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Financial Functions I
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TABLE OF CONTENTS
INTRODUCTION 1
TIMEVALUEOFMONEY REVIEW 1
SimplePresent&FutureValue 1
Present&FutureValuewithNonAnnualCompoundingPeriods 2
EffectiveandNominalRates 2
Present&FutureValuewithContinuousCompounding 3
Present
&
Future
Value
of
an
Annuity
3
AnAnnuitywithInfinitePeriodsAPerpetuity 4
NetPresentValueandtheInternalRateofReturn 4
CompoundAnnualGrowthRate 5
EXCELFINANCIALFUNCTIONS 6
PV,FV,RATE,NPER,PMT 6
EFFECT&NOMINAL 9
EXP 11
NPV,IRR&FVSCHEDULE 11
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INTRODUCTION
Webeginwitha reviewofsimpleTimeValueofMoney formulas,and thendiscuss
howtheseformulasmaybeappliedinMicrosoftExcel.Pleasenotethatthisdocument
assumesbasic familiaritywithTimeValueofMoneyconcepts,andwith thebasicsof
Excelitself.
If
you
are
already
familiar
with
Time
Value
of
Money
concepts
and
simply
wish toapply them inExcel, clickhere. Ifyouneed a refresher inExcelbasics, click
here.
TheTimeValueofMoneyreviewbelowisbasedonmaterialfromQuantitativeMethodsforInvestmentAnalysis,SecondEditionbyR.A.DeFusco,D.W.McLeavey,J.E.Pinto,andD.E.Runkle,drawnfromVolumeIoftheCFALevelI2006ProgramCurriculum.
TIME VALUE OF MONEY REVIEW
Thefollowingtimevalueofmoneyconceptswillbereviewed:simplePresent&Future
Value,Present
&
Future
Value
with
Non
Annual
Compounding
Periods,
Effective/Nominalrates,ContinuousCompounding,AnnuityandPerpetuityformulas,
Net Present Value (NPV) & Internal Rate of Return calculations, and Compound
GrowthRates.
Simple Present & Future Value
Thefollowingformulasareused todetermine thesimplePresentandFuturevalueof
aninvestment:
( ) n
NPV FV r
= +1
( )n
NFV PV r= +1
Inbothoftheseformulas,rrepresentstheinterestrate(ordiscountrate)tobeusedand
nrepresentsthenumberofperiods.
Example 1: You will receive $15,000 in 5 years time. You are able to borrow
and lend at a rate of 4% per year. What is the Present Value of the
Investment?
Answer: PV = $15,000(1.04)-5= $12,328.91
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Present & Future Value with Non-Annual Compounding Periods
In many Time Value of Money applications, the act of compounding occurs more
frequentlythanonanannualbasis.ThefollowingformulasmodifythesimplePresent
&Futurevalueformulastocorrectformorefrequentcompoundingperiods:
( )smn
r
N mPV FV
= +1
( )smn
r
N mFV PV = +1
wherem is equal to the number of compounding periods per year, rs is the stated
annual rate, or nominal rate in Excel terminology (aswillbe discussed in the next
section),andnnowstands for thenumberofyears.Assuch,mn represents the total
numberofcompoundingperiods.
Example 2: You will receive $15,000 in 5 years time. You are able to borrow
and lend at an annual rate of 4%, compounded monthly. What is the Present
Value of the Investment?
Answer: PV = $15,000(1+(0.04/12))-60= $12,285.05
Note that the simplePresent& FutureValue formulas in theprevious section are a
specialcaseoftheseformulaswheremisequalto1.
Effective and Nominal Rates
Asanextensionofthepreviousformula,whenastatedornominalannualrateis
quotedwithnonannualcompoundingperiods, the following formulacanbeused to
determinetheeffectiveannualrate:
Effective Rate = ( )msr
m+ 1 1
where,again,misequaltothenumberofcompoundingperiodsperyear,andrsisthe
nominalrate.
Example 3: You are charged an annual interest rate of 8.75% on your
personal line of credit with your bank, compounded daily. What is the effective
annual interest rate?
Answer: (1+(0.0875/365))365-1 = 0.0914, or 9.14%.
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Present & Future Value with Continuous Compounding
While the above formulas deal with discrete compounding intervals, a number of
applications inFinancedealwithcontinuouscompounding,whenan infinitenumber
ofcompoundingperiodsarepresent.ThisconventionisespeciallyprominentinOption
pricingmodels.
( )sr nNPV FV e=
( )sr nNFV PV e=
where e is a mathematical constant equal to the base of the natural logarithm
(2.718281828).Theformulaiscalculatedonayearlybasis.
Example 4: You have $12,280.96 in your bank account. You are able toinvest the full amount for 5 years, continuously compounded at a rate of 4%.
What is the Future Value of the investment?
Answer: FV = $12,280.96(e0.2) = $15,000
Present & Future Value of an Annuity
Whiletheaboveformulasareusedtocalculatethefairvalueofalumpsuminvestment,
separateformulasexisttocalculatethePresent&FutureValueofanequalseriesofcash
flows,
referred
to
as
Annuities.
These
formulas
are
shown
below:
( )+
=
nr
PV Ar
1
11
( ) + =
n
N
rFV A
r
1 1
whereA is the cash flow per period, r is the interest rate per period, and n is the
numberof
periods.
Example 5: You are offered the option of choosing between an immediate,
one-time, lump sum payment of $12,000, and $1,500 per year for 10 years.
You are able to borrow and lend at an annual rate of 4%. Which option
should you choose?
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Answer: PV = $1,500[(0.3244)/0.04] = $1,500(8.1108) = $12,166.34.
Therefore, you should choose the annuity because it is worth about $166
more in todays dollars than the lump sum payment.
Keep
in
mind
that
these
formulas
assume
the
same
cash
flow
per
period.
If
the
amount
ofthecashflowchangesovertimebutisstablewithinintervals(i.e.,$1,500foryears1
5,$2,000foryears610,etc.),anannuitycalculationcanbeperformedforeachinterval
atthebeginningoftheinterval,andthendiscountedbacktopresentvalue.Otherwise,
ifthecashflowsvaryfromperiodtoperiod,usingtheNetPresentValue(NPV)method
willbenecessary,asdiscussedbelow.
An Annuity with Infinite Periods A Perpetuity
When the number of periods in the Present Value version of the annuity formula
approachesinfinity,theannuityformulareducestoamuchsimplerversion:
APV
r=
In effect, a perpetual stream of cash flows, a perpetuity, canbe valued simplyby
dividingtheperiodiccashflowbytheperiodicinterestrate.Avariantofthisformula,
thegrowingperpetuity(thedenominatorbecomesrg,whereg is thegrowthrate), is
often an essential element in estimating the terminal value of an investmentwhen
conductingDiscountedCashFlowAnalysis.
Net Present Value and the Internal Rate of Return
Asmentionedabove,theNetPresentValue(NPV)methodisusedtoproperlydiscount
aseriesofunequalcashflowsbacktopresentvalue.TheNPVformulaisasfollows:
( )
N
t
t
t
CFNPV
r==
+
0 1
whereCFtequalstherelevantcashflowattimet,andrisequaltothediscountrate.The
rateobtainedbysettingNPVto0andsolvingforrisreferredtoastheInternalRateof
Return.
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Compound Annual Growth Rate
Thelastformulatoberevieweddealswithcalculatingacompoundannualgrowthrate
foraninvestmentoraccountingvalue.
Rearranging
the
simple
Future
Value
formula
to
solve
for
r,
we
obtain
the
following:
N
NFV
rPV
=
1
1
Inmanycases,youmay see the interest rateabove labelledasg (forgrowth rate)or
CAGR (for Compound Annual Growth Rate). This isbecause the formula is often
appliedtosituationswherethevariablecannotbecorrectlyviewedasaninterestrate.
Forexample, change inaccountingvalues (suchasTotalAssets,Sales,etc.)areoften
measuredwiththisformula.
Example 6: Company ABC Inc. had $150 and $175 million in Total Assets as
at December 31st, 2001 and 2006 respectively. Calculate the Compound
Annual Growth Rate in total assets over the 5 year period.
Answer: CAGR = (175/150)0.2-1 = 0.0313 or 3.13%.
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EXCEL FINANCIAL FUNCTIONS
Pleasenote that thissectionwillmakereferences tothesolvedexamplesonpages15
above.
PV, FV, RATE, NPER, PMT
When calculating Present and Future value using a computing aide (spreadsheet
program or a financial calculator), there are always five variables involved in the
calculation: thepresentvalue, futurevalue, interest rate,numberofperiods, and the
periodicpayment(orannuity).Onecansolveforanyofthesevariablesiftheotherfour
areprovided,andinExcel,afunctionexistsforcalculatingeach.Thefivefunctionsare:
PV,FV,RATE,NPERandPMT.
Withtheexceptionofthecontinuouscompoundingformula,thesefunctionswillwork
for
any
of
the
Present
and
Future
value
formulas
that
weve
discussed
above,
including
annuities.
Asanexample,Figure1illustratesthesyntaxforthePVfunction:
FIGURE 1
Typing in the beginning of the function prompts the entry of the four remaining
variablesplus the [type]optionwhichcanbespecified to indicatewhetherpayments
aremadeatthebeginning(1)ortheendoftheperiod(0).Ifyouomitavaluefor[type],
Excelassumes
that
payments
are
made
at
the
end
of
the
period.
When
calculating
simplePresentandFuturevalues,novalueforpaymentexists.Toindicatethis,entera
valueofzeroforPMT.
RecallingExample1above,RATE=0.04,NPER=5,PMT=0,andFV=15000.Applying
thisexamplewithExcelsPVfunctionisillustratedinFigure2below:
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FIGURE 2
Youwillnotice that theresultmatches theanswergiven inExample1,however, it is
displayedasanegative.ThisisaworthnotingwhenperformingTimeValueofMoney
calculationsusingExcel,andisimportanttounderstand.
Whencalculating
Present
and
Future
values
in
Excel
(or
financial
calculators
for
that
matter), investmentsare treatedasnegativecash flows,whereas the returnofcapital
(plusanygains)istreatedasapositivecashflow.Inthiscase,sincethe$15,000isagain,
it istreatedasapositivecashflow,andthepresentvalueofthatgainisviewedasan
investment (negativecash flow). If the$15,000wasowedat theendof5years, i.e.,a
loanat4%interestcompoundedannually,thentheFutureValuewouldbenegativeand
thePresentValuewouldbepositive,asshowninFigure3:
FIGURE 3
Nonannualcompoundingperiodsarehandled in thesame fashion,except theRATE
andNPER
values
must
be
represented
on
aperiodic
basis.
Figure
4illustrates
the
calculationofExample2aboveinExcel:
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FIGURE 4
Asyou can see,youdonotneed toactuallyperform theperiodic rate calculation in
advance,itcansimplybeenteredintoExcelasamathematicaloperation,(0.04/12)and
(5*12)inthiscase,andtheprogramwillcalculateitforyou.
Asfor
calculating
annuities,
the
only
difference
in
the
function
syntax
is
that
apayment
is present (PMT is the same asA in the annuity formulas discussed in the review).
Figure5belowillustratesthecalculationinExcelofExample5above:
FIGURE 5
Usingthesameexample,onecouldsolvefortheothervariablesusedwhencalculating
PresentValue,asshownbelow:
Inputting:
=RATE(10,1500,-12166.34,0) will return 4%
=NPER(0.04,1500,-12166.34,0) will return 10=PMT(0.04,10,-12166.34,0) will return $1,500
=FV(0.04,10,1500,-12166.34) will return 0
Lastly,RATEcanalsobeusedtocalculateaCompoundAnnualGrowthRate.Recalling
Example6,NPER=5,PMT=0,PV=150,FV=175.ApplyingthisexamplewithExcels
RATEfunctionisillustratedinFigure6below:
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FIGURE 6
EFFECT & NOMINAL
Excel functions also exist to calculate the Effective orNominal rate. In order to use
them,youmustfirstmakesurethattheExcelAnalysisToolPakaddinisinstalledand
enabled.
To
do
so,
click
on
Tools
in
the
Menu
bar
and
then
select
Add
Ins,
as
shown
in
Figure7:
FIGURE 7
ThiswillbringupthewindowdisplayedinFigure8.Ifitisntalreadychecked,check
theboxtotheleftoftheAnalysisToolPakoptionandthenclickonOk.
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FIGURE 8
OncetheToolPakhasbeenenabled,youwillbeabletousetheEFFECT&NOMINAL
functionsinExcel.Thesyntaxforbothfunctionsisasfollows:
=EFFECT(nominal_rate,npery)
=NOMINAL(effect_rate,npery)
wherenperyrepresentsthenumberofcompoundingperiodsperyear.
RecallingExample
3above,
NOMINAL_RATE
=0.0875
and
NPERY
=365.
Applying
thisexamplewithExcelsEFFECTfunctionisillustratedinFigure9below:
FIGURE 9
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EXP
NodirectfunctionexistsinExceltocalculatecontinuouscompounding.However,the
EXPfunctioncanbeusedtogeneratethediscountfactor,asitcalculatesex.
Recalling
Example
4
above,
PV
=
$12,281
and
e0.04(5)
=
1.2214.
Applying
this
example
with
ExcelsEXPfunctionisillustratedinFigure10below:
FIGURE 10
NPV, IRR & FVSCHEDULE
As an example of how to use theNPV, IRR and FVSCHEDULE functions in Excel,
assume the following:aprojecthasconstruction,machineryandmaintenancecostsof
$45,000 in the firstyear and $2,000peryear afterwards (toyear 5).Labour costs are
$60,000peryearforallfiveyears,andrevenuesare$100,000peryearstartinginyear2.
Alsoassumethatinflationisnotanissue.
Figure11givesanexampleofhowthecashflowsfromthisprojectcanbestructuredin
Excel:
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FIGURE 11
InFigure11,rows2through4representthecashflowsasdescribedabove.Noticethat
costs,orcashoutflows,arelistedasnegativenumbers.ThisisacrucialelementofNPV
analysis,andisinlinewiththediscussionaboutthePVfunction.Row5representsthe
sumofeachof theyearlycash flows. Inrow7,aseriesofpossiblediscountrateshas
been listed, andNetPresentValueshavebeen calculated in row 8 for each of these
rates.AnexampleofthisisincellB8,wheretheNPViscalculatedbasedonanassumed
6%discountrate.ThesyntaxfortheNPVfunctionisasfollows:
=NPV(rate,value1,[value2],)
For cellB8, the rate is6% (cellB7),and the rangeof cash flows isdisplayed in cells
B5:F5.Thesamecalculationisperformedbasedontheassumptionofan8%,10%,12%
and14%discountrateincellsC8throughF8,andwecanseethattheNPVispositivein
eachcase.TheNPVwouldcontinuetobepositiveuptoandincludingadiscountrate
of16.63%,aswecanseeusingtheIRRfunction(seeFigure12below).
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FIGURE 12
The IRR syntax is simpler as it only requires the range of cash flows that theNPV
calculationisbasedon.Asmentionedpreviously,theIRRisthediscountratethatsets
theNPVequaltozero.AscanbeseeninFigures11&12,theNPVvaluesdeclineasthe
discount rate increases,andwouldbeequal tozeroat the rate calculatedby the IRR
function.Pleasenote that thereare severalwellknownproblemswithusing the IRR
methodologyasaselectioncriteriaformultipleprojects1.
Anotherway
to
calculate
the
NPV
of
aseries
of
cash
flows
is
to
discount
them
manually
using theFVSCHEDULEfunction(which, like theEFFECTandNOMINALfunctions,
requiretheAnalysisToolPaktobeenabled).Whiletherewouldbenoneedtodothisin
the case of the above example, this function can prove usefulwhen correcting for
reinvestmentrateassumptionsandwhenconductingdiscountedcashflowanalysis.
ThesyntaxfortheFVSCHEDULEfunctionisasfollows:
=FVSCHEDULE(principal,schedule)
Theschedule
variable
in
this
case
can
be
asingle
interest
rate,
or
can
be
aseries
of
rates.
Ifyouareenteringvaluesintotheformulamanually,theymustbeenteredinasarrays
(withcurlybrackets).Ifyouusecellreferencesfortheschedule,youcandosowithout
anarray.Considerthefollowingexample:
1Seehttp://www.investopedia.com/ask/answers/05/irrvsnpvcapitalbudgeting.asp
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Entering =FVSCHEDULE(5000,{0.05,0.06,0.07}) returns $5,954.55, which is
the same as $5,000*(1.05)*(1.06)*(1.07)
Now assume that 0.05, 0.06 and 0.07 are entered as values in cells A1, B1
and C1 respectively. Entering =FVSCHEDULE(5000,A1:C1) also returns
$5,954.55.
In terms of calculating NPV, we will use just one interest rate and employ the
FVSCHEDULEfunctiontocalculatediscountfactors.Anillustrationofthisisshownin
Figure13below:
FIGURE 13
Here,thefunctionhasbeenusedsimplytocalculatethefirstyeardiscountfactor,based
onanassumeddiscountrateof14%.Sinceacellreferenceisusedfortheschedule,curly
bracketsarenotrequired.Figure14showshowthiscanbeextendedfortheremaining
discountfactors:
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FIGURE 14
NotethattheprincipalvalueforthefunctionasenteredincellC8isdifferentthanthat
incellB8.Here,cellB8issetastheprincipal,andwhiletherateisstillfoundincellB7,
it is an absolute reference as opposed to a relative cell reference.As a result, the
functionincellC8canbedraggedallthewaytocellF8,becausetheprincipalvalueis
alwaystakenfromthecelltotheleftofthefunction.Hadwenotincludedtheabsolute
reference for cell B7, the functionwould have referenced cells C7, D7, etc. for the
discountrate.
Oncethediscountfactorsarecalculated,row9simplydividesthecashflowsinrow5
by therelevantdiscountfactor,andcellB10sumsup thesediscountedcashflows.As
wecansee,thevalueisexactlythesameastheNPVcalculatedasshowninFigure11.
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