Chapter2answersWord文件下载.docx
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Chapter2answersWord文件下载.docx
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Pn=P0(1+r)nwheren=numberofperiods,Pn=futurevaluenperiodsfromnow(indollars),P0=originalprincipal(indollars)andr=interestrateperperiod(indecimalform).Insertinginourvalues,wehave:
P4.5=$10,400,000(1.0625)4.5=$10,400,000(1.313651676)=$13,661,977.43.Thus,itwillbeshort$13,661,977.43–$14,000,000=–$338,022.57.
3.Answerthefollowingquestions.
(a)Theportfoliomanagerofatax-exemptfundisconsideringinvesting$500,000inadebtinstrumentthatpaysanannualinterestrateof5.7%forfouryears.Attheendoffouryears,theportfoliomanagerplanstoreinvesttheproceedsforthreemoreyearsandexpectsthatforthethree-yearperiod,anannualinterestrateof7.2%canbeearned.Whatisthefuturevalueofthisinvestment?
Attheendofyearfour,theportfoliomanager’samountisgivenby:
Pn=P0(1+r)n.Insertinginourvalues,wehaveP4=$500,000(1.057)4=$500,000(1.248245382)=$624,122.66.Inthreemoreyearsattheendofyearseven,themanageramountisgivenby:
P7=P4(1+r)3.Insertinginourvalues,wehave:
P7=$624,122.66(1.072)3=$624,122.66(1.231925248)=$768,872.47.
(b)SupposethattheportfoliomanagerinQuestion3,parta,hastheopportunitytoinvestthe$500,000forsevenyearsinadebtobligationthatpromisestopayanannualinterestrateof6.1%compoundedsemiannually.IsthisinvestmentalternativemoreattractivethantheoneinQuestion3,parta?
Attheendofyearseven,theportfoliomanager’samountisgivenbythefollowingequation,whichadjustsforsemiannualcompounding.Wehave:
Pn=P0(1+r/2)2(n).Insertinginourvalues,wehaveP7=$500,000(1+0.061/2)2(7)=$500,000(1.0305)14=$500,000(1.522901960)=$761,450.98.Thus,thisinvestmentalternativeisnotmoreattractive.Itislessbytheamountof$761,450.98–$768,872.47=–$7,421.49.
4.Supposethataportfoliomanagerpurchases$10millionofparvalueofaneight-yearbondthathasacouponrateof7%andpaysinterestonceperyear.Thefirstannualcouponpaymentwillbemadeoneyearfromnow.Howmuchwilltheportfoliomanagerhaveifshe
(1)holdsthebonduntilitmatureseightyearsfromnow,and
(2)canreinvestalltheannualinterestpaymentsatanannualinterestrateof6.2%?
Attheendofyeareight,theportfoliomanager’samountisgivenbythefollowingequation,whichadjustsforannualcompounding.
Wehave:
whereA=couponratetimesparvalue.Insertinginourvalues,wehave:
+$10,000,000=$700,000[9.9688005]+$10,000,000=$6,978,160.38+$10,000,000=$16,978,160.38.
5.Answerthefollowingquestions.
(a)Ifthediscountratethatisusedtocalculatethepresentvalueofadebtobligation’scashflowisincreased,whathappenstothepriceofthatdebtobligation?
Afundamentalpropertyofabondisthatitspricechangesintheoppositedirectionfromthechangeintherequiredyield.Thereasonisthatthepriceofthebondisthepresentvalueofthecashflows.Astherequiredyieldincreases,thepresentvalueofthecashflowdecreases;
hencethepricedecreases.Theoppositeistruewhentherequiredyielddecreases:
Thepresentvalueofthecashflowsincreases,andthereforethepriceofthebondincreases.
(b)Supposethatthediscountrateusedtocalculatethepresentvalueofadebtobligation’scashflowisx%.Supposealsothattheonlycashflowsforthisdebtobligationare$200,000fouryearsfromnowand$200,000fiveyearsfromnow.Forwhichofthesecashflowswillthepresentvaluebegreater?
Cashflowsthatcomeearlierwillhaveagreatervalue.Aslongasx%ispositiveandtheamountisthesame,thepresentvaluewillbegreaterforthe$200,000fouryearsfromnowcomparedtofiveyearsfromnow.Thiscanalsobeseenbynotingthatifx>
0then
.Thelatterinequalityimplies
willhold.
6.Thepensionfundobligationofacorporationiscalculatedasthepresentvalueoftheactuariallyprojectedbenefitsthatwillhavetobepaidtobeneficiaries.Whyistheinterestrateusedtodiscounttheprojectedbenefitsimportant?
Thepresentvalueincreasesasthediscountratedecreasesanddecreasesasthediscountrateincreases.Thus,inordertoprojectthebenefitsaccurately,weneedanaccurateestimateofthediscountrate.Ifweunderestimatethediscountratethenwewillbeprojectingmoreavailablepensionfundsthanwewillactuallyhave.
7.Apensionfundmanagerknowsthatthefollowingliabilitiesmustbesatisfied:
YearsfromNow
Liability(inmillions)
1
2.0
2
3.0
3
5.4
4
5.8
Supposethatthepensionfundmanagerwantstoinvestasumofmoneythatwillsatisfythisliabilitystream.Assumingthatanyamountthatcanbeinvestedtodaycanearnanannualinterestrateof7.6%,howmuchmustbeinvestedtodaytosatisfythisliabilitystream?
Tosatisfyyearone’sliability(n=1),thepensionfundmanagermustinvestanamounttodaythatisequaltothefuturevalueof$2.0millionat7.6%.Wehave:
=
=$1,858,736.06.
Tosatisfyyeartwo’sliability(n=2),thepensionfundmanagermustinvestanamounttodaythatisequaltothefuturevalueof$3.0millionat7.6%.Wehave:
=$2,591,174.80.
Tosatisfyyearthree’sliability(n=3),thepensionfundmanagermustinvestanamounttodaythatisequaltothefuturevalueof$5.4millionat7.6%.Wehave:
=$4,334,679.04.
Tosatisfyyearfour’sliability(n=4),thepensionfundmanagermustinvestanamounttodaythatisequaltothefuturevalueof$5.8millionat7.6%.Wehave:
=$4,326,920.42.
Ifweaddthefourpresentvalues,weget$1,858,736.06+$2,591,174.80+$4,334,679.04+$4,326,920.42=$13,111,510.32,whichistheamountthepensionfundmanagerneedstoinvesttodaytocovertheliabilitystreamforthenextfouryears.
8.Calculateforeachofthefollowingbondsthepriceper$1,000ofparvalueassumingsemiannualcouponpayments.
Bond
CouponRate(%)
YearstoMaturity
RequiredYield(%)
A
8
9
7
B
20
C
6
15
10
D
14
A.Considera9-year8%couponbondwithaparvalueof$1,000andarequiredyieldof7%.GivenC=0.08($1,000)/2=$40,n=2(9)=18andr=0.07/2=0.035,thepresentvalueofthecouponpaymentsis:
=
=$$527,587.
Thepresentvalueoftheparormaturityvalueof$1,000is:
=$538,361.Thus,thepriceofthebond(P)=presentvalueofcouponpayments+presentvalueofparvalue=$527,587+$538,361=$1,065.95.
B.Considera20-year9%couponbondwithaparvalueof$1,000andarequiredyieldof9%.GivenC=0.09($1,000)/2=$45,n=2(20)=40andr=0.09/2=0.045,thepresentvalueofthecouponpaymentsis:
=
$45[18.401584]=$828.071.
=$171.929.Thus,thepriceofthebond(P)=$828.071+$171.929=$1,000.00.[NOTE.Wealreadyknewtheanswerwouldbe$1,000becausethecouponrateequalstheyieldtomaturity.]
C.Considera15-year6%couponbondwithaparvalueof$1,000andarequiredyieldof10%.GivenC=0.06($1,000)/2=$30,n=2(15)=30andr=0.10/2=0.05,thepresentvalueofthecouponpaymentsis:
=
$30[15.372451]=$461.174.
231.377.Thus,thepriceofthebond(P)=$461.174+$231.377=$692.55.
D.Considera14-year0%couponbondwithaparvalueof$1,000andarequiredyieldof8%.GivenC=0($1,000)/2=$0,n=2(14)=28andr=0.08/2=0.04,thepresentvalueofthecouponpaymentsis:
$0[16.66306322]=$0.[NOTE.Wealreadyknewtheanswerbecausethecouponrateiszero.]
$333.48.Thus,thepriceofthebond(P)=$0+$333.48=$333.48.
9.Considerabondsellingatpar($100)withacouponrateof6%and10yearstomaturity.
(a)Whatisthepriceofthisbondiftherequiredyieldis15%?
Wehavea10-year6%couponbondwithaparvalueof$1,000andarequiredyieldof15%.GivenC=0.06($1,000)/2=$30,n=2(10)=20andr=0.15/2=0.075,thepresentvalueofthecouponpaymentsis:
$30[10.1944913]=$305.835.
235.413.Thus,thepriceofthebond(P)=$305.835+$235.413=$541.25.
(b)Whatisthepriceofthisbondiftherequiredyieldincreasesfrom15%to16%,andbywhatpercentagedidthepriceofthisbondchange?
Iftherequiredyieldincreasesfrom15%to16%,thenweh
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