ACTSCI 331 - Life Contingencies I
Exercises (Chapter 8)
1. A special fully discrete whole life insurance is issued to (40). The death bene°t payable at the end of the
year of death and the premium paid by the insured at the beginning of the year (if alive) are respectively
b
K
+1
=
°
1000,
K
=
0, 1, ..., 24
500,
K
=
25, 26, ...
and
°
k
=
8
<
:
3
°; k
=
0, 1, ..., 14
2
°
,
k
= 15
, 16, ...29
0,
k
= 30
, 31, ...
:
Given that
k
A
40+
k
A
1
40+
k
:
25
°
k
j
°
a
40+
k
:
15
°
k
j
°
a
40+
k
:
30
°
k
j
0
0.202
0.114
10.005
13.414
10
0.316
0.149
4.386
10.806
20
0.458
0.112
±
6.929
40
0.744
±
±
±
,
(a) de°ne the loss r.v.
j
L
for all possible values of
j
(b) compute
°
under the equivalence principle and the bene°t reserves
10
V
,
20
V
and
40
V
. (Answers:
4.29, 120.97, 225.55, 372)
2. A special whole life insurance is issued to
(35)
with the following embedded components:
(1) death bene°t payable at the end of the (
k
+ 1
) year =
b
k
+1
= 50(
k
+ 1
),
k
= 0, 1, ...;
(2) level bene°t premium payable at the beginning of each year for at most 25 years (if alive)
Using
(1)
10
V
35
= 0.111;
(2)
(
IA
)
35
= 3.352;
(3)
A
35
= 0.142;
(4)
(
IA
)
45
= 4.191;
(5)
10
V
35:
25
j
= 0.227.
,
calculate the bene°t reserve at time 10. (Answer: 198.62)
3. A fully discrete whole life insurance issued to
(30)
has the following characteristics :
(1)
b
k
+1
=
e
0
:
06(
k
+1)
,
k
= 0, 1, 2, ...;
(2)
v
k
+1
=
e
°
0
:
08(
k
+1)
,
k
= 0, 1, 2,
... .
.
The bene°t premium is paid as long as the insured is alive at the beginning of each year.
Under a
speci°c mortality assumption, we have
10
V
30
A
30
A
40
@
±
= 2%
0.094
?
0.688
@
±
= 6%
0.045
0.359
?