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

WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGES 99While the proof is rather abstract and the algebra of our numerictrick can be confusing, it is easy to see how it all works in anysimple example.EXAMPLE 1:' S3(r) · ' S2(r):ProofConsider·S3;r¸6E =E[2¢ S3 3 ;6r]=E[(T 1 + T 2 + T 3 )+(T 1 + T 2 + T 3 );6r]Thus we have=E[(T 1 + T 2 )+(T 1 + T 3 )+(T 2 + T 3 );6r]¸ E[T 1 + T 2 ;2r]+E[T 1 + T 3 ;2r]+E[T 2 + T 3 ;2r]=3E[S 2 ;2r]:·S3;r¸ ·S2;r¸E ¸ E3 2and the desired conclusion follows.It is important to understand that we have not proved that anyway of adding risks together reduces the charge. For example,if we had a portfolio of independent risks with small charges,and then added another risk with a large charge function, theaddition of that risk could well result in a new larger portfoliowith a larger charge. However, that would violate our assumptionthat the risks were identically distributed. Also, if we had twoidentically distributed risks, initially independent, and then addeda third risk, but while doing so combined their operations so thatall the risks were now strongly correlated, the charge might wellincrease. This is not a counterexample to our result, because ourconstruction does not allow one to change correlations in themiddle of the example.