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IAI Actuarial Core Principles · Economic Modelling · Role of insurance in reducing or removing risk

Individual claims on a portfolio follow an exponential distribution with mean Rs 50,000. An insurer has an individual excess of loss reinsurance with retention Rs 40,000. Which expression gives the expected retained amount per claim?

The insurer's expected retained amount is the mean of the claim capped at 40,000, which equals 50,000 times (1 minus e to the power minus 0.8). This is about Rs 27,534. The Rs 22,466 figure is instead the reinsurer's expected payment.

  1. A50,000 × (1 − e^(−0.8)) = Rs 27,534Correct
  2. B50,000 × e^(−0.8) = Rs 22,466
  3. C40,000 × (1 − e^(−0.8)) = Rs 22,027
  4. D50,000 − 40,000 = Rs 10,000
  5. 40,000 × e^(−0.8) = Rs 17,973

Explanation

Retained claim is min(X,M). E[min(X,M)] = mean × (1 − e^(−M/mean)) = 50,000 × (1 − e^(−0.8)). e^(−0.8)=0.4493, so 1−0.4493=0.5507 and the result is Rs 27,534. Option with 50,000 e^(−0.8) is the expected reinsurer payment (Rs 22,466), which is the wrong side of the layer.

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