FRM Exam Part I · External and Internal Credit Ratings

Rating Transition Matrices and Default Probabilities Explained

Updated 11 October 2026 · Fact-checked

A rating transition matrix gives the probability that a borrower moves from one rating to another over a period, with default as a final state. For multi-year default probability, multiply the matrix by itself and read the default column. Marginal default is the extra probability in one year; cumulative is the total to date.

Understand Rating Transition Matrices and Default Probabilities

A rating transition matrix shows how credit quality changes over time. Each row is a starting rating. Each column is the rating at the end of the period, usually one year. The entry in row i, column j is the probability of moving from rating i to rating j. Each row sums to 100% because the borrower must end somewhere.

The last column is default. Default is an absorbing state: once a borrower defaults, it stays defaulted. So the default row is 0, 0, ..., 100%. The default column of a row gives the one-year default probability for that rating. Some published matrices also have a 'not rated' or 'withdrawn' column. It is a real outcome and the row still sums to 100%.

To get a two-year matrix you assume the one-year matrix is the same each year and that next year's move depends only on this year's rating (a Markov chain). The two-year matrix is then the one-year matrix times itself (matrix multiplication, not squaring each entry). The n-year matrix is the one-year matrix raised to the power n. The default column of that matrix is the cumulative default probability over n years.

Three related measures matter. Cumulative PD is the probability of defaulting at any time up to year t. Marginal PD for year t is the probability of defaulting in year t as seen from today: cumulative PD(t) − cumulative PD(t−1). Conditional (forward) PD for year t is the probability of defaulting in year t given survival to the start of year t: marginal PD(t) ÷ (1 − cumulative PD(t−1)).

In historical data, higher-rated bonds have lower default rates at every horizon. For investment-grade names, annual default rates typically rise with the horizon, because a strong credit has room to deteriorate. For speculative-grade names, conditional default rates typically fall with the horizon, because weak credits that survive early years are the stronger ones. Real ratings also depend on the economic cycle, so the constant-matrix assumption is an approximation.

Key formulas to remember

Row sum check: Σj P(i → j) = 100% for every starting rating i; Includes the default column and any not-rated column. Use it to find a missing entry.
Two-year transition probability: P2(i → k) = Σj P(i → j) × P(j → k); Matrix multiplication of row i with column k. This assumes a time-homogeneous Markov chain.
n-year matrix: Mn = M × M × ... × M (n times); Default column of Mn gives cumulative PD for each starting rating.
Two-year cumulative PD: CPD2(i) = Σj P(i → j) × CPD1(j), with CPD1(Default) = 100%; Fast way: you only need the one-year default column, not the whole matrix.
Marginal PD: MPD(t) = CPD(t) − CPD(t−1); Probability of defaulting in year t, measured from today.
Survival probability: S(t) = 1 − CPD(t); Probability of no default up to year t.
Conditional (forward) PD: CondPD(t) = MPD(t) ÷ S(t−1) = [CPD(t) − CPD(t−1)] ÷ [1 − CPD(t−1)]; Default in year t given survival to the start of year t.
Constant annual hazard: CPD(t) = 1 − (1 − h)^t, so h = 1 − (1 − CPD(t))^(1/t); Use only when told to assume a constant annual default probability. Continuous form: CPD(t) = 1 − e^(−λt).

How to solve Rating Transition Matrices and Default Probabilities questions

Use this method for any question on transition matrices, cumulative PD, marginal PD or conditional PD.

1Identify what is asked: a migration probability, a cumulative PD, a marginal PD or a conditional PD. Note the starting rating and the horizon.
2Find the starting rating's row. Rows are 'from' and columns are 'to'. Check that the row sums to 100%.
3If the horizon is one year, read the entry directly. If it is two or more years, multiply the one-year matrix by itself. Use the default column shortcut if only default is needed.
4For a two-year PD from rating i, multiply each entry in row i by the matching one-year default probability of the destination rating, then add. Default destination counts as 100%.
5Build the cumulative PD series for each year you need: CPD1, CPD2, CPD3.
6For marginal PD, subtract consecutive cumulative values. For conditional PD, divide that marginal PD by 1 − the previous cumulative PD.
7Sanity check: cumulative PD must rise with time and stay at or below 100%. Lower ratings should have higher PDs. Convert to percent and keep at least 3 decimals until the last step.

Quickest way: Default-column shortcut

When to use it: Use when the question asks only for two-year (or three-year) default probability from a given rating and gives a full one-year matrix.

Write down the one-year default column as a list: one value per rating, with Default = 100%.
Take the row of the starting rating. Multiply each entry by the default-column value of its destination rating and add up. That is CPD2.
For three years, multiply the same row by the CPD2 column of each rating. That needs CPD2 for every rating, so for long horizons use a calculator matrix function or work out only the ratings with non-zero probabilities.
Marginal PD = CPD2 − CPD1. Conditional PD = marginal ÷ (1 − CPD1).
Skip any entries that are zero in the row. They add nothing and save time.

Common mistakes in Rating Transition Matrices and Default Probabilities

Reading a column as the starting rating instead of a row.
Tables are often shown with 'to' labels on top and people read down by habit.
Fix: Rows are 'from', columns are 'to'. The row must sum to 100%. If a column sums to 100% instead, the table is probably transposed, so check the labels.
Squaring each entry to get the two-year matrix.
Confusing matrix multiplication with element-wise powers.
Fix: Use row-by-column multiplication: add up P(i→j) × P(j→k) over all intermediate ratings j, including the path through default.
Forgetting that default is absorbing, or leaving out the default path in the two-year sum.
Students multiply only through surviving ratings.
Fix: Treat the default destination as having a one-year default probability of 100%. A borrower who defaults in year 1 is counted as defaulted by year 2.
Computing cumulative PD as t × annual PD, or marginal PD as the cumulative PD.
Treating annual PDs as simply additive and mixing up the definitions.
Fix: Cumulative PD comes from the matrix power or from 1 − (1 − h)^t under a constant hazard. Marginal is the difference between two cumulative values.
Using marginal PD where conditional PD is asked, without dividing by survival.
The two sound alike and both describe a single year.
Fix: Marginal is measured from today. Conditional is given survival to the start of that year, so divide by 1 − CPD(t−1).
Ignoring the not-rated column or assuming the rows must add up without it.
Published matrices often omit it from the discussion.
Fix: Include every column when checking that a row sums to 100%. If the question tells you to ignore withdrawn ratings, renormalise the row by dividing by 100% minus the withdrawn share.

Worked examples

Example 1

A one-year transition 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. Assume the matrix is constant. For a borrower rated A today, find (a) the two-year cumulative default probability, (b) the marginal default probability in year 2, and (c) the conditional default probability in year 2.

Show the solution

Check rows: A row 90 + 8 + 2 = 100%. B row 10 + 80 + 10 = 100%.
One-year default column: A = 2%, B = 10%, D = 100%.
(a) CPD2(A) = P(A→A) × CPD1(A) + P(A→B) × CPD1(B) + P(A→D) × 100%.
= 0.90 × 0.02 + 0.08 × 0.10 + 0.02 × 1 = 0.018 + 0.008 + 0.020 = 0.046, or 4.6%.
(b) MPD(2) = CPD2 − CPD1 = 4.6% − 2.0% = 2.6%.
(c) Survival to the end of year 1 = 1 − 0.02 = 0.98. Conditional PD(2) = 2.6% ÷ 98% = 2.653%.

Answer: (a) 4.6%. (b) 2.6%. (c) about 2.65%.

Example 2

For a BBB-rated issuer, cumulative default probabilities are 0.50% at one year, 1.30% at two years and 2.40% at three years. Find (a) the marginal default probability in year 3, (b) the conditional default probability in year 3, and (c) the probability of surviving three years.

Show the solution

(a) MPD(3) = CPD3 − CPD2 = 2.40% − 1.30% = 1.10%.
(b) Survival to the end of year 2 = 1 − 0.013 = 0.987.
Conditional PD(3) = 1.10% ÷ 0.987 = 1.1145%, about 1.11%.
(c) S(3) = 1 − CPD3 = 1 − 0.024 = 0.976, or 97.6%.
Check: the conditional PD (1.11%) is larger than the marginal PD (1.10%) because the survival base is below 100%.

Answer: (a) 1.10%. (b) about 1.11%. (c) 97.6%.

Exam tips

Read the question for the exact measure: cumulative, marginal or conditional. Examiners often put the wrong ones among the answer options.
Use the default-column shortcut for two-year PD. It cuts the work to one short sum.
Always check that cumulative PD rises with the horizon and that a lower rating has a higher PD. This catches most row or column slips.
Keep the formula for conditional PD ready: marginal ÷ (1 − previous cumulative). A common wrong option divides by the current survival or by 100%.
Know the conceptual points: default is absorbing, matrices assume constant, memoryless behaviour, and real migration varies with the economic cycle.

Practice questions from External and Internal Credit Ratings

Rating Transition Matrices and Default Probabilities 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 Probabilities: frequently asked questions

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

Raise the one-year matrix to the power n and read the default column for your starting rating. For two years, add up P(i→j) × one-year default probability of j over all j, with default counted as 100%. This assumes the same matrix applies each year.

What is the difference between marginal and cumulative default probability?

Cumulative PD is the probability of defaulting at any time up to year t. Marginal PD is the probability of defaulting in year t alone, measured from today, equal to CPD(t) − CPD(t−1). Conditional PD further divides by the survival probability to the start of the year.

Why does each row of a transition matrix sum to 100%?

A borrower must end the period in some state: a rating, default, or not rated. The probabilities of all possible end states therefore add to 100%. If a row does not sum to 100%, an entry is missing or you are reading a column.

Is the constant transition matrix assumption realistic?

Only roughly. Ratings tend to move with the economic cycle, and past migration can affect future migration, which the Markov assumption ignores. Exam questions usually tell you to assume a constant matrix, but you should know these limits.