Answer the following questions and explain them fully.

Write down a single term expression to represent the present value of a deferred annuity-due of 1 pa payable to a life now aged x , with a deferment period of n years. Hence derive a formula for the variance of the present value of this contract (hard!).An impaired life aged 40 experiences 5 times the force of mortality of a life of the same age subject to standard mortality. A two-year term assurance policy is sold to this impaired life, and another two-year term assurance is sold to a standard life aged 40. Both policies have a sum assured of $10,000 payable at the end of the year of death. Calculate the expected present value of the benefits payable to each life assuming that standard mortality is AM92 Ultimate and interest is 4% pa.Solve the following constrained optimization problem: max ro + 2y, subject to: x2 + y? $5, y20, and find optimal (x, y)