Title: Finite Sum Evaluation of the Negative Binomial-Exponential Model
Abstract: 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 X 1 , X 2 , X 3 , … 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 M X (t), M N (t) and M S (t) are the associated moment generating functions, then