Financial Management I Chapter Three.pdf

Chapter Three: The Time Value of Money
Discussion Points:
Introduction
Compound Interest and Future Value
Future Value of An Annuity
Discounting Techniques and Present Value
Annuities – A Level Stream

1. Chapter Three
Time value of money (TVM)
5.1.Introduction
Businessorganizationsdealswithinterestrateswhenitmakesboth
financingandinvestmentdecisions.
▪Acompanycan,therefore,earnarateofreturnonitsinvested
fundsandarateofinterestonthefundsitlenttoborrowers.
▪Thekeyconceptthatunderliesthisisthetimevalueofmoney:that
“abirrtodayisworthmorethanabirrreceivedayearfromnow”.
▪Thisisbecauseifyouhaditnow,youcouldinvestthatbirrorgiven
asloansduringtheyearandearnedareturnoraninterestonit.
▪Ingeneralbusinessterms,interestisdefinedasthecostofusing
moneyovertime.

2. ▪Thisdefinitionisincloseagreementwiththedefinitionusedby
economists,whoprefertosaythatinterestrepresentsthetimevalue
ofmoney.
▪Ignoringtheeffectsofinflation,adollartodayisworthmorethana
dollartobereceivedayearfromnow.
▪Inotherwords,wewouldallprefertoreceiveaspecificamountof
moneynowratherthanonsomefuturedate.
✓Thispreferencerestsonthetimevalueofmoney.
TVMisalsodescribedasdiscountedcashflow(DCF).
✓DCFisatechniqueusedtodeterminethepresentvalueofa
certainamountofmoneyreceivedatafuturedate.
▪Theinterestrateisusedasthediscountingfactor,whichcanbe
foundbyusingapresentvalue(PV)table.

3. ❑Inflowsofdollarsonvariousfuturedatesshouldnotbeadded
togetherasiftheywereofequalvalue.
▪Thesefuturecashinflowsmustberestatedattheirpresentvalues
beforetheyareaggregated.
Theconceptofthetimevalueofmoneytellsusthatmore
distantcashinflowshaveasmallerpresentvaluethancash
inflowstobereceivedwithinashortertimespan.
“Thelattermoneyisreceived,thelessvalueitholds,andBr.1
todayisworthmorethanBr.1receivedatadateinthefuture”.
❑Similarreasoningappliestocashoutflows.
Beforeweaddtogethercashoutflowsonvariousfuturedates,we
mustrestatetheseoutflowsattheirpresentvalues.
Themoredistantthedateofacashoutflow,thesmallerisits
presentvalues.

4. ▪Asasimpleexampleofthisconceptofpresentvalue,assumethat
youaretryingtosellyourcarandyoureceiveoffersfromthree
prospectivebuyers.
❖Buyers“A”offersyouBr.8000tobepaidimmediately.Buyer“B”offersyouBr.8,200
tobepaidoneyearfromnow.Buyer“C”offersthehighestprice,Br.9,200butthis
offerprovidesthatpaymentwillbemadeinfiveyears.Assumingthattheoffersby“B”
and“C”involvesnocreditriskandthatmoneymaybeinvestedat5%interest
compoundedannually,whichofferwouldyouaccept?
YoushouldaccepttheofferofBr.8000tobereceivedimmediately,
becausethepresentvalueoftheothertwooffersislessthanBr.
8000.
✓IfyouweretoinvestBr.8000today,evenatthemodestrateof
interestof5%,yourinvestmentwouldbe(8,400)morethanBr.
8200inoneyearandconsiderably(10,000)morethanBr.9,200in
fiveyears.

5. ▪Thisexamplesuggeststhatthetimingofcashreceiptsandpayments
hasanimportanteffectontheeconomicworthandtheaccounting
valuesofbothassetsandliabilities.
▪Consequently,investmentandborrowingdecisionsshouldbemade
onlyafteracarefulanalysisoftherelativepresentvaluesofthe
prospectivecashinflowsandoutflows.

6. 5.2.SimpleInterestandCompoundInterest
•Interestistheexcessofresources(usuallycash)receivedorpaidover
theamountofresourcesloanedorborrowedatanearlierdate.
•Businesstransactionssubjecttointereststatewhethersimpleor
compoundinterestistobecalculated.
1)Simpleinterestisthereturnonaprincipalamountforonetime
period.
•Wemayalsothinkofsimpleinterestasareturnformorethanone
timeperiodifweassumethattheinterestitselfdoesnotearna
return,butthiskindofsituationoccursrarelyinthebusinessworld.
•Simpleinterestisusuallyapplicableonlytoshort-terminvestment
andborrowingtransactionsinvolvingatimespanoflessthanone
year.

7. •Interestisexpressedintermsofanannualrate.
•Theformulaforsimpleinterestis:
I=p*r*t(Interest=principalxannualrateofinterestxnumberof
yearsorfractionofayearthatinterestaccrues).
Or,usingalternativeapproach,
F=P+IThen,substituteI=P*i*nintheexpressiontoobtain
F=P+Pin
F=P(1+in)
Forexample,interestonBr.10,000at8%foroneyearisexpressedasfollows:
I=p.r.t
I=Br.10,000x0.08x1
I=Br.800
▪Theamounttoberepaidattheendoftheyearisthematurity(future)valueof
thespecifiedmoney.
Accordingly,F=P+I
F=10000+800
F=Br.10,800
I = p * r * t
F = P + I
F = P (1 + in)

8. Exercise:
1.Salon borrowed $5,000 at per year simple interest of 5% for
two years to buy new hair dryers. How much interest must be
paid?
2.Marcus Logan can purchase furniture with a two-year simple
interest loan at 9% interest per year. What is the maturity
value for a $2,500 loan?

9. 2)Compoundinterestisthereturnonaprincipalamountfortwoor
moretimeperiods,assumingthattheinterestforeachtimeperiod
isaddedtotheprincipalamountattheendofeachperiod,and
earnsinterestinallsubsequentperiods.
•Iscalculatedontheprincipalamountandontheaccumulated
interestofpreviousperiod(andthereforeberegardedas“intereston
interest”)
•Becausemostinvestmentandborrowingtransactionsinvolvemore
thanonetimeperiod,businessexecutivesevaluateproposed
transactionsintermsofperiodicreturns,eachofwhichisassumedto
bereinvestedtoyieldadditionalreturns.
Forexample,ifinterestat8%iscompoundedquarterlyforoneyearona
principalamountofBr.10,000thetotalinterest(compoundinterest)
wouldbeBr.824.32,ascomputedbelow:

10. Period PrincipalxRatexTime=CompoundInterestAccumulatedAmounts
1stquarter----Br.10,000x0.08x¼ Br.200 Br.10,200
2ndquarter-------10,200x0.08x¼ 204 10,404
3rdquarter--------10,404x0.08x¼ 208.08 10,612.08
4thquarter-----10,612.08x0.08x¼ 212.24 10,824.32
Interest--------------------------------------------------------------824.32
N.B,
•Aperiod,forthispurpose,canbeanyunitoftime.
•Ifinterestiscompoundedannually,ayearistheappropriate
compoundingorconversionorinterestperiod.
•Ifitiscompoundedmonthly,amonthistheappropriateperiod.
•Itisimportanttoknowthatthenumberofcompoundingperiod/s
withinayearis/areusedinordertofindtheinterestrateper
compoundingperiodsanditisdenotedbyiintheaboveformula.

11. ▪Consequently,whentheinterestrateisstatedasannualinterestrate
andiscompoundedmorethanonceayear,theinterestrateper
compoundingperiodiscomputedbytheformula:
Where;j=isannualquotedornominalinterestrate
m=numberofconversationperiodsperyearorthe
compoundingperiodsperyear
Where;t=isthenumberofyears
❑Inthecomputationofcompoundinterest,theaccumulatedamount
(interest)attheendofeachperiodbecomestheprincipalamount
forpurposesofcomputinginterestforthefollowingperiod.
i= j / m
n = m x t

12. •Incomputingsimpleinterest,thenumberofyearsortime,n,can
bemeasuredindays.
•Insuchcase,therearetwowaysofcomputingtheinterest.
1)TheExactMethod:ifayearisconsideredas365days,theinterest
iscalledexactsimpleinterest.
•Iftheexactmethodisusedtocalculateinterest,thenthetimeis
n=numberofdays/365
2)TheOrdinaryMethod(Banker’sRule):ifayearisconsideredas
360days,theinterestiscalledordinarysimpleinterest.
•Thetime,iscalculatedas
n=numberofdays/360

13. 5.3FutureandPresentValues
▪Futurevalueinvolvesacurrentamountthatisincreasedinthe
futureasaresultofcompoundinterestaccumulation.
▪Presentvalue,incontrast,involvesafutureamountthatisdecreased
tothepresentasaresultofcompoundinterestdiscounting.
▪Discounting,ineffect,extractstheinterestfromafuturevalue
therebyreturningtotheprincipalamount.
▪Thefactthatinvestmentshavestartingpointsandendingpoints
makesiteasiertounderstandpresentandfuturevalues.
▪Presentvalueingeneralreferstodollar(birr)valuesatthestarting
pointofaninvestment,andfuturevaluereferstoend-pointdollar
(birr)values.

14. ▪Ifthedollar(birr)amounttobeinvestedatthestartisknown,the
futurevalueofthatamountattheendcanbeprojected,providedthe
interestrateandnumberofinterestcompoundingperiodsarealso
specified.
▪Similarly,ifthedollar(birr)amountavailableattheendofan
investmentperiod(futurevalue)isknown,theamountofmoney
neededatthestartoftheinvestmentperiod(presentvalue)canbe
determined,againiftheinterestrateandnumberofinterest
compoundingperiodsareknown.
❑Presentvalueandfuturevalueapplytointerestcalculationsonboth
singlepaymentamountsandperiodicequalpaymentamounts
(annuities)

15. 5.3.1.FuturevalueofaSingleSum
•Theaccumulatedamountofasingleamountinvestedatcompound
interestmaybecomputedperiodbyperiodbyaseriesof
multiplication,asillustratedaboveforBr.10,000investedforone
yearat8%compoundedquarterly.
•Ifnisusedtorepresentthenumberofperiodsthatinterestistobe
compounded,iisusedtorepresenttheinterestperperiod,andpis
theprincipalamountinvested,theseriesofmultiplicationsto
computetheaccumulatedamount(FV)intheexampleabove
determinedasflows:
FV=p(1+i)
n
FV=Br.10,000(1.02)4
FV=Br.10,000[(1.02)(1.02)(1.02)(1.02)]
FV=Br.10,824.32
i= j/m=8%/4 = 2%
n= m*t=1*4 = 4

16. Exercise:
1.Find the future value of a $10,000 investment at 2% annual
interest compounded semiannually for three year.
2.A loan of $2,950 at 8% is made for two years compounded
annually. Find the future value (compound amount) of the
loan. Find the amount of interest paid on the loan.
3.Davis invested $20,000 that earns 6% compounded monthly
for four years. Find the future value of Davis’s investment.

17. Illustrationofcomputationoffutureamount
▪Ifonthedayherdaughterwasborn,BetheldepositedBr.10,000in
asavingsaccountthatguaranteestoaccumulateinterestquarterlyat
10%ayear.
▪Whatwillbetheamountinthesavingsaccountonherdaughter’s
18thbirthday?
❑Solution:Theamountinthesavingsaccountonherdaughter’s18th
birthdaywillbeBr.10,000(1+0.025)
72
.BecauseTabledoesnotgo
beyond50periods,theamountinthesavingsaccountonthe
daughter’s18thbirthdaymaybecomputedasfollows:
FV=Br.10,000(1+0.025)
50
x(1+0.025)
22
FV=Br.10,000(3.437109)x(1.721571)
=Br.59,172
i= j/m=10%/4=2.5%
n= m*t=18*4=72

18. OtherApplicationoffuturevalueamountofsinglepayment:
▪Thefuturevalueequationforsinglepaymentstatedinthismaterial
canalsobeusedtofindinterestrates,aswellas,numbersofyears
thatwillbeneededforthecompoundedamounttoequalthedesired
value.
FindingtheInterestrate:
•Ausefulapproachistotreattheinterestrateasanimplicitinterest
rateandfoundbyusingtheinteresttable(futurevaluetableofsingle
payment).

19. ▪Toillustrate,assumethatyouhaveinvested15,000Birrtodayata
bankwhereitcangrowtothefuturevalueof17,900Birrwithin
threeyearsfromnowintothefuture.
✓Whatistheinterestratethatthebankshouldpayforyour
accountinordertofulfillyourdesire?
▪Toanswerthisquestion,substitutingthosevaluesintothefuture
valueofsinglepaymentequation,youget:
FV
3= PV (1+i)
3
Fv/Pv= (1+i)
3
FVIF
i,
3= (1+i)
3
= 193.1
000,15
900,17
=

20. ▪Thefuturevalueinterestfactorintheinterest(futurevalueofsingle
paymenttable)correspondingtotheunknowninterestrate(i)anda
periodof3yearsn=3)is1.193.
▪Hence,lookupthethreeyear(n=3)rowandreadhorizontallyuntil
youfindthetablevalue(futurevalueinterestfactor)thatisequalor
theclosesttothecomputedvalueof1.193.
▪Thereisnotablevaluethatisexactlyequalto1.193.
▪Thetablevalueof1.191isfoundtobetheclosestvalueto1.193
anditcorrespondsto6percent.
▪Therefore,theinterestthatthebankactuallyhastopaytoyour
accountisslightlygreaterthan6percent.

21. Findingthenumberofyears:
▪Thefuture(compound)valueofsinglepaymentequationcanbeused
toestimatethenumberofyearsthatarerequiredforagivenamount
ofmoneydepositedataspecificinterestratetoproduceordesired
compoundamount.
▪Forexample,adepositof1000Birrismadeinaninterestbearing
accountthatpays10percentcompoundedyearly.
▪Yourgoalasadepositoristocollect1,500Birrafteranunknown
numberofyears.
▪Howmanyyearsshouldyouwaitforthedesiredamounttobe
realized?

22. Bysubstitutingthevaluesintothefuturevalueofsinglepaymentequation,
youget:
FVn=1000(1+I)
n
1,500=1000(1+0.1)
n
=1000(1.1)n
(1.1)
n
=1500/1000
=1.5,byusinglogarithm
n=log1.11.5=log1.5/log1.1=0.176/0.041=4.29years
•Againitispossibletolookupthe10percentcolumninthefuture
valueinterestfactors(futurevalue)tableandreadverticallyuntilyou
findatablevaluethatisequalto1.5orclosesttoit.
•Theclosesttablevalueis1.611,whichcorrespondstofiveyears(n=5).
•Thatmeansifthe1000Birriskeptintheaccountthatpays10percent
forfiveyears;theresultingcompoundingamountwillbe1,611Birr.
•Thisamountexceedsthedesiredamountof1,500Birr.Ifthe1000
Birriskeptintheaccountonlyforfouryears,thetablevalueis1.464.
Hence,the1000Birrhastobekeptintheaccountforaperiodslightly
greaterthan4years.

23. 5.3.2.PresentValueofaSingleSum
▪Manymeasurementandvaluationproblemsinfinancialaccounting
requirethecomputationofthediscountedpresentvalueofa
principalamounttobepaidorreceivedonafixeddate.
▪Thepresentvaluerepresentsthediscountedamount(interest
excluded)thatwillaccumulatetothefutureamount(interest
included).
▪Thepresentvalueofafutureamountisalwayslessthanthatfuture
amount.
▪Thecomputationofthepresentvalueofasinglefutureamountisa
reversaloftheprocessoffindingtheamounttowhichapresent
amountwillaccumulate.

24. FV=p(1+i)
n
,andwhenwesolveforpbydividingbothsidesofthe
equationby(1+i)
n
,wehave
PV=()
n
i
FV
+1
Therefore,theformulaforthePVofFVdueinnperiodsatirateof
interestperperiodis
PV=
Example:IfwewantanamountofBr.30,000after12yearsbymaking
asingledepositinasavingaccountwhichwillpay16%interest
compoundedquarterly,whatshouldtheamountofinitialdepositbe?()
n
i
FV
+1
FV = future amount
P = present value
i= interest rate per period
n = number of compounding period

25. Solution:ThepresentvalueisBr.30,000discountedat4%for48periods.
Thepresentvalueis
Exercise:
Abnetneeds$20,000infiveyearstobuyanewmachine.Howmuch
mustheinvestatthepresentifhereceives5%interestcompounded
semi-annual?()
n
i
FV
+1 ( ) 570528.6
000,30.
04.01
000,30.
48
BrBr
=
+
PV=
=
=Br.4565.84
=
i= j/m=16%/4=4%
n= m*t=12*4=48

26. Annuities
▪Anannuityisaseriesofuniformpaymentsorreceiptsoccurringat
uniformintervalsoveraspecifiedinvestmenttimeframe,withall
amountsearningcompoundinterestatthesamerate.
▪Paymentsofanytypeareconsideredasannuitiesifallofthe
followingconditionsarepresent:
i.Theperiodicpaymentsareequalinamount
ii.Thetimebetweenpaymentsisconstantsuchasayear,half
ayear,aquarterofayear,amonthetc.
iii.Theinterestrateperperiodremainsconstant.
iv.Theinterestiscompoundedattheendofeverytime.

27. ▪Annuitiesareclassifiedaccordingtothetimethepaymentismade.
▪Accordingly,wehavetwobasictypesofannuities.
1)Ordinaryannuity:isaseriesofequalperiodicpaymentismadeat
theendofeachintervalorperiod.
✓Inthiscase,thelastpaymentdoesnotearninterest.
2)Annuitydue:isatypeofannuityforwhichapaymentismadeat
thebeginningofeachintervalorperiod.
❖AmountofAnnuity:Amountofanannuityisfuturevaluesofa
seriesofequalreceiptsorpayments(rents)madeatregulartime
intervalsandatthesamerateofinterestcompoundedeachtime
thereceiptsorpaymentsaremade.

28. 5.4.1.FutureValueofanordinaryAnnuity
▪Anordinaryconsistsofaseriesofequalpaymentmadeattheendofeach
period.
▪Ifyoudeposit100Birrattheendofeachyearforthreeyearsinasaving
accountthatpays5percentperyear,howmuchwillyouhaveattheend
ofyearthree(n=3)?.
▪Toanswerthisquestion,youmustfindfuturevalueofanannuity(FVA
n
.)
▪Eachpaymenthastobecompoundedouttotheendofperiodn,andthe
sumofthecompoundedpaymentsgivesyouFVA
n
.
▪Thiscanbeshownbyusingthefollowingtimeline.
0 1 2 3
100Birr 100Birr 100Birr
105.00Birr=100(1.05)
1
110.25Birr=100(1.05)
2
FVA
3
=315.25Br.

29. FVA
n
=PMT(1+i)
0
+PMT(1+i)
1
+PMT(1+i)
2
+----+PMT(1+i)
n-1
•Inthisequationthefirstterm(i.e.PMT(1+i)
0
)isthecompounded
valueofthepaymentattheendoflastyear,oryearnofthe
annuitypayments;whilethelasttermintheequation(i.e.PMT
(1+i)
n-1
)isthecompoundedamountofthepaymentmadeatthe
endofyearone(n-1).
•Theaboveequationcanfurtherbesimplifiedto:
•Usingthisfuturevalueofanannuityequation,thefuturevalueofthe
100Birrdepositsmadeattheendofeachyearforthreeyearsatan
interestrateof5percentwouldbe:
FVA
3
=(100)




 −+
05.0
1)05.01(
3
=(100)(3.1525)=315.25Birr




 −+
i
i
n
1)1(
FVA
n= (PMT)
PMT = Per period payment amount

30. Example:Computetheamountofanordinaryannuityof16rentsof
Br.100at2%
Solution:
FVAn=Br.100( )
93.1863.
02.0
372786.0
100
02.0
1372786.1
02.0
102.01
16
Br=











 −
=




 −+

31. Exercise:
1.Find the value of an ordinary annuity after two years of
$1,500 invested semiannually at 4% annual interest?

32. 5.4.2.Present(Discounted)valueofordinaryAnnuity:
•Supposethatyouareofferedtwoalternatives:
(1)athree-annuitywithpaymentof100Br.attheendofeachyear
overthecomingthreeyears,or(2)a:lump-sumpaymenttoday.
1)Youhavenoneedforthemoneyduringthenextthreeyears,soif
youaccepttheannuity,youwouldsimplydepositthepaymentsin
thesavingaccountatDashenBankthatpays5percentinterestrate
compoundedyearly.
2)SimilarlytheLump-Sumpaymentwouldbedepositedinthesame
accountasyoudon'thaveotheroption.
•HowlargeshouldtheLump-sumpaymenttodayinorderforitbe
equivalenttotheannuity?

33. Toanswerthisquestion,letusstart-upwiththeaidofthetimeline.
0 1 2 3
100 100 100
95.24
90.70
86.38
PVA
3
=272.32Br.
▪Thepresentvalueoftheseriesofpaymentsof100Birrforthreeyears
annuity,PVA
3
is272.32Birrasshownwiththehelpofthetimeline.
▪Thesameproblemcanbeexpressedbyusingmathematicalequation.
▪Thegeneralmathematicalequationthatcanbeusedtofindthe
presentvalueofanordinaryannuityisshownbelow:

34. ()
n
i
PMT
i
PMT
i






+
+−−−−+





+
+





+ 1
1
1
1
1
1
21 













+
−−−−+





+
+





+
+





+
n
iiii 1
1
1
1
1
1
1
1
211 PVA
n
=(PMT)
SincePMTiscommonforallterms,theaboveequationcanbere-writtenas:
PVA
n
=(PMT)
Again the equation can berewritten as:
PVA
n
= (PMT) 













+

=
t
n
t i1
1
1 i
i
n
,)1(
1
1
+
− i
i
n−
+− )1(1
Thesummationterminthisequationiscalledthepresentvalueinterest
factorforanannuity(PVIFA)anditequivalentto: or











+
−





 +−
−
i
i
PMTor
i
i
nn
)1(
1
1
,
)1(1
PVA
n= (PMT)

35. ▪Byusingthismathematicalequation,thepresentvalueofanannuity
oftheproblemunderconsiderationcanbecomputedasfollows.
▪Youaregivenanannuitypayment,PMTof100Birr,interestrate,5
percentcompoundedyearlyandanannuityperiod,nofthreeyears.
▪Substitutingthesevaluesintotheaboveequation,yougetthepresent
valueofanannuityof272.32Birr,whichwasthesameasthe
amountcomputedbysummingtheindividualdiscountedvaluedon
thetimeline.
PVA
3= (100) 











+
−





+−
−
05.0
)05.01(
1
1
)100(,
05.0
)05.01(1
3 n
or
=(100)(2.7232)
=272.32Birr

36. 5.4.3.FuturevalueofAnnuityDue
▪Theamountofanannuitydue(orannuityinadvance)isthetotal
amountondeposit/paymentoneperiodafterthefinal
deposit/payment.
▪Similartoordinaryannuitiesexceptthatpayment/depositsaremade
atthebeginningofeachdeposit/paymentperiods.
Example.Ifthecooperativeunionsdeposit100Birrinterestof10%for
3years
Solution;FVA(Due)3=
=
=
=

37. 5.4.4PresentvalueofOrdinaryAnnuity
•Presentvalueofordinaryannuityisthediscountedvalueofaseriesof
futurerentsonadateoneperiodbeforethefirstpayment/deposit.
▪Youaregivenanannuitypayment,PMTof100Birr,interestrate,10
percentcompoundedyearlyandanannuityperiod,nofthreeyears.
CalculatethePVA(DUE)n?
Solution:
PVA (DUE)n = Br. 273.55

38. 5.4.5.PresentvalueofPerpetualAnnuity
▪Whenanannuityisexpectedtooccurindefinitely,itiscalled
perpetualannuity.
▪Perpetuityisastreamofequalpaymentsexpectedtocontinue
forever.
▪Thepresentvalueofperpetuityis:
PVP=
Payment =PMT
Interest rate k
Forexample,iftheinterestratewere12percent,perpetuityof$1,000a
yearwouldhaveapresentvalueof=$8,333.33.
PVp=Payment/Interestrate
=PMT/k
=$1,000/0.12
=Br.$8,333.33

39. Exercise:
1.What is the future value of an annuity due with an
annual payment of $1,000 for three years at 4% annual
interest? Find the total investment and the total interest
earned.
2.If you make six monthly payments of $50 to an annuity
due and receive 6% annual interest compounded
monthly, how much will you accumulate?

40. End of Chapter –Three
Thank You!!!