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FRM Part I · FRM Exam Part I · Multivariate Random Variables

A risk manager replaces a bivariate normal model of two asset returns with a bivariate Student's t model with few degrees of freedom, keeping the same means, variances and correlation. What is the main implication for joint extreme losses?

A multivariate t with few degrees of freedom has fatter tails and tail dependence, so extreme losses in both assets occur together more often than under a bivariate normal with the same correlation. Correlation alone does not capture joint tail behaviour.

  1. AJoint extreme losses become more likely because the t distribution exhibits tail dependenceCorrect
  2. BJoint extreme losses become less likely because the t distribution has thinner tails
  3. CJoint extreme losses are unchanged because the correlation is the same
  4. DJoint extreme losses become impossible because the t distribution is bounded

Explanation

The multivariate t has fatter tails than the normal and positive tail dependence, so large moves in both assets tend to occur together more often than under the normal, even with identical correlation. The normal has asymptotically no tail dependence. Matching correlation does not match the joint tail behaviour. The t distribution is not bounded.

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