This question is on applied microeconomics

1 (i) Define the following symbols in words, and give a formula in terms of an integral for each of them: (a) A (b) Az (c) ayr (ii) Consider the following sets of payments: (1) fl immediately on the death of (@) if (y) dies before (r), and (2) an income of 26 p.a. payable continuously to (r) after the death of (y), plus f1 immedia ately on the death of (r) if this occurs after that of (y). Prove that the present values (at force of interest $ pa.) of (1) and (2) are equal. Hence write down a relationship involving A ,, A, and ayr. .2 A special life policy on 2 lives aged r and y respectively provides cash sums of f10.000 and f20,000 immediately on the first and second deaths respectively. In addition, an annuity at the rate of 21,000 per annum will be paid continuously, commencing immediately on the first death and ceasing immediately on the second death. Obtain an expression for the mean present value of the benefits in terms of joint-life and single- life annuity functions and the force of interest. Ignore expenses. 3 An office issues a policy on the lives of a woman aged 60 and her husband aged 64. Under this policy, level premiums are payable annually in advance for 20 years or until the first death of the couple, if earlier. On the first death of the couple, the survivor will receive an annuity of f 10.000 per annum, payable weekly, beginning immediately on the first death. Calculate the annual premium if the office uses the basis given below: Mortality males: a(55) males ultimate females: a(55) females ultimate Expenses: 20% of the first premium 5% of each premium after the first Interest: 6% per annum. 4 A special annuity, payable yearly in arrear, is effected on the lives of a man aged r and his wife aged y. The conditions of payment are: (a) so long as both survive the rate of payment will be 23,000 per annum; (b) if the wife dies first, the rate of payment will be 22,000 per annum until the man's death; (c) payments at the rate of 23.000 per annum will continue for six years certain after the death of the husband, the first payment being at the end of the year of his death, and will be reduced thereafter to f1,500 per annum during the lifetime of the wife. Obtain an expression for the present value of this annuity in terms of single and joint-life annuity factors, life table and compound interest functions. Assume that the same (non- select ) table of mortality is appropriate for the two lives..5 A single-premium policy provides the following benefits to a husband and wife each aged 40. (1) An annuity of 25,000 per annum, payable continuously, commencing on the husband's death within 25 years. or on his survival for 25 years, and continuing so long as either husband or wife is alive. (2) A return of half the single premium without interest immediately on the death of the husband within 25 years, provided that his wife has already died. The office issuing the contract uses the following basis: mortality A1967-70 ultimate interest 4% per annum expenses are ignored. Calculate the single premium. .6 A husband and wife, aged 70 and 64 respectively, effect a policy under which the benefits are (1) a lump sum of f10.000 payable immediately on the first death, and (2) a reversionary annuity of 25,000 p.a. payable continuously throughout the lifetime of the surviving spouse after the death of the first. Level premiums are payable annually in advance until the first death. Calculate the annual premium on the undernoted basis: Males' Mortality: A(55) males ultimate Females' Mortality: a(55) females ultimate Interest: 8% p.a. Expenses: 10% of all premiums Ignore the possibility of divorce