Examples Solutions (Pp 61-80)

Examples

6 1.

For a 10-payment, 20-year term insurance of 100,000 on Pat:

(i) Death benefits are payable at the moment of death.

(ii) Contract premiums of 1600 are payable annually at the beginning of each year for 10 years. (iii) i = 0.05

(iv) L is the loss random variable at the time of issue. Calculate the minimum value of L as a function of the time of death of Pat.

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Page 61 of 100

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Examples

62.

Company ABC sets the contract premium for a continuous life a m i t y of 1 per year on (x) equal to the single benefit premium calculated using:

(i) 6 = 0.03 (ii) p,(t) = 0.02,

t2

0

However, a revised mortality assumption reflects future mortality improvement and is given by

Calculate the expected loss at issue for ABC (using the revised mortality assumption) as a percentage of the contract premium.

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Page 62 of 100

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Examples

63.

For a special fully discrete 30-payment whole life insurance on (45), you are given: (i) The death benefit of 1000 is payable at the end of the year of death. (ii) The benefit premium for this insurance is equal to 1000P45for the fust 15 years followed by an increased level premium of rr for the remaining 15 years. (iii) Mortality follows the Illustrative Life Table. (iv) i=0.06 Calculate n.

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Page 63 of 100

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Examples

64.

For a special fully discrete 2-year endowment insurance of 1000 on (x), you are given;

(i) The first year benefit premium is 668. (ii) The second year benefit premium is 258.

Calc.ulatethe level annual premium using the equivalence principle.

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Page 64 of 100

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Examples

65.

For a special hlly discrete 5-year deferred whole life insurance of 100,000 on (40), you are given;

(i) The death benefit during the 5-year deferral period is return of benefit premium paid without interest. (ii) Annual benefit premiums are payable only during the deferral period. (iii) Mortality follows the Illustrative Life Table.

(iv) z = 0.06

Calculate the annual benefit premium.

E

*

T== h

P,,: g-l = X,,

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ymkikh

n ~ d

E' 5 9s

11 c

& 95

Page 65 of 100

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Examples

66.

For a special fully discrete whole life insurance of 1000 issued on the life of (75), increasing premiums, nk, are payable at time k, for k = 0,1,2,. .. You are given: (i) nk = n O ( l + i ) k (ii) Mortality follows DeMoivre's Law with

OI = 105.

(iii) i = 0.05 (iv) Premiums are calculated in accordance with the equivalence principle. Calculate n, . A) 33

pf7?,~. iib&J

'

B) 36

fi -)APV (Seehefit( -

8

7 s"

/OOO

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I

30 4

A,,

-

I

1000~ 5

Page 66 of 100

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Examples

67.

For a fully discrete 3-year endowment insurance of 1000 on (x), you are given: (i) ,L is the prospective loss random variable at time k. (ii) i = 0.10

(iv) Premiums are determined by the equivalence principle.

Calculate ,L ,given that (x) dies in the second year after issue. A) 540

q-2

fve*.lr'Lk~.a

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Page 67 of 100

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Examples

68.

For a fully continuous whole life insurance of 1 on (30), you are given: (i) The force of mortality is 0.05 in the first 10 years and 0.08 thereafter. (ii)

s = 0.08

Calculate the benefit reserve at time 10 for this insurance.

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Page 68 of 100

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Examples

69.

For a whole life insurance of 1 on (x), you are given: (i) Benefits are payable at the moment of death.

(ii) Level premiums are payable at the beginning of each year. (iii) Deaths are uniformly distributed over each year of age. (iv) i = 0.10

(vi)

=6

Calculate the lob year terminal benefit reserve for this insurance.

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Page 69 of 100

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Examples

70.

For a fully discrete 3-year endowment insurance of 1000 on (x):

(i) i= 0.05

Calculate the second year terminal benefit reserve.

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Page 70 of 100

Examples

7 1.

For a special fully discrete 2-year endowment insurance on (x): (i) The pure endowment is 2000. (ii) The death benefit for year k is (1000k) plus the benefit reserve at the end of year k, k = 1,2. (iii) n is the level annual benefit premium. (iv) i = 0.08

(v) pxlR-I=0.9, k = 1,2. Calculate z . OeneCb

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'ioos

1 I

d

d0W

1

Page 7 1 of 100

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Examples

72.

For a fully discrete 5-payment 10-year decreasing term insurance on (60), you are given:

(i) b,,, =1000(10-k), k=O, 1,

..., 9

(ii) Level benefit premiums are payable for five years and equal 218.15 each

(iv) i = 0.06 Calculate ,v, the benefit reserve at the end of year 2.

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Page 72 of 100

Examples

73.

For a fully discrete whole life insurance of 1000 on (45), you are given:

Calculate 1000,,V,, .

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Page 73 of 100

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Examples

74.

You are given: (i) Ax.nl- = 0.20 (ii) d = 0.08 Calculate .-,V x:nl- . A) 0.78

B) 0.81

C) 0.84

I?) 0.87 E) 0.90

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Page 74 of 100

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Examples

75.

You are given:

(iii)

p:;;l = 0.002 11

(iv) P,,, = 0.01426 Calculate i. A) 5.33%

fi

V

X

=

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f2 :3 d

px:~

,06676 -- A o a l I ,07166 - ~ o ~ l l i

Page 75 of 100

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Examples

76.

You are given:

(i) p3, ( t )= 0.03 , t 2 0 (ii) s = 0.05 (iii) (iv)

I O O O A ; ~=.324.25 ~~ a35:ij

= 8.735 1 15-

-

Calculate 1000-5 V(A35:q).

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Page 76 of 100

Examples

77.

For a fully discrete 15-payment whole life insurance of 100,000 on (x) you are given:

(i) The expense-loaded level annual premium using the equivalence principle is 4669.95. (ii) 100,00OA, = 5 1,481.97

(iv) d = 0.02913

(v) Expenses are incurred at the beginning of the year. (vi) Percent of premium expenses are 10% in the first year and 2% thereafter.

(vii) Per policy expenses are Kin the fvst year and 5 in each year thereafter until death. Calculate K.

g

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Up ~~

- YC~~,Y&*

h l ' a h-*

Page 77 of 100

Examples

78.

For a fully discrete whole life insurance of 1000 on (60), you are given:

(i) The expenses, payable at the beginning of the year, are:

I

Expense Type % of Premium Per Policv I

First Year 20% 8

Renewal Years

I

6% 2

I

(ii) The level expense-loaded premium is 4 1.20. (iii) i = 0.05 Calculate the value of the expense augmented loss variable, L, ,if the insured dies in the 3" policy year.

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Page 78 of 100

Examples

79.

For a filly discrete whole life insurance of 100,000 on (35) you are given:

(i) Percent of premium expenses are 10%per year. (ii) Per policy expenses are 25 per year. (iii) Per thousand expenses are 2.50 per year (iv) All expenses are paid at the beginning of the year.

Calculate the level annual expense-loaded premium using the equivalence principle.

& P .i!cvel

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E~peare.kea:NllucC

P I P I ( I I ~ ~

Page 79 of 100

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Examples

80.

For a fully discrete whole life insurance of 1000 on (SO), you are given: (i) The annual per policy expense is 1. (ii) There is an additional first year expense of 15. (iii) The claim settlement expense of 50 is payable when the claim is paid (iv) AU expenses, except the claim settlement expense, are paid at the beginning of the year. (v) Mortality follows De Moivre's Law with

w = 100.

(vi) i = 0.05 Calculate the level expense-loaded premium using the equivalence

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Page 80 of 100

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