Attempts to develop bankruptcy prediction models began seriously
sometime in the late 1960’s and continue through today. At least three
distinct types of models have been used to predict bankruptcy:
• statistical models (primarily, multiple discriminate analyses-
MDA), and conditional logit regression analyses,
• gambler’s ruin-mathematical/statistical models,
• artificial neural network models.
Most of the publicly available information regarding prediction
models is based on research published by university professors. Commercial
banks, public accounting firms and other institutional entities (bond
ratings agencies, for example) appear to be the primary beneficiaries of
this research, since tthey can use the information to minimize their
exposure to potential client failures.
While continuing research has been ongoing for almost thirty years,
it is interesting to note that no unified well-specified theory of how and
why corporations fail has yet been developed. The available statistical
models derive merely from the statistical optimization of a set of ratios.
As stated by Wilcox, the “lack of conceptual framework results in the
limited amount of available data on bankrupt firms being statistically
‘used up’ by the search before a useful geeneralization emerges.”
How useful are these models? Almost universally, the decision
criterion used to evaluate the usefulness of the models has been how well
they classify a company as bankrupt or non-bankrupt compared to the
company’s actual status known after-the-fact (that is ex post). Mo
studies consider a type I error as the classification of a failed company
as healthy, and consider a type II error as the classification of a healthy
company as failed. In general type I errors are considered more costly to
most users than type II errors. The usefulness of fail/nonfail prediction
models is suggested by Ohlson (Ohlson, J.A., “Financial Ratios and the
Probabilistic Prediction of Bankruptcy,” Journal of Accounting Research,
“.real world problems concern themselves with choices which have
a richer set of possible outcomes. No decision problem I can think of has a
payoff space which is partitioned naturally into the binary status
bankruptcy versus non-bankruptcy.I have also refrained from making
inferences regarding the relative usefulness of alternative models, ratios
and predictive systems. Most of the analysis shhould simply be viewed as
descriptive statistics – which may, to some extent, include estimated
prediction error-rates – and no “theories” of bankruptcy or usefulness of
financial ratios are tested.”
Subject to the qualifications expressed above, bankruptcy
prediction models continue to be used to predict failure.
The early history of researchers’ attempts to classify and predict
business failure (and bankruptcy) is well documented in Edward Altman’s
seminal 1983 book, Corporate Financial Distress. There appears to be no
consensus on what constitutes business failure. However, most businesses
are considered to have failed once they h
A Short Z-Score History
In 1966 Altman selected a sample of 66 corporations, 33 of which
had filed for bankruptcy in the past 20 years, and 33 of which were
randomly selected from those that had not. The asset size of all
corporations ranged from $1 million to $26 million.approximately $5
million to $130 million in 2005 dollars.
Altman calculated 22 common financial ratios for all 66
corporations. (For the bankrupt firms, he used the financial statements
issued one year prior to bankruptcy.) His goal was to choose a small number
of those ratios that could best distinguish between a bankrupt firm and a
To make his selection Altman used the statistical technique of
multiple discriminant analysis. This approach shows which characteristics
in which proportions can best be used for determining to which of several
categories a subject belongs: bankrupt versus nonbankrupt, rich versus
poor, young versus old, and so on.
The advantage to MDA is that many characteristics can be combined
into a single score. A low score implies membership in one group, a high
score implies membership in the other group, and a middling score causes
uncertainty as to which group the subject belongs.
Finally, to test the model, Altman calculated the Z Scores for new
groups of bankrupt and nonbankrupt firms. For the nonbankrupt firms,
however, he chose corporations that had reported deficits during e
years. His goal was to discover how well the Z Score model could
distinguish between sick firms and the terminally ill.
Altman found that about 95% of the bankrupt firms were correctly
classified as bankrupt. And roughly 80% of the sick, nonbankrupt firms were
correctly classified as nonbankrupt. Of the misclassified nonbankrupt
firms, the scores of nearly three fourths of these fell into the gray area.
The Z Score Ingredients
The Z Score is calculated by multiplying each of several financial
ratios by an appropriate coefficient and then summing the results. The
ratios rely on these financial measures:
• Working Capital is equal to Current Assets minus Current Liabilities.
• Total Assets is the total of the Assets section of the Balance Sheet.
• Retained Earnings is found in the Equity section of the Balance Sheet.
• EBIT (Earnings Before Interest and Taxes) includes the income or loss
from operations and from any unusual or extraordinary items but not
the tax effects of these items. It can be calculated as follows: Find
Net Income; add back any income tax expenses and subtract any income
tax benefits; then add back any interest expenses.
• Market Value of Equity is the total value of all shares of common and
preferred stock. The dates these values are chosen need not correspond
exactly with the dates of the financial statements to wh
value is compared.
• Net Worth is also known as Shareholders’ Equity or, simply, Equity. It
is equal to Total Assets minus Total Liabilities.
• Book Value of Total Liabilities is the sum of all current and long-
term liabilities from the Balance Sheet.
• Sales includes other income normally categorized as revenues in the
firm’s Income Statement.
Use balance sheet figures from the end of the reporting period for
all Z Score calculations.
The following table shows how these measures are used to calculate
the three versions of the Z Score. The table is explained below.
In other words, the three Z Score versions (described below) are
calculated as follows:
• Z = 1.2*X1 + 1.4*X2 + 3.3*X3 + .6*X4 + X5
• Z1 = .717*X1 + .847*X2 + 3.107*X3 + .42*X4A + .998*X5
• Z2 = 6.56*X1 + 3.26*X2 + 6.72*X3 + 1.05*X4A
Reasons for Multiple Versions
Two of the ratios shown in the figure have tended to limit the
usefulness of the original Z Score measure.
One of these ratios is X4, the Market Value of Equity divided by
Total Liabilities. Obviously, if a firm is not publicly traded, its equity
has no market value. So private firms can’t use the Z Score.
The other problem is X5, Assets Turnover. This ratio varies
significantly by industry. Jewelry stores, for example, have a low asset
turnover while grocery stores have a high turnover. But since the Z Score
expects a value that is common to manufacturing, it could be biased in such
a way that a healthy jewelry store looks sick and a sickly grocery store
To deal with these problems, Altman used his original data to
calculate two modified versions of the Z Score, shown above. The Z Score is
for public manufacturing companies; the Z1 Score is for private
manufacturing companies; and the Z2 is for general use.
Therefore, according to the table, if a company’s Z2 score is
greater than 2.60, it’s currently safe from bankruptcy. If the score is
less than 1.10, it’s headed for bankruptcy. Otherwise, it’s in a gray area.
How to Interpret the Z Score
The Z Score is not intended to predict when a firm will file a
formal declaration of bankruptcy in a federal district court. It is instead
a measure of how closely a firm resembles other firms that have filed for
bankruptcy. It is a measure of corporate financial distress, a measure of
How accurately does the Z Score measure economic bankruptcy? The
original model has drawn several statistical objections over the years. The
model uses unadjusted accounting data; it uses data from relatively small
firms; and it uses data that today is nearly 60 years old.
And yet, despite these concerns, the original Z Score model is the
best-known and most widely used measure of its kind. This measure is far
from perfect, but it’s easy to calculate in Excel and many users continue
to find it useful. At last count, for example, Google offered 308,000 links
to the phrase, “Z Score”.
The Z Score model is a tool that can complement your other
analytical tools. Seldom, however, should you use any of the Z Score
measures as your only means of analysis.
In other words:
• Z-SCORE ABOVE 3.0 – The company is safe based on these
financial figures only.
• Z-SCORE BETWEEN 2.7 and 2.99 – On Alert. This zone is an area
where one should exercise caution.
• Z-SCORE BETWEEN 1.8 and 2.7 – Good chances of the company
going bankrupt within 2 years of operations from the date of
financial figures given.
• Z-SCORE BELOW 1.80- Probability of Financial embarassment is
Other Statistical Failure Prediction Models
Many additional bankruptcy prediction models have been developed
since the work of Beaver and Altman. Lev (Lev, B. “Financial Statement
Analysis, A New Approach,” Englewood Cliffs, N.J.: Prentice-Hall, 1974.) ,
Deakin (Deakin, E.B., “A Discriminant Analysis of Predictors of Business
Failure,” Journal of Accounting Research, March, 1972.), Ohlson (Ohlson,
J.A., “Financial Ratios and the Probabilistic Prediction of Bankruptcy,”
Journal of Accounting Research, Spring 1980.), Taffler (Taffler, R., and
Houston, “How to Identify Failing Companies Before It Is Too Late,”
Professional Administration, April 1980.), Platt & Platt (Platt, J.D. and
Platt, M.B., “Development of a Class of Stable Predictive Variables: The
Case of Bankruptcy Prediction,” Journal of Business Finance and Accounting,
Spring 1990.), Gilbert, Menon, and Schwartz (Gilbert, L.R., Menon, K., and
Schwartz, K.B., “Predicting Bankruptcy for Firms in Financial Distress,”
Journal of Business Finance and Accounting, Spring 1990), and Koh and
Killough (Koh, H.C. and Killough, L.N., “The Use of Multiple Discriminant
Analysis in the Assessment of the Going-concern Status of an Audit Client,”
Journal of Business Finance and Accounting. Spring 1990) among others have
continued to refine the development of multivariate statistical models.
Almost all of these traditional models have been either matched-pair multi-
discriminate models (such as Altman’s) or logit models (such as Ohlson’s).
A 1997 study by Begley, Ming and Watts concludes:
”Given that Ohlson’s original model is frequently used in academic research
as an indicator of financial distress, its strong performance in this study
supports its use as a preferred model.”
Alternative Failure Prediction Models – The “Gambler’s Ruin” Models
Wilcox, Santomero, Vinso and others have adapted a gambler’s ruin
approach to bankruptcy prediction. Under this approach, bankruptcy is
probable when a company’s net liquidation value (NLV) becomes negative. Net
liquidation value is defined as total asset liquidation value less total
liabilities. From one period to the next, a company’s NLV is increased by
cash inflows and decreased by cash outflows during the period. Wilcox
combined the cash inflows and outflows and defined them as “adjusted cash
flow.” All other things being equal, the probability of a company’s failure
increases, the smaller the company’s beginning NLV, the smaller the
company’s adjusted (net) cash flow, and the larger the variation of the
company’s adjusted cash flow over time. Wilcox uses the gambler’s ruin
formula (Feller, 1968) to show that a company’s risk of failure is
dependent on 1) the above factors plus 2) the size of the company’s
adjusted cash flow “at risk” each period (i.e., the size of the company’s
Using an alleged more robust statistical technique, Vinso extended Wilcox’s
gambler’s ruin model to develop a safety index. Based on input concerning
the variability of “expected contribution margin amounts,” the index can be
used to predict the point in time when a company’s ruin is most likely to
occur (called first passage time).
The statistics used in gambler’s ruin approaches are somewhat formidable
(especially to the average business reader). However, both Wilcox and Vinso
richly describe some of the factors which most affect business failure. For
example, Wilcox states:
The (cash) inflow rate . can be increased through higher average
return on investment. However, having a major impact here usually requires
long-term changes in strategic position. This is difficult to control over
a short time period except by divestitures of peripheral unprofitable
businesses.The average outflow rate is controlled by managing the
average growth rate of corporate assets. Effective capital budgeting .
requires resource allocation emphasizing those business units which have
the highest future payoff.
The size of the bet is the least understood factor in financial risk. Yet
management has substantial control over it. Variability in liquidity flows
governs the size of the bet. This variability can be managed through
dividend policy, through limiting earning variability and investment
variability, and through controlling the co-variation between profits and
investments.True earnings smoothing is attained by control of exposure to
volatile industries, diversification, and improved strategic position.
Vinso supports Wilcox’s emphasis on cash flow processes and
stresses the importance of debt capacity:
Before deriving a mathematical model for determining the risk of
ruin, it is necessary to describe the process. (First), a firm has some
pool of resources at time = 0 of some size U0, which are available to
prevent ruin (similar to Wilcox’s beginning NAV). (Then), earnings come to
(the) firm from revenue(s).less the costs incurred in producing (the
revenues). There are two types of costs to be considered: variable, which
change according to the stochastic nature of the revenue sources, and fixed
costs, which do not vary with revenue but are a function of the period. So,
revenue less variable costs.can be defined as variable profit (which is
available to pay fixed costs). If Ut is less than zero, ruin occurs because
no funds are available to meet unpaid fixed costs.These definitions,
however, ignore debt capacity, if available, which must be included as the
firm can use this source without being forced to confront shareholders,
creditors, a third party or a bankruptcy court.debt holders or other
creditors will force reorganization if a firm is unable to meet contractual
obligations because working capital is too low and the firm cannot obtain
more debt. (Emphasis added)
While Wilcox and the other gambler’s ruin researchers have made a
substantial contribution to business failure prediction, they do not appear
to have developed the generally accepted conceptual framework of business
failure that had been hoped for by Wilcox.
Alternative Models – Artificial Neural Networks
Since 1990, another promising approach to bankruptcy prediction,
based on the use of neural networks, has evolved. Artificial Neural
Networks (ANN) are computers constructed to process information, in
parallel, similar to the human brain. ANN’s store information in the form
of patterns and are able to learn from their processing experience. Unlike
MDA and logit analyses, ANN’s impose less restrictive data requirements
(the requirement for linearity, for example) and are especially useful in
recognizing and learning complex data relationships. However, ANN’s are
“black boxes” in that they do not reveal how they weigh independent
variables. Thus, the individual role each of the various variables plays
cannot be determined.
In general, the classification accuracy of ANN’s is considered comparable
to logit and MDA models.
Altman, Edward I. / Hotchkiss, Edith
Corporate Financial Distress and Bankruptcy
Predict and Avoid Bankruptcy, Analyze and Invest in Distressed Debt
This Third Edition of the most authoritative finance book on the
topic updates and expands its discussion of corporate distress and
bankruptcy, as well as the related markets dealing with high-yield and
distressed debt, and offers state-of-the-art analysis and research on the
costs of bankruptcy, credit default prediction, the post-emergence period
performance of bankrupt firms, and more.
[pic]Edward I. Altman
A Short Z-Score History
The Z Score Ingredients
Reasons for Multiple Versions
How to Interpret the Z Score
• Other Statistical Failure Prediction Models
• Alternative Failure Prediction Models – The “Gambler’s Ruin”
• Alternative Models – Artificial Neural Networks