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(a) Ignoring expenses, give a formula for the annual premium. (b) Would you advise the life office to issue the policy with premiums payable as suggested? Give reasons for your answer. 12.6 Estimate, by the trapezoidal rule or another suitable rule for approximate integration, the single premium for a temporary contingent assurance of 250.000 payable immediately on the death of Mrs Smith (aged 60), provided that this event occurs within 5 years and that her husband (aged 50) is alive at the date of her death. Mrs Smith is subject to the mortality of a(55) ultimate (females) and Mr Smith is subject to the mortality of A1967 - 70 ultimate. An interest rate of 7.5% p.a. is to be used, and allowance is to be made for expenses of 6% of the single premium. (Note A very accurate answer is not expected.) 12.7 Define the following functions in words, and give an expression for each of them in terms of an integral. (i) 2091 (ii) As (hi) Al 8 The chief of a certain tribe holds that office until age 50 or earlier death, and may be succeeded only by a person aged from 36 to 45. (A person aged exactly 36 is eligible, but a person aged exactly 45 is not.) The customs of the tribe require that a chief's successor be his oldest eligible brother; if there is no eligible brother, the position of chief is given to someone from outside the previous chief's family, who are then permanently debarred from becoming chief. The present chief is aged exactly 47 and has two brothers, aged exactly 37 and 33 respectively. The chief and his brothers may be regarded as independent lives subject to the mortality of a given table. Obtain an expression, in terms of quantities of the form ape, agi only, for the probability that (33) will become chief. 12.0 Your life office has been asked to quote a single premium for a contingent assurance policy providing 2300.000 immediately on the death of a woman now aged 80 within 15 years. provided that at the date of her death a man now aged 60 has died. Your office uses the following basis: mortality : males - a (55)ultimate (males) females - a (55) ultimate (females) interest : 8% per annum expenses : 10% of the single premium. (i) Assuming that the two lives are independent, write down a formula for the single premium in terms of an integral. (ii) State a suitable non-repeated rule of approximate integration for evaluating this integral. (You are NOT required to carry out the evaluation.) (iii) Would you subject the male life to stringent underwriting procedures? Give brief reasons for your answer.12.1 Which (if any) of the following statements are correct? A. An = 21, B. Arr = 21, C. Anr = 24, - 1 D. And = Al + A 12.2 Adams (aged 40) and Brown (aged 50) are two business partners. Adams wishes to provide for the sum of 280.000 to be paid immediately on Brown's death if Brown predeceases him within ten years, and effects a policy providing this benefit by single premium. The life office issuing the contract employs the following basis: Mortality (both lives) : A1967 - 70 ultimate Interest : 6% Expenses: 2% of the single premium. Using Simpson's rule, or otherwise, estimate the single premium payable by Adams. 12.3 Estimate the value of soft on the basis of A1967 - 70 ultimate mortality. (Assume /109 = 0, so that the integral is over a range of 24 years. Break this into 3 sub-intervals and use Simpson's Rule over each.) 12.4 (1) Express An in terms of ay, air, and the rate of interest. (fi) Smith and Jones are both aged 60. A life office has been asked to issue a special joint-life assurance policy providing 210.000 at the end of the year of death of the first to die of these two lives. In addition, if Smith is the second to die, a further 25,000 will be payable at the end of the year of his death. The policy is to have annual premiums payable during the joint lifetime of Smith and Jones. Calculate the annual premium on the following basis: A1967 - 70 ultimate mortality 4 interest expenses are 5% of all premiums, with an additional initial expense of f100. (ii) Write down (but do NOT evaluate) formulae for the reserve at duration 10 years (imme- diately before payment of the premium then due) on the premium basis, if (a) both Smith and Jones are alive; and (b) Jones has died but Smith is alive. 12.5 A policy providing the sum of 2100.000 immediately on the death of (r) if she dies before (y) s to be issued by a life office to a group of trustees. (i) Ignoring expenses, write down an expression for the single premium in terms of an integral. (ii) The trustees suggest that level annual premiums should be payable in advance until the death of the last survivor of (r) and (y)