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FRM Part II · FRM Exam Part II · Financial Correlation Modeling - Bottom-Up Approaches

Under a one-year horizon, two firms A and B each have a cumulative default probability of 2.275%, so the corresponding standard normal threshold is -2.0 (N(-2) = 2.275%). A Gaussian copula with correlation ρ is used. Which expression correctly gives the joint probability that both firms default within one year?

The joint default probability is the bivariate standard normal cumulative distribution evaluated at the two thresholds of -2 with correlation ρ, M(-2,-2;ρ). The simple product of marginals applies only if ρ is zero, and the sum minus M gives the probability of at least one default.

  1. AM(-2, -2; ρ), the bivariate standard normal cumulative distribution at (-2, -2) with correlation ρCorrect
  2. BN(-2) + N(-2) - M(-2, -2; ρ)
  3. CN(-2) × N(-2) regardless of ρ
  4. DM(2, 2; ρ), the bivariate normal cumulative distribution at (2, 2)

Explanation

In the Gaussian copula, each default time is mapped to a standard normal variable via N^-1 of its cumulative probability. Joint default means both variables fall below -2, so the probability is the bivariate normal CDF M(-2,-2;ρ). Option 3 holds only when ρ = 0, and option 2 is the probability of at least one default.

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