Chapter 9
Chapter 9
1. You are given that mortality follows the Standard Ultimate Life Table with i 0.05 . Assuming
that lives are independent and that deaths are uniformly distributed between integral ages, calculate:
a. q 10 50:60
Solution:
q 10 50:60
110
p50:60
110
p50 10
p60
1 l60 l50
l70 l60
1 l70 l50
1 91,082.4 98, 576.4
0.07602
b. q 10| 50:60
Solution:
10
|q50 :
60
10
p50 :
60 (1
p60
: 70 )
(1
0.076022)(1
l61 l60
l71 ) l70
(1
0.076022)(1
96,305.8 96, 634.1
90,134.0) 91, 082.4
0.012727
c. A60:60
Solution:
60:60 0.36906
d. The net annual premium for a fully discrete joint whole life on (60) and (60) with a death benefit of 1000.
Solution:
1000P60:60
1000 60:60 a60:60
1000
0.36906 13.2497
27.85
e. A60:60
Solution:
60:60
i
60:60
(1.02480)(0.36906)
0.37821
February 9, 2022 Copyright Jeffrey Beckley 2012, 2015, 2016, 2018, 2019, 2022
f. The net annual premium rate for a fully continuous joint whole life on (60) and (60) with a death benefit of 1000.
Solution:
P(60:60 )
60:60 a60:60
i
60:60
1
i
60:60
(1.02480)(0.36906) 12.74411
0.029678
0.029678*1000 29.68
g. p 10 50:60
Solution:
p 10 50:60
10 p50 10 p60 10 p50:60
l60 l50
l70 l60
l60 l50
l70 l60
96, 634.1 91, 082.4 91, 082.4 98,576.4 96, 634.1 98,576.4
0.998865
h. Calculate the probability that the survivor of (50) and (60) dies in year 11.
Solution:
10
p 50:60
p 11 50:60
0.998865 (11
p50
p 11 60
p 11 50 11
p60 )
0.998865
96, 305.8 98, 576.4
90,134.0 96, 634.1
96, 305.8 98, 576.4
90,134.0 96, 634.1
0.998865 0.998451 0.000414
i. Calculate the probability that exactly one life of (50) and (60) is alive after 10 years.
Solution:
10
p 50:60
10
p50:60
0.998865 [1 0.076022]
0.074887
j.
A 60:70
Solution:
A 60:70
A60
A70
A60:70
0.29028
0.42818 0.46562
0.25284
February 9, 2022 Copyright Jeffrey Beckley 2012, 2015, 2016, 2018, 2019, 2022
k.
Solution:
a60 a60 a60:60 (2)(14.9041) 13.2497 16.5585
l.
Solution:
a 60:70
a60
a70
a60:70
14.904112.0083 11.2220
15.6904
or
1 60:70
1 0.25284
15.6904
d
0.05 1.05
m. The net annual premium rate for a fully discrete survivor whole life on (60) and (70) with a death
benefit of 1000. Assume that the premium is paid until the second death.
Solution:
P
60:70
0.25284
0.016114
a 60:70
15.6904
60:70
0.016114*1000 16.11
n.
Solution: a70|60 a60 a60:70 14.904111.2220 3.6821
February 9, 2022 Copyright Jeffrey Beckley 2012, 2015, 2016, 2018, 2019, 2022
2. A joint annuity on (50) and (60) pays a benefit of 1 at the beginning of each year if both annuitants are alive. The annuity pays a benefit of 2/3 at the beginning of each year if one annuitant is alive.
You are given:
i. Mortality follows the Standard Ultimate Life Table. ii. (50) and (60) are independent lives.
iii. i 0.05
Calculate the actuarial present value of this annuity.
Solution:
a=23
b 23
c
1
a
b
1 3
aa50
ba60
ca50:60
2 3
(17.0245)
2 3
(14.9041)
1 3
(14.2699)
16.5291
3. A joint annuity on (50) and (60) pays a benefit of 1 at the beginning of each year if both annuitants are alive. The annuity pays a benefit of 2/3 at the beginning of each year if only (50) is alive. The annuity pays a benefit of ? at the beginning of each year if only (60) is alive.
You are given:
i. Mortality follows the Standard Ultimate Life Table. ii. (50) and (60) are independent lives.
iii. i 0.05
Calculate the actuarial present value of this annuity.
Solution:
a=23
b
=
1 2
c
1
2
3
1 2
1 6
aa50
ba60
ca50:60
2 3
(17.0245)
1 2
(14.9041)
16 (14.2699)
16.4234
February 9, 2022 Copyright Jeffrey Beckley 2012, 2015, 2016, 2018, 2019, 2022
4. You are given the following mortality table:
x
l x
q x
p x
90
1000
0.10
0.90
91
900
0.20
0.80
92
720
0.40
0.60
93
432
0.50
0.50
94
216
1.00
0.00
95
0
Assume that deaths are uniformly distributed between integral ages and the lives are independent.
Calculate at i 4% :
a.
A91:92
Solution:
l90:91 = (1000)(900) = 900, 000 l91:92 = (900)(720) = 648, 000 l92:93 = (720)(432) = 311, 040 l93:94 = (432)(216) = 93, 312 l94:95 = (216)(0) = 0
91:92
(648,
000
311,
040)( 1.104
)
(311,
040
93,
312)(
1 1.04
)
2
648, 000
(93, 312)(1.104)3
0.93867
b.
Solution:
A1
90:91:3
(900,
000
648,
000)(1.104
)
(648,
000
311,
040)(1.104
)2
(311,
040
93,
312)(
1 1.04
)3
900, 000
0.83045
c.
Solution:
February 9, 2022 Copyright Jeffrey Beckley 2012, 2015, 2016, 2018, 2019, 2022
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