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

WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGES 103ProofApplying the definition of decomposability, we deriveF T¹(t)=Pr(T ¹ · t) ¸ Pr(T ¹ + T ¢¹ · t)=F T¹ +T ¢¹(t)=F T¹+¢¹(t):F is thus a decreasing function of the risk size and Equation(3.3a) follows.To prove the partial exceeds zero in Equation (3.3b), we usethe decomposability property to deriveE[T ¹+¢¹ ;t] ¡ E[T ¹ ;t]=E[T ¹ + T ¢¹ ;t] ¡ E[T ¹ ;t] ¸ 0:To prove the partial is less than unity, we similarly deriveE[T ¹+¢¹ ;t] ¡ E[T ¹ ;t]=E[T ¹ + T ¢¹ ;t] ¡ E[T ¹ ;t]· E[T ¹ ;t]+E[T ¢¹ ;t] ¡ E[T ¹ ;t]=E[T ¢¹ ;t] · ¢¹:It follows thatE[T ¹+¢¹ ;t] ¡ E[T ¹ ;t]· 1¢¹and this leads immediately to our result.As for Equation (3.3c), we claim that with our continuousdifferentiability assumption, it suffices to show that E[T ¹+¢¹ ;t] ¡E[T ¹ ;t] is a decreasing function of ¹ for any ¢¹ > 0. This issufficient because, if it is true, we can then use an argumentbased on the Mean Value Theorem to show that the first partialderivative with respect to risk size is decreasing. A decreasingfirst partial derivative forces the second partial to be less than orequal to zero.To show E[T ¹+¢¹ ;t] ¡ E[T ¹ ;t] is decreasing, we first use theadditivity and independence assumptions to write the convolutionformula:F T¹+¢¹(t)=F T¹ +T ¢¹(t)=Z t0dF T¹(s) ¢ F T¢¹(t ¡ s):