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

Two obligors each have a one-year default probability of 2%. Under a Gaussian copula their latent variables have correlation ρ. Compared with a Student-t copula with the same ρ and the same marginals, which statement is correct?

The Gaussian copula lacks asymptotic tail dependence when correlation is below one, while the Student-t copula has positive tail dependence. So for the same marginals and correlation, the t copula typically assigns a higher probability to joint extreme defaults, a key reason it is used for stress modelling.

  1. AThe Gaussian copula shows tail dependence, so joint extreme defaults are more likely than under the t copula
  2. BBoth copulas give identical joint default probabilities because the marginals and ρ are the same
  3. CThe Gaussian copula has no asymptotic tail dependence for ρ < 1, so it generally gives lower probability of joint extreme defaults than the t copulaCorrect
  4. DThe t copula has no tail dependence, so it gives lower joint default probability

Explanation

The Gaussian copula has zero asymptotic tail dependence when ρ is below 1, whereas the Student-t copula has positive tail dependence that rises as degrees of freedom fall. With identical marginals and ρ, the t copula therefore typically produces more joint extreme defaults. The first option reverses this.

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