# Learning Objectives. Chapter 2 Pricing of Bonds. Future Value (FV)

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1 Leaig Objectives Chapte 2 Picig of Bods time value of moey Calculate the pice of a bod estimate the expected cash flows detemie the yield to discout Bod pice chages evesely with the yield Leaig Objectives (cotiued) The elatioship betwee pice ad yield of a optio-fee bod is covex The elatioship betwee coupo ate, time to matuity, bod yield, ad bod pice the picig of floatig-ate ad ivese-floatig-ate secuities accued iteest, dity pice, ad clea (quoted) pice Review of Time Value Futue Value (FV) The FV (P ) of ay sum of moey ivested today is: P = P 0(1+) = umbe of peiods P = futue value peiods fom ow (i dollas) P 0 = oigial picipal (i dollas) = iteest ate pe peiod (i decimal fom) (1+) epesets the futue value of f\$1 ivested dtoday fo peiods at a compoudig ate of

2 Review of Time Value (FV) Moe tha oe time pe yea, both the iteest ate ad the umbe of peiods must be adjusted aual iteest ate umbe of iteest payout pe yea umbe of times payout pe yea umbe of yeas FV iceases with the umbe of compoudig pe yea: eflects the geate oppotuity fo eivestig the iteest paid. Time Value of Moey Peiodic compoudig (If iteest is compouded m times pe aum) FV = PV 1 + m Cotiuous compoudig m FV = PVe m 1 1 lim(1 + ) = e lim(1 + ) = lim(1 + ) = e t t m m m m/ t m Simple compoudig 2-5 Review of Time Value (Auity) Review of Time Value (Auity) FV fo Odiay Auity Auity: Ivestig the same amout of moey peiodically. Odiay auity: Fist ivestmet occus oe peiod fom ow. The equatio fo the futue value of a odiay auity (P ) is: ( 1+ ) 1 P = A A = the amout of the auity (i dollas). = aual iteest ate umbe of times iteest paid pe yea = umbe of times iteest paid pe yea times umbe of yeas Example of Futue Value of a Odiay Auity Usig Aual Iteest: If A = \$2,000,000,,, = 0.08, ad = 15, the P =? ( ) 1+ 1 P = A ( ) 15 1 = \$2,000, P P = \$2,000,000 [ ] = \$54,

3 Review of Time Value (Auity) Example of Futue Value of a Odiay Auity Usig Semiaual Iteest: If A = \$2,000,000/2 = \$1,000,000, = 0.08/2 = 0.04, ad = 15(2) = 30, the P =? P ( ) 1+ 1 P = A ( ) 30 1 = \$1,000, Review of Time Value (PV) Peset Value(PV) PV is the FV pocess i evese. We have: P 1 = ( 1+ ) = aual iteest ate umbe of times iteest paid pe yea = umbe of times iteest paid pe yea times umbe of yeas PV deceases whe iteest ate teds to be highe o the time to paymet date teds to be loge. P = \$1,000,000 [56.085] = \$56,085,000> \$54, Review of Time Value (PV) PV of a Seies of FVs Calculate l the PV of each FV by discoutig The these peset values ae added togethe to obtai the peset value of the etie seies of futue values. Review of Time Value (Auity) Peset Value of a Auity due Whe the fist paymet is immediate, the auity is called a auity due. The PV of a auity due is: 1 (1 + ) C (1 + ) The PV of a odiay auity is 1 1/ ( 1+ ) PV = A A

4 Review of Time Value (cotiued) Example of Peset Value of a Odiay Auity (PV) Usig Aual Iteest: If A = \$100, = 0.09, ad = 8, the PV =? 1 1/ ( 1+ ) PV = A 1 1/ ( ) 8 + PV = \$ Picig a Bod Evaluatig a fiacial istumet equies a estimate of: i. the expected cash flows ii. the appopiate equied yield that eflects the yield i. fo fiacial istumets with compaable isk ii. alteative ivestmets The cash flows fo a bod that the issue caot etie pio to its stated matuity date cosist of i. peiodic coupo paymets to the matuity date ii. the pa (matuity) value at matuity PV = \$100 [ ] = \$ Value of a Coupo Bod The bod pice P ca be computed usig the followig fomula: P C M = t t t + = t= 1( 1+ ) ( 1+ ) Assume the coupo is paid semiaually. P = pice (i dollas) = umbe of peiods (umbe of yeas times 2) t = time peiod whe the paymet is to be eceived C = semiaual coupo paymet (i dollas) = peiodic iteest ate (equied aual yield divided by 2) M = matuity value A Example of Picig a Coupo Bod PV of coupos: Coside a 20-yea 10% coupo bod with a pa value of \$1,000 ad a equied yield of 11%. Give C = 0.1(\$1,000) 000) / 2 = \$50, = 2(20) = 40 ad = 011/ = 0.055, PV of the coupo paymets is: 1 1/ ( 1+ ) C 1 1/ ( ) 40 = \$ = \$50 [ ] = \$

5 A Example of Picig a Coupo Bod PV of pa value. The PV of the pa o matuity value of \$1,000 is: M \$1,000 = = ( ) ( ) 40 \$ Cotiuig the computatio fom the pevious slide: The pice of the bod (P) = PV(coupo paymets )+ PV( matuity value) = \$ \$ = \$ The Value of a Zeo Coupo Bod Fo zeo-coupo bods, iteest is the diffeece betwee the matuity value ad the puchase pice. P M = t ( 1+ ) P = pice (i dollas) M = matuity value = peiodic iteest ate (equied aual yield divided by 2) = umbe of peiods (umbe of yeas times 2) A Example of Picig a Zeo-Coupo Bod Zeo-Coupo Bod Example Coside the pice of a zeo-coupo bod (P) that matues 15 yeas fom ow, if the matuity value is \$1,000 ad the equied yield is 94% 9.4%. Give M = \$1,000, = / 2 = 0.047, 047 ad = 2(15) = 30, what is P? M \$, 1000 P = t = 30 = ( 1 + ) ( ) \$ Pice-Yield Relatioship Pice chages i the opposite diectio fom the chage i the equied yield Maximum Pice ce Pic P C M t= 1( 1+ ) ( 1+ ) = t t t Yield 2-20

6 Exhibit 2-1 Pice-Yield Relatioship fo a 20-Yea 10% Coupo Bod Yield Pi Pice (\$) Yield Pi Pice (\$) Yield Pi Pice (\$) , , , , , , , , , \$ , , Relatioship Betwee Coupo Rate, Requied Yield, ad Pice Whe yields ise above the coupo ate, the pice of the bod falls so that a ivesto buyig the bod ca ealizes capital appeciatio. The appeciatio epesets a fom of iteest to a ew ivesto to compesate fo coupo ate <equied yield. Whe a bod sells below its pa value, it is said to be sellig at a discout. A bod whose pice is above its pa value is said to be sellig at a pemium Relatioship Betwee Bod Pice ad Time Fo a bod sellig at pa value, coupo ate= equied yield. Bod pice emais pa as the bod moves towad the matuity date. The pice of a bod will ot emai costat fo a bod sellig at a pemium o a discout. Exhibit 2-3 (Next slide) shows the time path of a 20-yea 10% coupo bod sellig at a discout ad the same bod sellig at a pemium as it appoaches matuity. The discout bod iceases i pice as it appoaches matuity, assumig that t the equied yield does ot chage. Fo a pemium bod, the opposite occus. Fo both bods, the pice will equal pa value at the matuity date. Exhibit 2-3 Time Path fo the Pice of a 20-Yea 10% Bod Sellig at a Discout ad Pemium as It Appoaches Matuity Pice of Discout Bod Pice of Pemium Bod Yea Sellig to Yield 12% Sellig to Yield 7.8% \$ \$1, , , , , , , ,

7 Reasos fo the Chage i the Pice of a Bod The pice of a bod ca chage fo thee easos: i. chage i the equied yield owig to chages i the cedit quality of the issue ii. chage i the pice of the bod sellig at a pemium o a discout, without ay chage i the equied yield, simply because the bod is movig towad matuity iii. chage i the equied yield owig to a chage i the yield o compaable bods (i.e., a chage i the yield equied by the maket) Complicatios The famewok fo picig a bod assumes the followig: 1) the ext coupo paymet is exactly six moths away 2) the cash flows ae kow 3) the appopiate equied yield ca be detemied 4) oe ate is used to discout all cash flows Settle betwee Coupo Paymet Dates Puchases a bod whose ext coupo paymet is due i less tha six moths: C M P = 1 ( ) v ( ) t 1+ ( ) v ( ) t 1 t= whee v =(days betwee settlemet ad ext coupo) divided by (days i six-moth peiod) C v C 1 2 Complicatios Cash Flows May Not Be Kow callable bod floatig ate bod Detemiig the Appopiate Requied Yield Teasuy T yields as bechmak. Decompose the equied yield fo a bod ito its compoet pats. Oe Discout Rate Applicable to All Cash Flows discout with yield ate A bod ca be viewed as a package of zeo-coupo bods, each cash flow is discouted with zeo ate Settlemet date

8 Picig Floatig-Rate ad Ivese-Floatig-Rate Secuities The cash flow fo eithe a floatig-ate o a ivesefloatig-ate secuity depeds o the futue efeece ate Pi Pice of a Floate The coupo ate of a floatig-ate secuity (o floate) = efeece ate + spead (o magi). The pice of a floate depeds o i.the spead ove the efeece ate ii.ay estictios imposed o the esettig of the coupo ate i.ex: a floate may have a maximum coupo ate called a cap o a miimum coupo ate called a floo. Picig Ivese-Floatig-Rate Secuities Pice of a Ivese-Floate Ceated fom a fixed-ate secuity called collateal Fom the collateal two bods ae ceated: a floate ad a ivese floate. Collateal (Fixed-ate bod) Floatig-ate Bod ( Floate ) Ivese-floatig-ate bod ( Ivese floate ) The pice of a ivese floate equals the collateal s l pice mius the floate s pice Pice Quotes ad Accued Iteest Pi Pice Quotes A bod sellig at pa is quoted as 100, meaig 100% of its pa value. A bod sellig at a discout will be sellig fo less tha 100. A bod sellig at a pemium will be sellig fo moe tha 100. Pice Quotes ad Accued Iteest (cotiued) Tades quotig the bod pice as a pecetage of pa value. Exhibit 2-5 i ext slide shows how a quote pice is coveted ito a dolla pice. Whe a ivesto puchases a bod betwee coupo paymets, the ivesto must compesate the selle the coupo iteest eaed fom the last coupo date to the settlemet e t date of the bod. This amout is called accued iteest. Fo copoate ad muicipal bods, accued iteest is based o a 360-day yea, with each moth havig 30 days

9 Exhibit 2-5 Pice Quotes Coveted ito a Dolla Pice (1) Pice Quote (2) Coveted to a Decimal [= 1)/100] (3) Pa Value (4) Dolla Pice [= (2) (3)] 80 1/ , , / ,000, , / , , ,000 50, ,000 1, / , , / ,000 26, Pice Quotes ad Accued Iteest (cotiued) The amout that the buye pays the selle is the ageed-upo pice plus accued iteest. Called full pice o dity pice. The bod pice without accued iteest is called the clea pice. The exceptios ae bods that ae i default. Such bods ae said to be quoted flat, that is, without accued iteest Bloombeg quotes U.S. T-bod, with coupo ate 3.5% ad matue o 11/15/2006 Quote pice & quote yield Bloombeg quotes Gema T-bod, with coupo ate 4% ad matue o 07/04/2009 Bod atig Pi Pices ae quote i 32 d, 96-5 is 96 ad 5/

10 Pice quotes bod pices ae quoted as a pecetage of pa value Examples Pice quotes Bod quoted pice Clea pice = goss pice accued iteest = the sum of PV of the futue cash flows accued iteest Goss pice is the pice that bod buye must pay Goss/Dity/full pice= clea pice + accued iteest Accued Iteest Whe a ivesto puchases a bod betwee coupo paymets the iteest paymet pevious bodholde should eceive the quoted pices do ot iclude accued iteest buyes must pay the quote bod pice + accued iteest C C C C A factio of this coupo paymet belogs to the pevious bod holde days sice last coupo paymet Accued Iteest = C days betwee coupo paymets Accued Iteest Calculatio (Semi-aual coupo paymet ) Bod quoted pice Bloombeg quotes it actual umbe of days sice last coupo paymet accued iteest = 2 actual umbe of days i coupo peiod Actual umbe of days sice last coupo paymet Actual umbe of days i coupo peiod Last Coupo Paymet Bod Settlemet Date Next Coupo Paymet 39 40

11 Bloombeg quotes Bod quoted yield Goss, clea pice ad accued iteest A vaiety of yields is quoted Bod quoted spead Bloombeg quotes Bod yield expessed as the speads ove some bechmak yields 41 42

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