FRM Exam Part II · Credit Scoring and Rating

Rating Transition Matrices and Default Rates for FRM Part 2

Updated 11 October 2026 · Fact-checked

A rating transition matrix gives the probability that a borrower moves from one rating to another over a set horizon, with default as an absorbing state. Multiply the matrix by itself to get multi-year probabilities. The default column gives cumulative default probability. Marginal default is the extra default in one year.

Understand Rating Transition Matrices and Default Rates

A rating transition matrix (migration matrix) is a table. Each row is a starting rating. Each column is the rating at the end of the horizon, usually one year. Each cell is the probability of moving from the row rating to the column rating. Every row sums to 100%.

The last column is default. Default is an absorbing state: once a borrower defaults, it stays there. So the default row is 0, 0, ..., 1. Most of the probability sits on the diagonal, because most borrowers keep their rating over one year.

To get a multi-year matrix, multiply the one-year matrix by itself. This assumes a time-homogeneous Markov chain: next year's move depends only on today's rating, not on the past, and the matrix is the same each year. The default column of the n-year matrix is the cumulative default probability for each starting rating.

The marginal default probability (marginal default rate) in year n is the probability of defaulting in year n, given by cumulative PD(n) minus cumulative PD(n−1). The conditional (hazard) default rate divides that by the survival probability to the start of year n.

Real ratings are not Markov or constant. Rating momentum means a downgrade makes a further downgrade more likely than the matrix implies. Cyclicality means downgrades and defaults rise in recessions and fall in expansions. A through-the-cycle average matrix understates risk in downturns and overstates it in booms. Matrices conditioned on the economic state fix part of this.

Key formulas to remember

Row sum: Σj P(i → j) = 1; Each row of the matrix sums to 1, including the default column.
Multi-year matrix: P(n) = P^n (matrix power); Valid under the time-homogeneous Markov assumption. Use matrix multiplication, row times column.
Two-year transition probability: P2(i → k) = Σj P(i → j) × P(j → k); Sum over every intermediate rating j, including paths through default.
Cumulative default probability: CPD(n) = P^n(i → Default); Probability of defaulting at any time up to year n.
Marginal default probability: MPD(n) = CPD(n) − CPD(n−1); Unconditional probability of defaulting in year n only.
Conditional (hazard) default rate: h(n) = MPD(n) ÷ [1 − CPD(n−1)]; Default in year n given survival to the start of year n.
Cumulative from hazards: 1 − CPD(n) = Π [1 − h(t)], t = 1 to n; Survival is the product of one-year survival rates.

How to solve Rating Transition Matrices and Default Rates questions

Use this order for any question on transition matrices or default rates.

1Identify the starting rating and the horizon. Find the row for that rating.
2Check what the question asks for: transition probability, cumulative PD, marginal PD or conditional (hazard) PD.
3For a two-year result, multiply the starting row by the matrix. For each target column, sum row entry times that column's entry across all intermediate ratings.
4Remember that default is absorbing. Its row contributes 1 to the default column, so earlier defaults carry forward into cumulative PD.
5For marginal PD, subtract the previous cumulative PD from the current one.
6For conditional PD, divide the marginal PD by survival to the start of the year, which is 1 minus the previous cumulative PD.
7Sanity check: cumulative PD never falls, rows sum to 1, and lower ratings have higher PD.
8Add any qualitative point if asked: momentum and cyclicality mean the true PD can differ from the matrix.

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

When to use it: When the question gives a small matrix and asks for two-year cumulative PD of one rating.

Write the starting row as a vector.
Take the default column of the matrix.
Two-year CPD = sum of (row entry × that rating's one-year PD), with default's own entry counting as 1.
Equivalent: CPD(2) = PD1 + Σ (non-default migration to j × PD of j).
Marginal year 2 = CPD(2) − PD1. Do not build the full squared matrix.

Common mistakes in Rating Transition Matrices and Default Rates

Squaring the matrix element by element instead of using matrix multiplication.
Squaring each cell feels like the natural way to get two years.
Fix: Use row-times-column multiplication. Sum over all intermediate ratings.
Doubling the one-year PD to get the two-year PD.
Treating default as a simple constant rate.
Fix: Use CPD(2) = PD1 + (1 − PD1 paths that survive and then default). Include migration to weaker ratings.
Confusing marginal and cumulative default rates.
Both are described as default over time.
Fix: Cumulative covers all years up to n. Marginal covers year n alone and equals the difference of consecutive cumulative values.
Using marginal PD as the conditional rate.
Forgetting that firms which defaulted earlier cannot default again.
Fix: Divide marginal PD by survival probability 1 − CPD(n−1) to get the hazard rate.
Ignoring the default row or leaving defaulted names out of the calculation.
The default row looks trivial.
Fix: Keep default as absorbing with probability 1 of staying. This is why CPD rises with time.
Treating the average matrix as valid in every year.
Assuming time homogeneity and no momentum.
Fix: State that cyclicality and momentum break the assumptions. Downturn migration is worse than the through-the-cycle average.

Worked examples

Example 1

A one-year matrix has three states: A, B and Default. From A: A 90%, B 8%, Default 2%. From B: A 10%, B 80%, Default 10%. Default is absorbing. Find the two-year cumulative default probability for a firm rated A, and the marginal PD in year 2.

Show the solution

One-year PD for A = 2%.
Two-year paths from A: stay A (0.90) then default from A (0.02) = 0.018.
Move to B (0.08) then default from B (0.10) = 0.008.
Default in year 1 (0.02) then stays in default = 0.02.
CPD(2) = 0.018 + 0.008 + 0.02 = 0.046 = 4.6%.
Marginal PD year 2 = 4.6% − 2.0% = 2.6%.

Answer: Cumulative two-year PD is 4.6%; marginal year-2 PD is 2.6%.

Example 2

Using the same matrix, a firm rated B has cumulative PDs of 10% after one year and 18.2% after two years. Find the marginal PD in year 2 and the conditional (hazard) default rate in year 2.

Show the solution

Check CPD(2) for B: stay B then default = 0.80 × 0.10 = 0.08.
Move to A then default = 0.10 × 0.02 = 0.002.
Default in year 1 = 0.10.
Sum = 0.08 + 0.002 + 0.10 = 0.182, which matches 18.2%.
Marginal PD year 2 = 18.2% − 10% = 8.2%.
Survival to start of year 2 = 1 − 0.10 = 0.90.
Hazard = 0.082 ÷ 0.90 = 0.0911 = 9.11%.

Answer: Marginal year-2 PD is 8.2%; conditional year-2 default rate is about 9.11%.

Exam tips

Read whether the question asks for cumulative, marginal or conditional PD. Examiners offer all three as answer options.
Expect to compute only two years by hand. Use the row-vector shortcut and skip full matrix squaring.
Check that the default row is absorbing before you start.
For conceptual items, link downgrades to momentum and recessions to cyclicality, and say the average matrix understates downturn risk.
Always check that your cumulative PD is at least the one-year PD.

Practice questions from Credit Scoring and Rating

Rating Transition Matrices and Default Rates 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 and Default Rates: frequently asked questions

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

Raise the one-year matrix to the power n by matrix multiplication. Read the default column in the row of the starting rating. That value is the cumulative PD over n years.

What is the difference between marginal and cumulative default rate?

Cumulative default rate is the probability of default at any time up to year n. Marginal default rate is the probability of default in year n alone, found as CPD(n) minus CPD(n−1).

Why is default an absorbing state?

Once a borrower defaults, the model does not let it return to a rating. The default row is 0 in every column except default, which is 1. This makes cumulative PD increase over time.

How do cyclicality and rating momentum affect transition matrices?

Cyclicality means migration and default rise in recessions and fall in expansions, so an average matrix misstates risk in any given year. Momentum means recently downgraded firms are more likely to be downgraded again, which breaks the Markov assumption.