7. Jack and Jill, who are ages 50 and 60 respectively and have independent future lifetimes whose mortality follows the ILT, purchase a fully discrete 20 year term insurance policy. The policy pays a benefit of $250,000 upon the second death. Annual premiums are payable as long as both Jack and Jill are alive. Use i = 6%. (a) Show that A 1 50:60:201 +A = A1 60 : 60:20 50:201 + 41 60:20 (Hint: Consider the PVs of the underlying RVs for the four possible cases: (1) Tz 20; (3) T, > 20 and Ty > 20 and Ty > 20. Write an equation corresponding to the underlying RVs. Then we can argue that if the two sides of this equation are equal for all possible values of TY and Ty, then their expected values must be equal. This completes the proof. Or if you find a different/better way to prove it, that's fine too.) (b) Calculate the net annual premium for this policy. [1459] (c) Calculate the probability that this policy pays a benefit. [0.136] (d) Calculate the net premium reserve at time 10, under the assumption that: (i) Only Jack is living. [34196] (ii) Only Jill is living. [73752] (iii) Jack and Jill are both living. [3645]