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

128 WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGESAPPENDIX ABASIC INSURANCE CHARGE THEORYLet T be a non-negative random variable having finite positivemean, ¹, cumulative distribution function F, and tail probabilityfunction G =1¡ F. Define the normalized random variable Rassociated with T via R = T=¹.Perhaps the most compact mathematical definitions of thecharge and saving can be given by taking the expected valuesof “min” and “max” operators.A.1. Charge and Saving Functions Defined using Min and MaxExpectationsCharge:'(r)=Saving:E[max(0,T ¡ r¹)]¹Ã(r)==E[max(0,r¹ ¡ T)]¹E[T ¡ min(T,r¹)]¹=1¡ E[T;r¹]¹(A.1a)= r ¡ E[T;r¹] : (A.1b)¹The definitions can be simplified even further by using the normalizedrandom variable.A.2. Charge and Saving Definitions using the NormalizedRandom VariableCharge:'(r) = E[max(0,R ¡ r)] = E[R ¡ min(R,r)] = 1 ¡ E[R;r](A.2a)Saving:Ã(r) = E[max(0,r ¡ R)] = r ¡ E[R;r]: (A.2b)