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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