Two lives aged x and y take out a policy that will pay 15,000 immediately 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] Write down an integral expression (in terms of single integrals only) for the expected present value of the benefit. [3] Prove that the expected present value is equal to: 15,000 SPxxts.y - v15 [v> 15Px +15y] [3] (iv) Describe the appropriate premium payment term for this policy, assuming premiums are to be paid annually in advance. [2] [Total 10]