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

For a compound Poisson aggregate claims S with Poisson parameter λ and claim size X with moment generating function M_X(t), which expression gives the MGF of S?

The MGF of a compound Poisson variable is exp{λ(M_X(t) − 1)}. It comes from substituting ln M_X(t) into the Poisson MGF exp{λ(e^u − 1)}, and it correctly equals 1 at t = 0.

  1. Aexp{λ(M_X(t) − 1)}Correct
  2. Bλ(M_X(t) − 1)
  3. Cexp{λ M_X(t)}
  4. DM_X(t)^λ
  5. exp{λ(M_X(t) + 1)}

Explanation

M_S(t) = M_N(ln M_X(t)). For Poisson M_N(u) = exp{λ(e^u − 1)}, so substituting e^u = M_X(t) gives exp{λ(M_X(t) − 1)}. Omitting the −1 fails to give M_S(0)=1.

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