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

WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGES 107Proof It suffices to show ' Tn¹! ' 0 as n approaches 1 forarbitrary fixed ¹. We use a CV argument. Consider Var(T n¹ )=nVar(T ¹ ) for a decomposable model. Thus,CV 2 (T n¹ )=Var(R n¹ )= Var(T ¹)n¹ 2 ! 0 as n !1:By Equation (A.11), this impliesZ 10dr' Rn¹(r) ! 1 2 :Using R 10 dr' 0 (r)= 1 2, the result follows.It is not difficult to construct a risk-size model in whichcharges decrease with size, even though it is not decomposable.EXAMPLE 2: Charges Decrease by Size in a Non-DecomposableModelDefine T ¹ to be the distribution having probability mass p =¹=(¹ +1), at t = ¹ + 1 and mass 1 ¡ p, att =0. It follows thatE[T ¹ ]=¹ and' T¹(r)=1¡ r r ¹for 0 · r