FRM Exam Part II · Credit Risk Management

Credit Ratings and Rating Transition Matrices Explained

Updated 11 October 2026 · Fact-checked

A credit rating ranks a borrower's default risk. A rating transition matrix gives the probability of moving from each rating to every other rating, including default, over a period. Multiply the matrix by itself to get multi-year probabilities. The default column of the result is the cumulative default probability.

Understand Credit Ratings and Rating Transition Matrices

A credit rating is an ordinal ranking of default risk. External ratings come from agencies such as S&P, Moody's and Fitch. Internal ratings are produced by a bank's own models and analysts. They cover borrowers that agencies do not rate, and under the Basel internal ratings-based approach they feed PD estimates used for regulatory capital.

Rating philosophy matters. A through-the-cycle (TTC) rating looks past the current stage of the economic cycle. It aims to be stable and changes only when the borrower's long-run risk changes. A point-in-time (PIT) rating uses current conditions. It moves more as the cycle moves. So a TTC rating gives a stable rating but a PD that varies with the cycle. A PIT rating gives a PD that is close to stable for each grade but a rating that migrates often. External agency ratings are generally described as closer to TTC. Internal systems are often closer to PIT or a hybrid.

A rating transition matrix is a table. Each row is the starting rating. Each column is the ending rating after a fixed horizon, usually one year. Each cell is the probability of moving from the row rating to the column rating. Each row sums to 100%. The default state is absorbing: once in default, the borrower stays there, so its row is 0, 0, ..., 1. A 'not rated' column may also appear. Probabilities are usually largest on the diagonal, because most borrowers keep their rating.

The matrix serves two purposes. First, the default column gives the one-year PD for each rating. Second, if you assume the process is Markov (next period depends only on the current rating) and the matrix is constant over time, the n-year matrix is the one-year matrix raised to the power n. The default column of that matrix gives cumulative default probabilities. Migration also drives credit VaR: a downgrade widens the spread and lowers bond value even without default.

The Markov and time-homogeneity assumptions are approximations. Real ratings show momentum: downgraded issuers are more likely to be downgraded again. Transitions also vary with the cycle, and default rates are higher in recessions.

Key formulas to remember

Row sum condition: Σj P(i → j) = 1 for every starting rating i; Includes the default column. Use it to find a missing entry.
Absorbing default state: P(D → D) = 1, P(D → any other) = 0; Default has no exit in a standard matrix.
Two-year transition (Markov, constant matrix): P(i → k, 2 years) = Σj P(i → j) × P(j → k); This is the row of the matrix times the column of the matrix.
n-year matrix: Pⁿ = P × P × ... × P (n times); Needs the Markov and time-homogeneity assumptions.
Cumulative default probability: CPD(i, n) = Pⁿ(i → D); The default column of the n-year matrix.
Marginal (conditional) default probability in year n: CPD(n) − CPD(n − 1) = unconditional default probability in year n; Dividing by the survival probability (1 − CPD(n − 1)) gives the conditional rate.
Survival probability: S(n) = 1 − CPD(n); Probability of no default through year n.

How to solve Credit Ratings and Rating Transition Matrices questions

Use this method for any question on ratings, philosophy or transition matrices.

1Identify what is asked: a one-year PD, a migration probability, a multi-year cumulative default probability, or a concept (TTC vs PIT, internal vs external).
2For concept questions, decide what is stable. TTC: rating stable, PD varies with the cycle. PIT: rating varies, PD per grade stable.
3For matrix questions, find the starting row. Confirm the row sums to 1 and that default is absorbing.
4For a one-year PD, read the default column of that row.
5For two years, add up all paths: start to every intermediate rating j, then j to the target. Include default as a target by using the default column.
6Write each path as a product and sum the products. Do not forget the path through default itself, which has probability P(i → D) × 1.
7For marginal default in a later year, subtract: CPD(n) − CPD(n − 1). For a conditional rate, divide by survival at the start of that year.
8State the assumption (Markov, constant matrix) and sanity-check that probabilities lie between 0 and 1 and CPD rises with n.

Quickest way: Two-year default using one row and the default column

When to use it: When you need a two-year cumulative default probability for one starting rating.

Take only the starting row. Do not square the whole matrix.
Take the one-year default column, which holds each rating's PD.
Multiply entry by entry: row value × that rating's PD, and sum across all ratings. Default's own PD is 1.
The sum is the two-year CPD. Check it exceeds the one-year PD and is below 1.

Common mistakes in Credit Ratings and Rating Transition Matrices

Confusing TTC and PIT by saying TTC ratings move more with the cycle.
The word 'cycle' suggests the rating follows it.
Fix: TTC ratings ignore the cycle, so they are stable. Their realised default rates per grade vary with the cycle. PIT ratings migrate more.
Leaving out the path through default when computing two-year default.
Students sum only over surviving ratings.
Fix: Include the term P(i → D) × 1. Default is absorbing, so those borrowers stay defaulted.
Doubling the one-year PD to get the two-year PD.
It feels like a simple scaling.
Fix: Use the matrix product. Double-counting ignores that a defaulted borrower cannot default again and that ratings migrate.
Treating the matrix as fixed across the economic cycle.
Textbook examples use one constant matrix.
Fix: State that constant matrix and Markov are assumptions. Recession matrices have higher downgrade and default rates.
Reading a column instead of a row.
Rows and columns look alike in a table.
Fix: Rows are the starting rating and columns are the ending rating. The row sums to 1. Columns do not.
Assuming internal ratings are always just as reliable as agency ratings.
Banks know their borrowers well.
Fix: Internal ratings need validation, backtesting and governance. Agencies give an external benchmark, but it may lag and may be TTC.

Worked examples

Example 1

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

Show the solution

Row A: P(A→A) = 0.90, P(A→B) = 0.08, P(A→D) = 0.02. The row sums to 1.
Year-one PDs: A = 0.02, B = 0.10, D = 1.
Two-year CPD = 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 the two-year cumulative default probability and the conditional probability of default in year 2, given survival through year 1.

Show the solution

Row B: P(B→A) = 0.10, P(B→B) = 0.80, P(B→D) = 0.10.
Two-year CPD = 0.10 × 0.02 + 0.80 × 0.10 + 0.10 × 1.
= 0.002 + 0.08 + 0.10 = 0.182.
Year-1 CPD = 0.10, so survival through year 1 = 0.90.
Unconditional year-2 default = 0.182 − 0.10 = 0.082.
Conditional = 0.082 ÷ 0.90 = 0.0911.

Answer: Two-year CPD = 18.2%. Conditional year-2 default probability ≈ 9.11%.

Exam tips

Know the TTC and PIT direction cold. Questions often ask which approach gives stable ratings and which gives stable PDs per grade.
In matrix questions, check the row sum first. A missing entry is often 1 minus the others.
For two-year questions, use the row-times-default-column shortcut. It saves time.
Expect interpretation: say that Markov and constant-matrix assumptions may fail because of ratings momentum and the economic cycle.
Link migration to credit VaR: downgrades lower value through wider spreads even without default.

Practice questions from Credit Risk Management

Credit Ratings and 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.

Credit Ratings and Rating Transition Matrices: frequently asked questions

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

A through-the-cycle rating ignores the current stage of the cycle and changes slowly. A point-in-time rating reflects current conditions and changes often. With TTC, the PD for a grade moves with the cycle. With PIT, the PD for a grade is steadier.

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

Raise the one-year matrix to the power n, assuming the Markov property and a constant matrix. Read the default column for your starting rating. For two years, you can sum row entries times the one-year PDs.

Why is default an absorbing state?

Once a borrower defaults, the standard matrix has no way back. Its row is zero everywhere except default, which is 1. This makes cumulative default probability rise with time.

What is the difference between internal and external ratings?

External ratings come from agencies and are public. Internal ratings come from a bank's own models and judgement and cover unrated borrowers. Under the Basel IRB approach, internal ratings drive PD estimates for capital, so they require validation and governance.