PROCEEDINGS May 15, 16, 17, 18, 2005 - Casualty Actuarial Society

94 WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGESwill then extend this result to cover the type of parameter risk inseverity and frequency that is modeled in the CRM.A reader versed in stochastic-process theory might observethe concept of decomposability is quite similar to the idea of“infinite divisibility” used in connection with Levy processes([1], [7]). However, after some thought, we decided not to employthe terminology or results of stochastic theory. Though thereis an analogy between increasing the risk size in a size-of-riskmodel and increasing the time in a stochastic process, we wishto maintain relevant distinctions between the two operations. Instochastic processes, the major concern is how a random variablechanges over time [9] and the cumulative effect of possiblejumps over a time interval. In short, it is the study of samplepaths. For example, N(t) might be the number of times a particularevent has occurred as of time t and we might assumeN(t) is a right continuous function of t. The distribution of N(t)would be a probabilistic summary of the number of events thathave occurred as of time t, averaged over the space of samplepaths (equipped with appropriate measure). In risk-size models,we are concerned with how risks of different size relate to oneanother; but there is no real analogue to the space of samplepaths. This is not to say that many of the results to be presentedhere could not have been proven by applying stochastic-processtheory after appropriately accounting for the distinction betweentime dependent paths and risk size. However, we will leave thatwork as a task for others who are more knowledgeable aboutstochastic-process theory. Also, we will not adopt the terminologyof stochastic process theory since this might confuse thediscussion of risk size. As well, in keeping with our actuarialfocus, we will tend to make whatever reasonable assumptionswe need, even though some of these could possibly be provedfrom previous assumptions or from more minimalist hypotheses.For instance, some of the assumptions that will be made aboutdifferentiability of our models with respect to risk size mightbe replaceable with more general and less restrictive statements