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PREDICTING FINANCIAL DISTRESS OF COMPANIES:
REVISITING THE Z-SCORE AND ZETA®^ MODELS
Edward I. Altman*
July 2000
*Max L. Heine Professor of Finance, Stern School of Business, New York University. Thispaper 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
Background 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
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. Discriminant Analysis 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.
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 Sample Selection 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 $ 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. Variable Selection 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
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. X 1 , 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. X 2 , 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
firm will probably show a low RE/TA ratio because it has not had time to build up its cumulative profits. Therefore, it may be argued that the young firm is somewhat discriminated against in this analysis, and its chance of being classified as bankrupt is relatively higher than that of another older firm, ceteris paribus. But, this is precisely the situation in the real world. The incidence of failure is much higher in a firm’s earlier years. In 1993, approximately 50% of all firms that failed did so in the first five years of their existence (Dun & Bradstreet, 1994). In addition, the RE/TA ratio measures the leverage of a firm. Those firms with high RE, relative to TA, have financed their assets through retention of profits and have not utilized as much debt. X 3 , Earnings Before Interest and Taxes/Total Assets (EBIT/TA). This ratio is a measure of the true productivity of the firm’s assets, independent of any tax or leverage factors. Since a firm’s ultimate existence is based on the earning power of its assets, this ratio appears to be particularly appropriate for studies dealing with corporate failure. Furthermore, insolvency in a bankrupt sense occurs when the total liabilities exceed a fair valuation of the firm’s assets with value determined by the earning power of the assets. As we will show, this ratio continually outperforms other profitability measures, including cash flow. X 4 , Market Value of Equity/Book Value of Total Liabilities (MVE/TL). Equity is measured by the combined market value of all shares of stock, preferred and common, while liabilities include both current and long term. The measure shows how much the firm’s assets can decline in value (measured by market value of equity plus debt) before the liabilities exceed the assets and the firm becomes insolvent. For example, a company with a market value of its equity of $1,000 and debt of $500 could experience a two-thirds drop in asset value before insolvency. However, the same firm with $250 equity will be insolvent if assets
should be included as 10.0% and not 0.10. Only variable X 5 (sales to total assets) should be expressed in a different manner: that is, a S/TA ratio of 200% should be included as 2.0. The practical analyst may have been concerned by the extremely high relative discriminant coefficient of X 5. This seeming irregularity is due to the format of the different variables. Table 1 illustrates the proper specification and form for each of the five independent variables. Over the years many individuals have found that a more convenient specification of the model is of the form: Z = 1.2X 1 + 1.4X 2 + 3.3X 3 + 0.6X 4 + 1.0X 5. Using this formula, one inserts the more commonly written percentage, for example, 0.10 for 10%, for the first four variables (X 1 -X 4 ) and rounds the last coefficient off to equal 1.0 (from 0.99). The last variable continues to be written in terms of number of times. The scores for individual firms and related group classification and cutoff scores remain identical. We merely point this out and note that we have utilized this format in some practical application, for example, Altman and LaFleur (1981). Table 1 Variable Means and Test Significance Bankrupt Nonbankrupt Variable Group Meann^ Group Meann^ F Ration X 1 -6.1% 41.4% 32.50* XX 2 -62.6% 35.5% 58.86* X^3 -31.8%^ 15.4%^ 26.56* X^45 40.1%1.5X 247.7%1.9X 33.26*2. N = 33. F1.60 (0.001) = 12.00; F1.60 (0.01) = 7.00; F1.60 (0.05) = 4. *Significant at the 0.001 level. Variable Tests A test to determine the overall discriminating power of the model is the F-value which is the ratio of the sums-of-squares between-groups to the within-groups sums-of-squares. When this ratio is maximized, it has the effect of spreading the means (centroids) of the groups apart
and, simultaneously, reducing dispersion of the individual points (firm Z-values) about their respective group means. Logically, this test (commonly called the F-test) is appropriate because the objective of the MDA is to identify and utilize those variables which best discriminate between groups and which are most similar within groups. The group means of the original two-group sample are: Group 1 = -0.29 F = 20. Group 2 = +5.02 F4n (0.01) = 3. The significance test therefore rejects the null hypothesis that the observations come from the same population. Variable means measured at one financial statement prior to bankruptcy and the resulting F-statistics were shown in Table 1. Variables X 1 through X 4 are all significant at the 0.001 level, indicating extremely significant differences in these variables among groups. Variable X 5 does not show a significant difference among groups and the reason for its inclusion in the variable profile is not apparent as yet. On a strictly univariate level, all of the ratios indicate higher values for the nonbankrupt firms. Also, all of the discriminant coefficients display positive signs, which is what one would expect. Therefore, the greater a firm’s distress potential, the lower its discriminant score. It is clear that four of the five variables display significant differences between groups, but the importance of MDA is its ability to separate groups using multivariate measures. Once the values of the discriminant coefficients are estimated, it is possible to calculate discriminant scores for each observation in the samples, or any firm, and to assign the observations to one of the groups based on this score. The essence of the procedure is to compare the profile of an individual firm with that of the alternative groupings. The
understandable because impending bankruptcy is more remote and the indications are less clear. Nevertheless, 72% correct assignment is evidence that bankruptcy can be predicted two years prior to the event. The Type II error is slightly larger (6% vs. 3%) in this test, but still it is extremely accurate. Further tests will be applied below to determine the accuracy of predicting bankruptcy as much as five years prior to the actual event. Table 3 Classification Results, Two Statements Prior to Bankruptcy Number Percent Percent Predicted_________ Correct Correct Error n Actual Group 1 Group 2 (Bankrupt) (Non-Bankrupt) Group 1 23 9 Type 1 23 72 28 32 Group 2^2 Type II 31 94 6 33 Total 54 83 17 65
Potential Bias and Validation Techniques When the firms used to determine the discriminant coefficients are reclassified, the resulting accuracy is biased upward by (1) sampling errors in the original sample; and (2) search bias. The latter bias is inherent in the process of reducing the original set of variables (22) to the best variable profile (5). The possibility of bias due to intensive searching is inherent in any empirical study. While a subset of variables is effective in the initial sample, there is no guarantee that it will be effective for the population in general. The importance of secondary sample testing cannot be overemphasized. One type of secondary sample testing is to estimate parameters for the model using only a subset of the original sample, and then to classify the remainder of the sample based on the parameters established. A simple t-test is then applied to test the significance of the results. Five different
replications of the suggested method of choosing subsets (16 firms) of the original sample are tested. The test results reject the hypothesis that there is no difference between the groups and substantiate that the model does, in fact, possess discriminating power on observations other than those used to establish the parameters of the model. Therefore, any search bias does not appear significant. Secondary Sample of Bankrupt Firms In order to test the model rigorously for both bankrupt and nonbankrupt firms, two new samples are introduced. The first contains a new sample of 25 bankrupt firms whose asset size range is similar to that of the initial bankrupt group. On the basis of the parameters established in the discriminant model to classify firms in this secondary sample, the predictive accuracy for this sample as of one statement prior to bankruptcy is described in Table 4. The results here are surprising in that one would not usually expect a secondary sample’s results to be superior to the initial discriminant sample (96% vs. 94%). Two possible reasons are that the upward bias normally present in the initial sample tests is not manifested in this investigation and/or that the model, as stated before, is not optimal.
Table 4 Classification Results, Secondary Sample of Bankrupt Firms Bankrupt Group (Actual) Predicted_________ Number Percent Percent Correct Correct Error Bankrupt Non-Bankrupt 24 1 Type I (Total) 24 96 4 n = 25
F I G U R E 1
A v e r a g e Z - S c o r e s : U S I n d u s t r i a l F i r m s
Mar-75Mar-77Mar-79Mar-81Mar-83Mar-85Mar-87Mar-89Mar-91Mar-93Mar-95Mar-97Mar-
M e a n
Source: Osler and Hong, 2000.
Secondary Sample of Nonbankrupt Firms Up to this point, the sample companies were chosen either by their bankruptcy status (Group I) or by their similarity to Group I in all aspects except their economic well-being. But what of the many firms which suffer temporary profitability difficulties, but actually do not become bankrupt? A bankruptcy classification of a firm from this group is an example of a Type II error. An exceptionally rigorous test of the discriminant model’s effectiveness would be to search out a large sample of firms that have encountered earning problems and then to observe the Z-Score’s classification results. In order to perform the above test, a sample of 66 firms is selected on the basis of net income (deficit) reports in the years 1958 and 1961, with 33 from each year. Over 65% of these firms had suffered two or three years of negative profits in the previous three years. The firms are selected regardless of their asset size, with the only two criteria being that they were manufacturing firms which suffered losses in the year 1958 or 1961. The companies are then evaluated by the discriminant model to determine their bankruptcy potential. The results show that 14 of the 66 firms are classified as bankrupt, with the remaining 52 correctly classified. Therefore, the discriminant model correctly classified 79% of the sample firms. This percentage is all the more impressive when one considers that these firms constitute a secondary sample of admittedly below-average performance. The t-test for the significance of the result is 5=4.8; significant at the 0.001 level. Another interesting facet of this test is the relationship of these “temporarily” sick firms’ Z-Scores and the “zone of ignorance.” The zone of ignorance is that range of Z-Scores where misclassification can be observed. Of the 14 misclassified firms in this secondary sample, 10 have Z-Scores between 1. and 2.67, which indicates that although they are classified as bankrupt, the prediction of their
