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

WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGES 101the limited expected value and charge for the model M at size¹. To simplify notation when working with a unique size model,we may sometimes write F T¹(t) in place of F M (t j ¹).Next we define the notions of closure under independent summation,and decomposability in a unique size model.3.1. Definitions of Independence, Closure and Decomposabilityin a Unique Size ModelGiven a unique size model, M, and assuming ¹ 1 > 0, ¹ 2 > 0,we sayM is closed under independent summation if T ¹12 Mand T ¹22 M implies their independent sum, T ¹1+ T ¹2,is also in M. Note these could well be independentsamples of the same random variable.(3.1a)M is arbitrarily decomposable if for any positive¹, ¹ 1 ,and¹ 2 with ¹ = ¹ 1 + ¹ 2 ,thereexistT ¹ 2 M,T ¹12 M, andT ¹22 M such that the independentsum, T ¹1+ T ¹2has the same distribution as T ¹ . (3.1b)Unless there is need for greater specificity, we will usuallysay “closed” instead of “closed under independent summation.”In a closed complete model, we can add identically distributedrandom samples of any given risk in the model and still stay inthe model.Arbitrary decomposability is a strong condition. It says thatany way of splitting the mean of a risk into a sum leads to a decompositionof that risk into the independent sum of smallerrisks in the model. To simplify terminology when no confusionshould ensue, we may hereafter refer to “arbitrarily decomposable”models as simply “decomposable.” We will showthat charges decrease with size in a decomposable unique model.