Example of an Excel spreadsheet that uses Altman Z-score to predict the probability_that_a_firm_will_go_into_bankruptcy_within_two_years_.html" ;"title="bankruptcy.html" ;"title="probability that a firm will go into bankruptcy">probability that a firm will go into bankruptcy within two years ">bankruptcy.html" ;"title="probability that a firm will go into bankruptcy">probability that a firm will go into bankruptcy within two years The Z-score formula for predicting bankruptcy was published in 1968 by Edward I. Altman, who was, at the time, an Assistant Professor of Finance at New York University. The formula may be used to predict the probability that a firm will go into bankruptcy within two years. Z-scores are used to predict corporate defaults and an easy-to-calculate control measure for the
financial distress Financial distress is a term in corporate finance used to indicate a condition when promises to creditors of a company are broken or honored with difficulty. If financial distress cannot be relieved, it can lead to bankruptcy. Financial distres ...
status of companies in academic studies. The Z-score uses multiple corporate income and balance sheet values to measure the financial health of a company.

The formula

The Z-score is a linear combination of four or five common business ratios, weighted by coefficients. The coefficients were estimated by identifying a set of firms which had declared bankruptcy and then collecting a matched sample of firms which had survived, with matching by industry and approximate size (assets). Altman applied the statistical method of
discriminant analysis Linear discriminant analysis (LDA), normal discriminant analysis (NDA), or discriminant function analysis is a generalization of Fisher's linear discriminant, a method used in statistics and other fields, to find a linear combination of features ...
to a dataset of publicly held manufacturers. The estimation was originally based on data from publicly held manufacturers, but has since been re-estimated based on other datasets 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 < $1 million were eliminated. The original Z-score formula was as follows: : ''Z'' = 1.2''X''1 + 1.4''X''2 + 3.3''X''3 + 0.6''X''4 + 1.0''X''5. : ''X''1 = ratio of working capital to total assets. Measures liquid assets in relation to the size of the company. : ''X''2 = ratio of retained earnings to total assets. Measures profitability that reflects the company's age and earning power. : ''X''3 = ratio of earnings before interest and taxes to total assets. Measures operating efficiency apart from tax and leveraging factors. It recognizes operating earnings as being important to long-term viability. : ''X''4 = ratio of market value of equity to book value of total liabilities. Adds market dimension that can show up security price fluctuation as a possible red flag. : ''X''5 = ratio of sales to total assets. Standard measure for total asset turnover (varies greatly from industry to industry). Altman found that the ratio profile for the bankrupt group fell at −0.25 avg, and for the non-bankrupt group at +4.48 avg.


Altman's work built upon research by accounting researcher William Beaver and others. In the 1930s and on, Mervyn and others had collected matched samples and assessed that various accounting ratios appeared to be valuable in predicting bankruptcy. Altman Z-score is a customized version of the discriminant analysis technique of R. A. Fisher (1936). William Beaver's work, published in 1966 and 1968, was the first to apply a statistical method, ''t''-tests to predict bankruptcy for a pair-matched sample of firms. Beaver applied this method to evaluate the importance of each of several accounting ratios based on univariate analysis, using each accounting ratio one at a time. Altman's primary improvement was to apply a statistical method, discriminant analysis, which could take into account multiple variables simultaneously.

Accuracy and effectiveness

In its initial test, the Altman Z-score was found to be 72% accurate in predicting bankruptcy two years before the event, with a Type II error (false negatives) of 6% (Altman, 1968). In a series of subsequent tests covering three periods over the next 31 years (up until 1999), the model was found to be approximately 80%–90% accurate in predicting bankruptcy one year before the event, with a Type II error (classifying the firm as bankrupt when it does not go bankrupt) of approximately 15%–20% (Altman, 2000). This overstates the predictive ability of the Altman Z-score, however. Scholars have long criticized the Altman Z-score for being “largely descriptive statements devoid of predictive content ... Altman demonstrates that failed and non-failed firms have dissimilar ratios, not that ratios have predictive power. But the crucial problem is to make an inference in the reverse direction, i.e., from ratios to failures.” From about 1985 onwards, the Z-scores gained wide acceptance by auditors, management accountants, courts, and database systems used for loan evaluation (Eidleman). The formula's approach has been used in a variety of contexts and countries, although it was designed originally for publicly held manufacturing companies with assets of more than $1 million. Later variations by Altman were designed to be applicable to privately held companies (the Altman Z'-score) and non-manufacturing companies (the Altman Z"-score). Neither the Altman models nor other balance sheet-based models are recommended for use with financial companies. This is because of the opacity of financial companies' balance sheets and their frequent use of off-balance sheet items. Modern academic default and bankruptcy prediction models rely heavily on market-based data rather than the accounting ratios predominant in the Altman Z-score.

Original Z-score component definitions

: ''X''1 = working capital / total assets : ''X''2 = retained earnings / total assets : ''X''3 = earnings before interest and taxes / total assets : ''X''4 = market value of equity / total liabilities : ''X''5 = sales / total assets Z-score bankruptcy model: : ''Z'' = 1.2''X''1 + 1.4''X''2 + 3.3''X''3 + 0.6''X''4 + 1''X''5 Zones of discrimination: : ''Z'' > 2.99 – "safe" zone : 1.81 < ''Z'' < 2.99 – "grey" zone : ''Z'' < 1.81 – "distress" zone

Z-score estimated for non-manufacturers and emerging markets

: ''X''1 = (current assets − current liabilities) / total assets : ''X''2 = retained earnings / total assets : ''X''3 = earnings before interest and taxes / total assets : ''X''4 = book value of equity / total liabilities Z-score bankruptcy model (non-manufacturers): : ''Z'' = 6.56''X''1 + 3.26''X''2 + 6.72''X''3 + 1.05''X''4http://people.stern.nyu.edu/ealtman/IRMC2014ZMODELpaper1.pdf Z-score bankruptcy model (emerging markets): : ''Z'' = 3.25 + 6.56''X''1 + 3.26''X''2 + 6.72''X''3 + 1.05''X''4 Zones of discrimination: : ''Z'' > 2.6 – "safe" zone : 1.1 < ''Z'' < 2.6 – "grey" zone : ''Z'' < 1.1 – "distress" zone


See also

Standard score In statistics, the standard score is the number of standard deviations by which the value of a raw score (i.e., an observed value or data point) is above or below the mean value of what is being observed or measured. Raw scores above the mean ...
* ''Z''-test * Z-factor *
Ohlson O-score The Ohlson O-score for predicting bankruptcy is a multi-factor financial formula postulated in 1980 by Dr. James Ohlson of the New York University Stern Accounting Department as an alternative to the Altman Z-score for predicting financial distres ...


The Use of Credit Scoring Modules and the Importance of a Credit Culture
by Dr. Edward I Altman, Stern School of Business, New York University.

Further reading

* Caouette, John B; Edward I Altman, Paul Narayanan (1998). ''Managing Credit Risk – the Next Great Financial Challenge'',
John Wiley & Sons John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, journals, and encyclopedias, in ...
: New York. * {{cite journal , last1= Mare , first1= Davide , last2= Moreira , first2= Fernando , last3= Rossi , first3= Roberto , date= 2016 , title= Nonstationary Z-score measures , ssrn= 2688367, journal= European Journal of Operational Research , volume = forthcoming , pages= 348–358 , doi= 10.1016/j.ejor.2016.12.001 , hdl= 20.500.11820/0a77ff25-1f7c-40a0-8eb4-57f7b19ba48a , url= https://www.research.ed.ac.uk/en/publications/0a77ff25-1f7c-40a0-8eb4-57f7b19ba48a , hdl-access= free

External links

Altman Z-score calculator
Financial ratios Bankruptcy Credit risk Credit scoring 1968 introductions