An insurance policy sold to (x) and (y) provides for a payment of 2 units on the first death and 3 units on the second death. Level annual premiums are payable for as along as either individual is living, with the premiums reducing by one-third upon the first death. Find an expression for the initial premium in terms of A_x, A_y, A_xy, a_x, a_y, and a_xy. 10 years, the payments remain at 12. However, during the second 10-year period, the payment reduces to 6 if (x) only is alive, or to 8 if (y) only is alive. Write the present value in terms of single and joint-life annuities.