Answer the following microeconomics questions

9.1 (i) In the context of profit-testing, explain the difference between the "profit vector" and the "profit signature". (ii) A certain life office sells assurance policies with term 3 years to lives aged 70. For each policy, the profit vector is estimated to be (-50, 30, 30). Given that the mortality of the policy holder is expected to follow A1967-70 ultimate, calculate (a) the profit signature per policy sold; b) the net present value of the profit to the office on the basis of a risk discount rate of 8% per annum. 9.2 A life office issues a 5-year guaranteed bonus endowment assurance policy to a life aged 60, with basic sum assured 130.000. The sum assured, with attaching bonuses, is payable at the end of the year of death or at maturity. Level premiums are payable annually in advance. The office holds net premium reserves, using the basis A1967-70 ult at 3% p.a. Interest on premiums and reserves is expected to be earned at an effective rate of 8% p.a. Bonuses will be declared annually at a rate of 3% of the basic sum assured. Bonuses vest at the start of each policy year. Expenses of 40% of the first year's premiums and 5% of subsequent years' premiums will be incurred. Mortality is expected to follow A1967-70 ultimate. Withdrawals may be ignored. i) For each policy year, calculate the total sum assured. (ii) For each duration t = 1, 2, 3,4 years, calculate the reserve (V) immediately before pay- ment of the premium then due, given the following values on A1967-70 ultimate, 3% p.a. interest: (IA)60: = 4.1853 (IA )61:a = 3.4681 (IA )62:m = 2.6973 (IA)3:n = 1.8672 (IA)un = v = 0.97087 (iii) (Difficult.) Calculate the annual premium (P) required for the shareholders/with profits policy holders to achieve an internal rate of return of 12% p.a. on the sale of this contract. Ten years ago a life office issued a large block of 10-year without profits endowment assurances to lives then aged 30. Each policy was effected by annual premiums, and had a sum assured of f40,000, payable on survival or at the end of the year of death. The office's premium basis WAS A1967-70 ultimate 5% interest expenses of 2% of all premiums with additional initial expenses of 0.5% of the sum assured. It was found that mortality , interest and expenses followed these assumptions, but there were surrenders just before payment of the premiums due at durations 1,2 and 3 years. At each of these times, 3% of the surviving policyholders surrendered their contracts, and were given a surrender value equal to the office's reserve, which was calculated on the premium basis, minus a surrender penalty of f40, which the office transferred to the surplus account. By using a profit-testing approach, or otherwise, calculate the surplus accruing to the office at the end of each of the first three policy years, per policy sold. Hint. Note that (PRO), = 0, so profits arise only from surrenders.0.4 Your office is considering the issue of 3-year annual-premium endowment assurance policies without profits to lives aged 62. In respect of a policy with sum assured f10, 000, payable at the end of the year of death (if within 3 years) or on maturity, calculate the net present value of the profit signature on the following assumptions: premium basis: mortality: A1967-70 ultimate interest: 6% p.a. expenses: 3% of all premiums. reserve basis: net premium method using A 1967-70 ultimate, 4% p.a. interest rate of interest to be earned in life fund: 8 p.a. expenses: 3% of office premiums mortality: A 1967-70 ultimate risk discount rate: 10% p.a. 0.5 A life office issues 3-year term assurance policy to a man aged exactly 59. The sum assured is 15,000, payable at the end of the year of death. Level premiums are payable annually in advance. Expenses are expected to be as follows: initial expenses: E10 renewal expenses: E2 incurred at the beginning of the 2"" and each subsequent policy year. It is assumed that interest of 7% per annum will be earned on the life funds, and that mortality follows the A1967-70 ultimate table. The risk discount rate used by the office is 15% per annum. The office calculates the annual premium by requiring that the net present value of the expected profit on each policy is equal to 20% of one office premium. Calculate the office premium on each of the following reserving bases: (i) The office holds zero reserves at each year-end. (ii) The office sets up a reserve at each year-end (except the last) equal to 80% of one office premium