Teaching Bond Valuation: A Differential Approach Demonstrating Duration and Convexity

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1 JOURNAL OF EONOMIS AND FINANE EDUATION olume Number 2 Wier Teachig Bod aluaio: A Differeial Approach Demosraig Duraio ad ovexi TeWah Hah, David Lage ABSTRAT A radiioal bod pricig scheme used i iroducor fiace exs is simple eough bu o ecessaril iuiive. The differeial approach suggesed here preses premiums (discous as coupo ieres over- (uder- pames o make bod pricig damics more iuiive. The primar pedagogic beefi is he differeial approach demosraes he more sophisicaed bod valuaio coceps of duraio ad covexi. Iroducio The basic bod valuaio formula is radiioall preseed as a sraigh forward discoued cash flow applicaio. The relaioship bewee required ield ad price is geerall saed as Bod prices ad ields are iversel relaed: as ields icrease, bod prices fall; as ields fall, bod prices rise (Bodie, Kae ad Marcus pg. 54. A secod bod pricig observaio is A icrease i a bod s ield o mauri resuls i a smaller price chage ha a decrease i ield of equal magiude (Bodie, Kae ad Marcus pg. 54. Boh observaios are aribuable o Malkiel (962. A pedagogic difficul is sudes ma be able o solve for he bod price give a chage i ieres raes, bu ma o full udersad he fiacial damics for bod price premiums ad discous. Furher, he differeial impac of a icrease (decrease i ield o mauri o bod price ma be eve more uclear. This paper firs demosraes he radiioal bod valuaio approach. Secod, a differeial bod price premium ad discou cocepual approach is described. Third, he radiioal ad differeial approaches are compared. Fourh, he differeial approach pedagogic isigh io he more sophisicaed bod valuaio duraio ad covexi cocep is demosraed. A appedix proves he differeial approach is equivale o he radiioal approach equaio. Tradiioal Approach Bod pricig is a relaivel sraigh forward applicaio of he Discoued ash Flow (DF valuaio mehod, ad commo amog iroducor fiace ex books (Berk ad DeMarzo 200, Block ad Hir 2006, Breale, Mers ad Marcus 200, Brigham ad Houso 200, Ross, Weserfield ad Jorda 200, ad Keow, Mari, Pe ad Sco A geeral bod pricig equaio is oupo. ( bod ( ( where ime o mauri, required ield, oupo coupo (ieres rae (, ad ( par value $,000 i U.S. Aubur Uiversi Mogomer, Deparme of Accouig & Fiace, P.O. Box , Mogomer, AL

2 JOURNAL OF EONOMIS AND FINANE EDUATION olume Number 2 Wier The firs erm is he sum of prese values of fuure coupo pames or ieres pames. The secod erm is he prese value of he face value or he pricipal. The bod cash flows icludig he coupo ieres pames, ad he face value or pricipal are deermied a issue ad fixed, wih he excepio of floaig rae bods. Equaio ( is simple, bu o ecessaril iuiive. Iroducor fiace exs geerall refer o he fixed cash flows ad he varig required ield i equaio (. As such, par bods are defied as sellig a face or par value whe he required ield is equal o he coupo rae. Discou bods are bods sellig below face value whe he required ield is greaer ha he coupo rae. Premium bods are bods sellig above face value whe he required ield is below he coupo rae. Discou ad premium bod prices deviae from he face value or pricipal because of he chage i he required ield. The amou of he discou (premium is foud b subracig he from he radiioal formula Marke Price. Several auhors have oiced he pedagogic limiaios of he radiioal approach. Boles, Secres, ad Bure (2005 expaded he above radiioal bod pricig formula o bewee coupo pame daes. Yag (2002 ackowledges he above exbook approach, ad suggess he emphasis o he iverse relaioship bewee ields ad bod prices overlooks he impac of he loaable fuds marke o he behavior of bod prices. Our paper exeds he radiioal pedagogic approach b providig more iuiio behid bod pricig. Differeial Approach The differeial approach calculaes he bod price as equal o value of he bod plus he prese value of uder-paid or over-paid coupo pames from he ivesors view. Uder-paid coupo pames would exis whe ivesors perceived fair level of ieres rae is higher ha he coupo ieres rae he bod is paig. I his case, he price of he bod would be lower ha face value ($,000 b exacl he prese value of uder-paid coupo ieres pames over he life of he bod corac. Over-paid coupo pames would exis whe ivesors perceived fair level of ieres rae is lower ha he coupo ieres rae he bod is paig. I his case, he price of he bod would be higher ha face value ($,000 b exacl he prese value of over-paid coupo ieres pames over he life of he bod corac. Needless o sa, if ivesors perceived fair level of ieres is he same as he coupo ieres rae he bod is paig, here will be o uder-paid or over-paid coupo pames from ivesors view ad he bod price will be he same as face value ($,000. The deviaios of marke prices from face value, as oed above, ca be foud oce oe calculaes he bod price usig eiher a fiacial calculaor or a spreadshee program, followig equaio (. However, deermiig he deviaio usig he resulig bod price accordig o equaio ( does o offer he same cocepual pedagogic advaages as he suggesed differeial approach. ompariso of Tradiioal ad Differeial ocepual Approach Iroducor fiace exs ofe defie a bod sellig a a discou or a a premium, bu o more. Iroducor fiace exs geerall do o discuss how much price deviaio from face value or pricipal here should be for discou bods or premium bods. This ma be easil illusraed usig a differeial wa o fid he value of a bod. Time ash Flows $60 $60 $60 $60 $60 $60 $60$,000 Firs, le s hik of a -ear bod paig aual coupo ieress of $60, i.e., he coupo ieres rae is 6%. Assumig he required ield for his bod i he marke is 6%, he radiioal bod pricig equaio above will ield a bod price of $,000. B lookig a he cash flows, i should be clear ha ivesors are willig o pa $,000 i his siuaio. The required ield of 6% meas ha ivesors wa 6% aual ieres pames whe he, i a sese, led heir moe o he issuer of he bod. If he are paid 6% ieres ever ear, ad ge heir pricipal back a he ed of he bod corac, he prese value of he bod should equal o pricipal or face value.

3 JOURNAL OF EONOMIS AND FINANE EDUATION olume Number 2 Wier Nex, le s hik of a discou bod case, assumig he required ield is %. The bod price will be $946. usig he radiioal bod pricig equaio (. I his case, he required ield of % meas ivesors wa % aual ieres pames o led $,000 o he bod issuer. If he bod has a 6% coupo ieres, he he bod price should be adjused due o he differece bewee required ield ad coupo ieres rae. We ca see he price should be discoued bu do o see b how much. Iuiivel, he bod price discou should be sum of he prese values of uder-paid coupo pames. Thus ivesors will calculae he prese value of uder-paid coupo ieres pames ad add (deduc sice he value is egaive ha from $,000. As a resul of his adjusme, he bod price will be ($,000 he sum of prese values of uder-paid coupo pames. ash flows i he ime lie below idicae uder-paid coupo pames from he perspecive of ivesors. Time ash Flows -$0 -$0 -$0 -$0 -$0 -$0 -$0 Remember he ivesors required ield is % bu he bod pas 6%. Thus ivesors are uder-paid b % ($0 i dollar erms ever ear. This meas ivesors should exrac he % uder-paid coupo pames. Because he bod ideure ses he fixed coupo ieres rae ad face value, he adjusme ca ol be made i he bod price. Hece bod ivesors will require a discou i he bod price. Naurall, he appropriae discou amou will be sum of prese values of uder-paid coupos pames. I his example, he discou amou i he bod price will be -$53.89 (sum of prese values of $0 for seve ears a discoued a required ield of %. So he bod price will be $000 - $53.89 $946.. Or, for he discou bod example: Tradiioal Bod Pricig equaio: bod oupo $60 $,000 ( ( ( 0.0 ( 0.0 $946. Differeial Bod Pricig equaio: bod i Uder Paid oupo $0 $,000 ( ( 0.0 $,000 $53.89 $946.. Now le s cosider he premium bod case. Assume he bod pas 6% coupo ieres pames each ear. Bu ivesors hik he appropriae required ield is 5%. The ivesors receive higher ieres pames ha he require, compee for his bod ad are willig o pa more ha he face value. The quesio agai is, how much more are he willig o pa? Followig he same logic from above, he exac premium ivesors are willig o add o he face value will be he sum of prese values of over-paid coupo ieres pames. I his example, he over-paid aual coupo ieres is $0 i dollar erms. ash flows i he ime lie below idicae over-paid coupo pames from he perspecive of ivesors. Time ash Flows $0 $0 $0 $0 $0 $0 $0 Thus he exac premium ivesors are willig o add o he face value i deermiig he bod price will be $5.86. Hece he bod price i his example will be $000 $5.86 $, Or, for he premium bod example: Tradiioal Bod Pricig equaio: bod oupo $60 $,000 ( ( ( 0.05 ( 0.05 $,05.86

4 JOURNAL OF EONOMIS AND FINANE EDUATION olume Number 2 Wier Differeial Bod Pricig Equaio: bod Over Paid oupo $0 $,000 ( ( 0.05 $,000 $5.86 $, More geerall, his differeial approach o value bods ca be described as follows: bod (. ( where ime o mauri, required ield, oupo coupo (ieres rae (, ( par value $,000 i U.S. Though purposel o ideified as such, he relaioship bewee required ield ad price as discussed above is of course he fiacial bod pricig cocep of duraio (Macaula 938, Bierwag 98. Agai, as Bod prices ad ields are iversel relaed: as ields icrease, bod prices fall; as ields fall, bod prices rise (Bodie, Kae ad Marcus pg. 54. Addiioal pedagogic i-class exercises wih varig uder-paid ad over-paid coupos would reiforce he ield o mauri (required ieres rae iverse relaioship ad sesiivi of bod prices. Bod aluaio ovexi Recall, he secod bod pricig observaio is A icrease i a bod s ield o mauri resuls i a smaller price chage ha a decrease i ield of equal magiude (Bodie, Kae ad Marcus pg. 54. This differeial bod price impac from a icrease (decrease i ield o mauri refers o he well esablished fiacial bod pricig cocep of covexi (Macaula 938, Bierwag 98. Uforuael, bod price covexi ma have a greaer iuiive pedagogic difficul ha he duraio - bod price ield relaioship (Brooks ad Livigso 992, Shirvai ad Wilbrae. 2002, Foruael, he suggesed differeial approach ma also be used o demosrae bod valuaio covexi. The examples from above are show i he Table : Resuls Summar below. The % icrease i required ield reduced he bod price b $53.89 (i.e % decrease i price, while he same % decrease i ield icreased he bod price b $5.86 (i.e. 5.86% icrease i price. Reviewig he respecive differeial bod pricig equaios from above, sudes ca easil see he same $0 coupo ieres cash flow is discoued wo differe raes, respecivel % ad 5%. Buildig o heir prior ime value of moe kowledge, sudes ca see he prese value of he coupo cash flows a % will be smaller ha he prese value a 5%. A icrease (decrease i ield o mauri meas a higher (lower discou rae, hus he prese value of he uder- (over- paid coupo ieres cash flows mus be smaller (greaer. B emphasizig he prese value of he coupo cash flows, he differeial bod pricig approach offers a addiioal isigh io he bod price valuaio fiacial cocep of covexi. The differeial resuls are furher demosraed i Figure : ovexi - % Differeial Bod Price hage. Table Resuls Summar Required Yield Tradiioal Approach Differeial Approach Perce Price hage Marke Price / P Pames % (ield icreases $946.- $,000 - $53.89 $,000- $53.89 $ % 53.89/,000 5% (ield decreases $, $,000 $5.86 $,000 $5.86 $, % 5.86/,000

5 JOURNAL OF EONOMIS AND FINANE EDUATION olume Number 2 Wier 2008 Figure : ovexi - % Differeial Bod Price hage Bod Price 5.86% % Yield o Mauri oclusio Teachig bod valuaio wih he differeial approach usig discous ad premiums has hree pedagogic beefis. Firs, he bod price discou or premium is calculaed direcl raher ha as a secodar calculaio afer he bod price is deermied. Secod, bod price valuaio emphasizig discous (uder-paid ad premiums (over-paid coupo ieres reiforces he bod ideure fixed aure of he coupo ieres rae ad par value. Thus, uder- (over- paid coupos mus be accoued for i he bod price, or chages i ields lead o chages i bod price, demosraig he fiacial cocep of duraio or ieres rae sesiivi. Third, he more iuiivel difficul fiacial cocep of he differeial bod price respose o icreases (decreases i ield o mauri, covexi, is show as a sraighforward resul of usig a higher (lower discou rae for he uder- (over- paid coupo ieres cash flow. Refereces Berk, Joaha ad Peer DeMarzo orporae Fiace. New York: Pearso/Addiso Wesle. Bierwag, Gerald O. 98. Duraio Aalsis: Maagig Ieres Rae Risk. ambridge: Balliger. Block, Sale B., ad Geoffre A. Hir Foudaios of Fiacial Maageme. New York: Irwi- McGraw-Hill. Bodie, Zvi, Alex Kae, ad Ala J. Marcus Ivesmes, New York: Irwi- McGraw-Hill. Boles, Gerald., Thomas W. Secres, ad Rober B. Bure The Pricig of Bods bewee oupo Pames: From Theor o Marke Pracice. Joural of Ecoomics ad Fiace Educaio 4(2: 6-68.

6 JOURNAL OF EONOMIS AND FINANE EDUATION olume Number 2 Wier Breale, Richard.A., Sewar. Mers, ad Alle J. Marcus Fudameals of orporae Fiace. New York: McGraw-Hill - Irwi. Brigham, Eugee F. ad Joel F. Houso Fudameals of Fiacial Maageme. Maso: Thomso Souh-Weser. Brooks, Rober ad Miles Livigso Relaive Impac of Duraio ad ovexi o Bod Price hages, FPE Fiacial Pracice ad Educaio 2(, Keow, Arhur, Joh. Mari, William. Pe, ad David. Sco, Jr Foudaios of Fiace. Boso: Pearso Educaio. Macaula, F. R Some Theoreical Problems Suggesed b he Movemes of Ieres Raes, Bod Yields, ad Sock Prices i he Uied Saes sice 856, New York: Naioal Bureau of Ecoomic Research. Malkiel, Buro G Expecaios, Bod Prices, ad he Term Srucure of Ieres Raes, Quarerl Joural of Ecoomics 6: 9-28 Ross, Sephe A., Ross W. Weserfield, ad Bradford D. Jorda Esseials of orporae Fiace. New York: McGraw-Hill - Irwi. Shirvai, Hassa, ad Barr Wilbrae "Two Pedagogical Simplificaios of he ocep of Duraio." Joural of Ecoomics ad Fiace Educaio : Shirvai, Hassa, ad Barr Wilbrae " Duraio ad Bod Price olaili: Some Furher Resuls." Joural of Ecoomics ad Fiace Educaio 4(: -6. Yag, Bill Z A Two-Dimesioal Dual Preseaio of Bod Marke: A Geomeric Aalsis. Joural of Ecoomics ad Fiace Educaio (2: -.

7 JOURNAL OF EONOMIS AND FINANE EDUATION olume Number 2 Wier Appedix Proof : The differeial wa ields he same bod pricig equaio. bod oupo ( ( ( ( ( ( ( ( ] ( ( [ ( ( ime o mauri, required ield, oupo coupo (ieres rae (, ( par value $,000 i U.S