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Unformatted text preview: Tutorial 3—ACTSC 232, Fall, 2011
The ﬁrst two questions use this select table of mortality
l[x]
[ x]
[40] 100000
[41] 99802
[42] 99597
[43] 99365
[44] 99120 l[x]+1 l[x]+2 l[x]+3
99899 99724 99520
99689 99502 99283
99471 99268 99030
99225 99007 98747
98964 98726 98429 lx+4
99288
99033
98752
98435
98067 x+4
44
45
46
47
48 1. (a) Calculate the probability that a life age 42 dies between ages 45 and 46 given that
the life was selected at age 42.
(b) Calculate the probability that a life age 42 dies between ages 45 and 46 given that
the life was selected at age 41.
(c) Calculate the probability that a life age 42 dies between ages 45 and 46 given that
the life was selected at age 40.
2. The Index of selection is deﬁned as
q[x]+k
I (x, k ) = 1 −
qx+k
Calculate the index of selection at all durations for a life selected at age 44. (That is
I (44, k ) for all k .) Comment brieﬂy on the result.
¯
3. (a) Write A 1 x :n ¯
+ n Ax in single Actuarial notation. (b) Consistent with the actuarial notation seen in class, ¯ n A 1
x:m would represent the expected present value of $1 paid immediately on death of x over what interval?
¯
¯
=c·A 1
and give the value of c.
(c) Show that n A 1
x :m x+n:m 4. Describe in words the insurances with the present values given (without consulting your
notes!). You should specify how much beneﬁt is paid under every possible contingency.
T is the future lifetime r.v. for a life age x. 0,
T ≤5 T
20, 000 v ,
T ≤ 15,
5 < T ≤ 15
(a) Y1 =
(b) Y2 = 10, 000 v T , 10, 000 v 15 ,
T > 15, 10, 000 v 15 ,
T > 15
5. For Y1 and Y2 above, express the expected present value of the beneﬁt (i.e. E[Y ]) using
‘A’ type actuarial functions.
¯
6. You are given that, at an eﬀective rate of interest of 6% per year, Ax = 0.166117,
¯
¯
Ax+5 = 0.20718, Ax+15 = 0.314208. You are also given that lx = 93132, lx+5 = 91641,
lx+15 = 86409.
Calculate the expected values of Y1 and Y2 .
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 Fall '08
 MATTHEWTILL

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