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 ofPrecedents
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" zoneZ-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" zoneExamples
See also
* Standard score * ''Z''-test * Z-factor * Ohlson O-scoreReferences
Further reading
* Caouette, John B; Edward I Altman, Paul Narayanan (1998). ''Managing Credit Risk – the Next Great Financial Challenge'',External links