A DISCRIMINANT ANALYSIS OF THE BANKRUPTCY
OF SECURITIES BROKERAGE/DEALER FIRMS
By
Iraj Afkham
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1975
Copyright
by
Iraj Afkham
1975
To my family:
Maryam, my wife, and
Ladan and Mahnoosh, my daughters
ACKNOWLEDGMENTS
The author wishes to express his grateful apprecia
tion to his supervisory committee for their invaluable
thoughts, suggestions, and criticism throughout the course
of completing this dissertation.
A special word of appreciation is due to Dr. Robert
C. Radcliffe who took a far greater part in this endeavor
than could be expected from a person in his capacity. Even
the subject was proposed by him, but the author is the one
who is solely responsible for any defects in its contents.
Dr. Radcliffe generously and willingly spent numerous hours
in individual discussions and various meetings, and injected
invaluable insights into the study. His encouragement and
support at the time when they were most needed cannot be
forgotten.
The author is indebted to Dr. Henry R. Fogler and
Dr. Christopher B. Barry who generously gave their time,
thoughts, and enthusiasm during the previous abortive
research on some other dissertation topics.
This opportunity is taken as the most pleasant one
for extending the author's heartfelt gratitude to the
following persons who have, in one way or another, enriched
his education during his course of study at the University
of Florida: Drs. Fred D. Arditti, Sanford V. Pera, Clement H.
Donovan, Richard A. Elnicki, Jack M. Feldman, Frederick O.
Goddard, John H. James, Richard R. Jesse, Milton Z. Kafoglis,
Gary J. Koehler, Antal Majthay, John B. McFerrin, W.W. Menke,
Richard H. Pettway, James G. Richardson, Jack R. Vernon, Albert
R. Wildt, and William V. Wilmot.
Sincere appreciation is also extended to Dr. Donald
F. Chichester for his editorial assistance, Mr. Thomas A.
Readinger of SIPC for providing the author with some of the
data, Mr. William Ingram, III for assisting in computer
program troubleshooting, and Mrs. Mary Watson McCoy for
her library assistance.
Finally, profound thanks should be extended to Ms. Kitty
Hinton for her dependable work and typing the manuscript under
most trying circumstances.
TABLE OF CONTENTS
Page
iv
ACNKOWLEDGMENTS . . . . . . . . . .
LIST OF TABLES. . . . . . . . . .. xii
LIST OF FIGURES . . . . . . . . . xiii
LIST OF CHARTS . . . . . . . . ... xiv
ABSTRACT. . . . . . . . . ... . xv
CHAPTER
I INTRODUCTION . . . . . . . . 1
The Backbone of the Capital Market . . 1
Literature Review . . . . . . 2
Review of SEC Records of the Demise
of Selected Broker/Dealers . . . 5
Study of Unsafe and Unsound Practices
of Brokers and Dealers . . . .. 6
Securities Industry Study by the
Subcommittee on Securities . . . 7
Securities Industry Study by the Sub
committee on Commerce and Finance . 8
SIPC Study . . . . . . . . 9
The Early Warning Requirement of the
SIPC Act . . . . . . . . 10
The Existing Early Warning Systems . . 12
The Early Warning System of the SEC. . 13
The Early Warning System of the Self
regulatory organizations . . .. 14
The NASD Failed Firms Project. . . ... 16
Differences Between the NASD Study and
the Present Work . . . . . .. 18
The A'alytic Framework .... . . 18
The Variable Sets. . . . . . ... 19
The Sources of Data . . . . . 19
TABLE OF CONTENTS (CONTINUED)
Page
Data Set . . . . . . . . 20
The Results. ...... . . ....... 20
Different Approaches . . ... . . 21
The Deterministic Approach . . . .. 21
The Probabilistic Approach . . . .. 23
The Blend Approach . . . . . .. 26
The Purpose of the Study . . ... . . 27
The Problem Encountered . . . . .. 27
The Organization of the Dissertation . 28
Notes . . . . . . . . . 30
II THE GENERAL ANALYTIC FRAMEWORK . . .. 32
Introduction . . ... . . . 32
The Hypothesis . . . . . . . 33
Net CashFlow Per Dollar of Assets . .. 39
Revenue . . . . . . 40
Internal Factors . . . . . . 40
External Factors . . . . . .. 44
Expenses . . . . .  . 48
Internal Factors . . . . . . 48
External Factors . . ... . . .. 49
Risk . . . .. . . . . . . 50
Definition ...... . . * 51
Common Measures of Risk . . . .. 52
Variance or Standard Variation of a
Single Variable . . . . .. 52
SemiVariance . . . . . . 53
Third Moment . . ... . . 55
Rate of Growth . . ... . . . 56
Confidence Level . . . . . . 58
Covariance . . . ... . . 59
Correlation Coefficient. ....... 59
Variance of a Group of Variables . 60
The Probability of Disaster Level. ... 63
TABLE OF CONTENTS CONTINUEDD)
Page
Risk Factors. . . . . . 65
Risk Associated with Cashflow. ... . 65
Risk Associated with Revenue. . . ... 67
Risk Associated with Product Lines. . 70
Risk Associated with Places Occupied. 7
Risk Associated with Exchange Rate ... 80
Risk Associated with Age . . . 81
Risk Associated with Excenses . ... 83
Risk Associated with Labor. . . .. 86
Risk Associated with Liquid Assets. . 87
Risk Associated with Securities ... .90
Risk Associated with Credit Extending .. 94
Risk Associated with Inventories. . . 97
Risk Associated with Fixed Assets ... 99
Risk Associated with Patents. . . ... 100
Risk Associated with Debt . . ... 101
Risk Associated with Changes in Economic
and Market Factors . . . ... 103
Mismanagement . . . . . . . . 105
Summary and Conclusion . . . . . 107
Notes . . . . . . . . . 110
III MULTIVARIATE DISCRIMINANT ANALYSIS. ... 114
Introduction. . . . ... . . 114
General Assumptions . . . . . .. 117
Historical Development. . . . . . 119
Test of Equality of Dispersion Matrices . .121
Mathematical Formulations for Sample Use. . 126
The TwoGroup Case. . . . . ... 127
Equal Dispersion Matrices . . . .. 127
Unequal Dispersion Matrices . . .. 131
The kGroup Case. . . . . . ... 134
Equal Dispersion Matrices . . . .. 134
Unequal Dispersion Matrices . . .. 138
Relation to Regression Analysis . . .. 141
The Reliability Tests . . . . . .. 143
The Original Sample Method. . . . .. 143
The Holdout Sample Method . . . . 143
vi ii
TADLE OF CONTENTS (CONTINUED)
Page
The Lachenbruch oneataTime Holdoat
Method. . . . . . . . . 144
The Scrambled Yethod . . . . . .144
The Relative Importance of Variables. ... .146
Compute; Programs . . . . . . . 149
Cooley and Lohnes' Program. . . . ... 149
BMD04M. . . . . . . . . .. .149
BMD05M. . . . . . . . . .. 149
BMD07M. . . . . . . . . .. 150
SAS DISCRIM Procedure . . . . ... .150
SPSS DISCRIMINANT Program . . . ... .151
Eisenbeis and Avery's MULDIS Program. . 151
Application of MDA in Financial Area. . . 151
Notes . . . . . . . . . 158
IV THE EMPIRICAL TEST OF THE HYPOTHESIS AND
PRESENTATION OF RESULTS . . . . ... .165
Introduction. . . ... . . . . .165
Source of Data. . . . . . . .. .167
Selection of Samples. . . . . . .. .169
Failed Firms. . . . . . . .. .169
Nonfailed Firms . . . . . . .. .172
Data Set . . . . . . . . 172
Test I One Year in Advance of Bankruptcy. 175
Introduction. . . . . . . .. .175
Variable Set. . . . . . . .. .176
Discriminant Analysis . . . . .. 180
Methodology . . . . . . ... 180
The Selected Variables ....... . 184
H1 Test of Equality of Dispersion
Matrices. . . . . . . . . 188
The Discriminant Function and Its
Analysis .............. 189
The Differences Between the Group Means 199
The Overlap Between Groups. . . .. .200
Validity Tests. . . . . . . .. 201
Classification of the Original Sample . 204
Classification cf the Holdout Sample. . 208
TABLE OF CONTENTS CONTINUED')
Pa:
Potential Biases. . . . . . .. 212
Potential Biases Due to the Sample. . 212
Potential Biases Due to Data. . . .. .216
Potential Biases Due to the Variation
in Reporting Daes. . . . . .. 219
Potential Biases Due to the Variable
Selection . . . . . . ... 222
Results . . . . . . . . . 222
Test II Two Years in Advance of Bankruptcy. 224
Introduction. . . . . . . .. .224
Variable Set. . .. .. ...... . 226
Discriminating Analysis . . . . .. .226
Methology ................ 226
The Selected Variables. . . . .. .227
HI Test of Equality of Dispersion
Matrices. . ...... ..... . 229
The Discriminant Function and Its
Analysis. . . . . . . .. .229
The Differences Between the Group Means 234
The Overlap Between Groups. . . .. .234
Validity Tests. . . . . . . ... 235
Classification of the Original Sample 235
Classification of the Holdout Sample. 235
Results . . . . . . . ... 239
Notes . . . . . . . . . . 243
V SUMMARY AND CONCLUSION. . . . . ... 247
Summsry . . . . . . . . . 247
Conclusion. . . . . . . . ... 254
Implications. . . . . . . . .. .255
Suggestions for Further Research . . . 257
Notes . . . . . . . . . . 258
APPENDIX
A Failed Firms in the Original Sample . .. 259
B Nonfailed Firms in the Original Sample. .. .261
TABLE OF CONTENTS (CONTINUED)
Page
C Failed Firms in the Holdout Sample ... 263
D Nonfailed Firms in the Holdout Sample .. 265
E List of the Data Set Items . . . .. 267
F List of the Variables . . . . ... 270
G The Total Correlation Matrix of Test I. .. 274
H The Variable Structure of Total
Observations in One Year Prior to
Bankruptcy. . . . . . . .. 288
I A Profile of the Selected Variables for
the Reduced Sample. . . . . . 292
J The Linear and Quadratic Equations for
the Reduced Sample. . . . . .. 293
K Classification of the Reduced Sample Using
the Standard Method .......... 295
L Classification of the Reduced Sample Using
the Lachenbrucn Method. . . . .. 296
M Classification of the Original Sample
Using the "Normalized" Model. . . .. 297
N Classification of the Holdout Sample Using
the "Normalized" model. . . . .. 298
BIBLIOGRAPHY . . . ... . . . . . . 299
BIOGRAPHICAL SKETCH. . . . . . .... . . 308
LIST OF TABLES
Table Page
1. Some Selected Statistic Which Provide a
Brief Profile of the Securities Broker/Dealer
Firms for Years 196974.......................... 3
2. The FailedFirm Population by Years in
Business........................................ 170
3. Results of the Forward and Backward Selections
in Runs V and VI of Test I........................ 183
4. A Profile of the Selected Variables of Test I... 192
5. The Overlap Between Groups in Test I............. 200
6. Classification of the Original Sample of Test I. 207
7. Classification of the Holdout Sample of Test I.. 211
8. A Profile of the Selected Variables in Test II.. 231
9. The Overlap Between Groups in Test II........... 234
10. Classification of the Original Sample of Test
II .............................................. 238
11. Classification of the Holdout Sample of Test
II .............................................. 240
LIST OF FIGURES
Figure Page
1. A Pattern Classification System................. 115
2. Hotelling's T2; One Minus Significance Level
As a Function of Sample Size..................... 123
3. Hotelling's T2; One Minus Significance Level
As a Function of d for Several Values of r...... 123
xiii
LIST OF CHARTS
Page
Chart
1. Classification of the Original Sample
of Test I ....................................... 209
2. Classification of the Holdout Sample of
Test I... ....................................... 213
Abstract of Dissertation Presented to the Graduate
Council of the Univers ity of Florida in Partial
Fulfillment of the Requirements for the
Degree of Doctor of Philosophy
A DISCRIMINANT ANALYSIS OF THE BANKRUPTCY
OF SECURITIES 3ROKE:.GE/DEAT,ER FIRMS
By
Iraj Afkham
December, 1975
Chairman: Henry R. Fogler
CoChairman: Robert C. Radcliffe
Major Department: Management
This dissertation is an attempt to meet the following
objectives: (1) to identify potential economic and non
economic factors which might differentiate between finan
cially distressed firms and those which appear to be
economically viable; (2) to statistically determine a set of
variables which has the highest discriminatory power; and
(3) to suggest both public and managerial action which might
be taken in light of the statistical results obtained in
point 2.
To begin the analysis, it was hypothesized that the
probability of financial distress of a firm at time t is a
function of the net cashflow per dollar of assets at time
t, the extent of the risk exposure, and the mismanagement of
the firm.
Three poLtntial approaches for predicting failure
are discussed in the dissertation: deterministicc," prob
abilistic, and blend approach. The deterministicc" approach
was meant to define the traditional method which uses the
financial ratios in conjunction with the MDA technique.
The probabilistic approach was stated to be the one which
does not treat the financial ratios and other related forces
as factors whose values can be determined with certainty at
any point of time in the near future. Rather, it attaches
some probability distribution to each factor, based upon its
historical performance. And, instead of using the MDA
technique, it relies upon the modern probability theories.
The probabilistic approach suggested in this study,
although intriguing, requires a great deal of research be
fore it can be put to work. The blend approach, used in the
present study, was therefore proposed to act as a bridge
between the traditional and the probabilistic approaches.
It retains most of the favorable properties of the traditional
approach, specially with respect to the MDA technique, yet
incorporates some facets of the probabilistic approach.
After presenting an analytic framework for utilization
of the blend approach and providing a groundwork for the ap
plication of the MDA technique in two successive chapters,
an effort was made to empirically test the hypothesis. Due
to the confidentiality of a part of financial data, however,
the hypothesis had to be modified by substituting the cash
flow element with profitability.
Sixty SIPC failed and 60 nonfailed firms were selected
and the original and holdout samples were formed by halving
the entire observations and assigning each half randomly to
one sample. The main thrust of the empirical part rests
upon Test I whose predictive span is about one year. An
attempt was also made to extend the analysis to two years
prior to bankruptcy, in Test II.
Test I examines 83 variables and selects 12 of them
as an "optimal" subset by a multistage stepwise selection
procedure. The discriminant model developed in this test
successfully differentiates between failed and nonfailed
groups. The quadratic classification procedure correctly
classifies 98.33 percent of the original sample and mis
classifies, on average, 13.33 percent of the holdout sample.
These results, along with the results of applying the
scrambled method, Lachenbruch classification procedure, and
"normalized" model, reveal the relatively high accuracy and
validity of the model.
The 12 selected variables measure profitability, risk,
and mismanagement of a firm. The relative discriminatory
powers of these variables in the model reveal that the risk
element is the most significant determinant of the firm's
survival. Profitability and mismanagement are in the second
and third places. This may help managements in making
decisions when considering risky ventures.
xvii
Contrary to Test I, the result of Test II was
expectedly less than successful. This was partly due to the
nature of the securities business and partly due to the
type of data available.
CHAPTER I
INTRODUCTION
The Backbone of the Capital Market
During 1974, the market value of all security
transactions on the United States registered exchanges
amounted to $124.7 billion. The significance of the figure
may become more apparent if it is considered in conjunction
with the nation's Gross National Product in that year. The
GNP in 1974 amounted to $1,397.4 billion and the value of
the securities transactions was as large as an amount equal
to about 9 percent of the GNP. Although this is the lowest
percentage in the past few years (the highest is 23, in 1968),
it strikingly reveals the important role that the securities
broker/dealers play in flowing funds in the nation's capital
market.
The huge volume of securities transactions in each
year is handled by several thousands of broker/dealer firms.
In 1973, for example, the number of registered broker/dealer
firms was 4,088. These firms had employed a total of about
349,000 employees to channel the savings of individuals to
corporate users and to transfer the ownership of outstanding
securities from one investor to another. The total assets
marshalled by the securities firms having gross securities
income of $20,000 and over amounted to $25.9 billion.
In order to present a brief profile of the securities
broker/dealer firms, similar statistics as those stated above
are tabulated in Table 1 for the period 19691974. This may
aid in understanding the general position of the securities
industry in the overall economy.
Literature Review
The maintenance of fair and orderly markets for
securities has long been one of the highly desired objectives
of the legislatures, administrators, courts, and various self
regulatory organizations. The Securities Act of 1933 and
1934 along with the creation of the Securities and Exchange
Commission (SEC) to administer the acts, as well as the
pursuant amendment acts of 1939, 1940, 1964, 1970, 1974, and
1975 partially reflect the main concern of these authorities
in providing conditions designed to create securities mar
kets capable of meeting the nation's need for investment
capital while operating in the public interest and serving
to protect the public investor.
For a long time, the going presumption was that the
efforts directed toward preventing fraudulent, deceptive
and manipulative acts in the securities.markets would pro
tect the individual investors and the public interest. It
was not until 1970 that the hardship and sometimes disaster
which were faced by the individual investors whose broker/
dealers went bankrupt or underwent liquidation were widely
S1.'L 1. '..!I[ CLECTELL[ STATISTICS WHICH PROVIDE A BRIEF PROFILE OF THE SECURITIES BROKER/DEALER FIRMS FOR THE
,C'.I' lI' *4. (WHERE NOT SPECIFIED, DOLLARS ARE IN MILLIONS.)
No. of Firms ( (9*
(2) (3) (5) (6) Operating Return on
Volume of GNP Firms with No. of (7)* Income Assets (Be
(1) Securities (in Billions, (4) $20,000 or Employees Total (Before fore Taxes)
Year Transactions, $ Current $) Total More Income (in Thousands) Assets Taxes) (8)i(7)
1969 179,798 725.6 4,681 2,619 366 24,213 561 .023
1970 135,889 722.5 4,614 2,332 338 24,442 271 .011
1971 193,821 745.4 4,553 2,539 353 28,705 1,279 .044
1972 213,548 790.7 4,255 2,512 386 33,728 976 .029
1973 186,173 839.2 4,088 2,164 349 25,948 57 .002
1974 124,709 821.2 3,982 NA NA NA NA NA
*The figures in this column are related to the firms having gross securities income of $20,000 and over, as
reported in column (5).
IA indicates data not available.
Note: Figures in column 2 are for the U.S. registered exchanges only.
Source: For columns (2) and (3), 1974 Statistical Abstracts of the United States.1
For columns (4), (5), (6), (7), and (8), 1974 Securities and Exchange Commission Annual Reports.
realized. The substantial operational and financial
problems that the securities industry experienced during
the period from 1967 through 1970 had significantly dimin
ished the investors' confidence in the markets; and the need
for urgent corrective action was felt everywhere.
In December 1970, Congress passed the Securities
Investor Protection Act to restore public confidence in the
safety of the securities markets. This legislation estab
lished the Securities Investor Protection Corporation (SIPC).
And, the corporation provided a limited insurance for the
customer accounts of the securities industry.
During its four years of existence, SIPC has placed
a total of 109 failed firms in liquidation. It has received
a total of 97,230 customer claims. And, as of December 31,
1974, it has distributed securities and cash with an aggregate
value of $241 million to satisfy these claims. Of this
amount, approximately $202 million was derived from debtor
estates. The remaining $39 million, however, was provided
by SIPC through its advances to the trustees. SIPC's
assessments on member firms during the fouryear period
totaled $106 million.3
To minimize the exposure of the SIPC fund, which is
backed by a billion dollars of taxpayers' monies, Congress
directed the SEC, at the time of passing the Securities
Investor Protection Act, to compile a list of unsafe and
unsound practices by broker/dealers and to report on the
corrective steps being taken. Before the SEC report is
reviewed, the results of another study which was published
prior to the completion of this report is presented.
Review of SEC Records of the Demise of
Selected Broker/Dealers
This is the title of a staff study undertaken for
the Special Subcommittee on Investigations of the Committee
on Interstate and Foreign Commerce of the House of Repre
sentatives.4 The need for this study was stimulated by the
crisis of the securities industry over the period 196770
which had brought disaster to numerous broker/dealers and
financial hardship for the public investors. The study is
a review of the files of the commission's headquarters offi
ces for 46 selected broker/dealers who went into some form
of liquidation during the 3year period ending December 31,
1970. It also incorporates additional information obtained
from other sources; and includes case studies for all 46
firms.
The review concluded, among other things, that there
were many reasons for the failures of the subject firms and
the factors were both internal and external to the firms.
Internal factors included capital deficiency, operational
losses, mismanagement and inefficiency in operating the
firms' business, overexpansion of operations, inoperable
and overly expensive modernization programs, backoffice
difficulties, irresponsible actions by principals, and
and fraud. External factors included a reduced volume of
trading and a large decline in prices of many stocks during
the period.
Study of Unsafe and Unsound Practices
of Brokers and Dealersa
This study was undertaken by the regular staff of
the SEC pursuant to Section 11(h) of the SIPC Act. Like the
previous study, this one is also on the financial and opera
tional crisis of 196770, but on a larger scale and in a more
comprehensive way. It examines industry conditions and
practices which permitted the crisis to develop. To name
some, but not all, the report cites the following unsound
practices:
o Inadequacy and inpermanence of capital.
o Overemphasis on sales and trading activities
at the expense of operational resources.
o Absence of control of securities traffic.
o Absence of an effective early warning system
in selfregulatory organizations and their
inability to respond to the crisis with
meaningful corrective measures.
o Lack of experience of principal members of
many, mainly small, firms.
The study reviews corrective measures already
instituted, and suggests further actions to be taken by the
broker/dealer firms, selfregulatory organizations, and the
SEC. The corrective measures detailed in the report include,
among other things, increasing capital requirements, revis
ing or proposing rules to strengthen control over securities
and to provide greater protection for customers' free credit
balances and securities left with brokers, and taking measures
to provide the commission and selfregulatory authorities with
more effective early warning systems.
Securities Industry Study by the
Subcommittee on Securitiesb
During consideration of the SIPC legislation, the
Committee on Banking, Housing, and Urban Affairs of the
Senate concluded that a complete study of the securities in
dustry and the securities markets should be undertaken by
its Subcommittee on Securities. And, after authorization by
the Senate in June 1971, the study was launched.
This study is the culmination of the analysis of
the information obtained from a variety of sources. Some
of the information was obtained directly from industry and
government sources including responses to a detailed ques
tionnaire on institutional membership sent to 104 firms,
organizations and individuals. Other information was ob
tained through holding 18 days of hearings and receiving
testimony from 78 witnesses. It includes also four detailed
case studies of the operation of the regulatory system in
the securities industry.
The contents of the study give the impression that
the subcommittee tends to rely more on the market mechanism
and its competitive nature than on regulations. It also
looks beyond the current problems to the longrange struc
tural features of the industry, markets, and regulatory
system.
The basic conclusions of the study include, but are
not limited to, the adaptation of greater flexibility in the
industry, the redefinition of the division of responsibility
between SEC and selfregulatory organizations, the provision
of fair, orderly, and open procedure in reaching important
decisions by these organizations, and giving more weight
to competition than to regulation. In the words of the sub
committee report:
...Regulation, by government and by industry
groups, is an essential element in protection
of investors, but is not an effective substitute
for competition in assuring a flexible and
healthy industry.
The concept of industry selfregulation, sub
ject to SEC oversight, is welladapted to
dealing with problems of conduct and ethics,
but it is not welladapted to dealing with
general economic questions involving competi
tive interrelationships among firms within
the industry....7
Securities Industry Study by the Subcommittee
on Commerce and Financed
This study was undertaken in fulfillment of the
commitment made by the chairman of the committee on Inter
state and Foreign Commerce with respect to the committee's
intention to make a complete study of the securities market
and exchanges.
The study is largely the outgrowth of an extensive
number of panel hearings, witness testimonies, views of
industry leaders, interviews in virtually every section of
the country, and reviews of previous and current studies on
the securities industry.
It includes chapters on what went wrong in the
securities business during the period 196770, criminal en
forcement of securities laws, standards for entry into the
securities business, capital for broker/dealers, protection
of customers, accounting and auditing procedures, opera
tional difficulties, cooperative regulations, central market
system, commission rates, membership qualification for
exchanges, antitrust aspects, and the SEC budget.
Space does not allow the enumeration of the large
number and variety of conclusions reached, but to provide a
feeling about the nature of this study, a sentence from the
related letter of transmittal signed by J.E. Moss, Chairman
of the subcommittee, is quoted here.
...This study is, to my knowledge, the only
comprehensive congressional examination
of the securities industry to have been
completed within the time span of one
Congress....
SIPC Study
SIPC has also made inquiries of the trustees and
reviewed the reports of the staffs of the selfregulatory
organizations participating in the cases of the 94 broker/
dealer firms liquidated as of December 31, 1973, in an effort
to determine the cause of failure.
The study indicates that the matter of possible fraud
and manipulation which has surfaced in many of these liquida
tions must be recognized as a major factor in these failures.
Large concentrations of speculative issues in the trading
activities contributed to other failures. Inadequate, in
accurate, and sometimes nonexistent books and records were
another significant factor. In many cases, the operating
management did not have the qualifications or experience
needed to operate a general securities business. Lack of
adequate capital continues to be mentioned by the trustees
as a major factor in firm failures. According to the study,
this condition results from a number of factors including too
small a capital base, temporary illiquidity, overcommitment
in particular issues, inability to absorb an adverse market
movement, too rapid expansion, improper controls, and operat
ing losses due to reduced volume. The study states that
there was overreliance on subordinated capital in a number
of instances. In other cases, the subordination agreement
was allegedly improperly executed, fraudulently induced or
improperly withdrawn placing the firm in net capital viola
tion.
The Early Warning Requirement of the SIPC Act10
Section 5(a)(1) of the SIPC Act requires that the
commission or the selfregulatory organizations notify SIPC
immediately upon discovery of facts which indicate that a
broker or dealer subject to their regulations is in or
approaching financial difficulty.
After recei"'in such notification, SIPC determines
whether to apply for a court decree to initiate liquidation
or to defer action until further information is obtained and
doubt is removed.
There are five conditions specified in Section 5(h)
of the Act. At least one of these conditions must be found
to exist by SIPC to apply for a court decree and by the court
to issue the decree in every case. The court shall issue a
decree if it finds that the member firm:
a. Is insolvent within the meaning of Section 1(19)
of the Bankruptcy Act, or is unable to meet its obligations
as they mature, or
b. Has committed an act of bankruptcy within the
meaning of Section 3 of the Bankruptcy Act, or
c. Is the subject of a proceeding pending in any
court or before any agency of the United States or any state
in which a receiver, trustee, or liquidator for such member
has been appointed, or
d. Is not in compliance with applicable requirements
under the 1934 Acts or rules or regulations of the commission
or any selfregulatory organizations with respect to financial
responsibility or hypothecation of customers' securities, or
e. Is unable to make such computations as may be
necessary to establish complaince with such financial respon
sibility or hypothecation rules or regulations.
As was mentioned above, the SIPC Act requires that,
when it appears to the SEC or any selfregulatory organiza
tion that a brokerdealer is in or approaching financial
difficulty, SIPC is to be notified immediately. The statute,
however, does not define "financial difficulty." Neither has
it seemed practicable for SIPC to attempt to define these
terms. In view of the number of selfregulatory organiza
tions involved and the differences in their rules, procedures,
problems, and the manner in which the various reporting and
surveillance systems operate, SIPC has considered it approp
riate, up to this point at least, to work with each organiza
tion separately.
The regulators fulfill the Act's requirement through
regulatory procedures which integrate inspection and report
ing programs with an early warning procedure for notifying
SIPC. The primary objective of these programs is the early
identification of those members which are in or approaching
financial or operational difficulty and the initiation of
action necessary to protect customers.
The Existing Early Warning Systems
All over the entire securities industry emphasis is now
on the early warning systems by which the financial difficulty
of the broker/dealers is to be detected in advance. The two
main purposes of such early warning systems are: (1) to
afford a selfregulatory organization an opportunity to
initiate procedures so that the broker/dealer in question
does not become a SPIC casualty, and (2) to provide a basis
for notification to SIPC. Inasmuch as these warning systems
are directly related to the subject matter of the present
study, further review of the related material seems approp
riate.
The early warning system of the SEC. The SEC, in
discharging its duties, periodically reviews through onsite
inspections and inhouse studies the early warning surveil
lance tools of the selfregulatory organizations to ensure
that they constitute sound and effective programs.
The commission's early warning and surveillance tools
include Rule 17all, Rule 17a5(j), and Rule 17a10. Rule
17all was adopted to provide the commission and the self
regulatory authority with an adequate and timely flow of
information on the financial and operational condition of
broker/dealers. Among other provisions, it requires a
broker/dealer to notify, by telegram, the commission and the
appropriate selfregulatory organization if it breaks
through certain specified financial or operational parameters.
Rule 17a5(j) requires a broker/dealer to notify the commis
sion if its exchange membership is terminated and becomes
subject to the commission's net capital rule. Rule 17a10
requires a broker/dealer to file Form X17A10 annually with
the commission.
An additional financial responsibility rule, Rule
15c33, titled "Customer ProtectionReserves and Custody of
Securities," became effective on January 15, 1973. Failure
to comply with the requirements of this rule has, in a
number of instances, been the basis of the statutory condi
tion.
In addition to these rules, the commission, working
together with the NASD and the selfregulatory authorities,
has developed a monthly, combined early warning list and a
Rule 17all list for each regional office. Each month the
selfregulatory organizations submit, for the combined list,
data pertaining to the financial condition of troubled firms
within each commission region. The list is then speedily
transmitted to each regional office chief examiner for his
verification within 24 hours.
Finally, in many cases, the commission's application
for an injunction and the appointment of a receiver has
typically been based on violations of the net capital rules
or such inadequacy of books and records that the firm is un
able to make such computations as may be necessary to
establish compliance with the rules concerning financial
responsibility or hypothecation of customers' securities.1
The early warning system of the selfregulatory
organizations. Over the years, the exchanges and the NASD
each has developed a complex system of surveillance both on
its own initiative and in compliance with Federal laws. As
a result, many different early warning systems have emerged.
Space does not allow a review of all of these systems indi
vidually, here. It may, however, be relatively safe to
generalize that a selfregulatory organization places a mem
ber firm under a special surveillance if its net capital is
deficient, the ratio of aggregate indebtedness to net capital
exceeds the related rules or guidelines, or if the firm's
operating losses in a given period exceed certain criteria.
As an example, the New York Stock Exchange has the following
early warning criteria for carrying organizations:13
1. Rule 325 Capital Ratio at, or in excess of
1,000 percentas of the date of the Exchange's
Early Warning Report.
2. Cumulative losses during the most recent three
month period equal to, or in excess of, 50 per
cent of the organization's tentative net capital.
3. Any firm likely to begin liquidation within the
next six months because of losses, or scheduled
withdrawals of capital, which would jeopardize
their financial viability.
The final point to be noted relates to the case of
NASD. The organization has recently attempted to add a new
and significant feature to its operating early warning system.
Due to the similarity that exists between this feature and
the theme of the present study, it will be discussed in the
following section in more detail. At this point, however,
suffice it to say that, while all the previouslyreviewed
studies and the warning procedures base their analysis on
the individual effects of a criterion in a selected set of
criteria, the NASD's additional feature and the present work
consider the collective effects of such factors. The two
approaches may bring about different solutions to the same
problem. But, the superiority of the second approach has
been well established.14
As an example, consideration of a violation of the
net capital rule on the basis of its individual effect may
lead to a different action from that which may be induced
when its effect is considered collectively, in combination
with some other factors) such as net income, for example.
As once noted by SIPC, although in a little different con
text,
...A violation of the net capital rule might
not portend as serious a situation from the
point of view of customer protection as
originally feared. This rule basically is a
test of liquidity as of a particular time.
It does not necessarily follow that a tem
porary or possibly inadvertent failure to
comply with the net capital rule makes losses
to customers inevitable.15
The NASD Failed Firms Project16
When the present study was halfway through, it was
found cut that NASD has also undertaken a similar study, but
it has not been published yet.1 It was conducted by E.J.
Dervan and E.I. Altman. It applies the Multivariate Dis
criminant Analysis (MDA) technique as a tool for predicting
the failure of firms.
The Failed Firms Project uses a sample of two group
firms: The Failed Group included 40 firms which were under
SIPC liquidation; the Active Group consisted of 113 firms.
None of the firms was a member of the New York Stock
Exchange or American Stock Exchange. The financial data
on these firms were obtained from the related Annual Finan
cial Reports, Form X17A10.
The project had been launched to determine if the
reported financial experience of failed firms prior to
their failure could be utilized to establish an early warn
ing system for identifying potential failures. For this
purpose, it was reported that a total of 82 financial ratios
and nonfinancial indicators such as age and type of organiza
tion were examined. The following six variables were selected
as doing the best job, collectively, in discriminating be
tween active and failed firms:
1. Net Income After Taxes/Total Assets.
2. (Total Liabilities + Subordinated Loans)/Owner's
Equity.
3. Total Assets/Adjusted Net Capital.
4. (Ending CapitalCapital Additions)/Beginning Capital.
5. Scaled Age.
6. Composite (based on the ten separate variables
treated in a dichotomous way).
Among conclusions of the report were that a majority
of failures arise from a relatively small subset of NASD
member firms; the constructed model proved to be highly accu
rate by classifying 90.1 percent of an original sample of
failed and active firms; the model is highly reliable, and
it appears to provide significant lead time in which to
initiate corrective action.
The discussion about the results of the NASD study
and their comparison with thosa of the present study is
partly relegated to the next section and partly to Chapter
IV, where the results of this study are presented in detail.
Differences Between the NASD Study
and the Present Work
Although the two studies are similar in a number of
ways, including the application of the MDA technique, they
differ from each other in analytic framework, source of
data, data set, and results. More specifically, the dif
ferences lie in the following areas:
The Analytic Framework
The NASD study uses the traditional approach, which
may be dubbed deterministicc" to differentiate from the
"probabilistic" approach being presented in the following
section, in building up the prediction model. Whereas, the
approach taken in the present study is a blend of "determinis
tic" and "probabilistic" approaches. This point will become
more clear when the following section on "Different Ap
proaches" is reviewed.
The Variable Sets
Although the complete list of the mentioned 82
variables in the NASD study has not been reported, it is most
likely that the two variable sets used in building up the
models differ from one another. The difference is partially
due to the introduction of the "probabilistic" variables in the
present study, such as standard deviation of some financial
ratios and a version of Tchebycheff's inequality for estimat
ing the probability that the value of some variables fall
below a disaster level, and partly due to the sources of
data which will be discussed below.
The Sources of Data
Form X17A10 was the source of data for the original
model of the NASD study. This form contains most of the
financial data which are ordinarily included in Income State
ment and Balance Sheet. The present study, however, has used
information contained in Form X17A5. This form contains
only items which are ordinarily included in Balance Sheet and
lacks any incomestatement item. The choice of the source
for the present study was inevitably due to the inavalibility
of Form X17A10 to the public.
Data Set
In the NASD study only an original sample has been
employed and it has consisted of 40 failed firms and 113
nonfailed firms. In the present study, two independent
samples, i.e., the original sample and the holdout sample,
are used; each sample has 30 failed firms and 30 nonfailed
firms.
The Results
It is very difficult, if not impossible, to compare
the results of the two studies. The first reason is the lack
of the very important financial elements, i.e., the income
statement items, in the source of data of the present study.
The second reason is the absence of similar counterparts in
the two studies, especially with respect to the model which
predicts the bankruptcy two years in advance. The third, but
not the last, is the lack of the holdout sample in the NASD
study which might have been due to the nonexistence of suffi
cient numbers of observations at the time of the study.
Given these limitations, a comparison will be made in Chapter
IV, wherever it is feasible.
Different Approaches
In predicting the financial position of a firm in a
near future, there are at least three potential approaches:
The first one, which has already been put to use, is finan
cial ratio analysis, previously on the univariate basis and
recently on the multivariate scale. The approach may be
dubbed deterministic. The second one, which has not yet been
developed is a probabilistic approach. The reasons for con
sidering such an approach, along with a brief illustration of
the concept, will be given in the following paragraphs. Then,
the difficulties associated with its implementation are dis
cussed. As a consequence, the third potential approach,
which is a compromise between the first and second, emerges.
This last approach is the one which has been adopted in the
present study.
The Deterministic Approach
In the early years of the 1930's, Smith and Winakorl8
undertook a study of 133 financially troubled companies and
concluded that Working Capital/Total Assets, Net Worth/Total
Debt, and Net Work/Fixed Assets were effective indicators of
financial difficulties in firms. In 1931, in another study,
Fitzpatrickl9 employed 13 different ratios and concluded that
all the ratios predicted failure to an extent, and that the
best predictors were Net Worth/Total Debt and Net Profit/Net
Worth. Similar efforts were later made by Merwin,20 Tamari,21
22
and Beaver. But, none of these researchers came to the
same conclusion. What they had in common, however, was
their approach to the problem which was based on the uni
variate analysis of the financial ratios.
Later, in 1967, Altman23 improved upon the above
approach by taking advantage of the properties of the multi
variate discriminant analysis technique. This technique,
which drastically reduces the analyst's space dimensionality
and automatically assigns appropriate weights to appropriate
variables, removed some of the ambiguities, confusions, and
difficulties in the interpretation of the traditional ratio
analysis. Since then this approach has been utilized in
predicting the financial position of firms in various
industries.24
Although the recent method is both theoretically and
operationally superior to the traditional ratio analysis, it
still shares some drawbacks with the latter: Both methods
treat the process in such a way as though the property of
cause and effect relation perfectly holds in a deterministic
manner. The implied assumption is that a specific stimulus
or a set of stimuli (i.e., changing the financial ratios)
produces a specific response from the firm (i.e., failure).
And, the required effort to predict the firm's state is
solely to identify such stimuli. Furthermore, it is
implicitly assumed that the firm is in isolation and no
external influence (such as changes in the market and eco
nomic conditions) affects the future financial position of
the firm. In other words, the implied assumption is that
the state of the firm at some time in the future is determin
istically ascertained by its current and past states.
In the face of widely prevalent uncertainty around
every aspect of the business world, it is obvious that
reliance on such faroutofreality assumptions is not
plausible. In fact, the firm is a system that can be re
sembled to a "grey box," about which we have only a limited
knowledge. The numerous subsystems, including each indi
vidual, with his own capricious and everchanging charac
teristics, constituting the firm, and numerous systems and
subsystems building up the socioeconomic environment in
which the firm exists makes any certain inference or pre
diction about the firm's behavior very difficult, if not
impossible. The complexity of actions and interactions
between these many systems and subsystems are such that we
may never be able to assert with certainty that a specific
input to the firm yields a specific output. Under such
conditions, the best way that we may expect is a vector of
possible outcomes (rather than a single.certain outcome)
with some probabilities attached to each of them.
The Probabilistic Approach
This approach, which has not been developed yet, is
based upon the same concept which drove the statistician to
replace the complexity by uncertainty. The statistician who
knew that in tossing a coin the prediction of the ultimate
position of the coin is conceivably possible by using the
laws of mechanics (given sufficient information about its
initial position, the way in which it is thrown, and the
specifications of the surface on which it falls) concluded
that it is far too difficult a problem for contemporary science
to be solved in this way. He therefore suggested a more op
erational way of predicting the result of the tossingthe
0.5 probability of getting heads if the coin is a fair one.
Yet, the prediction of the firm's financial position
is not as simple as that of a fair coin. The firm's posi
tion is a combinatorial phenomenon which is the result of yet
unexplored special combinations of numerous internal and ex
ternal (to the firm) factors, each of which has a special
probability distribution of outcome. The concept, neverthe
less, is straightforward.
To illustrate, suppose that the probability of the
firm's financial distress is affected by only three proba
bilistic factors A, B, and C, whose a priori probability
distributions may be estimated through past experiments. If
these three factors were mutually exclusive, the probability
of financial distress would equal to the probability of get
ting any one of these three events. That is:
P(FD) = P(A or B or C) = P(A) + P(B) + P(C) (1)
If A, B, and C are not mutually exclusive, then equation (1)
would not hold. Depending upon the anticipated influence of
these factors on the probability of financial distress, there
are some other choices available. If we have reasons to
believe that each factor has effects individually, the more
relevant equation may be the following:
P(FD) = P(A or B or C) = P(A) + P(B) + P(C) P(A & B)
P(B & C) P(A & C) (2)
On the other hand, if the available evidence suggests that the
effect comes through the joint occurrence of all of the three
factors, then, equation (2) does not explain the case. If
A, B, and C are statistically independent, the proper equa
tion may be the following:
P(FD) = P(A & B & C) = P(A)P(B)P(C) (3)
Otherwise, equation (4) may be more relevant:
P(FD) + P(A & B & C) = P(A)P(BIA)P(CIA & B) (4)
The crux of the problem lies in the fact that we know
neither all of the probabilistic factors affecting the
probability of financial distress, nor the relationships be
tween such factors. While some of these factors, such as
longterm debt and shortterm debt, may be mutually exclu
sive, other factors such as revenues from different product
lines may not necessarily be so, and still other factors
such as cashflow and expenses may be statistically dependent.
Furthermore, it is conceivable that the conditional
probabilities of two factors may not be equal, i.e. P(AIB) f
P(BIA). Such a case may arise when A represents, for ex
ample, the revenue of the firm and B the general level of
economic activities. Thus, the relevant probability equa
tion of the firm cannot be a simple equation such as any one
of those mentioned above. Conversely, it is likely that the
function is a combination of a great number of such equations,
embracing a wide range of relevant factors.
The Blend Approach
Although the probabilistic approach seems to be
intriguing, it requires a great deal of research before it
can be actually put to work. More specifically, the research
needed comprises: (1) identifying all the probabilistic
factors affecting the probability of the firm's financial
distress. (2) determining the interrelationships between
these factors. (3) finding a suitable way to estimate the a
prior probabilities of such factors, and finally (4) for
mulating the probabilistic model.
Faced with such a huge work, one may think about the
ways of reducing the load. One way to do this, is to bridge
between the deterministic and the probabilistic approach. And
this is the path taken in the present study.
While the blend approach retains all the favorable
properties of the deterministic approach, especially with
respect to the utilization of the MDA technique, it also
incorporates some facets of the probabilistic approach. Such
a midway selection may ultimately be helpful in the develop
ment of the potentially more comprehensive probabilistic
approach.
The Purpose of the Study
The purpose of this study is threefold:
1. To identify potential economic and noneconomic
factors which might differentiate between finan
cially distressed securities brokerage/dealer
firms and those which appear to be economically
viable.
2. To statistically determine a set of variables,
from among the factors in point 1, which has the
highest discriminatory power.
3. To suggest both public and managerial action
which might be taken in light of the statisti
cal results obtained in point 2.
The Problem Encountered
When the work on the present study had been initiated,
it was hoped that no serious problem would be encountered with
respect to the data needed to test the suggested hypothesis.
Unfortunately, however, when the work on the theoretical part
of the dissertation was finished, and the time for collecting
the data came, the hope was replaced by regret.
The Public Reference Office of the SEC, although co
operative in supplying the financial Forms X17A5 of the
selected companies, denied the accessibility of Forms X17A
10, on the ground that they were confidential.
Since Form X17A5 contains only items similar to
those of Balance Sheet and lacks about half of the highly
valuable financial information which is contained in Form
X17A10, especially with respect to the revenue and expense
items, a full test of the suggested hypothesis is not pos
sible with the available data. It is hoped that in evalua
tion of the results of the study, this limitation is kept in
mind. The results might hava been better if all the finan
cial information could have been included in the analysis.
The Organization of the Dissertation
After the introductory chapter, the analytic frame
work is set forth in Chapter II which discusses the suggested
hypothesis and determines the potential factors affecting
the probability of financial distress by expanding the
hypothesized function. Included in this chapter is a
section on risk which explores the possible measures of
risk and risk factors.
Chapter III provides a groundwork for the use of the
MDA technique in the analysis. It presents an overall pic
ture of the MDA technique in application in a summary form.
Chapter IV integrates the materials included in
Chapters II and III into an empirical experiment consisting
of Test I and Test II. This chapter describes the sources
of data and data set, the procedure used in selecting samples,
and the research methodology employed to empirically test the
hypothesis. The greater part of this chapter is devoted to
Test I.
Finally, Chapter V comprises a summary of the
dissertation, the conclusion reached by the empirical ex
periment, the implication of the results found, and the
suggestions for further research.
NOTES
1. U.S. Department of Commerce, Statistical Abstracts of the
United States 1974, Washington, D.C.: U.S. Government
Printing Office, 1975.
2. Securities and Exchange Commission, Annual Report 1974,
Washington, D.C.: U.S. Government Printing Office, 1975.
3. Securities Investor Protection Corporation, Fourth
Annual Report, 1974, pp. 2 and 34.
4. U.S. House of Representatives, Review of SEC Records of
the Denise of Selected BrokerDealers, Staff Study for
the Subcommittee on Investigations (Subcommittee Print),
Washington, D.C.: U.S. Government Printing Office, July
1971.
5. U.S. House of Representatives, Study of Unsafe and Un
sound Practices of Brokers and Dealers, House Document
No. 92231 of the 92nd Congress, 2nd Session, Washington,
D.C.: U.S. Government Printing Office, 1971.
6. U.S. Senate, Securities Industry Study, Report of the
Subcommittee on Securities, Committee on Banking,
Housing, and Urban Affairs (Committee Print), Washing
ton, D.C.: U.S. Government Printing Office, February
1973.
7. Ibid., pp. 23.
8. U.S. House of Representatives, Securities Industry
Study, Report of the Subcommittee on Commerce and
Finance, (Subcommittee Print) Washington, D.C.: U.S.
Government Printing Office, August 23, 1972.
9. This section draws from SIPC, Third Annual Report 1973,
pp. 2425, and SIPC, Second Annual Report 1972, pp. 25
26.
10. This section draws from SIPC, Third Annual Report 1973,
which contains an excellent review of the manner in
which SIPC's activities mesh with existing regulatory
and selfregulatory organizations and procedures (see
pp. 68 and 1819 of the report).
11. This section draws from SEC Annual Report 1974, p. 59,
and SEC Annual Report 1973, p. 57.
12. SIPC Third Annual Report 1973, p. 7.
13. The New York Stock Exchange, Early Warning and Surveil
lance Procedures, Unpublished, July 1974, p. 3.
14. See the Subsection on the Deterministic Approach on
Page 21.
15. SIPC Third Annual Report 1973, p. 7.
16. E.J. Dervan, Jr. and E.I. Altman, The Failed Firms
Project, National Association of Securities Dealers,
Inc., Unpublished, August 31, 1973.
17. Thanks are due to J.L. Peterman of the SIPC who
brought to the author's attention the existence of such
a valuable study and provided a copy.
18. R.F. Smith and A.H. Winakor, Changes in the Financial
Structure of Unsuccessful Corporations, University of
Illinois: Bureau of Business Research, 1935.
19. P.J. Fitzpatrick, Symptoms of Industrial Failures,
Washington: Catholic University of American Press,
1931.
20. C. Merwin, Financing Small Corporations, New York:
Bureau of Economic Research, 1942.
21. M. Tamari, "Financial Ratios as a Means of Forecasting
Bankruptcy," Management International Review, Vol. IV
(1966), pp. 1521.
22. W.H. Beaver, "Financial Ratios as Predictors of Failure,"
Empirical Research in Accounting, Selected Studies, 1966,
Institute of Professional Accounting (January 1967), pp.
71111.
23. E.I. Altman, "Financial Ratios, Discriminant Analysis
and Prediction of Corporate Bankruptcy," Journal of
Finance, Vol. XXIII (1968), pp. 589609.
24. See, for example, R.O. Edmister, "An Empirical Test of
Financial Ratio Analysis for Small Business Failure
Prediction." Journal of Financial and Quantitative
Analysis (March 1972 pp. 14771493; R.R. Dince and
J.C. Fortscn, "Tne Use of Discriminant Analysis to
Predict the Capital Adequacy of Commercial Banks,"
Journal of Bank Research (Spring, 1972), pp. 5462; and
J.S. Trieschmann and G.E. Pinches, A Multivariate
Model for Predicting Financially Distressed Property
Liability Insurance Companies, unpublished.
CHAPTER II
THE GENERAL ANALYTIC FRAMEWORK
Introduction
This chapter presents an analytic framework for
identifying potential determinants of a firm's financial
distress. The analytic framework is presented here in a
general form, potentially applicable to all profit
oriented private organizations. Its application to secu
rities brokerage/dealer firms will be discussed in Chapter
IV.
The analysis begins with the hypothesis that the
probability of financial distress of a firm is a function
of net cashflow per dollar of assets, risk exposure, and
mismanagement of the firm. By decomposing the cashflow
equation, internal and external factors affecting revenue
and expenses are examined in some detail. Risk is then
defined, and after discussion of several measures of risk,
risk factors associated with different facets of a firm
are explained. Finally, a brief discussion is given of
factors comprising the "mismanagement" element in the
hypothesized function.
The analysis results in a widely expanded general
function that contains a large number of factors, each of
which possibly affects the probability of a firm's finan
cial distress. What is needed, however, is a parsimonious
function which contains only a few important factors with
specific relations among them. Such a desired specific
function may be then derived from the expanded general
function by utilizing the multivariate discriminant analy
sis described in Chapter III.
The Hypothesis
Every industry has its own problems and peculiari
ties. So does each firm within each industry. Machine
tool companies suffer from significant swings in scales,
steel companies undergo labor strikes, and oil companies
are highly dependent on geological discoveries. Food
companies demonstrate sales stabilities, public utilities
are regulated by governments, and business machine com
panies are characterized by growth.
In such diverse surroundings any specific analytic
framework for identifying potential determinants of a
firm's financial distress which fits a certain industry
will not be of much help to another industry. What is
needed, then, is a general analytic framework which may
be applicable to any profitoriented private organization,
large or small, manufacturing or servicing.
The effort undertaken in this chapter hopefully
will meet such a need. It is necessary, however, to treat
most factors as being equally important. This is due to
the fact that a factor which is relatively less important
in one industry may be of greater importance in another
one. Such a treatment, however, may not be harmful, be
cause, in applying the analytic framework to a specific
industry, appropriate weights will be assigned to the rela
ted factors.
To start the analysis, it is hypothesized that the
probability of financial distress of a firm at time t,
P(FDt), is a function of the net cashflow per dollar of
assets at time t (CASHFAt), the extent of the risk assumed
by and imposed upon the firm (RISK), and the mismanagement
of the firm (MISMGT).
P(FDt) = f (CASHFAt, RISK, MISMGT) (1)
The inclusion of these elements in equation (1) is
the consequence of the following line of reasoning;
With respect to the net cashflow per dollar of
assets, it is both an index of the profitability of the
firm and that of the availability of funds to meet the
fixed obligations in time. This index, which has been
relieved from size effect of the firm, can in fact be
divided into two parts, the first part being the rate of
return on equity capital and the second part being a fund
ratio added to or deducted from this rate due to the changes
in assets (including decrease in the amount of fixed assets
and increase in cash due to the depreciation allowances or
assets sales) and liabilities.
The rate of return on equity capital should have a
direct effect on the probability of financial distress. If
the firm has a high rate of return (compared with either
its cost of capital or the market riskfree rate of inter
est), it has a lower probability to become financially
distressed than otherwise, other things held constant. The
reason is that the high rate of return indicates that the
firm, in addition to wholly or partially satisfying the
expectations of the equity owners, depending on the height
of the rate and the level of the risk, has been able to
meet the requirements of the owners of debt and preferred
capital, thus alleviating situations which threaten the
existence of the firm. By contrast, a low rate of return,
and in extreme, a negative rate of return, signifies im
pending inability of the firm to meet its obligations and/or
probable depletion of the equity capital in shock absorbing
the adverse situations. Such situations may develop very
rapidly in some industries such as the securities busi
ness where the value of most assets is subject to the
changes in market and business activities and the ratio
of equity to debt capital is as high as 1 to 15. There
fore, the lower is the rate of return on equity capital,
the higher is the probability of financial distress, other
things being equal.
Although the rate of return on equity capital
indicates the profitability of the firm, there have been
occasions that the profitable firm (as indicated by this
rate) has been unable to meet its fixed obligations in
time, due to unavailability of sufficient cash in its
account. On some other occasions the reverse has been
observed. That is, a temporarily unprofitable firm which
has had access to sufficient funds has escaped from finan
cial distress.
Therefore, it is expected that cashflow per dollar
of assets has a higher impact on the probability of finan
cial distress than the corresponding rate of return on
equity.1
Regarding the risk element in equation (1), it is
a complex factor which reflects the overall risk exposure
of the firm. It is a combination of the risk factors
assumed deliberately by the firm itself and those imposed
upon it by external forces. The factors range from those
associated with liquidity, borrowing and lending, to
business activities, and to changes in market and the
economy in general. This element will be discussed later
in more detail. But, at this point, it is necessary to
present the reason of its inclusion in equation (1).
By any relevant definition of risk in this context,
one would expect to observe some unfavorable, as well as
favorable, impact of risk factors on the performance of the
firm. Thus, it seems to be relevant to relate the prob
ability of financial distress to risk. To illustrate the
point, consider two such risk factors, i.e., variance of
the rate of return and financial leverage.
Since the income of a firm is subject to the changes
in the level of general business activities (as well as to
many other factors), its rate of return is a random vari
able. Therefore, the rate of return may vary from a low
point to a high point over time. If the range of this
variation is very wide, other things being equal, there is
a higher probability of facing financial distress, as a
result of several consecutive occurrences of low rates,
than otherwise.
Similar reasoning applies for the dependency of
the probability of financial distress upon the financial
leverage. The owners' equity capital acts as a shock
absorber in a car. If the equity capital constitutes a
larger part of the total capital of the firm, there is a
broader base to support the burden of adverse situations.
That is, when several consecutive losses occur, the fixed
charges of debt capital and other liabilities can be paid
through digging into the equity capital. But, if the
equity portion of the capital is not sufficiently large,
such obligations cannot be met adequately.
With respect to the mismanagement element (MISMGT)
in equation (1), the word may not suitably explain the
purpose. It was, however, intended to embrace and describe
all factors, such as fraud and poor books and records,
which do not fall within the domain of the other two ele
ments in the equation but have some impact on the prob
ability of the firm's financial distress.
In equation (1) the relationships between the
independent and dependent variables were considered only
in a general form. It is, however, desirable to obtain
at least the approximation of the true relations in a
specific form. Since each of the above independent vari
ables is under the influence of numerous factors with some
possible complex interactions among them, error in com
puting the approximation through their direct measurements
(which is only possible for CASHFA) may highly be magni
fied. It seems, therefore, more appropriate to break down
the general relations, as far as possible, into several
simpler ones. This may be done by expanding equation (1)
in terms of all factors which seemingly underly the
independent variables. Then, measuring each of these
factors and statistically analyzing their joint effects
on the probability of financial distress may lead to a
better result.
Net CashFlow per Dollar of Assets
Starting the expansion of equation (1) by decom
posing net cashflow per dollar of assets at time t, it
is known that, by definition, CASHFAt is the ratio of the
available net cashflow at time t, ACFt, to corresponding
assets, ASSETt. Thus:
CASHFAt = ACFt/ASSETt (2)
Available cashflow itself, however, is determined
by several other factors.2 To be more exact:
ACFt = (NIAS + DEP) GASSET + GLIAB (3)
where NIAS = Net income available to shareholders,
DEP = Accounting depreciation and depletion
allowances put aside from operating
income to compensate the part of the
assets used up in generating the
revenue,
GASSET=Growth in assets, and
GLIAB = Growth in liabilities.
From equation (3), it is seen that available cash
flow is directly related to net income available for
shareholders as well as to depreciation allowances. An
increase in the assets, however, would have a negative
effect, whereas a decrease in the assets has a positive
effect on the available cashflow. In contrast, an in
crease in the liabilities would have a positive effect,
while a decrease in the liabilities has a negative effect
on the available cashflow.
In order to be able to continue the analysis, first
equation (2) is restated in the following general form:
CASHFAt = g(NIAS, EQC, GASSET, GLIAB, DEP) (4)
Subsequently observe that net income is a function of
revenue (REV) and expenses (EXP).
NIAS = h(REV, EXP) (5)
Revenue
Revenue is the bloodstream of the firm. It is,
however, subject to numerous internal and external factors.
These factors may be grouped accordingly under INTREV and
EXTREV variables, respectively, as in equation (6).
REV = i(INTREV, EXTREV) (6)
Internal factors. The elements of this group
consist of factors which are more or less under control of
the firm. When managed appropriately, these factors have
positive effects on revenue, otherwise, their impacts are
negative. Such factors include the following:
Number of product lines (NPL) As number of
product or service lines increases, even though they may
not be complementary, they reach a broader market, thus
causing an increase in sales and presumably more revenue,
and vice versa.
Number of places it occupies (NOPLOC). An explana
tion similar to that made for the number of product lines
applies to the number of places that a firm occupies.
Number of years in the business (NYR). The number
of years that a firm has been in the business should play
a role in its profitability. Nothing can replace the
experience obtained in practice. An established firm is
more likely to be a profitable one. Its mere lasting
existence is the evidence of its being profitable. The
relationship, however, may not be linear.
Research and development (R&D). In some industries,
research and development programs play a vital role in reve
nue generation. More importantly, in many instances, the
mere existence of a firm in some industries depends on the
extent to which it focuses on R&D acitivites.
Number of patents (NOPAT). Yhe number of patents
and franchises that a firm has in its possession likely
affects revenue. These factors give a monopolistic power
to the firm having them, and leads to potential exploita
tion.
Cash availability (CASHAV). Although availability
of cash is an absolute necessity to run most types of
businesses, and even in some instances taking advantage
of possible opportunities depends on this factor, excess
cash bears an opportunity cost. The income which could
be earned by putting the excess cash into a productive use,
is the associated opportunity cost. Leaving the excess
cash idle does not contribute to the profit; rather it de
ducts from the potential revenue.
Securities marketability (SECMKT). Marketability
of some securities is a feature which provides the pos
sibility of replacing a part of the required cash, which
brings no direct yield, with securities such as US
government bonds, which bring some additional yield at
practically no extra risk. If, however, the amount of this
item is in excess of what is necessary, in relation to cash
and other assets, the cost of unripe selling to obtain cash,
or the differential yield forgone by not putting the excess
amount of securities into a more productive investment, is
simply a deduction from actual or potential revenue,
respectively.
Credit easiness (CREDIT). A mild credit extension
may improve sales, but overeasiness may bring about such
large baddebt accounts and costs that it may actually
overoffset the increment revenue.
Inventory movement (INVENT). Similarly, a balanced
inventory may facilitate sales, whereas too much investment
in this item deducts from potential profit.
While the effects of CASH and SECMKT on REV are
negative after some point, the relations of CREDIT and
INVENT to REVare positive over a range.
Promotional efforts (PROMOT). Another factor which
is under control of the firm and has some effect on revenue
is promotional efforts. The relationship, however, is a
fuzzy one. The number of psychological and environmental
factors which act and react simultaneously is so many that it
makes clearcut inferences too difficult, if not impossible.
Yet, the large promotional expense incurred every year in
all industries is an indication of its positive effect in
practice.
Accounts payable (ACCPAY). Accounts payable nor
mally contribute to revenue. This is due to the poten
tially better use of the funds relieved by such accounts.
Usage of accounts payable, however, is not without limi
tations. It can reach a point beyond which its associated
costs overoffsets its potential benefits.
Debt capital (DBTCAP). In the world of existing
taxes, the tax treatment of interest expenses encourages
greater use of debt capital. If the firm is operating
profitably and the rate of interest on its debts remains
constant or rises slightly as the debt is increased, the
higher is the leverage, the higher becomes the profit.
Even under the above conditions, however, debt may not
always be profitable. Protectivecovenant constraints
such as aminimumworking capital, a percentage limitation
of longterm debt to working capital, and capital expendi
ture restrictions, which are imposed by debt owners, may
have a significant influence on the firm's profits. Sev
eral theories have been advanced as to the optimal level
of debt.4 The controversy about them has not, however,
been solved yet.
Debt mix. The optimal mix of shortterm (SHRTDT)
in relation to longterm (LONGDT) financing is also of
considerable importance. This is due to both interest
rate differential and other factors such as flexibility,
timing and etc.
There are of course some other internal factors,
but their effects on REV, individually and perhaps col
lectively, is likely to be minor, relative to the
previouslymentioned factors. Therefore, the above dis
cussion may be summarized in the following equation:
INTREV = j(NPL, NOPLOC, NYR, R&D, NOPAT, CASHAV,
SECMKT, CREDIT, INVENT, PROMOTE, ACCPAY,
DBTCAP, SHRTDT, LONGDT) (7)
The relative importance of each of the above
factors in the profitability of a firm differs from one
industry to another and within subdivisions of each
industry. Therefore, the explanatory variables for
describing the impacts of such factors on the probability
of financial distress of the firm should fit to specific
industry and its subdivision to which the firm belongs.
External factors. The elements of this group
consist of factors over which the firm does not have any
control, yet they have some effects on the revenue of the
firm. Among these factors, general business activity has
a special place.
General business activity (GBA). General business
and economic activity is subject to recurrent but non
periodic fluctuations, called business cycles, over a long
period of time. These fluctuations occur in aggregate
variables such as prices, GNP, investments, employment,
and income. The variables move at about the same time and
in the same direction, but their rates of change are dif
ferent. Although the effect of these fluctuations on a
firm differs from one industry to another, no firm remains
immune from their impact. In general, over any period of
time, the revenue of a firm is a function of the level of
general business activity.
Interest rate (INTRR). The market prime rate of
interest has also some effects on the revenue of the firm.
This, of course, relates to the lending position of the
firm. For a firm which has a borrowing position the ef
fect of interest rates on its performance comes through
expenses. If the firm is both lender and borrower at the
same time, the effect cones through both revenue and
expense sides. The net effect, then, may be either posi
tive or negative, depending upon the net lending or
borrowing position of the firm.
Population (POPUL). Population is another impor
tant external factor. Not only the number of inhabitants
in the related area matters; the population mix has also
bearing on the revenue. A firm which produces youth
products is more profitable in an area where the young
constitute a higher percentage of the population than in
other areas, other things being equal.
Per capital income (PCINC). Per capital income of
the inhabitants in the area where the firm runs its busi
ness is another determinant of the firm's revenue. Per
capital income, which is an index of the purchasing power
of the people residing in an area, differs from one region
to another. The variation is due to difference in the mix
of industries and in the prevalence of large cities, where
wages are higher, than in small towns and rural areas.
Competition (COMP). Competition, if it exists,
diffuses the potential market power of the firm. Then,
the price which it puts on its products and, as a con
sequence, its revenue, is governed by the market. In fact,
the effect of competition, which is the foundation of
capitalism, is to give rewards, in the form of profits,
to the firms which are able to survive, and penalties,
in the form of losses or bankruptcy, to those which are
not.
Competition, in a broad sense, embraces all
factors of productionland (which includes all natural
resources), capital, labor, and cntrcpreneurshin. In
contrast to the firm's desire to have a monopoly position
in selling products and monopsony power in buying supplies
and services of labor, land, and capital, the actual situa
tion which it faces may be exactly the opposite one. Ex
istence of labor unions, monopoly in some raw materials,
branch banking systems in some states, and exclusively
governmental purchases are commonplace facts which are faced
by many firms. All these factors have bearings on the
firm's revenue. Indeed, wherever appropriate, COMP factor
should be broken down into the above four dimensions (cor
responding to the factors of production) in evaluating the
firm's position with respect to competition.
Regulations (REGUL). Rules and regulations are
imposed by either governmental regulatory agencies or
selfregulatory organizations. They are intended to regu
late various aspects of the activities of firms in dif
ferent industries. These rules and regulations signifi
cantly affect the firm's profit. Rules and regulations
governing the rates of public utility companies and rail
roads can be mentioned as examples. The rules and regula
tions sometimes go to extremes. In banking, for example,
the regulations require a relatively low marginal rate of
return on capital. Each industry has a set of regulations
relevant to itself. Therefore, in evaluating the position
of the firm in a certain industry with respect to the
prevalent rules and regulations in that industry, factor
REGUL should be broken down into an appropriate number of
dimensions.
The above discussion can be mathematically ex
pressed:
EXTREV = k(GBA, INTRR, POPUL, PCINC, COMP,
REGUL) (8)
Expenses
The expense element (EXP) in equation (5) may also
be broken down into the part originated by internal fac
tors (INTEXP), and that caused by external factors (EXTEXP).
Thus:
EXP = 1(INTEXP, EXTEXP) (9)
Internal factors. Most of the internal factors
affecting the overall expenses of a firm can be categorized
into selling expenses (SELEXP), general expenses (GENEX),
and administrative expenses (ADMEXP). Although all firms
have these three types of expenses, the relative importance
of each type differs from one industry to another. Even in
the same industry the percentage of each type differs
among firms. This very difference may differentiate between
firms operating optimally or near optimally and those which
are run suboptimally. In fact, there can always be found
one or more expense items which are conspicuous in a
specific industry. As an example, the communication
expenses of firms in the securities industry is prominent.
Evaluation of a firm's position with respect to such an
item may reveal some characteristics of the firm's per
formance. To obtain such items, it is useful to break down
the above three types of expenses into more detailed com
ponents peculiar to each industry. In any event, deprecia
tion expenses, which may belong to one or more of the above
categories, should be singled out for calculating the net
cashflow, as stated in equation (3). To summarize the
above discussion:
INTEXP = m(SELEXP, GENEXP, ADMEXP) (10)
External factors. Although a little fuzzy, all
external factors which affect a firm's profit other than
through the revenue side fall in this category. One of
the most important elements in such factors is taxes.
Taxes (TAX). Aside from the economic purposes
of taxes, i.e., stabilization of economy, redistribution
of wealth, and reallocation of resources, businessmen
look at them as a compulsory payment to government. Indeed,
they reduce the firm's profit by a large percentage, called
tax rate. The legal tax rate varies over a period of time.
Whenever the rate is lower, the income remaining for share
holders is larger, other things held constant, and vice
versa.
Interest rate (INTRR). The role of the interest
rate in the performance of the firm was discussed in the
revenue section. To avoid exclusion of such an important
factor from among the determinants of expenses, however,
it is reintroduced in this section as well. In fact,
since debt usually constitutes a main part of the capital
structure of most firms, the impact of the interest rate
coming through the expense side seems to be more important
for most firms. In any event, the higher is the interest
rate, the larger is the interest charge as part of the
firm's total expenses, and vice versa.
Environmental factors (ENVIRN). Other factors
such as antipollution expenses, benevolence expenses, and
the like may be grouped under environmental factors. Firms
manufacturing defense products may incur opportunity costs.
These costs, which are equal to the differential profit
forgone by not employing the resources in a more productive
way should rightly be considered as expenses. In summary:
EXTEXP = n(TAX, INTRR, ENVIRN) (11)
Risk
It is now necessary to refer back to equation (1)
and make an effort to identify the components of the
overall risk of the firm which may play important roles
in determining the probability of financial distress.
Before going into details, however, a digression on de
finition and measures of risk seems appropriate.
Definition
Risk has been defined differently in various
sources, both in view of context and in regard to content.
In Webster's Dictionary, it has been defined as the pos
sibility of loss, injury, disadvantage, or destruction.6
Sloan and Zurcher defined risk as the possibility of loss
from some particular hazard, as fire risk, war risk,
credit risk.7
In the McGrawHill Dictionary, risk has been de
fined as the exposure of an investor to the possibility of
gain or loss of money. Machol and Lerner defined risk as
the cumulative probability of the return falling below some
9
level of ruin. And, in the stateofnature approach to
evaluating risk, it is implicitly defined as the possibility
that the desired, rather than the expected, return will not
be achieved.10
Although all of these definitions suit their own
places, there is need for some definition which fits the
context under study. For that purpose, risk is here
defined as the possibility that actual outcome of a
random variable will deviate from that which is expected.
The random variable can, of course, represent such di
verse factors as cashflow, rate of return, expenses,
bad debts, or even the business activity level.
Common YMeasjres of Risk
Having risk thus defined, then, a measure of risk
is a means by which the extent of risk is ascertained.
There is, however, no single measure of risk which can well
explain the actual risk associated with any factor under
consideration. Facing this difficulty, the authors in the
field of economic and business, and especially in the por
folio literature, have adopted various means as measures
of risk, each with its own virtues and limitations. So
far, the most popular measure of risk has been variance
(or standard deviation) of a random variable around its
mean.
Variance or standard deviation of a single
variable. Tobin, in adopting the standard deviation as a
measure of risk, although in a portfolio context, made the
following statement in his celebrated paper on "Liquidity
Preference as Behavior Towards Risk:"12
The risk attached to a portfolio is to
be measured by the standard deviation of R,
OR. The standard deviation is a measure
of the dispersion of possible returns
around the mean value uR. A high standard
deviation means, speaking roughly, high
probability of large deviations from uR,
both positive and negative. A low standard
deviation means low probability of large
deviations from u; in the extreme case,
a zero standard deviation would indicate
certainty of receiving the return 4R. Thus
a highoR portfolio offers the investor the
chance of large capital gains at the price
of equivalent chances of large capital
losses. A lowR portfolio protects the
investor from capital loss, and likewise
gives him little prospect of unusual gains.
Although it is intuitively clear that the
risk of a portfolio is to be identified with
the dispersion of possible returns, the
standard deviation is neither the sole mea
sure of dispersion nor the obviously most
relevant measure....
The variance of a random variable is the expected
value of the squared deviations from the mean:
n 2
V(R) = Z (R. R) P. (12)
i=l
where V(R) is the variance of the random variable R, n is
the number of possible outcomes, R. is the ith possible
outcome, R is the expected value of all possible outcomes,
and Pi is the probability or the relative frequency,
whichever is appropriate, of occurrence of the ith event.
The standard deviation is simply the square root
of the variance:
oR = \V(R) (13)
The variance obtained by equation (12) is in an
absolute term. For comparing the variability of different
distributions, however, there is need for a measure of
relative dispersion. This need is met by forming the
coefficient of variation as expressed below:
CV(R) = V(R) / R (14)
Semivariance. Markowitz, who pioneered the modern
theory on portfolio selection, considered the standard
deviation, the semivariance, the expected value of loss,
the expected absolute deviation, the probability of loss,
and the maximum loss as candidate measures of risk.1
Among these measures, however, he concentrated more on the
14
first two measures. One of the reasons for considering
semivariance as a candidate measure of risk is its ability
to measure the skewness of the probability distribution of
the random variable, say, return.
Unlike the variance, which ignores the shape of the
probability distribution, semivariance takes into account
the downside or upside tendency of the probability distri
bution. Since most economic units associate risk with the
possibility of loss, downside fluctuations in return, and
not dispersion per se, have considerable importance. A
distribution skewed to the left, as indicated by semi
variance, would involve more chance for low or negative
returns than a distribution skewed to the right, all other
things being the same.
Semivariance can be expressed in the same way as
variance was stated in equation (12) with the exception
that n and i now represent, respectively, only total num
ber of those outcomes and the ith outcome whose values
are smaller (or greater, according to one's purpose) than
R, the overall mean value:
n
SV(R) = E (Ri R)2 P, (15)
i=l
Markowitz suggested the ratio of V(R)/23V(R) as a
15
measure of relative skewness. For symmetric distributions
this ratio equals one, but it is greater or less than one if
distribution skewed to the right or left, respectively.
Third moment. The measure of skewness is not limi
ted, of course, to semivariance. Skewness, in absolute or
relative sense, may be measured also by the third moment of
the probability distribution of the random variable. The
third moment about the mean may be expressed by:
n
m3 = (Ri R) P (16)
S i=l1
which is similar to equation (12), with the exception that
the deviations from mean, the term (R R), is now cubed
rather than squared. Cubing the term does not change the
sign of the deviations, but it magnifies the extreme devia
tions, thus making the risk measurement possible. Against
this advantage, the third moment has the same disadvantage
that other moments have, i.e., it is greatly affected by
a few extreme deviations. In fact, since the absolute
amount of skewness depends on the dispersion of data, a
given value of m3 could be the result of slight asymmetry
together with large dispersion, or great asymmetry with
small dispersion. Its usefulness cannot be doubted, how
ever, especially when it is combined with other moments.16
Arditti, in an empirical study, "Risk and the
Required Return on Equity,"7 employed second and third
moments along with some other risk factors, and concluded,
among other things, that the market required return had
direct relation with third moment, as was expected from a
measure of risk.
Third moment may become more useful in comparing
among several distributions if it is expressed in a rela
tive term. This is done by dividing the moment by the cube
of the standard deviation of the random variable and is
called standardized third moment:
a3 = m 3/o (17)
Rate of growth. It was discussed in previous para
graphs that third moment about the mean is greatly affected
by a few extreme deviations. In fact, the sign of the
moment may be largely attributable to a single large nega
tive (or positive, as appropriate) deviation. Further
more, what a measure of skewness shows is the existence of
asymmetry in the frequency distribution. It can not iden
tify how this asymmetry occurred, i.e., whether it was due
to regular or irregular changes in the values of the random
variable. It is obvious that a steady trend in one direc
tion removes most of the uncertainty (or risk) associated
with predicting a certain value for the variable, whereas
irregular fluctuations make such prediction very difficult.
In order to detect the regular changes in one direc
tion, the concept of rate of growth, whether positivee or
negative, can be utilized. There are two ways to get the
rate of growth: the first one is through applying a geo
metric mean method. The second is through utilizing the
regression analysis concept. The latter method, which is
discussed below, seems to have more advantage than the
former. This is due to its ability to provide some addi
tional information, such as R the coefficient of deter
mination of the regression line.
The model which is discussed here has originated
from that which is used for computing the continuously
19
compounded rate of interest.9 It is expressed as:
V = Aert (18)
where Vt is the value of the variable after t years, A is
the initial value, e = 2.71828, r is the instantaneous rate
of growth, and t is total years under consideration.
To calculate r, first take the natural log from
both sides of equation (18):
In Vt = In A + rt (19)
Then, by utilizing equation (19), regress In Vt
2
against t, and obtain r as well as In A and R the coef
ficient of determination, of the regression line. R
which is bounded between zero and one, is a useful measure
of the goodness of fit of the regression line. The closer
is R2 to one, the better is the fit.20
In analyzing risk, it seems more useful to use R
along with r, the rate of growth, and both R2 and r in con
junction with other measures of risk such as variance and
semivariance, because each of these measures captures only
a portion of total risk associated with the variation of the
random variable.
Confidence level. Baumol, who was also concerned
about the downside fluctuations of a random variable, such
21
as return, proposed still another measure of risk.2 His
measure of risk resembles a lower bound or confidence
limit on the expected outcome and is expressed by:
L = R kaR (20)
where L is the suggested measure of risk, R is the mean of
the random variable, k is a positive parameter that re
presents the number of standard deviations below the mean
value, and oR is the standard deviation. Unlike the
previous measures of risk, however, this one is not in
dependent of the evaluator's preferences. That is,
before any value can be found for L, a value should be
assigned by the evaluator to k. The dependence of this
measure of risk on the evaluator's utility function detracts
from its usefulness for the purpose, here.
Covariance. One of the useful measures of risk is
covariance between two random variables, say, the firm's
net income and the changes in business activity. It is a
measure of the extent to which the value of the two vari
ables tend to move up and down together. It can be ex
pressed by the following equation:
n
COV(R,Q) = Z (Ri R)(Qi Q) P. (21)
i=l 1
where R and Q are two random variables with expected values
of R and Q, respectively, R. and Q. are the ith possible
outcomes of R and Q, n is the total possible outcomes of
each variable, and P. is the joint probability that R. and
Q. will occur simultaneously. As is seen from equation
(21), when both Ri and Q. are above or below their related
means for most of the times, the covariance becomes posi
tive, whereas, when one of them is above its mean more
frequently while the other is below its mean for most of
the times, the covariance becomes negative.
Correlation coefficient. Although the covariance
formula is a good mean to explain the relative movements
of the random variables, its absolute value is difficult
to interpret. A better measure which is directly related
to covariance and can be easily interpreted is correlation
coefficient. It may be expressed by:
COR(R,Q) = COV(R,Q) (22)
CR "' Q
If the two random variables move up and down in
perfect unison, the correlation coefficient is one (they
are perfectly correlated). A correlation coefficient of
zero, on the other hand, indicates that their movements
are completely independent from each other (they are uncor
related). If the two variables move in reverse direction
in perfect unison, the correlation coefficient is minus
one (I). Any other imperfect correlation lies somewhere
between +1 and 1.
Variance of a group of variables. In the port
folio context, where the return on portfolio is dependent
upon the returns on the constituent securities, it has
long been known that the variance of the portfolio return
is a function of the variances of the comprising securities
as well as the correlation coefficients between every pair
of these securities. More exactly:
n n
V(P) = E E X. X. COR(i,j) o. o. (23)
i=1 j= 1 3
where n is total number of securities in the portfolio, X.
and Xj are the percentages of funds invested in securities
i and j, COR(i,j) is the correlation coefficient between
these securities, and oi and .q are standard deviations of
securities i and j.
The concept may be extended to contexts other than
portfolio, where determination of a variation of a single
random variable, such as operating revenue of a firm, in
terms of variations of several other random variables, for
example, revenue from each line of product, is also
desirable. All elements of equation (23) can then be
easily redefined to fit such cases, except for X. and X.
1 3
which should be interpreted in a slightly different way.
In retrospect, when past series data are used, one may
interpret X. and X. as the shares of random variables i
and j, respectively, in the overall picture. For example,
if talking about the variation in the company's revenue,
one may interpret X. and X. as average proportions of the
total revenue which are contributed by product lines i and
j, respectively.
Computation of the collective variance by equation
(23) is easy when the number of constituent random vari
ables is small. For a large number of such variables the
computation in this way becomes cumbersome. The diagonal
22
model presented by Sharpe2 has solved this problem. In
the portfolio context, the following steps should be taken
to calculate the variance of portfolio:
First, a suitable market index such as GNP is
selected and then by using the following linear equation
for each security, individually, various returns of each
security are regressed against the corresponding value
of the index:
R.. = A. + B.I. + e.
13 1 1 3 1
where R. is the jth observation of the randcr variable
i, I. is the jth observation of the index, A., B. and e.
are yintercept, slope, and random error related to the
regression line of the variable i, respectively.
Then, by using the results obtained from this
step in the following equation, variance of the portfolio
is calculated:
n 2 2 2 2
V(P) = Xi i + Xn+ln+l (25)
i=1 1 n+ n+
where n is the total number of securities in the portfolio,
X. is the percentage of funds invested in security i, a.
is the standard deviation of e. in (24), a+l is the stan
dard deviation of the index, and Xn+ is a "pseudo" secu
rity defined as
n
X+1 = E X. B. (26)
n+l = i (26)
i=l
The expected value of the portfolio, which is
needed for computing the coefficient of variation, is cal
culated by the following equation:
n
E(P) = E XiA. + X A1 (27)
i=l 1 i n+l n+l
i=l
where A n+ is the expected value of the index. The co
efficient of variation is:
CV(P) = V(P)/E(P) (28)
Again, these formulations may be applicable to
contexts other than portfolio, as will be shown later.
The probability of disaster level. In the economic
world where the outcomes of most activities are subject to
uncertainty there is always a range of outcomes for each
activity over which the economic unit is drastically af
fected. The upper bound of the range where the vulner
ability to the chance event just begins may be referred to
as disaster level.
Examples of disaster level can be found abundantly
in the firm's various facets. If the firm's net cashflow
drops below a certain point, for example, the firm may
face a hard time. When the sales shrink to a specific
level, it may experience large losses. Or, if the total
asset value declines to a level which is less than the
total liabilities, the firm becomes insolvent.
If a means can be found to show the chance of a
random variable being at the disaster level or below,
then it can be used as an appropriate measure of risk for
dealing with such cases. Fortunately, following Roy,2
one can apply Tchebycheff's inequality in obtaining such
probability. This, of course, assumes possession of in
formation about the mean and variance as well as the
disaster level of the random variable. The Tchebvcheff
inequality may be expressed as the following in this
situation:
P( R El E d) u2/(E d)2 (29)
or,
P(R d) $ 22/(E d)2 (30)
where P is the probability operand and R is a random vari
able whose expected value, variance, and disaster level
are, respectively, E, o and d.
One of the advantages of the Tchebycheff inequality
is that it does not require the probability distribution of
the random variable to be normal. There is, however, a
difficulty in applying equation (30) in the context used
here. This difficulty is due to the unavailability of
disaster level (d) in any of the related cases. For
tunately, however, a designated d need not be very exact,
because firms usually are able to manage their affairs over a
range of each variable. And this range may not be rela
tively too narrow.
A sufficiently close approximation of d may be
obtained by averaging the variable over all available
firms of similar size and nature which have actually ex
perienced the financial distress. The implied assumption
here is, of course, the dependence of the financial dis
tress upon the disaster level of the variable.
If data unavailability precludes grouping the
firms of similar size and nature for this purpose, one may
resort to some appropriate ratio of the variable. The
ratio could first be averaged over all available firms
which have experienced financial distress and the average
could then be transformed to an absolute value for each
firm by a simple calculation. As an example, suppose that
the average disaster ratio of sales to assets turns out to
be 0.5, then the disaster level of sales for a firm having
a total assets of one million dollars can be obtained by
multiplying one million by 0.5 which equals $500,000.
Risk Factors
Risk associated with cashflow (RSKCF). The im
portance of sufficient cash inflow in meeting the firm's
obligations cannot be overstated. Since cashflow is a
random variable, at any point of time, it may take some
value between two possible extreme points relevant to the
firm. But, at some level of cash inflow which is of
course different for each firm, the firm may not be able
to run the business smoothly. The possibility that the
cash inflow falls below this disaster level, then,
constitutes a risk factor which may play an important
role in determining the possible financial distress of
the firm.
If it is possible to estimate such a disaster
level for each firm, it may then be possible to estimate
the probability of its occurrence by resorting to the
Tchebycheff's inequality, as reproduced here.
2
P(ACF : d) oCF/(ACF d) (31)
where P is the probability operand, ACF is the available
cashflow variable whose mean value and variance are ACF
and o2 respectively, and d is the disaster level for
ACE'
available cashflow.
To find the disaster level of each firm it is
highly desirable to have access to data of as many failed
firms as possible which have the same characteristics,
particularly with respect to size and business nature, as
those of the firm under consideration. But, since finding
such firms is not always possible, a reasonable approxima
tion may be obtained by first averaging the ratio of ACF
to total assets of all failed firms in the same industry,
and then transforming this average ratio to an absolute
value through its multiplication by total assets of the
firm.
To find the probability that the cashflow of the
firm may fall below this calculated disaster level, it is
necessary to calculate the mean value and variance of
cashflows of the firm over a recent period of time to be
used, along with the calculated disaster level, in the
Tchebycheff's inequality. The probability obtained in this
way :s a conservative one, yet it may be a useful approxi
mation to risk assumed by the firm with respect to its
cashflow procedure.
Donaldson has proposed that the main concern of a
company is to see whether the cash balances fall below
zero or not, and to determine such possibility, he suggests
examining the cashflows under recession condition.24
If that is the case, ACF in equation (31) may be
redefined to be net cashflow. Then, the disaster level,
d, changes to zero, and the probability that the net cash
flow falls below zero becomes equal to or less than the
ratio of the coefficient of variation to the mean value of
the net cashflow:
P(ACF 5 0) 5 CV(ACF)/ACV (32)
The preference of (31) to (32), or the other way
around, as a determinant of the firm's risk exposure
depends, of course, on the results of empirical checks.
Risk associated with revenue (RSKREV). Risk
associated with revenue is the possibility that actual
revenue will deviate from that which is expected.
Following the discussion above of the measure of
risk section with respect to the properties of the variance
of a random variable, one measures the risk associated with
total revenue by finding the variance of the values taken
by revenue variable over the period under consideration.
Equation (33) may be used for such purpose:
n
1 2
V(REV) (REV. REV) (33)
n i
i=l
where V(REV) is variance of revenue, n is total number of
observations on revenues, REV. is the ith observation, and
1
REV is the mean value of total observations on revenue.
This equation provides an absolute value for the vari
ability of revenue. Its relative value, however, which
is obtained by coefficient of variation, may be more useful
in discriminant analysis:
CV(REV) = V(REV)/REV (34)
As was discussed in previous section, however, the
variance alone does not capture all the risk associated
with revenue. If the frequency distribution of revenue is
skewed to the left, there is higher possibility of loss,
and as a consequence, higher possibility of financial
distress. It may, therefore, be helpful to supplement the
coefficient of variation with a measure of relative skew
ness. For this purpose it is possible to use either semi
variance or third moment of revenue. But to choose between
them by detecting the discriminatory power of each of them,
it seems appropriate to use both in this analysis. The
semivariance of revenue may be expressed by:
1 n
SV(REV) (REV. REV) (35)
n i=l I
where n represents the number of values below the mean,
REV, and REV. is the ith observation of such values.
The relative skewness, using semivariance, is
thus obtained by the following ratio:
CSV(REV) = V(REV)/2SV(REV) (36)
The third moment about the mean of the frequency
distribution of revenue may be expressed by:
1 3
m (REV. REV) (37)
3 n i=l I
where righthand symbols are the same as those in equation
(33).
The standardized third moment which indicates the
relative skewness of revenue frequency distribution is
then obtained by:
3
a3 = m3 / V(REV)2 (38)
For the reasons explained in the measures of risk
section, it may be useful to consider the rate of growth
of revenue in conjunction with the abovementioned measures
of revenue dispersion and skewness. It is necessary to re
define the variables in equation (19), first:
In Vt = In A + rt (39)
where A is the initial total revenue, Vt is total revenue
after t years, r is the rate of growth, and t is the tth
year of T, the total years under consideration.
Then.a regression line is fitted to the points
obtained by plotting the natural log of total revenue of
each year (Vt) against the time (t). The rate of growth
and R2 obtained from this line is then used in the analysis.
Risk associated with product lines (RSKPL). The
risk associated with revenue changes with the number of
product lines that the firm offers. The reason is based on
the same principle applied to diversification in investment.
Indeed, having several product lines may be necessary but
it is certainly not sufficient to reduce risk. If all the
product lines are perfectly positively correlated, with
respect to revenue, with each other, then there accrues no
benefit from such diversification. Their collective per
formance resembles that of any constituent individual line,
except on a larger scale.
Negative correlations between revenues generated by
each product line, on the other hand, reduce the risk as
sociated with revenue. The higher is the degree of diver
sification of product lines with such properties, the lower
is the related risk. The recent great wave of conflomera
25
tion seems to be partially, at least, in pursuit of such
concept.26
Existence of some negative correlations, however,
does not guarantee that the collective risk is changed
significantly. If the analysis of the historical data
indicates that the product line with negative correlation
contributes only a small percentage to the total revenue,
its riskreducing effect would be minor, and vice versa.
All the above points are accounted for if, follow
ing the portfolio concept, the variance of total revenue is
expressed by the following function:
n n
V(REV) = Z E X. X. COR(i,j)c. o. (40)
i=l j=l
where n is the number of product lines, Xi, Xj are the
historical average proportions of total revenue which come
from lines i and j, respectively, COR(i,j) is the correla
tion between revenues from lines i and j, and ai, oj are
standard deviations of revenues from lines i and j,
respectively.
Under a certain condition, it may be possible to
calculate the risk associated with product lines. This
condition is that the firm should either have only one
store, or else, if it is a multistore firm, all the stores
should be, revenuewise, perfectly positively correlated
to each other. If this condition is not met, however,
the calculated risk represents the combined risks of both
product lines and stores.
To find the risk associated with product lines,
assume first that all the lines are perfectly positively
correlated with each other; then all correlation terms in
equation (40) would, by definition, be equal to one. This
constitutes the unper limit of the variance of total reve
nue. On the other hand, if all the product lines were
perfectly negatively correlated to each other, all correla
tion terms in equation (40) would, by definition, equal
minus one. This constitutes the lower limit of the vari
ance of total revenue. For any imperfect correlation,
either positive or negative, between product lines, each
correlation term in equation (40) assumes a value between
1 and +1. The weighted average of these correlations
then determines the position of the variance of total
revenue between its upper and lower limits.
Bearing in mind the above discussion, one may
express the upper limit of the variance of total revenue
by the following equation:
n n
V(REV) = Z E X. X. a. J. (41)
i=l j=l 1 3 l 3
The case for any position between upper limit and
lower limit could, of course, be expressed by equation
(40) itself. The deduction from variance of total revenue
which resulted from imperfectly correlated product lines,
or equivalently, the risk (although negative) associated
with product lines, is then equal to the negative of the
difference between equations (40) and (41).
n n n n
L = [ : X. X. . :: X X. COR(i,j) i oj]
i=l j=l 1 1=1 j=l
The first term in the bracket can be rewritten as:
n n n
n X + z E X. X. a. a., for all i/j
i=l 1 i=l j=l 1
The second term in the bracket can be rewritten as:
n n n
E X o. + E X. X. COR(i,j) o. o., for all i/j
i=l i=l j=l 1
Replacing the terms in brackets with their equivalents,
n 2 2
omitting E X o, and factoring common factors, there
i=l
is obtained
n n
RSXPL = 2 F X. X. c. o. [1 COR(i,j)] for all ifj
i=l j=l1 1
(42)
The negative sign in front of the E notation
indicates that multiple product lines have a riskreducing
effect, but to compare the effects of various product lines
one should compare their negative effects on reducing risk
associated with total revenue. The larger is the absolute
value of RSKPL, the larger is the deduction from the vari
ance of total revenue as a result of such diversification,
and vice versa. Therefore, the larger is the absolute
value of RSKPL, the more desirable is the product line, and
in the discussed risk context, the smaller is the risk
associated with it. Conversely, the smaller is the absolute
value of RSKPL, the larg'r is the risk associated with
product line. The lower bound of the absolute value of
RSKPL is zero, and it occurs just when all the product
lines are perfectly positively correlated, i.e., correla
tions between all product lines are equal to one and thus
all [lCOR(i,j)] terms in equation (42) become zero. This,
of course, means that no benefit is accrued from such di
versification.
The risk expressed in equation (42) is in an absolute
term. In order to make it in a relative term, and bring
it into a line with the coefficient of variation, divide
both sides of (42) by the mean value of total revenue.
Thus, denoting the relative risk associated with product
line by RRSKPL, there is obtained:
n n
RRSKPL =  XX. o. o. [1 COR(i,j), for
REV i=l j=l 1 3 1 3
all i / j (43)
If the product lines are many and calculation of
the covariance terms is cumbersome, it may be helpful to
apply the indexing method in computing the risk associated
with product lines. In that case, GNP or another suitable
market index is selected and variables in equations (24)
through (27), which are reproduced here, are redefined to
fit the revenue context. Thus:
REV.. = A. + B.I. + e.
13 1 1 ] 1
where REV.. is the jth observation of the revenue gener
ated by product line i, I. is the jth observation of the
index, and A., B., and e. are yintercept, slope,and random
error related to the regression line of the product line
i, respectively.
n 2 2 2 2
V(REV) = X + X+1 on+ (45)
i=l n+l n+l
2
where V(REV) is variance of total revenue,o is the vari
1
ance of ei, an+ is the variance of the index, and Xn+ is
"pseudo" product line defined as
n
Xn+1 = Xi B. (46)
i=l
And the mean value of total revenue is obtained by:
n
REV = E X Ai + Xn+ An+ (47)
i=l
Where An+1 is the expected value of the index.
Now, if it is assumed that all the product lines
are prefectly positively correlated to each other, then B
B2 = ... = B = B*, where B* is the slope of the regression
line fitted to past series data on total revenue and the
index values. In that case:
n n
Xn+= X.B. = B* Z X. = B* (48)
i=l i=l1
Replacing (48) in (45), gives the risk associated with
total revenue in the absence of any advantage from diversification.
n 2 2 2
V(REV) = X o2 + B* 3 (49)
i= 1 n+
Reduction in this risk as a result of combining product
lines which are not perfectly positively correlated to
each other is obtained by calculating the difference be
tween equations (49) and (45). Again, the negativeof this
difference is the risk associated with product lines.
n n
RSKPL= [ o + B*2 2 (z X2 + 2+
i=l =n+l il n+I n+
2= (*2 2n
_n+B n+l
or,
2 2 n 2
n (50)
RSKPL =n+ [B*2 ( E XiBi)2] (50)
n+ i=
To get the relative risk associated with product
lines, divide both sides of equation (50) by the mean
value of total revenue.
o2 n
RRSKPL = n+ [B*2 ( X.B.) 2 (51)
REV i=l
Risk associated with places occupied (RSKPO).
Another factor which may have riskreducing effect on the
firm's position, is the number of places that it occupies.
The effect comes through the same principle as applied to
product lines. Over any period of time, many factors,
such as changes in the competitive forces, changes in the
buying population, and changes in the purchasing power of
the inhabitants of the location where the firm has its
business act and react on the firm's overall performance.
Supposedly, diversification into several places
deducts from the dependency of the firm's performance on a
set of factors peculiar to a certain location. This may
mean reduction in the business risk of the firm, but this
reduction may not materialize if some conditions, similar
to those of product lines, do not exist. To name only a
few, these conditions include existence of negative corre
lations between each pair of stores, as well as the rela
tive importance of the contribution that the stores with
negative correlations make to total revenue.
It seems also plausible to expect that the reduc
tion in risk through multiplication of stores is, ordi
narily, in addition to that which is resulted from
diversification in product lines. Otherwise, the mul
tiplication of stores would have a negative effect.
The provision for calculating risk associated with
number of stores is similar to that for product lines.
That is, the firm should either have only one product
line, or else, all product lines should be perfectly
positively correlated to each other.
To calculate RSKPO, the procedure is the same as
that applied to the product lines case. First, suppose
that the stores in different places are perfectly
positively correlated wiLh each other. Thus:
n n
V(REV) = E E Y Y.QQ. (52)
i1 j=l 1
where V(REV) is variance of total revenue, n is the number
of stores, and Y. and Y. are the historical average pro
portions cf total revenue which come from stores i and j,
respectively, and Q. Q. are standard deviations of reve
nues from stores i and j, respectively.
Then, consider the case in which the stores are
not, revenuewise, perfectly positively correlated with
each other. In that case:
n n
V(REV) = Z Y.Y.COR(i,j)QiQ (53)
i=1 j=l 1
where COR(i,j) is the correlation between revenues from
stores i and j.
The deduction from variance of total revenue as a
result of the imperfectly correlated stores is the differ
ence between equations (53) and (52). Thus, the risk
associated with stores (although negative) is the negative
of such difference:
n n n n
RSKPO = [ 7 Y.Y.Q.Q. E Y.Y.COR(i,j)Q.Q.] (54)
i=l j=l 3 i=1 j=1l 3
This expression, after some manipulation similar to that in
the product lines case, can be rewritten as the following:
n n
PSK O  2 Y.Y.Q.Q. [1 COR(i,j)1, for all ij (55)
i=l j=l 1
Which can be put in a relative form by dividing by the
mean value of total revenue:
n n
PSKPO = Y.Y.QQ .[lCOR(i,)], for all i/j (56)
R V i=l j= 1
At this point, it should be noted that when the
firm has more than one product line, or, otherwise, when
the product lines are not perfectly positively correlated
to each other, the calculated RSKPO above represents the
total risk associated with diversification in both dimen
sions, product lines and stores. For such companies, the
value of this combined risk should conceivably equal that
obtained through product lines analysis.
The risk associated with places that the firm
occupies is not limited to the one calculated above. It
can be extended to cover firms which operate internation
ally. Companies which are multinational in nature, are
constantly facing changes in exchange rate, changes in
foreign governments' policies, and changes in the eco
nomic conditions of these countries. The risk associated
with some of these factors can be quantitatively accounted
for in the analysis, but for other factors qualitative
measures may be more applicable.
Risk associated with eox:oanle rite (RSKXR) The
risk associated with exchange rate is suitable for treating
quantitatively. A part of this risk comes through the net
income before taxes contributed by the firm's divisions in
foreign countries. Another part comes through assets of
these divisions physically placed in such countries. Ac
cordingly, we may express this risk by the following equa
tions:
n n
RSKXR1 = E X.CV(XR). + Z Y.CV(XR). (57)
i=l i=1
where n is the number of countries in which the firm has a
division, X. is the proportion of net income before taxes
1
which comes from the division in ith country, CV(XR) is
the coefficient of variation of the exchange rate in the
ith country over some period of time, and Yi is the pro
portion of the assets placed in the ith country.
The first term in equation (57) is a weighted
average of coefficient of variations of exchange rates of
the related countries, whose weights are the relative
importance of the contribution of each division in foreign
countries to total net income before taxes. The second
term is another weighted average of such coefficients whose
weights are proportions of assets placed in each country.
The more is the company's dependency on the income provided
by a foreign division, and the more it has engaged its
assets there, the higher is its risk with respect to the
exchange rate. From equation (57), it follows that, for a
firm which operates only domestically, X 's and Y 's are
equal to zero, and thus it does not bear any exchange rate
risk.
It may be more useful to supplement the above
measure with the relative measure of skewness, the stan
dardized third moment about the mean of each of the ex
change rates. The negative sign of the moment may indicate
depreciative tendency of the changes in the exchange rate,
whereas a positive sign may signify appreciative propensity.
Since there are no positive correlations between all ex
change rates at all times, some component parts of the risk
may be cancelled out, if the weights are in appropriate
proportions. To express the above points in another way:
n n
RSKXR2 = Z X (a3 i + Z Yi(a )i (58)
i=l i=1
where (a3,) is the standardized third moment of the exchange
rate frequency distribution of the ith country which can
be obtained by applying equations (16) and (17).
The risk associated with exchange rate is then a
function of equations (57) and (58). Thus:
RSKXR = e(RSKXRl, RSKXR2) (59)
Risk associated with age (RSKAGE). The relation
ship between age and revenue of a firm was discussed
previously. This relationship can now be extended to the
extreme, the firm's financial distress. It has been an
observable fact, as shown by any listing of failed com
panies, that business failures occur mere frequently in a
certain age group. Conceivably, this age group differs
from industry to industry due to the peculiarities and the
complexities involved in and the experience required by
each industry. In general, however, younger firms are more
apt to fail than older ones. But the relationship is not
linear. In fact, over a certain range of the earlier years
the function is increasing until it reaches a peak point
and then it diminishes. Age, thus, seems to be a rela
tively important determinant in business failures.
The possibility that a firm faces financial dis
tress is, therefore, partially dependent upon its abso
lute age as well as its relative age position in a popula
tion of failed firms in its industry. To find the risk
that a firm bears by being in a certain age group is, then,
to find the probability that the age of the firm is the
age of failure. To find this probability, take advantage
of the Tchebycheff inequality:
P(A : d) $ 2/(E d)2 (60)
where A is defined here as failure age variable whose mean
2
value and variance are E and a respectively, and d is
any specified level of age within the range of A.
