Chapter 8

Chapter 8

1. Mayfawny purchases a whole life insurance policy.

There are three ways that Mayfawny¡¯s policy can terminate:

a. Death (1)

b. Diagnosis of a critical illness (2); and

c. Lapse (3).

The policy pays a death benefit of 10,000 at the moment of death. The policy will also pay a

critical illness benefit of 20,000 if Mayfawny is diagnosed with a critical illness. Only one benefit

will be paid.

There is no benefit paid upon lapse.

You are also given:

(1)

i. ? x ? 0.01

(2)

ii. ? x ? 0.015

(3)

iii. ? x ? 0.06

iv. ? ? 0.035

Mayfawny pays a net premium continuously for her lifetime as long as the policy is inforce. The

net premium is determined using the equivalence principle.

Calculate the net premium paid by Mayfawny.

February 17, 2015

Copyright Jeffrey Beckley 2012

Solution:

PVP ? PVB ??? Pax00 ? 10, 000 Ax01 ? 20, 000 Ax02 ? 0 Ax03

t

?

?

ax00 ? ? vt ?t px(? ) ? dt ? ? e ?? t ? e

0

?

?e

?0.035t

?e

?0.085t

?

? dt ? ? e

?

A ? ? v ?t p

t

(? )

x

0

?

A ? ? v ?t p

t

Pa

? dt ? ? e ?0.035t ? e

? ? ? dt ? ? e ?0.035t ? e ?0.085t ? 0.01? dt ?

??

? dt ?

? e ?0.12t ?

1

? dt ? ?

?

?

? ?0.12 ? 0 0.12

?

?

(2)

x

0

0

0

(? )

x

?

? (0.01? 0.015? 0.06)?ds

(1)

x

0

00

x

t

?

?

?0.12t

0

02

x

0

0

0

01

x

?

? ? x(?? s) ?ds

0.01

0.12

? dt ? ? e ?0.035t ? e ?0.085t ? 0.015 ? dt ?

0

0.015

0.12

10, 000 Ax01 ? 20, 000 Ax02 ? 0 Ax03

? 10, 000 A ? 20, 000 A ? 0 A ??? P ?

?

ax00

01

x

02

x

03

x

? 0.01 ?

? 0.015 ?

10, 000 ?

? ? 20, 000 ?

??0

? 0.12 ?

? 0.12 ?

? 400

? 1 ?

?

?

? 0.12 ?

February 17, 2015

Copyright Jeffrey Beckley 2012

2. Jeff is receiving a salary paid continuously for as long as he is in employed at Purdue. Jeff can

leave employment through death (1), retirement (2), or disability (3). Once Jeff leaves

employment, the salary stops.

You are given:

i.

ii.

iii.

iv.

The salary pays at an annual rate of 70,000 per year.

? ? 0.05

Jeff is currently age 59.

Jeff will retire at age 65 if he is still teaching. He will not retire prior to age 65.

(1)

v. ?59?t ? 0.01 ? 0.001t

(3)

vi. ?59?t ? 0.03 ? 0.001t

Calculate the present value of Jeff¡¯s future earnings while employed at Purdue.

Solution:

6

(? )

PV ? 70, 000? vt ?t p59

? dt

The limits on the integral are determined by Jeff's retirement date.

0

t

(? )

vt ? e ?? t ? e ?0.05t ;t p59

?e

6

? PV=70,000 ? v ?t p

t

0

(? )

59

?

(? )

? ?59

? s ?ds

0

t

?e

t

?

(1)

(2)

? ( ?59

? s ? ?59? s )?ds

0

6

? dt ?70, 000 ? e

0

?0.05t

?e

?e

?0.04 t

t

?

? (0.01? 0.001t ? 0.03? 0.001t )?ds

0

6

?e

? dt ? 70, 000 ? e ?0.09t ? dt ?

6

? e ?0.09t ?

? e?0.09(6) ? 1?

70, 000 ?

? ? 70, 000 ? ?0.09 ? ? 324,529.14

? ?0.09 ? 0

?

?

February 17, 2015

Copyright Jeffrey Beckley 2012

0

?

? 0.04?ds

0

? e?0.04t

3. You are given the following table where decrement (1) is death, decrement (2) is lapse, and

decrement (3) is diagnosis of critical illness:

x

55

56

57

58

59

qx(1)

0.02

0.03

0.04

0.05

0.06

qx(2)

0.15

0.06

0.04

0.03

0.02

qx(3)

0.010

0.015

0.020

0.025

0.030

qx(? )

px(? )

a. Complete the table using a radix of 10,000.

See answers at end.

b. Calculate:

i.

3

(? )

p55

Solution:

3

ii.

(? )

55

p

l58(? ) 6605.1

? (? ) ?

? 0.66051

l55 10,000

(2)

2 56

q

Solution:

(2)

2 q56 ?

iii.

d56(2) ? d57(2) 492 ? 293.56

?

? 0.0958

l56(? )

8200

(3)

1|2 55

q

Solution:

(3)

1|2 q55 ?

d56(3) ? d57(3) 123 ? 146.78

?

? 0.026978

l55(? )

10,000

February 17, 2015

Copyright Jeffrey Beckley 2012

lx(? )

dx(1)

dx(2)

dx(3)

iv. The probability that a person age 55 will decrement from death or critical illness

before age 60.

Solution:

d55(1) ? d56(1) ? ... ? d59(1) ? d55(3) ? d56(3) ? ... ? d59(3) 2136.763305

?

? 0.21368

l55(? )

10, 000

c. Assuming uniform distribution of each decrement between integer ages, calculate:

i.

(2)

0.25 55

q

Solution:

(2)

0.25 55

q

ii.

0.5

? 1500 ?

(2)

? (0.25)(q55

) ? (0.25) ?

? ? 0.0375

? 10,000 ?

(? )

p56

Solution:

iii.

0.5

(? )

(? )

(? )

p56

? 1 ?0.5 q56

? 1 ? (0.5)(q56

) ? 1 ? (0.5)(0.105) ? 0.9475

0.5

(? )

p56.8

Solution:

(? )

0.5 p56.8 ?

iv.

(? )

l57.3

(0.7)l57(? ) ? (0.3)l58(? ) (0.7)(7339) ? (0.3)(6605.1)

?

?

? 0.94776

(? )

l56.8

(0.2)l56(? ) ? (0.8)l57(? )

(0.2)(8200) ? (0.8)(7339)

(1)

0.5 55.6

q

Solution:

(1)

0.5 55.6

q

?

? d56(1)

(1)

0.4 55.6 0.1

(? )

55.6

d

l

?

(0.4)(200) ? (0.1)(246)

? 0.01173

(0.4)(10,000) ? (0.6)(8200)

February 17, 2015

Copyright Jeffrey Beckley 2012

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