FRM Exam Part II · Estimating Default Probabilities

Rating Transition Matrices for FRM Part II

Updated 11 October 2026 · Fact-checked

A rating transition matrix lists the probability that a borrower moves from each starting rating to every other rating, including default, over one period, usually a year. Each row sums to 1. For a two-year matrix, multiply the one-year matrix by itself. The two-year default probability is in the default column.

Understand Rating Transition Matrices

A rating transition matrix (also called a migration matrix) summarises how credit quality changes. Rows are the rating at the start of the year. Columns are the rating at the end. The entry in row i, column j is the probability that a borrower rated i at the start is rated j at the end.

Every row is a full set of outcomes, so each row sums to 1 (100%). The diagonal holds the probability of staying put, and it is usually the largest number in each row. Default is normally treated as an absorbing state: once a borrower defaults, it stays in default. So the default row is 0, 0, ..., 1. Some matrices also include a 'not rated' column. Read the question to see whether it is there.

Agencies estimate these probabilities from history. In the simple cohort approach you take all borrowers with a given rating at the start of a year and count how many ended in each rating. You then average across many years. Default rates rise sharply as ratings fall, and downgrades from low grades are more likely than from high grades.

To move from a one-year to a multi-year matrix, assume migration follows a Markov process: next year's move depends only on today's rating, not on the path or the past. The n-year matrix is then the one-year matrix multiplied by itself n times, written P^n. This lets you see that a borrower can default in year 2 after first being downgraded in year 1. In practice, ratings show momentum and depend on the economic cycle, so the Markov assumption is only an approximation.

Two more ideas matter. Through-the-cycle (TTC) ratings aim to look past the business cycle and change slowly, so they are more stable and have lower migration. Because grades stay fixed while conditions change, the default probability per grade varies with the cycle. Point-in-time (PIT) ratings reflect current conditions and change more often, so migration is higher, while the default probability per grade stays more stable across the cycle. Agency ratings are generally described as closer to TTC; internal bank models are often closer to PIT. Moody's and S&P publish their own matrices on different scales: Moody's uses Aaa, Aa1, Aa2 and so on, with Baa3 as the lowest investment grade; S&P uses AAA, AA+, AA and so on, with BBB- as the lowest investment grade. Their migration patterns are broadly similar but not identical, because methodologies, samples and treatment of withdrawn ratings differ.

Key formulas to remember

Row sum condition: Σj P(i, j) = 1 for every row i; Use it to check a matrix or to find a missing entry. The default row is (0, ..., 0, 1) when default is absorbing.
Two-year transition matrix: P(2) = P × P; Matrix multiplication under the Markov assumption. Row of the first matrix times column of the second.
Entry of a two-year matrix: P2(i, k) = Σj P(i, j) × P(j, k); Sum over every intermediate rating j after year 1, including default (j = D) when k is default.
n-year matrix: P(n) = P^n; The default column of P^n gives the cumulative default probability over n years for each starting rating.
Marginal default probability in year 2: PD(year 2, unconditional) = PD(2 yr cumulative) − PD(1 yr); Probability of defaulting in year 2 and not before.
Conditional default probability in year 2: PD(year 2 | survived year 1) = [PD(2 yr) − PD(1 yr)] ÷ [1 − PD(1 yr)]; Divide by the survival probability. This is the figure comparable to a hazard rate.
Constant-hazard shortcut (no migration): PD(n yr) = 1 − (1 − q)^n; Only if the annual default probability q is the same each year. A matrix with migration gives a different answer, so use P^n when asked.

How to solve Rating Transition Matrices questions

Use this method for any question on transition matrices, whether it asks for a two-year probability, a default probability or an interpretation.

1Identify the states and the starting rating. Find the starting row and note whether default is a column.
2Check the row sums to 1. If an entry is missing, set it equal to 1 minus the others in that row.
3Decide what is asked: stay probability, migration to a specific rating, cumulative PD, or a year-2 conditional PD.
4For two years, take the starting row of P and multiply it by each column of P. Write each term as (year 1 move) × (year 2 move) and add them.
5If the target is default, remember that the default row of P is (0, ..., 1). A borrower already in default after year 1 stays in default, so that path counts with probability 1.
6Check the answer: the two-year row should still sum to 1, and the two-year PD should be at least the one-year PD.
7For marginal or conditional PD, subtract the one-year PD from the two-year PD, and divide by 1 minus the one-year PD for the conditional figure.
8State the interpretation: the assumption is Markov, and TTC ratings give a more stable matrix than PIT ratings.

Quickest way: Row-times-column shortcut for a single two-year cell

When to use it: When the question asks for just one number, such as the two-year default probability for one rating. Do not build the full matrix.

Write the starting row of P as a list of probabilities.
Write the target column of P as a list, with the default row's entry last (1 for the default column).
Multiply the two lists term by term and add the products.
Sanity check: the result must be at least the one-year probability for the default column, and between 0 and 1.
If the question gives answer options, estimate the size first. Default paths through the 'stay' rating are usually small, so a number close to the one-year PD times 2 is a quick plausibility check for high grades.

Common mistakes in Rating Transition Matrices

Squaring each entry instead of multiplying matrices
Students treat P² as a cell-by-cell square because the notation looks like a power of a number.
Fix: Use row-times-column multiplication. Each two-year cell is a sum over all intermediate ratings.
Leaving out the default path in year 1 when computing two-year PD
Students only multiply year-1 migration to a live rating by its year-2 PD and forget that a year-1 default carries on to year 2.
Fix: Include the term P(i, D) × 1. Because default is absorbing, the two-year PD is the one-year PD plus the paths that default in year 2.
Doubling the one-year PD to get the two-year PD
It feels like a quick approximation.
Fix: Use P² or, if given, a constant-hazard formula 1 − (1 − q)². Doubling ignores survival and, for low ratings, the chance of downgrade before default.
Confusing the unconditional and conditional year-2 default probability
Both numbers are 'the probability of default in year 2' in everyday language.
Fix: Unconditional is PD(2) − PD(1). Conditional on survival divides that by 1 − PD(1). Read for the words 'given that it survived'.
Reading the matrix columns as the starting rating
Students scan down a column instead of across a row.
Fix: Rows are 'from', columns are 'to'. Probabilities across a row sum to 1; down a column they do not.
Treating TTC as 'more accurate' or PIT as 'worse'
Students think one approach must be better.
Fix: They serve different purposes. TTC ratings are stable and have lower migration but respond slowly to deterioration. PIT ratings react fast, migrate more and make PD per grade less variable, but they make capital more cyclical.

Worked examples

Example 1

A one-year transition matrix has three states. From A: 90% to A, 8% to B, 2% to Default. From B: 10% to A, 80% to B, 10% to Default. Default is absorbing. Find the two-year probability that a borrower rated A today is in default, and the full two-year row for A.

Show the solution

Row A of P = (0.90, 0.08, 0.02). Default column of P = (0.02, 0.10, 1) for rows A, B, D.
Two-year A to Default = 0.90 × 0.02 + 0.08 × 0.10 + 0.02 × 1 = 0.018 + 0.008 + 0.020 = 0.046.
A to A = 0.90 × 0.90 + 0.08 × 0.10 + 0.02 × 0 = 0.81 + 0.008 = 0.818.
A to B = 0.90 × 0.08 + 0.08 × 0.80 + 0.02 × 0 = 0.072 + 0.064 = 0.136.
Check: 0.818 + 0.136 + 0.046 = 1.000.

Answer: The two-year default probability for an A-rated borrower is 4.6%. The two-year row for A is (81.8%, 13.6%, 4.6%).

Example 2

Using the same matrix, a borrower is rated B today. Find the two-year probability of being in default, the probability of being rated A after two years, and the probability of defaulting in year 2 given it survived year 1.

Show the solution

Row B of P = (0.10, 0.80, 0.10).
Two-year B to Default = 0.10 × 0.02 + 0.80 × 0.10 + 0.10 × 1 = 0.002 + 0.080 + 0.100 = 0.182.
Two-year B to A = 0.10 × 0.90 + 0.80 × 0.10 + 0.10 × 0 = 0.09 + 0.08 = 0.17.
Check the row: B to B = 0.10 × 0.08 + 0.80 × 0.80 = 0.008 + 0.640 = 0.648. Then 0.17 + 0.648 + 0.182 = 1.000.
Unconditional year-2 default = 0.182 − 0.100 = 0.082.
Conditional on surviving year 1: 0.082 ÷ (1 − 0.10) = 0.082 ÷ 0.90 = 0.0911.

Answer: Two-year default probability is 18.2%. Probability of being rated A after two years is 17.0%. The conditional year-2 default probability, given survival of year 1, is about 9.11%.

Exam tips

Questions are usually small matrices (two to four states). Write the row and column as lists and compute only the cell asked for.
Always include the default-in-year-1 path with weight 1. Wrong answer options are often built by leaving it out.
Read carefully whether the question wants cumulative, marginal (unconditional) or conditional default probability.
Be ready for conceptual MCQs: rows sum to 1, default is absorbing, the Markov assumption is an approximation, and TTC ratings migrate less than PIT ratings.
Do not quote specific historical agency migration rates from memory. Questions give the numbers; learn the direction of the patterns instead: default risk rises sharply as ratings fall, and the diagonal (stay) probabilities are usually the largest entries in each row.

Practice questions from Estimating Default Probabilities

Rating 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.

Rating Transition Matrices: frequently asked questions

How do you compute a two-year transition matrix from a one-year matrix?

Multiply the one-year matrix by itself using row-times-column multiplication. Each entry is the sum over all intermediate ratings after year 1 of the year-1 move times the year-2 move. This relies on the Markov assumption.

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

Through-the-cycle ratings aim to look past the economic cycle, so they change slowly and show lower migration. Point-in-time ratings respond to current conditions, so they change more often and their default probability per grade is more stable across the cycle. Agency ratings are generally seen as closer to through-the-cycle.

Why is default treated as an absorbing state?

Once a borrower defaults, the matrix does not let it return to a live rating. The default row is therefore all zeros with a 1 in the default column. This is what makes cumulative default probabilities rise with the horizon.

How do Moody's and S&P rating migrations differ?

The scales are labelled differently: Moody's uses Aaa, Aa1 and so on with Baa3 as the lowest investment grade; S&P uses AAA, AA+ and so on with BBB- as the lowest investment grade. Their migration patterns are broadly similar, but exact figures differ because of methodology, sample and treatment of withdrawn ratings. In the exam you work with the numbers given.