Mortgage backed securities. In this exercise we look at some features of the optimal time of exercise the prepayment option embedded in mortgage pools. To keep things simple, we consider a short maturity (only 5 years, instead of 30) and semi-annual payments (instead of monthly). More specifically, consider a 5-year to maturity mortgage pool with average semi-annually compounded mortgage rate r™2 = 4%. Assume semi-annual payments of the coupon C. Let market rates evolve according to a simple BDT model, with volatility a = 20%, and 8t obtained to fit the zero coupon bonds in Table 12.12.
(a) The mortgage:
i. Compute the value of the coupon C and the stream of scheduled interest payments, principal payments, and the remaining principal over the 10 semesters.
ii. From the estimates of a and (t for the simple BDT model, construct an interest rate tree at semi-annual frequency.
iii. Compute the evolution on the tree of the net present value of future coupon payments without the prepayment option.
Maturity Rate
3 month LIBOR.........................5.37442%
6 month LIBOR.........................5.38063%
1 year swap...............................5.3340 %
2 year swap...............................5.1325 %
3 year swap...............................5.0740%
4 year swap...............................5.0665%
5 year swap...............................5.0765%
7 year swap...............................5.1155%
10 year swap..............................5.1690%
iv. Compute the value of the American option implicit in the mortgage.
v. Compute the option-adjusted value of the mortgage.
A. When is the prepayment option going to be exercised? Show all the nodes when the exercise of the option would occur (if interest rates reach that far).
B. Is there a path of interest rates whereby the prepayment option is only exercised in year 3 or later?
C. Given your estimates and disregarding default, is the mortgage fairly priced?
D. How would you expect the value of the mortgage to change if the homeowner does not exercise the option optimally (that is, he/she forgets to exercise when he/she should or vice versa, prepay when he/she should not)?
(b) The mortgage backed securities:
i. Consider a 3.5% pass-through security that is collateralized by the same mortgages as above (in average). Compute the value of the pass-through security, as well as its spot rate duration.
ii. Now make two independent trees for the interest rate only (IO) strip and the principal only (PO) strip, and compute their values at time zero.
A. Is the sum of the values of the IO and PO strips equal to the value of the pass-through security computed earlier?