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

244 WHEN CAN ACCIDENT YEARS BE REGARDED AS DEVELOPMENT YEARS?Then the forecast may be writtenĉ ij =[A ij =A i,j¡1 ]c i,j¡1=[(A i,j¡1 + b ij )=A i,j¡1 ]c i,j¡1=[(A + B)=A]C=[1+B=A]C:Similarly, the incremental forecast isˆp ij = ĉ ij ¡ c i,j¡1=[1+B=A]C ¡ c i,j¡1=[1+B=A]C ¡ C= B:C=A:That is, the forecast of the incremental observation is the productof (the sum of the incrementals above it) and (the sum of theincrementals to its left) divided by (the sum of all the incrementalsthat are both above and to the left). Note that this is symmetricin B and C (and also A)–interchanging i and j merelychanges the role of B and C. Thus we see that the chain laddermay be neatly defined directly in terms of the incremental paidloss amounts. See the appendix for a more formal proof of theabove symmetry.Itmayhelptogiveanexample.Imaginewehaveanincrementalpaid loss array as follows, and we wish to predict theincremental paid loss cell labeled P: