FRM Exam Part I · Measuring Credit Risk

Credit Ratings and Rating Transition Matrices Explained

Updated 11 October 2026 · Fact-checked

A credit rating ranks a borrower's default risk on an ordinal scale. A transition matrix gives the probability of moving from each rating to every other rating, including default, over one period. To get multi-year default probabilities, multiply the matrix by itself and read the default column.

Understand Credit Ratings and Transition Matrices

A credit rating is an opinion on how likely a borrower is to default and how much a lender may lose. Agencies such as S&P, Fitch and Moody's use letter scales. S&P and Fitch run from AAA down to D (Fitch also uses RD for restricted default). Moody's runs from Aaa down to C. Ratings of BBB- (S&P/Fitch) or Baa3 (Moody's) and above are investment grade. Anything below is speculative grade (high yield).

Ratings are ordinal. They rank risk but do not give an exact default probability. Agencies publish historical default rates for each rating, and these rise sharply as ratings fall.

A transition matrix (migration matrix) shows, for a one-year horizon, the probability that a borrower starting in rating i ends the year in rating j. Each row is a starting rating and sums to 100%. Default is usually an absorbing state: once a borrower defaults, it stays in default, so the default row is 0, 0, ..., 1. The diagonal is the probability of keeping the same rating and is typically the largest entry. The default column shows the one-year default probability for each starting rating.

For longer horizons, assume the process is Markov. That means next period's move depends only on the current rating, not on the past. Then the n-year matrix is the one-year matrix raised to the power n. The default column of that matrix is the cumulative default probability. Real ratings show momentum and cycle effects, so the Markov assumption is only an approximation.

Through-the-cycle (TTC) ratings aim to be stable across the business cycle. They look past temporary conditions and aim to reflect the borrower's long-run, cycle-neutral (often stress-aware) credit quality. Agency ratings are mostly TTC. They change rarely, so transition matrices show high diagonals. Point-in-time (PIT) ratings reflect current conditions and update quickly, as in many internal models and market-based measures. They move borrowers across grades more as conditions change, so the matrix has lower diagonals and more migration. Because borrowers migrate between grades as conditions change, the PD attached to each PIT grade tends to stay roughly stable over time. TTC ratings migrate less. The PD for a TTC grade is a long-run average, but the realized default rate within that grade moves with the cycle.

Key formulas to remember

Row sum property: Σj P(i → j) = 1 for every starting rating i; Each row of the matrix covers all outcomes, including default, so it must total 100%.
Absorbing default state: P(D → D) = 1, P(D → any other) = 0; Used in standard exam matrices. Default is never exited.
Two-year transition matrix: P(2) = P × P; Valid under the Markov and time-homogeneity assumptions. Use matrix multiplication, not squaring each entry.
n-year matrix: P(n) = Pⁿ; The default column of Pⁿ gives the cumulative n-year default probability for each starting rating.
Two-year default probability (one starting rating): PD(2) = Σk P(i → k) × P(k → D); Sum over every rating k reached after year 1, including D itself where P(D → D) = 1.
Marginal (unconditional) default probability: Marginal PD in year n = Cumulative PD(n) − Cumulative PD(n−1); This is the unconditional probability of defaulting in year n. It is not conditional on surviving to year n. The conditional PD in year n = [Cumulative PD(n) − Cumulative PD(n−1)] ÷ [1 − Cumulative PD(n−1)].
Survival probability: S(n) = 1 − Cumulative PD(n); Probability of not having defaulted by the end of year n.

How to solve Credit Ratings and Transition Matrices questions

Use this sequence for any question on ratings, transition matrices or cumulative default.

1Identify the starting rating and the horizon. Find the matching row.
2Check that default is an absorbing state and that rows sum to 100%. If a row does not sum to 100%, look for a missing column such as NR or a rounding note.
3For one year, read the default column of the starting row. That is the one-year PD.
4For two years, multiply the starting row by the default column: sum of P(i → k) × P(k → D) over all k, including k = D.
5For three or more years, repeat with the matrix multiplication, or compute the row vector after each year and then take its default entry.
6If asked for the marginal default probability in a year, subtract the earlier cumulative PD from the later one.
7If asked about the cycle, link the answer to the type of rating: TTC means stable and low migration, PIT means responsive and higher migration.
8Check the answer: a cumulative PD cannot fall as the horizon rises, and it must lie between 0 and 1.

Quickest way: Row-vector shortcut for two-year default

When to use it: Use when the question gives a small matrix (3 to 4 states) and asks for a two-year default probability from one starting rating.

Write the starting row as the probabilities after year 1.
Take the default column of the matrix as a vector of one-year PDs, with 1 for the default state itself.
Multiply each year-1 probability by its one-year PD and add. Include year-1 default directly.
You never need the full matrix square. One row times one column is enough.
Sanity check: the result should be at least the one-year PD.

Common mistakes in Credit Ratings and Transition Matrices

Squaring each matrix entry to get the two-year matrix.
It feels like 'multiply the matrix by itself'.
Fix: Use matrix multiplication: row times column, summing over intermediate ratings.
Forgetting the default-to-default term when computing two-year PD.
Students sum only over surviving ratings and drop the borrowers that defaulted in year 1.
Fix: Add P(i → D) × 1. Cumulative PD includes year-1 defaulters.
Treating the marginal PD of year 2 as the cumulative PD.
The question wording 'probability of default in year 2' is read as 'by year 2'.
Fix: Cumulative PD(2) − PD(1) gives default in year 2 only. 'By the end of year 2' means cumulative.
Saying agency ratings are point-in-time.
Ratings change after bad news, so students assume they track conditions.
Fix: Agency ratings are mainly TTC and move slowly. PIT ratings are typical of internal and market-based systems.
Assuming the default probability of a rating grade is constant under TTC.
Stable ratings sound like stable defaults.
Fix: TTC ratings are stable, but realized default rates per grade still vary with the cycle. PIT grades aim for steadier PD per grade.
Reading ratings as cardinal and adding or averaging letters.
Notches look like numbers.
Fix: Ratings are ordinal. Use the PD attached to each grade, not the notch count.

Worked examples

Example 1

A one-year transition matrix has states A, B and D (default). From A: to A 90%, to B 8%, to D 2%. From B: to A 10%, to B 80%, to D 10%. D is absorbing. Find the cumulative two-year default probability for a borrower starting in A.

Show the solution

Year-1 distribution from A: A 0.90, B 0.08, D 0.02.
Default probabilities in year 2 by position: from A 0.02, from B 0.10, from D 1.
Cumulative PD(2) = 0.90 × 0.02 + 0.08 × 0.10 + 0.02 × 1.
= 0.018 + 0.008 + 0.020 = 0.046.

Answer: 4.6%

Example 2

Using the same matrix, a borrower starts in B. Find (a) the two-year cumulative default probability and (b) the probability of defaulting during year 2 only.

Show the solution

Year-1 distribution from B: A 0.10, B 0.80, D 0.10.
Cumulative PD(2) = 0.10 × 0.02 + 0.80 × 0.10 + 0.10 × 1.
= 0.002 + 0.080 + 0.100 = 0.182.
Year-1 default probability is 0.10.
Default in year 2 only = 0.182 − 0.100 = 0.082.

Answer: (a) 18.2%; (b) 8.2%

Exam tips

Questions usually give a small matrix. Do row-times-column arithmetic carefully and show each product.
Always include the default state's own 1 when summing for multi-year PD.
Read the wording: 'by year 2' is cumulative, 'in year 2' is marginal.
For TTC versus PIT questions, link each to stability of ratings, migration frequency and procyclicality of capital.
Remember that the Markov assumption is a simplification. Rating momentum and cycle dependence break it in practice.

Practice questions from Measuring Credit Risk

Credit Ratings and Transition Matrices in other exams

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

Credit Ratings and Transition Matrices: frequently asked questions

How do I calculate cumulative default probability from a transition matrix?

Multiply the matrix by itself for each extra year and read the default column for your starting rating. For two years, you can also sum P(i → k) × P(k → D) over all k, including default itself. This assumes a Markov, time-homogeneous process.

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

Through-the-cycle ratings look past the current phase of the business cycle and change rarely. Point-in-time ratings reflect current conditions and change more often. Agency ratings are mainly TTC, while many internal models are PIT.

Why is default an absorbing state in a transition matrix?

Once a borrower defaults, the model treats it as staying in default. So the default row has 1 in the default column and 0 elsewhere. This makes cumulative default probabilities rise with the horizon.

Where is the investment grade cut-off on the agency scales?

Investment grade runs down to BBB- on the S&P and Fitch scales and Baa3 on Moody's. Ratings below that are speculative grade, also called high yield.