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FRM Exam Part II · Estimating Default Probabilities

Credit Scoring and Altman Z-Score Explained

Updated 11 October 2026 · Fact-checked

Credit scoring uses statistical models to turn borrower data into a default-risk score. Altman's Z-score is a discriminant model that weights five accounting ratios into one number. Compute Z, compare it with the cut-offs (below 1.81 distress, above 2.99 safe for the original model), and read the zone.

Understand Credit Scoring and Altman Z-Score

A credit scoring model converts borrower characteristics, usually accounting ratios, into a single score that ranks default risk. The idea is simple. Firms that failed in the past looked different from firms that survived. A model finds which ratios separate the two groups best.

Altman's Z-score (1968) uses multiple discriminant analysis (MDA). MDA finds the linear combination of variables that maximises the gap between the average scores of defaulters and non-defaulters, relative to the spread within each group. The result is Z = a weighted sum of five ratios covering liquidity, cumulative profitability, operating profitability, leverage and asset turnover.

You read Z against cut-offs. In the original model for public manufacturers, Z above 2.99 is the safe zone, Z below 1.81 is the distress zone, and between them is the grey zone. A lower Z means higher default risk. Z gives a ranking and a zone. It is not itself a probability of default.

Logit and probit models are the main alternatives. They regress a 0/1 default indicator on explanatory variables and output a probability between 0 and 1. Logit uses the logistic function, probit uses the normal CDF. They need no assumption that the variables are multivariate normal, which MDA assumes, and their output is a PD directly.

Limits matter for the exam. Accounting ratios are backward looking and reported infrequently. Models are fitted to a specific sample and period, so performance can decay. Weights and cut-offs differ for private firms, non-manufacturers and emerging markets. Model errors are Type I (a defaulter classed as safe) and Type II (a healthy firm classed as risky). Type I is usually costlier for a lender.

Key formulas to remember

Original Altman Z-score (public manufacturers)
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 ÷ book value of total liabilities; X5 = sales ÷ total assets. Ratios are entered as decimals.
Original zone cut-offs
Z > 2.99 safe; 1.81 ≤ Z ≤ 2.99 grey; Z < 1.81 distress
Lower Z means higher default risk. These apply to the original model only.
Z'-score (private firms)
Z' = 0.717·X1 + 0.847·X2 + 3.107·X3 + 0.420·X4 + 0.998·X5
X4 uses book value of equity instead of market value. Cut-offs are about 1.23 and 2.90. Know that the version changes weights and cut-offs.
Logit model
PD = 1 ÷ (1 + e^−(b0 + b1·x1 + … + bn·xn))
Output lies between 0 and 1. A positive coefficient on a variable raises PD as that variable rises.
Probit model
PD = N(b0 + b1·x1 + … + bn·xn)
N is the standard normal CDF. Same idea as logit with a different link function.

How to solve Credit Scoring and Altman Z-Score questions

Use this routine for any question on Z-scores or credit scoring models.

  1. 1Identify the model type: discriminant (Z-score) or regression (logit or probit). This decides whether the output is a score or a probability.
  2. 2For Z-score questions, check which version is given (original, Z' or others) and use its own weights and cut-offs.
  3. 3Compute each ratio as a decimal from the data given. Check the definition of X4 (market equity ÷ total liabilities).
  4. 4Multiply each ratio by its weight and sum them to get Z.
  5. 5Compare Z with the cut-offs and name the zone. State that lower Z means higher default risk.
  6. 6For logit or probit, compute the linear index first, then apply the logistic or normal function to get PD.
  7. 7Interpret in context: ranking versus probability, Type I versus Type II error, and limits such as backward-looking inputs.

Quickest way: Weighted-sum shortcut with a sanity check

When to use it: Use it for numeric Z-score MCQs when options are far apart and time is short.

  1. Write the five weights 1.2, 1.4, 3.3, 0.6, 1.0 next to the five ratios.
  2. Compute the heaviest term first, 3.3 × EBIT/TA, since it often dominates.
  3. Add the remaining terms and round to two decimals.
  4. Compare with 1.81 and 2.99 to pick the zone.
  5. If the question asks about a change, only recompute the changed term and apply the weight to the change.

Common mistakes in Credit Scoring and Altman Z-Score

  • Treating Z as a probability of default.

    Both are risk numbers and the text calls Z a predictor of failure.

    Fix: Z is a score that places the firm in a zone. Only logit and probit give a PD directly.

  • Entering ratios as percentages, such as 15 instead of 0.15.

    Ratios are quoted in percent in many data tables.

    Fix: Convert to decimals before multiplying by the weights.

  • Using book value of equity in X4 for the original model.

    Book equity is easier to find on a balance sheet.

    Fix: Original Z uses market value of equity ÷ book value of total liabilities. Book equity belongs to Z'.

  • Reading a higher Z as higher risk.

    Most risk measures rise with risk.

    Fix: Higher Z means safer. Below 1.81 is distress.

  • Saying logit and probit assume multivariate normal predictors.

    Mixing up the MDA assumptions with regression models.

    Fix: MDA assumes normally distributed predictors with equal covariance across groups. Logit and probit do not need that and give probabilities.

  • Applying the original cut-offs to banks or private firms without comment.

    Memorising one set of numbers.

    Fix: Remember the original model was built on public manufacturers. Other versions use different weights and cut-offs.

Worked examples

Example 1

A listed manufacturer has total assets of $500m, working capital $75m, retained earnings $100m, EBIT $60m, market value of equity $400m, total liabilities $250m and sales $600m. Using the original Altman model, find Z and the zone.

Show the solution
  1. X1 = 75 ÷ 500 = 0.15
  2. X2 = 100 ÷ 500 = 0.20
  3. X3 = 60 ÷ 500 = 0.12
  4. X4 = 400 ÷ 250 = 1.60
  5. X5 = 600 ÷ 500 = 1.20
  6. Z = 1.2(0.15) + 1.4(0.20) + 3.3(0.12) + 0.6(1.60) + 1.0(1.20)
  7. Z = 0.18 + 0.28 + 0.396 + 0.96 + 1.20 = 3.016
  8. 3.016 is above 2.99.

Answer: Z ≈ 3.02, which is in the safe zone, though only just above the 2.99 cut-off.

Example 2

A bank's logit model gives the index b0 + Σbixi = −2.20 for a borrower. Find the PD. Use e^2.2 = 9.025.

Show the solution
  1. PD = 1 ÷ (1 + e^−(−2.20))
  2. The exponent is −(−2.20) = +2.20, so PD = 1 ÷ (1 + e^2.20)
  3. e^2.20 = 9.025
  4. PD = 1 ÷ 10.025 = 0.0998

Answer: PD ≈ 9.98%, about 10%.

Exam tips

  • Know the five ratios and their weights cold. The 3.3 on EBIT/TA is the one most often tested.
  • Expect conceptual questions comparing MDA with logit and probit: assumptions, output type and interpretability.
  • Check that the question gives market or book equity so you pick the right Z version.
  • Be ready to link Type I and Type II errors to cut-off choice. A higher cut-off catches more defaulters but flags more healthy firms.
  • Cite limits when asked: backward-looking data, sample dependence, and accounting manipulation.

Practice questions from Estimating Default Probabilities

Credit Scoring and Altman Z-Score in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Credit Scoring and Altman Z-Score: frequently asked questions

What is the Altman Z-score formula?

For public manufacturers, Z = 1.2·X1 + 1.4·X2 + 3.3·X3 + 0.6·X4 + 1.0·X5. The ratios are working capital, retained earnings and EBIT over total assets, market equity over total liabilities, and sales over total assets.

How do I interpret the Z-score?

In the original model, Z above 2.99 is the safe zone, below 1.81 is the distress zone, and in between is the grey zone. A lower score means higher default risk.

How is logit different from discriminant analysis?

Discriminant analysis builds a linear score and does not give a probability directly. It assumes normally distributed predictors. Logit models the default probability itself through the logistic function and avoids that normality assumption.

Does Z-score give a probability of default?

No. It classifies firms into zones. To get a PD you must map scores to observed default rates or use a logit or probit model.