Skip to content

FRM Exam Part II · Fundamentals of Credit Risk

Credit Ratings and Transition Matrices for FRM Part 2

Updated 11 October 2026 · Fact-checked

A rating transition matrix shows the probability that a borrower moves from one rating to another over a fixed period, including to default. To get multi-year default probability, multiply the matrix by itself and read the default column. Cumulative default is not the sum of annual rates.

Understand Credit Ratings and 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 built by the bank for its own borrowers, using financial ratios, qualitative judgement and scorecards. Under the Basel internal ratings-based (IRB) approach, approved banks use their own rating systems to estimate risk parameters such as PD, and in the advanced approach LGD and EAD as well.

Ratings change over time. A transition matrix records this. Each row is the 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. Each row sums to 100%. Default is normally an absorbing state: once a borrower defaults, it stays there, so the default row is 0, 0, ..., 1.

The diagonal holds the probability of staying put. It is usually the largest number in each row. Moves from investment grade to default in one year are tiny. Lower grades have much higher default and more downgrade risk. The default column gives the one-year PD for each rating.

To look further out, use the Markov assumption: next period's move depends only on the current rating, and the matrix is constant. Then the two-year matrix is the one-year matrix multiplied by itself (P²), and the n-year matrix is Pⁿ. The default column of Pⁿ is the cumulative default probability. Real ratings show momentum and cycle effects, so the Markov assumption is only an approximation.

Ratings philosophy matters. Through-the-cycle (TTC) ratings aim to be stable across the economic cycle, looking past temporary conditions. Agency ratings are closer to TTC. Point-in-time (PIT) ratings reflect current conditions and move quickly with the cycle. Internal ratings are often closer to PIT. TTC ratings give stable capital but PDs that vary within a grade over the cycle. PIT ratings give stable PDs per grade but volatile migration and capital, which makes capital procyclical.

Agency ratings have limitations: they are slow to react, they are issuer-paid (conflict of interest), they are ordinal and not a precise PD, they show cliff effects when a downgrade triggers collateral or investment rules, and structured product ratings failed badly in the 2007-2009 crisis.

Key formulas to remember

Row sum of a transition matrix
Σⱼ P(i → j) = 1 for every starting rating i
Includes the default column. Use it to find a missing cell.
Two-year transition probability
P²(i → k) = Σⱼ P(i → j) × P(j → k)
Sum over every intermediate rating j, including default if k is default.
n-year matrix
Pⁿ = P × P × ... × P (n times)
Assumes a time-homogeneous Markov chain. Default column gives cumulative PD.
Cumulative PD with absorbing default
Cumulative PD(t) = 1 − Probability of no default by t
Cumulative PD never falls as t increases.
Marginal (unconditional) default probability
Marginal PD in year t = Cumulative PD(t) − Cumulative PD(t−1)
This is the unconditional probability of defaulting in year t, not conditional on survival. The conditional PD given survival to t−1 is [Cumulative PD(t) − Cumulative PD(t−1)] ÷ [1 − Cumulative PD(t−1)].
Annualised PD from cumulative PD
Annual PD = 1 − (1 − Cumulative PD)^(1/t)
Constant-hazard approximation.

How to solve Credit Ratings and Transition Matrices questions

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

  1. 1Identify what is asked: one-year PD, multi-year PD, probability of a specific migration, or a ratings concept.
  2. 2Locate the starting rating row. Check that it sums to 100% and that default is absorbing.
  3. 3For a one-year question, read the cell directly. For one-year PD, read the default column.
  4. 4For two years, compute only the cells you need: Σ P(start → j) × P(j → target). Do not build the full P².
  5. 5For three or more years, step forward the vector of the starting row: multiply the row by the matrix repeatedly.
  6. 6Compute marginal PD as the difference between cumulative PDs, or annualise with 1 − (1 − PD)^(1/t) if asked.
  7. 7For concept questions, decide TTC versus PIT, or IRB versus agency, and state the consequence for stability, procyclicality or capital.
  8. 8Sanity check: probabilities between 0 and 1, cumulative PD rising with horizon, and weaker ratings with higher PD.

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 default of one starting rating.

  1. Write the starting row as a vector of probabilities over the states.
  2. Take each probability and multiply it by that state's one-year default probability.
  3. Add the products. Add the one-year default probability from the start, because defaulted names stay defaulted.
  4. Two-year cumulative PD = PD₁(start) + Σ over non-default j of P(start → j) × PD₁(j).
  5. Check that the answer exceeds the one-year PD and does not exceed 1.

Common mistakes in Credit Ratings and Transition Matrices

  • Adding annual default probabilities to get cumulative PD.

    Adding looks natural and works as a rough approximation for tiny PDs.

    Fix: Use the matrix: cumulative PD(2) = PD₁ + Σ P(start → j) × PD₁(j). Add the first-year default and the second-year defaults of survivors, weighted by migration.

  • Leaving out the default state's own contribution in the second year.

    Students sum only over non-default ratings and forget that defaulted names remain defaulted.

    Fix: Include the one-year default probability of the start row, or use P²(start → D) with P(D → D) = 1.

  • Confusing TTC and PIT: calling agency ratings PIT.

    Both react to credit news, so the difference seems small.

    Fix: TTC ratings are stable across the cycle and smooth out temporary shocks. PIT ratings reflect current conditions and migrate with the cycle. Agency ratings lean TTC, internal ratings often lean PIT.

  • Reading a row of the matrix as a column or mixing up start and end rating.

    Tables are read quickly on a phone and the axis labels are missed.

    Fix: Rows are the starting rating, columns the ending rating. Check that the row you use sums to 1.

  • Treating transition probabilities as constant and exactly Markov.

    The textbook method makes it look like a fact.

    Fix: State that Pⁿ relies on the Markov and time-homogeneity assumptions. In practice there is rating momentum and cycle dependence, so downgrades tend to be followed by more downgrades.

  • Treating ratings as exact PDs.

    Each grade has a published average default rate.

    Fix: A rating is an ordinal ranking. Default rates in a grade vary with the cycle, especially for TTC ratings.

Worked examples

Example 1

A simplified one-year matrix has three states. From A: 90% stay A, 8% move to B, 2% default. From B: 10% move to A, 80% stay B, 10% default. Default is absorbing. Compute the two-year cumulative default probability for a borrower rated A today.

Show the solution
  1. Check rows: A row 0.90 + 0.08 + 0.02 = 1. B row 0.10 + 0.80 + 0.10 = 1.
  2. One-year PD from A = 0.02.
  3. Second-year default from survivors: stay A (0.90) × 0.02 = 0.018.
  4. Move to B (0.08) × 0.10 = 0.008.
  5. Two-year cumulative PD = 0.02 + 0.018 + 0.008 = 0.046.

Answer: 4.6%. Simply doubling the one-year PD would give 4.0%, which understates risk because of the migration to B.

Example 2

Using the same matrix, a borrower is rated B today. Find the two-year cumulative PD, the marginal PD in year 2, and the constant-hazard annualised PD.

Show the solution
  1. One-year PD from B = 0.10.
  2. Second-year default from survivors: move to A (0.10) × 0.02 = 0.002.
  3. Stay B (0.80) × 0.10 = 0.080.
  4. Two-year cumulative PD = 0.10 + 0.002 + 0.080 = 0.182.
  5. Marginal PD in year 2 = 0.182 − 0.100 = 0.082.
  6. Annualised PD = 1 − (1 − 0.182)^(1/2) = 1 − √0.818 = 1 − 0.9044 = 0.0956.
  7. Note that the annualised PD is only an average equivalent that assumes a constant hazard. This borrower's PD is not constant: it is 10% in year 1 and 8.2% in year 2 (marginal), so 9.6% matches neither year.

Answer: Cumulative two-year PD is 18.2%, marginal year-2 PD is 8.2%, and the annualised PD is about 9.6%. The annualised figure is an average equivalent and differs from the marginal PDs of 10% and 8.2%.

Exam tips

  • Expect a small matrix and a two-year question. Use the row-vector shortcut and skip full matrix multiplication.
  • Always check that the default state is absorbing before you add default probabilities.
  • For TTC versus PIT, tie the answer to a consequence: capital stability, procyclicality or PD variation within a grade.
  • On agency limitations, name the specific flaw: conflict of interest, lag, cliff effects, ordinal nature or structured product failures.
  • If options include the simple sum of annual PDs, it is usually a trap. That sum ignores migration. The correct cumulative figure depends on migration and on the second-year PD of the states reached, so compute it from the matrix.

Practice questions from Fundamentals of 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 compute 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, add the one-year PD to the sum of each migration probability times the next-year PD of that rating. This works only if default is absorbing.

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

Through-the-cycle ratings look past the current economic phase and stay stable. Point-in-time ratings reflect current conditions and change more often with the cycle. TTC gives steadier capital, while PIT gives PDs per grade that are more constant but capital that is more procyclical.

What are the main limitations of credit rating agencies?

Ratings can lag market information, the issuer-pays model creates conflicts of interest, and ratings are ordinal rather than exact PDs. Downgrades can trigger forced selling or collateral calls. Structured finance ratings performed poorly in the 2007-2009 crisis.

What is the internal ratings-based approach?

Under Basel, the IRB approach lets approved banks use their own rating systems to estimate risk parameters such as PD, and under the advanced approach LGD and EAD, for regulatory capital. Supervisors require the systems to meet governance, data and validation standards.