Predicting Bankruptcy

This entry is part 2 of 4 in the series Risk of Bankruptcy


The Z-score formula for predicting bankruptcy was developed in 1968 by Edward I. Altman. The Z-score is not intended to predict when a firm will file a formal declaration of bankruptcy in a district court. It is instead a measure of how closely a firm resembles other firms that have filed for bankruptcy.

The Z-score is classification method using a multivariate discriminant function that measures corporate financial distress and predicts the likelihood of bankruptcy within two years.1

Others like Springate2, Fulmer3 and the CA-SCORE model4 have later followed in Altman’s track using step-wise multiple discriminant analysis to evaluate a large number of financial ratio’s ability to discriminate between corporate future failures and successes.

Since Altman’s discriminant function only is linear in the explanatory variables, there has been a number of attempts to capture non-linear relations thru other types of models5 6. Even if some of these models shows a somewhat better predicting ability, we will use the better known Z-score model in the following.

Studies measuring the effectiveness of the Z-score claims the model to be accurate with >70% reliability. Altman found that about 95% of the bankrupt firms were correctly classified as bankrupt. And roughly 80% of the sick, non-bankrupt firms were correctly classified as non-bankrupt7. However others find that the Z-score tends to misclasifie the non-bankrupt firms8.

The Z-score combines four or five common business ratios using a linear discriminant function to determine the regions with high likelihood of bankruptcy. The discriminant coefficients (ratio value weights) were originally based on data from publicly held manufacturers, but have since been modified for private manufacturing, non-manufacturing and service companies.

The original data sample consisted of 66 firms, half of which had filed for bankruptcy under Chapter 7. All businesses in the database were manufacturers and small firms with assets of <$1million was eliminated.

The advantage of discriminant analysis 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.

The original score was as follows:

Z = 1.2 WC/TA + 1.4 RE/TA + 3.3 EBIT/TA +0.6R ME/BL +0.999 S/TA

WC/TA= Working Capital / Total Assets, RE/TA= Retained Earnings / Total Assets
EBIT/TA = EBIT/ Total Assets, S/TA = Sales/ Total Assets
ME/BL = Market Value of Equity / Book Value of Total Liabilities

From about 1985 onwards, the Z-scores have gained acceptance by auditors, management accountants, courts, and database systems used for loan evaluation. It has been used in a variety of contexts and countries, but was designed originally for publicly held manufacturing companies with assets of more than $1 million. Later revisions take into account the book value of privately held shares, and the fact that turnover ratios vary widely in non-manufacturing industries:

  1. Z-score for publicly held Manufacturers
  2. Z’-score for private Firms
  3. Z’’-score for Manufacturers, Non-Manufacturer Industrials & Emerging Market Credits

The estimated discriminant coefficients for the different models is given in the following table:

Discriminant coefficients

Factor Z Z' Z''
Working Capital to Total Assets 1.2 0.717 6.56
Retained Earnings to Total Assets 1.4 0.847 3.26
EBIT to Total Assets 3.3 3.107 6.72
Market Value of Equity to Book Value of Total Liabilities 0.6 - -
Book Value of Equity to Book Value of Total Liabilities - 0.420 1.05
Sales to Total Assets 0.999 0.998 -

and the accompanying borders of the different regions – risk zones – are given in the table below.

Discriminant scores and regions of risk

Risk zone Z Z' Z''
Safe 2.99 < z 2.9 < z 2.6 < z
Grey 1.8 < z < 2.99 1.23 < z < 2.9 1.1 < z < 2.6
Distress z < 1.80 z < 1.23 z < 1.1

In the following calculations we will use the estimated value of equity as a proxy for market capitalization. Actually it is the other way around since the market capitalization is a guesstimate of the intrinsic equity value.

In our calculations the Z-score metrics will become stochastic variables with distributions derived both from the operational input distributions for sale, prices, costs etc. and the distributions for the financial variables like risk free interest rate, inflation etc. The figures below are taken from the fifth year in the simulation to be comparable with the previous Z-index calculation that gave a very low probability for insolvency.

We have in the following calculated all three Z metrics, even when only the Z-score fits the company description.


Using the Z-score metric we find that the company with high probability will be found in the distress area – it can even have negative Z-score. The last is due to the fact that the company has negative working capital – being partly financed by its suppliers and partly to the use of calculated value of equity – which can be negative.

The Z’’-score is even more somber giving no possibility for values outside the distress area:


The Z’’-score however puts most of the observations in the gray area:


Before drawing any conclusions we will in the next post look at the time series for both the Z-index and the Z-scores. Nevertheless one observation can be made – the Z metric is a stochastic variable with an event space that easily can encompass all three risk zones – we therefore need the probability distribution over the zones to forecast the risk of bankruptcy.


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  1. Altman, Edward I., “Financial Ratios, Discriminant Analysis and the Prediction of Corporate Bankruptcy”. Journal of Finance, (September 1968): pp. 589-609. []
  2. Springate, Gordon L.V., “Predicting the Possibility of Failure in a Canadian Firm”. Unpublished M.B.A. Research Project, Simon Fraser University, January 1978. []
  3. Fulmer, John G. Jr., Moon, James E., Gavin, Thomas A., Erwin, Michael J., “A Bankruptcy Classification Model For Small Firms”. Journal of Commercial Bank Lending (July 1984): pp. 25-37. []
  4. “C.A. – Score, A Warning System for Small Business Failures”, Bilanas (June 1987): pp. 29-31. []
  5. Berg, Daniel. “Bankruptcy Prediction by Generalized Additive Models.” Statistical Research Report. January 2005. Dept. of Math. University of Oslo. 20 Mar 2009 <>. []
  6. Dakovic, Rada,Claudia Czado,Daniel Berg. Bankruptcy prediction in Norway: a comparison study. June 2007. Dept. of Math. University of Oslo. 20 Mar 2009 <>. []
  7. Altman, Edward I.. “Revisiting Credit Scoring Models in a Basel 2 Environment.” Finance Working Paper Series . May 2002. Stern School of Business. 20 Mar 2009 <>. []
  8. Ricci, Cecilia Wagner. “Bankruptcy Prediction: The Case of the CLECS.” Mid-American Journal of Business 18(2003): 71-81. []


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S@R develops models for support of decision making under uncertainty. Taking advantage of recognized financial and economic theory, we customize simulation models to fit specific industries, situations and needs.

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