Answer the following microeconomics questions
Example 5.6.1. A life office sells policies each providing a cash sum at age 65. Premiums of f1, 000 are payable annually in advance during the deferred period. On the death of the policyholder during the period of deferment, the premiums paid are returned immediately without interest. In respect of a life now aged 45, find the cash sum at age 65, given that the office uses the following basis: A1967-70 select 4% p.a. interest expenses are ignored5.1 A life aged 40 effects a 25-year without profits endowment assurance policy with a sum assured of 250.000 (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 5.2 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 f10,000, 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; 1 p.a. interest; expenses are 5 of each amual premium including the first, with additional initial ex- penses of 1% of the sum assured. Calculate the annual premium for each policy. 5.3 A 5-year temporary assurance, issued to a woman aged 55, has a sum assured of 250,000 in the first year, reducing by 210,000 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 1% p.a. interest expenses are 10% of all office premiums 5.4 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. 5.5 (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 (r) with sum assured (payable immediately on death if this occurs within a 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 f10,000 and the premium is f260 per annum, payable continuously during the term. The sum assured is payable immediately on death, if death occurs within the term of the policy.Assuming that the office will earn 4% interest per annum, that the future lifetime of the lives may be described statistically in terms of the A1967-70 ultimate table, and that expenses may be ignored, find (a) the mean present value of the profit to the office on the block of policies, and (b) the standard deviation of the present value of this profit. [ A35:2 = 0.15646 on A1967-70 ultimate 8.16%] 5.6 A life office issued a certain policy to a life aged 40. The benefits under this contract are as follows: On death before age 60: an immediate lump sum of f1, 000 On survival to age 60: an annuity of 6500 p.a., payable continuously for the remaining lifetime of the policyholder. Level annual premiums are payable continuously until age 60 or earlier death. Premiums are calculated according to the following basis: Mortality: English Life Table No. 12-Males Interest: 4% p.a. Expenses: Nil Calculate the annual premium
