1、1Chapter 2Time Value of Money2Time Value TopicsnFuture valuenPresent valuenRates of returnnAmortization3Time lines show timing of cash flows.CF0CF1CF3CF20123i%Tick marks at ends of periods,so Time 0 is today;Time 1 is the end of Period 1;or the beginning of Period 2.4Time line for a$100 lump sum due
2、 at the end of Year 2.100012 Yeari%5Time line for an ordinary annuity of$100 for 3 years1001001000123i%6Time line for uneven CFs100 50 750123i%-507FV of an initial$100 after3 years(i=10%)FV=?012310%Finding FVs(moving to the righton a time line)is called compounding.1008After 1 yearFV1=PV+INT1=PV+PV(
3、i)=PV(1+i)=$100(1.10)=$110.00.9After 2 yearsFV2=FV1(1+i)=PV(1+i)(1+i)=PV(1+i)2=$100(1.10)2=$121.00.10After 3 yearsFV3=FV2(1+i)=PV(1+i)2(1+i)=PV(1+i)3=$100(1.10)3=$133.10In general,FVn=PV(1+i)n.11Three Ways to Find FVsnSolve the equation with a regular calculator.nUse a financial calculator.nUse a sp
4、readsheet.12Financial calculator:HP10BIInAdjust display brightness:hold down ON and push+or-.nSet number of decimal places to display:Orange Shift key,then DISP key(in orange),then desired decimal places(e.g.,3).nTo temporarily show all digits,hit Orange Shift key,then DISP,then=13HP10BII(Continued)
5、nTo permanently show all digits,hit ORANGE shift,then DISP,then.(period key)nSet decimal mode:Hit ORANGE shift,then./,key.Note:many non-US countries reverse the US use of decimals and commas when writing a number.14HP10BII:Set Time Value ParametersnTo set END(for cash flows occurring at the end of t
6、he year),hit ORANGE shift key,then BEG/END.nTo set 1 payment per period,hit 1,then ORANGE shift key,then P/YR15Financial calculators solve this equation:FVn+PV(1+i)n=0.There are 4 variables.If 3 are known,the calculator will solve for the 4th.Financial Calculator Solution16310-100 0NI/YR PV PMTFV 13
7、3.10Clearing automatically sets everything to 0,but for safety enter PMT=0.Set:P/YR=1,END.INPUTSOUTPUTHeres the setup to find FV17Spreadsheet SolutionnUse the FV function:see spreadsheet in“CF2 Ch 02 Mini Case.xls”n=FV(Rate,Nper,Pmt,PV)n=FV(0.10,3,0,-100)=133.101810%Whats the PV of$100 due in 3 year
8、s if i=10%?Finding PVs is discounting,and its the reverse of compounding.1000123PV=?19Solve FVn=PV(1+i)n for PVPV=FVn(1+i)n=FVn()11+in PV=$10011.10 =$100 0.7513 =$75.13.3203 10 0100N I/YR PV PMTFV -75.13Either PV or FV must be negative.HerePV=-75.13.Put in$75.13 today,take out$100 after 3 years.INPU
9、TSOUTPUTFinancial Calculator Solution21Spreadsheet SolutionnUse the PV function:see spreadsheet in“CF2 Ch 02 Mini Case.xls”n=PV(Rate,Nper,Pmt,FVn=PV(0.10,3,0,100)=-75.132220%2012?-1 FV=PV(1+i)n Continued on next slideFinding the Time to Double23Time to Double(Continued)$2=$1(1+0.20)n (1.2)n=$2/$1=2n
10、LN(1.2)=LN(2)n=LN(2)/LN(1.2)n=0.693/0.182=3.8.24 20 -1 0 2NI/YR PV PMTFV 3.8INPUTSOUTPUTFinancial Calculator Solution25Spreadsheet SolutionnUse the NPER function:see spreadsheet in“CF2 Ch 02 Mini Case.xls”n=NPER(Rate,Pmt,PV,FV)n=NPER(0.10,0,-1,2)=3.826?%20123-1 FV=PV(1+i)n$2=$1(1+i)3 (2)(1/3)=(1+i)1
11、.2599=(1+i)i=0.2599=25.99%.Finding the Interest Rate273 -1 0 2NI/YR PV PMTFV25.99INPUTSOUTPUTFinancial Calculator28Spreadsheet SolutionnUse the RATE function:n=RATE(Nper,Pmt,PV,FV)n=RATE(3,0,-1,2)=0.259929Ordinary AnnuityPMTPMTPMT0123i%PMTPMT0123i%PMTAnnuity DueOrdinary Annuity vs.Annuity Due30Whats
12、 the FV of a 3-year ordinary annuity of$100 at 10%?100100100012310%110 121FV=33131FV Annuity FormulanThe future value of an annuity with n periods and an interest rate of i can be found with the following formula:=PMT(1+i)n-1i=100(1+0.10)3-10.10=33132Financial Calculator Formula for AnnuitiesnFinanc
13、ial calculators solve this equation:FVn+PV(1+i)n+PMT(1+i)n-1i=0.There are 5 variables.If 4 are known,the calculator will solve for the 5th.333 10 0 -100 331.00NI/YRPVPMTFVHave payments but no lump sum PV,so enter 0 for present value.INPUTSOUTPUTFinancial Calculator Solution34Spreadsheet SolutionnUse
14、 the FV function:see spreadsheet.n=FV(Rate,Nper,Pmt,Pv)n=FV(0.10,3,-100,0)=331.0035Whats the PV of this ordinary annuity?100100100012310%90.9182.64 75.13248.69 =PV36PV Annuity FormulanThe present value of an annuity with n periods and an interest rate of i can be found with the following formula:=PM
15、T1-1(1+i)ni=1001-1(1+0.10)30.10=248.6937Have payments but no lump sum FV,so enter 0 for future value.3 10 100 0NI/YRPVPMTFV-248.69INPUTSOUTPUTFinancial Calculator Solution38Spreadsheet SolutionnUse the PV function:see spreadsheet.n=PV(Rate,Nper,Pmt,Fv)n=PV(0.10,3,100,0)=-248.6939Find the FV and PV i
16、f theannuity were an annuity due.100100012310%10040PV and FV of Annuity Due vs.Ordinary AnnuitynPV of annuity due:n=(PV of ordinary annuity)(1+i)n=(248.69)(1+0.10)=273.56nFV of annuity due:n=(FV of ordinary annuity)(1+i)n=(331.00)(1+0.10)=364.141310 100 0 -273.55 NI/YRPVPMTFVINPUTSOUTPUTPV of Annuit
17、y Due:Switch from“End”to“Begin42310 0 100 -364.1 NI/YRPVPMTFVINPUTSOUTPUTFV of Annuity Due:Switch from“End”to“Begin43Excel Function for Annuities DuenChange the formula to:n=PV(10%,3,-100,0,1)nThe fourth term,0,tells the function there are no other cash flows.The fifth term tells the function that i
18、t is an annuity due.A similar function gives the future value of an annuity due:n=FV(10%,3,-100,0,1)44What is the PV of this uneven cash flow stream?010013002300310%-504 90.91247.93225.39-34.15530.08 =PV45Financial calculator:HP10BIInClear all:Orange Shift key,then C All key(in orange).nEnter number
19、,then hit the CFj key.nRepeat for all cash flows,in order.nTo find NPV:Enter interest rate(I/YR).Then Orange Shift key,then NPV key(in orange).46Financial calculator:HP10BII(more)nTo see current cash flow in list,hit RCL CFj CFjnTo see previous CF,hit RCL CFj nTo see subsequent CF,hit RCL CFj+nTo se
20、e CF 0-9,hit RCL CFj 1(to see CF 1).To see CF 10-14,hit RCL CFj.(period)1(to see CF 11).47nInput in“CFLO”register:nCF0 =0nCF1 =100nCF2 =300nCF3 =300nCF4 =-50nEnter I=10%,then press NPV button to get NPV=530.09.(Here NPV=PV.)48Excel Formula in cell A3:=NPV(10%,B2:E2)ABCDE1012342100300300-503530.0949N
21、ominal rate(iNom)nStated in contracts,and quoted by banks and brokers.nNot used in calculations or shown on time linesnPeriods per year(m)must be given.nExamples:n8%;Quarterlyn8%,Daily interest(365 days)50Periodic rate(iPer)niPer=iNom/m,where m is number of compounding periods per year.m=4 for quart
22、erly,12 for monthly,and 360 or 365 for daily compounding.nUsed in calculations,shown on time lines.nExamples:n8%quarterly:iPer=8%/4=2%.n8%daily(365):iPer=8%/365=0.021918%.51The Impact of CompoundingnWill the FV of a lump sum be larger or smaller if we compound more often,holding the stated I%constan
23、t?nWhy?52The Impact of Compounding(Answer)nLARGER!nIf compounding is more frequent than once a year-for example,semiannually,quarterly,or daily-interest is earned on interest more often.53FV Formula with Different Compounding PeriodsFV =PV 1 .+imnNommn 54$100 at a 12%nominal rate with semiannual com
24、pounding for 5 years=$100(1.06)10 =$179.08FV =PV 1 +imnNommn FV =$100 1+0.1225S2x5 55FV of$100 at a 12%nominal rate for 5 years with different compoundingFV(Annual)=$100(1.12)5=$176.23.FV(Semiannual)=$100(1.06)10=$179.08.FV(Quarterly)=$100(1.03)20=$180.61.FV(Monthly)=$100(1.01)60=$181.67.FV(Daily)=$
25、100(1+(0.12/365)(5x365)=$182.19.56Effective Annual Rate(EAR=EFF%)nThe EAR is the annual rate which causes PV to grow to the same FV as under multi-period compounding.57Effective Annual Rate ExamplenExample:Invest$1 for one year at 12%,semiannual:FV=PV(1+iNom/m)m FV=$1(1.06)2 =1.1236.nEFF%=12.36%,bec
26、ause$1 invested for one year at 12%semiannual compounding would grow to the same value as$1 invested for one year at 12.36%annual compounding.58Comparing RatesnAn investment with monthly payments is different from one with quarterly payments.Must put on EFF%basis to compare rates of return.Use EFF%o
27、nly for comparisons.nBanks say“interest paid daily.”Same as compounded daily.59EFF%=-1(1+)iNommm =-1.0(1+)0.1222 =(1.06)2-1.0 =0.1236=12.36%.EFF%for a nominal rate of 12%,compounded semiannually60Finding EFF with HP10BIInType in nominal rate,then Orange Shift key,then NOM%key(in orange).nType in num
28、ber of periods,then Orange Shift key,then P/YR key(in orange).nTo find effective rate,hit Orange Shift key,then EFF%key(in orange).61EAR(or EFF%)for a Nominal Rate of of 12%EARAnnual=12%.EARQ=(1+0.12/4)4-1=12.55%.EARM=(1+0.12/12)12-1=12.68%.EARD(365)=(1+0.12/365)365-1=12.75%.62Can the effective rate
29、 ever be equal to the nominal rate?nYes,but only if annual compounding is used,i.e.,if m=1.nIf m 1,EFF%will always be greater than the nominal rate.63When is each rate used?iNom:Written into contracts,quoted by banks and brokers.Not used in calculations or shownon time lines.64iPer:Used in calculati
30、ons,shown on time lines.If iNom has annual compounding,then iPer =iNom/1=iNom.When is each rate used?(Continued)65When is each rate used?(Continued)nEAR(or EFF%):Used to compare returns on investments with different payments per year.nUsed for calculations if and only if dealing with annuities where
31、 payments dont match interest compounding periods.66AmortizationnConstruct an amortization schedule for a$1,000,10%annual rate loan with 3 equal payments.67Step 1:Find the required payments.PMTPMTPMT012310%-1,0003 10 -1000 0 INPUTS OUTPUT NI/YRPVFVPMT402.1168Step 2:Find interest charge for Year 1.IN
32、Tt=Beg balt(i)INT1=$1,000(0.10)=$100.69Repmt=PMT-INT =$402.11-$100 =$302.11.Step 3:Find repayment of principal in Year 1.70Step 4:Find ending balance after Year 1.End bal=Beg bal-Repmt=$1,000-$302.11=$697.89.Repeat these steps for Years 2 and 3to complete the amortization table.71Amortization TableY
33、EARBEG BALPMTINTPRIN PMTEND BAL1$1,000$402$100$302$6982698402703323663366402373660TOT1,206.34206.341,00072Interest declines because outstanding balance declines.73nAmortization tables are widely used-for home mortgages,auto loans,business loans,retirement plans,and more.They are very important!nFina
34、ncial calculators(and spreadsheets)are great for setting up amortization tables.74Fractional Time PeriodsnOn January 1 you deposit$100 in an account that pays a nominal interest rate of 11.33463%,with daily compounding(365 days).nHow much will you have on October 1,or after 9 months(273 days)?(Days
35、given.)75iPer=11.33463%/365=0.031054%per day.FV=?0122730.031054%-100Convert interest to daily rate76 FV=$100 1.00031054 =$100 1.08846=$108.85.273273Find FV77273-100 0 108.85INPUTSOUTPUTNI/YRPVFVPMTiPer=iNom/m=11.33463/365=0.031054%per day.Calculator Solution78Non-matching rates and periodsnWhats the
36、 value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%,compounded semiannually?79Time line for non-matching rates and periods01100235%456 6-mos.periods 10010080Non-matching rates and periodsnPayments occur annually,but compounding occurs each 6 months.nSo we cant u
37、se normal annuity valuation techniques.811st Method:Compound Each CF01100235%456100100.00110.25121.55331.80FVA3=$100(1.05)4+$100(1.05)2+$100=$331.80.822nd Method:Treat as an annuity,use financial calculator Find the EAR for the quoted rate:EAR=(1+)-1=10.25%.0.1022833 10.25 0 -100 INPUTS OUTPUT NI/YR
38、PVFVPMT331.80Use EAR=10.25%as the annual rate in calculator.84Whats the PV of this stream?010015%2310010090.7082.2774.62247.5985Comparing InvestmentsnYou are offered a note which pays$1,000 in 15 months(or 456 days)for$850.You have$850 in a bank which pays a 6.76649%nominal rate,with 365 daily compo
39、unding,which is a daily rate of 0.018538%and an EAR of 7.0%.You plan to leave the money in the bank if you dont buy the note.The note is riskless.nShould you buy it?86iPer=0.018538%per day.1,0000365456 days-850Daily time line87Three solution methodsn1.Greatest future wealth:FVn2.Greatest wealth toda
40、y:PVn3.Highest rate of return:Highest EFF%881.Greatest Future WealthFind FV of$850 left in bank for15 months and compare withnotes FV=$1,000.FVBank=$850(1.00018538)456=$924.97 in bank.Buy the note:$1,000$924.97.89456-850 0 924.97INPUTSOUTPUTNI/YRPVFVPMTiPer=iNom/m=6.76649%/365=0.018538%per day.Calcu
41、lator Solution to FV90Find PV of note,and comparewith its$850 cost:PV=$1,000/(1.00018538)456=$918.95.2.Greatest Present Wealth91 456 .018538 0 1000 -918.95INPUTSOUTPUTNI/YRPVFVPMT6.76649/365=PV of note is greater than its$850 cost,so buy the note.Raises your wealth.Financial Calculator Solution92Fin
42、d the EFF%on note and compare with 7.0%bank pays,which is your opportunity cost of capital:FVn=PV(1+i)n$1,000=$850(1+i)456Now we must solve for i.3.Rate of Return93 456-850 0 1000 0.035646%per day INPUTSOUTPUTNI/YRPVFVPMTConvert%to decimal:Decimal=0.035646/100=0.00035646.EAR=EFF%=(1.00035646)365-1 =13.89%.Calculator Solution94P/YR=365NOM%=0.035646(365)=13.01 EFF%=13.89Since 13.89%7.0%opportunity cost,buy the note.Using interest conversion
