Answer the question below

Question 3.20 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 VS SPY Antsy - V'S 15Px Ax+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] Question 3.21 A level assurance is payable immediately on the death of a man aged 40 provided (1) he dies after his 50th birthday, and (2) his death occurs after, but within 15 years of, the death of another life now aged exactly 45. Derive an expression in terms of assurance functions payable on first death for valuing this benefit, assuming that both lives are subject to the same mortality table. [7]