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FRM Part II · FRM Exam Part II · Portfolio Credit Risk

A bank holds a very large homogeneous loan portfolio with a one-year PD of 2.28% (so N^-1(PD) = -2.00) and asset correlation ρ = 0.20. Using the Vasicek large-portfolio formula, what is the default rate that is not exceeded with 99.9% confidence (N^-1(0.999) = 3.09)? Use N(-0.4472... ) given below. Conditional default rate = N[(N^-1(PD) + sqrt(ρ)·N^-1(0.999)) / sqrt(1-ρ)]. Take sqrt(0.2)=0.4472 and sqrt(0.8)=0.8944. Approximate N(-0.70)=0.242, N(-0.70 is the closest value of the result).

The worst-case default rate is about 24.2%. Compute (-2.00 + 0.4472×3.09)/0.8944, roughly -0.69 to -0.70, and apply the normal CDF to get approximately 0.242. This is far above the 2.28% unconditional PD because of asset correlation.

  1. AAbout 24.2%Correct
  2. BAbout 2.28%
  3. CAbout 9.8%
  4. DAbout 75.8%

Explanation

Numerator: -2.00 + 0.4472×3.09 = -2.00 + 1.382 = -0.618. Divide by 0.8944 gives -0.691, about -0.70. N(-0.70) ≈ 0.242, so the 99.9% worst-case default rate is about 24.2%. The 2.28% option is just the unconditional PD, ignoring the correlation effect; 75.8% results from a sign error (N(+0.70)).

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