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

474 A NEW METHOD OF ESTIMATING LOSS RESERVESlieves, however, that the pattern of incremental incurred loss isdifferent, then tables of loss development factors to ultimate andannual loss development factors that are consistent with the specifiedpattern must be derived.To be precise, for j =0,1,:::,N let j denote the order ofthe set of order statistics Y (1) ,Y (2) ,:::,Y (N+1) that is used to definec ij .Notethat 0 , 1 ,:::, N is a permutation of the elementsof the set f1,2,:::,N +1g. (For example, equation (9) implies j = N +1¡ j. As another example, the actuary may specificallybelieve that 0 = N ¡ 1, 1 = N, 2 = N +1, j = N +1¡ jfor j =3,:::,N, which implies c i0 · c i1 · c i2 ¸ c i3 ¸¢¢¢¸c iN .) Itfollows that c ij and C ij are defined asc ij = S i Y (j ), and (18)C ijS i = P jk=0 Y : (19)( k )for j =0,1,:::,N.In general, the loss development factors can be obtained viaa simulation of sample size M as follows:STEP 1. For given settlement period of N + 1 years, setTEMP (1)j,N= 0 and TEMP(2)j,N=0forj =0,1,2,:::,N.STEP 2. Create an (N + 1) = 2 dimensional permutation vector =( 0 , 1 ,:::, N ) containing the actuary’s specified pattern ofincremental incurred losses.STEP 3. Generate N random variables U 1 ,:::,U j ,:::,U N from theactuary’s specified random splitting distribution, F U (u).STEP 4. Order the sampled U j ’s as U (1) · U (2) :::· U (N) .STEP 5. For j =1,2,:::,N +1, define Y j = U (j) ¡ U (j¡1) , withU (0) =0andU (N+1) =1.