Microeconomics questions give answer

An explorer aged exactly 57 has just made a proposal to a life office for a whole life assurance with a sum assured of f10.000 payable at the end of the year of death. For lives accepted at normal rate, level annual premiums are payable until death under this policy. The explorer is about to undertake a hazardous expedition which will last three years. The life office estimates that during these three years the explorer will experience a constant addition of 0.02871 to the normal force of mortality, but after three years will experience normal mortality. The life office quotes a level extra premium payable for the first three years. Calculate this level extra premium on the following basis: normal mortality: A1967-70 ultimate interest: 3% per annum expenses: none An impaired life aged exactly 55 wishes to effect a without profit endowment assurance for a sum assured of f1.000 payable at the end of 10 years or at the end of the year of earlier death. Level annual premiums are payable throughout the term of the policy. Special terms are offered on the assumption that the life will experience mortality which can be represented by: (a) for the first five years, a constant addition of 0.009569 to the normal force of mortality, and (b) for the remaining five years, the mortality of a life 8 years older. The life office quotes a level extra premium payable throughout the term. Calculate this level extra premium. Basis: normal mortality: A1967-70 ultimate interest: 3% per annum expenses: none 3 A group of impaired lives now aged 40 experience mortality according to A 1967-70 ultimate with an addition to the force of mortality. The addition is 0.0005 at age 40, increasing linearly to 0.0025 at age 60, at which level the addition remains constant. Find the probability that an impaired life aged exactly 40 (i) will die within 20 years; (ii) will die within 30 years; (iii) will die between 20 and 30 years from the present time. 4 (i) Can you envisage circumstances under which an office could offer an impaired life who wishes to pay the "normal" premiums a level debt but not a diminishing debt? Hint Consider a life who is very severely impaired, and think of an approximate rela- tionship between (a) the level debt and(ii) A certain proposer, aged 40. for a 20-year endowment assurance by annual premiums with sum assured f40.000 without profits (payable at the end of the year of death) is considered by your office to be subject to the mortality of the normal table with an addition of 10 years to the age. Your office's basis for calculating premiums for "normal" lives is A 1967-70 ultimate 4 p.a. interest expenses of 6% of all premiums. (a) The proposer asks to pay the annual premium for a "normal" life aged 40, and suggests that the office should make the normal death benefits subject to a deb which decreases in arithmetical progression to zero in the final year. Calculate the initial debt. (b) Suppose that the proposer also asks your office to quote a level debt. Calculate the bevel debt. 5 The mortality of a certain impaired life aged exactly 60 may be represented as follows: (1) at ages up to exact age 65, there is a constant addition of 0.009569 to the "normal" force of mortality, which is that of A1967-70 select; and (2) at ages between exact age 65 and exact age 70, mortality follows that of a "normal" (ultimate) life 4 years older. Suppose that this life requires a 10-year endowment assurance without profits with sum assured f10, 000 payable on maturity or at the end of the year of earlier death. The life wishes, however to pay the "normal" level annual premium for his age. Calculate the level debt which would be deducted from the normal death benefit, using the basis given below: mortality: as given above interest: 4% p.a. expenses: $% of all premiums, with an additional initial expense of 1% of the sum assured (before deducting any debt.)