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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Introduction to copulas

An insurer finds the empirical probability P(U1 > 0.99 | U2 > 0.99) is about 0.60 and P(U1 > 0.999 | U2 > 0.999) is about 0.05 for pseudo-observations of two claim types. Which conclusion is most appropriate?

The conditional probability drops sharply from 0.60 to 0.05 as the threshold approaches 1, suggesting the limit is zero. This indicates asymptotic tail independence, as in a Gaussian copula, even though dependence is clear at moderate thresholds.

  1. AUpper tail dependence is about 0.60 and a Gumbel copula is clearly needed
  2. BDependence appears to fade in the extreme tail, suggesting tail independence such as a Gaussian copulaCorrect
  3. CThe variables are independent since the second value is below 0.5
  4. DThe Clayton copula is needed because lower tail dependence is shown
  5. The conditional probabilities must be equal for any copula, so the data are in error

Explanation

Under genuine upper tail dependence these conditional probabilities would settle near a positive limit. Here they fall sharply as u moves toward 1, which points to a limit of 0, as for a Gaussian copula. Independence would give about 0.01 and 0.001, so the dependence is not absent at moderate levels. The data say nothing about the lower tail.

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