PAPER CODE
MATH 373
EXAMINER: Dr. C. Boado-Penas
DEPARTMENT: Mathematical Sciences
CLASS TEST 2 2020-2021
Life Insurance Mathematics II
Time allowed: 70 minutes plus 15 minutes uploading
INSTRUCTIONS TO CANDIDATES:
In addition to this paper you should have available the 2002 edition
of the Formulae and Tables.
By submitting solutions to this assessment you affirm that
you have read and understood the Academic Integrity Policy
detailed in Appendix L of the Code of Practice on Assessment and
have successfully passed the Academic Integrity Tutorial and Quiz.
The marks achieved on this assessment remain provisional until
they are ratified by the Board of Examiners in June 2021.
Paper Code MATH 373 Page 1 of 3 CONTINUED
1. In this question, β is the last digit of your University of Liverpool ID number,
i.e. it is 0,1,2,3,4,5,6,7,8 or 9.
A 2-year insurance contract on a life aged 60 pays 1, 000 + β × 1, 000 at the
end of the year of death if death is due to accidental causes. Level premiums
are payable in advance if the individual is alive. You are given the following
information from a double decrement table, where decrement 1 represents
accidental death and decrement 2 represents death from all other causes:
x lx d
(1)
x d
(2)
x
60 10,000 440 360
61 380 375
62 450 390
If the annual interest rate is equal to 3%, calculate the annual premium
[5 marks]
2. A defined benefit pension plan provides its members, upon retirement at age
65, an annual pension equal to 2% of final salary per year of service. Final
salary is defined to be the salary earned in the calendar year immediately
preceeding retirement. The pension benefit will be paid in the form of a life
annuity payable at the beginning of each month. The monthly benefit is the
annual benefit divided by 12.
Tom joined the pension plan at exact age 30 on January 1, 2015 with a
salary of £33,000. He expects this salary to increase by 3% each year on his
birthday. As of January 1, 2021, Tom’s annual salary has always increased
at that rate.
You are given that a¨
(12)
65 = 11.5, 29p36 = 0.9 and the annual interest rate is
equal to 2%.
(i) Calculate monthly accrued benefit at age 65 using the Traditional Unit
approach. [2 marks]
(ii) Calculate monthly accrued benefit at age 65 using the Projected Unit
approach. [2 marks]
(iii) Calculate the accrued liability on January 1, 2021, of Tom’s retirement
benefit using both the Traditional Unit and the Projected Unit approach.
Briefly comment. [3 marks]
(iv) Assuming that Tom works until age 65. Calculate the replacement ratio
using the Projected Unit credit method. Briefly comment on the value
of replacement ratio. [3 marks]
Paper Code MATH 373 Page 2 of 3 CONTINUED
3. A life insurance company issues a three-year unit-linked policy to a life aged
58 exact under which level premiums of £3,000 are payable annually in ad-
vance throughout the term of the policy or until earlier death. The premium
allocation rate (%) to the unit-fund at time t is given by:
[75 + 20t] where t = 0, 1 and 2
The units are subject to a bid-offer spread of 5%. An annual management
charge of 0.75% of the bid value of units is deducted at the end of each policy
year before any death, surrender or maturity benefits are paid.
If the policyholder dies during the term of the policy, the death benefit of
£9,000 or the bid value of the units if higher, is payable at the end of the
policy year of death. The policyholder may surrender the policy only at the
end of each policy year. On surrender at the end of the policy year or on
survival to the end of the term, the current bid value of the units is payable.
The company uses the following assumptions in carrying out profit tests of
this contract:
• Mortality: AM92 Select
• Unit fund interest rate: 4% per year
• Non-unit fund interest rate: 2% per year
• Initial expenses £275
• Renewal expenses: £70 per year on the second and subsequent premium
dates
• Initial commission: 5% of the first premium
• Renewal commission: 2% of the second and subsequent years’ premiums
• Surrender: 10% at the end of first, second and third policy years
• Risk discount rate: 6% per year
(i) Project the unit fund and the non-unit fund (profit vector) for the next
3 years. Briefly show your workings so that the derivation of each item
in the table can be seen. [7 marks]
(ii) Calculate the profit margin for the policy. [3 marks]
4. The commands below show some functions using the package lifecontingencies
in R. State, using traditional actuarial notation, what it is being calculated.
• axn(actuarialtable=soa08Act,x=25,n=10)
• axn(actuarialtable=soa08Act,x=25,m=10)
• Axn(actuarialtable=soa08Act, x=25, n=65-25, k=12)
[5 marks]