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Outline how rating transition matrices are used to derive credit VaR.

FRM® Part II · Credit Risk Measurement and Management · Credit Value at Risk · FRM2.CR.10.c

The idea

A rating transition matrix gives, for each starting rating, the probability of being in each rating class, including default, at the end of the horizon. Each row sums to 100 percent. CreditMetrics-style credit VaR uses the matrix in steps. For each possible end rating, revalue the bond using the forward credit spread curve for that rating. If the bond defaults, use its recovery value instead. Attach the transition probability to each value, giving a distribution of bond values. The expected value is the probability-weighted sum of those values. The credit measure is then VaR = expected value − value at the chosen percentile. Multi-year migration is found by multiplying the matrix by itself, with default treated as absorbing. The matrix alone says nothing about correlation between obligors.

Formulas

Credit VaR from a rating-migration value distribution

credit VaR = mean value − percentile value

mean value
the probability-weighted value of the bond across all end ratings, default included
percentile value
the value at the chosen percentile, counting cumulative probability up from the worst state

Measured from the mean, not from the current or the unchanged-rating value. If default alone holds more probability than the tail, the percentile value is the recovery value.

Two-year default probability from a one-year matrix

PD₂ = Σ pⱼ × PDⱼ

Σ
the sum over every end-of-year-one state, default included
pⱼ
the one-year probability of moving to state j
PDⱼ
the one-year default probability from state j, which is 1 for default itself

Default is absorbing, so the year-one defaults count in full. Doubling the one-year PD ignores migration and is wrong.

A practice question

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

In a one-year rating transition matrix, the row for a bond's current rating shows:

  1. the probability that each other rating class migrates into the bond's current rating.Why not: Reads down a column instead of across a row; the row describes where this bond may go.
  2. the probability that the bond defaults in each of the next several years.Why not: A one-year matrix gives one-year probabilities, not a term structure of default probabilities.
  3. the loss given default for the bond in each rating class.Why not: Transition matrices contain probabilities, not recovery or loss severities.
  4. the probability that the bond ends the year in each rating class, including default. - correct

Each row gives the distribution of end-of-horizon ratings for a starting rating and sums to 100%, with default as one of the states.

A harder one

A one-year transition matrix has three states. From investment grade (IG): 95% stay IG, 4% move to speculative grade (SG), 1% default. From SG: 10% move to IG, 80% stay SG, 10% default. Default is absorbing. The probability that an IG issuer defaults within two years is closest to:

  1. 2.00%Why not: Doubles the one-year PD, ignoring that defaults in year two come from the year-end rating distribution.
  2. 2.35% - correct
  3. 1.35%Why not: Counts only the paths through IG and SG in year one and omits that an issuer already in default stays in default.
  4. 1.40%Why not: Adds the default in year one and the IG-to-SG-to-default path but skips the IG-to-IG-to-default path.

Two-year PD = 0.95 x 0.01 + 0.04 x 0.10 + 0.01 x 1 = 0.0095 + 0.004 + 0.01 = 2.35%. Multi-year migration uses matrix multiplication with default absorbing.

The hardest tier

A 3-year, 6% annual-coupon bond (face 100) is rated A. After one year, immediately after the coupon is paid, two cash flows remain. Forward zero rates for the remaining two years are 4.0% and 4.7% for an A rating and 5.8% and 6.6% for BBB. The loss in value at the horizon if the bond is downgraded to BBB, relative to staying A, is closest to:

  1. 98.95Why not: Reports the BBB value itself rather than the loss relative to the A value of 102.47.
  2. 1.05Why not: Compares the BBB value to par (100 - 98.95) rather than to the value if unchanged.
  3. 3.51 - correct
  4. 1.83Why not: Discounts all cash flows at the one-year forward rate only, ignoring the two-year rate for the final payment.

A value = 6/1.04 + 106/1.047² = 102.47. BBB value = 6/1.058 + 106/1.066² = 98.95. Loss from downgrade = 102.47 - 98.95 = 3.51.

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