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

WHEN CAN ACCIDENT YEARS BE REGARDED AS DEVELOPMENT YEARS? 255APPENDIXDerivation of the Incremental Chain LadderIn the following, when an unavailable term appears on theright hand side of an equation, it is replaced by its predictedvalue. The usual form of the chain ladder predictions is givenby:ĉ ij = ˆ¯jc i,j¡12 3Xs¡j= 4 c hjh=15, 2 Xs¡j4h=1c h,j¡135:c i,j¡10 , 2 31Xs¡j= @1+c 4s¡j,j c 5 h,j¡1h=1A:c i,j¡10 2 3Xs¡j= @1+ 4 p 5, 2 31Xs¡jXj¡1Xj¡14 hj p 5 hkA: p ik :h=1h=1h=1 k=0h=1 k=0k=0k=0Hence0 2 3Xs¡jˆp ij = @1+ 4 p 5, 2 31Xs¡jXj¡1Xj¡1Xj¡14 hj p 5 hkA: p ik ¡023Xs¡j= @ 4 p hjh=1=(B=A):C,= B:C=AP j¡15, 2 Xs¡j431Xj¡1Xj¡1p 5 hkA:h=1 k=0(seeFigure4)where A = P s¡jh=1 k=0 p hk , B = P s¡jh=1 p hj and C = P j¡1k=0 p ik .Notethe symmetry in the subscripts.The Transpose Invariance PropertyThe symmetry immediately establishes the transpose invarianceproperty. Equivalently, refer to Figure 4, and note that thek=0p ikk=0p ik