Chapter-4-answers

56CHAPTER4BONDPRICEVOLATILITYANSWERSTOQUESTIONSFORCHAPTER4(Questionsareinboldprintfollowedbyanswers.)1.Thepricevalueofabasispointwillbethesameregardlessiftheyieldisincreasedordecreasedby1basispoint.However,thepricevalueof100basispoints(i.e.,thechangeinpricefora100-basis-pointchangeininterestrates)willnotbethesameiftheyieldisincreasedordecreasedby100basispoints.Why?Theconvexrelationshipexplainswhythepricevalueofabasispoint(i.e.,thechangeinpricefora1-basis-pointchangeininterestrates)willberoughlythesameregardlessiftheyieldisincreasedordecreasedby1basispoint,whilethepricevalueof100basispointswillnotbethesameiftheyieldisincreasedordecreasedby100basispoints.Moredetailsarebelow.Whentheprice-yieldrelationshipforanyoption-freebondisgraphed,itexhibitsaconvexshape.Whenthepriceoftheoption-freebonddeclines,wecanobservethattherequiredyieldrises.However,thisrelationshipisnotlinear.Theconvexshapeoftheprice-yieldrelationshipgeneratesfourpropertiesconcerningthepricevolatilityofanoption-freebond.First,althoughthepricesofalloption-freebondsmoveintheoppositedirectionfromthechangeinyieldrequired,thepercentagepricechangeisnotthesameforallbonds.Second,forverysmallchangesintheyieldrequired(like1basispoint),thepercentagepricechangeforagivenbondisroughlythesame,whethertheyieldrequiredincreasesordecreases.Third,forlargechangesintherequiredyield(like100basispoints),thepercentagepricechangeisnotthesameforanincreaseintherequiredyieldasitisforadecreaseintherequiredyield.Fourth,foragivenlargechangeinbasispoints,thepercentagepriceincreaseisgreaterthanthepercentagepricedecrease.2.Calculatetherequestedmeasuresinparts(a)through(f)forbondsAandB(assumethateachbondpaysinterestsemiannually):BondABondBCoupon8%9%Yieldtomaturity8%8%Maturity(years)25Par$100.00$100.00Price$100.00$104.055(a)WhatisthepricevalueofabasispointforbondsAandB?ForbondA,wegetabondquoteof$100.00forourinitialpriceifwehavea2-yearmaturity,an8%couponrate,andan8%yield.Ifwechangetheyieldonebasispointsotheyieldis8.01%,thenwehavethefollowingvariablesandvalues:C=$40,y=0.0801/2=0.04005andn=2(2)=4.57Insertingthesevaluesintothepresentvalueofthecouponpaymentsformula,weget:111n+rP=Cr=4111.04005$400.04005=$145.179.Computingthepresentvalueoftheparormaturityvalueof$1,000gives:1nMr=4$1,000(1.04005)=$854.640.Ifweaddabasispointtotheyield,wegetthevalueofbondAas:P=$145.179+$854.640=$999.819withabondquoteof$99.9819.ForbondAthepricevalueofabasispointisabout$100–$99.9819=$0.0181per$100.Usingthebondvaluationformulasasjustcompletedabove,thevalueofbondBwithayieldof8%,acouponrateof9%,andamaturityof5yearsis:P=$364.990+$675.564=$1,040.554withabondquoteof$104.0554.Ifweaddabasispointtotheyield,wegetthevalueofbondBas:P=$364.899+$675.239=$1,040.139withabondquoteof$104.0139.ForbondB,thepricevalueofabasispointis$104.0554–$104.0139=$0.0416per$100.(b)ComputetheMacaulaydurationsforthetwobonds.ForbondAwithC=$40,n=4,y=0.04,P=$1,000andM=$1,000,wehave:Macaulayduration(halfyears)=12121111nnCCnCnM++...++yyyyP=12441($40)2($40)4($40)4($1,000)1.041.041.041.04$1,000++...++=000,1$09.775,3$=3.77509.Macaulayduration(years)=Macaulayduration(halfyears)/2=3.77509/2=1.8875.ForbondBwithC=$45,n=10,y=0.04,P=$1,040.55andM=$1,000,wehave:Macaulayduration(halfyears)=12121111nnCCnCnM++...++yyyyP=5812441($45)2($45)10($45)10($1,000)1.041.041.041.04$1,040.55++...++=55.040,1$2929.645,8$=8.3084.Macaulayduration(years)=Macaulayduration(halfyears)/2=8.3084/2=4.1542.(c)Computethemodifieddurationforthetwobonds.TakingouranswerfortheMacaulaydurationinyearsinpart(b),wecancomputethemodifieddurationforbondAbydividingby1.04.Wehave:modifiedduration=1.8875/(1.04)=1.814948.TakingouranswerfortheMacaulaydurationinyearsinpart(b),wecancomputethemodifieddurationforbondBbydividingby1.04.Wehave:modifiedduration=4.1542/(1.04)=3.994417.[NOTE.WecouldgetthesameanswersforbothbondsAandBbycomputingthemodifieddurationusinganalternativeformulathatdoesnotrequiretheextensivecalculationsrequiredbytheprocedureinparts(a)and(b).Thisshortcutformulais:modifiedduration=12100111nnnC/yC1yyyP.whereCisthesemiannualcouponpayment,yisthesemiannualyield,nisthenumberofsemiannualperiods,andPisthebondquotein100’s.ForbondA(expressingnumbersintermsofa$100bondquote),wehave:C=$4,y=0.04,n=4,andP=$100.Insertingthesevaluesinourmodifieddurationformula,wecansolveasfollows:12100111nnnC/yC1yyyP=4524$100$4/0.04$411(1.04(1.04))0.04$100=($362.98952+$0)/$100=3.6299.ConvertingtoannualnumberbydividingbytwogivesamodifieddurationforbondAof1.8149whichisthesameanswershownabove.ForbondB,wehaveC=$4.5,n=20,y=0.04,andP=$104.055.Insertingthesevaluesinourmodifieddurationformula,wecansolveasfollows:5912100111nnnC/yC1yyyP=101124$100$4.5/0.04$4.511(1.04(1.04))0.04$104.055=($912.47578–$81.19762)/$104.055=7.988834orabout7.99.ConvertingtoannualnumberbydividingbytwogivesamodifieddurationforbondBof3.994417whichisthesameanswershownabove.](d)ComputetheapproximatedurationforbondsAandBusingthesh