The Green Mortgage Company has originated a pool containing $75$ ten$-$year fixed interest rate mortgages with an average balance of $$100,000$ each. All mortgages in the pool carry a coupon of $12$ percent. $($For simplicity, assume that all mortgage payments are made ann$$ally at $12\%$ interest.$)$ Green would now like to sell the pool to FNMA.

a$.$ Assuming a constant annual prepayment rate of $10$ percent $($for simplicity, assume that pre$-$payments are based on the pool balance at the end of each year$),$ what will be the price that Green should obtain on the date of issuance if market interest rates were $(1)11$ percent? $(2)12$ percent? $(3)9$ percent?

b$.$ Assume that five years have passed since the date in $($a$).$ What will the pool factor be$?$ If market interest rates are $12$ percent, what price can Green obtain then?

c$.$ Instead of selling the pool of mortgages in $($a$),$ Green decides to securitize the mortgages by issuing $100$ pass$-$through securities$.$ The coupon rate will be $11\mathrm{.}5$ percent and the servicing and guarantee fee will be $\mathrm{.}5$ percent. However, the current market rate of return is now $10\mathrm{.}5$ percent. How much will Green obtain for this offering of MPTs$?$ What will each purchaser pay for an MPT security, assuming the same prepayment rate as in $(a)?$

d$.$ Assume now that immediately after purchase in $(c),$ interest rates fall to $9$ percent and that the prepayment rates are expected to accelerate to $20$ percent per year, beginning at the end of the first year. What will the MPT security be worth now?