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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Compound distributions and their applications in risk modelling

Aggregate claims S are compound Poisson with parameter lambda = 2, and claim amounts X take the value 1 with probability 0.5 and the value 2 with probability 0.5 (units of Rs 1 lakh). What is P(S = 2)?

S equals 2 either with one claim of size 2 or with two claims of size 1. The probabilities are e^-2 and 0.5e^-2 respectively, giving P(S=2) = 1.5e^-2, which is about 0.203.

  1. Ae^-2 x 1.5
  2. Be^-2 x 1.0
  3. Ce^-2 x 0.75
  4. De^-2 x 1.25Correct
  5. e^-2 x 2.0

Explanation

S = 2 arises from N=1 with X=2, or N=2 with both X=1. P(N=1)=2e^-2, so the first contributes 2e^-2 x 0.5 = e^-2. P(N=2)=2e^-2, so the second contributes 2e^-2 x 0.25 = 0.5e^-2. The total is 1.5e^-2, the first option, not the 1.25 listed. Check by Panjer recursion: g2 = (lambda/2)(1 x f1 x g1 + 2 x f2 x g0); g0=e^-2, g1=2 x 0.5 x e^-2 = e^-2, so g2 = 1 x (0.5e^-2 + 2 x 0.5 x e^-2) = 1.5e^-2.

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