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

Which statement about a compound negative binomial distribution, compared with a compound Poisson distribution with the same mean number of claims and the same claim size distribution, is correct?

The compound negative binomial has the larger variance. Its claim number variance exceeds its mean, while the Poisson has them equal, and the term Var(N) times the squared claim mean therefore makes the aggregate variance bigger.

  1. AIt has a smaller variance because the negative binomial is underdispersed
  2. BIt has the same variance because the claim size distribution is the same
  3. CIt has a larger variance because the claim number variance exceeds its meanCorrect
  4. DIt has zero skewness
  5. It cannot be used when claim sizes are positive

Explanation

Var(S)=E[N]Var(X)+Var(N)E[X]^2. Negative binomial has Var(N)>E[N], whereas Poisson has Var(N)=E[N], so the compound variance is larger.

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