Today is June 7, 2007. Freddie Mac is issuing a 10-year Bermudan note on June 15 2007 under the terms described in Table 12.1410. A Bermudan security is like a callable security, but with call dates defined ex-ante in the term sheet. Prices of Treasury bills and for the same date are in Table 12.15.
Issue Date......................................... June 15, 2007
Maturity Date.................................... June 15, 2017
Subject to Redemption......................... Yes. The Medium-Term are redeemable at our [Freddie Mac] option, upon notice of not less than 5 Business Days, at a price of 100% of the principal amount, plus accrued interest to the Redemption Date
Redemption Date(s)...........................Semi-annually, on June 15 and December 15, commencing on June 15, 20090
Interest Rate Per Annum ..................... 6%
Frequency of Interest Payments ............ Semiannually, in arrears, commencing December 15, 2007
Interest Payment Dates .................... June 15 and December 15
Principal Payment .......................... At maturity, or upon redemption
CUSIP Number............................. 3128X6CD6
(a) From the Treasury data obtain the discount curve Z(0, T) up to June 2017. Plot your results. (You can use bootstrap, or curve fitting, or any other methodology.
(b) From the discount curve Z(0,T), fit a simple semi-annual BDT model, as in Chapter 11. Assume a constant a. The volatility a can be estimated from the variation of 6-month T-bills (see data at the Federal Reserve Web site
www.federalreserve.gov/Releases/hl5/data.htm).
(c) i. Use the simple BDT binomial tree to value the Freddie Mac callable medium term note. What price did you obtain? (Recall that the note becomes callable on June 15, 2009.)
ii. If the price is not equal to the quoted price (100), you can compute the option adjusted spread (OAS) of this security. What OAS did you obtain?
iii. What volatility a would you need to make OAS = 0?
(d) Price again the callable bond, but use the Ho-Lee model instead of the simple BDT model. Is the price the same as with the simple BDT model? Comment.
(e) Compute the spot rate duration of the callable bond.
(f) Compute the spot rate convexity of the bond. This convexity can be defined analogously to the spot rate duration, and it is given by

Where D$ denotes the dollar spot rate duration. On a tree, we can approximate the spot rate convexity in time/node (I, j) as follows
i. Compute the dollar spot rate duration at nodes (i + 1, j) and (i + 1, j + 1) and denote them Ds i+1j and Dsf + 1, j, respectively.
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Issue Date ................................... June 7, 2007
Maturity Date ............................... June 7, 2028
Subject to Redemption.................. Yes. The Medium-Term are redeemable at our [Freddie Mac] option, upon notice of not less than 5 Business Days, See "Redemption" below, We [Freddie Mec will redeem all of the Medium Term Notes if we exercise our option.]
Redemption Date(s).................... June 7, 2010, June 7, 2013, June 7, 2016, June 7, 2019, June 7, 2022, June 7, 2025, and June 7, 2028
Interest Rate per Annum ....... None
Principal Payment .............. at maturity, or upon redemption
Payment CUSIP Number...... 3128X6BZ8
Redemption............................................ Call Price Percentage
June 7, 2010 ............................................32.316531
June 7, 2013 ............................................39.011018
June 7, 2016 ............................................47.092292
June 7, 2019 ............................................56.847631
June 7, 2022 ............................................68.623824
June 7, 2025 ............................................82.839498
June 7, 2028 ............................................100.00000

ii. Compute
(g) How does your result depend on the lockout period? Recompute the convexity assuming that the note is callable starting June, 2008. Is it the same?

Transcribed Image Text:

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