PREDICTING FINANCIAL DISTRESS OF COMPANIES:
REVISITING THE Z-SCORE AND ZETA® MODELS
Edward I. Altman*
*Max L. Heine Professor of Finance, Stern School of Business, New York University. This
paper is adapted and updated from E. Altman, “Financial Ratios, Discriminant Analysis and the
Prediction of Corporate Bankruptcy,” Journal of Finance, September 1968; and E. Altman, R.
Haldeman and P. Narayanan, “Zeta Analysis: A New Model to Identify Bankruptcy Risk of
Corporations,” Journal of Banking & Finance, 1, 1977.
Predicting Financial Distress of Companies:
Revisiting the Z-Score and ZETA® Models
This paper discusses two of the venerable models for assessing the distress of industrial
corporations. These are the so-called Z-Score model (1968) and ZETA® 1977) credit risk model.
Both models are still being used by practitioners throughout the world. The latter is a proprietary
model for subscribers to ZETA Services, Inc. (Hoboken, NJ).
The purpose of this summary are two-fold. First, those unique characteristics of business
failures are examined in order to specify and quantify the variables which are effective indicators
and predictors of corporate distress. By doing so, I hope to highlight the analytic as well as the
practical value inherent in the use of financial ratios. Specifically, a set of financial and
economic ratios will be analyzed in a corporate distress prediction context using a multiple
discriminant statistical methodology. Through this exercise, I will explore not only the
quantifiable characteristics of potential bankrupts but also the utility of a much-maligned
technique of financial analysis: ratio analysis. Although the models that we will discuss were
developed in the late 1960’s and mid-1970’s, I will extend our tests and findings to include
application to firms not traded publicly, to non-manufacturing entities, and also refer to a new
bond-rating equivalent model for emerging markets corporate bonds. The latter utilizes a version
of the Z-Score model called Z”. This paper also updates the predictive tests on defaults and
bankruptcies through the year 1999.
As I first wrote in 1968, and it seems even truer in the late 1990’s, academicians seem to
be moving toward the elimination of ratio analysis as an analytical technique in assessing the
performance of the business enterprise. Theorists downgrade arbitrary rules of thumb (such as
company ratio comparisons) widely used by practitioners. Since attacks on the relevance on
ratio analysis emanate from many esteemed members of the scholarly world, does this mean that
ratio analysis is limited to the world of “nuts and bolts?” Or, has the significance of such an
approach been unattractively garbed and therefore unfairly handicapped? Can we bridge the gap,
rather than sever the link, between traditional ratio analysis and the more rigorous statistical
techniques which have become popular among academicians in recent years? Along with our
primary interest, corporate bankruptcy, I am also concerned with an assessment of ratio analysis
as an analytical technique.
It should be pointed out that the basic research for much of the material in this paper was
performed in 1967 and that several subsequent studies have commented upon the Z-Score model
and its effectiveness, including an adaptation in 1995 for credit analysis of emerging market
corporates. And, this author has co-developed a “second generation” model (ZETA) which was
developed in 1976.
Traditional Ratio Analysis
The detection of company operating and financial difficulties is a subject which has been
particularly amenable to analysis with financial ratios. Prior to the development of quantitative
measures of company performance, agencies were established to supply a qualitative type of
information assessing the credit-worthiness of particular merchants. (For instance, the
forerunner of the well-known Dun & Bradstreet, Inc. was organized in 1849 in Cincinnati, Ohio,
in order to provide independent credit investigations). Formal aggregate studies concerned with
portents of business failure were evident in the 1930’s.
One of the classic works in the area of ratio analysis and bankruptcy classification was
performed by Beaver (1967). In a real sense, his univariate analysis of a number of bankruptcy
predictors set the stage for the multivariate attempts, by this author and others, which followed.
Beaver found that a number of indicators could discriminate between matched samples of failed
and nonfailed firms for as long as five years prior to failure. He questioned the use of
multivariate analysis, although a discussant recommended attempting this procedure. The Z-
Score model did just that. A subsequent study by Deakin (1972) utilized the same 14 variables
that Beaver analyzed, but he applied them within a series of multivariate discriminant models.
The aforementioned studies imply a definite potential of ratios as predictors of
bankruptcy. In general, ratios measuring profitability, liquidity, and solvency prevailed as the
most significant indicators. The order of their importance is not clear since almost every study
cited a different ratio as being the most effective indication of impending problems.
Although these works established certain important generalizations regarding the
performance and trends of particular measurements, the adaptation of the results for assessing
bankruptcy potential of firms, both theoretically and practically, is questionable. In almost every
case, the methodology was essentially univariate in nature and emphasis was placed on
individual signals of impending problems. Ratio analysis presented in this fashion is susceptible
to faulty interpretation and is potentially confusing. For instance, a firm with a poor profitability
and/or solvency record may be regarded as a potential bankrupt. However, because of its above
average liquidity, the situation may not be considered serious. The potential ambiguity as to the
relative performance of several firms is clearly evident. The crux of the shortcomings inherent in
any univariate analysis lies therein. An appropriate extension of the previously cited studies,
therefore, is to build upon their findings and to combine several measures into a meaningful
predictive model. In so doing, the highlights of ratio analysis as an analytical technique will be
emphasized rather than downgraded. The questions are (1) which ratios are most important in
detecting bankruptcy potential, (2) what weights should be attached to those selected ratios, and
(3) how should the weights be objectively established.
After careful consideration of the nature of the problem and of the purpose of this
analysis, I chose multiple discriminant analysis (MDA) as the appropriate statistical technique.
Although not as popular as regression analysis, MDA has been utilized in a variety of disciplines
since its first application in the 1930’s. During those earlier years, MDA was used mainly in the
biological and behavioral sciences. In recent years, this technique has become increasingly
popular in the practical business world as well as in academia. Altman, et.al. (1981) discusses
discriminant analysis in-depth and reviews several financial application areas.
MDA is a statistical technique used to classify an observation into one of several a priori
groupings dependent upon the observation’s individual characteristics. It is used primarily to
classify and/or make predictions in problems where the dependent variable appears in qualitative
form, for example, male or female, bankrupt or nonbankrupt. Therefore, the first step is to
establish explicit group classifications. The number of original groups can be two or more.
Some analysts refer to discriminant analysis as “multiple” only when the number of groups
exceeds two. We prefer that the multiple concepts refer to the multivariate nature of the analysis.
After the groups are established, data are collected for the objects in the groups; MDA in
its most simple form attempts to derive a linear combination of these characteristics which “best”
discriminates between the groups. If a particular object, for instance, a corporation, has
characteristics (financial ratios) which can be quantified for all of the companies in the analysis,
the MDA determines a set of discriminant coefficients. When these coefficients are applied to
the actual ratios, a basis for classification into one of the mutually exclusive groupings exists.
The MDA technique has the advantage of considering an entire profile of characteristics
common to the relevant firms, as well as the interaction of these properties. A univariate study,
on the other hand, can only consider the measurements used for group assignments one at a time.
Another advantage of MDA is the reduction of the analyst’s space dimensionally, that is,
from the number of different independent variables to G-1 dimension(s), where G equals the
number of original a priori groups. This analysis is concerned with two groups, consisting of
bankrupt and nonbankrupt firms. Therefore, the analysis is transformed into its simplest form:
one dimension. The discriminant function, of the form Z = V1X1 + V2X2 +…+ VnXn transforms
the individual variable values to a single discriminant score, or z value, which is then used to
classify the object where V1, X2, . . . . Vn = discriminant coefficients, and
V1, X2, . . . . Xn = independent variables
The MDA computes the discriminant coefficient; Vi while the independent variables Xi are the
When utilizing a comprehensive list of financial ratios in assessing a firm’s bankruptcy
potential, there is reason to believe that some of the measurements will have a high degree of
correlation or collinearity with each other. While this aspect is not serious in discriminant
analysis, it usually motivates careful selection of the predictive variables (ratios). It also has the
advantage of potentially yielding a model with a relatively small number of selected
measurements which convey a great deal of information. This information might very well
indicate differences among groups, but whether or not these differences are significant and
meaningful is a more important aspect of the analysis.
Perhaps the primary advantage of MDA in dealing with classification problems is the
potential of analyzing the entire variable profile of the object simultaneously rather than
sequentially examining its individual characteristics. Just as linear and integer programming
have improved upon traditional techniques in capital budgeting, the MDA approach to traditional
ratio analysis has the potential to reformulate the problem correctly. Specifically, combinations
of ratios can be analyzed together in order to remove possible ambiguities and misclassifications
observed in earlier traditional ratio studies.
As we will see, the Z-Score model is a linear analysis in that five measures are
objectively weighted and summed up to arrive at an overall score that then becomes the basis for
classification of firms into one of the a priori groupings (distressed and nondistressed).
Development of the Z-Score Model
The initial sample is composed of 66 corporations with 33 firms in each of the two
groups. The bankrupt (distressed) group (Group 1) are manufacturers that filed a bankruptcy
petition under Chapter X of the National Bankruptcy Act from 1946 through 1965. A 20-years
period is not the best choice since average ratios do shift over time. Ideally, we would prefer to
examine a list of ratios in time period t in order to make predictions about other firms in the
following period (t+1). Unfortunately, it was not possible to do this because of data limitations.
Recognizing that this group is not completely homogeneous (due to industry and size
differences), I attempted to make a careful selection of nonbankrupt (nondistressed) firms.
Group 2 consists of a paired sample of manufacturing firms chosen on a stratified random basis.
The firms are stratified by industry and by size, with the asset size range restricted to between $1
and $25 million. The mean asset size of the firms in Group 2 ($9.6 million) was slightly greater
than that of Group 1, but matching exact asset size of the two groups seemed unnecessary. Firms
in group 2 were still in existence at the time of the analysis. Also, the data collected are from the
same years as those compiled for the bankrupt firms. For the initial sample test, the data are
derived from financial statements dated one annual reporting period prior to bankruptcy. The
data were derived from Moody’s Industrial Manuals and also from selected annual reports. The
average lead-time of the financial statements was approximately seven and one-half months.
An important issue is to determine the asset-size group to be sampled. The decision to
eliminate both the small firms (under $1 million in total assets) and the very large companies
from the initial sample essentially is due to the asset range of the firms in Group 1. In addition,
the incidence of bankruptcy in the large-asset-size firm was quite rare prior to 1966. This
changed starting in 1970 with the appearance of several very large bankruptcies, e.g., Penn-
Central R.R. Large industrial bankruptcies also increased in appearance, since 1978. In all,
there have been at least 100 Chapter 11 bankruptcies with over $1 billion since 1978 (the year of
the existing Bankruptcy Code's enactment).
A frequent argument is that financial ratios, by their very nature, have the effect of
deflating statistics by size, and that therefore a good deal of the size effect is eliminated. The Z-
Score model, discussed below, appears to be sufficiently robust to accommodate large firms.
The ZETA model did include larger sized distressed firms and is unquestionably relevant to both
small and large firms.
After the initial groups are defined and firms selected, balance sheet and income
statement data are collected. Because of the large number of variables found to be significant
indicators of corporate problems in past studies, a list of 22 potentially helpful variables (ratios)
was complied for evaluation. The variables are classified into five standard ratio categories,
including liquidity, profitability, leverage, solvency, and activity. The ratios are chosen on the
basis of their popularity in the literature and their potential relevancy to the study, and there are a
few “new” ratios in this analysis. The Beaver study (1967) concluded that the cash flow to debt
ratio was the best single ratio predictor. This ratio was not considered in my 1968 study because
of the lack of consistent and precise depreciation and cash flow data. The results obtained,
however, were still superior to the results Beaver attained with his single best ratio. Cash flow
measures were included in the ZETA model tests (see later discussion).
From the original list of 22 variables, five are selected as doing the best overall job
together in the prediction of corporate bankruptcy. This profile did not contain all of the most
significant variable measured independently. This would not necessarily improve upon the
univariate, traditional analysis described earlier. The contribution of the entire profile is
evaluated and, since this process is essentially iterative, there is no claim regarding the optimality
of the resulting discriminant function. The function, however, does the best job among the
alternatives which include numerous computer runs analyzing different ratio profiles.
In order to arrive at a final profile of variables, the following procedures are utilized: (1)
observation of the statistical significance of various alternative functions, including
determination of the relative contributions of each independent variable; (2) evaluation of
intercorrelations among the relevant variables; (3) observation of the predictive accuracy of the
various profiles; and (4) judgment of the analyst.
The final discriminant function is as follows:
Z = 0.012X1 + 0.014X2 + 0.033X3 + 0.006X4 +0.999X5
where X1 = working capital/total assets,
X2 = retained earnings/total assets,
X3 = earnings before interest and taxes/total assets,
X4 = market value equity/book value of total liabilities,
X5 = sales/total assets, and
Z = overall index.
Note that the model does not contain a constant (Y-intercept) term. This is due to the
particular software utilized and, as a result, the relevant cutoff score between the two groups is
not zero. Other software program, like SAS and SPSS, have a constant term, which standardizes
the cutoff score at zero if the sample sizes of the two groups are equal.
X1, Working Capital/Total Assets (WC/TA).
The working capital/total assets ratio, frequently found in studies of corporate problems,
is a measure of the net liquid assets of the firm relative to the total capitalization. Working
capital is defined as the difference between current assets and current liabilities. Liquidity and
size characteristics are explicitly considered. Ordinarily, a firm experiencing consistent
operating losses will have shrinking current assets in relation to total assets. Of the three
liquidity ratios evaluated, this one proved to be the most valuable. Two other liquidity ratios
tested were the current ratio and the quick ratio. There were found to be less helpful and subject
to perverse trends for some failing firms.
X2, Retained Earnings/Total Assets (RE/TA).
Retained earnings is the account which reports the total amount of reinvested earnings
and/or losses of a firm over its entire life. The account is also referred to as earned surplus. It
should be noted that the retained earnings account is subject to "manipulation" via corporate
quasi-reorganizations and stock dividend declarations. While these occurrences are not evident
in this study, it is conceivable that a bias would be created by a substantial reorganization or
stock dividend and appropriate readjustments should be made to the accounts.
This measure of cumulative profitability over time is what I referred to earlier as a “new”
ratio. The age of a firm is implicitly considered in this ratio. For example, a relatively young