Hello,help with the following questions

.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 policyholder 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. .2 A life office issues a 5-year guaranteed bonus endowment assurance policy to a life aged 60, with basic sum assured 30,000. 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 )60:51 = 4.1853 (IA )61:7 = 3.4681 (IA)62:31 = 2.6973 (IA ) 63:21 = 1.8672 (IA )64:1 = = 0.97087 (Difficult.) Calculate the annual premium (P) required for the shareholders/ with profits policyholders to achieve an internal rate of return of 12% p.a. on the sale of this contract. .3 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.You are given the following data from two populations: MADEUPTOWN STANDARD POPULATION Sex Occupation Population Deaths Population Deaths Office worker 20,000 100 10,000 50 Male Manual worker 60,000 500 20,000 300 Other 20,000 250 20,000 500 Office worker 20,000 50 10,000 50 Female Manual worker 30,000 200 10,000 100 Other 50,000 900 30,000 500 Total 200,000 2,000 100,000 1,500 Calculate the directly standardised mortality rate for Madeuptown, standardising by, (a) occupation, (b) sex, and (c) occupation and sex