MTH 214 Life Insurance Mathematics II 2021/22 open book final exam
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MTH 214
2nd SEMESTER 2021/22 open book final exam
Life Insurance Mathematics II
Questions
Q 1. A life insurance company sells a policy with a 20–year term that provides a benefit of $100,000
payable immediately on death or on earlier diagnosis of a critical illness. No further benefit is paid in the event of death after a critical illness claim has been paid. Premiums of P are paid annually in advance throughout the term or until a claim, if earlier. The company prices the policy using the following multiple state model using the forces of transition σ北 ,ρ北 ,µ北 ,ν北 for a life age x.
Healthy (H)
|
北 |
Critically ill (I)
|
|
||
北 |
Basis:
µ北 = 0.035 for all ages
σ北 = 0.020 for all ages
Interest rate is 3% per annual effective.
(i) Determine, and specify the reason, the values for ρ北 and ν北 that should be used in the pricing model for this policy. [2 marks]
(ii) Calculate the expected present value of the benefits for this policy. [5 marks]
(iii) Calculate the annual premium for this policy using equivalent principle. [5 marks]
Round your final solution to 2 decimal places.
[Total 12 marks]
Q 2. A Type B Universal Life policy was issued to a person (58). You are given:
. Policyholder’s account value AV4 = $650.
. The additional death benefit is $600,000.
. The COI (Cost of Insurance) rate in year 5 is $20 per $1,000 of insurance
. Expense charges are $300 per year plus 5% of premiums.
❼ Credited interest rate applied: ic = 5%
❼ The policy does not have a no-lapse guarantee.
Calculate the minimum premium to be paid at the beginning of year 5 so that the policy does not lapse before the next premium is paid at the beginning of year 6.
[6 marks]
Round your final solution to 2 decimal places
Q 3. You are given the following three-decrement service table for modeling employment.
x |
ℓx |
d |
d |
d |
60 |
10,000 |
300 |
150 |
25 |
61 |
9,525 |
360 |
120 |
50 |
62 |
8,995 |
390 |
115 |
75 |
(i) Calculate 3p6(0)0(1) and 2
[4 marks]
(ii) Calculate the EPV of a benefit of $10,000 payable at the end of the year of exit, if a
life aged 60 leaves by decrement 3 before age 63. Use an effective rate of interest of 4% per year. [5 marks]
(iii) Calculate the EPV of an annuity of $1,000 per year payable at the start of each of the
next three years if a life currently aged 60 remains in service. Use an effective rate of interest of 4% per year. [5 marks]
(iv) Calculate q6(*), assuming a constant force of transition for each decrement. [5 marks] (v) Calculate the revised service table for age 62 (i.e. the value of d for i = 1, 2, 3) if q6(*) is increased to 0.1 with the other independent rates (q6(*), q6(*)) remaining unchanged.
You should also assume a constant force of transition for each decrement.
[6 marks] [Total 25 marks]
Q 4. You are given the following mortality records for a standard population and the population
in province A.
Province A Standard Population |
||||
Age group |
Population at 30 June 2021 |
Deaths in 2021 |
Population at 30 June 2021 |
Deaths in 2021 |
0– 14 |
510,000 |
133 |
5,600,000 |
1,560 |
15-39 |
720,000 |
570 |
9,200,000 |
7,730 |
40-59 |
700,000 |
4360 |
6,200,000 |
53,960 |
60-79 |
730,000 |
33,930 |
4,500,000 |
267,550 |
80+ |
100,000 |
23,230 |
1,000,000 |
200,000 |
Total |
2,760,000 |
62,223 |
26,500,000 |
530,800 |
(i) Calculate the crude death rate for Provinces A and the standard population. [2 marks] (ii) Calculate the directly standardized mortality rate for province A, using standard pop-
ulation as reference.
Round your final solutions to 5 decimal places
Q 5. A pension scheme provides a pension of 1/80th of final pensionable salary on retirement, due
to age or ill-health, for each year of service (past years included). Final pensionable salary received is average salary over the three years before retirement. Normal retirement age is 65. Members contribute 5% of pensionable salary each year.
(i) Calculate the expected present value of the combined past and future benefits for a member aged 35 exact with 10 years of past service and salary in the previous year of $45,000. [7 marks]
(ii) Calculate the present value of the member’s future contributions. [8 marks]
Basis: Pension Scheme Table with annual interest rate 4% in the Formulae and Tables for Examinations.
Round your final solution to 1 decimal place.
[Total 15 marks]
Q 6. A special joint life annuity is issued to a male life now aged 65 exact and a female life now
aged 62 exact. The annuity is payable monthly in advance and is subject to the following conditions:
. The amount of the annuity while both lives survive is $100,000 per year.
PAPER CODE: MTH 214/2021/22/open book final exam Page 4 of 6
❼ If the male life dies first leaving the female life surviving the annuity reduces to $75,000
per year payable until she dies.
❼ If the female life dies first, no further annuity payments.
❼ In addition if both lives are alive at the 20th anniversaries of the policy, a cash lump
sum of $20,000 is paid on that date.
Calculate the expected present value of the annuity. [10 marks]
Basis :
Mortality: PMA92C20 and PFA92C20
Interest: 4% per annum
Initial Expense: None
Assumption: Use UDD assumption for m)
Two lives are assumed to be independent
Note: you should keep at least 4 decimal places throughout the calculation.
Round to 1 decimal place in your final solution
Q 7. A female aged 45 buys a 3-year equity-linked endowment insurance under which level pre-
miums of $2,500 are payable annually in advance throughout the term of the policy or until earlier death. The allocated premium (AP) is 90% for the first premium and 99% for the rest premiums, which are invested in policyholder’s fund.
The death benefit, which is payable at the end of the year of death, is the greater value between $7,500 (GMDB) and the fund value (Ft ).
The policyholder has the right to surrender (lapses) only at the end of each year. If the policyholder surrender, then 100% the policyholder’s fund value (Ft ) at the end of the year (after management charge deductions) is payable.
If the policyholder holds the contract to the maturity date, she receives the 101% of policy- holder’s fund value (Ft ) (after management charge deductions).
The insurance company incurs pre-contract expenses of $120 plus renewal expenses of $50 at the start of each policy year. At the end of each policy year a fund management charge of 0.5% of the value of the policyholder’s fund is transferred to the insurer’s fund. Note that management charges are deducted from fund before any benefit payments.
The mortality probability follows AM92 Ultimate. Surrender probability is 10% for the policy at the end of the first year, 5% for the policy at the end of the second year and no surrender is allowed in third year.
Policyholder’s fund are assumed to earn investment return at 8% per annual, insurer’s fund are assumed to earn interest at 5% per annual. Risk discount rate used to calculate NPV is
10%.
(i) Calculate and project the policyholder’s and insurer’s fund for the contract with the assumption that the company does not zeroise future expected negative cash flows and finally give the profit signature. [10 marks]
(ii) Calculate the NPV based on the profit signature obtained above. [5 marks]
(iii) Calculate potential required zeroised reserve at start of each year; and revise the profit
signature and NPV for this policy when considering zeroised reserve. [10 marks]
Round your final solution to 2 decimal places.
[Total 25 marks]
2022-07-30