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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Introduction to extreme value theory

Annual maximum flood losses are modelled by a Gumbel distribution H(x) = exp(-exp(-(x - 50)/10)), with x in Rs crore. What is the probability that the annual maximum exceeds Rs 70 crore?

The probability is about 0.1266. The Gumbel distribution function at 70 is exp(-e^-2), about 0.8734, and the exceedance probability is the complement of this, 1 minus 0.8734.

  1. Aexp(-e^-2) ≈ 0.8734
  2. Be^-2 ≈ 0.1353
  3. C1 - exp(-e^-2) ≈ 0.1266Correct
  4. D1 - exp(-e^2) ≈ 0.9994
  5. exp(-e^2) ≈ 0.0006

Explanation

z = (70-50)/10 = 2. H(70) = exp(-e^-2) = exp(-0.1353) = 0.8734. Exceedance probability = 1 - 0.8734 = 0.1266. The value 0.8734 is the non-exceedance probability.

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