10-year term assurance with a sum assured of 500,000 payable at the end of the year of death,

is issued to a male aged 30 for a level annual premium of 330.05. Calculate the prospective and

retrospective reserves at the end of the fifth policy year, ie just before the sixth premium has

been paid, assuming AM92 Ultimate mortality and 4% pa interest. Ignore expenses.

21.2 An annual premium conventional with-profits endowment assurance policy is issued to a life

aged 35. The initial sum assured is 50,000, the gross premium is 1,500 and the term of the

policy is 25 years. The death benefit of the sum assured and attaching bonuses is payable at the

end of the year of death. The office declares compound reversionary bonuses, vesting at the end

of each policy year. Given that bonuses of 3% pa have been declared for each year of the

contract so far, calculate the prospective gross premium reserve at the end of the fifth policy

year.

Basis: Future bonuses of 1.92308% pa compound

AM92 Ultimate mortality

6% pa interest

Renewal expenses of 5% of each premium

Claim expenses of 350

21.3 A 10-year regular-premium term assurance policy is issued to a life aged 40. The sum assured is

20,000 and is payable at the end of the year of death. Expenses of 72 are assumed to be

incurred at the start of each year in which the policy is in force, except at the start of the first year

when the expense is 425. The gross premium is 1,700 pa.

Write down an expression for the gross premium retrospective reserve immediately before the

6th premium is due.

21.4 A temporary annuity of 3,000 pa payable annually in arrears for a term of 10 years was

purchased one year ago by a life then aged exactly 60 by the payment of a single premium. Show

algebraically that the current retrospective and prospective gross premium reserves are equal,

assuming that the pricing and reserving bases are the same. The company assumes that each

policy incurs initial expenses of 200 and annual expenses of 1.5% of each annuity payment.

21.5 A special ten-year increasing endowment assurance policy payable by level annual premiums

provides a sum assured of 10,000 during the first year, which increases by 1,000 in each

subsequent year. The sum payable on maturity at age 60 is 25,000. Write down an expression

for the net premium reserve immediately before the 5th premium is paid.

21.6 An annual premium whole life assurance policy provides a sum assured of 30,000 payable

immediately on death. Write down an expression for the gross premium retrospective reserve

after 20 years in respect of a life aged 30 at entry, who is paying a gross annual premium of 250.

Expenses are 100 payable initially, with renewal expenses of 5% of each premium except the

The table below gives an extract of the data of a mortality investigation, which examines the mortality of lives aged between 70 and 71. For each life, you are given the age at which they were first observed, the age at which they ceased to be observed and their reason for leaving the investigation. Life Age at entry Age at exit Reason for leaving 1 70 years 0 months 71 years 0 months censored 70 years 0 months 70 years 11 months died W 70 years 0 months 70 years 8 months censored 70 years 1 month 70 years 3 months died UI 70 years 2 months 71 years 0 months censored 6 70 years 2 months 70 years 10 months censored 7 70 years 6 months 70 years 10 months died 8 70 years 6 months 70 years 8 months censored (i) Calculate the Kaplan-Meier estimate of the survival function STO (1) = P(770 > t) and sketch a graph of this function. [5] (ii) Construct a 90% confidence interval for 970 based on your estimate in part (i). [3] (iii) Calculate the total central exposed to risk and the total initial exposed to risk at age 70 last birthday for the group of lives given in the table above. [3] (iv) Calculate the actuarial estimate of 970 - [1] (v) Calculate the Poisson estimate of 97, and construct a 90% confidence interval for 970 based on this estimate. [5] (vi) Comment on the differences between the estimates of 970 calculated in (i), (iv) and (v). [4] [Total 21]