Three formulas, three sets of cutoffs
Altman re-estimated the model when replacing market equity with book equity; changing only that input in the original formula does not produce Z-prime. The four-factor Z-double-prime removes sales/assets to reduce industry effects. The zone ranges below are those reported by Cındık and Armutlulu (2021) for the conventional versions. The formulas use decimal ratios: 20% enters as 0.20. The original Z convention shown retains the 0.999 sales coefficient; some implementations round it to 1.0.
- The upper zone is often called the safe zone. That historical label does not establish that a company, bond or stock is safe.
- This guide puts exact cutoff values in the grey zone. Compare unrounded scores, and keep the convention consistent across the screen.
- The emerging-market bond-rating adaptation adds 3.25 to Z-double-prime. Do not compare that shifted score with the unshifted 1.10 and 2.60 cutoffs.
- Use one model within a comparison group. Differences between Z, Z-prime and Z-double-prime are not changes in a company over time.
| Model and intended use | Formula | Common distress / grey / upper zones |
|---|---|---|
| Z: publicly traded manufacturers | Z = 1.2A + 1.4B + 3.3C + 0.6D + 0.999E; D uses market equity | Below 1.81 / 1.81 to 2.99 / above 2.99 |
| Z-prime (Z'): private manufacturers | Z' = 0.717A + 0.847B + 3.107C + 0.420D + 0.998E; D uses book equity | Below 1.23 / 1.23 to 2.90 / above 2.90 |
| Z-double-prime (Z''): non-manufacturing, non-financial firms | Z'' = 6.56A + 3.26B + 6.72C + 1.05D; D uses book equity; no E | Below 1.10 / 1.10 to 2.60 / above 2.60 |