FRM Exam Part II · Introduction to Credit Risk Modeling and Assessment

Credit Ratings and Transition Matrices Explained

Updated 11 October 2026 · Fact-checked

A credit rating ranks a borrower's default risk. A transition matrix shows the probability of moving from each rating to every other rating, including default, over a fixed period. To get multi-year default probability, multiply the matrix by itself and read the default column, assuming a Markov process.

Understand Credit Ratings and Transition Matrices

A credit rating is a grade for the creditworthiness of a borrower or a debt issue. External ratings come from agencies such as S&P, Moody's and Fitch. S&P and Fitch use AAA down to D. Moody's uses Aaa down to C. BBB-/Baa3 and above is investment grade. Anything below is speculative grade. Internal ratings are built by the bank for its own borrowers, using financial ratios, qualitative judgement and scorecards. Under the Basel internal ratings-based approach, banks use internal ratings to estimate risk parameters such as PD.

Ratings can follow two philosophies. A through-the-cycle (TTC) rating looks past the current stage of the economy and focuses on the borrower's likely position in a stress. It is stable and changes slowly. A point-in-time (PIT) rating reflects current conditions and updates quickly. PIT ratings give PDs that rise in recessions. TTC ratings give PDs that stay close to a long-run average. Agency ratings are generally closer to TTC. Many internal and market-based models are closer to PIT.

A rating transition matrix records how ratings change over a horizon, usually one year. Each row is the starting rating. Each column is the ending rating. Each cell is the probability of moving from the row rating to the column rating. Each row sums to 100%. The last column is default, and default is an absorbing state: once there, the borrower stays there, so the default row is 0, 0, ..., 1.

The default column of a one-year matrix gives one-year PDs by rating. For longer horizons, assume the process is Markov: next period's move depends only on the current rating, and the matrix is the same each year. Then the n-year matrix is the one-year matrix multiplied by itself n times. The default column of that product gives the cumulative PD. The marginal PD (unconditional) for year n is the cumulative PD to year n minus the cumulative PD to year n-1. Dividing the marginal PD by survival to year n-1 gives the conditional PD.

The Markov assumption is a simplification. In practice ratings show momentum: a recently downgraded issuer is more likely to be downgraded again. Migration also varies with the business cycle and across industries. Investment-grade default rates are very low, so their estimates are noisy.

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.
Default as absorbing state: P(D → D) = 1; P(D → any other) = 0; The default row never changes in matrix multiplication.
n-period transition matrix: P(n) = P(1)^n (Markov, time-homogeneous); Multiply the matrix by itself. For two years, P(2) = P × P.
Two-year cumulative PD from rating i: PD2(i) = Σk P(i → k) × P(k → D); Sum over all intermediate ratings k, including D itself where P(D → D) = 1.
Survival probability: S(n) = 1 − cumulative PD(n); Probability of no default through year n.
Marginal PD in year n: marginal PD(n) = cumulative PD(n) − cumulative PD(n−1); Unconditional probability of defaulting in exactly year n.
Conditional PD in year n: conditional PD(n) = marginal PD(n) ÷ S(n−1); Probability of default in year n given survival to the end of year n−1.

How to solve Credit Ratings and Transition Matrices questions

Use this order for any question on ratings, matrices or PDs.

1Identify what is asked: one-year PD, multi-year cumulative PD, marginal PD, conditional PD, or a rating concept (TTC vs PIT, internal vs external).
2Locate the starting rating row. Read across columns; confirm that the row sums to 1 and find any missing entry.
3For a one-year PD, read the default column of that row. Stop.
4For a two-year cumulative PD, multiply each entry in the starting row by the default probability of the column rating, then add. Include the D column, where the default probability is 1.
5For longer horizons, compute the row vector times the matrix step by step, or use the P(n) = P^n result. State the Markov and time-homogeneity assumptions.
6Convert to the form asked: survival = 1 − cumulative PD; marginal = difference of cumulative PDs; conditional = marginal ÷ prior survival.
7Check sense: PDs rise with horizon, fall with better ratings, and lie between 0 and 1.
8For concept questions, tie the answer to cyclicality: TTC is stable, PIT moves with conditions.

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

When to use it: Use for any multi-year default probability question with a small matrix, where only one starting rating matters.

Write the starting row as a vector of probabilities.
Take the default column as a second vector, with 1 for D.
Multiply the two vectors entry by entry and add. That is the two-year cumulative PD.
Alternatively, add one-year PD to the probability of surviving into each rating times that rating's one-year PD. Same result.
The year-two marginal PD is the sum over non-defaulted intermediate ratings k of P(i → k) × P(k → D). Divide it by one-year survival to get the conditional PD.

Common mistakes in Credit Ratings and Transition Matrices

Doubling the one-year PD to get the two-year PD.
It looks like a simple extension of the horizon.
Fix: Use matrix multiplication. The second year adds default only from borrowers still alive, and they may have migrated to a different rating.
Forgetting that default is absorbing, so leaving out the D term when summing two-year PD.
The sum is done over non-default ratings only.
Fix: Include P(i → D) × 1. This captures the one-year defaults already booked.
Confusing marginal and conditional PD.
Both describe default in a single year.
Fix: Marginal is measured from today. Conditional is measured given survival. Divide the marginal by survival to the prior year.
Treating rows and columns the wrong way round.
Tables are read quickly on a phone screen.
Fix: Rows are the starting rating, columns are the ending rating. Check that the chosen row sums to 1.
Saying agency ratings are point-in-time.
Ratings seem to update with news.
Fix: Agency ratings aim to be through-the-cycle: stable and slow. PIT models respond quickly to current conditions and give more cyclical PDs.
Treating the Markov assumption as a fact.
The maths works so cleanly.
Fix: State it as an assumption. Real migrations show momentum, cyclicality and industry effects, so multiplied matrices can misstate multi-year PDs.

Worked examples

Example 1

A simplified one-year transition matrix has three 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 two-year cumulative PD for a borrower rated A.

Show the solution

Starting row for A: A 0.90, B 0.08, D 0.02. Row sums to 1.00.
Default column: from A 0.02, from B 0.10, from D 1.
Two-year PD = 0.90 × 0.02 + 0.08 × 0.10 + 0.02 × 1.
= 0.018 + 0.008 + 0.020.
= 0.046.

Answer: The two-year cumulative PD for A is 4.6%.

Example 2

Using the same matrix, a B-rated borrower has a one-year PD of 10%. Find the two-year cumulative PD for B, and the conditional PD in year two given survival through year one.

Show the solution

Starting row for B: A 0.10, B 0.80, D 0.10.
Two-year PD = 0.10 × 0.02 + 0.80 × 0.10 + 0.10 × 1.
= 0.002 + 0.080 + 0.100 = 0.182.
Marginal PD in year two = 0.182 − 0.100 = 0.082.
Survival to end of year one = 1 − 0.10 = 0.90.
Conditional PD in year two = 0.082 ÷ 0.90 = 0.0911, about 9.11%.

Answer: Two-year cumulative PD for B is 18.2%. The conditional year-two PD is about 9.11%.

Exam tips

Questions usually give a small matrix and ask for a two-year PD. Practise the row-times-default-column calculation until it takes under a minute.
Check the row sum first. Many questions leave one entry blank, and you must fill it before using it.
For TTC vs PIT, link each to behaviour: TTC is stable and gives lower cyclicality in capital; PIT is responsive and more procyclical.
Expect interpretation items: higher-rated issuers show lower PDs but PDs rise with horizon; speculative grade shows larger migration spreads.
Learn the Markov and time-homogeneity caveats. Options often test whether you know when the matrix multiplication shortcut can mislead.

Practice questions from Introduction to Credit Risk Modeling and Assessment

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.

FRM Part ICredit Ratings and Transition Matrices

Credit Ratings and Transition Matrices: frequently asked questions

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

Raise the one-year matrix to the power n, then read the default column for your starting rating. For two years, sum the probability of moving to each rating times that rating's one-year default probability, including 1 for default itself. This relies on the Markov and time-homogeneous assumptions.

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

A through-the-cycle rating assesses the borrower across the whole economic cycle, so it changes slowly. A point-in-time rating reflects current conditions and moves faster. PIT PDs are more cyclical. TTC ratings are more stable.

Why is default an absorbing state?

Once a borrower defaults, the matrix treats it as staying in default. The default row is therefore zeros with a 1 in the default column. This also lets default probabilities accumulate through repeated multiplication.

What is the difference between internal and external ratings?

External ratings are issued by agencies and are public. Internal ratings are built by a bank for its own borrowers and used for credit decisions, pricing and, under the Basel IRB approach, regulatory capital inputs such as PD.