Provide answers to the questions below.thanks

Question 1

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 A1967-70 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. (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 (b) the initial debt when debts decrease linearly to zero.A pension scheme provides a pension on age retirement of 1/60th of final pensionable salary for each year of service, with part years counting proportionately. Fina pensionable salary is defined to be the average salary over the 3 years prior to retirement. Members contribute 6% of their salaries to the pension fund. One member aged exactly 50 has 18 years of past service and earned $45,000 in the las year. Using the Pension Scheme Tables from the Actuarial Formulae and Tables calculate: (i) the expected present value of this member's past service benefit [2 (ii) the expected present value of this member's future service benefit [2 (iii) the expected present value of this member's future contributions. [2 [Total 6