How do credit migration matrices work, and how do you use transition probabilities in portfolio risk?

A credit migration (or transition) matrix is a square matrix showing the probability that a borrower with rating X at the start of a period will have rating Y at the end of the period. It is foundational to credit portfolio models like CreditMetrics.

Structure

Rows represent the starting rating, columns represent the ending rating, and each row sums to 100%.

Example 1-Year Transition Matrix (simplified):

How to Use It

Step 1 — Start with current ratings

Willowtree Capital holds a $10M BBB-rated bond issued by Harborfield Industries with 5 years to maturity.

Step 2 — Read the BBB row

Over 1 year, the bond has:

• 84.0% chance of staying BBB
• 4.5% chance of upgrading to A
• 7.5% chance of downgrading to BB
• 0.45% chance of defaulting

Step 3 — Compute value under each scenario

Using credit spreads by rating, reprice the bond under each possible migration:

Step 4 — Compute expected value and credit VaR

E[Value] = weighted average across all scenarios

Credit VaR = E[Value] - Value at the worst percentile

Key Points for FRM

1. Matrices are estimated from historical data — rating agencies publish them annually
2. Multi-year transitions can be approximated by raising the 1-year matrix to a power (matrix exponentiation)
3. Through-the-cycle vs. point-in-time — agency matrices tend to be TTC (smoothed), while internal models may be PIT (more volatile)
4. The absorbing state — once a borrower defaults, it stays in default (the Default row is [0,0,...,0,1])

For more credit risk modeling, check our FRM Part II course.