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

WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGES 131Further, the insurance charge must always be concave up. Thismeans the insurance charge curve never goes above a straight linedrawn between any two points on the curve.A.8.Insurance Charge is Concave Up'(wr +(1¡ w)s) · w'(r)+(1¡ w)'(s) for 0· w · 1:(A.8)ProofUsing the general property of the “min” operatormin(A + B,C + D) ¸ min(A,C)+min(B,D)we derivemin(R,wr +(1¡ w)s) ¸ min(wR,wr)+ min((1 ¡ w)R,(1¡ w)s):Factoring out w and (1 ¡ w) respectively, yieldsmin(R,wr +(1¡ w)s) ¸ w ¢ min(R,r)+(1¡ w)min(R,s):Using Equation (A.2) repeatedly, we find'(wr +(1¡ w)s)=1¡ E[R;wr +(1¡ w)s]· w +(1¡ w) ¡ wE[R,r] ¡ (1 ¡ w)E[R;s]= w(1 ¡ E[R;r]) + (1 ¡ w)(1 ¡ E[R;s])= w'(r)+(1¡ w)'(s):Though the charge and saving functions are continuous, theyneed not always be differentiable. However, when they are, thefollowing formulas hold.