Answer the following question

Two lives aged x and y take out a policy that will pay out $15,000 on the death of (x) provided that (y) has died at least 5 years earlier and no more than 15 years earlier. (i) Express the present value of this benefit in terms of the random variables denoting the future lifetimes of (x) and (y). [2] (ii) Give an integral expression (in terms of single integrals only) for the expected present value of the benefit. [3] (iii) Prove that the expected present value is equal to: 15,000 15 15 Px x+15:y [3] (iv) State the appropriate premium payment term for this policy, assuming premiums are to be paid annually in advance. [2] [Total 10]An employer provides the following benefits for his employees: immediately on death in service, a lump sum of $20,000 immediately on withdrawal from service (other than on death or in ill health), a lump sum equal to f1,000 for each completed year of service immediately on leaving due to ill health, a benefit of $5,000 pa payable monthly in advance for 5 years certain and then ceasing, and on survival in service to age 65, a pension of $2,000 pa for each complete year of service, payable monthly in advance from age 65 for 5 years certain and life thereafter. The independent rates of decrement for the employees are as follows: Age X 62 0.018 0.10 0.020 63 0.020 0.15 0.015 64 0.023 0.20 0.010 where d represents death, / ill-health retirement and w withdrawal. Each decrement occurs uniformly over each year of age in its single decrement table. (i) Construct a multiple decrement table with radix (al)62 = 100,000 to show the numbers of deaths, ill-health retirements and withdrawals at ages 62, 63 and 64, and the number remaining in employment until age 65. [6] (ii) Calculate the expected present value of each of the above benefits for a new entrant aged exactly 62. Assume that interest is 6% pa before retirement and 4% pa thereafter, and that mortality after retirement follows the PMA92C20 table. [10] [Total 16]