FRM Exam Part II · Credit Risk

Credit Scoring and Rating Systems for FRM Part II

Updated 11 October 2026 · Fact-checked

Credit scoring and rating systems rank borrowers by default risk. Scores come from statistical models; ratings come from agencies or internal grades. A rating transition matrix gives the probability of moving between grades over a period, and multiplying matrices gives multi-year default probabilities. TTC ratings are stable; PIT ratings move with conditions.

Understand Credit Scoring and Rating Systems

A credit rating is an ordinal grade of a borrower's default risk. External ratings come from agencies such as S&P, Moody's and Fitch. Internal ratings are assigned by the bank's own system, often by combining model output with analyst judgement. Under Basel's internal ratings-based approach, banks use internal ratings to estimate PD for capital.

A credit scoring model turns borrower data into a number. Examples are the Altman Z-score and logistic regression scorecards. Inputs are financial ratios, payment history and similar variables. The score is then mapped to a rating grade or a PD. Scoring is common in retail lending, where there are many small borrowers and much data.

A rating transition matrix shows, for each starting grade, the probability of ending in each grade (including default) after a fixed horizon, usually one year. Each row sums to 100%. Default is an absorbing state: once there, the borrower stays there. This is why the default column rises as you extend the horizon.

To get a multi-year PD, multiply the one-year matrix by itself. For two years, use the matrix squared. This assumes migration is a Markov process: next period's move depends only on today's grade, and the matrix is constant over time. Real ratings show momentum and cycle effects, so this is an approximation.

Through-the-cycle (TTC) ratings look past the current economic phase. They assume stress conditions and change slowly, so PDs per grade are stable. Point-in-time (PIT) ratings use current information and respond quickly to the cycle, so grade PDs and migrations vary. Agencies lean TTC. Many internal models lean PIT. PIT ratings make capital more procyclical.

Key formulas to remember

Row sum of a transition matrix: Σj P(i → j) = 100% for each starting grade i; Includes the default column. Use it to find a missing entry.
Two-year transition probability: P(i → k, 2 years) = Σj P(i → j) × P(j → k); This is the matrix product of the one-year matrix with itself. Assumes a constant Markov matrix.
Cumulative PD from the default column: PD(i, 2 years) = Σj P(i → j) × P(j → D), with P(D → D) = 100%; Default is absorbing. Include the direct one-year default and the paths through other grades.
Marginal (unconditional) default probability: Marginal PD in year 2 = Cumulative PD(2) − Cumulative PD(1); This is the unconditional probability of defaulting in year 2 specifically. A conditional PD would divide it by the probability of surviving year 1, which is 1 − Cumulative PD(1).
Logistic scorecard PD: PD = 1 ÷ (1 + e^−(b0 + b1x1 + … + bnxn)); Maps a linear score to a probability between 0 and 1.

How to solve Credit Scoring and Rating Systems questions

Use this order for transition-matrix and rating-system questions.

1Identify the type of question: matrix arithmetic, scoring model, or TTC versus PIT concept.
2For matrices, read the starting grade row and check that it sums to 100%. Fill any missing entry.
3Note the horizon asked for. If it is more than one year, plan to multiply matrices or sum over paths.
4List all paths from the starting grade to the target state in two steps, with the intermediate grade each time. Remember default stays default.
5Multiply probabilities along each path and add the paths.
6For marginal default in a later year, subtract the earlier cumulative PD.
7For TTC versus PIT, link the answer to stability, cyclicality and capital procyclicality.
8Sanity-check: probabilities must be between 0 and 1, and cumulative PD must not fall as the horizon lengthens.

Quickest way: Path-sum shortcut for two-year default

When to use it: Use when asked for a two-year PD from a small matrix with few grades.

Take only the starting grade's row and the default column.
Two-year PD = Σ P(start → j) × P(j → D). The term for j = D is P(start → D) × 1.
Skip any paths with zero probability.
Add the products and check the result exceeds the one-year PD.

Common mistakes in Credit Scoring and Rating Systems

Squaring each entry of the matrix instead of doing a matrix product.
Students treat the matrix like a list of independent numbers.
Fix: Multiply row by column and sum over intermediate grades.
Forgetting that default is absorbing and omitting P(D → D) = 100%.
The default row is often not shown.
Fix: Always add the direct one-year default path with weight 1 for a two-year PD.
Reporting cumulative PD when the question asks for default in year 2 only.
Both are called PD in notes.
Fix: Subtract the one-year cumulative PD from the two-year cumulative PD for the marginal figure.
Saying TTC ratings are more accurate at predicting next-year default.
Students confuse stability with accuracy.
Fix: PIT ratings track near-term default risk better. TTC ratings are stable and less procyclical.
Treating the Markov assumption as a proven fact.
Textbook calculations use it without comment.
Fix: State it as an assumption. Real migration shows momentum and depends on the cycle.

Worked examples

Example 1

A one-year transition matrix has three states. From A: 90% stay A, 8% move to B, 2% default. From B: 10% move to A, 80% stay B, 10% default. Default is absorbing. What is the two-year cumulative PD of a borrower rated A?

Show the solution

Paths from A to default in two years: A→A→D, A→B→D, A→D→D.
A→A→D: 0.90 × 0.02 = 0.018.
A→B→D: 0.08 × 0.10 = 0.008.
A→D→D: 0.02 × 1 = 0.020.
Sum: 0.018 + 0.008 + 0.020 = 0.046.

Answer: 4.6%

Example 2

Using the same matrix, a borrower rated B has a one-year PD of 10%. Compute its two-year cumulative PD and the marginal PD in year 2.

Show the solution

Paths: B→A→D, B→B→D, B→D→D.
B→A→D: 0.10 × 0.02 = 0.002.
B→B→D: 0.80 × 0.10 = 0.080.
B→D→D: 0.10 × 1 = 0.100.
Two-year cumulative PD: 0.002 + 0.080 + 0.100 = 0.182.
Marginal PD in year 2: 0.182 − 0.100 = 0.082.

Answer: Two-year cumulative PD is 18.2%; marginal year-2 PD is 8.2%.

Exam tips

Write out the paths explicitly. Most errors come from skipping the default-to-default path.
Read whether the question wants cumulative or marginal PD before calculating.
For TTC versus PIT, expect conceptual MCQs on procyclicality, rating stability and Basel capital.
Remember agency ratings lean TTC, while internal ratings and scoring models often lean PIT.
Check that each row sums to 100% before you start; a missing entry is often hidden in the question.

Practice questions from Credit Risk

Credit Scoring and Rating Systems 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 Rating Systems: frequently asked questions

What is a rating transition matrix?

It is a table of probabilities that a borrower moves from one rating grade to another, or to default, over a set period, usually one year. Each row sums to 100%. It is built from historical rating data.

What is the difference between through-the-cycle and point-in-time ratings?

TTC ratings assess risk across a full economic cycle and change slowly. PIT ratings reflect current conditions and change quickly. PIT ratings make capital requirements more procyclical.

How do I estimate default probability from rating migration?

Use the one-year matrix to read the one-year PD from the default column. For longer horizons, multiply the matrix by itself and read the default column. This assumes a constant Markov process.

What are the main credit scoring models?

Common ones are the Altman Z-score, logistic regression scorecards and other statistical models. They convert borrower data into a score that maps to a rating or a PD.