The compound negative binomial distribution with exponential claim amounts (severity) distribution is shown to be equivalent to a compound binomial distribution with exponential claim amounts (severity) with a different parameter. As a result of this, the distribution function and net stop-loss premiums for the Negative Binomial-Exponential model can be calculated exactly as finite sums if the negative binomial parameter α is a positive integer.
The result is a generalization of Lundberg (1940).
Consider the distribution of
where X1, X2, X3, … are independently and identically distributed random variables with common exponential distribution function
and N is an integer valued random variable with probability function
Then the distribution function of S is given by
If MX(t), MN(t) and MS(t) are the associated moment generating functions, then