Migration matrices and transition rates

How do you read a migration matrix?

Each row is a starting grade and each column an ending grade one year later. A grade B borrower has an 85% chance of staying in B, a 5% chance of upgrading to A, an 8% chance of downgrading to C, and a 2% chance of defaulting. Each row adds up to 100%. Default is absorbing: a defaulted borrower stays in default.

How do you get multi-year PDs from a migration matrix?

Multiply the one-year matrix by itself. The two-year probability of moving from B to default adds up every path: B to A then default, B to B then default, B to C then default, and B to default in year 1. Multiplying again gives three years.

Grade C's cumulative PD rises more slowly than three times its one-year PD, because survivors tend to improve; grade A's rises faster than three times, because strong borrowers can only drift down. The Migration sheet of the Roll rate model (Excel) performs the multiplication.

What does multiplying the matrix assume?

That migration probabilities depend only on the current grade and stay the same every year, a Markov chain. In practice, borrowers recently downgraded are more likely to be downgraded again, and migration rates worsen in downturns. Lenders adjust the matrix for the economic outlook, or use different matrices for each scenario, to make the PDs point-in-time.

How are migration matrices used for staging?

The same structure can track movements between IFRS 9 stages: the share of stage 1 loans that move to stage 2 or 3 in a period, and the share that cure back. Stage transfer matrices help explain movements in the allowance, test whether SICR criteria are working, and are a natural basis for the IFRS 7 reconciliation of the loss allowance by stage.

An example: a stage transfer matrix

A lender finds that, over a year, 92% of stage 1 loans stay in stage 1, 6% move to stage 2 and 2% to stage 3, while 30% of stage 2 loans cure back to stage 1, 55% stay and 15% move to stage 3. Applying these rates to the opening balances explains most of the year's movement in each stage's allowance, and a sudden change in the rates, such as cures drying up, is an early warning that credit quality is turning.

Where do migration matrices come from?

From the lender's own rating history, ideally covering a full credit cycle, or from rating agencies' published studies of how rated companies migrate over time. External matrices describe rated, mostly large companies and need adjusting for other portfolios. A matrix estimated from fewer than a few hundred rated borrowers can be unstable, with empty or erratic cells, so smaller lenders often blend their own data with published studies.

How does a migration matrix differ from roll rates?

Roll rates track movement through delinquency buckets month by month; migration matrices track movement between credit grades, usually annually. Roll rates suit consumer and trade receivables; migration matrices suit rated corporate and wholesale exposures. See roll rate analysis, lifetime PD and PD from external ratings.

This guide is general information. It is not tax or legal advice for your situation.