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

If S is compound Poisson with parameter lambda and the claim size X has moment generating function M_X(t), which expression gives the MGF of S?

The MGF of a compound Poisson variable is exp{lambda(M_X(t) - 1)}. It comes from substituting the log of the claim-size MGF into the Poisson MGF, because M_S(t) equals M_N evaluated at ln M_X(t).

  1. Aexp{lambda(M_X(t) - 1)}Correct
  2. Blambda(M_X(t) - 1)
  3. Cexp{lambda M_X(t)}
  4. D(1 - lambda + lambda M_X(t))
  5. exp{lambda(e^t - 1)} M_X(t)

Explanation

M_S(t)=M_N(ln M_X(t)). With Poisson M_N(u)=exp{lambda(e^u-1)}, substituting e^u=M_X(t) gives exp{lambda(M_X(t)-1)}. Option 3 resembles a Bernoulli-type form and is wrong.

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