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

An insurer has a loss-per-payment random variable after introducing an ordinary deductible d on a loss X with a heavy-tailed Pareto distribution. Compared with the ground-up loss, how does the mean excess over d, e(d) = E[X-d | X>d], behave as d rises for a Pareto (alpha>1)?

The mean excess function increases linearly with the deductible. For a Pareto with alpha above 1 it equals (lambda + d)/(alpha - 1), reflecting the heavy tail, whereas the exponential gives a constant mean excess.

  1. AIt is constant, as for the exponential
  2. BIt decreases linearly in d
  3. CIt increases linearly in dCorrect
  4. DIt is always zero
  5. It falls to the policy limit

Explanation

For a Pareto with parameters alpha and lambda, e(d) = (lambda + d)/(alpha - 1), which is linear and increasing in d. Constant e(d) belongs to the exponential, the memoryless case.

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