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

96 WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGESNext, we use the subadditivity property of the “max” operator toget max(0,A + B) · max(0,A) + max(0,B). We apply this to theprevious equation and do some simple algebra to find:' T1 +T 2(r) · ¹1 E[max(0,(T 1 ¡ r¹ 1 )]¹ 1 + ¹ 2 ¹ 1+ ¹ 2¹ 1 + ¹ 2E[max(0,(T 2 ¡ r¹ 2 )]¹ 2= ¹ 1¹ 1 + ¹ 2' T1(r)+ ¹ 2¹ 1 + ¹ 2' T2(r):Note this result applies even if T 1 and T 2 are not independent, as itfollows from subadditivity of the max operator and the linearityof the expectation operator. This result leads directly to the proofthat the summation of two identically distributed risks leads to asmaller insurance charge.2.2. Summation of Two Identically Distributed VariablesReduces the ChargeSuppose T 1 and T 2 are identically distributed and let T denotea random variable with their common distribution. Then' T1 +T 2(r) · ' T (r): (2.2)ProofFrom 2.1 it follows that' T1 +T 2(r) · 12 ' T 1(r)+ 1 2 ' T 2(r)=' T (r): (2.3)Note that if T 1 and T 2 were perfectly correlated, then A.6would apply and summing would be equivalent to doubling andthis would not change the charge. In Exhibit 2, we show a discreteexample with two cases: one in which the two variablesare independent and the other in which they are correlated. Notsurprisingly, the sum of independent variables does have a lowercharge than the charge for the sum when the variables are corre-