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

WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGES 129The charge and saving can also be expressed in terms of integrals.A.3. Insurance Charge and Saving Functions Defined usingIntegrals'(r)= 1 ¹Z 1r¹dF T (t)(t ¡ r¹)=Z 1rdF R (s)(s ¡ r)= 1 ¹Ã(r)= 1 ¹= 1 ¹Z 1r¹Z r¹0Z r¹0dtG T (t)=Z 1rdsG R (s)Z rdF T (t)(r¹ ¡ t)= dF R (s)(r ¡ s)0dtF T (t)=Z r0dsF R (s):(A.3a)(A.3b)When the random variable is discrete, these are viewed asReimann integrals and are interpreted as sums.Many basic properties can be proved directly from the definitionsusing simple properties of integrals, minimum operators,and expectations.A.4.Insurance Charge: Basic Properties' is a continuous function of r.' is a decreasing function of r, which is strictlydecreasing when '(r) > 0.'(0) = 1 and, as r !1, '(r) ! 0.' 0 (r) · '(r) · 1, where ' 0 (r) = max(0,1 ¡ r).(A.4a)(A.4b)(A.4c)(A.4d)With the definitions in A.2, one can show the charge and savingare related by a simple formula.