
5.84'
6.92
7.50
6.47
6.41
7.74 '
11.50
3.73 '
9.73
8.09
6.47
5.58
6.05
5.89
4.72
7.26
5.88
6.20
4.77
6.65
0.17
7.88
2.13
(10)
CRPJ
0.87
0.46
0.33
0.50
0.62
0.38
0.92
0.41
0.85
0.28
0.67
0.75
0.99
0.72
0.62
-0.05
0.65
0.37
0.62
0.54
0.68
0.66
0.83
0.58
0.30
0.42
0.19
0.21
0.86
0.43
0.11
0.28
0.54
0.26
based on month-by-month rates of return for the
periods January 1945 to June 1970 (security betas)
and January 1960 to December 1971 (mutual fund
betas). The returns were calculated using Equation
(la).
It is customary to convert the observed rates of
return to risk premiums. Section 6 showed that risk
premiums are obtained by subtracting the rates of
return that could have been achieved by investing
in short-maturity risk-free assets, such as treasury
bills or prime commercial paper. This removes a
source of "noise" from the data. The noise stems
from the fact that observed returns may be higher
in some years simply because risk-free rates of in-
terest are higher. Thus, an observed rate of return
of eight per cent might be regarded as satisfactory
if it occurred in 1960, but as a relatively low rate
of return when interest rates were at all-time highs
DESCRIPTION OF COLUMNS IN TABLES 3 AND 4
Column
Number Symbol Description
1
2
3
4
5
6
7
8
9
10
NOBS
ALPHA
SE.A
BETA
SE.B
SE.R
R**2
ARPJ
SD-R
CRPJ
Number of Monthly Returns
The Estimated Alpha Value
Standard Error pf Alpha
Estimated Beta Coefficient
Standard Error of Beta
Standard Error of the Regression —
an Estimate of the Unsystematic
Risk
R! Expressed in Percentage Terms
Arithmetic Average of Monthly
Risk Premiums
Standard Deviation of Monthly Risk
Premiums
Geometric (Time-Weighted) Average
of Monthly Risk Premiums
3-33
-------
TABLE 4.
REGRESSION STATISTICS FOR 49 MUTUAL FUNDS*
January 1960 - December 1971
SECURITY
1 McDonnell Fund
2 Value Line Spec. Sit.
3 Keystone S-4
4 Chase Fund of Boston
5 Equity Progress
6 Oppenheimer Fund
7 Fidelity Trend Fund
8 Fidelity Capital
9 Keystone K-2
10 Delaware Fund
11 Keystone S-3
12 Putnam Growth Fund
13 Scudder Special Fund
14 Energy Fund
15 One William Street
16 The Dreyfus Fund
17 Mass. Investors Gr. Stk.
18 Windsor Fund
19 Axe-Houghton Stock
20 S&P 500 Stock Index
21 T. Rowe Price Gr. Stk.
22 Mass. Investors Trust
23 Bullock Fund
24 Keystone S-2
25 Eaton & Howard Stock
26 The Colonial Fund
27 Fidelity Fund
28 Invest. Co. of America
29 Hamilton Funds - HDA
30 Affiliated Fund
31 Keystone S-1
32 Axe-Houghton Fund B
33 American Mutual Fund
34 Pioneer Fund
35 Chemical Fund
36 Stein R&F Balanced Fd.
37 Puritan Fund
38 Value Line Income Fd.
39 Geo. Putnam Fd. Boston
40 Anchor Income
41 Loomis-Sayles Mutual
42 Wellington Fund
43 Massachusetts Fund
44 Natlon-Wide Sec.
45 Eaton & Howard Bal. Fd.
46 American Business Shares
47 Keystone K-1
48 Keystone B-4
49 Keystone B-2
50 Keystone B-1
51 30 Day Treasury Bills
. . .Mean Sec. Values
. . .Standard Deviations
0)
NOBS
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
144.00
0.0
(2)
ALPH
0.58
0.02
0.03
0.11
-0.54
0.42
0.79
0.41
0.08
0.18
0.18
0.21
0.39
0.06
0.13
0.17
0.15
0.18
0.39
0.0
0.05
-0.02
0.09
0.04
-0.05
0.06
0.15
0.26
-0.12
0.08
0.03
0.01
0.20
0.24
0.57
0.06
0.19
0.07
0.07
-0.03
0.05
-0.12
0.04
-0.32
-0.07
0.12
0.01
0.12
0.05
-0.08
0.0
0.12
0.22
•Based on monthly data, regression results sorted
(3)
SE.A
0.82
0.40
0.28
0.33
0.41
0.24
0.29
0.24
0.22
0.19
0.19
0.19
6.28
0.18
0.22
0.14
0.16
0.16
0.30
0.0
0.14
0.14
0.19
0.12
0.13
0.19
0.11
0.20
0.23
0.10
0.10
0.20
0.20
0.16
0.25
0.10
0.15
0.17
0.10
0.13
0.10
0.13
0.11
0.15
0.12
0.09
0.11
0.13
0.10
0.10
0.0
0.19
0.12
by beta
(4)
BETA
1.50
1.48
1.43
1.42
1.26
1.23
1.23
1.20
1.17
1.15
1.14
1.13
1.12
1.10
1.06
1.04
1.03
1.03
1.02
1.00
0.98
0.97
0.96
0.96
0.95
0.95
0.95
0.95
0.93
0.90
0.88
0.86
0.85
0.84
0.83
0.79
0.78
0.78
0.77
0.74
0.74
0.72
0.72
0.67
0.62
0.53
0.53
0.30
0.16
0.07
0.0
0.93
0.30
(column 4).
(5)
SE.B
0.22
0.11
0.08
0.09
0.11
0.06
0.08
0.06
0.06
0.05
0.05
0.05
0.07
0.05
0.06
0.04
0.04
0.04
0.08
0.0
0.04
0.04
0.05
0.03
0.03
0.05
0.03
0.05
0.06
0.03
0.03
0.05
0.05
0.04
0.07
0.03
0.04
0.04
0.03
0.04
0.03
0.03
0.03
0.04
0.03
0.02
0.03
0.03
0.03
0.03
0.0
0.05
0.03
(6)
SE.R
9.76
4.78
3.38
3.94
4.85
2.89
3.52
2.81
2.63
2.32
2.32
2.25
3.33
2.18
2.66
1.69
1.96
1.95
3.62
0.0
1.72
1.72
2.32
1.45
1.52
2.27
1.31
2.40
2.73
1.22
1.21
2.44
2.38
1.88
3.03
1.21
1.79
2.01
1.18
1.60
1.22
1.54
1.26
1.78
1.46
1.10
1.32
1.51
1.16
1.21
0.0
2.32
1.42
(7)
R**2
25.18
57.62
71.77
64.78
48.89
72.16
63.39
72.17
73.90
77.62
77.50
78.19
61.93
78.39
69.33
84.40
79.65
79.87
52.96
0.0
82.08
82.07
71.10
86.12
84.75
71.24
88.08
68.79
62.55
88.55
88.18
63.68
64.35
73.85
51.50
86.05
72.89
67.96
85.75
75.24
83.96
75.60
82.16
66.45
71.62
76.96
69.59
35.82
22.03
4.43
0.0
69.25
17.50
(8)
ARPJ
1.13
0.57
0.55
0.63
-0.08
0.88
1.24
0.85
0.51
0.60
0.60
0.62
0.80
0.46
0.52
0.55
0.52
0.56
0.76
0.37
0.41
0.34
0.44
0.39
0.30
0.41
,0.50
0.61
0.22
0.41
0.35
0.32
0.51
0.55
0.88
0.35
0.48
0.36
0.35
0.24
0.32
0.14
0.30
-0.08
0.16
0.31
0.21
0.23
0.11
-0.06
0.34
0.46
0.27
(9)
SD.R
11.24
7.32
6.34
6.61
6.77
5,46
5.80
5.31
5.13
4.90
4.88
4.80
5.37
4.67
4.78
4.26
4.34
4.33
5.26
3.76
4.06
4.04
4.29
3.89
3.89
4.23
3.79
4.29
4.44
3.59
3.51
4.03
3.97
3.67
4.33
3.22
3.43
3.54
3.12
3.21
3.04
3.11 ;
2.98
3.07
2.74
2.28
2.39
1.88
1.31
1.23
0.12
4.25
1.64
(10)
CRPJ
0.67
0.30
0.35
0.41
-0.31
0.73
1.07
0.71
0.38
0.48
0.48
0.51
0.66
0.35
0.41
0.46
0.43
0.47
0.62
0.30
0.32
0.26
0.35
0.31
023
032
043
0.51
0.12
0.34
0.29
0.24
0.43
0.48
0.79
0.30
0.42
0.29
0.30
0.19
0.27 ,
0.09
0.26
-0.12
0.12
0.29
0.18 ,
0.21
0.10
-0.07
0.34
0.36
0.23
3-34
-------
in 1969. Rates of return expressed as risk
premiums will be denoted by small r's.
The market model of Equation (6), when ex-
pressed in risk-premium form, is the basic equation
used to estimate beta. The market model in risk-
premium form is given by
r= a + /3rm+
(16a)
The use of risk premiums instead of returns as in
Equation (6) simply changes the interpretation of
alpha, leaving beta unchanged. In the return form,
the expected value of alpha as given by the CAPM
is Rfe, column (7) the R2 in
percentage terms, columns (8) and (9) the arith-
metic average of monthly risk premiums and the
standard deviation, and column (10) the geometric
mean risk premium. The results are ranked in
terms of descending values of estimated beta. The
table includes summary results for the NYSE
market index and the prime commercial paper
risk-free rate.1 The last two rows of the table give
average values and standard deviations for the
sample. The average beta, for example, is 1.05,
slightly higher than the average of all NYSE stocks.
The beta value for a portfolio can be estimated
in two ways. One method is to compute the beta of
all portfolio holdings and weight the results by
portfolio representation. However, this method has
the disadvantage of requiring beta calculations for
each individual portfolio asset. The second method
is to use the same computation procedures used for
stocks, but to apply them to the portfolio returns.
In this way we can obtain estimates of portfolio
betas without explicit consideration of the port-
folio securities. We have used this approach to
compute portfolio and mutual fund beta values.
3-35
-------
Exhibit 9 shows the plot of the monthly risk
premiums on the 100-stock portfolio against the
NYSE index for the same 1945-1970 period. As in
the case of National Department Stores, the best-
fit line has been put through the points using
regression analysis. The slope of the line ($) is
equal tb 1.10, with a standard error of 0.03. Note
the substantial reduction in the standard error term
compared to the security examples. The estimated
alpha is 0.14, with a standard error of 0.10. Again,
we cannot conclude that the true alpha is different
from zero. Note that the points group much closer
to the line than in the National Department Store
plot. This results, of course, from the fact that
much of the unsystematic risk causing the points to
be scattered around the regression line in Exhibit 8
has been climated. The reduction is evidenced by
the R2 measure of 0.87 (versus 0.27 for National
Department Stores). Thus the market explains
more than three times as much of the return
variation of the portfolio than for the stock.
Table 5 gives regression results -for a sample of
49 mutual funds. The calculations are based on
monthly risk premiums for the period January
1960 to December 1971. The market is repre-
sented by the Standard & Poor's 500 Stock Index.
Average values and standard deviations for the 49
funds in the sample are shown in the last two rows
of the table. The average beta value for the group
is 0.93—indicating, on the average, that the funds
were less risky than the market index. Note the
relatively low beta values of the balanced and bond
funds, in particular the Keystone Bl, B2, and B4
bond funds. This result is due to the low systematic
risk of the bond portfolios.
Up to this point we have shown that it is a
relatively easy matter to estimate beta values for
stocks, portfolios, and mutual funds. Now, if the
RE
Stock
25.90
18.52
11.15
3.77
-3.60
-10.98
-18.35
-25.73
Exhibit 8
TURNS ON NATIONAL DEPARTMENT STORES VS. NYSE INDEX (% PER MONTH)
January 1945 - Juris 1970
— * —
X-
* * **
1C
* * *
* 2 2 -
+ * * *
* * * * *
* 2 8 *
_ 28 *** * * —
* ***** **
* ** ** 3 2 ****** *
** *2 **** 28*
2 * 2 *2 8*2 28 -
* * * * 28 2 8 * *
* ** * **3M83 3 8 * * 2 *
* *«*** 2883 *288* *
* "*2 8*228828288 ******
— ******228*288*28** ** —
* * * + * 2 * *** * * 2 8**
* 2 *22 88 ** ** * 2 8 * 2
* * * * * * *
- * * * * —
* * *2 88
* * * * *
* * *
1*1 1 1 1 1 1 1
NOBS = 306
a = -0.05%
SEa= 0.45%
£-1.26
SEo=0.12
df = 7.73%
R* = 0.27
7 = 0.81%
a(r) =9.04%
g = 0.41%
-8.1140 -5.4383 -2.7625 -0.0868 2.5890 5.2647 7.9405 10.6162
MKT.
3-36
-------
beta values are to be useful for investment decision
making, they must be predictable. Beta values
based on historical data should provide con-
siderable information about future beta values if
past measures are to be useful. How predictable
are the betas estimated for stocks, portfolios of
stocks, and mutual funds? Fortunately, we have
empirical evidence at each level.
Robert A. Levy [13]* has conducted tests of the
short-run predictability (also referred to as
stationarity) of beta coefficients for securities and
unmanaged portfolios of securities. Levy's results
are based on weekly returns for 500 NYSE stocks
for the period December 30, 1960 through Decem-
ber 18, 1970 (520 weeks). Betas were developed
for each security for ten non-overlapping 52-week
periods. To measure stationarity, Levy correlated
* References appear at end of article.
the 500 security betas from each 52-week period
(the historical betas) with the 52-week betas in the
following period (the future betas). Thus nine cor-
relation studies were performed for the ten periods.
To compare the stationarity of security and
portfolio betas, Levy constructed portfolios of 5,
10, 25, and 50 securities and repeated the same
correlation analysis for the historical portfolio
betas and future beta values for the same portfolios
in the subsequent period. The portfolios were con-
structed by ranking security betas in each period
and partitioning the list into portfolios containing
5, 10, 25 and 50 securities. Each portfolio con-
tained an equal investment in each security.
The results of Levy's 52-week correlation studies
are presented in Table 5. The average values of the
correlation coefficients from the nine trials were
0.486, 0.769, 0.853, 0.939, and 0.972 for port-
folios of 1, 5, 10, 25, and 50 stocks, respectively.
Exhibit 9
RETURNS ON 100 STOCK PORTFOLIO VERSUS NYSE INDEX (% PER MONTH)
January 1949 - June 1970
Stock
13.00
9.71
6.33
2.95
-0.43
-3.82
-7.20
-10.58
* *
*
*
* * *
*
* *** *
* * * ***** *
* 28* —
** 4 *3 * **
* 32 * 3 **2* *
* * **3 223* * *
***** * 2 8 2 *
* 3 2 3233 2882 —
* 3 3828823*2
* ** 4822.8422382 2 8
**2 5tf * 38
* * **3 828 2 *
* + *2882 8 32 -
2 * 288 «2 8* *
* 32*22222 *
* * * '38 * *
28 **3 8 *
* 2882
* **28*** *
* *
* *28 **
* .*28 —
* *
* *
*
_ **
6 *
*
q
I I 1 I I I I I
NOBS = 306
3= 0. 1 4%
SEa= 0.10%
l5-1.11
SEo = 0.03
O'e 1.64%
R2' = 0.87
7 = 0.91 %
<7(r) = 4.46%
9 = 0.81%
-8.1140 -5.4383 -2.7625 -0.0868 2.5890 5.2647 7.9405 10.6162
MKT.
3-37
-------
Correspondingly, the average percentages of the
variation in future betas explained by the historical
betas are 23.6, 59.1, 72.8, 88.2, and 94.5.
The results show the beta coefficients to be very
predictable for large portfolios and progressively
less predictable for smaller portfolios and in-
dividual securities. These conclusions are not af-
fected by changes in market performance. Of the
nine correlation studies, five covered forecast
periods during which the market performance was
the reverse of the preceding period (61-62, 62-63,
65-66, 66-67, and 68-69). Notably, the betas were
approximately as predictable over these five rever-
sal periods as over the remaining four intervals.5
The question of the stability of mutual fund beta
values is more complicated. Even if, as seen above,
the betas of large unmanaged portfolios are very
predictable, there is no a priori need for mutual
fund betas to be comparatively stable. Indeed, the
betas of mutual fund portfolios may change sub-
stantially over time by design. For example, a port-
folio manager may tend to reduce the risk exposure
of his fund prior to an expected market decline and
raise it prior to an expected market upswing.
However, the range of possible values for beta will
tend to be restricted, at least in the longer run, by
the fund's investment objective. Thus while one
does not expect the same standard of predictability
as for large unmanaged portfolios, it may never-
theless be interesting to examine the extent to
which fund betas are predictable.
Pogue and Conway [21] have conducted tests
for a sample of 90 mutual funds. The beta values
for the period January 1969 through May 1970
were correlated with values from the subsequent
period from June 1970 through October 1971. To.
test the sensitivity of the results to changes in the
return measurement interval, the betas for each
sub-period were measured for daily, weekly, and
monthly returns. The betas were thus based on
very different numbers of observations, namely
357, 74, and 17, respectively. The resulting
correlation coefficients were 0.915, 0.895, and
0.703 for daily, weekly, and monthly betas. Corre-
spondingly, the average percentages of variation in
second-period betas explained by first-period
values are 84, 8'1, and 49, respectively. The results
support the contention that historical betas contain
useful information about future values. However,
the degree of predictability depends on the extent
to which measurement errors have been eliminated
from beta estimates. In the Pogue-Conway study,
the shift from monthly to daily returns reduced the
average standard error of the estimated beta values
from 0.11 to 0.03, a 75 per cent reduction. The
more accurate daily estimates resulted in a much
higher degree of beta predictability, the correlation
between sub-period betas increasing from 0.703 to
0.915.6
Exhibit 10 shows a Pogue-Conway plot of the
first-period versus second-period betas based on
daily returns. The figure illustrates the high degree
of correlation between first- and second-period
betas.
In summary, we can conclude that estimated in-
dividual security betas are not highly predictable.
TABLE 5, CORRELATION OF 52-WEEK BETA FORECASTS WITH MEASURED VALUES
FOR PORTFOLIOS OF N SECURITIES
1962-1970
Forecast for
52 Weeks
Ended
12/28/62
12/27/63
12/25/64
12/24/65
12/23/66
12/22/67
12/20/68
12/19/69
12/18/70
Quadratic
Mean
Source: Robert A. Levy
1
.385
.492 -
.430
.451
.548
.474
.455
.556
.551
.486
[13], Table 2, p.
Product
5
.711
.806
.715
.730
.803
. .759
.732
.844
.804
.769
57.
Moment Correlations: N=
10
.803
.866
.825
.809
.869
.830
.857
.922
.888
.853
25
.933
.931
.945
.936
.952
.900
.945
.965
.943
.939
50
.988
.963
.970
'.977
.974
.940
.977
.973
.985
.972
3-38
-------
EXHIBIT 10
INTERPERIOD BETA COMPARISON:
DAILY DATA FOR 90 MUTUAL FUNDS
Second
Period
(June '70
to 1-50
Oct. '71)
1.00
0.50
"I
•
•
$
*v
~.
•
*-
•\°
•
.'
•
0 0.50 1.00 1.50 2.00
Beta - - First Period (Jan '69 to May '70)
Source: Pogue and Conway [21]
not necessarily zero for any single stock or single
period of time. After the fact, we would expect to
observe
Levy's tests indicated that an average of 24 per
cent of the variation in second-period betas is ex-
plained by historical values. The betas of his port-
folios, on the other hand, were much more predict-
able, the degree of predictability increasing with
portfolio diversification. The results of the Pogue
and Conway study and others show that fund
betas, not unexpectedly, are not as stable as those
for unmanaged portfolios. Nonetheless, two-thirds
to three-quarters of the variation in fund betas can
be explained by historical values.
The reader should remember that a significant
portion of the measured changes in estimated beta
values may not be due to changes in the true
values, but rather to measurement errors. This ob-
servation is particularly applicable to individual
security betas where the standard errors tend to be
large.
8. Tests of The Capital Asset
Pricing Model *
The major difficulty in testing the CAPM is that
the model is stated in terms of investors' ex-
pectations and not in terms of realized returns. The
fact that expectations are not always realized in-
troduces an error term, which from a statistical
point of view should be zero on the average, but
The material in this section was also prepared as an ap-
pendix to testimony to be delivered before the Federal
Communications Commission by S.C. Myers and GA.
Pogue.
/3;(Rm-R/)
(17a)
where Ry, Rm, and R/ are the realized returns on
stock j, the market index, and the riskless asset;
and ej is the residual term.
If we observe the realized returns over a series of
periods, the average security return would be given
by
Ry= Rf
.(lib)
where R;, RM, and RF are the average realized
returns on the stock, the market and the risk-free
rate. If the CAPM is correct, the average residual
term, e,-, should approach zero as the number of
periods used to compute the average becomes
large. To test this hypothesis, we can regress the
average returns, R/, for a series of stocks (j== 1, .. .,
N) on the stocks' estimated beta values, $/> during
the period studied. The equation of the fitted line
is given by
R/= yo + y£j + pj, (iga)
where ya and yl are the intercept and slope of the
line, and /u,y is the deviation of stock j from the
line. By comparing Equations (17b) and (18a), we
infer that if the CAPM hypothesis is valid, ^
should equal e, and hence should be smalL Fur-
thermore, .it should be uncorrelated with /3,, and
hence we can _also infer that ya and y^ should
equal RF and RM - RF respectively.
The hypothesis is illustrated in Exhibit 11. Each
plotted point represents one stock's realized return
versus the stock's beta. The vertical distances of
the points from the CAPM theoretical line (also
called the "market line") represent the mean
residual returns, e/. Assuming the CAPM to be
correct, the e/ should be uncorrelated with the /3,
and thus the regression equation fitted to these
points should be 0) lineaj, (2) upward sloping with
slope equal to RM - RF, and:(3) should pass
through the vertical axis at the risk-free rate.
Expressed in risk-premium form, the equation of
the fitted line is
iy= y» + y&j + /*/. (isb)
where 7/ is the average realized risk premium for
stock / Comparing Equation (18b) to the CAPM
3-39
-------
in risk-premium form [Equation (15)], the pre-
dicted values for y0 and y_i are_0 and rm. the
mean market risk premium (RM - Rf). Thus shift-
ing to risk premiums changes the predicted value
only for y0, but not for y,.
Other Measures of Risk
The hypothesis just described is only true if beta
is a complete measure of a stock's risk. Various
alternative risk measures have been proposed,
however. The most common alternative hypothesis
states that expected return is related to the stand-
ard deviation of return—that is, to a stock's total
risk, which includes both systematic and unsystem-
atic components.
Which is more important in explaining average
observed returns on securities, systematic or unsys-
tematic risk? The way to find out is to fit an ex-
panded equation to the data:
A A
Ry = 7o + 7l/3y + ya(SEy) + /Ay . (19)
Here /3y is a measure of systematic risk and SEy a
measure of unsystematic risk.7 Of course, if the
CAPM is exactly true, then y2 will be zero—that
is, SEy will contribute nothing to the explanation
of observed security returns.
Tests of the Capital Asset Pricing Model
If the CAPM is right, empirical tests would show
the following:
1. On the average, and over long periods of time, the
securities with high systematic risk should have high
rates of return.
2. On the average, there should be a linear relationship
between systematic risk and return.
3. The slope of the relationship (y,) should be equal to
the mean market risk premium (RM - RF) during
the period used.
4. The constant term (y0) should be equal to the mean
risk-free rate (R/r).
5. Unsystematic risk, as measured by SEy, should play
no significant role in explaining differences in
security returns.
These predictions have been tested in several
recent statistical studies. We will review some of
the more important ones. Readers wishing to skip
the details may proceed to the summary at the end
of this section. We will begin by summarizing
results from studies based on individual securities,
and then we will turn to portfolio results.
Results of Tests Based on Securities
The Jacob Study-."The Jacob study [9] deals with
the 593 New York Stock Exchange stocks for
Exhibit 11
RELATIONSHIP BETWEEN AVERAGE
RETURN (ftp AND SECURITY RISK
Average
Security
Return
Theoretical Line
X X _ X
X " X x
XX .
y X ^^
X Fitted Line
X X
X ' XX XX
xx x x
X
X
| I I I
-A
0.5 1.0 1.5 2.0 s
Risk
which there is complete data from 1946 to 1965.
Regression analyses were performed for the 1946-
55 and 1956-65 periods, using both monthly and
annual security returns. The relationship of mean
security returns and beta values is shown in Table
6. The last two columns of the table give the
theoretical values for the coefficients, as predicted
by the CAPM.
The results show a significant positive relation-
ship between realized return and risk during each
of the 10-year periods. For example, in 1956-65
there was a 6.7 per cent per year increase in
average return for a one-unit increase in beta.
Although the relationships shown in Table 6 are all
positive, they are weaker than those predicted by
the CAPM. In each period yl is less and y0 is
greater than the theoretical values.
The rviilier-Scholes Study. The Miller-Scholes
research [19] deals with annual returns for 631
stocks during the 1954-63 period. The results of
three of their tests are reported in Table 7. The
tests are (1) mean return versus beta, (2) mean
return versus unsystematic ri$!$ (SEr)2? and (3)
mean return versus both beta arid unsystematic
risk.
3-40
-------
The results for the first test show a significant pie, a substantial positive correlation exists lie-
positive relationship between mean return and tween beta and ($£/. Thus unsystematic risk
beta. A one-unit increase in beta is associated with
a 7.1 per cent increase in mean return.
The results for the second test do not agree with
the CAPM's predictions. That is, high unsystematic
risk is apparently associated with higher realized
returns. However, Miller and Scholes show that
this correlation may be largely spurious (i.e., it may
be due to statistical sampling problems). For exam-
will appear to be significant in tests from which
beta has been omitted, even though it may be
unimportant to the pricing of securities. This sort
of statistical correlation need not imply a causal
link between the variables.
Test number (3) includes both beta and (SEy)2 in
the regression equation. Both are found to be
significantly positively related to mean return. The
TABLE 6. RESULTS OF JACOB'S STUDY
f/ = TO + yfij + M/
Tests Based on 593 Securities
Period Return
Interval
46-55 Monthly
Yearly
56-65 Monthly
Yearly
(a) Coefficient units are: monthly
(b) Standard error.
Source: Jacob [9], Table 3, pp.
Regression Results'8' Theoretical Values
To
0.80
8.9
0.70
6.7
data,
7, R* 7o=0 r, = RM-RF
0.30 0.02 0 1.10
(0.07)(b)
5.10 0.14 0 14.4
(0.53)
0.30 0.03 0 0.8
(0.06)
6.7 0.21 0 10.8
(0.53)
percent per month; annual data, percent per year.
827-828.
TABLE 7
Regression
. RESULTS OF THE MILLER AND SCHOLES STUDY
R/ = y0 + y^i + yi(SEy)2 + fi/
Annual Rates of Return 1954-1963
Tests Based on 631 Securities
Results'8' Theoretical Values
7» y, y. R2 yo y, y,
12.2
( 0.7)(b)
16.3
( 0.4)
12.7
( 0.6)
7.1
(0.6)
4.2
(0.6)
0.19 2.8 8.5
39.3 0.28 2.8 8.5
( 2.5)
31.0 0.33 2.8 8.5
( 2.6)
0
0
0
(a) Units of Coefficients: per cent per year.
(b) Standard error.
Source: Miller and
Scholes [19],
Table 1B, p. 53.
3-41
-------
inclusion of (SE";)2 has somewhat weakened the
relationship of return and beta, however. A one-
unit increase in beta is now associated with only a
4.2 per cent increase in mean return.
The interpretation of these results is again com-
plicated by the strong positive correlation between
beta and (S§/, and by other sampling problems.8
A significant portion of the correlation between
mean return and (SIi,)2 may well be a spurious
result. In any case, the results do show that stocks
with high systematic risk tend to have higher rates
of return.
Results for Tests Based on Portfolio Returns
Tests based directly on securities clearly show
the significant positive correlation between return
and systematic risk. Such tests, however, are not
the most efficient method of obtaining estimates of
the magnitude of the risk-return tradeoff. The tests
are inef ient for two reasons.
The first problem is well known to economists.
It is called "errors in variables bias" and results
from the fact that beta, the independent variable in
the test, is typically measured with some error.
These errors are random in their effect—that is,
some stocks' betas are overestimated and some are
underestimated. Nevertheless, when these esti-
mated beta values are used in the test, the measure-
ment errors tend to attenuate the relationship be-
tween mean return and risk.
By carefully grouping the securities into port-
folios, much of this measurement error problem
can be eliminated. The errors in individual stocks'
betas cancel out so that the portfolio beta can be
measured with much greater precision. This in turn
means that tests based on portfolio returns will be
more efficient than tests based on security returns.
The second problem relates to the obscuring ef-
fect of residual variation. Realized security returns
have a large random component, which typically
accounts for about 70 per cent of the variation of
return. (This is the diversifiable or unsystematic
risk of the stock.) By grouping securities into port-
folios, we can eliminate much of this "noise" and
thereby get a much clearer view of the relationship
between return and systematic risk.
It should be noted that grouping does not distort
the underlying risk-return relationship. The rela-
tionship that exists for individual securities is
exactly the same for portfolios of securities.
Friend and Blume Studies. Professors Friend and
Blume [3,8] have conducted two interrelated risk-
return studies. The first examines the relationship
between long-run rates of return and various risk
measures. The second is a direct test of the CAPM
In the first study [8], Friend and Blume con-
structed portfolios of NYSE common stocks at th<
beginning of three different holding periods. The
periods began at the ends of 1929, 1948, and 1956
All stocks for which monthly rate-of-return data
TABLE 8. RESULTS OF FRIEMD-BIUME STUDY
Returns from a yearly revision policy for
stocks classified by beta for various periods.
Holding Period
Port-
folio
No.
1
2
3
4
5
6
7
8
9
10
Monthly
Source:
Beta
0.19
0.49
0.67
0.81
0.92
1.02
1.15
1.29
1.49
2.02
1929-1969
Mean
Return
%
0.79
1.00
1.10
1.28
1.26
1.34
1.42
1.53
1.55
1.59
1948-1969
Beta
0.45
0.64
0.76
0.85
0.94
1.03
1.12
1.23
1.36
1.67
Mean
Return
%
0.99
1.01
1.25
1.30
1.35
1.37
1.32
1.33
1.39
1.36
1956-1969
Beta
0.28
0.51
0.66
0.80
0.91
1.03
1.16
1.30
1.48
1.92
i
Mean
Return
%
0.95
0.98
1.12
1.18
1.17
1.14
1.10
1.18
1,15
1.10
arithmetic mean returns
Friend and
Blume [8], Table 4, p. 10.
3-42
-------
could be obtained for at least four years preceding
the test period were divided into 10 portfolios. The
securities were assigned on the basis of their betas
during the preceding four years—the 10 per cent
of securities with the lowest betas to the first port-
folio, the group with the next lowest betas to the
second portfolio, and so on.
After the start of the test periods, the securities
were reassigned annually. That is, each stock's
estimated beta was recomputed at the end of each
successive year, the stocks were ranked again on
the basis of their betas, and new portfolios were
formed. This procedure kept the portfolio betas
reasonably stable over time.
The performance of these portfolios is sum-
marized in Table 8. The table gives the arithmetic
mean monthly returns and average beta values for
each of the 10 portfolios and for each test period.
For the 1929-69 period, the results indicate a
strong positive association between return and
beta. For the 1948-69 period, while higher beta
portfolios had higher returns than portfolios with
lower betas, there was little difference in return
among portfolios with betas greater than 1.0. The
1956-69 period results do not show a clear rela-
tionship between beta and return. On the basis of
these and other tests, the authors conclude that
NYSE stocks with above average risk have higher
returns than those with below average risk, but that
there is little payoff for assuming additional risk
within the group of stocks with above average
betas.
In their second study [3], Blume and Friend
used monthly portfolio returns during the 1955-68
period to test the CAPM. Their tests involved fit-
ting the coefficients of Equation (18a) for three
sequential periods: 1955-59, l%0-&4. and l%5-
68. The authors also added u factor to the
regression equation to test for the linearity of the
risk-return relationship."
The values obtained for y0 and y, are not in
line with the Capital Asset Pricing Model's predic-
tions, however. In the first two periods, ?„ is sub-
stantially larger than the theoretical value. In the
third period, the reverse situation exists, with y0
substantially less than predicted. These results im-
ply that ylt the slope of the fitted line, is less than
predicted in the first two periods and greater in the
third.10 Friend and Blume conclude that "the com-
parisons as a whole suggest that a linear model is a
tenable approximation of the empirical relation-
ship between return and risk for NYSE stocks over
the three periods covered.""
Black, Jensen, and Scholes. This study [1] is a
careful attempt to reduce measurement errors that
would bias the regression results. For each year
from 1931 to 1965, the authors grouped all NYSE
stocks into 10 portfolios. The number of securities
in each portfolio increased over the 35-year period
from a low of 58 securities per portfolio in 1931 to
a high of 110 in 1965.
Month-by-month returns for the portfolios were
computed from January 1931 to December 1965.
Average portfolio returns and portfolio betas were
computed for the 35-year period and for a variety
of sub-periods. The results for the complete period
are shown in Table 9. The average monthly port-
folio returns and beta values for the 10 portfolios
are plotted in Exhibit 12. The results indicate that
over the complete 3 5-year period, average return
increased by approximately 1.08 per cent per
month (13 per cent per year) for a one-unit in-
TABLE 9. RESULTS OF BLACK-JENSEN-SCHOLES STUDY
1931-1965
Tests Based on 10 Portfolios
(Averaging 75 Stocks per Portfolio)
Regression Results'3'
Theoretical Values
70 =
-y, = RM-RF
0.519
(0.05)
(b)
1.08
(0.05)
0.90
0.16
1.42
(a) Units of Coefficients: per cent per month.
(b) Standard error.
Source: Black, Jensen, and Scholes [1], Table 4, p. 98, and Figure 7, p. 104.
3-43
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crease in beta. This is about three-quarters of the
amount predicted by the CAPM. As Exhibit 12
shows, there appears to be little reason to question
the linearity of the relationship over the 35-year
period.
Black, Jensen, and Scholes also estimated the
risk-return tradeoff for a number of subperiods.12
The slopes of the regression lines tend in most
periods to understate the theoretical values, but are
generally of the correct sign. Also, the subperiod
relationships appear to be linear.
This paper provides substantial support for the
hypothesis that realized returns are a linear func-
tion of systematic risk values. Moreover, it shows
that the relationship is significantly positive over
long periods of time.
Fama and MacBeth. Fama and MacBeth [6] have
extended the Black-Jensen-Scholes tests to include
two additional factors. The first is an average of
the /?/ for all individual securities in portfolio p,
designated /§P2. The second is a similar average of
the residual standard deviations (Sfiy) for all stocks
in portfolio p, designated S£P. The first term tests
for nonlinearities in the risk-return relationship,
the second for the impact of residual variation.
The equation of the fitted line for the Fama-
MacBeth study is given by
RP = y» + yJP + yjip* + y3S§p + Mp , (20)
where, according to the CAPM, we should expect
y2 and ya to have zero values.
The results of the Fama-MacBeth tests show that
while estimated values of y2 and ya are not equal
to zero for each interval examined, their average
values tend to be insignificantly different from
zero. Fama and MacBeth also confirm the Black-
Jensen-Scholes result that the realized values of y0
are not equal to Ry, as predicted by the CAPM.
Summary of Test Results
We will briefly summarize the major results of
the empirical tests.
1. The evidence shows a significant positive relationship
between realized returns and systematic risk.
However, the slope of the relationship (yj is usually
less than predicted by the CAPM.
2. The relationship between risk and return appears to
be linear. The studies give no evidence of significant
curvature in the risk-return relationship.
3. Tests that attempt to discriminate between the effects
of systematic and unsystematic risk do not yield
definitive results. Both kinds of risk appear to be
positively related to security returns. However, there
Exhibit 12
RESULTS OF BLACK, JENSEN
AND SCHOLES STUDY
1931 - 1965
.11
.10
.08
03
Z
DC
jjj .06
in
05
<
GC
III
.04
.02
.00
-.02
INTERCEPT = 0.00519
STD. ERR. = 0.00053
SLOPE
STD. ERR.
'0.01081
•• 0.00050
X
• X X
-x
X X'
0.0 0.5 1.0 1.5
SYSTEMATIC RISK
2.0
Average monthly returns versus systematic
risk for the 35-year period 1931-1965 for
ten portfolios and the market portfolio.
Source: Black, Jensen, and Scholes [1],
Figure 7, p. 104.
is substantial support for the proposition that the re-
lationship between return and unsystematic risk is at
least partly spurious—that is, it partly reflects
statistical problems rather than the true nature of
capital markets.
Obviously, we cannot claim that the CAPM is
absolutely right. On the other hand, the empirical
tests do support the view that beta is a useful risk
measure and that high beta stocks tend to be priced
so as to yield correspondingly high rates of return.
9. Measurement of Investment Performance
The basic concept underlying investment perform-
ance measurement follows directly from the risk-
return theory. The return on managed portfolios,
such as mutual funds, can be judged relative to the
3-44
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returns on unmanaged portfolios at the same
degree of investment risk. If the return exceeds the
standard, the portfolio manager has performed in a
superior way, and vice versa. Given this, it remains
to select a set of "benchmark" portfolios against
which the performance of managed portfolios can
be evaluated.
Performance Measures Developed
from the Capital Asset Pricing Model
The CAPM provides a convenient and familiar
standard for performance measurement; the bench-
mark portfolios are simply combinations of the
riskless asset and the market index. The return
standard for a mutual fund, for example, with beta
equal to PF, is equal to the risk-free rate (RF) plus
ftp times^he average realized risk premium on the
market (RM - RF)._Thus the return on the per-
formance standard (R5) is given by
Rs =
/3P(RM-RF),
(21)
where RM and Rf are the arithmetic average
returns on the market index and riskless asset
during the evaluation period. The performance
measure, designated ap, is equal to the difference
in average returns between the fund and its stan-
dard; that is,
a = p-
(22)
where RF is the arithmetic average return on the
fund. Under the CAPM assumption, the expected
values of RP and R5 are the same; therefore the ex-
pected value for the performance measure &p is
zero. Managed portfolios with positive estimated
values for ap have thus outperformed the standard,
and vice versa. Estimated values of alpha (&p) are
determined by regressing the portfolio risk
premiums on the corresponding market risk
premiums.
The interpretation of the estimated alpha must
take into consideration possible statistical
measurement errors. As we discussed in Section 7,
the standard error of alpha (SEJ is an indication
of the extent of the possible measurement error.
The larger the standard error, the less certain we
can be that measured alpha is a close ap-
proximation of the true value.13
A measure of the degree of statistical
significance of the estimated alpha value is given
by the ratio of the estimated alpha to its standard
error. The ratio, designated as ta, is given by
ta = ap I SE«
(23)
Exhibit 13
RELATIONSHIP BETWEEN THE
JENSEN AND TREYNOR MEASURES
OF INVESTMENT PERFORMANCE
Average
Portfolio R"
Return
A1
Lines of
Constant
Tl Values^
~The Market Line
0.5
1.0
1.5
Risk
Symbols: R; = Return on Market Index
Hp = Risk-free Rate of Interest
A,B = Managed Portfolios
A' = Portfolro A Levered to Same Beta as
Portfolio B
The statistic ta gives a measure of the extent to
which the true value of alpha can be considered to
be different from zero. If the absolute ^value of trt is
large, then we have more confidence that the true
value of alpha is different from zero. Absolute
values of ta in excess of 2.0 indicate a probability
of less than about 2.5 per cent that the true value
of alpha is zero.
These methods of performance measurement
were originally devised by Michael Jensen [10,11 ]
and have been widely used in many studies of in-
vestment performance, including that of the recent
SEC Institutional Investor Study '[21 ].
A performance measure closely related to the
Jensen alpha measure was developed by Jack L.
Treynor [24]. The Treynor performance measure
(designated TI)14 is given by
TI = a/j3,.;
(24)
The difference between the a and TI performance
measures is simply that the fund alpha value has
3-45
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been divided by its estimated beta. The effect,
however, is significant, eliminating a so-called
"leverage bias" from the Jensen alpha measures.
This is illustrated in Exhibit 13.
Funds A and B in Exhibit 13 have the same
alpha values. (The alphas are equal to the vertical
distance on the diagram between the funds and the
market line.) By combining portfolio B with the
riskless rate (that is, by borrowing or lending at
Rf), any return-risk combination along line Y can
be obtained-. But such points are clearly dominated
by combinations along line X—attainable by
borrowing or lending combined with fund A. As
Exhibit 13 shows, the alpha for fund A, when
levered to the same beta as fund B (Point A'),
dominates the latter's alpha value.
The Treynor measure eliminates this leverage ef-
fect. All funds which lie along a line (such as X or
Y) have the same Tl value; therefore borrowing or
lending combined with any fund outcome will not
increase (or decrease) its performance measure.
The Treynor measure thus permits direct perform-
ance comparisons among funds with differing beta
values.
Problems with the Market Line Standard
The tests of the CAPM summarized in Section 8
indicate that the average returns over time on
securities and portfolios tend to deviate from the
predictions of the model. Though the observed
risk-return relationships seem to be linear, they are
generally flatter than predicted by the CAPM, im-
plying that the tradeoff of risk for return is less
than predicted.
This evidence raises some question as to whether
the CAPM market line provides the best bench-
mark for performance measurement and suggests
instead that other benchmark portfolios may be
more appropriate. For example, under certain con-
ditions, the "empirical" risk-return lines developed
by Black, Jensen, and Scholes [1] and others
would seem to be a reasonable alternative to the
CAPM market line standard. This might .be the
case if the portfolio for which performance is being
measured were restricted to exactly the same set of
investment options used to create the empirical
standard, that is, if the portfolio were fully invested
in common stock and could not use leverage to in-
crease its beta value. For such a portfolio it would
seem appropriate to measure performance relative
to the empirical line, as opposed to the market
line.
A comparison of these standards is illustrated in
Exhibit 14
MEASUREMENT OF INVESTMENT
PERFORMANCE: MARKET LINE VERSUS
EMPIRICAL STANDARD
Average
Portfolio
Return
Empirical Line
Market Line
0.5
1.0
1.5
Risk
Symbols: R.. = Return on Market Index
RZ = Return on Zero Beta Portfolio
R_ = Risk-Free Rate of Interest
X = Investment Portfolios
O = Market Index
Exhibit 14. The market line performance measure
(designated as a t in Exhibit 14) is equal to the ver-
tical distance from the portfolio to the market line.
The empirical line measure (designated a2) is the
vertical distance from the portfolio to the empirical
line. Since ideally all the stocks used to develop the
empirical line are contained in the market index,
the empirical line, like the market line, would be
expected to have a return equal to market return,
RM, for beta equal to 1.0. The intercepts on the
return axis, however, are typically different for the
two lines. The market line intercept, by definition,
is equal to the average risk-free rate. The empirical
line intersects the return axis at a point different
from- RF, and typically above it. This intercept
equals the average_return on a portfolio with "zero
beta", designated Rz. The existence of a long-run
average return on the zero beta portfolio that dif-
fers from the riskless rate is a clear violation of the
predictions of the CAPM. As of this time, there is
no clear theoretical understanding of the reason for
this difference.
To summarize, empirically based performance
3-46
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standards could, under certain conditions, provide
alternatives to those of the CAPM market line
standard. However, the design of appropriate em-
pirical standards requires further research. In the
interim, the familiar market line benchmarks can
provide useful information regarding performance,
although the information should not be regarded as
being very precise.15 •
Footnotes
From this point on, systematic risk will be referred to simply
as "risk." Total risk will be referred to as "total risk."
We use the term portfolio in a general sense, including the
case where the investor holds only one security. Since port-
folio return and (systematic) risk are simply weighted
averages of security values, risk-return relationships which
hold for securities must also be true for portfolios, and vice
versa.
The sample was picked to give the broadest possible range
of security beta values. This was accomplished by ranking
all NYSE securities with complete data from 1945-70 by
their estimated beta values during this period. We then
selected every 25th stock from the ordered list. The data
was obtained from the University of Chicago CRSP (Center
for Research in Security Prices) tape.
The commercial paper results in Table 3 are rates of return,
not risk premiums. The risk premiums would equal zero by
definition.
Correlation studies of this type tend to produce a con-
servative picture of the degree of beta coefficient
stationarity. This results from the fact that it is not possible
to correlate the true beta values but only estimates which
contain varying degrees of measurement error.
Measurement error would reduce the correlation coefficient
even though the underlying beta values were unchanged
from period to period.
These results are consistent with those found by N. Mains in
a later study [16]. Mains' correlated adjacent calendar-year
betas for a sample of 99 funds for the period 1960 through
1 97 1 . The betas were based on weekly returns. The average
correlation coefficient for 1 1 tests was 0.788, with in-
dividual values ranging from a low of 0.614 to a high of
0.871.
SE/ is an estimate of the standard error of the residual term
in Equation (17a). Thus it is the estimated value for cr(ey),
the unsystematic risk term defined in Equation (8). See
column (6) of Tables 3 and 4 for typical values for
securities and mutual funds.
For example, skewness in the distributions of stock returns
can lead to spurious correlations between mean return and
SE,. See Miller and Scholes [19], pp. 66-71.
Their expanded test equation is
5.
10.
11.
12.
1 3.
where, according to the CAPM, the expected value of y, is
zero.
Table 1 , p. 25, of Blume and Friend [3] presents period-by-
period regression results.
Blume and Friend [3], p. 26.
Figure 6 of Black, Jensen, and Scholes [1], pp. 101-103,
shows average monthly returns versus systematic risk for 17
nonoverlapping 2-year periods from 1932 to 1965.
See columns 2 and 3 of Table 4 for typical mutual fund Si
and SEa values.
14. Treynor's work preceded that of Jensen. In a discussion of
Jensen's performance measure [26], Treynor showed that
his measure (as originally presented in [25]) was equivalent
to
TI = Rf - ot//3.
Since Rf is a constant, the TI index for ranking purposes is
equivalent to that given in Equation (24).
15. There are a number of excellent references for further study
of portfolio theory. Among these we would recommend
books by Richard A. Brealey [4], Jack Clark Frances [7],
and William F. Sharpe [24]. For a more technical survey of
the theoretical and empirical literature, see Jensen [12].
References
[1 ] Black, Fischer, Jensen, Michael C, and Scholes, Myron S.
"The Capital Asset Pricing Model: Some Empirical
Tests.' Published in Studies in the Theory oj Capital
Markets, edited by Michael Jensen. (New York: Pracger.
1972), pp. 79-121.
[2] Blume, Marshall E. "Portfolio Theory: A Step Toward Its
Practical Application." Journal of Business. Vol. 43
(April 1970), pp. 152-173.
[3] Blume, Marshall E., and Friend. Irwin. "A New Look at
the Capital Asset Pricing Model." Journal of Finance.
Vol. XXVIII (March 1973). pp. 19-33.
[4] Brealcy, Richard A. An Introduction to Risk and
Return from Common Slocks. (Cambridge. Mass.: MIT
Press, 1969.)
[5] Fama. Eugene F. "Components of Investment Per-
formance." The Journal of finance. Vol. XXVII (June
1972), pp. 551-567.
[6] Fama, Eugene F., and MacBcth, James D. "Risk. Return
and Equilibrium: Empirical Tests." Unpublished Working
Paper No. 7237. University of Chicago. Graduate School
of Business, August 1972.
[7] Francis, Jack C. Investment Analysis anil
Management. (New York: McGraw-Hill, 1972.)
[8] Friend, Irwin. and Blume, Marshall E. "Risk and the
Long Run Rate of Return on NYSE Common Stocks."
Working Paper No. 18-72. Wharton School of Commerce
and Finance, Rodney L. White Center for Financial
Research.
[9] Jacob, Nancy. "The Measurement of Systematic Risk for
Securities and Portfolios: Some Empirical Results." Jour-
nal of Financial and Quantitative Analysis. Vol. VI
(March 1971). pp. 815-H34.
[10] Jensen, Michael C. "The Performance of Mutual Funds in
the Period 1945-1964." Journal oj'Finance, Vol. XXIII
(May 1968), pp. 389-416.
[II] Jensen, Michael C. "Risk, the Pricing of Capital Assets,
and the Evaluation of Investment Portfolios." Journal of
Business. Vol. 42 (April l'969), pp. 167-247.
[12] Jensen, Michael C. "Capital Markets: Theory and
Evidence." The Hell Journal of Economics and
Management Science, Vol. 3 (Autumn 1972), pp. 357-
398.
[13] Levy, Robert A "On the Short Term Stationarily of Held
Coefficients." Financial Analysts Journal, Vol. 27
(November-December 1971). pp. 55-62.
[14] Lintncr, John. "The Valuation of Risk Assets and the
Selection of Risky Investments in Stock Portfolios and
Capital Budgets." Review of Economics and Statistics,
Vol. XLV11 (February 1965), pp. 13-37.
[15] Lintner. John, "Security Prices, Risk, and Maximal Gains
3-47
-------
from Diversification." Journal of Finance, Vol. XX
(December 1965). pp. 587-616.
116] Mains, Nonmin I-'. "Are Mutual Fund Hota Coeliieients
Stationary'.1" Unpublished Working Paper. Investment
Company. Institute, Washington, D.C., October 1972.
[17] Markowit/, Harry M. "Portfolio Selection." Journal of
Finance, Vol. VII (March 1952), pp. 77-91.
[18] Mnrkowitz, Harry M. Portfolio Selection: Efficient
Diversification of Investments. (New York: John Wiley
and Sons, 1959.)
[\9] Miller, Merton H., and Scholcs, Myron S. "Rates of
Returns in Relation to Risk: A Roexamination of Recent
Findings." Published in Studies in the Theory of Capital
Markets, edited by Michael Jensen. (New York: Pracger,
1972), pp. 47-78.
[20] Modigliani, Franco, and Poguc, Gerald A. A Study of In-
vestment Performance Fees. (Lexington, Mass.: Heath-
Lexington Books, Forthcoming 1974.)
[21 ] Pogue, Gerald A., and Conway, Walter. "On the Stability
of Mutual Fund Beta Values." Unpublished Working
Paper. MIT. Sloan School of Management, June 1972.
|22| Securities and Exchange Commission, Institutional In-
vestor Sluely Report of the Securities anil Kxi'hnnxt
Commission, Chapter 4, "Investment Advisory Com
plexes", pp. .125-.147. (Washington, !).('.: U.S. (iovern
incut Printing Office, 1971.)
[23] Sharpe, William F. "Capital Asset Prices: A Theory <>
Market Equilibrium under Conditions of Risk." Joitrnu
of Finance, Vol. XIX (September 1964), pp. 425-442
[24] Sharpe, William F. Portfolio Theory and Capita
Markets. (New York: McGraw-Hill, 1970.)
[25] Trcynor, Jack L. "How to Rate the Management of In-
vestment Funds." Harvard Business Review, Vol. XLIII
(January-February 1965), pp. 63-75.
[26] Treynor, Jack L. "The Performance of Mutual Funds in
the Period 1945-1964: Discussion." Journal of Finance
Vol. XXIII (May 1968), pp. 418-419.
[27] Wagner, Wayne H., and Lau, Sheila. "The Effect o
Diversification on Risk." Financial Analysts Journal
Vol. 26 (November-December 1971), pp. 48-53.
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IV. THE RELATIONSHIP BETWEEN REAL
AND FINANCIAL MEASURES OF RISK
Section IV is concerned with the real determinants of
the risk of common stocks as measured by their beta values.
By real determinants are meant variables which can be measured
directly from a firm's balance sheets and income statements.
The beta risk measure is the key variable of the Capital Asset
Pricing Model (CAPM) developed by Sharpe [26J, Lintner [9],
and Mossin [17]. Their model is given by
E(R. ) = RF + BjtECiy RF) (3-5)
where
E(R.) = the expected one-period rate of return
on stock j
E(RM) = the expected one-period return on the
market portfolio
RP = the one-period risk-free rate of interest
B. = the beta value for stock j
The model states that the expected rate of return on the firm's
common stock (a financial asset) is a linear function of its
market risk measure, beta. The model deals only with the
returns on financial assets. It says nothing about how the
expected returns or beta values are related to characteristics
of firms' real assets that can be measured directly from
3-49
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accounting (i.e., book) data. In other words, the CAPM says
nothing about the real determinants of beta.
As will be shown below, there have been several empirical
investigations of the real or book characteristics of high-
risk firms. The results are mostly consistent with intuition,
in that (various proxies for) earnings variability, growth,
cyclicality, and financial leverage are significantly
correlated with beta. However, there is not much theory having
more than intuitive content. The missing link is a dynamic
model of a firm's earnings behavior, growth, and market
valuation. The development of a fully rigorous and operational
theory of the real determinants of asset values is a difficult
task which, to date, has not been accomplished.
Section IV is divided into three segments. The first,
segment (a) presents a very complete survey of the empirical
literature through mid-1973. It was prepared by Professor
Stewart C. Myers of MIT as Part I of a paper entitled, "The
Relationship between Real and Financial Measures of Risk and
Return" [22]. In segment (b) this survey is brought up to
date (mid-1975). In segment (c) the current state of this
research is summarized. Readers interested only in the
conclusions and not the details can skip directly to segment
(c).
3-50
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(a) Empirical Studies on the Real Determinants of Beta
(Through Mid-1975)* (Excerpt from a Paper [22] by
Professor Stewart C. Myers)
This part reviews and summarizes all the empirical work
known to me relating to the real determinants of stocks'
systematic risk. However, it is obviously impossible to
provide a full description of each study's experimental
design. The following discussion is limited to the main ideas
and results.
Beaver, Kettler, and Scholes
Beaver, Kettler and Scholes (BKS [2]) were the empirical
pioneers, although Ball and Brown's work [1] preceded theirs.
However, the Ball and Brown paper is mostly devoted to ether,
broader issues, and the part investigating the real determinants
of systematic risk is much less ambitious and informative than
the BKS study. In any event, Ball and Brown's results were
confirmed by BKS.
BKS used seven accounting variables that intuition or
tradition suggest are associated with high-risk firms.
1. Dividend payout, defined as total dividends per
share paid by the firm over a nine-year period,
divided by total earnings per share over the same
* Footnotes and references for this section appear at the end
of the article. Reproduced with permission of Author
and Publisher.
3-51
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period. Two arguments can be made for including
payout as an independent variable. First, most firms
attempt to achieve stable (or stable growth in)
dividends, and in normal times regard them almost
as fixed liability. Nevertheless, earnings have to
cover dividends, at least in the long run. The
higher the variance of earnings, the lower must the
normal dividend be set in order to keep the proba-
bility of "trouble" (earnings less than the normal
dividend) acceptably low. Thus dividend payout
should be a proxy for management's uncertainty about
future earnings. Second, firms which grow rapidly
usually retain a greater proportion of earnings.
Since there is a long tradition in the literature
associating high growth with high risk, low dividend
payout should be a proxy for high risk.
2. Growth, measured by the log of the five-year change
in net book assets. Again, the rationale is the
traditional association between rapid growth and
high business risk.
3. Financial leverage, measured by the ratio of book
debt to net book assets. Given business risk, the
CAPM predicts a positive linear relationship between
beta and the ratio of the market value of debt to
the total market value of the firm (debt plus equity).
3-52
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However, this relationship will be difficult to
observe if firms with low business risk find they can
issue more debt. In the limiting case where firms
keep the sum of business and financial risk constant,
there will be no evident relationship between the
debt ratio and beta (until other variables successfully
account for differences in business risk).
4. Liquidity, measured by ratio of current assets to
current liabilities. This is widely used as measure
of solvency by creditors.
5. Size, measured by the log of net book assets. Casual
observation suggests a relationship between size and
safety, and to the extent that size reflects diversi
fication of activity, theory predicts that large firms
will have lower total risk. There is no rigorous
theory predicting that large firms have lower
systematic risk, however.
6. Earnings variability, measured by the standard
deviation of earnings per share over nine years,
where each year's net earnings available to common
is normalized by dividing by the value of the firm's
stock at the end of the preceding year. It makes
sense to predict that firms with highly volatile
earnings will have highly volatile stock prices.
3-53
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This means high total risk, however, not necessarily
high systematic risk. There is an empirical coriela
tion between total and systematic risk, but this is
to be expected, simply because the former includes
the latter. It is not true that increased total
risk necessarily means increased systematic risk.
Also, note that this variable is partly dependent on
variable 3, financial leverage: the more debt the
firm uses, the greater the variance of earnings
available to common stockholders.
7. The "Earnings Beta", which is the slope coefficient
of the regression of the firm's net earnings (again
normalized by the preceding period's stock value)
on the average normalized earnings for the entire
sample of firms. Thus it is a measure of cyclicality,
that is, the extent to which fluctuations in the firm's
earnings are correlated with fluctuations in earnings
of firms generally. Since stock prices clearly respond
to earnings, both individually and generally, the
earnings beta and the stock beta ought to be strongly
related.
I suspect that most readers will find this list of
variables intuitively reasonable. The intuitive concept of
risk, however, includes many things that are not necessarily
3-54
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related to risk as defined by the CAPM. A rigorous a priori
hypothesis can be made with respect to only one of the seven
variables, namely financial leverage. A less rigorous but
very plausible case can be made for variable 7, the "Accounting
Beta". As for the other five variables, there is no theory.
BKS tested the relationships between these variables and
beta by cross-sectional tests on a sample of 307 firms for
which complete accounting and stock price data were available
for the period 1947-65. Actually, four separate tests were run
First, the data were split into two subperiods, 1947 56 and
1957-65. Within each period, tests were performed both on
individual securities and on five-security portfolios. The
portfolios were formed by ranking the stocks on basis of the
relevant accounting variable, and then assigning firms ranked
1 - 5 to portfolio 1, those ranked 6 10 to portfolio 2, etc.
Tests were then performed on portfolio averages of the
4
accounting variables and betas.
For each of the four cases, pair-wise rank correlations
were calculated between each of the seven variables and
contemporaneous firm or portfolio betas. The results are shown
in Table 1. It is gratifying that leverage and the accounting
beta show the strong positive correlations expected on
theoretical grounds. (In this case a rank correlation of
+0.14 is significant at approximately the one percent confi
dence level.) Earnings variability and payout also have
3-55
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Table J
CONTEMPORANEOUS ASSOCIATION BETWEEN MARKET-DETERMINED
MEASURE OF RISK AND SEVEN ACCOUNTING RISK MEASURES3
\l Q ~Y* 1 Q i\ 1 (^
V a. L JL d. L) J. C
Payout
Growth
Leverage
Liquidity
Size
Earnings
Variability
Accounting
Beta
Period
(1947-
Individual
Level
One
56)
Portfolio
Level
.49
( -50)
. 27
(.23)
.23
(.23)
-.13
( -13)
.06
( .07)
.66
(.58)
. 44
(.39)
-. 79
( -77)
. 56
(.51)
.41
(.45)
-.35
( -44)
- .09
(-.13)
.90
(.77)
.68
(.67)
Period Two
(195
Individual
Level
7-65)
Portfolio
Level
- .29
( -24)
.01
(.03)
.22
(.25)
.05
( -01)
.16
( -16)
.45
•"*" (.36)
. 23
(.23)
. 50
( -45)
.02
(.07)
.48
(.56)
.04
(-.01)
. 30
( -30)
.82
(.62)
.46
(.46)
Rank correlation coefficients appear in top row,
and product-moment correlations appear in
parentheses in bottom row.
The portfolio correlations are based on 61
portfolios of 5 securities each.
Source: Beaver, Kettler and Scholes [2], p. 669.
3-56
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significant correlations in the expected direction. Size has
a somewhat weaker correlation. Finally, growth and liquidity,
although they were significantly correlated with beta in the
first subperiod, are totally unrelated in the second. In view
of this, it is hard to put much faith in these two variables.
Viewed as a whole, the results are encouraging. Theory,
what there is of it, is supported, and intuition is confirmed.
However, one can obtain significant associations between
variables without actually explaining very much, and to some
extent that is the case here. BKS ran a second test in which
they attempted to predict the 1957-65 betas by using the
accounting variables from the earlier 1947-56 period. The
best explanator of the 1947-56 betas, as determined by
regression analysis, is a linear combination of dividend payout,
asset growth,and earnings variability. This procedure explains
44.7 percent of the variance in the first period's betas.
However, when this equation is used to predict the 1957-65 betas,
using account variables from the first period, only 24 percent
of the variance is explained. This is not too impressive,
considering that 21 percent is explained by the "naive" predic-
tion that the 1957-65 beta would be equal to the 1947-56 beta.
Nevertheless, it is some improvement. Moreover, it must be
remembered that 100 percent prediction accuracy is impossible.
Even if we knew the exact, true betas of all firms for the second
3-57
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period, we could not predict the me a s u r e d betas, for that would
require an exact a priori prediction of the sampling errors
encountered in estimation.
White
White [15] followed broadly the same methodology as BKS,
but with important differences in variable definition and in
the way the tests were carried out. He hypothesized three main
factors characteristic of high-beta firms:
1. High debt ratio. This variable's effect on beta
is predicted by theory and supported by the BKS
results. However, White defined the debt ratio ii
terms of market values, which is the specificatiDn
called for by theory. BKS considered only variables
that could be derived from accounting data, and so
restricted their tests to the book debt ratio.
2. Rapid growth in sales or operating earnings,
measured by the log of the relative change of the
variable over the period examined. BKS examined
growth in book assets but obtained mixed results.
3. High asset beta, defined as the slope coefficient
of a regression of each firm's percentage change
in sales on the contemporaneous percentage- change
in gross national product. This is analogous to
BKS's "accounting beta" in that each measures firms'
3-58
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cyclicality. However, there are two important
differences. First, the index used is a national one,
not simply an average taken over the firms in the
sample under consideration. Since the investor is
concerned with the covariance of each stock's return
with the rate of return on the entire market (and in
principle with its covariance with changes in aggregate
national wealth), he should likewise be concerned with
the covariance of each firm's income with national
income. It is true that the average earnings of any
reasonably large sample of firms will be highly
correlated with national income, but errors will
necessarily enter if the sample average is used.
The second difference between White and BKS's
definition of cyclicality is that White used sales
rather than earnings. It is true that investors are
more concerned with earnings than sales per se. But
it is expected earnings which determine stock prices,
and changes in sales may be a better proxy for
changes in expected earnings than is the one-period
change in accounting earnings. Accounting procedures
induce various lags and biases in reported income,
for example.
Aside from differences in variable definition, White's
study differs from BKS's in the use of relatively large port
3-59
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folios of securities as the basis for his cross-sectional
tests. This has become standard operating procedure in tests
of the CAPM, since it vastly reduces errors in measurements of
individual firms' betas; these errors otherwise attenuate the
observed relationship between firms' betas and their average
returns. The aim in the present context is to reduce errors
both in measuring beta and in measuring the three hypothesized
determinants of beta.
The regressions summarized in Table 2 are typical of White's
results. In these tests the 210 securities in White's sample
were grouped into 10 portfolios of 31 securities each. (The
31 stocks with the highest betas were put in portfolio 1, the
31 with the next highest betas in portfolio 2, etc.) Then
four cross-sectional regressions were run, using as independent
variables various combinations of the portfolios' asset betas,
debt ratios, and rates of sales or earnings growth. As is
obvious from the table, White found significant relationships,
in the expected direction, for all variables. He also explained
a much higher proportion of the variance in the betas, although
it must be remembered that in this test there are only 10 port-
folio betas to be explained.
These tests were successfully repeated for various sub-
periods within the 1951-68 period,7 but I will not attempt
to present all the results here. It should be noted, however,
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Table 2
WHITE'S REGRESSION RESULTS l;0!l 10 PORTFOLIOS
OF 31 SECURITIES EACH, 1951-68
Regression
1
(R2 = .71)
2
(R2 = .85)
3
(R2 = .96)
4
(R2 = .97)
Variable
Constant
0- 505
(4.54)a
0.153
(0.969)
-0.347
( 2.31)
-0. 294
( 2.53)
Asset
Beta
0.325
(4.38)
0.247
(3.88)
0.274
(7.65)
0.220
(7.22)
Market
Debt Ratio
2.45
(2.62)
2. 35
(4.56)
1.69
(3.62)
Sales
Growth
7.16
(4.10)
Earnings
Growth
7.68
(5.03)
T-statistics in parentheses below coefficients
Source: White [15], Table 4.6.
3-61
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that the coefficients of the growth variables dropped to
insignificant levels when White repeated his tests for the
1961-68 subperiod. This matches BKS's findings. However,
since the asset beta—the "cyclicality" variable—and the debt
ratio both have a strong positive relationship with beta, it
is fair to say that White's results support the conclusions of
the BKS study, and of course vice versa.
Gonedes
Gonedes's work [4] provides additional, and somewhat
discouraging, evidence on the importance of BKS's accounting
beta as a determinant of firms' stock betas. He dealt with a
random sample of 99 firms for the period 1946-67, and three
seven-year subperiods, 1946-52, 1953-59, and 1961-67. (The
data for 1960 were reserved for various statistical tests
which will not be reviewed here.) The accounting betas were
measured by regressions of the form
X. - a + blXm + C.Xj + 3. (2)
where:
Oi
X. = firm j's earnings
•\j
XM = aggregate earnings of all firms for which
complete data were available over the
1946-68 period
3-62
-------
a/
Xj = that part of the aggregate earnings of
all other firms in j ' s industry which
could not be explained by XM
•v
e- = an error term reflecting unsystematic risk
The 99 firms in the sample were randomly selected from industries
for which adequate indexes could be constructed.
Regressions were run using net income (after interest and
taxes), net income normalized by net book assets, and first
differences of these two variables. As it turned out only the
first differences gave significant results.
Gonedes tested for correlation between stock betas and
accounting betas measured in terms of scaled net income and
found no significant relationship for any subperiods . This is
exactly counter to the BKS results. However, some positive
relationship was found for accounting betas derived from first
differences in scaled income. This is again counter to BKS,
who found that this procedure gave poorer results for their
s amp 1 e . v
Table 3 shows the correlations obtained using accounting
betas derived from first differences of scaled net income.
Significant relationships are found for the 1946-52 and 1953-59
subperiods when accounting betas are derived from the full 21
years of data, but the relationship disappears in the 1960's.
There is no relationship at all when the accounting betas are
measured from seven years' data, but this probably indicates
only that accounting betas are hard to measure with a handful
of observations.
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Table 3
CORRELATION BETWEEN ACCOUNTING AND STOCK BETAS
—GONEDES'S RESULTS
(Accounting betas derived from first differences
in net income scaled by net book assets.)
Accounting Beta
Measured over:
1946-68
(Excluding 1960)
1946-52
1953-59
1961-68
Stock Beta Measured Over:
1946-52
.32b
(.22)a
.29b
(.05)
1953-59
.41b
(-34)b
.0
(.04)
1961-68
.08
(.0)
.07
(.0)
Significant at 5 percent confidence level.
Significant at 1 percent confidence level.
Source: Gonedes [4], Table 5, pp. 434-35.
3-64
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Gonedes also tested for correlation between the coeffi
^
cients of determination (R~'s) of the equations used to estimate
the accounting and stock betas. The proposition is that if the
earnings index explains a large proportion of the variance of
a firm's earnings, then the stock market index ought to explain
a large proportion of the variance in returns on the firm's
stock. Gonedes found this to be true, so long as accounting
betas were derived from first differences in accounting income.
This is certainly evidence of an association between the cyclic-
ality of a firm's earnings and its stock's market prices.
However, correlating R 's and correlating betas are not the
same thing, since a firm can have a low R and high beta, or
conversely. Thus it is no paradox to find significant correla-
7
tion of R°'s but insignificant correlation of accounting and
stock betas.
There is no obvious reason for the differences between
q
BKS and Gonedes. Particularly in view of White's results, it
seems fair to claim that cyclicality, somehow measured, is a
determinant of beta. But the measurement of cyclicality and
the exact specification of its relationship to beta still pose
difficult problems.
Rosenberg and McKibben
One interesting aspect of the tests described so far is
that they have uniformly assumed (1J substantial cross - sectional
variance of betas and the real determinants of beta, but
3-65
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(2) stability of these variables across time for each firm.
Assumption (2) is implicit in the way the betas and the inda-
pendent variables are measured. However, it is obviously
questionable. There is no reason to expect the determinants of
beta to be strictly constant across time, and if the determinants
change, beta ought to change too.
Rosenberg and McKibben [14] use an interesting test which
explicitly assumes that beta shifts over time. Space is
insufficient for a full presentation, but the idea is this.
Suppose beta at time t is linearly related to certain real
determinants W .. For firm i,
nt J
This relationship is assumed to apply to all firms and to be
constant over time. However the real determinants vary across
firms and time, and thus so will beta.
Substituting Equation (3) into Equation (1) we have:
Normally beta is estimated by fitting the regression equation
S\ /S l~\j XV
"jt = aj + Vmt + 6jt C5)
where the rt's are returns in excess of the risk-free rate.
3-66
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But if Equation (4) is correct, (5) is incorrect, since it
assumes the true value o£ beta is constant. Instead we should
'run the regression
N
rit = ai + I b- (W. . r ^J + e.. (6J
It ] = i J 3nt mt it
where the independent variables are (W. r ) , the products of
the market excess returns and the hypothesized determinants of
beta. This procedure allows a direct measurement of the b.'s in
Equation (3). This one-step procedure is in contrast to the three
studies cited above, in which a two-step test was required—
one step to estimate the betas and their hypothesized determi
nants , and another to see whether the hypothesized variables
explain the cross - sectional differences in firms' betas.
The results of most interest here were obtained by pooling
time series data (yearly observations) and cross-sectional data
for 558 firms over the 1954-66 period. (The data have to be
pooled because the b.'s in Equations (3) and (6) are assumed to
be the same for all firms.) Unfortunately, the results are hard
to compare with those of BKS, White or Gonedes, since Rosenberg
and McKibben tried thirty-one independent variables, of which
eleven are based on stock market data. Thus, if we find that
a variable which we expect to work does not, this may simply
be due to its collinearity with one or more of the other 30
variables. There are four direct measures of firm growth,
for example, and at least three other variables which might
3-67
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reasonably be expected to proxy for growth. What is one to
conclude from the failure or success of any one of these
variables ?
Moreover, the inclusion of variables derived from stock
market data hampers our quest for the real determinants of
beta, since the market behavior of the firm's stock responds
to the real variables and to some extent proxies for them. In
any case, these financial variables are not usually available
for specific real assets. (They are for firms, which are
collections of assets, but in many instances that is not much
help.)
This is actually unfair to Rosenberg and McKibben, who
were largely concerned with forecasting betas. Nevertheless,
their paper strikes me in the same way that Chinese cooking
strikes some people who are brought up on meat, potatoes and
gravy. You may be served nine courses, each one delicious,
but a half hour after leaving the restaurant you're hungry
again. Here we have 31 plausible variables, but a half hour
after reading the paper it's hard to say what's been learned
about the real determinants of beta.
Of course this is an exageration, because Rosenberg and
McKibben do confirm the results of BKS, White and Gonedes in
several respects. First, financial leverage is again found
to have a significant positive relationship with beta. Second,
a positive and highly significant relationship is found for
3-68
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volatility in earnings, measured by the standard deviation of
changes in earnings per share over time. Third, strong positive
relationships are found for growth in sales and earnings (con-
firming White and BKS).
On the other hand, no significant relationship is found
for either the accounting beta or the dividend payout ratio,
which is generally contrary to the other authors I have cited.
Whether this is an actual failure of these variables given the
Rosenberg-McKibben sample and methodology, or whether it is
simply a matter of multicollinearity, is hard to say.
There were several other variables tested. Most were
insignificant and some of those that were significant had
unexpected signs. Since I could make no sense out of these
variables, they will not be reviewed here. Rosenberg and
McKibben note that the pattern of signs does not correspond to
their a priori expectations.
A Tentative Summary
There are several other studies which shed light on the
real determinants of beta, but it seems inefficient to lay out
the results of each. Instead I will propose a tentative
summary at this point, and then note whether the other studies
support or weaken this view- It seems safe to identify four
factors which generally contribute to high stock betas.
1. Cyclicality. This broad term is meant to include
BKS's "accounting beta" and the corresponding risk
3-69
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measures used in the other studies discussed above.
Specific definitions vary, but each study started
from the same hypothesis, namely that beta depends
on the covariance between swings in the firm's
earnings and swings in earnings in the economy
generally. This hypothesis is not based on a fully
rigorous model, but it is so much in the spirit of
the CAPM that it is hard not to hold it a priori.
The empirical results of BKS and White strongly
support the idea and Gonedes's work partly supports
it. Rosenberg and McKibben do not find the accounting
beta to be significantly related to the stock beta,
but I am inclined to think that the effect was stolen
by one of their 30 other independent variables.
Earnings Variability. Both BKS and Rosenberg-McKibben
find earnings volatility to be strongly related to
beta. This is mildly disturbing from a theoretical
point of view since earnings volatility represents
the total, not the systematic risk of earnings; we
would expect it to be less important than the
"accounting beta" or some other measure of cyclicality
Nevertheless, earnings variability corresponds closely
to the popular, intuitive idea of firm risk, and it is
sensible proxy for cyclicality. Thus, it certainly
belongs on any tentative list of real factors
associated with beta.
3-70
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However, it is unfortunate that no studies have
separated the variance of firms' earnings into
systematic and unsystematic components, and tested
which component is more strongly related to beta.
If the CAPM is right, the systematic component ought
to be more important.
3- Financial Leverage. Theory specifies a relationship
between the market debt ratios and beta, but the
effect comes through strongly even when the book
debt ratio is used.
4. Growth, which can be measured in a variety of ways.
BKS, White and Rosenberg-McKibben all found growth
to be important, although the effect seems less
strong in the 1960's than earlier. BKS also found
dividend payout to be important, which is consistent,
since rapid growth normally is associated with low
payout, and there is no reason for dividend policy
per se to affect risk.
It is not at all clear how best to measure
growth. Moreover, the studies cited have not drawn
a clear distinction between growth as expansion and
growth defined as the opportunity to invest in projects
offering expected rates of return exceeding the cost
of capital.
Several other comments should be made before turning to
other studies. First, none of the four investigations described
3-71
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above has a complete theoretical base. The authors would
of course agree, and note their willingness to use one as
soon as it is discovered. But we must bear in mind that the
puzzles they encountered may be due to use of the wrong
functional specification. Rosenberg and McKibben do at least
assume a specific, plausible specification and develop consis-
tent tests, but they would agree that their choice of inde-
pendent variables is not based on any clear theory.
In the absence of theory there are many intuitively
appealing variables and several plausible ways to measure each
of them. This naturally makes it difficult to compare the
studies cited, except by thinking in terms of general factors
like "growth" or "cyclicality."
Other Studies
Several other empirical studies investigate the real
determinants of beta. Since there are limits both to space
and to the reader's patience, I have chosen not to review them
in detail. Instead I will briefly describe whether they
support or weaken the case for the four general determinants
of beta suggested just above.
Cyclicality. The work of Pettit and Westerfeld [13] and
Gordon and Halpern [6] deals primarily with cyclicality as a
I
possible determinant of beta. The former derived an "earnings
beta" by first fitting a trend line to firms' earnings per share,
scaled by average earnings per share over the period investi
3-72
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gated, and also to the scaled earnings of the Standard and
Poor's 500 Stock Index. Then the earnings beta was defined
as the covariance of the residuals from these trend lines.
A highly significant pairwise correlation was found between
this statistic and the firms' stock betas. The relationship
persisted when the earnings beta was combined with several
other independent variables in a multiple regression. Finally,
these tests were successfully repeated with a different
earnings beta based on operating income.
Gordon and Halpern provide additional evidence as a by-
product of their application of the CAPM to the problem of
estimating the cost of capital for a division of a firm.
Their earnings beta was based on the covariance between the
rates of growth of firms' earnings per share and growth rates
for economy-wide corporate profits. Again, a strong positive
correlation between earnings betas and stock betas was found.
Earnings variability. Lev and Kunitsky [10] find that
stock betas are significantly related to the degree of
"smoothness" of various operating and financial series,
principally earnings, dividends, sales and capital expenditure.
Smoothness is measured as the mean absolute deviation of the
actual values of the variable from its trend. (If smoothness
is S, then a perfectly smooth series will have S = 0.)
Clearly this is another way of defining volatility; thus, the
discovery of a significant positive relationship between S and
the stock beta confirms the results reported earlier in this
paper.
3-73
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On the other hand, it is not obvious why the smoothness
of dividends, sales and capital expenditures should matter at
all once earnings volatility is accounted for. There are two
possible lines of response. The first, and simplest, is to
say that any simple measure of earnings volatility is an
imperfect representation of investors' actual uncertainty
about the firm's future earnings. In this case we can regard
the other smoothness measures as useful additional proxies.
Lev and Kunitsky see something deeper, however. They argue
that firms' managers are engaged in a continual battle to
control the environment, and that the greater their success,
the lower their firms' stocks' betas. Control of environment
is effected by reducing the variance of each of the firm's
major activities, not just the variance of the end result,
earnings. Thus, smoothness of sales would have a place even
if there were no problems in measuring earnings volatility
directly. Unfortunately, these explanations are not mutually
exclusive, and there is no way to distinguish between them
from the results at hand.
The role of earnings volatility is also supported by
Lev [9 ] who shows that firms with high operating leverage
(that is, high ratios of fixed to total costs) tend to have
high betas. This is sensible since high operating leverage
is by definition associated with high earnings volatility and
high earnings betas.
3-74
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Both Pettit and Westerfeld [13] and Melicher [11] confi m
that firms with low dividend payout ratios tend to have high
betas. This too supports the case of earnings volatility, since
low dividend payout is a natural response of a mangement that
is uncertain about future earnings.
The only authors who do not find a reliable positive
association between earnings volatility and beta are Breen and
Lerner [3]. However, their results can be attributed to an
inappropriate measure of volatility and to their attempt to
measure volatility from severely limited time series data.
Financial Leverage. The debt ratio performs somewhat
erratically in the studies of Pettit-Westerfeld, Breen and Lerner,
and Melicher. In the first case, the book debt ratio is not
significant in a multiple regression with several other inde-
pendent variables. In Breen and Lerner, the debt ratio is
significant but inexplicably of the wrong sign in two out of
twelve subsamples. Melicher found it significant only when
the square of the debt ratio was also introduced.
These results are not fatal to the theoretical prediction
of a positive relationship between the debt ratio and stock
beta, other things being equal. You can argue that other
things are not equal, so that other variables may be obscuring
or proxying for the hypothesized effect. Nevertheless, these
results introduce the seed of doubt.
Fortunately for theory, Hamada [8 ] devined an indirect
test which isolates the effect of financial leverage on beta.
3-75
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There is, of course, substantial variance in the cross - sectional
distribution of stocks' betas, even for firms in the same
industry. Since debt ratios also vary, theory would predict
that part of the dispersion is due to different debt policies.
Therefore,Hamada grouped a sample of firms into nine reasonably
homogenous industries and computed "unlevered betas" for each
firm. The unlevered betas were estimates—based on theory, the
stock beta and the observed debt ratio—of the beta the firms
would have had if their debt ratios were zero. Hamada's predic-
tion was that the dispersion of the firms' betas would be reduced
by this unlevering process. (If the stock beta were unrelated
to leverage, then the dispersion would be unaffected by unlevering
Since each industry group was chosen to hold other determinents
of beta roughly constant, the unlevered beta estimates should
cluster more closely around some true "industry beta.")
Hamada's prediction is confirmed by his results, which justify
substantial confidence in the link between financial leverage
and the stock beta.
Growth. Growth is the weakest of the four candidates
which I tentatively proposed as real determinants of beta.
Its weaknesses show up again in Pettit and Westerfeld's study.
Although they find a significant pairwise correlation between
the growth rate of earnings per share and the stock beta, the
variable is insignificant in multiple regressions. Breen and
Lerner find a generally significant relationship, but again it
has the wrong sign (negative) in two of the twelve subsamples
examined.
3-76
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On the other hand, the dividend payout ratio is negative
and significant in these two studies, as well as in Mclicher's.
Perhaps this ratio is simply a better proxy for the firm's
growth prospects. Unfortunately, none of the studies cited in
this paper have attempted to determine what the dividend payout
ratio is really proxying for, or whether it has an independent
effect on beta.
Summary
The empirical evidence on the real determinants of beta
can be summed up in two sentences. At least three real factors
can be identified with relative confidence, namely financial
leverage, cyclicality and volatility of operating earnings
(although the third may be proxying for the second). Growth
is a possible fourth variable, but its performance in the
empirical work cited has been erratic.
The summary has to be brief, because there is very little
to say once the four factors are noted. despite the substantial
number of studies cited, there is scant agreement on how the
factors are to be measured and the exact specification of their
effects on beta. Theory gives no guidance (except in the case
of financial leverage) , so there is little to say beyond the
generalities in the paragraph just above. Further progress in
understanding the real determinants -of beta depends on the
development of a theory which specifies the relevant variables
and how they should in principle be measured. A modest
beginning is made in the next section.
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FOOTNOTES
FOR S. C. MYERS' ARTICLE
1. The best-known proponent of this view is Gordon [5].
2. Hamada [8].
3. Dyer has found evidence of this effect. See J. Dyer,
"Financial Leverage, Business Risk and the Tax Subsidy
I
to debt: An Empirical Study." Unpublished M.Sc. Thesis,
MIT, 1974.
4. Errors in observing true values of beta and the accounting
variables ought to be reduced by forming portfolios.
5. The explanatory power was only slightly less when the
accounting beta was substituted for earnings volatility.
6. Actually, White used the ratio) of the book value of debt
to the market value of equity. It is the divergence
between book and market values of equity which causes
most of the error in the book debt ratio.
7. And also for various portfolio sizes, variable definitions,
etc.
8. Beaver, Kettler .and Scholes [2], p. 667.
9. It might be noted that Gonedes's sample was smaller than
that of BKS or White, and moreover that his sample was
concentrated in a few specific industries. Gonedes
3-78
-------
properly criticizes BKS for normalizing the earnings of
firms in their sample by past market values of equity.
Firms with high betas will tend to have more volatile
equity values, and under BKS's procedure, will auto-
matically tend to have more volatile earnings and higher
accounting betas. Thus BKS are not correlating beta solely
with accounting variables, but also with stock price
volatility.
10. Rosenberg and McKibbin [14], p. 327.
11. This follows from Miller and Modigliani's well known
proof of the irrelevance of dividend policy [12].
12. They first fitted a trend line to five years' earnings
per share data, and then measured volatility as the
proportion of the variance explained by the trend. This
is a poor procedure, not only because of the small number
of earnings observations, but because there is no necessary
2
relationship between R and the standard deviation of the
series. A firm's earnings could have a low standard devia-
2
tion but no trend and an R of essentially zero. Breen
2
and Lerner would take the low R as evidence of high
volatility in earnings.
3-79
-------
REFERENCES FOR EXCERPT FROM
S. C. MYERS' ARTICLE
1. Ball, R. and P. Brown, "Portfolio Theory and Accounting,"
Journal of Accounting Research, 7 (Autumn 1969), 300-323.
2. Beaver, W. H., P. Kettler and M. Scholes, "The Association
Between Market Determined and Accounting Determined Risk
Measures," Accounting Review, XLV (October 1970), 654-82.
3. Breen, W. J. and E. M. Lerner, "Corporate Financial
Strategies and Market Measures of Risk and Return,"
Journal of Finance, XXVIII (May 1974), 339-52.
4. Gonedes, N. J., "Evidence on the Information Content of
Accounting Numbers: Accounting-Based and Market-Based
Estimates of Systematic Risk," Journal of Financial and
Quantitative Analysis, VIII (June 1973), 407-44.
5. Gordon, M. J., The Investment, Financing and Valuation of
the Corporation (Homewood, Illinois: Richard D. Irwin,
1962).
6. and P- Halpern, "Cost of Capital for a
Division of a Firm," Journal of Finance, XXIX (September
1974), 1153-64.
3-80
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7. Hamada, R. , "Portfolio Analysis, Market Equilibrium
and Corporate Finance," Journal of Finance, XXIV
(March 1969) , pp. 13-31.
8. , "The Effect of the Firm's Capital Structure
on the Systematic Risk of Common Stocks," Journal of
Finance, XXVII (May 1972), 435-52.
9. Lev, B., "On the Association Between Operating Leverage
and Risk," Journal of Financial and Quantitative Analysis,
IX (September 1974), 627-42.
10. and S. Kunitsky, "On the Association Between
Smoothing Measures and the Risk of Common Stocks,"
Accounting Review, XLIX (April 1974), 259-70.
11. Melicher, R. W., "Financial Factors Which Influence Beta
Variations Within a Homogeneous Industry Environment,"
Journal of Financial and Quantitative Analysis, IX (March
1974), 231-43.
12. Miller, M. H. and F. Modigliani, "Dividend Policy, Growth
and the Valuation of Shares," Journal of Business,
34 (October 1961), 411 53.
3-81
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13. IVttit, R. arid R. Wester f eld, "A Model of Capital Asset
Risk," Journal of Financial and Quantitative Analysis,
7 (March 1972), 1649-68.
14. Rosenberg, B. and W. McKibben, "The Prediction of
Systematic and Specific Risk in Common Stocks,"
Journal of Business and Quantitative Analysis, VIII
(March 1973), 317-34.
15. White, R., "On the Measurement of Systematic Risk,"
Unpublished Ph.D. Dissertation, MIT, 1972.
(End of Excerpt from S. C. Myers' Article.)
3-82
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(b) Recent Studies
Two new studies have been published since the Myers
survey. The first is by Be.iver and Manegold, the second by
Gonedes.
Beaver and Manegold
The Beaver and Manegold Study [l] concentrates exclusively
on the relationship between market and accounting betas.
The market betas were computed in the usual manner by
regressing monthly stock returns on the corresponding returns
for the NYSE index. Three accounting betas were computed for
each stock using different definitions of accounting return.
(Note that the lack of a theoretical model leaves the specifi
cation of the accounting return unclear.) The returns used
were: (1) net income to total assets, (2) earnings to net worth,
and (3) earnings to market value of net work (shares x price).
These returns were computed for each sample firm (254 companies)
for each test year (1951-69). The first two return measures
are based solely on book data, the third combines book and
market data. For each year, the returns on the index portfolios
were unweighted averages of the 254 stock returns.
The data base for the study consisted of 254 firms (all
those on the Compustat Annual Industrial Tape with complete
data over the 1951-69 test period). Betas were computed for
the total period and for two subperiods (1951-60, 1961-69).
In addition to the procedures summarized here, several other
3-83
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refinements were employed to reduce sampling errors in the
estimated market and accounting betas (for example, a Bayesian
correction to the market betas; also accounting betas were
produced using first differences of accounting returns).
The end result of these various testing refinements was
378 sets of market and accounting betas. A correlation
coefficient was measured for each set. Of the 378 total, 363
of the coefficients were statistically significant bevond the
0.01 level, while 375 (all but threej were significant beyond
the 0.05 level. The implications of this finding are that
across a wide variety of specifications for accounting betas
and market betas, there is a statistically significant
relationship between the two. Alternatively stated, there
appears to be a significant relationship between risk as
reflected in security prices and risk as reflected in
accounting earnings data, and the existence of the relation-
ship can be detected in a variety of forms.
Apart from the issue of statistical significance is
the magnitude of the correlation coefficients. For the
total period, cross-sectional differences in the 254
accounting betas explained only about 15 to 20 percent of
the cross section variation in market betas. For the first
period, the proportion explained is approximately 20 to 25
percent. For the second period the account betas based
on the first two accounting return measures explain only 3
to 6 percent and the third form of the accounting betas
explains about 25 percent.
3-84
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A sample scattergram (one of 378 possible) is shown in
Figure 3-4. The figure shows the relationship between total
period market betas (vertical axis) versus total period
accounting betas. The accounting betas are based on the net
income to total assets return measure. The fact that a
relationship exists between the market and accounting betas
is evident from Figure 3-4. It is also evident, however, that
accounting betas do not explain much of the variation in
market betas.
The small proportion of variation accounted for in the
Beaver Manegold Study is due to a combination of two reasons.
First, the accounting betas have substantial measurement
errors (high standard errors) due to the small number of
annual observations available (compared with the market betas
which are based on monthly data). Second, the accounting beta
is only one of several possible factors that determine a
securities systematic risk. While both factors are obviously
involved, it is not possible to determine their relative
importance in this study.
Gonedes
A second article by Gonedes was published in June 1975 [5]
updating his earlier results [4]. The topic again was the
relationship between market and accounting betas.
The model used to obtain market betas was the familiar
market model
Rit = Oi + 0. Rmt + eit (3-6)
3-85
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1.7Jo«
SCATTER DIASgAH
MAKKEI BETA VEBSUS ACCOOBTTJIC BETA
TOTAL
PERIOD,
NOHBAYKSIAN
MARKET
BETA
* * * u * *
u » • *
» e * *
C. S /4
TOTAL PERIOD, EWNBAYESIAN, REGULAR
NL/TA ACCOUNTING BETA
Figure 3-4.
Scatter diagram: market beta
versus accounting beta. (Source:
Beaver, W. and Manegold, J., [1],
Figure 5, page 259.)
3-86
-------
where
it = ^e ra^e °f return on security i during
period t (tilde (^) denotes random variable)
the rate of return on the market portfolio
R
mt
in period t
An analogous model was used to measure the accounting betas
(designated 6 in Equation (3-7)):
where
•\j
V-. = the rate of return, based on accounting numbers,
for firm i in period t
a.
V = the market-wide index of rates of return, based
on accounting numbers
The specific measure of accounting return used by Gonedes
was the ratio of net income (for year t) to the book value of
common equity at the beginning of year t. The market-wide
index of returns used was a common-equity-weighted index of
the accounting returns for the firms in the sample. (The
method of index construction was one of the principal
differences between the two Gonedes studies.)
In addition, accounting betas were produced using first
differences of the stock and index accounting returns.
The data sample contained 316 firms (as opposed to
99 in the first study). The accounting betas were computed
3-87
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from annual data for the period 1940-69 (excluding 1953, 1960,
and 1968, which were reserved for various predictive tests
not described here). The market betas were computed using
monthly data for three intervals, 1946-52, 1953-60, and 1961-68.
The 316 accounting betas were then correlated with the market
betas from each of the subperiods. The results were somewhat
more encouraging than in the first study. The accounting betas
produced from Equation (3-7) tend to be significantly relat3d to
the market betas, which was not the case in the earlier stuiy.
Further, the difference in the results for the ordinary and
first difference accounting returns is now almost negligibls
(which is more consistent with the findings of the other
studies). However, Gonedes still finds the proportion of the
variation in market betas explained by the accounting betas
to be smaller than in other studies. For example, the propor-
tion of cross-sectional variation explained in the various
Gonedes regressions ranged from only I to 2 percent very
small indeed.
(c) Summary
The research into the nature of the real determinants of
beta is almost entirely of an empirical nature. Much of the
confusion that exists with respect to the measurement of
i
variables, the specification of tests, and the interpretation
of empirical findings results directly from the lack of an
acceptable theoretical framework. The empirical studies have
3-88
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identified four factors which seem to be systematically related
to beta: cyclicality and variability of earnings, leverage,
and growth.
i- Cyclicality of Earnings
Cyclicality is the tendency for corporate earnings
to move with earnings in the economy generally. Earnings
have been measured in different ways in the various studies.
However, the accounting return on equity is commonly used
(earnings divided by book value of equity). Cyclicality is
measured by an accounting beta, computed by regressing these
returns on market-wide index of accounting returns (see
Equation (3-7)). The accounting betas are typically found to be
related to the market betas in a statistically significant
manner. However, the authors disagree on the proportion of
the cross - sectional variation in market betas explained by the
accounting betas; percentages range from 1 percent to 40
percent, depending on the sample of firms and time periods
involved. A best guess would put the central tendency between
10 and 20 percent.
ii. Earnings Variability
Earnings variability is measured by the standard
deviation of historical earnings. It is a measure of the
total risk-of book earnings, while cyclicality is a measure
of systematic earnings risk. Several studies have found
3-89
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variability as well as cyclicality to be strongly related lo
the market beta. This result is somewhat surprising, since
portfolio theory would predict that only the systematic
component of earnings risk and not the total should be important,
It may well be, however, that the variability is serving as a
proxy for cyclicality. It is hoped that future studies will
more carefully segregate the systematic and nonsystematic
components of book earnings risk.
iii. Leverage
Financial leverage (the ratio of debt to total
assets) shows up significantly in almost all the studies where
it is considered. This is true whether the debt ratio is
defined in book (i.e., accounting) or market value terms.
Leverage is the one variable for which a specific theoretical
relationship exists. As given by Hamada [6],
BL = B0 (l + (1 TC) f | (3-8)
where
BL = the market beta for the stock of a levered
firm
BO = the unlevered beta (i.e., the beta of the
stock if the firm was all-equity financed)
g- = the market value debt- tto- equity ratio
Thus, for firms with similar business risks (as reflected by
BQ), the use of leverage will directly increase the risk of
3-90
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the common stock (as given by $T). This relationship, while
]_j
difficult to test directly, has been tested in indirect ways.
The results support the hypothesis (see Hamada [7]).
iv. Growth
Growth is measured in various ways in the different
studies (e.g., growth in earnings or sales) and captured by
various proxies (e.g., dividend payout ratio). The growth
variables were the most erratic of these four determinants.
It is commonly held that high-growth firms are riskier (i.e.,
they have higher market betas). This hypothesis was not
conclusively borne out in the studies reviewed. In some of
the multiple regressions growth was significant, but sometimes
the relationship was the reverse to that predicted.
3-91
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FOOTNOTES FOR PART 3
1. The summary of risk-return concepts contained in
Section II is based on a presentation by Stewart C
Myers and Gerals A. Pogue [20], pp. A3-A20.
3-92
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ESTIMATION OF THE COST OF CAPITAL
FOR MAJOR UNITED STATES INDUSTRIES
WITH APPLICATION TO POLLUTION-CONTROL INVESTMENTS
PART 4. ESTIMATION
Dr. Gerald A. Pogue
4 Summit Drive
Manhasset, New York 11030
November 1975
-------
I. INTRODUCTION l
Part 4 contains the estimation methodology and empirical
results. It is divided into seven sections.
Section II shows how the discounted cash flow cost of
capital models and the risk-return concepts of Parts 2 and 3,
respectively, can be combined to produce cross-sectional esti
mation equations. Section III describes the primary and
secondary company samples used in the study. Section IV
presents the equity risk measures for sample firms. Section V
describes the procedures used to estimate the cross - sectional
regression variables and presents the cost of equity capital
estimates. Section VI contains the methodology used to
estimate weighted average costs of capital and presents the
results. Finally, Section VII shows how the weighted costs
are likely to change during the 1975-84 period given estimates
of future interest rates.
4-1
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11. METHODOLOGY
(a) Combining the Cost of Capital and Risk-Return Models
In Part 2 three security valuation models were presented
which permitted estimates of a firm's cost of equity capital.
The models differ only in their assumptions regarding the firm's
future investment opportunities. The models are:
1. Perpetual Growth Model
Di
R = / + g (2-8)
2. No (Real) Growth Model
R = P^ (2-10)
3. Finite-Period (Real) Growth Model
E, I1
R = p1 + pi (r - R)T (2-12)
0 0
where the variables in the three models are collectively
defined below:
R = the cost of equity capital
PO = the current share price of the firm's common stock
E^ = the expected earnings per share during the
next year
D^ = the expected dividends per share during the
next year
4-2
-------
1^ - the expected equity investment per share during
the next year (i.e., retained earnings)
g - the expected growth rate of earnings and dividends
r = the rate of return on reinvested earnings
T = the number of years the firm is expected to have
"real" growth opportunities (i.e., r ^= R)
These models are typically used to produce estimates of the
cost of equity for individual firms. However, the market's
expectations for the various variables required by the
models cannot be precisely estimated using simply historical
data. Thus, the models will estimate equity costs with error.
But, as described in Part 2, the combined estimates of models 1
and 2, and the estimates of model 3 should be reasonably
unbiased (if noisy") proxies for the true costs.
If all of the firms in the data sample had the' same equity
risk, it would be a straightforward matter to estimate the cost
of equity capital. The optimal estimate would be a weighted
average of the individual values; individual company estimates
with lower measurement error would be weighted more heavily in
the average cost figure.
Unfortunately, the samples do not have the same risk.
Sample industries cover a wide range of equity risks. Even
within industries the companies differ significantly in risk.
What is needed is an estimation procedure that explicitly
allows for differences in company risk. The risk-return theory
4-3
-------
described in Part 3 provides a basis for such a procedure.
The Capital Asset Pricing Model (CAPM) specifies a relationship
between capital costs and equity risk, as measured by beta:
R = RF + e(Rm - RF) (3-3)
where
R = the cost of equity capital (i.e., the required or
expected rate of return on equity)
3 = the beta value for the firm's stock
R = the expected rate of return on the market
m r
Rp = the risk-free rate
A generalization of this model (the two-factor model) was
described in Part 3 as well (see Equation (3-4)). Both models
hypothesize a linear relationship between the cost of capital
(R) and risk (B).
It is not assumed in this study that the CAPM is empirically
valid. The tests of the model summarized in Part 2 indicate
that the slope of the actual risk-return line tends to be flatter
than that predicted by the CAPM. However, the linearity of the
risk-return relationship has been strongly established in several
studies. In this research this result is used. That is, the
relationship between the cost of capital and beta is assumed to
be given by
R = a + b(B) (4-1)
4-4
-------
where a and b are constants. Comparison of Equations (3-3) and
(4-1) shows that if the CAPM was empirically valid, then the
estimated values of a and b should approximate the risk-free
rate (Rp) and the market risk premium (R Rp) respectively.
(b) Estimation Equation (1 5)
The risk-return relationship (Equation (4-1)) provides no
insight as to how R should be estimated. Procedures for
estimating R for individual companies are given by the three
cost-of-capital models. Thus, combining Equation (4-1) with
the three models will result in estimation equations in which
all of the variables are directly observable or estimable.
Each of these equations will be dealt with in turn.
1. Perpetual-Growth Estimation Equation (Model 1)
'D-,
=± + g = a + b(B-) + e- (4-2)
0 Jj 3 ]
j = 1, ... , n
The j subscript designates the company. The
regression coefficients are estimated cross sectionally
using "point-in-time" estimates for the n sample firms.
The e. term represents the measurement error
resulting from the use of Equation (2-8) to estimate
RT. The e- is the difference between the true
J 3
4-5
-------
(unobservable) cost of capital and the estimated
2
value from Equation (2-8).
Since the variables in Equation (4-2) are
estimates of the "true" values, it is common practice
to put hats (") over them to so indicate (e.g., 3-).
This has not been done, however, for reasons of
notational simplicity.
2. No-Growth Estimation Equation (Model 2)
P
o
b(B.j) + e.j (4-3)
j = 1, ..., N
For Models 1 and 2, the estimated cost of capital
^ /\ ^ ^ ^
for firm j is given by a + b(3-) where a and b are
the estimated values for a and b resulting from
cross-sectional regression analysis for a specified
date (e.g., December 1974). (The cross-sectional
procedures are described in subsection (d).)
3. Finite-Growth Estimation Equation (Model 5)
The finite-growth model is more complicated and
several difficulties must be resolved before it can
be used to estimate R. These involve the T parameter
and the recursive nature of Equation (2-12).
The T parameter measures the duration of real
growth. It is not directly observable and must be
4-6
-------
estimated from the regression analysis. Conceivably,
every sample firm would have a different T value.
However, this would prohibit estimation of any T
values. Another assumption would be to assume all
firms have the same T value. But this would be
unrealistic. A compromise assumption which has
considerable merit is to assume the T is reasonably
constant for firms in an industry group. The data
base of the study consists of eight primary sample
groups (see Section III for a description of the
data base used). Thus, a duration parameter is
defined for each of the eight industries (T, ,
k = 1, ... , 8). Since the duration variables are
not directly observable (i.e., they must be estimated
from the regression analysis), they are replaced by
regression coefficients (C, , k = 1, ... , 8). These
coefficients will be estimated from the cross-sectional
regression along with a and b.
Each firm belongs to only one industry. To
separate the firms into groups, a series of eight
"dummy" variables (Z,, k = 1, ... ,8) are defined.
For each sample firm only one of the Z, variables
will equal 1, the remainder will be zero. The form
of the estimation equation is given by
4-7
-------
b(p.) + I C
k=l
(r-R) ^
Z
k
0,
j = 1, ... , N (4-4)
Note that Equation (4-4) has nine independent
variables: 3- and eight growth terms. For each
firm only the growth term corresponding to its
group membership will be non-zero.
The next problem with the finite-growth model
is its recursive nature. The term recursive refers
to the fact that R in Equation (2-12) is defined in
terms of itself. As a result, the growth terms in
Equation (4-4) contain R. While the other variables
in the growth terms (P~, r, I,) can be either
observed or estimated prior to running the regression
analysis, R obviously cannot be; R is the object of
the whole exercise.
The solution is to perform the regression analysis
for Equation (4-4) on an iterative basis. That is,
initialize R in Equation (4-4) with a best guess,
estimate the a, b, and C^ (k = 1, ... , 8) parameters,
re-estimate R using Equation (4-4) (i.e., R = a + b(3-))
and proceed as before. The iterative process will
continue until the estimated parameters stabilize.
The next question is how to initialize the
process; what should be the first "best guess" for
4-8
-------
the R_. values? The obvious answer is to initialize
with the perpetual-growth (Equation (2-8)) or no-
growth (Equation (2-10)) values. If the iteration
process is valid, it should not depend on which
estimate is used for initialization. This is the
procedure used in this study.
(c) Model 4—The Long-Run Risk-Return Line
It would be possible to estimate the cost of equity capital
directly from the Capital Asset Pricing Model (CAPM)- According
to the model (see Equation (5-5)), the rate of return expected
by stockholders is the sum of the risk-free rate (R,-) plus a
risk premium that is proportional to the stock's beta. However,
strict application of this formula poses several difficulties.
First, the long-run relationship between risk and return is
flatter than the CAPM predicts. Second, what is the appropriate
risk-free rate? Should we use the rate on short-term or long-run
government bonds? Third, what returns do investors expect on
the market—i.e., what is R ?
In view of these problems and particularly the first, it
would be unwise to rely solely on the CAPM to estimate R. Never-
theless, it would seem useful to consider the implications of
the long-run relationship between realized stock returns and risk
The fourth estimation equation is an extended version of
the CAPM which permits a flatter risk-return line than the
4-9
-------
original model (and thus eliminating the first difficulty with
the CAPM). It is given by
RF) (Sj - 1) (1 - 6) (Rm - RF)
Rearranging terms, we obtain
R., - [RF - (Rm - RF) (1 - 5)] + 6j(Rm - RF)-6
(4-5)
where 6 is the ratio of the actual to the CAPM-predicted slope
of the long-run risk-return line. 6 was estimated as follows.
The slope of the relationship between return and risk for all
stocks listed on the New York Stock Exchange between March
1931 and June 1970 was 0.827 percent per month. The predicted
value from the CAPM was 1.171 percent per month. The slope
attenuation function, 6, is the ratio of these two numbers, or
0.706. In its predictions of R., this extended model will
use a relationship between return and risk which is consistent
with long-run experience, as opposed to the one predicted by
the CAPM.
The second and third difficulties with the CAPM remain;
that is, what values should we use for Rp and Rm? Unfortunately,
no complete solution exists, and one can only try to do what
seems reasonable. For Rp the 1-year government rate for the
12 months following the estimation date was used. The expected
market return, Rm, was estimated by applying the perpetual-
growth model, Equation (2-8), to the market index. The index
4-10
-------
used was the Standard and Poor's 500 Stock Composite Index.
Since the index represents the market as a whole, there is no
a priori reason to believe that Equation (2-8) would produce
a seriously biased estimate of R . (The values used for the
1971 74 estimation equation are given in Table 4-8.j
(d) Cross-Sectional Regression Procedures
Cross-sectional regression analysis was used to estimate
the coefficients of Equations (4-2) through (4-4) for four
dates—December 1971 through December 1974. This procedure
pools the data from the N sample firms at each date to estimate
the relationship between equity cost and risk existing at that
time.
Prior to performing the cross - sectional regressions,
preliminary time series studies were performed for each firm
to estimate the necessary dependent and independent variables
(e.g., 3., g.). The details of the time series variable
estimation are described in Sections IV and V.
Weighted Regression Procedures
The three discounted cash flow (DCF) cost-of-capital
models are, in effect, used to prepare preliminary estimates
of the equity cost of each sample firm. These estimates are
then combined with the beta measures in cross-sectional
Equations (4-2) through (4-4) to produce new estimates for R..
4-11
-------
(i.e., R. " a + b(3.))- This procedure combines the informa-
tion in all N observations to produce more efficient estimates
s\
of R. than can be obtained individually by the DCF models.
This is the rational for the cross-sectional procedure.
The estimates of R. from the DCF models contain different
amounts of information. For some companies the historical data
permit estimates of R. having relatively little uncertainty
(i.e., small measurement error). For others the error will be
so large that estimates of R. will convey little information
about the true value.
Thus, when estimating the risk-return relationship at a
/\ /s /".
given time (i.e., a, b, and C, for December 1974), it would not
y\
be useful to give each DCF R. equal weight in the cross-sectional
regression. It is preferable from the statistical point of
view to weight each company estimate by its information content.
The appropriate weighting factors are related to the
standard deviations of the measurement errors of the DCF
estimates. The larger the standard deviation, the smaller
the amount of information about the true R. conveyed by the
estimate. Thus, the weighting factors would be the reciprocal
of the standard errors. Each of the three cross-sectional
equations would thus use different (but obvious highly related)
sets of weights.
In this research the weights used for all three cross-
sectional equations are the reciprocals of the standard error
4-12
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of past earnings growth. This weight is used as a proxy for
the weight described above which is not observable.
The standard error of past earnings growth is hypothesized
to be a good proxy for the unobservable measurement error
standard deviation. The degree of confidence we can have in
individual DCF R. estimates depends on our ability to infer
the future from historical data. In some cases, where past
trends are well defined, we can feel somewhat confidence
regarding our estimates. However, where the historical pattern
is murky, the time-series estimates will tend to be unreliable
predictors of what the market expects.
Further, since the valuation equations rely on the same
set of historical data for estimation, the measurement errors
of the R. estimates produced by Equations (2-8), (2-10), and
(2-12) will tend to be of comparable size. Thus, as an approxi
mation, it is not unreasonable to consider the same set of
weights to be applicable for all three DCF estimates. In this
study the same set of weights has been used for all three
cross-sectional regressions.
Details on estimation of the weights are contained in
Section V(b).
Weighted Cross - Section Estimates of R.
Estimates of R- based on the cross-sectional results were
prepared for each of the three estimation equations as follows:
4-13
-------
(4-6)
/N /N
where the a and b are regression estimates for a and b, and
the £ superscript indicates the model number (£ = 1, 2, 3).
/\
Additionally, for Model 3 the C, estimates indicate the
duration of real growth for each of the eight industry groups
Details of the results are left to Section V.
(e) Combining the Results of Models 1 through 4
The next step is to combine the four estimates into a
single estimate. This was done by average the four values.
An unweighted average was used since no compelling reason
existed for weighting the results of one model more than
3 4
another. The combined estimate, R., is given by
(4-7)
j = 1, . . . , N
It now remains to estimate an error range for the combined
/\
equity cost estimates, R.. This is accomplished by first
examining the relationship between the cost of equity capital
and beta produced when the four models are combined. That is,
R. = a + b^) + EJ (4-8)
4-14
-------
where the a and b terms are unweighted averages of the values
/N X*.
from the four equations (see Table 4-9 for estimates of a and b)
Now, the uncertainty in the estimated cost derives from the
variation in the four estimates, plus the standard error of the
beta coefficients (SE-g). The standard error associated with
the equity cost estimates, a., is measured by
a -
12
(4-9)
where sT, the variance of the e- variable, was estimated from
~ £
the variance of the R. estimates (I = 1, 2, 3, 4).
4-15
-------
III. DEFINITION OF THE PRIMARY
AND SECONDARY SAMPLES
The primary purpose of the study was to estimate the
costs of capital for six basic American industries: pulp and
paper, chemicals, petroleum refining, iron and steel, non-
farrous metals, and utilities.
The data base for the study was obtained from the Compustat
PDE tape. This tape contained 314 firms in the six groups
listed above. The data tape is organized along industry lines
as defined by the Standard Industry Codes (SIC). The six groups
are spanned by eight two-digit SIC code groups. The distribution
of firms in these two-digit industries is shown in Table 4-1.
The 314 firms were divided into primary and secondary
samples.
The primary sample consists of all companies with
sufficient historical data on the tape to permit inclusion in
the equity cross-sectional regressions. To be included, a
company must have at least 7 years of earnings data on the
tape. The primary sample included 208 of the 314 companies.
The distribution by group is shown in Table 4-1.
The secondary sample is a further classification of the
314 firms into the original six groups plus 26 subgroups.
(The subgroup names are given in Table 4-3.) The subgroups
4-16
-------
represent an attempt to build smaller homogeneous groups for
the purpose of later analysis. The major stratifying factors
were: size of operation (petroleum refining), degree of
resource control (papers), product similarities (chemicals,
steel, non-ferrous metals), and accounting convention (utilities)
The secondary sample included 207 firms. A high degree of
overlap existed between the two samples, but several firms in
the secondary sample were not contained in the primary sample.
For these firms, cost of equity capital estimates were imputed
from the results of the primary sample by substituting their
estimated betas into Equation (4-8)
4-17
-------
TABLE 4-1
DEFINITION OF SAMPLE SIZES
NUMBER OF COMPANIES
GROUP NAME
MINING *
FOREST PRODUCTS **
PAPER **
CHEMICALS
REFINING
STEEL
NON-FERROUS *
UTILITIES
TOTAL
SIC
CODES
1000-
1031
2400
2600-
2650
2801
2803
2911-
2913
3310-
3317
3331
3350
4911-
4912
TOTAL
ON TAPE
35
21
49
54
43
50
27
35
314
PRIMARY
SAMPLE
18
9
31
36
34
28
17
35
208
SECONDARY
SAMPLE
6
5
49
36 ***
36
27 ***
13
35 ***
207
* Combined as Non-Ferrous Metals group in
secondary sample.
** Combined as Pulp and Paper in secondary sample.
*** All companies in secondary sample are also in
primary sample.
4-18
-------
IV. MEASUREMENT OF EQUITY RISK
(a J Calculation of Betas
The basic data for estimating common-stock betas are
monthly rates of return during the February 1962 through
December 1974 period. The stock returns were computed from
the Compustat PDE tape.
The beta for a security is calculated by regressing the
monthly security risk premiums on the observed risk premiums
for the market. The risk premiums are formed by subtracting
the 30-day treasury bill rate from both the stock and market
returns. It is customary to convert the rates of return to
risk premiums to remove a source of "noise" from the return
data. The noise stems from the fact that observed returns
may be higher in some years simply because the risk-free rates
of interest are higher. Thus, an observed rate of return of
eight percent might be regarded as satisfactory in 1960, but
as a relatively low rate of return when interest rates were
at all-time highs in 1969. The form of the estimation
equation is given by:
r = a + Br . + e (4-101
t mt
t = 1, ... , 155
4-19
-------
where r. and r . are the observed risk premiums on the stock
t mt
and market index respectively during month t.
Betas were estimated for all stocks in the primary and
secondary samples. The market index used was the Standard and
Poor's 500 Stock Composite Index (with dividends reinvested).
The risk-free rate used to convert returns to risk premiums
was the 30-day treasury bill series.
The data tape began in February 1962 and ended in December
1974. For stocks with complete histories on the tape, the
full period was used in computing beta (155 months of returns).
For non-complete companies, all available data was used. All
companies in the primary sample had at least 5 years of
monthly return data.
Betas for groups and subgroups were computed by averaging
the betas for the companies in the group. This results in the
same beta as if the returns were first pooled into a group
portfolio and the portfolio beta computed as described above.
Beta Results
The average betas and standard errors of beta for the
eight primary groups are shown in Table 4-2 (first two numerical
columns). In addition, the standard error of the mean beta
is shown in parentheses under the mean. The data show forest
product common stocks to be the most risky (mean beta = 1.37)
4-20
-------
and, as expected, utilities the least risky (mean beta = 0.70)
When comparing mean betas, the reader must keep the standard
errors in mind. Differences in sample means can only be
considered statistically significant if large relative to the
standard errors of the means.
Similar results for the 26 secondary sample subgroups are
given in Table 4-3 (first two numerical columns). The
highest subgroup beta is 1.75 for the smallest category of
integrated petroleum refiners (10 - 30 thousand barrels per
day). However, this group contains only one company, and
the standard error of the measured beta is large (0.26).
Hence, this result is suspect. The lowest risk subgroup is
flow through earnings public utilities with mean beta equal
to 0.68.
4-21
-------
TABLE 4-2
STOCK RISK AND CAPITAL STRUCTURE DATA
PRIMARY GROUP AVERAGES
GROUP NAME
MINING (18)
FOREST PRODUCTS (9)
PAPER (31)
CHEMICALS (36)
REFINING (34)
STEEL (28)
NON-FERROUS (17)
UTILITIES (35)
TOTAL (208)
STOCK RISK
AVG.
BETA
0.94
(0.05)
1.37
(0.08)
0.97
(0.05)
1.22
(0.06)
1.06
(0.06)
1.00
(0.05)
1.07
(0.07)
0.70
(0.01)
1.01
(0.02)
AVG.
SE • 3
0.18
0.18
0.17
0.17
0.17
0.17
0.17
0.09
0.16
MARKET CAPITALIZATION
%
DEBT
10.3
22.0
20.6
19.9
14.6
25.5
24.1
44.1
23.6
\
LEASE
3.1
11.1
13.1
17. 2
19.4
10.1
11.2
0.0
11.1
%
PREF
1.2
2.0
2.7
2.6
2.6
3.1
0.7
9.6
3.6
%
COM.
85.4
65.0
63.6
60.4
63.4
61.3
64.4
46.3
61.7
( ) = Standard error of mean beta.
4-22
-------
TABLE 4-3
STOCK RISK AND CAPITAL STRUCTURE DATA
SECONDARY GROUP AVERAGES
GROUP
PULP
AND
PAPER
CHEMICALS
PETROLEUM
REFINING
IRON AND
STEEL
NON-
FERROUS
METALS
•UTILITIES
SUBGROUP
, LARGE MULTIPRODUCT
COMPANIES (S)
, MEDIUM WITH RESOURCE
Z CONTROL (1?)
, SMALL-MEDIUM W/0
•> RESOURCE CONTROL (10)
. FOREST PRODUCTS
* COMPANIES (5)
S CONVERTERS (12)
6 CHEMICALS — MAJOR (11)
, CHEMICALS —
INTERMED.(6)
„ CHEMICALS —
0 SPECIALTY (19)
Q INTEG. MAJORS (14)
* (200 + B/D)
ln INTEG. LARGE (S)
IU (200 + B/D)
,, INTEG. (3)
11 (100 - 200 B/D)
,, REFINERS (1)
(100 - 200 B/D)
., INTEG. .(2)
• (70 - 100 B/D)
, INTEG. (2)
14 (30 - 70 B/D)
., INTEG. CAN. (2)
li (30 - 70 B/D)
., REFINERS (2)
10 (30 - 70 B/D)
17 REFINERS (3)
l/ (10 - 30 B/D)
... INTEG. (1)
18 (10 - 30 B/D)
19 STEEL — MAJOR (7)
20 STEEL — MINOR (20)
21 PRIMARY COPPER (8)
,, PRIMARY LEAD AND
r-L ZINC (3)
23 PRIMARY ALUMINIUM (4)
24 SECONDARY SMELTING (4)
25 UTILITIES FLOW THRU -(IS)
26 UTILITIES .NORMALIZED (16)
STOCK RISK
AVG BETA
1.03
(0.06)
0.87
(0.06)
0.73
(0.15)
1.20
(0.05)
1.13
(0.10)
1.12
(0.04)
1.23
(0.08)
1.28-
(0.10)
0.91
(0.04)
o: 87
(0.06)
1.31
(0.06)
0.9.6
(NA)
1.69
• (NA)
1.33
(NA)
0.79
(NA)
1.24
(NA)
1.68
(0.26)
1.75
(NA)
0.94
(0.05)
1.02
(0.06)
1.12
(0.07)
1.22
(0.32)
1. 15
(0.13)
1.10
(0.11)
0.68
(0.03)
0. 72
(0. 04)
AVG SE-B
0.13'
0.17
0.29
0.23
0.31
0.11
0.19
0.19
0.11
0.17
0.18
0.25
0.36
0. 27
0.24
0.33
0.31
0.26
0.13
0. 18
0.15
0. 18
0.14
0. 26
0.09
0.09
MARKET CAPITALIZATION
\ DEBT
22.7
22.1
17. 5
20.7
19.1
18.1
21.8
20.3
14.2
14.0
16.7
32. 2
25.3
29.0
31.2
19.4
13.4
11.0
30.8
24.3
16.8
25.4
40. 4
32.6
45. 5
42. 5
% LEASE
11.7
13.9
13.6
7.2
20.2
14.8
18.9
17.9
24.0
22.7
24.6
7.2
41.2
11.8
14.0
18.0
12.2
5.9
10.3
10.0
3.8
4.6
16.1
16.9
0. 0
0.0
% PREF
2.6
3.3
2.1
2.9
2.3
1.3
6.1
2.2
3.2
4.2
1.4
5.5
3.6
0.6
12.5
0.0
2.7
6.8
2.7
3.4
0.8
6.3
2.8
1. 3
10.3
8.8
% COMMON
62.9
60.7
66.9
69.1
58.4
65.8
53.2
59.6
58.6
59.1
57.3
55.1
29.9
58.6
42.3
62.6
71.7
76.3
56.1
62.3
78.6
63.6
40. 7
49.1
44.2
48.7
( ) • Standard error of mean beta.
NA • Not Available.
4-23
-------
V. COST OF EQUITY CAPITAL: EMPIRICAL RESULTS
This section describes the procedures used to estimate
the variables for the cross-section regressions (with the
exception of 3 which was dealt with in Section IV), summarizes
the regression equations, and presents the cost of equity
capital estimates.
(a) Variable Estimation for Cross-Sectional Regression
Models 1, 2, and 5
Most of the variables required by estimation models 1, 2,
and 3 were estimated using trend projections. The statement that
a variable is a trend value means that it is equal to the
estimated value for the current period derived from a
regression analysis of the variable against time. This
smoothing is done to reduce potential errors in measurement.
The reasoning is that annual observations of a variable can
be viewed, to a first approximation, as observations about a
trend consisting of a true component and a random element.
If these random elements have zero mean and are serially
uncorrelated, use of the trend value of the variable, rather
than the current observed value, will tend to reduce errors
in measurement. In computing the trend values, all available
data up to the estimation date are included (a maximum of 13
observations for 1974 estimates).
4-24
-------
The variables to be estimated are: (i) the future growt <
rate of dividends, g; (ii) the dividend and earnings yields,
DT/PQ and E^/P^; (iii) the rate of return on reinvested
earnings, r; and (iv) the investment ratio 1,/P^.
i. Growth Rate of Dividends (gj_
What is required is an estimate of how the
market expects dividends per share will grow in the future.
However, we have no choice but to try to impute this figure
from past growth rates. Since dividends tend to be smoothed
by management, it is customary to measure instead the growth
of past earnings. This procedure will typically produce a
more reliable estimate of how dividends will grow in the
future. The model of earning growth typically used is given
by:
Et = E61(l + g)1 (4-11)
t = 1, ... , maximum 13
where E is the earnings per share in year 1961 + t (1962 is
the starting date on the PDE tape). Taking logarithms of both
sides results in a regression equation which is linear in t.
log Et = Y0 + Y! • t (4-12)
4-25
-------
where YA = log Efi, and YI = log (1 + gj . Liquation (4-1.2) was
fitted to the historical earnings data for each company and
estimates of the growth rates obtained. In order to be
included in the primary sample, at least four positive annual
earnings observations were required prior to the estimation
date. Years with negative earnings were deleted from the
regression analysis.
In addition to the estimate of future growth rates,
the above regression provides another important estimate the
standard error of the growth rate. The standard error is a
measure of the amount of information in our forecasts of
future dividends. A large standard error indicates that very
little can be learned from past growth about future growth.
Hence, a cost-of-capital estimate based on projection of past
trends may be subject to wide error. Conversely, if the
standard error of the past growth rate is small, our cost-of-
capital estimate will likely be a reasonable proxy for the
market's required rate of return. The standard error is
consequently used as a weighting factor in the cross-sectional
regressions. The application of these weights is discussed in
subsection (b) below.
ii. Dividend and Earnings Yields, (D^/P^) and (E-,/Pnj
JL U ™ J. U ~-
Respectively
These values are the trend values of dividends and
earnings per share divided by the price as of the estimation
4-26
-------
date. For example, the 1974 cross - section regressions use
the trend value 1975 estimate for D and E divided by the
price per share as of December 31, 1974.
iii. The Rate of Return on Reinvested Earnings, r
The only practical alternative for estimating r is
through the book rate of return figures. The values used were
the trend values of earnings per share divided by the book
value per share at the beginning of the year.
iv. Investment Ratio, (!-/?„)
The values used were the trend values of the change
in book value per share during the year to the stock price at
the beginning of the year. The change in book value is
essentially the retained earnings during the year.
(b) Estimation of Weights for Cross-Sectional Regression
Equations
As described in Section II, a weighted regression
procedure was used to estimate the coefficients of Equations
(4-2) through (4-4). The weights are the reciprocals of the
standard errors of the estimated earnings growth rates. They
are estimated along with g from Equation (4-12).
Exhibit 4-1 shows a plot of the December 1974 weights
versus stock betas. The weights have been normalized (i.e.,
4-27
-------
1.318 XSD« 3.325
ISTE9CE»T« 1.889 SUO'E" -3.883
J.330 . YSO« 1.227 CORR,« -3.234 N03S» 238
CELL SIZES., X« 0.0237 y. a, 2322
oo
9.2 a
«.34 X
6.33
4. 55 i
3.42 X
2.?«
*
2 »•
»*»?*2
** *
2* *
X
».
1,29 x
» * * 3* 22
2 « 4 ** 2*» 32
» * * 3*. 233 *4
*» 2 29* »»?» 3*»* 4
3 * * *
32** *3
24 « 3 « *
3. * 2+ 3* * * *
«2 « »» 2,
SETA
-------
the sum of the weights is equal to the number of companies in
the primary sample, 208) to preserve the usual intuitive inter-
pretation of means and regression coefficients.
(c) Weighted Regression Results
The results of the cross-sectional regressions are
summarized in Tables 4-4 through 4-8. The tables give the
XN S\
estimated parameter values (a, b) , t statistics for the
parameter estimates, and R coefficients. The regression
results are presented for year-end 1971 through 1974 estimation
dates, plus pooled results which combine data from the four
periods.
i. Model 1 (see Table 4-4)
The regressions explain roughly 80 percent of the
variation in the raw Model 1 cost-of- capital estimates. The
intercepts (a) tend to understate the risk-free rates, and
the slopes overstate the likely market risk premiums. (See
the last two rows of Table 4-10 for an estimate of the
expected market return and the actual risk-free rate.)
ii. Model 2 (see Table 4-5)
The explanatory power of the Model 2 cross - sectional
regressions is somewhat less than for Model 1, averaging 66
percent over the four test dates. The intercepts, while some-
4-29
-------
TABLE 4-4
WEIGHTED REGRESSION COEFFICIENTS
MODEL 1
D.
a + b
j = 1, ..., 208
YEAR
1971
1972
1973
1974
POOLED
INTERCEPT
A
a
1.27
1.01
0.28
1.17
0.93
t Stat.
2.5
2.0
0.5
1.4
2.7
SLOPE
s\
b
11.50
11.62
14.0
16.6
13.42
t Stat.
32.3
32.3
34.4
27.8
56.3
R2
0.84
0.84
0.85
0.79
0. 79
4-30
-------
TABU- 4-5
WEIGHTED REGRESSION COEFFICIENTS
MODEL 2
a + b(8,) + y.
j = 1, ... , 201
YEAR
1971
1972
1973
1974
POOLED
INTERCEPT
/-\
a
2. 78
2.72
1. 83
4.11
2. 85
t Stat.
5.4
4.9
2. 7
3.5
6.2
SLOPE
^v
b
6.89
7.11
11.27
15. 72
10.25
t Stat.
19.2
18. 3
23. 5
19.2
31. 8
R2
0.64
0.62
0. 73
0.64
0. 55
4-31
-------
what larger than for Model 1, still tend to understate the
government bond rates. The slopes still appear to overstate
the market risk premiums, but by less than for the Model 1
results.
iii. Model 3 (see Table 4-6)
The cross-sectional estimation procedures for
Model 3 are more complicated, as they involve successive
iterations with revised initial estimates of R.. The results
shown in Table 4-6 are from the fourth iteration. After four
iterations, the estimated parameters had not significantly
changed from the third iteration, and the process was stopped.
Two sets of iterations were carried out, one starting with the
DCF Model 1 cost-of-capital estimate, Equation (2-8), and the
other with the Model 2 estimate, Equation (2-10). The
iteration results are summarized in Table 4-7. Table 4-7
shows how the average-cost-of- capital estimate (i.e., the
/s
mean R.) changed from iteration to iteration. Note that after
four iterations the average cost of capital is independent of
the starting value.
/S
The intercept (a) appears to reasonably approximate
s\
the risk-free rate. However, the slope (b) seems low relative
to a priori expectations about market risk premiums, particu-
larly in 1971 and 1972. The percent of variation explained
is approximately 75 percent over the four test dates.
4-32
-------
TABLE 4-6
WEIGHTED REGRESSION RESULTS
MODEL 3
a + 6(0.) +
8
I
k=l
r - R*)
I.
j - 1 208
Zk= 0 or 1
YEAR
1971
1972
1973
1974
POOLED
/\
a
(t)
5.24
[12. 2)
5.29
(11.2)
4.78
(7.5)
7.72
(5.8)
5. 24
(11.1)
b
CO
2.30
- (5.1)
2.35
(4.8)
6.15
(10.3)
10.43
(9.7)
5.70
(13.0)
DUMMY VARIABLE COEFFICIENTS (Ck)
MINE .
-5.22
(-1.7)
-5.24
(-1.7)
-2.53
(-1.0)
-1.82
(-0.8)
-0.55
(-0.4)
FOR
-3.86
(-1.1)
-2.44
(-0.9)
-0.82.
(-0.6)
-0.25
(-0.2)
0.16
(0.2)
PAPER
-4.19
(-1.4)'
-3.54
(-1.3)
-1.91
(-0.9)
-1.16
(-0.6)
0.01
(0.0)
CHEM.
-0.16
(-0.1)
-0.41
(-0.2)
-0.60
(0.0)
-1.01
(-0.5)
1.71
(1.4)
REF.
-4.11
(-3.0)
-3.14
(-2.0)
-1. 50
(-0.9)
0.01
(0.0)
0.20
(0.2)
STEEL
-2.95
(-1.1)
-3.50
(-1-4)
-0.25
(-0.1)
-0.86
(-0.5)
0. 34
(0.3)
NON-FER
-3.58
(-1.4)
-3.78
(-1.7)
-1.31
(-0.5)
-2.96
(-0.9)
-0.90
(-0.5)
UTIL.
-9.04
(-13.1)
-8.79
(-12.7)
-9.30
(-11.6)
-9.63
(-7.0)
-8.95
(-14.6)
R2
0.82
0.80
0.84
0.71
0.65
-------
TABLH 4-7
MODEL 3 ITERATION SUMMARY
Average Cost of Equity Capital Estimate from
Model 3 Cross-Sectional Regression.
STARTING VALUE
FOR R.
FOR ITERATION 1
Dl
+ CT
po g
El
po
EST.
DATE
1971
1972
1973
1974
1971
1972
1973
1974
AVERAGE COST (%/YEAR)
ITERATION NUMBER
1
9.39
9.88
13.78
18.42
8.78
9.07
13.08
18.70
2
7.86
8.00
11.20
18.08
7.76
7.88
11.13
18.13
3
7.61
7.72
11. 01
18.24
7.60
7.70
11.00
18.24
4
7.57
7.67
10.99
18.25
7.57
7-67
10.99
18. 25
4-34
-------
iv. Model 4 (see Table 4-8)
The parameters used in Model 4 are summarized in
Table 4-8. The model, Equation (4-5), along with the estimates
of Rp and Rm were used to produce estimates of the cost of
equity capital for each firm in the primary sample for the
four test dates plus the pooled interval.
v. Combination Model 1 through Model 4 (see Table 4-9)
Table 4-9 gives the resulting relationship between
cost of equity capital and beta when the results of the four
cross-sectional models were combined.
Equation (4-8) and Table 4-9 parameter estimates were
used to compute the final cost-of-equity capital estimates for
the 208 firms in the primary sample. Additionally, Equation (4-8)
was used to compute the equity capital costs for firms in the
secondary sample that were not included in the primary sample.
(d) The Cost of Equity Capital: Results
The estimated costs of equity capital for the eight
primary sample groups are given in Table 4-10. The cost esti
mates for the 208 firms in the sample were computed from the
regression results for Equation (4-8) which are summarized in
Table 4-9. The group estimates are unweighted averages of
the results for the stocks in the groups. In addition,
4-35
-------
TABU: 4-8
PARAMETERS FOR MODEL 4
R.
RT
m
(3j 1) (1 - 6) (Rj,
6 =
Ratio of actual to predicted slope of
return-beta line in CAPM tests
0.706 (January 31 June 1970)
YEAR
1971
1972
1973
1974
POOLED
EXPECTED MARKET
RETURN (%/YR)
R *
m
13.70
13.05
14.14
17.14
14.51
EXPECTED RISK-FREE
RATE (%/YR)
D **
KF
4.35
5.62
7. 21
7.01
6.05
D.
for the Standard and Poor's 500 Stock
Composite Index.
** Twelve-month treasury bills, from January.
Source: Solomon Brothers
4-36
-------
TABLE 4-9
RELATIONSHIP BETWEEN COST OF EQUITY CAPITAL
AND BETA RESULTING FROM COMBINING
MODELS 1 THROUGH 4
+ b(B-)
Imputed risk-free rate
Expected risk premium on market (R - Rp]
YEAR
1971
1972
1973
1974
POOLED
INTERCEPT
a = (RF)
4.11
4. 22
4.05
5.76
4. 41
SLOPE
b E (Rm - RF)
6.81
6.57
9.07
12.45
8. 83
4-37
-------
TABLE 4-10
COST OF EQUITY CAPITAL (% PER YEAR): YEAR END 1971 - 1974
PRIMARY GROUPS
GROUP NAME
MINING (18)
FOREST PR. (9)
PAPER (31)
CHEMICALS (36)
REFINING (34)
STEEL (28)
NON-FER. (17)
UTILITIES (35)
TOTAL (208)
EXPECTED MKT.
RETURN
12-MO. RISK-
FREE RATE
SIC
CODES
1000-
1031
2400
2600-
2650
2801-
2803
2911-
2913
3310-
3317
3331-
3350
4911-
4912
BETA
(62-74)
0.94
(0.05)
1.37
(0.08)
0.97
(0.06)
1.22
(0.06)
1.06
(0.06)
1.00
(0.05)
1.07
(0.07)
0. 70
(0.02)
1.01
(0.02)
1.0
0.0
YR END 1971
COST
10. 5
13.4
10.7
12.4
11.4
10.9
11.4
8.9
11.0
10.9
ERROR
1. 8
2.8
1.9
2.3
2.0
1.9
2.1
1.4
1.8
1.8
4. 35
YR END 1972
COST
10.4
13.2
10.6
12.2
11.2
10.8
11.2
8. 8
10.9
10.8
ERROR
1. 7
2.6
1. 7
2.1
1. 9
1. 7
2.0
1.3
1.6
1.6
5.62
YR END 1973
COST
12.6
16.5
12.9
15.1
13.7
13.1
13.7
10.4
13. 2
13.1
7.21
ERROR
1.2
1.9
1.2
1.5
1.4
1.2
1.5
1.1
1.2
1. 2
YR END 1974
COST
17.5
22.8
17.8
21.0
19.0
18. 2
19.1
14.5
18.3
18. 2
ERROR
1.0
1.9
1.0
1.4
1.2
1.0
1.3
0.8
1.0
1.0
7.01
POOLED
(71-74)
COST
12. 7
16.5
13.0
15.2
13.8
13.2
13.8
10.6
13.3
13. 2
ERROR
1. 2
1.9
1. 2
1.5
1.4
1.2
1.5
1.0
1.2
1. 2
6.05
(Standard Error of Mean)
-------
Table 4-10 gives the error ranges associated with the group
means. The error ranges for the individual companies and
groups were computed using Equation (4-9).
The last two rows of Table 4-10 give the expected market
return and risk-free rate as of the end of the test year.
The market return, having been produced from Equation (4-8)
with beta equal to 1-0, is consistent with the industry costs
of capital by construction. The risk-free rates, however,
are independent and are measured by the 12-month treasury bill
rates. These values are all substantially less than the
expected market returns, as would be expected.
The results for the pooled interval are shown graphically
in Exhibits 4-2 and 4-3. Exhibit 4-2 shows the estimates for
the 208 firms versus their beta coefficients. Since the
combined equity estimates were produced from Equation (4-8),
the resulting scattergram is a straight line as shown. The
error ranges in Exhibit 4-3 show a more complex pattern. the
error ranges tend to decrease as beta moves from 0 to 1, and
they increase sharply. Firms with the highest betas tend to
have the highest standard errors of beta and hence larger
error ranges.
4-39
-------
X1E»N« 1,313 XSO* 3.325 YME»M« 13.323 YSD» 2.875 CORR," 1.300 NOSSi 208
INTERCE»T» a.425 StO?E» 8.825 CELL SIZES,. X« 0.0237 r» 0.464B
AE3»
27.82 X * *" * * * *
25.53 X
23,13 X
20.85 X ]
• * 0 <
2
• * ,
16.53 X 02 ]
02
2
32
.4.
IS.23 X .40 ;
22
48,
343
454
13.33 X 583
M3
X5
545
572
11,55 X 46
9*
- *53
S3*
222
9,23 / »5
3
2.
•
-2. , ,
••••*••• 5t 535J"" 3, 7721*'" i.339i" "*i. 2491 "" 1 . 4831 ""'1°. 7231* "" I . 9571*'" 2.1 94l" ' " 2?43i 1* ' "l!
BETA
Exhibit 4-2. Pooled estimated cost of equity capital versus beta.
-------
1.313 X3D» 3.323 VMEAV» 2,215 ySO» 3,768 CORR." ?.727 N03S* 288
INTERCEPT" 3.^83 SLO"E« 1.714 CEtL SIZES.. X" 0.0237 Y» 3.1322
7,33
6,57 X
6.31 X
3,35 X
X
9
I*
4,55 X
4,33 X
3,37 x
2,73
00
3
3 323
3 33 0
. , 2 .
0
3 .
>,a«
* *»»*
2 . ,, 2 ,3 ,
2 V. 2])*1
* *»»» *2*33 642*2
t *2****»233
3223 3*23222
3,5351 3.7721 1,3391 1,2451 1,4831 1.7231 1.9571 2.1941 2.4311 2.9881
__ SETA
Exhibit 4-3. Error range for pooled equity estimate versus beta.
-------
VI. THE WEIGHTED AVERAGE COST OF CAPITAL:
EMPIRICAL RESULTS
Section VI describes the methods used to estimate
the cost of debt and preferred stock capital for sample
firms, to compute the market value proportions of the firms'
capital structures, and to compute the weighted average
costs of capital. Empirical results are presented for the
8 primary and 26 secondary groups .
(a) The Cost of Debt and Preferred Stock Capital
The cost of debt capital to a firm is simply the yield
to maturity on its outstanding debt. This assumes, of course,
that the firm would be able to borrow additional debt capital
at this rate.
It is a fairly straightforward matter to compute the
yield to maturity on any bond which is traded on an exchange.
For example, assume a bond has a market value Vn , principal
value P, maturity date T years in the future, and annual
interest payments of C dollars per year. Then the yield to
maturity, i, is found by solving the equation
T
VQ = I - - - F + - ? - f (4-13)
U t=l (1 + i)t( (1 + i)1
In principal, this calculation could be performed for each
4-42
-------
bond issue and firm in our sample. But the data collection
requirements of this approach make it impractical.
A second approach would be to use the embedded debt cost
to the firm. Embedded debt costs (^.e., interest paid
divided by the book value of debt) are available in machine-
y
readable form on the Compustat Annual Industrial Tape. But
these debt costs will understate the cost of debt capital
since most corporate debt has been issued in times of lower
interest rates (and hence the market values of debt outstanding
are less than the book values).
The approach used in this study is midway between these
extremes. An aggregate bond risk rating was prepared for each
sample firm as of the years ending 1971 through 1974. The bond
ratings were obtained from the January issues of the Standard
and Poor's Bond Guides. The S^P bond risk ratings ranged
from AAA (the highest quality) to BB (the lowest quality) for
bonds of the sample firms. When a firm had two or more bond
issues with different S§P ratings (e.g., A and BBB), a weighted
rating was constructed based on the relative amount of the
bonds outstanding. When a firm's bonds were not rated by
Standard and Poors, they were assigned the average rating of
firms in their primary group. For example, a paper company
with unrated bonds would be assigned the rating of the average
of paper companies with ratings available.
Once a bond risk rating was assigned to each sample firm,
the cost of debt capital was then given by the yield to
4-43
-------
maturity on the corresponding Standard and Poor's Bond Index.
For example, a utility with an AA risk rating was assigned
the current yield on the S^P AA public utility index. When
a firm had a composite bond risk index, a composite bond
yield was computed using the same proportions.
The advantages of this approach are that it is feasible
to produce statiscally unbiased estimates of debt capital
costs without an enormous effort. The degree of approximation
is substantially lessened as we shift our attention to groups
of firms.
For preferred stock, a similar process was contemplated.
However, since preferred stock is a relatively insignificant
portion of the capital structure of all the industry groups
with the exception of utilities, the significant effort
required would have borne small return. Further, most of
the issues tend to be of high grade, particularly in the case
of public utilities. Thus, for preferred stock a simpler
approach was used. An index of yields on high-grade corporate
and public utility stocks was used to measure the current
market costs of preferred stock for all firms.
(b) The Capital Structure Proportions
Computing the proportions of various types of financing
in the firm's capital structure would be relatively simple
if market values were available for each type of capital.
But this is not the case.
4-44
-------
The Compustat Annual Industrial Tape, which provides the
data base for the capital structure calculations, has market
values only for the common equity portion (number of shares
outstanding time price per share). The tape contains book
values for debt and preferred stock. The problem is to
convert these latter figures to market values.
The market values corresponding to the book values were
estimated by using the relationship between the embedded
financing cost and the current cost, as determined in Section
Vl-a. A factor was computed for each firm and type of financing
which converted the book values to estimated market values.
The factor, F, is given by
F - I —*-^ + -i^ (4-14)
t=l (1 + i)r (1 + i)1
where d is the embedded debt cost per dollar of book debt,
T is average maturity of the firm's debt, and i is the current
cost of capital. For preferred stock, T is typically infinite;
for bonds we used the average maturity of all outstanding
corporate bonds (about 11 years). This procedure is again
statistically unbiased and will produce better estimates for
groups rather than individual stocks.
Another thorny problem involves the question of leases.
Leases provide an alternative form of debt financing. As such,
4-45
-------
they should be capitalized and considered as an integral part
of the debt structure of the firm. This was done in the study
by assuming that current lease payments would continue
indefinitely, and then capitalizing these payments at the cost
of debt capital. This was done for each of the four test years
for every sample firm with lease payments.
The capital structure proportions were computed for each
firm in the primary and secondary samples for each of the four
test years. Additionally, a pooled value was computed for
each firm by averaging the four annual values. These pooled
values are shown in Table 4-2 for the 8 primary group averages,
and in Table 4-3 for the 26 secondary subgroup averages (last
four columns).
(c) The Weighted Average Cost of Capital
I
The groundwork has now been fully laid for the calculation
of the weighted average cost of capital, which we have designated
as p (the } subscripts denoting the firm number have been deleted
for ease of exposition). For each firm, the formula for p*, as
given in Part 3 is
n * = in T1 — + i fl T1 + V r -i- P ^
P i U i J v + i U ! ' v p ? R V'
(See Part 2, Equation (2-27) for definition of symbols.)
4-46
-------
The cost of equity capital R was measured in Section V;
the remaining variables were estimates, as previously discussed
in this section.
Further, we can estimate an error range for the weighted
average cost of capital estimates. The major source of error
in these estimates is the error of the cost of equity
estimates. The errors associated with estimation of the other
variables in the p*formula are relatively smaller, and tend
to wash out as we shift our attention to groups of stocks.
The error range for a (which we shall designate as a^) is
given by
(4-15)
where a is the error range for the cost of equity capital
(see Equation (4-9)).
The empirical results are presented in Tables 4-11 and 4-13
Table 4-11 presents the cost estimates and error ranges for the
eight primary sample groups. Table 4-13 (first four numerical
columns) gives the weighted average cost estimates for the
26 secondary subgroups.
The pooled total cost estimates for the 208
primary sample firms are plotted versus their common stock
betas in Exhibit 4-4. The weighted cost estimates tend to
4-47
-------
TABLE 4-11
WEIGHTED AVERAGE COST OF CAPITAL (% PER YEAR): YEAR END 1971 - 1974
PRIMARY GROUPS
GROUP NAME
MINING (18)
FOREST PR. (9)
PAPER (31)
CHEMICALS (36)
REFINING (34)
STEEL (28)
NON-FER. (17)
UTILITIES (35)
TOTAL (208)
EXPECTED MKT.
RETURN
12-MO. RISK-
FREE RATE
SIC
CODE
1000-
1031
2400
2600-
2650
2801-
2803
2911-
2913
3310-
3317
3331-
3350
4911-
4912
MKT VALUE
DEBT/
TOT. ASSETS
13.4
33.0
33.7
37.0
34.0
35.6
35.2
44.1
34.7
YR END 1971
COST
9.3
10.1
8.1
9.1
8.4
8.1
8.4
6. 5
8.3
10.9
ERROR
1.5
1.8
1.2
1.4
1.3
1.1
1.3
0. 7
1. 2
1.8
4.35
YR END 1972
COST
9.3
9.8
8.2
9.0
8.5
8.2
8.1
6.3
8.2
10.8
ERROR
1.4
1.7
1.1
1.3
1.2
1.1
1. 2
0.6
1.1
1.6
5.62
YR END 1973
COST
11.2
11. 7
9.3
10.1
10.3
9.6
9.4
6.8
9. 5
13. 1
ERROR
1.0
1.2
0.7
0. 8
0.9
0.7
0.8
0.4
0.8
1. 2
7.21
YR END 1974
COST
15.4
15. 8
12. 5
13. 5
13.9
12.8
12.7
8. 7
12.7
18. 2
ERROR
0.8
1. 2
0.6
0. 8
0.8
0.6
0.8
0.3
0.7
1.0
7.01
POOLED
f"l— 4)
COST
11. 5
1 2 . 0
9. 8
10. S
10. :
9. &
10. 2
" . 5
q _ c
ERROR
1.0
1. 2
1
0.8
i
; 0.9
\
0. 9
' 0.8
0.9
i
0. 5
"0.8
13 2
1. 2
6.05
-------
XHCAN* 1.313 X30' 3.33S V M E 4 N • S.S82 V3D» 2 . «< 4 3 CORR.« 3.^17 N03S» Z38
INTERCEPT- 4.445 sto»E« 5.332 CELL SIZES.. x« 0,0237 y» 3.2833
TC.B
* * 1* *..'. *' *.,'.....*'. * *.,'..'.....*
18.53 X * X
• *•
* •
a
17,15 X * J
•
• * * » •
15.73 X. * I
' * •
•
14.33 X If X
* * * •
m •
* * .
• ** * * .
12,39 X + * + X
% ***** „
• *«?+* .
• ****** .
" ****•***• 4 «
11.45X ****** ),
• *+ * * 2 .
• ****** .
* '******* ,
* « »2 *2 2, ** * * .
13,32 X * *^*vt * _ X
- ********* * »
*» * 2 * .
• 2****** „
» ************* ,
8,53x +t**2+**** * X.
>* *****2*« .
• 3**»»* „
•* ***2***** * .
» ******* * ,
7.17 x 2*3**+ X
«• * 3* * »
*3 *
-* 2 * 3 .
-£» + * •
3.5351 3,7721 1,31331 1,2461 1,4331 1,7231 1,9571 2,l5«l 2,4Jlt 2.5531
8ET4
Exhibit 4—4. Pooled estimated total cost of capital versus beta.
-------
increase with stock betas, but the correlation is less than
perfect, as would be expected. The error ranges for these
cost estimates are shown in Exhibit 4-5.
4-50
-------
XMEHN" 1.313 XSO* 3.325 Y1E4>J»
INTERCEPT" 3.313 SUO»E« 1.356
1.383 YSO» 3.660 CORR." 3.S21 N03S» 238
CELL SIZES.. X« 0.0237 r» 3.3863
1,37
3,34 X
3,51 X
3.Z3 X
I-
X
2,55 X
2,22 X
1,79 X
•
1,35 X
0,92 X *
*
•Z
»**.*
2 * *
4 *
* ** *
* 4 * * 2 *» *
* * * V * * *
* *
* ** ***4***2 *
4 * 4 ***** *
4 4 ** 2* ** ***. 2
t * 4 t5 * ** *
3 * * *3 * *** *2 » **
**+ * * * * <•
** 3 3 ** * * 2 2* *• *
* 2 *2* * *** *2 »**
* * * 4 *
.
3.5351 3,7721 1,3351 1.2061 1.4631 1.7201 1,9571 2; 1941
Exhibit 4-5. Error range for pooled total cost estimate versus beta.
2,<13M 2.6681
BET*
-------
VII. FORECASTING THJi WHIGHTED AVERAGE
COST OF CAPITAL
Forecasting the manner in which capital costs are likely
to change over time is an extremely hazardous business. Many
problems are involved, not the least of which is the need to
predict general movements in stock market prices.
The total cost of capital for a firm is usually considered
as having three components: the real rate of interest, an
inflation premium, and a risk premium. That is,
Real Rate
of
Interest
* = of + Inflation + Risk
Premium Premium ^ J
The government bond rates, and in particular the shorter
term rates, are usually assumed to be estimates of the first
two components. Thus, given forecasts of government bond
rates, we would have at least part of the problem solved.
This leaves the really hard part how to predict changes
in risk premiums over time. There is no satisfactory way
to do this. About all that can be done is to assume that
future values will reflect an average of past risk premiums.
Forecasts of future capital costs were prepared using
Equation (4-16) and are presented in Tables 4-12 (the 8 primary
groups) and 4-13 (the 26 secondary subgroups). These forecasts
are conditional on the following assumptions:
1. The interest rate forecasts will reasonably
reflect changes in the real rate of interest
4-52
-------
and the inflation during the 1975-1984 period
for which forecasts are being made. The rate
forecasts were for 12-month government bonds
and were supplied by Chas< Econometrics.
2. The risk premiums required by investors during
the forecast period are equal to the average
estimated premiums during the 1971 1974 period.
These premiums are contained in the pooled cost
estimates and are estimated by subtracting the
12-month interest rates. This assumption may
not seem so unrealistic when market conditions
during the 1971 1974 period are considered. In
the early years, market levels were high and
risk premiums relatively low. At the end of
1974, market levels were low and observed risk
premiums appeared to be at an all-time high. Thus,
the pooled risk premium estimates will reflect
somewhat average conditions and provide a not-
unrealistic basis for the forecasts.
The reader should keep in mind that the forecasts given
in Tables 4-12 and 4-13 are not for 1-year periods but long-run
estimates as of the estimation date. For example, the 12.41
forecast made for Forest Products as of December 31, 1975 (see
Table 4-12, column 1, row 2) is an estimate of the long-run
cost of capital. It is not possible, given the current state
of finance theory, to say how this aggregate rate can be
4-53
-------
broken down into a series of annual discount rates (i.e., a
rate for 1975, for 1976, etc.). This would require a term
structure theory for stock prices which, to date, has not
appeared.
4-54
-------
TABLE 4-12
PROJECTED WEIGHTED AVERAGE COST OF CAPITAL (% PER YEAR): YEAR END 1975 - 1984
PRIMARY GROUPS
GROUP NAME
MINING (18)
FOREST PR. (9)
PAPER (31)
CHEMICALS (36)
REFINING (34)
STEEL (28)
NON-FER (17)
UTILITIES (35)
TOTAL (208)
EXPECTED MKT.
RETURN
EXPECTED RISK-
FREE RATE
1975
COST
11.8
12.4
10.1
11.1
10.5
10.2
10.5
7.7
10.2
13.6
ERROR
1.5
1.8
1.1
1.3
1.2
1.1
1.3
0.7
1.2
6.40
1976
COST
13.1
13.7
11.4
12.4
11.8
11.5
11.8
9.0
11.5
14.9
ERROR
1.8
2.2
1.4
1.6
1.5
1.3
1.6
0.8
1.4
7.71
1977
COST
13.2
13.8
11.5
12.5
11.9
11.6
11.9
9.1
11.6
15.0
ERROR
2.1
2.5
1.6
1.8
1.8
1.5
1.9
0.9
1.6
7.80
1978
COST
14.5
15.1
12.8
13.8
13.2
12.9
13.2
10.4
12.9
16.3
ERROR
2.3
2.8
1.7
2.0
2.0
1.7
2.1
1.1
1.8
9.10
1979
COST
14.8
15.4
13.1
14.1
13.8
13.2
13.5
10.7
13.2
16.6
ERROR
2.5
3.0
1.9
2.2
2.2
1.9
2.3
1.2
2.0
9.40
1980
COST
13.7
14.3
12.0
13.0
12.4
12.1
12.4
9.6
12.1
15.5
ERROR
2.7
3.3
2.1
2.4
2.3
2.0
2.5
1.3
2.2
8.30
1981
COST
12.5
13.1
10.8
11.8
11.2
10.9
11.2
8.4
10.9
14.3
ERROR
2.9
3.5
2.2
2.6
2.5
2.1
2.7
1.3
2.3
7.10
1982
COST
11.9
12.5
10.2
11.2
10.6
10.3
10.6
7.8
10.3
13.7
ERROR
3.1
3.7
2.3
2.7
2.6
2.3
2.8
1.4
2.5
6.50
1983
COST
12.1
12.6
10.4
11.4
10.8
10.4
10.8
7.9
10.5
13.8
ERROR
3.2
3.9
2.5
2.9
2.8
2.4
3.0
1.5
2.6
6.65
1984
COST
12..5
13.0
10,8
11.8
11.2
10.8
11.2
8.3
10.9
14.2
ERROR
3.4
4.1
2.6
3.0
2.9
2.5
3.1
1.6
2.7
7.05
-------
TABLE 4-13
WEIGHTED AVERAGE COST OF CAPITAL (* PER YEAR)
SECONDARY GROUPS
GROUP
PULP
AND
PAPER
CHEMICALS
PETROLEUM
REFINING
SUBGROUP
, LARGE MULTI PRODUCT
COMPANIES (8)
7 MEDIUM WITH RESOURCE
CONTROL (19)
, SMALL-MEDIUM W/0
RESOURCE CONTROL (10)
. FOREST PRODUCTS
4 COMPANIES (5)
5 CONVERTERS (12)
6 CHEMICALS — MAJOR (11)
7 CHEMICALS — INTERMED.(6)
8 CHEMICALS — SPECIALTY (19
q INTEG. MAJORS (14)
y (200 + B/D)
,n INTEG. LARGE (5)
10 (200 + B/D)
,, INTEG. (3)
11 (100 - 200 B/D)
19 REFINERS (1)
L* (100 - 200 B/D)
,, INTEG. (2)
I'i (70 - 100 B/D)
INTEG. (2)
14 (.30 - 70 B/D)
EXPECTED MKT
RETURN
12-MO RISK-FREE
RATE
ESTIMATED
1971
8.3
7.4
7.2
9.7
8.5
9.1
8.3
9.3
7.4
7.5
8.6
7.7
7.2
9.0
10.9
4.4
1972
8.4
7.5
7.3
9.3
8.7
8.9
8.5
9.2
7.4
7.6
9.5
6.3
7.0
9.2
10.8
5.6
1973
10.0
8.6
8.0
11.4
8.7
10.7
9.4
10.0
9.2
9.1
11.4
8.2
8.61
10.7
13.1
7.2
1974
13. S
11.6
10.8
15.4
11.5
14.5
12.4
13.3
12.4
12.3
15.5
10.7
11.1
14.3
18.2
7.0
POOLED
10.1
9.0
8.7
11.7
10.0
10.7
10.2
11.0
8.9
8.9
10.8
9.0
8.3
11.2
13.2
6.1
PROJECTED
1975
10.4
9.3
9.0
12.0
10.4
11.1
10.5
11.4
9.3
9.3
11.1
9.3
8.7
11.6
13.6
6.4
1976
11.8
10.6
10.3
13.3
11.7
12.4
11.8
12.7
10.6
10.6
12.4
10.6
10.0
12.9
14.9
7.7
1977
11.8
10.7
10.4
13.4
11.8
12.5
.11.9
12.8
10.7
10.7
12.5
10.7
10.1
13.0
15.0
7.8
1978
13.1
12.0
11.7
14.7
13.1
13.8
13.2
14.1
12.0
12.0
13.8
12.0
11.4
14.3
16.3
9.1
1979
13.4
12.3
12.0
15.0
13.4
14.1
13.5
14.4
12.3
12.3
14.1
12.3
11.7
14.6
16.6
9.4
1980
12. 3
11.2
10.9
13.9
12.3
13.0
12.4
13.3
11.2
11.2
13.0
11.2
10.6
13.5
15.5
8.3
1981
11.1
10.0
9.7
12.7
11.1
11.8
11.2
12.1
10.0
10.0
11.8
10.0
9.4
12.3
14.3
7.1
1982
10.5
9.4
9.1
12.1
10.5.
11.2
10.6
11.5
9.4
9.4
11.2
9.4
8.8
11.7
13.7
6.S
1983
10.7
9.6
9.3
12.3
10.6
11.3
10.7
11.6
9.5
9.5
11.3
9.5
8.9
11.8
13.8
6.7
1984
11.1
10.0
9.7
12.7
11.0
11.7
11.1
12.0
9.9
9.9
11.7
9.9
9.3
12.2
14,2
7.1
-------
WEIGHTED AVERAGE COST OF CAPITAL (% PER YEAR)
SECONDARY GROUPS
GROUP
PETROLEUM
REFINING
(CONT.)
IRON AND
STEEL
NON-
FERROUS
METALS
UTILITIES
SUBGROUP
.. INTEG. CAN. (2)
i:> (30 - 70 B/D)
,, REFINERS (2)
10 (.30 - 70 B/D)
17 REFINERS (3)
17 (10 - 30 B/D)
R INTEG. (1)
(10 - 30 B/D)
19 STEEL — MAJOR (7)
20 STEEL — MINOR (20)
21 PRIMARY COPPER (8)
PRIMARY LEAD AND
/Z ZINC (3)
7, PRIMARY
ALUMINIUM (4)
?, SECONDARY
Z4 SMELTING (4)
UTILITIES
5 FLOW THRU (19)
7fi UTILITIES
NORMALIZED (16)
EXPECTED MKT
RETURN
12-MO RISK-FREE
RATE
ESTIMATED
1971
6.3
8.8
11.8
13.6
7.3
8.3
9.6
8.7
6.5
7.1
6.3
6.7
10.9
4.4
1972
6.6
8.1
10.4
13.2
7.5
8.4
9-6
8.5
6.4
6.9
6.1
6.5
10.8
5.6
1973
7.8
10.3
12.7
16.5
8.8
9.8
11.8
10.2
8.1
8. 5
6.7
7.0
13.1
7.2
1974
10.3
13.9
17.1
22.5
11.8
13.1
16.1
13.7
10.6
11.3
8.4
9.0
18.2
7.0
POOLED
7. 5
11.0
14.4
16.3
9.0
10.1
12.0
10.5
8.2
9.0
7.1
7.6
13.2
6.1
PROJECTED
1975
7.7
11.4
14.7
16.7
9.3
10.4
12.4
10.9
8.6
9.3
7.5
8.0
13.6
• 6.4
1976
9.0
12.7
16.0
18.0
10.6
11.7
13.7
12.2
9.9
10. 7
8.8
9.3
14.9
7.7
1977
9.1
12.8
16.1
18.1
10.7
11.8
13.8
12.3
10.0
10.7
8.9
9.4
15.0
7.8
1978
10.4
14. 1
17.4
19.4
12.0
13.1
15.1
13.6
11.3
12.0
10.2
10.7
16.3
9.1
1979
10. 7
14.4
17. 7
19.7
12. 3
13.4
15.4
13.9
11.6
12.3
10.5
11.0
16.6
9.4
1980
9.6
13.3
16.6
18.6
11.2
12.3
14.3
12.8
10.5
11.2
9.4
9.9
15.5
8.3
1981
8.4
12.1
15.4
17.4
10.0
11.1
13.1
11.6
9.3
10.0
8.2
8.7
14.3
7.1
1982
7.8
11.5
14.8
16.8
9.4
10.5
12.5
11.0
8.7
9.4
7.6
8.1
13.7
6.5
1983
7.9
11.6
15.9
16.9
9.6
10.7
12.6
11.1
8.8
9.6
7.7
8.2
13.8
6.7
1984
8.3
12.0
16.3
17.3
10.0
11.1
13.0
11.5
9.2
10.0
8.1
8.6
14.2
7.1
-------
•'OOTNOTHS I;OU PART
1. The numerical results presented in Part 4 are taken
from an unpublished working paper by Professor Gerald
A. Pogue entitled "The implications of Modern Finance
Theory for Estimating the Cost of Capital" [25], dated
August 1975.
There is no need to assume that the variances of e-
/\
are the same for all firms. Indeed, the R. estimates
from the discounted cash flow models will convey
different amounts of information about the true values,
R.. This results in a heteroscedasticity problem which
is handled by using weighted regression analysis (see
Section II(d)).
This averaging procedure will reduce the measurement
s\
error for R. as long as the error variances of the four
individual estimates are roughly comparable in size and
not perfectly correlated. The optimal weighting scheme
requires knowledge of the (unobservable) error covariable
~ I?
matrix for the four estimates (R., £ = 1, 2, 3, 4).
With this knowledge an optimal set of weights can be
derived for each company to minimize the error of the
combined estimate R..
4-58
-------
4. An alternative weighting procedure to (1/4, 1/4, 1/4, 1/4)
would recognize the complementary nature of estimates 1
and 2 and give equal weight to estimates 3 and 4 and a
combination of 1 and 2 (e.g., 1/6, 1/6, 1/3, 1/3).
5. The PDE tape is produced by Investors Management
Sciences, Inc. (a subsidiary of Standard and Poor's
Corporation). The January 1975 version of the tape
was used. The tape contains monthly data for approxi
mately 3,000 corporations for the 1962-1974 period.
The data items include stock prices, dividends, earnings,
and common stock book values.
6. This effect is largely due to construction. The data
means for the four estimation equations tend to cluster
near 3 = 1.0. Thus the dispersion of the four R.
estimates tends to increase as |B• l| increases.
7. The Compustat Annual Primary Industrial File contains
balance sheet and income statement data for approxi
mately 3,000 industrial and utility firms. It is
produced by Investors Management Sciences, Inc.,
a subsidiary of the Standard and Poor's Corporation.
8. An average of the four risk premiums is a more efficient
estimator of future risk premiums than the last (1974)
4-59
-------
observation. The four observations can be viewed as
independent draws from the underlying risk premium
population. Thus, the best predictor of the next
value is an average rather than the last value.
4-60
-------
ESTIMATION OF THE COST OF CAPITAL
FOR MAJOR UNITED STATES INDUSTRIES
WITH APPLICATION TO POLLUTION-
CONTROL INVESTMENTS
PART 5. APPLICATIONS
Dr. Gerald A. Pogue
4 Summit Drive
Manhasset, New York 11030
November 1975
-------
I. INTRODUCTION
The purpose of Part 5 is to show how the cost-of- capital
estimates developed in Part 4 can be used for making capital
budgeting decisions in practical situations. Section II
reviews and extenda the capital budgeting procedures intro-
duced in Part 2. Section III develops the basic structure for
an analytical framework which can be used for measuring the
financial impact on corporations of investments in pollution-
control devices.
5-1
-------
II. CAPITAL BUDGETING
(a) The Weighted Average Cost of Capital
In finance theory it is usually assumed that the goal
for financial decision-making is to increase the wealth of
corporate stockholders. This results in the following decision
rule for evaluating corporate investment alternatives: Accept
only projects which have positive net present values. The
net present value (NPV) is the discounted sum of net project
cash flows (inflows minus outflows) . The discount rate is the
weighted average cost of capital.
T C
NPV(p*) = I - - ±— (5-1)
t=0 (1+*
where
C = the expected net cash flow in year t
p* = the weighted average cost of capital
T = the economic lifetime of the project
The decision rule for corporate managers is
Accept Project if NPV(p*) > 0 (5-2)
5-2
-------
The principal unknown in Equation (5-1) is the weighted
average cost of capital, p* As discussed in Part 2, p*
under certain circumstances can he estimated using a rule known
as the textbook formula.
P* = i(l Tc) • £ + R • | (5-3)
where
i = the current interest rate on the firm's bonds
R = the cost of equity capital
D = the market value of debt (bonds plus leases)
E = the market value of equity
T = the corporate tax rate
V = D + E
Equation (5-3) provides a reasonable estimate of the weighted
average cost of capital only under certain conditions. The
major requirements are (1) the project does not change the
business risk of the firm—it must be an average-risk invest-
ment, (2) the project does not shift the firm's debt ratio—it
must be financed with the same proportion of debt and equity
as. the overall firm, and (3) the project must make a permanent
contribution to debt capacity—the project must be a perpetuity
and support perpetual debt. (See Part 2, Section IV-b for
discussion of these assumptions.)
The Adjusted-Present-Value Approach
An alternative to Equations (5-1) and (5-2) is the adjusted
present-value (APV) approach. The adjusted present value is
defined by
-------
T C
APV = ----- L + PVTS (.T. 4
where
C = the expected net project cash flow in year t
(same as in Equation (5-1))
T = economic lifetime of the project (years)
PVTS = the present value of the tax shield on
debt supported by the project
P0 = the all-equity cost of capital — the cost of
capital if the project were all-equity financed
The capital budgeting decision rule is
Accept Project if APV > 0 (5-5)
The APV rule accomplishes in two parts what the NPV approach
attempts in one. The APV is the sum of the project's net
present value under the assumption of all-equity financing,
plus the present value of the tax shield on debt. Since both
components are dealt with separately, no restrictive assumptions
on the amount and duration of the project and debt cash flows
are required. The NPV rule, on the other hand, attempts to deal
with both components at once and results in a complicated
definition of p*. Only under restrictive assumptions can p* be
simply estimated. For example, p* can be estimated using the
textbook rule if the assumptions described in Section Il-a are
valid.
5-4
-------
While the APV is a somewhat more complicated calculation
than the NPV, it can result in improved decision-making.
Compared with the textbook rule for estimating p*—and hence
NPV(p*) the APV has three distinct advantages. It allows
treatment of projects with different business risks, that is,
the all-equity rate p can be tailored to the project risk.
Second, there is no need for the projects to be perpetuities.
Third, there is no requirement that the project be financed in
a predetermined way, nor that the debt be permanent.
One aspect of the formula left unresolved is the estimation
of the all-equity rate. We shall deal with this matter in
subsection c.
Example
Long Island Duck Farms (LIDF) has the opportunity to invest
$100,000 (t = 0) and expects after-tax cash returns of $60,000
at t = 1 and $70,000 at t = 2. The project will last for two
years only. The cost of capital assuming all-equity financing
is 121 (pnJ. The borrowing rate (i) is SI, and the firm's debt
ratio for a project of this type is 0.30. (The target debt
ratio is in book value terms.) The problem is to determine
if this project has a positive adjus-ted present value.
We start by computing the project's net present value under
the base case assumptions (i.e., all-equity financing).
5-5
-------
2
I
t = 0
100,000 H. > + 70>QOO
= $9,375
Next we determine the value of the tax shield associated with
the debt (PVTS). The book value of the project at t = 0 is
$100,000. Thus, $30,000 extra debt can be issued at t = 0 ,
and $15,000 repaid at t = 1 (assuming straight-line deprecia-
tion of the asset). Thus,
T iDn T iD,
PVTS = — + — -
0.5(0.08) (30,000) + 0.5(0.08) (15,000)
1.08 (1.08)2
= 1,111 + 514
= $1,625
Thus, the APV of the project is $9,375 + $1,625 = $11,000.
Suppose we wished to estimate the weighted average cost
of capital for this project such that the net present value
5-6
-------
as computed by liquation (5-1) would also equal $11,000
direct way is to solve for p* from Equation (5-1).
The
11,000 = 100 000 + 60'"?0 + 70,000
(lT^ (l+p*)2
Solving ,
P* = 11.61
Applying the textbook formula in a straightforward (but
inappropriate) fashion yields p* = 10.2%. If this estimate
were used along with Equation (5-1), the resulting NPV would
obviously be higher than 11,000. This results because the
textbook rule assumes perpetual debt, and thus overstates the
present value of the two years of tax shields. Further, the
textbook formula assumes the 0.30 debt ratio is in market terms,
not book terms, and further overstates the value of debt.
(c) The All Equity Cost of Capital
At this point it is reasonable to wonder why the textbook
formula for computing p* is used at all. One answer is that
it provides a helpful starting point for estimating the all-equity
rate pQ.
The textbook formula can be used to estimate p* for the
firm as a whole. Since the firm is assumed to have an indefinite
life, and since debt ratios remain reasonably constant, it fits
the textbook assumptions better than individual projects.
r _ 7
o /
-------
This rate can then be substituted into the MM cost-of-
capital formula to obtain an estimate of p.. for the firm as a
whole. The process proceeds as follows:
Step 1—Estimate the textbook p* for the overall firm,
P* = i^1 V £ + R £
Step 2—Substitute p* into the MM formula to obtain p,,,
Tc
• • p,
(This pn is applicable only to average-risk
proj ects . )
Step 3 — For projects satisfying the MM formula assumptions,
use this rule to estimate their p? (j denotes project)
P] = P0(l TcL1) (5-7)
where
L- = the debt- financing ratio for project j
This p* can now be substituted into Equation (5-1] to estimate
the NPV(p*) for project j.
For projects not satisfying the MM cost-of- capital rule,
use the APV approach.
C-t
PVTS. (5-8)
5-
-------
Note that we have been careful only to consider average-risk
projects for which the pQ is applicable. A thornier problem
arises when a project is of different riskiness. It is then
necessary to estimate pQ.—the all-equity rate applicable to
project j. There is no simple practical answer to this
problem.
When a market risk measure (beta) is available for the
project, then the p~. can be estimated using the procedures
and results of Part 4. That is, given the beta of the project,
an estimate of its cost of capital can be estimated. Beta
measures, however, are typically not available for individual
projects. About the best that can be done is to use the
(unlevered) beta for the common stock of a company whose
business risk resembles that of the project. For example, a
chemical firm contemplating investment in the paper industry
could use the average beta for a suitably chosen group of paper
companies as an estimate of the market risk of the project.
Given the risk estimate, the p~. follows as described in Part 4.
If this procedure is used, caution must be taken to ensure
that the betas used are unlevered. A company's beta reflects
not only its business risk but also its financial risk. To
estimate p • we require the all-equity beta (BQ) which reflects
only business risk.
The all-equity beta can be estimated from the firm's market
beta (3) and the market-debt - to-equity ratio -
D
as follows.
5-9
-------
§}
(5 - 9)
When market risk measures are not available, a heuristic
solution is to position the risk of new projects relative to
the average risk. The PQ-'S for individual projects are then
subjectively increased or decreased from the average value.
For example, a firm might categorize its investments into
three categories—high, average, and low risk—and subjectively
adjust the p.. for the high and low groups. While this approach
is not very satisfying, it is the one typically used in practice
Example
Suppose a firm has a market debt ratio of 0.20 and can
borrow at 8 percent. The management estimates investors'
required rate of return, R, at 12 percent. Then by the textbook
formula
P* = iCi - Tc) £ + R|
= 0.08 (0.50)(0.20) + (0.12) (0.80)
= 0.104
This 10.4-percent rate would be the correct hurdle rate for
average-risk perpetual projects with debt financing ratios
of 0.20.
5-10
-------
The MM formula can be used to estimate pn for the average
risk project
o* - "of1 Tcf
0.104
075(0. 20)
= 0.116
This is a starting point for using the MM formulas or the APV
approach for average-risk projects.
5-11
-------
III. MEASURING THE FINANCIAL IMPACT
OF INVESTMENTS IN POLLUTION-CONTROL DEVICES
Pollution-control requirements involve long-run sequences
of capital investments and annual operating expenses. Conse-
quently, there is a need to measure the magnitude of these
requirements relative to the firm's ability to carry them out.
Given the nature of pollution-control investments, their
imposition will tend to reduce the value of a firm's assets.
How significant is this reduction in value? How much would the
return on other corporate assets have to be increased to offset
this decline?
The purpose of this section is to introduce a framework
for analyzing these questions. This framework will provide a
first step toward analyzing the financial impact on corporations
of required pollution-control investments. It is not the
intention to provide a detailed analysis of such questions, but
to point the direction in which such an analysis might proceed.
Indeed, many significant questions are left unasked as well as
unanswered.
(a) A Financial Impact Index
Pollution-control programs require long-run expenditures
by corporations. These include periodic investments in facilities
plus annual operating costs. As with other investments, the
5-12
-------
current economic value of n pollution-control investment (PCI)
is the present value of its future cash flows (i.e., the APV).
For PCI's, the present values are negative (since all cash flows
are negative) and hence they will be undertaken by corporations
only if required by law.
A measure of the relative financial impact of a PCI is
obtained by comparing its present value to the present value
of the firm's other assets. The present value of the firm's
assets is given by the market value of the firm (VR) , that is,
the market value of the firm's equity and debt obligations
prior to adoption of the PCI (Vr> = Dr> + EB). The relative
D D D
impact of the PCI can be measured by the ratio of the APV to
the value of the firm's other assets. The ratio is designed
as the financial impact index (FIT).
FII = (5-10)
V
£>
where
APV| = the absolute value of the (negative) PCI-
adjusted present value
Note that the value of the firm after adoption of the PCI (V^
will equal the prior value VR plus the APV of the investment.
If the FII is small (e.g., 0.02), then the impact of the
PCI on the corporation would similarly be small. However, as
the FII increases, the ability of the firm to successfully
absorb the PCI will diminish until finally bankruptcy occurs.
5-13
-------
Exactly when the critical value of the FII occurs is a difficult
question requiring further research.
(b) The Risk and Financing of Pollution-Control Investments
The riskiness and method of financing of pollution-control
investments will depend on case-by-case analysis. However, it
is possible to define a set of boundary conditions which will
delineate the possibilities that will likely exist in practice.
The Risk of PCI's
The maximum risk of a PCI is probably the risk of the
firm's other assets. While it is possible the PCI risk would
be higher, it is unlikely. It is more likely that the PCI
cash flows will not be more variable or less predictable than
the firm's other operating cash flows. To the extent that
pollution-control costs tend to vary with the firm's business
activities, the PCI's will be of average risk.
On the other hand, the minimum possible riskiness will
resemble the risk of the firm's debt obligations. Even if the
pollution-control costs were contractual, the firm can default
by going out of business. Hence, the minimum risk can be no
less than that of the firm's debt service obligations.
The Financing of PCI's
The typical PCI will have predominantly negative cash
flows (e.g., investment outlays, operating expenses, etc.).
5-14
-------
While there will be some favorable side effects (e.g., tax
savings from investment depreciation, sale value of biproducts
such as sulphuric acid, etc.), these will typically be
insufficient to produce either a positive net present value
for the PCI or even a significant number of positive annual
cash flows.
Given the above, the PCI's will have no inherent debt
capacity of their own. In fact, a PCI will reduce the debt
capacity of the rest of the firm. This follows because the
negative net present values of the PCI's will reduce the ability
of the overall firm to support debt service requirements. Thus,
the PCI's have negative debt capacity.
For example, a firm which has debt outstanding with a loan
covenant requiring maintenance of a certain debt coverage ratio
may well have to reduce its debt level given the reduction in
operating earnings caused by the PCI.
If a PCI is partially or entirely financed by debt issues,
this is possible only because the firm has enough unused debt
capacity associated with its other assets (even after the
reduction in aggregate debt capacity caused by the PCI) to
support the new debt. We must be careful, however, not to
attribute the tax benefits of this new debt to the PCI. The
firm could have obtained these benefits by using the debt
l
capacity to finance its other projects.
5-15
-------
Consider now two limiting cases relating to the firm's
overall capital structure, all-equity financed and, the
situation where the firm has fully utilized the debt capacity
of its existing assets, maintaining a debt ratio equal to
(D/V).
In the first case, the reduction in debt capacity caused
by adoption of the PCI will have no effect. Since these firms
make no use of debt, the present value of tax savings on the
debt capacity reduction caused by the PCI (PVTS) has a zero
value.
In the second case, the reduction in debt capacity will
result in a reduction in the value of the firm. Assuming the
firm has fully utilized the debt capacity of its other assets,
the adoption of the PCI will require a reduction of the firm's
debt level. This follows because, in order to maintain a target
debt ratio of, say, 30%, a reduction in the value of the firm's
assets due to the PCI will necessitate a reduction in debt
level in order to maintain the target debt ratio. The present
value of the tax savings on the debt displaced by the PCI is
positive or, viewed from the point of view of the PCI, the PVTS
of the PCI is negative.
The matrix of possible risk and capital structures is
illustrated in Table 5-1. Cases 1 and 2 assume average-risk
investments, Cases 3 and 4 assume "low risk". To compute the
5-16
-------
TABLE 5-1
POSSIBLE RISK AND CAPITAL STRUCTURE CONSIDERATIONS
FOR POLLUTION-CONTROL INVESTMENTS
Investment
Risk
Average Risk
(Same as
Other Assets)
Low Risk
(Same as
Firm's Bonds)
Capital Structure of Firm
All Equity
Case 1
PCI
P0 = P0
PVTS = 0
Case 3
PCI
P0 = i
PVTS = 0
Equity $ Debt
Case 2
PCI
P0 = P0
PVTS < 0
Case 4
PCI
P0 = i
PVTS < 0
= all-equity rate for pollution-control investments
pn = all-equity rate of firm's other assets
i = interest rate on firm's bonds
PVTS = the present value of the tax shield on the
PCI debt capacity
5-17
-------
APV's for these cases, we require estimates of the all-equity
PC I
financing rate. In Cases 1 and 2 the all-equity rate, pQ ' ,
is equal to the all-equity rate for the firm as a whole
P
'0
PC I
In Cases 3 and 4, pn is the rate of interest on the firm's
bonds.
Cases 1 and 3 assume the firm is all-equity financed,
while Cases 2 and 4 assume the firm fully utilizes its debt
capacity, maintaining a target debt ratio equal to (D/V).
It is not possible to further specify the nature of the
PCI APV's without further definition of the PCI cash flows.
If we assume, for example, an initial investment I~ at t = 0,
and a perpetual stream of annual after-tax payments -C beginning
at t = 1, then the general expression for the PCI APV is given
by
APV = ~TCT :0 + PVTS (5-11)
P0
where
PCI
PQ = the all-equity rate for the PCI
-C = the annual operating cost (after
corporate tax)
IQ = the investment at t = 0
PVTS = the present value of the tax savings on
the debt capacity reduction caused by the
PCI (zero for Cases 1 and 3)
5-1;
-------
Assuming that perpetual debt has been issued by the i'i rm
in Cases 2 and 4, the present value of the tax savings on the
debt capacity reductions caused by a PCI is equal to T -DPCI,
where TC is the corporate tax rate z id D is the reduction
in debt capacity caused by the PCI. In effect, if the firm
was making no other investments other than the PCI, it would
PCI
be forced to retire D dollars of perpetual debt on acceptance
of the PCI. In practice, however, firms will be carrying out
other investment and financing programs along with the pollution
PCI
control requirements. Rather than retiring D dollars of
debt, the new financing mix would simply be changed to issue
more equity and less debt than otherwise. The effect, however,
is the same.
Calculation of the PVTS for the PCI's is complicated in
Cases 2 and 4 by the dependence of the reduction in debt
capacity on the (negative) value of the PCI, which in turn
depends on the present value of the tax savings. This can be
more clearly seen from the following steps.
PCI
From Equation (5-4) the APV is given by
APVPCI = NPV(p0) + PVTS
Assuming that all debt is perpetual, and that the firm
plans to maintain the target debt ratio equal to (D/V), the
adoption of the PCI would require that the debt level be
PfT
reduced by D dollars, where
5-19
-------
°
PCI
APV
.PCI
(5-12)
The present value of the tax savings associated with this
reduction is given by
PVTS = Tc[|] (APVPCI) (5-13)
Substituting from Equation (5-13) into Equation (5-4) we have
F)
= NPV(pQ) + Tc-H
or, rearranging terms,
rPCI
APV1
1-T •-
1 c V
m • NPV(pQ) (5-14)
Thus, reduction in market value of the firm will increase with
increasing debt ratios. Specific forms of Equation (5-14) for
the four cases are given in Table 5-2.
(c) The Offset Rate of Return
How much would the rate of return on the firm's other
assets have to increase to offset the negative impact of the
PCI? The effect would be offset when the increased present
value resulting from the increase in return was just equal to
the (absolute) value of the PCI APV. That is,
AAPV
•FIRM
APV
PCI
(5-15)
To further specialize Equation (5-15), further assumptions
regarding the firm's assets are required.
5-20
-------
TABLE 5-2
ADJUSTED PRESENT VALUE FOR EXAMPLE
POLLUTION-CONTROL INVESTMENT
Cash
Flow
.,0 L
Time 0 1
(Years)
-C -C
2 3
Case
Eq.
Eq.
Eq.
Eq.
1
(5-14a)
2
(5-14b)
3
(5-14c3
4
(5-14d)
Adjusted Present Value
(APV)
-c z
P0 0
1 I"'0 I 1
(i Vf) Lp° °J
i 0
1 r~c 1 1
KV?) Li :°J
5-21
-------
Assume, for example, the market value of the firm's assets
is V. [prior to adoption of the PCI). The assets generale ;
B
a
perpetual after-tax return of r percent per year. Also assume
the return can be increased by Ar percent per year; hence, the
increased annual cash flow is Ar VR dollars per year. Assume
further the firm will adjust its capital structure to maintain
its target debt ratio (and hence the appropriate discount rate
rate of p*) .
Ar ' V7
P*
APV
PCI
Therefore ,
APVPCI
Ar = J 77
• p*
¥B
or
Ar = (FII) p* (5-16)
where
p* = the (textbook) weighted average cost of capital
Thus, given the above assumptions, the necessary increase in
the return on the firm's other assets is equal to the financial
impact index (FII) times the weighted average cost of capital.
(Note that for an all-equity financed firm, p* = p~ and hence
Ar = (FII) pQ.)
5-22
-------
(d) Example
Consider three firms (A, B, and C) with market values of
assets prior to adoption of PCI's (Vfi) of 1000, 500, and 100,
respectively. Each firm generates a perpetual return on the
market value of its assets of 10 percent per year. The
all-equity rate for average-risk projects is 10 percent, and
the firms can borrow at a 6-percent rate. The corporate tax
rate (TC is 50 percent. The firms are assumed to be identical
except for the size of assets.
Each firm is required to install a polution-control
device at t = 0. The required investments are 100, 75, and 40
for the three firms respectively. Annual operating costs are
10, 5, and 1.
For each firm we will consider the implications of the
four cases defined in Table 5-1. In Cases 1 and 3 we assume
PCI
the PCI's to be of average risk. Therefore, pQ = 0.10.
In Cases 2 and 4 we assume the PCI's to be of minimal risk
PCI
(same risk as the firm's bonds). Therefore, PQ = 0.06.
In Cases 1 and 3 it is assumed that the three firms are
all-equity financed (and plan to remain so in the future). Thus,
PVTS = 0 and p* = PO = 0.10. In Cases 2 and 4 it is assumed
that the three firms have fully utilized all of the debt
capacity of existing assets. Each firm has a market value
debt ratio (D/V) equal to 30 percent which will be maintained
after the PCI. In order to achieve this, each firm will be
5-23
-------
required to retire debt equal to 30 percent of the (negative)
market value of the PCI through new equity issues 'see
Equation (5-12)). This is in addition to the equity issued
to finance the PCI initial investment (I~ dollars). Each firm
has a weighted average cost of capital of 8.5 percent p* = 0.08
The results for the four cases and three firms ai-e
summarized in Table 5-3. Row 7 of Table 5-3 gives the financial
impact index values; row 9 gives the offset rates of return.
While the numbers in the example were selected for
illustration only, the results clearly show the relationship
between the effect of pollution investments and the size cf
firm. It seems fairly clear a small firm such as C would be
forced out of business, while the largest firm A would haie the
best chance of successfully absorbing the PCI.
Note that the offset rates of return do not depend on the
capital structures of the firms (see row 9, Table 5-3). 1ven
though the initial impact of the PCI is more severe for the
debt firms (see row 7) due to the loss of valuable debt
capacity, it is exactly offset by the new debt capaciiy g-neiate
by the increased return (Ar) on the firm's other assets. This,
of course, assumes the offset returns can be achieved. I not,
the impact remains greater for the firms using debt.
5-24
-------
TABLE 5-3
EXAMPLE: THE IMPACT OF POLLUTION-CONTROL INVESTMENTS
ON FIRMS OF DIFFERENT SIZE*
Market Value of Firm (VB)
PC Investment (T = 0}
PC Annual Cost (t = 1,»)
Base Case NPV
PVTS
APV = NPV + PVTS
Impact Index
Offset AR (1)
Offset Return (1)
Average-Risk Investment (Cases 1 and 2)
Case 1
All Equity
Firm A
1000
-100
-10
-200
0
-200
.200
2.0
.12. 0
Firm B
500
-75
r
-12S
0
-125
.250
2.5
12. S
Firm C
100
-40
-1
-SO
0
-50
.500
5.0
15.0
Case 2
30% Debt Ratio
Firm A
1000
-100
-10
-200
-35.3
-235.3
.235
2.0
12.0
Firm B
500
-75
-5
-125
-22.0
-147. 0
.294
2.5
12.5
Firm C
100
-40
-1
-50
-8.8
-58.8
. 588
5.0
15.0
Low-Risk Investment (Cases 3 and 4)
Case 3
All Equity
Firm A
1000
-100
-10
-266. 7
0
-166.7
.267
2.7
12.7
Firm B
500
-75
-5
-158.3
0
-158.3
.317
3.2
13.2
Firm C
100
-40
-1
-56.7
0
-56. 7
.567
5.7
15.7
Case 4
30% Debt Ratio
Firm A
1000
-100
-10
-266.7
-47.1
-313.7
.314
2.7
12.7
Firm B
500
-75
-5
-Io3.3
-27.9
-186.2
.372
3.2
13.2
Firm C
100
-40
-1
-56.7
-16.7
-66.7
.667
5.7
15.7
Ni
U1
* The numbers are chosen for illustrative purposes only.
-------
The relationships between the impact index and size ar<
displayed in Figure 5-1. As shown in Figure 5-1, the relation
ship is very dependent on both the risk of the PCI and the
firms' capital structures.
5-26
-------
\
\
55
,50
V!
\
1 !
1
!
; Pi
i
V
V _.|
\: '
\
ki
LOt C
: 1
CA^
1 1
CASE
:-GAS-£
CASE
p ft c r
un o t
i
•
F rr^
i
FOR
E , :
i 1
2 '
3
•FUJ'J
ACT 'i
TAE}L
1
' ,
i :/^v
' - H~V
1 ;uc
• i ' r*
i -^,
' •
i
i
U^Ei 5-1
ND6X 'VERSUS
i.
E 5-3 EXAHPL
iii i i
!RllSl4
EJRAGEJ .RlSKl
ErRA^GE; RfrSK
iJ ;RI:S|K
i/» _ i\ j. ^ r\ ,
i ; i
;
FIrtM SIZ
E
|
CAPITAL
ALL E
-|30« D
ALL E
1 Q r\ q/ n
1
i
E
STRUCTURE
QUITY;
EBT
QUITY
p n, T
.
.45
35 h
,30
.25
,20
200
CASE 4
CASE 3
CASE 2
800
1000
SIZE OF FIRM (VB)
5-27
-------
FOOTNOTES FOR PART 5
The adjusted-present-value approach to capital
budgeting was developed by Professor Stewart C. Myers
and is described in detail in [21],
See Hamada [6], p. 20, Equations 15, 16, and 17.
The ratio of Equations (15) and (16) gives (in the
notation of this study)
E
J_
Bo
0
where E,-, is the total value of the unlevered equity,
E the levered equity. By Equation (17)
E,
'0
E + (1 Tc)D
or
- = 1
E L
(1 - T )
U E
Thus, combining these two results,
B
(1
as given in Equation (5-9).
o
-------
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-------
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R-4
-------
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