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1、CHAPTER4InterestRatesPracticeQuestionsProblem4.1.Abankquotesyouaninterestrateof7%perannumwithquarterlycompounding.Whatistheequivalentratewith(a)continuouscompoundingand(b)annualcompounding?(a) Theratewithcontinuouscompoundingis(0074Inl+二0.06944Jor6.94%perannum.(b) Theratewithannualcompoundingis(0.07
2、iec+亍j-1=0.0719or7.19%erannum.Problem4.2.ExpIainhowLIBORisdetermined1.lBoRistheLondonlnterBankofferedRatejtiscalculateddailyfromborrowingratesestimatedbyapanelofbanksProblem4.3.Thesix-monthandone-yearzeroratesareboth5%perannum.Forabondthathasalifeof18monthsandpaysacouponof4%perannum(withsemiannualpa
3、ymentsandonehavingjustbeenmade),theyieldis5.2%pra7HMw.H7zrz6/wd7ice?Wza”s泌e18-monthzerorate?AlIrateSareqUotedWithSemiannUalComPOUnding.SupposethebondhasafacevalueofS100.Itspriceisobtainedbydiscountingthecashflowsat5.2%,Thepriceis221021.0261.0262263=98.29Ifthe18-monthzerorateisR,wemusthave102(l+R2)3二
4、 98.29221.0251+0252whichgivesR=5.204%.Problen4.4.AninvestorreceivesSI,1OoinoneyeannreturnforaninvestmentofSI,OOOnow.Calculatethepercentagereturnperannumwitha)annualcompounding,b)semiannualcompounding,c)monthlycompoundingandd)continuouscompounding.(a) WithannualcompoundingthereturnisIl(X)-I二0.1100Oor
5、lO%perannum.(b) Withsemi-annualcompoundingthereturnisRwhere(R)210001+=HOO2)i.e.,R1+11二1.04882sothat/?=0.0976.Thepercentagereturnistherefore9.76%perannum.(c) WithmonthlycompoundingthereturnisRwhere(R)1210001+-=1100112J1 .e.二A=血皿沏sothatR=0.0957.Thepercentagereturnistherefore9.57%perannum.(d) Withconti
6、nuouscompoundingtheretumisRwhere:100OeR=IlOO1 .e.,eR=l.1SothatR=In1.1=0.0953.Thepercentagereturnistherefore9.53%perannum.Problem4.5.Supposethatzeroimerestrateswhhcontinuouscompoiindingareasfollows:Maturity(months)Raie(%perannum)33.063.293.4123.5153.6183.7Calculateforwardinterestratesforthesecond,thi
7、rd,fourth,fifth,andsixthquarters.TheforwardrateswithcontinuouscompoundingareasfbllowstoQtr23.4%Qtr33.8%Qtr43.8%Qtr54.0%Qtr64.2%Problen4.6.Assumingthatrisk-freeratesareasinProblem4.5,whatisthevalueofanFRAwheretheholderwillpayLIBORandreceive4.5%(quarterlycompounded)forathree-monthperiodstartinginoneye
8、aronaprincipalof$1,000,Ooo-TheforwardLIBORrateforthethreemonthperiodis5%quarterlycompounded.Fromcquation(4.9),thevalueoftheF7RAistherefore1,000,000X0.25X(0.045-0.050)eo.o36xi,25Zl-1J95or$l,195Problem4.7.Thetermstructureofinterestratesisupwardsloping-Putthefollowinginorderofmagnitude:(a) Thefive-year
9、zerorate(b) Theyieldonafive-yearcoupo-bearingbond(c) TheforwardratecorrespondingtotheperiodbetweenAJSandSyearsinthefutureWhatistheanswertothisquestionwhenthetermstructureofinterestratesisdownwardsloping?Whenthetermstructureisupwardsloping,cab.Whenitisdownwardsloping,Z?6rc.Problem4.8.Whatdoesduration
10、tellyouaboutthesensitivityofabondportfoliotointerestrates?WhatarethelimitationsOfthedurationmeasure?Durationprovidesinformationabouttheeffectofasmallparallelshiftintheyieldcurveonthevalueofabondportfblio.ThepercentagedecreaseinthevalueoftheportfolioequalsthedurationoftheportfoliomuItipliedbytheamoun
11、tbyWhichinterestratesareincreasedinthesmallparallelshift-Thedurationmeasurehasthefollowinglimitation.Itappliesonlytoparallelshiftsintheyieldcurvethataresmall.Problem4.9.Whatrateofinterestwithcontinuouscompoundingisequivalentto8%perannumwithmonthlycompounding?TherateofinterestisRwhere:(0.08)12r=1+Ii.
12、e.,1.mnfi+008A-nI%j=0.0797Therateofinterestistherefre7.97%perannum.Problem4.10.Adepositaccountpays4%perannumwithcontinuouscompounding,butinterestisactuallypaidquarterly.HoWmUChimereStW川bepaideachquarterona$10,00OdePOSit?TheequivalentrateofinterestwithquarterlycompoundingisRwhereo.04=orR二4(eoe-l)二0.0
13、402Theamountofinterestpaideachquarteristherefnre:0.040210,ooox-:二100.504or$100.50.Problem4.il.Supposethat6-month,12-month,18-month,24-month,and30-monthzeroratesare,respectively,4%2%,44%,4.6%,and4.8%perannumwithcontinuouscompounding.Estimatethecashpriceofabondwithafacevalueof100thatw川maturein30months
14、andpayacojponof4%perannumsemiannually.Thebondpays$2in6,12,18,and24months,and$102in30months.Thecashpriceis2e-0.04x0.5+2e-0.042xl.(H-2e-0.044xl.5+2e.(M6x2102-0.(M8x2.5=98.04Problem4.12.Athree-yearbondprovidesacouponof8%semiannuallyandhasacashpriceof104.Whatisthebondsyield?Thebondpays$4in6,12,18,24,and
15、30months,and$104in36months.Thebondyieldisthevalueofythatsolves4e-o.5y+4e-1.oy+4e-1.5y+4e-2.oy+4e-25y+104e-3.oy=104UsingtheSo/izen)rGoa/SeektoolinExcel,y=0.06407or6.407%.Problem4.13.Supposethatthe6-month,12-month,18-month,and24-monthzeroratesare5%,6%,6.5%,and7%respectively.Whatisthetwo-yearparyield?Usingthenotationinthetext,m=2,d=e-oo7x2=O.8694.AlsoA=-0.05x0.5+-0.06x1.0+-0.065x1.5+0.07x2.0=3.6935Thefbrmulainthetextgivestheparyieldas(Ioo-100XO.8694)-2-7创3.6935_*ToVer
