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

132 WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGESA.9.Insurance Charge and Saving First DerivativesIf ' or à is known to be differentiable at r, then td 'd'¹dr (r)=¡G R(r) and= ¡ G T (t)dt¹(A.9a) td Ãdùdr (r)=F R (r) and = F T(t)dt ¹ : (A.9b)Proof Apply Equation (A.4) and the Fundamental Theoremof Calculus.If R has a density function at a point r, one can take secondderivatives.A.10.Insurance Charge and Saving Second DerivativesIf R has a well-defined density f R at a point r, thend 2 'dr 2 (r)=f R(r)d 2 Ãdr 2 (r)=f R (r):(A.10a)(A.10b)Proof Take derivatives of the first derivatives shown in Equation(A.9a) and (A.9b).The variance can be expressed in terms of an integral of theinsurance charge.