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

WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGES 95or perhaps could be derived from the decomposability property.Since none of the assumptions put us outside the CRM, we leavethe more abstract development along these lines as a topic of futureresearch.To maintain focus in the main exposition, many basic definitionsand important foundational results have been relegated tothe Appendices. The reader may be well advised to review thesebefore proceeding much further.In the end, most actuaries will find nothing surprising in whatwe will prove. But we will have rigorously established that actuarialintuition about insurance charges does indeed hold truefor some fairly general classes of risk-size models, including theCRM.2. THE INSURANCE CHARGE FOR A SUM OF RANDOMVARIABLESWe start by studying inequalities for insurance charges ofsums. Our first result is that the insurance charge for the sum oftwo non-negative random variables is bounded by the weightedaverage of their insurance charges.2.1. Charge for Sum Bounded by Weighted Average of ChargesSuppose T 1 and T 2 are non-negative random variables withmeans, ¹ 1 and ¹ 2 , which are positive. Then, it follows that:' T1 +T 2(r) · ¹1'¹ 1 + ¹ T1(r)+ ¹ 2'2 ¹ 1 + ¹ T2(r): (2.1)2ProofApplying definition A.1 from the Appendix, we write' T1 +T 2(r)= E[max(0,(T 1 + T 2 ) ¡ r(¹ 1 + ¹ 2 ))]¹ 1 + ¹ 2= E[max(0,(T 1 ¡ r¹ 1 )+(T 2 ¡ r¹ 2 ))]¹ 1 + ¹ 2: