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Describe a rating transition matrix, calculate with it, interpret it, and explain what it is used for.

FRM® Part I · Valuation and Risk Models · External and Internal Credit Ratings · FRM1.VR.4.h

The idea

A rating transition matrix gives, for each starting rating in a row, the probability of being in each rating at the end of the horizon, usually one year. Each row sums to 100%, the diagonal shows the chance of keeping the same rating, and default is an absorbing state. Once defaulted, the issuer stays defaulted. Multiply the matrix by itself to get the two-year matrix. Sum over every intermediate rating k, including default, because the path through default counts. The two-year probability of moving from rating i to rating j, summed over all the paths, is P2 = Σ P(i,k) × P(k,j). This assumes the Markov property, that next year's move does not depend on past moves. It is only approximate because of ratings momentum. Matrices are used to build default probability term structures, to simulate migration for credit VaR and to price credit risk.

Formulas

Two-year rating transition probability

P₂(i, j) = Σ P(i, k) × P(k, j)

P(i, k)
the one-year probability of moving from rating i to rating k
P(k, j)
the one-year probability of moving from rating k to rating j

Sum over every intermediate rating k, including default, which is absorbing. It assumes Markov migrations, which ratings momentum breaks.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which statement about a one-year rating transition matrix is correct?

  1. Each column sums to 100% and default is an absorbing state.Why not: Columns do not sum to 100%; it is the rows (a given starting rating) whose probabilities add to one.
  2. Each row sums to 100% and defaulted issuers can return to investment grade.Why not: In a standard matrix the default row is 100% in default; defaulted issuers do not migrate back.
  3. Each row sums to 100% and default is an absorbing state. - correct
  4. The diagonal entries are the one-year default probabilities of each grade.Why not: Diagonal entries are probabilities of keeping the same rating; default probabilities are in the default column.

Rows are starting ratings and sum to 100%. Default is absorbing: once there, the issuer stays there (a 100% diagonal entry in the default row).

A harder one

The one-year transition matrix (percent, to A, B, C, D) is: from A: 90, 7, 2, 1; from B: 5, 85, 6, 4; from C: 1, 9, 75, 15; from D: 0, 0, 0, 100. Assuming migrations are independent from year to year, the probability that an A-rated issuer is rated C after two years is closest to:

  1. 3.72% - correct
  2. 2.00%Why not: Reports the one-year probability of moving from A to C, ignoring paths through other ratings.
  3. About 4.00%Why not: Doubles the one-year A-to-C probability of 2%, as if the two years were additive.
  4. 0.04%Why not: Squares the one-year A-to-C probability (2% x 2%), as if the issuer had to jump to C in both years.

Sum over the intermediate rating k of P(A to k) x P(k to C): 0.90 x 0.02 + 0.07 x 0.06 + 0.02 x 0.75 + 0.01 x 0 = 0.018 + 0.0042 + 0.015 = 3.72%.

The hardest tier

The one-year transition matrix (percent, to A, B, C, D) is: from A: 90, 7, 2, 1; from B: 5, 85, 6, 4; from C: 1, 9, 75, 15; from D: 0, 0, 0, 100. Assuming migrations are independent from year to year, the probability that a B-rated issuer has defaulted by the end of year 2 is closest to:

  1. 8.35% - correct
  2. About 8.00%Why not: Doubles the one-year B default probability (4% + 4%), ignoring that survivors can default after migrating.
  3. 7.84%Why not: Uses 1 - 0.96², which applies the B-grade default rate in year 2 to all survivors, ignoring migration to weaker grades.
  4. 4.35%Why not: Leaves out the 4% that has already defaulted by year 1 (the default-to-default path), so counts only new defaults in year 2.

Two-year PD = 0.05 x 0.01 + 0.85 x 0.04 + 0.06 x 0.15 + 0.04 x 1.00 = 0.0005 + 0.034 + 0.009 + 0.04 = 8.35%. The default state is absorbing, so the 4% already defaulted stays in.

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