Hi there, Can you please look at the sub-question i that how to work out the extra principal repayment of $74,899,017.65, and why it needs to add two cashflows with this repayment in this case?

An FI originates a pool of 500 30-year mortgages, each averaging $150,000 with an annual mortgage coupon rate of 8 percent. Assume that the GNMA credit risk insurance fee is 6 basis points and that the FI's servicing fee is 19 basis points.

a. What is the present value of the mortgage pool?

PV = $500 x $150,000 = $75 million

b. What is the monthly mortgage payment?

There are 360 monthly mortgage payments (30 years x 12 months). Monthly mortgage payments are $75,000,000 = PVA_{n=360, k=0.6667%}*R R = $550,323.43.

c. For the first two payments, what portion is interest and what portion is principal repayment?

For the first monthly payment, the monthly interest is 0.08/12 x $75 million = $500,000. Therefore, for the first monthly mortgage payment, $50,323.43 is repayment of principal.

For the second monthly payment, the principal outstanding is $75m $50,323.43 = $74,949,676.57. The monthly interest payment is $499,664.51. The principal payment in the second month is $550,323.43 $499,664.51 = $50,658.92.

d. What are the expected monthly cash flows to GNMA bondholders?

The GNMA bond rate is 0.08 (6 + 19) basis points = 7.75 percent. GNMA bondholders receive monthly payments of $75m = PVA_{n=360, k=0.0775/12}*R R = $537,309.18.

e. What is the present value of the GNMA passthrough bonds? Assume that the risk- adjusted market annual rate of return is 8 percent compounded monthly.

The discount yield is 8 percent annually, compounded monthly. The present value of the GNMA passthrough bonds is PV = $537,309.18*PVA _{n=360, k=0.6667%} = $73,226,373.05.

f. Would actual cash flows to GNMA bondholders deviate from expected cash flows as in part (d)? Why or why not?

Actual payments will equal expected payments if and only if no prepayments are made. If any mortgages are prepaid as a result of refinancing or homeowner mobility, then the monthly payments will change. In the month in which prepayments are made, monthly payments will increase to reflect the principal repayments. In all subsequent months, monthly payments will decline to reflect the lower face value of the passthrough bonds.

g. What are the expected monthly cash flows for the FI and GNMA?

GNMA and the originating bank share the difference between the monthly mortgage payments and the GNMA passthrough payments $550,323.43 $537,309.18 = $13,014.25. The originating bank gets 19 out of the 25 basis points (or 76 percent) for a payment of $9,890.83 monthly. GNMA receives the remaining 6 basis points (or 24 percent) for a payment of $3,123.42.

h. If all of the mortgages in the pool are completely prepaid at the end of the second month, what is the pool's weighted-average life? Hint: Use your answer to part c.

Time Expected Principal Payments Time x Principal

1 mo. $50,323.43 $50,323.43

2 mo. $74,949,676.57 $149,899,353.10

$75,000,000.00 $149,949,676.53

WAL = (149,949,676.53/75 million) = 1.9993 months

The principal payment in the first month is $50,323.43. If the loan is paid off after month two, the principal payment in month two is $75 million $50,323.43 = $74,949,676.57.

i. What is the price of the GNMA passthrough security if its weighted-average life is equal to your solution for part (h)? Assume no change in market interest rates.

The GNMA with a weighted average life of 1.9993 months has only two cash flows. The first month's cash flow is $537,309.18. The second month's cash flow is $537,309.18 plus the extra principal repayment of $74,899,017.65 = $75,436,326.83. The present value of the GNMA is PV = [$537,309.18/(1.006667)] + [$75,436,326.83/(1.006667)²] = $74,974,229.44, where the monthly discount rate is 1 + (.08/12).