So: [55]:10 a = 7.959 [1] We see here that the value of the select
annuity is actually lower than the value of the ultimate annuity. We
would normally expect the reverse since the (normally lighter) select
mortality would increase the expected present val
AM92 Ultimate mortality, 4% pa interest, no expenses [3] Question
2.11 A special ten-year increasing endowment assurance policy
provides a sum assured of 10,000 during the first year, which
increases by 1,000 in each subsequent year. The sum payable on
ma
5 10 100 150 50 150 50 m m m + + + + - + - t x xt t x xt t t t x xt
a p dt a a p dt a a p dt [1 ] The first integral is: ( ) 0.04 5 5 0.01 0 0 5
0.01 0.05 0 0.05 0.25 1 100 100 0.01 0.04 100 0.01 0.04 1 1 25 0.01
0.05 11.32683 m - - + - - - - - = = - - -
she is still alive at age x + n . [1] II is not correct. The n| a term on the
RHS will make payments even if the life has died. This should be
replaced with the temporary annuity factor | x:n a . [1] CT5: Q&A
Bank Part 1 Solutions Page 5 The Actuarial Edu
profit The present value of the profit to the insurance company is: EPV
Premiums EPV Benefits NPV = - The present value of the premiums is:
2 [60] [60] 1 62 [60] 2 EPV Premiums 3,000 103,735.4 103,061.1
102,040.8 3,000 103,735.4 3,000 2.8127 8,438.10 + +
deferred annuity is: 20 20 60 60 @0% 60 60 0 [40] [40] 200 1.05 200
= t t t l l v v p dt v a l l [2] We have: 20 20 60 [40] 9,287.2164 1.05
0.3552 9,854.3036 l v l - = = [1] The annuity factor at 0% is the same
as the complete expectation of life 60 e .
take care of your study material to ensure that it is not used or copied
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years is 10,000 pa, and thereafter it is 12,000 pa. The force of
mortality of this life is 0.03 pa between the ages of 60 and 65, and
0.04 pa between the ages of 65 and 70. Calculate the expected
present value of this annuity assuming a force of interest
Development Questions Question 2.1 Calculate the annual premium
payable in advance by a life now aged exactly 32 in respect of a
deferred annuity payable from age 60 for 5 years certain and for life
thereafter. The amount of the annuity is 400 pa, payable
2 Questions Page 3 The Actuarial Education Company IFE: 2013
Examinations Question 2.8 Consider each of the following policies,
issued to a life aged exactly x : (i) a regular premium whole life
assurance (ii) a single premium term assurance (t n < ) (iii
whole life annuity cannot be greater than the corresponding
perpetuity, which would be | a d 1/ 1.1/ 0.1 11 = = . [1] (iv) is true.
The probability that a 30-year old will die before age 55 is much less
than 30%. So the value of this function (even before
70 70 t t l t t e p dt dt t dt l + - = = = -= = D [2]
Alternatively, you could argue that for 0 70 < k : ( ) ( )( ) 30 30 1 30 30
1 30 30 70 70 1 1 70 70 k k k k l l PK k p p l k k + + + - = - = - - - = = So
if (30) is equally likely to die in each future
assurance policy provides a sum assured of 30,000 payable
immediately on death. Write down an expression for the retrospective
reserve after 20 years in respect of a life aged 30 at entry. Expenses
are 100 payable initially, with renewal expenses of 5% of
D EgK a D D D a a D D [3] [Total 4] Solution 1.8 The adjusted annual
rates of mortality will be: q q 65 65 = = = 2 2 0.014243 0.028486 [ ]
and: q q 66 66 = = = 2 2 0.015940 0.031880 [ ] The probability of
surviving for 2 years can then be calculated as: 2
pa interest. [7] (ii) During the first policy year 75 policyholders died.
Calculate the net premium reserve at the end of 2003 and hence the
mortality profit for the portfolio for calendar year 2003. [5] (iii) A
director of the company has calculated the
women aged exactly 60 are limited to 5 years. The annuity
commences at age 65, provided the policyholder is still alive at that
age. The annuity provides payments of 3,500 payable annually in
advance for 5 years certain (ie it continues to be paid for 5 y
Premium calculation The expected benefit outgo per policy is: 1 62
60:2| 60 62 60 50,000 50,000 802.40 50,000 0.45640 0.48458 882.85
50,000 0.0159775 798.88 D A AA D = - = - = =
[1] The variance per unit sum assured is equal to 21 1 2 60:2 60:2 | | A
A (
aged 32, buys a 12-year temporary assurance with a sum assured of
5,000 payable at the end of year of death. Calculate the annual
premium for this policy. Assume AM92 Select mortality, 4% pa interest
and that premiums are payable annually in advance. Allo
Actuarial Education Company IFE: 2013 Examinations and similarly:
( ) 0.45 65:5 1 1 4.02635 0.05 0.04 - = -= + a e [] So the expected
present value of the annuity is: (10,000 4.12100 12,000 0.67032
4.02635 73,597 + = ) ( ) [1] [Total 4] (ii)(a) EPV of ter
profit in part (ii) ignores the contingency loading in the premium
rates. This is the source of the profit. [1] Page 18 CT5: Q&A Bank Part 2
Solutions IFE: 2013 Examinations The Actuarial Education
Company If the director recalculated his figures with th
Part 2 Solutions IFE: 2013 Examinations The Actuarial Education
Company Solution 2.2 The premium equation is: 1 32:12 32:12 | | Pa A
= 5,000 [1] The factors are: 44 32:12| 32 44 32 1,747.41 21.520 19.075
9.725 2,825.89 D aa a D =- = - = [1] 1 44 32:12| 32
Rearranging, dividing by h , and letting h 0 , gives the differential
equation: () ( ) () V t P S Vt =- + + mxt xt + + m d or: ( ) ( ) [ ( )] V t P Vt
S Vt =+ - - d mx t + [1] The second form separates the savings
element (the growth in the reserve ignori
100 0.03 ( 1) [2] The factors are: 44 [32]:12| [32] 44 [32] 1,747.41
21.523 19.075 9.726 2,825.48 D aa a D =- = - = [1] 1 44 [32]:12| [32]
44 [32] 1,747.41 0.17218 0.26636 0.00745 2,825.48 D AA A D =- = - =
[1] So the premium equation becomes: 9.726 5,000
maturity value is 25,000. So the prospective net reserve is: 1 1 60 4
| | 54:6 54:6 54:6 54 13,000 1,000( ) 25,000 pro D V A IA Pa D = + +[2] where the premium is given by: 1 1 60 50:10 50:10 50:10 | | 50
9,000 1,000( ) 25,000 D Pa A IA D =+ + [2] [Total
2013 Examinations Hence the premium is: 60:2| 798.88 608.29
1,407.17 720.21 1.953825 P a + = = [1] [Total 7] (ii) Net premium
reserve and mortality profit The net premium reserve is equal to the
value of future benefits less the value of net premiums. The
result that follows from a uniform decrement assumption. [1] III is
false. The multipliers t and t -1 are transposed. For example, putting t
= 0 gives x x 1 l l = + , instead of x x l l = . [1] [Total 3] Page 4 CT5: Q&A
Bank Part 1 Solutions IFE: 2013 Exa
declared at the rate of 1.92308% of the sum assured, compounded
and vesting at the end of each policy year on the following basis:
Mortality: AM92 Select Interest: 6% pa Initial expenses: 114% of the
first premium and 2.5% of the basic sum assured Renewal