Using a one-year rating transition matrix, compute the two-year default probability of a grade B borrower and explain why it is more than twice the one-year figure.

The two-year PD is about 6.97%, more than the 6% you get by doubling the one-year 3%. Add the four paths: 3% default in year one, then 5% x 0.5% via A, 85% x 3% via B and 7% x 20% via C. The path through C adds 1.4 points on its own because borrowers who survive year one are no longer all grade B; some have migrated to a grade that defaults far more often.

Think of students in a middle section of a class. After a year, most are still in the middle, a few have moved up, a few have slipped to the bottom section, and some have dropped out. The dropout rate in year two depends on where each student now sits, not where they started. A two-year default is the sum of every path that ends in default: default in year one, or move to a grade in year one and default from there in year two. This is what multiplying the transition matrixA table giving, for each rating grade, the probability of moving to every other grade, or to default, over one period. by itself does, but on paper the tree is quicker and shows where the risk comes from.

If the borrower stayed a B, surviving year one at 97% and then defaulting at 3% gives 1 - 0.97 squared, about 5.9%, slightly less than double. It comes out higher because the survivors of year one are a riskier group than the borrowers who started: 7% of them now sit in grade C with a 20% default rate. Had those borrowers defaulted like B borrowers, the C path would add only 0.21 points instead of 1.40. The upgrades to A pull the other way, but only 5% move up and A's default rate is tiny, so the downgrade path wins. Given survival, the year-two default rate is about 4.1%, not 3%.

Pricing a two-year loan off 2 x 3% understates expected loss by about a sixth; over five years the gap is 6.4 points. Any multi-year measure, from lifetime provisions to the tenor limits on a grade, should come from the matrix raised to the horizon, not the one-year PD scaled by time. Then say the limits. The calculation assumes the matrix is the same every year and that a borrower's history does not matter. In practice recently downgraded borrowers are more likely to be downgraded again, and in a recession the whole matrix shifts towards default, so a matrix averaged over ten years understates risk just when it matters. A stressed matrix is the natural follow-up.