期权期货与其他衍生产品第九版课后习题与答案Chapter (4)

CHAPTER4InterestRatesPracticeQuestionsProblem4.1.Abankquotesyouaninterestrateof14%perannumwithquarterlycompounding.Whatistheequivalentratewith(a)continuouscompoundingand(b)annualcompounding?(a)Theratewithcontinuouscompoundingis0144ln1013764or13.76%perannum.(b)Theratewithannualcompoundingis401411014754or14.75%perannum.Problem4.2.WhatismeantbyLIBORandLIBID.Whichishigher?LIBORistheLondonInterBankOfferedRate.ItiscalculateddailybytheBritishBankersAssociationandistherateaAA-ratedbankrequiresondepositsitplaceswithotherbanks.LIBIDistheLondonInterBankBidrate.ItistherateabankispreparedtopayondepositsfromotherAA-ratedbanks.LIBORisgreaterthanLIBID.Problem4.3.Thesix-monthandone-yearzeroratesareboth10%perannum.Forabondthathasalifeof18monthsandpaysacouponof8%perannum(withsemiannualpaymentsandonehavingjustbeenmade),theyieldis10.4%perannum.Whatisthebond’sprice?Whatisthe18-monthzerorate?Allratesarequotedwithsemiannualcompounding.Supposethebondhasafacevalueof$100.Itspriceisobtainedbydiscountingthecashflowsat10.4%.Thepriceis23441049674105210521052Ifthe18-monthzerorateisR,wemusthave23441049674105105(12)Rwhichgives1042R%.Problem4.4.Aninvestorreceives$1,100inoneyearinreturnforaninvestmentof$1,000now.Calculatethepercentagereturnperannumwitha)annualcompounding,b)semiannualcompounding,c)monthlycompoundingandd)continuouscompounding.(a)Withannualcompoundingthereturnis11001011000or10%perannum.(b)Withsemi-annualcompoundingthereturnisRwhere21000111002Ri.e.,111104882Rsothat00976R.Thepercentagereturnistherefore9.76%perannum.(c)WithmonthlycompoundingthereturnisRwhere1210001110012Ri.e.1211110079712Rsothat00957R.Thepercentagereturnistherefore9.57%perannum.(d)WithcontinuouscompoundingthereturnisRwhere:10001100Rei.e.,11Resothatln1100953R.Thepercentagereturnistherefore9.53%perannum.Problem4.5.Supposethatzerointerestrateswithcontinuouscompoundingareasfollows:Maturity(months)Rate(%perannum)38.068.298.4128.5158.6188.7Calculateforwardinterestratesforthesecond,third,fourth,fifth,andsixthquarters.TheforwardrateswithcontinuouscompoundingareasfollowstoQtr28.4%Qtr38.8%Qtr48.8%Qtr59.0%Qtr69.2%Problem4.6.AssumethatabankcanborroworlendattheratesinProblem4.5.whatisthevalueofanFRAwhereitwillearn9.5%forathree-monthperiodstartinginoneyearonaprincipalof$1,000,000?Theinterestrateisexpressedwithquarterlycompounding.Theforwardrateis9.0%withcontinuouscompoundingor9.102%withquarterlycompounding.Fromequation(4.9),thevalueoftheFRAistherefore0086125[1000000025(0095009102)]89356eor$893.56.Problem4.7.Thetermstructureofinterestratesisupwardsloping.Putthefollowinginorderofmagnitude:(a)Thefive-yearzerorate(b)Theyieldonafive-yearcoupon-bearingbond(c)Theforwardratecorrespondingtotheperiodbetween4.75and5yearsinthefutureWhatistheanswertothisquestionwhenthetermstructureofinterestratesisdownwardsloping?Whenthetermstructureisupwardsloping,cab.Whenitisdownwardsloping,bac.Problem4.8.Whatdoesdurationtellyouaboutthesensitivityofabondportfoliotointerestrates?Whatarethelimitationsofthedurationmeasure?Durationprovidesinformationabouttheeffectofasmallparallelshiftintheyieldcurveonthevalueofabondportfolio.Thepercentagedecreaseinthevalueoftheportfolioequalsthedurationoftheportfoliomultipliedbytheamountbywhichinterestratesareincreasedinthesmallparallelshift.Thedurationmeasurehasthefollowinglimitation.Itappliesonlytoparallelshiftsintheyieldcurvethataresmall.Problem4.9.Whatrateofinterestwithcontinuouscompoundingisequivalentto15%perannumwithmonthlycompounding?TherateofinterestisRwhere:12015112Rei.e.,01512ln112R01491Therateofinterestistherefore14.91%perannum.Problem4.10.Adepositaccountpays12%perannumwithcontinuouscompounding,butinterestisactuallypaidquarterly.Howmuchinterestwillbepaideachquarterona$10,000deposit?TheequivalentrateofinterestwithquarterlycompoundingisRwhere401214Reor0034(1)01218ReTheamountofinterestpaideachquarteristherefore:0121810000304554or$304.55.Problem4.11.Supposethat6-month,12-month,18-month,24-month,and30-monthzeroratesare4%,4.2%,4.4%,4.6%,and4.8%perannumwithcontinuouscompoundingrespectively.Estimatethecashpriceofabondwithafacevalueof100thatwillmaturein30monthsandpaysacouponof4%perannumsemiannually.Thebondpays$2in6,12,18,and24months,and$102in30months.Thecashpriceis004050042100044150046200482522221029804eeeeeProblem4.12.Athree-yearbondprovidesacouponof8%semiannuallyandhasacashpriceof104.Whatisthebond’syield?Thebondpays$4in6,12,18,24,and30months,and$104in36months.Thebondyieldisthevalueofythatsolves05101520253044444104104yyyyyyeeeeeeUsingtheSolverorGoalSeektoolinExcel,006407yor6.407%.Problem4.13.Supposethatthe6-month,12-month,18-month,and24-monthzeroratesare5%,6%,6.5%,and7%respectively.Whatisthetwo-yearparyield?Usingthenotationinthetext,2m,007208694de.Also00505006100065150072036935AeeeeTheformulainthetextgivestheparyieldas(10010008694)2707236935Toverifythatthi