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 Ax, Ay, Axy, ax, ay, and axy. 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.