ACTSC 625 数学解题

ACTSC 625: Casualty and Health Insurance Mathematics
Winter 2022, Assignment 2
Consider the collective aggregate risk model
i=1 is a sequence of iid copies of X, which are all independent of N. The primary distribution
N  BIN(m; q) with m = 15 and q = 0:3 The secondary distribution X  EXP() with  = 15.
(1). Find an exact formula for the cdf of S.
(2). Discretize X with span h > 0, and denote the discretized random variable as Xb.
be the corresponding discretized aggregate risk. Use Panjerís recursion to compute the cdf FSb(x) at
x = nh for n = 0; 1; : : : ;
300
h. For h = 10; h = 5, h = 1, and h = 0:1, draw the absolute di§erence function
for x 2 f0; h; 2h; : : : ; 300g, respectively.
(3). For the discretized aggregate risk Sb, use the pmf method to compute the cdf FSb(x) at x = nh
for n = 0; 1; : : : ;
300
h
. Again, for h = 10; h = 5, h = 1, and h = 0:1, draw the absolute di§erence function
for x 2 f0; h; 2h; : : : ; 300g, respectively. Note: the pmf method involves the calculation of
inÖnite sum, which can be approximated by Önite sum, that is, you can impose some stopping rules.
(4). Draw some conclusion about the di§erences between these two approaches (Panjerís recursion and
pmf method) in terms of precision and computation time.