Lay out how Altman's Z-score method works and put it to use in forecasting default for a firm and in sorting a set of firms into credit quality classes.
The idea
Altman's Z-score is a linear discriminant function fitted to separate bankrupt from non-bankrupt manufacturing firms. It uses five ratios. X1 is working capital over total assets, X2 retained earnings over total assets, X3 EBIT over total assets, X4 market value of equity over book value of total liabilities, and X5 sales over total assets. For public companies the original score weights them as follows. Z = 1.2 × X1 + 1.4 × X2 + 3.3 × X3 + 0.6 × X4 + 1.0 × X5. A higher Z means lower default risk. The usual cut-offs are about 2.99 and 1.81. Above 2.99 is the safe zone, below 1.81 is the distress zone, and in between is a grey zone where no firm call is made. Z is a score, not a probability, so turning it into a default probability needs a mapping from historical experience. Variants exist for private firms and non-manufacturers.
Formulas
Altman Z-score
Z = 1.2 × X₁ + 1.4 × X₂ + 3.3 × X₃ + 0.6 × X₄ + 1.0 × X₅
- X₁
- working capital ÷ total assets
- X₂
- retained earnings ÷ total assets
- X₃
- EBIT ÷ total assets
- X₄
- market value of equity ÷ total liabilities
- X₅
- sales ÷ total assets
Higher Z means lower default risk; 1.81 and 2.99 bound the grey zone. Z is a score, so a PD needs a mapping from historical default frequencies.
A practice question
In Altman's original Z-score for public manufacturing firms, which ratio carries the largest coefficient, 3.3?
- Retained earnings divided by total assetsWhy not: Retained earnings over total assets (X2) has a coefficient of 1.4, not 3.3.
- EBIT divided by total assets - correct
- Market value of equity divided by book value of total liabilitiesWhy not: This is X4, with a coefficient of 0.6.
- Sales divided by total assetsWhy not: Sales over total assets (X5) has a coefficient of 1.0.
Z = 1.2 X1 + 1.4 X2 + 3.3 X3 + 0.6 X4 + 1.0 X5, where X3 is EBIT over total assets, the most heavily weighted variable.
A harder one
A bank sorts four manufacturers using Z-score cut-offs of 2.99 and 1.81. Their scores are P 3.15, Q 2.60, R 1.95 and S 1.70. Which classification is correct?
- P safe, Q and R grey, S distress. - correct
- P and Q safe, R grey, S distress.Why not: Moves Q into the safe zone; Q's 2.60 is below the 2.99 safe cut-off.
- P, Q and R safe, S grey.Why not: Treats 1.81 as the cut-off for the safe zone and ignores the distress zone.
- P distress, Q and R grey, S safe.Why not: Inverts the scale; higher Z means lower default risk.
Above 2.99 is safe (P), between 1.81 and 2.99 is grey (Q and R) and below 1.81 is distress (S).
The hardest tier
A manufacturer has a Z-score of 3.20 with a market value of equity of USD 600 million and total liabilities of USD 400 million. After a market sell-off its equity value halves and all other inputs are unchanged. What is the new Z-score and zone, using cut-offs of 2.99 and 1.81?
- 2.45, grey zoneWhy not: Omits the 0.6 coefficient on X4 when computing the fall in Z.
- 2.75, grey zone - correct
- 1.60, distress zoneWhy not: Halves the whole score instead of changing only the market-equity term.
- 3.20, safe zoneWhy not: Assumes Z uses only accounting data; X4 uses the market value of equity, so Z changes.
X4 falls from 1.50 to 0.75, so Z falls by 0.6 x 0.75 = 0.45, giving 2.75, below the 2.99 safe cut-off and above 1.81.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
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