Ch04SolutionsManual.docx
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Ch04SolutionsManual
Chapter4
TimeValueofMoney
ANSWERSTOEND-OF-CHAPTERQUESTIONS
4-1a.PV（presentvalue）isthevaluetodayofafuturepayment,orstreamofpayments,discountedattheappropriaterateofinterest.PVisalsothebeginningamountthatwillgrowtosomefuturevalue.Theparameteriistheperiodicinterestratethatanaccountpays.TheparameterINTisthedollarsofinterestearnedeachperiod.FVn（futurevalue）istheendingamountinanaccount,wherenisthenumberofperiodsthemoneyisleftintheaccount.PVAnisthevaluetodayofafuturestreamofequalpayments（anannuity）andFVAnistheendingvalueofastreamofequalpayments,wherenisthenumberofpaymentsoftheannuity.PMTisequaltothedollaramountofanequal,orconstantcashflow（anannuity）.IntheEARequation,misusedtodenotethenumberofcompoundingperiodsperyear,whileiNomisthenominal,orquoted,interestrate.
b.Theopportunitycostrate（i）ofaninvestmentistherateofreturnavailableonthebestalternativeinvestmentofsimilarrisk.
c.Anannuityisaseriesofpaymentsofafixedamountforaspecifiednumberofperiods.Asinglesum,orlumpsumpayment,asopposedtoanannuity,consistsofonepaymentoccurringnoworatsomefuturetime.Acashflowcanbeaninflow（areceipt）oranoutflow（adeposit,acost,oranamountpaid）.WedistinguishbetweenthetermscashflowandPMT.Weusethetermcashflowforunevenstreams,whileweusethetermPMTforannuities,orconstantpaymentamounts.Anunevencashflowstreamisaseriesofcashflowsinwhichtheamountvariesfromoneperiodtothenext.ThePV（orFVn）ofanunevenpaymentstreamismerelythesumofthepresentvalues（orfuturevalues）ofeachindividualpayment.
d.Anordinaryannuityhaspaymentsoccurringattheendofeachperiod.Adeferredannuityisjustanothernameforanordinaryannuity.Anannuityduehaspaymentsoccurringatthebeginningofeachperiod.Mostfinancialcalculatorswillaccommodateeithertypeofannuity.Thepaymentperiodmustbeequaltothecompoundingperiod.
e.Aperpetuityisaseriesofpaymentsofafixedamountthatlastindefinitely.Inotherwords,aperpetuityisanannuitywherenequalsinfinity.Consolisanothertermforperpetuity.ConsolswereoriginallybondsissuedbyEnglandin1815toconsolidatepastdebt.
f.Anoutflowisadeposit,acost,oranamountpaid,whileaninflowisareceipt.Atimelineisanimportanttoolusedintimevalueofmoneyanalysis;itisagraphicalrepresentationwhichisusedtoshowthetimingofcashflows.Theterminalvalueisthefuturevalueofanunevencashflowstream.
g.Compoundingistheprocessoffindingthefuturevalueofasinglepaymentorseriesofpayments.Discountingistheprocessoffindingthepresentvalueofasinglepaymentorseriesofpayments;itisthereverseofcompounding.
h.Annualcompoundingmeansthatinterestispaidonceayear.Insemiannual,quarterly,monthly,anddailycompounding,interestispaid2,4,12,and365timesperyearrespectively.Whencompoundingoccursmorefrequentlythanonceayear,youearninterestoninterestmoreoften,thusincreasingthefuturevalue.Themorefrequentthecompounding,thehigherthefuturevalue.
i.Theeffectiveannualrateistheratethat,underannualcompounding,wouldhaveproducedthesamefuturevalueattheendof1yearaswasproducedbymorefrequentcompounding,sayquarterly.Thenominal（quoted）interestrate,iNom,istherateofintereststatedinacontract.Ifthecompoundingoccursannually,theeffectiveannualrateandthenominalratearethesame.Ifcompoundingoccursmorefrequently,theeffectiveannualrateisgreaterthanthenominalrate.Thenominalannualinterestrateisalsocalledtheannualpercentagerate,orAPR.Theperiodicrate,iPER,istheratechargedbyalenderorpaidbyaborrowereachperiod.Itcanbearateperyear,per6-monthperiod,perquarter,permonth,perday,orperanyothertimeinterval（usuallyoneyearorless）.
j.Anamortizationscheduleisatablethatbreaksdowntheperiodicfixedpaymentofaninstallmentloanintoitsprincipalandinterestcomponents.Theprincipalcomponentofeachpaymentreducestheremainingprincipalbalance.Theinterestcomponentistheinterestpaymentonthebeginning-of-periodprincipalbalance.Anamortizedloanisonethatisrepaidinequalperiodicamounts（or"killedoff"overtime）.
4-2Theopportunitycostrateistherateofinterestonecouldearnonanalternativeinvestmentwithariskequaltotheriskoftheinvestmentinquestion.ThisisthevalueofiintheTVMequations,anditisshownonthetopofatimeline,betweenthefirstandsecondtickmarks.Itisnotasinglerate--theopportunitycostratevariesdependingontheriskinessandmaturityofaninvestment,anditalsovariesfromyeartoyeardependingoninflationaryexpectations.
4-3True.Thesecondseriesisanunevenpaymentstream,butitcontainsanannuityof$400for8years.Theseriescouldalsobethoughtofasa$100annuityfor10yearsplusanadditionalpaymentof$100inYear2,plusadditionalpaymentsof$300inYears3through10.
4-4True,becauseofcompoundingeffects--growthongrowth.Thefollowingexampledemonstratesthepoint.TheannualgrowthrateisIinthefollowingequation:
$1（1+I）10=$2.
Theterm（1+I）10istheFVIFforIpercent,10years.WecanfindIinoneoftwoways:
1.UsingafinancialcalculatorinputN=10,PV=-1,PMT=0,FV=2,andI/YR=?
.SolvingforI/YRyouobtain7.18%.
2.Usingafinancialcalculator,inputN=10,I/YR=10,PV=-1,PMT=0,andFV=?
.SolvingforFVyouobtain$2.59.Thisformulationrecognizesthe"interestoninterest"phenomenon.
4-5Forthesamestatedrate,dailycompoundingisbest.Youwouldearnmore"interestoninterest."
SOLUTIONSTOEND-OF-CHAPTERPROBLEMS
10%
4-1012345
||||||
PV=10,000FV5=?
FV5=$10,000（1.10）5
=$10,000（1.61051）=$16,105.10.
Alternatively,withafinancialcalculatorenterthefollowing:
N=5,I/YR=10,PV=-10000,andPMT=0.SolveforFV=$16,105.10.
7%
4-205101520
|||||
PV=?
FV20=5,000
Withafinancialcalculatorenterthefollowing:
N=20,I/YR=7,PMT=0,andFV=5000.SolveforPV=$1,292.10.
I/YR=?
4-3018
||
PV=250,000FV18=1,000,000
Withafinancialcalculatorenterthefollowing:
N=18,PV=-250000,PMT=0,andFV=1000000.SolveforI/YR=8.01%≈8%.
6.5%
4-40N=?
||
PV=1FVN=2
$2=$1（1.065）N.
Withafinancialcalculatorenterthefollowing:
I/YR=6.5,PV=-1,PMT=0,andFV=2.SolveforN=11.01≈11years.
12%
4-5012N–2N–1N
||||||
PV=42,180.535,0005,0005,0005,000FV=250,000
Usingyourfinancialcalculator,enterthefollowingdata:
I/YR=12;PV=42180.53;PMT=5000;FV=250000;N=?
SolveforN=11.Itwilltake11yearstoaccumulate$250,000.
4-6Ordinaryannuity:
7%
012345
||||||
300300300300300
FVA5=?
Withafinancialcalculatorenterthefollowing:
N=5,I/YR=7,PV=0,andPMT=300.SolveforFV=$1,725.22.
Annuitydue:
7%
012345
||||||
300300300300300
FVA5=?
Withafinancialcalculator,switchto“BEG”andenterthefollowing:
N=5,I/YR=7,PV=0,andPMT=300.SolveforFV=$1,845.99.Don’tforgettoswitchbackto“END”mode.
8%
4-70123456
|||||||
100100100200300500
PV=?
FV=?
Usingafinancialcalculator,enterthefollowing:
CF0=0;CF1=100;Nj=3;CF4=200（NotecalculatorwillshowCF2onscreen.）;CF5=300（NotecalculatorwillshowCF3onscreen.）;CF6=500（NotecalculatorwillshowCF4onscreen.）;andI/YR=8.SolveforNPV=$923.98.
TosolvefortheFVofthecashflowstreamwithacalculatorthatdoesn’thavetheNFVkey,dothefollowing:
EnterN=6,I/YR=8,PV=-923.98,andPMT=0.SolveforFV=$1,466.24.
4-8Usingafinancialcalculator,enterthefollowing:
N=60,I/YR=1,PV=20000,andFV=0.SolveforPMT=$444.89.
EAR=
–1.0
=（1.01）12–1.0
=12.68%.
Alternatively,usingafinancialcalculator,enterthefollowing:
NOM%=12andP/YR=12.SolveforEFF%=12.6825%.RemembertochangebacktoP/YR=1onyourcalculator.
6%
4-9a.01
||$500（1.06）=$530.00.
-500FV=?
6%
b.012
|||$500（1.06）2=$561.80.
-500FV=?
6%
c.01
||$500（1/1.06）=$471.70.
PV=?
500
6%
d.012
|||$500（1/1.06）2=$445.00.
PV=?
500
6%
4-10a.012345678910$500（1.06）10=$895.42.
|||||||||||
-500FV=?
12%
b.012345678910$500（1.12）10=$1,552.92.
|||||||||||
-500FV=?
6%
c.012345678910
|||||||||||$500（1/1.06）10=$279.20
PV=?
500
12%
d.012345678910
|||||||||||$500（1/1.12）10=$160.99
PV=?
500
7%
4-11a.?
||
-200400
Withafinancialcalculator,enterI/YR=7,PV=-200,PMT=0,andFV=400.ThenpresstheNkeytofindN=10.24≈10.
10%
b.?
||
-200400.
Withafinancialcalculator,enterI/YR=107,PV=-200,PMT=0,andFV=400.ThenpresstheNkeytofindN=7.27≈7.
18%
c.?
||
-200400.
Withafinancialcalculator,enterI/YR=18,PV=-200,PMT=0,andFV=400.ThenpresstheNkeytofindN=4.19≈4.
d.100%?
||
-200400.
Withafinancialcalculator,enterI/YR=100,PV=-200,PMT=0,andFV=400.ThenpresstheNkeytofindN=1.00≈1.
4-12
10%
a.012345678910
|||||||||||
400400400400400400400400400400
FVA10=?
Withafinancialcalculator,enterN=10,I/YR=10,PV=0,andPMT=-400
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