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

130 WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGESA.5.Relation Between Charge and Saving'(r) ¡ Ã(r)=1¡ r:(A.5)Proof Use Equations (A.2a) and (A.2b) to write '(r) ¡Ã(r)=1¡ E[R;r] ¡ (r ¡ E[R;r]) = 1 ¡ r.It is straightforward to see that multiplication of the underlyingrandom variable by a scalar does not change its insurancecharge.A.6.Scaling a Random Variable Does Not Change Its ChargeFor c>0,' T (r)=' cT (r):(A.6)ProofLeft as an exercise for the reader.The charge can never decline too rapidly between any two points.A.7.¡1Insurance Charge Slope Between Two Points Greater ThanIf s>r then'(s) ¡ '(r)s ¡ r¸¡1:(A.7)ProofUsing Equation (A.4), write'(s) ¡ '(r)=¡Therefore,'(s) ¡ '(r)s ¡ rZ sr= ¡(s ¡ r)+duG R (u)=Z sr¸¡1+ 1s ¡ rZ srduF R (u):Z srdu(¡1+F R (u))duF R (u) ¸¡1: