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FRM Part II · FRM Exam Part II · Structured Credit Risk

In a one-factor Gaussian copula model, the latent variable for firm i is A_i = sqrt(rho)*M + sqrt(1-rho)*Z_i, with M and Z_i independent standard normals. Each firm has a one-year default probability of 2.28%, so the default threshold is -2.0 (N(-2)=0.0228). With rho = 0.36 and the market factor M = -2.0, what is the conditional default probability of a firm, to the nearest percent? Use N(0)=0.50, N(0.5)=0.6915, N(1.0)=0.8413, N(1.5)=0.9332, N(2.0)=0.9772.

The conditional default probability is about 16%. Compute (-2.0 - 0.6 times -2.0) divided by 0.8, which equals -1.0, and N(-1.0) is 0.1587. A severe market factor lifts the default probability from 2.28% to roughly 16%.

  1. A16%
  2. B50%
  3. C84%Correct
  4. D97%

Explanation

Conditional PD = N((threshold - sqrt(rho)*M)/sqrt(1-rho)). sqrt(0.36)=0.6, sqrt(0.64)=0.8. Numerator: -2.0 - 0.6*(-2.0) = -2.0+1.2 = -0.8; divided by 0.8 gives -1.0. N(-1.0)=1-0.8413=0.1587, about 16%. Correction: the answer is 16%, the 84% option arises from forgetting the sign (N(+1.0)).

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