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On an Integral Equation for Discounted Compound – Annuity Distributions
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- Journal:
- ASTIN Bulletin: The Journal of the IAA / Volume 19 / Issue 2 / November 1989
- Published online by Cambridge University Press:
- 07 February 2018, pp. 191-198
- Print publication:
- November 1989
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- Article
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We consider a risk generating claims for a period of N consecutive years (after which it expires), N being an integer valued random variable. Let Xk denote the total claims generated in the kth year, k ≥ 1. The Xk's are assumed to be independent and identically distributed random variables, and are paid at the end of the year. The aggregate discounted claims generated by the risk until it expires is defined as
where υ is the discount factor. An integral equation similar to that given by Panjer (1981) is developed for the pdf of SN(υ). This is accomplished by assuming that N belongs to a new class of discrete distributions called annuity distributions. The probabilities in annuity distributions satisfy the following recursion:
where an is the present value of an n-year immediate annuity.
where υ is the discount factor. An integral equation similar to that given by Panjer (1981) is developed for the 
