《金融学（第二版）》讲义大纲及课后习题答案详解第八章

《⾦融学（第⼆版）》讲义⼤纲及课后习题答案详解第⼋章CHAPTER8VALUATIONOFKNOWNCASHFLOWS:BONDSObjectivesToshowhowtovaluecontractsandsecuritiesthatpromiseastreamofcashflowsthatareknownwithcertainty.Tounderstandtheshapeoftheyieldcurve.Tounderstandhowbondpricesandyieldschangeovertime.Outline8.1UsingPresentValueFormulastoValueKnownCashFlows8.2TheBasicBuildingBlocks:PureDiscountBonds8.3CouponBonds,CurrentYield,andYieldtoMaturity8.4ReadingBondListings8.5WhyYieldsfortheSameMaturityDiffer8.6TheBehaviorofBondPricesoverTimeSummaryAchangeinmarketinterestratescausesachangeintheoppositedirectioninthemarketvaluesofallexistingcontractspromisingfixedpaymentsinthefuture.Themarketpricesof$1tobereceivedateverypossibledateinthefuturearethebasicbuildingblocksforvaluingallotherstreamsofknowncashflows.Thesepricesareinferredfromtheobservedmarketpricesoftradedbondsandthenappliedtootherstreamsofknowncashflowstovaluethem.Anequivalentvaluationcanbecarriedoutbyapplyingadiscountedcashflowformulawithadifferentdiscountrateforeachfuturetimeperiod.Differencesinthepricesoffixed-incomesecuritiesofagivenmaturityarisefromdifferencesincouponrates,defaultrisk,taxtreatment,callability,convertibility,andotherfeatures.Overtimethepricesofbondsconvergetowardstheirfacevalue.Beforematurity,however,bondpricescanfluctuateagreatdealasaresultofchangesinmarketinterestrates.SolutionstoProblemsatEndofChapterBondValuationwithaFlatTermStructure1.Supposeyouwanttoknowthepriceofa10-year7%couponTreasurybondthatpaysinterestannually.a.Youhavebeentoldthattheyieldtomaturityis8%.Whatistheprice?b.Whatisthepriceifcouponsarepaidsemiannually,andtheyieldtomaturityis8%peryear?c.Nowyouhavebeentoldthattheyieldtomaturityis7%peryear.Whatistheprice?Couldyouhaveguessedtheanswerwithoutcalculatingit?Whatifcouponsarepaidsemiannually?SOLUTION:c.Price=100.Whenthecouponrateandyieldtomaturityarethesame,thebondsellsatparvalue(i.e.thepriceequalsthefacevalueofthebond).2.AssumesixmonthsagotheUSTreasuryyieldcurvewasflatatarateof4%peryear(withannualcompounding)andyouboughta30-yearUSTreasurybond.Todayitisflatatarateof5%peryear.Whatrateofreturndidyouearnonyourinitialinvestment:a.Ifthebondwasa4%couponbond?b.Ifthebondwasazerocouponbond?c.Howdoyouranswerchangeifcompoundingissemiannual?SOLUTION:aandb.Step3:Findratesofreturn:Rateofreturn=(coupon+changeinprice)/initialprice4%couponbond:r=(4+84.74-100)/100=-0.1126or-11.26%Zero-couponbond:r=(0+23.71-30.83)/30.83=-0.2309or-23.09%.Notethatthezero-couponbondismoresensitivetoyieldchangesthanthe4%couponbond.c.Step2:Findpricesofthebondstoday:Step3:Findratesofreturn:Rateofreturn=(coupon+changeinprice)/initialprice4%couponbond:r=(2+84.66-100)/100=-0.1334or-13.34%Zerocouponbond:r=(0+23.30-30.48)/30.48=-0.2356or-23.56%.Notethatthezero-couponbondismoresensitivetoyieldchangesthanthe4%couponbond.BondValuationWithaNon-FlatTermStructure3.Supposeyouobservethefollowingpricesforzero-couponbonds(purediscountbonds)thathavenoriskofdefault:a.Whatshouldbethepriceofa2-yearcouponbondthatpaysa6%couponrate,assumingcouponpaymentsaremadeonceayearstartingoneyearfromnow?b.Findthemissingentryinthetable.c.Whatshouldbetheyieldtomaturityofthe2-yearcouponbondinParta?d.Whyareyouranswerstopartsbandcofthisquestiondifferent?SOLUTION:a.Presentvalueoffirstyear'scashflow=6x.97=5.82Presentvalueofsecondyear'scashflow=106x.90=95.4Totalpresentvalue=101.22maturity.CouponStripping4.Youwouldliketocreatea2-yearsyntheticzero-couponbond.Assumeyouareawareofthefollowinginformation:1-yearzero-couponbondsaretradingfor$0.93perdollaroffacevalueand2-year7%couponbonds(annualpayments)aresellingat$985.30(Facevalue=$1,000).a.Whatarethetwocashflowsfromthe2-yearcouponbond?b.Assumeyoucanpurchasethe2-yearcouponbondandunbundlethetwocashflowsandsellthem.i.Howmuchwillyoureceivefromthesaleofthefirstpayment?ii.Howmuchdoyouneedtoreceivefromthesaleofthe2-yearTreasurystriptobreakeven?SOLUTION:a.$70attheendofthefirstyearand$1070attheendofyear2.b.i.Iwouldreceive.93x$70=$65.10fromthesaleofthefirstpayment.ii.Tobreakeven,Iwouldneedtoreceive$985.30-$65.10=$920.20fromthesaleofthe2-yearstrip.TheLawofOnepriceandBondPricing5.Assumethatallofthebondslistedinthefollowingtablearethesameexceptfortheirpatternofpromisedcashflowsovertime.Pricesarequotedper$1offacevalue.UsetheinformationinthetableandtheLawofOnePricetoinferthevaluesofthemissingentries.Assumethatcouponpaymentsareannual.SOLUTION:FromBond1andBond4,wecangetthemissingentriesforthe2-yearzero-couponbond.Weknowfrombond1that:1.0092=0.06/1.055+1.06/(1.055)2.Thisisalsoequalto0.06/(1+z1)+1.06/(1+z2)2wherez1andz2aretheyieldstomaturityonone-yearzero-couponandtwo-yearzero-couponbondsrespectively.Frombond4,wehavez1,wecanfindz2.1.0092–0.06/1.0526=1.06/(1+z2)2,hencez2=5.51%.TogetthepricePper$1facevalueofthe2-yearzero-couponbond,usingthesamereasoning:1.0092–0.06x0.95=1.06xP,henceP=0.8983Tofindtheentriesforbond3:firstfindtheprice,thentheyieldtomaturity.Tofindtheprice,wecanusez1andz2foundearlier:PVofcouponpaymentinyear1:0.07x0.95=0.0665PVofcoupon+principalpaymentsinyear2:1.07x0.8983=0.9612BondFeaturesandBondValuation6.Whateffectwouldaddingthefollowingfeatureshaveonthemarketpriceofasimilarbondwhichdoesnothavethisfeature?a.10-yearbondiscallablebythecompanyafter5years(comparetoa10-yearnon-callablebond);b.bondisconvertibleinto10sharesofcommonstockatanytime(comparetoanon-convertiblebond);c.10-yearbondcanbe“putback”tothecompanyafter3yearsatpar(puttablebond)(comparetoa10-yearnon-puttablebond)d.25-yearbondhastax-exemptcouponpaymentsSOLUTION:a.Thecallablebondwouldhavealowerpricethanthenon-callablebondtocompensatethebondholdersforgrantingtheissuertherighttocallthebonds.b.Theconvertiblebondwouldhaveahigherpricebecauseitgivesthebondholderstherighttoconverttheirbondsintosharesofstock.c.Theputtablebondwouldhaveahigherpricebecauseitgivesthebondholderstherighttoselltheirbondsbacktotheissueratpar.d.Thebondwiththetax-exemptcouponhasahigherpricebecausethebondholderisexemptedfrompayingtaxesonthecoupons.(Couponsareusuallyconsideredandtaxedaspersonalincome).InferringtheValueofaBondGuarantee7.Supposethattheyieldcurveondollarbondsthatarefreeoftheriskofdefaultisflatat6%peryear.A2-year10%couponbond(withannualcouponsand$1,000facevalue)issuedbyDafoltoCorporationisratesB,anditiscurrentlytradingatamarketpriceof$918.Asidefromitsriskofdefault,theDafoltobondhasnootherfinanciallysignificantfeatures.HowmuchshouldaninvestorbewillingtopayforaguaranteeagainstDafolto’sdefaultingonthisbond?SOLUTION:valueofaguaranteeagainstdefault:1073.3-918=$155.3TheimpliedValueofaCallProvisionandConvertibility8.Supposethattheyieldcurveonbondsthatarefreeoftheriskofdefaultisflatat5%peryear.A20-yeardefault-freecouponbond(withannualcouponsand$1,000facevalue)thatbecomescallableafter10yearsistradingatparandhasacouponrateof5.5%.a.Whatistheimpliedvalueofthecallprovision?b.ASafecoCorporationbondwhichisotherwiseidenticaltothecallable5.5%couponbonddescribedabove,isalsoconvertibleinto10sharesofSafecostockatanytimeuptothebond’smaturity.Ifitsyieldtomaturityiscurrently3.5%peryear,whatistheimpliedvalueoftheconversionfeature?SOLUTION:istheimpliedvalueofthecallprovision:1062.3–1000=$62.3Notethatthecallprovisiondecreasesthevalueofthebond.Thisbondhasthesamefeaturesasthe5.5%defaultfreecallabl