$3.$ Consider $$100$ million of $30-$year mortgages with a coupon of $5\%$ per year quarterly.

a$.(3$ points$)$ What is the quarterly mortgage payment?

b$.(10$ points$)$ What are the interest and principal repayments over the first year of the life of the mortgages? Calculate for each quarter in year $1.$

c$.(3$ points$)$ Construct a $30-$year GNMA CMO using this mortgage pool as collateral. The pool has three tranches, where tranche A offers the least protection against prepayment and tranche C offers the most protection against prepayment. Tranche A of $$25$ million receives quarterly payments of $4\%$ per year, tranche B of $$50$ million receives quarterly payments at $5\%$ per year, and tranche C of $$25$ million receives quarterly payments at $6\%$ per year.

d$.(30$ points$)$ Assume no prepayments. What are the total promised coupon payments to the three classes in each quarter of the first year? What are the principal payments to each of the three classes in each quarter for the first year?

e$.(30$ points$)$ If over the first year, the trustee receives quarterly prepayments of $$5$ million on the mortgage pool, how are these funds distributed? Show amortization schedules for each tranche for the first year.

f$.(15$ points$)$ How are the cash flows distributed if prepayments in the first half of the second year are $$10$ million quarterly? Show amortization schedules for each tranche for quarters $5$ and $6$ only.