Describe what a rating migration matrix is, and calculate the marginal, cumulative and ordinary probabilities of default from it.
The idea
A rating migration (transition) matrix shows, for each starting rating, the probability of ending a year in each rating, including default. Every row sums to 100 percent. Default is an absorbing state, so its row is zero everywhere except a 100 percent chance of staying in default. The one-year default probability is read from the default column. Multi-year probabilities come from multiplying the matrix by itself, so the n-year matrix is the one-year matrix raised to the power n. Its default column gives the cumulative probability of default by year n. Write C(n) for that cumulative figure. In year n, the marginal (unconditional) probability of default is the increase in the cumulative figure, Marginal = C(n) − C(n−1). The conditional probability divides the marginal figure by the survival probability, so Conditional = Marginal ÷ (1 − C(n−1)). Squaring the one-year default probability ignores migration and is the classic error.
Formulas
Multi-year migration matrix
n-year matrix = Mⁿ
- M
- the one-year migration matrix, including an absorbing default state
- n
- the number of years
Its default column is the cumulative default probability to year n. Squaring a single default probability ignores migration.
Marginal default probability in year n
marginal PD = C(n) − C(n−1)
- C(n)
- the cumulative default probability by year n
- C(n−1)
- the cumulative default probability by year n − 1
Unconditional, as seen today. It is smaller than the conditional figure, which divides it by 1 − C(n−1).
Conditional default probability in year n
conditional PD = marginal PD ÷ (1 − C(n−1))
- marginal PD
- the unconditional probability of default in year n
- C(n−1)
- the cumulative default probability by year n − 1
Given survival to the start of year n. Forgetting the division reports the marginal figure instead.
A practice question
What does a rating migration matrix show?
- For each rating, the probability of default in each future year, with each column summing to 100 percent.Why not: Describes a cumulative default term structure, not a migration matrix; the rows, not columns, sum to one.
- The correlation between rating changes of every pair of rating categories.Why not: A migration matrix holds transition probabilities, not correlations.
- The recovery rate assigned to each rating category on default.Why not: Recovery rates are not part of a migration matrix.
- For each starting rating, the probability of ending the period in each rating category, including default, with each row summing to 100 percent. - correct
Each row is a probability distribution over end-of-period ratings (including default) for a borrower starting in that rating.
A harder one
Using the same type of matrix, a Speculative borrower has one-year transition probabilities of 10% to Investment, 80% to Speculative and 10% to Default, while an Investment borrower has a 1% one-year default probability and a 7% chance of moving to Speculative. What is the marginal (unconditional) probability that a Speculative borrower defaults during year two?
- 18.10%Why not: Gives the two-year cumulative probability by including the year-one default.
- 9.00%Why not: Gives the conditional probability by dividing by the survival probability.
- 8.10% - correct
- 10.00%Why not: Reuses the one-year default probability, ignoring migration during year one.
Marginal PD in year two = 0.10 x 0.01 + 0.80 x 0.10 = 8.1%; cumulative would be 18.1%; the conditional figure would be 8.1% / 0.90 = 9.0%.
The hardest tier
A one-year migration matrix over states A, B, C and D (default) has these rows: A to A 90.0%, B 8.0%, C 1.5%, D 0.5%; B to A 5.0%, B 85.0%, C 7.0%, D 3.0%; C to A 1.0%, B 9.0%, C 70.0%, D 20.0%; D stays in D with 100%. A two-year loan is made to a B-rated borrower. What is its cumulative two-year probability of default?
- 5.91%Why not: Squares the one-year survival probability and ignores migration.
- 3.98%Why not: Gives the year-two marginal probability only and drops the year-one default.
- 6.00%Why not: Doubles the one-year default probability, ignoring both migration and survival.
- 6.98% - correct
Cumulative PD = 0.03 + 0.05 x 0.005 + 0.85 x 0.03 + 0.07 x 0.20 = 0.03 + 0.00025 + 0.0255 + 0.014 = 6.98%.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
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