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

WHEN CAN ACCIDENT YEARS BE REGARDED AS DEVELOPMENT YEARS? 241FIGURE 2Two Incremental Arrays to which an “Across” (thestandard chain ladder) and a “Down” Version of theChain Ladder are Appliedtio r s¡i+1 , and recursively projecting those forecasts in turn bymultiplying by the next ratio.Now imagine a version of the chain ladder working in theother direction (“down” rather than “across”)–where you cumulatedownward, take ratios running down (accident-year-toaccident-yearratios), project down into the future, and differenceback to incrementals, as in Figure 2. It turns out that the incrementalforecasts for both the usual chain ladder (the version thatruns across) and this new “down” version of the chain ladder arethe same.2. THE INCREMENTAL CHAIN LADDERTo see that the “across” and the “down” versions of the chainladder are the same, we will first write the chain ladder purelyin terms of incrementals (which we call the incremental chainladder).Consider that we are attempting to forecast a cumulative paidloss amount, c ij , in the next calendar year. Let A ij = P k c kj ,thatis, A ij is the sum of all the cumulatives in the column above