A distress-zone Z-score does not mean a 70% chance of bankruptcy

The Altman Z-score is the most widely used bankruptcy formula ever published. Score below 1.81 and you are in the distress zone, and the sentence that follows — in textbooks, on finance sites, in our own calculator until this week — is some version of this:

About 70% of firms in the distress zone went bankrupt within two years.

That is not what the number measures. It is the same statistic read backwards, and reading it that way requires roughly one in five American industrial companies to go bankrupt every two years.

What the 72% actually counts

Altman's 1968 study used 66 firms: 33 that went bankrupt and 33 that did not. The number everyone quotes comes from his classification results two statements before bankruptcy:

The 72% is 23 divided by 32: of the firms that did go bankrupt, this share had already scored in the distress zone. It is a property of the failures, measured after the fact. It says nothing about what happens to a firm scoring low today.

The two questions are distinct, and they have different answers:

• Sensitivity — of the firms that failed, how many did we flag? That is Altman's 72%.
• Predictive value — of the firms we flag, how many fail? That is the number a reader actually wants, and it is not in the Type I column.

Even inside Altman's own sample, the answer is not 70%

The in-sample predictive value is computable from the same table. The firms the model placed in the distress zone are the 23 bankrupt firms it caught, plus the 2 healthy firms it wrongly flagged — 33 healthy firms at 94% correct leaves 2 errors. So:

23 ÷ (23 + 2) = 0.92

⚠️ So in Altman's own sample the distress zone was 92% predictive. The popular 70% is not a rounded version of the right number; it is a different cell of the table, relabelled.

And 92% cannot leave the sample either

Here is the part that matters more than the mix-up. Altman's sample was built as 33 bankrupt and 33 healthy by construction — a matched design, chosen so the model could be fitted. That embeds a 50% bankruptcy rate into the sample before a single ratio is computed.

Real economies do not have a 50% corporate bankruptcy rate. Any predictive value read off a matched sample is tied to that sample's artificial base rate. Exporting it to a general population is the same invalid step that makes a 99%-accurate test for a rare disease produce mostly false positives.

What the popular reading would require

We can put a number on how wrong it is, using only figures Altman published himself. Writing π for the share of firms that actually go bankrupt in a two-year window, Bayes gives:

P(bankrupt | Z < 1.81) = 0.72 × π ÷ P(Z < 1.81)

Altman reports that by 1999, "the proportion of U.S. industrial firms … that had Z-Scores below 1.81 was over 20%". Taking that 20%:

P(bankrupt | Z < 1.81) = 0.72 × π ÷ 0.20 = 3.6 π

Set that equal to the claimed 70% and solve:

π = 0.70 ÷ 3.6 = 0.194

⚠️ 19.4% of US industrial firms would have to go bankrupt within two years for the sentence to be true. Nothing close to that has ever happened, in any period, including the ones we name after their bankruptcies.

What the number roughly is instead

Run the same formula with a plausible rate. If 1.5% of listed industrial firms fail in a given two-year window:

0.72 × 0.015 ÷ 0.20 = 0.054

About 5%. A distress-zone score raises the odds of bankruptcy roughly fivefold against a small base rate; it is a screening result, not a verdict. A fivefold lift is a useful signal, but it is not a coin flip weighted towards disaster.

What this does not mean

It does not mean the Z-score is broken. It separated bankrupt from healthy firms remarkably well on the data it was fitted to, it has survived fifty years of out-of-sample testing better than most models of anything, and Altman himself flagged the drift: he notes the false-positive rate rose substantially, with "as much as 15-20% of all firms" scoring below 1.81 by the late 1990s. A model whose author publishes its degradation is being used honestly.

It does not mean low scores are safe to ignore either. A fivefold lift on bankruptcy risk is exactly the kind of thing a screen is for.

The distress zone is a reason to look, not a finding. Treating it as a probability requires inverting a conditional, and the inversion always runs in the direction that overstates alarm.

What we changed on our own tool

Our Altman Z-score calculator made this exact error. Its zone legend read "Z < 1.81: DISTRESS (~70% of these firms filed bankruptcy within 2 years in Altman's sample)" — the inversion, stated as ours. It now reports the 72% as the sensitivity it is, and says plainly that it is not the share of low scorers who fail.

We removed something else in the same pass. The page claimed "48% at 5-year" accuracy, and we could not verify from a primary source which horizon that figure belongs to. Rather than swap in a different unsourced number — which would be the same mistake wearing a correction's clothes — we took it out and kept what is sourced: 95% at one year, 72% at two, and Altman's own note that accuracy declines "with the one exception of the fourth and fifth years, when the results are reversed from what would be expected". His decay is not smooth, so no single long-horizon figure describes it honestly.

Where this comes from, and what will date it

The sample design, the classification table and the two quoted sentences are from Altman's own July 2000 restatement of the 1968 paper, Predicting Financial Distress of Companies: Revisiting the Z-Score and ZETA Models, read from an archived copy of the author's university page. Every calculation above reproduces from those figures and is shown in full rather than asserted.

⚠️ Two limits. The 1.5% failure rate in the worked example is illustrative, not measured — it is there to show the shape of the answer, and the argument does not depend on it, because the 19.4% requirement is derived from Altman's numbers alone. And we read the author's restatement rather than the 1968 journal article itself; the tables are his, but a reader wanting the original should say so.

This will not date. It is an argument about what a conditional probability means, made against a table published in 1968, and neither half is going to move. What could change is the base rate in the worked example, which would shift the 5% up or down without touching the reasoning.