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

A NEW METHOD OF ESTIMATING LOSS RESERVES 469The marginal distribution of Y (j) is given by Maurer and Margolin[12, equation (4.4)] asà !N+1 Xm ¡ 1Pr[Y (j) >y]= (¡1) m¡(N+2¡j)m=N+2¡jN +1¡ jà !½N +1£ (1 ¡ my) N I y< 1 ¾mmwhere IfAg is an indicator of the occurrence of the event A. Inaddition, the moments of Y (j) satisfy the recursion(n ¡ j)¹ (k)j:n + j¹(k) j+1:n = n¹(k) j:n¡1(16)where ¹ (k)j:n = E[Yk (j)] for sample size n.Due to the difficulties in deriving the inverse moments neededin LDF (2)j,N , however, Monte Carlo simulations6 areusedtodetermineboth sets of loss development factors. Table A1 in the Appendixshows the loss development factors LDF (1)j,Nand LDF(2)j,N ,respectively.4. NUMERICAL EXAMPLESExample 1: A Simple Data SetSuppose a new product was introduced on January 1, 2000and the loss development data as of December 31, 2002 are givenin Table 2.The total paid loss to date is thus 1,719 + 2,573 + 1,761 =6,053. To estimate the total ultimate loss, we must first specifyN. If we assume N = 3, then the total estimated ultimate loss for6 The uniform (0,1) random number generator run in Press et al. [13, chapter B7, page1142] is used to perform all simulations.