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FRM Part I · FRM Exam Part I · External and Internal Credit Ratings

A bond rated B has a cumulative default probability of 10% over one year. The one-year transition matrix says a B-rated issuer defaults with probability 10% and CCC is not involved. Assuming a time-homogeneous Markov process in which only B (stay) and D (default, absorbing) are possible, with P(B to B)=90% and P(B to D)=10%, what is the cumulative two-year default probability?

The two-year cumulative default probability is 19%. Under the Markov assumption, survival for two years is 0.90 squared, or 81%, so default is the remaining 19%. Simply doubling 10% to 20% double counts, since a defaulted issuer cannot default a second time.

  1. A20.0%
  2. B19.0%Correct
  3. C10.0%
  4. D81.0%

Explanation

Survival over two years is 0.9 x 0.9 = 0.81, so cumulative default is 1 - 0.81 = 19%. Adding the one-year probabilities (20%) ignores that defaulted issuers cannot default again. 81% is the survival probability, not default.

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