Determine E[(S − 3)+ ] with Aggregate Risk and Panjer's

School: University of Waterloo - Course: ACTSC 363 - Subject: Accounting

ACTSC 363 - Weekly Problem Set 9 In addition to the exercise provided below, the following exercises are recommended from theLoss Models (5th Edition)textbook (which is available online for free at the UW Library): Textbook Exercises 9.14, 9.15, 9.16. Exercise 5.10.Suppose the insurer has the aggregate risk S= ( ∑ N i=1 Yi ,N >0 0,N= 0 whereN∼POI(3) andYi ∼BIN(0.5,5). DetermineE[(S-3)+ ]. 19

Solution.Denotingg(n) =P(Yi =n) and using the binomial pmf, we can easily find g(0) = 0.55 = 0.03125,g(1) =g(4) = 0.15625,g(3) =g(2) = 0.3125. Also, sinceNis an (a, b,0) member witha= 0 andb= 3, using Panjer's recursion, we have P(S= 0) =E(f0 )N=e3[g(0)-1] = 0.05468 P(S= 1) = 3g(1)P(S= 0) = 0.02563, P(S= 2) = 1.5g(1)P(S= 1) + 3g(2)P(S= 0) = 0.05727. Hence, the stop-loss premium can be computed recursively using E[S] =E[N]E[Yi ] = 3(5)(0.5) = 7.5 E[(S-1)+ ] =E[S]-[1-FS (0)] =E[S]-[1-P(S= 0)] ≈6.5547, E[(S-2)+] =E[(S-1)+ ]-[1-FS (1)] =E[(S-1)+ ]-[1-P(S= 0)-P(S= 1)] ≈5.6350 E[(S-3)+] =E[(S-2)+ ]-[1-FS (2)] =E[(S-2)+ ]-[1-P(S= 0)-P(S= 1)-P(S= 2)] ≈4.7726.