Structure of fixed icome securities Bod Valuatio The Structure of fixed icome securities Price & ield to maturit (tm) Term structure of iterest rates Treasur STRIPS No-arbitrage pricig of coupo bods A Fixed Icome Securit promises to pa fixed coupo pamets at a prespecified dates ad a fixed pricipal amout (the face value) at the maturit date. Whe there are o coupo pamets the the bod is called a zero coupo bod or a pure discout bod. Pamets of a N ear bod with aual coupo C ad face value F pamets: C C C CF Time: 0.. N- N Coupo Bods The coupo pamets o coupo bods are tpicall stated as a percetage of the pricipal (or face value) paid per ear. If coupo pamets are made times per ear, the the coupo amout is (c x F)/, where c is the coupo rate ad F is the face value. Uderstadig the terms: U.S. Treasur Notes ad Bods are tpicall issued with face value of $0,000, ad pa semi-aual coupos. Assume a 30 ear coupo rate of 7.5%. What pamets do ou receive from buig this bod? You receive a coupo pamet of (0.075 x $0,000)/ $375 twice a ear ad o the maturit date ou will receive the coupo of $375 plus the pricipal of $0,000. The U.S. govermet issues bods Default free bods issued b the govermet: Treasur bills have a iitial maturit less tha oe ear ad are all discout bods Treasure otes have iitial maturities betwee oe ad te ears ad pa coupos Treasur bods have iitial maturities loger tha te ears ad pa coupos Default free bods issued b govermet sposored agecies Faie Mae: Federal Natioal Mortgage Associatio Giie Mae: Govermet Natioal Mortgage Associatio Freddie Mac: Federal Home Loa Mortgage Corporatio Federal Home Loa Bak
Price & Yield to Maturit The Yield to Maturit (tm) is the (costat) discout rate which equates the discouted cash flows to the price. YTM o a ear Zero Coupo Bod with face value F ad price P F P ( ) Example: Suppose that a 5 ear discout bod with face value $00 is sellig for $95. What is the ield to maturit o this bod? 0. 00 00 95.03% 5 ( ) 95 Calculatig the YTM of a Coupo Bod Example: A 7% coupo bod that pas auall, has ears to maturit, face value of $00 ad price of $98.. Fid the bod s tm. 7 07 98. 8% ( ) What if the price was $00? 7 07 00 7% ( ) What if the coupo rate was 0% (price still $98.)? 0 0 98. % ( ) The Yield Curve/Term Structure of Iterest Rates So fare we have assumed that iterest rates are costat across time I fact, iterest rates var cosiderabl depedig o how log ou are goig to borrow/led: Log term iterest rates reflect ivestors expectatios of future real iterest rates ad iflatio i comig ears If either iflatio or the real iterest rate are expected to chage i the future, the log term rates will differ from short term rates. smartmoe.com Treasur STRIPS & the Yield Curve U.S. Treasur STRIPS are the stripped coupos from T-Bills ad Bods. The prices of these STRIPS are reported as price per $00 of face value. Deote the price of a ear STRIP bod b B The tm of a ear STRIP bod,, is 00 B ( ) 00 B Example: The prices of a, ad 3 ear STRIPS are $96, $9 ad $85. Fid the ields to maturit of these bods? 00 4.75%, 96 00 00 3 4.83%, 3 5.57% 9 85
Prices U.S. Govermet Strips (004) Yield Curve U.S. Govermet Strips (004) Chage i Yield Curve U.S. Govermet Strips (000 vs. 004) Usig market bod prices to discout cash flows Corporatios use bod prices whe evaluatig a stream of certai cash flows. Example: Suppose that ou are cosiderig a ivestmet i a project that geerates the followig cash flows with perfect certait: Date 3 4 Cash flow $50 $00 $00 $00 You also kow the discout bod (STRIPS) prices Date 3 4 Price B j $98 $95 $9 $88 What is the PV of the cash flows from the project? 3
There are two was to calculate the PV: Method I: use the discout bod (STRIPS) prices directl The PV of $ received i ears is B /00. 98 95 9 88 PV 50 00 00 00 34 00 00 00 00 Method II: use ields to maturit PV Date 3 4 j.04%.60%.8% 3.5% 50.004 00 00 3 (.060) (.08) (.035) 00 34 4 No-Arbitrage Pricig of Coupo Bods We ca value coupo bods usig the prices of STRIPS Example: Suppose that ou are give the optio to purchase a 5 ear coupo bod with aual coupo rate 0% ad face value of $,000. Also ou have the followig Treasur STRIPS data: Years to Maturit 3 4 5 STRIPS Price $98 $95 $9 $89 $85 What must the price of the bod be? 98 95 9 89 PV 00 00 00 00 00 00 00 00 85 85 00 000 $,309 00 00 Pricig Coupo Bods with STRIPS prices I geeral the price of a bod is, Bod price C i i Bi 00 Where C i is the i th pamet (icludig coupos ad pricipal) ad B i is the time matched STRIPS price. Term Structure Implied b Coupo Bods We have showed how to price coupo bods usig the prices of STRIPS. STRIPS are priced usig the prices of coupo bods. Example: Cosider the followig coupo bod prices: Years to maturit Face value Coupo Rate Curret price Bod A $,000 5% $997.5 Bod B $,000 8% $,048 What are the prices ad tm s of the oe ad two ear discout bods? 4
Method I: The oe ear tm, The two ear tm, Method II: 80 000 80 048 5.4% B $90 ( ) The price of a oe ear STRIP, The price of a two ear STRIP, 000 50 997.5 5.6% 00 B $95 B 997.5 050 B 00 B B 048 80 080 B 00 00 90 95 Sthetic Discout Bods How could ou stheticall costruct ad discout bods (with face value $00) from the two previous coupo bods? Sthetic ear zero coupo bod: Bod A pas $050 at time. So bu 00/0500.0953 uits of bod A. the price is 0.0953 0953 x 997.5 $95 Sthetic ear zero coupo bod: Bod B pas $080 at time. So bu 00/080 0.0959 uits of bod B. But the ou also get a coupo i the amout $80 x 0.0959 $7.407. To elimiate this sell 7.407/050 0.00705 uits of bod A. Total cost is 0.0959 x 048 0.00705 x 997.5 $90 5