完整word版投资学题库Chap011.docx
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完整word版投资学题库Chap011.docx
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完整word版投资学题库Chap011
Chapter11
ManagingBondPortfolios
1.Durationcanbethoughtofasaweightedaverageofthe‘maturities’ofthecashflowspaidtoholdersoftheperpetuity,wheretheweightforeachcashflowisequaltothepresentvalueofthatcashflowdividedbythetotalpresentvalueofallcashflows.Forcashflowsinthedistantfuture,presentvalueapproacheszero(i.e.,theweightbecomesverysmall)sothatthesedistantcashflowshavelittleimpact,andeventually,virtuallynoimpactontheweightedaverage.
2.Alowcoupon,longmaturitybondwillhavethehighestdurationandwill,therefore,producethelargestpricechangewheninterestrateschange.
3.Arateanticipationswapshouldwork.Thetradewouldbetolongthecorporatebondsandshortthetreasuries.Arelativegainwillberealizedwhenratespreadsreturntonormal.
4.-25=-(D/1.06)x.0025x1050…solvingforD=10.09
5.d.
6.Theincreasewillbelargerthanthedecreaseinprice.
7.Whileitistruethatshort-termratesaremorevolatilethanlong-termrates,thelongerdurationofthelonger-termbondsmakestheirratesofreturnmorevolatile.Thehigherdurationmagnifiesthesensitivitytointerest-ratesavings.Thus,itcanbetruethatratesofshort-termbondsaremorevolatile,butthepricesoflong-termbondsaremorevolatile.
8.Computationofduration:
a.YTM=6%
(1)
(2)
(3)
(4)
(5)
TimeuntilPayment(Years)
Payment
PaymentDiscountedat6%
Weight
Column
(1)
×
Column(4)
1
60
56.60
0.0566
0.0566
2
60
53.40
0.0534
0.1068
3
1060
890.00
0.8900
2.6700
ColumnSum:
1000.00
1.0000
2.8334
Duration=2.833years
b.YTM=10%
(1)
(2)
(3)
(4)
(5)
TimeuntilPayment(Years)
Payment
PaymentDiscountedat10%
Weight
Column
(1)
×
Column(4)
1
60
54.55
0.0606
0.0606
2
60
49.59
0.0551
0.1101
3
1060
796.39
0.8844
2.6531
ColumnSum:
900.53
1.0000
2.8238
Duration=2.824years,whichislessthanthedurationattheYTMof6%
9.Thepercentagebondpricechangeis:
–Duration´
ora3.27%decline
10.Computationofduration,interestrate=10%:
(1)
(2)
(3)
(4)
(5)
TimeuntilPayment(Years)
Payment
(inmillionsofdollars)
PaymentDiscounted
At10%
Weight
Column
(1)
×
Column(4)
1
1
0.9091
0.2744
0.2744
2
2
1.6529
0.4989
0.9977
3
1
0.7513
0.2267
0.6803
ColumnSum:
3.3133
1.0000
1.9524
Duration=1.9524years
11.Thedurationoftheperpetuityis:
(1+y)/y=1.10/0.10=11years
Letwbetheweightofthezero-couponbond.Thenwefindwbysolving:
(w´1)+[(1–w)´11]=1.9523Þw=9.048/10=0.9048
Therefore,yourportfolioshouldbe90.48%investedinthezeroand9.52%intheperpetuity.
12.Thepercentagebondpricechangewillbe:
–Duration⨯
ora0.463%increase
13.
a.BondBhasahigheryieldtomaturitythanbondAsinceitscouponpaymentsandmaturityareequaltothoseofA,whileitspriceislower.(Perhapstheyieldishigherbecauseofdifferencesincreditrisk.)Therefore,thedurationofBondBmustbeshorter.
b.BondAhasaloweryieldandalowercoupon,bothofwhichcauseittohavealongerdurationthanthatofBondB.Moreover,BondAcannotbecalled.Therefore,thematurityofBondAisatleastaslongasthatofBondB,whichimpliesthatthedurationofBondAisatleastaslongasthatofBondB.
14.Choosethelonger-durationbondtobenefitfromaratedecrease.
a.TheAaa-ratedbondhastheloweryieldtomaturityandthereforethelongerduration.
b.Thelower-couponbondhasthelongerdurationandmoredefactocallprotection.
c.Thelowercouponbondhasthelongerduration.
15.
a.Thepresentvalueoftheobligationis$17,832.65andthedurationis1.4808years,asshowninthefollowingtable:
Computationofduration,interestrate=8%
(1)
(2)
(3)
(4)
(5)
TimeuntilPayment(Years)
Payment
PaymentDiscounted
at8%
Weight
Column
(1)
×
Column(4)
1
10,000
9,259.26
0.5192
0.51923
2
10,000
8,573.39
0.4808
0.96154
ColumnSum:
17,832.65
1.0000
1.48077
b.Toimmunizetheobligation,investinazero-couponbondmaturingin1.4808years.Sincethepresentvalueofthezero-couponbondmustbe$17,832.65,thefacevalue(i.e.,thefutureredemptionvalue)mustbe:
$17,832.65´(1.08)1.4808=$19,985.26
c.Iftheinterestrateincreasesto9%,thezero-couponbondwouldfallinvalueto:
Thepresentvalueofthetuitionobligationwouldfallto$17,591.11,sothatthenetpositionchangesby$0.19.
Iftheinterestratefallsto7%,thezero-couponbondwouldriseinvalueto:
Thepresentvalueofthetuitionobligationwouldincreaseto$18,080.18,sothatthenetpositionchangesby$0.19.
Thereasonthenetpositionchangesatallisthat,astheinterestratechanges,sodoesthedurationofthestreamoftuitionpayments.
16.
a.PVofobligation=$2million/0.16=$12.5million
Durationofobligation=1.16/0.16=7.25years
Callwtheweightonthefive-yearmaturitybond(withdurationof4years).Then:
(w´4)+[(1–w)´11]=7.25Þw=0.5357
Therefore:
0.5357´$12.5=$6.7millioninthe5-yearbond,and
0.4643´$12.5=$5.8millioninthe20-yearbond.
b.Thepriceofthe20-yearbondis:
[60´Annuityfactor(16%,20)]+[1000´PVfactor(16%,20)]=$407.12
Therefore,thebondsellsfor0.4071timesitsparvalue,sothat:
Marketvalue=Parvalue´0.4071
$5.8million=Parvalue´0.4071ÞParvalue=$14.25million
Anotherwaytoseethisistonotethateachbondwithparvalue$1000sellsfor$407.11.Iftotalmarketvalueis$5.8million,thenyouneedtobuy:
$5,800,000/407.11=14,250bonds
Therefore,totalparvalueis$14,250,000.
17.
a.Thedurationoftheperpetuityis:
1.05/0.05=21years
Letwbetheweightofthezero-couponbond,sothatwefindwbysolving:
(w´5)+[(1–w)´21]=10Þw=11/16=0.6875
Therefore,theportfoliowillbe11/16investedinthezeroand5/16intheperpetuity.
b.Thezero-couponbondwillthenhaveadurationof4yearswhiletheperpetuitywillstillhavea21-yearduration.Tohaveaportfoliowithdurationequaltonineyears,whichisnowthedurationoftheobligation,weagainsolveforw:
(w´4)+[(1–w)´21]=9Þw=12/17=0.7059
Sotheproportioninvestedinthezeroincreasesto12/17andtheproportionintheperpetuityfallsto5/17.
18.MacaulayDurationandModifiedDurationarecalculatedusingExcelasfollows:
Inputs
FormulaincolumnB
Settlementdate
5/27/2010
=DATE(2010,5,27)
Maturitydate
11/15/2019
=DATE(2019,11,15)
Couponrate
0.07
0.07
Yieldtomaturity
0.08
0.08
Couponsperyear
2
2
Outputs
MacaulayDuration
6.9659
=DURATION(B2,B3,B4,B5,B6)
ModifiedDuration
6.6980
=MDURATION(B2,B3,B4,B5,B6)
19.MacaulayDurationandModifiedDurationarecalculatedusingExcelasfollows:
Inputs
FormulaincolumnB
Settlementdate
5/27/2010
=DATE(2010,5,27)
Maturitydate
11/15/2019
=DATE(2019,11,15)
Couponrate
0.07
0.07
Yieldtomaturity
0.08
0.08
Couponsperyear
1
1
Outputs
MacaulayDuration
6.8844
=DURATION(B2,B3,B4,B5,B6)
ModifiedDuration
6.3745
=MDURATION(B2,B3,B4,B5,B6)
Generally,wewouldexpectdurationtoincreasewhenthefrequencyofpaymentdecreasesfromonepaymentperyeartotwopaymentsperyear,becausemoreofthebond’spaymentsaremadefurtherintothefuturewhenpaymentsaremadeannually.However,inthisexample,durationdecreasesasaresultofthetimingofthesettlementdaterelativetothematuritydateandtheinterestpaymentdates.Forannualpayments,thefirstpaymentis$70paidonNovember15,2010.Forsemi-annualpayments,thefirst$70ispaidasfollows:
$35onNovember15,2010and$35onMay15,2010,sotheweightedaverage“maturity”thesepaymentsisshorterthanthe“maturity”ofthe$70paymentonNovember15,2010fortheannualpaymentbond.
20.
a.Thedurationoftheperpetuityis:
1.10/0.10=11years
Thepresentvalueofthepaymentsis:
$1million/0.10=$10million
Letwbetheweightofthefive-yearzero-couponbondandtherefore(1–w)istheweightofthetwenty-yearzero-couponbond.Thenwefindwbysolving:
(w´5)+[(1–w)´20]=11Þw=9/15=0.60
So,60%oftheportfoliowillbeinvestedinthefive-yearzero-couponbondand40%inthetwenty-yearzero-couponbond.
Therefore,themarketvalueofthefive-yearzerois:
$10million´0.60=$6million
Similarly,themarketvalueofthetwenty-yearzerois:
$10million´0.40=$4million
b.Facevalueofthefive-yearzero-couponbondis:
$6million´(1.10)5=$9.66million
Facevalueofthetwenty-yearzero-couponbondis:
$4million´(1.10)20=$26.91million
21.ConvexityiscalculatedusingtheExcelspreadsheetbelow:
22.
a.Interestrate=12%
TimeuntilPayment(Years)
Payment
PaymentDiscountedat12%
Weight
Time
×
Weight
8%coupon
1
80
71.429
0.0790
0.0790
2
80
63.776
0.0706
0.1411
3
1080
768.723
0.8504
2.5513
Sum:
903.927
1.0000
2.7714
Zero-coupon
1
0
0.000
0.0000
0.0000
2
0
0.000
0.0000
0.0000
3
1000
711.780
1.0000
3.0000
Sum:
711.780
1.0000
3.0000
Atahigherdiscountrate,theweightsofthelaterpaymentsofthecouponbondfallandthoseoftheearlierpaymentsrise.Sodurationfalls.Forthezero,theweightofthepaymentinthreeyearsremainsat1.0,anddurationthereforeremainsat3years.
b.Continuetouseayieldtomaturityof12%:
TimeuntilPayment(Years)
Payment
PaymentDiscountedat12%
Weight
Time
×
Weight
8%coupon
1
120
107.143
0.1071
0.1071
2
120
95.663
0.0957
0.1913
3
1120
797.194
0.7972
2.3916
Sum:
10
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