#3. Use the following short rate tree. Short rate can increase or decrease in 6 months by equal probability. Use semi-annual compounding. Time 0 10% Now, consider a 1 year mortgage (semi-annually paid) with a semi-annual mortgage rate of 10% and initial principal balance of $10,000. The mortgage is divided into three sequential pay tranches. Tranche A receives the first $3,000 of principal, tranche B receives the next $3,000 of principal, and tranche C receives the remaining $4,000 of principal. (a) Fill in the amortization schedule of the whole mortgage below. Beginning Scheduled Principal Semi-annual Balance Payment Date (Year) Date (Year) 0.5 1 0.5 (i) Time 0.5 12% 1.0 8% (b) Fill the scheduled payments to the A, B, and C tranches, assuming no prepayment. (ii) Semi-annual Interest tranche A? Scheduled Principal Ending Principal Repayment Balance A principal A interest B principal B interest C principal C interest 6 (c) According to the interest tree, calculate 6-month discount factor for each node. Moreover, compute 1-year discount factor today. (d) Assuming that there are no prepayments, what is the value of the whole mortgage?