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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Loss distributions, with and without risk sharing

A motor insurer's ground-up loss X follows an exponential distribution with mean Rs 50,000. A policy has an ordinary deductible of Rs 20,000. What is the expected payment per loss (including losses that produce no payment)?

The expected payment per loss is about Rs 33,516. For an exponential loss, the expected excess over a deductible d equals the mean times e^(-d/mean), which is 50,000 × e^(-0.4). Simply subtracting the deductible from the mean is wrong.

  1. ARs 50,000 × e^(-0.4) = Rs 33,516 (approximately)Correct
  2. BRs 50,000 − Rs 20,000 = Rs 30,000
  3. CRs 50,000 × (1 − e^(-0.4)) = Rs 16,484 (approximately)
  4. DRs 20,000 × e^(-0.4) = Rs 13,406 (approximately)
  5. Rs 50,000 × e^(-2.5) = Rs 4,104 (approximately)

Explanation

For an exponential, E[(X-d)+] = ∫ from d of S(x)dx = mean × e^(-d/mean). Here d/mean = 0.4, so the result is 50,000 × e^(-0.4) = 50,000 × 0.67032 = Rs 33,516. Subtracting the deductible from the mean (Rs 30,000) ignores that many losses fall below the deductible.

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