Conditional Probability and the Collective Risk Model

Abstract. One of the most powerful and profound tools of casualty actuarial science is the collective risk model S = X +K+ X N 1 . It is widely used by casualty actuaries, especially by those in the field of reinsurance. Nearly one hundred pages of one standard textbook (Klugman, [1998], Chapter 4) hardly suffice to survey the ingenuity with which actuaries and scholars have analyzed it. Much of their analysis proceeds from the application of conditional probability to the so-called individual risk model n S = X +K+ X 1 . This paper penetrates deeper into both conditional probability and the collective risk model, deriving new insights into higher moments and their generating functions. Particular attention is devoted to the fourth moment of the collective risk model, for which no formula seems previously to have been published. An appendix extends conditional probability to a novel technique of loss development.

Keywords: conditional probability, moments, cumulants, collective risk model.