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FRM Exam Part II · Credit Scoring and Rating

Credit Scoring Models and Scorecards for FRM Part II

Updated 11 October 2026 · Fact-checked

Credit scoring models turn borrower data into a default score or probability. Logit and probit regressions give a probability between 0 and 1. Discriminant analysis, such as the Altman Z-score, gives a linear score that you compare with cut-offs. To solve questions, compute the score, map it to a zone or probability, then interpret it.

Understand Credit Scoring Models and Scorecards

A credit scoring model uses past borrower data to rank new borrowers by default risk. You pick a set of inputs, such as leverage, profitability, liquidity and payment history. The model combines them into one number. That number is either a score or a probability of default (PD).

Discriminant analysis builds a linear score, Z = a1x1 + a2x2 + ..., chosen to separate defaulters from non-defaulters as well as possible. The Altman Z-score is the best-known example. You compare Z with cut-offs to place a firm in a safe, grey or distress zone. It ranks firms. It does not directly give a PD.

Logit and probit are regression models for a yes/no outcome (default or no default). Both compute a linear index z = β0 + β1x1 + ... and then squeeze it into a probability between 0 and 1. Logit uses the logistic function. Probit uses the standard normal cumulative distribution N(z). Results are usually very close. Logit has fatter tails and its coefficients are easier to read through odds. A plain linear probability model can give probabilities below 0 or above 1, which is why it is not preferred.

A scorecard is a logit model turned into points. Each borrower characteristic is grouped into bands, and each band earns points. The total score maps to log-odds of good to bad. Lenders set cut-offs for approve, refer and decline. Scorecards are widely used in retail lending because they are transparent and easy to explain to regulators.

A model is only useful if it ranks well and stays stable. You test discrimination with the ROC curve, AUC, the Gini coefficient (2 × AUC − 1) and the KS statistic. You also test calibration, meaning predicted PDs match observed default rates. Testing on out-of-sample data guards against overfitting.

Key formulas to remember

Altman Z-score (original, public manufacturers)
Z = 1.2X1 + 1.4X2 + 3.3X3 + 0.6X4 + 1.0X5
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.
Altman Z-score zones (original model)
Z > 2.99 safe; 1.81 ≤ Z ≤ 2.99 grey; Z < 1.81 distress
Revised versions for private firms or non-manufacturers use different coefficients and cut-offs. Use the ones given in the question.
Logit probability
PD = 1 ÷ (1 + e^(−z)), where z = β0 + Σ βi·xi
A higher z means a higher PD when coefficients are set so that default is the outcome being modelled.
Log-odds and odds
ln[PD ÷ (1 − PD)] = z; odds = PD ÷ (1 − PD) = e^z
A one-unit rise in xi multiplies the odds by e^βi. It does not add βi to the PD.
Probit probability
PD = N(z), where N is the standard normal CDF
Same linear index as logit but a different link function.
Scorecard points scaling
Score = Offset + Factor × ln(odds good:bad); Factor = PDO ÷ ln 2
PDO is points to double the odds. Each PDO points added doubles the good:bad odds.
Gini coefficient from AUC
Gini = 2 × AUC − 1
AUC of 0.5 means no discrimination (Gini 0). AUC of 1 means perfect ranking (Gini 1).

How to solve Credit Scoring Models and Scorecards questions

Use the same routine for any credit scoring question. It stops you applying the wrong model or misreading the output.

  1. 1Identify the model type: discriminant (Z-score), logit, probit, or a points scorecard.
  2. 2Write down the inputs and check units. Ratios are usually decimals, not percentages, unless the question states otherwise.
  3. 3Compute the score or index: for Z-score, multiply each ratio by its coefficient and sum; for logit or probit, compute z = β0 + Σ βi·xi.
  4. 4Convert if needed: logit PD = 1 ÷ (1 + e^(−z)); probit PD = N(z); scorecard points via Offset + Factor × ln(odds).
  5. 5Interpret: compare Z with the zone cut-offs, or read PD against the lender's cut-off. For odds, use e^β.
  6. 6Sanity check: PD must lie between 0 and 1, and the direction must make sense (more leverage should not lower risk).
  7. 7If the question is about performance, name the right metric: AUC or Gini or KS for ranking, calibration for accuracy of PD levels.

Quickest way: Compute, convert, compare

When to use it: Use this for numerical MCQs where you are given coefficients and ratios and must pick a score, PD or zone.

  1. Compute the linear sum first. Do not touch the other options until you have it.
  2. For logit, remember z = 0 gives PD = 50%. A negative z gives PD below 50%, a positive z above 50%. This often eliminates two options at once.
  3. For Z-score, compare with 1.81 and 2.99 only after the sum is done.
  4. For odds-ratio questions, calculate e^β. A coefficient of 0.7 means odds rise by a factor of about 2.01, not 70%.
  5. Spot trap options: 1 − PD, the odds instead of PD, or Z in the wrong zone.

Common mistakes in Credit Scoring Models and Scorecards

  • Treating the Z-score as a probability of default

    Both are numbers used to judge default risk, so they look alike.

    Fix: A Z-score is a ranking score. You can map it to a PD only with a separate calibration. Zones are safe, grey and distress.

  • Mixing up the Altman X4 definition

    Students remember 'equity' and forget which equity and which denominator.

    Fix: In the original model X4 is market value of equity ÷ book value of total liabilities. Revised models for private firms use book equity instead.

  • Adding the logit coefficient to the PD

    Linear regression habits carry over.

    Fix: The coefficient acts on log-odds. A one-unit rise in x multiplies the odds by e^β. The change in PD depends on where you start.

  • Stating that logit and probit give very different answers

    Students focus on the formulas, not the results.

    Fix: Both use a linear index and give similar PDs. Logit has fatter tails and gives odds-ratio interpretation. Probit uses the normal CDF.

  • Confusing discrimination with calibration

    AUC and Gini sound like overall quality measures.

    Fix: AUC, Gini and KS measure ranking power only. A model can rank well and still understate PD levels, so calibration must be tested separately.

  • Using percentage figures in Z-score ratios

    Data are quoted as '20%' and are entered as 20.

    Fix: Convert to decimals (0.20) unless the question states the model uses percentage inputs.

Worked examples

Example 1

A listed manufacturer has working capital ÷ total assets = 0.20, retained earnings ÷ total assets = 0.30, EBIT ÷ total assets = 0.10, market value of equity ÷ book liabilities = 1.50 and sales ÷ total assets = 1.20. Using the original Altman Z-score, which statement is correct? A) Z = 2.45, grey zone. B) Z = 2.99, safe zone. C) Z = 3.09, safe zone. D) Z = 1.75, distress zone.

Show the solution
  1. Use Z = 1.2X1 + 1.4X2 + 3.3X3 + 0.6X4 + 1.0X5.
  2. 1.2 × 0.20 = 0.24.
  3. 1.4 × 0.30 = 0.42.
  4. 3.3 × 0.10 = 0.33.
  5. 0.6 × 1.50 = 0.90.
  6. 1.0 × 1.20 = 1.20.
  7. Sum: 0.24 + 0.42 + 0.33 + 0.90 + 1.20 = 3.09.
  8. 3.09 is above 2.99, so the firm is in the safe zone.

Answer: C) Z = 3.09, safe zone.

Example 2

A bank's logit model for default is z = −3.0 + 0.8 × (debt ÷ EBITDA) − 1.5 × (interest cover ratio ÷ 10). A borrower has debt ÷ EBITDA = 2.5 and interest cover ratio ÷ 10 = 1.0. What is the model PD? A) 2.5%. B) 7.6%. C) 12.2%. D) 92.4%.

Show the solution
  1. Compute z = −3.0 + 0.8 × 2.5 − 1.5 × 1.0.
  2. 0.8 × 2.5 = 2.0, so z = −3.0 + 2.0 − 1.5 = −2.5.
  3. PD = 1 ÷ (1 + e^(−z)) = 1 ÷ (1 + e^2.5).
  4. e^2.5 ≈ 12.1825, so PD = 1 ÷ 13.1825 ≈ 0.0759.
  5. Check: z is negative, so PD should be below 50%. 7.6% fits. 92.4% is the probability of no default (1 − PD).

Answer: B) about 7.6%.

Exam tips

  • Memorise the five Altman coefficients and the 1.81 and 2.99 cut-offs. Questions often give ratios and expect you to supply the weights.
  • Check the sign of z first in logit questions. It tells you whether PD is above or below 50% and removes trap options.
  • Know the difference between discrimination (AUC, Gini, KS) and calibration. Case questions often ask which test addresses which weakness.
  • Expect conceptual items on logit versus probit versus linear probability, and on why scorecards are favoured for retail portfolios: transparency and ease of explaining decisions.
  • Read what the question asks: a score, a PD, odds, or a zone. Each needs a different final step.

Practice questions from Credit Scoring and Rating

Credit Scoring Models and Scorecards: frequently asked questions

What is the Altman Z-score formula and how do I interpret it?

The original formula is Z = 1.2X1 + 1.4X2 + 3.3X3 + 0.6X4 + 1.0X5, built from five financial ratios. A Z above 2.99 is the safe zone, between 1.81 and 2.99 is grey, and below 1.81 is distress. Revised versions for private or non-manufacturing firms have different coefficients and cut-offs.

What is the difference between logit and probit credit scoring?

Both model default as a binary outcome using a linear index z. Logit converts z with the logistic function, PD = 1 ÷ (1 + e^(−z)). Probit uses the standard normal CDF, PD = N(z). They give similar PDs, but logit coefficients read directly as changes in log-odds.

How do you build a credit scorecard using logistic regression?

Collect historical borrower data with a default flag, group variables into bands, and fit a logistic regression on a development sample. Convert coefficients into points using the Offset and Factor scaling, then set cut-offs. Finally, validate on out-of-sample data for ranking power and calibration.

Why not use a linear probability model for default?

It can output probabilities below 0 or above 1, and its errors are not constant in size. Logit and probit keep PDs between 0 and 1. That is why they are the standard choice.