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

Annual maximum flood losses at a site are modelled by a GEV distribution with μ = 100, σ = 20 and ξ = 0 (Gumbel), in Rs crore. What is the probability that next year's maximum loss exceeds 140?

The standardised value is (140-100)/20 = 2, so the Gumbel distribution function is exp(-e^(-2)) ≈ 0.873. The probability of exceeding 140 is the complement, about 0.127. Giving 0.873 would confuse the non-exceedance probability with the exceedance probability.

  1. A1 - exp(-e^(-2)) ≈ 0.127Correct
  2. Bexp(-e^(-2)) ≈ 0.873
  3. C1 - exp(-e^(2)) ≈ 1.000
  4. Dexp(-e^(-0.5)) ≈ 0.545
  5. 1 - exp(-e^(-0.5)) ≈ 0.455

Explanation

For Gumbel, G(x) = exp(-exp(-(x-μ)/σ)). Here (140-100)/20 = 2, so G(140) = exp(-e^(-2)) = exp(-0.13534) ≈ 0.8734. The exceedance probability is 1 - 0.8734 ≈ 0.127. Option B is the non-exceedance probability, a wrong-tail error.

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