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On an Integral Equation for Discounted Compound – Annuity Distributions

Published online by Cambridge University Press:  07 February 2018

Colin M. Ramsay
Colin M. Ramsay*
Affiliation:
Actuarial Science, University of Nebraska, Lincoln NE, USA68588-0307, (402) 472-5823.
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Abstract

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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.

Keywords

Annuity distributionsintegral equationaggregate discounted claims
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 19 , Issue 2 , November 1989 , pp. 191 - 198
Copyright
Copyright © International Actuarial Association 1989

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