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 eexample) appear to be the primary beneficiaries of
this research, since they 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 generalization 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). Most of the
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 tthe relative usefulness of alternative models, ratios
and predictive systems. Most of the analysis should 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 have entered formal bankruptcy
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 wwhich 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 earlier
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 which the market
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
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