A life aged 40 effects a 25-year without profits endowment assurance policy with a sum assured

of 50,000 (payable at the end of the year of death or on survival to the end of the term). Level

premiums are payable annually in advance throughout the term of the policy or until earlier

death of the life assured. Calculate the level premium, P, using the following premium basis

Mortality: A1967-70 Ultimate;

Interest: 6% p.a.

Expenses: none

An office issues a large number of 25-year without-profit endowment assurances on lives aged

exactly 40. Level annual premiums are payable throughout the term, and the sum assured of

each policy is 10,000, payable at the end of the year of death or on survival to end of the

term. The office's premium basis is:

A1967-1970 ultimate;

4% p.a. interest;

expenses are 5% of each annual premium including the first, with additional initial expenses of 1% of the sum assured.

Calculate the annual premium for each policy.

A 5-year temporary assurance, issued to a woman aged 55, has a sum assured of 50,000

in the first year, reducing by 10,000 each year. The sum assured is payable at the end of the

year of death. Level premiums, limited to at most 3 years' payments, are payable annually in

advance. Calculate the annual premium.

Basis:

A1967-1970 select mortality

4% p.a. interest

expenses are 10% of all office premiums

A life office sells immediate annuities, using English Life Table No. 12 - Males, 4% p.a. interest

with no expenses as the premium basis. Assuming that the mortality of annuitants does follow

this table, that investments will earn 4% per annum, and that expenses are negligible, find the

probability that the office will make a profit on the sale of an annuity payable continuously to

a life aged 55.

(i) Let g(T) be the present value of the profit to the life office, at the issue date, in respect

of an n-year without profits endowment assurance to (x) with sum assured (payable

immediately on death if this occurs within n years) and premium P per annum, payable

continuously for the term of the policy. Expenses are ignored in all calculations.

(a) Write down an expression for g(T).

(b) Derive expressions for

(1) the mean, and

(2) the variance of g(T).

(c) For what value of P is the mean of g(T) equal to zero?

(ii) An office issues a block of 400 without profits endowment assurances, each for a term of

25 years, to lives aged exactly 35. The sum assured under each policy is 10, 000 and the

premium is 260 per annum, payable continuously during the term. The sum assured is

payable immediately on death, if death occurs within the term of the policy

Following archaeological excavations at a site in Egypt, ten samples of wood were carbon-dated and their ages x (years) estimated as: 4,900 4,750 4,820 4,710 4,760 4,570 4,300 4,680 4,800 4,670 Ex =46,960 [x2 =220,772,800 (i) Calculate a 95% confidence interval for the true mean age of the wood found at this site. [3) (ii) Present these data values graphically and comment on the validity of the confidence interval calculated in part (i). [2) (iii) Ideally the archaeologist would like the 95% confidence interval for the true mean age, calculated in (i) above, to have a width of no more than 200 years. Calculate the minimum sample size needed. [3 (iv) At a second site, eight samples of wood gave the following results: Ey=36,000 Ey? =162,280,000 Calculate a 95% confidence interval for the difference between the mean ages of the wood found at the two sites. [3 (v) Obtain a 90% confidence interval for the ratio of the underlying variances in the ages of the two samples of wood. Hence comment on the validity of the confidence interval given in part (iv). [4)