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Foundations of Casualty Actuarial Science

we assume that the claims are Poisson-distributed then
variance = mean. So, Var(N/) = .

Substituting this value on the above equation, we get,

Var (N) = E() + VarE()

Since we know that E() = 1, and Var() = , we get
Var (N) =  + 2 , this follows because  is scalar with
respect to integration over .

This formula has been called the ' excess variance'
formula. If the claim frequency varies among insured,
the total variance of the claim counts will be higher than
the average claim frequency for the population . The
excess variance is related to the structure variance. This
formula also demonstrates the point made above, that
the total variance in costs consist of two parts, a process
variance  and a structure variance 2.

It should be noted that the resulting claim count
distribution is a negative binomial distribution with
parameters  &

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