excel financial functions i

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Excel

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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BA C K TO TA BLEOF C ON TEN TS

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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http://www.investopedia.com/ask/answers/05/irrvsnpvcapitalbudgeting.asphttp://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.

15of15 Lastmodified:11May2007

excel financial functions i

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