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

116 WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGESshow@F(n j )= ¡f(n j )@@E[T();n]= F(n ¡ 1 j )@@ 2 E[T();n]@ 2 = ¡f(n ¡ 1 j ):Suppose the prior on is an exponential with mean ¹, sothat1 ¡ H() = exp(¡=¹). Applying Equation (4.3), we derive Z n 1' T(£) = d e ¡=¹ ¢ e ¡ n¡1 ¹ n¹ 0(n ¡ 1)! = :¹ +1To see this is correct, we apply the prior to the conditional densityand integrate to obtain the unconditional densityf T(£) (n)= 1 ¹ n:¹ +1 ¹ +1We recognize this as a Geometric density. It is an exercise insummation formulas to then verify the insurance charge associatedwith this density is in fact the same as the one just derivedusing Equation (4.3).We may now put the results from Equations (4.2) and (4.4)together to show that decreasing charges by risk size for thepriors acting on a differentiable decomposable conditional familylead to unconditional charges that decrease with risk size.4.5. Charges Decrease by Size for Model Based onDecomposable Conditionals with Priors that Decrease by SizeSuppose M = fT() j > 0g is a differentiable decomposablerisk-size model and let Q be a risk-size model with unique randomvariables, f£ ¹ g such that E[£ ¹ ]=¹.If ' £2 · ' £1when ¹ 1